Rotation and Accretion
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Rotation and Accretion
Powered Pulsars
WORLD SCIENTIFIC SERIES IN ASTRONOMY AND ASTROPHYSICS Editor: Jayant V. Narlikar Inter-University Centre for Astronomy and Astrophysics, Pune, India
Published: Volume 1:
Lectures on Cosmology and Action at a Distance Electrodynamics F. Hoyle and J. V. Narlikar
Volume 2:
Physics of Comets (2nd Ed.) K. S. Krishna Swamy
Volume 3:
Catastrophes and Comets* V. Clube and B. Napier
Volume 4:
From Black Clouds to Black Holes (2nd Ed.) J. V. Narlikar
Volume 5:
Solar and Interplanetary Disturbances S. K. Alurkar
Volume 6:
Fundamentals of Solar Astronomy A. Bhatnagar and W. Livingston
Volume 7:
Dust in the Universe: Similarities and Differences K. S. Krishna Swamy
Volume 8:
An Invitation to Astrophysics T. Padmanabhan
Volume 9:
Stardust from Meteorites: An Introduction to Presolar Grains M. Lugaro
Volume 11:
Find a Hotter Place!: A History of Nuclear Astrophysics L. M. Celnikier
*Publication cancelled.
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World Scientific Series in Astronomy and Astrophysics – Vol. 10
@]bObW]\O\R/QQ`SbW]\
>]eS`SR>cZaO`a Pranab Ghosh
Tata Institute of Fundamental Research, India
World Scientific NEW JERSEY
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LONDON
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SINGAPORE
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BEIJING
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SHANGHAI
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HONG KONG
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TA I P E I
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CHENNAI
Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE
British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.
Cover figure: Chandra X-ray image of the Crab pulsar and nebula. Reproduced with permission from CXC Education/Outreach Co-ordinator.
World Scientific Series in Astronomy and Astrophysics — Vol. 10 ROTATION AND ACCRETION POWERED PULSARS Copyright © 2007 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.
For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher.
ISBN-13 978-981-02-4744-7 ISBN-10 981-02-4744-3
Printed in Singapore.
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To the Memory of My Parents
Verba docent, exempla trahunt.
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Preface
“The truth, the whole truth, and nothing but the truth,” observes a character in Aldous Huxley’s The Genius and the Goddess, “All witnesses take the same oath, and testify about the same events. The result, of course, is fifty-seven varieties of fiction.” We forget all too often that this describes not only witnesses and artists, but scientists as well. We describe what is piously believed to be objective truth, but the outcome is necessarily our view of that truth. This book grew out of my desire to communicate to a wide scientific readership in a simple, accessible way the unifying overview we have begun to achieve in recent years of rotation- and accretion-powered pulsars — the phases that rotating, magnetic neutron stars go through during their lives. The basic facts are scattered far and wide over professional scientific literature, but the manner in which I collect and present them reflects my own overview of the subject, as it must. For a long time, it has been clear that, while there were classic, pioneering text-books on rotationpowered pulsars, and excellent individual chapters on X-ray binaries within collective volumes, the unified view was not really available as an advanced text book. And yet, this overview is one of the milestones in modern highenergy astrophysics. It is my hope that this book will fill that void. It has been my pleasure and privilege to learn from many fine physicists over the years, and their gifts to my mind will be felt throughout the book. I am proud of this. Among them, the name of Prof. Frederick K. Lamb is one that I always mention with great pleasure and honor. While writing the book, I have marveled at how various branches of physics blend beautifully to give us a comprehensive description of the physics of rotating, magnetic neutron stars and their environments. While this synthesis may be found to a certain extent in many parts of astrophysics, that of rotation- and accretion-powered pulsars seems to be a remarkably strong example. vii
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For motivation and fruition of this project, appreciation and acknowledgment of a deep debt of gratitude go to my late parents, and to my wife Surita. That thirst for knowledge which the former propagated into me has been the ultimate motive power behind all my work. This book owes a great deal to Surita’s constant inspiration and encouragement, and to her unflinching support and endless patience while it was being written.
Pranab Ghosh 2006
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Acknowledgments
Permissions for reproduction of previously-published figures have been obtained as follows. Permissions were obtained from sole/lead authors (and/or other authors where appropriate), the author names being given in the respective figure captions, and full references, e.g., journal/proceedings/book names, volume and page numbers, given in the Bibliography. Permissions were also obtained from the appropriate publishers/permission-granting authorities of these publications, as detailed in the following table. In case none of the authors of a publication was available or alive, only the latter permission could be obtained. We thank all such authors and publishers/permission-granting authorities for their kind permission, and display the appropriate copyright sign and/or other material in respective figure captions for those who require it as per terms and conditions. Those authors who showed further kindness by providing us with copies of their figures are much appreciated, and mentioned in the respective figure captions.
Table 1
PERMISSIONS
Figure(s)
Publisher/Permission-granting authority
Book cover
Chandra X-ray Center Education/Outreach Co-ordinator
2.1,2.4,7.13,7.15,7.16,13.4 3.1,4.18,4.19,6.1,9.1,9.12,9.13 9.23 2.3 9.7,9.8,9.22,9.18 4.9,4.10,4.11,4.14,7.19,B.1,D.1
Blackwell Publishing, Oxford Annual Reviews, Palo Alto Revista Mexicana de Astron. y Astrofis. Taylor & Francis Ltd, Oxford Astronomical Society of Japan Elsevier B.V., Amsterdam & Oxford
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Table 1
PERMISSIONS (continued)
Figure(s)
Publisher/Permission-granting authority
1.4,7.10 6.8 6.6,11.2 2.5,4.8,4.12,4.15,4.16,5.5,5.7,5.9,6.4,7.23 7.14,7.20,7.22,11.1 7.1,7.2,7.3 7.24,10.30 8.1 9.26 14.1,8.3,8.4,8.5,5.1,5.3,10.6,10.16 5.6,10.20,9.19,9.21,7.6,5.11,5.12 5.13,9.11,10.5,10.15,10.17,5.14 10.21,11.4,6.3,5.10,7.12,1.5,9.10 5.2,5.4,5.15,1.10,1.9,11.8,10.22 10.23,10.29,9.15,9.24,11.6,11.7 9.20,7.6,10.12,10.26,10.27,10.28 1.2,1.3,9.25,10.2,10.8,10.9,7.23 9.3,1.7,7.4,7.5,7.19,7.22,9.6,9.9 7.17,10.10,10.13,10.18,10.19,12.1 12.2,12.3,12.4,10.3,10.4,10.14,12.6
IEEE Astronomical Society of the Pacific International Astronomical Union American Physical society The Royal Society, London W.H. Freeman & Co., New York Societ` a Astronomica Italiana Progress of Theoretical Physics M. Sako & G. Branduardi-Raymont
American Astronomical Society
6.16 9.5,9.14,9.16,9.17,7.8,10.1,13.5,6.10 7.11
AAAS Cambridge University Press Koninklijke Nederlandse Akad. Wetensch.
6.2,6.7,6.9,6.11,6.12,6.13,9.2,8.2 10.24,11.5,10.3,10.4,10.14,12.6
Springer Science and Business Media
6.14,6.15,13.3 5.8 B.3 6.5 1.1 11.3 C.1 12.5 A.2
MacMillan Publishers Ltd H. Heiselberg M. van der Sluys K. Kifonidis The Nobel Foundation The University of Chicago ATNF Outreach Indian Acad. Sci. Princeton University Press
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Contents
Preface
vii
Acknowledgments
ix
1. The 1.1 1.2 1.3
Discovery of Pulsars Rotation-Powered Pulsars: Radio Discovery . . Accretion-Powered Pulsars: X-Ray Discovery . . Pulsars as Neutron Stars . . . . . . . . . . . . . 1.3.1 Rotation-Powered Pulsars . . . . . . . . 1.3.2 Accretion-Powered Pulsars . . . . . . . 1.4 Powering Pulsars by Rotation and Accretion . . 1.4.1 Rotation Power . . . . . . . . . . . . . . 1.4.2 Accretion Power . . . . . . . . . . . . . 1.5 Galactic Distributions of Pulsars . . . . . . . . . 1.5.1 Rotation-Powered Pulsars . . . . . . . . 1.5.1.1 Pulsar distances . . . . . . . 1.5.1.2 Galactic electron distribution 1.5.1.3 Galactic pulsar distribution . 1.5.1.4 Selection effects . . . . . . . . 1.5.2 Accretion-Powered Pulsars . . . . . . . 1.6 Period Distributions of Pulsars . . . . . . . . . .
2. Physics of Neutron Stars — I. Degenerate 2.1 Historical Notes on Neutron Stars . 2.2 Degenerate Stars . . . . . . . . . . . 2.2.1 Degeneracy . . . . . . . . . 2.2.2 Electron Degeneracy . . . . xi
Stars . . . . . . . . . . . . . . . .
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2.3
2.4
2.5
2.6
2.2.3 Neutron Degeneracy . . . . . . . . . . . . . . . 2.2.4 Complete Degeneracy . . . . . . . . . . . . . . The Landau Arguments . . . . . . . . . . . . . . . . . . 2.3.1 Degenerate Stars: Rough Mass Limits . . . . . 2.3.2 Degenerate Stars: Mass-Radius Relations . . . White Dwarfs . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 The Stoner-Anderson Work . . . . . . . . . . . 2.4.2 Polytropes . . . . . . . . . . . . . . . . . . . . 2.4.3 Chandrasekhar’s Work . . . . . . . . . . . . . . 2.4.4 Modern Work . . . . . . . . . . . . . . . . . . . Neutron Stars . . . . . . . . . . . . . . . . . . . . . . . 2.5.1 The Oppenheimer-Volkoff Work . . . . . . . . 2.5.2 A General-Relativistic “Toy” Neutron Star . . Landau Arguments: General-Relativistic Modifications
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51 51 53 58 59 60 61 65 68 75 80 81 88 91
3. Physics of Neutron Stars — II. Physics of Dense Matter-1 3.1 Matter at Low Densities: Electronic Energy . . . . . . . 3.1.1 Wigner-Seitz Cells . . . . . . . . . . . . . . . . . 3.1.2 The Thomas-Fermi Approximation . . . . . . . . 3.2 Dense Matter Below Neutron-Drip Density: Equilibrium Nuclei . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Effects of Lattice Energy . . . . . . . . . . . . . 3.2.2 Nuclear Shell Effects . . . . . . . . . . . . . . . . 3.3 Neutron Drip . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Matter Above Neutron-Drip Density: Nuclei Surrounded by Neutrons . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.1 Nuclear Surface Energy: A Primer . . . . . . . . 3.4.2 BBP Results . . . . . . . . . . . . . . . . . . . . 3.4.3 More Accurate Results . . . . . . . . . . . . . . 3.4.4 Simple Scalings . . . . . . . . . . . . . . . . . . .
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4. Physics of Neutron Stars — III. Physics of Dense Matter-2 4.1 Above Nuclear-Matter Density: Uniform Nuclear Matter 4.1.1 The Goldstone Expansion . . . . . . . . . . . . . 4.1.2 Goldstone Diagrams . . . . . . . . . . . . . . . . 4.1.2.1 First-order diagrams . . . . . . . . . . 4.1.2.2 Second-order diagrams . . . . . . . . . 4.1.3 Brueckner’s Reaction Matrix . . . . . . . . . . . 4.1.4 Correlations and “Healing” . . . . . . . . . . . .
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4.1.5 4.1.6
4.1.7
Brueckner-Bethe-Goldstone (BBG) Theory . . . The Variational Method . . . . . . . . . . . . . . 4.1.6.1 Cluster diagrams . . . . . . . . . . . . 4.1.6.2 Modern calculations . . . . . . . . . . Recent Developments in BBG and Variational Approaches . . . . . . . . . . . . . . . . . . . . . 4.1.7.1 Relativistic effects in Brueckner theory 4.1.7.2 Relativistic effects in variational methods . . . . . . . . . . . . . . . . 4.1.7.3 Recent results . . . . . . . . . . . . . .
5. Physics of Neutron Stars — IV. Mass, Radius and Structure 5.1 Masses and Radii . . . . . . . . . . . . . . . . . . . . . . 5.1.1 Insensitivity of Radius to Mass . . . . . . . . . . 5.2 Internal Structure . . . . . . . . . . . . . . . . . . . . . . 5.2.1 Crustal Properties: Mass and Radius . . . . . . 5.3 Non-Spherical Shapes: Rods and Plates . . . . . . . . . . 5.3.1 Turning Nuclei “Inside Out” . . . . . . . . . . . 5.3.2 Physical Insights: Frustrated Fission . . . . . . . 5.3.3 Uncertainties . . . . . . . . . . . . . . . . . . . . 5.4 The Maximum Mass of Neutron Stars . . . . . . . . . . . 5.4.1 The Maximum Compactness . . . . . . . . . . . 5.4.2 The Maximum Mass . . . . . . . . . . . . . . . . 5.5 Rotating Neutron Stars . . . . . . . . . . . . . . . . . . . 5.5.1 The Hartle-Thorne Approximation . . . . . . . . 5.5.2 Arbitrary Rotation . . . . . . . . . . . . . . . . . 5.5.3 Maximum and Minimum Rotation . . . . . . . . 5.5.3.1 The mass-shed limit . . . . . . . . . . 5.5.3.2 The gravitation-wave limit . . . . . . . 5.5.3.3 The r-mode . . . . . . . . . . . . . . . 5.5.3.4 The supramassive-collapse limit . . . . 5.5.3.5 Maximum angular velocity of neutron stars . . . . . . . . . . . . . . . . . . . 5.5.4 Hartle-Thorne Approximation: Reprise . . . . . 5.5.5 Moments of Inertia . . . . . . . . . . . . . . . . . 5.5.5.1 Crustal moment of inertia . . . . . . .
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6. Origin and Evolution of Neutron Stars 221 6.1 Binary Stellar Evolution . . . . . . . . . . . . . . . . . . . 221
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6.1.1 6.1.2
6.2
6.3
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Cases A, B, and C . . . . . . . . . . . . . . . . . . Orbital Changes . . . . . . . . . . . . . . . . . . . 6.1.2.1 Conservative mass transfer . . . . . . . 6.1.2.2 Non-conservative mass transfer . . . . . 6.1.3 Stellar Evolution . . . . . . . . . . . . . . . . . . . 6.1.3.1 Conservative evolution . . . . . . . . . . 6.1.3.2 Common-envelope (CE) evolution . . . Supernovae: Birth of Neutron Stars . . . . . . . . . . . . . 6.2.1 Final Evolution of Helium Stars and Cores . . . . 6.2.2 Core Collapse: Neutron Star Formation . . . . . . 6.2.3 Core Bounce: Supernova Explosion . . . . . . . . 6.2.3.1 Explosion mechanisms . . . . . . . . . . 6.2.3.2 Orbital changes due to supernova explosions . . . . . . . . . . . . . . . . . 6.2.4 Evolution of Proto-Neutron Stars . . . . . . . . . . Rotation Power in Young Pulsars . . . . . . . . . . . . . . 6.3.1 Missing Links: Ante-Deluvian Systems . . . . . . . 6.3.2 Probes of Be-Star Outflow: PSR B1259-63 . . . . 6.3.2.1 The Shvartsman surface . . . . . . . . . 6.3.2.2 Propeller spindown . . . . . . . . . . . . 6.3.2.3 Tilted Be-star disks . . . . . . . . . . . 6.3.2.4 Recent work: further orbital dynamics . Accretion Power in Middle-Aged Pulsars . . . . . . . . . . 6.4.1 Evolution to Massive X-Ray Binaries . . . . . . . 6.4.2 Evolution to Intermediate-Mass X-Ray Binaries . 6.4.3 Evolution to Low-Mass X-Ray Binaries . . . . . . 6.4.3.1 CVs . . . . . . . . . . . . . . . . . . . . 6.4.3.2 LMXBs . . . . . . . . . . . . . . . . . . Rotation Power in Old, Recycled Pulsars . . . . . . . . . . 6.5.1 Final Evolution of HMXBs and Recycling . . . . . 6.5.1.1 Recycled pulsars: single or with degenerate companions . . . . . . . . . . 6.5.2 The Double Pulsar Binary J0737-3039 . . . . . . . 6.5.3 Final Evolution of LMXBs and Recycling . . . . . 6.5.3.1 Gravitational radiation . . . . . . . . . . 6.5.3.2 Magnetic braking . . . . . . . . . . . . . 6.5.3.3 The minimum period and the period gap 6.5.3.4 Donor evolution and expansion . . . . . 6.5.3.5 Recycled pulsars from LMXBs . . . . .
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6.5.4 6.5.5 6.5.6 6.5.7
Missing Links: Accreting Millisecond Pulsars Irradiation of Low-Mass Companions . . . . Missing Links: “Black Widow” Pulsars . . . Pulsars with Planets . . . . . . . . . . . . . .
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7. Properties of Rotation Powered Pulsars 7.1 Pulse Properties . . . . . . . . . . . . . . . . . . . . . . 7.1.1 Integrated Pulse Profiles . . . . . . . . . . . . 7.1.1.1 Pulse shapes . . . . . . . . . . . . . 7.1.1.2 Interpulses . . . . . . . . . . . . . . 7.1.1.3 Polarization . . . . . . . . . . . . . . 7.1.1.4 The rotating vector model . . . . . . 7.1.1.5 Frequency dependence and stability 7.1.2 Individual Pulses . . . . . . . . . . . . . . . . . 7.1.2.1 Intensity variations and nulling . . . 7.1.2.2 Subpulse drifting . . . . . . . . . . . 7.1.2.3 Micropulses . . . . . . . . . . . . . . 7.1.2.4 Giant pulses . . . . . . . . . . . . . . 7.2 Timing Properties . . . . . . . . . . . . . . . . . . . . . 7.2.1 Pulsar Timing . . . . . . . . . . . . . . . . . . 7.2.2 Secular Period Changes . . . . . . . . . . . . . 7.2.2.1 Characteristic age . . . . . . . . . . 7.2.2.2 Braking index . . . . . . . . . . . . . 7.2.2.3 Higher derivatives . . . . . . . . . . 7.2.3 Irregular Period Changes . . . . . . . . . . . . 7.2.3.1 Timing noise . . . . . . . . . . . . . 7.2.3.2 Millisecond pulsars as stable clocks . 7.2.3.3 Limits on cosmic gravitational wave background . . . . . . . . . . . . . . 7.2.3.4 Glitches . . . . . . . . . . . . . . . . 7.2.3.5 Glitches: starquakes . . . . . . . . . 7.2.4 Timing Rotation Powered Pulsars in Binaries . 7.2.4.1 PSR 1913+16: historical notes . . . 7.2.4.2 Binary rotation-powered pulsars: general systematics . . . . . . . . . . 7.2.4.3 PSR 1913+16: nursery of relativistic gravity . . . . . . . . . . . . . . . . . 7.2.4.4 PSR 1534+12 and similar “relativity laboratories” . . . . . . . . . . . . .
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The double-pulsar binary as relativity laboratory . . . . . . . . . . . . . . . . . 373 A related “fundamental” effect . . . . . 375
8. Superfluidity in Neutron Stars and Glitch Diagnostics 8.1 Superfluidity in Neutron Stars . . . . . . . . . . . . . 8.1.1 Pairing . . . . . . . . . . . . . . . . . . . . . 8.1.2 Gap Energy . . . . . . . . . . . . . . . . . . . 8.1.3 Rotating Superfluids and Quantized Vortices 8.2 Post-Glitch Relaxation: Two-Component Theory . . 8.3 Glitches: Vortex Pinning . . . . . . . . . . . . . . . . 8.3.1 The Pinning Force . . . . . . . . . . . . . . . 8.3.2 Strong and Weak Pinning . . . . . . . . . . . 8.3.3 The Magnus Force . . . . . . . . . . . . . . . 8.4 Post-Glitch Relaxation: Vortex Creep . . . . . . . . . 8.4.1 Rotational Dynamics: Steady State . . . . . 8.4.2 Approach to Steady State . . . . . . . . . . . 8.5 Stellar Parameters from Glitch Data . . . . . . . . . . 8.6 Glitches: Recent Developments . . . . . . . . . . . . .
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9. Properties of Accretion Powered Pulsars 9.1 Binary Characteristics . . . . . . . . . . . . . 9.1.1 Displaying Binary Systematics . . . . 9.1.2 Newtonian Apsidal Motion . . . . . . 9.1.3 Orbital Period Changes . . . . . . . . 9.1.4 Neutron-Star and Companion Masses 9.2 Pulse Profiles . . . . . . . . . . . . . . . . . . 9.3 Secular Period Changes . . . . . . . . . . . . . 9.4 Timing Noise . . . . . . . . . . . . . . . . . . . 9.4.1 Power-Density Spectra . . . . . . . . . 9.4.1.1 Observed power spectra . . 9.4.2 Time-Domain Analysis . . . . . . . . 9.5 Quasi-Periodic Oscillations (QPOs) . . . . . . 9.5.1 QPOs in LMXBs . . . . . . . . . . . . 9.5.1.1 Low-frequency QPOs . . . . 9.5.1.2 High-frequency QPOs . . . 9.5.1.3 LF QPO diagnostics . . . . 9.5.1.4 HF QPO diagnostics . . . . 9.5.2 QPOs in HMXBs . . . . . . . . . . . .
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X-Ray 9.6.1 9.6.2 9.6.3 9.6.4
Spectra . . . . . . . . . . . . . . . . . . . . . . . Continuum Emission . . . . . . . . . . . . . . . Pulse-Phase Spectroscopy . . . . . . . . . . . . Cyclotron Features . . . . . . . . . . . . . . . . Emission Lines: Fluorescence, Recombination, Resonance . . . . . . . . . . . . . . . . . . . . 9.6.4.1 Fluorescence . . . . . . . . . . . . . 9.6.4.2 X-ray ionization of stellar winds . . 9.6.4.3 Observations . . . . . . . . . . . . . 9.6.4.4 Recombination . . . . . . . . . . . . 9.6.4.5 Recent observations . . . . . . . . . 9.6.4.6 Wind diagnostics . . . . . . . . . . . Mutants: Anomalous X-Ray Pulsars (AXPs) . . . . . .
10. Pulsar Magnetospheres 10.1 Magnetospheres of Accretion-Powered Pulsars . . . . 10.1.1 Exterior Flow and Plasma Capture . . . . . . 10.1.1.1 Capture from stellar winds . . . . 10.1.1.2 Fluctuations in stellar winds . . . 10.1.1.3 Capture from Roche-lobe overflow 10.1.2 Formation of Magnetospheres . . . . . . . . . 10.1.2.1 Lengthscales of magnetospheres . . 10.1.3 Radial Flow . . . . . . . . . . . . . . . . . . . 10.1.3.1 Size of the magnetosphere . . . . . 10.1.3.2 The shock . . . . . . . . . . . . . . 10.1.3.3 Shape of the magnetosphere . . . . 10.1.3.4 Magnetospheric boundary with accretion flow . . . . . . . . . . . . 10.1.4 Disk Flow . . . . . . . . . . . . . . . . . . . . 10.1.5 Thin Keplerian Accretion Disks . . . . . . . 10.1.5.1 One temperature disks . . . . . . . 10.1.5.2 The α-model of disk viscosity . . . 10.1.5.3 The nature of disk viscosity . . . . 10.1.5.4 The structure of α-disks . . . . . . 10.1.5.5 Two temperature disks . . . . . . . 10.1.6 The Disk-Magnetosphere Interaction . . . . . 10.1.6.1 Basic electrodynamic processes . . 10.1.6.2 Steady flow models . . . . . . . . .
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10.1.7
Disk-Magnetosphere Boundary Layer . . . . . . 10.1.7.1 Boundary-layer behavior . . . . . . . . 10.1.7.2 Boundary-layer: nature and structure 10.1.8 Inner Edge of the Disk . . . . . . . . . . . . . . . 10.1.9 Outer Transition Zone . . . . . . . . . . . . . . . 10.1.10 Further Work . . . . . . . . . . . . . . . . . . . . 10.1.11 Plasma Entry into Magnetospheres: Radial Flow 10.1.11.1 Entry via Rayleigh-Taylor instability . 10.1.11.2 Cusp entry . . . . . . . . . . . . . . . 10.1.11.3 Entry by other modes . . . . . . . . . 10.1.12 Plasma Entry into Magnetospheres: Disk Flow . 10.1.13 Accretion Flows Inside Magnetospheres . . . . . 10.1.13.1 Field-aligned flow . . . . . . . . . . . . 10.1.13.2 Flow “between” field lines . . . . . . . 10.1.14 Accretion on Stellar Surface: Stopping Mechanisms . . . . . . . . . . . . . . . . . . . . . 10.1.14.1 Radiative stopping . . . . . . . . . . . 10.1.14.2 Collisional stopping . . . . . . . . . . 10.2 Magnetospheres of Rotation-Powered Pulsars . . . . . . . 10.2.1 The Goldreich-Julian Argument . . . . . . . . . 10.2.2 The Aligned Rotator . . . . . . . . . . . . . . . . 10.2.2.1 The pulsar equation . . . . . . . . . . 10.2.2.2 Convergence . . . . . . . . . . . . . . . 10.2.2.3 Results . . . . . . . . . . . . . . . . . 10.2.3 Problems with the Standard Model . . . . . . . 10.2.4 Vacuum Gaps . . . . . . . . . . . . . . . . . . . 10.2.4.1 Ruderman-Sutherland gaps . . . . . . 10.2.5 The Oblique Rotator . . . . . . . . . . . . . . . 10.2.6 The Double-Pulsar Binary as Magnetospheric Probe . . . . . . . . . . . . . . . . . . . . . . . . 11. Pulsar Emission Mechanisms 11.1 Emission by Rotation-Powered Pulsars 11.1.1 Coherent Emission . . . . . . . 11.1.2 Emission by Bunches . . . . . 11.1.3 Maser Emission . . . . . . . . 11.1.4 Relativistic Plasma Emission .
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11.2 Emission by Accretion-Powered Pulsars . . . . . . . . 11.2.1 Radiation Transport in Strongly Magnetized Plasmas . . . . . . . . . . . . . . . . . . . . . 11.2.1.1 Basic radiative processes . . . . . . 11.2.1.2 Transport calculations . . . . . . . 11.2.1.3 Results . . . . . . . . . . . . . . . 11.2.2 Pulse Shapes and Spectra . . . . . . . . . . . 12. Spin Evolution of Neutron Stars 12.1 Spin Evolution of Rotation-Powered Pulsars 12.1.1 Electromagnetic Spindown . . . . . 12.1.2 Propeller Spindown . . . . . . . . . 12.1.3 Other Spindown Torques . . . . . . 12.1.4 Spindown of PSR B1259-63 . . . . . 12.2 Spin Evolution of Accretion-Powered Pulsars 12.2.1 Torques on Disk-Fed Pulsars . . . . 12.2.2 Comparison with Observations . . . 12.2.3 Torques on Wind-Fed Pulsars . . . . 12.3 Pulsar Period — Magnetic Field Diagram . . 12.3.1 The Spinup Line . . . . . . . . . . . 12.3.2 Disk Diagnostics: Spinup Lines . . .
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13. Neutron Star Magnetic Fields 13.1 Exotic Atoms in Strong Magnetic Fields . . . . . . . . 13.1.1 Further Exotica: Molecular Chains . . . . . . 13.2 Origin of Neutron-Star Magnetic Fields . . . . . . . . . 13.2.1 Fossil Fields . . . . . . . . . . . . . . . . . . . 13.2.2 Thermo-Magnetic Effects . . . . . . . . . . . . 13.2.3 Dynamos in Young Neutron Stars: Magnetars 13.3 Evolution of Neutron-Star Magnetic Fields . . . . . . . 13.3.1 Accretion-Induced Field Decay . . . . . . . . . 13.3.1.1 Ohmic decay . . . . . . . . . . . . . 13.3.2 Field Decay or Hiding/Burial? . . . . . . . . .
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14. Strange Stars 14.1 EOS of Strange Matter . . . . . . . . . . . . . . . . . . . . 14.2 Structure of Strange Stars . . . . . . . . . . . . . . . . . . 14.3 Search for Strange Stars . . . . . . . . . . . . . . . . . . .
693 694 694 696
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Appendix A Astronomical Preliminaries A.1 Units . . . . . . . . . . . . . . . . . . A.2 Astronomical Co-Ordinate Systems . A.2.1 Equatorial Co-Ordinates . . A.2.2 Precession of Equinoxes . . . A.2.3 Galactic Co-Ordinates . . . . A.2.4 Co-Ordinate Transformation A.2.5 Time Keeping . . . . . . . . A.2.6 Stellar Classification . . . . . A.2.6.1 Color Index . . . .
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Appendix B Binary Dynamics B.1 Binary Orbit . . . . . . . B.1.1 Orbital Elements B.1.2 Mass Function . B.1.3 Position in Orbit B.2 Roche Lobes . . . . . . .
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Appendix C Single Star Evolution C.1 The Hertzsprung-Russell Diagram C.2 Stellar Evolution . . . . . . . . . . C.2.1 Hydrogen Burning . . . . C.2.2 Helium Burning . . . . . C.2.3 Advanced Burning Stages C.2.4 Essential Timescales . . . C.2.5 Mass-Radius Relations . . C.2.6 Brown Dwarfs . . . . . .
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Appendix D The Two-Nucleon Potential 725 D.1 Skyrme Interaction . . . . . . . . . . . . . . . . . . . . . . 728 Appendix E Tables of Pulsars 731 E.1 Accretion Powered Pulsars and AXPs . . . . . . . . . . . . 731 Bibliography
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Chapter 1
The Discovery of Pulsars
1.1
Rotation-Powered Pulsars: Radio Discovery
There is no doubt today that the 1967 discovery of rotation-powered pulsars at radio frequencies was the single most significant event responsible for the recognition of neutron stars as a physical reality rather than a brainchild of brilliant physicists. The idea of neutron stars was conceived in the 1930s following Chadwick’s discovery of the neutron1 , the possibility of their birth in supernovae proposed [Baade & Zwicky 1934a], and the first calculation of their masses and radii performed [Oppenheimer & Volkoff 1939]. After 1939, there was little interest in neutron stars for about two decades because (a) the original motivation for the study of neutron cores of stars as possible stellar energy sources had vanished with the acceptance of stellar thermonuclear reactions as this source of energy, and, (b) it was mistakenly thought (see Sec. 2.1) that neutron stars would not be able to form in the last stages of stellar evolution, because of a paradox which was not resolved until 1959 [Cameron 1959b]. A revival of interest in neutron stars in the 1960s was inspired largely by the discovery of discrete X-ray sources outside the solar system in 1962 [Giacconi et al. 1962], because these X-rays were thought at first to be thermal emission from newborn, hot neutron stars. Even in 1965, however, physicists writing papers on neutron stars tended to be somewhat skeptical at times, as illustrated by the following quote from Bahcall and Wolf (1965b): “In order to prevent the investigation of neutron stars from degenerating into a philosophical discussion, it is necessary to concentrate on those aspects of the theory which are at least in principle connected with observation.” 1 For
a brief historical account of neutron stars before 1967, see Sec. 2.1. 1
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Then, in August-September 1967, a doctoral student named Jocelyn Bell at the University of Cambridge, who had just started operating a newly-constructed large dipole array (2048 dipole antennae spread over an area of 4.5 acres, operating at 81.5 MHz) for studying interplanetary scintillation of compact radio sources with her thesis supervisor Anthony Hewish, found “a bit of ‘scruff’ on the records” [Bell-Burnell 1977] of her chart recorder. Radio telescopes are extremely sensitive to man-made electromagnetic interference (from, e.g., automobile ignition or electrical farm equipment), but this did not look like interference; in addition, it came repeatedly from the same part of the sky [Bell-Burnell 1977]. On a recorder with a short response time ∼ 0.05 seconds, Bell’s “scruff” or fluctuations came out on November 28, 1967, to be a train of astounding, precisely periodic, pulses at a period of 1.337 seconds. Fig. 1.1 shows these pulses. Bell and Hewish investigated the pulses, painstakingly eliminating one possibility after another2 . The pulses were not from radar beams reflected off the moon. They were not from man-made satellites in unusual orbits, or deep space probes. In fact, since the pulses came about 4 minutes earlier
Fig. 1.1 Pulses from PSR 1919+21, the first rotation-powered pulsar, discovered in 1967. Note that this pulsar is referred to as CP 1919 in the figure, as per the customs of the time: CP stands for Cambridge pulsar. Reproduced with permission by the Nobel c 1975 The Nobel Foundation. Foundation from Hewish (1975). 2 Bell’s foresight and perseverance was instrumental to this, as the custom at the time in radio-astronomy circles was to dismiss such signals as man-made [Lyne & GrahamSmith 1990]. Bell’s own delightful account of the discovery [Bell-Burnell 1977] contains these words: “I contacted Tony Hewish who was teaching in an undergraduate laboratory in Cambridge, and his first reaction was that they must be man-made. This was a very sensible response under the circumstances, but due to a truly remarkable depth of ignorance I did not see why they could not be from a star.” We may well wonder where we would be today but for Bell’s “depth of ignorance”.
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each night, i.e., kept to the sidereal time (see Appendix A), their source had to be far outside the solar system. An immediate and interesting possibility was that of communication signals deliberately sent out by an extraterrestrial civilization—the proverbial “little green men” of early science-fiction. But this alternative became improbable when careful Doppler-effect analysis of the period of the pulses revealed only our Earth’s motion around our Sun, and no trace of the further Doppler shift expected from the motion of the planet inhabited by these distant intelligent beings around their sun [Bell-Burnell 1977]. By late December, 1967, another pulsing radio source was discovered in a different part of the sky, making it clear that the phenomenon being observed was indeed natural radio emission. Studies of the dispersion of the radio signals established that the pulses were coming from within our galaxy, but outside the solar system. The first pulsar results were announced to the world in the journal Nature on February 24, 1968 [Hewish et al. 1968], and in a seminar that Hewish gave in Cambridge a few days before that. “Every astronomer in Cambridge, so it seemed, came to the seminar,” recalled Bell later, “and their interest and excitement gave me a first appreciation of the revolution we had started” [Bell-Burnell 1977]. The universal question was: what was the nature of these cosmic pulsing radio sources? Since a period ∼ 1.3 seconds was too short to be associated with the oscillations of a large, low-density, “normal” star, Hewish et al. immediately called attention to compact, high-density stars like white dwarfs and neutron stars, which are among the end-points of stellar evolution (see Chapter 6 and Appendix C), and whose frequencies of radial oscillation ran from about ten seconds to a small fraction of a second. A significant historical point, to which we come back in Sec. 2.1, is that the pulsar-discovery paper [Hewish et al. 1968] considered oscillations, not rotations, of compact stars as the probable cause of radio pulses. Radio astronomers all over the world now directed their telescopes and their attention to the discovery of more of these new pulsing radio sources, or “pulsars”, as they came to be called. New pulsars were found in profusion, in particular two fast-rotating pulsars in supernova remnants: the Crab pulsar (with a period of 33 milliseconds) and the Vela pulsar (with a period of 89 milliseconds). This opened up a new, rich field of study in astronomy and astrophysics. Identification of pulsars with rotating neutron stars, which was established within about a year and a half (see Sec. 1.3) following the seminal paper by Gold (1968), led to torrential theoretical work in the 1970s on every conceivable aspect of neutron stars: internal structure, surface properties, origin, and evolution (see Chapters
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2 and 3). In addition, attempts to understand the properties of rotationand accretion-powered pulsars (see Chapters 7, 8 and 9) led to fundamental work on the electrodynamics and plasma (i.e., ionized gas) dynamics near rotating, magnetic, compact stars (see Chapter 10), and on theories of gravitation . In recognition of the profound impact on physics of the discovery of rotation-powered pulsars, Hewish shared3 the 1974 Nobel Prize in Physics: he was cited for “his pioneering research in radio astrophysics, particularly the discovery of pulsars”. More than 1700 rotation-powered pulsars are known today: as it is not practicable to list their essential parameters in this book, we refer the reader to the Australia Telescope National Facility (ATNF) Pulsar Catalogue, available at the website: http://www.atnf.csiro.au/research/pulsar/psrcat. A uniform way of naming rotation-powered pulsars has emerged now. Each pulsar is identified by its position on the sky in terms of two angles, right ascension and declination (see Appendix A). The sequence of numbers specifying the angles is preceded by the letters PSR (for Pulsar, obviously, replacing older prefixes like CP for Cambridge Pulsar, etc). Thus, the first pulsar discovered [Hewish et al. 1968] is PSR 1919+21, the Crab pulsar is PSR 0531+21, the (Hulse-Taylor) binary pulsar is PSR 1913+16, the first millisecond pulsar discovered is PSR 1937+21, and so on. A slight modification has arisen in recent years because of the transition from the original 1950 coordinates to the current 2000 coordinates, which is explained in Appendix A. To distinguish between the two, the letter J is appended to the prefix PSR for the 2000 coordinates, and the letter B for the old 1950 ones, the nomenclature and the transformation between the two being given in Appendix A.
1.2
Accretion-Powered Pulsars: X-Ray Discovery
X-ray astronomy is a product of the space age, as X-rays from celestial sources are absorbed by the Earth’s atmosphere, and the only way of observing them is through detectors sent above the atmosphere in space vehicles4 . The subject started in 1949 with the detection of X-ray emission from the Sun by Geiger counters flown on a V-2 rocket captured at the end of the Second World War. Solar X-ray flux was so weak, however, 3 With Ryle, who won it also for achievements in radio astrophysics which lie outside the scope of this book. 4 High-altitude balloons can also be used to detect hard X-rays.
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that observable X-ray emission from sources outside the solar system was not expected, based on the assumption that sun-like stars are the dominant X-ray sources in the universe. The absurdity of this assumption may be obvious to us today, but the atmosphere of skepticism was so persuasive in 1962 that the avowed purpose of the rocket flight of Giacconi and co-workers [Giacconi et al. 1962] had to be the detection of solar X-rays bouncing off the moon, while the real purpose was a search for extrasolar X-ray sources [Rossi 1973]. That Aerobee rocket flight became history: the counters on board detected a very bright X-ray source in the constellation of Scorpius (in addition to the expected X-ray fluorescence from the moon), giving birth to extrasolar X-ray astronomy, and showing, in Rossi’s (1973) words, “the boundless wealth and complexity of nature”. Between 1962 and 1970, about 50 cosmic X-ray sources were discovered by rocket and balloon flights: most of these were inside our Galaxy, but the first extragalactic X-ray source—the giant elliptical galaxy M87 in the Virgo cluster—was also discovered in the same era. It became customary to name the galactic X-ray sources by the constellation they were in, and the order in which they were discovered. Thus the first extrasolar source ever detected was called Scorpius X-1, or Sco X-1 for short, and so on. Accretion is a word which etymologically means “the process of growth or enlargement” by various means, in particular “by external addition or accumulation, as by external parts or particles”5. This meaning is slightly adapted in astrophysics to signify the gravitational capture by a star of the matter surrounding it (either matter through which it is passing, such as the interstellar medium or moecular clouds, or matter which is flowing toward it, such as that coming from its companion in a binary system), leading to an increase in the star’s mass. Study of accretion by the Sun and similar stars during their passage through diffuse gas, in contexts which are no longer of interest to us, goes back to Eddington (1930), who mistakenly thought that the geometrical size of the star was the capture radius, inside which all particles of matter (following trajectories somewhat “focused” toward the star by its gravitational attraction) were intercepted by the star and accreted by it. Hoyle and Lyttleton (1939a,b) introduced the essential physics, namely that at all gas densities of astrophysical interest, the collisions between particles greatly reduced their transverse momenta, much enhancing the effective capture radius, which depended basically on the stellar mass M and the relative velocity v between the star and the 5 From
the Latin accretus, past participle of the verb accrescere, to increase.
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streaming matter, scaling as GM/v 2 , and usually exceeding the stellar radius by a large factor. This radius, now called the accretion radius, is a key parameter in the theory of accretion. The effects of the thermal energy content of the streaming gas on the accretion radius were included later. The potential significance of accretion by compact stars was emphasized in 1965 by Zel’dovich and Novikov, before the discovery of either rotationor accretion-powered pulsars. They clearly expressed their appreciation of the crucial point about accretion by neutron stars as follows: “However, the main stimulus for the study of accretion lies in the energy released during accretion. Particles incident on the surface of a neutron star give up to (0.2– 0.3) c2 of energy per gram, which is much more than can be obtained from nuclear reactions.” Zel’dovich and Novikov (1965) then went on to make the prophetic remark that “Connected with the accretion phenomenon is the very possibility of observing neutron and cooled stars...” (“cooled stars” was the name these authors used for black holes). As we know today, the 1962 discovery of Sco X-1 had, in fact, been the first clear detection of a neutron star powered by accretion. But this was not realized in 1965. The 1967 discovery of radio pulsars, powered by rotation, not accretion, and their rapid identification with neutron stars led to the first acceptance of the reality of these stars (see Sec. 1.1). The 1971 discovery of binary X-ray pulsars (see below) and their identification with accreting neutron stars finally made us aware of the abundant occurrence in nature of the phenomenon that Zel’dovich and Novikov had speculated on in 1965. For objects which are candidates for being black holes, emission powered by accretion remains our best observational probe to this day. An optical counterpart for Sco X-1 was found in 1966 [Sandage et al. 1966], a faint, blue star with excess ultraviolet emission, somewhat resembling an old nova. Novae were well-known to be binary stars, with evidence for gas streams flowing between the two stars. In 1967, Shklovskii proposed that Sco X-1 “is a neutron star forming a comparatively massive component of a close binary system”, and that “a stream of gas flowing out of the second component is permanently incident on the neutron star”, X-ray emission occurring from the hot gas in the stream. This pioneering6 6 Curiously, in a brief 1966 note, Zel’dovich and Guseynov had advocated the use of known single-line spectroscopic binaries to search for black holes and neutron stars, arguing that the optically-unseen companions in some of these binaries could be such objects. They had also made a casual comment that gas moving in the strong gravitational field of black holes could show spectral features in the X-rays. But these authors had not described sources powered by accretion, emitting primarily X-rays. This is amazing in the context of the 1965 Zel’dovich-Novikov work on accretion described in the previous paragraph.
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scenario contained most of the essential ingredients of what later became the standard model of compact X-ray sources and accretion-powered pulsars, namely, (1) a neutron star or a black hole forming a binary stellar system with a “normal” companion star still burning thermonuclear fuel, (2) the companion transferring mass to the compact object by one or more of several possible mechanisms, and, (3) the gravitational energy released by matter accreting onto the compact object converted into electromagnetic radiation, primarily at X-ray wavelengths. In 1969, Zel’dovich and Shakura published the first quantitative, detailed calculation of the process of spherical accretion by neutron stars, and the spectrum of the X-rays emitted in the process, neglecting the magnetic field of the star. These authors started their paper with the remarkable statement “Neutron stars were born at the tip of the theoretician’s pen over 30 years ago7 , but a convincing identification of such a star has yet to be made”, although their paper was published in Russian in March-April 1969, by which time the rotating neutron-star model of radio pulsars was becoming accepted, as we described in Sec. 1.1. Zel’dovich and Shakura considered various mechanisms for deceleration and stopping of matter near the stellar surface, showing that the emergent X-ray spectrum depended both on this mechanism and on the accretion rate. The calculated spectra, which differed considerably from the Planck spectrum of a black body (particularly when the deceleration mechanism was collective plasma oscillations), was compared by them with the X-ray spectrum of Sco X-1 known at the time, after which they stated that only through such comparisons “might one be able to ascertain whether the observed point X-ray sources are in fact neutron stars, and whether neutron stars are entitled to pass from the realm of hypothetical objects into the class of reliably identified stars”. Apparently, these authors did not believe that rotation-powered pulsars already entitled neutron stars to join “the class of reliably identified stars”; the verdict of history turned out to be exactly contrary. X-ray astronomy really came into its own in 1970 with the launch of NASA’s first astronomy satellite Uhuru8 totally dedicated to surveying the X-ray sky in the energy range 2–20 keV, obtaining accurate positions (for its time) of the X-ray sources, and studying their spectra and time-variability. Uhuru discovered some 300 X-ray sources during its twenty-seven month 7 See
footnote 1. Swahili word meaning “freedom”: the satellite was launched off the coast of Kenya, on the independence day of that country. 8A
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life-span, establishing a new branch of astronomy which was subsequently brought to its full glory of abundance and maturity by successive generations of X-ray satellites. In January 1971, about a month into the Uhuru mission, Giacconi and his co-authors noticed that a previously known X-ray source in the constellation of Centaurus, Cen X-3, was exhibiting periodic pulses. Based on the Uhuru data obtained in Januaray and April, 1971, these authors obtained a pulse period of 4.84 seconds, and published their result in July, 1971 [Giacconi et al. 1971]. This was the discovery of the first X-ray pulsar: Fig. 1.2 shows the Cen X-3 pulses. A second X-ray pulsar, Her X-1, was discovered in the constellation of Hercules by the end of 1971: it had a period of 1.24 second, and became one of the most widely studied accretion-powered pulsars [Tananbaum et al. 1972]. The binary nature of Cen X-3 soon became clear from the X-ray data gathered by Uhuru between January and December, 1971: the source underwent “regular changes in intensity between two distinct levels with a period of 2 days”, and its pulse-period underwent sinusoidal variations with the same period, correlated with the intensity variations [Schreier et al. 1972], as shown in Fig. 1.3. Schreier et al. wrote: We interpret this effect as due to an occulting binary system. The changes in intensity are then due to occultation of the X-ray source by a large massive companion, and the sinusoidal variations in the period of the 4.8 s pulsations are due to Doppler effect. The orbital period of the X-ray source is 2.08712 ± 0.00004 days. This interpretation became the standard picture of binary X-ray pulsars. The “large massive companion” in Cen X-3 was optically identified by Krzeminski (1973) to be an evolved supergiant star of spectral class late O (see Appendix A): it is now called Krzeminski’s star, or V779 Cen. Similar results followed quickly for Her X-1, with a binary period of 1.7 days
Fig. 1.2 Pulses from Cen X-3, the first accretion-powered pulsar, discovered in 1971. Reproduced by permission of the AAS from Schreier et al. (1972): see Bibliography.
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Fig. 1.3 Orbital variation of the pulse-period and intensity of Cen X-3, indicating its binary nature. Reproduced by permission of the AAS from Schreier et al. (1972): see Bibliography.
[Tananbaum et al. 1972], and a relatively low-mass evolved (spectral class late A / early F) optical companion named HZ Her. Within about a year of their discovery, the X-ray emitting stars in these binary X-ray pulsars were argued to be neutron stars (see Sec. 1.3.2 for details), largely carrying over the expertise obtained a few years earlier in debating the case of rotation-powered pulsars. The idea that they were powered by accretion onto the neutron stars also came naturally [Pringle & Rees 1972; Davidson & Ostriker 1973; Lamb et al. 1973] in the wake of the discussion of accreting neutron stars which we have summarized above, and which had existed in the scientific literature for about seven years by then. Thus, the pioneering Shklovskii (1967) scenario for Sco X-1 blossomed into a full-fledged model for accretion-powered X-ray pulsars, which has become universally accepted today. Ironically, the binary nature of Sco X-1 itself was not established until 1975, and no X-ray pulses have yet been detected from this source. About 110 accretion-powered pulsars are known now, their periods ranging from 1.7 milliseconds to 9860 seconds: some of their essential parameters are lised in Table E.1. These form a subset of binary, accretionpowered, compact X-ray sources known today from several generation of
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10
X-ray satellites over the last three decades, starting with Uhuru, and coming all the way to the current satellites RXTE, BeppoSAX, Chandra, and XMM-Newton. It is no longer convenient to list them by the constellations, and the modern nomenclature system for X-ray sources closely follows that for rotation-powered pulsars (see Sec. 1.1). The position on the sky is specified by right ascension and declination, and these numbers are preceded by an abbreviation for (usually) the name of the satellite that generated the catalog of X-ray sources (and, occasionally, the version of that catalog). Thus, Cen X-3, the first accretion-powered pulsar discovered, was named 4U 1119-60 in the 4th Uhuru catalog, and the recently-discovered 2.5 ms pulsar (which we discuss later at length because of its importance as an evolutionary link; see Sec. 6.5.4) was named SAX J1808.4-3658 or XTE J1808-359 after the satellites BeppoSAX and RXTE respectively. As position determinations become more accurate with succeeding generations of satellites, the numbers change slightly, and more significant digits are added. An alternative earlier system of using galactic co-ordintes (see Appendix A), in which the numbers were galactic longitude and latitude, preceded by the letters GX, has also been largely superseded. We may use it in this book occasionally for historical reasons. In recognition of his pioneering efforts in X-ray astronomy, his monumental efforts in developing this field to its present level of advancement, and his direct involvement in many of the finest discoveries in this field (including that of accretion-powered pulsars, which we are discussing in this book), Giacconi shared9 the 2002 Nobel Prize in Physics: he was cited for “pioneering contributions to astrophysics, which have led to the discovery of cosmic X-ray sources”.
1.3
Pulsars as Neutron Stars
1.3.1
Rotation-Powered Pulsars
In a Nature paper published almost exactly three months after the Hewish et al. (1968) discovery paper of radio pulsars, Gold (1968) made the pioneering, definitive suggestion that these objects were rotating neutron stars, the pulse period being the rotation period of the neutron star10 . Over a 9 With
Davis and Koshiba, who won it for their work in neutrino astrophysics. 1967, before the discovery of pulsars, Pacini had already described the emission of electromagnetic radiation from rotating, magnetized neutron stars in a related but different context, which is discussed in Sec. 2.1. 10 In
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duration of about a year, this suggestion took hold, grew in detail as more properties of radio pulsars were established, and eclipsed all other suggestions to become what is now the universally-accepted physical model of pulsars. Gold initially (1968) appealed to the extreme stability of the periods (1 in 108 ) to argue the case for a massive, compact, rotating object (as opposed to stellar vibrations, or oscillations of plasma configurations), and predicted a “slowing down of the observed repetition frequencies”, which was subsequently confirmed. The fact that the rate of loss of rotational energy of the Crab pulsar was roughly the same as that required to power the Crab nebula (essentially repeating the 1967 scenario of Pacini), as Gold argued in a subsequent (1969) paper, clinched the issue. We now summarize the arguments in favor of the rotating neutron-star model and against the competing models, as they stand today, rather than following the historical development. The crucial observational point used in these arguments are that (1) the periods of the radio pulsars are so short (∼ milliseconds to several seconds), (2) radio pulsars are exremely accurate clocks (measurements upto precisions ∼ 1 part in 1014 have been possible), and (3) the periods of radio pulsars always increase slowly and gradually, except for occasional, discontinuous decreases or “glitches” (which we treat in some detail in Chapters 7 and 8). The first point implies that rotations or vibrations of compact, dense stars, e.g., white dwarfs or neutron stars, (or orbital motions around such stars) must be involved rather than those of large, low-density, “normal” stars, which occur on much longer timescales, since all these characteristic times scale roughly as ρ−1/2 (see below), where ρ is the average density of the star. Further, since the shortest known period among radio pulsars is ∼ 1.6 ms (see Sec. (1.6)), and light travels ∼ 500 km in this time, this distance must be an upper limit to the size of the emitting region. Because of the great accuracy of the pulsar clock mechanism, we can argue that this must also be a rough upper limit to the size of the star itself, since it is basically impossible to justify such accuracy for an emitting region much smaller than the star and unrelated to the size of the latter: the two must be closely coupled. Of the various compact-star phenomena, consider first oscillations of white dwarfs or neutron stars. Periods of fundamental modes of oscillations of white dwarfs are in the range 2–10 s, and those of neutron stars are in the range 1–10 ms [Meltzer & Thorne 1966]. It is thus clear that a single class of objects will not account for the entire range of pulsar periods. White-dwarf oscillations at higher harmonics will, of course, have shorter periods than above, but it requires special circumstances to excite these
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harmonics, and the sharpness of the pulses (one of their major observed characteristics) will then be destroyed by mode-mixing due to any small nonlinearity (always present in real stars). In any case, the periods of wellknown pulsars like Crab (33 ms) and Vela (89 ms), which lie in the “gap” between the above ranges for white dwarfs and neutron stars, are almost impossible to account for in this way. Finally, if pulses were emitted by this mechanism, the energy-loss from the oscillating system would lead to a decrease in the pulse period, the exact opposite of what is observed. Thus, oscillations are ruled out. Consider next rotating white dwarfs. The shortest period P , or the highest angular velocity Ω, at which a star of mass M and radius R can rotate without being torn apart by centrifugal forces is given by Ω2 R ∼
GM . R2
(1.1)
In terms of the average density ρ of the star, Eq. (1.1) for the breakup angular velocity can be recast in the approximate form Ω ∼ (Gρ)1/2 ,
(1.2)
which is a very useful, general result for a variety of dynamical timescales associated with a star. White dwarfs with ρ ∼ 107 –108 g cm−3 therefore have breakup rotation periods P = 2π/Ω ∼ 1–10 s, a result which rules out rotating white dwarfs. Note also the similarity between this range of periods and that given above for white-dwarf oscillations. This is a consequence of the virial theorem [Chandrasekhar 1935], according to which the period of the fundamental oscillation mode of a star is of the same order of magnitude as its breakup period11 [Gold 1968], and so given roughly by Eq. (1.2), as we anticipated above. Now consider orbital motion around compact stars, either (a) of a small object in a close orbit around the star, or, (b) of a close binary system of compact stars around each other. In the former case, balancing the centrifugal force against the gravitational attraction of the star again yields an equation like Eq. (1.1) with the orbital radius r replacing the stellar radius R, and since r ∼ R for a close orbit, we recover Eq. (1.2). In the latter case, we again get an equation of the form 1.1, with M replaced by the total mass (= 2M for two identical stars of mass M ) of the system, and 11 Elastic
restoring forces, which augment the usual gravitational restoring forces in solid or partly-solid stars like white dwarfs and neutron stars, shorten the oscillation periods somewhat, but the scaling with density in Eq. (1.2) is still roughly valid.
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the orbital radius r replacing the stellar radius R as before. With r ∼ 2R for two identical stars of radius R, we again roughly recover Eq. (1.2). This further illustrates the universal significance of the dynamical timescale. The earlier arguments given for the range of periods thus roughly apply to orbital motion as well. Additional, severe difficulties arise. First, such a close binary system containing a neutron star would be a copious emitter of gravitational waves, the two objects spiralling into each other in a very short time due to energy-loss through these waves. This timescale for a binary system of two neutron stars, each of mass M , and a binary period of P is roughly Tspiral ∼ 10
−3
P 1s
8/3 yr,
(1.3)
an absurdly short lifetime. In reality, pulsar periods are observed to change typically on timescales ∼ 107 yr. Furthermore, orbit decay due to gravitational radiation would cause the period to decrease, contrary to what is observed12 . For a planet or satellite of mass M in a close orbit around a neutron star of mass M ( 1), the spiral-in timescale is again given by Eq. (1.3), but with an extra factor of −1 included in its right-hand side. Observed period-change timescales of pulsars would thus imply tiny masses ( ∼ 10−10 ) of the satellite. Even under the unlikely assumption that such an insignificant object could produce the observed radio pulses, we immediately encounter the problem that such a satellite would be pulverized by the enormous tidal forces in the strong gravitational field close to a neutron star. The high radiation field of the pulsar only adds to the difficulty, as it would tend to melt or evaporate the satellite13 . This rules out orbital periods. Finally, consider rotation periods of neutron stars. As applications of Eq. (1.2) show, neutron stars with ρ ∼ 1014 g cm−3 have breakup rotation periods ∼ 1 ms. Thus, rotation periods > ∼ 1 ms are all possible for neutron stars; indeed, the shortest pulse period ∼ 1.6 ms known for radio pulsars is rather close to (but, of course, longer than) the numerical 12 Double neutron-star binaries are known, of course, the classic example being PSR 1913+16, the famous Hulse-Taylor pulsar discovered in 1974. The pulse period of this pulsar is P ≈ 59 ms, the orbital period is Porb ≈ 7.7 hours, and the rate of decrease of the latter due to gravitational radiation has been measured accurately. See Sec. 7.2.4 for a full discussion. 13 Indeed, the radiation field of the binary pulsar PSR 1957+20, the so-called “black widow” pulsar, is believed to be slowly evaporating its low-mass white dwarf companion. See Sec. 6.5.5 for details.
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value of the breakup period calculated from detailed neutron-star models. All the other objections against the alternative scenarios detailed above also do not apply to the rotating neutron-star scenario. Thus the rotating neutron-star model survives as the only viable scenario. The accuracy of the pulsar clock relates well to the rotation of a massive, compact star with a solid surface, the lengthening of the pulse period to the slowing down of the rotation (or spindown, as it is called) due to loss of rotational energy through electromagnetic emission at the rate expected for canonical magnetic fields (∼ 1012 G) of neutron stars, and the above energy loss-rate to the (apparent) energy supply-rate to canonical supernova remnants like the Crab nebula. Many details of the observed features of pulsars, which we describe in later chapters, have generally found satisfactory explanations in the rotating neutron-star model, although our understanding of the pulseformation mechanism is still not as good as it could be (see Chapter 11). Because of the universal agreement today that the pulsars originally (and still being) discovered at radio frequencies are powered by the rotational energy of neutron stars, we shall refer to these as rotation-powered pulsars throughout the rest of the book. As these pulsars sometimes have emission in other wavebands (optical, X-ray, γ-ray,...) as well, a categorization which is more meaningful physically is in terms of their source of energy, which sets them apart from the other class of pulsars with which we are concerned in this book. 1.3.2
Accretion-Powered Pulsars
Let us summarize the arguments identifying binary X-ray pulsars as rotating, accreting neutron stars, again as they would stand today, rather than historically. Arguments favoring rotating neutron stars follow much the same route as those used for rotation-powered pulsars, using the observational facts that (1) the pulse periods range from ∼ 2 milliseconds to ∼ 104 seconds, and, (2) the periods are quite stable after correction for the Doppler effect due to binary motion [Davidson & Ostriker 1973]. The first point implies that, again, the only kind of stars capable of having dynamical phenomena over this whole range of periods are compact stars—white dwarfs, neutron stars, and black holes—and, of these, the only single class able to account for the entire range is that of neutron stars. This follows from the arguments about dynamical timescales in terms of the average stellar density ρ summarized in the previous subsection, particularly Eq. (1.2). Rotating black holes are ruled out because non-axisymmetric
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black holes, such as would be required for producing pulses, can have only a transient existence, if any. For neutron stars, only rotation can cover this entire range of periods, oscillation periods being far too limited in range (∼ 1–10 ms), as explained in the previous subsection. The second point, the stability of the pulses, also argues in favor of rotations rather than oscillations as the basic cause of modulation at the pulse period. Again, the argument is that the rotation of a compact, massive body, isolated except for the gravitational influence of its binary companion, is far stabler by nature than a train of oscillations excited by any conceivable mechanism. By the same argument, plasma oscillations are ruled out even more strongly. Finally, orbital periods around neutron stars are untenable because (a) the range of observed periods cannot be accounted for, and, (b) such periods would change on absurdly short timescales due to emission of gravitational radiation (see Eq. (1.3)). Thus, we are left with rotation periods of neutron stars as the only viable model. That the source of power for the X-ray emission from binary X-ray pulsars could not be the rotational energy of the neutron stars was clear from the beginning, since the typical observed X-ray luminosities of these sources (∼ 1037 erg s−1 ) were far too high to be accounted for in this manner, even if we assumed, for the sake of argument, that these sources were all spinning down. This is easily seen by noting that the slowingdown time of a binary X-ray pulsar of period P and luminosity L is [Lamb et al. 1973]: Tslow ∼ Erot /L ∼ 60I45
P 1s
−2
L−1 37 yr,
(1.4)
where Erot = 12 IΩ2 is the rotational energy of the neutron star, I being its moment of inertia. In Eq. (1.4), L37 is L in units of 1037 erg s−1 , and I45 is I in units of 1045 gm cm2 : this convention of expressing a physical variable in units of its typical order of magnitude in a given problem is standard in astrophysics, and we shall be using it throughout this book. This timescale is absurdly small for the vast majority of the sources with periods in the range 1–1000 seconds. Actually, periods of binary X-ray pulsars show a complex behavior with time, consisting of (a) a secular component which is spinup (i.e., P decreasing with time) for some pulsars, spindown (i.e., P increasing with time) for some, and and no significant change for yet others, superposed on (b) a fluctuating “noise” component.
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So the spindown argument is irrelevant to these pulsars as a general class14 . Accretion remains the only viable source of power to the class of binary X-ray pulsars as a whole [Pringle & Rees 1972; Davidson & Ostriker 1973; Lamb et al. 1973]. The vast quantity of energy released by matter falling into the deep gravitational potential wells of neutron stars of radii R ∼ 106 cm and masses M ∼ M , amounting to = GM/R ∼ 1020 ergs per gram of accreted matter, i.e., about a tenth of its rest-mass energy (∼ 0.1c2 ), makes accretion an ideal source of power, as was recognized by Zel’dovich and coauthors long ago [Zel’dovich & Novikov 1965; Zel’dovich & Shakura 1969]. To generate X-ray luminosities ∼ 1037 erg s−1 typical of bright, binary Xray pulsars, accretion rates ∼ 1017 gm s−1 ∼ 10−9 M yr−1 are therefore required. The binary companion of the neutron star is a natural, and more than adequate, source of supply of the matter to be accreted, as stars lose mass through a variety of processes during the course of their evolution, particularly if they are in binary systems. Massive, young, O and B stars drive sizable outflows of matter called “stellar winds”, and rapidly-rotating Be stars shed, in addition, rings of matter from time to time. At several stages of their thermonuclear evolution, stars expand by large factors to become “giants” or “supergiants” (see Appendix C): if they are in close binaries, they lose mass copiously during these stages as they attempt to expand beyond their limiting gravitational equipotentials, which are called Roche lobes (see Appendix B). Finally, the X-rays emitted by the neutron star can heat the companion’s surface, causing mass loss in a so-called “selfexcited wind”. All this matter is available for accretion by the neutron star, which the latter can do through several types of accretion flows, ranging from spherical inflow (for matter with negligible angular momentum relative to the neutron star) to inflow through an accretion disk (for matter with so much angular momentum that it can go into orbits around the neutron star obeying Kepler’s laws). We discuss these processes in detail in later chapters. 14 Historically,
the prototype binary X-ray pulsars Her X-1 and Cen X-3, which were used to start building models for these pulsars, were soon found to have secular spinup, at which point rotation-power for emission became irrelevant and was forgotten. In the last few years, this chapter has been reopened for a small subclass of X-ray pulsars with periods ∼ 5–12 seconds: these show secular spindown, and their binary nature has never been really established. The idea that these Anomalous X-ray Pulsars (AXPs) could be rotating neutron stars with unusually, but not impossibly, high magnetic fields, and that the observed spindown is by the same electromagnetic torques that brake rotationpowered pulsars, is being explored. We return to AXPs in Chapter 9.
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There is universal agreement today that binary X-ray pulsars are powered by accretion. Additional supporting evidence has accumulated over the years, for example from rates of secular spinup of binary X-ray pulsars [Lamb et al. 1973; Rappaprt & Joss 1977], and from strong evidence for the presence of accretion disks in these and similar binaries through probes at other wavelengths, e.g., optical. In close analogy with what we described for the radio pulsars in the previous subsection, all major details of the properties of binary X-ray pulsars have found satisfactory explanation in the framework of the accreting neutron-star model. A particular triumph of this model has been the natural explanation of the evolutionary link between young, high magnetic-field, rotation-powered pulsars (e.g., the Crab pulsar) and old, low magnetic-field, “recycled” rotation-powered pulsars spun up to millisecond periods, whose discovery started in 1982 with the ∼ 1.6 millisecond pulsar PSR 1937+21. As this is a dominant theme of this book, we shall return to evolutionary questions repeatedly, particularly in Chapter 6. Finally, in analogy with the nomenclature introduced in the previous subsection, we shall refer to binary X-ray pulsars as accretionpowered pulsars throughout the rest of the book, despite the fact that they were, and are still being, discovered by X-ray satellites. The reason, again, is that they have emission at other wavelengths—optical, UV, IR, radio, and the like—and studies at some of the other wavelengths is often crucial (such as optical studies for the clarification of binary properties), whereas their source of energy is their key physical feature distinguishing them from the rotation-powered pulsars described earlier.
1.4 1.4.1
Powering Pulsars by Rotation and Accretion Rotation Power
Consider a neutron star of moment of inertia I, rotating with an angular velocity Ω, so that its rotational energy is given by Erot =
1 2 IΩ ≈ 2 × 1046 I45 P −2 (s) erg, 2
(1.5)
where I45 is I in units of 1045 g cm2 , as before, and P = 2π/Ω is the ˙ its rotational rotation period. If the star’s spin is changing at a rate Ω, energy is changing at a rate ˙ 2 ≈ IΩΩ˙ = −4 × 1032 I45 P −3 (s)P˙−14 erg s−1 . ˙ + 1 IΩ E˙ rot = IΩΩ 2
(1.6)
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Here, I˙ is the rate of change, if any, of the star’s moment of inertia during the spin-change process, due to a variety of possible reasons we discuss later, and the approximate version of the right-hand side of Eq. (1.6) is obtained by neglecting the inertia-change term, which we shall be using most frequently. Loss of rotational energy is then inferred from spindown, ˙ < 0, or, equivalently, P˙ > 0, as is the case for rotation-powered pulsars. Ω In Eq. (1.6), P˙−14 is P˙ in units of 10−14 second per second (s s−1 ), a typical order of magnitude for rotation-powered pulsars, roughly comparable to a nanosecond per day (ns d−1 ), which is also used as a unit for P˙ . How does such a pulsar convert its rotational energy into electromagnetic energy and radiate it? This was the problem addressed by Pacini (1967) and Gold (1969). Young neutron stars have very large magnetic 12 fields (B > ∼ 10 G), the origins of which we discuss in Chapter 13. As a first approximation, we can represent this field by that of a magnetic dipole, since any magnetic field configuration in the star can be represented by a combination of magnetic multipoles (see, e.g., Jackson 1975), as seen by an outside observer, and the dipole component is dominant when the observer is sufficiently far away from the star. Consider a neutron star of magnetic dipole moment µ oriented at an angle α with respect to its rotation axis. (The magnetic field at the stellar surface of radius R is given roughly by Bs ∼ µ/R3 , and the field at either of the magnetic poles is given exactly by Bp = 2µ/R3, so that typical orders of magnitude for young pulsars are Bs ∼ 1012 G and µ ∼ 1030 G cm3 .) This rotating oblique dipole appears as a time-varying one from the point of view of a distant observer [Jackson 1975], and so radiates energy at a rate 2µ2 Ω4 sin2 α 2 E˙ mdr = ≈ 1031 B12 R66 P −4 (s) sin2 α erg s−1 , 3c3
(1.7)
as magnetic dipole radiation at a frequency Ω. Here, B12 is the polar field Bp in units of 1012 G, and R6 is R in units of 106 cm. A comparison of Eqs. (1.6) and (1.7) shows how readily magnetic dipole radiation accounts for the spindown rates of rotation-powered pulsars. In fact, the observed spindown rate of a given pulsar can be used to estimate its magnetic field through the relation we obtain by combining Eqs. (1.6) and (1.7), namely,
B12 ≈ 6
I45 P˙−14 P (s) R63 sin α
,
(1.8)
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and is the standard method for doing so. The angle of obliquity α is, of course, not known from the spindown observations alone, but we can either (a) set sin α = 1 in Eq. (1.8), and interpret the result as the magnetic field due to that component of the magnetic moment which is perpendicular to the rotation axis, or, (b) use a statistically averaged value of sin α expected for a collection of pulsars, unless we have clues to the value of α from other observation, e.g., that of polarization. Note that the magnetic field ˙ strength inferred in this way scales as P P . Thus, because of its short period (P ≈ 33 ms) and high spindown rate (P˙−14 ≈ 42), the Crab pulsar has a higher energy, Erot ≈ 2 × 1049 erg, and higher energy-loss rate, E˙ rot ≈ −5 × 1038 erg s−1 , than the canonical values appearing in Eqs. (1.5) and (1.6), but the inferred magnetic field strength, B12 ≈ 7, is canonical, since the factor of increase in P˙ largely cancels the factor of decrease in P . In this argument, we have used the canonical estimate ∼ 1 for I45 and R6 , as we shall do throughout this book. It was the remarkable agreement between the above estimate of E˙ rot for the Crab pulsar and the inferred energy requirements of the Crab nebula that proved instrumental for identifying rotation-powered pulsars with neutron stars [Gold 1969]. Although E˙ rot is the rate of loss of rotational energy from the neutron star, sometimes called the spindown power, only a small fraction of it appears in the radio pulses. For example, the fraction of the above spindown power for the Crab pulsar that goes into radio pulses is ∼ 10−7 , i.e., absolutely tiny, as is the typical value ∼ 10−5 characteristic of the known population of rotation-powered pulsars. Where does the bulk of the energy go? It is generally thought that much of this energy goes into accelerating charged particles15 to high energies, which then deposit their energy in the surrounding nebula (supernova remnant) in the case of young pulsars like the Crab pulsar to make the nebula a strong emitter of electromagnetic radiation (radio, optical, X-ray...). Note also that the magnetic dipole radiation we discussed above is clearly not what we observe as pulses from the pulsars, since this radiation is emitted at the stellar rotation frequency (ν ≡ Ω/2π ∼ 1–1000 Hz), and has an approximately sinusoidal pulse shape. The observed pulses are modulations at radio frequencies (ν ∼ 10 MHz— 10 GHz), or even higher frequencies of electromagnetic radiation, and their shapes resemble narrow spikes, covering a small fraction (∼ 10−2 –10−1) of the total pulse period (see Chapetr 7). Thus, the magnetic dipole radiation is merely a process that taps the reservoir of the neutron star’s rotational 15 γ-ray pulses, which have been detected from a small number of rotation-powered pulsars so far, have been observed to carry upto ∼ 0.3 of the spindown power.
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energy; the pulse-generation process is far more complex, to which we return in Chapter 11. Whatever this process is, it must be clear that it is very strongly tied to the stellar rotation, since the pulses come with such stable periodicity, and it produces electromagnetic radiation in extremely sharp, narrow beams (unlike magnetic dipole radiation). Charged particles moving along the stellar magnetic field lines, which are corotating with the star, are therefore an essential ingredient of any such process, as we shall see. In the universally-accepted model of rotation-powered pulsars, such a process generates a very narrow beam of radiation rotating with the star, as shown schematically in Fig. 1.4, and when this beam sweeps by the earth, rather like a lighthouse beam, we see the pulses. 1.4.2
Accretion Power
Consider a neutron star of mass M and radius R accreting mass at a rate M˙ . Since each unit of accreted mass releases an amount of gravitational potential energy GM/R on reaching the stellar surface, the lumnosity (i.e., energy per unit time) generated by the accretion process is given by [Zel’dovich & Shakura 1969]
Fig. 1.4 Basic model of rotation-powered pulsars. Reproduced by permission of IEEE c 1991 IEEE. from Taylor (1991), Proc. IEEE, 79, 1054.
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L=
GM M˙ ≈ 1.3 × 1037 M˙ 17 (M/M )R6−1 erg s−1 , R
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(1.9)
where M˙ 17 is M˙ in units of 1017 g s−1 . Thus, accretion rates of bright accretion-powered pulsars with L ∼ 1037 erg s−1 are ∼ 1017 g s−1 ∼ 10−9 M yr−1 , as we mentioned before, and the accreted mass comes from the binary companion of the neutron star. How, again, is this energy converted into electromagnetic radiation— primarily X-rays—that we see in the pulses? The basic process is thermal emission from the heated stellar surface and the hot plasma (ionized matter) close to it, the source of heat being the gravitational potential energy released as described above, as the accreting matter falling towards the stellar surface is brought to a halt by a variety of dissipative processes. As a first approximation, we can treat this as blackbody emission (complexities due to radiative transfer in the presence of intense neutron-star magnetic fields are discussed in Chapter 11) at an effective temperature T from that part of the stellar surface which receives the accreting matter, and whose area is A. By Stefan’s law, the luminosity is then L = σAT 4 ,
(1.10)
which must equal that given by Eq. (1.9) in a steady state. Here, σ is the Stefan-Boltzmann constant. What is the accretion area A on the neutron star? Note first that if we equated this to the entire stellar surface area, As = 4πR2 , we would obtain the case of spherical accretion onto the neutron star, such as was considered by Zel’dovich and Shakura (1969). While interesting in its own right, such a situation does not produce pulses (as must be obvious since the stellar rotation produces no modulation in the outgoing radiation) and so is irrelevant for accretionpowered pulsars, in which the intense magnetic field of the neutron star channels the flow of the accreting matter (which, being completely ionized, must move along the magnetic field lines, a situation which is called “flux-frozen” flow; see, e.g., Jackson 1975) towards the magnetic poles of the star [Davidson & Ostriker 1973; Lamb et al. 1973]. Matter thus accretes on two polar caps at the stellar surface, producing two “hot spots” which emit radiation, as shown in Fig. 1.5: matter raining down on a hot spot forms a column-like structure called “accretion column”16 . If the magnetic axis of the star is oblique with respect to its rotation axis 16 Also
called “accretion funnel” because of its shape on a larger scale: see Fig. 1.5.
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Fig. 1.5 Basic model of accretion-powered pulsars. Reproduced by permission of the AAS from Lamb et al. (1973): see Bibliography.
(as for rotation-powered pulsars), this is a natural mechanism for modulating the emitted radiation at the stellar rotation frequency, as observed. The area of these polar caps, which is what enters into Eq. (1.10), can be estimated by the following method [Davidson & Ostriker 1973; Lamb et al. 1973]. Far away from the neutron star, the flow of the accreting matter is not influenced by the stellar magnetic field, and is determined, rather, by the details of how it is lost by the companion star, and by the dynamics of the binary system (see Chapter 10). Close to the neutron star, on the other hand, the flow is completely dominated by the stellar magnetic field, matter falling onto the star along the field lines: this region is called the magnetosphere of the neutron star. Of crucial importance is the transition boundary between the magnetospneric flow and the outer flow: its location can be calculated from various physical arguments involving the accreting matter and the magnetic field, which we discuss in Chapter 10. All these arguments give a boundary radius rm which depends basically on the accretion rate M˙ and the surface magnetic field Bs of the neutron star, as may
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be expected: for typical values appropriate for bright accretion-powered pulsars (M˙ 17 ∼ 1, B12 ∼ 1), rm ∼ 108 cm (see Chapter 10 for detailed expressions for various estimates of rm ). The size of the polar cap is then roughly estimated by the simple, but useful, geometrical argument that the field lines passing through the outer edge of this cap must be the the outermost stellar field lines that can possibly close within rm (see Fig. 1.5). As a first approximation, we assume a dipolar magnetic field for the star (as we did for rotation-powered pulsars, and as we shall do throughout this book): the equation for the field lines of a magnetic dipole in polar coordinates is [Jackson 1975] r = r0 sin2 θ,
(1.11)
where r0 is a field line ‘label’, identifying the particular field line by its value of r on the equatorial plane of the dipole (θ = π/2). Since r0 = rm for the outermost field line we are concerned with here, the value of θ for the footpoint of this field line on the stellar surface (r = R) given from Eq. (1.11) by θa ≈ (R/rm )1/2 , which is also the semi-angle subtended by the polar cap at the center of the star (see Fig. 1.5). Thus the total area of the two polar caps (one at each pole of the star) is given by Ap ≈ 2πR2 θa2 = 2πR3 /rm ≈ 6R63 (rm /108 cm)−1 km2 ,
(1.12)
which is much smaller than (in fact, only about 0.5 % of) the total surface area, As ≈ 1260 R62 km2 , of the neutron star. Note that even if the stellar field configuration was a perfect dipole to begin with, the accretion process itself is likely to distort it, so that the above picture is only approximate. Nevertheless, this simple geometrical estimate, which can be regarded as an upper limit to the size of the hot spots, makes a good beginning, which can be improved upon by detailed considerations of the accretion flow pattern (see Chapter 10). The blackbody emission temperature of the hot spots can now be obtained by combining Eqs. (1.9), (1.10), and (1.12). The result, 1/4 T ≈ 4 × 107 M˙ 17 (M/M )1/4 R6−1 (rm /108 cm)1/4 K,
(1.13)
readily shows why the bulk of the emission is in the X-rays, since the mean energy of the photons emitted by a blackbody of this temperature is ¯ ≈ 2.7kT ≈ 9 keV for canonical values of the parameters in Eq. (1.13), k being
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Boltzmann’s constant17 . Furthermore, since the blackbody temperature is only a lower bound on the actual emission temperature, the latter may be higher. In chapter 11, we discuss how additional effects may modify the shape of the emergent spectrum of an accretion-powered pulsar from that of a blackbody. While a rotating hot spot does generate periodic modulations due to a varying angle of viewing, further complications are introduced into the anisotropy of the emergent radiation due to the stellar magnetic field, since the opacity of matter in such fields is highly energy-dependent. The study of the latter effect was pioneered by Lamb et al. (1973), who identified three regimes of magnetic effects, depending on whether the stellar magnetic field was above, below, or comparable to a critical field strength ∼ 2 × 1011 G. Far below the critical value, the magnetic field has little effect on the opacity of the matter. Since the integrated density of matter, and so its opacity, is much greater along the accretion column than across it, the radiation emerges preferentially through the sides of the column, producing a fanshaped beam. Far above this value, the magnetic field drastically reduces the opacity of matter, essentially making the accretion column transparent, so that the viewing-angle effect mentioned above produces the modulation. Finally, when the magnetic field is comparable to the critical value, interesting opacity effects are possible, leading to complex pulse shapes. We discuss radiative transfer and pulse-shaping in accretion-powered pulsars in Chapter 8. Note that the observed pulse profiles of accretion-powered pulsars (see Chapter 9) are generally much smoother—covering most or all of the pulse period—than those of rotation-powered pulsars. This is entirely consistent with the fact that the above modulation mechanisms tend to produce smooth, gradual variations, and shows that pulse-shaping mechanisms in rotation-powered pulsars must be very different indeed.
1.5 1.5.1
Galactic Distributions of Pulsars Rotation-Powered Pulsars
Fig. 1.6 shows the positions of ∼ 1500 rotation-powered pulsars in the sky, in galactic coordinates (i.e., galactic latitude and longitude; see Appendix A). The concentration of pulsars towards the galactic plane which is 17 Here, keV stands for kilo-electron volts, an electron volt being the energy gained by an electron in a potential difference of 1 volt, which is approximately 1.6 × 10−12 erg. A handy relation between energy E in electron volts (eV) and temperature T in degrees K is T (K)≈ 1.16 × 104 E(eV), which follows directly from E = kT .
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Fig. 1.6 Distribution of rotation-powered pulsars in galactic coordinates (see Appendix A), using a Hammer-Aitoff (equal-area) projection. Shown are the 1509 pulsars known roughly as of December 2004: more have been discovered since. Note that the figure includes rotation-powered pulsars, as well as AXPs and SGRs (see Chapter 9). Categories of rotation-powered pulsars as indicated. Reproduced by permission of the AAS from Manchester et al. (2005): see Bibliography.
apparent from the figure is a real effect: pulsars are galactic objects. There is also a real tendency for pulsars to be concentrated in the longitude quadrant toward the galactic center, which suggests that the density of pulsars increases radially towrads the galactic center, at least in the vicinity of the sun, since most of the observed pulsars lie in this neighborhood (because of the difficulty in detecting pulsars far away from us due to their faintness, which is an example of selection effects discussed later). In addition to these two angles, we need to know a distance, say that of the pulsar from the galactic center, r, projected onto the galactic plane, in order to completely specify the position of the pulsar in the galaxy. Actually, the galactic distribution of pulsars is usually quoted in terms of the two cylindrical coordinates r and z, the former being the projected radius just defined, and the latter the vertical distance from the galactic plane. Azimuthal symmetry is assumed. The distances r and z cannot be measured directly, of course. What we can estimate by various methods is the distance d of the pulsar from us, which is also its distance from the Sun for all practical purposes. In terms of d, the galactic longitude l, and the galactic latitude b, the required distances are then given by the trigonometric relations: 2 + d2 cos2 b − 2r d cos b cos l, z = d sin b. (1.14) r = r
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Here, r ≈ 8.5 kiloparsec (kpc) is the distance of the Sun from the galactic center (parsec is a unit of distance defined in Appendix A). 1.5.1.1
Pulsar distances
But how do we measure d? Foremost among the methods for doing this is that of dispersion measure, which makes use of the dispersion of the radio pulses during propagation through the ionized interstellar medium, because of which pulses from a given pulsar arrive slightly later at lower frequencies than at higher frequencies. The group velocity vg of propagation of electromagnetic waves of frequency ν through a plasma with electron density ne is given (see, e.g., Krall & Trivelpiece 1973) by:
νp2 vg = c 1 − 2 ν
1/2 .
(1.15)
Here, νp ≡ ne e2 /πme is the plasma frequency of the medium, e and me being respectively the charge and the mass of the electron. Since the time of travel d of a pulse of frequency ν from a pulsar at a distance d to us is ta ≡ 0 dl/vg , the dispersion in ta , i.e., the difference ∆ta between the times of arrival of pulses at two frequencies around ν differing by ∆ν is found from Eq. (1.15) to be ∆ta = −
∆ν ν
e2 πme cν 2
d
ne dl = − 0
∆ν ν
DM (cm−3 pc) 2 120.5νGHz
(1.16)
d correct to the first order in νp2 /ν 2 . Here, DM ≡ 0 ne dl is called the dispersion measure of the pulsar, which is the integrated density (or column density, as it is called in astrophysics) of electrons along the line of sight to the pulsar, conventionally expressed in the hybrid unit of cm−3 pc. In Eq. (1.16), ∆ta is in seconds, and νGHz is ν in GHz. Now, since
d
ne dl = ne d,
DM =
(1.17)
0
in terms of the average electron density ne in the direction of the pulsar, we can find d if we know ne , or the electron-density distribution, in the galaxy. The latter is usually obtained from dispersion measurements of pulsars whose distances are independently known by one of the following methods.
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First, for very nearby pulsars (within about 1 kpc from the Sun), it is possible to measure distances from observations of the annual parallax (i.e., the apparent annual cyclic movement of position due to the Earth’s orbital motion around the Sun), as is done for nearby stars. For pulsars, this is achieved through very long baseline radio interferometry (VLBI). However, the number of such extremely close pulsars is very small. Secondly, when pulsars at low galactic latitudes are observed through the spiral arms of the galaxy, their spectra show marked absorption at the famous 21-cm line of neutral hydrogen (HI). The velocity of the HI cloud can be determined from the Doppler shift in the absorption line, which yields its distance with the aid of our knowledge of the rotation velocity of matter as a function of distance from the center of the galaxy, which is called the rotation curve of the galaxy. This method is by far the most widely applicable one for pulsar distance determination independent of dispersion measure [Manchester & Taylor 1977]. Finally, when a pulsar is associated with a supernova remnant (e.g., the Crab and Vela pulsars) or a globular cluster (e.g., pulsars in 47 Tuc), the distances to which are known from optical observations, we have an independent distance measure: this method is also applicable to only a relatively small number of pulsars. Distances inferred by these methods generally range from ∼ 40 pc to ∼ 20 kpc, and the average electron densities ne so obtained are in the range 0.02–0.07 cm−3 , a typical value ∼ 0.03 cm−3 being representative of much of our galaxy. Corresponding values of DM are in the range 3–800 cm−3 pc. 1.5.1.2
Galactic electron distribution
The above procedure gives a first, rough map of the distribution of ne in various directions in the galaxy, which is used for a first determination of distances to those pulsars for which only the dispersion-measure method is possible. This model of electron-density distribution is then refined from the pulsar measurements themselves, using an iterative scheme. Starting with the model of Manchester and Taylor (1981), successive refinements by Lyne et al. (1985), Taylor and Cordes (1993), and Lazio and Cordes (1998) have now yielded an electron-density distribution capable of providing pulsar distances accurate to ∼ 25% or better. Galactic ne -distribution models consist of the following components. The first is a smooth, axisymmetric component representing the smooth distribution of ionized hydrogen (HII) in the galactic disk. This consists of two parts, representing the inner and the outer disk. The second is a large HII region close to the Sun known as
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the Gum Nebula, which is located at galactic longitude l = 260◦ and latitude b = 0◦ . Because of its close proximity (d ≈ 500 pc) and large angular diameter (∼ 40◦ ), the effects of the nebula are prominent on the dispersion measures of a significant number of pulsars located behind it. The third contribution comes from the four spiral arms of the galaxy, each of which has prominent HII regions (around newborn, massive OB stars) lined along it [Taylor & Cordes 1993], which particularly affect the dispersion measures of pulsars seen by us along tangent directions to these spiral arms. In order to convey the flavor of typical ne -distribution models used in modern work on rotation-powered pulsars, we now summarize the distribution advocated by Taylor and Cordes (1993). Each of the smooth, axisymmetric components is assumed to factorize, or be separable, in the coordinates r and z defined earlier. This is reminiscent of the separable nature the galactic distribution of pulsars (see below), which is supported by the observed lack of correlations between r and z in that distribution. The outer-galaxy component is (r, z) = 0.02 nouter e
sech2 (r/20) sech2 (z/0.88) cm−3 sech2 (r /20)
(1.18)
Here, r and z are in units of kpc, and r ≈ 8.5 kpc is the distance of the Sun from the galactic center, as before. The distribution in Eq. (1.18) has been so normalized as to reproduce the known electron density, ne ≈ 0.02 cm−3 near the Sun. The choice of the squared hyperbolic secant [sechz ≡ 1/ cosh z ≡ 2/{exp(z) + exp(−z)}] is not unique, but plausible and convenient. Its use in the z-direction is prompted by the facts that (a) it is more appropriate than the exponential distribution exp(−|z|) used earlier for vertical distributions, since it has the physically expected zero derivative at z = 0, and (b) the density profile of an isothermal gas layer is expected to have this functional form [Spitzer 1942]. Extension to r and other coordinates is largely by analogy. The inner-galaxy component is (r, z) = 0.1 exp{−[(r − 3.5)/1.8]2}sech2 (z/0.15) cm−3 . ninner e
(1.19)
The radial part of this distribution is an annulus with the maximum density at a radius 3.5 kpc and a Gaussian half-width of 1.8 kpc. This ring-like structure is prompted by a similar structure seen in molecular line observations [Burton 1988], which suggest enhanced star formation and consequent creation of HII regions within such a ring.
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The Gum Nebula contribution is modeled as 1, u ≤ 0.13 GN −3 ne = 0.25 cm exp{−[(u − 0.13)/0.05]2}, u > 0.13
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(1.20)
where u is the distance in kpc from the center of the Gum Nebula to the point at which ne is being evaluated. The nebula’s contribution is constant = 0.25 cm−3 out to a distance of 130 pc from its center, and its at nGN e edges have been “softened” as a Gaussian fall-off with a 50 pc length scale. Finally, the contribution from each of the four spiral arms is again modeled by a factorisable, or separable, distribution of the form = 0.08sech2 (z/0.3) exp[−(s/0.3)2 ]gspine(t) cm−3 . nspiral e
(1.21)
The first distribution factor on the right-hand side of Eq. (1.21) is the standard Taylor-Cordes sech2 -distribution in the z-direction, i.e., perpendicular to the galactic plane. Within the plane, the distribution is described naturally in terms of two following coordinates. Imagine a curved line following a spiral arm along its central and densest parts (see Fig. 1.7), which we can call the “spine” of that spiral arm18 . Then a (curvilinear) coordinate t following the spine, and a coordinate s transverse to t specify the planar distribution completely. The transverse s-distribution is taken as a Gaussian with a scale length of 0.3 kpc, as Eq. (1.21) shows. What about the t-distribution along the spine, gspine(t)? Taylor and Cordes approximated it as follows. The spine can, of course, be described as a curve in the galactic plane in terms of the galactocentric radius r and a galactocentric angle θ measured, say, counterclockwise from galactic longitude l = 0. These authors took gspine = f (θ)R(r), using simple functions for f and R. For the radial distribution they took a constant value inside a cutoff radius of 8.5 kpc, and a fall-off outside this following a sech2 -law with a scale length of 2 kpc, in order to roughly account for the observed number densities of H II regions. The θ-distribution they kept as simple as possible, using a constant value unless the observed distribution of H II regions along the spine showed large variations in some angular region of the spiral arm, in which case that was accounted for in a simple way. We refer the reader to the original paper for further details. 18 We deliberately avoid using the word “axis” in this context [Taylor & Cordes 1993], as it often means something entirely different for spiral structures, namely an axis perpendicular to the plane of the spiral, passing through the central point of the whole spiral.
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Fig. 1.7 Electron distribution in the galactic plane (z = 0) in gray-scale representation. The electron density is roughly maximum, ne ∼ 0.25 cm−3 , near the center of the Gum Nebula, shown by the large dark spot. For comparison, ne ∼ 0.019 cm−3 at the position of the Sun. Reproduced by permission of the AAS from Taylor & Cordes (1993): see Bibliography.
The Taylor-Cordes ne -distribution in the galactic plane (z = 0) is shown in Fig. 1.7. The largest electron densities are found in the outer parts of spiral arm 2 and in the Gum Nebula (ne ∼ 0.2 cm−3 ), the density in the solar neighborhood being only a tenth of this. Further refinements have recently been added to the model by Lazio and Cordes (1998) in the treatment of the outer galaxy, by including such effects as flaring (i.e., scale height increasing with r) and warping (i.e., the disk not being strictly planar) of the galactic disk, and investigating whether the radial distribution is better described by the sech2 -distribution of Eq. (1.18) or by a disk truncated at about 15 kpc.
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Galactic pulsar distribution
We can now discuss the distribution of rotation-powered pulsars in terms of the galactic coordinates r and z, and the pulsar luminosity L. This distribution ρ(r, z, L) is taken to be separable in r, z, and L, because of the lack of correlations between these variables demonstrated in the early work on pulsars [Manchester & Taylor 1977], so that ρ(r, z, L)drdzdL = ρr (r)drρz (z)dzΦ(L)
dL . L
(1.22)
Note that the L-distribution is defined to be that over a unit logarithmic interval. Consider the galactic distribution first. The distribution perpendicular to the galactic plane is usually described in terms of an exponential, ρz (z) ∝ exp(−|z|/h), with a scale height of h. Recent estimates of h are in the range 400–500 pc [Lyne & Graham-Smith 1990; McLaughlin & Cordes 2000]. In view of the discussion in the previous subsection, it must be clear that a sech2 -distribution can also be used. Note that the scale height for pulsars is much higher than that of galactic Population I objects (massive OB stars which produce neutron stars through supernovae; see Appendices A and C, and Chapter 6) which is ∼ 50–100 pc. This is understood in terms of the high pulsar velocities: it is believed that they have traveled away from the galactic plane, their birthplace, due to such velocities received as “kicks” at their births in supernova explosions. The luminosity distribution is described over a considerable range (∼ 3 decades) by a power-law, Φ(L) ∝ L−n , with n ≈ 1. Note that, for rotation-powered pulsars, the “luminosity” is often defined by L = S400 d2 , where S400 is the mean flux density at 400 MHz. Since S is measured in units of milliJansky or mJy (1 Jansky ≡ 10−23 erg s−1 cm−2 Hz−1 ), and d in kpc, the units of L are then mJy kpc2 . To convert this into the more conventional units of erg s−1 (which are used for accretion-powered pulsars), we need to integrate over the bandwidth of radio frequency and the solid angle of the pulsar beam. Assuming a bandwidth of 400 MHz and a conical beam of width 10◦ , 1 mJy kpc2 translates into 3.4×1025 erg s−1 [Manchester & Taylor 1977]. Limiting sensitivities of pulsar surveys are typically ∼ 0.1–0.3 mJy kpc2 , which are sufficient to show clearly that the distribution at very low luminosities falls well below that predicted by the above power-law. This is as expected, since a distribution ∝ L−2 dL at arbitrarily low luminosities would give a divergent total population upon integration.
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The distribution in r (galactocentric radius projected onto the galactic plane) shows a monotonic decrease outward from r ∼ 4 kpc, with very few pulsars detected beyond 15 kpc. Inside 4 kpc, the distribution is poorly determined. It is not entirely clear if (a) there is really a large inward drop in the pulsar density from this radius, or, (b) if pulsars are harder to detect in this direction because the limiting flux at which they can be detected increases inward due to (i) the higher noise-level in a radio telescope looking towrad the galactic center because of a high level of background radiation, and (ii) much stronger interstellar scattering in the innermost parts of the galaxy, which disperses the pulses more and makes them harder to detect. In reality, all of these effects could act simultaneously. In any case, there is some indication that the pulsar density does actually decrease inward inside ∼ 4 kpc. A simple distribution used in recent work [McLaughlin & Cordes 2000] is a superposition of two Gaussians, ρ(r) ∝ exp[−(r/6)2 ] + 0.25 exp[{−(r − 4)/1.5}2],
(1.23)
the first a disk of radial scale length 6 kpc, and the second an annulus at 4 kpc with a width of scale 1.5 kpc. This will remind the reader of the radial distribution of galactic electron density described in the previous subsection. Indeed, the annular distribution is inspired by the molecular ring mentioned there. Integration over the above distributions gives an estimate of ∼ 5 × 104 − −1 × 105 for the total number of observable pulsars in the galaxy above a threshold luminosity corresponding to the typical limiting sensitivity quoted earlier. To obtain the total of number of active pulsars in this luminosity range, we have to correct this by a beam-width factor, yielding a number ∼ a few times 105 . The fact that the r-distribution is poorly determined at small radii (r < 4 kpc) does not have a large effect on this estimate, since the integration over area is weighted as rdr, and the area inside the above limiting radius is not a large fraction of the total area (r ≤ 15 kpc, say) of the effective disk [Manchester & Taylor 1977]. The surface density of observable pulsars in the galactic disk in the vicinity of the Sun (above the same luminosity threshold) is estimated to be ∼ 70 pulsars per square kpc. 1.5.1.4
Selection effects
Selection effects are observational biases of various kinds introduced by limitations or discriminatory choices inherent in our detection procedures.
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A universal, and almost trivial, example is luminosity bias. All detection equipment has a limiting sensitivity, so that our observations are necessarily cut off at that limit: if we were to conclude from this that no sources exist below that luminosity, we would be completely wrong. Like other astronomical observations, pulsar observations are plagued by several selection effects, of which we give a few examples here. The classic example is that of the distribution of pulsars projected onto the plane of our galaxy (i.e., a distribution in r and θ, in terms of the coordinates defined earlier). This shows a huge concentration of pulsars around the Sun [see, e.g., Fig. 8.2 in Lyne & Graham-Smith 1990], which is, of course, not real: it merely reflects the fact that pulsars are harder to detect when they are farther away from us, since the flux detected at Earth goes down as the inverse square, d−2 , of the pulsar’s distance from us. Indeed, the observed pulsars represent only that small minority of the galactic population of pulsars which have very high luminosity and so are detectable from the Earth. Conversely, although only a small percentage of the pulsars discovered by our radio telescopes have low luminosities (∼ 0.3– 3 mJy kpc2 , say), they represent the overwhelming majority of the galactic population [Lyne & Graham-Smith 1990]. In order to determine the true distribution, this selection effect has to be taken into account. Another well-known example can be seen in Fig. 1.6, namely the apparent concentration of pulsars near the galactic longitude l = 50◦ . This is also not real, but only reflects the fact that the famous Arecibo survey of pulsars was very deep (i.e., went down to very low limiting fluxes), but covered only this region of the sky, so that it preferentially detected a large number of pulsars only in this region. A diagnostic use of the cumulative number of pulsars N (d) with distances ≤ d serves as a further example. The reader can easily show that, if the pulsars were distributed uniformly through a galactic disk of half-thickness h, then N should scale as d3 for d < h, and as d2 for d > h. The observed N (d) vs. d distribution [Manchester & Taylor 1977; Beskin et al. 1993] roughly follows this scaling upto a limiting distance dl , at which point the curve drops below the d2 -line. This only means that a significant fraction of pulsars begins to become undetectable beyond dl due to the limiting sensitivity. Entirely as expected, dl is found to be larger for pulsar surveys which are more sensitive (i.e., which have lower thresholds of detection): Beskin et al. (1993) illustrate this point with the fact that the Jodrell Bank I survey, with a threshold flux density ∼ 15 mJy, had
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dl ∼ 1 kpc, while the more sensitive Massacusetts-Arecibo I survey, with a tenfold lower threshold, had dl ∼ 5 kpc. 1.5.2
Accretion-Powered Pulsars
Of the ∼ 110 known accretion-powered pulsars today, ∼ 2 are in our neighboring Andromeda galaxy (M31), ∼ 1 in M33, ∼ 32 are in the Small Magellanic Cloud (SMC), ∼ 6 in the Large Magellanic Cloud (LMC), and ∼ 68 in our Milky Way. Thus, some ∼ 40 such pulsars have been found in the Magellanic Clouds: these pulsars have been discovered in profusion only in the last few years, before which there were only about 3 such known [Nagase 2001], although the discovery of the first such source, SMC X-1, goes back to 1976, i.e., relatively early days of X-ray astronomy. Some essential parameters of these pulsars are listed in Table E.1 in Appendix A. Note that the SMC/LMC pulsars, and a few well-known individual pulsars like Her X-1 and X Per, are significantly away from the galactic plane, but the majority of the accretion-powered pulsars are in, or close to, the galactic plane. Actually, it is the Massive (or High-Mass) X-ray Binaries (HMXBs for short; see Chapter 9), which follow the disk population of massive stars in our galaxy, since these are their progenitors. The LowMass X-ray Binaries (LMXBs for short; see Chapter 9) are largely a bulge population, following the distribution of old, low-mass stars in the bulge of our galaxy. Since HMXBs are, by far, the dominant population among accretion-powered pulsars, the whole population also roughly follows the galactic disk. The distribution of accretion-powered pulsars in galactic coordinates is shown in Fig. 1.8.
1.6
Period Distributions of Pulsars
The distribution of the pulse periods of rotation-powered pulsars is shown in Fig. 1.9 [Manchester et al. 2005]. As these authors point out, the distribution is clearly bimodal, with a dichotomy between “normal” pulsars in the period range of about 0.1 second to 10 seconds, and recycled “millisecond” pulsars in the period range of about 1 to 10 milliseconds. Evolutionary origins of this, and related phenomena, will occupy us over much of this book. Further, as these authors also stress, binary rotation-powered pulsars have pulse periods predominantly in the millisecond range, a major key to the
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Accretion Powered Pulsars
Her X-1
LMC
SMC
Fig. 1.8 Distribution of accretion-powered pulsars in the Galaxy, in the Small Magellanic Cloud (SMC), and in the Large Magellanic Cloud (LMC) in galactic co-ordinates (see Appendix A), using a Hammer-Aitoff (equal-area) projection. Asterisks: pulsars in HMXBs. Diamonds: pulsars in LMXBs (including systems like Her X-1; see Chapter 9). Squares: pulsars in binaries whose natures are not certain yet (see Table E.1). Labeled are some sources significantly off the galactic plane: Her X-1, SMC, and LMC. The numerous pulsars in SMC appear as a cluster, and similarly for LMC. Accretion-powered millisecond pulsars (see Chapter 6) form a cluster around the galactic center, and some have been omitted for clarity. Not shown are AXPs (see Chapter 9), and pulsars in M31 and M33. Data partly from Nagase (2003).
idea of recycling, which we discuss in Chapter 6, and elsewhere in the book. Expected rotation periods of nascent neutron stars are considered in the discussion in Chapter 6 of the birth of neutron stars in supernovae. The distribution of the pulse periods of accretion-powered pulsars is shown in Fig. 1.10 [Laycock et al. 2005], wherein pulsars in the Galaxy, SMC, and LMC are included, as indicated. Barring one or two very short and long-period pulsars, the whole distribution runs from about half a second to several thousand seconds. The galactic pulsar population appears skewed toward longer periods compared to the SMC population, which could be a selection effect, but is not really clear at present, as discussed by these authors. Evolutionary and dynamical origins of the pulse periods
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observed in accretion-powered pulsars are discussed in Chapters 6, 9, and 12. One of the major consequences of the recycling scenario, as discussed in Chapter 6 and further elaborated on in Chapter 12, is a comprehensive account of how a spinning neutron star in a binary functions first as a rotation-powered pulsar somewhere in the right-hand part of the distribution of Fig. 1.9, then as an accretion-powered pulsar somwhere in the distribution of Fig. 1.10, and finally, after recycling, again as a rotationpowered pulsar, but now somewhere in the left-hand part of Fig. 1.9. This is, undoubtedly, a major road-marker in today’s high-energy astrophysics.
Fig. 1.9 Distribution of the pulse periods of rotation-powered pulsars and AXPs (see Chapter 9). Reproduced by permission of the AAS from Manchester et al. (2005): see Bibliography.
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Fig. 1.10 Distribution of the pulse periods of accretion-powered pulsars in the Galaxy, the Small Magellanic Cloud (SMC) and the Large Magellanic Cloud (LMC), as indicated. Reproduced by permission of the AAS from Laycock et al. (2005): see Bibliography.
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Chapter 2
Physics of Neutron Stars — I Degenerate Stars
2.1
Historical Notes on Neutron Stars
While reminiscing at a Solvay Conference in 1973 a few months before his death, Rosenfeld (1974) recalled a lively discussion in Bohr’s institute on that day in 1932 when the news of Chadwick’s discovery of the neutron reached from Cambridge to Copenhagen. During this discussion, Landau, who was visiting the institute, proposed the idea of Unheimliche Sterne (Strange1 Stars), which Rosenfeld identified with neutron stars. Landau thought that these stars would normally be invisible, showing their existence only by causing explosions when they collided with normal stars, and that these explosions could be supernovae. It is unclear if Landau was really thinking about what we call neutron stars today. According to Rosenfeld, Landau then published a paper in which he mentioned neutron stars and (mistakenly) thought that they were systems “to which quantum mechanics would not be applicable”. Actually, the Landau (1932) paper to which Rosenfeld was apparently referring does not mention neutron stars at all. It is essentially a re-derivation of Chandrasekhar’s (1931a,b,c) celebrated result on the maximum mass (∼ 1.4M) of white dwarfs (i.e., stars which have exhausted their nuclear fuel and are supported by the pressure of degenerate electrons; see Sec. 2.4), the Chandrasekhar limit, above which white dwarfs cannot exist, and matter collapses further (see below). Landau’s arguments, which we summarize in Sec. 2.3, were brilliant in their intuitive appeal, and provided the first physical explanation of the mass 1 The
adjective unheimlich normally translates as “weird” or “sinister”: Rosenfeld chose the first meaning. In the neutron-star context, a more suitable adjective would have been “strange” or “exotic”, whose German equivalent would be closer to unheim. Why Landau applied a negative connotation is unknown, assuming that the Rosenfeld quotation is accurate. 39
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limit to white dwarf stars which is now named after Chandrasekhar. However, there appears to be no mention of neutronized matter in stars in the papers of either Landau2 or Chandrasekhar until about 1938. The name “neutron star” was coined by Baade and Zwicky in 1934 while attempting a connection between the intensity of cosmic rays observed on the Earth, and the energy released in supernova3 explosions of stars, whose study they had pioneered. They made the following, justifiably famous, statement: “With all reserve we advance the view that supernovae represent the transitions from ordinary stars into neutron stars, which in their final stages consist of extremely closely packed neutrons.”, which has been called “prescient” [Shapiro & Teukolsky 1983]. Indeed, it appears today that the statement did not come from an understanding of what we believe now to be the correct supernova mechanism (see below and Chapter 6). Due to a gross overestimate of the energy output of supernovae, these authors came to the (incorrect) conclusion that “in the supernova process mass in bulk is annihilated” (Baade & Zwicky 1934a,b,c) and converted into the energy of supernova explosions, and went on to suggest that the remainder of the star’s mass would become a neutron star. Further, it also seems that Baade and Zwicky had no inkling of the physical reason why stellar matter must consist mostly of neutrons at extremely high densities4 . Ironically, the fundamental physics that makes the process of neutronization inevitable at very high densities of stellar (or other) matter, namely, that it is then energetically favorable for electrons to combine with the protons in the nuclei to produce neutrons (a process called inverse β-decay), had already been discovered in 1933 by Sterne, but not appreciated fully. Sterne (1933) constructed a statistical theory of the equilibrium nuclear abundances in stellar matter, based on the properties of nuclei and the 2 Landau did state in his 1932 paper that “all stars heavier than 1.5M certainly pos sess regions in which laws of quantum mechanics (and therefore of quantum statistics) are violated”, and did suggest that this might “occur when the density of matter becomes so great that atomic nuclei come in close contact, forming one gigantic nucleus”. However, this had nothing to do with neutronized matter. While writing his 1932 paper (which was done before Chadwick’s discovery of the neutron in the same year), Landau, like most physicists of his time, still thought that atomic nuclei were composed of protons and electrons, as a glance at his paper will show. Perhaps we should take Landau’s speculations metaphorically, not literally. 3 The name “supernova” was also coined by Baade & Zwicky. 4 In fact, Baade and Zwicky (1934c) made such statements as: “If neutrons are produced on the surface of an ordinary star they will “rain” down towards the center...”.
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neutron known at the time. He expressed one “conclusion of a simple nature” from the graphical display of his results, reproduced in Fig. 2.1, as: “At these high densities, matter at low temperatures would be literally squeezed together into the form of neutrons.”, which, in retrospect, we can only call profound: it was not speculative, but the result of quantitative calculations. Naturally, Sterne’s estimate of the lower bound on the density above which matter would be extremely 10 −3 neutron-rich (ρ > ∼ 2.3 × 10 g cm ) is roughly consistent with the results of modern calculations (see Sec. (3.3)). Even his words for describing the neutronization process, as quoted above, were so accurately picturesque that Gamow (1939) later adopted them almost verbatim to write “electrons are, so to speak, squeezed into the nuclei”, an imagery we continue to use to this day. But the profound astrophysical implications of this result were lost on Sterne himself. In his excellent 1936 review entitled “Matter under very high pressures and temperatures”, Hund gave a thorough discussion of Sterne’s results, and displayed, in a pioneering fashion, the equation of state (i.e., the pressure vs. density relation) of high-density matter, complete with the “kink” in it that accompanies neutronization. Hund also coined the term Dichte
Fig. 2.1 Equilibrium nuclear abundance calculated by Sterne (1933). Shown are the “abundance lines” for various species. x is a rough measure of the density, and −y a rough measure of the abundance in completely degenerate (T → 0) matter. Since the line for neutrons has the highest slope, it follows that neutrons will dominate the matter at sufficiently high densities. Reproduced by permission of Blackwell Publishing from Sterne (1933): see Bibliography.
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Sterne, which roughly translates as “Compact Stars” – a name which is universally used today for degenerate stars, and he discussed the significance of the Landau mass scale (see Sec. 2.3) and the gravitational radius associated with it. Ironically, again, Hund failed to recognize the possibility of a new kind of stars supported by the pressure of degenerate neutrons, considering neutronization only as a perturbation in the dense regions of white dwarfs, and discussing the collapse of more massive stars into black holes (see below). It remained for Gamow (1936) to make the brilliant astrophysical connection, suggesting in the following words that neutron cores would be formed at the centers of sufficiently massive stars after exhaustion of their nuclear fuel: “For still higher densities electrons will probably be absorbed by the nuclei (an inverse β-decay process) and the mixture will tend to a state which can be described very roughly as a gas of neutrons. For densities of the order of magnitude ρ ∼ 1012 gm/cm3 (average density of atomic nuclei) nuclear exchange forces between the gas particles will come into play and the conditions in the ‘gas’ will become analogous to the conditions inside an atomic nucleus. Such an extreme state of matter we shall call the ‘nuclear state’ and the region of the star occupied by such nuclear matter the ‘stellar nucleus’.” In 1938, Landau considered neutron cores of stars, estimating the minimum mass above which neutronized matter would be more stable than “normal” matter consisting of nuclei and electrons, and stating that he had in 1932 “already shown that the formation of a core must certainly take place in a body with mass greater than 1.5M ”. Today, Landau’s 1938 statement could be dubbed fanciful (see above), were it not for the crucial historical fact that it inspired Oppenheimer and Volkoff (1939) to do their seminal work on neutron stars, which has since served as the point of departure for all modern work. Still thinking of neutron cores rather than neutron stars, Oppenheimer and Volkoff scrutinized the ∼ 1.5M Landau limit in the light of the most accurate treatment of the essential physics possible at the time. They noted first the minor point that the Landau limit should really be ∼ 5.7M for neutrons, and not the ∼ 1.4M value relevant for white dwarfs (Gamow had made the same error in 1936), since the limit goes as the inverse square of the “mass per particle obtained by spreading out the total mass over only those particles which essentially determine the pressure of the Fermi gas”: this mass is approximately twice the nucleon mass for white dwarfs (since, on the average, there are roughly
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two nucleons per electron in white dwarf matter, and degenerate electrons supply the pressure), but equal to the nucleon mass for neutron stars (since their matter is mostly neutrons, which themselves supply the pressure). But it was the masterly handling of two major points of the physics of neutron stars by Oppenheimer and Volkoff that made their paper a milestone. These were: (1) that at the enormous densities of neutron stars, Newtonian gravitational theory was not adequate, and general relativistic effects must be included, and, (2) that the neutrons, because of their large mass, would be non-relativistically degenerate, in contrast to the relativistically degenerate electrons (see Sec. (2.4)) in white dwarfs, and the correct equation of state must be used. The inclusion of these effects (actually, Oppenheimer and Volkoff used an equation of state valid for both non-relativistic and relativistic regimes) brought the above limit down to ∼ 0.7M: above this mass, no equilibrium configuration would be possible, and the core would continue to collapse indefinitely. Below this limit, which now bears the name of Oppenheimer and Volkoff, their quantitative calculations of the masses and radii of neutron stars—the first ever done—stood for twenty years before they were improved upon. These improvements came from one crucial point of physics that was still missing from their treatment, namely, the interaction between the neutrons: by assuming a Fermi gas of free neutrons, Oppenheimer and Volkoff had neglected this completely. In an era when children learn in school that stars produce energy by converting hydrogen to helium through thermonuclear reactions (see Appendix C), it may be difficult for us to believe that the main motivation behind the original interest of Landau, Gamow (1936), and others in neutron cores was their possible role as stellar energy source. In the period 1937-39, however, the stellar-energy problem was satisfactorily solved by the pioneering and classic works of von Weizs¨ acker, Bethe, and others on stellar thermonuclear reactions. After this, the above motivation vanished, and, with it, most of the interest in neutron stars. More disastrous was the fact that Oppenheimer, who, by 1939, had accepted the central role of thermonuclear energy generation in stars, and was studying neutron cores for exploring their role in stellar evolution after the exhaustion of nuclear fuel, himself came to the (manifestly incorrect) conclusion that “it seems unlikely that static neutron cores can play any great part in stellar evolution” [Oppenheimer & Volkoff 1939], and never returned to the subject. This conclusion was based on the fact that the Oppenheimer-Volkoff limit, as calculated by these authors themselves (∼ 0.7M), was considerably lower than the Chandrasekhar limit (∼ 1.5M ). Since the latter limit had to be
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exceeded before the collapse could proceed further towards the formation of a neutron core, and since such a neutron core would be unstable because it was above the former limit, it was not clear how neutron cores would ever be produced in nature. It was to be two decades before this paradox would be resolved by the inclusion of neutron-neutron interactions (see below), which showed that the Oppenheimer-Volkoff limit was, in fact, above the Chandrasekhar limit, so that there was no problem in forming neutron cores or stars. Indeed, we are well aware today of the key roles played by neutron cores and neutron stars in the late stages of stellar evolution, without which there would be no need to write this book. Physicists will be thankful today for Zwicky’s missionary service for the cause of neutron stars, but for which there was little mention of these Unheimliche Sterne anywhere until the late 1950s. When progress did come to the field in 1958, it was actually as a sidelight of a “resurvey” of the basic aspects of gravitational collapse by Harrison, Wakano, and Wheeler (1958), in which they attempted to “connect up” (for purposes of completeness) the Chandrasekhar limit and the Oppenheimer-Volkoff limit. However, the resulting construction of the equation of state of matter from ordinary to super-nuclear densities (spanning about twenty orders of magnitude) was pioneering, as were the computations of the masses and radii of collapsed objects over this entire range of densities (also see Harrison, Thorne, Wakano, and Wheeler 1965, henceforth HTWW, and Wheeler 1966). These partly inspired the next stage of development, and the detailed work on neutron stars in the post-pulsar “modern” era (see below) has often used these early models as reference points. In 1959, Cameron took the crucial next step of introducing neutronneutron interactions into neutron-star calculations (also see Ambartsumyan and Saakyan 1960). Using the effective nuclear potential (then newly) proposed by Skyrme (1959), Cameron (1959b) showed that the OppenheimerVolkoff limit was raised to ∼ 2M , i.e., above the Chandrasekhar limit, aptly calling the result “very gratifying”, since it solved the Oppenheimer paradox5. Cameron (correctly) speculated that the latter result would be a general one, and consequently that there would be no difficulty in forming neutron stars in supernovae: modern calculations have borne this out. Further, confirming the visionary comments of Gamow (1939), Cameron described the basic scenarios (1959a) for (a) the chain of nuclear reactions in the stellar core preceding a supernova, (b) the collapse of the neutron 5 After Saakyan (1963) corrected an error in Cameron’s treatment of energy density, the limit came down to ∼ 1.7M , but still remained above the Chandrasekhar limit.
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core, and, (c) the expulsion of the rest of the star by the energy released in the collapse. These appear to be the seeds from which the subsequent, detailed, computational scenarios for the birth of neutron stars in supernova explosions and core collapses, such as those pioneered by Colgate and co-authors in the 1960s [Colgate & White 1966], have grown. Research on neutron-star models was continued in the mid-1960s by Tsuruta and Cameron (1965), who used the more elaborate Levinger-Simmons nuclear potential, by Saakyan and co-authors [Saakyan & Vartanyan 1964], who used their own nuclear potential, and by Salpeter and co-authors [Hamada & Salpeter 1961]. It was primarily the discovery of cosmic X-ray sources outside the solar system [Giacconi et al. 1962], a major event in high energy astrophysics (see Sec. 1.2), that sustained practical interest6 in neutron stars during the above era. The principal motivation at the time was the possibility that the observed X-rays from the discrete X-ray sources could be thermal radiation from young, hot neutron stars, which prompted calculations of the cooling rates of such neutron stars [Bahcall & Wolf 1965a; Tsuruta & Cameron 1965]. As we know today, the X-ray sources discovered in this era were largely powered by either (a) the release of gravitational energy during accretion onto neutron stars, for point-like sources, or, (b) thermal and nonthermal emission from hot plasmas containing magnetic fields in supernova remnants (like the Crab nebula), for extended sources. However, these cooling calculations did prove useful later. Another major event, namely, Schmidt’s 1963 discovery of quasars, also spurred brief interest in neutron stars, because of a mistaken belief. It was thought at first that the large redshifts of quasar spectral lines could be of general-relativistic origin, arising in the strong gravity on the surfaces of neutron stars, similar to what Zwicky had (also mistakenly) proposed in the 1930s as the cause of the line-shifts in the spectra of supernovae. This idea was soon abandoned as it became clear that the largest quasar redshift observed by then (small compared to the largest value known today) already exceeded the maximum redshift possible from neutron-star surfaces. It remained for the discovery of rotation-powered pulsars in 1967 and their rapid identification with neutron stars (see Chapter 1) to finally catalyze an era of intense research on the physics of neutron stars: this can be called the “modern” era of neutron-star research. An historical irony regarding the identification of neutron stars with pulsars is famous: this 6 See
the Bahcall-Wolf comment quoted in Sec. 1.1.
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concerns Pacini’s classic 1967 paper, which had appeared about two weeks before Bell detected the first pulses from a radio pulsar (see Sec. 1.1). Pacini (1967) had, in this paper, estimated the rate of emission of electromagnetic radiation from a rotating neutron star with a dipolar magnetic field. Yet, his paper was unknown not only to the radio-astronomers who discovered the first pulsars [Hewish et al. 1968], and did not even consider the possibility in their discovery paper that the rotation of neutron stars could be the cause of the pulses, but also to Gold (1968), who did make the definitive connection between pulsars and rotating neutron stars in his seminal paper. At first, this may sound unbelievable, since all of the above scientists published their works in the same journal (Nature) within a period of about seven months, and further since Pacini and Gold were apparently at the same academic institution at the time [Lyne & Graham-Smith 1990]. However, we must not forget that no one was expecting observable periodic pulses of electromagnetic radiation from neutron stars in 1967. Pacini (1967) was only interested in the consequences of the absorption of electromagnetic radiation from neutron stars by their supernova remnants, such as the Crab Nebula. Intrigued by an earlier suggestion that vibrations of a magnetized neutron star could be this source of electromagnetic radiation, Pacini was actually looking for an alternative source because the original suggestion had been found to be untenable by then, and found this in the magnetic dipole radiation from a rotating neutron star. Radio astronomers have explored possible causes for why pulsars had not been discovered before 1967, although radio telescopes capable of doing so had existed since the 1950s [Manchester & Taylor 1977; Lyne & Graham-Smith 1990]. A similar attempt to explore possible causes of the inadequacy of human imagination, which had failed to predict that periodic electromagnetic pulses from neutron stars could reach our detectors on (or near) the Earth before such pulses were actually discovered, would be futile. The fact remains that these clock-like pulses across the electromagnetic spectrum, from radio to γ-rays, are the best signature for identifying neutron stars today: these pulses tell us the rotation periods of neutron stars, and the changes in the periods of the pulses give us valuable information on the magnetic field and internal structure of neutron stars, and on the flow of matter around these stars when they occur in accreting binary systems. The pulses are, of course, the origin of the name “pulsar”, which has now become almost synonymous with the words “neutron star”.
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2.2
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Degenerate Stars
A normal star like the Sun generates thermal energy through thermonuclear reactions in its interior. Its size is determined by a balance between the selfgravitational attraction of the stellar mass, which tends to contract the star to its center, and the thermal pressure, which tends to expand it outward. When its nuclear fuel is exhausted completely, the star has no further source of energy. It turns colder and contracts, converting gravitational potential energy released in the infall into thermal energy, and radiating it away (see Appendix C). But the star cannot go on contracting indefinitely in this manner, extracting its own gravitational potential energy and shrinking to arbitrarily small radii. At sufficiently high densities, stellar matter becomes degenerate7 , following quantum statistics (i.e., Fermi-Dirac statistics for particles with half-integral spin, called fermions, which the stellar matter is composed of) instead of the classical Maxwell-Boltzmann statistics.
2.2.1
Degeneracy
In statistical mechanics, the volume in the six-dimensional phase space (i.e., the space of coordinates and momenta) available to structureless particles of mass m with momenta between p and p + dp is 4πV p2 dp, where V is the physical volume occupied by the particles. For fermions with spin 12 , such as electrons, protons, and neutrons, this expression is multiplied by a factor 2, corresponding to the two possible orientations of such a spin. According 7 The use of the word ‘degenerate’ here is in its original etymological sense of “deviant from the ideal situation” (with an implied derogatory connotation), the ‘ideal’ in this case being the classical Maxwell-Boltzmann statistics. There is nothing ‘ideal’ about Maxwell-Boltzmann statistics, of course; it is only a useful limiting case. Further, there is an unfortunate confusion between this usage and that of the same word ‘degenerate’ for a wholly different quantum-mechanical phenomenon, in which the eigenvalues corresponding to two or more distinct eigenstates are identical. This confusion arose from an indiscriminate use of the words ‘degenerate’ and ‘degeneracy’ (actually, their German equivalents entartet and entartung respectively, as much of the original work on both quantum statistics and quantum mechanics was published in German) in both contexts in the early days of quantum mechanics. The usage in quantum statistics was apparently pioneered by Fermi (1926), and continued extensively by Sommerfeld (1928) in his classic study of degenerate electrons in metals. The usage in quantum mechanics, itself more emotive than scientific, can be clearly seen in the pioneering work of Born, Heisenberg and Jordan (1926) on quantum-mechanical perturbation theory. In his famous book on quantum mechanics, Dirac (1958) remarked about this latter usage that “these words are not very appropriate from the modern point of view”, but no one apparently took heed.
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to the simple description of quantum statistics pioneered by Einstein8 , the phase space can be pictured as being divided into ‘cells’ of volume h3 , h being Planck’s constant, and the three factors of h corresponding to the three translational degrees of freedom, since each such degree yields dpdq ∼ h according to quantum mechanics. The number a of cells available to particles in this energy range is thus a = 4πV p2 dp(2/h3 ), and the problem of distributing s such particles into these cells becomes a combinatorial one, the result of which depends on what rules we follow about how many particles can go into a given cell. Two situations arise naturally, viz., (1) there is no limit to the number of particles that a cell can contain, so that each cell can have 0, 1, 2, ... s particles, or, (2) a cell can contain no more than one particle (Pauli’s exclusion principle), so that each cell can only have 0 or 1 particle. The former situation leads to Bose-Einstein statistics, applicable to particles with zero or integral spin, e.g., photons (which are described by symmetric quantum-mechanical wave functions), and the latter situation to Fermi-Dirac statistics, applicable to particles with half-integral spin, e.g., electrons, protons, and neutrons (which are described by antisymmetric wave functions). The classical Maxwell-Boltzmann statistics is the limit of a s: both Bose-Einstein and Fermi-Dirac statistics then reduce to the classical one, since the above rules for putting particles into cells make little difference in this limit, as must be clear on a little reflection. For a given statistics, the distribution function f is the average occupation number of a cell (i.e., the average number of particles in it), as defined above, in that statistics. The distribution function is normally expressed in terms of the kinetic energy E ≡ p2 c2 + m2 c4 − mc2 rather than the momentum p. In the non-relativistic (NR) r´egime, where p mc, E reduces to E ≈ p2 /2m, while in the extremely relativsitic (ER) r´egime, where p mc, E is approximately given by E ≈ pc. For the classical Maxwell-Boltzmann case, the distribution function has the form fMB (E) ∝ exp(−E/kT ), where k is Boltzmann’s constant, and T is the temperature of the system. In Fermi-Dirac statistics, f (E) is given by fFD (E) ≡
1 A
exp
1
E kT
+1
,
(2.1)
where A is the degeneracy parameter introduced by Fermi (1926) and Sommerfeld (1928) to facilitate comparison with the classical MaxwellBoltzmann statistics. For A 1, if we multiply the numerator and the 8A
more rigorous approach is through the Schr¨ odinger equation; see Fowler 1927.
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denominator of the right-hand side of Eq. (2.1) by A and neglect higher powers of A, we recover the classical statistics: fMB (E) = A exp[−E/kT ].
(2.2)
Small values of A thus correspond to classical, or non-degenerate, behavior, and large values of A to highly degenerate behavior. The boundary between the two r´egimes is approximately determined by the condition A = 1. The degeneracy parameter is often recast in terms of a reference potential µ as A ≡ exp(µ/kT ), µ being called the chemical potential, and the Fermi-Dirac distribution written in the standard form fFD (E) ≡
1 1 + exp
E−µ kT
.
(2.3)
The functional form on the right-hand side of Eq. (2.3) is now called the Fermi function; it is also useful in other contexts in physics, as we shall see later in the book. Classical behavior then corresponds to large negative chemical potentials, and highly degenerate behavior to large positive ones, the transition region being given by | µ | kT . The degeneracy parameter A is a function of the mass m and the number density n of the particles, and of the temperature T . The value of A can be determined by noting that the total number, N = nV , of the particles in the volume V is obtained by multiplying the number of phase-space cells a in the momentum range p to p+dp, as given above, by the distribution function f given in the previous paragraph, and integrating over all momenta. For the classical distribution (given by Eq. [2.2]) in the NR r´egime, this yields A=
nh3 . 2(2πmkT )3/2
(2.4)
Strictly speaking, this equation is valid only in the limit A 1 and in the NR r´egime, but it often serves as an excellent estimate of A for deciding whether a given system is degenerate or not9 . If A, as calculated from Eq. (2.4), is 1, the system is non-degenerate; if it is 1, the system is degenerate. It is clear from Eq. (2.4) that systems at low densities and high temperatures are non-degenerate, while those at high densities and 9 The exact value of A for the Fermi-Dirac distribution of Eq. (2.1) can, of course, be determined by numerical integration. For the NR r´egime, Sommerfeld (1928) has given an analytic series solution for A 1. The corresponding result for the ER r´egime has been given by Chandrasekhar (1931c).
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low temperatures are degenerate. Further, at the same number density and temperature, lighter particles are more degenerate than heavier ones. 2.2.2
Electron Degeneracy
The classic use of the above degeneracy criterion is the 1928 demonstration by Sommerfeld that the electron gas in metals is highly degenerate even at normal temperatures, because of the smallness of the electron mass. This follows immediately from Eq. (2.4) on substituting the electron mass for m: we get A ≈ 2 × 10−16ne T −3/2 , where ne is the number density of electrons. If we approximate the number density of conduction electrons by that of the metal atoms, nM = ρNA /A, where ρ is the metal’s density in g cm−3 , A is its atomic weight, and NA is Avogadro’s number, we get ne ≈ 6 × 1022 cm−3 for silver with ρAg = 10.5 and AAg = 107.9. Thus, at T = 300 K, the degeneracy criterion for silver stands at AAg ≈ 2.4 × 103 , so that the conduction electrons in it are very degenerate indeed. What about the degeneracy of electrons in stars? Consider a spherical star of total mass M and radius R in hydrostatic equilibrium, the condition for which in the radial direction is: Gm(r) dP (r) = −ρ(r) . dr r2
(2.5)
Here, P is the total pressure, ρ is the density, and m(r) is the mass contained within a radius r from the center of the star, all of which are functions of r in general. From Eq. (10.35), which is a local equation valid at every point inside the star, we can obtain a global condition which is approximately valid for the star as a whole. This very useful estimate, which is widely used in astrophysics, proceeds as follows. We replace the local variables P and ρ by their average values P¯ and ρ¯ ≡ 3M/4πR3 over the whole star, and the derivative d/dr by a dimensional estimate ∼ 1/R of its average value. If, further, we consider a star whose pressure is dominated by that of its gas content, we can write P¯ ∼ 2¯ nk T¯ , and Eq. (10.35) yields k T¯ ∼
GM mp . 2R
(2.6)
Here, n ¯ = ρ¯/mp is the average number density of protons, mp being the proton mass, and we are considering for simplicity a gas consisting predominantly of hydrogen. Eq. (2.6) is the useful dimensional estimate of the average stellar temperature we were seeking. We now return to Eq. (2.4), which gives the degeneracy parameter for electrons with m = me , n = ne ,
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and T = Te . Using ne ≈ n ¯ and Te ≈ T¯ as given by Eq. (2.6), we obtain an estimate of the average electron-degeneracy parameter of the star: 1/2 −1 ρ¯ M (3ρ¯)1/2 h3 A∗ ∼ ∼ . (2.7) 5/2 3/2 2 × 104 g cm−3 M 2π 2 G3/2 M mp me The crucial role of stellar density in deciding electron degeneracy in stellar matter is evident from Eq. (2.7). A normal star like the Sun with ρ¯ ∼ 1 g cm−3 has A∗ ∼ 10−2 , and so is non-degenerate, as expected. A white dwarf star with ρ¯ ∼ 106 g cm−3 (see below) and mass comparable to that of the Sun has A∗ ∼ 10, so that the electrons in it are degenerate. Indeed, it is the pressure of the degenerate electrons that balances the self-gravitational attraction of matter in a white dwarf, leading to an equilibrium stellar configuration, as we shall see in detail in Sec. (2.2.3). The transition from non-degenerate to degenerate electrons, roughly given by A∗ ∼ 1, occurs at an average stellar density ∼ 2 × 104 g cm−3 for solar-mass stars. 2.2.3
Neutron Degeneracy
Sidestepping, for the moment, the details of the circumstances under which stellar matter may come to consist mostly of neutrons (to which we return in later sections), we can now ask the question: can such neutronized stellar matter become degenerate, and, if so, at what densities? Eq. (2.4) readily provides a crude but useful answer. Setting m = mn in it, where mn is the mass of the neutron, and following a procedure otherwise identical to that used above for electron degeneracy, we obtain the following estimate of the average neutron-degeneracy parameter: 1/2 −1 ρ¯ M (3ρ¯)1/2 h3 ∼ . (2.8) A∗ ∼ 2 3/2 1014 g cm−3 M 2π G M m4n It is clear from Eq. (2.8) that neutronized stellar matter becomes degenerate 14 −3 at ρ¯ > ∼ 10 g cm , which are just the typical average densities of neutron stars (see below). Thus, neutron-star matter is expected to be degenerate. 2.2.4
Complete Degeneracy
The extreme case of strong degeneracy occurs in the limit T → 0, which is called complete degeneracy. It is clear from Eq. (2.4) that, for given values of n and m, the degeneracy parameter becomes arbitrarily large in this limit. What happens to the Fermi-Dirac distribution then? To see this,
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note first that the chemical potential µ, defined above by A ≡ exp(µ/kT ), remains finite in this limit, while A → ∞, of course. Call this limiting value µ0 . Note next that the Fermi-Dirac distribution, as given by Eq. (2.3) becomes very simple in the limit T → 0. If E < µ0 , fF D = 1, and if E > µ0 , fF D = 0. Thus, µ0 is the limiting kinetic energy of the fermions in this limit, and so of key importance. It is customary to define a limiting momentum, called pF , corresponding to µ0 by the the Fermi momentum 2 2 2 2 4 relation µ0 = pF c + m c − mc , to write the Fermi-Dirac distribution as fF D =
1 , p < pF , 0 , p > pF
(2.9)
and to state that all momentum states upto pF are completely filled, and all states above pF are completely empty. It is also customary to define a total 2 energy (i.e., including the rest energy mc of the fermion) corresponding 2 2 2 to pF by the relation EF ≡ pF c + m c4 , and call it the Fermi energy. All states of total energy upto the Fermi energy are then fully occupied in the limit of complete degeneracy, and all states above this energy are completely empty. We shall be using the complete-degeneracy limit throughout the next few sections while describing the basic properties of degenerate stars, not only because they were first calculated in this limit, but also because this limit contains the essence of the basic physics of degenerate stars, the finitetemperature effects being largely corrective factors, most important near the (low-density) surface layers of these stars, where matter passes from degenerate to non-degenerate state. In these T → 0 calculations, a relation between the Fermi momentum pF and the number density n that will be constantly useful is n=
8πp3F . 3h3
(2.10)
Equation (2.10) follows from the fact that n is the obtained by multiplying the number of phase-space cells a in the momentum range p to p + dp, as given earlier, by the Fermi-Dirac distribution in the complete-degeneracy limit, as given by Eq. (2.9), and integrating over all momenta. The Fermi momentum, or Fermi energy, is thus a direct measure of the number density of the completely degenerate fermions.
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The Landau Arguments
A key property of degenerate matter is that its kinetic energy and pressure do not vanish in the T → 0 or complete-degeneracy limit, in sharp contrast to the situation for non-degenerate matter following classical statistics. The residual values at T = 0, which are called zero-point energy and zero-point pressure (nullpunktsenergie and nullpunktsdruck respectively in the original German; see Fermi 1926 and Sommerfeld 1928), are crucial for determining the essential physics of degenerate stars like white dwarfs and neutron stars. It is this nullpunktsdruck that balances the self-gravitational attraction in these ‘dead’ stars, which have ceased to generate energy inside them by thermonuclear reactions, and makes equilibrium stellar configurations possible even for arbitrarily cold stellar matter. For a non-relativistic (NR) gas of fermions like electrons or neutrons, the zero-point kinetic energy Enull per particle and the zero-point pressure Pnull can be calculated from the completely degenerate Fermi-Dirac distribution of Eq. (2.9), and are given by [Fermi10 1926, Sommerfeld 1928]: Enull
3 = 40
2/3 2 2/3 3 h n , π m
Pnull
1 = 20
2/3 2 5/3 3 h n . π m
(2.11)
Here, n is the number density of the fermions and m is the fermion mass. For a highly relativistic (ER) fermion gas, on the other hand, the expressions for zero-point kinetic energy per particle and zero-point pressure are [Stoner 1930; Chandrasekhar 1931b,c]: Enull =
3 8
1/3 3 hcn1/3 , π
Pnull =
1 8
1/3 3 hcn4/3 , π
(2.12)
showing that these quantities are independent of the fermion mass in this limit. We shall derive the exact, general expression for zero-point kinetic energy per particle in Sec. 2.4.1, and that for zero-point pressure in Sec. 2.4.3: these expressions will be valid all the way from the NR to the ER r´egime. Consider now that the self-gravitational energy of a star of mass M and radius R, which is negative and of the form: Egrav = −αGM 2 /R,
(2.13)
where α is a structure factor depending on the density profile of the star. For a uniform density, ρ = ρ¯, we have R = (3M/4π ρ¯)1/3 , and α = 3/5, so 10 The
spin factor 2−2/3 was missing in Fermi’s original expressions.
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that Egrav = −γGM 5/3 (¯ ρ)1/3 , where γ = (3/5)(4π/3)1/3 . For other density profiles, the form of Egrav remains the same with different values11 for the numbers α and γ, with ρ¯ signifying the average density of the star. Let the star be supported against self-gravity by the zero-point pressure of fermions whose total number is N , and whose number density is n. To relate these to M and ρ¯, we now define µ as the number of pressure-supporting fermions per nucleon in the star. This variable µ is of central importance in the subject: we shall use it constantly in this chapter, and elsewhere in the book. For white dwarfs, these fermions are the electrons, and the stellar matter consists of nucleons and electrons with µ ≈ 1/2, since most elements contain about two nucleons per electron. For neutron stars, on the other hand, these fermions are the neutrons, and since the stellar matter consists almost wholly of these neutrons themselves, µ ≈ 1. In general, we have N = M µ/mN and n = ρµ/mN . We can now define a self-gravitational energy per pressure-supporting fermion, Egrav = Egrav /N , which is given by n)1/3 , Egrav = −γ(mN /µ)4/3 GM 2/3 (¯
(2.14)
where n ¯ is the average value of n. For brevity, we shall henceforth omit the bars on n and ρ throughout the rest of this section. The scaling of the energies Egrav and Enull (in both the r´egimes described above) with the number density n can thus be expressed as: Egrav = −agrav n1/3 ,
NR 2/3 Enull = aNR , null n
ER 1/3 Enull = aER , null n
(2.15)
where the superscripts NR and ER respectively signify the non-relativistic and extreme-relativistic r´egimes, as before. The coefficients are given by: 4/3 2/3 2 1/3 mN h 3 3 3 3 ER , a GM 2/3 , aNR = = hc. agrav = γ null µ 40 π m null 8 π (2.16) These scalings had already been described by Stoner in 1930. But it remained for Landau to make use of them in 1932 to give a set of simple and brilliant arguments which showed that the mass of a self-gravitating star consisting of zero-temperature fermions could not exceed a limit which depended essentially only on the fundamental constants G, c, h, and the mass mN of a nucleon, quite independent of the details of how a degenerate star is put together. The heart of the Landau argument is a study of the behavior of the total energy, E = Egrav + Enull , per pressure-supporting fermion 11 For
a polytrope (see Sec. 2.2.3.2) of index s, α = 3/(5 − s).
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in the star. (Considering the behavior of the total energy of the whole star, i.e., E = EN , would give the same results, since the total number N of these fermions remains constant during all of the variational processes we might consider.) This gives us the existence and character of possible equilibrium configurations of the star, as we now show. Consider non-relativistic fermions first. The total energy E is given by 2/3 − agrav n1/3 , E NR = aNR null n
(2.17)
so that E NR is negative at small densities and scales as n1/3 , while it is positive at large densities and scales as n2/3 . As the reader can easily verify by sketching E NR as a function of n, as given by Eq. (2.17), there must always be a minimum in E NR , which is the (stable) equilibrium configuration of the star. The density at which this occurs is obtained by setting the derivative ∂E NR /∂n to zero, which gives 3 n = (agrav /2aNR null ) .
(2.18)
Thus, a stable equilibrium is always possible between the inward gravitational force and the outward zero-point pressure of degenerate nonrelativistic fermions: the star merely expands or contracts to achieve this equilibrium density, given by Eq. (2.18). Next consider extremely relativistic fermions. E is now given by 1/3 , E ER = (aER null − agrav )n
(2.19)
and the situation is completely different, since the monotonic variation of E ER with n makes any extremum impossible: the magnitude of the total energy always increases with density as n1/3 . This means that, if aER null > agrav , E ER is positive, and the star tries to minimize its energy by expanding ER is indefinitely to lower and lower densities. Conversely, if aER null < agrav , E negative, and the star tries to minimize its energy by collapsing indefinitely to higher and higher densities. Thus, no equilibrium is possible for extremerelativistic fermions in either of the above two cases. The transition between ER = 0 at all the two cases occurs when aER null = agrav , in which case E densities, and there is a neutral equilibrium. In reality, the fermions change from the NR to the ER r´egime at sufficiently high densities, the transition density nrel being given by the condition that the Fermi momentum (see above) of the degenerate fermions is comparable to mc, m being the fermion mass (so that the relativity
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parameter x ≡ pF /mc becomes ∼ 1), or equivalently by the condition that NR ER and Enull , as given by Eq. (2.15), are comparable, which yields: Enull 3 5 1 mc 3 nrel = . (2.20) 4 3π 2 This corresponds to an average stellar mass density ρrel ∼ 2 × 106 g cm−3 when the fermions are electrons, and ρrel ∼ 1 × 1016 g cm−3 when the fermions are neutrons, as can be seen by substituting the appropriate fermion mass in Eq. (2.20). For stars of average densities ρrel , the pressure-supporting fermions are non-relativistic, while for stellar densities ρrel , they are highly relativistic. Thus, the electrons in white dwarfs of average density ρ ∼ 106 g cm−3 are mildly relativistic, while the neutrons in neutron stars of average density ρ ∼ 1014 g cm−3 are non-relativistic [Oppenheimer & Volkoff 1939], as we mentioned in §2.1. In an actual star, therefore, an appropriate combination of the above NR and ER results applies, as Landau argued. If aER null > agrav , the star expands until its density falls below nrel , at which point NR results apply, and the star achieves equilibrium, as described above. On the other hand, if aER null < agrav , the star keeps on collapsing without limit, reaching higher and higher densities, and making the fermions more and more ER: an equilibrium is never possible. Stars supported against self-gravity by the zero-point pressure of degenerate fermions can only exist, therefore, if aER null ≥ agrav . Use of Eq. (2.16) turns this condition into an upper limit on the stellar mass: 3/2 c 1 ML ≡ . (2.21) M ≤ Mc ≡ µL ML , G m2N Here, ML is the Landau mass scale, which depends only on universal constants and has a numerical value ≈ 3.69 × 1033 g ≈ 1.85M, and µL ≡ (9π/8)γ −3/2 µ2 is a number ∼ 1 whose value depends on the density profile of the star through γ. For the uniform-density case considered above, µL ≈ 3.72µ2 . With the aid of the exact expression for the zero-point kinetic energy given in Sec. 2.4.1, we display in Fig. 2.2 the variation of the total energy E with density (or, equivalently, with the relativity parameter x), extending from the NR to the ER r´egime, for a range of stellar masses M . For M < Mc , there is always a minimum in the energy corresponding to a stable equilibrium configuration, which is the actual configuration of the degenerate star of mass M . As M approaches Mc , this energy-minimum
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57
Landau arguments Newtonian
0.1
0.5
0.5
0.9
0.0 -0.5 -1.0 0.01
5.0 1.5
1.1
0.10 1.00 10.00 Relativity Parameter (x)
1.0
100.00
Fig. 2.2 Original Landau arguments for Newtonian gravity. Shown is the total energy per pressure-supporting fermion (E) in units of this fermion’s rest-mass energy for a uniform-density degenerate star vs. its relativity parameter x, the latter being a measure of its density (see text). Curves are given for various values of M/Mc , each curve labeled by its value of M/Mc . Dashed line: critical curve corresponding to M/Mc = 1, where the minima in the curves disappear. Curves corresponding to higher masses have no extrema, so that no equilibria are possible.
moves to higher and higher densities, i.e., higher and higher values of x, which simply means that higher stellar masses require higher densities for stable degenerate stars. In the limit M → Mc , this minimum occurs at arbitrarily high densities. For M > Mc , no minimum (indeed, no extremum of any kind) appears in the energy, and no equilibrium is possible. The total energy is then always negative, and its magnitude increases monotonically as density increases, indicating a collapse to arbitrarily high densities. Thus, degenerate stars supported by the zero-point pressure of a given kind of fermions (electrons, say) have a mass limit Mc they cannot exceed. A configuration of higher mass will not be supported by these fermions, and so will collapse, either indefinitely or until degenerate fermions of another kind (neutrons, say) can supply adequate zero-point pressure at sufficiently higher densities, if the latter situation is possible. To give a specific example, white dwarfs, which are supported by degenerate electrons, cannot exceed a mass limit named after Chandrsekhar (1931a,b,c). Masses above
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the Chandrasekhar limit collapse until neutrons become degenerate at sufficiently high densities (see Sec. 2.1.2), and supply zero-point pressure. Then, if the mass does not exceed the limit which corresponds to degenerate neutrons and which is named after Oppenheimer and Volkoff (1939), a neutron star forms. Masses above the Oppenheimer-Volkoff limit collapse without limit, into a black hole. The wonderful thing about the Landau arguments is that they are quite general, equally applicable to electrons, neutrons, and any other degenerate fermions one can think of, although Landau himself did not stress this in his 1932 paper. The only feature missing in them is general relativity (since Landau used Newtonian theory of gravitation throughout his work), to which we shall return at the end of this chapter. Even in their Newtonian form, however, Landau’s arguments readily suggest a deeply significant feature of the maximum mass Mc of degenerate stars, namely that this mass is, crudely speaking, independent of the nature of the degenerate fermions providing the pressure support. This is so because the Landau mass scale ML depends only on the fundamental constants G, c, h, and mN , and so is independent of the details, e.g., the mass, of the pressure-supporting fermions. This, in turn, is a direct consequence of the fact that the zeropoint pressure of ER degenerate fermions is independent of their mass (see Eq. (2.12) and the discussion below it). The fermionic details enter only through the factor µ2 occurring in µL (see above): µ being the number of pressure-supporting fermions per nucleon introduced earlier. For different fermions, µ will be somewhat different in general (e.g., µ ≈ 1/2 for electrons, and µ ≈ 1 for neutrons; see above), but it is ∼ 1 within a factor of about 2 in all situations of practical interest, so that Mc for different fermions would be the same within a factor of about 4 (we shall see later that general-relativistic effects considerably reduce this factor of difference). This fundamental property of Mc seems not to have been appreciated by Stoner (1930) and Chandrasekhar (1931b) when they gave their derivations of the mass limit of white dwarfs: the clue came undoubtedly from Landau’s 1932 work, and was fully appreciated by 1938-39 [Landau 1938; Oppenheimer & Volkoff 1939]. 2.3.1
Degenerate Stars: Rough Mass Limits
The simple arguments summarized above would give Mc ≈ 1.72M and ≈ 6.89M for white dwarfs and neutron stars respectively. Refinements can now be incorporated. First, instead of the uniform-density approxi-
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mation, we can use a polytropic approximation (see Sec. 2.2.3.2), in which the equation of state has the power-law form P ∝ nΓ , Γ being the polytropic exponent. As Eqs. (2.11) and (2.12) show, the polytropic approximation holds accurately in both NR and ER limits, with ΓNR = 5/3 and ΓER = 4/3. We can then use Emden’s (1907) extensive results on polytropic gas spheres, as Chandrasekhar (1931a,b,c) and Landau (1932) did. The resultant changes are twofold: (1) the gravitational-energy structure factor γ in Eq. (2.14) changes appropriately from the uniform-density value given above, and, (2) the simple condition aER null = agrav for neutral equilibrium changes to an outer boundary condition on the polytrope for the existence of well-behaved solutions of the polytropic equation (see below, Sec. 2.2.3, for details of polytropes) in the ER case. The final result in Landau’s treatment of the problem is that the constant of proportionality between µL and µ2 changes from the value ≈ 3.7 in the simple case given above to a value ≈ 3.1 [Landau 1932]. This gives Mc ≈ 1.4M for white dwarfs [Landau 1932], and Mc ≈ 5.7M for neutron stars [Oppenheimer & Volkoff 1939]. In later sections, we shall describe some of the next refinements to the calculations, which are the inclusions of (a) the exact equation of state for degenerate fermions, as opposed to polytropic approximations, (b) the effects of the general theory of relativity, where appropriate, and, (c) the interaction between the fermions. Inclusion of the first two effects (of which the general-relativistic effect is by far the dominant one) lowers the mass limit of neutron stars to ≈ 0.7M (see Sec. 2.1 and Oppenheimer and Volkoff 1939), but has only a small effect on that of white dwarfs. The third effect raises the neutron-star mass limit to ≈ 2M , which proved historically crucial for resolving the Oppenheimer paradox, as we explained in Sec. 2.1. 2.3.2
Degenerate Stars: Mass-Radius Relations
To show the power and beauty of the Landau arguments, we now calculate the radii of degenerate stars from the simple estimates for uniformdensity stars given above. For a given stellar mass M < Mc , there is only one possible configuration in stable equilibrium, whose density n is given by Eq. (2.18). Its radius R is therefore given by the relation R = (3M µ/4πnmN )1/3 , which, upon substitution of Eqs. (2.18) and (2.16), yields the following mass-radius relation for degenerate stars:
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M ML
1/3 R = µR RG (ML )
m mN
−1
≈ 10.1µ5/3
m mN
−1 km,
(2.22)
where µR ≡ (9π/4)2/3 µ5/3 ≈ 3.7µ5/3 , and RG (ML ) ≡ GML /c2 is the gravitational radius corresponding to the Landau mass scale defined earlier (see Eq. (2.21)). The gravitational radius is a length of central importance in the general theory of relativity, to which we shall return repeatedly in this book. The nice thing about this mass-radius relation is, again, its generality. It holds for all pressure-supporting fermions, the transparent scaling R ∝ m−1 with the fermion mass m showing that, unlike the mass limits, the radii are strongly influenced by m, the radius of a neutron star being ∼ 10−3 of that of a white dwarf of the same mass. Further, as before, the use of polytropic models would leave the form of Eq. (2.22) unaltered, changing only the numerical constant in the relation between µR and µ. Substituting into this equation the numerical value of ML from Eq. (2.21), we obtain
M M
1/3
R ≈ 12µ
5/3
m mN
−1 km ≈
7.1 × 103 km, white dwarfs . 12 km, neutron stars (2.23)
In subsequent sections, we shall compare the radii given by Eq. (2.23) with those obtained from (a) the uniform-density calculations of Stoner (1929) and the polytropic calculations of Chandrasekhar (1931a) for white dwarfs, and, (b) the numerical calculations of Oppenheimer and Volkoff (1939) for neutron stars to emphasize the value of these simple estimates.
2.4
White Dwarfs
Observationally, white dwarfs had been known as stars with masses comparable to that of the Sun (inferred from, e.g., binary dynamics for Sirius B, the binary companion to the prominent optical star Sirius: for the basics of binary dynamics, see Appendix B), but with radii much smaller than those of Sun-like stars, as inferred from Stefan’s law of blackbody emission, L = 4πR2 σT 4 (see Eq. (1.10) and Appendix A), the effective surface temperature T being obtained from spectral measurements. Their discovery goes back to 1910, when Russell, Pickering, and Fleming (see Russell 1944) found that the faint star 40 Eridani B had a spectral class A (see Appendix
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A), i.e., an unusually high temperature, or a very “white” color, for such low-luminosity stars, which normally have a spectral class M (i.e., a low temperature and a “red” color), and so are called M- or red-dwarfs (the reader will immediately see from Stefan’s law that low luminosities and high temperatures imply small radii). By the mid-1920s, it was established from the early, rough determinations of masses and radii of white dwarfs like Sirius B and 40 Eridani B that the average densities of these stars 4 5 −3 were > ∼ 5 × 10 − 10 g cm . This prompted Eddington (1926) to argue in his classic book The Internal Constitution of the Stars that such dense, ionized matter would exhibit paradoxical behavior at low temperatures if it followed classical statistics, since, after having radiated away a sufficient amount of its thermal energy, it would reach a state where its energy content would be less than that of the same matter in the form ordinary unionized atoms at zero temperature, and it was not clear how the former state would “cool down” to the latter, as it was expected to do according to the laws of physics. The answer came from Fowler: the pioneering papers of Fermi (1926) and Dirac (1926) on Fermi-Dirac statistics had just been published, and Fowler (1926) made the first application of their results to astrophysics, proposing the profound idea that the properties of stellar matter at such high densities as expected in white dwarfs would be governed by quantum, not classical, statistics. “This discovery,” Chandrasekhar was to write in his 1945 obituary of Fowler, “which must be certainly counted among the more important of the astronomical discoveries of our time, could not have been made except by one whose grasp of the theories both of physics and of astrophysics was of the highest.” This not only solved the Eddington paradox readily, since high-density degenerate matter obeying Fermi-Dirac statistics has a large energy-density even at zero temperature (this is the nullpunktsenergie we discussed in Sec. 2.2.2), but also introduced the very concept of the degenerate stars that we discuss in this book. 2.4.1
The Stoner-Anderson Work
The pioneering application of Fowler’s idea to white dwarfs was made in 1929-30 by Stoner, who investigated uniform-density model stars, using energy arguments essentially identical to those which we have given in Sec. 2.2.2. Stoner (1929) first considered stars pressure-supported by NR degenerate electron gas, obtaining the equilibrium density from Eq. (2.18), NR )/∂n = 0 which gives rise to this and describing the condition ∂(Egrav + Enull equation (see Sec. 2.2.2) in the picturesque and physically apt words “the
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gravitational energy released just supplies the energy required to squeeze the electrons together”. Stoner gave the result not as a mass-radius relation (see Sec. 2.2.2.2), but as a density-mass relation, which is easily obtained from Eq. (2.23) by using R = (3M/4πρ)1/3 , and is12 ρ ≈ 4.12 × 104 µ−5
M M
2
g cm−3 ≈ 1.32 × 106
M M
2
g cm−3 . (2.24)
Relations like Eq. (2.24) first established the correct scale of white-dwarf densities, ρ ∼ 106 g cm−3 , showing these to be considerably higher than thought before. In a now-forgotten critique of the Stoner’s above work, Anderson (1929) made the first attempt to include relativistic effects in the theory of degenerate stars by incorporating these effects into the calculation of Enull . √ Whereas the NR theory gives M ∝ ρ (see Eq. [2.24]), Anderson found that the relativistic effects reduced the mass at a given density below the value given by the NR relation, the deviation so increasing with increasing density that M attained a limiting constant value at very high densities. This last result, which apparently went unnoticed by Anderson himself although the density-mass tables in his 1929 paper clearly showed it, was the first appearance of that mass limit to white dwarfs which we call the Chandrasekhar limit today. Unfortunately, Anderson’s approximate treatment of relativity was unacceptable: Stoner immediately recognized this, as also the existence of the mass limit. In his reply to Anderson’s critique, Stoner (1930) gave the pioneering exact derivation of Enull for a degenerate Fermi gas valid for all momenta of the fermions, and used it to obtain a densitymass relation for uniform-density model white dwarfs that was valid all the way from the NR to the ER r´egime. We obtain Enull by averaging the exact, general expression for the kinetic energy per fermion over the Fermi-Dirac distribution in the completedegeneracy limit (Eq. [2.9]): 1 pF 2 2 2 [ p c + m2 c4 − mc2 ] 3 4πp2 dp, (2.25) Enull = n 0 h 12 Stoner’s original number in the second expression on the right-hand side of Eq. (2.24) was somewhat different, viz., ≈ 4×106 (M/M )2 g cm−3 , because he had used µ = 1/2.5, as did Chandrasekhar after him in the 1930s, as per the custom of the time. The custom goes back to Jeans (1928), and this value of µ, which is representative of heavy elements in the periodic table around lead, is only an historical curiosity now. As we know today, and as we describe in later subsections, µ ≈ 1/2 gives an excellent representation of the elements that the bulk of the white-dwarf matter is thought to be composed of, e.g., C, O, Ne, and Mg.
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where pF is the Fermi momentum introduced earlier, and n = 8πp3F /3h3 is the number density of the fermions (see Sec. 2.2.4). The integral in Eq. (2.25) is performed by defining the dimensionless momenta x ≡ pF /mc (x is called the relativity parameter; see Sec. 2.3) and y ≡ p/mc, the result being conveniently expressed in units of the rest-mass energy, mc2 , of the fermion as: Enull = mc2 fE (x),
fE (x) ≡
3 fS (x) − 1, 8x3
where fS (x) is the Stoner (1930, 1932) function, defined by x fS (x) ≡ 8 1 + y 2 y 2 dy = x 1 + x2 (1 + 2x2 ) − sinh−1 x.
(2.26)
(2.27)
0
We leave it as an exercise for the reader to show that the energy function fE is given by
3 fE (x) = 3 x 1 + x2 (1 + 2x2 ) − sinh−1 x − 1, (2.28) 8x that the NR (x 1) limit of this function is13 fENR =
3 2 3 x − x4 · · · , 10 56
and that its ER (x 1) limit is14 3 1 fEER = x + · · · − 1. 4 x
(2.29)
(2.30)
The reader will also see that, on substituting the value of x in terms of n, i.e., using the relation 8π mc 3 3 n= x , (2.31) 3 h NR (as given by the leading terms in fENR and fEER above readily yield Enull ER Eq. [2.11]) and Enull (as given by Eq. [2.12]) respectively, as they should. 13 Note the difference with the corresponding result in Shapiro and Teukolsky [1983], even after conversion from their energy-density to our energy per particle. The reason for this is that the Shapiro-Teukolsky expressions are for the total energy, i.e., that including the rest-mass energy of the fermions, while ours are for the kinetic energy, i.e., that excluding the rest-mass energy. Our choice makes the differences from classical statistics more transparent. 14 We neglect here the logarithmic term, −(3/8x3 ) ln(2x), which arises from the x 1 √ limit of the function sinh−1 x = ln(x + 1 + x2 ), as its contribution is unimportant.
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Using this exact Enull in the energy-extremum condition given earlier, Stoner (1930) obtained the exact mass-density relation for uniform-density white dwarfs shown in Fig. 2.3, where the Anderson (1929) result is also shown. The asymptotic approach to a limiting maximum mass is clear in the ER r´egime at high densities, this limiting mass MS of Stoner being exactly equal to the mass limit Mc given by Eq. (2.21), i.e., MS = Mc ≡ µL ML ,
µL ≡ (15/8)3/2 (2π/3)1/2 µ2 ≈ 3.72µ2 ,
(2.32)
Fig. 2.3 Stoner’s mass-density relation for uniform-density white dwarfs (solid line). Shown also are Anderson’s results (dashed line). Notice the asymptotic approach to a limiting maximum mass. Positions of a few well-known white dwarfs are also indicated. Reproduced with permission by Taylor & Francis Ltd (http://www.tandf.co.uk/journals) from Stoner (1930): see Bibliography.
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where ML is the Landau mass scale defined in Sec. 2.2.2. Hence, for µ ≈ 1/2, as we adopt for electrons, MS ≈ 1.72M , while Stoner (1930) himself obtained a value15 MS ≈ 1.10M using the customary value µ ≈ 1/2.5 of his time, as explained above. At first sight, the equality of the two results may not be surprising, since we used uniform-density model stars throughout our simple calculations of Sec. 2.2.2. But the reader should appreciate that Stoner’s (1930) mass limit was given without the benefit of Landau’s (1932) arguments on the deep significance of a natural limit to masses of degenerate stars. Stoner (1930) apparently had no inkling of the special case of neutral equilibrium for aER null = agrav (see Sec. 2.2.2), ER (both going as n1/3 ) and interpreted the similar scaling of Egrav and Enull as implying that “there will obviously be no equilibrium” in the ER limit, totally unperturbed by the fact that he was obtaining an equilibrium solution in this limit, namely his mass limit. What was actually happening is clear to us today: if we assume that the energy-extremum condition ∂(Egrav + Enull )/∂n = 0 always gives us an equilibrium solution, and look for this solution, we shall get a solution for n (e.g., Eq.[2.18] in the NR limit) as long as Egrav and Enull have different scalings with n. But when Egrav and Enull have identical scalings with n, as the n1/3 scaling in the ER limit, the only possible solution will simply be the condition that the coefficient of n1/3 is zero, i.e., that aER null = agrav , which just gives the mass limit on degenerate stars, as we saw above. This is what Stoner (1930) did. In reality, white dwarf stars have a radial variation in density, and so in pressure and in the zero-point energy Enull per electron. A quantitative theory of white dwarfs must take this variation into account. Chandrasekhar did this in the period 1931-35, treating the NR and ER limits in the polytropic approximation in a series of papers in 1931 before attempting a general numerical calculation valid for all r´egimes in 1935. We summarize below the essential properties of polytropes before describing Chandrasekhar’s work. 2.4.2
Polytropes
We introduced polytropes in Sec. 2.2.2.1 as those systems for which pressure and density are connected by a relation of the form: P = KρΓ ,
(2.33)
15 Anderson’s (1929) mass limit, obtained from his approximate treatment of relativistic effects, was ≈ 0.69M .
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K and Γ being constants. It is customary to write the exponent Γ in the form: 1 Γ≡1+ , s
(2.34)
and call s the polytropic index. As explained in Sec. 2.2.2.1, a degenerate Fermi gas has Γ = 5/3 when it is non-relativistic, and Γ = 4/3 when it is extremely relativsitic. The corresponding values of the polytropic index s are sNR = 3/2 and sER = 3 respectively. In the early years of the twentieth century, polytropes were thoroughly studied for obtaining the first indications of the internal structures of stars. Apart from its computational convenience (see below), the polytropic form has the significance that numerous limiting physical situations of basic importance in stars lead to a pressure-density relation of this form: this is why its use has survived to this day. For example, the (non-degenerate) matter in normal stars obeys a polytropic relation with s = 3/2 if the star is fully convective and its pressure is dominated by the gas pressure. On the other hand, a normal star in which the ratio of gas pressure to radiation pressure is constant throughout the star, and the energy-transport is primarily by radiative transfer (i.e., the so-called standard model), is closely represented by a polytrope with s = 3. For degenerate stars, the polytropic form applies in both NR and ER limits, as indicated above. The structure of a polytropic gas sphere in hydrostatic equilibrium is determined by first combining Eq. (10.35) for hydrostatic equilibrium with an equation satisfied by the mass m(r) contained within a radius r from the center of a spherically symmetric distribution, namely, dm(r)/dr = 4πr2 ρ(r). This yields: 1 d r2 dP = −4πGρ. (2.35) r2 dr ρ dr In combining Eq. (2.35) with the polytropic relation (Eqs. (2.33) and (2.34)) to obtain the polytrope-structure equation, it is convenient to define two dimensionless variables. The first is a density-related variable θ, obtained by scaling in terms of the central density ρc of the polytrope: ρ = ρc θ s ,
(2.36)
the choice of the exponent refelecting the fact that ρ ∝ T s (T is the temperature) in a polytrope of index s obeying classical statistics, which the reader can readily demonstrate by combining the classical equation of state,
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P ∝ ρT , with Eqs. (2.33) and (2.34). Thus, θ is a temperature-like variable. The second is a dimensionless radial co-ordinate ξ:
( 1 −1)
(s + 1)Kρc s a≡ 4πG
r = aξ,
12 .
(2.37)
In terms of ξ and θ, the polytrope-structure equation reduces to a simple, ordinary differential equation called the Lane-Emden equation, given by: 1 d 2 dθ (2.38) ξ = −θs , ξ 2 dξ dξ and its solution with appropriate boundary conditions is called the LaneEmden function of index s. These boundary conditions are: θ = 1 and θ = 0 at ξ = 0, where θ ≡ dθ/dξ. The first condition follows when we consider Eq. (2.36) at the center of the polytrope. The second comes from a consideration of Eq. (10.35) in the neighborhood of r = 0: approximating ρ by ρc and m(r) by 4πρc r3 /3 in this neighborhood, we see that dP/dr ∝ r, so that dP/dr → 0 as r → 0, and so do dρ/dr and dθ/dξ. The Lane-Emden equation is simple, but nonlinear except in the special cases s = 0 and 1, so that the vast array of analytical techniques available for solving linear differential equations is inapplicable to it in general. Indeed, numerical techniques have to be used for all cases except s = 0, 1, and 5, integrating Eq. (2.38) outward from ξ = 0, subject to the two boundary conditions described above. For all values of s less than 5, one finds that θ(ξ) decreases outward monotonically, vanishing at a finite radius ξ = ξ1 , which must, therefore, correspond to the surface of the polytropic star, since p = 0 = ρ there. The physical radius of the star follows from Eq. (2.37) as:
(s + 1)K R = aξ1 = 4πG
12
1−s
ρc 2s ξ1 .
(2.39)
R The mass of the polytropic star, M ≡ 0 4πr2 ρdr, is obtained by substituting Eq. (2.36) in the integrand and making use of Eq. (2.38), yielding:
(s + 1)K M = 4π 4πG
32
3−s
ρc2s ξ12 | θ (ξ1 ) | .
(2.40)
Elimination of ρc between Eqs. (2.39) and (2.40) yields the mass-radius relation for polytropic stars:
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3−s
M R s−1 = 4π
(s + 1)K 4πG
s s−1
s+1
ξ1s−1 | θ (ξ1 ) |,
(2.41)
which shows that the scale for this relation is set by the constant K occurring in the polytropic relation (which is determined by the character of the stellar matter; for degenerate matter, K depends on the fundamental constants h, c, mN , and also on the mass m of the pressure-supplying fermion in the NR case: see Sec. 2.2.2) and, of course, by the gravitational constant G. The detailed numbers on the right-hand side of Eq. (2.41) depend on the numbers ξ1 and | θ (ξ1 ) | for the particular polytropic index s we are interested in: these are obtained from a numerical solution of the LaneEmden equation, and have been tabulated16 extensively by Emden (1907) and Chandrasekhar (1939). 2.4.3
Chandrasekhar’s Work
Chandrasekhar’s (1931a,b,c) application of the theory of polytropic gas spheres [Emden 1907] to degenerate stars, undoubtedly inspired both by Eddingtion’s (1926) earlier application of this theory to normal stars and by Sommerfeld’s classic (1928) work17 on electron degeneracy in metals (see Sec. 2.2.1.1), stands in scientific history as the first construction of a quantitative theory of white dwarfs. The first result Chandrasekhar (1931a) gave was for the NR degenerate electron gas, which corresponds to a polytropic index s = 3/2. The constant K is then obtained from Eq. (2.11) as K
NR
1 = 20
5/3 2/3 2 µ 3 h , π me mN
(2.42)
and Eq. (2.41) reduces to M R3 = (4π)−2 (5K NR /2G)3 η1 in this particular case, where η1 ≡ [ξ15 | θ (ξ1 ) |]3/2 , the subscript 3/2 on the square brackets indicating that solution of the Lane-Emden equation for the s = 3/2 case is to be used. The mass-radius relation then becomes M R3 =
16 Actually,
6 9π 2 µ5 η1 . 3 128 G m3e m5N
(2.43)
it is customary to tabulate ξ12 | θ (ξ1 ) | instead of | θ (ξ1 ) |. recalled in 1989 the inspiring personal interaction with Sommerfeld in 1928: see Chandrasekhar (1989), p. xii. 17 Chandrasekhar
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In order to calculate η1 , we use the tables in Chandrasekhar’s (1939) classic text An Introduction to the Study of Stellar Structure, which give ξ1 ≈ 3.654 and ξ12 | θ (ξ1 ) |≈ 2.714 for s = 3/2, which yield η1 ≈ 132.4. The massradius relation finally becomes M 3 R = 2.161 × 1028 µ5 cm3 , M
(2.44)
which is essentially identical to Chandrasekhar’s (1931a) corresponding result18 , the slight difference in the number on the right-hand side being due to the fact that, in his 1931 works, Chandrasekhar used the older tables in Emden’s (1907) classic book Gaskugeln. The corresponding relation between mass and average density ρ¯ is 2 2 M M g cm−3 ≈ 0.70 × 106 g cm−3 , (2.45) ρ¯ ≈ 2.20 × 104 µ−5 M M where we have used µ ≈ 1/2 in the second relation. Note that Chandrasekhar, like Stoner before him, used µ ≈ 1/2.5 throughout his 1931 work, as was customary then (see Sec. 2.2.3.1). It is instructive to compare these results with those of Stoner (see Sec. 2.2.3.1), particularly Eq. (2.45) with Eq. (2.24). We can also compare the above results with those of Sec. 2.2.2.2 by combining Eq. (2.43) with the definitions of the Landau mass scale ML and the gravitational radius RG (ML ) corresponding to ML , given in Secs. 2.2.2 and 2.2.2.2 respectively. These yield: 2 1/3 1/3 −1 9π η1 me M 5/3 R= µ RG (ML ) (2.46) ML 128 mN −1 me km, (2.47) ≈ 12.4µ5/3 mN which is to be compared with Eq. (2.22). The closeness of numerical values with those given in Eq. (2.23) is clear when we rewrite the above as:
M M
1/3
R ≈ 15µ
5/3
me mN
−1 km ≈ 8.7 × 103 km,
(2.48)
where we have substituted µ ≈ 1/2 for white dwarfs, as before (see Sec. 2.2.2). This is remarkable in view of the simplicity of our estimates of Sec. 2.2.2.2, and shows that the essential physics of the equilibria of white 18 Note
that Chandrasekhar (1931a) defined his µ to be the reciprocal of ours.
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dwarfs, as contained in the Landau (1932) arguments, had been already brought out almost entirely in Stoner’s (1929) uniform-density calculations. Chandrasekhar’s next result (1931b) was on the ER degenerate electron gas, which corresponds to a polytropic index s = 3. The constant K in this case is seen from Eq. (2.12) to be K ER =
1 8
4/3 1/3 µ 3 hc , π mN
(2.49)
and Eq. (2.41) then gives the remarkable result that the mass of the star is in this case independent of its radius (and, therefore, of its density), being given simply by M = (4π)(K ER /πG)3/2 χ1 , where χ1 ≡ [ξ12 | θ (ξ1 ) |]3 , the subscript 3 on the square brackets indicating that solution of the LaneEmden equation for the s = 3 case is to be used. This unique value of the mass, given by √ 3/2 3π c 1 2 µ χ1 , (2.50) MCh = 2 G m2N was immediately identified by Chandrasekhar (1931b) as the polytropic analogue of the white-dwarf mass limit described earlier by Stoner (1930) for uniform-density white dwarfs. It is called the Chandrasekhar limit today. The occurrence of the Landau mass scale in Eq. (2.50) is obvious: we can recast it as MCh ≈ 3.1µ2 ML ≈ 1.4M ,
(2.51)
using χ1 ≈ 2.018 from Chandrasekhar’s (1939) tables. The first form of Eq. (2.51) is exactly that which was given by Landau (1932) in his rederivation of the Chandrasekhar limit. The final numerical value originally obtained by Chandrasekhar (1931b), MCh ≈ 0.9M, was somewhat different from ours, again because he used µ ≈ 1/2.5 (see above). The reader can compare Eq. (2.51) with the Stoner limit, MS ≈ 3.7µ2 ML ≈ 1.7M , which is identical to the results summarized in Sec. 2.2.2. Like Stoner (1930) before him, Chandrasekhar (1931b) also did not explain why there should be a limit to the mass of white dwarfs: the explanation came from Landau in 1932. In 1935, Chandrasekhar took the next logical step of generalizing the polytropic approach to calculate the equilibrium properties of white dwarfs all the way from the NR to the ER r´egime. For this purpose, he used the exact, general equation of state of a degenerate Fermi gas, valid for all
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momenta of the fermions, derived19 earlier by Stoner (1932). We obtain this equation of state by noting that the usual NR kinetic-theory result that the average transfer of momentum for a particle of velocity v andmomentum p is vp/3 generalizes in the relativistic case to βcp/3 = p2 c2 /3 p2 c2 + m2 c4 (Stoner 1932), so that the pressure of degenerate fermions is obtained by integrating this over the Fermi-Dirac distribution: 1 P = 3
pF
0
p 2 c2 2 4πp2 dp, 2 2 2 4 h p c +m c 3
(2.52)
The integral in Eq. (2.52) is performed, as before, by defining the dimensionless momenta x ≡ pF /mc and y ≡ p/mc, the result being conveniently expressed in units of P0 ≡ (8π/3)(m4 c5 /h3 ) as: P = P0 fP (x),
fP (x) ≡ x3
3 1 + x2 − fS (x), 8
(2.53)
where fS (x) is the Stoner function defined by Eq. (2.27). Note that the reference pressure P0 does indeed have the dimensions of pressure (which are the same as those of energy density), since it can be expressed as P0 = n0 mc2 , where n0 ≡ (8π/3)(mc/h)3 is a reference number density. (This becomes clear on comparing the expression for n0 with the relation between the fermion number density and Fermi momentum, n = (8π/3)(pF /h)3 , given in Sec. 2.2.3.1. Indeed, n0 for electrons can be alternatively expressed as ne0 = 1/3π 2 λ3e , where λe ≡ /me c is the electron Compton wavelength.) We again leave it as an exercise for the reader to show that the pressure function fP is given by 3 2x2 −1 2 fP (x) = − 1) + sinh x , x 1+x ( 8 3
(2.54)
that the NR (x 1) limit of this function is fPNR =
x7 x5 − ···, 5 14
(2.55)
19 Chandrasekhar (1935) gave a re-derivation of the general equation of state of a degenerate Fermi gas, in which there is an error in the ER limit corresponding to Eq. (2.56), which was repeated in Schatzman’s (1958) book White Dwarfs. The correct expression is given here, in Stoner (1932), and in Shapiro and Teukolsky (1983).
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and that its ER (x 1) limit is20 fPER =
x2 x4 − ···. 4 4
(2.56)
Once again, the reader will see that, on substituting the value of x in terms of n (i.e., using Eq. [2.31]), the leading terms in fPNR and fPER above NR ER (as given by Eq. [2.11]) and Pnull (as given by Eq. [2.12]) readily yield Pnull respectively, as they must. Equations (2.53) and (2.54), together with the relation between x and the density ρ (which readily follows from Eq. [2.31]), namely, 8π mc 3 mN 3 x , (2.57) ρ= 3 h µ constitute a parametric form of the general equation of state of a completely degenerate Fermi gas, which we shall henceforth call the StonerChandrasekhar equation of state. In order to obtain the general equilibria of white dwarfs, Chandrasekhar (1935) combined the equation of hydrostatic equilibrium, Eq. (2.35), with the above general equation of state, instead of the earlier polytropic relations. With characteristic mathematical skill, he cast the resultant structure equation, the Chandrasekhar equation, into a Lane-Emden-like form 1 d 2 dθ (2.58) ξ = −(θ2 − ζc2 )3/2 , ξ 2 dξ dξ which made its connections with the NR and ER polytropic limits transparent. In Eq. (2.58), the variable θ is now the Fermi energy, EF = p2F c2 + m2 c4 , of the electrons at any radius, scaled in terms of the Fermi energy, EFc , of the electrons at the center of the star, i.e., at r = 0, and the constant ζc is the reciprocal of EFc in units of the electron rest-mass energy: θ ≡ EF /EFc ,
ζc ≡ me c2 /EFc .
(2.59)
The dimensionless radial variable ξ is still defined by r ≡ aξ, as before, with the scale a now being given by: −1 me 3π a≡ RG (ML ) ≈ 7.72 × 103 µζc km. (2.60) µζc 4 mN 20 Again, we neglect here the logarithmic term, (3/8) ln(2x), which arises from the large-x limit of sinh−1 x.
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The reader will immediately recognize this scale as being essentially that occurring in Eqs. (2.22) and (2.23), execpt for the factor ζc , which shows the basic importance of this lengthscale, as we indicated earlier. The NR limit (EFc → me c2 ) corresponds to ζc → 1, and the ER limit c (EF me c2 ), to ζc → 0, so that the entire collection of degenerate stars from the one limit to the other is described by the range of values 0 ≤ ζc ≤ 1. It is straightforward to show that the Chandrasekhar equation does, indeed, reduce to the appropriate polytropic Lane-Emden equations in these two limits, i.e., to a polytrope of index s = 3/2 in the ζc → 1 limit, and to a polytrope with s = 3 in the ζc → 0 limit21 . The stellar radius R is defined, as before, as the point where the density vanishes, i.e., where n = 0 = pF , so that EF = me c2 , and θ = ζc (see Eqs. [2.59] and [2.31]). If this happens at ξ = ξ1 , then R = aξ1 . The stellar mass is calculated in a way entirely analogous to that which was used in Sec. 2.2.3.2 for polytropes, i.e., by integrating over the density profile and utilizing Eq. (2.58). This yields √ 3π ML µ2 ξ12 | θ (ξ1 ) | , M= 2
(2.61)
showing, as expected, the fundamental significance of the Landau mass scale ML . The reader will notice that Eqs. (2.61) and (2.50) look identical, and should not interpret this to mean that they really are so. The Chandrasekhar limit of Eq. (2.50) only applies in the ER limit, corresponding to the polytrope with s = 3, and it gives a constant mass. On the other hand, Eq. (2.61) is a general result valid for all white dwarfs, and it does not give a constant mass except in the ER limit. The complications are all hidden in the boundary-value quantity ξ12 | θ (ξ1 ) |, which in Eq. (2.61) depends on ζc , and so on EFc and the central density (since the Chandrasekhar equation, Eq. [2.58], involves ζc ), and is no longer a simple number charcteristic of a polytrope of a given index s. As noticed by Chandrasekhar (1935), this destroys the homology property of polytropes, which is that, independent of their masses, all polytropes of a given index have the same density profile (i.e., the same relative density in units of the central density at the same relative radius in units of the stellar radius). Each chosen value of the central density ρc now gives a distinct density profile, a distinct value of ξ12 | θ (ξ1 ) |, and so distinct values for both M and R on integrating the Chandrasekhar equation numerically. 21 The ER limit is trivial. The NR limit is obtained by noting that both θ and ζ then c have Taylor-series expansions around unity, and rewriting the Chandrsekhar equation in terms of modified definitions of θ and ξ. For details, see Chandrasekhar (1935).
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Fig. 2.4 Chandrasekhar’s mass-radius relation for white dwarfs. The mass M3 is the Chandrasekhar limit, obtained when the density becomes arbitrarily large, and the radius approaches zero. Reproduced with permission by Blackwell Publishing from Chandrasekhar (1935): see Bibliography.
This is what Chandrasekhar (1935) did: the result was a mass-radius relation for white dwarfs (or, alternatively, one between their masses and average densities ρ¯, defined as before), which is shown in Fig. 2.4, and a comparison with Fig. 2.3 again makes it quite obvious how much of the basic physics was already contained in the simple, uniform-density, StonerAnderson results. In the ER (high ρ) limit, M approaches the constant value which is, of course, the Chandrasekhar limit, and in the NR (low ρ) limit, the relation reduces to the M − ρ¯ relation for a polytrope22 of index of index s = 3/2, i.e., Eq. (2.45). 22 The reader may be surprised by this, as the boundary-value quantity ξ 2 | θ (ξ ) | 1 1 should then reduce to a constant number characteristic of a polytrope of index 3/2, and Eq. [2.61] should imply a constant mass, instead of a mass-radius relation. This is not so because of the way in which the Chandrasekhar equation reduces to the Lane-Emden equation in the NR limit, as explained in the previous footnote. The modified function θ1 which satisfies the Lane-Emden equation does not obey the standard central boundary condition θ c = 1 at ξ = 0. Rather, its central value θ1c depends on the central density in such a way that the boundary-value quantity ξ12 | θ (ξ1 ) |, which occurs in Eq. [2.61] and which involves the original θ, exactly reproduces Eq.[2.40] for the mass of a polytrope of index s = 3/2. See Chandrasekhar (1935) for a more detailed explanation of this point.
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We close this account of our foundations of the understanding of white dwarfs with two brief historical points. First, it took quite a while for the ER limit of degenerate-matter physics, which we have described above, to be fully understood and accepted by astrophysicists, since the conclusion that it logically led to, namely, that stars above a limiting mass would go into indefinite collapse, and produce what we call black holes now, was somewhat challenging philosophically at the time. Today, of course, we accept the idea of black holes and look for their observational signatures, and we regard the above idea of relativistic degeneracy and the consequent mass limit on degenerate stars as one of the cornerstones of modern astrophysics, together with Fowler’s brilliant, pioneering suggestion that dense stars should be degenerate. Second, for his achievements in astrophysics, Chandrasekhar shared23 the 1983 Nobel Prize in Physics, being cited for “his theoretical studies of the physical processes of importance to the structure and evolution of the stars”.
2.4.4
Modern Work
Even in the limit of complete degeneracy, i.e., without considering finitetemperature effects, stellar matter is subject to a variety of physical effects of electrostatic and nuclear origin which were not included in the early works described above, and which began to be investigated in the 1950s. In 1961, Salpeter published a comprehensive treatment of these effects which has become the point of departure for modern work. Consider first the effects of electrostatic or Coulomb interactions between the electrons and nuclei in the fully-ionized degenerate matter obtained from an element of atomic number Z and mass number A. Foremost among these is the attractive interaction between the nuclei and the electrons: at the densities of interest in white dwarfs, the nuclei of charge Ze form a lattice immersed in a sea of electrons, and this attraction wins over the mutual repulsion between the electrons, causing a decrease in the outward pressure. Thus, this effect reduces the zero-point energy and zero-point pressure of degenerate matter, yielding a smaller radius and higher density for a white dwarf of given mass, as compared to those given by the Chandrasekhar model. The 23 With W. A. Fowler (not R. H. Fowler, whose pioneering contributions to the physics of degenerate stars we have discussed above, and who had passed away in 1944: see Chandrasekhar’s obituary of Fowler referred to in Sec. 2.4), who won it for his work in nucleosynthesis.
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calculation of the Coulomb effects proceeds by first making the WignerSeitz approximation, in which each lattice cell is replaced by a uniform negatively-charged sphere of total charge −Ze (representing the electrons) with a nucleus of positive charge Ze at its center: we describe this calculation in Sec. 3.1. The “classical” Coulomb energy so calculated is, in fact, the largest electrostatic correction by far: (smaller) modifications made to it take into account (a) the non-uniform distribution of the negative electronic charge, which is done according to the Thomas-Fermi prescription (see Sec. 3.1), (b) the energy of quantum-mechanical exchange interactions, which occur because of the antisymmetric wave function of the electrons, and, (c) even smaller, higher-order corrections to the electron-electron interaction energy. In contrast to the situation for non-degenerate matter, the Coulomb corrections decrease with increasing density for degenerate matter (see Sec. 3.1), and so are the most important for the lightest white dwarfs, which have the smallest density. Consider now the effects of inverse β-decay, which become important at high densities, and cause a qualitative change from the Chandrasekhar results, as opposed to the detailed, quantitative changes due to Coulomb interactions described above. At sufficiently high densities, the nuclei of the element (or elements) that the degenerate white-dwarf matter is composed of undergo inverse β-decay, in which protons in these nuclei capture elctrons to produce neutrons: p + e− → n + ν .
(2.62)
This is the process of neutronization, which is crucial for eventually producing neutron stars at very high densities, and which we shall discuss further in later sections of this chapter. Note first that inverse β-decay is energetically possible whenever an electron has more total energy than the difference between the rest-mass energies of the neutron and the proton, (mn − mp )c2 ≈ 1.29 MeV, but the process is effective only when the reverse process, i.e., the β-decay of neutrons into protons, n → p + e− + ν¯ ,
(2.63)
which is always energetically possible by itself, is blocked because all the possible electronic energy states in which the electron of Eq. (2.63) may be emitted are fully occupied in the degenerate electron plasma (see Sec. 2.2.1.3). This happens when the Fermi energy EF exceeds the total
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energy of the highest energy-state that the electron can be emitted into, which, in turn, happens only at sufficiently high densities, as a glance at Eq. (2.10) shows. Each individual inverse β-decay converts a nucleus (Z, A) (i.e., atomic number Z, mass number A) into a nucleus (Z − 1, A). The resulting phase transition from one nuclear species to another “softens” the stellar matter, i.e., increases its compressibility ( the technical words used in the subject are that it softens the equation of state). This is so because compression and consequent density increase do not now increase the pressure and the Fermi energy as they normally would, but, rather, cause nuclear phase transition. The polytropic exponent Γ (see 2.2.3.2), which was already close to 4/3 because the electrons are highly relativistic in this density range, now falls below 4/3, which makes the situation unstable, and the star subject to gravitational collapse. The density at which this happens first is, therefore, the maximum density that a white dwarf can have: this density depends on the chemical composition of the white-dwarf matter, as we describe below. Thus, for a given composition (Z, A), the sequence of possible white dwarfs on the density vs. mass plot does not go to arbitrarily high densities and the Chandrasekhar limit, but terminates at a finite, maximum density ρmax and a corresponding maximum mass Mmax , which is less than the Chandrasekhar mass MCh for this composition. This is the qualitative difference from the Chandrasekhar result we referred to above. Computationally, it appears in the following way. If we compute the masses and radii of equilibrium white dwarf configurations after including the effects of inverse β-decay, we find that, while M increases with increasing ρ at low and moderately high densities, it reaches a maximum Mmax at a very high density ρmax , after which M falls with increasing ρ. It is this falling branch that contains the unstable white dwarfs with Γ < 4/3, stars which cannot occur in nature. This phenomenon was clearly noticed in 1958 both by Schatzman and by Harrison, Wakano, and Wheeler. In 1961, Hamada and Salpeter gave a detailed computation of the effect, which we summarize next. If the white-dwarf matter were pure hydrogen, so that the nuclei in question were just protons, the criterion for the onset of inverse β-decay would be EF ≥ (mn − mp )c2 , which corresponds to a density threshold of 7 −3 ρ> ∼ 1.2 × 10 g cm . But, of course, the matter in real white-dwarf stars is predominantly not hydrogen: in the process of their evolution through thermonuclear reactions (see Appendix C), normal stars convert most of
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their hydrogen into heavier elements, so that the white dwarfs produced at the end of this evolution have cores (which contain most of the dwarf’s mass) made of the heavier elements, surrounded by lighter, outer envelopes that contain the remnant hydrogen, if any. Which elements dominate the chemical composition of the white-dwarf core depends on the initial mass of the star that produced the white dwarf (see below). Consider, for simplicity, a white-dwarf core made entirely of degenerate electrons and the nuclei (Z, A), as above, which undergo inverse β-decay into (Z − 1, A) when the electron Fermi energy satisfies EF > Z , where Z is the elctron-emission β-decay energy of nucleus (Z − 1, A) (see above). Now, since nuclei with both an even number of protons and an even number of neutrons (called even-even nuclei), and therefore with both Z and A even are generally more stable than odd-odd nuclei, the original (Z, A) nuclei are likely to be even24 28 56 even ones, possiblities24 being 126 C, 168 O, 20 10 Ne, 12 Mg, 14 Si, . . . 26 Fe. Note that the number of electrons per nucleon µ = Z/A for all of these elements upto Si is exactly µ = 1/2, while that for Fe is µ = 26/56 ≈ 1/2.154. When such an even-even nucleus (Z, A) becomes an odd-odd nucleus (Z − 1, A) by an inverse β-decay, it generally turns out that the elctronemission β-decay energy Z−1 of the next even-even nucleus (Z − 2, A) is less than Z and therefore less than EF . Hence (Z − 1, A) immediately undergoes another inverse β-decay into (Z − 2, A), which means that, in effect, an even-even nucleus undergoes a rapid double β-decay into the next even-even nucleus with the same A. Examples are 126 C → 125 B → 12 24 24 24 4 Be, and 12 Mg → 11 Na → 10 Ne. Generally, this next even-even nucleus is stable against further inverse β-decay until a considerably higher density and higher Fermi energy is reached. The density thresholds for inverse βdecay for the nuclei listed above are in the range ∼ 1 × 109 – 4 × 1010 g cm−3 , the corresponding Fermi energies are in the range ∼ 4 – 14 MeV, and the corresponding values of the relativity parameter x are in the range ∼ 8 – 28, showing that the elctrons are extremely relativistic at this point. Hamada and Salpeter (1961) computed white-dwarf models for the pure (initial) compositions listed above, including all the Coulomb and inverse β-decay effects just described. Figure 2.3(b) shows their results for 126 C, 24 28 56 12 Mg, 14 Si, and 26 Fe white dwarfs: in each case, the reader can clearly see 24 Of these, iron is the most tightly bound nucleus, the standard endpoint of the chain of complete thermonuclear burning in stars, which does not mean, of course, that whitedwarf matter can necessarily reach this endpoint. Indeed, we believe today that it does not: see next page.
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the occurrence of a maximum mass Mmax below MCh and the unstable stellar models beyond this maximum density on the falling branch of the curve. Modern calculations of stellar evolution indicate that most white dwarfs form from progenitor stars of original mass ∼ 1 – 10 M . After the completion of helium burning in the stellar core, such a star “climbs” the Asymptotic Giant Branch (AGB) in the Hertzsprung-Russell diagram, with increasing luminosity and decreasing temperature (see Appendix C). At this stage, the star consists of a degenerate core composed of C and O (and also Ne and Mg for the heaviest progenitors), surrounded by thin shells where thermonuclear burning of H and He is still going on (see Iben 1991). The degenerate core contains a mass ∼ 0.5 – 1.4 M in a region of the size of the Earth: this is what will eventually become the white dwarf. The rest of the stellar mass lies in a cool, hugely extended envelope (radius several hundred times that of the Sun) of very weak gravitational binding, which is expelled during the AGB evolution, particularly due to large energy inputs from periodic pulsations in the burning shells due to thermonuclear runaway (the so-called “shell flashes”). The degenerate core thus exposed is a white-hot dwarf star—the white dwarf—which then cools slowly to oblivion (unless the white dwarf is in a mass-transfer binary, and emits radiation due to accretion of matter from its companion, as we describe in Chapter 6). For progenitors of initial mass ∼ 1–8 M , the cores consist of C and O, and the resulting degenerate stars are called CO white dwarfs, while for progenitors of mass ∼ 8–10 M , the cores consist principally of O, Ne, and Mg, and we get ONeMg white dwarfs. CO white dwarfs range in mass from ∼ 0.5M to ∼ 1.3M (scaling roughly linearly with the original progenitor mass; see Iben and Truran 1978), and ONeMg white dwarfs have a narrow range of masses around ∼ 1.35M (Nomoto 1984). It thus appears that µ = 1/2 should provide an excellent description of the matter in white-dwarf cores. Eggleton (1982) has given a simple analytic approximation to the results of numerical calculations of the mass-radius relation for white dwarfs (see Truran and Livio 1986),
M M
1/3
R ≈ 9.6 × 10 km 1 − 3
M MCh
4/3 ,
(2.64)
which is very useful in modern work for a variety of rapid estimates, and which can be compared with the mass-radius relations given earlier.
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As this is not a book on white dwarfs, we conclude our account of these degenerate stars here, and return to these in later sections or chapters only with reference to neutron stars. For the reader interested in further study of white dwarfs, there is Schatzman’s (1958) pioneering book White Dwarfs, many excellent review articles, e.g., those by Weidemann (1968, 1990) and Liebert (1980), and numerous conference proceedings, all of which have references to the original papers. Evolution and structure of white dwarfs, their luminosity function, their cooling and crystallization, their atmospheres, and their magnetic fields have all blossomed into important specific areas of research, as has the study of their pulsations through novel techniques devised recently. The reader can get a feel for the progress made in these areas in the last ten to fifteen years from the proceedings of such conferences as White Dwarfs: IAU Colloquium 114 (Wegner 1989), or White Dwarfs: 9th European Workshop (Koester and Werner 1995). Binary stellar systems containing white dwarfs is a rich field of study to which we shall return at appropriate points in this book, as it will be of direct interest to us.
2.5
Neutron Stars
After inverse β-decay makes white dwarfs unstable, cold degenerate matter continues to collapse, and becomes more and more neutronized through a set of transformations which form the subject of the next sections. Long before they were fully appreciated, however, it was correctly guessed that matter would consist mostly of neutrons at sufficiently high densities (see Sec. 2.1), that these neutrons would become degenerate (see Sec. 2.2.1.2), and that the zero-point pressure of this cold, degenerate neutron gas would again be able to balance its own self-gravitational attraction in a stable fashion, producing a new kind of degenerate star, viz., the neutron star. It was on this basis that Oppenheimer and Volkoff (1939) performed their pioneering calculation of the masses and radii of neutron stars. The main concern of these authors was the correct mass limit for neutron stars—the analogue of the Chandrasekhar limit for white dwarfs—above which the cold, degenerate neutrons cannot support a stable configuration. Not satisfied with the simple-minded scaling of the Chandrasekhar mass limit to µ = 1 for neutrons (see Sec. 2.2.2), which gives Mc ≈ 5.7M (see Sec. 2.2.2.1) for neutron-star mass limit, Oppenheimer and Volkoff correctly anticipated that the neglect of general-relativistic effects, as done quite justifiably for
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white dwarfs, would not be justified at all at the enormous densities of neutron stars. The inclusion of these effects changed the results grossly, bearing out this anticipation fully. We summarize the Oppenheimer-Volkoff work below, before describing more modern and more detailed understanding of neutron-star matter in the next chapters. 2.5.1
The Oppenheimer-Volkoff Work
What essential changes enter into the description of a degenerate star when we include the effects of general relativity? First of all, mass-density ρ in our intuitive, Newtonian sense ceases to be the most basic quantity: rather, it is generalized to a total energy-density divided by c2 , ρ ≡ /c2 , of which our usual mass-density is the Newtonian limit. Here, includes the rest-mass energy of the particles involved, as well as all other forms of energy they have, since all forms of energy are are equivalent to mass according to the theory of relativity. For a completely degenerate Fermi gas, we can express in terms of the zero-point kinetic energy Enull per particle, introduced in Sec. 2.2.2, as = n(Enull + mc2 ) ,
(2.65)
m being the fermion mass, and n the number density of the fermions, as before. Of course, m means the neutron mass mn in all discussions of neutron stars. What happens to the Stoner-Chandrasekhar equation of state for completely degenerate matter (see Sec. 2.2.3.3) because of this? Nothing of substance, except that we have to re-express the pressure P (x) parametrically in terms of the energy density (x) now, x ≡ pF /mc being the relativity parameter introduced earlier. This is easy, since the pressure is already given in this form by Eqs. (2.53) and (2.54), and the energy density is readily obtained by combining Eq. (2.65) with Eqs. (2.26) and (2.27), which gives: =
3 P0 fS (x) , 8
(2.66)
where fS (x) is the Stoner function, and P0 ≡ (8π/3)(m4 c5 /h3 ) is the reference pressure, or energy-density, introduced earlier. Actually, Oppenheimer and Volkoff (1939) introduced a modified relativity parameter t related to the usual one x by the transformation x = sinh(t/4), in terms of which they expressed their results as:
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P =
t P0 (sinh t − 8 sinh + 3t) , 32 2
≡ ρc2 =
3P0 (sinh t − t) . 32
(2.67)
The algebra leading from Eqs. (2.53), (2.54), (2.66), and (2.27) to Eq. (2.67) is straightforward, which we leave to the reader as an exercise. The most significant changes enter in the description of the hydrostatic equilibrium of the degenerate star, as one might expect, since it involves the self-gravitational field of the star, and general relativity is a relativistic theory of gravitation, as opposed to Newton’s theory, which we have been using so far in this book. How is Eq. (10.35) modified by general relativity? Newtonian theory, or, equivalently, Poisson’s equation relating the second derivative of the Newtonian gravitational potential to the usual mass-density, is now replaced by Einstein’s equation relating the curvature tensor (which involves the derivatives of the co-ordinate metric; see,e.g., Weinberg 1972) to the energy-momentum tensor. For a spherically symmetric star in hydrostatic equilibrium, the gravitational field outside the star is given by the exterior Schwarzschild solution, and at every radius r inside the star, the condition for hydrostatic equilibrium becomes √ dP (r)/dr = (ρ(r) + P (r)/c2 )(−d ln −gtt /dr), i.e., (ρ + P/c2 ) replaces the √ mass-density in Eq. (10.35), and ln −gtt replaces the Newtonian gravitational potential. Here, gtt is the “time-time” part of the metric inside the star. Using the Einstein equations (see Weinberg 1972), we can eliminate the derivative of the metric in the above hydrostatic-equilibrium condition, and so obtain the equation describing the internal structure of spherically symmetric, static stars in general relativity, the Tolman-OppenheimerVolkoff (TOV) equation25 , as
P (r) 4πr 3 P (r) dP (r) Gm(r) 1 + ρ(r)c2 1 + m(r)c2
= −ρ(r) . dr r2 1 − 2Gm(r) rc2
(2.68)
Here, m(r), defined either by the differential equation dm(r)/dr = 4πr2 ρ(r) ,
(2.69)
25 Tolman not only laid the foundations of this field through his influential (1934) book Relativity, Thermodynamics and Cosmology, but also discussed various analytic solutions of TOV equations, and associated matter distributions, in a paper (1939) accompanying the Oppenheimer-Volkoff paper.
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r or, equivalently, by the integral m(r) ≡ 0 4πr2 ρ(r)dr, is the total massenergy inside radius r, including the gravitational self-energy. This is a crucial point, which needs to be fully appreciated by the reader before proceedR ing further. The mass of the star, defined by M ≡ m(R) ≡ 0 4πr2 ρ(r)dr, contains both the matter-energy (which includes the rest-mass energy and the zero-point kinetic energy, both of which have been counted already in our general-relativistic definition of ρ given above, as well as thermal energy, if we are considering non-zero temperatures) and the energy of the self-gravitational field of the star. This mass M is what is measured by a distant observer as the star’s mass, as inferred from its external gravitational field. (In more technical words, M is what describes the exterior Schwarzschild solution: see Weinberg 1972.) Thismay seem paradoxical to R the reader, since the simple-looking integral M = 0 4πr2 ρ(r)dr appears to be only a sum over the matter (or material) energy, and yet we are claiming that it includes both material and gravitational energy. However, it is the appearance which is wrong here, because the proper volume element dv in general relativity is not simply 4πr2 dr, but involves the space-time metric, and is (1 − 2Gm(r)/c2 )−1/2 4πr2 dr in this case, so that M , as defined by the above expression, is not an integral simply over the element of material energy ρdv. As a matter of fact, M does include both material and selfgravitational energy. It is the superficial similarity of Eq. (2.69) with the corresponding Newtonian equation (see Sec. 2.2.3.2) that tends to mislead us. In the TOV equation, the general-relativistic modifications can be seen as the factors within square brackets26 . The origins of these modifications are clear from the discussion given above Eq. (2.68): the first square bracket in the numerator comes from the fact that (ρ + P/c2 ) now replaces the mass-density of the Newtonian formulation, representing the components of the energy-momentum tensor (remember that ρ is now the energy-density, ρ ≡ /c2 , as explained earlier). The rest of the brackets are the effects of the metric, the Schwarzschild metric in this case. We expect, therefore, that when the relevant metric changes, e.g., for describing a rotating neutron star, the latter two factors will change, while the former will remain the same. 26 In
general relativity, it is convenient to use natural or geometrized units, in which c = G = 1. However, since we are introducing general-relativistic effects at this point, we have displayed the TOV equation in conventional units, to familiarize the reader with the way in which factors of c and G enter. Later, we shall use the geometrized units when appropriate.
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Oppenheimer and Volkoff integrated Eqs. (2.67), (2.68) and (2.69) to obtain the masses and radii of neutron stars. The integration had to be numerical, of course, since even the Newtonian limit of these equations, which were combined by Chandrasekhar into Eq. (2.58), had to be integrated numerically, and, indeed, even the Lane-Emden equation, of which the Chandrasekhar equation can be thought of as a generalization, has to be so integrated except for very special values of the polytropic index s (which lie outside the range of s-values relevant for degenerate stars; see Sec. 2.2.3.2). It should come as no surprise to the reader, therefore, that all subsequent work on neutron-star masses and radii, which incorporated a variety of further physical effects into the equation of state and which we shall describe in the next chapters, necessarily involved numerical integration of the TOV equation. This is a standard procedure today, indeed rather easy for modern computers. We choose a central density (in the above generalized sense) ρc , and start integrating the above equations from the center, r = 0, where the other boundary condition is m(0) = 0. At each step, the equation of state relates the value of P to that of ρ: for the more complete, modern equations of state, the P –ρ relation becomes so complicated that it is customary to store it as a numerical P vs. ρ table in the computer, instead of the analytic relation given by Eq. (2.67). We stop the outward integration where P = 0: this is the surface of the neutron star, so that corresponding value r = R is the stellar radius, and m(R) = M is the stellar mass. Actually, Oppenheimer and Volkoff chose to work in terms of their modified realtivity parameter t rather than the density ρ (the relation between the two is, of course, Eqs. [2.67]), and so displayed their results, shown in Fig. 2.5, not as a relation between M and ρc , but as one between M and t0 , where t0 = tc ≡ t (r = 0) is the central value of t. (From the definition of t given earlier, the reader can easily show that t0 is closely related to the relativity parameter, and so to the Fermi energy and momentum, at the stellar center: xc = sinh[t0 /4], pcF = mcxc and EFc = mc2 cosh[t0 /4].) The stellar mass increases at first with increasing t0 (and therefore increasing ρc and xc ), representing stable neutron stars, until t0 ≈ 3 is reached, corresponding to xc ≈ 0.82 and ρc ≈ 4 × 1015 g cm−3 , where M passes through a maximum value Mmax = MOV ≈ 0.71M, the Oppenheimer-Volkoff mass limit. After this, M decreases with increasing ρc , xc , and t0 , representing unstable neutron stars which cannot occur in nature. The limiting stable configuration with mass MOV had a radius R ≈ 9.5 km according to the Oppenheimer-Volkoff calculation, which can be compared with the radius given by the simple estimate of Eqn. (2.23).
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Fig. 2.5 The Oppenheimer-Volkoff relation between a neutron star’s mass and the parameter t0 . The latter is a measure of its central density: see text. Reprinted with perc 1939 mission by the APS from Oppenheimer & Volkoff (1939), Phys. Rev., 55, 374. American Physical Society.
Note well that the physical origin of the Oppenheimer-Volkoff maximum in the neutron-star mass in Fig. 2.5 is different from that of the Chandrasekhar limit (1931b,c; 1935) for white dwarfs or its equivalent for neutron stars proposed by Landau (1938). Chandrasekhar and Landau worked entirely within the domain of the Newtonian theory of gravitation, and the Chandrasekhar-Landau limit comes from a physical cause which was first identified by Landau (1932), and which we have discussed earlier. It is that extremely relativistic degenerate fermions cannot provide an equilibrium stellar configuration by balancing the star’s self-gravitation with their zero-point pressure, if the star’s mass exceeds the Chandrasekhar-Landau limit. This limit does not appear as a maximum mass in the density-mass (or mass-radius) relation, but, rather, is appraoched asymptotically as the stellar density becomes very high, and the fermions become extremely relativistic, as we saw earlier. By contrast, Oppenheimer and Volkoff worked in a fully general-relativistic theory of gravitation, where new pieces of essential physics are added, as described above. As the density of the neutron star increases, general-relativsitic effects become important long before the neutrons are extremely relativistic and the Landau limit for neutrons has any relevance. These effects make the neutron star unstable to perturbations beyond a finite critical density (corresponding to a relativity
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parameter x ≈ 1 for neutrons: see below), and so terminate the sequence of physically viable neutron stars at this point. This is the OppenheimerVolkoff limit, where the stellar mass passes through a maximum. Beyond this limit, equilibrium configurations are still possible (on the falling branch of the curve in Fig. 2.5, where stellar mass decreases with increasing density), but they are unstable and so do not occur in nature. There is, thus, a basic difference between the characters of the two limits. We illustrate the basic features of the Oppenheimer-Volkoff limit with the aid of a simple, but general-relativistic, “toy” model of neutron stars at the end of this chapter. We give there the essential variations in energy with density (or, equivalently, the relativity parameter x), i.e., the generalized Landau arguments for the general-relativistic case. As we shall see there, general relativity introduces two extrema in the energy curves as opposed to the one seen in Fig. 2.2 — a minimum corresponding to the Newtonian one already described earlier, and a maximum which is entirely general-realtivistic. The former is a stable equilibrium where the star is actually found, while the latter is an unstable equilibrium (corresponding to the falling branch in Fig. 2.5) which cannot occur in nature. As the stellar mass M increases, the two extrema approach each other, and coalesce at an inflection point for a maximum mass Mmax , which is the Oppenheimer-Volkoff limit (see Fig.2.7). At higher masses, there are no extrema, and no equilibria are possible. In this sense, then, we could think of a formal similarity between the Newtonian and the general-relativistic case, but it is only formal. The physical reasons for the disappearance of the extrema in the two cases are quite different: in the Newtonian case, it happens because the fermions become extremely relativsitic (ER) in their zero-point motion, as explained earlier, while in the GR case, it happens because of general-relativistic effects at high densities, which have no Newtonian analogue, and which occur, indeed, at densities where the zero-point motion of the neutrons is still only mildly relativistic (x < ∼ 1). Thus, the Chandrasekhar-Landau limit is completely Newtonian, while there appears to be no Newtonian analogue of the Oppenheimer-Volkoff limit. The reader will remember that we described earlier that white-dwarf masses also pass through a maximum value at a finite density when the effects of inverse β-decay are taken into account, terminating the sequence of physically viable white dwarfs at this maximum mass, since white dwarfs at higher densities are unstable, lying on a branch of the curve where stellar mass decreases with increasing stellar density. Naturally, the reader might wonder what connection there is between this phenomenon and the
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Oppenheimer-Volkoff limit. The answer, of course, is: none whatever. The white-dwarf maximum mass at a finite density occurs because of the detailed nuclear physics of white-dwarf matter, i.e., inverse β-decay of a given nuclear species above a density threshold, while the Oppenheimer-Volkoff maximum mass is a basic result of general relativity, completely independent of the details of neutron-star matter. The former occurs when electrons are extremely relativistic, with the relativity parameter at the stellar center xec ≡ peF /me c ∼ 10 – 30 (see Sec. 2.4.4), while the latter occurs when the neutrons are just becoming relativistic, with xnc ≡ pnF /mn c < ∼ 1, as we saw above. In fact, the Oppenheimer-Volkoff (1939) treatment of neutron stars is exactly at the same level of detail as the Chandrasekhar (1935) treatment of white dwarfs as far as the description of the stellar matter is concerned: each neglects the interactions between the fermions concerned, and each uses an exact, completely degenerate, equation of state for these non-interacting fermions. Modifications of the asymptotic approach to the Chandrasekhar limit enter only at the next higher level of detail, as we described in Sec. 2.4.4, when we put in the nuclear interactions. We stress this point particularly because there is a tendency in scientific literature to apply the name “Chandrasekhar limit” to the above maximum mass of white dwarfs at a finite density, which can cause confusion about the underlying physics. The correct analogue for neutron stars of the inverse β-decay effects on the white-dwarf mass would be the effects of any further nuclear transformations in neutronized matter at extremely high densities, e.g., production of muons and then a variety of hyperons, which we summarize later in the book. The other natural question, of course, is if there is an analogue of the Oppenheimer-Volkoff limit for white dwarfs at all. In other words, if we included general relativity in our description of stars supported by the zero-point pressure of degenerate electrons and neglected all other corrections to the Stoner-Chandrasekhar equation of state that we described in Sec. 2.4.4, would there be a point where the general-relativistic effects alone are sufficient to make the star unstable, and where the density is not yet high enough to make the neutrons degenerate? The answer is yes [Chandrasekhar 1964]: the onset of general-relativistic instability in white dwarfs occurs at a density ∼ 2.6 × 1010 gm cm−3 when its constituent elements have µ = 1/2, as is the case for C, O, Ne, Mg — the elements we now believe to be the principal constituents of white dwarfs existing in nature (see Sec. 2.4.4). This density is either greater than or comparable to that for the onset of inverse β-decay of these elements (see Sec. 2.4.4).
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Thus, depending on the elemental composition, the point where the whitedwarf sequence terminates is determined entirely by the nuclear properties of its constituents, or by a combination of these and general-realtivistic effects, which are difficult to disentangle in practice. It appears at this time, therefore, that the effects of general relativity on white-dwarf stars are of secondary importance for practical purposes. 2.5.2
A General-Relativistic “Toy” Neutron Star
In order to clarify the essential effect of general relativity on the masses of neutron stars without the complication of numerical work (which sometimes tends to obscure the physics involved), we now present a simple, analytic, “toy” model of a neutron star which still retains general relativity in full, and so displays its effect transparently. The “toy” neutron star has a uniform density, in much the same spirit as our Newtonian uniform-density models of degenerate stars of Sec. 2.3, and Stoner’s white-dwarf models of Sec. 2.4.1. Amazing as it may seem, the TOV equation then has a simple, analytic solution, which was discovered by Schwarzschild in 1916 in his famous application of Einstein’s general-relativity theory to uniform spheres, and which we return to in Chapter 5. The reader can easily obtain the Schwarzschild (1916) solution by writing m(r) = (4/3)πr3 ρ in Eq. (2.68) and changing the radial variable from r to (1 − 8πGρr2 /3c2 ). The solution is P (r) 2 = −1 , 1−8πGρr 2 /3c2 ρc2 3− 1−2GM/Rc2
(2.70)
where M ≡ m(R) = (4/3)πR3 ρ is the stellar mass, R being the stellar radius. In terms of the central pressure Pc ≡ P (0), the stellar radius (where P (R) = 0) is given by R= and the stellar mass by M=
3Pc 2πρ2 G
6 π
1/2 1/2 1 + 2Pc /ρc2 , (1 + 3Pc /ρc2 )
Pc Gρ4/3
3/2 3/2 1 + 2Pc /ρc2 . (1 + 3Pc /ρc2 )3
(2.71)
(2.72)
We emphasize that, by setting ρ to a constant value independent of the radial coordinate, we are not claiming that the stellar matter is perfectly
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incompressible, in which case the equation of state for degenerate matter (Eq. [2.67]) would not apply. Rather, we are crudely mimicking the actual situation for neutron stars, particularly the heavier ones, where the density is nearly constant over most of the star (which accounts for almost all of the stellar mass), before falling off rapidly in the outermost layers, as we shall see later. While this simple approach is not strictly self-consistent, since P varies with r in Eqn. (2.72) but ρ does not, the results will, hopefully, display the essential features: this is the idea of a “toy” model. We now need to relate the central pressure to the central density, using an appropriate equation of state. Again in the spirit of the toy model, and remembering that neutrons in neutron stars are non-relativistic or mildly relativistic, we use the non-relativistic equation of state Pc = K NR ρ5/3 ,
(2.73)
where the constant K NR for the neutrons (compare with Eq. [2.42]) is given by K
NR
1 ≈ 20
2/3 2 3 h . 8/3 π mN
(2.74)
Combining Eqs. (2.72), (2.73), and (2.74), we obtain the mass vs. density relation for our toy neutron stars. We can now use the Landau mass scale ML ≡ (c/G)3/2 /m2N ≈ 1.85M and the Landau length scale RL ≡ RG (ML ) = GML /c2 defined in Sec. 2.2.2, as they are the natural units for the problem, and furthermore define a Landau density scale ρL 3 by the relation ρL ≡ ML /[(4/3)πRL ] ≈ 4.29 × 1016 g cm−3 . In these units, the density-mass relation is: M 9π = √ ML 5 10
ρ ρL
3/2 1/2 1 + 2β (ρ/ρ )2/3 L
3 , 2/3 1 + 3β (ρ/ρL )
(2.75)
where β ≡ (9π/4)2/3 /5 ≈ 0.737. If we neglect the deviation of the quantities within square brackets in Eq. (2.75) from unity, we recover the familiar √ density-mass relation (M ∝ ρ) for a non-relatvistic degenerate Fermi gas of neutrons (a polytrope with s = 3/2) in Newtonian gravity. Thus, the general-relativistic effects are contained entirely (and transparently) in the square brackets, the strength of these effects being measured by the parameter (Pc /ρc2 ) = β(ρ/ρL )2/3 . It must be clear to the reader that, as ρ
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increases and so does this parameter, M given by Eq. (2.75) passes through a maximum, and declines at still higher densities. Actually, it is instructive to rewrite our results in terms of the relativity parameter xnc for the neutrons at the stellar center introduced above, which we shall abbreviate as x in further discussions of this toy model. x is, of course, a measure of the importance of the effects of special relativity, by definition. Note, however, that we can express it as 1/3 1/3 1/3 ρ ρ 9π ≈ 1.92 , (2.76) x= 4 ρL ρL with the aid of Eq. (2.57), so that we can rewrite the general-relativity parameter introduced above as: 2/3 1 ρ Pc =β = x2 . (2.77) ρc2 ρL 5 This shows that x is also a measure of the general-relativistic effects: we shall display our final results in trems of x, for ease of comparison with those of Oppenheimer-Volkoff and others. We rewrite Eq. (2.75) as 6 M = ML 5
π 3/2 x g(x) , 10
g(x) ≡
3/2 1 + 2x2 /5 (1 + 3x2 /5)3
,
(2.78)
and display the toy neutron-star’s mass as a function of x in Fig. 2.6. The reader can easily show that the maximum mass is Mmax ≈ 0.51M, that it occurs at x ≈ 1.07, and that the corresponding value of the density is ρ ≈ 7.4 × 1015 g cm−3 . These values can be compared with the corresponding Oppenheimer-Volkoff values given earlier. Rather than investigating the details of the deviations of the toy-star properties from those of the numerical Oppenheimer-Volkoff models, we should note the remarkable similarity between the two despite the simplifying assumptions that went into the former, and ask ourselves how this happened. The answer is clear: the uniform-density profile is close enough to the actual profiles of numerically calculated neutron-star models, and the NR approximation is adequate enough for the equation of state of neutrons in range of densities relevant for stable neutron stars (although not so, of course, for x 1), that no gross errors were made on the stable branch of the density-mass relation. The toy model is very useful because it shows us directly where and how the general-relativistic effects come in, viz., through the quantities in square brackets in Eq. (2.75). Each of these
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Uniform Density "Toy"
Neutron Star Mass
0.3 0.2 0.1 0.0 0.01
0.10 1.00 Relativity Parameter (x)
Fig. 2.6 Mass of our uniform-density “toy” neutron star vs. its relativity parameter x, the latter being a measure of its density (see text). Note the similarities with the Oppenheimer-Volkoff results shown in Fig. 2.5.
arises from a mixture of the two effects we described in Sec. 2.5.1: (a) the replacement of ρ by ρ + P/c2 , and, (b) the deviation of the metric gtt from its flat spacetime value.
2.6
Landau Arguments: General-Relativistic Modifications
Beautiful as they are, the Landau arguments as given in Sec. 2.3 still lack one essential piece of physics, since they use Newtonian theory of gravity, and so miss the general-relativistic effects. We have already seen in the last section what happens to the hydrostatic equlibrium and the densitymass relation of degenerate stars when the Newtonian theory is replaced by the general theory of relativity. What happens to Landau’s energy arguments under this replacement is also highly interesting, as we describe now: the modified Landau arguments immediately show the existence of the branch of unstable equilibria in a transparent way, again underscoring their inherent power. The essential modifications to the Landau arguments are entirely in the self-gravitational energy Egrav , the Newtonian expression for which,
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as given by Eq. (2.13), must now be replaced by an appropriate generalrelativistic expression. By comparison, the slight modification in the zeropoint energy is only formal, involving merely a re-definition of the massenergy as explained above, and of little consequence. Even then, we shall include it in the explicit calculations presented below. These calculations are done for a slightly more sophisticated version of the uniform-density toy neutron star introduced above, in which we use the exact equation of state for the degenerate neutrons instead of its NR limit as we did in the last section. This makes the comparison with the Newtonian Landau arguments presented in Sec. 2.3 more transparent. Of course, this slightly more sophisticated toy has properties which are slightly different from those of the simple toy of last section. For example, the maximum mass now occurs at Mmax ≈ 0.85M, as we shall see. We calculate the self-gravitational energy Egrav of our toy star by noting that it is the difference, Egrav = (M − M0 )c2 , between the toR tal “gravitational” mass of the star, M ≡ 0 4πr2 ρ(r)dr, as seen by a distant observer, and the total matter-content (including both restmass energy) of the star, which is the integral M0 ≡ R and zero-point 2 −1/2 4πr2 ρdr, as explained in Sec. 2.5.1. Using m(r) = 0 (1 − 2Gm(r)/c ) 3 (4/3)πr ρ and M = (4/3)πR3 ρ for our toy star, we get Egrav = −G(a), M c2
G(a) ≡ 3 0
1
y 2 dy −1 , 1 − ay 2
(2.79)
where a ≡ (2GM/Rc2 ) is called the compactness parameter of the star, and sometimes also its redshift parameter, since it is closely connected to the general-relativistic redshift of the radiation emitted from the stellar surface, as we shall see later in the book. The integral in Eq. (2.79) is elementary and yields G(a) =
3 −1 √ sin a − a(1 − a) −1 . 2a3/2
(2.80)
The reader can easily work out the Newtonian (a 1) limit of G(a), viz., G(a) =
9 3 a + a2 · · · , 10 56
(2.81)
which shows that we recover the Newtonian result Egrav = −(3/5)GM 2 /R from the leading term in Eq. (2.81), as we must.
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Consider now the modified definition of density, ρ ≡ n(mN +Enull /c2 ) as explained above, n being the number density of the neutrons, and note that the total number N of neutrons in the star is related to its total mattercontent M0 introduced above as M0 = N (mN + Enull /c2 ) = M mN [1 + fE (x)], where we have used Eq. (2.26) in the last step. Accordingly, the self-gravitational energy per neutron, Egrav ≡ Egrav /N , can be expressed as Egrav G(a) [1 + fE (x)] , =− m N c2 1 + G(a)
(2.82)
and the total energy per neutron, E ≡ Egrav + Enull can therefore be written, using Enull = mN c2 fE (x), as: E mN
c2
=
fE (x) − G(a) . 1 + G(a)
(2.83)
Observe next that the compactness parameter, which can be written as a = 2(4π/3)1/3 GM 2/3 ρ1/3 /c2 for a uniform-density star, can be expressed in terms of the relativity parameter x with the aid of the expression for the modified mass-density ρ given in the last paragraph, and Eq. (2.31). The result is most conveniently expressed in terms of the Newtonian mass limit Mc we introduced in Sec. 2.3 (see Eq.[2.21]), identical to Stoner’s mass limit MS discussed in Sec. 2.4.1 (see Eq.[2.32]), and is given by a=
5 2
M Mc
2/3 x[1 + fE (x)]1/3 .
(2.84)
For neutron stars, Mc = MS ≈ 6.89M corresponding to µ ≈ 1, as explained before. The quantitative energy results of our general-relativistic toy neutron star are contained in Eqs. (2.83) and (2.84). Before discussing them, however, we extract their Newtonian limit, which we used in Sec. 2.3 for the Newtonian energy curves shown there. This is the limit a 1, in which we use the appropriate limit given by Eq. (2.80), and the usual Newtonian definition of density, and so obtain 3 ENewton = fE (x) − 2 mN c 4
M Mc
2/3 x,
(2.85)
which can, of course, be directly obtained from the Newtonian expressions for energy given in earlier sections.
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Total Energy (ε)
0.10
Landau arguments General Relativistic
0.07
0.05
0.085
0.00
0.1
-0.05 -0.10 0.01
0.13 0.8
0.4
0.2
0.10 1.00 Relativity Parameter (x)
Fig. 2.7 Generalized Landau arguments including general relativity. Shown is the total energy per neutron (E) in units of the neutron’s rest-mass energy for a uniform-density general relativistic “toy” neutron star vs. its relativity parameter x, the latter being a measure of its density (see text). Curves are given for various values of M/Mc , each curve labeled by its value of M/Mc . Dashed line: critical curve corresponding to M/Mc ≈ 0.13 (or the maximum mass M ≈ 0.89M possible for this toy neutron star; see text), where the maxima and minima in the curves coalesce into an inflection point. Curves corresponding to higher masses have no extrema, so that no equlibria are possible. Note the difference from the Newtonian case shown in Fig. 2.2.
Consider, finally, the variation of the general-relativistic energy with density or relativity parmeter x, as given by Eqs. (2.83) and (2.84), and as displayed in Fig. 2.7. The essential new feature is clear: in addition to the minimum in the energy, as in the Newtonian case, there is now also a maximum at higher densities, corresponding to an unstable equilibrium of a stellar configuration of the same mass. This corresponds to the falling, unstable branch in Fig. 2.6. The physical cause of this maximum is clear from a glance at Eqs. (2.80) and (2.81). General-relativistic effects increase the magnitude of the gravitational binding energy of a star, i.e., the selfgravitational energy of a star of a given mass and radius is more negative when these effects are included. At sufficiently high densities, this effect is so large that, even for those stellar masses where the (positive) zeropoint energy dominated at high densities in the Newtonian case, so that E was increasing monotonically there after passing through a minimum,
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the energy is now dominated by the (negative) gravitational component at very high densities, which stops the above increase in energy and reverses it, producing a maximum. Thus, for a given stellar mass M , there are two possible equilibria—one stable and one unstable—corresponding to the two extrema, and only the stable one occurs in nature. As M increases, the two extrema approach each other, and at a maximum mass Mmax , they merge at an inflection point, corresponding to the maximum mass seen in the mass-density plots. For M > Mmax , there are no extrema in the energy, and no equilibrium configuration is possible, the situation leading to gravitaional collapse. For our “sophisticated” toy neutron star of this section, Mmax ≈ 0.89M, and the corresponding value of x is x ≈ 0.63. These values can be compared with the corresponding ones for either (a) the numerical models of Oppenheimer and Volkoff or (b) the simple, analytical toy considered above. The energy arguments thus show their power once again, displaying in an elementary and transparent way how the character of the equilibria of degenerate stars changes due to the “stronger” gravitational forces (or, strong field effects, as they are often called) that apply when general relativity is invoked. We emphasize that the above differences in energy variation from the Newtonian case are qualitative and therefore essential, not just a matter of detail. Further discussion on this point has been given in Sec. 2.5.1.
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Chapter 3
Physics of Neutron Stars — II Physics of Dense Matter-1
The essence of more realistic calculations of the basic properties of degenerate stars is the recognition that the matter in them is actually far more complicated than the simple, non-interacting, completely degenerate matter discussed in the last chapter. The complications come from the electromagnetic, weak, and strong interactions between the constituent particles (electrons, protons, neutrons, and a variety of mesons and hyperons produced at very high densities, and possibly quark matter at even higher densities), and we now consider the basic physics of dense matter when these interactions are included. It is essential to understand this physics well, since this is what determines how the equation of state (henceforth abbreviated as EOS) of matter changes as its density increases, which, in turn, determines the internal structure, the mass-radius relation, and the maximum mass of degenerate stars, particularly neutron stars. We have already introduced some of these effects (particularly those which are important at the relatively low densities found in white dwarfs) qualitatively in Sec. 2.4.4. In this and the next chapter, we give a comprehensive discussion of the essential physics of dense matter, keeping in mind that our main concern in this book is with neutron stars, and therefore with matter at very high densities.
3.1
Matter at Low Densities: Electronic Energy
At low densities of matter, the most stable nucleus is iron in the form of 56 26 Fe, as we stated in Sec. 2.4.4. Consider the physical reasons for this. If the only force between the protons and neutrons in a nucleus were the strong nuclear forces, which are attractive (except at extremely short distances), nuclei would grow to unlimited size, i.e., to aggregates of nucleons with 97
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arbitrarily large mass numbers A. The Coulomb force of repulsion between the protons prevents this from happening, since very large nuclei would undergo spontaneous fission into smaller fragments due to the repulsive Coulomb forces. The point of stable equilibrium between these opposing forces, which occurs at the minimum of the total energy, determines the mass number A and the atomic number Z of the stable nucleus. At relatively low densities, by which we mean all the way from terrestrial densities to those at which inverse β-decay starts in the cores of white dwarfs (See Sec. 2.4.4), the most stable, i.e., the most tightly bound, nucleus is 56 26 Fe. The ground state of cold matter in nuclear equilibrium at these densities would, therefore, be expected to be a lattice of such iron nuclei immersed in an electron ‘sea’ or fluid1 . At very low densities, the behavior of cold, crystalline matter is determined by solid-state forces which depend on the properties of the (strongly interacting) electron fluid. Calculation of electronic energies is the most difficult at the lowest densities of this range [Pethick and Ravenhall 1991], due to complicated effects of the electronic shell-structure in atoms, but the problem becomes easier with increasing density, as (a) the relative importance of elctronic Coulomb interactions decreases (see below), and, (b) a statistical treatment of the atomic electrons by the Thomas-Fermi method (see below) becomes possible. 3.1.1
Wigner-Seitz Cells
Consider the basics of Coulomb interactions of electrons with nuclei and electrons in matter composed of nuclei of atomic number Z, mass number A, and number density nN , immersed in a fluid of electrons with number density ne : clearly, ne = ZnN by the requirement of overall charge neu1 This does not mean, of course, that terrestrial, planetary, or white-dwarf matter really consists wholly of iron, but rather that the lowest-possible equilibrium state of cold matter at their densities would be obtained for iron. The actual composition of an object which is not cold and not in nuclear equilibrium depends on its formation history and thermal structure. Planets (which are generally not cold systems) are believed to have formed from left-over material in accretion disks after stellar formation, and their matter, which is not in nuclear equilibrium, contains a wide variety of elements. Even the cores of cold white dwarfs, whose elemental compositions are determined by the history of thermonuclear burning in their progenitors, are now believed to consist of elements much lighter than Fe, e.g., C, O, Ne, Mg (See Sec. 2.4.4). However, it is essential for our purposes here to clarify the nature of the ground state of cold matter at these densities before discussing that of matter at much higher densities in the interiors of neutron stars, since an understanding of the progression of the essential physics with rising density is crucial.
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trality of matter. It is useful to define a length rc such that the volume per nucleus is equal to that of a sphere of radius rc . This length is given by nN =
1 . 4πrc3 /3
(3.1)
The physical picture we have in mind is the Wigner-Seitz approximation (see Sec. 2.4.4), in which each nucleus is surrounded by Z electrons, producing a neutral sphere of radius rc . In this simple, pioneering view of solids, which goes back to the 1930s, the lattice of ions immersed in the electron fluid is roughly approximated by an arrangement of these spherical Wigner-Seitz “cell”s. Of course, rc is a measure of the lattice spacing. In a similar way, we can define a length re for the electrons by the relation ne = 1/(4πre3 /3). The length re , which is a measure of the average distance between the electrons, is seen to be related to rc as rc = Z 1/3 re on using the above relation between nN and ne . The total Coulomb energy of interaction between all the electrons and ions, which determines the properties of matter and so its EOS at these densities, is easy to calculate in the Wigner-Seitz approximation, as each of the above spherical cells is neutral, so that interactions between the cells can be neglected in a first approximation, and we need only calculate the Coulomb energy of a single cell, followed by a sum over all cells. To make a simple estimate of this, imagine that the total charge −Ze of the Z electrons in a Wigner-Seitz cell is uniformly distributed over the volume of the sphere of radius rc , surrounding the nucleus of charge Ze, regarded as a point charge at the center of the sphere. Then the Coulomb energy of mutual repulsion of the electrons is just the elctrostatic self-energy of the sphere excluding the nucleus, wee =
3 Z 2 e2 , 5 rc
(3.2)
while that of the attraction between the nucleus and the electrons is given by: 3 Z 2 e2 ρe d3 r =− , (3.3) wei = Ze r 2 rc where ρe ≡ −Ze/(4πrc3 /3) is the electronic charge density. Note that wei and wee are generally of the same order, but the magnitude of the former always exceeds that of the latter, yielding an overall negative Coulomb energy per cell,
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wc = wee + wei = −
9 Z 2 e2 , 10 rc
(3.4)
as expected. It is a well-known, remarkable fact that the Coulomb energy per electron, Ec ≡ wc /Z = −(9/10)(4π/3)1/3Z 2/3 e2 n1/3 e ,
(3.5)
obtained in this simple way is very close (within 1%) to the results of the actual calculations for body-centered, face-centered, or simple cubic lattices, which implies, of course, that Ec depends little on the details of the lattice structure. Indeed, the energy is close to the above value even if the ions form a cold liquid, i.e., one whose thermal energy is small compared to the above energy [Pethick and Ravenhall 1991]. As the density rises, so does the kinetic energy of an electron, and when it much exceeds the magnitude of the above Coulomb energy, the electrons behave essentially as free electrons, i.e., unaffected by the Coulomb interactions described above. With the aid of Eq. (2.11) for the kinetic energy per electron, Enull , of cold electrons, this criterion, namely, Enull | Ec |, yields the condition ne (4/π 3 )(Z 2 /a30 ) on the electron density, where a0 ≡ 2 /me e2 is the Bohr radius. The corresponding condition on the matter density, ρ = Amb ne /Z, is then ρ ρfree ≡
4 mN AZ 3 ≈ 1.4 AZ g cm−3 , π3 a0
(3.6)
showing the natural density scale of the problem to be mb /a30 ≈ 11 g cm−3 . Here, mb is the mass of a nucleon (baryon). For hydrogen, ρfree ∼ 1 g cm−3 , 4 −3 and for iron it is ∼ 2 × 103 g cm−3 . At densities ρ > ∼ 10 g cm , we thus expect the Coulomb corrections to the EOS to be insignificant, and the Stoner-Chandrasekhar EOS for degenerate electrons, described in detail in the previous chapter, to be a sufficiently accurate description, until other physical effects enter at much higher densities. Note that electrons are still non-relativistic at densities where Coulomb corrections are significant (ρ < ∼ ρfree ), as explained in Sec. 2.3 (see discussion below Eq. (2.20)): they 6 −3 become relativistic only at ρ > ∼ 10 g cm ρfree . At densities below ρfree , then, there is a significant contribution to the total energy per electron, E = Enull + Ec , from the electronic Coulomb energy Ec , which reduces it from its zero-point value, since Ec < 0. The corresponding pressure can be obtained from the general relation
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P = n2b
∂(u/nb ) , ∂nb
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(3.7)
where nb is the number density of baryons and u is the total energy density: we shall use this relation repeatedly in this chapter, and throughout the book. In this case, using E = (u/nb )(A/Z) and nb = ne (A/Z), we can write P = n2e (∂E/∂ne ), which, together with Eqs. (2.11) and (3.6), yields 1/3 ρfree P = Pnull 1 − , (3.8) 8ρ showing that the pressure is reduced by the Coulomb interactions, as expected. Indeed, Eq. (3.8) would imply that the pressure vanishes at a density ρ = ρfree /8 ≈ 0.18 AZ g cm−3 . Although this result is not surprising in view of the above way in which we estimated the pressure reduction, the crudeness of that estimate becomes evident from the following considerations, first given by Salpeter (1961). Consider solid iron, 56 26 Fe. If we argue that the equilibrium state of this solid in terrestrial laboratories is roughly the point at which the (outward) zero-point pressure and the (inward) overall Coulomb self-attraction balance, i.e., where the pressure given by Eq. (3.8) vanishes, we would get a density ρ ≈ 260 g cm−3 , to be contrasted with the actual terrestrial density of iron, ρ ≈ 7.9 g cm−3 ! 3.1.2
The Thomas-Fermi Approximation
The major piece of missing physics in the above estimate is the nonuniformity of the electron density in the Wigner-Seitz cell, whose effects simply cannot be ignored at low densities. While these effects are extremely complicated at terrestrial densities, a statistical description of the electronic orbits becomes viable at somewhat higher densities, e.g., at ρ > ∼ 15 g cm−3 for iron. This is the Thomas-Fermi approximation, in which one improves on the assumption of constant electron density ne inside a Wigner-Seitz spherical cell of radius rc by approximating it with a spherically symmetric function of radius, ne (r). This radially-varying electron density is determined by assuming that, inside each Wigner-Seitz cell, the electrons ‘see’ only a slowly-varying, spherically symmetric potential φ(r), and relating φ(r) to ne (r) through Poisson’s equation, as we summarize below. The assumption of a slowly-varying φ(r) is crucial, because it implies that the (electrostatic) forces, −dφ(r)/dr, generated from it are relatively
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small, which means that we can apply the usual Fermi-Dirac statistics. It also shows the inherent limitation of the Thomas-Fermi approximation: it is basically a perturbation theory, and will not work when Coulomb interaction energy is comparable to zero-point energy, i.e., at very low densities. The Fermi energy EF , defined in Sec. 2.2.4, now needs to be modified by the inclusion of the potential energy −eφ(r), and it is this total EF which determines the statistical properties of electrons in a Wigner-Seitz cell. In complete degeneracy, all energy-states upto EF are occupied, and all those above it empty. Note that this total EF must be constant over a cell, for if it had different values at two radii, electrons would move from the higher value to the lower value. However, both ne and φ do vary with radius over the extent of a cell: in fact, they do so in roughly similar ways (see below), so as to keep EF constant. The Thomas-Fermi calculation is straightforward, although requiring (a small amount of) numerical work. Noting that EF = me c2 + p2F (r)/2me − eφ(r) for non-relativistic electrons, we choose the kinetic energy Ek (r) = p2F (r)/2me = EF − me c2 + eφ(r) as the variable used for the computation, rewriting Eq. (2.10) as ne (r) = (8π/3h3 )(2me Ek (r))3/2 . Poisson’s equation relating φ(r) to the charge-density in the Wigner-Seitz cell, i.e., ∇2 φ(r) = 4πene (r) − 4πZeδ(r), where the second term on the right-hand side represents the point-like nucleus of charge Ze, reduces to 32π 2 e2 1 d2 [rE (r)] = [2me Ek (r)]3/2 k r dr2 3h3
(3.9)
everywhere outside the nucleus, eliminating φ(r) with the aid of the above relation between it and Ek (r). Equation (3.9) is to be solved subject to the boundary conditions (a) that the potential must correctly reduce to that due to the nucleus as r → 0, i.e., [rφ(r)]r→0 = Ze, and (b) that the electrostatic forces must vanish at the cell boundary, r = rc , since each cell is electrically neutral, i.e., (dφ/dr)rc = 0. Equation (3.9) can be readily cast in the standard, dimensionless, Thomas-Fermi form by identifying the natural scales for length and energy in this problem, the former being the Fermi length λ≡
9π 2 128Z
1/3 a0 ,
(3.10)
and the latter being the reference Coulomb energy Ze2 /r. In Eq. (3.10), a0 ≡ 2 /me e2 is the Bohr radius, as before. Introducing the dimensionless
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energy ψ ≡ rEk /Ze2 and Fermi’s dimensionless radial co-ordinate x ≡ r/λ, we obtain from Eq. (3.9) the Thomas-Fermi equation: ψ 3/2 d2 ψ = 1/2 , 2 dx x
(3.11)
and similarly translate the above boundary conditions on φ into the following ones on ψ: (a) ψ(0) = 1, and (b) (dψ/dx)xc = ψ(xc )/xc . Equation (3.11) is simple but nonlinear, and has to be solved numerically: this was done by Feynman, Metropolis and Teller in 1949 in order to obtain the modified EOS. The cell boundary xc is obtained from the numerical integration as the radius where the second boundary condition is satisfied2 . We then calculate the resultant pressure of the system by noting that it is the electron pressure Pc at the cell boundary which balances the external pressure on the system, since electrostatic forces vanish at that boundary by definition, and there are no interaction between cells. Therefore, the observed pressure is this Pc , given by Eq. (2.11), with the relevant electron density now being nce ≡ ne (xc ), i.e., that at the cell boundary. This pressure can be compared with that obtained earlier for a uniform electron distribution and no Coulomb effects, which we denoted above by Pnull : this is given by Eq. (2.11) with an electron density related to the observed matter density ρ by ρ = AmN ne /Z, as discussed above. We drop the subscript c on P henceforth, and express the ratio of the pressures, P/Pnull = (nce /ne )5/3 , in terms of the solution of the Thomas-Fermi equation by the following method. The local relation between ne and Ek (given above Eq. [3.9]) holds everywhere: by applying it to the cell boundary, and combining the result with the value at this boundary of the dimensionless energy introduced earlier, Ekc = Ze2 ψc /λxc , the reader can show with the aid of Eq. (3.10) that nce /ne = (ψc xc )3/2 /3, where ψc ≡ ψ(xc ). This yields the “pressure correction factor” of Feynman, Metropolis and Teller (1949): 1 P = 5/3 (ψc xc )5/2 , Pnull 3
(3.12)
and so their EOS, which is generally adequate for this low-density r´egime upto ρ ≈ 104 g cm−3 . At the latter density, the pressure correction factor 2 The original Thomas-Fermi model had the defect that the zero-pressure case, corresponding to free atoms, yielded xc → ∞, indicating an infinitely large radius for such atoms. This defect was remedied by including the Dirac exchange term . In this Thomas-Fermi-Dirac model, the right-hand side of Eq. (3.11) acquires an extra factor of (1 + x/ψ)3 , and that of Eq. (3.12) an extra factor (1 + xc /ψc )5 [1 − (5/4)( + ψc /xc )−1 ]. Calculation of the exchange energy is mentioned in the text in brief.
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of Eq. (3.12) becomes essentially unity, corresponding to the fact that the elctronic Coulomb contributions to the total pressure become insignificant at and above this density, as discussed earlier. Thus, the direct influence of 4 the electronic, or lattice, energy on the EOS becomes negligible for ρ > ∼ 10 −3 g cm . However, it retains a significant indirect role in determining the nature of dense matter (through its role in determining equilibrium nuclides after the onset of inverse β-decay) upto much higher densities, as we shall see in Sec. 3.2. An alternative approach was adopted by Salpeter (1961), who calculated the Thomas-Fermi corrections to the energy, ETF , in a manifestly perturbative, linearized theory, arguing that the energies Enull , Ec , and ETF were successive terms in a perturbation expansion at the densities of interest. By contrast, the above Feynman et al. approach calculates the total energy at one go, in a nonlinear theory. Salpeter (1961) also calculated the (smaller) exchange energy Eex (see footnote), which takes account of the antisymmetric wave functions of the electrons, and also the (even smaller) correlation energy, all within his perturbative approach. In the non-relativsitic limit, only the simplest, Coulomb exchange interaction survives, yielding the re1/3 NR = −(3/4)(9/π 2)1/3 Z 1/3 e2 /rc = −(3/4)(3/π)1/3 e2 ne . In practisult Eex cal calculations, the above two approaches yield essentially identical results, and the Feynman-Metropolis-Teller EOS is often used for densities 10 g 4 −3 cm−3 < ∼ρ< ∼ 10 g cm . Above this upper limit, the Stoner-Chandrasekhar EOS (see Sec. 2.4.3 and Eq. [2.53]) with the simple Coulomb correction represented by Eq. (3.4) is adequate until the equilibrium nuclides start changing at ρ ≈ 8 × 106 g cm−3 (see Sec. 3.2).
3.2
Dense Matter Below Neutron-Drip Density: Equilibrium Nuclei
6 −3 (see After the electrons become relativistic at densities ρ > ∼ 10 g cm Sec. 2.3), their kinetic energies go into the MeV range, making inverse β-decay possible by blocking its reverse reaction, i.e., β-decay, as we explained in Sec. 2.4.4. Although the actual density thresholds for inverse β-decay of the nuclei of interest are typically in the range ρ ∼ 109 – 1010 g cm−3 (see Sec. 2.4.4), the presence of highly energetic electrons changes the properties of matter long before such densities are reached. This occurs because the weak interactions, which determine the balance between the protons and neutrons in the nuclei and the electrons surrounding them, play a crucial role in determining the equilibrium state of the nuclei, in
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addition to that played by the strong interactions between the nucleons in the nuclei. Equilibrium with respect to weak interactions determines the neutron/proton ratio in the nuclei, and the MeV-scale energy of electrons at these densities has a profound effect on this ratio. As we shall see below, the equilibrium condition is that the difference between the neutron chemical potential and the proton chemical potential equals the electron chemical potential (µn − µp = µe ), which physically means that it costs no energy to convert an electron and a proton into a neutron, or vice versa. With rising density, µe rises, leading to an increase in µn − µp , which means an increase in the neutron/proton ratio, or neutron richness, of the nuclei. For calculating the equilibrium nuclides, one borrows from the earlier, extensive research in nuclear physics on the semi-empirical mass formula for nuclei, the origins of which go back to the classic work of von Weizs¨acker (1935). This is a simple, liquid-drop model of the nucleus, in which the energy of an isolated nucleus (A, Z), including the rest-masses of its constituent protons (mp ) and neutrons (mn ), is given by [Pethick and Ravenhall 1991]: E(A, Z) = Evol + Esurf + ECoul ,
(3.13)
where Evol = (A − Z)mn c2 + Zmp c2 − Eb A + Es (1 − 2xp )2 A,
(3.14)
is the bulk energy, and the surface energy Esurf and Coulomb energy ECoul are given by Esurf = Eσ A2/3
,
ECoul = Ec x2p A5/3 .
(3.15)
Here, xp ≡ Z/A is the proton fraction, Eb is the bulk binding energy per nucleon, Es is the symmetry energy coefficient, and Eσ and Ec are the coefficients for surface and Coulomb energy respectively. Note that another very useful parameter related to the proton fraction is the neutron excess δ ≡ (N − Z)/(N + Z) = 1 − 2xp , in terms of which the symmetry energy has a quadratic form Es δ 2 A: this form is characteristic of laboratory nuclei with low to moderate neutron excess. We shall see later that the form may be inadequate for very neutron-rich nuclei, i.e., those with large values of δ. The physical origins for the scalings of the surface and Coulomb energies with A and xp become clear when we write these in their usual forms, i.e., Esurf = σS, and ECoul = (3/5)Z 2 e2 /rN , where σ is the surface tension,
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2 S = 4πrN is the surface area, and we have assumed, for simplicity, spherical nuclei of radius rN (thus improving on the point-nucleus assumption which we made while calculating the lattice energy in Sec. 3.1), with the protonic charges Ze distributed uniformly through this sphere. Since the expected, obvious scaling rN ≈ r0 A1/3 (i.e., nuclear volume ∝ A) is borne out by actual size measurements of nuclei, the above scalings follow, with Eσ = 4πσr02 and Ec = (3/5)e2 /r0 . Here, r0 ≈ 1.14 fm is the standard nuclear scale size, in terms of the unit of length 1 fm ≡ 10−13 cm named after Fermi and used universally3 in nuclear physics. Similarly, if we wish to bring out the scaling of the bulk energy Evol , we can express it as Evol = Eb A, in terms of the energy per nucleon (baryon) of uniform matter, Eb , and express the latter as
Eb = mn c2 − Eb − xp (mn − mp )c2 + Es (1 − 2xp )2 .
(3.16)
Note that σ depends on the proton fraction xp (or neutron excess δ), and Eb on both xp and the number density of nucleons (baryons) nb . The total energy per unit volume of the matter is, therefore, given by utot = nN E(A, Z) + ne Ee = nb (Eb + Eσ A−1/3 + Ec x2p A2/3 + xp Ee ), (3.17) where the energy per electron Ee includes (a) the rest mass me , (b) the zero-point kinetic energy (see Sec. 2.3), and (c) the complete Coulomb energy, including electron-nucleus and electron-electron contributions (see Sec. 3.1). At a given density, the equilibrium nuclide (A, Z) is that which minimizes utot with respect to both A and Z. It is convenient to minimize first with respect to A and then with respect to xp ≡ Z/A, and this two-step procedure has important physical significances, as stressed by Pethick and Ravenhall (1995). The first step is really an optimization with respect to the strong interaction process, which conserves not only the total number of baryons, but also protons and neutrons separately, and so also the number of electrons. Consequently, nb and xp remain unchanged under this process (and therefore Eb (nb , xp )), as does Ee . The minimization with respect to A then simply finds that value of A for which the nucleus has a minimum 3 Note
also that 1 fm = 10−15 m, which is called a femtometer. This unit, which is relevant for nuclear physics, is not to be confused, of course, with the much larger unit of length λ (given by Eq. [3.10]) which Fermi introduced in the Thomas-Fermi theory of the atom, and which we discussed in Sec. 3.1. λ is comparable to atomic sizes, and measured in units of the Bohr radius a0 ≈ 0.5 × 10−8 cm.
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value of (surface + Coulomb) energy. Straightforward differentiation of Eq. (3.17) yields A=
Eσ , 2Ec x2p
(3.18)
which means Eσ A2/3 = 2Ec x2p A5/3 , or, Esurf = 2ECoul
(3.19)
This simple and classic result, namely that the surface energy equals twice the Coulomb energy for a nucleus at equilibrium under strong interactions, is as expected, since the bulk (including symmetry) properties of nuclear matter drop out of this equilibration process, as explained above, leaving only those effects which depend on the actual size of the nuclei, i.e., the finite-size effects. It is very instructive, and indeed essential for later usage, to compare Eq. (3.19) with another classic finite-size effect of crucial importance in nuclear physics, namely the Bohr-Wheeler (1939) criterion for nuclear fission. The latter criterion states that an isolated nucleus undergoes spontaneous quadrupolar deformation and fission into smaller pieces if its Coulomb energy exceeds twice its surface energy, 0 0 ≥ 2Esurf , ECoul
(3.20)
where the superscript 0 stresses the fact that the nucleus is isolated. As 0 0 long as Esurf ≈ Esurf and ECoul ≈ ECoul , i.e., we can neglect the changes in these energies due to the presence of other nuclei, and of electrons (and also neutrons at sufficiently high densities, as we shall see below) surrounding the nuclei, we can infer that the equilibrium condition (3.19) guarrantees stability against sponatneous fission by a large margin. However, this is true only at low densities: we shall see later how drastically this may change at sufficiently high densities. Consider now the second step, which, in a similar vein, is really an optimization with respect to the weak interaction process, which conserves the total baryon number but changes the proton/neutron ratio. Specifically, the minimization with respect to Z is the condition of stability against both β-decay and inverse β-decay: the former raises Z while the latter lowers it, and this condition ensures that it would cost energy to do either. Note that, since changes in Z change the number of electrons, the electronicenergy term in Eq. (3.17) plays an essential role in the optimization process
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now. Operationally, in order to minimize utot with respect to xp , we first substitute the value of A from Eq. (3.18) into Eq. (3.17), which turns the sum of the surface- and Coulomb-energy terms in the latter equation into 2/3 the form (3/2)(2Eσ2 Ec )1/3 xp , and then differentiate with respect to xp . The resulting relation, , Ee = (mn − mp )c2 + 4Es (1 − 2xp ) − (2Eσ2 Ec )1/3 x−1/3 p
(3.21)
is really a statement of thermodynamic equilibrium, i.e., a condition of balance between the chemical potentials of electrons, protons, and neutrons: µe + µp = µn .
(3.22)
These chemical potentials, or Fermi energies, are calculated from the standard definition µi ≡ ∂utot /∂ni (where i = e, p, n), and include the restmass energy of the respective particle in the convention adopted henceforth. Physically, Eq. (3.22) means that, exactly at the equilibrium point, it costs no energy to convert an electron and a proton into a neutron, or vice versa. (The neutrinos that are emitted in these electron-capture and neutrondecay processes are assumed to leave with essentially zero energy, and so do not enter explicitly into the equilibrium condition.) We leave it as an exercise for the reader to explicitly calculate the chemical potentials from Eqs. (3.17) and (3.16) with the aid of the definition given above4, and show that these are given by: µe = Ee ,
(3.23)
1 µn = mn c2 − Eb + Es (1 − 4x2p ) + (2Eσ2 Ec )1/3 x2/3 p , 2 µp = mp c2 − Eb − Es (1 − 2xp )(3 − 2xp ) + 1 1 + (2Eσ2 Ec )1/3 x2/3 . p 2 xp
(3.24)
(3.25)
It can then be seen that Eqs. (3.21) and (3.22) are entirely equivalent. An immediate consequence of the above chemical-potential balance condition is that the proton fraction xp decreases, and so the nuclei become increasingly neutron-rich, as the density of matter rises. This is easily seen by noting first that the chemical potential of the electrons: µe = (3π 2 ne )1/3 c ∝ ρ1/3 ,
(3.26)
4 Note that u tot in Eq. (3.17) must be expressed entirely in terms of nn , np , and ne (using xp = np /nb and nb = nn + np ) before taking the partial derivatives.
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increases with increasing density for the relativistic electrons considered here, and then that an increasing Ee = µe in Eq. (3.21) implies a decreasing xp . The second point becomes clear when we recognize that the variation of the right-hand side of Eq. (3.21) is controlled by the symmetry-energy term for all values of the proton fraction xp of practical interest, and this term obviously increases with decreasing xp . This happens because, among the coefficients of symmetry, surface, and Coulomb enrgies, i.e., Es , Eσ , Ec , the first is the largest, typical values being Es ≈ 24 MeV, Eσ ≈ 17 MeV, and Ec ≈ 0.7 MeV [Bohr and Mottelson 1969]. These numbers readily show that the coefficient of the second term on the right-hand side of Eq. (3.21) is 4Es ≈ 96 MeV, larger by far than that of the third term (2Eσ2 Ec )1/3 ≈ 7.4 MeV5 . Physically, this means that, as the electron chemical potential rises with rising density and the weak-interaction equibrium condition (Eq. [3.21]) requires that the neutron-proton chemical potential difference rise in step with it, the way this occurs is by increasing the asymmetry between the neutron and proton numbers (i.e., moving away from the symmetric situation xp ≈ 1/2 which obtains for the very stable nuclei at low densities): a dropping proton fraction raises the neutron chemical potential and lowers the proton chemical potential (see Eqs. [3.25] and [3.25]), thus raising the difference between them. This decreasing proton fraction with rising density leads to another immediate conclusion from Eq. (3.18), namely that the mass number A of the equilibrium nuclide increases as the density increases, i.e., the nuclei get progressively heavier, larger, and richer in neutrons. This is as expected physically, since a decreasing proton fraction means a smaller (repulsive) proton-proton Coulomb force per nucleus, which shifts the balance in favor of the (attractive) nuclear forces, leading to a larger number of nucleons per nucleus. The numerical estimate obtained from Eq. (3.18) with the aid of the above numerical values of Eσ and Ec , A≈
12 , x2p
(3.27)
is very instructive [Pethick and Ravenhall 1995]. Consider first terrestrial laboratory conditions, under which stable nuclei have xp ≈ 1/2 (roughly equal number of protons and neutrons): Eq. (3.27) then yields A ≈ 50, 5 In
fact, the reader can easily show that the derivative of the right-hand side of −3 Eq. (3.21) with respect to xp is negative for xp > ∼ 10 . As we show in later sections, individual nuclei dissolve into a liquid of neutrons, with a small admixture of protons and electrons, long before such low proton fractions are reached, so that the considerations of this section are no longer relevant.
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Z ≈ 25, confirming the well-known fact that iron and elements near it in the periodic table are the most stable ones. Consider now what happens at the upper limit of the density r´egime being discussed in this subsection, namely, the neutron drip point, at which free neutrons first begin to appear in the very dense matter, as we indicate in the next subsection. The density corresponding to the onset of neutron drip is ρdrip ∼ 4 × 1011 g cm−3 , ≈ 0.32. as we show there, and the corresponding proton fraction is xdrip p The equilibrium nuclide corresponding to this has A ≈ 118, Z ≈ 38, as Eq. (3.27) readily shows. These are enormously neutron-rich isotopes of elements like Kr and Sr (see Sec. 3.3). The origins of the above approach for calculating equilibrium nuclides go back at least as far as van Albada’s (1947) work on the genesis of heavy elements. Further work was done primarily in connection with white-dwarf matter at first, such as those of Schatzmann (1958) and Salpeter (1961). In their studies of gravitational collapse, however, Harrison, Wheeler, and co-authors (1958, 1965) focussed on a complete account of the EOS over the entire density range from terrestrial values to the highest conceivable ones. Consequently, their work has often been a reference point for research on neutron star EOS in the post-pulsar era. The simple equilibrium-nuclide calculations we have given above are rather similar to the Harrison-Wheeler one, in which these authors used the version of the semi-empirical liquiddrop mass formula for the nuclear mass (i.e., the combination of Eqs. [3.13], [3.14], and [3.15]) due to Green, instead of the original mass formula of Fermi used in Wheeler’s (1955) earlier work. 3.2.1
Effects of Lattice Energy
In 1971, Baym, Pethick, and Sutherland (henceforth BPS) recalculated the the equilibrium nuclides using an improved semi-empirical mass formula due to Myers and Swiatecki (1966), and pointed out, for the first time, the significant role played by the lattice energy (i.e., the energy of Coulomb interactions between electrons and nuclei, and between the electrons themselves, which, for each Wigner-Seitz cell, is given by wc of Eq. [3.4]) in determining the composition of the equilibrium nuclides. This may surprise the reader at first, since wc leads to only a small correction in the electron pressure at these densities, as explained in Sec. 3.1. The situation becomes clear when we compare the Coulomb energy of an isolated nucleus, nuc = (3/5)Z 2 e2 /rN (see above), with wc , writing the total Coulomb enECoul cell nuc = ECoul + wc = (3/5)(Z 2 e2 /rN )[1 − (3/2)(rN /rc )]. ergy per cell as ECoul
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The correction term within the square brackets, (3/2)(rN /rc ), depends on (rN /rc ), i.e., the ratio of nuclear size to cell size, which is closely related to the fraction of the available space actually occupied by the nuclei, u ≡ (rN /rc )3 . We introduce the parameter u at this point, since we shall need it constantly in subsequent discussions. Note that u can be expressed in the form u = (ρ/ρnm ) in terms of nuclear-matter density ρnm ≡ (3/4π)(mb /r03 ) ≈ 2.8 × 1014 g cm −3 , corresponding to a number density nnm ≈ 0.16 fm−3 , at which nuclei begin to merge into one another, eventually forming a nuclear-matter liquid. Here, we have used the standard value for the nuclear scale radius r0 ≈ 1.14 fm introduced earlier, and mb is the mass of a nucleon (baryon). This is easily seen by demonstrating with the aid of the relations rN = r0 A1/3 , ρ = Amb nN , and Eq. (3.1) that u = 1 and rN = rc , i.e., the nuclei just “touch” one another when ρ = ρnm . At the neutron-drip density (see above), ρdrip ∼ 4 × 1011 g cm−3 ∼ 10−3 ρnm , the nuclei occupy only u ∼ 10−3 of the available volume, but (rN /rc ) is already ≈ 0.12 (i.e., the cell radius is ∼ 8 times the nuclear radius), and the correction term is ≈ 18%. This effect reduces the total Coulomb energy, and so the coefficient Ec , and hence produces larger nuclei with higher A at the same proton fraction, i.e., roughly the same density, as a glance at Eq. (3.18) shows. (The proton fraction xp is, to the first order, only a function of the density through Ee , as can be seen from Eq. [3.21], the last term on the right-hand side of which makes only a small contribution.) In fact, in the absence of other effects, e.g., nuclear shell effects (see below), this would make the equilibrium value of A about 18% bigger at a given density — a considerable increase in nuclear size. Indeed, BPS showed that their equilibrium nuclides, given in Table 3.1, were appropriately larger than those calculated earlier by Salpeter (1961), who had neglected this effect. In Table 3.1, b is the binding energy per nucleon of the nuclide, ρmax is the maximum density at which that nuclide is present, and µe is the electron chemical potential at that density. Values of ρ and µe at the neutron-drip point are shown within parentheses. Actually, the lattice energy wc that we used in the previous paragraph is incomplete for the finite-size nuclei we are considering here, since it comes from the calculation of Sec. (3.1) done for point nuclei. However, it is easy to complete the calculation, as we do now. The only modification is that in the energy of attraction between the nucleus and the electrons, wei , which is given for a finite nucleus with a charge-density distribution ρnuc (r) by
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Table 3.1 Nucleus 56 Fe
62 Ni 64 Ni 84 Se
82 Ge 80 Zn 78 Ni 76 Fe
124 Mo 122 Zr 120 Sr
118 Kr
EQUILIBRIUM NUCLEI (from BPS 1971)
b (MeV) 8.7905 8.7947 8.7777 8.6797 8.5964 8.4675 8.2873 7.9967 7.8577 7.6705 7.4522 7.2002
Z/A
ρmax (g cm−3 )
µe (MeV)
0.4643 0.4516 0.4375 0.4048 0.3902 0.3750 0.3590 0.3421 0.3387 0.3279 0.3166 0.3051
8.1×106
0.95 2.6 4.2 7.7 10.6 13.6 20.0 20.2 20.5 22.9 25.2 (26.2)
2.7×108 1.2×109 8.2×109 2.2×1010 4.8×1010 1.6×1011 1.8×1011 1.9×1011 2.7×1011 3.7×1011 (4.3×1011 )
wei =
φe (r)ρnuc (r)d3 r,
(3.28)
where φe (r) is the electrostatic potential due to the charge density ρe (r) of the electrons in the Wigner-Seitz cell. φe is obtained by solving Poisson’s equation, ∇2 φ(r) = −4πρe (r). At the densities under consideration in this subsection (and, of course, at higher densities), uniform electron density, ρe = −Ze/(4πrc3 /3), is an excellent approximation, Poisson’s equation reduces to r−1 (d2 /dr2 )(rφe ) = 3Ze/rc3 , and the reader can easily show that the solution of this equation, subject to the appropriate boundary conditions at r = 0 and rc , is 2 r Ze φe = −3 + . (3.29) 2rc rc Substitution of this potential into Eq. (3.28) reduces wei to a sum of two obvious parts: wei = −
3 Z 2 e2 Z 2 e2 2 + r nuc , 2 rc 2rc3
(3.30)
where we have used ρnuc (r)d3 r = Ze as the total charge of the nucleus, and (Ze)−1 r2 ρnuc (r)d3 r ≡ r2 nuc as the definition of the mean square nuclear charge radius r2 nuc . The first term on the right-hand side of Eq. (3.30) is the value of wei obtained in Sec. (3.1) in the point-nucleus
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approximation, and the second term is the correction due to the finite nuclear size. For evaluating the latter, we assume a uniform nuclear charge 3 /3), of course, as we did above for calculating density, ρnuc = Ze/(4πrN 2 , the Coulomb self-energy of an isolated nucleus. Then r2 nuc = (3/5)rN 2 2 2 2 2 3 and wei = −(3/2)(Z e /rc ) + (3Z e rN /10rc ). The lattice energy wc is, 2 therefore, given by wc = −(9/10)(Z 2 e2 /rc )+(3Z 2 e2 rN /10rc3 ), and the total Coulomb energy per cell by 3 Z 2 e2 u 3 cell = ECoul 1 − u1/3 + . (3.31) 5 rN 2 2 The last term on the right-hand side of Eq. (3.31) shows the effect that the finite nuclear-size contribution to the lattice energy has on the the total cell used in equilibrium nuclide calculations. Note that, Coulomb energy ECoul −3 since this term scales as u, and so is < ∼ 10 at densities below that of neu−2 tron drip (see above), this correction is very small, < ∼ 10 of the leading 1/3 (see above) at these densities, and therefore does correction term (3/2)u not affect the BPS arguments in any way. But as the density rises much above ρdrip , this term becomes quite significant: indeed, Eq. (3.31) shows cell that ECoul → 0 as u → 1, as it must, since nuclear and electronic charge densities exactly cancel each other in this limit, so that the net charge density is zero everywhere and the Coulomb energy vanishes. Before this can happen, however, nuclei dissolve into a uniform nuclear-matter fluid, as we shall see. Nevertheless, at high enough densities where nuclei are still present, the leading correction term above, (3/2)u1/3 , may lead to remarkably exotic effects through the Bohr-Wheeler (1939) condition introduced in Sec. 3.2. We shall return to this point. 3.2.2
Nuclear Shell Effects
An essential aspect of nuclear physics still missing from the simple liquid drop model described so far are the nuclear shell effects [Blatt & Weisskopf 1952], i.e., the effects of closing shells of neutrons and protons at special, or magic, values of N or Z on the determination of equilibrium nuclides. In the early post-pulsar era work of BPS, these were included by using a semi-empirical mass formula which had shell effects built into them [Myers & Swiatecki 1966] from the known properties of laboratory nuclei. As a result (see Table 3.1), these authors found nuclei with N = 50 (84 Se to 76 Fe), a “magic” neutron number (i.e., a closed neutron shell) for laboratory nuclei, for 8.2 × 109 ≤ ρ ≤ 1.8 × 1011 g cm−3 , and, above this density,
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Table 3.2 1994) Nucleus 56 Fe
62 Ni 64 Ni 66 Ni
86 Kr 84 Se
82 Ge 80 Zn 78 Ni
126 Ru
124 Mo 122 Zr 120 Sr
118 Kr
EQUILIBRIUM NUCLEI (from Haensel & Pichon
Z/A
ρmax (g cm−3 )
µe (MeV)
0.4643 0.4516 0.4375 0.4242 0.4186 0.4048 0.3902 0.3750 0.3590
8.0×106 2.7×108 1.3×109 1.5×109 3.1×109 1.1×1010 2.8×1010 5.4×1010 9.6×1010
0.95 2.6 4.3 4.45 5.7 8.5 11.4 14.1 16.8
0.3492 0.3387 0.3279 0.3167 0.3051
1.3×1011 1.9×1011 2.7×1011 3.8×1011 (4.3×1011 )
18.3 20.6 22.9 25.4 (26.2)
nuclei with N = 82 (124 Mo to 118 Kr), another magic neutron number for terrestrial nuclei, upto neutron drip density. The key question, however, was whether the shell effects were predicted accurately by this very large extrapolation from laboratory nuclei (with N ∼ Z) to extremely neutronrich nuclei (with N ∼ 2Z and above) without terrestrial analogue. In principle, this could be tested by calculations of the energy of a WignerSeitz cell through the Hartree-Fock method, using the Skyrme model of the nucleon-nucleon interaction (see Appendix D). Calculations done in the early 1970s seemed to suggest that the above extrapolation was reasonable. Recent calculations have demonstrated that the situation is more complex. In 1994, Haensel and Pichon did a calculation rather similar to that of BPS, in which they directly used the most accurate experimental masses of nuclei available at the time, as far as these went, and theoretical masses at higher neutron numbers from the latest mass formula of M¨ oller. Their results, given in Table 3.2, largely confirm those of BPS, since they also found nuclei with N = 50 (86 Kr to 78 Ni6 ) in the density range 3.1 × 109 ≤ ρ ≤ 9.6 × 1010 g cm−3 , and nuclei with N = 82 (126 Ru to 118 Kr) above that, upto neutron drip density. In Table 3.2, note that the notation is as in Table 3.1, the upper part of the table gives results which 6 This nuclide is doubly magic, since both Z = 28 and N = 50 are magic values, for protons and neutrons respectively.
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directly used the observed nuclear masses, and the lower part gives those which used the mass formula of M¨ oller. Two differences found by these authors were that (a) at low densities, 2.7 × 108 ≤ ρ ≤ 1.5 × 109 g cm−3 , the equilibrium nuclides were all isotopes of Ni, corresponding to the closure of a proton shell, i.e., the magic proton number Z = 28, of which there was only a faint trace in the BPS results, and (b) the effects of the N = 50 neutron-shell closure were reduced compared to most results obtained from semi-empirical mass formulas alone, including those of BPS. On the other hand, the 1989 results obtained from the Hartree-FockBogoliubov calculations of Haensel, Zdunik, and Dobaczewski were quite different. These authors used a Skyrme interaction (see Appendix D), the parameters of which had been fitted to observed properties (binding energy, radii, pairing, etc) of laboratory nuclei, and hoped that the actual shellstructure of neutron-rich nuclei would be an outcome of the calculation, rather than being built into a semi-empirical mass formula to begin with. What they found was a complete dominance of the shell-closure of protons, not neutrons. In the density range 2.7 × 108 ≤ ρ ≤ 3.3 × 1011 g cm−3 , the equilibrium nuclides were predominantly isotopes of Ni, corresponding to the proton magic number Z = 28. This means that, while at low densities, 2.7 × 108 ≤ ρ ≤ 1.5 × 109 g cm−3 , their results were essentially identical to those of Haensel and Pichon (1994), the two were completely different above ρ ≈ 2×109 g cm−3 . They were also completely different from ρ ≈ 3.3×1011 g cm−3 to the neutron-drip density, since Haensel et al. (1989) found all equilibrium nuclides here to be isoptopes of Zr, corresponding to the filled proton subshell Z = 40. Rather than attempting a post facto justification of why the above two approaches should lead to such completely different results, we might wonder about the true nature of the shell-closure effects in neutron-rich nuclei in cold, catalyzed matter. If the question is still open, as it appears to be, what will provide a reliable answer? An obvious, logical suggestion is that of Haensel and Pichon (1994), namely, that this will come from future experimental determination (from, e.g., heavy ion reactions) of the masses of more and more neutron-rich nuclei, although the current data are insufficient. A much more reassuring fact is that these details of nuclides lead to only very small changes (by a few percent in the most extreme cases) in the EOS, and to insignificant changes in the neutron-drip density (see below). Thus, nuclear shell effects on the basic structure of neutron stars (e.g., mass, radius, crust thickness) are almost negligible.
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Neutron Drip
As we explained above, the neutron chemical potential µn rises with rising density. When µn reaches the rest-mass energy mn c2 of the neutron, it becomes energetically favorable to have the next neutron as a free one rather than one bound in a nucleus. This is easily seen by remembering the definition of the chemical potential, µn ≡ ∂u/∂nn, which shows that µn , as calculated above, is the increase in energy of the system when the next neutron is added as a bound neutron to one of the (neutron-rich) nuclei in it. Hence, if µn > mn c2 , the system can attain a lower energy by having this neutron in a free, continuum state7 rather than in a bound one, and this is what it will do. This appearance of free neutrons surrounding the neutron-rich nuclei is described by the picturesque term neutron drip coined by Werner and Wheeler (1958). The point of neutron drip is defined, then, by µn = mn c2 .
(3.32)
Combining Eqs. (3.32) and (3.25), we obtain the equation for the proton fraction xdrip at the neutron-drip point: 1 2/3 Eb = Es (1 − 4x2drip ) + (2Eσ2 Ec )1/3 xdrip . 2
(3.33)
Using canonical values Eb ≈ 16 MeV, Es ≈ 24 MeV, Eσ ≈ 17 MeV, and Ec ≈ 0.7 MeV [see, e.g., Bohr and Mottelson 1969], we see that the last term on the right-hand side of Eq. (3.33), which involves surface and Coulomb energies, is a relatively small correction, since 12 (2Eσ2 Ec )1/3 ≈ 3.7 MeV. Neglecting this correction, we obtain an approximate value xdrip ≈ (1/2) 1 − Eb /Es ≈ 0.29 as a first estimate of the crucially important parameter xdrip entirely in terms of bulk energy parameters. Actually, the solution of the exact equation above is xdrip ≈ 0.32, showing that the correction term is only ∼ 10 %. Note, finally, that the idea of neutron drip is considerably older than the 1958 nomenclature described above: in their 1949 study of the origin of elements, G¨ oppert-Mayer and Teller described how, for a given Z, there is a maximum neutron number beyond which neutrons are no longer bound in nuclei, complete with a condition for this which was essentially equivalent to Eq. (3.32). In Wheeler’s (1955) study 7 Actually, the lowest continuum state of a neutron in the presence of these neutronrich nuclei has a total energy very slightly below mn c2 . However, since the nuclei occupy only ∼ 10−3 of the total volume at the neutron-drip point (see above), the correction, ∼ 10−2 MeV, is completely negligible. See Baym, Bethe and Pethick (1971).
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of nuclear stability, the name “neutron emission” was used for the same phenomenon, and the condition for it was given in a form very similar to Eq. (3.32). Not surprisingly, the values for xdrip that emerged from these early studies were rather similar to those which we have derived above. As explained in the discussion in Sec. (3.2) following Eq. (3.27), the equilibrium nuclide at the neutron drip point can be readily found from the value of xdrip with the aid of Eq. (3.27). The above value, xdrip ≈ 0.32, yields Adrip ≈ 120 and Zdrip ≈ 38, i.e., the nuclide 120 38 Sr. This was the nuclide at neutron drip found in Salpeter’s (1961) calculation, revising the earlier result of Wheeler and co-authors, namely, a neutron-drip nuclide of 122 39 Y, obtained from an older mass formula. Subsequent inclusion of lattice energy (and nuclear shell effects), as summarized in Secs. (3.2.1) and (3.2.2), brought changes the neutron-drip nuclide which are easily understandable. We argued in Sec. (3.2.1) that the inclusion of lattice energy reduces the total Coulomb energy, and so, in effect, the coefficient Ec . It follows immediately from Eq. (3.33) and the discussion below it that this reduces the value of xdrip (i.e., the correction term becomes smaller). Indeed, a value of xdrip ≈ 0.30 was found by BPS and in subsequent calculations of Haensel and Pichon (1994) and Haensel et al. (1989). The neutron-drip nuclide 118 36 Kr found by BPS was confirmed by Haensel and Pichon (1994), while the differences in the treatment of the nuclear-shell effects in Haensel et al. (1989) (see above) led to a heavier neutron-drip nuclide 134 40 Zr. The density ρdrip at the neutron-drip point is also a crucially important parameter, as is the electron chemical potential µdrip there, which is closely related to the former. Substituting our above value, xdrip ≈ 0.32, into Eqs. (3.21) and (3.24), we immediately obtain µdrip ≈ 25 MeV, which yields ρdrip ≈ 3.5 × 1011 g cm−3 . These values are very close to those obtained by Wheeler and by Salpeter (1961), while the inclusion of lattice energy by BPS modified them to µdrip ≈ 26 MeV and ρdrip ≈ 4.3 × 1011 g cm 3 . The latter values are essentially the same as those obtained in the works of Haensel and Pichon (1994) and Haensel et al. (1989) described above, and are used in current literature. Note that the neutron-drip density ρdrip is ∼ 10−3 of the nuclear matter density ρnm introduced in Sec. (3.2.1), as mentioned earlier.
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Matter Above Neutron-Drip Density: Nuclei Surrounded by Neutrons
At densities above the neutron-drip point, ρdrip , matter consists at first of neutron-rich heavy nuclei surrounded by neutrons, until nuclei dissolve into a uniform fluid of neutrons, protons and electrons as the nuclear-matter density ρnm is reached (see below). How do we describe the equilibrium nuclides in the density range between ρdrip and ρnm ? Consider first the additional pieces of physics that enter the picture due to the presence of “dripped” neutrons around the nuclei. The crucial effect is the decrease in the surface tension σ (or, equivalently, the surface-energy coefficient Eσ introduced earlier) of the nuclei. As the neutron excess (see above) δ of a nucleus increases, its surafce tension decreases below the typical value8 σ ∼ 1 MeV fm−2 for laboratory nuclei with symmetric (δ = 0) nuclear matter or with low neutron excess δ. Above ρdrip , the density of neutrons surrounding the nuclei increases with increasing density, and the neutron excess of the matter in the nuclei also increases, so that the matter inside the nuclei becomes increasingly similar to the neutron sea outside [Pethick and Ravenhall 1991]. Under these circumstances, the surface tension (strictly speaking, the interfacial tension between the nuclei and the neutron sea outside, closely related to the energy associated with the interface between the two) decreases rapidly, and falls far below the above value, vanishing finally when the matter inside and outside become identical at ρnm , and the nuclei dissolve into a uniform fluid. Because of this decrease in surface tension, increasingly larger and more neutron-rich nuclei are produced as the density rises. Further, since the total energy of the system now includes that of the neutrons outside the nuclei, this must be explicitly accounted for in the energy minimization process decsribed earlier. In the same vein, we must include the fact that the outside neutrons exert a pressure on the nuclei, which is balanced at equilibrium by the pressure of the matter inside the nuclei. Note that this external neutron pressure on the nuclei tends to reduce the nuclear volume, while the decrease in surface tension described above has the opposite effect, i.e., it tends to increase the nuclear volume (Baym, Bethe, and Pethick 1971, henceforth BBP). Finally, at very high values of the neutron-excess parameter δ, the quadratic approximation in δ of the bulk energy of the matter in the nuclei may not be adequate, as indicated in Sec. 3.2. 8 Using the relation E = 4πσr 2 given in Sec. 3.2, the reader can easily see that this σ 0 value of σ corresponds to the typical value of Eσ quoted there.
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Now consider the generalizations that need to be introduced into the liquid-drop model of Sec. 3.2 in order to incorporate the additional physics just described. First, the energy of neutrons outside the nuclei have to be included in the expression for the total energy of the system per unit volume utot given by Eq. (3.17). To do this consistently, note first that, in addition to the number density of nuclei, nN , and that of electrons, ne , introduced earlier, we now need the number density of neutrons outside the nucleus, nn , which is calculated in the following way. If VN is the volume of one nucleus, then a unit total volume of the system has only the volume (1 − nN VN ) available to the external (dripped) neutrons. Hence, if there are Nn neutrons in a total volume V , the external neutron density is given by nn =
Nn . V (1 − nN VN )
(3.34)
The electron density ne is given by the condition of charge neutrality, ne = ZnN , and the baryon density of interest is now the mean baryon density nb , given by nb = AnN + (1 − nN VN )nn .
(3.35)
If un is the energy-density of the external neutrons, which should be a function of their number density nn , the generalized form of Eq. (3.17) above the neutron-drip density is: utot = nN E(A, Z) + ne Ee + (1 − nN VN )un (nn ).
(3.36)
The surface-energy term Esurf in E(A, Z) (see Eq. [3.13]) must now be appropriately generalized to take into account the lowering of surface tension described above. We describe this in detail below, since these details proved crucial for deciding the progression of equilibrium nuclides with density above neutron drip. Following the pioneering work of BBP, let us now summarize the essential steps involved in the minimization of the above utot with respect to its arguments A, Z, nN , VN , and nn , subject to the constraint that the mean baryon density of Eq. (3.35) is kept fixed. This leads to four independent conditions, of course. The first is the optimization with respect to strong interactions, as explained in Sec. 3.2, which means finding the optimal number A of nucleons in a nucleus. In the present case, it means finding the
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optimal A with the condition that, per unit volume, the following quantities are fixed: the number of protons, nN Z, the number of neutrons inside the nuclei, nN (A − Z), the number of neutrons outside the nuclei, nn , and the fraction of volume occupied by the nuclei, nN VN . Mathematically, this means minimizing utot with respect to A at fixed nN Z, nN A, nN VN , and nn , which leads, again, to the well-known relation between surface and Coulomb energies, Esurf = 2ECoul , given in Sec. 3.2 (see Eq. [3.19]). For the Coulomb energy, we must now use the full expression given by Eq. (3.31), as the correction terms become very important at densities much above neutron drip: the result is 3 Z 2 e2 u 3 2 σ=2 1 − u1/3 + . (3.37) 4πrN 5 rN 2 2 The second step is also a repetition of what we did in Sec. 3.2, namely, the optimization with respect to weak interactions (or stability against β-decay and its inverse process), which means minimization of utot with respect to Z, now at fixed A, nN , VN , and nn . This, again, yields eaxctly the condition for thermodynamic equilibrium we found in Sec. 3.2, namely, the relation between the chemical potentials of the electrons and the protons and that of the neutrons inside the nuclei given by Eq. (3.22), which we now rewrite as ) µe + µp = µ(N n .
(3.38)
The superscript (N ) reminds us explicitly that these neutrons are inside the nuclei; we shall use the superscript (G) below to refer to the rest of the neutrons, which are in the neutron gas surrounding the nuclei. The physical interpretation of Eq. (3.38) has already been given in Sec. 3.2. Of course, the detailed expressions for the chemical potentials appearing in Eq. (3.38) are now more complicated than those given in Sec. 3.2, and we do not give them here. The third and fourth steps, which are new, describe the equilibrium between the nuclei and the surrounding neutron gas. The first of these is obvious, namely that there must be pressure balance between the nuclei and the outside neutron gas in order to have mechanical equilibrium: P (N ) = P (G) .
(3.39)
Here, P (N ) is the total pressure of a nucleus at its surface, and P (G) is the pressure of the outside neutron gas. Mathematically, this condition is obtained by minimizing utot with respect to VN at fixed A, Z, nN , and a
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fixed number of outside neutrons, nn (1 − nN VN ), in a unit volume of the system. We thus allow for a variable volume of a nucleus with given A and Z, because of which this generalized model is often called the compressible liquid drop model. The second of these is more subtle, and has to do with the thermodynamic equilibrium between the neutrons inside the nuclei and those in the outside neutron gas. This requires that ) = µ(G) µ(N n n ,
(3.40)
i.e., that the chemical potential of the neutrons inside the nuclei must equal that of those in the outside neutron gas. The physical interpretation of this is entirely analogous to that of the chemical potential balance condition for weak interactions described in Sec. 3.2, namely that, at the equilibrium point, it costs no energy to move a neutron from inside a nucleus to the outside, or vice versa. Mathematically, this is equivalent to minimizing utot with respect to A, but now9 at fixed Z, nN , and VN . 3.4.1
Nuclear Surface Energy: A Primer
In order to obtain the equilibrium nuclides and the EOS through the above steps, we need an explicit calculation of the surface energy of a nucleus in the presence of an external neutron gas, or, more accurately, one of the energy associated with the interface between the two. We now introduce the reader to a simple scheme, first given in 1971 by BBP, for estimating this energy, and so the surface tension associated with it. The formalism has the virtue of bringing out the essential physics in a transparent fashion, and therefore is ideally suited for a first look, although more elaborate 9 The reader may worry that this step looks rather similar to the first one, i.e., optimization with respect to strong interactions, but actually they are completely different because of the very different quantities held fixed in the two cases. It is physically more illuminating to think of the first step as minimization with respect to the number density of nuclei, nN , and the fourth one as minimization with respect to the number density, nn , of neutrons outside the nuclei. A little reflection will show that, per unit volume of the system, if we hold fixed the number of protons and neutrons in the nuclei, as we do in the first step, the only way of changing the number of nucleons per nucleus, A, is by changing nN . Similarly, if we hold nN fixed, as well as the number of protons per nucleus, Z, as we do in the fourth step, the only way of changing A is by changing the number of neutrons inside the nuclei, which is possible in this case only by changing the number of external neutrons (i.e., transferring neutrons between the nuclei and the neutron gas), since the total number of baryons in the system is always invariant. It is just mathematically more convenient to describe both steps as minimization with respect to A with different quantities held fixed.
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calculations need to be performed before accurate properties of nuclides can be obtained, as we shall see later. The BBP scheme follows the basic approach of the Thomas-Fermi theory of finite nuclei (Bethe 1968): in such a scheme, an interfacial energy is expressed in terms of the variation of the bulk energy across the interface. To see how this comes about, consider a nucleus of radius R, with a surface or “skin” of thickness ∆R R separating it from the external neutron gas. For studying phenomena near the interface, then, we can approximate it as a plane interface perpendicular to the z-axis, separating two semi-infinite regions. The interior bulk of the nucleus corresponds to z → −∞, and the exterior bulk of the neutron gas to z → ∞. The calculation proceeds in terms of the bulk energy of matter per nucleon, W, excluding the restmass energies. Compare this definition with the way we defined in Sec. 3.2 the bulk energy per nucleon Eb in the interior of a nucleus by Eq. [3.16]. For this case, the definition we use here yields Wi = −Eb + Es (1 − 2xp )2 , showing it to be basically the binding energy per nucleon of the nucleus, including the symmetry energy. Here and henceforth, subscripts i and o on a physical quantity respectively denote its values well inside the nucleus (z → −∞) and well outside it (z → ∞), in the bulk of the neutron gas. In the latter region, then, W can be expressed as Wo = (un /nn ) − mn c2 , since we included particle rest energies in our earlier definitions of the energydensity u. How do we define the interfacial energy in this scheme? First, we define an interface, i.e., a surface of sharp dicontinuity in physical variables, e.g., nucleon density n, energy per nucleon W, and so on, at z = z0 , such that these variables have their constant interior (subscript i) values everywhere in the region z < z0 , and constant exterior (subscript o) values everywhere in the region z > z0 . In the actual system, by contrast, each variable goes smoothly from its interior value to its exterior one over a region of size ∼ b in z, where b ∼ ∆R. It is then possible to define the interfacial energy in the following way with the aid of the laws of conservation of the total number of particles and the total energy of the system. Observe first that the total number of baryons in the system must be the same whether we describe it as the actual system with gradual variation, or our fictitous system with a sharp interface. The resulting condition,
z=∞
nd3 r = z=−∞
z=z0
z=−∞
z=∞
ni d3 r +
no d3 r, z=z0
(3.41)
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determines z0 , the location of the interface. Observe next that, once z0 is so determined, there are no more adjustable parameters in the interface picture. This implies that total energy of the system (exclud z=∞ ing the rest masses), which is z=−∞ W(n)nd3 r for the actual system, will be equal in general to that in the interfacial picture, namely, z=znot z=∞ 0 3 3 W i ni d r + z=z0 Wo no d r. We conserve energy by defining an inz=−∞ terficial energy which is the difference between these two, and which, in a physically consistent picture, must be associated with the interface itself. Following the usual convention, we shall henceforth call this the surface energy, and the amount of this energy per unit area the surface tension σ. In the above plane-surface approximation, the surface tension is then given by: z=∞ z=z0 [W(n)n − Wi ni ]dz + [W(n)n − Wo no ]dz. (3.42) σ= z=z0
z=−∞
In doing simple model calculations in the BBP scheme, it is convenient to describe the profiles of n(z) and W(z) by appropriate dimensionless functions of the dimensionless co-ordinate ζ ≡ (z − z0 )/b: n(ζ) = no + (ni − no )f (ζ), W(ζ) = Wo + (Wi − Wo )g(ζ).
(3.43) (3.44)
Of course, the dimensionless profiles f (ζ) and g(ζ) must satisfy the boundary conditions f (−∞) = g(−∞) = 1,
f (∞) = g(∞) = 0.
(3.45)
Illustrative, useful examples of such dimensionless profiles are (a) the linear profile ⎧ ζ ≤ − 21 ⎨1 , 1 f (ζ) = 2 − ζ , − 12 ≤ ζ ≤ 12 , (3.46) ⎩ 0, ζ ≥ 12 and (b) the Fermi-function profile f (ζ) =
1 . exp(ζ) + 1
(3.47)
∞ Substitution of Eq. (3.44) into Eq. (3.41) yields the condition 0 f dζ = 0 (1 − f )dζ ( a symmetry condition on the integral properties of f , the −∞ significance of which will be obvious if the reader draws a diagram of f (ζ)),
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the insertion of which into Eq. (3.42) leads to the evaluation of the surface tension. The algebra is straightforward, and the final result is: (3.48) σ = b(Wo − Wi ) ni f (1 − g)dζ − no g(1 − f )dζ . Note that all integrals here and henceforth in this subsection are understood to run from -∞ to ∞, and the limits are omitted. To further simplify matters, we shall henceforth assume that f = g, i.e., that n and W have the same (dimensionless) profile. This is tantamount to assuming that the energy per particle, W, is a linear function of the nucleon density, n, i.e., neglecting any higher order terms in the W - n relation. The surface tension is then σ = λb(Wo − Wi )(ni − no ), where λ is a second moment of the profile f defined by: λ ≡ f (1 − f )dζ.
(3.49)
(3.50)
Again, a diagram of f (ζ) will convince the reader that significant contributions to λ come only from the neighborhood of the interface, as expected. Further, the reader can readily evaluate the integral in Eq. (3.50) for the specific profiles given by Eqs. (3.46) and Eq. (3.47), and find that λ = 1/6 for the linear profile, and λ = 1 for the Fermi-function profile. Although the surface tension in Eq. (3.49) comes from simple considerations (known technically as the local density approximation), and should be corrected for higher-order effects (e.g., terms ∼ (∇n)2 ), we already see quite transparently how it arises from the difference between the bulk properties of matter inside and outside the nuclei, just as expected on physical grounds, as we argued earlier. In view of the uncertain validity of the standard ways of incorporating these higher-order effects into the simple, intuitive picture given above, BBP decided to retain the simple form of Eq. (3.49), and adjust the constant λ empirically. The reader will have noted that σ vanishes when no → ni , as it should, provided that the surface thickness b behaves properly in this limit. To settle the latter point, BBP devised the following, elementary quantum-mechanical way of estimating the skin thickness b of a nucleus in the presence of external neutron gas. They argued first that, in a simple picture, a nucleus in the absence of external neutrons can be thought of as free nucleons bound in a square-well potential, in which case the surface thickness of the system would scale
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with the Fermi wavelength of the nucleons inside the well, i.e., b ∼ 2π/kF , where kF ≡ pF /, and pF is given by Eq. (2.10), in terms of the nucleon density n in the nucleus. Generalize now to a nucleus with an internal nucleon density ni surrounded by an external neutron gas of density no , they argued next, and the scale generalizes to the Fermi wavelength of nucleons just at the top of the well, i.e., with zero kinetic energy outside, so that b=
ηπ , kc
kc3 = ni − no . 3π 2
(3.51)
Here, η is a number ∼ 1. It is clear that b becomes arbitrarily large in the limit no → ni , as intuitively expected, since the idea of a skin thickness loses meaning when the matter outside the nucleus becomes the same as that inside. Note, however, that the skin thickness given by Eq. (3.51) is wellbehaved in the sense of the previous paragraph. This is seen on evaluating the surface tension by combining Eqs. (3.49) and (3.51), which yields σ = (π/3)1/3 ηλ(Wo − Wi )(ni − no )2/3 ,
(3.52)
and therefore shows clearly that σ vanishes when no → ni , as promised. Since σ is closely related to the surface-energy coefficient Eσ (as defined in Sec. 3.2) by the relation Eσ = 4πσr02 , we can now evaluate this coefficient in the BBP model. To do this, we note that the volume VN of a nucleus 3 /3 = 4πr03 A/3, and also by VN = A/ni , so that is given by VN = 4πrN −1/3 . Substitution of the last condition in the above relation r0 = (4πni /3) between Eσ and σ yields, with the aid of Eq. (3.52), the surafce-energy coefficient: 2/3 no Eσ = (12π 2 )1/3 ηλ(Wo − Wi ) 1 − . (3.53) ni The scaling of Eσ is seen to be manifestly similar to that of σ. 3.4.2
BBP Results
In using the above model of nuclear surface energy in their calculational scheme, the BBP approach was to evaluate unknown coefficients like ηλ by comparison with known properties of nuclei and nuclear matter. Near the neutron-drip point, for example, Wo ≈ 0, and no /ni 1, and we have Eσ ≈ −(12π 2 )1/3 ηλWi = (12π 2 )1/3 ηλ[Eb − Es (1 − 2xp )2 ], using the value of Wi given earlier. This shows Eσ to be basically the bulk binding energy
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per nucleon (including the symmetry energy), apart from the numerical factor ηλ, and so leads to a direct determination of this factor, using the known values of Eσ , Eb , Es , and xp ≈ xdrip given earlier. Finally, since the bulk energy per nucleon of matter is a function of both its nucleon density n and its proton fraction xp , i.e., W = W(n, xp ), how does one describe it for matter inside and outside in a consistent way, making sure that Wo ≈ Wi when the density approaches the nuclear-matter density ρnm ? To achieve this, BBP devised an interpolation scheme for W(n, xp ) for bridging the gap between calculations for nearly symmetric matter, W(n, 1/2), and almost pure neutron matter, W(n, 0), that existed at the time, using a single function W(n, xp ) throughout. The final BBP results for the equilibrium nuclides showed that, as the matter density increased from ρdrip to ρ ∼ 1014 g cm−3 , Z increased from Zdrip ≈ 40 (see above) to Z ≈ 100, and A increased from Adrip ≈ 120 (see above) to A ≈ 700. The proton fraction decreased monotonically from its neutron-drip point value of xdrip ≈ 0.3 to xp ≈ 0.06 at the highest density studied by BBP, ρ ≈ 2.4 × 1014 g cm−3 . 3.4.3
More Accurate Results
Despite its intuitive appeal and transparency of essential physics, the BBP model of nuclear surface energy proved to be inadequate for an accurate calculation of the properties of equilibrium nuclides above neutron drip. It soon became clear that the BBP model underestimated the reduction in the surface tension σ due to the presence of the dripped external neutrons, and so predicted nuclei which had too much charge (i.e., too high Z) and were much too heavy (i.e., too high A). Detailed calculations of the energy of the plane interface (introduced in Sec. 3.4.1) in either Thomas-Fermi scheme or Hartree-Fock approximation with a nucleon-nucleon interaction of the Skyrme type (see Appendix D) showed in the early 1970s [Buchler & Barkat 1971; Ravenhall, Bennett & Pethick 1972] that the effective σ was considerably smaller (e.g., by a factor ∼ 4 at a density ∼ 1014 g cm−3 ) than the BBP estimate, and had a different trend of variation with the proton fraction xp (see below). Use of this surface tension in the above compressible liquid-drop model readily showed that equilibrium nuclides had much smaller values of Z and A than the BBP ones at the same density. Indeed, as shown in Fig. 3.1, the value of Z obtained from these detailed calculations remained roughly constant at Z ∼ 35 − 40, close to the neutron-drip value, as the density increased beyond the neutron drip, while
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Fig. 3.1 Atomic number Z of nuclei in neutron-star matter vs. density of matter, beyond neutron-drip point. BBP results are shown (see text) for comparison. RBP marks Hartree-Fock results of Ravenhall et al. (1972), and FPS marks the Pethick et al. (1995) results using FPS interaction (see text). Crosses indicate the Negele-Vautherin (1973) results, wherein shell effects lead to magic Z-values (see footnote). Also shown are the densities at which non-spherical nuclei may occur (see Chapter 5). Reproduced with permission by Annual Reviews from Pethick & Ravenhall (1995): see Bibliography.
the BBP value rose sharply with density. This has been generally confirmed by all subsequent calculations10 . The value of A obtained from these calcu10 Shell effects, when implicitly included in Hartree-Fock calculations [Negele & Vautherin 1973], have given values of Z jumping from the magic value of 40 to that of 50 when density is increased at first beyond neutron drip, but then jumping back to the magic values 40 and 28 at higher densities.
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lations rises with density, albeit much more slowly than in the BBP results, reaching A ∼ 200 – 300 at ρ ∼ 1014 g cm−3 . The fact that equilibrium nuclei are so heavy makes us check again the Bohr-Wheeler (1939) criterion (3.20) for spontaneous fission, introduced in Sec. 3.2, and the consequences are remarkable [Haensel 2001]. The densities are so high in this r´egime that the filling factor u (see Sec. 3.2.1) is not negligible any more, and we have to include corrections to surface and Coulomb energies, as well as those in the fission criterion (3.20) itself due to the presence of other nuclei. It then turns out that the leading correction is due to the term (3/2)u1/3 introduced in Sec. 3.2 as the leading correction to the Coulomb-energy term, in the folloing sense. The fission criterion remains Eq. (3.20) upto this order, since the leading correction is ∼ u [Pethick and Ravenhall 1995]. But the equilibrium condition (3.19) is now modified, upto this order, to11 0 0 Esurf = 2ECoul [1 − (3/2)u1/3 ]
(3.54)
Combining Eqs. (3.54) and (3.20), we arrive at the remarkable conclusion that the Bohr-Wheeler criterion would be satisfied when the densities are so high that u > 1/8, and the equilibrium nuclei would then be unstable to fission. Is this criterion satisfied in neutron stars? What happens then? The answers are closely connected with the possibilty of the occurrence of non-spherical, rod- or plate-like nuclei, which we shall deal with in Sec. 5.3.
3.4.4
Simple Scalings
The following simple, heuristic clarification of the differences between the BBP calculations and the more accurate ones brings out the essential scalings of the coefficients of surface and Coulomb energy (Eσ and Ec respectively) with the proton fraction xp above neutron drip. Recall first how we obtain the charge (proton) number Z and the mass number A of the equilibrium nuclides. The proton fraction, xp = Z/A, is determined by the density ρ from the weak-interaction thermodynamic equilibrium condition, Eq. (3.21), through the dependence of the electron chemical potential µe on 11 It
0 is understood that Esurf in Eqs. (3.54) and (3.20) now includes the effects of the reduction of surface tension due to the presence of dripped neutrons around nuclei, as described above.
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ρ. Given xp , A is determined by the ratio of Eσ to Ec , through Eq. (3.18). Hence, we can write Z=
Eσ , 2Ec xp
A=
Z , xp
(3.55)
as our algorithm for calculating Z and A, given xp , Eσ and Ec . Thus, the behavior of Z and A with density above neutron drip depends on the scaling of Eσ and Ec with xp : what are these scalings? Consider first the physical reasons why Eσ and Ec change as ρ rises and xp falls in the density r´egime above neutron drip. As ρ rises, Eσ decreases due to the decrease in surface tension of increasingly neutron-rich nuclei surrounded by increasingly dense neutron gas, as explained above in detail. Ec also decreases with increasing density, as we discussed in Sec. 3.2.1, since the contribution of the lattice energy to the total Coulomb energy, which tends to reduce the latter, increases with increasing density, or increasing values of the filling factor, u ≡ ρ/ρnm , introduced earlier (see Eq. [3.31]). Hence both Eσ and Ec decrease with decreasing xp : let us look at the actual scalings now. Take Ec first, and rewrite Eq. (3.31) as 2 u 3 1 Ec = Ec0 1 − u1/3 + 1 + u1/3 , (3.56) = Ec0 1 − u1/3 2 2 2 where Ec0 ≡ (3/5)e2 /r0 is the constant value below neutron drip introduced in Sec. (3.2). Now relate the filling factor u to the proton fraction xp through Eq. (3.21). We do not give the details of the derivation12 here, but only quote the final result: 1 (3.57) 1 − u1/3 , xp ≈ 2 which holds roughly in the region of interest (u 1, 0.1 ≤ xp ≤ 0.3) for standard nuclear-matter parameters. Eqs. (3.56) and (3.57) readily show that the leading term in the Ec - xp scaling is Ec ∼ x2p .
12 The
(3.58)
reader can supply the straightforward algebra. Using Eq. (3.26), rewrite 7/3 E ). Eq. [3.21] in terms of xp and µ as: 2xp + β(2xp u)1/3 ≈ 1, where β ≡ (µnm s e /2 is the electron chemical potential at ρ : show that β ≈ 1 for standard Here, µnm nm e nuclear-matter parameters. The approximate solution of this equation in the region of interest, i.e., xp ≈ (1/2)(1 − βu1/3 ), then reduces to Eq. (3.57).
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Take Eσ now, for which we quote results from the original work [Ravenhall, Bennett & Pethick 1972]. The BBP scheme and the more accurate, numerical schemes give the following rough scalings respectively: EσBBP ∼ x2p ,
Eσ ∼ x3p .
(3.59)
The scalings of Z and A for BBP and more accurate calculations follow immediately on substituting Eqs. (3.58) and (3.59) in Eq. (3.55). For the BBP scheme, in which surface and Coulomb energies decrease equally fast with rising density, Z BBP ∼ x−1 p ,
ABBP ∼ x−2 p .
(3.60)
By contrast, the more accurate calculations, in which the surface energy decreases faster than the Coulomb energy with rising density, give Z ∼ constant,
A ∼ x−1 p .
(3.61)
These scalings roughly describe the numerical results given in BBP and subsequent works, and show their physical underpinnings clearly.
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Chapter 4
Physics of Neutron Stars — III Physics of Dense Matter-2
4.1
Above Nuclear-Matter Density: Uniform Nuclear Matter
As we described in Sec. 3.2.1, nuclei begin to merge into one another as the number density approaches the nuclear-matter value nnm ≈ 0.16 fm−3 , thus forming a uniform nuclear-matter liquid consisting mainly of degenerate neutrons, with a small (∼ 5% by concentration just above ρnm ) admixture of degenerate protons and an equal admixture of degenerate electrons to maintain charge neutrality. How do we describe the properties of this nuclear matter? The clue comes from earlier research in nuclear physics again, although not from the work on the semi-empirical nuclear mass formula, unlike the case for the density range described in Secs. 3.2 and 3.4. Here, one adapts the earlier work on nuclear matter, which is a hypothetical system of equal numbers of protons and neutrons filling all space at uniform density, the Coulomb forces being assumed to be turned off. The translational invariance of such a system affords a far-reaching simplification in calculation, since it implies that the we shall be able to represent the single-particle basis states by plane waves in the quantum-mechanical description of such a system. Why study such a system? It was realized by the mid-1950s that, while a first-principles theory of finite nuclei based on the fundamental interaction between nucleons would be a formidable task at the time, a thorough understanding of this simpler, idealized system of infinite nuclear matter would be a valuable step towards constructing such a theory. To see the significance of this step, reconsider the semi-empirical mass formula given in Sec. 3.2 by Eqs. (3.13), (3.14), and (3.15), setting xp = 1/2, as appropriate for symmetric nuclear matter with N = Z, and switching off the
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Coulomb forces by setting Ec = 0, so that the symmetry and Coulomb energy terms both vanish. Now consider the limit of very large A, so that the surface energy Eσ A2/3 is negligible compared to the bulk binding energy Eb A. This is the limiting case of nuclear matter, described solely by the binding energy per baryon Eb , the empirical value of which is Eb ≈ 16 MeV, as indicated earlier. The main purpose of a theory of nuclear matter is to obtain an understanding of this value of Eb in terms of the effective interaction between the baryons. Specifically, the energy per baryon, −Eb (excluding the rest mass) can be calculated as a function of density, and the equilibrium state of nuclear matter at the observed density ρnm and the observed value of −Eb should then appear as the minimum of the calculated function. Even this calculation is beset with fundamental difficulties. What makes it so difficult? The answer reveals a crucial point about the interaction between nucleons, which must be fully appreciated before proceeding further. As the reader is perhaps aware, the two-nucleon interaction potential v is one of the fundamental reference points of all nuclear physics [Blatt & Weisskopf 1952]. We give a brief introduction to this potential in Appendix D, and use it throughout this section, and elsewhere in the book. The basic features of v that we need at this point are that it has (1) a strong, short-range repulsion, or a repulsive core, of radius c ≈ 0.4 fm, and (2) a long-range attractive part which depends on spin, parity, relative angular momentum, etc, of the two nucleons, and which reduces to the one-pion-exchange potential (OPEP; see Appendix D) for distances > ∼ 1.4 fm. It is this strong repulsive core which is at the heart of the difficulty, for it makes a simple, straightforward application of the usual calculational devices of quantum mechanics, e.g., perturbation and variational methods, quite impossible. This happens because the strong repulsive core, which prevents two nucleons from coming any closer to each other than a distance ∼ c, also makes the usual expectation values and matrix elements of v, which naturally occur in these quantum-mechanical methods, become extremely large, becoming infinitely large in the limit of an infinitely “hard” core (i.e., v attaining arbitrarily large values for r < c) which corresponds to hard spheres of radius c. Thus, a na¨ıve application of quantum mechanics is of no use: perturbation expansions (see below) do not converge, and variational calculations give large values which blow up in the hard-sphere limit. Rather, we have to first understand the essential physics involved, and then modify our depiction of the quantum-mechanical states and matrix elements accordingly. Indeed, in 1937, before the existence of the nuclear repulsive core was known, a pioneering attempt at the
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theory of nuclear matter was made by Euler on the basis of perturbation theory (as formulated in 1926 in the original work of Born, Heisenberg, and Jordan; also see Sec. 2.2), using a simple, Gaussian, attractive potential well. Such calculations would, of course, give completely absurd results when applied to two-nucleon potentials with repulsive cores. In the 1950s, ideas began to appear as to how one could formulate a consistent quantum-mechanical theory of such many-body systems that would lead to meaningful, finite results. They came from the realization that there was no difficulty of principle with a repulsive potential core, even the infinitely hard one described above. When the correct nature of the many-body states (or wave functions) which occur in the presence of such potentials was taken into account, the expectation values and matrix elements were well-behaved. For example, the wave functions must have “holes” or “wounds” (i.e., regions of small probability) in them corresponding to inter-nucleon distances < ∼ c, which must “heal” at appropriately large distances. We shall consider below the details of such constraints on the wave functions, or correlations, as they came to be called. The perturbative method, which was made viable by Brueckner and co-authors in the 1950s, and which used Goldstone’s diagrammatic summation scheme, was first called the Brueckner-Goldstone method, and, after crucial work done in the 1960s by Bethe and co-authors on the convergence of the perturbation series, the Brueckner-Bethe-Goldstone (henceforth BBG) method. At about the same time, Jastrow’s (1955) seminal work made the variational method viable, which was further developed in the 1960s by Feenberg, Clark and coauthors. In the post-pulsar era of intense research on neutron star matter in the 1970s, these methods were promptly adapted for extensive studies of the nuclear-matter fluid in the interiors of neutron stars. The variational method, in particular, underwent extensive development in the hands of Pandharipande and co-authors, and proved to be one of the most useful calculational tools for obtaining those physical parameters in the interiors of neutron stars which are crucial for determining their observable properties. Finally, the relativistic mean-field theory of nuclear matter, which also has its origins in the 1950s, but which adopts a completely different approach, has seen much development in the last two decades and been applied to neutron-star matter. In this section, we introduce the reader to some of the basic physical principles underlying the above methods, and to some their principal results, with particular reference to the beta-stable nuclear-matter liquid in the interiors of neutron stars.
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The Goldstone Expansion
The Goldstone (1957) expansion is a perturbation series for the groundstate energy of a many-body system. The major concern in a perturbation approach is the convergence of the series, and we shall see below how Goldstone tackled this issue. Consider a sytem of A identical nucleons, described by a non-relativistic Hamiltonian which is the sum of the kinetic energies of all the nucleons, plus the sum of all the two-body interactions between them: H=
1,A
Ti +
1,A
i
vij .
(4.1)
i<j
Here, Ti ≡ −2 ∇2i /2mi is the kinetic energy operator, and the two-nucleon potential v has been introduced above and is discussed in Appendix D. The Hamiltonian of Eq. (4.1) thus corresponds to a many-body Schr¨ odinger equation for this system of nucleons. In an attempt to obtain a convergent perturbative solution to this equation1 , Goldstone introduced the device of splitting the total Hamiltonian H, not into the two obvious parts suggested by Eq. (4.1), but rather into the following two parts: H = H0 + H 1 ,
H0 ≡
1,A i
(Ti + Ui ),
H1 ≡
1,A i<j
vij −
1,A
Ui .
(4.2)
i
Here, U is a single-particle potential, introduced to improve the convergence of the series. The motivation is obvious: by subtracting U , we hope to “reduce” the size of the perturbation H1 , and so to make the perturbation series converge faster. Of course, H is independent of U , and so, at least in principle, must be the value of the energy obtained by summing the perturbation series. Our task is to so optimize the choice of U that the summation converges in the fastest way possible. We return to this point later. The only additional point required for a straightforward application of standard perturbation theory to our many-body system is a formulation of many-particle states. For this, we first obtain the single-particle eigenfunctions φp of the unperturbed Hamiltonian H0 by solving the single-particle Schr¨ odinger equation (T + U )φp = Eφp . Note that, in the limit of infinite nuclear matter, A → ∞, these eigenfunctions are plane waves, as indicated 1 For an excellent introduction to Brueckner-Goldstone theory, see the classic review by Day (1967).
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above: φp = Ω−1/2 exp(ikp .rp ), where Ω is the volume of the system. We can now represent the unperturbed ground state of the A-particle system by a wave function Φ which is a product of A single-particle eigenstates φpi (i = 1 . . . A) of lowest energy – one state for each particle. It is always understood that this product has been antisymmetrized in all the co-ordinates (including spin) of identical particles if they are fermions, in order to take Pauli’s exclusion principle into account, and symmetrized, if they are bosons. As nucleons are fermions, we shall be dealing mostly with the latter in this book: for these the many-particle unperturbed ground state is given by Φ=A
1,A
φpi (ri ).
(4.3)
i
Here, A is the antisymmetrization operator, which ensures that Φ vanishes identically if two identical particles are in the same eigenstate. It is customary to write the antisymmetric wave function of Eq. (4.3) as a determinant called the Slater determinant , making natural use of the well-known property of a determinant that it vanishes when any two of its rows or columns are identical. For A = 2, the Slater determinant is φp1 (r1 ) φp2 (r1 ) (4.4) φp (r2 ) φp (r2 ) 1 2 and the generalization to larger numbers of particles is obvious. Φ, then, is an eigenstate of the unperturbed Hamiltonian, H0 Φ = E0 Φ: we assume this state to be non-degenerate (see Sec. 2.2). The corresponding eigenvalue is given by E0 =
1,A
Ep ,
(4.5)
p
in terms of the single-particle energies Ei of the occupied states. Let us now introduce some standard terminology of this subject, which we shall find useful throughout. The above collective state Φ consisting of A single-particle states of lowest energy is called the Fermi sea, and all states of higher energy are said to be above this Fermi sea. The motivation for the terminology is obvious from the discussion of completely degenerate fermions given in Chapter 2: in the A → ∞ limit, we have uniform nuclear matter, which is a degenerate Fermi sea of identical nucleons characterized by a Fermi momentum pF or a Fermi wave number kF ≡ pF /, such that
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all momentum states upto pF are completely filled, and all those above it are completely empty. Note that, for nuclear matter, kF is given in terms of the number density nnm ≈ 0.16 fm−3 of nuclear matter by kF =
3π 2 nnm 2
1/3
≈ 1.33 fm−1 .
(4.6)
Note further that Eq. (4.6) does not correspond exactly to Eq. (pfermichap2), but rather to one obtained by multiplying the right-hand side of the latter equation by a factor of 2. This is so because in nuclear matter, by hypothesis, the Coulomb interactions are turned off, so that a nucleon in a given momentum state has four possibilities: its spin can be either up or down, and, in each such spin state, it can be either a proton or a neutron. The former multiplicity is already included in Eq. (pfermichap2), and the inclusion of the latter one introduces the extra factor of 2. As the reader may be aware, the latter multiplicity can be looked upon as the two states of a formally-defined “isotopic spin”, in analogy with the usual spin: this representation is standard in nuclear physics (see, e.g., Blatt and Weisskopf 1955, and Appendix D). Continuing with our introduction to the terminology of Brueckner-Goldstone theory, vacancies in the Fermi sea are called holes, and occupied states above the Fermi sea are called particles. In this scheme, then, Φ is the state with no particles or holes in it, and any other state will be labeled by which particles and which holes it contains. Further, in order to abbreviate notation, we shall henceforth refer to the single-particle state φp simply as state p, and use Dirac’s (1958) standard ket notation |p, when necessary. Similarly, two-particle wave functions involving states p and q will be denoted by |pq, and so on. Application of standard quantum-mechanical perturbation theory [Schiff 1968] to the above system is straightforward. The actual ground state Ψ of the system is an eigenstate of the total Hamiltonian, HΨ = EΨ, and the perturbation expansion for the exact eigenvalue E in terms of the unperturbed value E0 , in powers of the perturbing Hamiltonian H1 , is: E = E0 + Φ|H1 |Φ + Φ|H1 (E0 − H0 )−1 P H1 |Φ + · · ·
(4.7)
Here, P ≡ 1 − |ΦΦ| is a projection operator that “projects off” the state Φ, making sure that it does not occur as an intermediate state in the matrix elements.
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Goldstone Diagrams
Goldstone devised a very useful diagrammatic scheme for representing and summing the terms in the above perturbation series, which we introduce now. For this, we first express the perturbation Hamiltonian H1 , given by Eq. (4.2), in terms of fermion creation and destruction operators a† and a [see, e.g., Schiff 1968]: a†s creates a particle in state s, and as destroys a particle in this state. In terms of the standard bracket notation of the anticommutator between two operators a and b, [a, b]+ ≡ ab + ba, these operators satisfy the following relations: [ar , as ]+ = 0 = [a†r , a†s ]+
[a†r , as ]+ = δrs
,
We can now write the perturbation as: pq|v|rsa†p a†q as ar − p|U |qa†p aq , H1 = pqrs
(4.8)
(4.9)
pq
and introduce the building blocks of Goldstone diagrams as follows.
Fig. 4.1
Representation of pq|v|rs a†p a†q as ar |Φ in Goldstone diagram.
Consider the first term on the right-hand side of Eq. (4.9). What happens when it operates on Φ ? It destroys particles in states r and s in the Fermi sea, i.e., produces two holes in it, and creates particles in states p and q above the Fermi sea. Fig. 4.1 represents this, introducing the following three rules for diagrams: (1) an outward line from a vertex represents a particle above the Fermi sea, (2) an inward line going into a vertex represents a hole in the Fermi sea, and, (3) a horizontal dashed line represents a matrix element, in this case that of the two-particle potential v between two-particle states, pq|v|rs, so that the dashed line runs between two vertices. Note that |p is associated with |r in the sense that they have the same arguments (co-ordinates, spin, . . . ), and so these appear at the same vertex. Similarly for |q and |s. Now consider what the second term on the right-hand side of Eq. (4.9) does to Φ. It creates a hole in state q and a particle in state p, as depicted in Fig. 4.2, which also introduces the fourth rule: the dashed line representing the matrix element of the one-particle
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potential U between one-particle states, p|U |q, has a bare end, where we put a cross.
Fig. 4.2
4.1.2.1
Representation of p|U |q a†p aq |Φ in Goldstone diagram.
First-order diagrams
We can now write the first-order perturbation term in Eq. (4.7), i.e., Φ|H1 |Φ diagrammatically: this is shown in. All we do for this is project the results of H1 |Φ, as described above, onto |Φ, which means filling up the hole(s) and destroying the particle(s) created by H1 , thereby taking us back to Φ. For the two-particle v-term, this is possible only if (a) r = p, s = q, or, (b) s = p, r = q. In the first case, each of the two destroyed particles is created right back in its original state again: this is shown in panel (a) of Fig. 4.3 by joining the upward and downward arrows at each vertex into a self-closing loop, or bubble. In the second case, shown in panel (b), each destroyed particle is created back into the original state of the other particle, so that the two vertices are joined in a closed loop, as shown. The one-particle U -term is particularly simple, as shown in panel (c), as the only possibility is q = p, i.e., restoring the single destroyed particle to its original state, depicted by the closed loop at the single vertex. This exhausts all possibilities for first-order Goldstone diagrams.
Fig. 4.3 First-order Goldstone diagrams. Φ|H1 |Φ .
The sum of the three contributions is
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The contribution of this order to the eigenvalue E is now easy to calculate. Panel (a) contributes an amount which is a sum over p and q of terms, each of which is a product of a matrix element pq|v|pq and an expectation value Φ|a†p a†q aq ap |Φ. The latter value is ±1 depending on the exact ordering of the fermion operators, and 1 in this particular case. So, this contribution is (1/2) pq pq|v|pq, where the factor 1/2 comes from the fact that we summed over only distinct matrix elements in the original perturbation series, whereas summation over all p and q double-counts each such element, since pq|v|pq = qp|v|qp. Similarly, Panel (b) contributes an amount −(1/2) pq pq|v|qp. Finally, Panel (c) contributes an amount − p p|U |p. To the first order, the energy given by Eq. (4.7) is then the sum of the unperturbed energy E0 and the three contributions given above. But note that E0 , as given by Eq. (4.5), can be written as E0 = p Ep = p|T |p + p|U |p, showing that the U -term in it exactly cancels the p p contribution from Panel (c) given above. Thus, to first order, the groundstate energy is: E=
p
1 p|T |p + ( pq|v|pq − pq|v|qp). 2 pq pq
(4.10)
This cancellation of U in the first order is automatic, independent of the choice of U . However, the right-hand side of Eq. (4.10) still depends indirectly on U , since the single-particle states p, q, . . . used in the calculation of the matrix elements depend on U . 4.1.2.2
Second-order diagrams
We describe in brief the calculation of the second-order perturbation term, Φ|H1 (E0 − H0 )−1 P H1 |Φ, in Eq. (4.7), as this completes our description of the rules for constructing and evaluating Goldstone diagrams: higherorder diagrams do not require any new rules. In the second order, we first operate on the results of H1 |Φ with the operator (E0 − H0 )−1 , it being understood (because of the projection operator P ) that we shall consider only those parts of H1 |Φ which are orthogonal to |Φ. Recall the form of H1 |Φ, and consider the v-term first. It contains particles in states p and q, and holes in states r and s, and therefore will give, when operated on by (E0 −H0 )−1 , the quantity −(Ep +Eq −Er −Es )−1 , which is called an energy denominator for the above intermediate state (with particles p, q and holes r, s) in the language of elementary perturbation theory. This gives the fifth
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rule, namely, that the energy denominator for an intermediate state is the sum of the particle energies minus the sum of the hole energies.
Fig. 4.4
A second-order Goldstone diagram.
We next operate with H1 , and project the results back onto |Φ: the only terms in H1 which are relevant now are those which carry the state a†p a†q as ar |Φ back into |Φ. One such term, shown in Fig. 4.4, is rs|v|pqa†r a†s aq ap , since it destroys the intermediate particles p, q and fills up the holes r, s. We show this pictorially by rejoining the upward and downward lines coming out of each vertex of Fig. 4.1 at another vertex above it, thus forming two closed loops. The contribution of this diagram to E is −(1/2) pqrs rs|v|pq(Ep + Eq − Er − Es )−1 pq|v|rs, where the factor 1/2 comes for the same reason as above, and the expectation value Φ|a†r a†s aq ap a†p a†q as ar |Φ (which is ±1 in general) equals 1 in this case. The last result comes from a handy rule for obtaining expectation values of fermion-operator chains: the sign is just (−)h+l , where h is the number of hole lines and l is the number of closed loops [Goldstone 1957; Day 1967].
Fig. 4.5
A second-order Goldstone diagram, showing the U-term.
Finally consider the U -term. It contains a particle in state p and a hole in state q, and therefore gives, when operated on by (E0 −H0 )−1 , the energy denominator −(Ep −Eq )−1 for the intermediate state a†p aq |Φ. To carry this
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state back to |Φ, consider a v-term in H1 , namely, qr|v|pra†q a†r ar ap . The result is shown in Fig. 4.5: the only non-trivial vertex of the U -interaction created particle p and hole q, which is now joined with one vertex of the v-interaction , where p is destroyed and q filled up, thus forming a joining loop, as above. The other vertex of the v-interaction creates a hole r and fills it up right back again, forming a self-closing loop, or bubble (see above). The contribution of this diagram to E is pqr qr|v|pr(Ep − Eq )−1 p|U |q, remembering the fermion-operator rule above, and noting that the negative sign of the energy denominator cancels that associated with the matrix element p|U |q (the U -interaction always occurs with a negative sign in H1 : see Eq. [4.9]). This brings us to the sixth and last rule: the sign of the contribution of any diagram is (−)h+l+e+u , where h is the number of hole lines, l is the number of closed loops, e is the number of energy denominators, and u is the number of U -interactions [Day 1967]. With the above rules, Goldstone diagrams of any order can be constructed and their contributions to the energy obtained with ease: this is important because, for application to nuclear matter, certain classes of diagrams need to be summed to infinite order. A virtue of the Goldstone expansion is that it explicitly displays the ground-state energy of a manybody system as a sum of terms, each of which is proportional to the number of particles A. This may seem self-evident, but the point is often difficult to demonstrate in many-body theories. Goldstone’s scheme achieves this by showing that those terms which contain square or higher power of A (corresponding to diagrams containing disjoint pieces, i.e., disconnected diagrams) completely cancel one another in each order. Thus, only connected diagrams survive, each of which is proportional to A [Day 1967]. 4.1.3
Brueckner’s Reaction Matrix
The Goldstone expansion, as described so far, does not converge for nuclear matter. This is so because the matrix elements of the two-nucleon interaction v, which occur throughout the above formalism, are large due to the presence of the strong repulsive core in v (see above and Appendix D). Indeed, they blow up in the limit of an infinitely hard core (i.e., v → ∞ for r < c), as indicated earlier. The achievement of Brueckner and co-authors was to show that, if we replace the matrix elements of v (between the various states described above) by those of a different, suitably-defined matrix G, then the latter are all well-behaved and finite even for an infinitely hard core. Thus, all individual terms in such an expansion in orders of G are
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finite and of reasonable size, so that the series is expected2 to converge. The matrix G is now called Brueckner’s reaction matrix, and the resulting expansion the Brueckner-Goldstone expansion. Why is this possible? Clearly, if the sum of a series is not finite to begin with, we cannot make it so by merely rearranging its terms in a different way. The resolution is that the sum, i.e., the eigenvalue E, is in fact finite; it is the na¨ıve application of perturbation theory which sets out the expansion scheme in an unfortunate way. In the Brueckner G-matrix approach, this is remedied by regrouping the Goldstone diagrams in such a fashion that the two-nucleon potential operator v is replaced by an operator G, which is an infinite series in powers of v. In a way, then, G corresponds to taking account of the two-nucleon interaction upto all orders of the two-nucleon potential, and it is the matrix elements of G which are well-behaved. But the next natural question would be: why does this work? How is it that matrix elements of v have poor convergence properties, but those of an infinite series in v do not? In answer, the following analogy with twonucleon scattering is often given, making use of the fact that this scattering is described by the same potential v. It is well-known that, because of the strong repulsive core, we get an incorrect, very large, scattering amplitude if we keep only the first-order term in v (i.e., make the Born approximation), while if we keep all orders in v, i.e., find an exact solution of the Schr¨ odinger equation, the correct scattering amplitude turns out to be much smaller and well-behaved. However, this analogy is not really apt because it refers to the scattering of two isolated nucleons, while we are considering here two nucleons immersed in an infinite sea of other nucleons, i.e., nuclear matter. The Pauli exclusion principle plays a dominant role here, as we shall see later, preventing scattering into occupied states, and “healing” the two-particle wave function rapidly at large distances [Bethe & Goldstone 1957]. The actual reason for the convergence of Brueckner’s G-matrix is this dominance of the Pauli principle, which prevents the occurrence of any singularity in the energy denominators introduced above, so that (in terms of more formal mathematics) there is no singularity in the kernel of either the integral equation that defines the G-matrix (see below) or the one that describes the scattering amplitude. Consider now the explicit formulation of the G-matrix. We illustrate the replacement of v by an infinite series in v with the aid of Fig. 4.6, which 2 We
shall see below that even this expectation later proved to be incorrect: further modifications had to be devised by Bethe (1965) for the convergence of higher-order terms.
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Fig. 4.6
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Illustrating the concept of Brueckner’s G-matrix.
shows an elementary part of a Goldstone diagram consisting of two particle lines (p, q) and two hole lines (r, s) interacting through v (the dashed line), the corresponding matrix element being pq|v|rs. To this, we now add an infinite sequence of higher-order interactions in v, such that the original single dashed line is replaced by a ladder of two, three, . . . dashed lines respectively. Naturally, this involves summations over intermediate states a, b for the second order, states a, b, c, d for the third order, and so on, as shown. Using the rules introduced earlier, the second-order matrix element is − ab pq|v|ab(Ea + Eb − W )−1 ab|v|rs, the third-order one is abcd pq|v|ab(Ea + Eb − W )−1 ab|v|cd(Ec + Ed − W )−1 cd|v|rs, and so on. Here, W , which is called the starting energy, depends only on the states p, q, r, s and generally also on the rest of the diagram. We now write the operator (or matrix) G as an infinite sum in orders of the operator v with the aid of the two following two-particle operators. The first is the Pauli operator Q, defined by the relation |pq, if p, q both above Fermi sea Q|pq = , (4.11) 0, otherwise which annihilates a two-particle state |pq unless both particles are above the Fermi sea. The second is the energy operator e, defined by e|pq = (Ep + Eq − W )|pq,
(4.12)
which gives the energy of the two-particle state minus the starting energy. In terms of these, we write G as: G(W ) = v − v
Q Q Q v + v v v + ···, e e e
(4.13)
showing explicitly that G is a function of the starting energy W . The reader can readily verify that the matrix elements of the operator given by
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Eq. (4.13) are exactly those described above. Diagrammatically, G represents the sum of the diagrams on the left-hand side of Fig. 4.6, and the standard Brueckner-Goldstone notation is to denote this sum by the diagram shown on the right-hand side of the same figure, in which the single wavy line replaces the infinite sequence of ladders of dashed straight lines. Thus, the first-order Goldstone diagrams in Fig. 4.3 are now replaced by the first-order Brueckner-Goldstone diagrams shown3 in Fig. 4.7, and the extension to higher-order diagrams is straightforward.
Fig. 4.7
First-order Brueckner-Goldstone diagrams.
To calculate G, use is made of the fact that Eq. (4.13) is equivalent to the integral equation G=v−v
Q G. e
(4.14)
The reader can verify this by a variety of methods, the easiest of which may be to transpose the second term on the right-hand side to the lefthand side, and then to substitute the expression for G from Eq. (4.13). As we shall see, this device of evaluating the sum of an infinite operatorseries (or, equivalently, an infinite sequence of operator diagrams) by solving a suitable integral equation for the sum is used widely: we shall return to it repeatedly in our discussions of Brueckner-Goldstone and variational methods. For actually computing G, one always calculates first the correlated two-particle wave function Ψpq , which is conceptually crucial to both Brueckner-Goldstone and variational methods, and which we study below in some detail. In the Brueckner-Goldstone formalism, we can define Ψpq in terms of the unperturbed two-particle wave function, |pq ≡ φp (r1 )φq (r2 ), introduced in Sec. 4.1.1, as: Ψpq ≡ |pq −
Q G|pq. e
(4.15)
It must be obvious that Ψpq takes into account the effects, or correlations, introduced into the two-particle wave function by the nucleon3 Note
that the diagram corresponding to the U -term remains as before, since only v is replaced by G. In any case, the U -term is always automatically cancelled in the first order, as explained earlier.
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nucleon interaction. By combining Eqs. (4.15) and (4.14), the reader can derive two alternative relations for Ψpq , namely, vΨpq = G|pq, and Ψpq = |pq − (Q/e)vΨpq , which are also useful. It is the character of Ψpq that we discuss next. 4.1.4
Correlations and “Healing”
In a brief but seminal paper, roughly contemporaneous with the beginnings of Brueckner-Goldstone theory, Jastrow (1955) made a profound observation on the nature of the eigenfunctions of a many-particle system (see Sec. 4.1.1) interacting via a nucleon-nucleon potential with a strong repulsive core. Jastrow was considering the variational method (see, e.g., Schiff 1968) for obtaining an upper bound on the energy eigenvalue of the ground state of such a system using a many-particle wave function of the form given by Eq. (4.3), and observed that the upper bound so obtained blew up in the limit of an infinitely repulsive “hard” core (i.e., a “hard sphere” potential; see Sec. 4.1). He realized that the trouble came from a missing piece of physics in the above description, which was that the correlations produced in the many-particle wave function by the nucleon-nucleon interactions have not been included in a simple wave function like that given by Eq. (4.3). For example, argued Jastrow, the correct A-nucleon eigenfunction Ψ(r1 , r2 , . . . , rA ) must vanish whenever two nucleons attempt to come within a distance c of each other, i.e., rij ≡ |ri − rj | < c, c being the radius of the hard core (see Sec. 4.1). He proposed to include these correlations in Ψ by writing it in the form4 Ψ=
f (rij )A
i<j
1,A
φpi (ri ),
(4.16)
i
where f (rij ) are suitable correlation functions that describe the above effects. Clearly, f (r) must vanish for r < c, and approach unity at large distances. An explicit form of f (r) introduced by Jastrow (1955) in his illustrative calculations for a “hard sphere” gas was the so-called Yukawa form: 0, r
4 For a Bose gas, i.e., a system of bosons, the antisymmetrization operator in Eq. [4.16] will, of course, be omitted.
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which is shown in Fig. 4.8. This simple figure illustrates a basic concept which is central to both Brueckner-Goldstone and variational methods: the correlations produce, in effect, a “hole” or a “wound” in the wave function at very small interparticle distances, which “heals” at large distances. This picturesque language was invented by Gomes, Walecka and Weisskopf (1958), as we shall see below, after the quantum-mechanical basis of the phenomenon was clarified in the classic work of Bethe and Goldstone (1957). Correlated many-particle wave functions of the kind given by Eq. (4.16) are now called Jastrow wave functions, and universally used in variational calculations of nuclear matter. In his original work, Jastrow calculated the energy eigenvalues of hard-sphere Fermi and Bose gases by minimization with respect to the parameter β in the fixed-form correlation function of Eq. (4.17). We shall show later how, in the modern variational calculations pioneered by Pandharipande and co-authors, the form of the correlation function was computed in a self-consistent manner. A rigorous quantum-mechanical description of the pheonomenon described by Jastrow was not long in coming. In 1957, Bethe and Goldstone
Fig. 4.8 Jastrow’s correlation function. Shown are the two-particle (dashed line) and three-particle (solid line) correlations, compared to the classical results for hard spheres, as indicated. Reprinted with permission by the APS from Jastrow (1955), Phys. Rev., c 1955 American Physical Society. 98, 1479.
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gave a clear (and now classic) account of how the interaction between two nucleons is modified when these are immersed in an infinite sea of other nucleons, i.e., nuclear matter. These authors showed that the essence of the modification was contained in Pauli’s principle. An isolated pair of nucleons in states p and q can undergo a scattering through the two-nucleon potential v into two different states r and s. In nuclear matter, however, all states in the Fermi sea are filled, as we have indicated earlier, so that, by Pauli’s exclusion principle, there are no states available for scattering in this sea except for the original ones p and q themselves, and no scattering will take place. Note first that this happens despite the fact that the nucleon-nucleon interaction is assumed to be entirely two-body in nature, i.e., there is nothing in v that corresponds to a simultaneous interaction between three or more nucleons. So the effects of the other nucleons on a given pair’s mutual interaction are completely indirect, and operate only through the Pauli principle. Note next that the result is a complete change of character of the two-particle wave function Ψpq at large interparticle distances. An isolated pair shows a phase shift in Ψpq at large distances with respect to the unperturbed wave function |pq, corresponding to scattering. Since there is no scattering in nuclear matter, this phase shift is zero, and Ψpq at large distances is just |pq. What happens at small distances? The mathematical description of the whole phenomenon proceeds through a generalization of the two-particle Schr¨ odinger equation which takes into account the above effects of the Pauli principle through a suitable Pauli operator. It is called the Bethe-Goldstone equation, and given by:
2 (∇21 + ∇22 ) + Epq Ψpq = QF pq {v(r12 )Ψpq } . 2m
(4.18)
Here, Epq is the energy eigenvalue, and QF pq is a Pauli operator that annihilates all states in Fermi sea except |pq, and allows all states above the Fermi sea: a formal expression for this Pauli operator in Dirac’s standard notation is QF pq ≡ ab>F |abab| + |pqpq|, where the sum extends over all states a, b above the Fermi sea. Note that if we replace the Pauli operator by unity, we recover the two-particle Schr¨ odinger equation, as we must. At large interparticle distances r12 , v(r12 ) → 0, and the right-hand side of Eq. (4.18) is negligible with or without the Pauli operator, but the asymptotic behaviors of the solutions in the two cases are entirely different, as explained above. At small values of r12 , on the other hand, the right-
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hand side is quite significant, but the Pauli operator has little effect. This is so because such small distances correspond to high momenta, which are above the Fermi momentum of the sea (k > kF ). Thus the relevant states are all above the Fermi sea, and therefore all allowed by the Pauli operator. In this case, thus, the right-hand side of Eq. (4.18) is essentially identical to what one obtains for an isolated pair of nucleons, and we recover the Schr¨ odinger equation. Ψpq at small distances is, therefore, unaffected by the presence of the Fermi sea, and shows the correlations induced by v, in particular the “hole” or “wound” produced by the strong repulsive core, as already inferred by Jastrow (1955) and described above. In a lucid summary of the Bethe-Goldstone work published in 1958, Gomes, Walecka and Wiesskopf introduced a language for describing the above phenomena which has become standard5 . These authors clearly explained that the Fermi sea severely restricts the range of correlations produced by v in Ψpq , so that the “wound” produced in the unperturbed wave function |pq at short distances (r ∼ c) “heals” rapidly at larger distances, on a length scale ∼ kF−1 ≈ 0.7 fm, leaving no trace whatever of the interaction at large distances. Here, kF ≈ 1.33 fm−1 is the Fermi wave number of nuclear matter, as discussed earlier. This piece of physics was new, not given by Jastrow’s intuitive arguments. In the same language, Gomes et al. pointed out the contrast with an isolated nucleon-pair: there the wound in the wave function has a permanent effect even at very large distances, in the form of the scattered wave and the phase shift. In Fig. 4.9, we reproduce an example of the healing phenomenon given by Gomes et al., which drives the point home. By displaying together (a) the unperturbed two-nucleon wave function |pq, (b) the wave function Ψ0pq for an isolated pair of nucleons interacting through a purely repulsivecore interaction, and, (c) the two-nucleon wave function Ψpq with the same interaction in the presence of a Fermi sea, the effects are clearly seen. The wound in Ψpq comes from the fact that it must vanish for r < c ≈ 0.4 fm, i.e., inside the hard core, while |pq does not. Just outside the core, Ψpq follows the isolated-pair wave function Ψ0pq closely upto r ∼ kF−1 ≈ 0.7 fm, as indicated above. But then Ψpq heals rapidly on a scale kF−1 to |pq, becoming almost identical to the latter by r ∼ 5kF−1 , while Ψ0pq remains phase-shifted by a constant amount relative to Ψpq upto arbitrarily large distances. Note that the addition of an attractive part to the potential at larger distances, as the actual two-nucleon potential contains, produces 5 This reference also serves as an excellent elementary introduction to the properties of nuclear matter as known in the 1950s.
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Fig. 4.9 “Healing” of a two-nucleon wavefunction due to correlations produced by the Fermi sea. Shown are (a) the unperturbed wave function (dashed line), (b) the wave function of an isolated nucleon-pair interacting through a repulsive-core potential (dotdashed line), and, (c) the wave function of the same interacting pair embedded in a Fermi distribution of nucleons (solid line). Reproduced with permission by Elsevier B.V. from Gomes et al. (1958): see Bibliography.
significant changes in Ψ0pq , but essentially none in Ψpq . This is so because the long-range part of the potential corresponds to small momenta which are within the Fermi sea (k < kF ), and all such states are disallowed by the Pauli operator, as explained above. Thus, Ψpq is almost the same as if only the repulsive core of the nuclear potential were present, and almost identical to the free-particle wave function |pq except in the core and its immediate neighborhood. In the above picturesque language, this is called the “stiffness” of the wave function against any influence of the potential. It must be clear now that this stiffness is caused entirely by the Fermi sea, which severely restricts the potential’s influence through the Pauli exclusion
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principle, allowing changes in Ψpq only at wavenumbers above the Fermi −1 sea, i.e., at distances < ∼ kF . We conclude this introduction to correlations and healing by mentioning a measure of the phenomenon introduced by Gomes et al. (1958) and widely used subsequently. It is the difference χ ≡ Ψpq − |pq, which (or a minor variation of which) is called the “defect” wave function, and which is displayed in Fig. 4.10 for a typical case. χ(r) oscillates around zero, and is quickly damped out on a scale ∼ kF−1 . The contrast of this with the defect of the isolated-pair wave function, χ0 ≡ Ψ0pq − |pq, which is also shown in Fig. 4.10, must be clear.
Fig. 4.10 The “defect” wave function for an isolated pair (dashed line), and one embeeded in a Fermi sea (solid line). Reproduced with permission by Elsevier B.V. from Gomes et al. (1958): see Bibliography.
4.1.5
Brueckner-Bethe-Goldstone (BBG) Theory
In the 1960s, Bethe and co-authors introduced the “reference spectrum method” for calculating the reaction matrix G by a very convenient kind of successive-approximations method[Bethe, Brandow & Petschek 1963], and Bethe (1965) resolved a residual difficulty (first suggested by Rajaraman’s 1963 work) with the convergence of the series expansion in G. The result is now known as Brueckner-Bethe-Goldstone (henceforth BBG) theory.
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The reference spectrum method proceeds by first approximating the actual energy spectrum E(k) by a simple, “reference” energy spectrum E R (k) ≡ k 2 /2m∗ + E0 ,
(4.19)
for which the reference correlated wave function ΨR pq and the reference reaction matrix GR can be easily calculated. Here, m∗ is the effective mass (constant), and E0 is a suitable constant. The calculation is easy because the energy operator e, defined above by Eq. (4.12), now has the simple form eR = −(2m∗ )−1 (∇21 + ∇22 ) + 2E0 − W . Furthermore, as we shall see, with a proper choice of the above quadratic energy spectrum, we can approximate the Pauli operator Q (see Eq. [4.11]) by unity without making a serious error. ΨR pq is then obtained by solving a relatively simple differential equation: ΨR pq ≡ |pq −
1 vΨR , eR pq
(4.20)
which follows from the relations given below eq. (4.15). Similarly, the reference G-matrix is given by GR = v − v
1 R G , eR
(4.21)
and the elements of GR are evaluated with the aid of the relation rs|GR |pq = rs|v|ΨR pq , which is obtained, again, from the relations given below eq. (4.15). Once GR is obtained, better approximations for G can be achieved by iterating the exact equation for G given earlier. The procedure converges rapidly. Indeed, for purely central potentials, GR is already a very accurate approximation to G (e.g., to within ∼ 5% for the diagonal elements for the matrix, corresponding to ∼ 2 - 3 MeV per particle). It is important to understand why this method works so well. First, with a suitable (Hartree-Fock) choice of the single-particle potential U (k) (see below), the typical energy spectrum E(k) is a monotonically increasing function of k, rising from negative values ∼ −100 MeV at k ≈ 0, and crossing over to positive values at k ≈ kF ≈ 1.4 fm−1 . With suitable choices for the parameters m∗ and E0 , this spectrum can be well approximated by the quadratic reference spectrum of Eq. (4.19) over limited ranges of k, in particular over the range 3 fm−1 ≤ k ≤ 5 fm−1 , which is crucial for determining the G matrix. In the latter range, a pure quadratic form (E0 ≈ 0) with m∗ ≈ mN works adequately. Secondly, for a given value of W , the energy spectrum in the Fermi sea does not enter at all into the calcula-
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tion of G. This is readily seen from the relation Ψpq = |pq − (Q/e)vΨpq given earlier: the Pauli operator Q allows e−1 to operate only on those states which have both nucleons above the Fermi sea (k > kF ). Hence only the energy spectrum above the Fermi sea is relevant for this calculation. Consequently, although the reference spectrum is a poor description of the energy spectrum within the Fermi sea (k < kF ), this does not matter at all. Thirdly, we can replace the Pauli operator by unity in the actual calculations without making too much of an error, which makes the computations so easy. This may seem strange at first, but it is quite true for the following reason: by replacing Q with unity, we admit in the calculations those states in the Fermi sea which Q would have rejected, but this introduces little error. To see why, note first that the total volume in phase space occupied by the Fermi sea is ∼ kF3 , which is a small fraction (∼ 1/50) of the total phase-space volume from which important contributions to G come, since the latter is ∼ (5 fm−1 )3 from the arguments given above. Hence the total contribution from the Fermi sea is correspondingly small, even if the individual contributions from different parts of the phase space are comparable. Note next that the contributions from individual two-nucleon states in the Fermi sea are typically small, since the energy denominators generated by (eR )−1 are rather large, as these contain the difference between the reference energies in the Fermi sea, which are ≈ k 2 /2mN and therefore positive, and the energies in the actual spectrum, which are ∼ -(50-100) MeV. Thus, the details of the Fermi sea do not matter for the calculation of G, with or without the Pauli operator. The reference wave function ΨR pq , the unperturbed wave function |pq, and the defect wave function χ ≡ ΨR pq − |pq obtained in typical calculations of the kind just described are very similar to those described in Sec. 4.1.4, which emphasizes the central role of correlations and healing in BBG theory. Pioneering work along these lines were done in the 1960s for the Moszkowski-Scott hard-core potential (also see Appendix D), given by v(r) =
+∞, rc v0 e
(4.22)
which was called the “standard” hard-core potential in that era. Although relatively simple, it reproduces some qualitative features of the (much more complicated) realistic modern potentials, and can still be used as a good “toy” potential for illustration. We describe later its role in the development of the variational method.
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Consider what an actual BBG-theory calculation involves. First, a suitable choice for the single-particle potential U has to be made, as mentioned earlier. The normal choice is the Hartree-Fock potential UHF , which, for a general many-particle system interacting via a two-particle potential v, has matrix elements of the form p|UHF |q ≡ r (pr|v|qr − pr|v|rq). This choice has the virtue of cancelling “self energy” diagrams. For infinite nuclear matter described by the G-matrix, this translates into the poten tial U (kp ) = q pq|G(W = Ep + Eq )|pq, where the summation is over the Fermi sea6 . Note the specific choice of the starting energy W , which is said to be on the energy shell. This choice of U removes many of the Brueckner-Goldstone diagrams from consideration, and was adopted universally in nuclear matter calculations of the 1960s. Next, note that, to get a first estimate of the energy eigenvalue, we need evaluate only the firstorder G-matrix diagrams, which is easy (recall that the U -diagram cancels out, as explained earlier). Note further that the second-order G-matrix diagrams all vanish. The ground-state energy is upto this order is therefore obtained by replacing v with G in Eq. (4.10), which yields: E=
m
1 m|T |m + ( mn|G|mn − mn|G|nm). 2 mn mn
(4.23)
In order to obtain the next iterate of the eigenvalue, it is necessary to evaluate the sum of the three-body G-matrix diagrams7. While attempting this, Rajaraman (1963) and Bethe (1965) came to the conclusion that the expansion in powers of G was divergent. Bethe (1965) resolved the difficulty by recasting the sum of the problematic series expansion as the solution of an integral equation, exactly as we have indicated in Sec. 4.1.3, showing that the three-body energy was really finite but the series had poor convergence behavior because the magnitude of the defect wave function |χ| was large at small radii. Just as the formulation of the integral equation for the G-matrix is related to two-particle correlations described by the Bethe-Goldstone equation, the formulation of the integral equation for this problem involves three-particle correlations. For describing the latter, Bethe formulated an equation which made use of the three-body techniques pioneered by Fadeev, and which is now called the Bethe-Fadeev equation. Obtaining an approximate solution to this equation, Bethe demonstrated that the three-body contribution to the energy was smaller than the two-body energy by a factor ∼ 3(c/r0 )3 ≈ 1/8. Here, 6 For
details, see the aforementioned review by Day (1967). mean here three-hole-line diagrams, not third-order diagrams: for an explanation of this technical point, see Day (1967). 7 We
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c ≈ 0.4 fm is the core radius, as before, and r0 is the internucleon spacing in nuclear matter, defined by: 1/3 3 ≈ 1.14 fm, (4.24) r0 ≡ 4πnnm with the nuclear-matter number density being nnm ≈ 0.16 fm−3 , as before. Note that this r0 is identical to the nuclear scale radius introduced in Secs. 3.2 and 3.2.1, and so we have used the same symbol for both. This identity is easily seen from the discussion given in Sec. 3.2.1 on the fillingfactor parameter u: the limit of u = 1 corresponds to nuclei merging into a uniform nuclear matter of density ρnm . For the two-body contribution to the energy per nucleon, BBG-theory results thus yielded a value ≈ −35 MeV, and, for the three-body contribution, a value ≈ −5 MeV. Magnitudes of the higher-order contributions were expected to be < ∼ 1 MeV. The energy per nucleon of nuclear matter E is the sum of these contributions plus the kinetic energy per nucleon in the degenerate Fermi sea of nucleons, the latter being T = 32 kF2 /10mN ≈ 23 MeV. Here, kF ≈ 1.33 fm−1 is the Fermi wave number of nuclear matter, as before. Thus, E ≈ −17 MeV, with an uncertainty of ∼ 1 MeV, which is to be compared with the observed binding energy of nuclear matter, Eb = −E ≈ 16 MeV, given in Sec. 4.1. Note that, in these results of the late 1960s, the two-nucleon potentials used were the “realistic” ones available at the time, e.g., the hard-core or soft-core Reid potential, or the Hamada-Johnston potential (see Appendix D), which today look rather simple compared to the most realistic ones currently in use. We shall return later to the essential novel features of modern BBG-type calculations, in particular the introduction of relativistic effects. 4.1.6
The Variational Method
It was realized very early in the post-pulsar era of neutron star research that the densities in the deep interiors of neutron stars might reach considerably above (∼ 5−10) the nuclear-matter density. Since a perturbation expansion method, like the BBG method, is plagued by convergence difficulties of the series as we proceed to higher and higher densities, the value of an alternative, non-perturbative method became clear. Pandharipande (1971a,b; also see Siemens & Pandharipande 1971) suggested that Jastrow’s (1955) variational method (see Sec.4.1.4), which had been developed for application to Fermi and Bose systems like He3 and He4 since Jastrow’s original work,
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was ideally suited for this. Although the method was originally launched [Pandharipande 1971a; Siemens & Pandharipande 1971] by arguing its connection with the BBG method (see below) and comparing its results with those of the latter, it rapidly developed in the hands of Pandharipande and co-authors into one of the most useful computational tools for studying matter at extremely high densities. While comparisons with the results of more advanced versions of BBG theory (see below) are still made, there is no doubt that the variational method stands today as a most valuable scheme in its own right. The variational method [Schiff 1968] proceeds by first obtaining the expectation value of the Hamiltonian given by Eq. (4.1) in a state described by a trial wave function which has the form given by Eq. (4.16) and so includes the correlations f (rij ) induced by the nucleon-nucleon interactions: E=
Ψ|H|Ψ . Ψ|Ψ
(4.25)
As this expectation value is, by definition, an upper bound on the actual eigenvalue of H, a good estimate of the latter is obtained by minimizing E with respect to the parameters included in the correlation function f (r), and, by refining our choice of the forms of Ψ and f (see below), this estimate can be made very accurate. In fact, as Pandharipande (1971a) proposed, a more satisfactory method would be to calculate f self-consistently, subject to appropriate constraints. The idea is to first choose general constraints in the spirit of Jastrow’s original decsiption of f , as given above. Pandharipande chose the boundary conditions: df = 0, (4.26) f (r > d) = 1, dr r=d where d is the healing distance, as described in Sec. 4.1.4. d is so chosen that, on the average, there is only one particle within a distance d of a given particle, expressed by the following condition8 in terms of the number density n: d f 2 d3 r = 1. (4.27) n 0 8 Actually,
the left-hand side of Eq. [4.27] is the number of particles within d in the lowest order, which is appropriate because we are describing lowest-order variational calculations at this point.
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The underlying physics is that the nearest neighbor of a given particle should give the dominant contribution to the correlations, and it should be possible to absorb the total effect of the more distant neighbors into an average potential. We can then treat this average part of v separately, defining the φp in Eq. (4.16) to be eigenfunctions of a Hamiltonian containing this part. This was, in fact, the separation scheme advocated earlier by Moszkowski and Scott, and the healing conditions given by Eq. (4.26) are basically the Moszkowski-Scott boundary conditions. How do we calculate f ? Pandharipande first did it in a lowest-order variational calculation, which follows from the idea of cluster expansion. We illustrate this idea in an elementary way by going back to Jastrow’s (1955) original work, and considering a system of bosons (instead of fermions, which nucleons are) at first for simplicity, since the antisymmetrization operator in Eq. (4.16) is omitted for bosons, which makes the calculation easier. For such systems, Jastrow showed that the expectation value of Eq. (4.25) can be considerably simplified in the following way, using the orthonormal plane-wave basis states φp = Ω−1/2 exp(ikp .rp ). The numerator 1,A Ψ|H|Ψ reduces as follows. The kinetic energy part, Ψ| i Ti |Ψ, sim f (rij )(−2 /2m)∇2 i<j f (rij ) k d3 rk , plifies first into the form A i<j 2 2 2 and then further into the form A2 ) k d3 rk i<j f (rij )(− /2m)(∇ f /f on noting that the cross terms vanish on integration. Here, k d3 rk ≡ energy part, Ψ| 1,A d3 r1 d3 r2 · · · d3 rA . Similarly, i<j vij |Ψ sim the potential 3 2 f (r )v(r ) d r , since A(A − 1)/2 is plifies into A(A − 1)/2 ij 12 k i<j k the number of independent particle-pairs interacting via the two-nucleon 3 2 potential. Finally, the denominator Ψ|Ψ is just i<j f (rij ) k d rk . Combining these, and going to the limit of large A, Jastrow wrote the energy per particle as 2 2 n ∇ f (r12 ) E = − + v(r12 ) g(r12 )d3 r12 , (4.28) A 2 m f (r12 ) in terms of the pair distribution function g(r12 ), which is obtained basically by integrating Ψ2 over all particles except 1 and 2: 2 3 3 3 i<j f (rij )d r3 d r4 · · · d rA 2 g(r12 ) = Ω . (4.29) 2 3 i<j f (rij ) k d rk The remarkable thing about Eqs. (4.28) and (4.29), noted Jastrow, was that they were identical in form to those that occur in the classical theory of imperfect gases, with the factor exp[−v(rij )/kT ] replacing our f 2 (rij ) in the classical analogue. He argued, therefore, that Mayer’s well-known
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scheme for a cluster expansion of g in ascending powers of the particle density n should be useful here, too. This expansion proceeds by defining F (rij ) ≡ f 2 (rij ) − 1, and noting that the pair distribution function can then be expanded as: g(r12 ) = f 2 (r12 )[g (2) (r12 ) + ng (3) (r12 ) + · · ·], where the terms in the expansion are given by: g (2) (r12 ) ≡ 1, g (3) (r12 ) ≡ F (r13 )F (r32 )d3 r3 ,
...
(4.30)
(4.31)
The physical basis for the cluster expansion is clear: the leading term in Eq. (4.30) describes the two-body correlations, for which f (r12 ) is sufficient; the next term describes three-body correlations, i.e., those arising from the presence of a third particle in the neighborhood of 1 and 2, and so on. For fermions, the idea is the same, but the mathematics is a bit more complicated because of the inclusion of the antisymmetrization operation. This means working with Slater determinants formed from plane-wave states, as described in Sec. 4.1.1, which introduces an extra set of integrals in the expansion terms given by Eq. (4.31) which differ from those given in that equation by having extra factors ∼ exp[i(kp − kq ).rpq ] in the integrands [Pandharipande & Bethe 1973]. The lowest-order variational calculation of Pandharipande corresponds to keeping only the leading term in the above cluster expansion, i.e., considering only two-body correlations, by setting g(r12 ) = f 2 (r12 ) in Eq. (4.28)9 . Minimization of energy with respect to variations in f then proceeds with the aid the standard Moszkowski-Scott way of separating long-range “tail” of the potential v(r): we assume that v(r > d) contributes only to the average field, which is consistent with the healing conditions (4.26), so that the integral in Eq. (4.28) is confined to the region r ≤ d. In the latter region, we make the simplifying and plausible assumption that the contribution of the average field to v is a constant λ. Then the variational condition on the energy reads d 2 − 2 2 2 f ∇ f + vf − λf d3 r = 0, (4.32) δ m 0 and directly leads, in the region r ≤ d, to an Euler-Lagrange type equation: −2 2 ∇ f + vf = λf. m 9 Note
(4.33)
that, in the lowest order, this result holds for both bosons and fermions.
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Fig. 4.11 Typical correlation functions f in various spin-states as indicated, for nuclear matter at kF = 1.4 fm−1 with the Reid soft-core potential. Reproduced with permission by Elsevier B.V. from Pandharipande (1972): see Bibliography.
Note that Eq. (4.33) looks somewhat similar to a two-body Schr¨ odinger equation: we shall call it the Pandharipande equation. For obtaining f , we need to solve Eqs. (4.33) and (4.27) simultaneously, subject to the boundary conditions (4.26). Shown in Fig. 4.11 is the typical form of f (r) obtained in Pandharipande’s early calculations [Pandharipande 1972] on nuclear matter at kF = 1.4 fm−1 with the Reid soft-core potential (see Appendix D). 4.1.6.1
Cluster diagrams
To facilitate the inclusion of higher orders in the cluster expansion, Pandharipande and Bethe (1973) introduced a diagrammatic represention, which we show in Fig. 4.12 for fermion systems. The idea is as follows. As we saw above, the pair distribution function gmn ≡ g(rmn ) has a denominator and a numerator which are integrals of i<j f 2 (rij ) over particle co-ordinates, the former being over all particles, and the latter over all except particles m and n. Using the above definition Fij ≡ F (rij ) + 1 ≡ f 2 (rij ), we can express the product i<j f 2 (rij ) as the sum 1+ Fij + Fij Fkl + · · · , (4.34) i<j
i<j,k
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and so expand both numerator and denominator as sums involving zero, one, two, . . . factors of the F function. These are precisely the cluster diagrams shown in Fig. 4.12.
Fig. 4.12 Pandharipande-Bethe denominator diagrams for Fermi fluids. Reprinted with permission by the APS from Pandharipande & Bethe (1973), Phys. Rev. C, 7, 1312. c 1973 American Physical Society.
The rules for the Pandharipande-Bethe cluster diagrams are rather similar to those for the Goldstone diagrams given earlier, namely, that (1) points represent particle co-ordinates, (2) dashed lines represent F functions, and, 2 . In (3) numerator diagrams always connect points m and n through fmn addition, in fermion systems: (4) full lines represent k of the plane-wave states, so that (5) km originates at the point m, (6) a km terminating at the point n represents a plane-wave state φm occupied by particle n in the final state Ψ| occurring in the expectation values (see above), and so gives rise to a factor exp(km .rmn ), and, finally, (7) the sign of a fermion diagram is (−)l+s , where l is the number of closed loops and s is the number of state lines. It will come as no great surprise, then, that the cluster diagrams appear rather similar to the Goldstone diagrams. In particular, note that the first two non-trivial denominator diagrams in the above figure
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are identical to the two first-order v-term Goldstone diagrams described in Sec. 4.1.2.1. It is instructive to compare the higher-order cluster diagrams with the corresponding Goldstone diagrams. To see how one sums cluster diagrams, note first that two points p and q can be connected in many ways, some of which are illustrated in Fig. 4.13 for the relatively easy case of boson systems. Simplest connections are those shown in the top line of the figure, which are called single chains. The next, relatively simple, connection, which nevertheless involves a double chain, is shown in the next line: its contribution is expected to be small [Pandharipande & Bethe 1973]. Particularly important, however, is the case shown at the bottom of the diagram, where two points of a chain are themselves connected by many chains: this is the hypernetted chain – HNC for short – so named because of the many “nets” of correlations generated. How does one sum all such HNC (and other) diagrams? The answer is, again, what we anticipated in Sec. 4.1.3: a suitable integral equation is found whose solution is the required sum of diagrams. For cluster diagrams, the necessary techniques were developed by van Leeuwen and co-authors, and by Feenberg (1969) and co-authors. To give a flavor of
Fig. 4.13
Examples of Pandharipande-Bethe cluster diagrams for Bose fluids.
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how the calculation proceeds, we give below a very simple example of the relevant integral equation for gmn in the case of boson systems, which is called the HNC equation [Feenberg 1969]: gmp gmn (4.35) ln =n gmp − 1 − ln (gmp − 1)d3 rp . 2 2 fmn fmp A comparison of this integral-equation approach to that used for the Gmatrix in Sec. 4.1.3 is instructive. Just as G was obtained there from the known two-nucleon potential v through Eq. (4.14), the pair distribution function gmn is obtained here from the known correlation function fmn (which is itself determined by v and the healing conditions, as explained above) through Eq. (4.35). There is thus a close analogy between the two cases: indeed, the relation between G and f was used originally by Pandharipande (1971a) to motivate the use of the variational method in this context. He combined Eqs. (4.16) and (4.3) with the relation GΦ = vΨ given above to argue that, crudely speaking, Ψ/Φ ∼ G/v ∼ f , so that10 working with the correlation function f in a variational scheme was analogous to doing a Brueckner G-matrix calculation in a simpler and quicker way. Pandharipande and Bethe (1973) adapted the above summation techniques for calculating the energy per particle of a neutron gas and so estimating the accuracy of the earlier lowest-order variational calculations. 4.1.6.2
Modern calculations
By the late 1970s, the variational method developed into a sophisticated scheme for state-of-the-art calculations of the properties of nuclear and neutron matter11 . Two directions of refinement in the method are of particular interest to us here. First, as a result of rapid advancements in nuclear physics, the two-nucleon interaction (which lies at the foundation of both variational and Brueckner calculations) could be modeled by the potential v with great accuracy and refinement, by fitting the low-energy (< ∼ 300 MeV) two-body scattering data and properties of light nuclei, as described in Appendix D. The simple, soft-core, central potential v c (r) due to Reid or to Bethe and Johnson, which had been the state of the art in the early 1970s, and in which the radial function v c (r) had usually been pa10 This relation works best when the matrix G(n, r ) is diagonal in the relative distance ij rij , which is nearly the case at low densities; see Pandharipande 1971a and references therein. 11 See the excellent 1979 review by Pandharipande and Wiringa for a thorough account.
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rameterized as a superposition of Yukawa potentials ∝ exp(−µr)/µr, was now upgraded to a much more complex one containing eight to ten operator components (see Appendix D). The first six operators were the simplest, namely, (1) the (old) central part v c , (2) the spin operator part v σ (σi .σj ), (3) the isospin operator part v τ (τi .τj ), (4) the spin-isospin operator part v στ (σi .σj )(τi .τj ), (5) the tensor operator part v t Sij , and, (6) the tensorisospin operator part v tτ Sij (τi .τj ). Here, σi are the Pauli spin matrices, τi are the analogous isospin matrices [Blatt & Weisskopf 1952], and the tensor operator Sij is given by Sij ≡ 3(σi .ˆr)(σj .ˆr) − σi .σj (see Appendix D). A potential including only these six terms was called a v6 potential. The next more complicated potential was the v8 , which contained two further terms, namely, (7) the spin-orbit operator v b (L.S)ij , and, (8) the spin-orbit-isospin operator v bτ (L.S)ij (τi .τj ). Variational calculations with these vn (n has gone up to 18 in the most recent calculations, as we shall see later) require a generalization of the correlation function f (rij ) to a correlation opreator of the form: Fij =
1,n
p βp f p (rij )Oij ,
(4.36)
p p where Oij are the operators described above. The radial functions f p (r), with p = 1, 2, . . . , n, are the generalization of the single function f (r) of the simple theory and βp ’s are the relative strengths of the terms. We can set the normalization β1 = 1 by comparison with the simpler case, but then βp>1 still need to be determined by minimization of energy (see below). The set of n radial functions f p (r) is obtained by a generalization of the procedure outlined in Sec. 4.1.6, i.e., by solving n Pandharipande equations. Note, however, that the first one of the boundary (or healing) conditions (4.26) is different for f 1 and f p>1 . While the central part f 1 = f c heals to 1 at r > d, all the f p>1 heal to 0. Fig. 4.14 shows an example of the 6 correlation functions obtained for a Reid v6 potential [Pandharipande & Wiringa 1979a]. After obtaining the f p (r, d), the energy E(ρ, d, βp>1 ) is calculated and minimized with respect to variations in d and βp at each density ρ or number density n. Secondly, the diagrammatic cluster expansions become more complip in the potential. Pandcated with the above inclusion of the opeartors Oij haripande and Wiringa (1979) evolved a diagrammatic scheme for this, generalizing and extending the earlier Pandharipande-Bethe scheme (see above). The HNC summation scheme described above had been extended
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Fig. 4.14 The six correlation functions for a Reid v6 potential. Reproduced with permission by Elsevier B.V. from Pandharipande & Wiringa (1979a): see Bibliography.
earlier to fermion sytems, where the diagrams get more complicated due to the inclusion of exchange effects, and the resulting scheme for summing the Fermi hypernetted chain, or FHNC, diagrams is appropriately more complicated. Wiringa and Pandharipande further extended the FHNC scheme to include the above operators in the single-operator chain, or SOC, approximation, in which each link in the chain can contain only one operator element or chain. The resultant FHNC/SOC scheme became a standard method for variational calculations of nuclear matter: with some modifications described below, it remains so today. We illustrate typical late-1970s results obtained for the v6 and v8 potentials from the BBG-theory and variational calculations of nuclear matter in Fig. 4.15 [Pandharipande & Wiringa 1979], which shows the energy per nucleon E/A vs. the Fermi wavelength kF for the Reid v6 potential for various calculations. The lowest-order Brueckner theory (LOBT) gives too high an energy, of course, and a good BBG calculation, with three- and four-body terms included, lowers it below the FHNC/SOC variational values. On the whole, the calculations gave minima of the E(kF ) curves whose energy values bracketed the experimental values, but the calculated values of kF (and so the densities) were too high.
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Fig. 4.15 Results for various calculations of E(kF ) (see text) for a Reid v6 potential. Reprinted with permission by the APS from Pandharipande & Wiringa (1979), Rev. c 1979 American Physical Society. Mod. Phys., 51, 821.
4.1.7
Recent Developments in BBG and Variational Approaches
The account given by Wiringa, Fiks and Fabrocini (1988) on further developments in variational and BBG approaches in the 1980s showed that the main refinemnts in this era had been in the potential models v of nucleonnucleon interaction. The v8 models of the previous decade had been extended to v14 , including six new operators which were (1) the L2 operator, (2) the L2 (σi .σj ) operator, (3) the quadratic spin-orbit operator (L.S)2 , and the three operators obtained by multiplying each of these three with the isospin part (τi .τj ) (see Appendix D). It also became customary to write the each radial function v p as a sum of (a) a long-range, one pion exchange part, (b) an intermediate-range, two-pion exchange part, and (a) a shortrange part, coming from the exchange of heavier mesons or an overlap of composite quark systems, as described in Appendix D. Such v14 potentials
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were evolved by research groups at various places, e.g., Urbana, Argonne, Paris, Bonn, and so on, and they came to be labeled by these place names, e.g., UV14, AV14, Paris V14, Bonn V14, and so on. In addition, first attempts were made to formulate and include an actual three-body potential in v, i.e., an interaction which is inherently between three nucleons, and which cannot be represented by an iteration of two-nucleon interactions. A form for such three-nucleon interaction (TNI) explored by the Urbana group was: Vijk =
U [u (rij )u (rik )P (cosθi )]cyc ,
(4.37)
where the variables for the three nucleons i, j, k are the interparticle distances rij , rik , and the angle θi between the two vectors rij , rik . In Eq. (4.37), U are strength parameters, u (r) is a suitable potential function of the interparticle distance r, and the subscript “cyc” indicates cyclic permutation of the indices i, j, k. In the variational calculations, the FHNC/SOC summation processes for the v14 potentials now became more complicated, of course, involving the simultaneous solution of many integral equations, and an automated search procedure for finding the best values of variational parameters in the parameter space. Nevertheless, these could be performed within reasonable CPU times on supercomputers available in the era [Wiringa, Fiks & Fabrocini 1988]. Figure 4.16 shows typical results from these calculations with the AV14 and UV14 potentials for nuclear matter. For the Urbana potential, the effects of adding either (a) a relatively simple three-nucleon potential named UVII, which consists of a long-range two-pion exchange part and an intermediate-range repulsive part, or, (b) a simple prescription for mimicking the effects of TNI (as introduced above) through densitydependent terms were also considered, and the results are shown in the figure. It became important to compare the above variational results with those obtained from BBG theory for the same, modern nucleon-nucleon interactions, at least over the (moderately high) density range where BBG calculations were still considered reliable. A feature of the “standard” 1960s choice of the single-particle potential U , namely that it was zero for particle states and it equalled the self-consistent Hartree-Fock potential for hole states, and so had an unphysical gap at the Fermi surface, came in for criticism in the 1970s. The remedy for this, the so-called “continuous
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choice”, was found by the Li`ege group [Jeukenne, Lejeune & Mahaux 1976] by extending the concept of Hartree-Fock energies to particle states. This was generally adopted, and modern forms of which are still in use. BBG calculations by Day and others in the 1970s included the three-body cluster contributions and estimates of the contributions of four-body cluster contributions. Although the actual calculation of the latter was very difficult, it became clear that this would be necessary for a < ∼ 10% accuracy in the binding energy. For comparison, results of the 1985 BBG calculations of Day and Wiringa are displayed in Fig. 4.16 for the Argonne AV14 potential, together with variational results for the AV14 and UV14 potential, as also for these potentials augmented by UVII. Clearly, there was already qualitative agreement in the 1980s between the variational and BBG calculations. Furthermore, variational results for the UV14 + TNI potential were running close the observational values. It also became clear from that the addition of the three-body potential reduces the saturation density of nuclear matter, i.e., the density at which the minimum in the E(ρ) curve occurs, and this is what cures the earlier
Fig. 4.16 Nuclear matter E(ρ) from variational calculations for several potentials, as indicated. Also shown are BBG results for the AV14 potential. Reprinted with permisc 1988 American sion by the APS from Wiringa et al. (1988), Phys. Rev. C, 51, 38. Physical Society.
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defect of obtaining minima at densities which were too high compared to the observed satuartion density (see Sec. 4.1.6.2). This means, then, that threenucleon interactions stiffen the EOS of nuclear matter. TNI also stiffen the EOS for pure neutron matter, and increase the difference between the EOS of nuclear and neutron matter, i.e., they increase the symmetry energy coefficient Es . Among the major developments in the 1990s and later, we shall concentrate here on (a) further refinements in the nucleon interaction potential, particularly those by the Nijmegen group (also see Appendix D), and, (b) introduction of relativistic effects into the nuclear Hamiltonian. In the early 1990s, the Nijmegen group [Stoks et al. 1994] took the important step of compiling all nucleon-nucleon scattering data below the pion production threshold of ∼ 350 MeV published since 1955, extracting a large number of “reliable” neutron-proton and proton-proton scattering data to form a state-of-the-art database, obtaining accurate values of phase shifts and mixing parameters from this database, and, finally, constructing “modern” nucleon-interaction model potentials by fitting to the latter. These model potentials came to be called Nijmegen I, Nijmegen II, Reid 93, and so on. In 1995, a similar v18 potential called AV18, the upgraded version of the previous Argonne potential AV14 described above, was given by Wiringa, Stoks, and Schiavilla, and the CD-Bonn potential was given in 1996. In order to achieve an accurate simultaneous fit to the np and pp data, these models had to include (a) a detailed description of the electromagnetic interactions, and (b) terms describing deviations from the isospin symmetry of strong interactions [Heiselberg & Pandharipande 2000]. We illustrate this with the AV18 potential, since it was designed partly with applications to many-body calculations of nuclear matter in mind [Wiringa, Stoks & Schiavilla 1995]. It has a complete electromagnetic potential, containing Coulomb, Darwin-Foldy, vacuum polarization, and magnetic moment terms with finite-size effects. The four new operators, which describe deviations from charge-independence and charge-symmetry (which all the earlier 14 operators constituting the v14 obeyed) are as follows. The first three oparators are charge-dependent: (1) the isotensor operator Tij , (2) the spin-isotensor operator (σi .σj )Tij , and, (3) the tensor-isotensor operator Sij Tij . The fourth operator, the z-component of isospin τ , is charge-asymmetric: (τi + τj )z . Here, the isotensor operator z)(τj .ˆ z) − τi .τj is defined in analogy with the tensor opeartor Sij Tij ≡ 3(τi .ˆ described above. Some comparative details of other modern potentials are given in Appendix D.
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Consider now the second point: our discussion of nuclear-matter theory has so far been entirely non-relativistic, since we started from a nonrelativstic nuclear many-body Hamiltonian. Let us see how the effects of the special theory of relativity can be and have been introduced into both BBG and variational calculations of nuclear matter. 4.1.7.1
Relativistic effects in Brueckner theory
The Brueckner approach lends itself rather easily to relativistic extension. Following the advent of relativistic theory of nuclear structure in the 1970s, such extensions, appropriately named Dirac-Brueckner approach, were suggested by Shakin and co-authors [Celenza & Shakin 1986], and extensively developed by Brockmann and Machleidt (1990), and by ter Haar and Malfliet (1987). Some of the original motivation12 for the DiracBrueckner efforts came from the failure of earlier BBG works to reproduce the observed nuclear saturation density (see above), and, indeed, it was soon demonstrated that a general feature of Dirac-Brueckner results was an additional, strongly density-dependent, repulsion between nucleons, which stiffened the EOS (just as TNI did, as explained earlier), and so improved agreement with observation. We illustrate the Dirac-Brueckner approach through the BrockmannMachleidt (1990) prescription, which is particularly transparent. As the name implies, the Dirac equation is now used to describe the motion of single nucleons in nuclear matter: ˜ s) = 0, (c p − M c2 − U )w(p,
(4.38)
in terms of the free-nucleon mass M , and an effective potential, U = Us + γ 0 Uv , consisting of an attractive scalar part Us and repulsive vector part Uv (the timelike component of a repulsive vector field), which are to be determined self-consistently. Here, γ µ (µ = 0, 1, 2, 3) are the Dirac matrices, and we use the standard Feynman notation p ≡ γ µ pµ (see, e.g., Schweber 1966). Equation (4.38) is the appropriate generalization of the free Dirac equation, and its solutions w(p, ˜ s) (which are functions of the momentum 12 We showed in Chapter 2 that degenerate neutron matter is non-relativistic at the average densities of neutron stars, and estimated the density at which it would become relativistic. Extending these arguments, the reader can easily show that, even if the central densities of neutron stars are ≈ 10ρnm , the zero-point motion of degenerate neutrons there is only mildly relativistic. The relativistic many-body effects under discussion here are more subtle.
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p and spin s = ± 21 ) are corresponding generalizations of the free Dirac spinors [Schweber 1966], namely: w(p, ˜ s) =
⎛ 2 ˜ ˜ E(p) + M c ⎜ ⎝ ˜ 2E(p)
˜ ˜ c2 ≡ M c2 + Us , E(p) ≡ Here, M Pauli spinors [Rose 1961]: χ 12 =
1 0
χs c(σ.p) ˜ ˜ 2 χs E(p)+ Mc
⎞ ⎟ ⎠ .
(4.39)
˜ 2 c4 , and χs are the standard p2 c2 + M
χ− 12 =
0 . 1
(4.40)
Clearly, the limit of the free Dirac nucleon is recovered in Eqs. (4.38) and ˜ = M , as expected, and the normalization for (4.39) by setting U = 0, M the Dirac-spinor nucleon states w in nuclear matter is w† w = 1. The G-matrix formalism proceeds as before, except that an appropriate relativistic version of Eq. (4.14) has to be used: Brockman and Machleidt chose the Thompson (1970) equation. The lowest-order energy is still given formally by Eq. (4.23), provided we remember that, in this equation, the following changes are to be understood: (a) a state |m is now a Dirac spinor w(pm , s) as above, (b) for the kinetic-energy operator T , the relativistic odinger form Dirac form (γ.pc + M c2 ) replaces the non-relativistic Schr¨ ˜ is still evaluated p2 /2M used earlier, (c) the Dirac-Brueckner matrix G ˜ on energy shell, but with the modified energies E(p) defined above, i.e., ˜n )|mn, where E ˜m ≡ E(p ˜ m ), ˜m + E individual terms look like mn|G(W = E and, finally, (d) we must remember (as we do in any relativistic calculation) to subtract the rest-mass energy M c2 per nucleon at the end to obtain the final result for E. The kinetic energy part of E is particularly simple in this ˜ c4 + p2m c2 )/E ˜m , where the sum formulation, being given by m
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Fig. 4.17 Dirac-Brueckner E(ρ) results for pure neutron matter (PNM; upper curves) and symmetric nuclear matter (SNM; lower curves) for several potentials, as indicated. Reproduced with permission by Annual Reviews from Hesielberg & Pandharipande (2000): see Bibliography.
4.1.7.2
Relativistic effects in variational methods
Incorporating relativistic effects in variational methods proved to be a more difficult task, and so took longer in coming, since these methods are deeply rooted in the non-relativistic Schr¨ odinger formalism. Here, the idea is to mimic the effects of relativistic boost by extra terms in the potential v in the center-of-mass frame, utilising the prescription given in the mid1970s by Friar, following the work of Krajcik and Foldy [Friar 1975]. When two nucleons i and j with momenta pi and pj interact, the interaction depends, in general, on both the relative momentum pij ≡ pi − pj and the total momentum Pij ≡ pi + pj . We normally consider only the former dependence by referring nucleon-nucleon interactions to the center-of-mas frame, where Pij = 0; this is the momentum dependence contained in the usual nucleon-nucleon potential vij discussed so far. How do we account for the relativistic boost interaction which depends on Pij ? The FriarKrajcik-Foldy prescription is to do this by adding an extra component δv
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to the potential which depends on Pij , i.e., v(pij , Pij ) = vij (pij )+δv(Pij ). The form of δv(P) given by these authors was δv(P) =
' 1 & −P 2 v + [P.rP.∇, v] + [(σi − σj ) × P.∇, v] , 2 8M
(4.41)
which took care of the boost interaction upto order P 2 . Here, v = vij is the usual center-of-mass potential discussed above, σi are the Pauli spin matrices as before, and [a, b] ≡ ab − ba is the commutator of operators a and b in the standard bracket notation [Schiff 1968]. A general validity of the above prescription was argued in the mid-1990s by Forest, Pandharipande and Friar (1995), and subsequent applications to variational calculations made [Akmal, Pandharipande & Ravenhall 1998]. With a static potential v s , the above boost term simplifies to δv(P) =
' 1 & −P 2 + P.rP.∇ v s , 2 8M
(4.42)
if we neglect the last term on the right-hand side of Eq. (4.41), which represents Thomas precession and other small effects. The two parts of the boost interaction represented by the two terms on the right-hand side of Eq. (4.42) have straightforward interpretations: the first comes from the relativistic expression for energy, and the second from Lorentz contraction. Studies of light nuclei, e.g., 3 H and 4 He, using this boost interaction indicate that its effect is identical to that of a repulsive term in the static potential. Specifically, its strength can be measured in terms of that of the repulsive part of the Urbana three-nucleon potential UIX, which is the descendant of the earlier UVII potential described above in connection with potentials of the v14 era of the 1980s. The strength of the boost interaction is ≈ 37% of the repulsive part of the Vijk of UIX, which implies that, when boost corrections are included, the repulsive part of the three-nucleon potential must be appropriately reduced in order to fit the nuclear data. This has been done in the latest Argonne v18 potential incuding relativistic boost, and the corresponding three-nucleon potential is called UIX∗ . For further details, we refer the reader to the review by Heiselberg and Pandharipande (2000). 4.1.7.3
Recent results
We illustrate in Fig. 4.18 the recent status of nuclear matter calculations by displaying together the Dirac-Brueckner and variational results for the same potential, namely the AV18, for both symmetric nuclear matter and
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Fig. 4.18 Comparison of recent variational (VCS; solid line) and Dirac-Brueckner (LOB; dashed line) E(ρ) results for pure neutron matter (PNM; upper curves) and symmetric nuclear matter (SNM; lower curves) for the AV18 potential. Reproduced with permission by Annual Reviews from Hesielberg & Pandharipande (2000): see Bibliography.
pure neutron matter [Heiselberg & Pandharipande 2000]. At densities < ∼ 4nnm ∼ 0.6 fm−3 , there is excellent agreement between the results from the two approaches. Generally, we expect the true eigenvalues to be below the variational upper bounds by, say, a few MeVs. Note the large difference between the two results for symmetric nuclear matter at very high densities, where the convergence of Brueckner expansion is questionable, as explained earlier.
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Chapter 5
Physics of Neutron Stars — IV Mass, Radius and Structure
Calculations of the masses, radii, and structures of “realistic” neutron stars proceed by numerical integration of the Tolman-Oppenheimer-Volkoff (TOV) equation, as described in Chapter 2, the input for which are the numerical tables of the EOS (i.e., the P − ρ relation) of dense matter through various ranges of high densities, the physics of which we described in Chapters 3 and 4. EOS in these density ranges are shown in Fig. 5.1 as an example of typical EOS proposed in the 1970s, and in Fig. 5.2 as an example of typical EOS proposed in the 1980s and ’90s. As detailed below, the internal structure of a neutron star consists of distinct layers exactly corresponding to these ranges of density, so that the properties of these layers are determined by the essential physics given in Chapters 3 and 4. Consider first the masses and radii of neutron star given by various equations of state.
5.1
Masses and Radii
We display calculated masses (M ) and radii(R) of neutron stars in Fig. 5.3 for various EOS of the 1970s, and in Fig. 5.4 for various EOS of the 1980s and ’90s. Roughly speaking, the EOS are differentiated by their choice of the prescription above the nuclear-matter density, ρ ≥ ρnm , since this is what determines the gross, overall properties of a neutron star. This is so because, as we shall see below, the (inner) part of the star with density in this range contains ∼ 99% of the star’s mass, and covers ∼ 90% of its radial extent. This does not imply, of course, that the outer, less dense, layers of the star are unimportant, as we shall also see below. But all modern neutron-star calculations use almost the same EOS for these latter regions, namely, (a) 173
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Fig. 5.1 Left panel: basic characteristics of the EOS of neutron-star matter over the entire density range, with different regions marked appropriately (see Chapters 3 and 4). Shown are (a) Oppenheimer-Volkoff (OV) free-neutron gas EOS (dashed line) for reference (Chapter 2), (b) (BPS + BBP) EOS (solid line), as detailed in Chapter 3. Right panel: magnification of the nuclear-matter region appearing as the rectangle on the top right corner of left panel, showing the various nuclear-matter EOS proposed in the 1970s, labeled as follows. Variational calculations are: A, B, C, D, M: Pandharipande & co-authors, using various potentials (see Appendix D). BBG calculations are E, F, G: various authors. Mean-field calculations are L, N: various authors. Free-neutron (OV) gas labeled H, and shown for reference. Reproduced with permission by the AAS from Arnett & Bowers (1977): see Bibliography.
Feynman-Metropolis-Teller EOS and Chandrasekhar EOS with Coulomb 6 −3 corrections for ρ < (see Chapter 3), (b) Baym-Pethick∼ 8 × 10 g cm Sutherland EOS, or more modern results like those of Haensel & Pichon 11 −3 (1994), for 8 × 106 g cm−3 < ∼ ρ ≤ ρdrip ≈ 4 × 10 g cm (see Chapter 3), and (c) the Negele-Vautherin (1973) results or FPS (1992) results for ρdrip ≤ ρ ≤ ρnm (see Chapter 3). Naturally, the calculated structures of these layers are also qualitatively all similar, except possibly for the question of appearance of non-spherical nuclei in the innermost of these layers, to which we shall return later.
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Fig. 5.2 Various nuclear-matter EOS proposed in the 1980s and 1990s, labeled as follows. Variational calculations are FP: Friedman & Pandharipande (1981), WFF1-3: Wirringa et al. (1988), AP1-4: Akmal & Pandharipande (1997). Dirac-Breuckner calculations are MPA1-2: M¨ uther et al. (1987), Engvik et al. (1996). Field theoretical calculations are: MS1-3: M¨ uller & Serot (1996), GM1-3: Glendenning & Moszkowski (1991), PCL1-2: Prakash et al. (1995). Reproduced with permission by the AAS from Lattimer & Prakash (2001) : see Bibliography.
It is clear that the values of M and R given above are in order-ofmagnitude agreement with the simple estimates for Newtonian neutron stars given in Chapter 2, and with the values given by the pioneering, general-relativistic Oppenheimer-Volkoff calculation for a gas of free neu-
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Fig. 5.3 Neutron star mass-radius relation for EOS of the 1970s, EOS labels as in Fig. 5.1. Reproduced with permission by the AAS from Arnett & Bowers (1977) : see Bibliography.
trons also described there. As expected, however, the details are quite different. For example, we show in Fig. 5.5 the mass-radius relations for EOS obtained from typical v14 potentials of the 1980s, which are moderately stiff, compared with those obtained from typical potentials of the 1970s, one of which is very soft (Pλ) and the other very stiff (TI), so that they roughly bracket the range of possibilities of that earlier era [Wiringa, Fiks & Fabrocini 1988].
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Fig. 5.4 Neutron star mass-radius relation for EOS of the 1980s and ’90s, EOS labels as in Fig. 5.2. Reproduced with permission by the AAS from Lattimer & Prakash (2001): see Bibliography.
While R decreases with increasing M in all cases, the modern calculations generally yield a roughly constant radius over most of the mass range, and a rapid decrease of R with increasing M only at the smallest masses: we return below to this question of the insensitivity of R to M . By contrast, the soft EOS leads to a more gradual decrease of R. Of course, all of these variations are much more complicated than the simple R ∼ M −1/3 scaling of Eq. (2.23). Fig. 5.4 shows some typical results for EOS of the 1990s, i.e., the most recent ones: these include (a) variational calculations, (b) DiracBrueckner calculations, both done with recent v18 potentials, and, (c) results from field theoretical calculations. The first two show the constant-R feature described above, and the last one shows a more gradual variation characteristic of a softer EOS.
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Fig. 5.5 Mass-radius relation for neutron stars following EOS obtained from various v14 potentials of the 1980s, as marked. Also shown are the corresponding relations for (a) a very soft EOS, the PΛ (Pandharipande’s hyperonic EOS), and (b) a very stiff EOS, the TI (tensor interaction), both dating from the 1970s. The latter bracket the former, as explained in text. Reprinted with permission by the APS from Wiringa et c 1988 American Physical Society. al. (1988),Phys. Rev. C, 51, 38.
Noteworthy general features of neutron-star masses and radii are: (a) stiffer EOS generally lead to higher maximum masses, (b) for a given stellar mass, a stiffer EOS generally leads to a larger radius and a smaller central density. The density profile ρ(r) of neutron stars is of considerable interest: this is displayed in Figs. 5.6 and 5.7.. The first shows the effects of varying the stellar mass M for a given EOS (in this case a very soft EOS, corresponding to the P Λ model for nuclear matter referred to above), while the latter shows those of varying the EOS for a given stellar mass of the canonical value 1.4M (these are same set of EOS as given in Fig. 5.5). For all but the lightest neutron stars, the density profile is remarkably simple, i.e., a rather uniform density throughout most of its extent, falling to small values near the surface. This makes for obvious simplifications in rough calculations to which we shall return later at appropriate places, and also supplies the justification for studying our “toy” neutron star in Chapter 2, as we mentioned there.
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Fig. 5.6 Density profile of neutron stars with PΛ EOS for nuclear matter. The effect of varying the stellar mass is shown. Reproduced with permission by the AAS from Baym et al. (1971): see Bibliography.
5.1.1
Insensitivity of Radius to Mass
To explain the insensitivity of the radius R to the mass M of neutron stars for many EOS, Lattimer and Prakash (2001; henceforth LP) noticed that this behavior was reminiscent of that of Newtonian polytropes of index s = 1, as described in Chapter 2. As we saw there, the radius is√independent of the mass and the central density in this case, and scales as K, K being the constant appearing in the polytrope equation P = Kρ2 for this case. When these authors further noticed that d ln P/d ln ρ was, in fact, roughly 2 for these EOS in the density range ρ ∼ ρnm − 2ρnm , they were encouraged, and tested the idea further in the following way. If we compare variables for various EOS at the same density, ρ = ρnm , say, then K scales as P , and so R should roughly scale as a power of P , and the exponent should be 1/2 if Newtonian scalings apply. What LP found was that there was, indeed, such a scaling, but the exponent was ≈ 1/4, the difference being attributable, no doubt, to the effects of general relativity. (LP argued the latter point by referring to an exact solution due to Buchdahl for general-relativistic s = 1 polytropes,
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Fig. 5.7 Density profile of 1.4M neutron stars, showing the effect of varying the EOS, from the very soft PΛ, through the moderately stiff EOS given by the v14 potentials of the 1980s (see above), to the very stiff TI EOS. Reprinted with permission by the APS c 1988 American Physical Society. from Wiringa et al. (1988),Phys. Rev. C, 51, 38.
which shows a similarity of the exponent.) For the large number of EOS considered, the LP scaling comes out to be roughly: R6 ≈ [P (MeV fm−3 )]1/4 ,
(5.1)
where the R6 is R in the standard units of 106 cm, as before, and P is expressed in its standard units of MeV fm−3 , as used in the literature. The constant used in Eq. (5.1) is that corresponding to the density ρ = ρnm . The above LP relation proves to be very useful for a fast, simple estimate of the essential physical properties of neutron-star crusts, as we shall see below. The essential physics in it is also significant: P at the nuclear-matter density is largely determined by the symmetry properties of the EOS, i.e., the symmetry energy term (see Chapter 3), which may make it possible to constrain this term [Lattimer & Prakash 2001]. 5.2
Internal Structure
The internal structure of a neutron star according to typical modern EOS is shown in Fig. 5.8.
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Fig. 5.8 Schematic cross-section of a 1.4M neutron star. Reproduced with permission from Heiselberg (2002).
If we go radially into the star, starting at the surface and going into denser and denser layers, we encounter the components described below. 1. A solid crust , composed of two parts: (a) An outer crust whose density ranges from terrestrial densities to the neutron-drip density ρdrip ≈ 4 × 1011 g cm−3 : we have described the essential physics of the matter in this range in Secs. (3.1) to (3.2). Upto a density ∼ 8 × 106 g cm−3 , it is a Coulomb lattice of Fe nuclei surrounded by a sea of electrons. The EOS describing this matter in the density range 10 4 −3 g cm−3 < ∼ρ< ∼ 10 g cm is the Feynman-Metropolis-Teller one, and above the latter density, it is the Chandrasekhar EOS with Coulomb corrections 6 −3 upto ρ < ∼ 8 × 10 g cm . Above that density, it is still a Coulomb lattice of nuclei in an electron sea, but the equilibrium nuclide changes, becoming progressively neutron-rich as the density increases, and the relevant EOS is the Baym-Pethick-Sutherland one, or slightly modified, modern versions thereof. (b) An inner crust whose density ranges from the neutron-drip density ρdrip ≈ 4 × 1011 g cm−3 to approximately the nuclear-matter density ρnm ≈ 2.8 × 1014 g cm−3 . Matter in this range consists of a lattice of nuclei surrounded by neutrons, the whole assembly immersed in an electron sea. The essential physics was described in Sec. (3.4), and the relevant EOS used in modern calculations is often the Negele-Vautherin or the FPS one. An interesting, modern development has been the exploration of the possibility
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that non-spherical nuclei, in the shapes of rods or plates, may occur at the lower edge of the inner crust, just before the lattice dissolves into uniform nuclear matter. A typical thickness of the outer crust is ≈ 0.3 km, and that of the inner crust ≈ 0.6 km, for a 1.4M neutron star following a modern EOS, so the total crust thickness is typically ∼ 1 km, i.e., ∼ 10% of the radius, R ∼ 10 km, of such a star. A typical mass for the whole crust of such a star is 0.012M, i.e., ∼ 1% of its total mass. 2. A liquid core, in which the density steadily rises inward from ρnm to the central density ρc which is 4–7 times ρnm for modern EOS [Heiselberg & Pandharipande 2000]. The core is thought to consist of the following parts: (a) An outer core, covering the density range from ρnm to ∼ 2ρnm , is thought to be composed of uniform nuclear-matter consisting mostly of neutrons with protons and electrons in a small proportion determined by the requirement of β-stability, and with the appearance of muons above the threshold. (b) An inner core, covering the density range from ∼ 2ρnm to ρc , whose nature is still uncertain after more than thirty years of neutron-star research in the post-pulsar era. However, the following possibilities and suggestions are noteworthy: (i) Hyperons, e.g., Σ− , ∆− , Λ, . . . are very likely to appear at densities > ∼ 2–3 times ρnm , but the details are uncertain as hyperon-nucleon interactions are much more poorly known than nucleon-nucleon ones. One plausible conclusion is that the effects of hyperons on the EOS are likely to be minor, while those on the electron chemical potential µe and the proton fraction xp may be significant. (ii) Meson condensates of various sorts, i.e., π − , π 0 , K − , may occur. − A π -condensate, i.e., a physical appearance of negative pions, may occur at densities > ∼ 1.5ρnm , when the electron chemical potential exceeds their rest-mass energy ≈ 139 MeV. A π 0 -condensate, which is a really a long range nucleon-nucleon correlation (a spin-density wave for neutron matter) with π 0 quantum numbers, was found to occur at densities 1.5 – 2 times ρnm for the v14 potentials [Wiringa, Fiks & Fabrocini 1988]. K − -condensates may occur when the kaon energy, which decreases with increasing density due to the attractive kaon-nucleon interaction, drops below the electron chemical potential. But various estimates put the threshold density for this at > ∼ 5ρnm , which is only attained very close to the center of the heaviest neutron stars following recent EOS.
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(ii) Quark matter may appear at densities > ∼ 2 – 6 times ρnm , if it (or a mixed phase of it with nuclear matter) has a lower energy than nuclear matter at such densities. Such matter may occur1 in the following ways: • A small quark core at the center of the star, where densities are > , in which case the star may be called a hybrid star. The size of 6ρ nm ∼ the quark core would depend on the EOS of quark matter: for large bag constants, for example, this core would vanish. • Mixed phases of nuclear and quark matter may occur in the inner core, starting at densities > ∼ 2 – 3 ρnm , in which case the star may be called a mixed star. One such possibility is shown in Fig. 5.8, whose details are perhaps to be taken as a bit more speculative than the rest of the uncertain physics of the inner core. Small amounts of quark matter may first appear as droplets in a sea of nuclear matter (and perhaps also as rods or plates, analogous to the situation for nuclei in a sea of neutrons at the bottom of the inner crust, as indicated above). Then the relative amount of quark matter would increase as we go in further, and, if the density is high enough near the very center of the star, the situation may reverse itself (i.e., turn “inside out”, a term coined in 1971 by Baym, Bethe and Pethick, as explained below), with nuclear matter as droplets (and also perhaps as rods or plates) in a sea of quark matter. Of course, depending on the stellar mass and EOS, some or most of this series of configurations may not appear, if densities are not high enough. An important point to note is that, generally speaking, occurrences of all of the above “exotic” forms of matter in the inner core tend to soften the EOS, with consequences for neutron-star structure which we have described above. However, some effects, e.g., those of meson condensates, may operate only over a very limited density range, and not affect the overall stellar properties very much.
5.2.1
Crustal Properties: Mass and Radius
Although the solid crust of a neutron star occupies a small fraction of the total stellar volume and its mass is a small fraction of the total stellar mass, as indicated above, many observational properties of neutron stars depend crucially on those of the crust, which we shall summarize at appropriate 1 The possibility of strange stars, i.e., stars made of quarks, bound together not by gravitational attraction but by strong interactions, is considered separately in Chapter 14, since the basic physics of such stars is entirely different from that of neutron stars or other degenerate stars bound by self-gravitational attraction.
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places. Here we summarize a simple, approximate method [Pethick and Ravenhall 1995] of estimating the mass and radius of the crust, which is extremely useful. The method actually utilizes r the above fact that, over the entire crust, the mass co-ordinate m(r) ≡ 0 4πr2 ρ(r)dr is almost equal to the total stellar mass M = m(R), and that the radial co-ordinate r can also be replaced by the stellar radius R in the argument of most other functions of position (except those which change by a large relative amount from one boundary of the crust to the other, i.e., ρ and P , and, of course, r itself) in the TOV equation. Further, the ratio P (r)/ρ(r)c2 , which is an increasing function of ρ, is quite small (< ∼ 1%) compared to unity throughout the lowdensity crust, and the ratio [4πr3 P (r)/m(r)c2 ] ≈ [P (r)/ρ(r)c2 ][3ρ(r)/ρ] is even smaller. (Here, ρ is the average density of the neutron star.) In the TOV equation, both square brackets in the numerator can therefore be replaced by unity, and it simplifies into the approximate form GM dP (r) ≈ −ρ(r) , dr (1 − a)R2
(5.2)
in terms of the compactness parameter a ≡ (2GM/Rc2 ) introduced in Chapter 2, or the closed related general-relativistic surface redshift given by Eq. (5.13). Equation (5.2) can be readily integrated over the crust’s extent (outer boundary: r = R, P ≈ 0, inner boundary: r = R − ∆Rcrust , P = PB ) to yield the crustal mass: R 4πR4 (1 − a) PB . ∆Mcrust ≈ 4πR2 ρ(r)dr ≈ (5.3) GM R−∆Rcrust This shows that the crucial physical parameter at the phase boundary B between the solid crust and the liquid core which determines the crustal mass is the pressure PB there. Using the standard units, MeV fm −3 , for PB introduced in Sec. 5.1.1, we can give a very useful numerical estimate of the relative crustal mass: −2 M PB ∆Mcrust −2 4 ≈ 2.2 × 10 R6 , (5.4) M 1.4M MeV fm−3 which clearly shows its size and essential scaling with stellar parameters. Here, R6 is the stellar radius in standard units of 106 cm, as before. In Eq. (5.4), we have used the value of the compactness parameter, a ≈ 0.42, corresponding to the canonical values M ≈ 1.4M , R6 ≈ 1 of stellar mass and radius. Since “realistic” EOS give 0.25 < ∼ 0.65 in the above ∼ PB <
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units [Lattimer & Prakash 2001], the numerical factor in Eq. (5.4) ranges between 0.55 × 10−2 and 1.4 × 10−2 , exactly the range found from detailed numerical integrations of the TOV equation. An alternative form for the approximate crustal mass, even more useful for rapid estimates if the stellar radius is accurately known, is obtained by combining Eq. (5.4) with the Lattimer-Prakash correlation (5.1) between PB and R (assuming, of course, that the crust-core boundary B occurs at ρ ≈ ρnm ), which yields: ∆Mcrust ≈ 2.2 × 10−2 R68 M
M 1.4M
−2 .
(5.5)
Although potentially extremely useful, this estimate is, of course, more approximate, since we have raised the rather rough correlation (5.1), which is subject to considerable scatter, to the fourth power in order to obtain Eq. (5.5). The extreme sensitivity of Eq. (5.5) to R makes it important that we know the stellar radius accurately, in order to obtain reliable answers. Consider now the thickness of the crust, an estimate of which can be obtained in a way similar to that given above, although the scalings are not quite as simple, as we shall see. Equation (5.2) can be written with the aid of the zero-temperature Gibbs-Duhem relationship, dP = ndµ, as dr = −dµ[(1 − a)R2 /GM mn ], using the fact that ρ ≈ mn n in the lowdensity crust, mn being the neutron mass. This can be trivially integrated between any two points A and B in the crust to yield the crust thickness, ∆RAB , between these two points as: ∆RAB ≈2 R
1 −1 a
µB − µA m n c2
.
(5.6)
The numerical factor in Eq. (5.6) being 2(a−1 − 1) ≈ 2.8 (we have used, again, the value of a given above for canonical stellar parameters), the scale of the entire effect is set by the difference of chemical potentials µB − µA in units of the neutron’s rest-mass energy mn c2 . Thus, the total thickness of the crust is determined by µB − µS , where B is the crust-core boundary, as above, and S is the stellar surface, and that of the inner crust is determined by µB − mn c2 , since the boundary between the inner and outer crusts is the neutron-drip point (see Sec. (3.3)), where µ = mn c2 . The fact that ∆R/R ∼ 0.1 for neutron stars, as indicated above, then merely means that µB − µS ∼ mn c2 /30, which comes from detailed calculations.
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Non-Spherical Shapes: Rods and Plates
Why should shapes matter at all, either for nuclei or for drops of nuclear or quark matter? Consider nuclei first, and recall from the liquid-drop model described in Chapter 3 that terms like the bulk energy or symmetry energy cannot have anything to do with this, since they have no dependence on shape. It can, therefore, only involve the surface and Coulomb energy terms, which do depend on shape, and indeed it does. By assuming earlier that this shape was spherical, we never explored what shape actually minimizes the energy, but we shall do so now. As we shall see, spheres actually have the lowest energy over most of the density range of the crust (so that our earlier results were, in fact, correct at those densities), but at the highest densities found near the boundary between the solid crust and the liquid core, non-spherical shapes may have lower energy, depending on the EOS, and so become the preferred shape. This has been a major result of the 1980s and ’90s, although Baym, Bethe, and Pethick alreday pointed out in 1971 in their pioneering paper that the spherical-drop picture of nuclei was certainly not valid beyond the point where nuclei began to touch each other near the crust-core boundary. Explicit calculations were initiated for hot, dense matter in stellar collapse by Ravenhall, Pethick, and Wilson in 1983 and continued by others, while those for cold, dense neutron-star matter started in the 1990s [Lorenz, Ravenhall & Pethick 1993] and are continuing [Douchin & Haensel 2000]. The basic physics of non-spherical shapes is simple and instructive [Pethick and Ravenhall 1995; Haensel 2001]. For the picture of nuclei at the centers of Wigner-Seitz cells arranged on a lattice, such as we used in Chapter 3 to describe neutron-star crusts, all we need to do now is to allow for the fact that these nuclei and cells need not be spheres any more. What other shapes are basic? This is related to the basic dimensionality d of the system. For d = 3, spheres are the natural symmetrical shapes. But for d = 2, i.e., when physics is independent of one of the dimensions, circular cylinders (whose axes are parallel to this redundant dimension) are the natural shapes. Simlarly, for d = 1, plane slabs are the natural shapes. These non-spherical natural shapes can be called rods and plates, or, if we prefer more “culinary” names [Haensel 2001], spaghetti and lasagna respectively [Ravenhall, Pethick & Wilson 1983]. The basic work then is simply to so generalize our expressions for surface and Coulomb energies, given in Chapter 3 for spherical (d = 3) geometry, that they now describe spheres, rods and plates. The quantitative description still proceeds in terms of the
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length scales rN and rc for the nuclei and the Wigner-Seitz cells introduced in Chapter 3, it being now inderstood that they refer to radii of spheres for d = 3, to radii of cylinders (which nuclei and Wigner-Seitz cells are) for d = 2, and to half-spacings between plane slabs (which nuclei and WignerSeitz cells are) for d = 1. In terms of these lengths, the generalization of the filling factor (i.e., the fraction of all available space actually occupied by the nuclei) introduced for spherical (d = 3) nuclei in Chapter 3, is now u ≡ (rN /rc )d : this is a crucial parameter, as we shall see below. The general expressions for Esurf and ECoul of a nucleus in any of the above dimensionalities have the following, characteristic dependence on rN : −1 Esurf = Csurf rN ,
2 ECoul = CCoul rN ,
where the coefficients C are given by σ Csurf ≡ ud, CCoul ≡ (2πnN x2p e2 )ufd (u), nN
(5.7)
(5.8)
Here, nN is the number density of nuclei, xp is the proton fraction, and σ is the surface tension, as before, and fd is a dimensionless function of the filling factor u which has the essence of the geometrical information contained in the dimensionality of the system, as far as Coulomb effects are concerned. We give the general form of fd [Ravenhall, Pethick & Wilson 1983] for reference: 1 2 2 u d fd (u) ≡ 1 − u1− d + . (5.9) d+2 d−2 2 2 Actually, it is physically more illuminating to consider the specific cases2 separately:
2 , 5 1 rods : f2 ≡ 14 [− ln u − 1 + u] − ln(eu) , 4
−1 1 1 , plates : f1 ≡ 3 u − 2 + u 3u
spheres : f3 ≡
2 5
1 − 32 u1/3 +
u 2
(5.10) (5.11) (5.12)
where the approximate expressions are valid in the low-density limit u 1. Equation (5.10) recovers the well-known results for spherical geometry, e.g., Eq. (3.31), given in Sec. 3.2.1, as it must. But note how enormously different the low-density behaviors of the three geometries are. 2 For details, we refer the reader to the original paper by Ravenhall, Pethick, and Wilson (1983).
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Minimization of (Esurf + ECoul ) with respect to rN readily leads to the classic condition given by Eq. (3.19), showing that this condition is independent of the dimensionality of the system, and reflects only the fact that space is really three-dimensional [Pethick and Ravenhall 1995]. This is why the equilibrium size of nuclei is given by rN = (Csurf /2CCoul )1/3 independent of whether they are spheres, rods, or plates. But the coefficients C are, of course, shape-dependent, as indicated above, and so yield different nuclear size and total energy for different shapes. Consider the low-density limit first, where fd and therefore CCoul is progressively higher in going from spheres to rods to plates, which produces progressively smaller nuclei, i.e., a finer and finer subdivision, which leads to unfavorable surface-to-volume ratios, and so to higher energies. For example, at the neutron-drip density ρdrip , u ∼ 10−3 as shown in Chapter 3, so that rN ∼ 5, 3, and 0.5 fm for spheres, rods, and plates respectively, and the latter two shapes have energies higher by ∼ 0.6 and 12 MeV per nucleon respectively compared to spheres [Pethick and Ravenhall 1995]. Naturally, spheres are the preferred shapes at these (relatively) low densities. By contrast, the situation may reverse itself at the highest densities found in neutron-star crusts, as the detailed computations of Lorenz, Ravenhall, and Pethick (1983) show in Fig. 5.9. For a relatively small range of densities near the inner edge of the crust, corresponding to number-densities 0.066 fm−3 < ∼ n < ∼ 0.096 −3 fm , non-spherical shapes actually have lower energies than spheres if the EOS is that given by the Friedman-Pandharipande-Skyrme (FPS for short) interactions, and so would be expected to occur in neutron-star crusts. 5.3.1
Turning Nuclei “Inside Out”
As we mentioned above, it was pointed out by Baym, Bethe, and Pethick in 1971 that, when nuclei occupy more than half of all available space, i.e., u > 1/2, it is energetically favorable to turn them “inside out”, producing a situation in which bubbles (in spherical or non-spherical shape) of neutron gas would be surrounded by a sea of nuclear matter, reversing the earlier situation where nuclei (in spherical or non-spherical shape) were surrounded by a sea of neutrons. The shapes in the inverted situation may be called bubbles, tubes, and plates, the last situation being its own inverse, as a little reflection will show. If we prefer culinary terms, as above, we can perhaps use names like swiss cheese, anti-spaghetti, and lasagna, since lasagna is its own inverse. Indeed, this was also found in the calculations of Lorenz et al. at sufficiently high densities, just before the crust dissolved
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Fig. 5.9 Energy per unit volume of various non-spherical phases as a function of baryon density, measured with reference to a two-phase state with no Coulomb and surface contributions. Nuclear interactions considered are (a) SKM: the Skyrme interaction (see Appendix D) used by, e.g., Bonche & Vautherin (1981), and (b) FPS: the generalized Skyrme-type interaction of Friedman & Pandharipande (1981): see text. Reprinted with c 1993 permission by the APS from Lorentz et al. (1993), Phys. Rev. Lett., 70, 379. American Physical Society.
into the uniform liquid of the core. The sequence of phase transitions with increasing density found in these calculations was spheres → rods → plates → tubes → bubbles, as may have been expected. Formally, Eqs. (5.7) (5.12) still hold for the inside-out phase, with u replaced by 1 − u, and with the understanding that rN now stands for the size of the (spherical or non-spherical) bubbles. Note that this entire sequence of first-order phase transitions of shape occurs with very small energy differences between the various phases, because the finite-size (surface and Coulomb) terms are small compared to the bulk terms in this density range. 5.3.2
Physical Insights: Frustrated Fission
Formal energy considerations apart, we can ask what physical reason would lead to non-spherical shapes for nuclei at sufficiently high densities. The answer is precisely the question we asked ourselves at the end of Sec. 3.4.3,
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namely, what happens when the Bohr-Wheeler criterion (3.20) is satisfied at sufficiently high densities, because the filling factor u becomes large enough to satisfy the condition u > 1/8 ? Consider spherical nuclei first. The quadrupolar deformation leads to elongated, cigar-shaped nuclei, which would fission into two pieces for an isolated nucleus: this is the classic BohrWheeler picture. But in a close lattice of nuclei in very dense matter, we can argue plausibly that, before such fission can occur, neighboring nuclei elongating along the same lattice direction would join up to form long, rod-like structures, exactly as envisioned above for rod-like nuclei [Pethick and Ravenhall 1995]. Similar pictorial arguments about rods coalescing into plate-like structures at higher densities are not hard to devise, and are left as an exercise for the reader. We coin the term frustrated fission to describe this scenario for producing rods and plates from highly deformed nuclei which are unable to fission because of the extreme close proximity of other nuclei at high densities. It would seem that this frustrated fission is a general feature which should occur whenever matter is so dense that the nuclei occupy more than ∼ 1/8 of its total volume, and yet not so dense that nuclei dissolve into a continuous fluid. Therefore, it should apply equally well to neutronstar matter, matter in stellar collapse, and any laboratory situations, e.g., heavy-ion reactions, where we are able to explore this density range, if only for a short time. Indeed, the explicit neutron-star calculations advocate the required density range to be exactly that given above, the upper end of the range being that of the crust-core boundary. Estimates of the latter density can be made from two approaches: either by considering the point where nuclei dissolve into a uniform nuclear-matter fluid as density is increased (as we have argued repeatedly in this book so far), or by exploring where uniform nuclear matter becomes unstable to clumping of protons as the density is decreased [Pethick, Ravenhall & Lorenz 1995]. The relevant range of densities is rather small in all cases, its extent depending on the EOS of matter in this density r´egime. 5.3.3
Uncertainties
The real uncertainty about the occurrence of non-spherical nuclei in neutron stars is not so much that the relevant density range is small, but rather that it is non-existent for some EOS. Whereas the original Lorentz et al. (1993) calculations with the FPS interaction did show the existence of such a range, those with the SKM interaction found that spherical nuclei were favored all
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the way to the inner edge of the crust. Both of these were Skyrme interactions, obtained by fitting a Skyrme-like energy-density functional to the detailed results of a numerical EOS calculation, which was that of Friedman and Pandharipande (1981) for the FPS interaction, and that of Siemens and Pandharipande (1971) for the SKM interaction. Oyamatsu applied the Extended Thomas-Fermi method, which is a semi-classical, approximate version of the full, quantum-mechanical many-body problem discussed in earlier chapters, to this problem, again using approximate energy-density functionals obtained from the above Friedman-Pandharipande calculations, and found non-spherical nuclei. On the other hand, the recent calculations of Douchin and Haensel (2000), in which they used the SLy (Skyrme Lyon) interaction obtained by fitting to (a) observed properties of neutron-rich nuclei for ρ < ρnm , and, (b) nuclear matter calculations of Wiringa, Fiks, and Fabrocini (1988), as described in Chapter 4, for ρ > ρnm , found spherical nuclei to have the lowest energy throughout the crust. The basic feature which decides the occurrence or otherwise of nonspherical nuclei is simple, involving the interplay between the density at which they first appear, where u ≈ 1/8, and that at which the solid crust dissolves into the liquid core. If the former density is less than the latter one, rod- and plate-like nuclei are possible; if not, spherical nuclei will continue to the inner edge of the crust. Which situation obtains for a given EOS depends on its details, particularly on the relative strength of the finite-size terms (surface and Coulomb energies) discussed earlier. If these are relatively small compared to the bulk terms, the crust dissolves at relatively low densities into the core, and the required threshold value of u ≈ 1/8 may never be reached in the crust, in which case non-spherical nuclei will not appear. On the other hand, if they are sufficiently large, the crust-core boundary occurs at a relatively high density, so that the above threshold is reached at some radius in the crust, and non-spherical nuclei will appear between this radius and that of the crust-core boundary. It seems fair to say that the existence of rod- and plate-like nuclei in neutron star crusts remains a fascinating possibility, if not a certainty, at this time. If these do occur, a major effect would be on the transport properties of the crust, which are very different for rod- and plate-like geometries from what they are for lattices of usual, spherical nuclei.
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The Maximum Mass of Neutron Stars
We have seen by now that all EOS for dense nuclear matter lead to a maximum mass, Mmax , of neutron stars, above which such stars are unstable. The essential physical basis for this maximum was clarified in Chapter 2, and the value of Mmax obtained from the pioneering numerical work of Oppenheimer and Volkoff (1939), as well as those obtained from simple but instructive analytic “toy” models of neutron stars, were given there. In general, it must be clear that the value of Mmax so obtained depend on the EOS. A question of basic physical significance, which we take up now, is the following: can we put upper bounds on Mmax on general grounds that do not depend on the details of EOS, which are still rather uncertain at very high densities? The answer also has great practical significance, since it is crucial for distinguishing between neutron stars and black holes in accreting X-ray binaries, and for determining the outcome of such astrophysical processes as (a) gravitational collapse of stellar cores following supernovae, and (b) mergers of binary neutron stars.
5.4.1
The Maximum Compactness
Closely connected to the above question is that of the maximum value of the compactness parameter, a ≡ (2GM/Rc2 ), which we introduced in Chapter 2, and which is related to the general-relativistic redshift, zs ≡ ∆λ/λ, of the radiation emitted from the surface of the neutron star by zs = (1 − a)−1/2 − 1.
(5.13)
Is there an upper bound amax to the compactness? The reader will recognize the immediate connection of this to the mass-radius relation of neutron stars shown earlier. For stable neutron stars, R decreases with increasing M until Mmax is reached, at which point stars become unstable, and do not exist. Thus amax for any EOS is, in fact, that obtained for the maximum mass for that EOS, and we shall give numerical illustrations below. The key question is: can we set an upper bound to the compactness of stars independent of the details of EOS, i.e., from the structure of the Tolman-Oppenhemier-Volkoff (TOV) equation (see Chapter 2), or, at one remove, from that of the Einstein equations, from which the TOV equation has been derived? In 1916, Schwarzschild discovered such an absolute upper bound for uniform-density stars,
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a < aabs max ≡
8 , 9
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(5.14)
which was later (see Weinberg 1972) shown to hold for any star satisfying some very general, plausible conditions described below, simply as a consequence of the general theory of relativity. We shall call this the Schwarzschild limit. This assumes, of course, that the general theory of relativity is the correct theory of gravity, which we do throughout this book. To see the origins of this bound, it is helpful to first reconsider our “toy” neutron star of Sec. 2.5.2 — a sphere of uniform density which is an exact solution of TOV or Einstein equations. This solution was first given by Schwarzschild, of course, in his pioneering and classic 1916 application of Einstein’s theory of general relativity to the gravitational field of an incompressible fluid sphere. A look at Eq. (2.70) shows that the pressure P would become infinitely large at a radius r∞ given by: 27c2 8 2 ≡ r∞ a− , (5.15) 8πGρ 9 and the metric would also become singular at this radius. This is impossible 2 must be negative for in any physically viable model for a star, so that r∞ all admissible models. This leads readily to Eq. (5.14). In Schwarzschild’s (1916) prophetic words, there is a limit to the concentration, above which a sphere of incompressible fluid cannot exist. Using this result, Weinberg (1972) showed that this upper bound on a holds not only for uniform-density stars, as Schwarzschild had already shown in 1916, but any star which satisfies the two conditions that (1) its density must not increase outward anywhere, i.e., ∂ρ/∂r ≤ 0 for all r ≤ R, and, (2) that the metric coefficient grr must not be singular anywhere inside r the star, which implies that the mass-energy co-ordinate, m(r) ≡ 0 4πr2 ρ(r)dr, defined in Sec. 2.5.1, satisfies the condition that m(r) < (rc2 /2G) for all r ≤ R. These conditions are very general and plausible, as it is difficult to imagine a stable stellar model which could violate these. Keeping the mass M and the radius R of the star fixed, Weinberg studied the behavior of the metric coefficient gtt , i.e., the “timetime” part of the metric inside the star (see Sec. 2.5.1). Actually, the √ quantity studied by Weinberg was ζ(r) ≡ −gtt , which is closely related to the general-relativistic analogue of the Newtonian gravitational potential in the hydrostatic balance equation, as explained in Sec. 2.5.1. He first argued that ζ(r) must be positive -definite everywhere in the star if we are
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to avoid the occurrence of a zero in ζ(r) and so a singularity in P (r), and then derived an upper bound on ζ(0), which led to Eq. (5.14) again, showing that it holds for all stars satisfying the above conditions. For details of the derivation, we refer the reader to Weinberg’s (1972) excellent book Gravitation and Cosmology. How do the compactnesses of model neutron stars calculated from various EOS compare with the Schwarzschild limit given above? For this, it is convenient to write a as: a≡
(M/M ) 2GM ≈ 0.3 , Rc2 R6
(5.16)
where R6 is the stellar radius in units of 106 cm = 10 km, as before. Consider first the values of amax given by some specific EOS, the maximum value for each EOS being that corresponding to the maximum mass possible for that EOS, as explained above. For the original Oppenheimer-Volkoff (OV) EOS (free neutrons), Mmax /M ≈ 0.7, and the corresponding radius is 9.6 km (see Chapter 2), so that aOV max ≈ 0.22, i.e., only about a quarter of the Schwarzschild limit. Modern EOS, which are moderately stiff, give much higher values of Mmax and amax . For example, the recent A18 + δv + UIX∗ potential gives Mmax /M ≈ 2.2 and the corresponding radius as 10.1 km [Heiselberg & Pandharipande 2000], so that aA18 max ≈ 0.65. Alternatively, we can consider the compactness of canonical 1.4M neutron stars: for such a mass, the above A18 potential gives a radius of 11.5 km, so that a ≈ 0.36, and an example of the v14 potentials of the 1980s, namely, the UV14 + TNI potential [Wiringa, Fiks & Fabrocini 1988], gives a radius of 10.86 km, so that a ≈ 0.38. Consider, finally, the general-relativistic redshifts on the surfaces of the above model neutron stars, which are now proving to be excellent diagnostic tools in combination with observed X-ray spectra of neutron stars, as we shall describe later. With the aid of Eqs. (5.13) and (5.14), we see that the Schwarzschild limit for the surface redshift of any star is zsmax (abs) = 2, which is a remarkable result by itself. Compare this with the maximum value for the Oppenheimer-Volkoff model, zsmax (OV) ≈ 0.13, and with that for the above A18 model, zsmax (A18) ≈ 0.69. For the typical 1.4M neutron star, the above A18 model gives zs ≈ 0.25, and the above UV14 model gives zs ≈ 0.27. Clearly, gravitational redshifts that can reasonably be expected in the (X-ray) spectra of neutron stars are far below the Schwarzschild limit.
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The Maximum Mass
The question, then, is: can we put an upper bound on Mmax in the above way, independent of the details of EOS and entirely from the structure of Einstein’s equations, as we just did for the compactness? The answer is, unfortunately, no. To see how a straightforward attempt would fail, it is instructive to consider our uniform-density “toy” neutron star again. Expressing the radius of the maximum-mass star as R = (3Mmax /4πρ)1/3 , we can rewrite the Schwarzschild limit, Eq. (5.14), as a mass limit: Mmax ≈
11.3M , √ ρ14
(5.17)
where ρ14 is the (uniform) density in units of 1014 g cm−3 , as before. But this limit is density-dependent, and tells us little until we know ρ. We can say that we expect the density to be ∼ ρnm ≈ 2.8 × 1014 g cm−3 , the nuclear-matter density, but that is not precise enough. If we take the density to be ρnm , we get Mmax ≈ 6.7M , if we take it to be 2ρnm , we get Mmax ≈ 4.8M , and if we take the density as 4ρnm, we get a limit of Mmax ≈ 3.4M . Which one is relevant? In an influential paper published in 1974, Rhoades and Ruffini introduced a way of obtaining more general bounds on Mmax , showing how we can circumvent this arbitrariness of density to some extent. The key idea was to use the fact that the EOS is quite reliably known upto a maximum density ρ0 , and use the most general principles of physics possible for densities above ρ0 in order to obtain bounds on Mmax . The general principles proposed by Rhoades and Ruffini, and still in wide use, are the following two conditions on the speed of sound in matter at ρ > ρ0 , i.e., the uncertain density r´egime: 1. The causality condition that the speed of sound must not be “superluminal”, i.e., exceed the speed of light3 dP ≤ c2 . dρ
3 This
(5.18)
condition is physically plausible in view of the special theory of relativity, but there are subtle points here which have not been settled yet. Sound velocity depends on frequency, dP/dρ obtained from the EOS of a star at equilibrium is essentially the zero-frequency limit of this, and there is apparently no general proof at this time that the ground state of matter must obey dP/dρ ≤ c2 . See Hartle (1978).
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2. The local stability condition, also known as Le Chatelier’s principle, that the sound speed dP/dρ must be real everywhere, or, equivalently, that P must be a monotonically non-decreasing function of ρ: dP ≥ 0. dρ
(5.19)
Violation of this condition would lead to spontaneous collapse of matter locally. Subject to the two constraints above, Rhoades and Ruffini (1974) calculated the mass of the neutron star by integrating the TOV equation with the EOS of Harrison, Wheeler and co-authors (see Chapter 3) upto the fiducial density ρ0 , and maximised the mass with respect to the unknown EOS in the ρ > ρ0 r´egime, using control-theory techniques, with a suitable function of density as the control variable. Their result was that the maximum mass is obtained for that EOS which maximizes, at each density, the velocity of sound, which is, of course, c by the first constraint above. Thus, the upper bound on M corresponds to a very simple EOS for ρ > ρ0 , i.e., P = P0 + (ρ − ρ0 )c2 , where P0 ≡ P (ρ0 ). This bound has a rough scaling with the fiducial density ρ0 at relatively low densities which is the same as that given by Eq. (5.17), but with a different constant4 : 6.7M Mmax ≈ 0 , ρ14
(5.20)
where ρ014 is ρ0 in units of 1014 g cm−3 . It is easy to understand this result. For relatively low values of the fiducial density ρ0 , the problem is rather similar to the “toy” problem discussed above, except that the causality condition now limits the sound velocity to be subluminal (≤ c), which means that matter cannot be arbitrarily “stiff” (i.e., have arbitrarily large values of dP/dρ: see Chapters 3 and 4). A less stiff EOS gives a lower Mmax , and this is what happens here: the constant is reduced by a factor ∼ 2, but the scaling remains roughly intact. Unfortunately, the arbitrariness of the fiducial density ρ0 still remains in Eq. (5.20). Rhoades and Ruffini took ρ0 ≈ 4.6 × 1014 g cm−3 ≈ 1.6ρnm, for reasons which are only of historical relevance5 today, instead of being phys4 Actually, we have used here the value of the constant found in the recent calculations of Kalogera and Baym (1996), which we describe later in some detail. The value found in the original Rhoades-Ruffini (1974) work was very slightly different. 5 These authors used Ruffini’s earlier result, namely, that, below ρ ≈ 1.6ρ nm , a free neutron-gas EOS gave a higher sound velocity than any EOS including nuclear interactions, to argue that this should be the fiducial density.
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ically compelling. Accordingly, the Rhoades-Ruffini limit Mmax ≈ 3.2M , obtained by using the above value of the fiducial density in Eq. (5.20), would appear to be rather uncertain, although it had been widely used in the past. The indication that the actual situation is not as bad as above came from further work in the 1970s, expertly summarized by Hartle (1978). The key point is that, as we increase the fiducial density ρ0 , the above simple scaling is no longer valid, and Mmax eventually becomes independent of ρ0 at high enough densities (for ρ0 > ∼ 4ρnm , say), showing that there is, indeed, a lowest upper bound on the mass of a neutron star from general physical principles. We illustrate this in Fig. 5.10, which shows the results of the recent calculations of Kalogera and Baym (1996) for the v14 EOS of Wiringa, Fiks, and Fabrocini (1988). The physics of this constancy of Mmax at very high fiducial densities is transparent: at such high values of ρ0 , the mass of the uncertain r´egime described above becomes small compared to that of known r´egime calculated from the given EOS, and the latter then determines the upper bound on Mmax . The practical difficulty, however, is in calculating this constant value of Mmax . Every known EOS becomes uncertain, or shows undesirable features,
Fig. 5.10 Maximum mass Mmax of a neutron star as a function of the fiducial density ρ0 (see text). The EOS used is two variations, (AV14 + UVII) and (UV14 + UVII), of the v14 type given by Wiringa et al. (1988). Reproduced with permission by the AAS from Kalogera & Baym (1996): see Bibliography.
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above an appropriate maximum density: if it did not, we would not need this general upper bound. As described in the previous chapter, EOS for superdense matter have become much more relaible today than they were in the 1970s. This comes to our aid, since we can now use a modern EOS to calculate Mmax as a function of ρ0 , and see how close we can get to the asymptotic value of Mmax by evaluating Mmax at the highest value of ρ0 at which the EOS is considered reliable. As we indicated in the previous chapter, some of the standard, modern EOS become superluminal, i.e., violate the above causality condition, above a certain density, and so cannot be considered reliable beyond this point, which then serves as an upper limit to the density upto which a given EOS should be considered reliable. For the Wiringa-Fiks-Fabrocini EOS, this point occurs at ρ > ∼ 5ρnm . As is clear from Fig. 5.10, the asymptotic value of Mmax ≈ 2.2M is reached at just about this point, the value at ρ0 = 2ρnm being ≈ 2.9M , and that at ρ0 = 4ρnm being very close to the above asymptotic value. Kalogera and Baym (1996) advocated the value at ρ0 = 2ρnm , i.e., Mmax ≈ 2.9M as their safest estimate. It appears, therefore, that (a) the Rhoades-Ruffini limit of Mmax ≈ 3.2M was reasonable, for reasons which are becoming clear only now, and, (b) for an accurate estimate of the least upper bound on the mass of a neutron star, we need only understand the properties of neutron-star matter thoroughly and reliably upto a density ∼ 4ρnm , which appears feasible in the foreseeable future.
5.5
Rotating Neutron Stars
Pulsars are rotating neutron stars, as we stressed at the beginning of this book, so that it is essential to explore how the basic properties of neutron stars, e.g., mass, radius, and shape, are modified by rotation. This is done by constructing equilibrium models of rotating neutron stars according to various EOS, and studying (a) how the mass of a star consisting of a given number of baryons changes as it rotates, and therefore how the maximum mass of a neutron star changes with rotation, (b) how the stellar shape changes from a sphere to an oblate form, and (c) the stability of such models to gravitational and rotational perturbations, and therefore the maximum possible rotation frequency for stable neutron stars. As we have already seen, construction of neutron-star models must be done in the formalism of general relativity, and so involves numerical computation even for nonrotating stars. For rotating stars, the computations necessarily become
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more intensive, since, in general relativity, stellar rotation changes the very structure of the spacetime around the neutron star, as opposed to what happens in Newtonian gravity. The metric becomes more complicated, its spherical symmetry for nonrotating stars being now lowered to only an axial symmetry about the axis of stellar rotation. While the former metric was diagonal, the latter acquires an essential off-diagonal component gtφ . The crucial piece of physics contained in this modified metric is that an observer in an inertial frame near the rotating star is found rotating around the stellar center, relative to the distant stars: this rotation vanishes at infinity, and becomes more rapid as the approach to the star becomes closer, as may be expected. This “dragging” of the local inertial frames by stellar rotation is called frame dragging or the Lense-Thirring effect . It changes the internal structure of the star, and can have remarkable influence on the motion of accreting matter near the neutron star, as we shall discuss in later chapters. The effect is contained, of course, in the above off-diagonal term, which can be written to the first order in the stellar angular velocity Ω as: gtφ = gφt ∝ gtt ω(r, θ).
(5.21)
Here, ω is the frame-dragging frequency, i.e., the angular velocity of rotation of local inertial frames, which was introduced above. ω depends only on r to this order, and scales as ω(r) ∝ Ω/r3 ; we shall give more complete expressions below. It is this frame dragging that complicates matters: for example, the centrifugal force on a fluid element inside the star now depends not on the stellar angular velocity Ω, which is constant for a uniformly rotating star, but on the difference ω ¯ ≡ Ω − ω(r, θ), which is not. Computing this relative angular velocity ω ¯ throughout the star, which is an essential component of the formalism, takes considerable effort. It will come as no surprise, then, that full calculations of stellar properties for arbitrary values of Ω, which involve numerical solutions of Einstein’s equations for stationary, axisymmetric gravitational field, coupled to the equation of hydrostatic equilibrium, require large computational resources, even for uniformly rotating stars. Such calculations are feasible in this era of supercomputers, and we shall describe some of their results later. But it is instructive at this stage to consider the limit of slowly rotating stars, in which case a considerable range of simplifications come into play, and make the computations much easier. Historically, this is how the subject started.
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The Hartle-Thorne Approximation
Hartle and Thorne pioneered the formulation of the general-relativistic description of slowly rotating stars [Hartle 1967; Hartle & Thorne 1968]. These authors essentially expanded the metric in a perturbation series of deviations from the spherically symmetric metric of a nonrotating star. The deviations consist of monopole and quadrupole terms, each term being a function of r and Ω. Use of this expansion on Einstein’s equation then leads to Hartle’s equation for the above relative angular velocity ω ¯ as a function of position inside the star: ω 4 dj 1 d 4 d¯ ω ¯ = 0. (5.22) r j + r4 dr dr r dr 0 inside the star and j(r) ≡ 1 outside Here, j(r) ≡ [1 − (2Gm(r)/rc2 )]/gtt 0 it, gtt being the “time-time” part of the unperturbed metric inside the r non-rotating star. The mass-energy co-ordinate m(r) ≡ 0 4πr2 ρ(r)dr was introduced in Chapter 2. It is easy to integrate the Hartle equation, which is linear, subject to the boundary conditions that (a) ω ¯ is well-behaved as r → 0, and (b) ω ¯ → Ω as r → ∞, and so obtain the relative angular velocity everywhere. The total angular momentum of the star is then given by: ω c2 4 d¯ R , (5.23) J= 6G dr r=R where R is the radius of the non-rotating star. The moment of inertia of the star is given by I = J/Ω, as in the non-relativistic case, or by the equivalent expression: ω ¯ ρ(r) + P (r)/c2 8π R . (5.24) dr r4 I= 0 [1 − (2Gm(r)/rc2 )] Ω 3 0 gtt The expression for the frame-dragging frequency ω is particularly simple outside the star, since the Hartle equation reduces there to the simple form (d/dr)(r4 d¯ ω/dr) = 0. The reader can readily solve this equation, subject to appropriate boundary conditions, and show that ω(r) = 2J/r3 = (2I/r3 )Ω in this region. Of prime interest is the fact that the mass of the rotating star is greater than that of the corresponding non-rotating star, even in the Hartle-Thorne approximation. We can consider it in two ways. First think of a star consisting of a given total number of baryons N . If we actually start with
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such a non-rotating star, and make it rotate, N does not change, and the mass M (N, Ω) of a star rotating with an angular velocity Ω is related to that of the corresponding non-rotating star M (N, 0) by: 1 M (N, Ω) = M (N, 0) + IΩ2 . 2
(5.25)
Alternatively, think of a star with a given central density ρc . If we actually start with such a non-rotating star, and make it rotate, ρc does change due to rotation: it is lowered. But we can still formally compare the masses of a rotating and a non-rotating star of the same central density ρc , and obtain the relation: 2 Ω , (5.26) M (ρc , Ω) = M (ρc , 0) + δM ΩK which shows that the leading correction is again ∼ Ω2 . Here, ΩK is the Keplerian angular velocity, GM (ρc , 0) , (5.27) ΩK ≡ R3 (ρc , 0) given by the Newtonian expression at the equator of the non-rotating star. This a rough measure of the limiting rotation rate at which the star begins to shed mass at its equator. We shall return below to a general-relativistic description of the Keplerian angular velocity, and so of this mass-shed limit. The mass-change δM has been calculated by Hartle and Thorne. Finally, the small deformation of the slowly-rotating star from spherical symmetry can also be calculated in the Hartle-Thorne approximation, and expressed either (a) as an eccentricity parameter , so defined that the actual eccentricity e of the stellar surface, which is an oblate spheroid (upto order ∼ Ω2 ) in the Hartle-Thorne approximation, is given by e ≡ (Ω/ΩK ), or, (b) as a mass quadrupole moment parameter q, so defined that the actual quadrupole moment Q is given by (Q/M R2 ) ≡ q(Ω/ΩK )2 . Expressions for and q are given by Thorne. In their pioneering work of 1971, Baym, Pethick and Sutherland (BPS; see Chapter 3) calculated and tabulated the quantities I, δM/M , , and q versus the central density ρc for neutron-star models calculated according to their EOS. Typical values found by BPS were δM/M ∼ 10−3 − 10−1 , ≈ 1, and q ∼ 10−3 − 10−2 . Note that these are the sizes of the coefficients of the appropriate powers of (Ω/ΩK ) (see above); the actual departures from the non-rotating values are much smaller.
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Arbitrary Rotation
The Hartle-Thorne approximation for slowly rotating stars, which leads to expansions in powers of the variable (Ω/ΩK ), must become invalid for fast rotation, i.e., above some critical value of this variable. The crucial question is: what is “fast”, or, equivalently, what is the critical value of (Ω/ΩK )? When the discovery of millisecond pulsars began in 1982, it became clear that neutron stars rotating at such high frequencies could not be operating in the r´egime of small values of (Ω/ΩK ), and the applicability of the HartleThorne approximation to these fast-rotating pulsars naturally came into question. What happens if we apply the Hartle-Thorne approximation to fast rotators anyway? It would appear that we should get results which are qualitatively correct, such as an increase in mass δM/M , but that the numerical values would be quantitatively incorrect compared to those given by an exact calculation performed in the manner indicated above. This is what was found when numerical calculations started in the mid1980s. Accordingly, such numerical calculations have become the standard procedure in the subject since then, and we shall now summarize the results of some of these calculations. It was only in the early 1990s that further clarification was obtained on what was “fast” rotation vis-` a-vis HartleThorne approximation, and ways were suggested for obtaining results of reasonable accuracy from this approximation even for neutron stars rotating at millisecond periods. We shall return to this point later. In 1986, Friedman, Ipser and Parker (henceforth FIP) published numerical neutron-star models with arbitrary but uniform rotation (subject to maximum rotational velocities implied by constraints described below) for a comprehensive collection of EOS available at the time, in response to the need existing then for rotating stellar models capable of describing the recently-discovered millisecond pulsars. They displayed the sequence of maximally-rotating neutron stars (see below) versus that of non-rotating neutron stars, showing that the mass of the rotating star was always higher than that of the non-rotating star of the same central density by an appropriate fraction δM/M . The maximum mass of the former sequence was higher than that of the latter sequence by an amount δMmax /Mmax which depended on EOS and was in the range ≈ 13 − 20%, stiffer EOS generally leading to higher values of fractional mass increase. These numbers were generally larger by a factor ≈ 2 than those obtained by a straightforward application of the Hartle-Thorne approximation to the maximally rotating stars.
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In 1994, Cook, Shapiro and Teukolsky (henceforth CST) published an updated set of numerical calculations for rotating neutron stars which incorporated (a) advances in numerical algorithms for treating fast-rotating, very distorted configurations accurately, and (b) treatment of the modern EOS which are of much interest to us today, but which were not proposed yet at the time of the FIP work. We display selected CST results now. In Fig. 5.11, we show mass-radius plot (i.e., analogs of earlier figures for nonrotating neutron stars), for rotating and non-rotating sequences together. CST found the mass increase due to rotation, δMmax /Mmax , to vary from ≈ 14% for soft EOS to ≈ 21% to stiff EOS. In the mass-radius plots, the “radius” for rotating, non-spherical models means the equatorial radius Re , which is relevant for the mass-shed limit introduced above. Although the shape of the stellar surface, as calculated by FIP, is not a spheroid for fast-rotating stars, we can define an effective
Fig. 5.11 Constant rest-mass sequences showing total gravitational mass M vs. equatorial radius Re of neutron stars with FPS EOS (see above). Selected sequences are labeled by the value of the rest mass M0 , the sequence corresponding to M = 1.4M being marked by an asterisk. The mass-shed limit (see text) is the dashed line. The static limit is the solid line roughly parallel to the mass-shed limit. Reproduced with permission by the AAS from Cook et al. (1994): see Bibliography.
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eccentricity e by the relation e ≡ 1 − Rp2 /Re2 , in terms of Re and the polar radius Rp . For slowly-rotating stars in the Hartle-Thorne approximation, the surface is actually a spheroid (see above), and e is its true eccentricity. For maximally-rotating stars, FIP and CST tabulated e for various EOS considered by them: the FIP values are in the range e ≈ 0.6 − 0.7, and the CST values in the range e ≈ 0.78 − 0.82, showing that the stars are quite distorted, as expected. Equilibrium sequences of neutron-star models in general relativity for a given EOS can be labeled and classified according to the following scheme introduced by CST, and shown in Fig. 5.12. Along each sequence, the rest mass of the star is kept constant, while its angular velocity or momentum is varied: this corresponds to studying the quasi-stationary evolution of a star of given baryonic rest mass as its spin-rate evolves, due to energy
Fig. 5.12 Constant rest-mass sequences showing angular velocity Ω vs. angular momentum J of neutron stars with FPS EOS. Labels and lines as in Fig. 5.11, with the addition that stable parts of the sequences are denoted by solid lines, and unstable parts by dotted lines. The inset shows an expanded view of the region around the maximum-mass model (open circle). Reproduced with permission by the AAS from Cook et al. (1994): see Bibliography.
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and angular momentum loss through, e.g., electromagnetic or gravitational radiation. A sequence is labeled by the value of its rest mass. A normal sequence is one which has a spherical, non-rotating model as a limiting member at one end. All normal sequences that connect to a stable, nonrotating model are themselves stable to radial perturbations. By contrast, all members of a supramassive sequence have rest masses exceeding that which corresponds to the maximum possible mass of spherical, non-rotating models. Thus, supramassive stars exist only by virtue of their rotation. Most, but not all, supramassive sequences contain a section that is unstable to radial perturbations.
5.5.3
Maximum and Minimum Rotation
These equilibrium sequences are bounded by several rotational limits of basic importance, which we discuss now. Consider first the lower limit to rotation. The non-rotating or static limit, Ω → 0, J → 0, which we have decsribed in detail in the earlier parts of the book, would seem to be the obvious choice for this. But this is so only for normal sequences. A supramassive sequence, the members of which must necessarily rotate in order to exist, has a non-zero lower limit to rotation, below which its self-gravitation cannot be supported any more, and it collapses, entirely analogous to what happens to a non-rotating star above the maximum mass, as discussed earlier. We shall call this the supramassive-collapse limit. Now consider the upper limit to roation, for which there are two candidates. The first is the mass-shed limit introduced above, which occurs when the angular velocity at the equatorial surface of the star equals the Keplerian angular velocity Ω = ΩK , at which point the star begins to shed mass from its equator. The second is the gravitation-wave limit, which occurs because non-axisymmetric perturbations of the star, which lead to radiation of gravitational waves, can become unstable at sufficiently high rotation rates. We discuss the two upper limits first, then the supramassive-collapse limit, and finally return to the question of the maximum possible rotation frequency of neutron stars, which has been of long-standing interest. 5.5.3.1
The mass-shed limit
While the physics of this limit is elementary, namely, that self-gravitational attraction is unable to keep matter bound to the equator of a sufficiently fast-rotating neutron star, the equation giving the Keplerian angular veloc-
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ity ΩK is much more complicated than its familiar Newtonian limit given by Eq. (5.27). In terms of the standard form of the metric, the Keplerian angular velocity is: ΩK = eν−ψ v + ω,
(5.28)
where ω is the frame-dragging frequency, as introduced earlier, and v is the orbital velocity of a particle in a circular orbit at the equator, as measured by a so-called zero-angular-momentum observer6 (ZAMO for short; see below). In terms of the above metric, v is given by (FIP): ⎡ 2 ⎤1/2 ω ψ−ν ⎦ ω ψ−ν ⎣ 2 ν e + c + e , (5.29) v= 2ψ ψ 2ψ where primes denote differentiation with respect to a radial co-ordinate. It is instructive to display the mass-shed limit versus the ratio t ≡ T /W of the rotational kinetic energy T to the gravitational binding energy W of the star, in analogy with well-known results for Newtonian stars. However, for relativistic calculations, we need to generalize the simple, well-known, Newtonian expressions, i.e., T = (1/2)IΩ2 and W = G dm1 dm2 /r12 , to: T =
1 JΩ, 2
W = M p c2 + T − M c 2 ,
(5.30)
where Mp is the proper mass of the star. The mass-shed limit can be seen in the Ω vs. t plot for the FPS EOS in Fig. 5.13 Normal sequences all connect to the origin, t = 0, Ω = 0, and supramassive sequences do not, as expected. The latter sequences also show characteristic unstable branches beyond the radial-stability limit (see below). Now consider the effect of varying EOS: a softer EOS, which produces a star of smaller radius for the same M0 , yields a higher value Ω for the same value of t, and so a higher mass-shed limit. Further, the value of t which corresponds to the mass-shed limit is ≈ 0.09−0.12 for most EOS. Similarly, Fig. Hb shows that t ≈ 0.10 − 0.12 for stellar masses > ∼ 1.4M for the FPS EOS, and inspection of CST results for other EOS generally confirms this. Thus, this range t-values is of basic significance in our understanding of rotational limits on neutron stars. 6 The
zero-angular-momentum observer (ZAMO) is at rest in the neutron star’s axisymmetric, stationary, gravitational field. For rotating stars, this implies that the ZAMO’s angular velocity must be such that the observer’s world line appears orthogonal to three-dimensional hypersurfaces of constant time. See Bardeen, Press and Teukolsky 1972.
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Fig. 5.13 Constant rest-mass sequences showing Ω vs. T /W of neutron stars with FPS EOS. Labels and lines as in Fig. 5.12. Reproduced with permission by the AAS from Cook et al. (1994): see Bibliography.
5.5.3.2
The gravitation-wave limit
Neutron stars have normal modes of oscillation, and some of these modes can become unstable in rotating stars when the stellar angular velocity exceeds certain critical values. The specific mechanism we describe here involves non-axisymmetric modes and the emission of gravitational waves [Chandrasekhar 1970; Friedman & Schutz 1978], and is now called the Chandrasekhar-Friedman-Schutz instability. Consider a star rotating with an angular velocity Ω, and imagine such a mode ∼ exp[i(mφ − ωt)] traveling backward with an angular velocity ω/m with respect to the direction of rotation of the star. The nonaxisymmteric oscillation leads to the emission of gravitational waves, which carry away angular momentum from the star. For a non-rotating or a slowly-rotating star, the effect of stellar rotation is unimportant, and the angular momentum carried off by the gravitational waves has the same direction as that of the oscillations, thus decreasing the latter’s angular momentum, and so damping the oscillations. Now consider a star rotating so fast that the oscillations are dragged along with stellar
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rotation, and so have a resultant angular momentum in the direction of the star’s rotation, as seen by a distant observer. The gravitational radiation will now carry off angular momentum in the direction of the star’s rotation, i.e., forward, while the angular momentum of the oscillations still remain backward. To conserve the total angular momentum of the oscillations and the gravitational wave, the amplitude of the former increases. Thus, the emission of gravitational waves now drives the oscillations instead of damping them, and the situation is unstable. The boundary between the two regions is the critical angular velocity in question. The m = 2 mode (Dedekind bar instability) was discovered first [Chandrasekhar 1970], but it has been argued that modes with higher values of m become unstable before this mode does (FIP). How do we find the point of onset of the instability, i.e., the critical value of Ω? The frequency ω of the non-axisymmetric mode described above has an imaginary part due to the combined action of (a) dissipation of the mode’s energy by viscous stresses in the star, and (b) either growth or decay of the mode’s energy by gravitational radiation, as described above. The energy thus grows or decays on a timescale τ , given by τ −1 = τv−1 + τg−1 , where τv is the viscous decay timescale, and τg the gravitational radiation reaction timescale [Glendenning 1996]. Of course, τg has the same sign as τv if gravitational radiation damps the mode, and opposite sign if it makes the mode grow. The point of onset of instability is where τ −1 changes sign. This determines the critical angular velocity Ωm for the mth mode, since the timescales τv and τg for a rotating star depend on its angular velocity. The indications obtained from calculations done for rotating polytropes (see Chapter 2), employing Newtonian or post-Newtonian physics, are interesting. It is generally expected that high-m modes will be damped out in neutron stars by viscosity, so that we need not consider modes with, say, m > 5. Modes with m = 3, 4 (and possibly also m = 5) would therefore set the gravitation-wave limit to rotation. These limits have been expressed, again, in terms of a limiting value of the parameter t ≡ T /W introduced above, suggesting that instability may set in at t ≈ 0.8 (FIP), thus emphasizing the importance of this range of t-values once again (see above). Since this limit is most relevant when viscosity is relatively small, it is reasonable to think that, for neutron stars, it is most relevant for hot stars, whose viscosities are low. This primarily means newborn stars. Old, “recycled” stars, spun up by accretion (see Chapter 6), are also heated up somewhat by accretion, but not nearly to such high temperatures as newborn stars have.
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5.5.3.3
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The r-mode
In the late 1990s, Andersson and co-authors, as also Friedman and Morsink (1998) and others, pioneered studies of the instability of the r-modes of neutron stars in the above context. What are r-modes? Consider Newtonian gravity first, in which a nonrotating perfect-fluid star has no (non-trivial) axial (also called toroidal) modes of pulsation, the familiar p-, g-, and f modes (see, e.g., Tassoul 1978) being all polar. However, there is a set of modes even for a nonrotating star which is degenerate at zero frequency, and as soon as the star rotates, these modes acquire non-zero frequencies [Andersson 1998]: these are the r-modes, analogous to the Rossby waves in terrestrial oceanography, which play a significant role in energy transport in the oceans. These waves are mainly velocity perturbations of the ocean’s fluid, the restoring force being the Coriolis force arising in rotating systems. Now consider general relativity, wherein these modes are those of the stellar fluid and the associated space-time metric, and the above authors calculated the properties of such modes. In the absence of viscous damping, i.e., for a perfect fluid, these r-modes are unstable at all stellar rotation rates due to the amplification of retrograde modes by the emission of gravitational radiation — the Chandrasekhar-Friedman-Schutz instability described in the previous subsection. In neutron stars, there is always (bulk and shear) viscosity, as explained above, and the resultant damping rate exceeds the above growth rate below a suitable critical frequency of stellar rotation, and so stabilizes these modes. Thus, one immediate conclusion might be that these are the modes that decide the maximum possible spin rate of a neutron star. Indeed, this was one of the strongest motivations for studying the r-modes, the other being the exciting possibility of detecting gravitational waves emitted by nascent neutron stars during this r-mode instability. The mode usually considered for this is that corresponding to l = 2, m = 2. This maximum spin rate refers, of course, to newborn neutron stars, since, as argued above, viscous damping of the r-modes was expected to be low for them, particularly before the solid crust had formed. By contrast, the old, recycled millisecond pulsars, which are much colder and have a solid crust (see Chapter 6), are expected to have much higher viscosity, so that these modes were not expected to play a significant role in determining the maximum spin-rate reached during recycling, and so the “spinup line” (see Chapter 12). Indeed, first estimates of shortest spin periods comparable to those of the Crab pulsar from such considerations for newborn neutron
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stars were encouraging. Further work, including such effects as superfluidity, boundary-layer effects at the crust-core interface, and the so-called “fall-back” of part of the material ejected in the supernova that produced the neutron star (see Chapter 6) has shown the situation to be much more complicated. For further detail, we refer the reader to the orginal works cited above.
5.5.3.4
The supramassive-collapse limit
What determines the minimum rotation below which a supramassive neutron star would collapse? The intuitive way of answering this follows a line of reasoning obtained by an obvious generalization of the one we gave above for non-rotating stars, and was originally given by Friedman, Ipser and Sorkin (1988). Consider the mass (M ) vs. central density (ρc ) plot for stars rotating with a given angular momentum J, as shown in Fig. 5.14. Then, as we would expect, and as was formally proved by the above authors, the limit of stability of stars with given J is the maximum point of the curve, where ∂M (ρc , J)/∂ρc = 0, just as it was in the non-rotating limit J → 0. As before, the mass at this point is the maximum that it can be for a neutron star rotating with an angular momentum J. Any heavier star would collapse. The locus of the maxima, as shown in Fig. 5.14, is then the supramassive-collapse limit. But this is not exactly the question we were asking. On a supramassive sequence, we hold the rest mass M0 constant, and we want to know what minimum rotation it must have in order not to collapse. The supramassivecollapse limit would then be the locus of these minima, as we vary M0 . To answer this question intuitively, again, we would expect that a supramassive sequence, such as shown in Fig. 5.11, would pass through a minimum in J, and this would be the lower limit we seek. But where is this minimum? Note that each supramassive sequence does pass through a minimum in M , where ∂M/∂ρc = 0. We suspect that this might be the point, but we need to prove this, as the derivative here is not the same as that given in the previous paragraph. The proof was given by Cook, Shapiro and Teukolsky in 1992, and we recount it here in brief. These authors used the fact that, along a sequence of constant rest mass M0 , changes in M and J are related by dM = ΩdJ, which had been proved earlier in the literature cited by them. Using this, we can express the derivative (∂M/∂ρc)M0 in two ways: either (a) as (∂M/∂ρc )M0 = Ω(∂J/∂ρc )M0 from the above result, or (b) as (∂M (ρc , J)/∂ρc )M0 = (∂M/∂ρc)J + (∂M/∂J)ρc (∂J/∂ρc )M0 , using the calculus of variations. Equating the two gives the result
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Fig. 5.14 Mass of rotating neutron stars vs. central density at constant angular momentum J. Curves labeled by value of J in units of 1049 g cm2 s−1 , and EOS is Bethe-Johnson (see Apeendix D). Those parts of the curves which lie to the right of the dash-dotted line represent secularly unstable configurations. Reproduced with permission by the AAS from Friedman et al. (1988): see Bibliography.
∂M ∂ρc
= Ω−
J
∂M ∂J
ρc
∂J ∂ρc
,
(5.31)
M0
which readily shows that (∂M/∂ρc )J = 0 implies (∂J/∂ρc )M0 = 0, and vice versa, provided Ω = (∂M/∂J)ρc . Thus, the turning points of M and J are one and the same, and this is the supramassive-collapse limit we were looking for. Pictorially, the line joining the maxima of M in Fig. 5.14, that joining the minima of M on the supramassive sequences, and that joining the turning points of J on the supramassive sequences in the Ω vs. J plot for the CST results, displayed in Fig. 5.12, represent the same limit, namely, the supramassive-collapse limit. We have thus found the lower rotational limit to supramassive stars. Note that the supramassive-collapse limit is also the boundary between stable and unstable stars on a given supramassive sequence, and that this is a slightly different concept from that explained above. This is best il-
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lustrated by looking at Fig. 5.14 first. For a given J, the stars on the higher-density side of the maximum are unstable to radial perturbations, just as they are in the static (J → 0) limit, as we explained earlier. This branch, then, contains possible equilibrium configurations (as opposed to configurations with masses above the maximum, which do not exist because no equlibrium is possible for them, as explained above) which do not exist in nature because they are unstable. The supramassive collapse limit can thus also be called the radial-stability limit, as it has been by CST: in Figs. 5.11, 5.12, 5.13, the stars on two sides of this limit on a given supramassive sequence have been marked differently to emphasize this. For our purposes in this section, however, it is more useful to study this limit as a lower bound to rotation, which extends the static, nonrotating limit of normal neutron stars appropriately to supramassive stars.
5.5.3.5
Maximum angular velocity of neutron stars
Except for a small range of masses inhabited by the supramassive stars, we do not have to consider non-zero (i.e., supramassive-collapse) lower limits to the rotation of a neutron star. But the maximum possible angular velocity of neutron stars is of immense practical interest in many ways, e.g., in connection with millisecond pulsars. Let us summarize here what recent models of rotating neutron stars tell us about this limit. As we shall see below, the supramassive-collapse limit does play an indirect role in determining the absolute maximum of the rotation frequency of neutron stars (for a given EOS), because of a curious behavior of supramassive stars at very high masses. The main question is whether the mass-shed limit or the gravitationwave limit determines the maximum rotation frequency. Unfortunately, a definitive answer cannot be given at this time, as the latter limit has not been calculated yet with the same degree of detail and sophistication as the former, the former limit being shown in Figs. 5.12 and 5.13. However, if the rough estimate of the latter limit, t ≈ 0.8 (see above), is borne out by detailed general-relativistic calculations, it will decide the maximum rotation frequency for all but the lightest neutron stars. In the absence of the above information, we can try to determine the absolute maximum value that the (uniform) angular velocity Ω of a neutron star following a given EOS can have, solely on the basis of the mass-shed and the supramassive-collapse limits. The idea is illustrated in Fig. 5.12. For normal sequences, the mass-shed limit of Ω increases monotonically with the
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mass M , as is evident from this figure. This is continued to higher masses at first with supramassive sequences, until the following complication sets in at the highest masses and angular velocities. At these extreme values, a star spins up, i.e., its Ω increases, as it loses angular momentum. This curious behavior, which may appear counter-intuitive at first, is due to the high oblateness of a fast-rotating star: since Ω = J/I, whether Ω increases or decreases as J decreases depends on whether I decreases (due to decreasing oblateness) faster than J or not. This happens in both Newtonian gravity and general relativity; for details see Cook, Shapiro and Teukolsky (1992), and references therein. Thus, supramassive sequences at the highest masses in Fig. 5.12 actually have higher Ω on the supramassive-collapse limit than on the mass-shed limit. The lower limit on rotation then actually overtakes the upper limit in this “exotic” region of fast-rotating stars, and the two intersect at a point which is the absolute maximum value that Ω, or the minimum value that the rotation period P = 2π/Ω, can have for this EOS. For the FPS EOS shown in Fig. 5.12, this minimum spin period is ≈ 0.53 ms. Note that this point is very close, but not identical, to that of the absolute maximum mass of rotating neutron stars according to this given EOS, where the set of supramassive sequences terminates: no higher mass is possible for this EOS, even with rotation. For many EOS, it is an interesting, empirical fact, noticed by several authors [Haensel & Zdunik 1989; Friedman & Ipser 1992], that the above absolute maximum angular velocity Ωmax for a given EOS is described very well by a simple relation in terms of the Newtonian expression (which is the same as the Schwarzschild expression) for the Keplerian angular velocity, ms 3 , for the maximum-mass non-rotating (static) star ΩK = GMms /Rms following that EOS, Mms and Rms being the mass and radius of this star. The relation is: 1/2 Mms −3/2 ms R6 , (5.32) Ωmax = ω0 ΩK = Ω0 M where R6 is Rms in units of 106 km, and ω0 and Ω0 are empirical constants, the former being a dimensionless number, and the latter having dimensions of angular velocity. We give the following results for the fits performed by CST to their numerical results: ω0 ≈ 0.68, and Ω0 ≈ 7840 s−1 . Rather than offering any “justification” of these empirical results at this point, we comment that a Keplerian velocity is the most natural unit in which stellar rotation limits can be expressed. Further, it is clear that the dimensionless factor ω0 has to be less than unity, since the small increase in Mmax in going
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from non-rotating to the maximally-rotating case is completely dominated by the large increase in Re3 , because of the highly oblate figure in the latter case (see above), even within the simple (Newtonian and Schwarzschild) expression for the Keplerian angular velocity given above. 5.5.4
Hartle-Thorne Approximation: Reprise
If we know the metric, as we do at the end of the numerical computation of a stellar model, we can and do calculate the Keplerian angular velocity from Eqs. (5.28) and (5.29) in a straightforward manner, as indicated above. The corresponding procedure in the Hartle-Thorne approach had a curious inconsistency in it, attention to which was first called by Glendenning and coauthors in the early 1990s [Glendenning 1996]. Since the perturbative approach of Hartle and Thorne involve expansions in powers of Ω/ΩK , the rotational corrections to the metric do depend on ΩK , so that a calculation of the metric corrections first, and then of ΩK from it, as was apparently done earlier, is not really justified. Rather, argued these authors, one has to calculate them simultaneously and self-consistently. Let us explain the point with a simple illustration given by Glendenning (1996). The “time-time” part of metric gtt , written as e2ν in the standard notation, which had the Schwarzschild value e2ν0 = 1 − (2GM/rc2 ) outside the star (r ≥ R) in the absence of rotation, now acquires both a monopole correction h0 and a quadrupole one h2 in the Hartle-Thorne approximation, and so has the general form e2ν = e2ν0 + h0 + h2 P2 (cos θ). Here, P2 (cos θ) is a Legendre polynomial. We shall keep only the monopole correction, h0 = −2GδM/rc2 + 2J 2 /r4 , to the exterior (r ≥ R) metric, δM being, of course, the rotational increase in the star’s mass introduced earlier, and J its angular momentum. The corrected metric is then e2ν = 1−(2GM/rc2 )+ 2J 2 /r4 upto the monopole level, M now being the mass of the rotating star. We can now evaluate this metric at the stellar surface r = R, and use it in Eqs. (5.28) and (5.29), remembering that the part of the metric e2ψ remain unchanged to this order. The point made by Glendenning and co-authors is that the equation for ΩK now involves ΩK itself on the right-hand side, i.e., it is a self-consistency relation. To this order, the relation can be written as: Ω2K + ω(R)ΩK − ω(R)2 =
GM , R3
(5.33)
where the frame-dragging frequency at the stellar surface, ω(R), is related to the Keplerian frequency by ω(R)/ΩK = 2I/R3 , I being the star’s mo-
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ment of inertia. Eq. (5.33) is very simple, but it illustrates the essential point. By performing detailed calculations to higher orders with “realistic” EOS, these authors demonstrated that properties like ΩK , e, ρc of a star of given mass could be obtained by this method to within ∼ 10% of the values obtained from the exact, numerical calculations for a star of the same mass, upto stellar rotation periods as short as ∼ 1 ms. For such applications, therefore, the “slow-rotation” Hartle-Thorne approximation can be thought of as being extendable with reasonable accuracy to rotations we normally consider “fast”, provided we use the above scheme of selfconsistent calculation. However, it does not appear that the increase of the maximum mass, δMmax /Mmax , for fast rotation, and similar problems, can be handled with the same accuracy by this scheme. 5.5.5
Moments of Inertia
We have given expressions earlier for the moments of inertia of slowly rotating neutron stars in the Hartle-Thorne approximation; we shall now summarize some essential features of this potentially observable property of neutron stars. Note first that, due to general-relativistic effects in Eq. (5.24), the moment of inertia are, in general, quite different from the Newtonian value for uniform spheres, I = 2M R2 /5, which would be our na¨ıve estiamte not only for the moment of inertia of our uniform-density “toy” neutron star of previous sections and chapters, but also for all model neutron stars except the lightest ones, since, as explained earlier, these models generally have density profiles which are almost uniform. This is illustrated in Fig. 5.15, which shows the moments of inertia I, in units of M R2 , of model neutron stars following a variety of EOS, displayed versus half the value of the compactness parameter introduced earlier, i.e., a/2 = (GM/Rc2 ) [Lattimer & Prakash 2001]. Clearly, I/M R2 depends primarily on the compactness of the star, and not on the details of the EOS. Indeed, stars following all modern EOS seem to lie in a narrow band on this plot, thereby almost defining a universal function of a with which we can describe I/M R2 . How do we understand this universal behavior? The clue comes from the fact that, on evaluating the Hartle-Thorne expression for I for known analytic solutions of the TOV equation, it is found that I/M R2 indeed depends primarily on the compactness a. We illustrate this with three examples. The first is our “toy” uniform-density neutron star discussed earlier (also see Chapter 2), for which the Hartle-Thorne results may be
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Fig. 5.15 Moment of inertia I in units of M R2 vs. compactness parameter M/R, for various EOS introduced in Fig. 5.2. Other curves labeled as follows. Inc: “toy” uniformdensity neutron star (see text and Chapter 2). T VII: the Tolman solution (see text). RP: the Ravenhall-Pethick approximation (see text). The inset shows details of I/M R2 for M/R → 0. Reproduced with permission by the AAS from Lattimer & Prakash (2001): see Bibliography.
approximated to within ∼ 0.5% by the analytic expression [Lattimer & Prakash 2001] : Iuni 2/5 ≈ , M R2 1 − 0.43a − 0.07a2
(5.34)
showing how I/M R2 increases from its Newtonian value 2/5 with increasing compactness. The second example utilizes one of the exact solutions of the TOV equation given by Tolman in 1939, in the seminal paper mentioned in Chapter 2. Largely overlooked for many years, this solution has been recently reconsidered in some detail by Lattimer and Prakash: it corresponds to a parabolic profile of mass-energy density, ρ = ρc [1 − (r/R)2 ], and its pressure profile is naturally a bit more complicated than that of our “toy” star given in Chapter 2 [Lattimer & Prakash 2001]. The moment of inertia for this “Tol-
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man” star can be similarly approximated to within the same accuracy as above by the analytic expression given by Lattimer and Prakash: 2/7 ITol ≈ . M R2 1 − 0.55a − 0.15a2
(5.35)
Note that the Newtonian limit for the Tolman star is the value 2/7 appropriate for a parabolic density profile, quite different from the well-known value 2/5 for uniform density. Finally, our third example is a simple, analytic approximation suggested by Ravenhall and Pethick (1994): 0.21 IRP . ≈ 2 MR 1−a
(5.36)
As we see from Fig. 5.15, the uniform-density star gives a poor description of the moments of inertia of “realistic” neutron stars. The Tolman star gives a remarkably accurate description for compactness values a > ∼ 0.2, but fails for smaller values. While the Ravenhall-Pethick approximation gives a cruder description of the above, “realistic” values of a, it works over a larger range, a > ∼ 0.06, of compactness values. The success of the Tolman solution has been attributed by Lattimer and Prakash to the fact that the parabolic density profile is rather similar to the actual profiles found for several EOS, particularly for the heavier stars. Note, however, that in the limit of small compactness, all these approximations are completely invalid, as I/M R2 becomes very small and, indeed, appears to → 0 as a → 0. This curious result is usually explained by the point that the adiabatic index of matter below nuclear density is close to, but less than, 4/3, and I/M R2 of such polytropes is known to become very small [Lattimer & Prakash 2001]. The reader will have noticed that the entire discussion so far has been about the moments of inertia of slowly rotating stars in Hartle-Thorne approximation. It is now important to indicate what happens for the fastrotating stellar models described earlier in this section. First, I must now be computed not from the Hartle-Thorne prescription, but from the general definition I ≡ J/Ω given earlier, using the numerical computations of J for rapidly-rotating models described earlier. Note first that the star becomes oblate as it rotates fast, so that, for a given mass, we would expect its moment of inertia to increase due to the increase in the equatorial radius. This is indeed the bulk of the effect (in Newtonian gravity as well as general relativity), leading to ∼ 60−70% increase in I at or near the mass-shed limit for neutron stars, as shown by FIP, much more than the rotational increase
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in M given above. For a given EOS, the model with highest moment of inertia is not the one with the highest mass and central density, for nonrotating as well as rotating stars. The reason for this is clear: as M and ρc increase, R decreases, so that I ∼ M R2 passes through its maximum at an intermediate value of M . The behavior of I/M R2 for fast-rotating stars is more subtle. For a given M , this ratio decreases with increasing rotation. The reason is that the large increase in radius at fast rotation affects primarily the low-density, outer parts of the star, while the mass distribution and the metric in the interior of the star are less affected. Thus, I does not increase quite as fast with rotation as M R2 does, and the ratio I/M R2 actually drops, e.g., by ∼ 25% for the stiffer EOS considered by FIP. Extending the universal behavior at slow rotation described above, it is clear that I/M R2 is now a function of both the compactness parameter a and the rotational fastness parameter Ω/ΩK . 5.5.5.1
Crustal moment of inertia
The moment of inertia ∆Icrust of the solid crust is of considerable interest in understanding many aspects of the rotational behavior of pulsars, e.g., glitches, as we shall describe later, since the coupling between the solid crust and the liquid core is not rigid in general, so that the character of the angular-momentum transfer between them depends crucially on the ratio ∆Icrust /I. Using methods analogous to those given in Sec. 5.2.1, but now considering the crustal limit of both the TOV equation and the HartleThorne equations for slowly-rotating stars, we can derive estimates for this ratio which are very useful [Ravenhall & Pethick 1994], although valid, of course, only within the bounds of the Hartle-Thorne approximation. We give only the final results here, referring the reader to Ravenhall and Pethick (1994) for details. It is convenient to express the ratio ∆Icrust /I in terms of crustal mass ratio given in Sec. 5.2.1. The expression is: 2 1 − aα ∆Mcrust ∆Icrust ≈ , (5.37) I 3α 1 − a M where α ≡ I/M R2 (we have described above the detailed behavior of α for various EOS), and a is the compactness parameter, as before. This shows, then, that both the basic size of ∆Icrust /I and its scaling with stellar parameters are determined by those of ∆Mcrust /M . Indeed, using
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the first form (5.4) for ∆Mcrust /M , the value of a ≈ 0.42 for canonical stellar parameters (as in Sec. 5.2.1), and an approximate value α ≈ 0.5 applicable to the canonical parameters M ≈ 1.4M, R6 ≈ 1, we get our first numerical form: −2 M PB ∆Icrust −2 4 ≈ 4 × 10 R6 . (5.38) I 1.4M MeV fm−3 The numerical factor in Eq. (5.38) ranges between 1 × 10−2 and 2.6 × 10−2 , corresponding to the canonical range 0.25 < ∼ 0.65 in the units given ∼ PB < above, as explained in Sec. 5.2.1. The second form (5.5) for ∆Mcrust /M gives our second form: −2 M ∆Icrust ≈ 4 × 10−2 R68 . (5.39) I 1.4M Of course, our discussion below Eq. (5.5) applies here as well, due to the extreme sensitivity to R.
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Chapter 6
Origin and Evolution of Neutron Stars
We saw in Chapter 2 that the concept that neutron stars are born in supernova explosions is almost as old as the concept of the neutron star itself. But our understanding of exactly how this might happen began to grow only in the 1960s and ’70s, and is still very much an area of active research. Intimately connected with this is the idea of how stellar evolution in binary stellar systems leads to supernova explosions of massive stars1 , the understanding of which also began to develop in the same time frame, and led to quantitative formulations of the origin and evolution of neutron-star binaries, as well as to those of binaries containing other compact objects, i.e., white dwarfs and black holes), which, taken together, form a rich astrophysical subject today. In this chapter, we introduce the essentials of binary stellar evolution, supernovae, and the formation of neutron stars, followed by accounts of how neutron stars in binaries evolve with time as the binary systems themselves evolve, so that the pulsars pass from a young, rotation-powered phase to a middle-aged, accretion-powered phase, and, finally, to an old, recycled phase, in which they are powered by rotation once again. It is this modern understanding of the various phases in the life-cycles of neutron stars that puts our unified description of rotation- and accretion-powered pulsars on a firm basis.
6.1
Binary Stellar Evolution
We have summarized the bare essentials of the evolution of single stars in Appendix C. The key additional feature that determines stellar evolution in binary systems is the fact that, for close binaries, the more massive star, 1 Single massive stars also undergo supernova, of course. However, we are focusing here on neutron-stars in binaries. See also Sec. (6.2).
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i.e., the primary, overflows its Roche lobe, i.e., the critical equipotential which we have discussed in Appendix B, at some stage of its evolution, and the matter thus lost from it is transferred to its binary companion, i.e., the secondary, and/or lost from the system. This can and does change the evolution of both the primary and the secondary — sometimes profoundly — and also the dynamics of the binary system. So the first question is: when is a binary sufficiently close that the above will happen? We shall be answering this question in detail below, but a first idea is given by the rough estimate that the binary period Pb must be < ∼ 10 years [van den Heuvel 1983]. This condition still leaves us with a huge number of binaries with a large range of possible values of the primary/secondary mass ratio, whose basic evolutionary features we summarize below. 6.1.1
Cases A, B, and C
Close binaries in the above sense are, then, those in which the two stars interact, and the manner of this interaction depends on (1) the evolutionary state of the core of the primary at the onset of its Roche-lobe overflow, (2) the structure of the primary’s envelope at that point, and (3) the mass ratio q ≡ M1 /M2 . Why is the primary so important? By definition, the primary is the heavier of the two stars forming the binary, and more massive stars evolve faster, as we have explained in Appendix C. Thus, it is always the primary star of a binary that reaches Roche-lobe overflow first, and starts the interaction between the two stars. Of the three criteria given above, it is the first one that proves critical for determining what kind of remnant will be finally left by the primary — a central question in the subject, to which we return later. In 1967, Kippenhahn and Weigert devised a classification scheme based on this criterion, which has now become the standard benchmark for discussing interacting binaries. We now explain the idea, which is simply an interplay between the evolutionary expansion of a single star, as sketched in Appendix C, and the scaling of the Roche-lobe radius with the binary separation (and therefore with the orbital period), as given in Appendix B. Roughly speaking, the overall radius of a single star increases slowly and mildly during each phase of nuclear burning (H, He, C. . . ) in the core, but it increases rapidly and strongly (i.e., the star ascends the giant branch) between the exhaustion of one nuclear fuel and the ignition of the next heavier one, since the envelope expands and the core collapses fast between these two points, until the next nuclear burning stage ignites [Verbunt & van den Heuvel 1995]. Of particular interest to
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us here are the first three stages of expansion, as shown in Fig. 6.1: (a) Case A — that between the ignition of H in the core on the main sequence (the so-called Zero Age Main Sequence, or ZAMS) and the end of core Hburning, (b) Case B — that between the end of Case A and the ignition of He in the core, and, (c) Case C — that between the end of Case B and the ignition of C in the core. This is the Kippenhahn-Weigert scheme: depending on whether the primary is in Case A, B, or C when it begins to overflow its Roche lobe, the close-binary evolution is classified as Case A, B, or C, respectively. Note that, apart from detailed effects like composition of the stellar material and uncertainties in our current knowledge of stellar evolution, the gross properties the radius evolution shown in Fig. 6.1 are basically determined by the mass of the star. What parameters, then, determine the point of onset of Roche-lobe overflow, where the stellar radius equals RL , and so decide whether the
Fig. 6.1 Close binary evolution in Cases A, B, & C. Shown is the evolution of the radius of a 5M star in a binary. Reproduced with permission by Annual Reviews from Paczy´ nski (1971): see Bibliography.
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case is A, B, or C? Remember from Appendix B that RL ∝ a, a being the separation between the centers of the two stars, and the proportionality constant being a function of the mass-ratio q alone. Now, the binary period Pb is related to a and the total mass Mt ≡ M1 +M2 of the system by Kepler’s third law, which we can write [Paczy´ nski 1971] as: log Pb (d) =
a Mt 3 1 log − log − 0.936. 2 R 2 M
(6.1)
Here, we have expressed Pb in days, and a and Mt in the usual solar units. Thus, the condition for the onset of Roche-lobe overflow roughly translates into a relation between Pb , M1 (primary mass), and Mt . The actual situation is even simpler, since Mt is dominated by the primary’s mass M1 by definition. The relation becomes basically one between Pb and M1 , with minor variations coming from those in the mass ratio q and in the chemical composition. In other words, given a primary mass, the boundaries between cases A, B, and C are simply given by bounding values of Pb , the above details causing minor variations in these bounds on Pb . In fact, much larger variations in these bounds on Pb are caused by the uncertainties in our knowledge of stellar evolution. Modern evolutionary calculations, including convective overshooting and mass loss, give a radius evolution substantially different from the above Paczy´ nski result, and indicate, for a primary mass M1 = 9M and a secondary mass M2 = 5M , the following bounds on the binary period as Case A: 0.65 < ∼ 3, Case B: 3 < ∼ Pb (d) < ∼ 900, ∼ Pb (d) < < (d) 2000 [Verbunt & van den Heuvel 1994], as the and, Case C: 900 < P ∼ ∼ b reader can verify with the aid of Eq. (6.1) and the Roche-lobe radii given in Appendix B. These bounds are somewhat different from those obtained previously for essentially the same system [van den Heuvel 1983] using earlier evolutionary calculations, which had neglected convective overshooting. Since Cases B and C together cover a very much wider range in Pb than does Case A, it is clear that the former cases will be, by far, the most common ones found in nature. What happens when Roche-lobe overflow starts? Matter reaching the inner Lagrangian point (see Appendix B) will flow to the companion star, i.e., the secondary, and be accreted by it. But if the primary expands so rapidly that its surface reaches the outer Lagrangian point (see Appendix B), matter flowing through this point may be lost from the binary system, or form a disk around it. We need to understand the effects of this mass transfer and/or loss on (a) the orbital dynamics of the binary, and (b) the
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evolutions of the primary and the secondary stars. We discuss these below in turn.
6.1.2
Orbital Changes
The changes in the binary orbit are simple, geometrical effects, immediately calculable from Newtonian dynamics. The key point is whether the total mass and angular momentum of the system are conserved (see above) during the mass-transfer process or not. If they are, we call the mass transfer conservative, and the orbit-change calculations — done with the aid of conservation laws — are exceedingly simple. If the mass transfer is non-conservative, we must specify the rates of loss of mass and angular momentum from the system to proceed further, but the calculations are still straightforward. We sketch both cases in brief. 6.1.2.1
Conservative mass transfer
Note first that, even if all mass lost by the primary is taken up by the secondary, this does not guarantee that the total angular momentum of the system is unchanged, unless we can assume that the rotational angular momenta of the two stars is negligible compared to the orbital angular momentum of the system, since matter in one star with a given angular momentum per unit mass (or specific angular momentum, as it is often called) about its rotation axis will not generally have the same specific angular momentum when transferred to the other star. But the above assumption is generally quite good, and we can write the two conserved quantities of a conservative system, viz., the total mass Mt and the total angular momentum J, in the following way, assuming a circular orbit for simplicity: Mt ≡ M1 + M2 = const,
J2 = G
M12 M22 a = const. Mt
(6.2)
Elimination of Mt between these two equations readily gives the law for the change of the orbit radius or stellar separation a as a(M12 M22 ) = const, which we can rewrite as af = ai
M1i M2i M1f M2f
2 ,
(6.3)
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where the superscripts i and f denote the initial and final values, respectively. Kepler’s third law then transforms Eq. (6.3) into that for the change Pf
in the orbital period, yielding Pbi = (M1i M2i /M1f M2f )3 . b The consequences of the above law are transparent: if mass is being transferred from the more massive star to the less massive one, the orbit shrinks, i.e., a decreases, and so the orbital period Pb also decreases, but if mass is being transferred from the less massive to the more massive star, the orbit expands, i.e., a increases and Pb increases. This is easily seen from Eq. (6.3) when we note that M1 + M2 = const, and remember the elementary result from differential calculus that, if the sum of two variables is kept constant, their product is a maximum when they are equal. It follows, then, that when mass is transferred from the more massive star to the less massive one, M1 and M2 move closer to equality, hence M1 M2 increases, and so a ∝ (M1 M2 )−2 decreases. Conversely, when mass is transferred from the less massive star to the more massive one, M1 and M2 move further away from equality, M1 M2 decreases, and a increases. The first mass transfer in the evolutionary history of a given binary always starts from the primary, i.e., the more massive star, to the less massive one, so that, at first, the orbit always shrinks and the orbital period decreases. As we shall see, however, as mass transfer continues, so much mass may eventually be transferred to the less massive star that it becomes the more massive one. At this stage, further mass transfer from the less massive star (i.e., the former primary) to the more massive one, which happens routinely, makes the orbit expand, so that Pb increases. Such phases may occur several times in the course of the evolutionary history of a binary. The beauty of Eq. (6.3) is that it allows us to make a connection between any initial and final states according to the general rule given below, irrespective of the details of what happened in between, provided only that the mass transfer had stayed conservative between the states. The general rule is this: the final orbital period is shorter or longer than the initial one according as the final mass-ratio of the two stars is closer to or farther away from unity, respectively, compared to the initial mass-ratio. The reason for this rule must be clear from the above discussion on the behavior of M1 M2 . 6.1.2.2
Non-conservative mass transfer
Actually, mass transfer is expected to be non-conservative in a variety of real cases. Massive stars are known to drive strong stellar wind, and so to lose a lot of mass in the process. Part of this mass will be lost from
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the system, and carry some angular momentum away with it. As another example, in very close binaries, angular momentum is lost from the system through the emission of gravitational waves (see below), even if there is no mass loss in the process of being transferred from one star to the other. This process, as well as magnetic braking, is of great importance in understanding the evolution of cataclysmic variables (CVs) and low-mass X-ray binaries (LMXBs), as we shall see later. Finally, an extreme example of non-conservative binary evolution is the common-envelope evolution, wherein the rapidly expanding envelope of the primary completely engulfs the secondary, the latter spirals inward due to the large frictional drag of this common envelope, and the large amount of heat generated in this process expels the envelope. This process, again, is believed to be of crucial importance in understanding the formation of CVs, LMXBs, close double neutron-star systems (e.g., the Hulse-Taylor pulsar; see Sec. (6.5.1)), and close double white-dwarf systems, and we return to it later. How do we handle such cases? We must have a prescription for describing both mass- and angular momentum-loss from the system. Suppose we describe the former by saying that a fraction α of the mass being lost at a rate2 M˙ from M1 is not transferred to M2 , but, rather, lost from the system. Further, we describe the latter by saying that the magnitude of the rate of loss of orbital angular momentum J is J˙orb . Then a differentiation of the expression for J given in Eq. (6.2) and the use of the above descriptions M˙ 1 = −M˙ , M˙ 2 = (1 − α)M˙ , and J˙ = −J˙orb readily yield: M˙ J˙orb a˙ q + 1/2 =2 , (1 − q) + αq −2 a M1 q+1 J
(6.4)
where q ≡ M1 /M2 is the mass ratio introduced earlier. This equation clarifies the basics of orbit evolution under non-conservative mass transfer [van den Heuvel 2001]. Recover first the old result for conservative mass transfer, given by the first term within the square brackets on the righthand side of the above equation, since the other terms vanish in this special case, as both α and J˙orb are zero by definition. The result, which is M˙ a˙ =2 (1 − q) , a M1 2 We
(6.5)
˙ to be positive here, in keeping with custom followed define the mass-loss rate M in binary-evolution literature. The definition used in van den Heuvel (2001) is slightly different.
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is just the differential form of Eq. (6.3), and it shows that the orbit shrinks if q > 1, i.e., mass is transferred from the more massive star to the less massive one, and widens if q < 1, i.e., the reverse case, as we saw above. Note next that the effect of mass loss by itself, given by the second term within the square brackets on the right-hand side of the above equation, is always to widen the orbit. Note finally that the effect of angular momentum loss by itself, given by the last term on the right-hand side of the above equation, is always to shrink the orbit. In general, the interplay of all these effects decides orbital evolution, after we have put in values of M˙ , α, and J˙orb appropriate for the case in question. A useful special case we shall discuss later in this chapter is that of angular momentum loss but no mass loss (α = 0), in which case Eq. (6.4) reduces to M˙ J˙orb a˙ =2 . (1 − q) − 2 a M1 J
(6.6)
A case of great importance is that of angular momentum loss by gravitational radiation in very close binaries (see above), wherein J˙orb is given by: J˙orb 32G3 M1 M2 Mt = . J 5c5 a4
(6.7)
It is clear from the strong inverse dependence of this loss rate on the orbit radius a that, for sufficiently narrow orbits, this effect will dominate all others in Eq. (6.4), and the orbit will shrink continually, forcing mass transfer from a Roche lobe-filling star. This is believed to be a major mechanism for driving mass transfer in CVs and LMXBs, as pointed out in 1971 by Faulkner. We shall return to this question later. 6.1.3
Stellar Evolution
What evolutionary changes occur in the primary and the secondary in response to the above mass transfer/loss, in addition to the usual, nuclear evolution that single stars show, as summarized in Appendix C? Two general evolutionary trends have been identified and discussed, their key difference coming from the nature of the mass transfer that ensues when the primary first overflows its Roche lobe, namely, (a) conservative mass transfer, which occurs when the primary has a radiative envelope and the mass ratio q ≡ M1 /M2 is not very large, q < ∼ 3, say, and, (b) highly non-conservative, common-envelope (CE for short) mass transfer, which occurs when the pri-
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mary has a convective envelope or the mass ratio q is sufficiently large, q> ∼ 3, say. We summarize these in turn. 6.1.3.1
Conservative evolution
If the primary at Roche-lobe contact has a radiative envelope, and loses a mass ∆M , say, it first re-adjusts itself to the new hydrostatic equilibrium corresponding to the new, lower mass on a dynamical timescale (see Appendix C), i.e., almost instantaneously. For a star with a radiative envelope, this new radius R of hydrostatic equilibrium is smaller than before (since R increases with M for such stars; see Appendix C), so the star shrinks at first. But the star is out of thermal equilibrium now, and, in order to restore this, the star begins to expand on the thermal, or Kelvin-Helmholtz, timescale τKH (see Appendix C). Now consider what has happened to the Roch-lobe radius RL as a result of this mass-transfer. As the primary is, by definition, the more massive star in the binary, the orbit has shrunk, and so has RL , according to the discussion given above. Therefore, the primary stays in Roche-lobe contact, and continues mass transfer3 on a timescale τKH , i.e., at a rate M˙ KH ∼ M1 /τKH ∼ 3 × 108 (M1 /M )3 M y−1 , as the reader can show, using the expressions for τKH given in appendix C. The process continues until the primary transfers so much mass that it becomes the less massive star in the binary, at which point further mass transfer widens the system, as we showed above, and so increases RL , so that R eventually drops below RL , and mass transfer stops. The actual situation for Case B evolution4 is shown in Fig. 6.1: the key point that emerges is that R drops below RL only after essentially all of its hydrogenrich envelope has been lost. Thus, all that is left of the former primary is basically its helium core (surrounded by a tenuous H-rich envelope of very small mass). The masses MHe of such He cores produced by Case B mass transfer have emerged from numerous evolutionary calculations done since 3 If R after hydrostatic re-adjustment is > R , more mass will be lost until R = R , L L and the subsequent thermal re-adjustment and expansion will start mass transfer again. On the other hand, if R after hydrostatic re-adjustment is < RL , mass transfer will stop briefly, and resume with thermal re-adjustment and expansion. See van den Heuvel (2001). 4 Detailed calculations for Case A show that, in this case also, R < R L becomes possible only after the primary transfers so much mass that it has become the less massive star in the binary. Subsequently, the former primary expands again on a slow, nuclear timescale (see Appendix C) as H-burning is still going on in its core, comes into Roche-lobe contact again, and transfers mass on this slow, nuclear timescale. Such slow ˙ ∼ M1 /tnuc ∼ 10−10 M y−1 is believed to be going on in mass transfer, at rates M Algol-type binaries.
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the 1970s: the resultant scaling of MHe with the initial primary mass M1 can be approximated [see van den Heuvel 1983 and references therein] well by simple analytic expressions5 of the form: MHe ≈ 0.073 M
M1 M
1.42 .
(6.8)
The reader can readily calculate from this equation the ratio of the He-core mass to the initial primary mass, mHe ≡ MHe /M1 , and note (a) that this 0.42 M1 ratio increases with increasing primary mass as mHe ≈ 0.073 M ,
and, (b) mHe ∼ 0.20 − 0.30 over the relevant range of primary masses (M1 ∼ (10 − 30)M , say). We shall use both of these facts later in this chapter. Subsequently, this He-core contracts further, until it reaches sufficiently high temperature and density to ignite He, at which point the star expels the last vestiges of its tenuous H-rich envelope, and becomes a He-burning pure He star. Such a He star has a very small radius, and lies, accordingly, deep inside its Roche lobe. Finally, what happens to the mass-gaining star, i.e., the former secondary which has accreted so much matter that it is the more massive star in the binary? The answer is: nothing much, as long as the initial mass-ratio q did not exceed the rough bound given above. It simply accepts the unprocessed H-rich material transferred from the envelope of the former primary, re-adjusts to appropriate, new, hydrostatic and thermal equilibria, slowly becoming a massive O/B star, and continuing its nuclear evolution. Indeed, as we shall see below, it is this ability of the mass-gaining star to re-adjust itself that makes conservative evolution possible in the first place. In 1967, Paczy´ nski pointed out that the system at the end of the conservative mass transfer described above, viz., a He star with a ∼ 2 − 4 times more massive O/B companion, exactly resembled the well-known WolfRayet binaries in our Galaxy and the Magellanic Clouds. Wolf-Rayet, or WR, stars have long been known to be H-deficient stars, highly overluminous for their masses (indeed, they have typically the same luminosity as their massive main-sequence O/B companions in these binaries), with remarkable emission spectra indicating strong stellar winds. Indeed, ∼ 60% of all known WR stars are found in close binaries of the above type, making the above identification most attractive and generally accepted today. 5 The precise expression depends on the composition: the one given here is for X = 0.70, Z = 0.03.
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Fig. 6.2 Evolutionary stages of a massive close binary which started with components of 20M and 8M , with an orbital period Pb = 4.7 d. Stages: a) initial state, b) & c) mass transfer by primary, leaving a He star, d) He-star supernova, leaving a compact 2M remnant at t ≈ 6 × 106 y, Pb ≈ 12.6 d, e) massive companion becomes a supergiant at t ≈ 10.4 × 106 y, driving a strong wind and turning on the HMXB, f) massive companion’s rapid mass transfer, extinguishing the X-ray source, and large mass-loss from system, g) massive companion leaves a second He star, h) He-star supernova at t ≈ 11 × 106 y: two compact stars produced, survival or disruption of binary. Reprinted with permission by Springer Science & Business Media from van den Heuvel (2001): see c Bibliography. 2001 Kluwer Acad. Publ.
6.1.3.2
Common-envelope (CE) evolution
The above scenario of conservative evolution fails when the secondary — the mass-gaining star — fails to re-adjust its structure at the rate dictated by the mass-transfer rate from the primary. Why should this happen?
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Recall that this re-adjustment has two parts, namely, the fast, almost instantaneous, hydrostatic part, and the slower thermal part. It is the second that causes the bottleneck and the consequent failure, if the mass supply (2) goes too fast, i.e., M2 /M˙ is much shorter than τKH , the Kelvin-Helmholtz timescale of the secondary. Under what circumstances is the mass-supply expected to be so fast? Two possibilities are well-known, and we summarize them. The first, and obvious, one is when the primary — the mass-losing star — has a deep convective envelope, since, on losing mass, such a star would do the fast, hydrostatic re-adjustment in the form of an expansion, since R increases with decreasing M for such a star (see Appendix C). By contrast, the Roche lobe RL of the star would shrink, of course, as explained above. The result would be a violently unstable phase of mass transfer on extremely short (∼ 102 − 103 y) timescales, far shorter than the Kelvin-Helmholtz timescale of any conceivable secondary. The second one, which is independent of the structure of the primary (and therefore more ubiquitous and more important) is when the KelvinHelmholtz timescale of the primary is much shorter than that of the secondary, since the primary transfers mass on its own Kelvin-Helmholtz (1) timescale τKH (if it has a radiative envelope, as explained above), or on the faster, dynamical timescale (if it has a convective envelope, as just explained), and adequate secondary re-adjustment is guaranteed to be impos(2) sible if τKH is much longer than even the longer of the above two timescales. This is what happens when the the mass ratio q ≡ M1 /M2 is sufficiently high, recalling from Appendix C the inverse scaling of τKH with M . In fact, it is clear that, for q > ∼ 3, the Kelvin-Helmholtz timescale of the primary is shorter than that of the secondary by an order of magnitude or more, and we can take this as an approximate criterion for failure of the re-adjustment of the secondary in step with the mass supply. What are the consequences of the above failure on the part of the secondary to re-adjust to new thermal equilibrium appropriate for the mass supply to it? Since the secondary is always out of thermal equilibrium in this situation, as explained above, it is reasonable to think that it will keep on expanding on its own Kelvin-Helmholtz timescale, overflowing its Roche lobe and thus becoming unable to accept any more matter from the primary. What will happen to the primary’s envelope, then, which keeps on overflowing its own Roche lobe? Again, it is reasonable to imagine that this primary envelope will keep on expanding, engulfing the secondary, thus forming a common envelope, inside which the secondary and the core of the
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primary will orbit each other. This is what Ostriker and Paczy´ nski suggested in the mid-1970s (in a context which is closely related but slightly different, namely, the formation of cataclysmic variable binaries, which we shall consider later), and the process has now acquired the generic name of common-envelope (CE for short) evolution, wherever it happens. What is the outcome of CE evolution? Detailed hydrodynamic computations performed by various groups over the years (see, e.g., the 2000 review by Taam and Sandquist) have given us a reasonably good idea of this. The basic physics can be clarified by a beautifully simple, energybased consideration pioneered by Webbink (1984), which is supported by detailed numerical calculations, and which we summarize in the next paragraph. The key process here is the large frictional drag of the CE on any orbital motion inside it, as a result of which the secondary star rapidly spirals inward, towards the primary’s core. In this process, a large amount of orbital binding energy is converted into heat, causing the CE to be expelled if the energy released is large enough. Qualitatively, then, there are two possible outcomes. First, if there is enough orbital binding energy available to be released, the envelope will be expelled, leaving a very close binary consisting of the secondary and the (evolved) core of the primary, whose parameters we shall presently estimate. This is expected to occur in all cases of the massive, relatively wide, primordial binary systems that we are concerned with in this section. Second, if there is not enough orbital energy available to expel the envelope, either because the orbit was rather narrow at the onset of the CE phase, or because the primary’s core is not sufficiently compact, or both, the stars will completely merge, forming a single star rotating rapidly. We shall describe in later sections of this chapter how such situations may well arise in more evolved stages of neutron-star binaries. One remarkable feature of CE evolution that has been amply demonstrated by the above numerical hydrodynamic calculations is the enormous speed with which it occurs, whichever way it goes, since the spiral-in is a dynamical process: typical timescales are ∼ 102 − 103 y, i.e., very small compared to the usual stellar evolution timescales. We focus now on the first outcome above, and invoke the energy considerations. Call the primary core mass M1c and envelope mass M1e , so that M1 = M1c + M1e . Also call the initial and final (circular) orbit radii ai and af , respectively. Then the initial and final orbit-binding energies are GM1 M2 /2ai and GM1c M2 /2af , and it is their difference which sets the scale for the energy used for expelling the CE. Let us say that the actual energy used for this purpose is α times the above difference in binding en-
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ergies and dub α the efficiency factor of CE evolution [Verbunt & van den Heuvel 1994]. Now, the energy needed to expel and disperse the envelope completely is the binding energy with which it was originally bound in the primary at the onset of mass transfer, which must be ∼ GM1 M1e /RL1 , in terms of the Roche lobe radius RL1 of the primary. The exact value depends on the primary’s structure, and it is customary to express it as GM1 M1e /λRL1 , in terms of a structure factor λ. Webbink’s (1984) energybalance equation is then, simply: GM1c M2 GM1 M1e GM1 M2 =α − , (6.9) λRL1 2af 2ai which the reader can immediately solve to obtain the final orbit radius in terms of the initial one as: M2 M1c af = . . ai M1 M2 + 2M1e λ−1 α−1 (RL1 /ai )−1
(6.10)
In Eq. (6.10), the value of RL1 /ai depends on the initial mass ratio q ≡ M1 /M2 alone, expressions for which are given in Appendix B. After the envelope is expelled, we are left with a very close binary, consisting of the evolved core of the primary, i.e., a He star, as above, and the secondary, which is a relatively low-mass (M2 ∼ a few M ), unevolved star in the CE case. (The latter is true because the usual way of having CE evolution is by starting with a large value of q, i.e., a massive primary with a relatively low-mass secondary, which accretes only a small amount of mass before CE evolution starts.) The classic model example of such a case is the scenario proposed originally by Sutantyo (1975) to explain the origin of the famous accretion-powered pulsar Her X-1, and similar binary systems. With an initial mass ratio q = 15M /2M = 7.5, CE evolution and envelope expulsion yield a very close binary with a 4M He star and a 2.5M secondary on the main sequence. We return to the subsequent evolution of such a system later, during our discussion of the evolution of Her X-1-type systems.
6.2
Supernovae: Birth of Neutron Stars
It must be clear from the above that, no matter which route the the initial evolution of a massive primordial binary takes — conservative or commonenvelope — we end up with a He star which is the evolved core of a massive
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star, with a relatively unevolved companion, which is a massive star in a relatively wide orbit for conservative evolution, and a relatively low-mass star in a relatively narrow orbit for common-envelope evolution. What happens next is determined by the further evolution of this He star: if it is sufficiently massive, it produces, as the final end-product of its evolution, a neutron star — the prime object of our study in this book. This is accompanied by the release of enormous amounts of gravitational binding energy, which produces spectacular stellar explosions called supernovae, which the inspired visions of Zwicky and Baade had correctly identified (see Chapter 2) as far back as the 1930s as birthplaces of neutron stars. 6.2.1
Final Evolution of Helium Stars and Cores
What is the result of further evolution of the above He star? Note first that this evolution is generally believed to be essentially unaffected by the loss of the extended H-rich envelope of the former primary, which means, conversely, that what we describe for the He star should also hold for the evolution of He cores in post-main-sequence massive stars. In other words, the way in which massive stars in binaries (which have transferred all of their hydrogen envelopes to their companions and so become bare He stars) evolve to finally produce various compact objects, including neutron stars, should be essentially identical to that in which single massive stars (which can retain much or some of their hydrogen envelopes depending on how much mass is lost from them through stellar winds at various evolutionary stages) of the same initial masses would do the same. While this appears to be generally supported by detailed evolutionary computations done in the 1980s and ’90s, there have also been indications of some specific differences, e.g., in the threshold for neutron-star formation, between the outcomes for single and binary massive stars: we shall return to this point below. Evolutionary calculations for bare He stars were pioneered in the 1970s by Paczy´ nski and by Arnett, and refined in the 1980s and ’90s by Nomoto and co-authors, by Habets, and by Pols. Those for the final evolution of massive, single stars have been performed by various groups over the 1980s and ’90s, e.g., Iben and co-authors, Woosley and co-authors, and GarciaBerro and co-authors. We summarize the essential story emerging from such calculations here, referring the reader for more details to the outstanding, recent review by Woosley, Heger and Weaver (2002; heneceforth WHW) and the references cited therein. The advanced stages of nuclear burning
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Fig. 6.3 Remnants of massive stars as function of initial mass and metallicity: see text. Reproduced with permission by the AAS from Heger et al. (2003): see Bibliography.
that occur in these He stars or He cores after completion of He burning6 , are the successive burning of C, Ne, O, and Si, each stage using as its fuel the ashes of the earlier stage (see WHW). The final outcome regarding the formation of a heavy-element (Z > 2) core is conveniently classified according to the initial mass M of the massive star, and displayed in Fig. 6.3. Below a lower critical mass M< ≈ 8M , the core is never able to ignite carbon, so that nuclear burning stops there, yielding a degenerate core consisting of carbon and oxygen, i.e., a C-O white dwarf. Above an upper critical mass M> ≈ 11M, on the other hand, carbon and neon are both ignited while the core is still in a non-degenerate state, and all advanced stages of nuclear burning (including silicon burning) are completed while the core is still in hydrostatic equibrium, generating an iron core of roughly the Chandrasekhar mass (see Chapter 2). This iron core subsequently collapses (see below), yielding a neutron star. It is in the transition region between M< and M> that complications arise in Cand Ne-burning due to the residual effects of degeneracy, and due to fuel ignition off-center (rather than at the center, as may be na¨ively expected, since the conditions for a given thermonuclear fusion reaction are normally 6 The
complications occurring in nuclear burning in the transition region between white-dwarf and neutron-star remnants, corresponding to initial massive-star masses of ∼ 8 − 11M , are summarized at the appropriate places.
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met first at the center of the core) on many occasions. The outcomes are also complicated, as may be expected. The C-O-Ne core is surrounded by a thin, pulsating He shell, which can drive “superwinds” (in the manner of Asymptotic Giant Branch, or AGB, stars, but stronger) with hefty mass losses. If sufficient mass is thus lost, the outcome will not be collapse to a neutron star (and the accompanying supernova), but rather an O-Ne-Mg white dwarf (see Chapter 2). On the other hand, if M ∼ 9 − 10M , then, as Nomoto and co-authors, as also Habets, have shown in the 1980s, Ne never burns stably in hydrostatic equilibrium. Rather, the core grows to the Chandrasekhar mass, electron capture at sufficiently high densities (see Chapter 2) ignites Ne, but this capture quickly becomes so copious as to remove pressure support drastically, resulting in collapse to a neutron star and the accompanying supernova. Above M> , neutron stars form for M < ∼ 25M in the manner described above. For more massive stars, the core mass exceeds the OppenheimerVolkoff (OV) limit (see above), and core collpase leads to black holes. This can happen in two ways. As M first exceeds the above limit, the core is not quite above the OV limit at formation, but the supernova that accompanies its collapse is not energetic enough to blow away its envelope completely: some of the ejecta fall back later on the core, and drive it over the OV limit. This is the delayed, so-called fallback mechanism. When M is sufficiently high, however, the core is over the OV limit at formation, and collapses promptly into a black hole: this is the so-called direct mechanism. The threshold for the latter depends on the metallicity. For zero-metallicity, primordial stars, it is estimated to be M ∼ 40M , but the situation may be very different for solar-metallicity stars (WHW). The natures and masses of compact remnants formed in various ranges of M according to our current understanding are summarized in Fig. 6.3. Finally, note that Wellstein and Langer (1999) have argued that there may be a significant difference in the neutron-star-formation threshold discussed above for single massive stars and those in binaries. Whereas we have quoted this threshold as ∼ 8 − 10M above from evolutionary calculations summarized there, the computations of these authors for Case B mass transfer in close binaries indicate that the threshold has a somewhat higher value ∼ 13M in this case. The reason (WHW) appears quite straightforward and physically appealing (albeit rather post facto): loss of the envelope by binary mass transfer early in the He-burning phase (a) halts any possible growth of the He core by hydrogen-shell burning in the
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H-rich envelope, and, (b) possibly removes, during the very last stages of the mass transfer, the outermost parts of the He core itself. 6.2.2
Core Collapse: Neutron Star Formation
Consider now the collapse of an iron core into a neutron star. The critical core mass for this is, of course, the Chandrasekhar mass (see Chapter 2), but its value, as given by Eqs. (2.50) and (2.51) needs modification appropriate for the present case. The major conceptual change is the inclusion of finitetemperature effects: whereas the discussion in Sec. (2.4.3) was on zerotemperature, completely degenerate matter, the iron core has a significant thermal content, which needs to be taken into account7 . This modifies the expression for the Chandrasekhar mass given by Eqs. (2.50) and (2.51), (0) which we can call MCh , to π2 k2 T 2 (0) MCh = MCh 1 + , (6.11) EF2 in terms of the core temperature T , and the Fermi energy EF of the electrons, introduced in Sec. (2.2.4). Using the expression for EF given there, together with Eq. (2.10), the reader can easily show that, for the extremely relativistic (ER) electrons under consideration here, the Fermi energy is given in terms of the matter density ρ and the number of electrons per baryon µ (introduced in Sec. (2.3)) in the core material as (WHW): 1/3
EF ≈ 1.1ρ7 µ1/3 MeV,
(6.12)
where ρ7 is ρ in units of 107 g cm−3 . Note that µ is often called the lepton fraction in modern literature, since the original Landau definition given in Sec. (2.3) can be generalized to include all leptons, in which case µ is the number of leptons per baryon. Other modifications to MCh are those of detail, e.g., Coulomb and relativistic effects, which we have touched upon in Sec. (2.4.4), as also that which takes into account the fact that the iron core in this case is surrounded by matter, so that its surface boundary condition is one of non-zero pressure, as opposed to the situation for an isolated star. What is the final value of MCh after all these effects are included? Note first that a typical iron 7 The whole concept of a maximum mass, above which equilibria maintained by electron-degeneracy pressure no longer exist (as discussed in Chapter 2), still works here because matter is still strongly degenerate in the situation under consideration, owing to the very effective cooling due to neutrino losses (WHW).
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core at the point of collapse has a lepton fraction of µ ≈ 0.42 (i.e., almost the value adopted by Chandrasekhar and others in their calculations in the 1930s; see Sec. (2.4.3)) at its center, which rises gradually to µ ≈ 0.48 (i.e., almost the value we used for modern estimates in Chapter 2) at its edge, as detailed computations have shown (WHW). Taking µ ≈ 0.45 as an average value, the above modifications yield MCh ≈ 1.34M as the effective Chandrasekhar mass for the iron core of a 15M star, and MCh ≈ 1.79M for a 25M star. Generally speaking, more massive stars tend to produce more massive iron cores and possibly more massive neutron stars as may have been expected. The core collapses, then, as its mass exceeds MCh , but the process is not abrupt: rather, it happens on a thermal timescale, as copious neutrino emission carries away the gravitational binding energy of the core. Two instabilities which enhance the collapse process are (a) electron capture by the iron-group nuclei in the core, which leads to increasingly neutronrich nuclei (see Chapter 3) and a decrease in µ, thus reducing pressure support and bringing the adiabatic index of core matter close to the point of instability, and, (b) photodisintegration of the nuclei into α particles8 , which becomes energetically favorable at sufficiently high temperatures and densities in the core (particularly important for the more massive stars), and which reduces the effective Chandrasekhar mass by cooling the electrons indirectly. Note, however, that a full disintegration into α particles does not occur. In Fig. 6.4, we show the profile of the radial infall velocity during core collapse. 6.2.3
Core Bounce: Supernova Explosion
The above core collapse does not continue indefinitely, however, if the core mass is below the OV limit (see above). The collapse is halted at an appropriate point, as the pressure of the degenerate neutrons (see Chapter 2) takes hold, and the core “bounces” back. We can visualize the situation along the lines of the popular paradigm for nuclear matter introduced in Chapter 3: as collapse progresses and the core density rises, the large, neutron-rich nuclei eventually begin to touch each other, and merge at 8 There is a curious historical point here: prior to 1979, it was thought that this photodisintegration might not stop at α particles but continue all the way to individual nucleons, i.e., a complete disintegration of the nuclei. However, Bethe et al. pointed out in 1979 that, since the nuclear partition function at high temperatures scales roughly as exp(bT )/T , so that the number of excited nuclear states populated at these temperatures increases essentially exponentially with T , total disintegration of the nuclei will not occur.
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Fig. 6.4 Collapse and bounce of the iron core in the supernova of a 13M star. Shown is radial-velocity vs. enclosed mass at -0.5 ms, 0.2 ms, and 2 ms (the lowest, middle, and highest curves respectively in the left-half of the diagram) with respect to bounce. Notice how infall inside the core halted, and outflow begins, while matter further out continues to fall in. Reprinted with permission by the APS from Woosley et al. (2002), c 2002 American Physical Society. Rev. Mod Phys., 74, 1015.
just below nuclear density into essentially one gigantic stellar-mass nucleus, which overshoots the nuclear density by a factor of several before bouncing back. The essential physics is aptly summarized by WHW in a simple, picturesque, mechanical analogy: “the repulsive hard-core potential of the nucleus acts as a stiff spring, storing up energy in the compressive phase, then rebounding as the compression phase ends”. This rebounce is shown in Fig. 6.4: notice how the radial-velocity profile in the inner part of the core changes drastically at bounce, with infall changing into outflow. This inner core is essentially the “homologous core”, which initially collapses roughly homologously with v ∝ r; it is also roughly that part of the core which stays in sonic communication throughout. Just outside this inner core, a shock wave develops immediately as the rebounding core encounters matter that is continuing to fall in, since the impact between the two is supersonic. We can call this prompt hydrodynamic shock the bounce shock.
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Explosion mechanisms
It was this bounce shock that originally looked like a good candidate for the mechanism that produces the supernova explosion. However, as soon as Colgate and White did their pioneering numerical hydrodynamic calculations in 1966 (see Chapter 2), the problem with this candidate became clear. The shock never reaches the outer layers of the star with enough energy to explode them, because of severe energy losses on the way. The major way in which this loss occurs is through photodisintegration9 : as the shock moves through infalling bound nuclei, it heats them and so tears them apart into the constituent nucleons, spending ∼ 1051 ergs per 0.1M in the process, and so becoming stalled before it can do any explosive damage. Accordingly, Colgate and White (1966) suggested that neutrinos were the key to supernova explosion: these are emitted copiously from the hot, collapsed core (and thereby cool the core, as we mentioned above, carrying away the gravitational binding energy released in the formation of the neutron star), and these authors suggested that these neutrinos would stream through, deposit some fraction of their energy and momentum in the outer stellar layers ahead of the shock, and explode them away. Unfortunately, detailed calculations in the 1980s and ’90s, with progressively improving treatments of the essential physics (e.g., EOS, neutrino transport) found no evidence for the direct neutrino-driven Colgate-White explosion, but they amply confirmed the above stalling of the prompt bounce shock. A major reason for the failure of this extremely fast transfer of neutrino energy and momentum to the nuclei in the outer stellar layers — essential for the direct Colgate-White explosion — was the enhanced coupling of the neutrinos to the outer-layer plasma due to weak neutral currents, as has been emphasized by Janka, Kifonidis and Rampp (2001; henceforth JKR). This strong coupling ensures that time spent by neutrinos in these layers before escaping is ∼ seconds (also see below), so that neutrino luminosities, and therefore energy-momentum transfer rates, are not high enough to cause an explosion. It were the neutrinos, however, which ultimately resolved this difficulty in another way, providing a mechanism which is currently believed to be the most likely one for the supernova explosion. In the early-to-mid 1980s, Wilson discovered that the above stalled prompt shock could be revived by 9 Another
process that cools the shock on a hydrodynamical timescale, immediately following core bounce, is the passage through it of neutrinos emitted from regions behind it, as the shock moves outward. We return to this point later.
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neutrino energy-deposition on a timescale ∼ 0.1 s in a gain layer behind the shock [see Wilson 1985; Bethe & Wilson 1985]. There is a hierarchy of three crucial timescales in this overall supernova scenario, whose importance has been stressed in a particularly lucid discussion by WHW: we summarize it here. The first, and shortest, timescale is the hydrodynamic one ∼ a few ms, on which the shock breaks out and reaches the edge of the core. On this timescale, the neutrinos actually extract energy from the shock, as explained above (see footnote), thus cooling it. The second, and intermediate, timescale is the Wilson timescale ∼ 0.1 s, on which the shock expands to a larger radius and cools, and energetic neutrinos, streaming out from smaller radii, can deposit a small fraction of their energy-momentum in the above gain layer. This fraction is typically a few per cent, which is much smaller than that envisaged in the above Colgate-White scenario, but the energy deposited is ∼ 1051 ergs, enough to power the supernova explosion of a massive star. The third, and longest, timescale is the Kelvin-Helmholtz timescale of the neutron star ∼ 3 − 10 s, on which the neutrinos carry away the bulk of its gravitational binding energy, as they diffuse through and escape from the outer stellar layers, as explained above. As may be expected, the Wilson mechanism has been studied extensively in the 1980s and 1990s, both by Wilson, Bethe, Mayle, and coauthors, and by numerous other groups, e.g., by Bruenn, Mezzacappa, and co-authors, by Woosley, Weaver, and co-authors, by Janka, M¨ uller, and co-authors, and so on. These studies have shown that, while the mechanism appears to be generally viable, the details of the outcome depend very strongly on the detailed post-bounce evolution of the collapsed core, which depends, in turn, on the details of (a) the nuclear EOS used, (b) the treatment of neutrino transport and neutrino-matter interactions, (c) the structure of the collapsing star, and, (d) general-relativistic effects. Of particular interest is the role of convection and mixing in the neutrino-heated regions — a subject of much study in the last decade or so through multidimensional simulations — since this enhances the neutrino luminosity of the nascent neutron star (or proto-neutron star; PNS for short), as also the explosion energy by increasing the efficiency of neutrino absorption. Results from a typical two-dimensional simulation are shown in Fig. 6.5. While there is little doubt at this point that convective overturn is likely to occur, what is not clear yet (JKR) is whether this is essential or incidental for the success of the Wilson mechanism. It appears fair to say that the Wilson mechanism is currently the most favored one, although numerical simulations have not converged yet to a
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Fig. 6.5 Mixing during the supernova explosion of a 15M red supergiant. Left panel shows density, and right panel elemental abundances (O, Si, Ni) on logarithmic scale. Reference scales are given. Reproduced with permission from Kifonidis et al. (2000). Figure kindly provided by K. Kifonidis.
universally acceptable description, and observational support for the mechanism is not yet conclusive (see below). Both positive and negative aspects (the latter including worries/uncertainties) of this neutrino-driven mechanism have been summarized admirably by JKR: we close our discussion of this subject by sketching them for the reader. Encouragement comes from the facts that (a) neutrinos with roughly expected properties are observed to be emitted in supernova explosions, e.g., Supernova (SN) 1987a (see below), (b) simulations do produce explosions under suitable conditions, (c) neutrinos can deposit the ∼ 1051 ergs required to power the canonical explosions, (d) the Wilson shock-revival timescale (see above) does roughly imply the observed neutron-star masses and supernova nucleosynthesis features, and also from the general argument that neutrino heating behind the stalled shock must occur at some point after core bounce. Worries come
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from (a) discrepancies between results from simulations by different groups, (b) the fact that the most advanced one-dimensional (i.e., spherically symmetric) models do not produce explosions, from uncertainties about the adequacy of (a) the relatively simple neutrino-transport treatment used in current multi-dimensional simulations, and, (b) the rotation of the core for explaining the large asphercities and anisotropies observed in some supernovae by this mechanism, and from doubts about the origin of (a) the observed kick velocities of pulsars, and, (b) hyperenergetic supernovae, i.e., those with kinetic energy releases of upto ∼ 50 times the value for ordinary explosions, in this mechanism. 6.2.3.2
Orbital changes due to supernova explosions
As the He star in a close mass-transfer binary explodes in a supernova, the parameters of the binary orbit undergo changes in way that is immediately calculable from classical mechanics. Consider first the simplest and easiest description, in which we assume that the only effect of the explosion is a sudden, instantaneous loss of mass of amount ∆M from the star of mass M1 , and that the initial orbit is circular. We need to relate (a) the initial orbit radius ai to the semi-major axis af of the final, eccentric orbit, and, (b) the initial and final orbital periods Pbi and Pbf , respectively, as also find the eccentricity e of the final orbit. This is done by invoking the basic conservation laws 10 , and the use of the notation of Sec. (6.1.2.1) proves most convenient. As shown by Flannery and van den Heuvel (1975), the above relations are expressed succinctly in terms of the ratio µ ≡ Mtf /Mti of the final and initial total masses of the system. (Of course, Mti ≡ M1i + M2i , Mtf ≡ M1f + M2f , and ∆M = M1i − M1f = Mti − Mtf .) The results are: µ af , = i a 2µ − 1
Pbf µ = , i Pb (2µ − 1)3/2
e=
∆M 1−µ = , µ Mtf
(6.13)
from which one conclusion is immediately obvious. The characteristic appearance of the factor 2µ−1 in the above expressions for the size and period of the orbit, it follows that the system becomes unbound (i.e., the binary is disrupted) if µ < 1/2, i.e., if more than half of the total mass of the system is lost in the explosion. 10 We
can cast these as the conditions that (a) the pre-explosion radius of the orbit equals the periastron distance in the post-explosion orbit, and, (b) the relative velocity of the two stars is unchanged by the explosion. See Verbunt and van den Heuvel 1994.
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This is a famous result, which follows from the virial theorem [van den Heuvel 2001], and which caused considerable worries during early attempts at understanding the birth of neutron stars in supernova explosions in massive binaries. Since most of the mass of the more massive primary would be lost in the explosion (remember from above that the He core has something like only a quarter of the total mass of the original primary), went the argument, µ < 1/2 will surely happen, and the system will be disrupted upon explosion. How would we ever have a neutron star in a binary? The resolution came from mass transfer, which we have described in earlier sections. The initial massive primary will transfer most of its mass to its companion, thereby becoming a He star — the less massive component of the system. The supernova explosion of this He star will cause only a modest amount of mass to be thrown out of the system, and µ will come nowhere near the critical value given above. Put in another way, the huge and massive envelope of the initial primary that would have been expelled from the system by the supernova, had there been no mass transfer, is now sitting safely on the other star, and so remains very much a part of the system. The reader must not get the impression, however, that the above resolution implies that no neutron-star producing supernova in a binary ever disrupts the binary. Indeed, the fact that a huge majority of the known rotation-powered pulsars are single suggests that, in addition to those which are born in the supernovae of single massive stars, some of these pulsars became single because the supernovae in which they were born disrupted the binaries. Another immediate consequence of the explosion is that the center of mass of the binary system is given a “kick” velocity vcm , so that the whole system becomes a “runaway” one. This velocity is given in terms of the orbital velocity v1 of the exploding star by vcm =
∆M Mtf
v1 = ev1 ,
(6.14)
showing that the largest possible runaway velocity is v1 . Of course, the above is a very simple description, and real supernova explosions are likely to be more complicated. In particular, dynamical effects of the explosion and the ejecta may be quite important. An effect whose importance has been realized for a considerable time now, and which has been studied in some detail in the hydrodynamical simulations of recent years, is that of an asymmetric explosion. If the explosion does not have a perfect spherical symmetry, which is quite possible, the proto-neutron star (PNS) will be given a net recoil momentum (or, as it is often called in the
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literature, a “kick”) — a sort of rocket effect — which the resultant neutron star will carry. Indeed, the observed, relatively large space velocities of young neutron stars make the idea of such kick velocities ∼ 100 km s−1 quite plausible. Furthermore, the frequent occurrence of considerable eccentricities in Be-star X-ray binaries also strongly suggests that appreciable natal kicks are given to neutron stars. We return to this point later. A different effect, which has also been known for a long time, is that of the explosion ejecta impinging on the companion star with a velocity vej , say, which (a) will give the companion a “kick” velocity vr along the radius vector joining the two stars, and, (b) can even ablate material off the companion’s surface, if vej exceeds the escape velocity vesc from the companion. These effects were explored by Colgate (1970), who showed that (a) vr is simply given by M2 vr = (Ω/4π)∆M vej from momentum conservation, Ω being the solid angle subtended by the companion at the center of the explosion, and, (b) the ablation effects are logarithmic, and so generally small, multiplying on the right-hand side of the above equation by a factor [1 + ln(vej /vesc )]. McCluskey and Kondo (1971) incorporated these effects into the above calculation of the changes in orbital parameters, showing that af /ai and e generally increased because of these effects, i.e., the post-explosion orbit became less bound and more eccentric, as may have been expected. 6.2.4
Evolution of Proto-Neutron Stars
“And so one is left”, write WHW, “about 10 ms after the core has bounced, with a hot dense proto-neutron star accreting matter at its outer boundary at a high rate.” Formally, we could say that a proto-neutron star (PNS for short) is born when the stellar remnant becomes gravitationally decoupled from the expanding ejecta [Prakash et al. 2001, henceforth P01], which happens somewhat later, at ∼ 0.5 − 1 s, say. The question then is: what happens next to this proto-neutron star (PNS for short)? How does it turn into the neutron star that we eventually see in rotation- and accretion-powered pulsars? In a few words, the answer is: it converts its (still large) content of electrons and protons into neutrons (i.e., neutronizes; see Chapter 2), and cools, first on short timescales (∼ 10 sec to 1 min, say) by the emission of neutrinos, and then on long timescales (∼ 10 − 106 years, say) by the emission of both photons and neutrinos. Note that, at the very beginning, the neutrinos are trapped inside the core of the PNS (in other words, this core is optically thick to neutrinos), and it
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is only when they begin to diffuse out that the evolution occurs, in terms of both the de-leptonization and the cooling of the core. There are several distinct stages in PNS evolution (P01) which are instructive to consider: we do so now. The first stage, which follows the core bounce and and the stalling of the prompt bounce shock at a radius ∼ 200 km, say, occurs on a hydrodynamic timescale, and has been aptly described by WHW, as quoted above. The PNS at this initial stage consists of (a) a dense, relatively cool core of mass ∼ 0.7M and radius ∼ 20 km, say, which contains trapped neutrinos, and, (b) a relatively low-density and hot mantle of roughly comparable mass and radius 200 km, which is still accreting matter, and losing energy by thermal neutrino emission. The second stage starts as the shock, revived on the Wilson timescale, blows off the stellar envelope, accretion onto the PNS stops, and the mantle of the PNS collapses onto the core after de-leptonization (and consequent loss of lepton pressure-support) and extensive neutrino losses. This occurs on a timescale ∼ 0.5 − 1 s, the PNS having a radius ∼ the core radius given above. If enough mass had been accreted already by the mantle, so that the (core + mantle) mass exceeds its Oppenhemier-Volkoff (OV) limiting mass (see above and Chapter 2), the whole PNS could collapse at this stage, and form a black hole. Since we are mainly concerned with neutron stars in this book, we do not consider here the details of this process any further. When similar possibilities arise in subsequent stages below, we mention them briefly again. It is during this stage that the PNS is thought to develop strong convection, particularly in low-latitude regions at intermediate radii (10 − 15 km; remember that the PNS has a radius ∼ 20 km at this stage): results from a typical two-dimensional simulation demonstrating this are shown in Fig. 6.6 [Janka 2004], wherein the convective cells can be clearly seen. This convection is driven by the negative gradient of the lepton fraction µ, i.e., ∂µ/∂r < 0, that exists at this stage. The role played by the rotation of the PNS on this convection is crucial and interesting, as may be expected, and is indicated in Fig. 6.6. The specific angular momentum j is roughly constant on cylinders parallel to the rotation axis (following the famous Taylor-Proudman theorem in hydrodynamics), and generally increasing as we go outward in a direction perpendicular to the rotation axis. But there is a region of almost constant j precisely in the range of intermediate radii mentioned above: this is where strong convection occurs. Interior to this, j has a steep gradient perpendicular to the rotation axis, which drastically
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Fig. 6.6 Convection in proto-neutron stars. Shown are meridional cuts through a nascent neutron star, 0.75 s after its formation. Rotation axis is the y-axis. Left panel: contours of specific angular momentum j, color-coded with color scale given. Right panel: contours of total velocity of fluid motion, similarly coded. Arrows indicate flow direction in the meridional plane. Reprinted with permission by the International Astronomical Union from Janka (2004), in Proc. IAU Symposium no. 218, Young Neutron Stars and Their Environments, eds. F. Camilo and B. M. Gaensler, Astronomical Society of the Pacific, p.3.
suppresses convection in this direction: the weak convective cells that do occur are much elongated along the rotation axis. By contrast, convection in a non-rotating PNS would occur in spherical shells. The third stage, which occurs on a timescale 10 − 15 s, is that of the crucial neutrino diffusion11 and escape from the core, which de-leptonizes it, and also generates a lot of heat in it, as energetic (∼ 200 − 300 MeV) neutrinos lose most of their energy in the core, before escaping with energies ∼ 10 − 20 MeV. Copious neutrino emission takes place in this stage from the PNS, carrying away most of the gravitational binding energy of the PNS: the effective surface from which this neutrino emission takes place is often called the neutrinosphere. This neutrino loss de-leptonizes the PNS, as mentioned above, with very interesting consequences, one of which is the possibility of appearance of strange matter — in the form of hyperons, pion or kaon condensates, and/or quark matter (see Chapter 5). Why would this happen at this stage? When neutrinos are trapped in the matter, as they have been before this stage, the threshold density for the appearance of 11 The diffusion timescale is ∼ R2 /cλ, R being the stellar radius and λ the neutrino mean free path.
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strange matter is extremely high, but it becomes considerably lower at this stage as neutrinos are emitted profusely. If strange matter does appear, it softens the EOS, as we mentioned in Chapter 5. This opens up another possibility for collpase to a black hole, since the OV limit is lower for softer EOS, and if the PNS mass exceeds this new lower limit, it will collapse. The fourth stage comes at ∼ 50 s, when the PNS becomes transparent to neutrinos. How does this happen? While profuse emission of thermal neutrinos continues to cool the PNS, the mean neutrino energy decreases, and so the neutron mean free path increases. At this stage, the mean free path becomes comparable to the PNS radius (which, by now, has contracted roughly to the canonical value 10 km for a neutron star), and the star consequently becomes transparent to neutrinos. The PNS is cooler now, which decreases the threshold density for appearance of strange matter further, and so raises the possibility of a late appearance of such matter, and the resultant possibility of collapse to a black hole at this stage, through the mechanism described above. The fifth stage, lasting ∼ 10 − 100 years (note the drastic change in timescale compared to the previous stages), sees cooling by neutrino emission, the crust staying hotter than the core throughout this stage, with surface temperatures Ts ∼ 3 × 106 K. The sixth stage starts when the entire body of the PNS comes to thermal equilibrium at ∼ 10 − 100 years, so that the whole star becomes isothermal, equalizing the crust and core temperatures, as a result of which the surface temperature drops very rapidly. During this stage, which lasts for ∼ 106 years, neutrino emission from the core dominates the cooling, and so determines the surface temperature of the star, which, by now, has turned into a canonical young neutron star. (We can define neutron stars with 5 ages < ∼ 10 years as being “young” for our purposes here.) Note, however, that the neutron-star evolution actually observable at this stage (as opposed to the direct neutrino signal observable, at least in principle, during the first four stages given above) is through its thermal photon emission, which comes in a waveband determined by Ts , which, in turn, is determined by the neutrino-cooling rate. Of course, we need hardly remind the reader that the most obvious photon emission from such young neutron stars — when operating as rotation-powered pulsars — is non-thermal, coming as pulses in the radio waveband, the properties and production mechanisms for which are completely different, and which occupy much of our attention elsewhere in this book, particularly in Chapters 7, 8, and 11.
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The neutrino-cooling rate is high when the URCA12 process for producing the neutrinos, i.e., the combination of alternate electron capture and beta decay (see Chapter 2) p+ e− → n+ ν and n → p+ e− + ν¯ — which was first invoked by Gamow and Sch¨ onberg in 1941 in the related, but somewhat different, context of energy-loss from the cores of pre-supernova stars, and which may be cast in the form p+e− → p+e− +ν + ν¯ to emphasize that this process changes nothing except for producing a neutrino-antineutrino pair which carries off energy13 — proceeds directly without the need for mediation by a “spectator” nucleon. Then the surface temperature Ts ∼ 105 K is relatively low, so that the X-rays emitted from the neutron star are very soft, and the emission is very weak (as the reader can readily show with the aid of Stefan’s law of blackbody radiation), making it hard to detect. On the other hand, the neutrino-cooling rate is low when the direct URCA process is blocked, and only the indirect one mediated by a “spectator” nucleon is possible, which we can write in the above vein as p + e− + (n, p) → p + e− + (n, p) + ν + ν¯. This leads to the so-called standard cooling scenario, as envisaged originally by Gamow and Sch¨onberg in their context of pre-supernova stellar cores. The temperature is then Ts ∼ 106 K, making the soft X-rays emitted by the young neutron star much more easily detectable. Around ∼ 106 years, photon emission from the stellar surface begins to dominate over the above neutrino emission from the core as the leading cooling mechanism, and the neutron star is, by this time, a typical (still relatively young) rotation-powered pulsar, as we discuss in the next section.
6.3
Rotation Power in Young Pulsars
A young neutron star starts operating as a rotation-powered pulsar if it is rotating fast enough, and if its magnetic field is large enough. How 12 The name URCA, coined by Gamow, is not really the pseudo-formal acronym Unidentified Regenerative Cooling Agent that Gamow himself had sometimes circulated to obscure the real story behind the name, as he explained later. The name is simply that of a famous casino in Rio de Janeiro, where Gamow once pointed out to Sch¨ onberg that the fast disappearance of thermal energy from stellar cores due to URCA-process neutrinos was rather like the fast disappearance of money at the roulette tables around them. 13 As Clayton beautifully explains in his 1968 book Principles of Stellar Evolution and Nucleosynthesis, the energy that the two neutrinos carry away has to come ultimately from the thermal energy of the system, given through the kinetic energy of the electron to the neutrinos in the situation we are considering here. This is the heart of the URCA cooling mechanism.
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the power of such pulsars comes from their rotation has been explained in Chapter 1, and the observed properties of such pulsars, as well as suggested mechanisms for their operation, are detailed in Chapters 7, 8 and 11. If such a pulsar is single, i.e., not in a binary, then either (a) it was born in the supernova of a massive, single star, or, (b) it was born in a binary, but the supernova that created it disrupted the binary. A large majority of the known rotation-powered pulsars is single, of course, so that the pulsar properties summarized in the above chapters complete a first description of these pulsars. Accordingly, we focus here on those young, rotation-powered pulsars which are in binaries, the companion being a “normal” star, either (a) a massive star in a relatively wide orbit (for conservative evolution), or, (b) a relatively low-mass star in a relatively narrow orbit (for commonenvelope evolution), as we explained in Sec. (6.2). The former type is the natural progenitor of HMXBs, and the latter, that of intermediatemass X-ray binaries like Her X-1, as we shall describe in the next section, where we discuss the mid-life behavior of these neutron-star binaries, during which phase accretion occurs from the companion to the neutron star, and the latter becomes an accretion-powered pulsar. These young, rotationpowered pulsars in binaries in the pre-accretion phase are sometimes called ante-deluvian, using biblical14 imagery [see van den Heuvel 2001]. At present, we know only about four such ante-deluvian systems. In three of these, the young, rotation-powered pulsar is in a highly eccentric, relatively wide orbit around a massive (∼ 10M ) O, B, or Be star. These are: (a) PSR B1259-63, with Pb ≈ 3.4 y, e ≈ 0.87, and a Be-type companion, (b) PSR J0045-7319, with Pb ≈ 51 d, e ≈ 0.81, and a B-type companion, and, (c) PSR J1740-3052, with Pb ≈ 231 d, e ≈ 0.58, and a possible B-type companion. The first two were discovered in the early- to mid-1990s [Johnston et al. 1992; Kaspi et al. 1994], and the third one recently [Stairs et al. 2001]. The fourth one, PSR B1820-11, with Pb ≈ 358 d, e ≈ 0.8, is thought to have a low-mass companion, and was also discovered in the mid-1990s [Arzoumanian 1995]: this may be a prototype for the likely progenitors of low-mass X-ray binaries, a point to which we shall return later. Binaries containing young, rotation-powered pulsars are not all ante-deluvian, however: we shall summarize the other categories later, but let us first consider these ante-deluvian systems in a bit more detail. 14 The original meaning of the word is “before the biblical flood or deluge”. Here, “deluge” has been liberally interpreted to mean mass flow onto the neutron star from its companion, which, as we shall see later, can run away for a massive companion under certain circumstances, engulfing the neutron star, and thus creating a literal deluge.
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Missing Links: Ante-Deluvian Systems
In our study of the evolution of neutron-star binaries, we shall encounter certain classes of binaries, each of which logically represents the intermediate stage between two well-known and much-observed classes of binaries, and yet did not have a single observed example for many years. In the parlance of biological evolution, such examples are called missing links, and we shall adopt the same name here, describing these missing links at appropriate places in this chapter. Ante-deluvian systems are a good example of such missing links. With the profusion of known, single, young, rotation-powered pulsars, and that of known X-ray binaries containing accretion-powered pulsars, it was clear for a long time that there must be pre-accretion — or ante-deluvian — bianries containing young, rotationpowered pulsars with main-sequence companions (most of which would be massive stars). Yet, before the discovery of PSR B1259-63 by Johnston et al. in 1992, not a single example was known. As indicated above, we know four such systems now. We cannot over-emphasize the crucial importance of the discovery of such missing links, since it is these that have put our overview of the evolution of neutron-star binaries on a firm basis, with every link in the evolutionary chain ringing true. We shall return later to another deeply significant missing link.
6.3.2
Probes of Be-Star Outflow: PSR B1259-63
The observed ante-deluvian systems turned out to be quite valuable in another way, namely, by yielding important diagnostics of the nature of the outflows from the massive companions in these systems as the radio pulses from the young pulsar probed the conditions of the plasma around the binary orbit, particularly near the periastron, where the outflow’s effects on pulses are the most pronounced. This point was brought home beautifully by PSR B1259-63 (the first ante-deluvian pulsar binary to be discovered) as soon as early observations of this binary [Johnston et al. 1992, 1994] clearly demonstrated how the radio pulses were scattered and “washed out” by the plasma as the periastron was approached, the pulsar being finally eclipsed for a duration ∼ 40 days during periastron passage. Let us consider PSR B1259-63 in a bit more detail. The massive companion in this binary is a Be star named SS2883, with a mass Mc ∼ 10M , and a radius R∗ ∼ 6R . As we discuss below, Be (or B-emission) stars rotate rapidly, and so eject matter from their equatorial regions, which forms
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an outflow in the shape of a cool, dense disk in the equatorial plane of the star. In addition, a hot, tenuous, more-or-less spherically symmetric wind, driven by radiation pressure, also blows from other stellar latitudes, in the usual manner of B stars. These two forms of outflow both interact with the pulsar as it moves around the binary orbit, and, in an eccentric orbit, the interaction is clearly the strongest at periastron, where the two stars are closest to each other. 6.3.2.1
The Shvartsman surface
The interaction process goes as follows. First, the above outflow from the Be star come into a balance with the outflow — or pulsar wind — from the young pulsar in a manner which was first considered in 1970 by Shvartsman, and which is determined essentially by the ratio of the momentum fluxes of the two flows. Numerous authors calculated the details of this process in the mid-1990s [see, e.g., Kochanek 1993; Lipunov et al. 1994; Tavani et al. 1994], showing how a shock is formed around the pulsar, such that, everywhere on the (curved) surface defined by the shock, the radiation pressure of the pulsar wind balanced the ram pressure of the Be-star outflow. For this surface, whose symmetry axis is along the line joining the two stars, and which is generally an oblate surface not centered on the pulsar [see Lipunov et al. 1994], Ghosh (1995) proposed the name Shvartsman surface, while Lipunov et al. proposed the name Shvartsman radius for the distance rs of the shock from the pulsar along the line joining the two stars. Clearly, rs is the radius of the Shvartsman surface at its apex or “nose”. Calculation of the Shvartsman radius is straightforward once we know the structure of the Be-star disk and wind outflows, and the latter are wellknown15 [see, e.g., Ghosh 1995 and references therein]. Summarizing, the density structures of both disk and wind are represented well by power-law forms: n R∗ , (6.15) ρ(r) = ρ∗ r where ρ∗ is the density at the stellar surface. For the disk, ρ∗ ∼ 10−13 − 10−11 g cm−3 , while that for the wind is ∼ 500 − 1000 times smaller. The exponent is n ≈ 3 − 4 for the disk, and n ≈ 2 for the wind. The velocity structure is quite simple: since the closest distance of approach of the two 15 For the wind, the velocity structure is essentially the same as that for massive O/B stars described below.
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stars (i.e., the periastron distance) in such systems is ∼ (10 − 100)R∗ (it is actually ∼ 25R∗ for PSR B1259-63), the velocities in winds and disks probed by the pulsar are the terminal velocities, which are 1000 km s−1 for the wind, and 150 − 300 km s−1 for the disk. Details of the calculation of the Shvartsman radius rs in terms of (a) the stellar separation d(θ) (here, θ is the true anomaly in the binary orbit; see Appendix B), (b) the efficiency of conversion of the spin-down luminosity of the pulsar into the luminosity of the pulsar wind, and, (c) the momentum flux (or ram pressure) Υ of the Be-star’s outflow, are given in Tavani et al. (1994), and further relevant modifications are given in Ghosh (1995). We only quote the final result from the latter author here, 1/2 R n2 −1 rs ∗ ≈ 7.2 , d(θ) Υ d(θ)
(6.16)
obtained under the approximation rs d(θ) and R∗ d(θ). In this 2 equation, Υ ≡ 1.4 × 104 ρ−11 v200 is a convenient, dimensionless measure of the outflow’s ram pressure, ρ−11 being ρ∗ in units of 10−11 g cm−3 , of course, and v200 being the terminal velocity in units of 200 km s−1 . Further, the spindown luminosity of PSR B1259-63, E˙ ≈ 8 × 1035 erg s−1 has gone into the above equation, making it specific to this pulsar. The second step in the interaction process is that crucial bit of physics which determines if the post-shock plasma from the Be-star outflow can couple effectively to the pulsar magnetosphere, which is essential for this plasma to have any influence on the pulsar’s emission and spin-rate. A sufficient condition for this [Ghosh 1995] is that the above Shvartsman surface lies partly or wholly inside the accretion radius ra defined and discussed in Sec. (10.1.1.1). This is so because, under these circumstances, the flow of the post-shock plasma inside those parts of the Shvartsman surface which lie inside ra will be dominated (by definition) by the pulsar’s gravity, and so arrive at the magnetospheric boundary rm essentially with the free-fall velocity, ready to interact with the pulsar’s magnetic field, and, through it, with the pulsar. To cast the above condition in a convenient form, Ghosh (1995) noticed that, since the Shvartsman radius rs scales with the stellar separation d(θ), and since dmin = a(1 − e) is the minimum separation at periastron, rs is also a minimum there. Suppose that we form the ratio γ ≡ rsmin /ra , of this minimum Shvartsman radius to the accretion radius, expressing it in the form:
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−1 ,
(6.17)
with the aid of Eqs. (6.16) and (10.2). Then, argued this author, the condition γ < 1 will ensure that at least a part of the Shvartsman surface can supply plasma that will couple effectively to the pulsar (and so influence its rotation rate, for example: see below). With typical outflow parameters quoted above, the reader can readily show, as Ghosh (1995) did, that the above condition is satisfied for disks (0.03 < ∼ γ < ∼ 0.7), but violated for < winds (50 < γ 450), in the PSR B1259-63/SS2883 system. ∼ ∼ What happens when this condition is violated? The reader may be surprised to learn that the answer is not fully known yet. Since the postshock plasma flow is then not readily dominated by the pulsar’s gravity, it would seem reasonable that the plasma would tend to flow “around” the pulsar instead of towards it, as was suggested in the mid-1990s [ Kochanek 1993; Tavani et al. 1994]. But what happens then? There can, of course, be no accretion onto the neutron star under these circumstances, but there would be none even when this condition is satisfied, if the pulsar is rotating at such fast rates as it is expected to have at this stage. As we indicate below and discuss in detail in Sec. (12.1.2), “propeller” torques exerted by the neutron star then transfer energy and momentum to the plasma, expelling it, and spinning down the neutron star in the process. What really happens can be determined only by detailed modeling of the plasma flow under these circumstances.
6.3.2.2
Propeller spindown
Following the discovery of PSR B1259-63, Ghosh (1995) proposed that such systems would be ideal for mapping out the propeller torque that had been proposed originally in the mid-1970s [Illarionov & Sunyaev 1975] as the mechanism which spins down a rotation-powered pulsar after its initial, extremely rapid rotation has been braked by the electromagnetic torque (see Sec. 12.1.1), but when the pulsar is still rotating too fast to accrete mater from its surroundings. We discuss the propeller torque quantitatively in Sec. 12.1.2, and summarize its essential manifestations in PSR B1259-63 system. Here, we stress that the role of this torque in ante-deluvian systems is particularly important, since it decides the specific manner in which a young, rotation-powered pre-accretion pulsar spins down, starts accreting,
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and so turns into an accretion-powered pulsar. The PSR B1259-63/SS2883 binary has been a famous prototype in this respect, as it has been argued in the literature [Cominsky 1993; Ghosh 1995] that the rotation-powered pulsar with a spin period P = 47 ms in this system, as it progressively spins down, may successively mimic such accretion-powered pulsars with Be-star companions as A 0538-66 (P = 69 ms), 4U 0115+63 (P = 3.6 s), EXO 2030+375 (P = 41.8 s), and A 0535+26 (P = 104 s). Such evolution in the spin- or rotation-period of a neutron star through various phases of its life is known as spin evolution, which we discuss in Chapter 12. 6.3.2.3
Tilted Be-star disks
The PSR B1259-63/SS2883 binary also proved instrumental for the first quantitative investigation of a feature of Be-star binaries which had been suggested earlier, but not been amenable to direct probe. We have described above that Be stars have equatorial outflow disks. The question is: in a given binary system, what is the orientation of this disk relative to the binary’s orbital plane? The answer is not as trivial as it might seem. In primordial binaries, the rotational angular momenta of both stars are invariably assumed, with very good justification (since the stars formed from the same rotating gas cloud), to be aligned with the orbital angular momentum, so that the Be-star disk would be in the orbital plane. But we are concerned with a neutron-star Be-star binary here, which is quite another story. The supernova that produces the neutron star may also play a role in deciding the post-supernova orientation of the orbital plane, if the neutron star receives an appreciable “kick” velocity at birth: we believe today that this is what happens in many cases, possibly depending on the supernova mechanism. The direction of this kick has nothing to do with the direction of the Be star’s rotation axis, of course, so that the resultant orbital plane will not, in general, remain in the plane of the Be-star’s disk. In fact, if the kick velocity is comparable to the orbital velocities of the stars, we can expect considerable tilts (∼ 10◦ − 70◦ , say) between the orbital plane and that of the Be star’s disk. In the context of the PSR B1259-63/SS2883 system, the idea of a tilted Be-star disk was introduced by Johnston (1994), by Ghosh (1995), and by Melatos et al. (1995), and further work was done by various authors, notably by Johnston and collaborators. A natural explanation for the remarkable fact that the radio emission from the pulsar in this system is eclipsed for ∼ 40 days around periastron passage is that, from our point of view, the pulsar
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disappears behind such a tilted disk for this duration, as shown in Fig. 12.1, and this is what the above authors invoked to infer the detailed properties of the Be-star disk. Ghosh (1995) focused on the propeller spindown effects during periastron passage, while Melatos et al. (1995) focused on the effects of this on the orbital variations of the dispersion measure and the rotation measure around periastron passage. Basic disk characteristics obtained by these approaches were rather similar: (a) a longitude of ωD ≈ 90◦ for the ◦ position of the disk (see Fig. 12.1), (b) a tilt angle of > ∼ 60 by the former ◦ ◦ method, and i ∼ 10 − 70 by the latter method, (c) a disk thickness, as measured by the half-opening angle, δθ ≡ h/r (h being the half-thickness of the disk at radius r), of δθ ∼ 1◦ − 2◦ by the former method, and δθ ∼ 0◦ .5 − 5◦ by the latter, although this parameter was not constrained very well, (d) a density scale ρ∗ which was on the low side of the canonical range given in Sec. (6.3.2.1), and, (e) a density power-law index n, as defined above, of n ∼ 3.5 − 4. The last three disk properties can be compared with those which had been either observationally inferred [Waters 1986] or theoretically calculated [Bjorkman & Cassinelli 1993] for single Be stars, and they do agree [Ghosh 1995]. Indeed, the parameterization used above was suggested by these works on single Be-star disks. But the first two specify the orientation of the disk relative to the binary system, for which the comparison would be with properties of Be-star disks in X-ray binaries, including those binaries which have been suggested above as descendants of systems like PSR B1259-63/SS2883. This is discussed in Ghosh (1995), to which we refer the reader for detail, mentioning here only one essential feature of tilted Be-star disks. While there is no particular reason to believe that there is a preferred range of tilt angles i in the immediate post-supernova binary, it is clear that i must subsequently decrease with time, i.e., the disk will become more aligned with the orbital plane, as the Be-star binary ages, due to tidal torques. Since the same torques also circularize the binary orbit, we would, then, expect ante-deluvian Be-star binaries like PSR B1259-63 to have both tilted disks and highly eccentric orbits, and, after accretion starts and the system becomes a Be-star X-ray binary, we would expect (a) the younger, shorter-spin-period (P ∼ 10 − 100 s) systems like EXO 2030+375 and A 0535+26 (see above) to still have tilted disks and eccentric orbits, since tidal effects have had a relatively short duration of operation, and, (b) the older, longer-spin-period (P ∼ 400 − 1000 s) systems like A1118-61 and X Per to have nearly circular orbits and aligned disks, since tidal effects have had enough time to go to conclusion.
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How do we test these predictions? One diagnostic is the behavior of Hα line — the emission site for which is believed to be the Be-star disk — as the neutron star passes around the periastron, accompanied by enhanced X-ray emission due to enhanced accretion. If the disk is in the orbital plane or close to it, the tidal torques, which become very powerful during periastron passage, will wreak such havoc on the Be-star disk that the Hα line profile will be grossly perturbed. On the other hand, if the disk is greatly tilted, the tidal torques will have relatively little effect on it, and Hα emission will remain essentially unchanged. Indeed, Hα emission is basically unaffected during the periastron passage of PSR 1259-63, as also during that of EXO 2030+375, but is much enhanced during the X-ray outburst of A1118-61 [Ghosh 1995 and references therein]. Further applications of such ideas to the massive X-ray binary GX 301-2 have been made by Pravdo and Ghosh (2001).
6.3.2.4
Recent work: further orbital dynamics
Since the discovery of PSR B1259-63, about 15 years of data, covering five periastron passages of 1990, 1994, 1997, 2000, and 2004, have now been collected, and, as Johnston and co-authors have described, long-term trends in its spindown behavior can be studied today. From an analysis of data upto the periastron passage of 2000, Wang et al. (2004) showed that these trends appear to be more complicated than they were earlier thought to be. The extra spindowns during periastron passage, in excess of that expected from pure electromagnetic braking, i.e., δP/δPEM in the notation of Sec. 12.1.2, are inadequate for describing the entire data-span satisfactorily, even when glitches (see Sec. 7.2.3.4), as expected in young pulsars, are allowed for. The implications of the recent values of δP/δPEM for propeller spindown are summarized in Sec. 12.1.2. We close our discussion with an intriguing result from the above Wang et al. analysis, namely that, if a prominent glitch in 1997 is taken into account, then the rest of the timing residuals (see Chapter 7) upto the 2000 periastron passage can be described well by postulating a series of jumps in the projected semi-major axis a sin i of the binary orbit, one during each periastron passage, the strengths of the jumps being adjusted, of course, to fit the data. While this is an empirical result, it is interesting to consider if viable physical processes could possibly lead to this. Actual jumps in the semi-major axis a of the orbit are ruled out, of course, as these authors note, since these would imply rather enormous changes in orbital period, which
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would be immediately detectable in this era of ultraprecise pulsar timing (see Chapter 7), and for which there is no evidence whatever. What about small jumps in i, the orbit inclination? Is that physically admissible? Here, we return again to orbital dynamics with a fast-roating, tilted Be star, and to the role of tidal coupling between them during close periastron passage, as discussed above, perhaps enhanced by possible resonances. Is it thus feasible to produce a small “jitter” in i during such passage, i.e., a sort of minute “nodding” of the orbit? The basic idea remains interesting but uncertain, and, as the authors themselves state, it may not work in this specific instance, owing to the rather large stellar separation at periastron in this system.
6.4
Accretion Power in Middle-Aged Pulsars
As the neutron stars in the above ante-deluvian systems spin down and rotate slower than the critical rate given in Chapter 12, accretion begins on the neutron star, and an X-ray binary is born. Such binaries are normally classified according to the mass of the companion star, viz., massive, or high-mass, X-ray binaries (HMXB), low-mass X-ray binaries (LMXB), and intermediate-mass X-ray binaries (IMXB), and the evolutionary status of the companion becomes rather crucial at this point, since there must be mass transfer from it at an adequate rate (M˙ ∼ 10−10 − 10−8 M y−1 , say) to the neutron star to make the latter a bright X-ray source. On the whole, this evolutionary status is such that the companion, whether massive, lowmass, or intermediate mass, fills (or very nearly fills) its Roche lobe. However, there are a few notable exceptions. First, bright O/B stars can drive sufficiently hefty stellar winds that gravitational capture from them can provide enough accretion to power bright HMXBs, as we describe in Chapter 10. Indeed, it was thought in the early days of X-ray binary studies that the well-known HMXB Cen X-3 (as also the famous black-hole binary Cyg X-1, to which similar arguments apply) and similar sources accrete in this manner. But now there is considerable evidence that there is Roche-lobe overflow in this system, so that the massive companion in Cen X-3, and possibly in other HMXBs, is at or close to the point of filling its Roche lobe, in addition to driving a stellar wind. Second, in Be-star X-ray binaries, the Be-star companion usually lies deep inside its Roche lobe (for reasons explained below), and the normal, radial stellar wind from such a companion is rather weak (see above). But these stars have cool, dense
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outflow disks, as discussed above, and most of the significant accretion onto the neutron star takes place during its passages through this disk. This is what leads to the transient behavior of Be-star X-ray binaries, i.e., the fact that they are detected in the X-rays only or largely during their outbursts. Finally, in very close LMXBs, the low-mass companion may fill its Roche lobe not because of its nuclear evolution and consequent expansion (see Appendix C), but because of orbit shrinkage (which shrinks the Roche lobe proportionally: see Appendix B) due to gravitational radiation (discussed in Sec. 6.5.3.1) and/or magnetic braking (discussed in Sec. 6.5.3.2). In the latter case, then, it is really the orbit dynamics — not the companion’s evolution — that drives mass transfer. We summarize now the basic evolutionary characteristics of the neutron star and its companion during the above accretion-powered X-ray binary phase, which we can call the “middle age” of the neutron star, as opposed to its early life as a rotation-powered pulsar in an ante-deluvian binary, as discussed earlier, and its old age as a “recycled” rotation-powered pulsar, to be discussed later. This middle age is shown as parts of the full evolutionary diagrams in Fig. 6.2 [van den Heuvel 1991, 1992, 2001]. 6.4.1
Evolution to Massive X-Ray Binaries
Consider high-mass X-ray binaries (HMXBs) first, in which this phase is relatively short-lived, as might be expected, since massive stars evolve fast (see Appendix C). Among HMXBs, we first take up the subcase of a neutron star with an O/B supergiant companion. How such a system may be produced is shown in Fig. 6.2. Suppose we start with a primordial binary of ∼ 20M and ∼ 8M with an initial orbital period of Pb ∼ 4.7 d. Then, through conservative mass transfer and the subsequent supernova of the He star, as described earlier, we arrive at a neutron-star binary with a ∼ 23M unevolved companion and an orbital period Pb ∼ 12.6 d in ∼ 6 × 106 y. This massive companion then evolves, finishes core H-burning, and expands to become an O/B supergiant in 107 y, driving a strong stellar wind. Accretion from this wind turns the neutron star into a bright X-ray source — an accretion-powered pulsar like Cen X-3 or Vela X-1 — as discussed above. But evolution is very fast at this stage: the massive companion fills its Roche lobe in another 104 − 105 y, after which mass transfer proceeds at such a huge rate that the X-ray source is totally engulfed by the transferred mass, and so extinguished from our view by absorption, and the system begins a common envelope (CE) phase of evolution, the essential properties of
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which we have discussed in Sec. 6.1.3.2. The consequences of CE evolution at this particular stage of HMXB evolution will be summarized in the next section. Thus, this X-ray binary phase lasts only 104 − 105 y. Towards the end of this duration, there will, of course, be both wind accretion and Roche-lobe overflow, as observed in sources like Cen X-3. Among HMXBs, consider next the subcase of a neutron star with a Be star companion. The possible origin of such a system is shown in Fig. 6.7. Suppose we start with a primordial binary of ∼ 13M and ∼ 6.5M with an initial orbital period of Pb ∼ 2.6 d. Then, through conservative mass transfer and the subsequent supernova of the He star, as described earlier, we arrive at a neutron-star binary with a ∼ 17M unevolved companion and an orbital period Pb ∼ 30.6 d in ∼ 1.5 × 107 y. This is the Be-star companion. The obvious question that arises immediately is: what is the difference between this case and that given in the previous paragraph? To
Fig. 6.7 Evolutionary scenario for formation of Be-star X-ray binaries through conservative mass transfer: see text for detail. Reprinted with permission by Springer Science & Business Media from van den Heuvel (2001) in The neutron star-black hole connection, c 2001 Kluwer Academic Publishers. eds. C. Kouveliotou et al..
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answer this, first note that Be-star X-ray binaries have generally wider orbits and lower-mass companions than canonical HMXBs with O/B supergiant companions, which is why the the former lie deep within their Roche lobes, while the latter fill their Roche lobe quickly as they begin to evolve off the main sequence (see above). A natural dynamical origin for this can be found if we propose, as above, that Be-star X-ray binaries started with significantly lower primary masses in the initial, primordial binary: this was done by van den Heuvel in 1983. The dynamical arguments are transparent, and we now summarize them. For conservative mass transfer, as shown in Sec. 6.1.3.1, the change in the orbital size (and so orbital period) is immediately calculable, and this contains the key, when we remember from the discussion below Eq. (6.8) that the ratio of the mass of the He-core to that of the original primary mass, mHe ≡ MHe /M1 , decreases with decreasing M1 . This is easily seen by rewriting Eq. (6.3) in terms of mHe and the initial mass ratio q ≡ M1i /M2i in the form af −2 = m−2 , He [1 + q (1 − mHe )] ai
(6.18)
and noticing that most of the variation of the right-hand side of the above equation would normally come from the first factor, while the factor within the square brackets would stay around ∼ 2.5 − 2.7 for the range of values of q and mHe of interest here (2 ≤ q ≤ 2.5 and 0.2 ≤ mHe ≤ 0.3). Recalling the scaling of mHe from Sec. 6.1.3.1, then, we get af /ai ∼ M1−0.84 , and so Pbf /Pbi ∼ M1−1.26 . This shows clearly why lighter initial primaries will have bigger increases in their orbital size and binary period due to conservative mass transfer. (Of course, the neutron-star producing supernova will cause a further increase in the orbital size, as explained earlier. This, and the fact that the different values of q in the two cases considered in the two previous paragraphs also contribute to the variations, account for the actual numbers quoted there.) With the above explanation for the wider orbits and lower companion masses in Be-star X-ray binaries, we still have to understand why the companions in these systems are Be stars, i.e., why they rotate fast and produce outflow disks, as described above. This understanding appears still to be qualitative, the basic idea being the obvious one that, if matter is transferred in a wider (or widening) orbit, it will have more angular momentum with respect to the mass-receiving star, and so will add more angular momentum to it and spin it up more. It is believed that this spinup
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is sufficient to make the companion a rapid rotator, i.e., a Be star, in the second case discussed above, and in similar cases. 6.4.2
Evolution to Intermediate-Mass X-Ray Binaries
Consider next intermediate-mass X-ray binaries (IMXBs), suggestions for the origins of which go back to the 1973 work of van den Heuvel and de Loore on binaries like Cyg X-3, and to the 1975 work of Sutantyo on binaries like Her X-1. The idea was mentioned briefly earlier. Suppose we start with a primordial binary of ∼ 15M and ∼ 2M , i.e., an extreme mass ratio, and an initial orbital period of Pb ∼ 200 − 600 d, say, i.e., a relatively wide binary [van den Heuvel 2001]. The key point, as introduced earlier, is that the secondary is unable to adjust to the fast and huge mass transfer from the primary following its Roche-lobe overflow (which occurs after ∼ 8 × 106 y of evolution of the primary), so that a common envelope (CE) forms, and the system goes into CE evolution, the essential properties of which are summarized in Sec. (6.1.3.2). In this case, the binary spirals in so that the orbit shrinks by a very large factor, af /ai ∼ 85 − 170, as an application of Eq. (6.10) with λ ∼ 1/2 and α ∼ 1 − 2 [van den Heuvel 2001] readily shows. The binding energy thus released expels the CE, and we end up with a quite close binary with a period of Pb ∼ 0.4 − 0.5 d, consisting of a ∼ 4M He star remnant from the original primary, and a ∼ 2.5M secondary (which has accreted a small amount during the above mass transfer). As pointed out earlier, CE evolution goes extremely fast. What happens next? In ∼ 6 × 105 y after spiral-in, the 4M He star undergoes supernova explosion, ejecting ∼ 2.4M of mass, which will not disrupt the system according to the criterion given in Sec. 6.2.3.2, even if the effects of the ejecta’s impact on the companion, possible ablation due to this, etc, as explained there, are taken into account. The result of this explosion, then, is a binary consisting of a neutron star and a 2.5M companion, in a very eccentric orbit with a period Pb ∼ 5.7 d. In addition, as discussed earlier, the system as a whole (i.e., its center of gravity) receives a kick velocity in the explosive mass loss, which makes it “run away” with v ∼ 140 km s−1 . Over the next ∼ 107 y, tidal forces circularize the binary orbit, and bring the period to Pb ∼ 1.7 d. After this, nothing really happens for a long time ∼ 5 × 108 y, which the 2.5M companion needs to evolve off the main sequence, and fill its Roche lobe. Then mass transfer from the companion to the neutron star via Roche lobe overflow starts, and the system becomes an intermediate-mass X-ray binary (IMXB), very much
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resembling the famous source Her X-1, complete with the runaway velocity inferred for this system from its rather large distance (z ∼ 3 kpc) above the galactic plane.
6.4.3
Evolution to Low-Mass X-Ray Binaries
“From a morphological standpoint,” wrote Webbink (1992), “the existence of both cataclysmic variables (CVs) and low-mass X-ray binaries (LMXBs) poses similar evolutionary problems. Both types of binary possess a lowmass star together with a stellar remnant in an orbit much too small to have accommodated the progenitor of that remnant.” In a nutshell, this is the problem, and the solution was also stated succinctly by Webbink: common envelope (CE) evolution, which we have introduced and discussed in Sec. 6.1.3.2. No matter what the details are — and we summarize them below — CE evolution must occur at some point in the history of these systems, as a “wholesale reduction ” (as Webbink puts it humorously) of the original orbit separation is essential for producing the narrow orbits (Pb ∼ 1h − 1d ) of these systems (see Fig. 6.8), and CE evolution does just that. To remind the reader the essentials in a few words: copious dissipation of orbit binding energy in the CE achieves both (a) severe shrinkage of the orbit, and, (b) ultimate ejection of the CE by the large energy deposited in it. 6.4.3.1
CVs
Consider CV systems first, wherein the compact star is a white dwarf, accreting matter from a low-mass companion. As we focus on neutron-star binaries in this book, we shall mention CVs only briefly as an introduction to LMXBs, and refer the reader for more detail to the excellent 1992 review by Webbink, and to the references therein. But this introduction is essential, as the importance of CE evolution in this context was first stressed by Paczy´ nski in 1976 in connection with CVs. As we have indicated above, wide initial orbits and extreme initial mass ratios are required for CE evolution which ends in expulsion of the CE, leaving an appropriate core of the primary in a narrow orbit with a low-mass companion. For CVs, initial orbital periods are expected to be in the range Pb ∼ 102 − 104 days, and initial mass ratios q ≡ M1 /M2 > ∼ 3.6. Let us briefly discuss the origin of these constraints [Webbink 1992], as they clearly indicate the basic constitution of CV systems. Consider first the former constraint: this comes from the
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Fig. 6.8 Current period distribution of cataclysmic variables (CVs). The sample contains 531 CVs. The 2 – 3 h period gap is shown in grey. The sharp cutoff at ∼ 80 min is also seen. Reproduced with permission by the Astronomical Society of the Pacific from G¨ ansicke (2005): see Bibliography. Figure kindly provided by B. G¨ ansicke.
observational fact the white dwarfs in CVs are, on the average, somewhat more massive than single white dwarfs, and the progenitors of such white dwarfs could only have been giant/asymptotic-branch giant (AGB) stars, from which it follows that the original binary must necessarily have been wide enough to accommodate such giant/AGB stars. Consider now the latter constraint, i.e., that on the initial mass ratio, whose origins are more complicated, partly involving the condition for stable operation of the CVs that form from these progenitors. These origins are twofold. First, there is the rather obvious requirement that the CE must be massive enough to be able to absorb the energy that would be deposited in it in the process of spiralling in from such wide orbits as above to such narrow orbits as CVs have (see above), before being expelled by this deposited energy. Otherwise, it would be driven off too soon, yielding a binary too wide to end up as a CV. This lower limit on the CE mass immediately translates into one on the total mass of the above giant/AGB star, and therefore into one on the mass ratio q ≡ M1 /M2 , for a given mass of the secondary. But the second requirement on the mass ratio is really the heart of the matter. Once the CV is formed, mass transfer from the
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secondary to the accreting white dwarf must proceed stably. Otherwise, the mass lost from the secondary would engulf the system again, leading to a further CE phase, which would end in total tidal disruption of the system. What does this stability require? For a donor star (i.e., secondary) on the main sequence with a convective envelope, which occurs for secondary masses M2 < ∼ 0.8M , this means that the mass transfer must be stable on the dynamical timescale, for which the mass ratio in the CV, defined as qCV ≡ MW D /M2 must be sufficiently large. Here, MW D is the mass of the white dwarf, i.e., the core of the initial primary. This lower limit on qCV can be well-approximated [Webbink 1992 and the references therein] by the analytic form: qCV
−1 3 M2 > , where ∼ 2 1 + f M − 0.43 0, x<0 f (x) = . 3.3x4/3 , x > 0
(6.19) (6.20)
On the other hand, for main-sequence secondaries with radiative envelopes, which occurs at larger secondary masses M2 > ∼ 0.8M, this means that the mass transfer must be stable on the thermal timescale, the condition for which is qCV > ∼ 4/5. From these conditions, the reader can easily show that, over the entire range of expected main-sequence secondaries, stable mass transfer in the CV requires that the secondary’s mass must not exceed somewhere between ∼ 2/3 and ∼ 5/4 of the white dwarf’s mass, the factor being larger for more massive secondaries. As the donor star evolves off the main sequence, the above constraint becomes more stringent: for a giant-branch donor, the above factor becomes ∼ 2/3, excluding all heavier secondaries, and all orbital periods of the CV exceeding ∼ 2d . It is basically this lower limit on qCV that translates into the lower limit on the initial mass ratio q described above. 6.4.3.2
LMXBs
Consider now LMXBs, for which we would expect a variation/extension of the above scenario to be applicable. This is indeed the case, as we shall see now. The two much-discussed evolutionary scenarios we summarize here are either (a) an obvious variation of the above one, with the white dwarf substituted by a neutron star originating from an appropriately heavier primary, or, (b) a possible further evolution of a CV (or closely related) system, wherein we appeal to an appropriate mechanism for driving the
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white dwarf over the Chandrasekhar limit, so that it collapses into a neutron star. However, an essential general point about LMXB-formation scenarios which we must always remember is that they describe a rare event. This point is so crucial that we cannot possibly over-emphasize it, and so we must first understand where it comes from. It is an observational fact that, in the disk of our galaxy, CVs outnumber LMXBs by a factor ∼ 104 , i.e., NCV /NLMXB ∼ 104 . Now, the expected mean lifetimes of CVs and LMXBs are very similar, since this lifetime τ is the duration of the accretion phase, the obvious rough estimate for ˙ , i.e., how long the secondary mass M2 would last if which is τ ∼ M2 /M transferred at a rate M˙ , and both M2 and M˙ are very similar for CVs and LMXBs. So, τCV ∼ τLMXB . Next, if we assume a steady state — at least approximately — for the populations of CVs, LMXBs, and related objects, as is customary, then it follows that the rates of formation and decay are equal for each of these populations. Now, since the decay rate of a population is its number N divided by its lifetime τ , it follows that its formation rate N˙ must also be given by N˙ = N/τ in a steady state. The above numbers for CVs and LMXBs then yield N˙ LMXB ∼ 10−4 N˙ CV , showing that the LMXB formation rate is very small compared to that for CVs. Thus, the formation of an LMXB is, indeed, a rare event, and a viable LMXB-formation scenario must, in Webbink’s (1992) words, “appeal to an improbable chain of events”. • Scenario 1: Helium Star Supernova Consider now the first LMXB-formation scenario listed above. The neutron star in such a system comes from a sufficiently massive (M1 > ∼ 12M , say) original primary, as explained earlier. If single, such a massive progenitor would have produced a core of mass MN S > MCh (MCh being the Chandrasekhar mass, which, after burning He and C non-degenerately, would have cooled to degeneracy, and collapsed into a canonical single neutron star, thus expelling the envelope in a canonical core-collapse supernova explosion, as explained earlier. In a binary in a CE evolutionary phase, however, this envelope has already become the common envelope surrounding the system, which consists of a He core and the low-mass original secondary. The key difference is that this He core (which may itself have a smaller C-O core, in which He-burning has been completed) is still non-degenerate when the CE is expelled, in contrast to the above CV-formation scenario, wherein the degenerate white dwarf has already formed at this point. Thus, the im-
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mediate post-CE binary we get in this case consists of a He-star with a low-mass companion. Subsequently, the He-star completes its nuclear evolution and collapses to a neutron star, the accompanying supernova causing relatively little mass loss, which is why the system does not come apart. Indeed, this puts a rather tight upper bound on the mass of the He-star, as we shall presently see. The first essential point about this scenario is that it requires even more extreme initial mass ratios, q ≡ M1 /M2 > ∼ 8, by an extension of the arguments given above for CVs, the lower bound roughly corresponding to M1 /MCh for the initial primary. Such extreme q-values imply that the initial orbital radius of the binary was so large as to be just about at the limiting value for which Roche-lobe overflow and mass transfer are still possible (we discussed this limit above). The second point is that the constraints on the mass MHe of the He-star are extremely tight. The lower bound of MHe > ∼ 3M comes from the fact that a less massive Hestar will expand so much during the He shell-burning phase [Habets 1986] that it will engulf the low-mass companion, initiating a second CE phase, which will lead to a complete merger of the two stars, i.e., a destruction of the binary system. The upper bound of MHe < ∼ 4M comes from the requirement — already hinted at above — that the mass loss in the Hestar’s supernova must not unbind the system. From the discussion given earlier, the reader can readily show that the condition that the mass loss must not exceed half the total mass of the pre-supernova binary leads to the bound MHe < ∼ (2MN S + M2 ), which yields the above numerical value for the canonical neutron-star mass, MN S ∼ 1.4M , and M2 ∼ M . Thus, the allowed range in MHe becomes very narrow indeed if M2 is sufficiently small. Also, since the allowed range never extends very far away from the upper bound required for binary survival, it follows from the earlier discussion that the immediate post-supernova orbit would be very eccentric in all cases. By the time the low-mass secondary reaches Roche-lobe contact and the system starts functioning as a LMXB, however, strong tidal forces in such a close binary will have largely circularized the orbit. Let us finally consider why this scenario represents a rather “fragile channel”, as Webbink (1992) puts it, for LMXB formation, since this is a general requirement, as we argued above. It is the combination of several constraints — generally working against each other — that severely reduces the total number of systems that would be able to satisfy all the constraints. For example, the above constraint that the initial orbit must be as wide as it can possibly be (i.e., that which makes the first Roche-lobe contact still
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possible) implies that CE ejection will fail — and the binary destroyed by merger — if the efficiency factor α is not sufficiently high. In fact, this rules out secondaries with M2 much below M , since the latter would require, for CE expulsion, spiral-in to such small orbital radii that the nuclear evolution and expansion of the He-star described above will destroy the system by engulfment and a second CE phase. On the other hand, there is an upper limit M2 < ∼ 1.5M from the conditions described above. Again, as decribed above, on the He-star’s mass there are very tight constraints indeed from both directions, which translates into correspondingly tight constraints on the original primary mass M1 , which the reader can work out with the aid of Eq. (6.8). Thus, both M1 and M2 are severely constrained, and the initial binary period Pbi is also constrained to be in a narrow range just below the upper limit for interacting binaries (see Sec. (6.1)), as explained above. The number of primordial binaries satisfying all these constraints simultaneously is quite small. Add to that the further requirement that, during the development of the above evolutionary sequence, the following condition on Pb must be satisfied around the time of the He-star supernova. On the one hand, both pre- and post-supernova binaries must be wide enough to accommodate the low-mass main-sequence secondary, but, on the other hand, the post-supernova binary, after circularization, must not be so wide that the secondary needs to evolve all the way to the giant branch in order to attain Roche-lobe contact, since that would lead to mass transfer on a dynamical timescale, as explained earlier, leading to a further CE phase, and so to a merger and destruction of the binary. It will come as no surprise, then, that the total number of systems satisfying all these constraints is tiny, and LMXB formation through this channel is a rare event. • Scenario 2: Accretion-Induced Collapse Consider next the second LMXB-formation scenario listed above, namely, accretion-induced collapse of a white dwarf in a CV or CV-like system into a neutron star. The idea again relates to what single, massive stars do, and how this changes in a binary. As explained earlier, a single star in a relatively narrow mass-range just below that given above for the He-star-supernova scenario (M1 ∼ 8 − 12M, say) produces a degenerate O-Ne-Mg core at the end of its nuclear evolution: this core grows in mass and collapses into a neutron star due to electron capture. In a binary, the collapse would be stopped if the envelope, i.e., the CE, is lost (by the
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mechanism discussed above) after the O-Ne-Mg core is created but before it has grown enough to collapse. Then we would get a CV consisting of the O-Ne-Mg white dwarf (whose mass is not far below MCh ) and the low-mass secondary. As mass transfer from the latter to the former starts and continues, the former’s mass MW D eventually exceeds MCh , and it collapses to a neutron star, yielding a LMXB. This scenario may, at first, seem a bit contrived, but immediate circumstantial evidence rallies in its favor: it is well-known that the CV systems known as classical and recurrent novae (so named because of their prominent nova outbursts, believed to be due to runaway thermonuclear burning of accreted material on the surface of the white dwarf) contain far more than their fair share of the massive O-Ne-Mg white dwarf, which are considerably less common than their less massive cousins, the C-O white dwarfs. This preferential occurrence of the most massive white dwarfs in novae is not, by itself, surprising, since it follows from a beautiful bit of basic physics, as stressed by Webbink (1992). A nova explosion occurs when the pressure p at the base of the accreted envelope of Me on the surface of the white dwarf exceeds the critical value pc ∼ 2 × 1019 dyne cm−2 at which hydrogen can undergo degenerate ignition. The quantitative condition is straightfor4 ward, viz., p = GMW D Me /4πRW D > pc (RW D being the radius of the white dwarf), and shows immediately that, since pc is constant, and since RW D decreases with increasing MW D , so does Me . Consequently, heavier white dwarfs require smaller critical values of Me , and so produce outburst more frequently, than lighter ones. As MW D → MCh , this variation is extremely steep, thus confining the interesting region for the recurrent novae to O-Ne-Mg white dwarfs very close to the Chandrasekhar limit. It is these recurrent novae that were thought in the late 1980s and early 1990s to be the prime candidates for systems that would produce LMXBs through this channel, since it was argued plausibly that (a) extremely massive white dwarfs, precariously close to the Chandrasekhar limit, would grow by these repeated episodes of accretion at a relatively low rate (M˙ ∼ a few 10−9 M yr−1 ) and a small outburst, while (b) lighter white dwarfs would actually be eroded by these episodes of accretion and large outbursts, with strongly supporting observational evidence [Webbink 1992 and references therein]. However, with the discovery of the so-called Supersoft Binary Xray Sources (SSS), an alternative candidate has also become quite plausible. These SSS are believed to have white dwarfs with companions whose initial masses (M2 ∼ (1.2 − 3)M , say) exceed that of the white dwarf. Then, as argued above, the mass transfer proceeds on the thermal timescale of the
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donor star, i.e., at a hefty rate M˙ ∼ 1 − 4 × 10−7 M yr−1 . The crucial point is that the accreted matter burns steadily at these accretion rates, leading to a continuous growth in MW D and eventual collapse [van den Heuvel 2001 and references therein] to a neutron star. Remembering the discussion on the possible CE-evolution origins of Her X-1-like systems, the reader will have noticed that some of such SSS binaries may also evolve into Her X-1 or similar IMXBs. Historically, the accretion-induced collapse scenario appeared to have a rather compelling aspect in the first half of the 1980s, when the inferred magnetic fields for the recycled millisecond pulsars (see below) then being discovered were found to be too large to be reconciled with the original idea of spontaneous decay of neutron-star magnetic fields, which had been in vogue since the early days of pulsar physics, as explained in Chapter 13. This conflict came because the recycling time, as estimated from the evolutionary calculations of recycling systems which began with a neutron star according to the He-star supernova scenario above, was convincingly shown to be so long that the pulsar’s magnetic field — if decaying spontaneously — could remain nowhere as large as was observed in recycled pulsars. Accretion-induced collapse of a white dwarf into a neutron star, which created the neutron star at a much later stage in the evolution, managed to side-step this issue. However, history took another turn by the second half of the 1980s. It became clear by then from various lines of evidence, including Kulkarni’s 1986 discovery of cool, old white dwarf companions in binary millisecond pulsars (which implied that these pulsars were actually quite old), that the notion of a spontaneous decay of neutron-star magnetic fields was untenable, and a reassessment of our understanding was essential. In an important 1986 paper, Taam and van den Heuvel argued (see Chapter 13) that most or all of the magnetic-field decay/reduction in a neutron star takes place only when it accretes. This is the so-called accretion-induced field decay, which has now become the accepted paradigm in the subject, and which we elaborate on in Chapter 13. It will come as no surprise to the reader, then, that the compulsion to invoke accretion-induced collapse has disappeared now. At the same time, it is also clear that such collapses must be occurring in some CVs under appropriate conditions, as described above, and that this channel very likely does account for some of the observed LMXBs, as well as for IMXBs similar to Her X-1. But we shall not enter here into a discussion of how rare this channel might be, referring the curious reader to Webbink’s excellent 1992 review.
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We close here our discussion of suggested evolutionary pathways for LMXB formation with a brief mention of another suggested pathway, namely, the triple-star evolution scenario of Eggleton and Verbunt (1986). This scenario is a variant of the standard HMXB evolution scenario, wherein such a system with an extreme mass ratio has a relatively distant, low-mass, third member, which is still within the reach of tidal interaction. After the formation of the HMXB, as described above, and the completion of the canonical wind-powered HMXB phase (see Sec. (6.5.1)), the massive companion evolves to fill its Roche lobe, whereupon rapid mass transfer on a thermal timescale engulfs the whole binary, because of the extreme mass ratio, thus leading to CE evolution. This is described in more detail in the next section, where we point out that, in some of these systems, the energy released by the binary’s spiral-in is inadequate for expelling the CE, and the neutron star then sinks to the center of the giant companion, producing a ˙ Thorne-Zytkow object. However, if there is a third low-mass member, it is also engulfed by this CE, and the extra energy released by its spiral-in may be adequate, under the correct circumstances, to eject the CE, producing a LMXB. Why do we need to invoke a third star? As Webbink (1992) clarifies, this scenario was originally devised to explain the existence of black-hole X-ray binaries with low-mass companions like A0620 − 00, in an era when accretion-induced collapse was the favorite formation scenario due to reasons explained above. Now, as the reader can see, accretion-induced collapse would never yield a black-hole X-ray binary with a black-hole mass above the maximum possible neutron-star mass (see Chapter 5), which is what systems like A0620 − 00 are. Hence the ingenious Eggleton-Verbunt triple-star device. In the contemporary context, of course, an extension of the He-star supernova scenario to even more massive initial primaries would naturally yield A0620 − 00-like systems without any trouble, so that the above device is not really essential. Nevertheless, initial triple systems, such as envisaged by Eggleton and Verbunt, do appear to exist in our galaxy, and actual LMXB formations by this channel may well occur.
6.5
Rotation Power in Old, Recycled Pulsars
We now consider the “old age” of a neutron star in a close, interacting binary, which follows the accretion-powered “middle age” described in the last section. With termination of accretion, the rotating neutron star becomes a rotation-powered pulsar again — if its rotation is fast enough, and its
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magnetic field still strong enough — whether in a binary or single, the latter being the case if the binary is disrupted or otherwise destroyed after its “middle age” by one of the processes described below. Thus, this transition may, in a way, be looked upon as a return to its original rotation-powered state, and the name “recycling” was coined for it. The reader must bear in mind, however, that such an old rotation-powered neutron star may be different from a newborn or young rotation-powered neutron star in many ways, just as, in everyday life, an object made from recycled material may not have quite the same properties as one made from the original material, so that the ecological analogy seems apt. Note further that, unless the neutron star is in the high-density core of a globular cluster, this recycling works only once, i.e., the above old, rotation-powered stage is, in fact, the last stage in the career of the neutron star as a pulsar: after that, it fades into oblivion as it spins down. In the core of a globular cluster, however, there is a chance (because of the frequent stellar encounters) that the old, single neutron star may again form a binary with a main-sequence or evolved star, either (a) through the tidal capture of this neutron star by a background star in the cluster core, or, (b) through the replacement of one of the two stars in a normal, primordial binary in the cluster core by this neutron star, i.e., an exchange encounter. This may lead to possibilities for double- or multiple-recycling. Consider now the essential features of neutron-star recycling in both HMXBs and LMXBs. Note that the term “recycling” was originally coined in the early 1980s to describe this process in LMXBs, as the focus at that time was on the evolutionary origins of the then recently-discovered millisecond pulsars, which are now widely accepted as products of long-term recycling in long-lived (∼ 108 − 109 y) LMXBs, which can spin the neutron star up to such short periods as milliseconds, and can reduce their magnetic fields to such low values (∼ 109 G) as are inferred for millisecond pulsars (see Chapter 12). But, in this book, we shall use the same term also for HMXBs, which are short-lived (∼ 106 − 107 y lifetime for the massive stars in them, and a few times 104 y duration for the HMXB phase, as explained earlier), and so can give only short-term recycling. The crucial point here is that there is only a quantitative — not qualitative — difference between the two cases: how an accreting neutron star responds to the accretion torque which spins it up (see Chapter 12), or to the processes which reduce its magnetic field (see Chapter 13), has basically no reference to its binary companion. Consequently, the term applies equally well, in principle, to both LMXBs and HMXBs. The product, in all cases, is a rotating,
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magnetized neutron star which will not accrete any more (unless it is in a globular cluster and acquires a fresh companion through stellar encounters: see above), and so will be left to its own devices, eventually completing its life as a pulsar and becoming unobservable. Only the exact conditions in which the product finds itself, e.g., its spin-rate and the magnetic field, whether it is single or in a binary, the nature of the binary companion in the latter case, and so on, are determined by the details of the previous evolutionary history of the binary in which it was recycled, as we shall see now. 6.5.1
Final Evolution of HMXBs and Recycling
As indicated above, a massive X-ray binary functions as an X-ray source because either (a) the companion is an O/B supergiant and the system is a so-called “standard” HMXB, in which case the neutron star accretes matter either (i) from the strong stellar wind of the companion or (ii) through Roche-lobe overflow of the tenuous outer atmospheric layers of the companion (sometimes called atmospheric Roche-lobe overflow), or, (b) the companion is a Be star, in which case the neutron star accretes matter primarily during its passages through the outflow disk of the fast-rotating Be star, and produces X-ray outbursts. This HMXB phase lasts for only a few times 104 years for standard HMXBs [Meurs & van den Heuvel 1989], although the evolutionary timescale of the corresponding O/B companion is a few million years. For Be-star HMXBs, on the other hand, this phase lasts essentially over the entire evolutionary timescale of the Be star, which is ∼ 107 years (remember that Be stars are less massive than typical O/B supergiants). What happens next to HMXBs? We described earlier how a massive star expands at the end of its core hydrogen burning phase, fills its Roche lobe, and starts copious mass transfer: this is what happens now to the massive companions in HMXBs. In a few million (for O/B companions) to 107 (for Be-star companions) years of evolutionary time, the companion in a massive X-ray binary invariably starts a full-scale Roche-lobe overflow. The result is also inevitable: because the mass ratio is quite extreme in such a system, since the neutron star has Mns ∼ 1.4M and the massive companion has M2 ∼ (8 − 40)M , the HMXB rapidly goes into a CE phase, as described earlier, wherein the neutron star and the He-core of the companion spiral in towards each other inside the common envelope, releasing orbital binding energy through the large frictional drag of the CE and thereby heating the latter. As described
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Fig. 6.9 Possibilities for the final evolution of HMXBs. Reprinted with permission by Springer Science & Business Media from van den Heuvel (2001) in The neutron star-black c hole connection. 2001 Kluwer Academic Publishers.
earlier, whether this deposition of energy in the CE is sufficient to expel it depends on how wide the binary orbit was at the onset of the CE phase: if it was sufficiently wide, then the difference between its binding energy at that stage and its binding energy after it becomes sufficiently narrow after spiral-in in the CE would be sufficient to expel the CE, and we would end up with a very close neutron-star He-star binary — whose orbital size corresponds to the latter state, as shown on the right-hand panel of Fig. 6.9. On the other hand, if it is not sufficiently wide for the above expulsion to occur, the neutron star will spiral in all the way into the core of the massive companion, producing a very interesting object, viz., a massive star with ˙ a neutron-star core. This is called a Thorne-Zytkow object , referring to ˙ the pioneering 1977 work of Thorne and Zytkow, and the structure of such objects has been further studied in 1991 by Cannon and by Biehle. This is shown on the left-hand panel of Fig. 6.9. As the reader can show with the aid of the quantitative details of CE evolution given in Sec. (6.1.3.2), and as is generally appreciated now, the
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boundary between the above two possible outcomes of CE evolution occurs at an initial HMXB orbital period Pb ∼ 100 − 200 days [van den Heuvel 1992], the exact value depending on the details of CE evolution, e.g., the efficiency parameter introduced in Sec. (6.1.3.2). Consequently, standard HMXBs (with O/B supergiant companions), which have orbital periods well below the above critical value, and those Be-star HMXBs which are close enough to have Pb below this critical value spiral in completely to form ˙ Thorne-Zytkow object. On the other hand, those Be-star HMXBs which have wide enough orbits to have Pb above this critical value will form very close He-star neutron-star binaries after expelling the CE. Since it is only the Be-star systems which contain the possibility of bifurcating into one or the other of the above two evolutionary branches depending on the orbital period, it would be interesting to estimate this branching ratio, or, in other words, estimate what fraction of Be-star HMXBs survives as binaries after the CE phase. If the Pb -distribution of Be-star HMXBs was accurately known, the answer would be trivial. But, even with a rather limited sample of such systems with known Pb , reasonable bounds can be readily obtained, as was shown by van den Heuvel in 1992. At that time, Pb had been measured for only 9 of 22 such known systems, and, among these, about 4 had Pb above the critical value given above. Thus, argued van den Heuvel, we can set bounds on the above fraction in a straightforward manner. The upper bound, 4/9 ≈ 45%, is obtained by assuming that the Pb -distribution for this small number of binaries with known Pb is, in fact, identical to that for the entire Be-star HMXB population. On the other hand, the lower bound, 4/22 ≈ 20%, is obtained by assuming that the above 4 systems with Pb above the critical value are, in fact, the only ones among the 22 known systems that have above-critical Pb -values. Thus, 20 − 45% of all Be-star HMXBs are expected to emerge from the CE phase as close He-core neutron-star binaries. We summarized the van den Heuvel arguments above to illustrate the idea: more such systems are now known, of course, but the final estimate has not changed grossly.
6.5.1.1
Recycled pulsars: single or with degenerate companions
It is the final evolution of the above two possible systems — the Thorne˙ Zytkow object or the close He-star neutron-star binary — that is of crucial importance, since it decides the final state in which we expect to find the recycled neutron star: as we shall see below, these states can only be (a) single, or, (b) binaries with degenerate companions, i.e., neutron stars or
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˙ white dwarfs, born during this final evolution. Consider the Thorne-Zytkow object first, whose fate is believed to be rather simple. Sitting at the center of its supergiant companion, the neutron star is in a position to accrete matter copiously, and that is what it is expected to do, releasing so much energy that it would drive away the entire companion in a short time (a few thousand years or so), leaving us with a single, recycled neutron star, which would have relatively low magnetic field (due to accretion-induced field reduction; see Chapter 13) and low space velocity (as opposed to the other case discussed below). This is shown on the left-hand panel of Fig. 6.9. Consider now the close He-star neutron-star binary, whose evolution can go in one of three following ways, all of which proved crucial for clarifying the origins of well-known types of recycled pulsars, binary and single. The primary parameter that selects which course the evolution takes is the mass M2 of the massive companion in the HMXB before the CE phase. The selection goes in two ways, viz., (a) whether the He-core will explode in a supernova and thereby produce a second neutron star in the system or not, and, in the former case, (b) whether the system will come unbound in the supernova explosion or not. How M2 decides the former selection has been explained earlier in this chapter. If M2 is below a lower bound M< ≈ 8M , the collapse of the He-core certainly does not lead to a supernova, but rather to the production of a massive white dwarf. On the other hand, if M2 is above an upper bound M> ≈ 12M, the core’s collapse certainly leads to a supernova and the accompanying production of a neutron star. For M2 between M< and M> , details of the physical processes involved decide whether a very massive white dwarf or a neutron star is the final outcome of the core collapse. In this selection, then, either a white-dwarf or a neutron-star companion is born to the old, recycled neutron star. If the former is the case, then we have a very close binary with circular orbit, consisting of a recycled neutron star and a massive white dwarf, as shown in the right-hand panel of Fig. 6.9. PSR 0655+64 is the prototype for this class of binaries. If the latter is the case, i.e., a neutron star is born from the core collapse, however, the accompanying supernova brings in the second selection above, namely, does the binary survive the supernova or not? The criterion for this has also been described earlier in the chapter: the mass-loss from the system, which is just Mc − Mns (Mc being the mass of the core of the massive companion of mass M2 ), must not exceed half the total mass of the pre-supernova system, which is Mc + Mns . This immediately yields
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Mc ≤ 3Mns ≈ 4.2M as the upper limit on the mass of the He core. With the aid of Eq. (6.8), which relates Mc to M2 , we can convert this into a upper bound on the mass of the massive companion in the HMXB, which reads M2 ≤ Mcr ≈ 17M. It is this critical value Mcr , then, which decides whether the post-supernova binary remains bound or not. If M2 < Mcr , we get an eccentric-orbit double-neutron-star binary consisting of the old, recycled neutron star and the newborn neutron star, the classic example and prototype of this class being the famous Hulse-Taylor pulsar PSR 1913+16, which proved to be such a magnificent proving ground for relativistic effects, as described in Chapter 7. This is shown in the right-hand panel of Fig. 6.9. If M2 > Mcr , on the other hand, the binary comes apart, yielding two “runaway” neutron stars, one old and recycled, the other young, as shown also in the right-hand panel of Fig. 6.9. Because of the three-way branching described above, we may naturally wish to estimate the branching ratios, as we did in the previous subsection. If the mass M2 of the original HMXB companion were the only consideration, we could immediately estimate this branching by assuming an appropriate M2 -distribution, i.e., an initial mass function (IMF) — as it is called — for massive stars, say ψ(M2 ) ∼ M2−2.5 , and then simply working out (a) how many massive companions would have M2 < ∼ (8 − 12)M , (b) how < < many have (8 − 12)M ∼ M2 ∼ 17M, and, (c) how many M2 > ∼ 17M , as explained above. We shall leave this as an exercise for the reader. What makes the situation more complex and interesting, however, is the essential role played by two other phenomena in deciding the second branching ratio, i.e., what fraction of the close He-star neutron-star binaries survives the second supernova in the system. The first phenomenon is the “kick” velocity imparted to the neutron star born in the second supernova explosion, which we have discussed earlier in the chapter, and whose crucial importance is fully appreciated today. The first and obvious point in this matter is that, since the close pre-supernova −1 binaries have high orbital velocities > ∼ 400 km s , these kicks, if they are −1 in the range 100 − 200 km s as they are supposed to be, should not alter very much the survival probability of systems that would survive anyway by the M2 -selection argument given above, except those which are close to the survival limit by this argument. The second and more interesting point is that, since the kick directions are expected to be random, and kicks retrograde with respect to the instantaneous orbital velocity tend to bind the system more tightly rather than unbind it, a fraction of the systems that we would expect to be disrupted by the M2 -selection argument given
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above will now survive [van den Heuvel 1992]. Assuming a ∼ 50% chance of retrograde kicks of the above magnitude, some ∼ 10 − 30% of such binaries with original Be stars of mass M2 > ∼ 17M are expected to survive the supernova. By the same token, ∼ 50% of those systems which have M2 ≈ 17M , and so are close to the survival limit (see above) are expected to survive due to retrograde kicks. The second phenomenon, whose importance does not seem to have been fully appreciated before the 1990s, is the likely occurrence of a second spiralin phase for a considerable fraction of the close He-star neutron-star binaries we are discussing here. From the extensive work on the evolution of He stars in the 1980s, it became clear that those with masses MHe < ∼ (3.5 − 4)M < (corresponding to M2 ∼ (14 − 15)M , say) will undergo a considerable radius expansion during He-burning in the core and in shells, which, in these very close binaries, will put the (old) neutron star within the He-rich envelope, and so start a second CE phase. As before, the ensuing spiral-in will expel the envelope, leaving us with a carbon core in an extremely close binary with a the old, recycled neutron star. Subsequently, the C-core will undergo collapse, producing the young neutron star with the accompanying supernova. As a typical example, van den Heuvel (1992) cites a 4M He star and an old neutron star in a binary with an orbital separation of 2R . Using the relations in Appendix B, the reader can show that the Roche-lobe radius of the He star is then about 0.95R , so that it will overflow its Roche lobe just prior to carbon ignition, and CE evolution will start. The outcome of this CE evolution will be a shrinking of the orbit by a factor ∼ 10 − 15, yielding a very narrow binary consisting of a 2.7M C-core and an old, recycled neutron star. This would be only an interesting detail, were it not for the fact that this process increases the survival probability in the second supernova in two ways. First, it reduces the mass of the exploding star considerably, as above, so that the mass ejection in the supernova is correspondingly reduced (since the mass of the remnant, newborn neutron star would remain roughly the same), which ensures that the post-supernova system will be more tightly bound, other things remaining the same. Second, it shrinks the binary and so leads to higher orbital velocities, which ensures (see above) that a given kick velocity will have a smaller effect on the orbit, other things remaining the same. Taken together, the above two phenomena may easily increase the survival probability in the second supernova explosion to > ∼ 50%, which would imply, from the branching-ratio estimate given in the previous subsection,
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that ∼ 10 − 20% of all Be-star HMXBs end up as double-neutron-star binaries like PSR 1913+16 [van den Heuvel 1992]. We refer the reader interested in more detail of the final fate of HMXBs to the series of definitive, superb reviews by van den Heuvel (1991, 1992, 2001), and conclude our discussion here with a comment on the nature of the supernovae in these systems. The statistics of galactic X-ray binaries imply an estimated rate of HMXB formation in our galaxy of a few times 10−3 per year: this is somewhere between a third and a half of the estimated total supernova rate in our galaxy. The key point about supernovae in HMXBs is that they are dim, as they must be, since they are explosions of bare helium or carbon cores without envelopes (which have been lost by mass transfer or through CE evolution), as opposed to supernovae of single massive stars, which eject their envelopes in spectacular displays of energy output. This is true for the first supernova in a massive X-ray binary (i.e., the one that produces the neutron star that eventually becomes the recycled one), and even more true for the second one if it occurs (i.e., the one that would produce the young neutron star that would either become the binary companion to the recycled one, or run away if the binary is disrupted, as described above) since, in this case, both the H-rich and the He-rich envelope may be lost in two stages of CE evolution, leaving a C-core, as we just saw. Using population statistics, we can estimate what fraction these bare He- or Ccore supernovae are expected to constitute of all supernovae, as Meurs and van den Heuvel did in 1989, and the answer is: somewhere between a half and two-thirds. This is to be compared with the above estimate, and also with that of Nomoto et al. (1990), who identified bare-core supernovae with SN Types16 Ib and Ic, and estimated that about a half of all massive-star supernovae belonged to these types.
6.5.2
The Double Pulsar Binary J0737-3039
We summarized above the evolutionary origins of double neutron-star binaries. Today17 , there are eight binaries which are either known or very 16 Supernovae are classified into the following types according to certain features of their optical spectra. The basic types are I and II, the latter showing H in their spectra, and the former not. Type I is further subdivided according to the presence or absence of Si and He in the spectrum. Type Ia shows Si in absorption, Type Ib shows no Si but He in emission, and Type Ic shows neither Si nor He. It is believed that SN Types Ib,c and II result from core collapse of massive stars, as described in this chapter, while Type Ia is electron-capture SN in massive white dwarfs, also mentioned here. 17 As of late 2005; see Manchester (2006).
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likely to be such double neutron-star systems, namely, PSRs J0737-3039, J1518+4904, B1534+12, J1756-2251, J1811-1736, J1829+2456, B1913+16, and B2127+11. All have eccentric orbits, short (∼ 20 − 100 ms) pulse periods, and low period derivatives (indicative of low magnetic fields, as explained in Chapters 1 and 7, and so of recycling), as expected. These systems lie in the lower-middle of the binary period vs. pulse period diagram of Fig. 7.19. However, in only one of them are both neutron stars observed as rotation-powered pulsars. This is the famous double-pulsar binary PSR J0737-3039, discovered in April 2003 during a high galactic-latitude pulsar survey of the southern sky, using the 64-m Parkes radio telescope in Australia [Burgay et al. 2003]. The first of the two pulsars to be discovered was J0737-3039A, a 22.7 ms pulsar, and the short binary period of Pb ≈ 2.4 hr with a mildly eccentric orbit (e ≈ 0.088) was also determined in early 2003. In October 2003, the other pulsar J0737-3039B was discovered, with a pulse period of 2.77 sec [Lyne et al. 2004]. The essential pulse parameters and the Keplerian orbital parameters of the system are shown in Table 7.4 [Lyne et al. 2004; Burgay et al. 2006]. It is interesting that, of all the known double neutron-star binaries, J0737-3039 has (a) the “millisecond” pulsar with the shortest period, (b) the shortest orbital period, and (c) the smallest orbital eccentricity [Dewi & van den Heuvel 2005]. Further, this system is seen nearly “edge-on” by us, i.e., our line of sight makes an angle i ≈ 87◦ (i has been estimated in two ways from the post-Keplerian (PK) parameters ω˙ and s and the mass ratio R of the two pulsars, yielding consistent results: see Chapter 7 for a description of these parameters) with the normal to the orbital plane of the binary. This gives us an invaluable tool (see below) for studying magnetospheric physics as the observed beam of radiation from one pulsar probes the magnetosphere of the other at conjunction (i.e., when the two pulsars are most closely aligned with our line of sight). It is now clear that J0737-3039 gives us a rather gratifying confirmation of the evolutionary scenario for double neutron-star systems sketched above [Dewi & van den Heuvel 2005], with PSR J0737-3039A being the firstborn neutron star, now an old, recycled “millisecond” pulsar with a weak magnetic field (B ∼ 6×109 G, spin-down age ∼ 2×108 yr), and PSR J07373039B the second-born (from the supernova of the He star: see above) neutron star, a “normal” pulsar with a strong magnetic field (B ∼ 1 × 1012 G, spin-down age ∼ 5 × 107 yr). Detailed evolutionary scenarios for J07373039 have ben considered by Dewi & van den Heuvel (2005) and others. In view of the evolutionary possibilities for the He-star in such systems
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discussed above, an essential point is that J0737-3039 appears to be the only one among the known double neutron-star binaries which could not have formed without a phase of mass transfer from the He-star. On the other hand, this mass transfer must not have gone into a dynamically unstable phase, which puts an upper limit MHe < ∼ 3.3M on the He-star’s mass. A lower limit on MHe > would come naturally from the requirement 2.1M ∼ that the He-star must be massive enough to produce a neutron star. These authors also estimated that a minimum kick velocity ∼ 70 km s−1 was required to explain the observed orbital eccentricity of J0737-3039. Among other points are of interest about the evolutionary aspects of this system, we mention two here. First, the masses of the two neutron stars, as determined from Keplerian and PK orbital parameters of the system and as displayed in Table 7.4, show that the mass of PSR J0737-3039A, MA ≈ 1.34M, is very similar to the canonical value of observed neutron-star masses, while that of PSR J0737-3039B, MB ≈ 1.25M , appears to be significantly below this value. Second, there was some concern about whether the difference of ∼ 150 Myr between the apparent ages of the two neutron stars (see above) was in agreement with our ideas of evolutionary scenarios for such systems. However, as argued by the above authors, spin-down ages may be subject to considerable errors because of the inherent uncertainty in our knowledge of the spin-period of a neutron star at birth, so that there does not appear to be any basic contradiction. The short orbital period of this double neutron-star system makes it an ideal laboratory for studying relativistic gravity, as we discuss in Chapter 7. In addition, a unique feature of J0737-3039 is that, since we have here two observable rotation-powered pulsars in a close binary which we are viewing almost edge-on, as pointed out above, the interaction of the (pulsed and beamed) emission from one of them with the magnetosphere of the other gives us a magnetospheric probe of unprecedented power: we turn to this in Chapter 10. 6.5.3
Final Evolution of LMXBs and Recycling
In discussing the operation of LMXBs after they form, during which the accreting neutron star undergoes a long period of recycling, and the final evolution of these LMXBs after accretion ceases, we note first a remarkably universal — indeed rather geometrical — property of the binary periods of LMXBs, which has been widely recognized [see Verbunt & van den Heuvel 1995], and which is simply that, if the low-mass companion to
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Table 6.1 MASS-RADIUS & MASS-ORBITAL PERIOD RELATIONS FOR COMPANIONS IN LMXBS (from Verbunt & van den Heuvel 1995) Companion Type
M2 − R2 Relation
M2 − Pb Relation
Main Sequence He Main Sequence White Dwarf
R2 /R ≈ M2 /M R2 /R ≈ 0.2M2 /M R2 /R ≈ 0.0115(M /M2 )1/3
Pb ≈ 8.9(M2 /M ) hr Pb ≈ 0.89(M2 /M ) hr Pb ≈ 40(M /M2 ) sec
the neutron star fills its Roche lobe (as it must to transfer matter), then the binary period Pb of the LMXB depends only on the average density, ρ¯ ≡ M2 /(4πR23 /3), of the companion. This may sound strange at first, but the relation which shows this, viz., −1/2
Pb = 8.9 (M2 /M )
3/2
(R2 /R )
hr = 8.9 (ρ2 /ρ )
−1/2
hr,
(6.21)
follows readily upon combining Kepler’s third law with the expression for the Roche-lobe radius of the low-mass companion given in Appendix B, as the reader can show. Verbunt obtained a very nice result that follows immediately from Eq. (6.21), namely that, since there are characteristic mass-radius relations that apply to various possible classes of low-mass donors, e.g., mainsequence stars, stars on the He-burning main sequence, white dwarfs, and so on (see Appendix C and Chapter 2), the use of these relations in Eq. (6.21) yields a characteristic relation between the binary period Pb and the Rochelobe-filling donor’s mass M2 for each of the above classes. These relations, summarized in Table 6.1 and displayed in the left panel of Fig. 6.10, both taken from Verbunt & van den Heuvel (1995; henceforth VH95), are very useful. For example, these tell us that LMXBs with main-sequence companions must have Pb < ∼ 9 hr, therefore implying that those with longer orbital periods must have a subgiant or giant companion, which has a larger radius. By contrast, very short periods, Pb < ∼ 1 hr, say, would indicate a He star or a white dwarf, since these have much smaller radii. The Pb -distributions of LMXBs and CVs are similar in many respects, as expected and as explained before. So, first consider the CV distribution for reference. The (rather sharp) cutoff in this distribution at its short-period end at Pb ∼ 80 min is determined basically by a competition between orbit shrinkage rate due to gravitational radiation and Kelvin-Helmholtz relaxation rate of the low-mass donor (see, e.g., Rappaport, Joss & Webbink 1982): at such orbital periods, the donor cannot be any larger than a
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Fig. 6.10 Left panel: Relation between orbital period Pb and donor mass M2 for LMXBs. Right panel: Mass-transfer rates driven by gravitational radiation alone vs. M2 . Reproduced with permission by Cambridge University Press from Verbunt & van den Heuvel (1995): see Bibliography.
main-sequence star if it is to fit inside its Roche lobe (as explained above), and so must be unevolved. At Pb > ∼ 9 hr, however, the donor must be an evolved star, according to the argument given above, which also holds roughly for CVs, as the reader can easily show. Finally, the clear paucity of CVs with Pb ∼ 2 − 3 hr — a “gap” in the distribution — is the wellknown period gap: both this and the minimum value of Pb given above are believed to be related to the essential properties of angular-momentum transport mechanisms in very close binaries. We shall return to this point below. Now consider the LMXB distribution, which spans a larger range of Pb -values, ranging from ∼ 10 minutes to ∼ 10 days. A large number of LMXBs do seem to have evolved donors by the Pb > ∼ 9 hr argument given above. Further, there seems to be no evidence for the above period gap in LMXBs, a point to which we shall, again, return below. As we have pointed out repeatedly in this book, mass transfer in CVs and LMXBs proceeds by Roche-lobe overflow, and we now summarize those
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processes which are believed to drive this overflow. A conceptually transparent way of introducing these is that stressed by VH95, namely, by appealing to the essential characteristics of orbital changes as a result of mass transfer in binaries and/or angular momentum loss from such systems, as described in Sec. (6.1.2.2), particularly by Eqs. (6.5) and (6.6). As we saw there, conservative mass transfer widens the binary, if mass is transferred from the less massive to the more massive star, as is the case for CVs and LMXBs. This increases the size of the Roche lobe of the low-mass donor (see Appendix B), so that, even if it was in Roche-lobe contact earlier, it will no longer be so. If mass transfer is to continue, so must Roche-lobe contact, and these authors argued that this can happen in one (or both) of the two following ways: either (a) the orbit (and so the Roche-lobe size) must shrink, which can occur if angular momentum is lost from the system by some suitable mechanism, as Eq. (6.6) shows, or, (b) the donor star must evolve and expand. We now turn to a discussion of these two possibilities. Consider angular-momentum loss first. The angular-momentum loss rate J˙orb and the mass transfer rate M˙ determine the orbit-change rate a˙ through Eq. (6.6). Of course, we have to remember that, in the current context, mass is being transferred from the low-mass companion of mass ˙ here. The rest M2 to the compact star of M1 , so that M˙ 1 = M˙ , M˙ 2 = −M of the algebra is as before, yielding: M˙ J˙orb 1 a˙ =2 , 1− −2 a M2 q J
(6.22)
where the first term on the right-hand side describes the widening of the binary (since 1/q ≡ M2 /M1 < 1) due to mass transfer, and the second term the shrinking due to angular momentum loss, as before. (Remember that both J˙orb and M˙ are positive by definition, as explained earlier.) As shown by VH95, the above equation can be recast in an instructive form by requiring that, whatever the angular-momentum loss mechanism, the Roche-lobe shrinkage exactly keeps pace with the changing stellar radius of the low-mass donor star at all times. This implies both R2 = RL2 and R˙ 2 = R˙ L2 , and the latter condition can be turned into a relation between the orbit-change rate and mass-transfer rate by using (a) the expression for the Roche-lobe radius RL2 of the low-mass companion given in of Appendix B (as above), and (b) a generic form, R2 ∝ M2n , for the mass-radius relation of the low-mass companion, which enables us to describe companions of various types by varying n, e.g., n = 1 for main-sequence companions,
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n = −1/3 for white dwarfs, and so on. We leave the straightforward algebra to the reader, giving the final result as: M˙ 1 a˙ = −n . (6.23) a M2 3 Combining the above equation with Eq. (6.22), we obtain the instructive form: 5 n 1 J˙orb M˙ , β≡ =β + − , (6.24) J M2 6 2 q which brings out the following consistency condition. Clearly, the above parameter β must be positive: otherwise the whole picture makes no sense. But what does this condition, which we can rewrite as n>
2 5 − , q 3
(6.25)
really mean? A look at Eqs. (6.23) and (6.22) shows us that, by locking the size of the low-mass donor to its Roche lobe here, we are effectively pitting two effects against each other: the mass-losing donor would normally contract (unless it is a white dwarf: see below) due to mass-loss, which means that the Roche lobe must also shrink, while mass transfer from the low-mass donor to the heavier compact star would normally widen the binary and so the Roche lobe. For consistency of the scenario, then, the latter effect must be stronger than the former, and the balance must be made up by the orbit (and so Roche lobe) shrinkage due to angular momentum loss. Eq. (6.25) is precisely a statement of the former inequality, and therefore a consistency condition18 . Is it satisfied in LMXBs? It is easy to show that the answer is yes for all non-degenerate donors. If we take M2 < ∼ 1.4, ∼ M as a conservative estimate for the donor’s mass, so that q > the upper bound on the right-hand side of inequality (6.25) is ≈ −0.3. Now, the left-hand side of this inequality, n, is > ∼ 1 for all non-degenerate donors, whether (H- or He-burning main sequence) main-sequence (see above) or subgiant/giant, so that the condition is satisfied by a large margin. Only for a white-dwarf donor with n = −1/3 (see above and Chapter 2) is there any possibility of violating the condition. What happens when it is violated? As indicated in the footnote, mass transfer can then proceed even without 18 It
is not really a stability condition, as it is sometimes thought to be, although, as β → 0, a given angular-momentum loss rate would formally imply an arbitrarily large mass-transfer rate. In reality, it merely means that the above two effects are just balancing each other, so that mass transfer can proceed without any angular momentum loss.
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angular-momentum loss: the donor white dwarf expands as it loses mass, and the binary widens in step with it. We shall not discuss this case here any further. Let us now consider the proposed mechanisms of angular momentum loss. 6.5.3.1
Gravitational radiation
We have already introduced this mechanism in Sec. (6.1.2.2), and have given the expression for J˙orb /J due to this process in Eq. (6.7). We stress again the characteristic ∼ a−4 dependence of this fractional loss rate, and so its dominance at small orbit sizes. A rough measure of this point of dominance is a binary period of about 0.5 day — in LMXBs with shorter orbital periods, gravitational radiation is expected to be the dominant driver of mass transfer. This fiducial value Pb ∼ 0d .5 is easy to calculate: we can recall the rate of orbital decay by gravitational radiation from Sec. (6.1.2.2), and rewrite it as one for the decay timescale tGR , arriving at Faulkner’s (1971) pioneering result: 8/3 Pb (m1 + m2 )1/3 yr, (6.26) tGR ≈ 4 × 107 m1 m2 1h .6 where m1 and m2 denote the masses of the two stars in solar units. From this, it follows directly that tGR is shorter than the Hubble time (∼ 1010 yr) only for orbital periods shorter then the above fiducial value. Under these circumstances, we can immediately calculate the masstransfer rate in the system by combining Eqs. (6.24) and (6.7), and the results are shown in the right panel of Fig. 6.10. This, then, is the rate at which gravitational radiation alone can drive mass transfer in LMXBs. The two panels of Fig. 6.10 show the interesting result that, for a main-sequence donor, the mass-transfer rate by this mechanism is held almost constant at M˙ ∼ 10−10 M yr−1 , as the donor mass M2 decreases due to mass transfer, while the donor radius and the binary period decrease, the latter being the Faulkner (1971) result given earlier. 6.5.3.2
Magnetic braking
The existence of luminous LMXBs with L ∼ 1037 − 1038 erg s−1 clearly indicates that much larger accretion rates, M˙ ∼ 10−9 − 10−8 M yr−1 (see Chapter 1), than the above value given by gravitational radiation are possible in some of these systems, and we are left looking for additional
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mass-transfer driver mechanisms. As we shall see below, for sufficiently wide LMXBs, angular-momentum loss mechanisms become ineffective, and expansion of the low-mass donor because of its nuclear evolution is the dominant driver of mass transfer, which is quite capable of sustaining the large rates of mass transfer quoted above. However, there are some sufficiently narrow LMXBs — say with orbital periods between about 0.5 and 1 day — where angular-momentum loss must be the driving mechanism, and yet gravitational radiation is inadequate for driving the mass-transfer rates inferred from their observed luminosities. Clearly, then, other mechanisms of angular momentum-loss should exist. In 1981, Verbunt and Zwaan suggested such a mechanism, namely, magnetic braking. The idea is as follows. It is well-known that late-type mainsequence stars lose angular momentum with age, and so rotate more slowly. It is believed that the strong magnetic fields of such stars (particularly when they rotate rapidly) are responsible for this: such magnetic fields force the stellar winds emitted by these stars to co-rotate with the star out to several stellar radii, so that the specific angular momentum carried away by the wind can be quite large. Consequently, even modest mass-losses can lead to large angular-momentum losses. Furthermore, in such close binaries as LMXBs, tidal forces can keep the late-type, low-mass donor locked in corotation with the the orbital revolution, so that the ultimate loss of angular momentum is from the orbit, driving the two stars toward each other. The original Verbunt-Zwaan (1981) formulation is straightforward and instructive, so we recount it here in brief. Consider a donor of mass M2 rotating with an angular velocity tidally locked to the orbital value Ωorb = GM/a3 , where M = M1 + M2 is the total mass, as before. The angular momentum of the low-mass donor due to its own rotation is then J2 = M2 k 2 R22 Ωorb , where kR2 is its radius of gyration (k 2 ≈ 0.1 for cool, mainsequence stars). These authors used the observed result that the rotation of such stars decrease with age t roughly as Ωorb ∼ t−1/2 , to argue that J˙2 ∼ t−3/2 ∼ Ω3orb . In this scenario, then, the orbital angular-momentum loss rate scales as J˙orb = J˙2 ∝ M2 k 2 R24 Ω3orb . With the aid of Eq. (6.2) for Jorb , the fractional angular-momentum loss rate by magnetic braking alone is given by J˙orb k 2 M 2 R24 = J0 , Jorb M 1 a5
(6.27)
where J0 is a constant. Two points are immediately noteworthy about Eq. (6.27). The first is the characteristic ∼ a−5 dependence of the frac-
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tional loss rate due to magnetic braking, according to the original VerbuntZwaan prescription. This implies that magnetic braking would dominate over gravitational radiation at sufficiently small orbit sizes. The second is that, in contrast to Eq. (6.7), the constant J0 in Eq. (6.27) is empirical, determined by the observed rotation-age relation for appropriate stars. As these authors showed, magnetic braking can, indeed, enhance the mass transfer rate considerably in LMXBs. Interpreted literally, however, the original Verbunt-Zwaan prescription would imply domination of magnetic braking at even shorter orbital periods than that of gravitational radiation, in contrast to what we expected above. The key point, as VH95 pointed out, is that the correct scaling laws for stellar winds and magnetic-field structures of late-type stars are not known observationally, so that plausible scaling laws proposed in the literature cannot be readily verified. It is quite possible that the actual scalings are d such that magnetic braking does dominate in the 0d .5 < ∼ Pb < ∼ 1 range [van den Heuvel 1991]. However, Verbunt has subsequently pointed out that, in view of recent studies of stellar rotation in young star-clusters, it is also possible that the efficiency of magnetic braking might have been overestimated earlier.
6.5.3.3
The minimum period and the period gap
We are now in a position to discuss the origins of (a) the minimum period in CVs and LMXBs, and (b) the period gap in CVs, both of which were introduced earlier. The reason for the period minimum is now completely obvious, if we look at Fig. 6.10 and Fig. 6.11. As the former figure shows, a companion of mass M2 ∼ M on the H-burning main sequence gives Pb ∼ 9h . As M2 decreases, Pb decreases linearly with it, as given above. As M2 falls below ∼ 0.3M, the mass-transfer timescale, τM˙ ≡ M2 /M˙ , falls below the thermal or Kelvin-Helmholtz timescale tKH introduced earlier. The star can then no longer adjust itself (i.e., cool) fast enough to the new configurations required by further mass transfer, and so is driven out of thermal equilibrium [van den Heuvel 1992]. Its radius becomes larger than the main-sequence radius, and the Pb − M2 relation deviates more and more from the linear main-sequence relation, as shown in Fig. 6.11. Pb passes through a minimum value of ∼ 80 minutes at M2 ∼ 0.1M , and rises again for smaller M2 , as the star moves to the degenerate track, where Pb ∼ M2−1 . This is a common feature of CV and LMXB evolution: the existence of this minimum Pb was pointed out by
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Fig. 6.11 Origin of the period gap in CVs. Shown is a schematic Pb − M2 relation. Reprinted with permission by Springer Science and Business Media from van den Heuvel (1992) in X-ray Binaries & Recycled Pulsars, eds. E. P. J. van den Heuvel & S. Rappac 1992 Kluwer Academic Publishers. port, p 233.
Faulkner in a pioneering 1971 paper, and its identification with the observed 80-min minimum was stressed by Paczynski and Sienkiewicz (1981). Indeed, no H-rich companions have been found in systems with Pb shorter than the above minimum value: as Fig. 6.10 leads us to expect, these systems should have H-poor companions and follow the track for He-stars, which is borne out by observations. Consider now the period gap in CVs, which is believed to be related to properties of magnetic braking. How? If this mechanism requires a firm anchoring of the magnetic field in the radiative core of the companion [Spruit & Ritter 1983], which seems a plausible assumption, then we could argue that the mechanism would suddenly cease to be effective as M2 falls below ∼ 0.3M , and the star becomes fully convective. A scenario based on this idea is sketched in Fig. 6.11 [van den Heuvel 1992]. At masses higher than the above critical value, magnetic braking would operate in addition to gravitational radiation, leading to large rates of mass transfer. This would shorten the mass-transfer timescale τM˙ , driving the star out of
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thermal equilibrium as explained above, so that it will expand, and Pb will follow a track above the main-sequence one, as shown. When the above critical value of M2 is reached, corresponding to Pb ∼ 3h , magnetic braking will lose hold and mass transfer will drop drastically. The companion will then relax back to thermal equilibrium, shrink, and recede inside its Roche lobe. Mass transfer will stop completely, and the system will become unobservable. However, gravitational radiation, which is operational throughout, will continue to make the orbit shrink, and so the companion’s Roche lobe. At Pb ∼ 2h , the Roche lobe will become small enough that the companion will fill it, and start mass transfer again, making the system observable again. Thus, there will be no observable systems in the range h 2h < ∼ Pb < ∼ 3 , explaining the period gap. The reader will have noticed that the nature of the accreting compact star did not enter the above arguments, so that we might expect LMXBs to have a similar period gap. Actually, LMXBs do not show a period gap: rather, the number of observed sources peters off below Pb ∼ 3h , and does not increase again below Pb ∼ 2h , there being a few very short-period ones with H-poor or degenerate companions. Why is this? The reason is believed to be the evaporation and destruction of the donor by the radiation from the fast-spinning neutron star (spun up by accretion torques during the earlier phase: see below), to which we return below.
6.5.3.4
Donor evolution and expansion
Now consider mass transfer driven by the nuclear evolution and consequent expansion of the donor, which makes it fill its Roche lobe. This is the dominant mechanism in relatively wide LMXBs, since the angular-momentum loss mechanisms described above are ineffective at large a, scaling as they do as strong negative powers of a. The point of dominance is roughly at Pb ∼ 1d , i.e., for LMXBs with larger orbital periods, angular-momentum loss has negligible effect. Indeed, we can treat the angular momentum Jorb roughly as a constant in this limit, i.e., recover the conservative masstransfer scenario introduced earlier, if we also assume that all mass lost by the donor is transferred to the neutron star. The quantitative description is then straightforward, using the equations given earlier, as explained in VH95. If we set J˙orb /J to zero in Eq. (6.22), it turns into a simple relation between a/a ˙ and M˙ /M2 . In this relation, we ˙ can then express a/a ˙ in terms of M /M2 and and R˙ 2 /R2 , using, as before, ˙ ˙ R2 = RL2 (i.e., the condition that, after coming to Roche-lobe contact, the
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donor stays in contact at all times), and using again the expression for the Roche-lobe radius RL2 of the low-mass companion given in Appendix B. The result is a relation between M˙ /M2 and R˙ 2 /R2 , namely, M˙ R˙ 2 5 1 − =2 , R2 M2 6 q
(6.28)
which transparently shows how donor expansion drives mass transfer. This statement is quite literally true, as the following physical arguments given by VH95 clearly show. As explained repeatedly in this chapter, mass transfer from the low-mass donor to the heavier neutron star widens the orbit by itself, and so the Roche lobe, until the donor goes just inside its Roche lobe. As the donor expands more, mass is transferred again, the Roche lobe widens again, and the above process repeats itself. It is ultimately the rate of expansion of the donor, therefore, that determines the rate of the whole mass-transfer process. However, an actual determination of the above rate would require a full stellar evolutionary calculation of the structure of the expanding donor — a subgiant. Webbink, Rappaport and Savonije (1983) and Taam (1983) introduced considerable simplification in these calculations by noting that the properties of such a subgiant depend mainly on the mass Mc of its He core. We summarize only the basic idea here, referring the reader to the original work for more detail. The results of detailed evolutionary calculations of both the radius R2 and the luminosity L2 of low-mass giants show that ln R2 and ln L2 can be described well by polynomials in y ≡ ln Mc . In practice, only terms upto y 3 need be kept, and the relevant coefficients are tabulated by Webbink et al. (1983), and by VH95. Furthermore, the luminosity obeys a rough proportionality law L2 ∝ M˙ c , reflecting the fact that, on the giant branch, the luminosity is almost entirely due to H shellburning, and the He thus produced just adds to the core mass (VH95). This makes for a very simple calculational scheme, as follows. Given the binary period and M1 and M2 (and therefore a) of the LMXB, the value of R2 at which the expanding donor fills its Roche lobe is known. Then the the core mass Mc is known by the above Webbink et al. scheme, and therefore L2 is also known. The latter, in turn, determines the core-growth rate M˙ c because of its above proportionality to L2 . Next, M˙ c determines the radius expansion rate R˙ 2 through the relation obtained by differentiating the above polynomial dependence of ln R2 on ln Mc (the reader can show this by assuming a relation of the form ln R2 = a0 + a1 y + a2 y 2 + a3 y 3 ). Finally,
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Fig. 6.12 Evolution of a LMXB like Cyg X-2 into a binary like PSR 1953+29, containing a recycled neutron star with a low-mass white dwarf companion in a wide, circular orbit. Reprinted with permission by Springer Science and Business Media from van den c Heuvel (1992) in X-ray Binaries & Recycled Pulsars, p 233. 1992 Kluwer Academic Publishers.
the mass-transfer rate M˙ is obtained from R˙ 2 with the aid of Eq. (6.28), completing the scheme. We conclude this topic by showing in Fig. 6.12 the evolution of a typical wide LMXB like Cyg X-2 through mass transfer as just described, and indicating the main results of the kind of calculations just summarized. First, M˙ is higher for systems with the longer binary periods, i.e., larger values of the orbit size a, since these systems accommodate larger giants, which expand faster than smaller giants (VH95). A useful analytic relation connecting the average mass-transfer rate19 M˙ and the orbital period P0 at the start of the mass transfer (remember that mass-transfer by this 19 Note
that the numbers in Eq. (6.29) would imply super-Eddington mass-transfer d rates for systems with P0 > ∼ 12 , which is not to be taken literally. Rather, this implies that the conservative evolution picture assumed in the above treatment is no longer valid, there being considerable mass loss from such systems. This is the case for the system PSR 0820+02 given in Table 6.3.
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mechanism always widens the binary, i.e., increases P , typically by a factor 10 − 20, as in Fig. 6.12) is: P0 (6.29) M˙ ≈ 8 × 10−10 M yr−1 1 day Next, most of giant’s envelope is transferred during the LMXB masstransfer phase, which lasts ∼ 108 − 109 yr generally, and ∼ 8 × 107 yr for the specific model system in Fig. 6.12. The small remainder is added to the degenerate He core. Finally, the strong tidal forces acting in systems with lobe-filling giants tend to circularize such systems rapidly. 6.5.3.5
Recycled pulsars from LMXBs
Mass transfer in LMXBs stops when the mass of the envelope drops below about 0.01M , at which point the envelope collapses (VH95) onto the degenerate He core, which cools into a low-mass (∼ (0.3 − 0.4)M ) He white dwarf. We are thus left with a recycled neutron star with a low-mass whitedwarf companion in a circular orbit. Such a system is precisely like the class of binary, recycled, rotation-powered pulsars shown in Fig. 6.12 [van den Heuvel 2001], the prototype of which is the system PSR 1953+29 containing a 6 ms pulsar, the first one of this class to be discovered [Boriakoff et al. 1983]. This channel for producing recycled pulsars was, naturally, suggested widely immediately after discovery [Joss and Rappaport 1983, Savonije 1983, Paczynski 1983], and formed one of the foundations for establishing the recycling picture as a viable scenario. Some immediate, useful connections between the masses and geometrical properties of the progenitor LMXBs and the descendant recycled-pulsar binaries can be easily made, and we give two examples. First, for the above conservative mass transfer driven by donor expansion which leads from relatively wide LMXBs (like Cyg X-2, as shown in Fig. 6.12) to wide, circular radio-pulsar binaries with low-mass white-dwarf companions, the relations given earlier between initial and final periods apply. Given the present orbital period P of the latter, we can therefore infer the original period P0 of the LMXB at the onset of mass transfer readily, as did Verbunt (1989) and van den Heuvel (1992). Results of model calculations for three systems are shown in Table 6.2, where P , P0 are shown, together with the final core mass Mcf at the end of the mass transfer (close to the mass of the white dwarf; see above), a neutron-star mass of 1.4M and an initial companion mass of 0.9M being assumed in all cases.
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Table 6.2 EVOLUTION OF LMXBS INTO BINARY ROTATION-POWERED PULSARS WITH LOW-MASS WHITE-DWARF COMPANIONS (after Verbunt 1989 & van den Heuvel 1992) System
P (d)
P0 (d)
Mcf /M
PSR 1953+29 PSR 1620-62 PSR 0820+02
120 191 1232
6.5 16 170
0.27 0.29 0.38
As mentioned earlier, P/P0 is typically in the range 10 − 20, i.e., there is a considerable widening of the binary orbit. Second, in this type of mass transfer, since the radius of the expanding donor is related to the core mass, as explained above, we can argue as follows. The orbital period P of the radio-pulsar binary is related by Kepler’s third law to the orbital radius a, and therefore, through the mass-ratio of the two stars (see Appendix B), to the Roche-lobe radius of the donor, which, in turn, is equal to the final donor radius before mass transfer ceases. Therefore, there should be a relation between P and the final core mass, and therefore with the white-dwarf mass Mwd , which is almost the same (to within ∼ 0.01M; see above) as the former mass. This is, indeed, the case, and the simple, useful expression given by Joss, Rappaport and Lewis (1987) is: 11/2
P ≈ 8.4 × 104 mwd day,
(6.30)
where mwd is the white dwarf’s mass in solar units, and the relation is valid for P > ∼ 2 days and circular orbits. Note the very strong dependence on mwd , because of which systems in Table 6.2 with white-dwarf companions of mass mwd ∼ 0.3 − 0.4 have orbital periods ∼ 102 − 103 days, as observed. As remarked earlier, the crucial feature about recycling in LMXBs is the long duration (∼ 108 − 109 yr) of the recycling phase. This makes for a long phase of spinup of the neutron star by accretion torques, a subject we take up in Chapter 12. It is this spinup that leads to the fast-spinning neutron stars with millisecond periods in LMXBs, which emerge as millisecond rotation-powered pulsars when accretion stops. During the same recycling phase of long accretion, magnetic fields of accreting neutron stars are reduced drastically from the canonical ∼ 1012 G to ∼ 109 G or lower. This is called accretion-induced field decay, which we summarize in Chapter 13, and the detailed mechanisms of which are not very well understood yet.
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Fig. 6.13 Types of binary, rotation-powered pulsars. Left panel: Binary pulsars of the PSR 1913+16 class. Right panel: Binary pulsars of the PSR 1953+29 class. See text for detail. Reprinted with permission by Springer Science and Business Media from van den c Heuvel (1992) in X-ray Binaries & Recycled Pulsars, p 233. 1992 Kluwer Academic Publishers.
In Fig. 6.13, we display the main classes of binaries containing recycled, rotation-powered pulsars. In one panel, we show recycled pulsars of the PSR 1913+16 class, which tend to have quite narrow and eccentric orbits, the companion being a neutron star or a “massive” (i.e., close to the Chandrasekhar limit) CO or ONeMg (see Chapter 2) white dwarf, as explained above, and as tabulated in Table 7.2. In the same panel, we also show unrecycled pulsars like PSR 2303+46, which have similar white-dwarf companions, and rather similar orbital characteristics. In the other panel, we show pulsars of the PSR 1953+29 class, which are all recycled, and which tend to have relatively wide, circular orbits, the companions being low-mass [∼ (0.2 − 0.4)M ], He white dwarfs. 6.5.4
Missing Links: Accreting Millisecond Pulsars
In 1998, Wijnands and van der Klis discovered the missing link that finally established the recycling scenario described in this chapter on a firm footing. The discovery of the first accretion-powered millisecond pulsar in the
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Fig. 6.14 Discovery of the first accretion-powered millisecond pulsar. Shown is the power spectrum of time-variations: the clear, strong peak at ∼ 400 Hz corresponds to a pulse period of ∼ 2.5 ms. Reprinted with permission by MacMillan Publishers Ltd from c MacMillan Publishers Ltd 1998. Wijnands & van der Klis (1998), Nature, 394, 344.
X-ray binary XTE J1808-369 was, without doubt, the crucial link between the accretion-powered LMXBs and rotation-powered millisecond pulsars (whether single or in binaries) that had been proposed widely but never seen before. Perhaps more than anything else, it is this link that makes the entire chain of evolutionary stages in the life-history of neutron stars presented in this chapter ring true today. As we have emphasized here repeatedly, understanding these stages in the lives of neutron stars forms the basis of modern appreciation of intimate connections between rotationand accretion-powered pulsars, and is a major motivation for writing this book. The above X-ray binary was known as a transient X-ray source, discovered in 1996 by in ’t Zand et al. with the X-ray observatory BeppoSAX, and classified as LMXB with a low magnetic field after the observation of X-ray bursts from it. The Rossi X-ray Timing Explorer (RXTE) detected it during an outburst in April 1998, and, in this data, Wijnands and van der Klis (1998) found a clear, strong, very stable periodicity at a period ∼ 2.5 ms, shown in Fig. 6.14 taken from the original discovery paper, which also shows the near-sinusoidal pulse shape of the accretion-powered pulsar in this system. The binary parameters of the system were also determined immediately afterwards by Chakrabarty and Morgan20 (1998), from the periodic Doppler delays observed in pulse-arrival times, the method being 20 Indeed, this paper appeared together with the Wijnands-van der Klis paper in the journal Nature.
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Table 6.3 PARAMETERS OF XTE J1808-369 (after Chakrabarty & Morgan 1998) Barycentric Pulse Frequency Projected Semi-major Axis Orbital Period Epoch of 90◦ mean logitude Pulsar Mass Function
400.9752106(8) Hz 62.809(1) lt-ms 7249.119(1) sec MJD 50914.899440(1) 3.7789(2) × 10−5
explained in Chapter 9, and the periodic delay being shown in Fig. 6.15 taken from the original discovery paper. The binary period was Pb ≈ 2 hours, and the orbit was circular, essential parameters being given in Table 6.4, where the numbers in parentheses are the 1σ uncertainties in the last significant figure. Epoch is given in Barycentric Dynamical Time (TDB; see Chapter 7), and MJD is the modified Julian date, defined as MJD ≡ JD – 2400000.5. Seven such systems are known today, with their neutron-star spin frequencies in the range 185 – 435 Hz, and their orbital periods in the range ∼ 40 minutes to ∼ 4 hours, i.e., quite short (see Table 6.5). In this table, Pb is the binary orbital period, as usual, νspin is the neutron-star spin frequency, ax sin i is the projected semimajor axis, fx is the pulsar mass function, and Mc,min is the minimum companion-mass for an assumed 1.4M neutron star. More details are given in reviews by Wijnands (2004), Lamb (2006) and Poutanen (2006). All of these systems are X-ray transients, i.e., systems that are dormant for a year or two, and then undergo outbursts lasting tens of days. Almost always, even their outburst luminosities are either significantly or considerably below those of the usual, bright LMXBs, and their average luminosities are, of course, quite tiny because of the long periods of low accretion when they are unobservable. (Indeed, it has been argued that the neutron star undergoes propeller spindown, as described earlier in this chapter, and in Chapter 12, at sufficiently low rates of mass transfer, expelling matter, so that there is no accretion: however, this is not clear yet.) This explains the practical difficulty in finding them, and, therefore, why it took so long to find this missing link. The accretion-powered pulsars in these systems are generally observed to be spinning down during outbursts, with frequency derivatives ν˙ ∼ 10−14 − 10−13 Hz s−1 . In addition to the above pulse periods, some of these sources exhibit quasi-periodic oscillations (QPOs; see Chapter 9) at high frequencies, and, in the case of XTE J1808-369, also at low frequencies. The
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Fig. 6.15 Discovery of the binary nature of the first accretion-powered millisecond pulsar. Shown are the pulse-arrival time delays due to ∼ 2 hr binary orbit, with respect to the expected time of arrival for a constant pulse frequency. Reprinted with permission by MacMillan Publishers Ltd from Chakrabarty & Morgan (1998), Nature, 394, 346. c MacMillan Publishers Ltd 1998.
difference between the pair of high-frequency (“kilohertz”) QPOs is closely related to the spin frequency of the neutron star: in XTE J1808-369, it is half the spin frequency, while in XTE J1807-294, it is roughly equal to the neutron-star spin frequency. Further, oscillations during thermonuclear bursts — similar to the X-ray bursts in some non-pulsing LMXBs — have now been observed from some of these pulsars at frequencies very close to the spin frequencies of the neutron stars in them, thus putting them clearly in the context of the collection of ∼ 100 LMXBs in our galaxy. Two crucial questions have come up in our general understanding of these missing links and their relation to other, “normal”, LMXBs, and we conclude our discussion with these. First, their evolutionary standing. The low-mass donors in these systems appear to have extremely low masses, M2 /M ∼ 0.01 − 0.05, so far as these have been measured. This puts them at the very end of the mass spectrum for LMXB donors, far below the
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Table 6.4 PARAMETERS OF ACCRETION-POWERED MILLISECOND PULSARS (after Poutanen 2006) System
Pb (min)
νspin (Hz)
ax sin i (lt-ms)
fx (M )
Mc,min (M )
SAX J1808.4−3658 XTE J1751−305 XTE J0929−314 XTE J1807−294 XTE J1814−338 IGR J00291+5934 HETE J1900.1−2455
121 42.4 43.6 40.1 257 147 83.3
401 435 185 191 314 599 377
62.809 10.113 6.290 4.75 390.3 64.993 18.39
3.779 × 10−5 1.278 × 10−6 2.9 × 10−7 1.49 × 10−7 2.016 × 10−3 2.813 × 10−5 2.00 × 10−6
0.043 0.014 0.0083 0.0066 0.17 0.039 0.016
∼ 0.08M limit for H-burning, and makes them rather like the companions in the “black widow” systems described later. Clearly, they have to be dwarfs of some sort, perhaps brown dwarfs (Appendix C), or He white dwarfs. Evolutionary scenarios leading to such systems were considered by Ergma and Antipova (1999), and by Bildsten and Chakrabarty (2001), the latter authors considering how the LMXB evolutionary scenarios considered above might lead to brown-dwarf companions, taking XTE J1808-369 as the prototype system. The idea follows directly from the scenario for mass transfer by angular-momentum loss described above: the system passes through the period minimum, as shown earlier, and thus passes into the degenerate branch: the companion is whittled down to a < ∼ 0.05M brown dwarf by the time a binary period ∼ 2 hours is reached. Heating of the dwarf by the X-rays emitted by the neutron star keeps its radius larger than it would be for a cold brown dwarf, as required by the condition of Rochelobe filling (see above). Similarly, a heated He dwarf has been suggested for the system XTE J1751-305. The low accretion rates, M˙ ∼ 10−11 M yr−1 , that occur in this r´egime of mass transfer by gravitational radiation both (a) agree with the typical average luminosities for these systems, and (b) provide a mechanism for accretion-disk instabilities that naturally account for the observed outbursts in these systems. Second, why do these systems show pulsations, while the bright, “normal” LMXBs do not? This is, of course, closely connected with the discussion on the latter question given in Chapter 9. While the explanation of the latter phenomenon in terms of a scattering cloud surrounding the system that “washes out” the pulses could perhaps be extended to the former phenomenon by arguing that such clouds would be much thinner, or absent, at the low accretion rates relevant for accreting millisecond pulsars,
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a different scenario has been explored in the early 2000s. The accretion is also the key parameter here, but the essential physics is related to the surface magnetic field of the neutron star, the argument proceeding as follows. As explained throughout this book, the pulsing in accretion-powered pulsars is due to channeling of the accretion flow to the magnetic poles of the neutron star by its magnetic field. In this scenario, the surface magnetic field of the neutron star is diagmagnetically screened — or “buried”, so to speak — by the freshly-accreted matter, above a critical accretion rate ∼ a few percent of the Eddington rate. Then the accreting matter approaching the neutron star feels little magnetic force, and is not channeled effectively at all. Hence no pulsation. The basic scenario is very much in line with the recent idea that reduction/decay of neutron-star magnetic fields is causally connected with accretion by neutron stars, which we discuss in Chapter 13. Further, there may be suggestive evidence for a stronger magnetic field in XTE J1808-369 than in non-pulsing LMXBs from the difference in the behaviors of their oscillations during thermonuclear bursts: the oscillation frequency drifts in different ways in the two systems as the burst develops (see Chakrabarty 2004 for further detail). 6.5.5
Irradiation of Low-Mass Companions
A major property of recycled millisecond rotation-powered pulsars is that many of them are single, as the first-discovered one PSR 1937+21. How can this be, since we decsribed their production in LMXBs above? What happens to the low-mass companion? The resolution of this paradox was worked out in the 1980s and 1990s, the answer being the expected one that the companion must be destroyed in some manner subsequent to recycling. We summarize the scenarios now. The basic “powerhouse” for this is the neutron star itself, through X-ray emission during the accretion phase, or through particle and electromagnetic radiation during the subsequent fastrotating, rotation-powered phase. X-ray heating of the companion in LMXBs have been considered since the 1970s, the basic effects being an expansion of the companion, accelerated mass transfer, and mass evaporation from the surface of the donor. Some of the essential effects of radiation on the structure of the donor were clarified in the early 1990s by Podsiadlowski (1991) and by Harpaz and Rappaport (1991). Irradiation of low-mass main-sequence stars suppresses convection in it, bringing the convective zones which usually dominate the structure of such stars into radiative equilibrium. During this change, the
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star expands, enhancing the mass-transfer rate. But once this is done, the star settles down in its new equilibrium and stops expanding, so that mass transfer proceeds again by angular-momentum loss, as above. On the other hand, if the companion is a giant/subgiant (see above), no such new equilibrium is possible, and mass transfer is accelerated, resulting in the transfer of the whole envelope of the companion to the neutron star. Thus, irradiation by accretion-generated X-rays will very likely not destroy the companion altogether, although it can modify the evolution of the LMXB, as also its observed properties in various spectral bands. However, when accretion stops and the recycled nueutron star turns on as a millisecond pulsar, the copious emission of pulsar wind from it, consisting of e+ e− pairs and electromagnetic radiation, can ablate and/or evaporate the low-mass companion, leading to its total destruction, and thus producing a single, recycled, millisecond pulsar. This scenario became very popular following the discovery of the millisecond pulsar PSR 1957+20 — a binary in which the recycled, rotation-powered pulsar appears to be evaporating its low-mass companion [Fruchter, Stinebring & Taylor 1988] so effectively as to have whittled it down to an incredibly low mass ∼ 0.025M — and came to be known, with somewhat gruesome humor, as the black widow 21 scenario. We consider this next.
6.5.6
Missing Links: “Black Widow” Pulsars
In 1988, Fruchter et al. discovered the 1.6 ms pulsar PSR 1957+20 with the short binary period Pb = 9.2 hours, whose properties proved to be crucial for establishing the above missing link between LMXBs and single recycled pulsars. While similar to other millisecond pulsars in many of its radio-pulse characteristics, this pulsar showed the remarkable properties that (a) it disappeared in eclipse behind the companion star for about 10% of each orbit, (b) the companion’s mass (established from the mass function of the binary and the argument that an eclipsing system is being viewed by us nearly edge-on, i.e., sin i ≈ 1) was ≈ 0.025M, and so, (c) the Roche lobe of the companion, with RL ≈ 0.3R , was much smaller than the eclipsing region, whose characteristic radius ≈ 0.75R followed from the above eclipse duration. The last point was the key, implying that ma21 The name refers to spiders of genus Latrodectus. The proverbial description is that the female of the genus is the one usually seen, as the male is often eaten by the female after mating. While the parallel with single, recycled pulsars is uncannily apt, the veracity of the above description is questioned by some modern experts.
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terial ablated from the companion surface must be continually replenishing the eclipsing medium [Fruchter et al. 1990]. The immediate suggestion was that a fraction of the spin-down luminosity of the pulsar in the form of a pulsar wind was doing the ablation (as also heating the illuminated face of the companion), thus completing the scenario given above. Further observed properties indicated that (a) the companion, probably a degenerate hydrogen dwarf of radius R2 ∼ 0.1R , was well inside its Roche lobe (but see below), and (b) the eclipsing material was “swept back” due to binary orbital motion — exactly like what happens to stellar winds in HMXBs, as described in Chapter 9 — as evidenced by asymmetries between ingress and egress properties of the eclipses. Two years later, Lyne and co-authors (1990) found a second example of this kind, namely the eclipsing, millisecond, binary pulsar PSR 174424A in the globular cluster Terzan 5. For this pulsar, the orbital period, Pb = 1.8 hr, was even shorter, and the companion mass was rather similar to that for PSR 1957+20. Eclipses covered a longer fraction of the orbit, suggesting an eclipsing region of size ∼ 0.9R , while the Roche-lobe size was RL ≈ 0.2R . About five such companion-ablating eclipsing millisecond pulsar binaries are known today, and Chakrabarty and Morgan (1998) made the natural suggestion that the missing-link accreting millisecond pulsars like XTE J1808-369 would turn, at later stages, into missing-link “blackwidow” pulsars, leading ultimately to isolated millisecond pulsars. A major evolutionary parameter, in view of the fact that the companions in these systems have already been ablated away to such small masses, is the timescale on which they are expected to be destroyed completely, pro7 ducing the isolated millisecond pulsar. This has been argued to be > ∼ 10 yr, on the basis of the rate of change of Pb [Fruchter et al. 1990]. Another major line of enquiry has gone, as expected, into the mechanism of the pulsar’s radio eclipses by the ablated wind from the companion. One idea is that of reflection or refraction of the pulsar signal by a dense plasma contained behind a contact discontinuity, as proposed by Phinney et al. (1988) and others. Another is that of free-free absorption in the plasma streaming off the companion, as proposed by Rasio et al. (1989) and others. A third is that the eclipsing material is largely confined within the companion’s magnetosphere, as proposed by Michel. Detailed comparisons of the predictions from these models were performed in Fruchter et al. (1990), and further diagnostics were done by Fruchter and Goss (1992) on the basis of radio-continuum studies of the eclipses of PSR 1957+20 performed by them. We refer the reader to these papers for more detail, and to the
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references contained therein. On the whole, free-free absorption seems to give a good account of the observations, and a rather surprising conclusion from that would be that the temperature of the ablated wind from the companion is quite low, Tw ∼ 300 K, in which case that wind would contain a substantial neutral component. This is not understood very well, unless the companion is considerably larger than suggested originally — almost as large as its Roche lobe. As Fruchter and Goss (1992) suggested, the latter is possible if distances have been underestimated because of inaccurate knowledge of galactic electron distribution (see Chapter 1). Soft x-ray emission, reported from PSR 1957+20, could serve as a good diagnostic tool for the wind conditions, in analogy with what we describe in Chapter 9 on HMXB winds. 6.5.7
Pulsars with Planets
In 1992, Wolszczan and Frail reported the discovery of two planets around the 6.2 ms pulsar PSR B1257+12. This was done through the detection of periodic variations in the pulse-arrival times (TOAs) of the millisecond pulsar due to the planets’ orbital motions, just as is done for binary-pulsar orbits, as described in Chapter 7. It was the extreme accuracy of pulsar timing, as well as the phenomenal stability of millisecond-pulsar clocks (see Chapter 7) that made this possible, since planetary variations in TOAs were very necessarily very small, amounting to only about ±1.5 ms, which, as Wolszczan (1994) pointed out, corresponded to a tiny amplitude vpsr ∼ 0.7 m s−1 of variation in the above pulsar’s radial velocity. The planetary masses were comparable to that of the earth, ≈ 3.4M⊕ and ≈ 2.8M⊕ (for edge-on planetary orbits, i.e., sin i ≈ 1), with orbital periods of 66.5 and 98.2 days respectively. This was the first discovery of earth-mass planets around a star other than the Sun, and therefore most exciting. Since optical techniques involving Doppler spectroscopy had, at that time, velocity resolutions which were at least one order of magnitude worse than above, they would not have been able to detect sub-Jovian planets (i.e., those with masses less than that of Jupiter) around sun-like stars [Wolszczan 1994]. This discovery was thus a great tribute to the accuracy of pulsar timing. As the reader can show from standard binary-dynamics results (Appendix B), the above amplitude of the pulsar’s velocity variation scales as −1/3 vpsr ∼ mpl Porb [Wolszczan 1994], mpl being the planet’s mass, and Porb the orbital period of the planet around the pulsar. The timing residuals τpl due to planetary motion — the crucial quantity for successful detection —
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goes as vpsr Porb , and so scales as mpl Porb with the planetary mass and orbital period [Wolszczan 1997]. Thus, more massive planets and those with longer orbital periods are easier to detect (although the latter scaling may appear counter-intuitive at first). More details are given in a lucid discussion by Wolszczan (1997), who also made the essential point that, other things being equal, heavier objects like Jovian or terrestrial planets should be detectable rather easily even around “normal”, slow pulsars, which show much timing noise (see Chapter 7), while lighter objects like “moons” or large asteroids can be detected only around millisecond pulsars, which have much lower timing noise and excellent stability, as mentioned above. Indeed, with more extensive timing, a small, third planet with a moon-like mass ≈ 0.015M⊕ and an orbital period of about 25.3 days was discovered around PSR B1257+12 by 1994. In Fig. 6.16, we show the periodic post-fit
Fig. 6.16 Post-fit residuals of the pulse-arrival times of PSR B1257+12, folded at the orbital periods of the three planets, viz., Planet A: 25.3 d, Planet B: 66.5 d, Planet C: 98.2 d. For each planet, the arrival-time variations due to the other two have been fitted out. Reprinted with permission by AAAS from Wolszczan (1994), Science, 264, 538. c 1994 AAAS.
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residuals in TOAs for each of these planets, taken from Wolszczan’s excellent 1994 review. By 1997, there were possible indications of a fourth planet around this pulsar, as also suggestions for Jovian or terrestrial planets around other pulsars. With greatly improved sensitivities of optical planetary-detection methods, and the profusion of planets — both Jovian and terrestrial — known today around “normal” stars other than the Sun (exoplanets in modern parlance), planets around pulsars may have now lost some of the unique novelty they once had. But their role in our understanding of the evolutionary origins of recycled pulsars continues to be important. Within the recycling scenario given in this chapter, we envisage the formation of accretion disks around the pulsars which are being recycled, and planets can condense out of such disks, just they are believed to do out of disks around protostars which eventually turn on as “normal” stars, sun-like or more massive, and so have planetary systems around them, like our solar system. In this view, the pulsar planets are vestiges of the accretion disks that spun them up (see Chapter 12), and, therefore, witnesses of the era when their low-mass companions were transferring mass to them, before these companions were destroyed in the manner suggested above.
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Chapter 7
Properties of Rotation Powered Pulsars
We give an overview of the basic observational properties of rotation powered pulsars in this chapter: pulsation properties, characteristics of the changes in pulsation periods and in the orbital periods of such pulsars in binaries, and so on. Theoretical frameworks for our attempts at understanding these properties are summarized as required, and developed in more detail in specific cases. An example of the latter is superfluidity in neutron stars, which we introduce and describe in the next chapter, since it now occupies a central place in our probes of the internal structure of neutron stars. 7.1
Pulse Properties
A basic fact about rotation powered pulsars is that, while individual pulses from a given pulsar show huge variations, the integrated (or mean) pulse profile for that given pulsar (integrated over, say, a few hundred pulses) is so remarkably stable in its properties that it serves as a “signature” of that particular pulsar. (Exceptions are the so-called mode-switching pulsars, a small minority class which we describe later.) It is thus useful to consider first the essential properties of integrated pulse profiles, as is customary and as we do here, before going into the further wealth of information on the variable characteristics of individual pulses.
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Integrated Pulse Profiles Pulse shapes
Integrated pulse profiles come in a wide variety of shapes, as shown in Fig. 7.1. The profiles are smooth, and most are dominated by a single component, but two- and multiple-component profiles are also widely known. Since the basic pulse shape usually remains the same as the (radio) frequency of observation changes, though there may be changes of detail in some cases with changing frequency, early shape-classification schemes, e.g., Types S (for simple = single component), C (for complex ), M (for multiple), were evolved [Manchester & Taylor 1977]. Correlations between these classes and other pulsar properties, such as periods and period derivatives, were also studied in the early days of pulsar research. As Fig. 7.1 shows, these components generally occupy only ∼ 2 − 10% of the pulse period P , i.e., rotation powered pulsars generally have very small duty cycles at radio frequencies. The duty-cycle, i.e., the fraction of P over which the observed energy-flux is significantly above the quiescent flux level, can be expressed either as a percentage, as above, or as an angular, longitudinal, or phase extent, with reference to a whole period P occupying 360◦ . Alternatively, an equivalent width W of the pulse is often defined by observers as a measure of this extent in seconds, either as (a) the ratio of total energy Ep delivered in a pulse to the peak intensity Imax of the pulse, W ≡ Ep /Imax , or as (b) the time-interval between the points on the two sides of the peak intensity where the intensity falls to some specified small fraction, say 10%, of the peak value. The two definitions generally yield rather similar properties for W , which is often displayed as a W versus P plot, as we show in Fig. 7.2. As we see, pulsars largely follow the behavior W ∝ P , with a constant of proportionality which corresponds to an angular or longitudinal extent ∼ 10◦ . Note that there is considerable scatter around this average relation, which is nevertheless valuable because it allows us to think of the pulsar emission as a process which occurs roughly over a constant (small) rotation angle of the pulsar.
7.1.1.2
Interpulses
Exceptions to the above rule of small-angular-extent pulses are the minority of pulsars which show much broader pulses, covering a large fraction of 360◦ . In the case of a few pulsars like PSR B0826-34, this wide pulse actually has fairly high intensity over essentially all of its large angular extent, perhaps
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Fig. 7.1 Examples of pulse profiles of rotation-powered pulsars at radio-frequencies ∼ 400 − 650 MHz, all plotted on the same scale of pulse phase. To show the smallness of typical duty-cycles, a bar indicating a 90◦ span (a quarter of a pulse period) is provided. Reprinted with permission by W. H. Freeman & Co. from Manchester & Taylor (1977): see Bibliography.
with a double- or multiple-peak structure. But a very interesting subclass of pulsars shows a roughly 180◦ extent, the total pulse basically consisting of two usual, small-angular-extent pulses at the two extremities of this ∼ 180◦ duration, with almost no signal in between. The weaker of the two pulses is called the interpulse. Examples of pulsars with interpulses are PSRs B0823+26 and B1702-19. This class of pulsars is of considerable interest because of the obvious possibility that the main pulse and the interpulse are emissions from the
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Fig. 7.2 Equivalent width vs. pulse period for rotation-powered pulsars: see text. The line represents a mean width of 10◦ in phase. Reprinted with permission by W. H. Freeman & Co. from Manchester & Taylor (1977): see Bibliography.
two magnetic poles of a magnetized neutron star, both poles being visible in these cases because the magnetic axis is nearly perpendicular to the rotation axis (i.e., a nearly orthogonal rotator), with our line of sight lying nearly in the equatorial plane of the rotation axis. By contrast, pulsars with only a main pulse, which constitute the majority, would then have their magnetic axes misaligned from their rotation axes by a significant, but not very large, angle, so that only one of the two magnetic poles is ever visible to us. Finally, pulsars with genuinely wide pulses like PSR B082634 can be interpreted as those with the magnetic and rotation axes nearly aligned, and with our line of sight also nearly along this common direction, so that we see only the emission from one pole, but over a large fraction of the total rotational extent of 360◦ [Lyne & Graham-Smith 1990]. Thus, the morphology of interpulses is of much help in clarifying the emission geometry of rotation powered pulsars. 7.1.1.3
Polarization
One of the most remarkable properties of rotation powered pulsars is the high degree of polarization often observed in their emission at radio frequen-
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cies, implying the importance of magnetic fields and non-thermal processes in the emission mechanism. For example, the integrated pulse profiles of some pulsars, e.g., PSR B0833-45 (also known as the Vela pulsar; see Chapter 1), show almost complete linear polarization. Statistically, the range of linear polarization does vary from a few per cent to almost 100% [Manchester & Taylor 1977], but larger percentages are found more frequently. Pulsars with drifting subpulses (see below) generally have low polarization in their integrated profiles, and pulsars of Type C (i.e., complex profiles; see above) usually show a maximum of ∼ 50-60% polarization in the midregion of the profile, and weak polarization in the wings (i.e., extremities) of the profile. These features are illustrated in Fig. 7.3, and we refer the reader to the book by Manchester and Taylor (1977) for more details. Circular polarization is also widely observed, but its strength rarely exceeds 20% in the integrated pulse profile. The position angle of the above linear polarization, conventionally defined as the orientation of the projection of the electric vector on the plane of the sky, has well-known, interesting “signatures”, some of which are shown in Fig. 7.3. This angle varies, usually smoothly, through the integrated pulse profile. Whereas the variation is roughly linear for some well-known pulsars of Type S (i.e., simple profile), e.g., the Vela pulsar, Type C pulsars have a characteristic S-shaped profile of position-angle variation, which is a classic signature, and which is shown for PSR B2045-16 in Fig. 7.3. As the name implies, the shape has rapid variation in the middle of the integrated pulse profile, and slow variation in the wings, the total swing in the posi◦ tion angle through the pulse profile being < ∼ 180 . The center of the S is identical, or very close, to the midpoint of the integrated pulse profile, and the absolute value of the maximum gradient in the position-angle profile (at the center of the pulse) is directly correlated with the total swing in the position angle: the higher the former is, the closer the latter is to 180◦ . These systematics of polarization agree very well with the predictions of the rotating vector model (see below), and so have served as guides for the construction of viable models for pulsar emission. Indeed, understanding the role of polarization in the structure of the pulses is a major step towards our understanding of the pulse emission mechanism. This is brought home by the fact that, even in complex integrated profiles consisting of many separate features which show large variations between individual pulses, as we shall see below, and which can therefore be interpreted as discrete sources without strong correlations with one another (although very likely belonging to the same emission region),
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Fig. 7.3 Examples of polarization characteristics of rotation-powered pulsars. In each case, the two curves under the total integrated pulse profile give the intensities of the linear and circular polarization, as marked. Position angles are also shown in the upper part of each panel. Reprinted with permission by W. H. Freeman & Co. from Manchester & Taylor (1977): see Bibliography.
the swing of the angle of linear polarization remains continuous through the whole profile. Thus, Lyne and co-authors have argued that polarization is perhaps the best diagnostic of the total emission region and its associated magnetic structure, while individual features can be regarded as small areas of excitation within this total region. From this argument, we can define a measure of the total angular extent of a pulse, or pulse-width W , in terms of the total swing in polarization, which serves as an alternative to the definitions given in Sec. (7.1.1.1). Lyne and Manchester (1988) have shown that this definition of W gives numerical values rather similar to those given by the earlier ones, but that its scaling with P is somewhat different. Instead of the earlier W ∝ P scaling, which gave a (roughly) constant angular extent ∼ 10◦ , we now get W ∝ P 2/3 roughly, corresponding to an angular
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extent ∼ 13◦ P (s)−1/3 , which decreases slowly with increasing period. Here, P (s) is the pulse period in seconds. 7.1.1.4
The rotating vector model
In 1969, Radhakrishnan and Cooke proposed a pioneering, geometrical model for interpreting the observed position angles which proved to be quite useful. Using the geometry of a simple, undistorted dipolar magnetic field, these authors utilized the fact that curvature radiation emitted by charged particles accelerated along open magnetic field lines (see Chapter 11) will be beamed along the local tangent to the field line of interest, and it will be polarized in the plane of curvature of this field line, which contains the dipole axis. Thus, it will be polarized parallel to the projected direction of the magnetic axis, the position angle ψ of polarization (measured with respect to the projected direction of the rotation axis of the pulsar) varying with the pulse phase or longitude φ (see above) according to the relation tan ψ =
sin α sin φ . sin ζ cos α − cos ζ sin α cos φ
(7.1)
Here, α is the angle between the rotation and magnetic axes of the pulsar, and ζ that between the rotation axis and our line of sight. This relation follows from elementary spherical trigonometry: we refer the reader to the original work for detail. A straightforward differentiation of the above equation shows that the rate of change of ψ goes through a maximum at φ = 0, where our line of sight crosses the meridian containing the magnetic axis. This maximum rate of change of the position angle is given by: sin α dψ . (7.2) = dφ max sin(ζ − α) The wonderful thing about the above simple relation is the accuracy with which it reproduces the characteristic S-shaped swings of position angle described in the previous subsection, which suggests that a basic geometrical feature of rotation-powered pulsar emission may be already incorporated correctly in it, independent of the details of the emission mechanism, which still present many difficulties, as we shall see in chapter 11. 7.1.1.5
Frequency dependence and stability
Although the gross aspects of the integrated pulse profile generally remain recognizable as the radio frequency of observation changes, as indicated
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above, several specific features show well-known frequency dependences. For example, the separation between two given components of a pulse profile generally decreases with increasing frequency roughly as ν −p , where p ∼ 0.20 − 0.25. There is also some indication that the widths of the individual features themselves decrease in a similar fashion with increasing frequency. As another example, the fractional polarization of integrated pulses also generally decreases with increasing frequency. Actually, the detailed behavior seems to come in three categories, namely, (a) the polarization decreases as ν −p , with p ∼ 0.4 − 1.2, (b) the polarization remains almost constant over the whole range of observed frequencies, and, (c) the polarization remains roughly constant upto some critical frequency, and decreases roughly as ν −1 above this frequency [Manchester & Taylor 1977]. The basic reason for the frequency dependence of the shape of the integrated pulse profile, when this is significant, is that the spectra of the different components of the profile can be different. Typical overall radio spectra of pulsars can be described roughly by piece-wise power laws, i.e., S ∝ ν −α , where S is the flux density, with the index α varying typically in the range 2 to 4, with a flattening of the power law at low frequencies, or even a turnover. Spectra of the individual components have indices which also lie in the same range, but different components can have different spectra, which results in a change in the integrated pulse shape as the frequency varies. How, then, should we quantify the stability of the integrated pulse profile of a given pulsar, the notion of which we introduced earlier? Obviously, this should be done with reference to a given observation frequency. At this frequency, we can first obtain a reference profile by summing all the available data. Then we can obtain the integrated profile of N pulses, and compute the average cross-correlation coefficient between this profile and the reference profile. Finally, we can increase N , and study how this coefficient approaches unity. Generally, the residual deviation from unity decreases as N −a , with a in the range ∼ 1.3 − 0.55. The fastest decrease (i.e., the largest values of a) is seen for pulsars with drifting subpulses (see below), intermediate for Type S pulsars, and the slowest for Type C pulsars. Note that the extreme cases of Type C (complex profile) pulsars show a convergence rate only slightly better than that expected for completely random distribution of components within a pulse, which is a = 0.5. Closely connected with this anomalously poor stability of the integrated pulse profiles of some rare Type C pulsars is the phenomenon of mode switching in these, originally discovered by Backer in 1970. At intervals which are not
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precisely predictable, but which have characteristic orders of magnitude, such pulsars change from the usual integrated profile to one quite different from it, due to an abrupt change in the relative strengths of the components of the Type C pulse profile. Most of the mode-switching pulsars show two modes, one normal, in which the pulsar spends most of its time, and the other abnormal. But a few pulsars like PSR B0329+54 show several possible modes.
7.1.2
Individual Pulses
There is a rich variation in the essential properties,e.g., intensity, shape, and polarization, of individual pulses from a given pulsar, although averaging over a few hundred of them often suffices to yield that stable pulse profile which we described above, and which is so reliable as to serve as a “signature” of the given pulsar. Let us now summarize some essential characteristics of individual pulses, whose components are often identifiable as subpulses, which we can think of as basic units of emission, possibly identifiable in the context of pulsar emission models as radiation from an isolated location, the summation over all relevant locations then representing the integrated pulse [Lyne & Graham-Smith 1990]. Subpulses have typical widths ∼ 1◦ - 3◦ , as compared with the ∼ 10◦ “window” of the integrated pulse (see above), and they may occur (a) randomly within this window, (b) show preferences for certain longitudes, or, (c) “drift” across the window slowly (see below). There can also be microstructure within the pulse on a much shorter scale, which we summarize below. 7.1.2.1
Intensity variations and nulling
The total energy contained in an individual pulse varies greatly from pulse to pulse, the systematics of which can be characterized in different ways. A straightforward number distribution readily shows some basic types of behavior, e.g., (i) a roughly normal distribution about a mean value, (ii) an essentially monotonically deceasing distribution, with most pulsars at low pulse-energies, and, (iii) a bimodal distribution, with one peak similar to that found for the first type, and another at zero energy. The last type is associated with the phenomenon of nulling, which we describe below. Alternatively, these variations can be described in terms of a fluctuation spectrum, i.e., the Fourier transform or power-density spectrum obtained from the time variation of the total pulse energy. Fluctuation spectra of
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pulsars, given in Fig. 7.4, show various combinations of the following essential features: (a) featureless flat spectra representing random fluctuations, often called white noise, (b) featureless spectra rising towards lower frequencies, i.e., red noise, (c) narrow or wide line features, representing periodic or quasi-periodic oscillations of the pulse energy. Features (a) and (b) are often found in short-period pulsars, while feature (c) is often found in longer-period pulsars, particularly of Type C, the (periodic or quasiperiodic) modulations being due to nulling and/or to the phenomenon of subpulse drifting (see below), or, sometimes, to the mode-switching phenomenon described above. The degree of modulation of pulse intensities can be quantified by a modulation index m, introduced by Taylor, Manchester and Huguenin (1975), and defined as: 2 − σ2 σon off . (7.3) m≡ E Here, σon is the rms fluctuation of the pulse energy about its mean value E, and σoff is the rms fluctuation in the (random) radio noise off the pulse (i.e., in a direction slightly offset from that of the pulsar, containing no other
Fig. 7.4 Examples of fluctuation spectra of rotation-powered pulsars: see text. Reproduced with permission by AAS from Taylor & Huguenin (1971): see Bibliography.
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radio sources). Typical values of m are in the range 0.5–2.5, m for a given pulsar almost always increasing with decreasing radio-frequency. Pulsars with high values of m usually correspond to a number distribution of class (ii) described above, while those with low m-values usually correspond to class (i) distribution above. The phenomenon of nulling has been known since early days of pulsar research: the intensity of individual pulses suddenly drops to a very low value, < ∼ 1% of the usual mean pulse energy, say, and stays this way for some duration before abruptly returning to the usual intensity level. For a given pulsar, the duration of the nulls and the intervals between their occurrences vary randomly about characteristic values. The ratio of these characteristic values varies widely among nulling pulsars, from a small ratio corresponding to a few null pulses every ∼ 100 pulses, to spending ∼ 50% of the time in the null state, or even ≈ 70% of the time in the extreme example of PSR B0826-34 cited by Lyne and Graham-Smith (1990), which makes the detection of such a pulsar difficult to confirm until its nulling behavior is appreciated. It is widely believed that nulling is related to the aging, and eventual cessation of pulsing, or “death”, of rotation powered pulsars. This is so because nulling seems to be a characteristic of older pulsars: indeed, most nulling pulsars are found near the so-called “death line” on the period derivative vs. period or magnetic field vs. period diagrams of pulsars (see Sec. 12.3). Therefore, a scenario which would appear very plausible would be that this pulsar “death” is simply the permanent form of nulling: as the pulsar ages and its rotation slows down, the fractional time it spends in the null state increases progressively, until it reaches 100%. Whether or not this is really true is not clear at present. A related, important point is that the total power radiated by a rotation powered pulsar does decrease with increasing age, whether nulling occurs or not. Thus, the latter decrease could represent an overall reduction in the strength in the pulse emission process of a rotation powered pulsar as its rotation rate decreases (and perhaps also as its magnetic field decreases somewhat, although current opinion does not favor this; see Chapter 13), while nulling could represent the failure to satisfy the threshold conditions required for the occurrence of certain specific local mechanisms (e.g., spark generation) involved in the pulse generation process, due to the same causes associated with aging, e.g., reduction in rotation rate.
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Subpulse drifting
As we mentioned above, the march or “drifting” of subpulses across the window of the integrated pulse profile is an interesting part of the phenomenology of rotation powered pulsars. Indeed, pulsars exhibiting this property are sometimes classified as Type D, extending the scheme introduced in Sec. 7.1.1.1. We illustrate the phenomenon schematically in Fig. 7.5: let P be the pulse period, as before, and let a given subpulse drift in longitude at such a rate that it returns to the same longitude after a repetition period Pr . Thus, the fluctuation spectrum shows a strong feature at a frequency Pr−1 , as indicated above: this feature can be a narrow line for such pulsars as PSR B0809+74 [Manchester & Taylor 1977], implying quite strict periodicity in the drift. It is customary to quote Pr in units of P , values of Pr /P usually lying in the range 3–20 [Manchester & Taylor 1977]. Note that Pr is also called the band spacing, for reasons obvious from Fig. 7.5. Another parameter specifying the drift properties is the so-called secondary period Ps , which is the spacing between two subpulses showing drift. Ps can be expressed either in units of time (usually milliseconds) or in longitudinal extent, taking P to be equivalent to 360◦ , as before. The latter measure usually gives values in the range 3◦ –20◦ . The ra-
Fig. 7.5 Drifting subpulses. Shown are pulse phase (or longitude) vs. time diagram, as well as the mean pulse profile, for several pulsars. Observed intensity at any phase is coded by a dot whose area is ∝ intensity. Time increases upward, and to the right. PSRs 0031-07 & 0809+74 show drifting subpulses, with a characteristic slanted band pattern. The other pulsars do not show this phenomenon. Reproduced with permission by AAS from Taylor et al. (1975): see Bibliography.
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tio D ≡ Ps /Pr is called the drift rate, since successive drifting subpulses are brought to the same longitude at this rate; it is also called the band slope for pictorial reasons clear from Fig. 7.5. From the numbers given above, it is clear that typical expected values of the drift rate in units of degrees/period would be ∼ 1, and, indeed, the observed drift rates lie in the range 0.5–3 or so, in these units [Manchester & Taylor 1977; Lyne & Graham-Smith 1990]. Drift rates can be both positive and negative, the former corresponding to a drift from the leading edge of the profile to the trailing edge, and the latter, a drift in the opposite direction. Some pulsars exhibit more than one drift rate in the following manner: the spacing Ps between the subpulses does not change, but the repetition period or band spacing Pr can have different values in different modes. An example is PSR B0031-07, where three distinct values of drift rate are found, corresponding to three distinct peaks (i.e., three distinct values of Pr ) in the fluctuation spectrum. By contrast, a few pulsars like PSR B1237+25 can exhibit multiple drift rates due to the opposite cause: while the repetition period Pr remains constant, so that the fluctuation spectrum shows only a single peak, the spacing Ps varies with longitude within the integrated profile. Note that drift rates generally increase with decreasing radio frequency, largely due to the fact that the spacing Ps between subpulses generally increases with decreasing radio frequency, the latter being a general property that we have indicated earlier. Finally, a curious property of drifting after nulling episodes is that the drift of a subpulse seems to have a “memory” of its condition before the null: as the null starts, drifting stops, and at the end of the null, it recovers roughly exponentially to its pre-null value [Lyne & Ashworth 1983]. For more details, including tables of drift rates, we refer the reader to the excellent books by Manchester and Taylor (1977) and by Lyne and Graham-Smith (1990). These detailed properties of drifts can serve as valuable guidelines for the construction of pulsar emission models. 7.1.2.3
Micropulses
Individual pulses of many pulsars show structure on timescales much shorter than those of the subpulses, viz., a few to a few hundred microseconds, at high time-resolution observations. This is, of course, most practicable for the brightest pulsars, since it involves the use of some techniques for removing part of the dispersion of the signal during propagation. These structures are called micropulses, and their relation to the subpulses is, in many ways,
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reminiscent of that of the subpulses to the pulses. Thus, micropulses can occur at random positions within the subpulses, although quasi-periodic groups of micropulses are also well-known, with typical periodicities ∼ 1 millisecond. A major property of the above microstructure is that its characteristics at different radio frequencies are generally very well-correlated. This phenomenon of wide bandwidth, as it is sometimes called [Lyne & GrahamSmith 1990], may be of central importance in building models of pulasr emission. The polarization of the micropulses is generally similar to that of the underlying subpulse. To show the range of essential timescales involved in the pulsing phenomenon of rotation-powered pulsars, it is instructive to compute the autocorrelation function (ACF) of a set of individual pulses, after dispersion removal. The typical form of the resulting ACF, shown schematically in Fig. 7.6, illustrates the hierarchy of basic timescales in a transparent manner. The longest times, ∼ 10−2 – 10 s, say, clearly correspond to the integrated pulse profiles. The next shorter range of timescales, ∼ 10−4 – 10−2 s, say, correspond to the subpulses. Next, in the range ∼ 10−6 – 10−4 s, say, come the micropulses. Of course, there is an overlap of categories at each of the boundaries described above, as the characteristic periods or timescales of the three structures can overlap to some extent, as seen
Fig. 7.6 Autocorrelation function (ACF), showing the hierarchy of essential timescales in the pulsing of rotation-powered pulsars. Reproduced with permission by AAS from Rickett (1975): see Bibliography.
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earlier. For example, pulsar periods ∼ 1 ms reach down into the region of widths of the shorter subpulses of typical long-period pulsars, and so on. However, for a given pulsar, the boundaries are quite well-defined, as they must be. Finally, at the shortest timescales, i.e., for time lags τ < ∼ τb , where τb ∼ 1/∆νb corresponds to the frequency bandwidth ∆νb sampled by the observation, the ACF rises steeply with decreasing τ , corresponding to random noise in the radio receivers.
7.1.2.4
Giant pulses
The Crab pulsar (PSR B0531+21; see Chapter 1) was known from the days of its discovery in the late 1960s to produce an occasional pulse hundreds of times brighter than a typical pulse: indeed, this fact was instrumental to its discovery. These were called giant pulses, and, until the mid-1980s, thought to be a unique property of the Crab pulsar, since the number distribution of the pulse energies of all other pulsars (see Sec. 7.1.2.1) known at the time was known to extend only upto energies ∼ 4 – 10 times the mean pulse energy E, never upto > ∼ 100E. By contrast, the energy of a giant pulse from the Crab pulsar can go upto > ∼ 2000E. Further, in contrast to the exponential fall-off at high pulse-energies in other pulsars, the tail of the Crab pulsar distribution was a power law, the fraction f of the pulses with pulse energies E ≥ SE (i.e., the cumulative distribution) scaling as f ∼ S −α . For “giant” main pulses, α ≈ 2.5, while, for the interpulse, the “giant” versions fell off with an exponent ≈ 2.8. The occurrence of giant pulses is broad-band, showing essentially the same phenomenon at different radio frequencies. In addition, these pulses are of short duration, typical widths being a few microseconds (with structure discernible upto nanoseconds). Following the discovery of the first millisecond pulsar PSR B1937+21 in 1982, indications came in the mid-1980s that this pulsar may also show giant pulses. These indications were confirmed in the mid-1990s, and a detailed study of the giant pulses from this millisecond pulsar was published by Cognard et al. in 1996, showing that such pulses with S as large as 300 were exhibited by this pulsar. The cumulative distribution of pulse energies was, again, found to be describable by a power law at high energies, with an exponent α ≈ 1.8, as shown in Fig. 7.7. Further, Cognard et al. (1996) showed that the above distributions corresponded to rather similar statistical rates of occurrence of giant pulses for the Crab and the millisecond pulsar, namely, one pulse with sufficiently
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Fig. 7.7 Cumulative ditribution of relative pulse energy S ≡ E/ E for the first millisecond pulsar PSR B1937+21. Clearly, giant pulses with S ∼ 300 − 400 are seen in this pulsar. Reproduced with permission by AAS from Cognard et al. (1996): see Bibliography.
high energy, S ≥ 100, say, every (1–3)×105 pulses. As with the Crab, giant pulses are broadband, with the duration even shorter for the millisecond pulsar. Even the average profile of a giant pulse is extremely narrow, with a very sharp rise on a timescale ≤ 5 µs. In addition, giant pulses from this pulsar show an unusual feature, namely, that many individual giant pulses show a very high degree of circular polarization. The Crab pulsar and the millisecond pulsar are remarkably different, the former being a young pulsar with a high magnetic field, and the latter being a recycled (see Chapter 6), old pulsar with a low magnetic field. Thus, the discovery of giant pulses from only these two pulsars was a puzzle at first. However, as Cognard et al. (1996) pointed out, one feature is common to these two pulsars, and may hold the key to understanding the mechanism of emission of the giant pulses: the strength of the magnetic field at the light cylinder (see Chapter 10), as estimated from the observed period and period derivative, are very close for these two, and the highest known among pulsars for which this estimate has been possible. Indeed, it appears that the above magnetic field strength shows relatively high values for two groups of pulsars, viz., (a) the young, fast-rotating pulsars at periods P ∼ 0.1 s, and, (b) the old, recycled pulsars, spun up to much faster rotation periods P ∼ 1 − 10 ms, i.e., the “millisecond” pulsars. Thus, the next logical step would be to investigate those members of the above groups which corresponded roughly to the next highest values of this field strength, after the highest value mentioned above.
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In the last few years, such investigations have been started by Johnston, Romani, and co-authors. From the millisecond pulsar group, giant pulses were indeed shown by PSR B1821-24, the recycled pulsar with the next highest value of the inferred magnetic field strength at the light cylinder, with strengths S > ∼ 50 (Romani and Johnston 2001). However, several other investigated millisecond pulsars did not show any evidence for giant pulses, down to appropriate limits. By contrast, young, relatively fast-rotating pulsars like the Vela pulsar show a curious phenomenon, whose relation to the giant pulses is not entirely clear yet. This was first noticed in the Vela pulsar by Johnston et al. (2001): while this pulsar does not show any giant pulses in the sense of S > ∼ 10, say, it does show extemely narrow and highly-polarized pulses, whose peak intensities exceed the integrated peak intensity by factors ∼ 40. These authors named them giant micropulses. The fact that these pulses have a power-law distribution like the giant pulses led to the interesting point of whether the extended tail of this distribution might not go into the “true” giant-pulse r´egime. This point is unresolved at present, but other young, Vela-like pulsars do show a general lack of giant pulses, with PSR B1706-44 showing giant micropulses on the trailing edge of the profile, similar to Vela. As Cognard et al. (1996) pointed out, other than their potential for clarifying the physics of the emission region in pulsars, giant pulses may also be useful for high-precision timing measurements. This latter potential is based on the very short rise time of these pulses indicated above, added to the facts that (a) they arrive at the same phase (which needs to be checked by studying the long-term stability of giant-pulse arrival phase), and, (b) they obviously have a very high signal-to-noise ratio. Submicrosecond accuracy may be achievable by this method, which would be based entirely on these pulses.
7.2 7.2.1
Timing Properties Pulsar Timing
Determination of time of arrival (TOA) of the pulses with great accuracy is perhaps the most crucial aspect of observational research on rotation powered pulsars: this has established that such pulsars are excellent clocks, the accuracy of the best of them (which are the millisecond pulsars, as we shall see) rivaling that of the best atomic clocks known. How is this timing done with radio telescopes? First, a very stable standard pulse profile
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with high signal-to-noise ratio is obtained by summing over a sufficiently large number (∼ a few thousands, say) of pulses. Next, the integrated pulse profile obtained from the stretch of data (of a few minutes’ duration, say) for which we need to determine the TOAs is cross-correlated with the standard profile, in order to determine the pulse phase with accuracy. The pulse phases are finally converted into TOAs with the aid of time standards, which, through the master clock of the radio astronomical observatory, are ultimately referred to the best atomic time standards on the Earth via radio or satellite links. For the 305-meter telescope of the Arecibo Observatory (in Puerto Rico, USA), for example, one of the time standards of reference has been that kept by the US Naval Observatory (in Washington, DC). Typical accuracies in TOA achieved in this way in the 1980s for the famous Taylor-Hulse binary pulsar, PSR B1916+13 (see Sec. 7.2.4.1), were < ∼ 20 µs [Taylor & Weisberg 1989]. These TOAs are, of course, those at the observing telescope on the Earth, i.e., topocentric, as they are often called1 , and so subject to variations due to the motion of the Earth around the Sun. We subtract these effects by calculating the TOAs at the center of mass, or barycenter , of the solar system – an inertial frame of reference – in the following way. In the process, we also take into account the effects of frequency-dependent dispersion of the pulsar signal (see Chapter 1). Let t be the topocentric TOA and tb be the corresponding barycentric TOA for infinite-frequency pulses (see below). The difference or delay between them is: tb − t =
d − |d − r| − D/ν 2 + ∆C + ∆E − ∆S , c
(7.4)
where d ≡ n ˆd is the vector distance of the pulsar from the solar system barycenter, and the vector r points from the barycenter to the telescope. In addition, ν is the radio frequency of observation, D is the dispersion constant of the pulsar, and ∆C is the offset between the observatory’s master clock and the reference atomic standard in Eq. (7.4). Furthermore, the relativistic corrections are contained in the last two terms on the righthand of Eq. (7.4): ∆E is the solar-system Einstein delay, and ∆S is the solar-system Shapiro delay, both of which are described below. Let us consider, in turn, each term in the above time-delay. The first is the obvious delay due to the different times of propagation of electro1A
topocentric value is one measured with respect to a reference system centered at some location on the Earth, the radio astronomical observatory in this case, or, the phase center of the radio telescope of that observatory, if we wish to be more precise.
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magnetic waves from the pulsar to the topocenter (i.e., telescope) and the barycenter because of the orbital motion of the Earth around the Sun, and is called the Rømer term or Rømer delay2 . A simple Taylor expansion shows the Rømer delay to be ∆R ≡
n ˆ.r (ˆ n.r)2 − r2 d − |d − r| ≈ + . c c 2dc
(7.5)
In older literature, only the first term in the above expansion was retained, but modern, high-precision timing work includes the second term as well (see, e.g., Ryba & Taylor 1991). (Note that there is some confusion about definition here, as the first-order term (ˆ n.r/c) is sometimes defined to be the Rømer delay, in analogy with Rømer’s historical work.) The rough size of the Rømer delay is ∼ (P⊕ /2π)(v⊕ /c), where P⊕ is the Earth’s orbital period, and v⊕ its velocity with respect to the solar-system barycenter. The second term D/ν 2 is the usual frequency-dispersion of the TOA of the pulses introduced in Chapter 1, except that the dispersion constant D occurring here is defined differently from the dispersion measure DM introduced there. D is measured in units of Hz, and is related to DM as D ≡ DM (e2 /2πme c) ≈ DM (cm−3 pc)/(2.41 × 10−16 ) Hz, where we have expressed DM in its usual units given in Chapter 1. Since the delay scales as 1/ν 2 , it is convenient to define a fictitious “infinite-frequency limit”, ν → ∞, for reference purposes, since the delay vanishes there. Consider now the Einstein delay ∆E , which represents the combined effects of gravitational redshift and time dilation due to motions of the Earth and other bodies in the solar system, and which is obtained by integrating the equation 2 v⊕ d∆E Gmi = + − constant . dt c2 ri 2c2 i
(7.6)
Here, mi are the masses of all bodies of significant mass in the solar system, excluding the Earth, and ri is the distance of mi from the Earth. Further, v⊕ is the velocity of the Earth with respect to the solar-system barycenter, as above, and the constant in Eq. (7.6) is so chosen that the long-time average of the right-hand side of this equation is zero. The size of the Einstein delay is ∼ e⊕ v⊕ /c times that of the Rømer delay, where e⊕ is the eccentricity of the Earth’s orbit. 2 The
Danish astronomer Rømer used a closely analogous difference between lightpropagation times from Jupiter’s satellites to two positions of the Earth in its orbit around the Sun in his pioneering determination of the velocity of light.
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Finally, consider the Shapiro delay ∆S , which is caused by the propagation of electromagnetic waves from the pulsar through the curved spacetime near the Sun, i.e., the well-known “bending” of light rays near a massive object. Its value is given by ∆S = −2T log(1 + cos θ),
(7.7)
where θ is the pulsar-Sun-Earth angle at the time of observation. In Eq. (7.7), the eccentricity of Earth’s orbit has been neglected, which is quite justifiable even for high-precision timing, and we have introduced the timescale T ≡ GM /c3 = 4.925490947 µs, which we shall find constantly useful in studying relativistic effects in pulsar timing3 , and whose value is known with great accuracy (since the product GM is known with higher accuracy than either G or M separately; see Taylor & Weisberg 1989). The size of the Shapiro delay is set by the above size of T , and it depends logarithmically on the angular impact parameter, as Eq. (7.7) shows, so that its maximum value is ∼ 120 µs, corresponding to the situation where radio waves from the pulsar just graze the limb of the Sun on their way to our Earth-bound telescope, i.e., when θ is close to 180◦), How are the above delays measured with sufficient accuracy in modern pulsar timing work? An account given by Ryba and Taylor (1991) of their work on PSR B1855+09 with the Arecibo telescope serves as a good example. The position and velocity of the telescope on the Earth (the topocenter), which occur in the Rømer, Einstein, and Shapiro delays, are determined by interpolating a solar system ephemeris and adding terms that account for measured irregularities in the Earth’s rotation. These authors used the PEP740R ephemeris of the Center for Astrophysics (in Cambridge, Massachusetts, USA) or the DE200 ephemeris of Jet Propulsion Laboratory (in Pasadena, California, USA), the Earth-rotation corrections being taken from the Bulletin of the International Earth Rotation Service. The clock correction ∆C was obtained by them from measurements made with the satellites of the Global Positioning System, while observations were in progress. Since these satellites were in simultaneous view of the Arecibo observatory and the U. S. National Institute of Standards and Technology (in Boulder, Colorado, USA), reference could be made ultimately to primary atomic time scales, using the known offsets of the standard time kept at the latter institute from this primary time reference. For the Einstein delay, 3 The reader will readily recognize T as the light travel time across the gravitational radius GM /c2 corresponding to one solar mass.
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the work is relatively easy, since a table of it is included in the ephemeris distributed by the Center for Astrophysics, and we only need to interpolate the table for obtaining the value at the time of observation. For isolated rotation powered pulsars, i.e., those which are not in binary systems, the above work is essentially all we need to investigate the properties of the pulses in the pulsar’s own rest frame, since the effects of binary motions do not need to be separated out. We now consider timing studies of isolated pulsars, and return later to the crucially important subject of timing rotation powered pulsars in binaries, which, as we describe later in the chapter, has produced fundamental results in physics over the last quarter of a century. For isolated pulsars, the barycentric time tb is equivalent to the proper time T in the rest frame of the pulsar, up to an additive constant and a (nearly) constant factor describing the Doppler effect and gravitational redshift. Hence, we can work with a proper time T ≡ tb − t0 , where t0 is a suitable reference epoch, as explained below. As we described in Chapter 1, a rotation powered pulsar spins down, i.e., its rotation period P increases, and its rotation frequency4 ν ≡ 1/P decreases because of the loss of rotational energy by emission of electromagnetic radiation and charged particles. The pulse phase φ(T ) then has two parts: (a) a deterministic part due to the spin-down process, and, (b) a stochastic part (T ), i.e., timing noise, which we shall describe in Sec. 7.2.3.1. The phase is thus given by 1 2 1 3 ˙ + ν¨T + (T ), φ(T ) = νT + νT 2 6
(7.8)
where dots over symbols denote time derivatives, as usual. Note that, with this definition of phase, φ increases by unity per pulse period, not by 2π. Thus, in the absence of noise ( = 0), the value of φ for a stretch of data containing an integral number of pulses should, ideally, be an integer. We shall see the significance of this below. Finally, by a suitable choice of the reference time t0 , we can make φ(T ) vanish when the barycentric TOA equals t0 : the reference time then represents a nominal infinite-frequency TOA at the barycenter. Timing of isolated pulsars proceeds in the following way. From the observed TOA, we compute φ(T ) with the aid of Eq. (7.8), neglecting the 4 We are obliged to use ν for pulsar rotation frequency in this book because of its widespread use in the literature to denote this quantity. We shall avoid any confusion with the radio frequency of observation by always stating the context clearly.
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stochastic part and assuming trial values for the parameters ν, ν, ˙ and ν¨. We then minimize the deviation of this computed value of φ(T ) from the integer nearest to it. The minimization is done with respect to the above three parameters, and also with respect to (i) parameters describing the pulsar’s position, viz., right ascension α and declination δ (see Appendix A), (ii) its parallax π, (iii) parameters describing its proper motion, viz., ˙ (iv) the reference epoch t0 described above, and, µα ≡ α˙ cos δ and µδ ≡ δ, (v) the dispersion constant D introduced above, thus making a total of 10 parameters. Parameters in categories (i), (ii), and (iii) occur in the transformation of the TOAs from the topocenter to the barycenter, and that in category (v), in accounting for propagation effects. The goodness of fit is measured by the usual χ2 statistic: χ2 =
2 1,N φ(Ti ) − ni i
σi /P
,
(7.9)
where N is the total number of TOAs, φ(Ti ) is the ith computed phase, ni is the integer closest to φ(Ti ), and σi is the estimated uncertainty in Ti . As pointed out by Ryba and Taylor (1991), it is possible to determine these parameters with very high precision because the integers ni , which are often extremely large, are known exactly. 7.2.2
Secular Period Changes
The secular, or long-term, changes in the period of a rotation powered pulsar are measured by the frequency derivatives ν, ˙ ν¨ obtained from the timing studies described above. Equivalently, the derivatives of the period P or the angular velocity of rotation of the neutron star, Ω ≡ 2π/P , can be used, these being related by: ˙ Ω ν˙ P˙ =− =− P Ω ν ¨ ν ν¨ ΩΩ P P¨ = =2− . 2 ν˙ Ω˙ 2 P˙ 2
(7.10) (7.11)
Consider the first derivative of the period or frequency. We showed in Chapter 1 that emission of magnetic dipole radiation by a rotation powered pulsar leads to a spin-down rate:
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Ω˙ = −KΩ3 ,
K≡
2µ2 sin2 α . 3Ic3
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(7.12)
Here, µ is the magnetic moment and I is the moment of inertia of the neutron star, and α is the angle of inclination between the rotation and magnetic axes. This leads to a natural, and very useful, generalization of the form of the spin-down equation to: Ω˙ = −KΩn ,
(7.13)
where the general exponent n (n = 3 for dipole radiation) is called the braking index . Different values of n would correspond to different processes of rotational energy loss, or braking, and the value of the constant K would, of course, also be different for these processes. 7.2.2.1
Characteristic age
The spin-down equation (7.13) can be readily integrated to yield the relation between the initial angular velocity Ωi at t = 0 and the current angular velocity Ω at time t. For n = 1, the relation is: n−1 Ω Ω t=− . (7.14) 1− ˙ Ω (n − 1)Ω i In the limit Ωi Ω, the square bracket in Eq. (7.14) is nearly unity, and t becomes the characteristic age or spin-down age, τ =−
1 P 1 ν 1 Ω = =− . n − 1 Ω˙ n − 1 ν˙ n − 1 P˙
(7.15)
This is a very useful measure of the time-scale on which the pulsar is spinning down. Clearly, τ is an upper limit of the actual age of the pulsar, since an initial angular velocity which is not much greater than the current one will lead to an age less than τ , as Eq. (7.14) readily shows. Thus, for the famous Crab pulsar, PSR B0531+21, τ ≈ 1250 y from spin-down measurements, while its actual age (obtained from the known time of birth of the neutron star in the well-recorded supernova event of 1054 AD) is ≈ 950 y. Any decrease of the pulsar’s magnetic field with age also produces the same effect, as the reader can show. Finally, note a curious, historical point. For magnetic dipole braking, n = 3 and τ = −ν/2ν. ˙ But this expression is traditionally taken as the definition of characteristic age, despite the fact
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that n can be different from 3, and is, in fact, so for all cases in which a stable, accurate value has been determined, as we shall see below.
Fig. 7.8 P − P˙ diagram of rotation-powered pulsars. Shown are: single pulsars (dots), binary pulsars (encircled dots) and loci of constant magnetic field B (dashed lines) and constant characteristic age (dotted lines). Also shown as dash-dotted lines are the socalled “death line” (see text) and the “spin-up line” for recycled pulsars (see Chapters 6 and 12). A few pulsars are marked individually. For reasons explained in Chapter 9, AXPs (triangles) and SGRs (stars) are also shown. Reproduced with permission by Cambridge University Press from Tauris & van den Heuvel (2003, using data from the ATNF Pulsar Catalog): see Bibliography.
We show in Fig. 7.8 the widely-used P − P˙ diagram, a log-log plot of P˙ vs. P for rotation-powered pulsars. As is evident from this plot, there is a wide scatter of points, roughly divisible into two clusters joined
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BRAKING INDICES
PSR
P
n
B0531+21 B0833-45 B0540-69 B1509-58 J1119-6127
33 ms 89 ms 50 ms 0.151 s 0.408 s
2.52 1.4 1.81 2.83 2.91
by a “bridge”. The first cluster, at relative slow rotation rates, i.e., long periods (P ∼ 0.1-1 s) and high period derivatives (P˙ ∼ 10−13 − 10−16 s/s), corresponds to neutron stars in early stages of their lives (see Chapter 6), functioning as rotation powered pulsars. The second cluster, at fast rotation rates, i.e., short “millisecond” periods, (P ∼ 1-10 ms) and low period derivatives (P˙ ∼ 10−19 − 10−20 s/s), correspond to old neutron stars which have been “recycled” by phases of mass accretion in binary systems, to function once again as rotation powered pulsars after accretion has stopped (see Chapter 6). For the entire sample of pulsars, the logarithmic range of P˙ is therefore about twice as large as that of P . We shall come back to this and related topics in Chapter 12, while discussing the evolution of neutron star spins and magnetic fields.
7.2.2.2
Braking index
The second derivative of the pulsar period or frequency makes a direct measurement of the braking index n possible. Consider the particular combination of first and second derivatives (and, of course, the frequency or the period) given by Eq. (7.11). Use of the spin-down equation (7.13) will immediately show that this combination is, in fact, identical to n. What are the measured values of n? Stable, accurate values of braking indices have been obtained for five pulsars so far, and we show these in Table. 7.1, which shows all the measured values of n to be less than 3, the canonical value for magnetic dipole braking (see above), which is also valid for particle loss from an aligned rotator by the Goldreich-Julian process (see Chapter 10). This can be attributed to non-dipolar components in pulsar magnetic fields, the presence of particles in the magnetosphere, change with time of the inclination angle α between rotation and magnetic axes, and decrease of magnetic field with time.
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We shall discuss some of these effects now, but let us first ask a very interesting question. What happens for the other pulsars, which form the vast majority? In general, they all have “measured” values of ν¨, of course, and formal braking indices calculated from the above expression can have either positive and negative sign, and magnitudes in the range 1 – 103 . But these are believed to be spurious values, caused by timing noise. This is so because these values are generally not stable, but vary widely between different observations of the same pulsar. This is a particular characteristic of the older pulsars, so that a meaningful measurement of n is thought to be possible only for young pulsars. Theoretically, the effects of alignment (dα/dt > 0) or misalignment (dα/dt < 0) between the rotation and magnetic axes, magnetic field decay, and so on, on n are easy to calculate (see Ghosh 1984, and references therein). For a dipolar field, n changes from its above canonical value of 3 by an amount 2τ (d ln sin2 α/dt) due to alignment or misalignment between the rotational and magnetic axes, where τ is the chracteristic age introduced above. Magnetic field decay on a time-scale τB adds a term 4τ /τB , and so on. An alternative approach, due to Blandford and Romani (1988), is to replace the constant K in Eq. (7.13) by a function of time, f (t), which contains these essential effects, and then to attempt to constrain f in the following way. A straightforward differentiation of the spin-down equation (7.13) with K replaced by f (t) shows that the braking index for a dipolar field is 3 − 2τ d ln f /dt. Thus, the observed Crab pulsar results (see above) give 2τ d ln f /dt ≈ 0.49 for the dimensionless first derivative of f . 7.2.2.3
Higher derivatives
Is it possible to measure the third and higher derivatives of the pulse fre... quency? Note first that the third derivative ν can be used to check the braking law, at least in principle, since the following combination of it with the first derivative and the frequency is readily seen from the spin-down equation (7.13) to be ... 2 ν
ν = n(2n − 1), ν˙ 3
(7.16)
which can be compared with the braking index obtained from ν¨ for consistency. In practice, this requires long years of data on a given pulsar, and so has been feasible so far only for the Crab pulsar, with ∼ 18 y of data (Lyne, Pritchard & Smith 1988). Even so, the measured value of this quantity, which we shall call the braking-variability index or superbraking
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index s, is known to within only about 10% accuracy. However, its value, s = 10 ± 1, is consistent with the observed braking index n ˜ given earlier, i.e., that which we would get by substituting n = n ˜ in the right-hand side of Eq. (7.16). In the Blandford-Romani approach, we can calculate superbraking index by a generalization of the method described above to the next order, n − 4) for a dipolar field. Thus, the diwhich yields s = 4τ 2 (f¨/f ) + 3(3˜ 2 ¨ mensionless second derivative 4τ (f /f ) can be obtained from the observed value of the superbraking index given above as s − 3(3˜ n − 4) ≈ −0.6 ± 1. (Notice the large error bar, which makes the result rather tentative, and shows the inadequacy of even a 10%-accuracy determination of the superbraking index for this purpose.) Clearly, this method can be thought of in principle as building a Taylor expansion of f term by term, but derivatives higher than the third would require prohibitively long times of observation. Alternatively, one can always check to see if canonical forms of f (t), e.g., exponentials or power laws (which naturally occur in descriptions of magnetic field decay; see Chapter 13), are consistent with these constraints.
7.2.3
Irregular Period Changes
After the secular period changes have been accounted for, which for single pulsars means the spin-down rate ν, ˙ and the second derivative ν¨ for a few young pulsars, we are left with the “residuals”, i.e., the irregular period changes contained in the term (t) introduced above. We now discuss the nature of these changes. These come in two categories: (i) random, “noisy” behavior, which is called timing noise, and, (ii) sudden, discontinuous increases in the frequency ν, which are called glitches. We consider these in turn. 7.2.3.1
Timing noise
Figure 7.9 illustrates the typical timing noise behavior: shown are the timing residuals for a collection of pulsars. Generally, younger pulsars show more timing noise than older ones, although it is a rather widespread phenomenon. How do we quantify this noise? As is customary in the study of noise processes, the rms value of the noise is used for this purpose, except that this has been traditionally (Cordes & Helfand 1980) normalized by the value obtained for the Crab pulsar (i.e., the rms value for the pulsar under study is divided by that of the Crab pulsar). The logarithm of this
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ratio was defined as the activity parameter , A, of the pulsar, and used for this study. It became clear that there was a good correlation between A and the period derivative P˙ of the pulsar. A related, alternative scheme for quantification of this was suggested by Arzoumanian et al. (1994), who defined a measure ∆ of the timing noise in the following way. These authors first found the default solution of Eq. (7.8) by keeping only the first derivative ν˙ on its right-hand side, and then found a phenomenological “second derivative” by adding only the next derivative ν¨ (and nothing else) to this equation, and solving it, in effect making the fitted value ν¨T 3 /6 a measure of the noise (T ). (This is meaningful if ν¨ is greater than the formal uncertainty found in it after fitting; otherwise the latter uncertainty is obviously a better measure of (T ).) The noise parameter was then defined as ∆(T ) ≡ log(¨ ν T 3 /6ν), and Arzoumanian et al. (1994) took T = 108 s, close to the length of their data span, naming the resultant quantity ∆8 . The method has the advantage of using absolute values for a given pulsar, instead of referring it to another stochastic quantity for the Crab pulsar, and these authors showed that the relation ∆8 = 6.6 + 0.6 log P˙
(7.17)
described pulsar data well, as shown in Fig. 7.9. A basic question is: random walk behavior in precisely which physical property of the pulsar is the origin of this noise? Some obvious candidates are the (a) the phase φ, (b) the frequency ν, and, (c) the frequency derivative or spindown rate ν. ˙ These are called phase, frequency, and derivative (or spindown) noise respectively. How can we identify them observationally? It is well-known from the theory of noise processes that the rms value of these noises scale differently with the time span T of the observations. The scaling is roughly ∼ T n , where n = 12 , 32 , 52 respectively for phase, frequency, and spindown noise. Equivalently, the power spectrum of the variations of the pulse phase φ has the characteristic power-law dependence f −2r , often called r-th order red noise, with r = n − 1/2, i.e., r = 0, 1, 2 respectively for phase, frequency, and spindown noise. The name “red” comes from the fact that, for r > 0, the power increases towards lower frequencies; noise with r = 0, i.e., zeroth order red noise, is called white noise, for obvious reasons. Note that a white noise in ν˙ translates into a red f −4 noise in φ, and so on. In 1970s, Groth pioneered the study of timing noise in rotation-powered pulsars, introducing the use of orthogonal polynomials for subtracting the systematic trends in the pulse phase φ prior to noise analysis, a well-known technique in signal-processing work
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Fig. 7.9 Noise parameter vs. period derivative for rotation-powered pulsars. The straight line is the relation proposed by Arzoumanian et al. (1994): see text. Reproduced with permission by AAS from Arzoumanian et al. (1994): see Bibliography.
[Groth 1975]. This is entirely natural for pulsar timing work, since Eq. (7.8) clearly shows that the linear and quadratic terms in T are, in fact, what we subtract in order to obtain the residual noise. The real question is, of course, what physical phenomena would lead to these three kinds of noise? Simple, intuitive pictures are not difficult to imagine, e.g., that, if the emission spot were to jitter about a mean position on the surface of the neutron star, phase noise would be a natural consequence. Similarly, any phenomenon that would cause the rotation frequency to fluctuate, e.g., a fluctuation in the moment of inertia of the neutron star, would lead to frequency noise. The latter could be due to, e.g., fluctuations
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in the coupling between the superfluid component (see Chapter 8) and the solid crust of the neutron star, since this coupling (which is thought to be through vortex pinning; see Chapter 8) determines the effective moment of inertia participating in the rotation. Again, any phenomenon that would cause fluctuations in the spindown torque on the star would lead to spindown, or derivative, noise. If this torque is electromagnetic, as it is thought to be, fluctuations in the magnetic field structure around the neutron star would be an obvious underlying cause. In general, there is no reason why two or more types of the above phenomena should not be simultaneously operational in the observed pulsars, and they probably are. Note also that, in a young, active (see above for the definition of the activity parameter) pulsar like the Crab, various other possibilities may need to be considered, e.g., (a) whether an unrecognized higher-order period derivative (associated with the spindown process) is still hiding in the timing noise, (b) whether a part (or even most) of the “noise” may actually be quasi-periodic phenomena, possibly related to lowfrequency oscillations in the neutron star, perhaps in its superfluid component – the Tkachenko oscillations. Note, finally, that the millisecond 9 pulsars, which are recycled (see Chapter 6) and very old (> ∼ 10 y), show little timing noise. This is what we would expect from the ∆ — P˙ correlation described above, since millisecond pulsars have small period derivatives (because of their low magnetic fields; see Chapters 12 and 13), and this is what puts them among nature’s most stable clocks, competitive with the best terrestrial atomic clocks. We take up this and related matters now. 7.2.3.2
Millisecond pulsars as stable clocks
That rotation powered pulsars are very good clocks has been known from early days of pulsar work. Their pulses come with great regularity, and when the long-term trends in their frequencies, as given by the secular derivatives on the right-hand side of Eq. (7.8) are measured and removed, the residuals are remarkably small (fractional stabilities ∼ 10−10 over many months were already implied by early pulsar measurements), indicating an underlying clock mechanism for the rotation which is, by far, the most stable one known for astrophysical objects. Following the discovery of the first rotation-powered millisecond pulsar PSR 1937+21 in 1982 (see Chapter 6), and subsequent timing studies of this and other such pulsars in the 1980s and ’90s, it became clear that in these recycled, millisecond pulsars, the clock stability attained an amazing excellence. After fitting to changes
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in the pulse frequency expected from known causes (see above), the post-fit residuals were so small — only a few µs over ∼ 5 − 10 years, as shown in Fig. 7.10 — as to imply that all major effects had actually been accounted for5 , and the inherent fractional stability of the millisecond pulsar clock was at least as good as a few parts in 1014 .
Fig. 7.10 Post-fit residuals for two millisecond pulsars. Reproduced with permission c 1991 IEEE from Taylor (1991), Proc. IEEE, 79, 1054.
This raised fascinating new questions about terrestrial time standards, which are our ultimate reference points for pulsar timing, as well as for all other timing work which needs to be very accurate. Atomic transitions had been the basis of these standards since 1955, and of the official definition of the second since 1967. Until the mid 1990s, the long-term stability of 5 It is possible for even such small residuals to show an apparent systematic trend for some pulsars, e.g., PSR 1937+21, as seen in Fig. 7.10. This suggests that some effects may have still not been modeled in such cases. Possibilities include (a) solar system ephemeris errors, (b) clock instabilities in the terrestrial time standards employed, (c) propagation effects, and (d) rotational irregularities of the pulsar itself. After a careful analysis of these for PSR 1913+16, Kaspi et al. (1994) came to the conclusion that the last possibility, i.e., intrinsic fluctuations in the pulsar’s own rotational frequency, was the most likely cause in this case.
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the best atomic clocks were ∼ 2 × 10−14 , i.e., about the same as that quoted above for millisecond pulsars, and so a natural issue arose as to where the ultimate accuracy would come from, and what role millisecond pulsars could play in this [Taylor 1991, Kaspi et al. 1994]. Interesting and valuable insights were gained in the process of tackling these issues, and we briefly summarize some of these below. The reader should be aware, however, that there has been much technical progress in realizations of terrestrial atomic times since about 1995, which has very likely improved their stability by about one order of magnitude (although a convincing demonstration of long-term stability takes ∼ 5 − 10 years), and stabilities ∼ 10−16 may not be impossible in future [Petit 1999]. At the same time, long-term monitoring of many millisecond pulsars discovered in the early 1990s has, by now, produced a valuable data bank, appropriate uses of which may improve the long-term stability figure for pulsars in the sense of an ensemble-average pulsar time (see below). Thus, the above issues remain interesting, but the situation is not quite what it was thought to be in the early 1990s. First, let us consider some major terrestrial atomic time standards. We have already mentioned one such standard, viz., that maintained by the U. S. National Institute of Standards and Technology (NIST). It is called AT1, and its reference atomic clock is an optically pumped cesium beam standard at NIST, the NIST-7 (which went into operation in 1994), or improved versions thereof. Consider now another international standard, which is of great formal and practical importance. In 1991, the International Astronomical Union defined the Terrestrial Time (TT) in terms of the Geocentric Co-ordinate Time. The International Atomic Time, or TAI, is a realization of TT, and the Bureau International des Poids et Mesures (BIPM) in S`evres, France is in charge of establishing and distributing TAI. It computes time retrospectively, using data from more than 200 atomic clocks spread world-wide, and makes it available with a delay of one month. Among these clocks, two major ones are (a) the cesium fountain at the LPTF in Paris, France (LPTF-FO1 has been operational since 1995), and, (b) the cesium beam standard at the Physikalisch-Technische Bundesanstalt (PTB) in Braunschweig, Germany, (PTB-CS3 has been operational since 1996). Every year, BIPM performs a new realization (i.e., a complete recalculation) of TT, using all available data from all primary frequency standards. The nomenclature is, for example, TT(BIPM99) for the realization of 1999, which covers the period from 1975.5 to 1999.0, and so on. Finally, the Universal Time Co-ordinated, or UTC, which is the basis for all legal time
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references, is so defined that it differs from TAI by an exact number of seconds. The question is one of comparing the long-term stability of atomic clocks with that of pulsar clocks, so we must next understand how stability is measured. In the study of accurate clocks, time-keeping, and frequency standards, which is a sub-field of metrology, the science of standardization of weights and measures, the stability of a clock is usually measured by a quantity called Allan variance, σy . This is a sort of standard deviation, obtained by subtracting a linear phase drift [i.e., keeping only the linear term in T in Eq. (7.8)] and doing a spectral analysis of the residual frequency differences. In the time domain, this amounts to obtaining a measure of the variance of the second differences of the timing residuals. This can be done in practice by fitting a set of orthonormal polynomials pj of order j = 0, 1, 2 to the residuals r(ti ) over the available times of observation ti , i = 1, . . . , N by the least-sqaures technique, i.e., minimizing the difference between r(ti ) 0,3 and a linear combination j Cj pj (ti ) of the polynomials with respect the values of the coefficients Cj . Of the coefficients so determined, C0 and C1 contain no stability information, as they are basically part of the linear trend we are subtracting6 , but C2 does contain stability information: in fact, C22 is an estimate of S1 , the residual power density — a measure of the variance we are seeking — at the lowest accessible frequencies, f ∼ 1/T , where T ≡ tN − t1 is our data span. For similar estimates Sm at higher frequencies mf , where m = 2, 4, 8, . . ., say, we need only divide the data span into two halves of roughly equal length, and repeat the above procedure, then subdivide each of these into halves again, and so on. The dimensionless variance of the Allan type is then basically σy2 (t) = (m/T )2 Sm , with t = T /m. In the 1990s, Taylor and co-authors [Stinebring et al. 1990, Taylor 1991, Kaspi et al. 1994] argued that a generalization of the above Allan variance was required for characterizing the stability of pulsars, since we subtract a second-order polynomial in time, i.e., keep linear and quadratic terms in T in Eq. (7.8), to obtain the residuals. In other words, we subtract a parabolic drift, instead of a linear one, for pulsar timing work. In the time domain, therefore, we should now obtain a measure of the variance of the third differences of the timing residuals, instead of the second. This is readily done by increasing the order of the above set of fitting polynomials by one, i.e., making j = 0, 1, 2, 3. Everything goes through as before: now C0 , C1 , 6 In
formal language, we say that they are covariant with the phase and the period.
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and C2 are basically part of the parabolic trend, and C3 contains stability information. We use C32 as an estimate of the spectral power density now, and write the dimensionless variance σz of this type, which we shall call the Taylor variance, as σz2 (t) = (m/T )3 Sm (note the characteristic increase in the power of m/T as we go to the higher order), with t = T /m again.
Fig. 7.11 Allan variance σy (τ ) vs. length of data span τ for various atomic clocks, and the millisecond pulsar PSR B1855+09. Curves marked as follows: 1: TAI (see text), 2: commercial cesium clock, 3: PTB-CS2 primary cesium clock, 4: LPTF-FO1 cesium fountain (see text), 5: passive hydrogen maser, 6: active hydrogen maser, 7: above millisecond pulsar. Reproduced with permission by Konin. Nederland. Akad. Wetensch. from Petit (1999): see Bibliography.
In practice, both Allan and Taylor variances have been used, as we show in Figs. 7.11 and 7.12. The former figure [Petit 1999] shows the Allan variances for several atomic clocks and time scales, and also for a millisecond pulsar (PSR B1855+09). The latter figure shows the Taylor variances for the two well-studied millisecond pulsars PSR 1937+21 and PSR 1855+09 [Kaspi et al. 1994]. These figures illustrate quantitatively the general points made earlier in the discussion. Note that, for small values of t (data span), the pulsar residuals are dominated by random measurement errors (which have a constant spectral density, i.e., a “white” noise), so that the Taylor variance scales as σz ∼ t−3/2 , and the Allan variance as σy ∼ t−1 , as the
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reader can show from their definitions given above, and as can be clearly seen in the figure. For PSR 1855+09, such errors seem to dominate over all available data spans, while for PSR 1937+21, an intrinsic “red” noise (i.e., noise whose spectral density increases with decreasing frequency) begins to dominate at low frequencies corresponding to t > ∼ 2 yr. As we indicated above, it is probable that this red noise is due to the pulsar’s own rotational instabilities.
Fig. 7.12 Taylor variance σz (t) vs. length of data span t for millisecond pulsars PSR B1855+09 and B1937+21. See text for detail. Reproduced with permission by AAS from Kaspi et al. (1994): see Bibliography.
The possibility of a pulsar-based timescale was suggested by Taylor and others in the early 1990s. We conclude our discussion by an indication of the scope and feasibility of this suggestion. First, it is not, and cannot be, a fundamental timescale, since, unlike atomic energy-level differences, the rotational characteristics of neutron stars are not fundamental constants of nature. We cannot hope to define the length of a second, for example, from these characteristics. Basically, a spinning pulsar is a heavy and enormously compact flywheel [Kaspi et al. 1994]. That this flywheel’s rotation should show such phenomenally good regulation is not to be attributed to any fun-
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damental natural constant, but to the extreme “cleanliness” (we return to this point later) of a rotation-powered pulsar, either when isolated or when in a binary with a compact companion. Except for the electromagnetic torques produced by its own magnetic field (see Chapters 1 and 12), which spin it down, and which we measure and account for with great accuracy, there is almost nothing to perturb its rotation either (a) externally, since it can have only gravitational interaction with a compact companion (which is like a point mass), which does not affect its rotation, or, (b) internally, since it is a superdense, degenerate, cold, “dead” star, where all internal motion due to the thermal energy generated by nuclear reaction has ceased long ago. Exceptions to the second category are small internal readjustments either in the structure or in the coupling between various stellar components, which lead to the “glitch” phenomenon we shall discuss below. But even these are absent in millisecond pulsars, which are old, recycled neutron stars, leading to such total lack of rotational disturbance, and therefore such rotational stability. What, then, is the motivation for a pulsar-based timescale? It is precisely the extreme stability we have been discussing. The idea is to monitor a number of millisecond pulsars over many years, and to use this data bank to define a joint timescale for the whole ensemble. The essential point is that since the noise processes in different pulsars are independent of one another (except for those associated with a common atomic timescale, say, used in all measurements), it may be possible to “average them out” in some sense, and so obtain a timescale with extremely high long-term stability. It is not clear if this stability can be made higher than that of the atomic time standards existing at the time, but, if so, this pulsar timescale can then be used to study relative merits of different realizations of terrestrial time. 7.2.3.3
Limits on cosmic gravitational wave background
Following a remarkable idea, originally suggested in the late 1970s, and explored in detail in the 1980s [see, e.g., Bertotti et al. 1983 and references therein], attempts can be made to convert the above timing residuals of millisecond pulsars into a powerful limit on the energy density of a possible cosmic background of low-frequency gravitational waves. Where would such a background come from? Cosmological models generally have some fraction of the energy going into this form during the radiation dominated era in the early universe, so that it is a relic background today. Its energy density ρ per unit logarithmic frequency interval is normally expressed as a
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fraction, Ωg ≡ ρ/ρc , of the closure density, ρc ≡ 3H02 /8πG, of the Universe. Here, H0 is the Hubble constant (see, e.g., Weinberg 1972). But how does such a background influence pulsar timing? The idea is that the Earth and the pulsar we are studying are two freely falling masses, whose positions respond to changes in the local spacetime metric. Passing gravitational waves perturb the metric, and so the positions of these two bodies, and therefore the travel time of radio signals between them, which shows up as a perturbation in the TOA of the pulses, i.e., a contribution to the timing residuals discussed above. (The principle is rather similar to that of laser interferometric gravitational wave detectors like LIGO, which depend on very accurate distance measurements.) The handling of the timing residuals is exactly as described above (indeed, the same sets of data on PSRs 1937+21 and 1855+09 were used), i.e., fitting polynomials, estimating noise spectral densities S(f ) at a frequency f ≡ 1/T , and at its higher octaves 2f, 4f, . . . by progressively subdividing the data span, and calculating the Taylor variances σz at these frequencies, or at the corresponding timescales T, T /2, T /4, . . . [Stinebring et al. 1990, Kaspi et al. 1994]. We need to remember only a few points about this specific application. First, this probe is most sensitive to gravitational waves of frequency f ≡ 1/T ∼ 0.1 − 0.2 yr−1 ∼ (3 − 6) × 10−9 Hz for a 5 – 10 year data span T , so we are thinking of very low-frequency gravitational waves here. Next, because of the high precision of TOA measurements with a fractional accuracy ∼ 10−14 (see above), this probe is potentially sensitive to gravitational-wave amplitudes of the same order. Finally, we need an explicit form for the expected power spectrum Pg (f ) of this gravitationalwave background. If the above energy density Ωg is roughly frequencyindependent, as suggested by cosmological theories, this power spectrum is [Bertotti et al. 1983]: Pg (f ) =
H02 Ωg f −5 ≈ 1.3 × 104 Ωg h2 f −5 µs2 yr. 8π 4
(7.18)
In the second form of the above equation, f is in units of yr−1 , and h is H0 in units of 100 km s−1 Mpc−1 — the customary way of expressing it. Note also the curious, but standard, units used for Pg (f ) in the second form. The steeply “red” f −5 spectrum above is a well-known signature of gravitational waves. We can now summarize the results obtained from timing residuals of PSRs 1937+21 and 1855+09 [Stinebring et al. 1990, Kaspi et al. 1994] in a straightforward manner. For roughly uniform sampling, the expectation
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values of the power-density estimators Sm introduced earlier, as obtained by Monte Carlo simulations (for details, see Kaspi et al. 1994), are approximately given by Sm g ∼ 5Pg (f ) in terms of the gravitational wave power-density given by Eq. (7.18). Use of this Sm leads to Taylor variances σz which are shown in Fig. 7.12 for two values of Ωg h2 , the quantity which measures the strength of the process which we are discussing here, and which is, therefore, bounded by exercises like this. Using the definition of σz given above, and Eq. (7.18), the reader can easily show that σz ∼ f −1 ∼ t for gravitational waves, as clearly seen in Fig. 7.12. To obtain observational bounds on Ωg h2 , we then attribute the entire observed value of Sm to this process, i.e., simply equate it to Sm g . The bound obtained −7 by Stinebring et al. in 1990 was Ωg h2 < ∼ 4 × 10 with 95% confidence, which was improved, with a longer data set, by Kaspi et al. in 1994 to −8 Ωg h 2 < ∼ 6 × 10 ,
(7.19)
with the same confidence. This is a stringent limit, in the sense that cosmological models which produce higher gravitational-wave backgrounds can be ruled out safely. However, the actual strength of any such background being sampled by observed millisecond pulsars is uncertain, and likely to be far below this. The Stinebring et al. (1990) estimate of the observed power spectrum of residuals in PSR 1937+21 suggested an average spectrum which was much flatter than the f −5 gravitational wave one, i.e., ∼ f −α , with α ∼ 1 − 3, which can come from clock instabilities (as the reader can readily verify from Fig. 7.11), or other causes. On the other hand, Fig. 7.12 suggests that the residuals of this pulsar are consistent at the lowest frequencies (i.e., largest values of t) with the f −5 signature of gravitational waves. However, as we mentioned earlier, Kaspi et al. (1994) have suggested that the dominant contributions to the timing residuals of this particular pulsar are, actually, from the pulsar’s own rotational instabilities. 7.2.3.4
Glitches
Glitches appear as small, sudden, discontinuous increases in the frequency ν of a pulsar, usually followed by a recovery to the pre-glitch frequency. A schematic representation of a glitch is given in Fig. 7.13. Well-known glitches in the Vela pulsar are shown in Fig. 7.14. Like timing noise, glitches also occur more frequently in younger pulsars. But the whole glitch phenomenon is rather rare. As Lyne (1992) pointed out,
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Fig. 7.13 The essential parameters of a pulsar glitch. The pulse frequency suddenly increases by an amount ∆ν0 , a fraction Q of which recovers exponentially on a timescale τ . There is often an accompanying permanent increase ∆ν˙ in the spindown rate. Reproduced with permission by Blackwell Publishing from Lyne et al. (2000): see Bibliography.
only 8 pulsars had exhibited glitches during the first 20 years of pulsar astronomy. Even though the subject is almost 35 years old today, only about 18 pulsars are known to have shown glitches, despite the fact that the situation improved greatly in the 1990s, due to the discovery of many young pulsars in modern surveys, and due to more extensive monitoring. Nevertheless, it is important to study this phenomenon, since it provided one of the first dynamical probes into the internal structure of a neutron star, and continues to hold our interest as a valuable probe of the physics of neutron star interiors, through a sort of “rotational seismology” [Lyne, Shemar & Smith 2000], as we shall see below. The first glitch was discovered in the Vela pulsar in March, 1969 [Radhakrishnan & Manchester 1969; Reichley& Downs 1969]: its frequency increased by ∆ν/ν ∼ 2 × 10−6 suddenly, on a timescale which was very likely < ∼ 1 day (this inference is drawn from the behavior of later glitches, which were monitored more closely). It has shown 10 more glitches upto now, and the order of magnitude of the first one remains typical of the largest glitches ever seen in this or any other pulsar. (The largest glitch seen so far has been in PSR B0335+54, with ∆ν/ν ∼ 4 × 10−6 , and the smallest glitches are of order ∆ν/ν ∼ 10−9 , below which it becomes rather difficult to distinguish them from timing noise.) By contrast, the young Crab pulsar
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Fig. 7.14 Pulse period history of the Vela pulsar over eight years, showing three glitches superposed on the regular spindown. Reproduced with permission by the Royal Society, London from Lyne (1992): see Bibliography. −8 shows much smaller glitches, ∆ν/ν < ∼ 10 . Generic features of a glitch are given in Fig. 7.13. The step-like change ∆ν is followed by the relaxation of a part Q∆ν of it in a roughly exponential manner, on a time-scale τ , with τ ranging between a few days and hundreds of days. Observed values of the fraction Q range from ∼ 10−3 to ≈ 0.9. In general, this relaxation process may have n exponential components, the ith one having an amplitude ∆νi and a time-scale τi . Quite often, there is also an accompanying step-like change ∆ν˙ in the first derivative, i.e., the magnitude of the spindown rate −ν˙ changes by an amount −∆ν: ˙ this change is almost always an increase. Observed magnitudes of ∆ν/ ˙ ν˙ range from 10−3 to 10−2 for “typical” glitches, then all the way up to ≈ 0.6 for some “giant” glitches, and values as small as 10−5 have also been (rarely) seen. Extensive listings
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Fig. 7.15 Histogram of glitch size ∆ν/ν: data from 48 glitches in 18 pulsars. Reproduced with permission by Blackwell Publishing from Lyne et al. (2000): see Bibliography.
of the above parameters are given in the review by Lyne, Shemar and Smith (2000). Let us discuss a few of the essential collective properties of glitches. Consider their sizes first. Fig. 7.15 shows a histogram of glitch sizes [Lyne, Shemar & Smith 2000]. There is a significant peak at ∆ν/ν ∼ 2 × 10−6 , just below the maximum observed value. Consider next the position of the glitching pulsars on the P − P˙ diagram, or, equivalently, the ν − ν˙ diagram. Clearly, glitches occur most frequently in young pulsars with large spindown rates. It is also clear that glitch sizes are generally larger for higher frequencies of rotation of the neutron star, since the glitching pulsars on the right-hand side of the cluster of points on the plot representing the positions of the pulsars, often called the “pulsar island”, generally display larger glitch sizes than those on the left-hand side of the island. There may be additional dependence of the glitch size on other stellar properties, e.g., magnetic fields. Now consider the cumulative effect of glitches over a long period of time, which we can call integrated glitch activity [Lyne, Shemar & Smith 2000], vis-` a-vis the integrated spindown over the same period of time, or, equivalently, the average spinup rate due to glitches over this period versus the average spindown rate. How do we calculate the former? As Lyne, Shemar and Smith (2000) explain, we can take all available data on glitches in all pulsars that have been monitored for this purpose over the years.
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(Their total sample contained 292 pulsars, of which 18 had shown glitches during the period of observation.) We first divide the data into a convenient number of bins covering the whole range of observed spindown rates −ν. ˙ In a given bin, suppose there are Np pulsars, of which Ng have glitched, and suppose that the ith glitching pulsar has had ni glitches, and that the jth glitch in the ith pulsar had a size ∆νij . Then the integrated glitch activity 1,N i in this bin is just i g 1,n ∆νij . Now suppose that the kth pulsar of the j whole collection of Np has been observed for a period Tk , so that the total 1,N cumulative time of observation is k p Tk . Then the average spinup rate 1,Ng 1,ni 1,N due to glitching is given by ν˙ glitch = ∆νij / k p Tk . Note i j that the analysis extends over all monitored pulsars, including those which have not glitched, but only the glitching ones contribute to the numerator of the right-hand side of the last equation, of course.
Fig. 7.16 Glitch spinup rate vs. average spindown rate: see text. Reproduced with permission by Blackwell Publishing from Lyne et al. (2000): see Bibliography.
This ν˙ glitch is then plotted against the average spindown rate −ν˙ of the bin: we show this in Fig. 7.16. There is, clearly, a strong positive correlation between the two, indicating a possible functional relation. In attempting to find this relation, we exclude the youngest and the oldest pulsars [Lyne,
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Shemar & Smith 2000] for the following reasons. The youngest pulsars, like the Crab, appear to be in a class by themselves, with distinct properties. The oldest, i.e., the millisecond pulsars with small spindown rates show very little glitching, so that, basically, upper limits are obtained in these bins with small values of −ν. ˙ Excluding these, rotation powered pulsars ˙ quite well. Thus, appear to follow a linear relation, ν˙ glitch ≈ −1.7 × 10−2 ν, −2 on the average, a fixed fraction ≈ 1.7 × 10 of the spindown amount of a pulsar is reversed by glitch activity, except in very young pulsars and in millisecond pulsars. We shall see later that this has crucial significance for neutron-star structure. Finally, consider the recovery fraction Q introduced above. There seems to be a strong correlation between Q and and the spindown rate −ν, ˙ again indicating a possible functional relationship [Lyne, Shemar & Smith 2000], which could put strong constraints on the theory of glitches. 7.2.3.5
Glitches: starquakes
About two months after the first glitch was discovered in the Vela pulsar, Ruderman (1969) proposed the first theory of the phenomenon, namely, starquakes. The name was coined in analogy with what happens to the Earth’s crust during an earthquake. Ruderman argued that a just-born, fast-rotating pulsar will have a very oblate shape because of its rotation, and that the solid crust will form almost immediately after birth, “frozen” into this shape. This is so because the melting temperature of the crustal Coulomb lattice of nuclei with charge Ze and number density nN 1/3 is Tm ∼ (Z 2 e2 nN /180kB ) ∼ 109 K for the range of crustal densities and equilibrium nuclides discussed in Chapter 3, and the neutron star will cool below such Tm in a matter of minutes after its birth, due to fast cooling by neutrino emission (see Chapter 6). Subsequently, as the pulsar spins down, i.e., its rotation becomes slower, its equilibrium shape (as determined by the balance between centrifugal and gravitational forces) would become less oblate and more spherical, but the solid crust will be held in its original shape by the Coulomb forces which give the lattice its rigidity. This imbalance will build up stresses in the solid crust as the pulsar slows down, until the stress σ exceeds the critical value σc which the crustal solid can withstand. At this point, the crust will suddenly crack, reducing its oblateness, and so relieving some of the stress. This is the starquake, as a result of which the crustal moment of inertia Ic (which is, of course, larger for more oblate shapes: see below) will decrease suddenly, and so, therefore,
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will the moment of inertia I of the whole star. In order to conserve angular momentum, the pulsar will consequently have to speed up suddenly (we emphasize that crust-breaking is instantaneous compared to the timescales, ts ∼ 103 − 106 years, on which the external spin-down torque operates on the pulsar). This is the glitch. The energy released in a large, Vela-type glitch is Eglitch = IΩ∆Ω ∼ 1043 ergs. This is only ∼ ∆Ω/Ω ∼ 10−6 of the total rotational energy Erot = (1/2)IΩ2 ∼ 1049 I45 Ω22 ergs of a young, rotation-powered pulsar like Vela or Crab, where I45 is the stellar moment of inertia I in units of 1045 g cm2 , and Ω2 is the stellar angular velocity Ω in units of 102 rad s−1 typical of these pulsars. But it is enormous by terrestrial standards: an earthquake of this strength would be of magnitude 15 – 16 on the (logarithmic) Richter scale! Of course, all this energy ultimately derives from the rotational energy of neutron star (in this as well as other models for glitches, as we shall see later), since other sources of energy have long since exhausted themselves [Anderson et al. 1982] in these cold, dead stars: nuclear reactions have concluded, gravitational collapse has reached its endpoint, and the total content of thermal and electromagnetic (associated with the stellar magnetic field, say) energies is far too small. The way this rotational energy is stored and released in the starquake scenario is also clear from the previous paragraph. As the pulsar spins down, most of its reduction in rotational energy is radiated away to the outside world through electromagnetic torques, but a tiny fraction of it is stored as elastic or strain energy in the crust, and subsequently released in the glitches. As we shall describe at this end of this section, this fraction proved, eventually, to be too tiny to explain the repetition rates of the large, Vela-type glitches, and so to be the undoing of the starquake model. However, the role of the rotational energy as being the ultimate energy source survives in subsequent glitch models, as it must. The reader can see now why the name “rotational seismology” [Lyne, Shemar & Smith 2000], quoted above, is rather apt for the study of pulsar glitches, although the word “seismology” should not, perhaps, be taken literally. A simple, elegant mathematical formulation of the starquake ideas was given by Baym and Pines (1971), which we now recount briefly, in order to appreciate the quantitative essentials of starquake. Consider an oblate, rotating neutron star, and describe its (quadrupolar) deformation in terms of a single parameter , defined as
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(0)
≡
Ic − Ic (0)
,
(7.20)
Ic
(0)
where Ic is the crustal moment of inertia, and Ic is the value that Ic would have if the star were not rotating. We can call the oblateness of the star. The first point to note is that, when we obtain the equilibrium value of by minimizing the total energy, we just balance the competition between (a) the rotational energy J 2 /2I plus the strain energy B(0 − )2 on the one hand, both of which decrease7 with increasing and so tend to make the star more oblate in a minimization procedure, and, (b) the gravitational deformation energy A2 on the other, which decreases with decreasing , and so tends to make the star less oblate in a minimization procedure. Here, A and B are constants whose values depend on the properties of the non-rotating star (see below), J and I are respectively the total angular momentum and total moment of inertia of the star, and 0 is the oblateness of the crust at the moment of its formation. The total energy can then be written as E = E0 +
J2 + A2 + B(0 − )2 , 2I
(7.21)
and a straightforward differentiation, holding the total stellar angular momentum constant, yields the equilibrium oblateness as =
∂I B Ω2 ∂I Ω2 + 0 ≈ , 4(A + B) ∂ A+B 4A ∂
(7.22)
where we have used the relation J = IΩ. The second, approximate form of Eq. (7.22) is obtained from the fact that B A. To appreciate this, note that A = αGM 2 /R is, dimensionally speaking, the self-gravitational energy of the non-rotating star (though α is smaller than the coefficient that occurs in the expression for this energy; see below), while B = µVc /2 is basically the strain energy of the crust, µ being the mean shear modulus of the crustal material, and Vc the crustal volume. For an incompressible sphere, for example, α = 3/25, so that A ≈ 4×1052 (M/M )2 R6−1 ergs, R6 being the stellar radius in units of 106 cm, as usual. What about B? The shear modulus of a crustal Coulomb lattice of 7 This
is obvious for the strain energy, since 0 is a constant. For the rotational energy, this becomes clear when we note that the minimization is done at constant J, and I increases with increasing oblateness by definition.
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4/3
nuclei with charge Ze and number density nN is µ ∼ Z 2 e2 nN ∼ 1030 dyne cm−2 for the relevant range of crustal densities and equilibrium nuclides, so that B ≈ 6 × 1047 µ30 R63 (∆R/R)−1 ergs, using Vc = 4πR2 ∆R, ∆R being the thickness of the crust. Here, (∆R/R)−1 is the relative thickness of the crust in units of its typical value 10−1 (see Chapter 5). Thus, B/A ∼ 10−5 for typical neutron-star parameters, i.e., quite negligible. Consider now such a rotating pulsar before its first starquake, and define the mean stress in the crust as σ ≡ µ(0 − ), since the mean strain is just (0 − ), the difference between the actual oblateness 0 , and the value it should have in order to have an exact equilibrium between gravitational and centrifugal forces, and therefore no strain. Call the latter quantity the equlibrium oblateness. With the aid of Eq. (7.22), we can write this stress as σ≈
µ ∂I 2 Ω0 − Ω2 , 4A ∂
(7.23)
where Ω0 is the angular velocity of the pulsar at the moment of the formation of its solid crust. Equation (7.23) shows that the stress builds up as the decrease in the square of the frequency as the star spins down. When it exceeds the critical value σc , the crust cracks for the first time, i.e., we have the first starquake, in which the actual oblateness decreases from 0 to a smaller value 1 , relieving some of the stress. The Baym-Pines (1971) idea was that the stress is only partly relieved in a starquake, i.e., in the above example, the oblateness does not reduce all the way to the equilibrium value , which would relieve all the stress, but only to some larger value 1 , which retains some of the stress. As the star spins down further, the stress builds up again, until it exceeds σc again, the crust cracks a second time, the actual oblateness decreases to 2 , and so on. It is easy to formulate these successive glitches mathematically. Call the decrease in the actual oblateness in a quake ∆0 ≡ 0 − 1 , and note first that the resultant decrease ∆Ic in the crustal moment of inertia and the increase in the angular velocity of the crust are given by ∆0 =
∆Ic (0) Ic
=−
∆Ω . Ω
(7.24)
Note next that the equilibrium oblateness also decreases by an amount ∆ due to the quake, but ∆ is negligible compared to ∆0 . This is readily seen from the exact version of Eq. (7.22): the change in due to a change in 0 , with other quantities remaining the same, is ∆ = [B/(A + B)]∆0 ∼
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10−5 ∆0 , i.e., entirely negligible. Hence the stress relieved in the quake is ∆σ ≈ µ∆0 . Following the quake, the star continues to spin down, i.e., Ω decreases at a rate Ω˙ = −Ω/ts , ts being the spindown timescale. This builds up the stress again, at a rate obtained by taking the time derivative of Eq. (7.23) and using the above expression for the spindown rate, which yields σ˙ =
µ ∂I Ω2 . 2A ∂ ts
(7.25)
When this build-up replenishes the above stress-release ∆σ in the quake, σ reaches σc again, and the crust cracks again. Thus, the time to the next ˙ which, with the aid of Eqs. (7.24) and (7.25), starquake is tq ≡ ∆σ/σ, becomes: 2A ∆Ω M 3 −2 ∆Ω ≈ 100 t4 R Ω yr. (7.26) tq = t s (∂I/∂)Ω2 Ω M 6 2 Ω −6 Here, t4 is the spindown time ts in units of 104 years, and (∆Ω/Ω)−6 is the magnitude of the glitch (∆Ω/Ω) in units of 10−6 . In obtaining the second form of Eq. (7.26), we have used ∂I/∂ ≈ I0 , I0 being the moment of inertia of the whole star when not rotating (this amounts to assuming that I of the whole star scales with Ic under shape change: see Pines 1971), approximated I0 as I0 ≈ (2/5)M R2, and used the value of A given above. The first point to notice in Eq. (7.26) is that the time to the next glitch is proportional to the size of the previous glitch, which is a well-known characteristic of relaxation oscillations. Next, we can now readily predict the glitch repetition timescales tq of the well-studied Vela and Crab pulsars. For Vela, using the recent value ts ≈ 2.2 × 104 yr, Ω ≈ 69 rad s−1 , and the typical size (∆Ω/Ω)−6 ≈ 2 of glitches in this pulsar [Lyne, Shemar & Smith 2000], we get tq ≈ 920 years. Thus, the large Vela glitches should be very rare phenomena. For Crab, using the recent value ts ≈ 2.4 × 103 yr, Ω ≈ 188 rad s−1 , and the much smaller glitches of sizes in the range (∆Ω/Ω)−6 ≈ 4 × 10−3 to 8 × 10−2 found in this pulsar [Lyne, Shemar & Smith 2000], we get tq ≈ 10 to 200 days. Thus, the small Crab glitches are expected to be quite frequent phenomena. Ironically, the same Vela pulsar which, through the discovery of the first glitch in 1969, had originally inspired the starquake idea also proved to be the undoing of the starquake model in the end. After the original 1969 glitch, Vela displayed 4 more glitches of comparable size in the next ∼ 12 years, indicating that the observed glitch repetition time of this pulsar was
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tq ∼ 3 years, in sharp contrast to the above predicted value. (This trend has also continued consistently after that, with 10 more glitches occurring in the first ∼ 25 years of Vela glitch observations, 9 of these glitches being of the same large magnitude as the original one, and one smaller [Lyne, Shemar & Smith 2000].) Thus, it became clear by the early 1980s that the starquake scenario could not account for the large Vela glitches. The elastic energy that could be stored in the strained crust in the ∼ 3 years between glitches was far too small (by a factor ∼ 1/300) to produce the observed glitches. The scenario could still account for the much smaller Crab glitches, of course, but then we would have to look for a different mechanism for the large, Vela-like glitches, which were later also found in many other pulsars, and which today constitute the majority of the observed glitches [Lyne, Shemar & Smith 2000]. Quite generally, a unified, single theory covering all essential aspects of a phenomenon is considered preferable to a multi-component, hybrid theory — an example of the principle often called Occam’s razor . Thus, the emphasis shifted in the 1980s to the exploration of other possible glitch mechanisms. Since these involve the superfluidity of the neutrons in neutron stars in a crucial way, we must introduce this superfluidity phenomenon first, as we do in the next chapter, before discussing such mechanisms. Before leaving the subject of starquakes, however, we emphasize that the idea is beautiful and interesting, and such quakes do undoubtedly occur in neutron-star crusts, along the lines summarized above. These quakes, and variations thereof, have been re-invoked in the context of various pulsar phenomena (including recent theories of glitches), as we shall see in the next chapter. Nevertheless, they appear untenable as a unified glitch scenario, because of their insufficient energy budget. 7.2.4
Timing Rotation Powered Pulsars in Binaries
Of the 1720 rotation powered pulsars known today, 126 are in binary systems. For such pulsars, we need to further transform tb , the solar-system barycentric TOA (see above), to the proper time T in the rest-frame of the pulsar. The transformation compensates for the binary orbital motion of the pulsar, and so contains Rømer, Einstein, and Shapiro delays, exactly as Eq. (7.4) did, but now for the pulsar’s binary orbit rather than that of the Earth, so that the binary period Pb , eccentricity e, velocity, and phase angle are those of the pulsar orbit, as are the masses m1 and m2 of the pulsar and its companion, respectively. These Keplerian orbit parameters
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occur in the above delays, as expected by analogy with Sec. 7.2.1. In fact, the five Keplerian parameters that specify the binary orbit (see Appendix B) are (1) the projected8 semi-major axis x ≡ a1 sin i, (2) the eccentricity e, (3) the binary period Pb , (4) the longitude of periastron ω, and, (5) the time of periastron passage T0 . We shall see below that several, critically important, post-Keplerian parameters also occur in these delays, describing the relativistic behavior of the pulsar’s binary orbit. It is these parameters that led to spectacular advances in finding testing grounds for relativistic gravity, and so to dramatic confirmations of Einstein’s theory in the strong-field r´egime. We write the above transformation as tb − t0 = T − ∆R − ∆E − ∆S − ∆A ,
(7.27)
where t0 is a reference epoch, from which all the pulses are numbered. Note the appearance of the extra term ∆A , which is the “aberration” delay, caused by the rotation (i.e., spin) of the pulsar while moving in its binary orbit. How do we relate these delays to the binary orbital parameters described above? The answer is: the relations are now much more complicated than those given in Sec. 7.2.1, where we could use the weakfield, slow-motion limit of the theory of general relativity, applicable to the solar system. This we cannot do for the dynamics of those close, relativistic binary pulsars which we discuss in this section. The post-Keplerian parameters mentioned above — about half-a-dozen of them — now play an essential role, as we shall see. As indicated in Chapter 1, the discovery of PSR 1913+16, the first rotation-powered pulsar in a binary, by Hulse and Taylor in 1975 (also see below) started the subject, and attempts to obtain useful formulations of the above relations began soon after that9 . In the mid-1980s, Damour and Deruelle evolved a general, elegant formulation which became the standard tool, particularly in the hands of Taylor and co-authors, and whose essentials we now summarize. Damour and Deruelle argued that, while Einstein’s general theory of relativity appeared viable and was generally believed, a more theoryindependent formulation of the above relativistic delays would be highly desirable for the analysis of extremely precise timing data, if we were to 8 That is, projected on the plane of the sky, i being the angle of inclination. See appendix B. 9 We do not describe the early formulations here. The interested reader can read the excellent account given in Taylor and Weisberg (1989), which has references to the original work.
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look in these data for any possible deviations from the general theory of relativity under strong-field conditions. To this end, they gave a formulation [Damour & Deruelle 1986, henceforth DD] valid under very general assumptions about the nature of gravity under strong-field conditions, based on their new method of solving the relativistic two-body problem to postNewtonian order. In the DD formulation, the delays in Eq. (7.27) depended on the above 5 Keplerian parameters, and, in addition, on 7 post-Keplerian (henceforth PK) parameters, the most significant of which are the following five: (1) orbit decay rate P˙b due to emission of gravitational radiation, (2) rate of relativistic advance of the longitude of periastron (i.e., relativistic apsidal motion) ω, ˙ (3) the Einstein parameter γ — a measure of the combined effects of gravitational redshift and time dilation (see Sec. 7.2.1) due to pulsar orbital motion — so defined that the amplitude of the Einstein delay ∆E is γ, and, finally, two parameters specifying (4) the amplitude or “range” r and (5) the “shape” s of the Shapiro delay ∆S . The roles of these 5 PK parameters can thus be conveniently grouped as follows. The Rømer delay depends on the first two PK parameters in addition to the Keplerian ˙ and its amplitude is ∝ x, as may parameters, i.e., ∆R (x, e, Pb , ω, t0 , P˙b , ω), have been expected from the size estimate given in Sec. 7.2.1. The Einstein delay has an amplitude γ, the third PK parameter, i.e., ∆E ∝ γ. Of course, both delays also depend on the orbital phase (see below). Finally, the Shapiro delay has an amplitude or “range” r, i.e., ∆S ∝ r, and the orbital-phase dependence of its logarithmic factor (see Sec. 7.2.1) is controlled by the shape parameter s = sin i, where i is the angle of inclination, as before. It will be useful to appreciate the orders of magnitude of the above relativistic effects at this point (also see Sec. 7.2.1). Consider the Einstein delay ∆E first. It consists of two parts: (a) the gravitational redshift due to the companion of mass m2 , which is ∼ Gm2 /r12 c2 , r12 being the distance between the pulsar and its companion, and (b) the time dilation, which is ∼ v 2 /c2 . Actually, both parts are ∼ v 2 /c2 , as a little reflection will show, and so is the total, and the whole effect is, of course, also ∝ e, as we have indicated earlier. Consider next the relativistic periastron advance ω, ˙ which 2 2 is ∼ v /c . Now consider the Shapiro delay ∆S , which is ∼ Gm2 /c3 (see Sec. 7.2.1). Finally, consider the orbit decay rate P˙ b due to emission of gravitational radiation, the order of magnitude of which has a remarkable behavior which has given it a high diagnostic value, as we shall see later. In Einstein’s theory of general relativity, it is ∼ v 5 /c5 , but, in alternative theories of relativistic gravity, it has a generic order of magnitude ∼ v 3 /c3 .
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The variation of the total orbital delay ∆R + ∆E + ∆S with orbital phase for the Hulse-Taylor pulsar PSR 1913+16 (see below), which has a very eccentric orbit, naturally has a very non-sinusoidal, asymmetric shape, as Taylor (1994) has shown. Although the Einstein and Shapiro delays are orders of magnitude smaller than the Rømer delay, they can still be measured separately because the precision of TOA measurements is enormously high — a tribute to the painstaking, careful work that led to advances in fundamental measurements in relativistic gravity, as we shall summarize presently. When the orbital plane of the binary pulsar is nearly parallel to the line of sight, i.e., i ≈ 90◦ , the shape parameter, s = sin i, of the Shapiro delay is about the largest it can be, so that the orbital modulation of ∆S is greatly magnified, and relatively easy to detect. This is the case for the binary millisecond pulsar PSR 1855+09, for which the measured orbital variation of the Shapiro delay is shown in Fig. 7.17. The sharp peak is, of course, at that orbital phase which corresponds to superior conjunction, where the radio waves coming from the pulsar to us pass closest to its companion, almost grazing it, and so maximizing the Shapiro delay. The general expressions for the above delays in the DD formulation are straightforward but lengthy. Instead of reproducing them here, we refer the reader to the original DD paper, or to Taylor and Weisberg (1989), and focus instead on a particular way of using the general DD framework which has been remarkably useful and popular since the late 1980s. Taylor and his co-authors [Taylor & Weisberg 1989; Ryba & Taylor 1991; Taylor 1992; Taylor 1994] pointed out that, within any specified relativistic theory of gravity (not necessarily Einstein’s), the measured value of each of the above PK parameters defines a curve in the (m1 , m2 ), or mass-mass, plane, m1 being the mass of the pulsar, and m2 that of its companion. This is most easily demonstrated for Einstein’s theory of general relativity, in which case the above 5 PK parameters are given in terms of m1 and m2 (and also, of course, the Keplerian parameters) by the 5 following equations: −5/3 5/3 m1 m2 T 192π Pb 37 73 , P˙b = − 1 + e2 + e4 5 2π 24 96 M 1/3 (1 − e2 )7/2 −5/3 2/3 2/3 T M Pb , ω˙ = 3 2π 1 − e2 1/3 2/3 T Pb m2 (M + m2 ), γ=e 2π M 4/3 r = T m2 ,
(7.28) (7.29) (7.30) (7.31)
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Fig. 7.17 Shapiro delay in the PSR 1855+09 system. The lower panel shows the residuals after accounting for this delay in the manner described in the text. Reproduced with permission by AAS from Ryba & Taylor (1991): see Bibliography.
Pb s=x 2π
−2/3
M 2/3 1/3
.
(7.32)
T m2
Here, M ≡ m1 + m2 is the total mass of the binary system, all masses are expressed in solar units, and T ≡ GM /c3 = 4.925490947 µs is the timescale introduced earlier. We see that the above system of equations is greatly over-determined, in the sense that we need to measure only 2 PK parameters to determine the masses m1 and m2 , which then leads to explicit predictions of the rest of the PK parameters. If independent measurements of any of the latter are also possible, this leads, therefore, to a check on the internal consistency of the theory, in this case Einstein’s, and also on the correctness of the theory as a description of relativistic gravity upto that point (e.g., fitting a third PK parameter would mean that the theory consistently describes those three relativistic effects, and so on). As Taylor (1994) put it in his Nobel
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lecture: If three or more post-Keplerian parameters can be measured for a particular pulsar, the system becomes over-determined, and the extra degrees of freedom transform it into a calibrated laboratory for testing relativistic gravity. This idea proved instrumental for the most accurate verification of Einstein’s theory of general relativity in the strong-field and radiative10 r´egimes that exists today. The binary pulsar with which all of this was done for the first time was none other than PSR 1913+16, the first binary pulsar ever discovered (Hulse & Taylor 1975), now known as the Hulse-Taylor pulsar. 7.2.4.1
PSR 1913+16: historical notes
In a research proposal entitled “A high-sensitivity survey to detect new pulsars”, submitted to the U. S. National Science Foundation in September, 1972, Taylor noted that it would be highly desirable “to find even one example of a pulsar in a binary system, for measurement of its parameters could yield the pulsar mass, an extremely important number”, although the main purpose of the survey would be to try to double or triple the number of known pulsars. “Little did I suspect that just such a discovery would be made”, said Taylor twenty-one years later in his 1993 Nobel prize lecture, “or that it would have much greater significance than anyone had foreseen!” [Taylor 1994]. As a matter of fact, the survey was carried out in 1973-74 by Hulse and Taylor, using the 305 m Arecibo radio telescope in Puerto Rico: about 40 new pulsars were discovered in the survey, including PSR 1913+16 in July, 1974 [Hulse & Taylor 1975]. Its position in galactic co-ordinates is as follows: close to the galactic plane, around a galactic longitude of 50◦ , and, as Taylor (1994) put it, “in a clump of objects” in that part of the sky which transits directly overhead at Arecibo’s latitude as the Earth rotates, it was nothing remarkable in itself. Similarly, its pulse period, P ≈ 59 ms, was short, but longer than that of the Crab pulsar, which had the shortest known period at the time, and therefore not really outstanding. What was most unusual about PSR 1913+16 was that it frustrated attempts to determine its period accurately by displaying apparent period changes that were quite enormous (∼ 80 µs from day to day, and sometimes ∼ 8 µs over 5 minutes) on the scale of secular period changes (∼ 10 µs per year ), or irregular period changes (orders of magnitude smaller than the secular changes), of all pulsars known at the time. It was quickly recognized that these changes were Doppler shifts due to binary orbital motion of the 10 That
is, where emission of gravitational radiation is important.
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Fig. 7.18 Velocity curve for the Hulse-Taylor pulsar, PSR 1913+16. Reproduced with permission by AAS from Hulse & Taylor (1975): see Bibliography.
pulsar, and the corresponding radial velocity curve was obtained within a couple of months of its discovery: this curve is shown in Fig. 7.18. The orbital period of the binary was short, Pb ≈ 7.75 hr, and the orbit was highly eccentric, e ≈ 0.62, as evident from the very non-sinusoidal shape of the radial velocity curve in Fig. 7.18, which also shows that the amplitude of radial velocity curve was ∼ 200 km s−1 ∼ 10−3 c, i.e., rather large. We give the essentials of Newtonian binary dynamics in Appendix B, where we introduce the binary mass function f (m1 , m2 , sin i), which can be determined from a knowledge of the Keplerian parameters Pb and x (see above). In the present context, we can express the mass function in units of M as: f (m1 , m2 , s) =
(m2 s)3 (x/c)3 = , 2 (m1 + m2 ) T (Pb /2π)2
(7.33)
where m1 and m2 are in units of M , as before, and we have used the shorthand notation s ≡ sin i introduced above. The mass function for PSR 1913+16 was found to be f ≈ 0.13 in the above units. Knowing the mass function alone does not give us m1 , m2 , and s separately, of course. But using plausible sets of values of s (guided by other observed properties, e.g., absence of eclipses, or the amplitude of the radial velocity curve), we can attempt to constrain the mass ratio m1 /m2 . This is what Hulse and Taylor did, coming to the conclusion that m1 ∼ m2 . From this, and the absence of eclipses, they concluded that the unseen compan-
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ion was a compact star, probably another neutron star. We do not recount these arguments here because they are only of passing interest today, since accurate determination of m1 , m2 , and s was possible later through observation of relativistic effects, as we shall see below. Indeed, we need only quote a prophetic line from the last section of the Hulse-Taylor paper (1975) to put PSR 1913+16 in its proper historical context: The binary configuration provides a nearly ideal relativity laboratory including an accurate clock in a high-speed eccentric orbit and a strong gravitational field. We shall return to this “nearly ideal relativity laboratory” below, after a brief, general summary of the observed binary rotation-powered pulsars. 7.2.4.2
Binary rotation-powered pulsars: general systematics
After the discovery of PSR 1913+16, other binary rotation-powered pulsars were found as the total number of rotation-powered pulsars increased steadily. By 1994, ∼ 600 pulsars were known, 40 of these were in binaries, and 35 had been studied well enough to obtain their basic parameters [Taylor 1994]. Today11 , 1720 rotation-powered pulsars are known, and 126 of these are in binaries: these are shown in the binary period vs. pulse period plot of Fig. 7.19, the anologous diagram for accretion-powered pulsars being given in Chapter 9. The double neutron-star binaries lie in the lower-middle of Fig. 7.19. They have eccentric orbits, the eccentricity coming from the supernova that produced the second neutron star, and their pulse periods are relatively short (∼ 20 − 100 ms) but not very much so, because their recycling has been only mild, i.e., of short duration, since they are descended from HMXBs with short evolution times, as explained in Chapter 6 [Manchester 2006]. When referreing to “basic parameters” of such binary rotation-powered pulsars, we normally mean Keplerian parameters, because post-Keplerian work has been possible on only a few pulsars, as we describe below. Indeed, only limits are available even for some Keplerian parameters of quite a number of pulsars. Nevertheless, some important trends have emerged, which we shall discuss in this chapter. Recall first from Chapter 6 the various basic types of binary systems (both recycled and unrecycled) in which rotation-powered pulsars occur, as summarized in Table 7.2. Now consider the mass function f which does not, by itself, give us the two masses m1 and m2 and the inclination sin i, as we mentioned earlier. 11 As
of late 2005. See Manchester (2006).
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Fig. 7.19 Binary period vs. pulse period for rotation-powered pulsars in binaries. Orbital eccentricity is indicated by the shape of symbol. Open symbols: binaries in Galactic disk. Filled symbols: binaries in globular clusters. An upward arrow marks those systems in which the companion mass exceeds 0.45M . Reproduced with permission by Elsevier B.V. from Manchester (2006): see Bibliography.
However, a study of f vs. e, the orbital eccentricity, is still very instructive. For statistical studies, we can convert f into an estimate of the companion mass m2 by using (a) an appropriate average for the inclination angle, the median value cos i = 0.5, say, and, (b) an estimate of the pulsar mass, m1 ≈ 1.4M , say, which is indicated by measured neutron-star masses. Then a classification of binary rotation-powered pulsars into three categories is possible [Taylor 1992, 1994], as follows. A plot of f vs. e, as suggested by Taylor, is useful for this, and an early version of it is shown in Fig. 7.20. 1. In the first category are binaries with relatively low eccentricities, e < 0.25, say, and low-mass (typically ∼ [0.1 − 0.3]M ) companions. This category contains most of the binary pulsars, and the companions are low-
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Table 7.2 TYPES OF BINARY ROTATION-POWERED PULSARS (After Tauris & van den Heuvel 2003) Category
Companion Type
Sub-type
Example
Binary Period
Unrecycled
Massive(B/Be)
PSR 1259-63
3.4 yr
Unrecycled Unrecycled
Main Sequence “Ante-deluvian” Main Sequence White Dwarf
Low-mass WD(CO) + NS
PSR 1820-11 PSR 2303+46
357 d 12.3 d
Recycled Recycled Recycled Recycled
“Massive” (0.5 − 1.4M ) “Low-Mass” (< ∼ 0.45M )
NS + NS NS + WD WD(He) + NS WD(He) + NS
PSR PSR PSR PSR
7.75 hr 1.03 d 1232 d 117 d
1913+16 0655+64 0820+02 1953+29
mass white dwarfs. Most of these have nearly circular orbits. (Indeed, the few with e more than a few percent are located in globular clusters, where their orbits have undoubtedly been perturbed by close interactions with other stars.) Evolutionary scenarios for such systems have been discussed in Chapter 6: this appears to be the most common channel for producing recycled pulsars in binaries. 2. In the second category are binaries with much larger eccentricities, and companions with masses ∼ M . These are the double neutron-star binaries like PSR 1913+16, i.e., the potentially “nearly ideal relativity laboratories” of Hulse and Taylor, as indicated above. Evolutionary scenarios for such systems, in particular the second-supernova origin of the large eccentricities, have been discussed in Chapter 6. This appears to be a rarer, but very important, channel for producing recycled pulsars in binaries. 3. In the third category is a small number of binaries in which the companion is a massive (∼ [10 − 30]M) main-sequence star. The eccentricities of these are generally quite high. These are the ante-deluvian systems discussed in Chapter 6, containing young neutron stars in binaries which have not been disrupted by the supernova that produced the neutron star. The first member of this category, PSR 1259-63, was discovered in 1992, and four such systems are known now [Manchester 2006]. Binaries in the second category prove particularly suitable for relativistic gravity work, not only because they have large mass functions and eccentricities, which make the relativistic effects we are looking for particularly large, but also because they are astrophysically “clean” (Taylor 1992), which means that these effects would not be masked by details of stellar
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Fig. 7.20 Mass function of binary pulsars vs. orbital eccentricity. Double neutron-star binaries cluster around the indicated position for PSR 1913+16. At the top right-hand corner are the “ante-deluvian” systems (see text) with massive companions. The rest are believed to have white-dwarf companions. Reproduced with permission by the Royal Society, London from Taylor (1992): see Bibliography.
or accretion phenomena. Clean means, in this context, that (a) there is negligible matter between the two stars (e.g., in the form of stellar winds, or of accreting matter, which will be present when the companion to the pulsar is not a neutron star, and mass transfer from it to the neutron star is under way), since these may cause non-relativistic changes in the orbital period due to mass loss and/or transfer, and, (b) the companion has no significant structure, since that would cause non-relativistic apsidal motion. A double neutron-star binary suits these requirements admirably, since it is, in effect, two point masses interacting only via gravitation, the above non-relativistic stellar effects being utterly negligible. However, the reader should not get the impression that binaries in the first category, i.e., those with low-mass white dwarf companions, are entirely hopeless in this regard. Under very favorable circumstances, they can yield useful PK parameters, as we shall mention at appropriate places.
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PSR 1913+16: nursery of relativistic gravity
The way PSR 1913+16 revolutionized our ideas about testing relativistic gravity was expressed by Damour in 1992 in the following way: Let us recall that, up to the discovery of binary pulsars, and apart from the qualitatively fascinating but quantitatively poor confirmations of general relativity coming from cosmological data, the only available testing ground for relativistic gravity was the Solar System. Damour went on to explain that, despite wonderful high-precision technologies brought to bear on the solar-system studies of relativistic gravity in the limit of weak fields and slow motions, and the consequently impressive measurements of minute deviations from Newtonian gravity, “Solar System tests have an important qualitative weakness: they say a priori nothing about how the correct theory of gravity might behave in presence of strong gravitational fields”. The question was not academic at all, since it was already known that some alternative theories of gravity coincided with Einstein’s general theory of relativity in the weak-field limit, but gave very different predictions in strong-field and/or rapidly-varying field r´egimes. We have indicated above that the discoverers of PSR 1913+16 already knew in 1975 that long-term studies of this pulsar were going to change all this. We have also sketched above the method of testing relativistic gravity using binary rotation-powered pulsars, with the aid of post-Keplerian (PK) parameters. We now summarize the results of Taylor and his co-authors on PSR 1913+16. By the time Taylor and Weisberg (1989) reported their work, 14 years of data on this pulsar were in hand, and the precision in the measurement of TOAs had become extremely high since about 1981, typically ∼ 15 µs in 5-minute observations. With such accuracy and data span, the Keplerian parameters of the orbit were determined to within a few parts per million or better [Taylor 1992]: these are shown in Table 7.2, together with some PK parameters. As before, the number in parentheses give the error in the last digit. The PK parameters ω, ˙ γ, and P˙b were determined to fractional accura−6 −4 cies of 4 × 10 , 7 × 10 , and 4 × 10−3 , respectively, and it is these that will dominate the whole story. The plan is as explained already: from ω˙ and γ, we can obtain the masses m1 and m2 (we actually obtain M and m2 first, as a look at Eqs. [7.29] and [7.30] will show the reader), and use these to calculate P˙ b with the aid of Eq. (7.28), which we then compare with the observed value of P˙ b . But each part of the story is remarkably instructive in itself, as we shall see now.
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Table 7.3 lor 1992)
PARAMETERS OF PSR 1913+16 (after Tay-
Orbital Period Pb Eccentricity e Periastron Advance Rate ω˙ Einstein Delay γ Predicted Period Decay P˙bGR Observed Period Decay P˙ bobs Kinematic Correction P˙ gal b
27906.9807807(9) sec 0.6171309(6) 4.22663(2)◦ yr−1 4294(3) µs −2.40258(4) × 10−12 −2.42(1) × 10−12 −0.017(5) × 10−12
Fig. 7.21 Advancement of the periastron longitude ω in PSR 1913+16 over the 14-year span 1975-89. See text for detail. Reproduced with permission by AAS from Taylor & Weisberg (1989): see Bibliography.
Consider relativistic apsidal motion, or the rate of relativistic advance of the periastron longitude, ω, ˙ first. This is a huge effect in PSR 1913+16, ◦ ω˙ ∼ 4 per year, as Hulse and Taylor had already estimated theoretically in 1975, and as the reader can readily compute from Eq. (7.29) by substituting the binary parameters and masses from the above table. Fig. 7.21 shows the advancement of the periastron longitude ω in this binary over a ∼ 14 year span, reported by Taylor and Weisberg in 1989: the line of apsides had rotated by nearly 60◦ during this time-span. This demonstration brings home the status of binary pulsars as proving ground for relativistic gravity
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in a dramatic fashion: the effect is so enormously strong and clear that it does not leave any room for doubt, and the proverbial “cleanliness” of double neutron-star systems, referred to above, is entirely obvious. Gravitational redshift and time dilation, measured by the PK parameter γ, was also a large and easily-detected effect (γ ∼ 4 ms), and the use of ω˙ and γ together determined the masses (in solar units) to be m1 = 1.442 ± 0.003 for the pulsar, and m2 = 1.386 ± 0.003 for its companion, i.e., the fractional accuracy was about 2 × 10−3 . With these in hand, the rate of orbital decay, P˙ b , could be calculated from Eq. (7.28) to an accuracy of a few parts in 103 , and then compared with the measured value of this PK parameter, which was also known with the same order of accuracy (see above). The observed value was P˙ bobs = (−2.425 ± 0.01) × 10−12 ,
(7.34)
which corresponded to a fractional rate of change P˙bobs /Pb ≈ −2.74 × 10−9 yr−1 . The result, reproduced from Taylor (1992) and shown in Fig. 7.22, is undoubtedly one of the landmarks of twentieth-century physics. In this figure, plotted against time (in years) and labeled “accumulated shift of orbital phase” is the difference between the value of the time of periastron passage T0 (one of the Keplerian parameters described earlier), and the value it would have if there were no orbital decay due to energy loss by the emission of gravitational radiation, i.e., if Pb were a constant. The data points correspond to the measured values of P˙b over ∼ 17 years, and the parabolic curve corresponds to the values of P˙b calculated from Eq. (7.28), i.e., Einstein’s theory, using the above values of the masses, which were obtained from independent measurements of the PK parameters ω˙ and γ over the same time-span, as explained above. The value thus calculated from the theory of general relativity was P˙bGR = (−2.40258 ± 0.00004) × 10−12 ,
(7.35)
the (much smaller) uncertainties in this value originating from those in the measured values of the masses, as above. Note that a kinematic effect, caused by the differential acceleration of the Solar system and the binary pulsar system in the gravitational field of our galaxy, increases the absolute value of the observed P˙ b (i.e., makes it appear to be more negative than it really is) by about 5 parts in 103 . Since this effect is significant at the high level of accuracy of PSR 1913+16 work, it was corrected for before making comparisons with the theoretical value. The effect depends on the size and
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Fig. 7.22 Orbital decay of PSR 1913+16 due to emission of gravitational radiation. See text for detail. Reproduced with permission by the Royal Society, London from Taylor (1992): see Bibliography.
the rotation rate of the galaxy and on the pulsar’s distance and proper motion (mathematical expressions are given in Damour and Taylor 1991): since all of these were known with adequate accuracy from independent observations, the correction could be calculated with the same accuracy, and subtracted from P˙bobs before making the comparison. After doing so, the agreement between the observed and calculated values of P˙b was better than 4 parts in 103 (i.e., equal to the accuracy of the measurements, as explained above) — an “extraordinarily stringent test” (as Taylor put it)
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of Einstein’s theory, and, therefore, an equally extraordinary verification of it, in a r´egime where relativistic theories of gravitation had never been testable before. In 1993, Hulse and Taylor shared the Nobel Prize in Physics, being cited for “their discovery of a new type of pulsar, a discovery that has opened up new possibilities for the study of gravitation”. We shall summarize below some outcomes from such “new possibilities”, but let us now introduce a succinct, graphical way of displaying relativistic binary-pulsar results which has great diagnostic value, and which was pioneered by Taylor and coauthors. The idea follows directly from the observation made earlier that, within any specified relativistic theory of gravity, the measured value of each of the 5 PK parameters defines a curve in the (m1 , m2 ), or mass-mass, plane, if the Keplerian orbit parameters are known. If the theory is selfconsistent and a correct description of the essential physics of relativistic gravity, and, furthermore, if we could measure all 5 PK parameters with arbitrarily high accuracy, all 5 curves should intersect at (at least) one common point (m1 , m2 ), which would give the masses of the pulsar and its companion. It is this diagnostic set of 5 curves for Einstein’s theory, which we can simply call the mass-mass plot, that has become very popular. The mass-mass plot for PSR 1913+16 [Taylor 1994] is shown in Fig. 7.23. The reader can easily derive the equation for these curves from Eqs. (7.28), (7.29), (7.30), (7.31) and (7.32). They are all fairly simple, and some of them extremely so, e.g., a given value of r just means m2 = const., a given ω˙ means m1 + m2 = const., and so on. In reality, there are always measurement errors, and we should have a band on the mass-mass plot for each measured PK parameter instead of a line. However, the errors in ω, ˙ γ, and P˙b are so small for PSR 1913+16 that the width of these bands would not be visible. The fact that the three curves do intersect at a single “point” (actually, a region of fractional size 4 × 10−3 , i.e., the fractional error quoted above) is thus a verification (of both the self-consistency and the physical correctness) of Einstein’s theory. Finally, as Taylor (1994) emphasized, one of the consequences of the highly precise agreement between the observed P˙b and that calculated from Einstein’s theory of general relativity was that gravitational radiation did exist and was quadrupolar in nature, i.e., its strength was ∼ v 5 /c5 to the leading order. This has great significance for alternative theories of relativistic gravity, since, as mentioned above, these theories generically give a leading order of magnitude ∼ v 3 /c3 for this effect, i.e., it has a dipolar nature (Esposito-Far`ese 1999). Actually, the dipolar emission in
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Fig. 7.23 Mass-mass plot of PSR 1913+16. See text for detail. Width of allowed bands too small to be visible. Reprinted with permission by the APS from Taylor (1994), Rev. c 1994 American Physical Society. Mod. Phys., 66, 711.
these theories typically depends on the difference between the masses of the pulsar and the companion, and so would completely dominate P˙b for binary rotation-powered pulsars with low-mass, white-dwarf companions, like PSRs 1855+09 and 0655+64, but would not be nearly as pronounced for double neutron-star binaries, where the masses of the two stars are very close to each other (see above). However, the mesaured values of P˙b for the former low-mass systems do seem to agree, within observational errors, not with the dipolar values, but with the much lower quadrupolar values given by Einstein’s theory. The viability of alternative theories of relativistic gravity vis-` a-vis binary-pulsar observations is a fascinating subject, in which constant use has been made of the diagnostic mass-mass plots described above. We shall not go into it here, but refer the interested reader to the extensive work done by Damour, Esposito-Far`ese and others in the 1990s, which has been expertly reviewed by Esposito-Far`ese (1999).
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PSR 1534+12 and similar “relativity laboratories”
In 1991, the binary rotation-powered pulsar PSR 1534+12 was discovered [Wolszczan 1991], and appeared at first to be an even more promising “relativity laboratory” than PSR 1913+16 [Taylor 1994]. The reason was a technical one: this pulsar, which had a spin period P ≈ 38 ms, an orbital period Pb ≈ 10.1 hr, an eccentricity e ≈ 0.27, and a mass function f ≈ 0.31 in solar units, had a stronger signal (it was a bright and relatively nearby pulsar) and a narrower pulse than PSR 1913+16, so that its TOAs could be determined with considerably higher accuracy than was possible for PSR 1913+16. The estimates for TOA accuracy of PSR 1534+12 from early Arecibo data were ∼ 3 µs for 5-minute observations, i.e., a factor of ∼ 5 better than the corresponding figure for PSR 1913+16 quoted above. Furthermore, the binary was seen nearly “edge-on”, i.e., the inclination angle i was nearly 90◦ , which greatly enhanced the orbital modulation of the Shapiro delay (see above), and so made its measurement easier. However, hopes of another spectacular demonstration of an extraordinarily accurate agreement between observed P˙b and that due to the emission of gravitational radiation, as calculated from Einstein’s theory, were belied by uncertainties in the knowledge of the distance to this pulsar. This distance occurs in the kinematic effect on P˙b described above, and the term containing it dominated the whole effect in this particular case [Stairs et al. 2002]. Actually, the real problem for PSR 1534+12 was that the kinematic effect amounted to a ∼ 40% correction to P˙b for this pulsar, unlike the tiny correction for PSR 1913+16 described above, because the overall size of P˙b for PSR 1534+12 was only ∼ 6 × 10−2 of that for PSR 1913+16, while the sizes of the kinematic corrections for the two pulsars were within a factor ∼ 3 of each other (the one for PSR 1534+12 being the larger). The situation was remedied by assuming the validity of Einstein’s theory, and using the measued value of P˙b to determine the pulsar’s distance. What could be verified, then? Using the accurately measured values of ω˙ and γ, the masses of the pulsar and its companion were determined to be m1 = 1.333 ± 0.001 and m2 = 1.345 ± 0.001 in solar units, as shown in the table. According to the scheme outlined earlier, this was then used to check consistency with the measured value of s = sin i, the Shapiro delay shape parameter, whose value was very close to unity (see above) for this pulsar, i.e., the maximum it can be, as explained earlier, and so could be measured very accurately. The results are shown in the mass-mass plot for PSR 1534+12 [Stairs et al. 2002], displayed in Fig. 7.24. There is agreement to within the frac-
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tional accuracy of the mass determinations and the measurement of s, i.e., about 10−3 . This can, therefore, be regarded as a very precise verification of the non-radiative (i.e., not involving gravitational radiation) aspects of Einstein’s theory of general relativity.
Fig. 7.24 Mass-mass plot of PSR B1534+12. See text for detail. Width of allowed bands visible for some PK parameters. Reproduced with permission by AAS from Stairs et al. (2002): see Bibliography.
Attempts at similar tests for other double neutron-star binaries, e.g., PSRs 2127+11C and 1820-11, have been going on, the best candidates for such studies being, of course, those binaries which have the largest mass functions and eccentricities, since the relativistic effects are the largest in these. But even neutron-star white-dwarf binaries, with relatively low-mass degenerate dwarf companions, can be valuable “relativity laboratories”, as the classic case of PSR 1855+09 [Ryba & Taylor 1991] has shown. This is a 5.4 ms pulsar in a 12.3 day binary orbit, consisting of a neutron star of mass m1 ≈ 1.50M and a degenerate dwarf of mass m2 ≈ 0.26M [Kaspi et al. 1994]. Its mass function is small, therefore, and since its eccentricity
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is also almost negligible (∼ 10−5 ), the relativistic effects represented by the first 3 PK parameters described above are somewhat problematic for the observer, as follows. The Einstein parameter γ, which is ∝ e (see Eq. [7.30]), is negligible, and ω˙ is difficult to measure with accuracy, because the periastron is difficult to identify for such extremely near-circular orbits. P˙b is also relatively small, but has been measured [Kaspi et al. 1994], if not very accurately. By contrast, the Shapiro delay, measured by the other 2 PK parameters r and s, is very prominent indeed: this is so because the orbital inclination of this binary system is extremely close to 90◦ , making the orbital modulation, i.e., the shape parameter s = sin i essentially unity — the largest it can be. We have already shown this result in Fig. 7.23, a clear, large effect, despite the fact that the range r of the Shapiro delay, which scales as the companion mass m2 (see Eq. [7.31]) is only ∼ 1.3 µs in this case, i.e., about a fifth of the usual neutron-star value r ∼ 6.8 µs found in PSR 1913+16 and other double neutron-star systems, since the mass of this degenerate dwarf is about a fifth of the canonical neutron star mass ∼ 1.3M found in recent measurements. 7.2.4.5
The double-pulsar binary as relativity laboratory
As mentioned in Chapter 6, the double-pulsar binary PSR J0737-3039 is an admirable relativity laboratory: this was clear from the beginning, when its (rather enormous) rate of periastron advance, ω˙ ≈ 16.9◦ yr−1 , was measurable after only the first few days of observation. Indeed, all five PK parameters for J0737-3039 have now been measured with high accuracy, as shown in Table 7.3 [Burgay et al. 2006]. As before, numbers in parentheses in the table denote standard errors in the last digit(s). Note that observed rate of orbital decay implies that the two neutron stars will spiral into each other and coalesce in ∼ 85 million years, which significantly increases estimates of the expected detection rate of gravitational waves from such systems [Lyne et al. 2004]. We show the mass-mass plot for J0737-3039 in Fig. 7.25. As pointed out by Burgay et al. (2006), the double-pulsar binary is the most overdetermined system to date for studying relativistic gravity, in view of the accurate PK parameters and an accurate determination of the ratio of the masses of the two pulsars R ≡ mA /mB = xB /xA , where xA , xB are the projected semi-major axes of the two pulsars, which have been already determined quite accurately (see Table 7.3). The key point here is that this method of determination of the value of R is expected to
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Table 7.4 al. 2006)
PARAMETERS OF PSR J7037-3039 (after Lyne et al. 2004 & Burgay et
Pulsar
PSR J0737-3039A
PSR J0737-3039B
PULSE PARAMETERS Pulse Period Period Derivative Spin-down Age Surface Magnetic Field
22.699378556138(2) ms 1.7596(2) × 10−18 210 Myr 6.3 × 109 G
2.7734607474(4) sec 0.88(13) × 10−15 50 Myr 1.2 × 1012 G
KEPLERIAN ORBIT PARAMETERS Orbital Period Pb Eccentricity e Projected Semi-major Axis x = a sin i/c Mass Ratio R = MA /Mb
0.1022515628(2) day 0.087778(2) 1.415032(2) sec
1.513(4) sec
1.071(1)
POST-KEPLERIAN PARAMETERS Periastron Advance rate ω˙ Einstein Delay γ Orbital Decay P˙ b Shapiro Delay parameter s = sin i Shapiro Delay parameter r Stellar Mass from (R, ω) ˙
16.900(2)◦ yr−1 0.39(2) ms −1.20(8) × 10−12 0.995(4) 6.2(6) µs 1.337(5)M
1.250(5)M
be valid for any “realistic” theory of gravity (see above), and the value so determined is independent of the strong-field effects, while the values of the PK parameters are not. In other words, the position of the R-line on the mass-mass plot is independent of the strong-field effects [Lyne et al. 2004]. Thus, as all the above authors have stressed, a new constraint for alternative theories of relativistic gravity emerges here, since any combination of masses derived from the PK parameters determined according to any such theory must be consistent with the observed value of R. Furthermore, as the accuracy of our knowledge of the PK parameters increases with time, higher-order effects in these should become measurable, e.g., the contribution of the spin-orbit coupling effect to ω. ˙ Since the latter effect depends on the moment of inertia I of the neutron star, this raises the fascinating
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Fig. 7.25 Mass-mass plot of the double-pulsar binary J0737-3039. See text for detail. Width of allowed bands visible for some PK parameters. Also shown is the constraint from the determined mass-ratio R ≡ mA /mB in this system. Shown inset is the area inside the small square in the main figure, with linear scales enlarged by a factor of 16. Reproduced with permission by Societ` a Astronomica Italiana from Burgay et al. (2006): see Bibliography.
possibilty of measuring I of a neutron star whose mass is known very accurately, and so constraining the EOS of neutron-star matter (see Chapter 5).
7.2.4.6
A related “fundamental” effect
The close agreement in PSR 1913+16 between the observed value of P˙bobs and the calculated value P˙btheo from Einstein’s theory (including the kinematic correction indicated earlier) opens the interesting possibility of utilizing the very small difference, δ P˙b ≡ P˙bobs − P˙btheo , between them to put upper bounds on the strengths of various putative causes for this difference. We now briefly mention one such “fundamental” effect whose size has been bounded in this way.
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This effect comes from the fascinating possibility of changes in the Newtonian gravitational constant G on very long timescales, originally suggested in 1937 by Dirac on the basis of his large-number hypothesis. The renewed theoretical interest in this in the 1980s was due to work on Kaluza-Klein and string theories, which naturally predicted changes in G (and other coupling constants) on a Hubble time12 , and, in the latter context, some of the interest still survives. Observationally, the best limits, −11 ˙ < yr−1 , available upto the mid 1980s were obtained from |G/G| ∼ 3 × 10 high-quality radar-ranging data between the Earth and Mars, using the Viking landers. In 1988, Damour et al. suggested that an upper bound ˙ on |G/G| could be found from high-precision binary pulsar timing data by ascribing the entire δ P˙b /Pb above to this effect, since the two would then be related by G˙ δ P˙ b =− , G 2Pb
(7.36)
as shown by the detailed, relativistic calculations of these authors. Using the PSR 1913+36 results available at the time, Damour et al. showed that ˙ the upper limit on |G/G| could be improved slightly from the above value. Using the more accurate PSR 1913+16 results quoted above, the reader −12 ˙ < can show from Eq. (7.36) that the limit now becomes |G/G| ∼ 5 × 10 −1 yr , improving by almost an order of magnitude over the radar-ranging value quoted above.
12 That is, the timescale of the Hubble expansion of the Universe: see, e.g., Weinberg 1972.
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Chapter 8
Superfluidity in Neutron Stars and Glitch Diagnostics
8.1
Superfluidity in Neutron Stars
Neutrons and protons in the nuclear matter found inside ordinary terrestrial nuclei are well-known to be superfluid, and the reason for this has been wellunderstood for a long time (see below). It will come as no surprise, then, that the neutron-rich nuclei in the crust of a neutron star have the same property, and the “dripped” neutrons in the high-density inner crust are also superfluid. In the core, where the density is above the nuclear-matter density, and matter is predominantly neutrons with an appropriately small admixture of protons and electrons (see Chapters 3 and 4), both neutrons and protons are superfluid, and the (charged) protons are superconducting as well. But why are they superfluid? Before that, precisely what do we mean by superfluidity in this context? The usual, laboratory notion of a superfluid is one that flows without significant viscous dissipation, either within the body of the fluid itself or in contact with the walls of the container, in contrast to what happens for ordinary fluids. This is how the phenomenon was discovered in the classic experiments on liquid Helium II by Kapitza and others in the 1930s, and this is how the long relaxation times of glitches made us aware in the early days of pulsar research that a superfluid component might indeed be at work here, although the original suggestion of the occurrence of superfluid neutrons in neutron stars by Migdal (1959) is much older. The microscopic physics that underlies this, and the closely related phenomenon of superconductivity (whose macroscopic manifestation is the familiar circulation of electric currents without any significant resistive dissipation in metals below a critical temperature), is what concerns us here. The understanding of this physics in terms of correlations, i.e.,
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in terms of the formation of pairs, in systems of fermions with attractive interactions has been one of the cornerstones of twentieth century physics, namely, the 1957 Bardeen-Cooper-Schriffer (BCS) theory of superconductivity, for which these authors received the Nobel Prize in 1972. These fermion pairs are known as Cooper pairs now. The basic idea is that, due to such pairing correlations, there is a major change in the low-energy spectrum of the system: a finite energy gap appears between its ground state and first excited states, and the system can undergo a phase transition to a superfluid or superconducting state below a critical temperature Tc . 8.1.1
Pairing
Although the original BCS work [Bardeen, Cooper & Schriffer 1957a,b] was concerned with the above kind of pairing of electrons in metals, the analogy with nuclei and nuclear matter followed almost immediately, in a brilliant conjecture by Bohr, Mottelson and Pines (1958)1. These authors pointed out the similarity of the unusually large energy gap between the ground state and the first excited intrinsic states of heavy nuclei of the even-even type (see Chapter 3) with that in the low-energy electron spectra of superconductors, and suggested that Cooper pairing of nucleons was the underlying cause. Whereas the attractive interaction between electrons in superconductors is mediated by the interaction between electrons and phonons, i.e., quantized vibrations of ionic lattices, that between nucleons is simply their mutual interaction, which is attractive at large distances (see Chapter 4 and Appendix D). Detailed calculations for strongly interacting systems of fermions by Cooper, Mills and Sessler (1959) followed rapidly. The first indications were that such systems without a repulsive core in the interaction (see Appendix D) always led to a superfluid state, while the presence of a repulsive core seemed to require a critical minimum strength of attraction. This conclusion was hardly surprising, but subsequent calculations showed that “realistic” nuclear interactions with repulsive cores actually led to superfluidity. Today, the evidence for superfluidity in nuclei and nuclear matter2 is overwhelming, and this is really the basis for our confidence that it certainly occurs at the same densities found in neutron stars [Anderson et al. 1982]. The first suggestion about superfluidity in neutron stars came in 1959 from Migdal, in a rather casual remark made 1 Also
see de Dominicis and Martin (1958). energy gap has the largest value for infinite nuclear matter, and is appropriately reduced for finite nuclei, as Bohr et al. (1958) had originally predicted. 2 The
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in a work that was really devoted to calculations of the reduction of nuclear moments of inertia due to superfluidity, an effect which is particularly direct and convincing. As pointed out in the original BCS work (1957a), pairing occurs essentially between fermion states near the Fermi surface. The most favorable situation arises when the two fermions have equal and opposite momenta, and opposite spins, since the (magnitudes of the) matrix elements are the largest in this condition. We can denote these fermion states as |k ↑, |−k ↓, and such pairs then form the ground state of the system. Because of pairing, it costs a finite amount of energy to excite a fermion from the ground state to the first excited state, the energy gap EG , or, equivalently, the transition temperature Tc ≡ EG /k being given by the celebrated BCS equation: −1 EG = kTc = EF exp . (8.1) N (EF )V Here, EF is the Fermi energy, as before (see Chapter 2), N (EF ) is the density of states at the Fermi surface, and V is the effective interaction in the many-fermion system under study. The formation of Cooper pairs in superfluids/superconductors is an excellent example of the quantum-mechanical phenomenon of condensation, which is interesting and instructive in itself: we can characterize condensation as macroscopic occupation of a single quantum state (Sauls 1989). What does this mean? In a superfluid, for example, the wave function of the condensate has the form ψ(R) ∼ |ψ(R)| exp[iθ(R)], i.e., it is coherent in the phase θ(R) over the whole fluid. In other words, if the condensate phase is known at any point R, we can predict the phase a macroscopic distance away. As far as the wave function of a given Cooper pair is concerned, it must be of the form ψs1 ,s2 (R, r), where s1 , s2 are the spin projections of the two fermions, R is location of pair’s center of mass, and r is their relative separation. Typical extent of this pair wave function in r — the orbital size of the Cooper pair, if we wish to put it that way — is ∼ 100 fm in neutron matter (Sauls 1989). This is much larger than the mean separation between the neutrons in the interior of neutron stars (see Chapter 4). The crucial point, however, is that the amplitude of this wave function is coherent over macroscopic distances, comparable to the dimensions of the fluid interior of the star, as indicated above. Stated differently, a macroscopic number of neutrons form pairs in the same pair wave-function state, independent of their center-of-mass co-ordinates R.
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Gap Energy
An accurate calculation of the gap energy EG is a rather difficult task, since it depends, as the BCS equation shows, exponentially on the effective interaction V in the neutron fluid in neutron stars, which is the case of interest to us in this book. V has two aspects that need particular attention: (a) a short-range correlation due to the repulsive core in the nuclear interaction, which tends to keep the neutrons from sampling too much of the core, and, (b) a polarization effect in the background fluid due to particle-hole correlation, which tends to screen out part of the long-range attractive interaction, and so to reduce the long-range pairing correlation [Pines 1991], and therefore EG . Early variational calculations, such as the pioneering work of Hoffberg et al. (1970) and Takatsuka (1972), took the first of the above effects into account, but not the second. Results are shown in Fig. 8.1. Since the nuclear interaction depends on the total spin S ≡ s1 + s2 of the two nucleons forming the pair (see Chapter 4 and Appendix D) and their relative orbital angular momentum l, so does the pairing interaction, and consequently EG . Which states of S and l dominate the pairing interaction depends on the density. The state of spin singlet, S = 0, and s-wave, l = 0, i.e., the 1 S0 state is the dominant one for neutron pairs at low neutron densities, ρn < ∼ 1.5 × 1014 g cm−3 , say, relevant for the “dripped” neutron fluid in the inner crust of a neutron star. This 1 S0 state is also the relevant one for electron pairing in most laboratory superconductors. The variation of EG with ρn for this 1 S0 pairing is shown in Fig. U. At low densities, the pairing interaction is ineffective because the density of states in the neutron-matter is low, and EG is small, as is readily seen from the BCS equation (8.1). At high densities approaching the nuclear density ρnm (see Chapter 3), the neutrons approach one another so closely that they probe the repulsive core extensively, and pairing interaction and EG drop drastically [Pines 1991]. Thus, the maximum of EG occurs somewhere in between, actually at a density ∼ ρnm /8 ≈ 3.5 × 1013 g cm−3 , with a maximum value of ∼ 2.5 MeV 14 in the Takatsuka (1972) calculations. At higher densities, ρn > ∼ 1.5 × 10 −3 g cm , relevant for the neutron fluid in the core of a neutron star, the dominant state is a spin triplet, S = 1, and p-wave, l = 1, one. The state is actually 3 P2 , with a total angular momentum J = 2. (The spin-singlet, p-wave states of superfluid He3 are related to this, but not identical.) For this pairing, EG again passes through a peak as the density increases, the maximum value being lower by a factor ∼ 3–5 from that for the earlier
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Fig. 8.1 Superfluid gap nergy for neutron matter vs. Fermi energy and density. 1 S0 state dominant at low densities, and 3 P2 at high densities. Solid lines: full calculation, including tensor interaction. Dashed lines: calculation neglecting tensor interaction. Reproduced with permission by Prog. Theor. Phys. from Takatsuka (1972): see Bibliography.
S0 case. Thus, typical gap energies are ∼ 1 MeV, setting the scale of the effect. Typical critical temperatures corresponding to such gap energies are 8 then Tc ∼ 1010 K, showing why neutron stars, with temperatures < ∼ 10 K, are sure to contain superfluid neutrons. A similar argument applies to the (small fraction of) protons in the fluid core of the star. The dominant pairing interaction between the protons there is 1 S0 , and the gap energies for these are rather similar to that for the neutrons there. Modern calculations have taken the polarization effects into account, using either a Fermi-liquid or a variational approach, and the results of the two approaches are in general agreement. As we stated above, these effects 1
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reduce the gap energy and critical temperature. We shall only mention the results of the Fermi-liquid calculations of Ainsworth, Wambach and Pines (1989): the profile of EG versus density remains roughly the same as before, so that the maximum value of the gap energy occurs at roughly the same density as described above, but all energy values are reduced by appropriate factors. To appreciate the meaning of this factor, we note first that, were the polarization screening effects (see above) neglected in these modern calculations, we would obtain gap energies which are larger by a factor ∼ 4 than those obtained in the above Hoffberg et al. or Takatsuka calculations. Screening effects are extremely important, reducing the gap energy by a factor ∼ 4, and we end up with a maximum value comparable to the original ones. 8.1.3
Rotating Superfluids and Quantized Vortices
What happens when superfluids rotate? The answer to this question, which is at the heart of much of the rotational dynamics of neutron stars, had been known for a long time before the era of pulsars, from studies of rotating superfluid He II. From elementary quantum mechanics, the velocity vs of a superfluid is given in terms of the (coherent) condensate phase θ introduced above by the expression vs =
∇θ, 2mn
(8.2)
where 2mn is the mass of a neutron pair. Since vs is the gradient of a scalar, we get the immediate result that the flow of a superfluid is irrotational, i.e., ∇ × vs = 0,
(8.3)
except possibly at an isolated set of singularities [Baym 1970]. These are actually line singularities, i.e., vortex lines. The way a rotating superfluid carries angular momentum is, then, by forming these vortex lines, as shown in Fig. 8.2, and the strength of such a vortex is measured by the circulation or vorticity around it, defined and evaluated as follows, using Eq. (8.2): , h ∆θ = n , (8.4) κ ≡ vs .dl = 2mn 2mn n being an integer. The wave function ψ is assumed to be single-valued, so that its phase θ must change by an integral multiple n of 2π in going
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Fig. 8.2 Vortex lines in a rotating superfluid. Reprinted with permission by Springer ¨ Science and Business Media from Sauls (1989) in Timing neutron stars, eds. H. Ogelman c and E. P. J. van den Heuvel, p. 457. 1989 Kluwer Academic Publishers.
around the line. Thus, the vorticity κ is quantized in units of h/2mn . In other words, quantized vortices must be present in rotating superfluids: this was pointed out long ago by Onsager (1949) and Feynman (1955), so that such vortices are sometimes called Onsager-Feynman vortices. It is now easy to calculate the superfluid velocity vs at a distance r from the vortex line, since vs .dl = 2πrvs because of cylindrical symmetry around this line. The result is: vs =
n . 2mn r
(8.5)
The energy per particle of a vortex is given, to a first approximation, by the kinetic energy of the fluid’s rotation: 1 1 Evort = mn vs2 nn d3 r, (8.6) Nn 2 where nn is the number density of neutrons, and Nn is the total number of neutrons present in the system [Baym 1970]. The above integral extends
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over the entire rotating system, except very close to the center of the vortex, where the condensate wave function ψ goes to zero, the phase coherence characteristic of superfluids (see Sec. 8.1.1) is lost, and matter becomes “normal”. The radial extent of this normal vortex core is of the order of the coherence length ξ in the neutron superfluid, which can be roughly defined as that distance from the center of the vortex at which the kinetic energy density (1/2)mn vs2 nn (see above) of the superfluid equals the condensation energy density, the latter being given by the BCS theory. An approximate expression for the coherence length is: (n)
ξ≈
2 kF , πEG mn
(8.7) (n)
where EG is the gap energy, as before, and kF is the Fermi wave-number of the neutrons, as discussed in Chapter 2. For typical values EG ∼ 1 MeV (n) (see above), and kF ∼ 1 fm−1 , we get ξ ∼ 10 fm, setting the scale size of the vortex core. An alternative, useful form for ξ given by Pines (1971) is: (n) EF 2 , (8.8) ξ≈ (n) EG πk F
(n)
(n)
where EF ≡ 2 (kF )2 /2mn is the Fermi energy of the neutrons, excluding the rest mass (see Chapter 2). Suppose we have neutron superfluid in a cylinder of radius R, and we make the cylinder rotate, starting from rest. What will happen? Note first that, if the vortex line is along the axis of the cylinder, each neutron-pair has an angular momentum 2mn vs r = n, with the aid of Eq. (8.5), so that each neutron has an angular momentum L = n/2. Similarly, each neutron has a rotational energy Evort given by Eq. (8.6), which, using Eq. (8.5) again, gives us R Ωc1 2 dr n2 2 = n , Evort = (8.9) 2 4mn R ξ r 2 where Ωc1 ≡ (/2mn R2 ) ln(R/ξ) is a lower critical angular velocity, whose physical significance we shall now see. Suppose that we are making the container rotate at this moment with an angular velocity Ω. The equilibrium configuration is obtained by minimizing the free energy Evort − ΩL. The reader can readily show from the above expressions for energy and angular momentum per particle that, when Ω < Ωc1 , the least value of the above quantity is obtained for n = 0, i.e., no vortices present in the superfluid
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at all. This, then, is the situation below the lower critical angular velocity Ωc1 : the container rotates but the superfluid sits still [Baym 1970]. The relevance to neutron stars must be evident by now: the “container” mimics the solid crust of the star, with R ∼ 10 km. It is then clear that, with ξ ∼ 10 fm as given above, Ωc1 ∼ 10−14 s−1 , i.e., a negligibly slow speed −3 −1 compared to the values Ω > ∼ 10 s for the slowest-rotating neutron stars known. As Ω exceeds Ωc1 , a single quantized vortex appears first, then more and more as Ω increases. When Ω Ωc1 and the spacing between the vortices becomes R, as shown in Fig. 8.2, the flow pattern looks on a macroscopic scale like a uniform rotation, characterized by a density of vortices per unit area (i.e., an areal density) nv , which is easily calculable in terms of the (uniform) angular velocity Ω and the vortex quantum h/2mn , introduced above, as follows. We write the circulation integral (see above), vs .dl, in two ways: (a) as 2πr¯ vs , in terms of the mean flow velocity v¯s at a radius r, and (b) as (nv πr2 ).(h/2mn ) (assuming a uniform distribution of vortices at these small vortex spacings), since the first bracket gives the total number of vortices within a radius r, and the second, the quantum of circulation. Equating the two, we readily get v¯s = (hnv /4mn )r : this shows that the flow is a uniform rotation, v¯s = Ωr, with an angular velocity Ω = (hnv /4mn ). We thus arrive at the Onsager-Feynman expression for the areal density of vortices: nv =
2Ω 4mn Ω = . h (h/2mn )
(8.10)
The second form of the Onsager-Feynman expression shows an alternative, a bit more formal, way of deriving it. From elementary hydrodynamics, the circulation per unit area, ∇ × vs , is just 2Ωˆz for a uniform rotation z×r about the z-axis, as the reader can easily show by substituting vs = Ωˆ in the above curl expression. The number of vortices per unit area must, therefore, be this amount of circulation, 2Ω, divided by the quantum of circulation given above, and, indeed, it is. Note that, if we use the pulsar rotation period P = 2π/Ω instead of the angular velocity Ω in the OnsagerFeynman relation, we obtain the convenient expression nv = 6.3×103P (s)−1 vortices/cm2, where P (s) is the pulsar period in seconds. For the Crab pulsar (P (s) ≈ 0.033), then, the spacing between the vortices is ∼ 10−3 cm, and similar values obtain for the Vela pulsar. Consider, finally, what happens at still faster rotation. When the vortex spacing approaches the coherence length ξ, so that vortex cores over-
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lap, superfluidity ceases, as must be clear from the earlier observation that the condensate wave function vanishes within vortex cores. From the expressions given above, the reader can immediately show that this happens above an upper critical angular velocity Ωc2 ∼ (/2mn ξ 2 ) ∼ 1020 s−1 . 3 −1 Again, this is enormously large compared to the values Ω < ∼ 10 s for the fastest-rotating “millisecond” pulsars known, and the limit of stability for rotating neutron stars (see Chapter 5) is only about an order of magnitude higher than the above highest value seen in millisecond pulsars. Thus, pulsars always “sit” very safely between Ωc1 and Ωc2 [Baym 1970], far away from either limit. Superfluidity and the occurrence of quantized vortices are guaranteed, as is the validity of the classical, macroscopic description in terms of uniform rotation, due to the close spacing of the vortices, for the entire range of pulsar periods.
8.2
Post-Glitch Relaxation: Two-Component Theory
The long, “macroscopic” relaxation times (∼ tens to hundreds of days; see Chapter 7) of the glitches made it clear from the beginning that superfluids were very likely involved in the process, since coupling through “microscopic” processes involving normal matter would be extremely tight, leading to timescales ∼ seconds. Accordingly, a phenomenological, twocomponent theory of post-glitch relaxation was proposed in 1969 by Baym et al., in which a “container” or c-component rotated at a slightly lower angular velocity from the “superfluid” or s-component it contained. The container component was actually the solid crust, plus any other (charged) component that was tightly coupled to the crust. The superfluid component was actually the neutron superfluid in the core of the neutron star, loosely coupled to the container. The two components were coupled by a “mutual friction” between them, with a long relaxation time τ of the order of magnitude given above, so that the coupling was described by a term ∝ (Ωs − Ωc )/τ in the equations of rotational dynamics, the subscripts here referring to the components labeled as above. The glitch was, of course, described in this theory as a starquake in the crust (see Chapter 7), as was customary at the time. Post-glitch relaxation then readily emerged from the theory as an exponential healing of a fraction Q of the glitch ∆Ω on a timescale ∼ τ , and the fraction Q (see Chapter 7 for its observed values) was calculable. The microscopic process underlying such long relaxation times as above was believed in the 1970s to be the scattering of electrons by the normal
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cores of the vortex lines in the neutron superfluid, and this seemed to be supported by detailed calculations performed in that era. In the early 1980s (i.e., about the same time when the inadequacy of the starquake model was becoming clear), however, re-investigations of this scattering process established that the dominant mechanism was, in fact, the scattering of electrons by the inhomogeneous magnetization surrounding vortex lines [Alpar et al. 1984], and the timescale was far shorter than imagined before. This magnetization, which is typically ∼ 1015 G, arises from the current of superconducting protons entrained by the superfluid neutron currents circulating around the vortex lines, and the timescale for the relaxation of the relative velocity between an electron and a vortex line is roughly τv ∼ Ω−1 2 second. Here, Ω2 is the pulsar’s angular velocity in units of 102 rad s−1 , as before. The corresponding dynamical coupling time τd between the crust and the superfluid neutron core is given roughly by τd ∼ (20 – 600)τv ∼ (1 – 10)Ω−1 2 minutes, i.e., essentially instantaneous compared to the observed relaxation times given above. Thus, the original two-component theory also became untenable at about the same time as the starquake model did, and we shall not discuss it any further. It is interesting, however, that a later incarnation of the theory came into vogue in the 1980s when the next successful model of the glitches was proposed, and post-glitch relaxation needed to be explained from an altogether different point of view. After introducing this model below, we shall give an overview of the modern two-component theory. As the reader will see, the constitution of the components become somewhat different there, and the nature of the coupling between them is entirely different.
8.3
Glitches: Vortex Pinning
Following the realization that starquakes were untenable as a unified model of the glitch phenomenon (see Chapter 7), a novel mechanism involving the rotating neutron superfluid in a neutron star was suggested in the early 1980s and thoroughly studied subsequently. The key idea for this was much older: in an excellent 1971 review of the essential condensed-matter physics of neutron stars, Pines considered the possibility that the above quantized vortices in the superfluid “dripped” neutrons in the inner crust of the neutron star could be pinned to the lattice of the heavy, neutron-rich nuclei which exists there, and which is surrounded by these dripped neutrons (see
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Chapter 3). Pinning means, as the word suggests, that the “normal” core regions of the vortex lines (see above) would prefer to pass through the positions of the crustal nuclei on the lattice sites, and would stay fixed in this manner until a stronger force could unpin them from these sites. Although Pines already suggested in his original (1971) work that vortex pinning could be “an important question in sorting out the response of the neutron system to sudden changes in crustal motion”, and although Anderson and Itoh briefly reconsidered the idea in 1975, making the pioneering connection between sudden vortex unpinning and pulsar glitches, and stating that this idea explained the observations “at least equally well” (compared with the starquake scenario, which was still the standard model at the time), it was not until the inadequacy of the starquake model was appreciated by about 1981 that extensive work on the vortex-pinning model of glitches began. As the reader may have guessed, pinning occurs when it is energetically favorable for the system to do so, and so the pinning force is closely related to the energy gained when vortices become pinned to the lattice of nuclei, as we shall show below. Historically, the idea of pinning in neutron stars came from an analogy with the behavior of laboratory superconductors and superfluids: from the work of Anderson and others, it had been known since the early 1960s that magnetic flux-vortices (or fluxoids, as they are sometimes called) in terrestrial superconductors tend to pin to defects in crystalline lattices or to impurity sites, because it is energetically favorable for them to do so. Similarly, vortices in rotating superfluid helium in a container tend to pin to defects in the wall of the container. The major difference in this case is that the coherence length ξ introduced in Sec. 8.1.3 — the effective width of the vortex core — is less than or comparable to the nuclear lattice spacing b over most parts of the inner crusts of neutron stars, in contrast to what happens in ordinary laboratory superconductors. Because of this, the vortex lines can pin to the lattice itself, not just to the imperfections in it [Anderson et al. 1982]. It is, in fact, instructive to compare the orders of magnitude of several length scales involved in the problem. First, there is the spacing be−2 P (s) cm, introduced in Sec. 8.1.3, which tween vortex lines, rv ∼ 10 is enormous compared to all other lengthscales of the problem (Pines 1971), and so irrelevant for the pinning criterion. Then, there is the coherence length for dripped neutrons discussed above, which we can express in the form ξ ∼ 40[EG (MeV)]−1 (ρ/ρnm )1/3 (ρdn /ρ)1/3 fm by us(dn) = (3π 2 ρdn /2mn )1/3 . Here, the label (dn) deing Eq. (8.7) and kF notes dripped neutrons, and the factor (ρdn /ρ) takes care of the fact
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that only a part of the crustal matter is in the form of dripped neutrons. Note that ξ < ∼ 1 fm near the outer edge of the inner crust (which is the neutron-drip point), since (ρdrip /ρnm ) ∼ 10−3 and (ρdn /ρ) 1 there, and that ξ ∼ 40 fm near the inner edge of the inner crust, where (ρ/ρnm ) and (ρdn /ρ) are both ∼ 1. Next, there is the lattice spacing b, for which we shall use the Wigner-Seitz approximation (see Chapter 3), b ∼ rc ∼ [3AmB /4π(ρ − ρdn )]1/3 ∼ 6(ρ/ρnm )−1/3 [1 − (ρdn /ρ)]−1/3 (A1/3 /5) fm. The lattice spacing is b ∼ 60(A1/3 /5) fm near the outer edge of the inner crust, then, and b ∼ 10(A1/3 /5) fm near the inner edge. Note that, for the equilibrium nuclei in the inner crust (see Chapter 3), A1/3 rises from ∼ 5 near the outer edge to ∼ 6 – 7 near the inner. Finally, there is the size of the nuclei at the lattice sites, rN ≈ r0 A1/3 ∼ 6(A1/3 /5) fm. For exploring the pinning criterion, we thus need only consider the relative values of ξ, b, and rN . Pines (1971) originally considered the ratio ξ/rN , which is interesting for other considerations (see Sec. 8.3.1), but it became clear subsequently that ξ/b was the ratio crucial to geometrical criterion for pinning. Since ξ ∼7 b
EG 1 MeV
−1
A1/3 5
−1
ρ ρnm
2/3
ρdn ρ − ρdn
1/3 ,
(8.11)
we see that ξ/b ∼ 10−2 near the outer edge of the inner crust, and ξ/b ∼ 7 near the inner edge. Thus, except in the innermost parts of the inner crust, ξ/b < ∼ 1, so that pinning to the lattice sites is a geometrically valid concept. Pinning will actually occur if it is also energetically favorable, which we shall show to be the case below. As we shall also indicate, pinning in this region can be divided into two r´egimes, strong and weak, depending on the strength of the pinning force relative to that of the lattice force. When ξ/b > ∼ 1, the geometrical notion of pinning begins to become somewhat vague, since a vortex line would then engulf more than one lattice site across its width: perhaps a superweak pinning r´egime is still possible under these circumstances [Alpar et al. 1984]. 8.3.1
The Pinning Force
How do we calculate the pinning force and energy? The latter is calculated by a method which follows straight from the qualitative idea of pinning introduced above, namely, that pinning occurs when the energy that obtains if the vortex line passes through a nucleus is less than that which obtains if it
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threads through the interstitial region between nuclei: the pinning energy is simply the difference between the two. Neutrons inside the nuclei are superfluid, as are the “dripped” neutrons outside the nuclei, as explained earlier. So, it is a matter of calculating the energy difference between creating a vortex core in the outer, dripped-neutron superfluid and creating one in the inner, nuclear superfluid. If the energy cost per particle to create the normal core of a vortex is , then the pinning energy per unit volume is (nn )out − (nn )in , where nn is the number density of superfluid neutrons, and the labels ‘in’ and ‘out’ are self-explanatory. The question is: what is ? This energy, which is of basic importance in the physics of superfluid vortices, is given in terms of the gap energy EG of the superfluid and the Fermi energy (excluding the rest mass) EF of the 2 /EF . Feibelman’s (1971) fermions constituting the superfluid by ∼ EG way of deriving this estimate is transparent and physically appealing, and we summarize it here. Consider the normal core of a vortex line, a cylinder of radius ξ. The momentum scale for motion perpendicular to the core axis is /ξ from elementary quantum mechanics, and the corresponding energy scale is 2 /2mξ 2 , where m is the fermion mass. This, then, sets the scale for fermion energy-level differences in the vortex core, and so is a reasonable measure of the energy per particle . Using Eq. (8.7) for ξ, we 2 2 /EF ) ∼ EG /EF . Here, EF ≡ 2 kF2 /2m is the readily obtain ≈ (π 2 /4)(EG Fermi energy of the fermions, excluding the rest mass (see Sec. 8.1.3). The pinning energy per nucleus Ep can now be written down in the form [Alpar 1977; Anderson et al. 1982; Alpar et al. 1984]: 3 Ep = 8
nn
2 EG(n)
−
(n)
EF
out
nn
2 EG(n)
V.
(n)
EF
(8.12)
in
Here, the label (n) denotes neutrons, as before, and V is the overlap volume between the vortex core and the nucleus. From the length scales described in Sec. 8.3, the reader can readily see that ξ/rN ∼ 7[EG (MeV)]−1 (ρ/ρnm )1/3 (ρdn /ρ)1/3 (A1/3 /5)−1 [Pines 1971], so that ξ > ∼ rN except in a small region near the outer edge of the inner crust. Thus, V is of the order 3 over most of the pinning region. In the of the nuclear volume (4π/3)rN 2 exceptional region, V ∼ πξ rN , as the reader can easily show. We show the variation of Ep [Alpar et al. 1984] with the density in the inner crust in Fig. 8.3: clearly, pinning is strongly favorable energetically (Ep is positive and > ∼ρ< ∼ 0.4ρnm , the pinning ∼ 0.5 MeV) in the density range 0.1ρnm < energy being a maximum around ρ ∼ 8 × 1013 g cm−3 ∼ ρnm /4.
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Fig. 8.3 Pinning energy Ep per nucleus vs. density in neutron-star crusts. Reproduced with permission by the AAS from Alpar et al. (1984): see Bibliography.
The pinning force Fp is obtained by dividing Ep by the length scale of the interaction between the vortex core and the nucleus, for which we can adopt the larger of the two lengths ξ and rN [Alpar 1977], which, in view of the above discussions, is just ξ over essentially all of the region where effective pinning occurs. So, Fp ≈ Ep /ξ is a very good estimate. Finally, the pinning force fp per unit length of the vortex line is given by fp = Fp /a, where a is the spacing between successive pinning centers along the vortex line. 8.3.2
Strong and Weak Pinning
We now describe the two r´egimes of pinning introduced in Sec. 8.3. If the vortex lines were oriented exactly along one of the axes of the nuclear lattice, the spacing a between successive pinning centers would equal the lattice spacing b for a simple cubic lattice. This would lead to the largest possible value of fp (see above), i.e., optimal pinning. But this would happen only if the direction of the angular velocity Ω of the star, which is the direction of the vortex lines, coincided with that of one of the lattice axes, which will
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not be the case, in general. This is so because the conditions under which the crust solidified and the lattice froze had basically nothing to do with stellar rotation. In general, then, the vortex lines will be at some angle with the lattice axes. The question is, what would optimize pinning under such conditions? The answer introduces us to the idea of strong pinning. Imagine that the pinning force is so strong that it can displace the nuclei from their equilibrium lattice sites by amounts sufficient to arrange them along the vortex line. This is optimal indeed, since the spacing a between pinning centers along a vortex line will then be ∼ a few times the lattice spacing b [Anderson et al. 1982], as the reader can visualize. In effect, the pinning is strong enough to “pull” nuclei from nearby lattice sites into the vortex line. The quantitative criterion for this strong pinning follows directly from its definition: the pinning force Fp must exceed the lattice displacement force FL . An adequate estimate for the latter force is the difference between the forces at two points in the vicinity of a lattice site displaced from each other by a length comparable to the length-scale of the pinning force, which we have shown above to be ∼ ξ. Thus, FL ∼ [(∂/∂r)(Z 2 e2 /r2 )]r=b ξ ∼ (Z 2 e2 /b3 )ξ ∼ (wc /ξ)(ξ/b)2 . Here, wc ∼ Z 2 e2 /rc ∼ Z 2 e2 /b is the Coulomb energy of a Wigner-Seitz cell, introduced in Chapter 3, where we gave the lattice energy both excluding and including the electrostatic self-energy of the finite-size nuclei, as also the corrections in the electron-nucleus interaction energy due to the finite size of the nuclei. The full expression given there is more complicated than above. However, as the reader can show, the simple expression used above is still a reasonable order-of-magnitude estimate in the pinning region, which extends over the density range described above. Exception comes very close to the inner edge of the crust, just before it dissolves into a nuclear fluid, but pinning is not expected to be significant there, as explained above. Hence, the strong-pinning condition, Fp > FL , reduces to the condition Ep > Ecrit on the pinning energy, where the critical energy is Ecrit ≡ wc (ξ/b)2 ∼ 1–2 MeV in the pinning region. As we go inward through the pinning region, the density rises, and so does Ecrit , since it scales as ξ 2 /b3 , and ξ increases and b decreases with increasing density (see above). On the other hand, Ep passes through a maximum and falls with rising density thereafter, as shown above. Hence, above a critical density ρcrit ≈ ρnm /3, the strong-pinning condition is no longer valid, and we reach the r´egime of weak pinning, where Fp < FL . What happens then? We might think at first that the vortex line would then bend as necessary to optimize pinning, since the pinning force is no
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longer strong enough to displace nuclei for this purpose. This is not possible, however, since the energy per unit length of vortex lines is EF kF > ∼ 100 MeV fm−1 , much larger than that which the forces relevant to the pinning problem can provide. What is believed to happen instead is that the lattice does deform, but now more smoothly on a length scale a, rather in a wavelike fashion, along the vortex line [Anderson et al. 1982]. Thus, a is also a measure of the distance between the effective pinning sites along the vortex line, the quantity introduced earlier, except that its value is now larger than it was in the strong-pinning r´egime. We can estimate a in this situation by equating the orders of magnitude of the pinning and and lattice-distortion energies, which must be valid at equilibrium. Per pinning site, the former energy is just Ep . To estimate the latter, note that a wave-like distortion of wavelength a and amplitude ξ (which is the range of the vortex-lattice interaction) leads to a relative displacement ∼ ξ/a in the lattice, and so to a lattice-distortion energy ∼ wc (ξ/a) per pinning site. Equating these and using the expression for the critical pinning energy Ecrit introduced above, we obtain Ecrit b a ∼ , b Ep ξ
(8.13)
which manifestly shows that a > ∼ b, as it must be, since Ep < Ecrit and ξ < b in this r´egime. In fact, as pinning becomes very weak, i.e., Ep Ecrit , the pinning sites become sparse, a b, as expected. Eventually, pinning becomes so weak and the lattice-distortion wavelength so large that we can assume the lattice to be effectively rigid. In this limit, the spacing a between the pinning sites can be obtained from a simple statistical argument, as follows [Anderson et al. 1982]. We imagine a vortex line as a cylinder of radius ξ oriented in an arbitrary direction with respect to a simple cubic lattice of spacing b, and ask ourselves the question: how many lattice sites N will fall into a length L of this cylinder? This is what we need, since these will be the only pinning sites on a length L of the vortex, and so the average spacing of pinning sites along the vortex line will be given by a = L/N . The answer for N is a statistical one, as stated above, and the counting method is simple: since each cell of the lattice occupies a volume b3 , the volume πξ 2 L of the cylinder must contain a number πξ 2 L/b3 of lattice cells, and since each lattice cell contains one lattice site on the average, the number of the latter must be N ≈ πξ 2 L/b3 on the average. Thus, the average spacing between pinning sites along the vortex line is given by:
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a≈
b3 b3 ∼ . πξ 2 ξ2
(8.14)
Finally, close to the point of dissolution of the crust, there may be a small region where ξ > ∼ b, and where superweak pinning may occur, as indicated earlier. We shall not consider this region any further, as it does not appear to be significant dynamically. 8.3.3
The Magnus Force
We now describe how the above phenomenon of pinning is relevant to glitches. A vortex line would tend to move with the ambient superfluid, unless there is some force in the system which acts to prevent this. Now, the ambient superfluid moves with an angular velocity Ωs which is determined by the areal density of vortices according to Eqn. (8.10). However, the vortex lines pinned to the crustal nuclei move with the solid crust, which generally moves at a lower angular velocity Ωc . This is so because the electromagnetic torques which slow down a rotation-powered pulsar act directly (through the stellar magnetic field) only on the crust (and also possibly on that small portion of the fluid core which is charged, i.e., the electrons and protons, if the core is magnetically coupled to the crust), and not on the superfluid of “dripped” neutrons in it. Thus, it is Ωc that is to be identified with the observed angular velocity of the star, since pulsar emission processes are believed to be related to the external surface of the solid crust, and the magnetic field threading through it. Ωs always lags behind it. ˆ between a Because of this difference of velocities, δv = r(Ωs − Ωc )φ, vortex line and its ambient superfluid, the former is acted upon by a force which is called the Magnus force, and which would neutralize the above velocity difference (and so make the vortex line move with the ambient superfluid, as indicated above) if there was no other force in the system to counteract it. Here, the angular velocities are oriented along the z-axis, and φ is the azimuthal angle around this axis. How the Magnus force would do this becomes clear when we write its value fM per unit length of the vortex line as: fM = −ρ(κˆz) × δv.
(8.15)
Here, κ = h/2mn is the vorticity quantum introduced earlier, so that κˆz is the vorticity vector. From Eq. (8.15) and the above expression for δv, it is
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clear that the Magnus force acts in the radially outward direction, i.e., it tends to push the vortex lines outward, so as to transfer some of the angular momentum of the superfluid to the crust, thus spinning up the latter, and reducing the above difference of velocities. The Magnus force is counteracted by the pinning force fp described above, which acts in a radially inward direction, and keeps the vortex lines pinned to the nuclei, as long as the pinning force exceeds the Magnus force, fp > fM . Given a strength of the pinning force (or pinning energy), the difference in angular velocities, δΩ ≡ Ωs − Ωc , thus builds up as the crust spins down, until it becomes so large that fp = fM . At this point, the vortex lines unpin from the nuclei suddenly, moving outward and transferring their angular momentum to the crust in a catastrophic manner, thus causing a sudden speed-up of the crust, i.e., a glitch. The critical angular-velocity difference, δΩcrit , at which this happens is readily obtained by inserting the expressions for fp and fM given earlier into the above condition of their equality at the unpinning point: δΩcrit =
EP . ξarρκ
(8.16)
We show the variation of δΩcrit with density from an early calculation [Anderson et al. 1982], using the original Hoffberg et al. (1970) results for the gap energy, and variations thereof. Clearly, the value of the critical angular-velocity difference depends sensitively on the gap energy.
8.4
Post-Glitch Relaxation: Vortex Creep
Between glitches, the vortex lines are believed to undergo a slow, thermally activated process called vortex creep, which is a quantum tunneling between adjacent sites which are geometrically suitable for pinning. The analogy is with the process of flux creep in Type II superconductors. In this picture, the creep process approaches a steady state when left to itself, a glitch perturbs it away from this steady state, and the post-glitch relaxation is a process of recoupling the creep to another steady state. A simple, potentialfield description in terms of the pinning-energy profile, given by Alpar et al. (1984), clearly explains the essential physics of vortex creep, and we recount it here in brief. Consider, schematically, the potential “seen” by a vortex line in a lattice in the pinning r´egime, i.e., where pinning is energetically favorable. The
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Fig. 8.4 Radial pinning-energy profile (a) without bias, and (b) with bias. Reproduced with permission by the AAS from Alpar et al. (1984): see Bibliography.
nuclei represent sites of lower energy — potential wells — and the spaces between them have higher energies, i.e., “barriers” between these wells, the difference between these being ∼ Ep . In the absence of any bias, the energy profile seen by the vortex line is the same in any direction, as shown schematically in Fig. 8.4). Ep is the order of magnitude of the amplitude of variation this quasi-periodic potential, whose periodicity in space is ∼ a, the average distance between pinning centers, as above. In this situation, the probability of thermally-activated barrier penetration is the same in all directions, and the net motion of a vortex line is zero. Now consider what happens because of the angular-velocity difference δΩ between the superfluid and the pinned vortex, as described above. The Magnus force
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per unit length fM = ρκrδΩˆr tends to push the vortex lines outward, as we argued above. We can now cast this into a potential description by first noting that the force per pinning site is FM ≈ fM a, and then that the energy difference corresponding to this force would be ∆Ep ∼ FM ξ. Thus, a site at a distance r from the rotation axis will have an energy higher than one at a distance ∼ a further from the axis by an amount ∆Ep ∼ ξarρκδΩ = Ep
δΩ , δΩcrit
(8.17)
where we have used Eq. (8.16) to obtain the last expression. This introduces a bias in the energy profile, as shown in Fig. 8.4: the profile, while still quasiperiodic with a spacing ∼ a and an amplitude ∼ Ep , now has an overall downward slope in the radially outward direction. This favors thermallyactivated barrier penetration in a radially outward direction over that in an inward direction, and so produces an overall outward drift of the vortex lines. This, precisely, is the vortex creep. We can calculate the radial creep velocity vr from barrier-penetration considerations in a straightforward manner [Alpar et al. 1984]. For motion in an outward direction, the energy barrier is lowered to Eout = Ep −∆Ep = Ep [1 − (δΩ/δΩcrit )], while for motion in an inward direction, it is raised to Ein = Ep + ∆Ep = Ep [1 + (δΩ/δΩcrit )]. Hence the overall creep velocity in the radially outward direction is: Eout Ein vr = v0 exp − − exp − , (8.18) kT kT where v0 is the velocity scale for microscopic motion of the vortex lines between pinning sites. v0 is determined by the Bernoulli force, and is estimated to be ∼ 107 cm s−1 . A form of Eq. (8.18) that we shall find of much practical use below is δΩ Ep vr = 2v0 exp − sinh kT δΩ δΩ Ep exp . (8.19) ≈ v0 exp − kT δΩ Here, we have defined a reference value of the angular-velocity difference, or lag, δΩ, by the relation δΩ ≡ (kT /Ep )δΩcrit . The second, approximate form of Eq. (8.19) applies when δΩ is close to δΩcrit , so that δΩ δΩ, since Ep ∼ 1 MeV, and kT < ∼ 1 keV. As we shall see below, this situation is of much practical importance.
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Rotational Dynamics: Steady State
An excellent approach to the rotational dynamics of the star according to the above scenario of vortex pinning, catastrophic unpinning, and gradual creep is in terms of a two-component theory, which we now describe. While the origins of this approach may lie in the original two-component theory of the starquake era, which we mentioned in Chapter 7, the components themselves, and the nature of the coupling between them, are quite different in this theory. The first component is again the “container” (labeled by c in what follows), i.e., those parts of the star which rotate quite rigidly together as one body, and are acted upon by directly by the external torque Next . But now this component actually consists of the solid crust, together with the neutron and proton superfluids in the core of the star. The reader may remember that research in the 1980s had shown that the entire superfluid in the core was very tightly coupled to the crust, contrary to what had been thought before. The second component is the more loosely coupled “superfluid” (labeled by s in what follows), which rotates with an angular velocity different from that of the above c-component. This s-component now actually consists of the crustal superfluid alone, which can be pinned or unpinned at a given instant in time, depending on the circumstances, as explained above. Note that, although the s-component contains only ∼ 10−2 of the total moment of inertia of the star, it is thus dynamically crucial in this scenario for explaining glitches, post-glitch relaxation, and associated phenomena [Alpar et al. 1984]. As we shall also see below, the coupling between the two components is now completely different from the frictional coupling postulated in the two-component theory of the starquake era. The rotation of the c-component is described quantitatively by: ˙ c = Next + Nint , ˙ s, Nint ≡ − dIs Ω (8.20) Ic Ω where the internal torque Nint arises from the s-component, the integration extending over all possible parts of that component. From observation (see above), we know that Nint ∼ 10−2 Next . On the other hand, the rotation of the s-component is obtained by combining the law of vortex conservation, ∂nv + ∇.(nv vr ˆr) = 0, ∂t
(8.21)
with a generalization of Eq. (8.4) to a situation where there can be differential rotation in the superfluid, so that both Ωs and nv can be functions of the radial co-ordinate r:
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, 2
vs .dl = 2πΩs (r)r = κ
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r
2πrnv (r)dr.
(8.22)
0
Here, nv is, of course, the areal density of vortices, and κ ≡ h/2mn is vorticity quantum, both introduced earlier. The differential form of Eq. (8.22), which reads κnv (r) = 2Ωs (r)+r(∂Ωs /∂r), can be combined with Eq. (8.21) to yield the required quantitative description of the rotation of the scomponent: κnv vr 2Ωs (1 + )vr ∂Ωs =− =− . ∂t r r
(8.23)
Here, (r) ≡ (r/2Ωs )(∂Ωs /∂r) is the differential-rotation parameter of the superfluid. For the radial creep velocity of vortices, we use Eq. (8.18), and this completes the description. We now summarize some essential features of the rotational behavior described by the solutions of the above equations: for more detail, the reader can refer to the original work described in Alpar et al. (1984), and in Alpar and Pines (1989). First, this description indicates the existence of a steady-state solution, corresponding to a constant asymptotic value of the lag δΩ∞ . Whether this steady state is actually achieved in practice or not, it remains a useful reference point for the description of vortex creep. It is easily obtained from the requirement that, in this steady state, the ˙c = Ω ˙ ∞, s- and c-components spin down at the same rate, ∂Ωs /∂t = Ω say, so that the lag stays constant at δΩ∞ . With the aid of Eq. (8.20), we ˙ c = Next /I, see that the steady-state spin-down rate is given by Ω˙ ∞ = Ω where I ≡ Ic + dIs is total moment of inertia of the star. Further, the steady-state creep velocity is obtained from Eq. (8.23), and expressed as: v∞ =
r |Ω˙ ∞ |r ≈ , 2Ωs (1 + ) 2ts
(8.24)
˙ ∞ |, in terms of an (observable) spin-down timescale of the star, ts ≡ Ωc /|Ω ˙ which can be closely approximated by ts ≈ Ωs /|Ω∞ |, since δΩ Ωs , as we described above. In obtaining the last approximation in Eq. (8.24), we have neglected the superfluid’s differential-rotation parameter , and we shall continue to do so henceforth, because of the reason explained earlier: although a rotating superfluid cannot, strictly speaking, sustain uniform rotation, it mimics such rotation very closely by creating a dense array of quantized vortices, for all rotational velocities found in pulsars.
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A simple description of the above steady state is thus that (a) the vortices “creep” over a distance comparable to the stellar radius in a spin-down time ts , such that (b) the crustal superfluid’s angular velocity just “keeps up” with the change in the angular velocity of the crust (and the core tightly coupled to it) produced by the external spin-down torque on the pulsar. It is now easy to see that the steady-state δΩ∞ must be close to δΩcrit , and so would the actual δΩ for expected rotational states of the pulsar, as stated earlier. We do this by applying the approximate form of Eq. (8.19) to steady-state creep, which yields 2ts v0 δΩ∞ kT 1− = ln , (8.25) δΩcrit Ep r with the aid of Eq. (8.24). Now, on the right-hand side of Eq. (8.25), the −3 logarithmic factor is ∼ 10 – 30, but kT /Ep < ∼ 10 , as explained above, so that the right-hand side is 1 quite generally, and consequently δΩ∞ is always close to δΩcrit . Glitches occur preferentially, of course, in those regions of the crust where δΩ∞ is closest to δΩcrit , since fluctuations can raise δΩ in these places to δΩcrit most easily. As shown below, the actual values of δΩ for pulsars, even when not in steady state, are not far away from their steady-state values, so that this approximate equality also holds for the latter. 8.4.2
Approach to Steady State
We now briefly sketch the next essential feature of the solutions of the above rotational equations, namely, the approach of the vortex creep to a steady state, say following a glitch, as δΩ tries to “heal” back to δΩ∞ . The equation for the evolution of the lag δΩ is obtained by combining Eqs. (8.20) and (8.23) with the expressions involving Ω˙ ∞ given above, and can be expressed in the dimensionless form 1 dy = 1 − sinh y, dx η
(8.26)
in terms of a dimensionless lag y ≡ δΩ/δΩ, and a dimensionless time co-ordinate x ≡ t/τ , the relevant timescale τ being defined by τ ≡ ˙ ∞ |) ≈ (Ic /I)(kT /Ep )(δΩcrit /Ωc )ts < 10−5 ts . In Eq. (8.26), (Ic /I)(δΩ/ |Ω ∼ η is the non-linearity parameter, defined as . Ep r|Ω˙ ∞ | η≡ exp . (8.27) 4Ωs v0 kT
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Thus, we expect the solution of Eq. (8.26) to have linear and nonlinear r´egimes of behavior, depending on the value of η. But just what is linearity in the context of vortex creep? The physical picture becomes clear when we recognize that the velocity scale for purely thermal motion of vortex lines through the unbiased random potential of Fig. 8.4 is given by vth ∼ v0 exp(−Ep /kT ), while that for the steady-state vortex creep dictated by the external spin-down torque is just v∞ , given by Eq. (8.24). The parameter η is now readily seen to be the ratio of the two velocity scales, η = (v∞ /2vth ), or, alternatively, that of the corresponding timescales, η = (tth /2ts ), where tth ≡ (r/2vth ) is the thermal transit timescale. Consider first what happens when η 1, which means v∞ vth , or, tth ts . The steady-state creep velocity required by the external torque is then small compared to the thermal velocity, and the system can easily set up such a creep rate with only a small bias or lag δΩ. This is easily seen by considering the steady-state value y∞ implied by Eq. (8.26), i.e., the value for which dy/dx = 0, which is obviously y∞ = sinh−1 η. In the η 1 limit, this reduces to y∞ ≈ η, which gives δΩ∞ ≈ ηδΩ. Thus, on a scale δΩ, the steady-state lag δΩ∞ is indeed small, and the η 1 r´egime can be appropriately called the linear r´egime of vortex-creep response. Consider now the opposite limit, η 1, in which v∞ vth , and tth ts . Now the external torques require a steady creep rate which is large compared to the thermal velocity, and the system’s response changes completely. It can set up such a creep rate only by creating a large bias or lag δΩ, which then dominates the whole situation: the system’s thermal processes are now driven strongly by the external torque, with interesting — and occasionally almost counter-intuitive — results [Alpar & Pines 1989], as we shall see below. This η 1 r´egime is the non-linear r´egime of vortex-creep response, in which the value of the the steady-state lag δΩ∞ is given, from the above expression, by δΩ∞ ≈ ln(2η)δΩ. We can express this steady lag in the form δΩ∞ ≈ δΩcrit [1 − (kT /Ep ) ln(2ts v0 /r)] with the aid of the definitions of η and δΩ given above: this form is, of course, identical to Eq. (8.25), showing that δΩ∞ ≈ δΩcrit , which is large on a scale δΩ, as anticipated above. From the discussion given below Eq. (8.25), we would expect most of the observed pulsars to lie in this r´egime, and this is indeed the case, as we shall presently see. The exact solution of Eq. (8.26), which can be readily obtained with the aid of standard tables [Gradshteyn & Ryzhik 1980], bear out the above conclusions in detail. Once again, we give only the essentials here. The initial condition to be satisfied by the required solution to the (first order)
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differential equation (8.26) is the specified value of the lag, δΩ0 , just after the glitch, i.e., at t = 0. This translates into the condition y = y0 ≡ δΩ0 /δΩ at x = 0. The constants appearing in the solution are then y0 and the steady-state value of y, y∞ = sinh−1 η, introduced above. The solution can be expressed in a compact and convenient form by defining a new dependent variable z as: z ≡ tanh y∞ tanh
y + sechy∞ , 2
(8.28)
With the aid of a bit of algebraic manipulation, which we leave as an exercise for the reader, the solution can be written as: exp(−x coth y∞ ) =
z − 1 z0 + 1 . , z + 1 z0 − 1
(8.29)
where z0 is the value of z at x = 0. The linear and non-linear limits of the solution (8.29) can now be extracted readily. In the linear r´egime, η ≈ y∞ 1, and we need to consider the limit in which the arguments of the hyperbolic functions in Eqs. (8.28) and (8.29) are all 1, so that tanh u ≈ u and sech u ≈ 1 − u2 /2 to the leading order. The algebra is straightforward, and the reader can easily show that Eqs. (8.28) and (8.29) respectively reduce to x y − y∞ y∞ (y − y∞ ), and exp − (8.30) ≈ z ≈1+ 2 y∞ y0 − y∞ in this limit. Rewriting the second equation in (8.30) as x y = y∞ + (y0 − y∞ ) exp − , η
(8.31)
we clearly see the characteristics of the solution in this manifestly linear r´egime. First, it depends linearly on the initial perturbation y0 , as it must [Alpar & Pines 1989]. Next, it is a simple exponential approach in time from the initial value y0 to the steady-state value y∞ , as shown in Fig. 8.5. Finally, the characteristic time of approach in this linear r´egime, τl = ητ , is shortened by a factor η with respect to the timescale τ introduced earlier: we shall return to this point below. The algebra involved in extracting the non-linear limit, η 1, is equally straightforward. Now the arguments of the hyperbolic functions in Eqs. (8.28) and (8.29) are all 1, so that the appropriate expressions
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Fig. 8.5 Internal torque vs. time (t), showing the characteristic Fermi-function healing. See text for detail. Reproduced with permission by the AAS from Alpar et al. (1984): see Bibliography.
to the leading order are tanh u ≈ 1 − 2 exp(−2u) and sech u ≈ 2 exp(−u), and Eqs. (8.28) and (8.29) respectively reduce to z ≈ 1 + 2 e−y∞ − e−y ,
and e−x ≈
e−y − e−y∞ e−y0 − e−y∞
(8.32)
We rewrite the second equation in (8.32) as exp(−y) = exp(−y∞ ) + [exp(−y0 ) − exp(−y∞ )] exp(−x),
(8.33)
and note the characteristics of the solution in the non-linear r´egime. First, it has a highly non-linear dependence on the initial perturbation y0 , as expected. Next, it has a much more complicated time-dependence, to which we shall return presently. Finally, the characteristic time of approach in this non-linear r´egime is just τnl = τ , the timescale introduced earlier3 . A remarkable piece of physics emerges from considerations of the response timescales in the linear and non-linear r´egimes, as has been emphasized by Alpar and Pines (1989). From the definitions of τ , η, ts and tth given above, we can write the timescale in the linear r´egime as τl = (1/2)(Ic /I)(kT /Ep )(δΩcrit /Ωc )tth , and that in the non-linear r´egime as τnl = (Ic /I)(kT /Ep )(δΩcrit /Ωc )ts . Apart from the common factors of −3 −2 (see above), then, the Ic /I ≈ 1, kT /Ep < ∼ 10 , and δΩcrit /Ωc < ∼ 10 3 Note that the non-linear limit, Eq. (8.33), can be obtained from the linear limit, Eq. (8.31), by replacing y with exp(−y) everywhere, and similarly for y0 and y∞ , and also replacing x/η by x in the exponential evolution term. This shows the formal similarity between the two limits; note also that z is close to unity in both limits.
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linear response timescale is determined by thermal timescale tth and the non-linear response timescale by the spin-down timescale ts required by the external torque. This is, of course, consistent with our earlier comment that the non-linear system is driven by the external torque. But the remarkable feature becomes clear when we compare the temperaturedependence of the two timescales. The linear response time scales as τl ∝ (kT /Ep ) exp(Ep /kT ) (remembering that tth = (r/2vth ) = (r/2v0 ) exp(Ep /kT )), so that its behavior is completely dominated by the exponential factor, τl becoming rapidly shorter as T increases, i.e., faster relaxation for hotter systems. This is, of course, the expected behavior of thermal relaxation processes. By contrast, the non-linear response time scales as τnl ∝ kT /Ep , implying faster relaxation for colder systems. This appears so counter-intuitive as to be disturbing at first, until we realize that the thermal processes are strongly driven by the external torque in this extreme non-linear limit, as indicated above. The colder the system is, the larger is the steady-state lag or bias required to drive it, as shown earlier, and, consequently, the faster is the relaxation towards this steady state. We have shown above that the linear response is an exponential healing of the lag, so the question now is: is there a correspondingly simple way of characterizing Eq. (8.33) for the non-linear response? It turns out that there is such a way, but in terms of the spindown rate, rather than directly in terms of the lag. If we express the spindown rate of the c-component, Ω˙ c (which is the observable spindown rate of the star) in terms of its steadystate value Ω˙ ∞ , their ratio shows a simple behavior in terms of the Fermi function introduced in Eq. (2.3). The derivation of this result with the aid of Eqs. (8.20), (8.26) and (8.33), as well as the relations given above Eq. (8.24), is straightforward, and we leave it as an exercise for the reader, quoting only the final result: ˙c 1 Is Ω
. =1+ ˙ I Ω∞ c 1 + exp t−t0 τnl
(8.34)
Here, t0 is an offset time [Alpar & Pines 1989], which we define as t0 ≡ ˙ ∞ |, and whose physical significance we discuss below. Note (δΩ∞ − δΩ0 )/|Ω first a simplification which we have used in the last step for obtaining Eq. (8.34). If we compare the orders of magnitude of t0 and τnl , we see ˙ ∞ | ≈ δΩcrit /|Ω ˙ ∞ | = (δΩcrit /Ωc )ts , while τnl is given by that t0 ∼ δΩ∞ /|Ω the expression in the previous paragraph. Thus, τnl /t0 ∼ (Ic /I)(kT /Ep ) < ∼
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10−3 (see above), i.e., t0 τnl , which we have used to simplify the righthand side of Eq. (8.34) into the form in which it appears above. To see the physical meaning of the offset time t0 , consider Eq. (8.34), i.e., dimensionless spindown rate vs. time. At the glitch-point, the spindown rate jumps to a value which is (1 + Is /Ic ) times the steady-state rate, and stays essentially constant at this value until a time ∼ t0 after the glitch. After this, it heals to the steady-state value on a timescale τnl , following the shape of the Fermi function. What happens physically is that the glitch leads to an almost complete stoppage of vortex creep at first. The crustal superfluid (the s-component) unpins, and so is uncoupled from the external torque, which then acts solely on the c-component for a time ∼ t0 , raising the spindown rate for this duration. As the above definition of t0 shows, the lag comes back close to δΩ∞ at the end of this duration, and the s-component repins on a timescale τnl . Vortex creep starts again, and the system again approaches a steady state. This persistent shift in the spindown rate following a glitch is thus a signature of strongly non-linear vortex-creep response.
8.5
Stellar Parameters from Glitch Data
Can we attempt to determine structural parameters of neutron stars, and/or parameters of the essential microscopic physics of their crusts and crustal superfluids, from the data on pulsar glitches in the context of the vortex pinning model described above, in analogy with what was attempted in the 1970s in the context of the starquake model (see Chapter 7)? The answer is yes, and we now indicate some results we get from such attempts. Consider first the diagnostic value of observed glitch “jumps” in essential ˙ Ω. ˙ The former jump does not, by itself, quantities, e.g., ∆Ω/Ω and ∆Ω/ give an essential structural parameter, unlike what happened for the starquake model, but it can yield valuable results in conjunction with other glitch data, as we shall argue below. ˙ Ω ˙ in the spindown rate is, however, a direct diagnostic in The jump ∆Ω/ the vortex-pinning model, since we have shown above that this jump is Is /Ic in this model, and so a direct handle on the fractional moment of inertia of ˙ Ω ˙ generally lie in the pinned crustal superfluid. The observed values of ∆Ω/ −3 −2 the range 10 – 10 , the large, Vela-type glitches, which constitute the majority, being associated with the largest values of the spindown jump. This indicates the typical moment of inertia of the crustal superfluid to be
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Is Is ∼ 10−2 , ≈ Ic I
(8.35)
which is a very interesting result. The above result is directly supported by the recent diagnostic found by Lyne, Shemar and Smith (2000), as described in Chapter 7, namely, that, averaged over the observation history of the known glitching pulsars, a fraction ≈ 1.7 × 10−2 of the spindown amount is reversed by the spinup due to glitch activity. In the vortex-pinning model, the latter spinup is due to the decoupling of the crustal superfluid’s moment of inertia Is from the external torque because of unpinning, and hence the above fraction is Is /I. This yields Is ≈ 1.7 × 10−2 , I
(8.36)
in agreement with Eq. (8.35). In fact, the above result can be combined with the observed values of ∆Ω/Ω to obtain diagnostics of the critical lag δΩcrit , as follows [Lyne, Shemar & Smith 2000]. As the lag δΩ increases with continuing spindown and reaches its critical value, the crustal superfluid unpins suddenly, transferring part or all of its excess angular momentum Is δΩcrit to the crust rapidly, causing a glitch, i.e., a jump ∆Ω in the stellar angular velocity. Hence a lower limit on the critical lag is given by: I ∆Ω ∆Ω δΩcrit −5 > ≈ 6 × 10 , (8.37) Ω ∼ Is Ω Ω −6 where we have used Eq. (8.36) in obtaining the second form of the righthand side, and (∆Ω/Ω)−6 is the glitch size in units of 10−6 , as before. For a value Ω ∼ 102 rad s−1 typical of pulsars like Vela or Crab, this yields a −1 lower limit δΩcrit > ∼ 0.006 rad s on the critical lag. We shall come back to this result below. Consider now the diagnostics possible from the observed post-glitch response of pulsars, and note first that it is not always easy to distinguish between linear and non-linear response of the vortex creep observationally. There is usually an uncertainty of typically a few weeks in determining the exact glitch-point: if this is ∼ t0 , a non-linear response may masquerade as a linear one, since, as Fig. 8.5 shows, the non-linear Fermi-function healing of the spindown rate can look remarkably similar to a linear, simpleexponential healing if the observations started ∼ t0 days after the glitch.
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Further, a given observation of post-glitch relaxation may require several components, linear and non-linear, for an adequate fit, and different combinations of components may work equally well. However, the persistent shift in Ω˙ c , which we argued above to be the most convincing signature of non-linear response, does appear to be present in all pulsars for which post-glitch relaxation has been studied in detail, most notably the Crab and Vela pulsars, PSR 0355+54, and PSR 0525+21. Thus, all pulsars may have non-linear response at least over some parts of their crusts, which brings us to the question of the transition point between linear and non-linear r´egimes. By definition, this point is given by η = 1, which, with the aid of the discussion below Eq. (8.27), means tth = 2ts , or, exp(Ep /kT ) = 4v0 ts /r. Using r ∼ R, the stellar radius, we can write the transition value of Ep /kT as:
Ep kT
= ln trans
4v0 ts R
≈ 30.2 + ln t4 + ln
v7 R6
,
(8.38)
where t4 is ts in units of 104 years, v7 is v0 in units of 107 cm s−1 , and R6 is R in units of 106 cm [Alpar & Pines 1989]. Only those pinning layers which have Ep /kT greater than this transition value will support non-linear response. Note that the transition criterion depends only logarithmically on the pulsar’s (spindown) age ts , but linearly on the temperature T of the inner crust: this determines the effects of aging on the extent of non-linear response in the following, straightforward way. If T remained constant as a pulsar ages, the transition value of Ep would increase slowly, so that, for a given profile of Ep vs. layer density, such as given in Fig. 8.3, the extent of the region of non-linear response would decrease slowly. However, the effects of cooling swamp this trend completely, since Ep decreases linearly ˙ and so on with T , and T has a power-law or similar dependence on |Ω| ts . Thus, the transition value of Ep actually decreases as the pulsar ages, so that the extent of the non-linearly responding region increases. Physically, as the aging pulsar becomes colder, thermal processes rapidly lose their efficiency, so that strong biases and non-linear response are required over ever-increasing portions of the pinning region in order to meet the demands of the external torque (see above), even though this torque itself is decreasing with time. Younger pulsars like Crab and Vela do, indeed, seem to possess some linearly responding regions; it is possible that these regions shrink with age, becoming negligible in old pulsars [Alpar & Pines 1989]. In what follows, we confine ourselves to non-linear response.
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The major diagnostic comes from the response time τnl : combining the earlier expression τnl = (Ic /I)(kT /Ep )(δΩcrit /Ωc )ts with Eq. (8.16), we obtain τnl =
kT 1 1 . . . ˙ |Ω∞ | ρκr aξ
(8.39)
If we have an observational limit on T , or a theoretical estimate (see below), a knowledge of τnl therefore gives a value of aξ, which constrains the physics of superfluidity and pinning. Alternatively, if we have a model for the latter, and so can calculate aξ, we get an estimate of the temperature T of the inner crust and so of the isothermal core of the neutron star (note that the inner crust becomes part of this isothermal core within ∼ 500 years of the birth of the neutron star). An illustrative example given by Alpar and Pines (1989) is instructive, and we sketch it briefly. If pinning is weak, we can use the statistical arguments for estimating a given in Sec. 8.3.2 to obtain a more explicit diagnostic form of Eq. (8.39), as we shall now show. But the question is: is pinning weak in observed pulsars? These authors argued that it was, since observed upper limits on the critical lag obtained are δΩcrit < ∼ 0.7 rad > 0.006 s−1 (see below). When combined with the recent lower limit δΩcrit ∼ rad s−1 [Lyne, Shemar & Smith 2000] described above, it does suggest that the expected values of the critical lag are much smaller than the early theoretical values described earlier, and far too small to correspond to such strong pinning forces as would be able to “pull” nuclei out of their lattice sites. Thus, pinning is probably in the weak r´egime over most of the pinned superfluid in the glitching pulsars. We can then use Eq. (8.14) for the spacing a between successive pinning centers along the vortex line, and cast Eq. (8.39) in a useful diagnostic form with aid of Eq. (8.7) for the coherence length ξ, the definition of the vorticity quantum κ given earlier, the rough estimate of the lattice spacing b given in the paragraph above Eq. (8.11), which we simplify here as ρb3 ∼ (3A/4π)mB , and a little straightforward algebra. The form is: T6 t4 kF (fm−1 ) days, EG (MeV)Ω2 R6 (n)
τnl ∼ 40
(8.40)
where T6 is T in units of 106 K, Ω2 is Ω in units of 102 rad s−1 , other symbols have their usual or previous meanings, and we have used a representative value A ∼ 200 for the inner crust. Equation (8.40) is a valuable, direct handle on the superfluid energy gap EG if T can be estimated independently, or vice versa.
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Finally, a diagnostic connection between the critical lag δΩcrit and the surface temperature Ts of the neutron star may be possible for old pulsars, in the following way. Vortex creep is a dissipative process, the major source of dissipation being the excitation of long-wavelength Kelvin waves, particularly at low creep velocities. Dissipation of energy during steady-state ˙ ∞ |. This rate would, creep occurs at a rate E˙ dis = Is δΩ∞ |Ω˙ ∞ | ≈ Is δΩcrit |Ω in general, be added to that of loss of the initial thermal energy content of the neutron star to account for the radiative loss rate 4πR2 σTs4 through the surface, and the two contributions would be difficult to separate. However, for old pulsars, we can argue that the initial thermal energy has already been radiated away, and equate the above dissipation rate to the above radiation rate. This yields R62 T64 t6 2πσR2 Ts4 ts δΩcrit Is ∼ 10−2 , ≈ Ωc Erot I E49 IIs −2
(8.41)
where Erot ≡ (1/2)IΩ2c is (roughly) the rotational energy of the star. In Eq. (8.41), T6 is Ts in units of 106 K, t6 is ts in units of 106 years, E49 is Erot in units of 1049 ergs, and the subscript −2 applied to (Is /I) means its value in units of 10−2 . Although the above equality of rates remains in some doubt, using Eq. (8.41) to convert upper limits on the thermal luminosities of radio pulsars to upper limits on the critical lag is a more secure procedure, as the reader can readily show by considering the nature of the inequalities. This has been done for PSR 1929+10, and has yielded −1 the result δΩcrit < ∼ 0.7 rad s quoted above. 8.6
Glitches: Recent Developments
The treatment of vortex pinning described above implicitly assumed a single nuclear lattice extending over the whole inner crust, since the vortexnucleus interaction described by Eq. (8.12) was just summed over the inner crust to get the total pinning energy or force. This amounts to viewing the entire inner crust as a giant single crystal, which is, without doubt, a great oversimplification. As the newborn neutron star cools, we expect, according to the standard theory of homogeneous nucleation, that the crystallization of the fluid of heavy, neutron-rich nuclei would start from a metastable, supercooled state of the liquid (typically, at ∼ 0.8 times the melting temperature of the Coulomb lattice that would form from it), thereby forming a polycrystalline structure. The typical size of the individual crystals would
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depend on such details as the cooling rate of the star and the growth velocity of the crystalline phase, and the orientations of their lattices would be in random directions with respect to vortex lines, if crystallization occurred before the dripped-neutron fluid underwent its phase transition to superfluidity. In the 1990s, Jones considered the effects of these complications, reassessing the strength of the pinning force (see Jones 1998 and the references therein), and so its relevance for pulsar glitches. Jones (1998) described a vortex line in the above polycrystalline structure as the z-axis, say, of a vortex-fixed reference frame passing through a series of crystals, such that the orientation of the lattice of the i-th crystal was specified by its set of reciprocal lattice vectors gi (see, e.g., Kittel 1970) referred to this frame, and the intercept of the vortex line in this crystal was of length bi . He then expressed the vortex-nucleus interaction potential energy (see Eq. [8.12]) in terms of a potential U per unit length of the vortex, and worked with the Fourier components Ug of this potential in terms of the reciprocal lattice vectors g, i.e., U (r) = g Ug exp(ig.r). We refer the reader to the original Jones (1998) paper for the details of the calculation, quoting only his final estimate for the pinning force fp per unit length in terms the above Fourier component, and an appropriate average value b of the intercept length over individual crystals, as fp ∼ |Ug |/b. From the above result, Jones argued for the inadequacy of the pinning force in the following way. First, he calculated the Fourier components Ug of the vortex-nucleon interaction described earlier, showing that |Ug | as a function of density had a profile qualitatively similar to that of Ep described earlier, i.e., that it increased with increasing density at first, passed through a maximum, and decreased rapidly thereafter with increasing ρ. But the maximum of |Ug | was at ρ ∼ 3 × 1013 g cm−3 ∼ ρnm /10. From this, and using a plausible value b ∼ 5 × 104 fm for the average linear size of a single crystal in the polycrystalline structure, Jones (1998) found that the pinning force was roughly adequate at the above maximum of |Ug | to balance the Magnus force inferred from the large glitches observed in the Vela pulsar, but that the pinning force decreased rapidly as the density increased (since |Ug | decreased rapidly, as explained above), falling far short of the inferred Magnus force in these regions. As the latter regions contain most of the moment of inertia Is of the s-component (see above), Jones concluded that the vortex-nucleus interaction for single crystals in the polycrystalline structure appeared to be too weak to account for the observed large glitches. Further, argued Jones, pinning in an exceptionally large single crystal may develop instabilities due to the formation of
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“kinks” in the vortex line. Finally, Jones pointed out that multi-crystal statistical effects can only serve to reduce the total pinning force from a simple sum of the forces for the individual crystals, due to any cancellations that may occur because of the different orientations of different crystals. His conclusion, therefore, was that vortex pinning in the neutron-star crust was generally inadequate for explaining the observed glitches. The Jones criticism must be understood better and verified independently, but it emphasizes, at the very least, the need to explore (a) other possible sites of vortex pinning in neutron stars, and, (b) other possible glitch mechanisms. The fluid core of the neutron star, where the neutrons form a rotating superfluid sustaining quantized vortex lines, as described earlier, and the protons form a type II superconductor sustaining quantized tubes of magnetic flux, had been suggested since the mid 1980s as a possible site of pinning between vortex lines and flux tubes. We shall come back to this idea in Chapter 13, in connection with the evolution of magnetic fields of neutron stars. It seems fair to say that the possibility of pinning occurring in this manner is a fascinating one, if somewhat uncertain, but its role in producing the observed pulsar glitches through the canonical mechanism of sudden, large-scale unpinning is far more questionable. However, it is not impossible to imagine scenarios in which an interaction of an appropriate kind between vortex lines and flux tubes in the core of a neutron star is instrumental for other, alternative glitch mechanisms operating in the crust. An example of this is the class of models explored by Ruderman and co-authors in recent years [Ruderman et al. 1998]. These authors envisage that, when the vortex tubes in the superfluid core move outward because the star is spinning down and losing angular momentum (as they also do in the crustal superfluid if they are not pinned, as we explained earlier) they push on the magnetic flux tubes, thereby either (a) forcing the latter to move with them, or, (b) “cutting” through the latter, if these cannot respond fast enough. This is reminiscent of pinning and unpinning, but it is not quite the same phenomenon. The critical velocity vc which marks the transition between the above −1 two r´egimes is estimated [Ruderman et al. 1998] to be vc = 10−6 βΩ2 B12 cm s−1 , where Ω2 is the pulsar’s angular velocity in units of 102 rad s−1 , and B12 is its magnetic field in units of 1012 G, as before. The parameter β ∼ 1 depends on the (uncertain) physics of the interaction between vortex lines and flux tubes, to which we shall return later. Since a vortex line at a perpendicular distance r⊥ from the stellar rotation axis moves radially outward with a velocity vV ∼ r⊥ /ts due to spindown on a timescale ts
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(compare Eq. [8.24]), the above critical velocity implies a critical radius, −1 cm, such vortex lines inside rc move outward slowly rc ∼ 106 t4 Ω2 B12 enough to be able to carry all flux tubes with them, while the faster-moving ones outside rc have to cut through flux tubes. Here, t4 is ts in units of 104 years, as before. Thus, for Vela-like and older pulsars, which have t4 > ∼ 1, the stellar radius, which means that all magnetic flux in such pulsars rc > ∼ can be pushed out from the core by the moving vortex lines. By contrast, young Crab-like pulsars, with t4 ∼ 0.1, have rc ∼ 1/10 of the stellar radius, showing that most of the magnetic flux in these pulsars cannot be efficiently pushed out of the core by the vortex lines. What happens to the magnetic flux thus pushed out of the neutron star’s core and into the inner crust? According to the view of Ruderman et al. (1998), it can play an essential role in both the evolution of the magnetic field of the neutron star and the glitch phenomenon. We return to the former point below, and indicate the latter here in brief, referring the reader to the original paper [Ruderman et al. 1998] for details. As the expelled magnetic flux collects in the inner crust, the magnetic field strength and associated magnetic stress increases there. When this stress exceeds the yield stress of the crust material, corresponding to the maximum possible strain, θmax ∼ 10−3 , that the crustal material can withstand, the crust relieves the stress by breaking or by undergoing plastic flow or creep. As long as the crusts of young, rotation-powered pulsars are still warm, plastic flow is possible. As the pulsar ages, however, the crust cools and becomes brittle, so that breaking is the only available means of relieving stress. The transition to brittleness is expected to occur at a temperature Tb , which is related to the melting temperature Tm of the Coulomb lattice constituting the crust roughly as Tb ∼ 0.1Tm ∼ 108 K. The crusts of pulsars even as young as the Crab are expected to have cooled to such temperatures already, not to mention those of the older Vela-like, or even older, pulsars. Thus, crust “cracking” is the only possibility for these pulsars, and the connection with glitches is immediately obvious, in view of our earlier account of starquakes. Indeed, in this scenario for glitches by crust-cracking, vortex unpinning in the crust, as described above, may still play a role, since such sudden crustal movements may well trigger an unpinning event. Various trigger mechanisms for glitches have been studied over the years.
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Chapter 9
Properties of Accretion Powered Pulsars
We give an overview of the basic observational properties of binary accretion-powered pulsars in this chapter, including their binary, pulse, and timing properties, as well as their major spectral characteristics in the X-rays. As we described in Chapter 6, the evolutionary history of these binary pulsars naturally accounts for the classes of binaries that contain the observed accretion-powered pulsars, viz., (1) low-mass X-ray binaries (LMXBs), in which the companions of the accreting neutron stars have low (∼ 0.1 − 2M , say) masses, and (2) high-mass, or massive, X-ray binaries (HMXBs), in which the companions have high (∼ 8 − 30M , say) masses. The latter class is the further subdivided into (a) binaries containing O/B supergiant companions, with the highest masses of this range, ∼ 15−30M, say, and, (b) binaries containing Be stars, with more moderate masses, ∼ 8 − 15M , say. Accretion in LMXBs proceeds almost invariably through Roche-lobe overflow (for discussions of Roche lobes, see Chapter 6 and Appendix B), while that in HMXBs proceeds predominantly through stellar winds driven by the massive companions, although a combination of stellar wind and Roche lobe overflow is also possible, if the orbital size of the HMXB is small enough. In addition, Be-star systems are almost always transient, and their binary parameters (see below), when measured, almost always reveal a considerable orbital eccentricity. After summarizing the basic binary characteristics of accretion-powered pulsars below, we focus on the essential properties of the accreting, rotating, and pulsing neutron stars in these systems, taking up, in turn, pulse profiles, secular and “noisy” variations in the pulse periods, and essentials aspects of the X-ray spectra, including the diagnostically valuable fluorescence lines of iron and other elements, and equally valuable cyclotron features. We also give a brief account of the highly interesting phenomenon of quasi-
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periodic oscillations (QPOs) in accreting X-ray binaries, which were first detected in their earliest known form in the mid-1980s, whose occurrence in various frequency ranges in the power spectra of LMXBs became clear in the 1990s, and whose potential as deep diagnostic probes into the close environs of neutron stars, where relativistic gravity plays an essential role, is widely appreciated now.
9.1
Binary Characteristics
Binary orbits of some well-known accretion-powered pulsars are shown in Fig. 9.1.
Fig. 9.1 Orbits of accretion-powered pulsars, drawn to scale. Companion stars (shaded circles) also drawn roughly to scale. Left Panel: low-mass companions. Right Panel: massive companions. Note that Her X-1, which has an intermediate-mass companion, is in both panels, appropriately expanded in the left panel, whose scale corresponds to the small orbital sizes of LMXBs, as indicated. Reproduced with permission by Annual Reviews from Bradt & McClintock (1983): see Bibliography.
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The five Keplerian parameters introduced in Chapter 7, i.e., (1) the binary period Pb , (2) the eccentricity e, (3) the projected semi-major axis of the neutron star1 ans sin i, (4) the longitude of periastron ω, and, (5) the time of periastron passage T0 , and, in addition, the mass function f (m1 , m2 , sin i) defined by Eq. (7.33), are also the ones normally listed in tables of binary parameters of accretion-powered pulsars. See, for example, the compilation by Bildsten et al. (1997), which lists these parameters for 18 such binary systems. LMXBs have circular, or very nearly circular, orbits, Be-star systems have very eccentric orbits, as indicated above, and O/B supergiant systems can have both low and high eccentricities. How do we determine these binary parameters? The answer has, of course, been given already in Chapter 7. First, we transform the TOAs of the pulses from the topocentric to the barycentric frame, as described in Chapter 7. The added complication for accretion-powered pulsars is that their observations are made by our X-ray detectors which are not on the Earth’s surface, unlike the radio telescopes used for observing rotationpowered pulsars, but rather on board an X-ray satellite, which is in orbit around the Earth. So the effects of the satellite’s motion around the Earth must be subtracted, in addition to those of the Earth’s motion around the Sun. The extra work is relatively easy, because an accurate tracking of our satellite (which is of our own making and under our control) is a standard, routine procedure. Next, we need to further transform the solar-system barycentric TOAs to the proper time T in the rest-frame of the accretion-powered pulsar, compensating for the binary orbital motion of the pulsar, as described in Chapter 7. This is formally still described by Eq. (7.27), but much simplification now obtains for accretion-powered pulsars, because we need normally keep only the Rømer delay term for them. This may mystify the reader at first. Surely, the Einstein and Shapiro effects, as described in Chapter 8 for rotation-powered pulsars, do not cease to exist for accretion-powered pulsars? Indeed, they do not, but it is not practicable to measure them in accreting X-ray pulsars, as these subtle relativistic effects are completely swamped by much larger non-relativistic effects of purely astrophysical origin, due to the presence of (a) normal stellar companions which are losing 1 Note
that we shall use ans and a1 interchangeably for the neutron star’s semimajor axis in this book. As the observations of accretion-powered pulsars are done predominantly in the X-rays, and as almost all the information on the neutron star’s orbital motion in these systems comes from X-ray observations, it is customary to denote the above length also by ax , which we may use as necessary.
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mass, and, (b) the interaction of the accreting matter with the neutron stars that are accreting it. Let us consider the situation in more detail. The Rømer delay is not really a relativistic effect, but rather a lighttravel time effect, which scales as a1 sin i/c ∼ Pb (v/c), as indicated earlier. By comparison, the truly relativistic effects, namely, the Einstein and Shapiro delays, are orders of magnitude smaller, as we indicated in Chapter 7. We pointed out there that the latter delays could still be measured separately for rotation-powered pulsars because the precision of TOA measurements was high enough. But an assumption is implicit in such a statement, namely, that these minute relativistic effects are not swamped within the binary itself by gross, non-relativistic, astrophysical effects which are difficult to calculate precisely. Otherwise, precise TOA measurements are of little consequence. The above assumption fails for accretion-powered pulsars: in contrast to the proverbially “clean” system that a double neutron-star binary is (see Chapter 7), an accreting X-ray binary is a remarkably “dirty” system. The gravitational interaction between the neutron star and its normal companion is complicated by the latter’s internal structure, which causes tidal torques and apsidal motion ω˙ even in Newtonian gravity. These effects, as well as mass-loss from the system, can change its binary period Pb at rates which are several orders of magnitude higher than the generalrelativistic orbit decay rate described in Chapter 7. (For example, Cen X-3 has P˙b /Pb ≈ −1.8 × 10−6 yr−1 , which can be compared with the generalrelativistic rate P˙bGR /Pb ≈ −2.7 × 10−9 yr−1 for PSR 1913+16, described in Chapter 7.) Finally, matter accreting on the neutron star disturbs its rotation, by exerting an accretion torque (see Chapter 12) which is extremely “noisy” in character, and so causes large intrinsic residuals in the TOA: we shall return to this point later. Thus, it is not the precision of TOA measurement that hinders us in the case of accretion-powered pulsars; indeed, the precision of measurement of TOAs of individual photons by modern X-ray detectors (large proportional counters, say) is typically a few µs. Rather, it is the “dirt” in these systems, i.e., their astrophysical complexities. But these complexities are extremely valuable, as we shall see, as they make it possible for us to study a wealth of fascinating and instructive astrophysical phenomena, which a “clean” double neutron-star binary never could. The form used for Rømer delay in timing accretion-powered pulsars can be expressed as ∆R = (a1 sin i/c)F (e, ω, T0 , φ), in terms of the projected semi-major axis a1 sin i, and a function F of the orbital phase, the orbit
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parameters e, ω, T0 being the parameters of F . The orbital phase can be expressed in terms of either the mean anomaly, φ ≡ 2π(t − T0 )/Pb , or the true anomaly θ, the relation between these being given in Appendix B. In terms of these, the function F is given [Nagase 1989] implicitly by sin(θ + φ) . F (e, ω, T0 , φ) = 1 − e2 1 + e cos θ
(9.1)
The fitting procedure is as described in Chapter 7. The pulse phase of the pulsar in its own rest frame is expressed by Eq. (7.8), and the parameters of the fit are the spin parameters ν, ν˙ (and also ν¨, if necessary) of the pulsar (or, equivalently, P, P˙ , P¨ , where P is the spin period of the pulsar), and the Keplerian orbit parameters described above. 9.1.1
Displaying Binary Systematics
There is a very instructive way in which we can display the binary systematics of accretion-powered pulsars, viz., in a plot of the spin period P vs. the orbital period Pb , as shown in Fig. 9.2, which has been very popular since the pioneering work of Corbet (1984), and which we shall call the Corbet diagram. Corbet (1984) noticed a strong, positive correlation between P and Pb of Be-star systems on these plots, which is clearly visible in Fig. 9.2, and which can be roughly described by a power-law relation P ∝ Pbα , with the exponent in the range α ∼ 1 − 1.5. By contrast, the other systems show either weak or no correlation. The supergiant systems can be further divided into two subclasses. In the first class, the supergiants fill their Roche lobes (see Appendix B): these, naturally, have to be rather close binaries (i.e., have to have rather short orbital periods, Pb < ∼ 4 days) and therefore have rather high accretion rates and luminosities. In the second class, the supergiants lie well inside (or underfill, as the commonly used expression in the subject is) their Roche lobes: these, naturally, have to be rather wide binaries (i.e., have to have rather long orbital periods, Pb > ∼ 4 days) and therefore have rather low accretion rates and luminosities. The Roche lobe-filling systems have rather short spin periods (P < ∼ 10 s), while the lobe-underfilling systems have very long spin periods (P > ∼ 100−1000 s). On the Corbet diagram, there appears to be no correlation whatever between P and Pb for the second class, i.e., the lobe-underfilling supergiants. For the first class, i.e., the lobe-filling supergiants, there may be a negative correlation between P and Pb , but this is not clear.
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Fig. 9.2 Corbet diagram, or pulse period (P ) vs. orbital period (Pb ) plot for accretionpowered pulsars. Shown are: Roche-lobe overflow systems (crosses), supergiant systems (open circles), Be-star systems (filled circles). Also shown for comparison are rotationpowered pulsars with Be-star companions. Dashed line: P − Pb relation proposed by Corbet (1984). Reprinted with permission by Springer Science and Business Media from van den Heuvel (2001) in The neutron star-black hole connection, eds. C. Kouveliotou c 2001 Kluwer Academic Publishers. et al..
For LMXBs, there used to be too little data to make meaningful statements on their behavior in the Corbet diagram (White et al. 1995), since they displayed no pulsations. With the discovery of the first accretionpowered millisecond pulsar SAX J1808-3586 in 1998 (see Chapter 7), and of six more such pulsars since then, the situation has become more interesting, but not simpler. The new accretion-powered millisecond pulsars, widely believed to be the “missing links” (see Chapter 6) between the usual, accretion-powered, galactic LMXBs, from which pulsations have not been detectable so far due to various speculated reasons, and the recycled, millisecond rotation-powered pulsars (which we described in Chapter 6), occupy the bottom left-hand corner of the Corbet diagram, i.e., a region which is very distinct and far away from that occupied by all other accretion-powered pulsars. This is not surprising when we compare this region with that occupied by the recycled, millisecond rotation-powered pulsars in binaries in the Corbet diagram.
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The natural explanation of the systematics of the Corbet diagaram is in terms of the torques that act on accreting, rotating neutron stars, since it is these torques that determine the long-term behavior of their spin periods: we shall discuss these in Chapter 12. 9.1.2
Newtonian Apsidal Motion
As we mentioned above, there is significant apsidal motion, i.e., advance of the periastron, in an eccentric binary even within the confines of Newtonian gravity, if the companion of the neutron star is a normal star, either on the main sequence or evolving off it. This is the “classical” apsidal motion, the rate of which is determined by the structure of the companion, and which is, therefore, an excellent probe of this structure. The value of the apsidal motion test as one of the few direct proving grounds of our models of stellar interiors was originally demonstrated and stressed by Russell in 1928 in connection with “classical” binaries containing two normal stars, before the idea of neutron stars was concieved (see Chapter 1), and detailed theory of apsidal motion was given by Cowling and by Sterne in the 1930s. For a superb account of the essential physics and the basic quantitative description, we refer the reader to Schwarzschild’s (1958) classic text Structure and Evolution of the Stars. In brief, the first star of the binary distorts the second’s structure by tidal forces, i.e., forces due to the slightly different gravitational attractions of the first on different parts of the extended structure of the second, creating a non-spherical shape. This modifies the mutual gravitational force between the stars from a simple inverse-square one, which would lead to a Keplerian elliptical orbit, to a more complicated one, which leads to a more complicated orbit, decsribable as a Keplerian ellipse whose major axis — or the “line of apses”, to use an old-fashioned name — precesses slowly. This is apsidal motion. To complete the story, we need only appreciate that the second star must also produce a similar tidal distortion of the first, adding to the above motion. This is the total apsidal motion. The mathematical decsription proceeds by noting that the lowest-order distortion in the stellar structure is a quadrupole term, which adds a term v2 P2 (cos θ)/r3 to the usual Keplerian r−1 potential (r being the distance between the centers of the two stars, and θ the angle measured with respect to the line of centers), and so causes apsidal motion. The rate ω˙ of apsidal advance is, of course, proportional to the strength of v2 of this additional term, and a suitable dimensionless form of it is called the apsidal motion
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constant , k. This constant k is, in turn, determined by the strength a2 of the above quadrupolar distortion term, the calculation of which involves the unperturbed stellar structure [Schwrazschild 1958]. This is how2 stellar structure enters the problem, and k can therefore be looked upon as a sort of “one-parameter description” of the stellar density profile [Kelley et al. 1981]. Indeed, for homogeneous stellar models, the apsidal motion constant crudely scales with the central condensation of the star as k ∼ (ρc /ρ)−1 , where ρc is the central density and ρ the average density [Stothers 1974]. Values of k for main-sequence stars were computed by Kushwaha and Schwarzschild in the 1950s, and by numerous other authors in the 1960s and ’70s, by numerical integration of Radau’s equation (see footnote). Schwarzschild’s (1958) values were in the range −2.1 < ∼ log k < ∼ −1.7 for < 10, with k increasing slowly with the stelstellar masses 2.5 < M/M ∼ ∼ lar mass. The work on apsidal motion in accretion-powered pulsars in the 1980s that we describe below utilized Stothers’ (1974) computations, which used the most recent opacity tables available at the time. Stothers (1974) < log k ∼ < −2.0, for stellar masses found somewhat lower values, −2.4 ∼ < 25, with k increasing slightly with mass at first, passing 2 < M/M ∼ ∼ through a maximum at M ∼ 10M and decreasing slightly thereafter. For stars evolving off the main sequence, k decreases, as we would expect from the scaling argument given above, since stars become more centrally condensed, i.e., ρc /ρ increases, during this evolutionary expansion. In an accretion-powered binary pulsar consisting of a neutron star and a normal companion, only one of the two above parts of the total apsidal motion is significant, namely, that due to the distortion of the companion by the neutron star. Any tidal distortion of the neutron star is utterly negligible; indeed, it can be regarded as a point mass in all considerations of binary orbital dynamics. The rate of periastron advance is given by 5 mns 1 + 32 e2 + 18 e4 2πk Rc 15 , (9.2) ω˙ = 5 Pb a mc (1 − e2 ) where Rc is the radius of the companion, and a is the full semi-major axis of the orbit. This means that a = ans +ac is the sum of the semi-major axes of 2 Actually,
k depends on the values of a2 (r) and its radial first derivative at the stellar surface, and a2 (r) satisfies a simple, second order differential equation called Radau’s equation. The coefficients of this equation depend on the unperturbed density profile of the star, because they involve ρ(r)/ ρ r , where ρ(r) is the local density and ρ r is the average density inside r. See Schwarzschild 1958, who gives references to original works.
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the orbits of the neutron star and the companion about the center of mass, being related to ans by a = ans (1 + q), where q ≡ mns /mc is the mass ratio. Here, mns ≡ m1 and mc ≡ m2 are, of course, the masses of the neutron star and the companion respectively. Note that, throughout this book, we shall use the symbols mns and m1 interchangeably for the pulsar’s mass, and the symbols mc and m2 similarly for its binary companion’s mass. Eq. (9.2) is valid for a non-rotating companion; for a rotating companion whose angular velocity is Ω in units of the orbital angular velocity Ωb ≡ 2π/Pb , we need to add a term Ω2 (1 + mns /mc )/(1 − e2 )2 to the expression within the square brackets [Kelley et al. 1981], provided that the rotational and orbital angular momenta are parallel, which we shall assume to be true. Attempts at utilizing the apsidal motion probe for accretion-powered pulsars were pioneered by Rappaport and co-authors in the early 1980s, using the well-known eccentric binaries as Vela X-1 (4U 0900-40) and 4U 0115+63 [Rappaport et al. 1980; Kelley et al. 1981]. These authors obtained upper limits on the apsidal motion in both cases. For Vela X-1 (whose orbital eccentricity, e ≈ 0.09, is rather modest), the upper limit was ω˙ < ∼ < −2.5 on the apsidal motion 3◦ .8 yr−1 , leading to un upper limit log k ∼ constant [Rappaport et al. 1980]. The latter limit was consistent with the computed value of k [Stothers 1974] for the optically identified companion3 HD 77581, whose spectral type was B0.5 Ib (see Appendix A). For the ◦ more eccentric (e ≈ 0.34) binary 4U 0115+63, the limit was ω˙ < ∼ 2 .1 −1 yr , which was used in an attempt to constrain the parameters of the Be-star companion V635 Cas in this system [Kelley et al. 1981]. In the mid 1980s, Nagase and co-authors, as well as Deeter and co-authors improved the upper limit on the apsidal motion in Vela X-1 by about a factor of two [Nagase et al. 1984, Deeter et al. 1987], using a longer baseline of timing data. The above upper limit for 4U 0115+63 was also improved to 0◦ .3 yr−1 by Ricketts et al. (1981), shortly after the above work of Kelley et al. (1981). The breakthrough came in 1992, when Tamura and co-authors succeeded in obtaining a measurement (not an upper limit) of the rate of periastron advance for 4U 0115+63 — the first ever for an accretion-powered binary pulsar — using the ∼ 20 yr database that had accumulated by then on this source [Tamura et al. 1992]. The measured value was ω˙ = 0◦ .030 ± 0◦ .016 yr−1 , with 95% confidence. With this, constraints could immediately be placed on the Be-star companion V635 Cas, as shown in the 3 This means that the companion star’s identification number in a catalog of optical stars is HD 77581. Here, HD stands for the Henry Draper catalog.
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Fig. 9.3 Constraints on the mass and radius of the companion in 4U 0115+63 from apsidal motion. Shown are contours of constant log k, from (a) measurement of apsidal motion (solid line), and (b) model calculations for a representative Be-star companion (dashed line). Also shown are the allowed regions as hatched areas. The panels correspond to different rotation rates of the companion, as marked. See text for more detail. Reproduced with permission by the AAS from Tamura et al. (1992): see Bibliography.
mass-radius plots in Fig. 9.3 for the companion, reproduced from Tamura et al. (1992). The first point is that a non-rotating star, or even a slowlyrotating star with Ω < ∼ 2 (with Ω as defined above), appears to be ruled out by the observations, since the allowed regions in the plot in these cases are inconsistent with the properties of main-sequence B stars. While this is not surprising, since Be stars are generally thought to be rapid rotators, and so is the companion in this particular system from its spectral properties (although its actual rotation rate is unknown), it is good to have independent confirmation. For faster-rotating stars, the sequence of panels in Fig. 9.3, arranged in order of increasing Ω, shows that consistency is possible, but that the allowed region shrinks as rotation increases, making the constraints tighter. If the rotation rate of V635 Cas is measured by future optical observations, its apsidal motion constant k can be determined fairly accurately.
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As Nagase (1989) had pointed out, and as we saw in the above examples, the crucial requirement for this purpose is to be able to extend the observational baseline sufficiently. This has been the stumbling block, since most of the prime candidates for this study are Be-star binaries, which are transient by nature, and so can be studied only during outbursts. Nevertheless, highly eccentric HMXBs like GX 301-2, V 0332+53 and EXO 2030+375, as well as several others, have the potential to yield definitive measurements of apsidal motion in the future. 9.1.3
Orbital Period Changes
Even in Newtonian gravity, orbital periods of accretion-powered pulsars can change very slowly due to various forces, and, as we indicated above, these forces usually dominate completely over the general-relativistic orbit decay described in Chapter 7. Measurements of orbital period changes can thus be a probe of phenomena that produce such forces, e.g., tidal torques, mass loss, and so on, in these systems, when interpreted properly. We illustrate the idea with the aid of a brief summary of the work on the classic, original, accretion-powered X-ray pulsar Cen X-3, with a binary period Pb ≈ 2.1 days [Kelley et al. 1983]. The average rate of orbital decay, P˙b /Pb ≈ −1.8 × 10−6 yr−1 , for this stem was determined in 1983 by Kelley et al. by analyzing the slow change in the times of its X-ray eclipses (i.e., eclipses of the neutron star by its supergiant companion V779 Cen, or Krzeminski’s star; see Chapter 1) over an interval ∼ 10 years, and confirmed by several authors over the 1980s. How do we interpret this observation, which implies that the orbital angular momentum of the system is decreasing? An obvious way of this happening is that mass is being lost from the system, and is carrying away angular momentum with it. Such mass loss is somewhat natural in an accreting binary system, since the entire mass lost by the mass-donor star (see Chapter 6) need not be accreted by the other star. The loss may take place either as a wind from the massive companion in the HMXB, or through the outer Lagrangian point L2 (see Appendix B) of the binary’s Roche lobe. The second mechanism carries away much more angular momentum per unit mass, and so is much more suitable for examination as the cause of orbit decay in this system. Even for this, however, Kelley et al. (1983) found that the required mass-loss rate, M˙ ∼ 3 × 10−6 M yr−1 , was absurdly large: such a large mass flow in the form of a stream (which would be a rough description of the manner of flow through L2 , as opposed to the
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quasi-spherical outflow of a stellar wind) would have been detected by its large, orbital phase-dependent, absorption in the X-rays and the optical. The above authors thus considered tidal interactions as a likely cause of the observed orbit decay in Cen X-3. We have summarized the essential physics of tidal distortion in our account of apsidal motion given above. The key point relevant here is that the tidal distortion (or bulge, as it is often called) produced by the neutron star on its companion “follows” the former as it goes around the binary orbit with an angular velocity Ωb . Seen from an inertial frame, the tidal bulge goes around the companion with an angular velocity Ωb , but since the companion itself is rotating, with an angular velocity which we expressed above as Ω in units of Ωb , in the stellar rest-frame the bulge is actually going around with an angular velocity Ωb (1 − Ω). This causes dissipation of energy in the companion, which ultimately comes from the energy of orbital motion of the neutron star around the companion: so the orbit decays, i.e., the neutron star spirals in very slowly, and Pb decreases. The only exception to this is when the companion rotates synchronously with the orbital motion (Ω = 1), because then the bulge is always “locked” in the same position in the stellar restframe, and there is no dissipation. Asynchronous rotation (Ω = 1) is, therefore, essential for this mechanism to be operational. How do we give a quantitative description of this mechanism? Consider first the simple but useful formulation given by Kelley et al. (1983), in which they express the rate of change of the neutron star’s angular momentum Jns as dJns Ic Ωb =− (1 − Ω), dt τ
(9.3)
in terms of moment of inertia Ic of the companion, and a single parameter τ , the synchronization time, taken as a constant in a first formulation. (We shall presently see how the formulation can be made more “realistic”.) Using Kepler’s third law for the binary orbit (see Appendix B), the above equation can be easily converted into one for the rate of orbital period change: 2 Rc Ic (mns + mc )2 1 − Ω P˙b = −3 , (9.4) Pb mc Rc2 a mns mc τ Rc and a having the same meanings as above. Combining Eq. (9.4) with the value of mc ≈ 19 for Cen X-3, and the typical values Rc /a ≈ 0.65 and (Ic /mc Rc2 ) ≈ 0.06 expected for a star like Krzeminski’s star in Cen X-3, and
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a typical synchronization time τ ∼ 105 yr, the reader can easily show that the observed rate of orbital period change quoted above can be accounted for if the rotation rate of Krzeminski’s star was asynchronous by ∼ 10% (i.e., 1 − Ω ∼ 0.1): this is what Kelley et al. (1983) argued. Actually, the synchronization time τ is itself expected to be depend on the tidal lag or asynchronism parameter 1 − Ω, details of which depend on the model used. In Zahn’s extensive study in the 1970s of radiatively damped dynamical tides, τ was found to scale as τ = τ0 (1 − Ω)−5/3 [Zahn 1977 and references therein], and this prescription was used by Kelley et al. to formulate a more sophisticated description. This is also straightforward, as the reader can readily show that it changes Eq. (9.4) only to the extent of replacing the factor (1 − Ω)/τ on its right-hand side by (1 − Ω)8/3 /τ0 , and the conclusions are almost as easy to obtain as before. Using τ0 ∼ 4 × 103 − 1 × 104 yr from Zahn’s (1977) calculations, Kelley et al. (1983) reached the conclusion that a somewhat higher amount of asynchronism, ∼ 20%, was required for Cen X-3 in this model. The crucial point is, of course, whether such asynchronism is actually expected in a system like Cen X-3, and this appears a difficult question to answer theoretically. On the one hand, since the orbit of Cen X-3 is highly circular (see Table 9.1), and since the timescales for orbit circularization and synchronization of the companion’s rotation are very similar (as both basically originate from the same tidal effects), we would expect Krzeminski’s star to rotate synchronously by now. On the other hand, Kelley et al. (1983) suggested two mechanisms which could maintain asynchronous rotation in Cen X-3, namely, (a) loss of rotational angular momentum from Krzeminski’s star by a stellar wind, and, (b) an orbital instability, originally proposed by Darwin in the nineteenth century and reconsidered by various authors in the 1970s and ’80s, which operates when the rotational angular momentum of the companion exceeds 1/3 of the total orbital angular momentum, as appears to be the case for Cen X-3. Observationally, the situation is less ambiguous: measurements of the projected rotational velocity v sin i have been possible for companions in several HMXBs, though not with great accuracy except for Vela X-1 [van Kerkwijk et al. 1995], and these serve as rough lower limits on v. When the above dimensionless angular velocity Ω of the companion (which we can also call its synchronism parameter , if we wish) is estimated from these data, the values for various systems turn out to be in the range 0 < ∼Ω< ∼ 1.5 — a fact to which we shall return below. For the specific case of Cen X-3, we can estimate Ω from the estimate v sin i ∼ 250 km s−1 . Combining this with the inferred values i ≈
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75◦ and Rc ≈ 12R given by Nagase (1989; also see below), the reader can readily estimate the angular velocity of Krzeminski’s star. Remembering that the orbital period of this system is Pb ≈ 2d .09, the synchronism parameter comes out to be Ω ≈ 0.89, which is consistent (with considerable uncertainties) with the theoretical estimates given above. In 1993, Levine et al. measured the rate of change in the orbital period (Pb ≈ 3.9 days) of SMC X-1 (which also has a very circular orbit; see Table 9.1) as P˙b /Pb ≈ −3 × 10−6 yr−1 , rather similar to the rate for Cen X-3. By contrast, the orbital period (Pb ≈ 1.7 days) of Her X-1, the well-known binary with a rather low mass companion and a very circular orbit, is decreasing at a much smaller rate P˙b /Pb ≈ −1.2 × 10−8 yr−1 [Deeter et al.1991]. Several other HMXBs, e.g., LMC X-4, which has a fairly circular orbit, and the eccentric binaries Vela X-1 and 4U0115+63 (whose apsidal motions we discussed above), are good candidates for similar measurements, as White et al. (1995) have pointed out, when their baselines of measurement become extensive enough. 9.1.4
Neutron-Star and Companion Masses
It is straightforward to obtain the mass function f (mns , mc , sin i) of a neutron-star binary from the observed binary period Pb and projected semimajor axis ans sin i of the neutron star, the relation [Nagase 1989] being: 3
4π 2 (ans sin i) GPb2 3 ans sin i ≈ 1.074 × 10−3 [Pb (d)]−2 , 1 lt−s
f (mns , mc , sin i) =
(9.5)
where the customary units have been used, i.e., solar mass for the mass function, day for the orbital period, and lt-s, or light-second (≈ 3×1010 cm, the distance traveled by electromagnetic waves in one second), for the semimajor axis. The reader can compare the above equation with Eq. (7.33) for binary rotation-powered pulsars to remember the specific combination of mns , mc , and sin i that is so determined. The problem, as we explained earlier, is that the mass function does not, by itself, yield the masses of the two stars and the inclination angle. Unlike the case of relativistic binaries containing rotation-powered pulsars discussed in Chapter 8, here we have no measurement of the post-Keplerian parameters for reasons explained at the beginning of this chapter (see Sec. 9.1), and so no additional information on the masses. How do we solve the problem?
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The normal companion comes to our rescue, by furnishing two additional clues. First, since it is an optical star, it shows optical spectral lines which are Doppler shifted by the binary motion, the shift, and so the radial velocity inferred from it, varying periodically with a period Pb . The semi-amplitude Kc of this optical Doppler velocity curve is related to the neutron star’s projected semi-major axis ans sin i [Nagase 1989] and the mass ratio q (see above) as: √ Kc Pb 1 − e2 mns ac sin i = q≡ = mc ans sin i 2πans sin i √ −1 2 ans sin i 1−e ≈ Kc (km s−1 )Pb (d) . (9.6) 21.8 1 lt−s This determines the mass ratio, and so is an additional constraint on the masses. Second, since the companion is a star of normal size (i.e., Rc ∼ 4R for the relatively low-mass companion in Her X-1, and Rc ∼ [10 − 30]R for supergiant companions), it eclipses the neutron star once every orbital period Pb , blocking off the X-rays from the latter star, provided the inclination angle i is large enough, i.e., our line of sight is close enough to the orbital plane. This has been known since the discovery of the first accretion-powered pulsar Cen X-3 (see Chapter 1). The duration of the eclipse, usually measured by the eclipse half-angle θe (such that the duration of each eclipse is [θe /π]Pb ), is then a measure of sin i. This is easily seen by writing [see Joss & Rappaport 1984 and Appendix B] the companion’s (average) radius Rc in terms of (a) the separation a between the centers of mass of the two stars (see above) during the eclipse, (b) the eclipse half-angle θe , and, (c) the inclination angle i, as: Rc = a cos2 i + sin2 i sin2 θe . (9.7) The question, then, is: how do we know Rc ? It is customary to parameterize it in terms of the size RL of the Roche lobe (see Appendix B) of the companion, since (a) there are good indications from the observed ellipsoidal variations of the optical emission from supergiant companions that Rc is not much less than RL , i.e., such companions nearly fill their Roche lobes, i.e., if we take Rc ≡ βRL , then β > ∼ 0.9, and, (b) we know that accretion must proceed through Roche-lobe overflow (see Appendix B) in relatively low-mass X-ray binaries like Her X-1, and so the companions in such systems must fill its Roche lobe, i.e., β = 1. The Roche lobe
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radius RL is, of course, known very well in terms of the stellar separation a from classical studies of “normal” binary stars, as we describe in Appendix B. The ratio RL /a is a function of the mass ratio q alone for a non-rotating companion: several good, analytic approximations to this function are available, and are given in Appendix B. For rotating companions, correction terms containing the synchronism parameter Ω introduced above are also known, and described in Appendix B. We can then invert Eq. (9.7) to obtain the following explicit expression for the inclination:
sin i =
2 1 − β 2 (RL /a) cos θe
,
(9.8)
where it is understood that the functions of q (and Ω if necessary) given in Appendix B are to be substituted for RL /a when using Eq. (9.8). It is now clear that Eqs. (7.33), (9.5), (9.7) and (9.8) together point to a straightforward way of determining the masses of both stars, as well as the inclination angle. This approach was pioneered in the early 1980s by Joss, Rappaport, and co-authors [Joss & Rappaport 1984]. These authors used a Monte Carlo error propagation technique to obtain the most probable values of mns , mc , sin i, and β from a large number of trial evaluations, in which (a) the values of ans sin i and Kc were chosen randomly with respect to Gaussian distributions around the observed mean values, (b) those of θe were chosen randomly and uniformly in the range implied by observations, and, (c) those β and Ω were also chosen randomly and uniformly, the former in the range 0.9 − 1.0, and the latter in the range 0− 1.5 (see above). Stellar masses, inclination angles, and radii of the companions were obtained by these authors for the six binaries Her X-1, Cen X-3, Vela X-1, SMC X-1, LMC X-4 and 4U1538-52. The main difficulty in obtaining accurate values of the above parameters is the uncertainty in the optical data, particularly that in Kc [Nagase 1989]. As optical observational techniques improved over the years, the above analysis on the above six binaries were repeated by Nagase (1989) and by van Kerkwijk et al. (1995). We present the latter results in below, and give an on-scale display of the binary orbits in Fig. 9.1. As Nagase (1989) pointed out, this analysis can be extended to at least seven more binaries when optical data on these are available. The inferred neutron-star masses for the above 6 accretion-powered pulsars are shown in Fig. 9.4[van Kerkwijk et al. 1995]. Neutron-star masses have now been determined with much greater accuracy for 19 neutron stars in binary rotation-powered pulsars, favoring a very narrow range
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Fig. 9.4 Masses of neutron stars in rotation-powered (below) and accretion-powered (above) pulsars, as of 1995. Reproduced with permission from van Kerkwijk et al. (1995): see Bibliography.
1.3M < ∼ mns < ∼ 1.4M of masses: see Fig. 9.5. (A similar trend is seen in Fig. 9.4, of course, but as van Kerkwijk et al. (1995) pointed out, while the total data set was consistent with a narrow range of masses ∼ (1.35−1.45)M, a much wider range of masses was also possible, because of the large uncertainties in the measured masses of accretion-powered pulsars, and these authors stressed that the situation would improve when the quality of the available optical data improved.) Charles and Coe (2003) have recently given an excellent compilation of neutron-star mass determinations in rotation- and accretion-powered pulsars, which is shown in Fig. 9.5, where determined masses of black holes in X-ray binaries are also shown for comparison.
9.2
Pulse Profiles
X-ray pulse profiles of several accretion-powered pulsars are shown in Fig. 9.6. The general features of these that immediately strike us are (1) large duty cycles > ∼ 50%, in sharp contrast to rotation-powered pulsars,
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Fig. 9.5 Masses of neutron stars (left) in rotation- and accretion-powered pulsars, and masses of black holes (right) in X-ray binaries of various types, as indicated. Reproduced with permission by Cambridge University Press from Charles & Coe (2003): see Bibliography.
whose radio pulses are extremely sharp, with typical duty cycles of a few percent or less (see Chapter 8), (2) modulation factors ranging from ∼ 10% to ∼ 90%, (3) shapes which are generally asymmetric, but can occasionally be quite symmetric. There is no obvious correlation between pulse shape and pulse period, but the correlations of this shape with X-ray energy E and X-ray luminosity Lx are interesting. As E increases, the pulse profile can become simpler or more complicated, or remain roughly unchanged, depending on the pulsar. Nagase (1989) has proposed the following classification scheme for this variation, examples of which can be found in Fig. 9.6:
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• Single sinusoid-like shape with little dependence on energy. Example: X Per, GX 304-1. • Sinusoid-like double peaks with little energy-dependence. Example: SMC X-1, GX 301-2. • Asymmetric single peak, sometimes with one or two extra features, and with small energy-dependence. Example: Cen X-3, GX 1+4. • Single sinusoid-like peak (sometimes with an extra feature) at both high and low energies, and close double-peaks at intermediate energy. Example: Her X-1, 4U 1626-67. • Double sinusoid-like peaks at high energy, and complex multiple-peak structure at low energy. Example: Vela X-1 (4U 0900-40), A 0535+26. Variation of pulse profile with Lx is quite fascinating, and has been recorded for large outbursts of transient Be-star binaries, particularly for the two pulsars EXO 2030+375 and A 0535+26 [White et al. 1995; Bildsten et al. 1997]: we refer the reader to the original work for detail. The former gives a truly remarkable account of the changes that occur as the outburst luminosity decays by a factor ∼ 100: the main pulse and the interpulse reverse roles completely, indicating a radical reversal of the beam pattern [White et al. 1995]. Attempts to understand the above pulse profiles and their changes start with the basic picture of accretion onto the magnetic poles of a neutron star along magnetic field lines, and beamed emission from these parts of the stellar surface (or their immediate vicinity), which we have introduced in Chapter 1 [Pringle & Rees 1972; Davidson & Ostriker 1973; Lamb et al. 1973]. The basic geometry of the problem involves the interplay between the three directions essential to the problem, viz., those of the rotation axis, the magnetic axis, and the line of sight, which are all distinct from one another in general. These are, of course, also the basic directions in the corresponding problem for rotation-powered pulsars. With these, and a postulated qualitative shape of the beam, e.g., “pencil” or “fan” (see Chapters 1 and 11), we can try to reproduce the essential, qualitative geometrical patterns underlying the observed pulse profiles. The actual situation is far more complex and interesting, involving the physics of the emission process, and also, quite crucially, that of the transfer of the emitted radiation through magnetized plasmas, which is highly anisotropic in character, the anisotropy depending on the energy of the emitted X-ray photons. The pulse profile thus naturally depends on both E and Lx . In Chapter 11, we shall return to the question of modeling these physical and geometrical effects.
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Fig. 9.6 Typical pulse-profiles of accretion-powered pulsars. Each profile shown over one-and-a-half pulse periods for clarity. Variation of profile with energy also shown. Reproduced with permission by AAS from White et al. (1983): see Bibliography.
9.3
Secular Period Changes
Within a few years of the discovery of the first two accretion-powered pulsars Cen X-3 and Her X-1, it became clear that their intrinsic pulse periods P (obtained after the subtraction of orbital Doppler effects, as explained
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earlier in this chapter) were showing a long-term decrease, or, equivalently, their pulse frequencies ν were showing a secular increasing trend. This was the famous spin-up of accretion-powered pulsars, which stood in sharp contrast to the spin-down of rotation-powered pulsars, which had already been established well and studied thoroughly by that time. The prompt conclusion was that the torque responsible for this was the accretion torque, entirely different from the electromagnetic torque responsible for the spindown of rotation-powered pulsars, and entirely rooted in the process of accretion of matter by the neutron star. Why does accretion exert a torque? Because, in a binary system with the two stars in orbital motion around each other, the matter that is being transferred to the neutron star from its companion always has some angular momentum with respect to the former star, which is deposited by this matter on the neutron star upon accretion. This is particularly pronounced when accretion is by Roche-lobe overflow (see Appendix B), which is the case for LMXBs and for close HMXBs like Cen X-3, but quite true even when accretion occurs from the stellar wind of a supergiant companion, as we shall see in Chapter 12. For Be-star binaries, most of the matter shed by the rapidly-rotating Be star is actually in the form of rotating rings, which spread predominantly in the plane containing the star’s rotational equator before being accreted by the neutron star (see Chapter 6), so that the matter has a great deal of angular momentum, and the situation is rather similar to Roche lobe overflow. The first estimates of the accretion torque [Pringle & Rees 1972; Davidson & Ostriker 1973; Lamb et al. 1973] recognized the first, and most important, bit of essential physics in the problem, namely, the angular momentum content of the accreting matter, as introduced above. The matter forms an accretion disk around the neutron star (see Chapter 10), in which it goes into Keplerian orbit, such that the Keplerian angular velocity at any radius r is given by the balance between the gravitational attraction of the neutron star and the centrifugal force as ΩK ≡ Gmns /r3 . The angular momentum per unit mass of matter, often called its specific angu√ lar momentum, is therefore given by = r2 ΩK = Gmns r at that radius. Sufficiently close to the star, this accretion disk is terminated by the force exerted by the magnetic field of the neutron star (see Chapter 10), and the matter accretes onto the neutron star along the stellar magnetic field lines, as indicated earlier. Let the inner edge of the disk have a radius rin . The region interior to rin and exterior to the neutron star itself, where the flow of matter is completely dominated by the stellar magnetic field, is called the magnetosphere, whose essential properties we describe in Chapter 10.
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In a first, simple description of the above accretion process, we can argue, as the above authors did in the early 1970s, that the matter simply √ carries its specific angular momentum at rin , i.e., in = Gmns rin , to the star and deposits it there, thereby exerting an accretion torque (9.9) N0 = M˙ Gmns rin , on the neutron star, where M˙ is the accretion rate. What happens to the star’s rotation in response to this follows directly from the conservation of angular momentum, J˙ns = N0 , where Jns ≡ Ins Ω is the stellar angular momentum. Consider first the general situation, where the accretion torque has any value N ≡ M˙ corresponding to a general value of the specific angular momentum deposited on the star, not necessarily the value in given above. Straightforward differentiation of Jns , given by the above expression, yields: ˙ M˙ Ω d ln Ins = − , (9.10) Ω mns ns d ln mns where ns ≡ Jns /mns = ΩRg2 is the specific angular momentum of the Ins /Mns being its radius of gyration [Ghosh et al. 1977]. star, Rg ≡ The physical meaning of Eq. (9.10) is quite obvious. Assuming that the neutron star’s rotational angular momentum is in the same direction as the orbital angular momentum (quite a safe assumption on evolutionary grounds, and no counter-examples are known), all angular momenta in the above equation have the same sign, and we readily see that Ω˙ > 0, i.e., the star spins up due to accretion, if /ns exceeds the logarithmic derivative (d ln Ins /d ln mns ). This is what we would intuitively expect, since /ns is the specific angular momentum added by accretion to the star per unit specific angular momentum it already has, (d ln Ins /d ln mns ) is the (logarithmic) increase in the stellar moment of inertia per unit increase in the stellar mass due to accretion, and the former effect, which tends to spin the star up, must exceed the latter, which tends to spin the star down. Consider now the specific case of an accretion torque given by Eq. (9.9), which means simply replacing by in in Eq. (9.10). Note first that the logarithmic derivative (d ln Ins /d ln mns ) ∼ 1 for essentially all neutronstar EOS, except at the smallest and largest possible neutron-star masses allowed by the EOS: examples are given in Ghosh et al. (1977). On the other hand, the ratio in /ns = [ΩK (rin )/Ω](rin /Rg )2 1, since rin ∼ 108 cm Rg ∼ 106 cm, as we already indicated, and shall describe in detail in
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Chapter 12, and further since the condition Ω < ΩK (rin ) must be satisfied for accretion to be possible, as we shall discuss in Chapter 12. Thus the spinup term completely dominates over the spindown one for this simple version of the accretion torque, and the neutron star always spins up due to accretion. Agreement between early observations of spinup rates of accretion powered pulsars and simple estimates of spinup torques like that given above [Rappaport & Joss 1977] were useful for building confidence in our understanding of the basic features of accretion by magnetic neutron stars, but it became clear even in the 1970s that the actual situation was more complicated. Accretion-powered pulsars like Her X-1 showed a spinup rate much smaller than that implied by Eq. (9.9), and pulsars like Cen X-3 occasionally showed short, transient episodes of spin-down, i.e., a decrease in Ω, or, equivalently, in the pulse frequency ν ≡ Ω/2π. The essential role of magnetic stresses in accretion torques was appreciated in attempts to understand these features. In the late 1970s, Ghosh and Lamb formulated the quantitative theory of accretion torques, to which we shall return in Chapter 12. However, it was clear by the mid 1980s, and completely obvious by the late 1980s [Nagase 1989], that reality was far more complex than a basic secular spin-up of all accretion-powered pulsars. The observed secular trends could be divided into the following classes: • Secular spin-up, i.e., secular increase in the pulse frequency ν. Example: Her X-1, Cen X-3. • Long stretches of secular spin-up and spin-down, i.e., secular increase and decrease in ν, alternating with each other. Example: GX 1+4, 4U 1626-67, Vela X-1. • No secular trend. Variations consistent with a random walk in ν. Example: 4U 0115+63. • Secular spin-down, i.e., secular decrease in the pulse frequency ν. Example: 1E 2259+586. In Figs. 9.7 and 9.8, we give examples of typical secular trends in the pulse frequency variations of accretion-powered pulsars belonging to the first three classes above. We do this because there is controversy today, as explained in Chapter 1, as to whether the pulsars of the fourth class above, which consists of six to eight secularly spinning-down pulsars with pulse periods in the range P ∼ 5 − 8 s, are really powered by accretion at all. As described later in this chapter, the causes of this controversy are various signs of apparent anomaly in the properties of these pulsars, of which one of the most important is that no companion to the neutron
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Fig. 9.7 Secular pulse-period changes in accretion-powered pulsars. Reproduced with permission by the Astronomical Society of Japan from Nagase (1989): see Bibliography.
star has been detected so far in any of these systems. Thus, they are currently called anomalous X-ray pulsars (AXPs), and one of the possibilities being explored now is that they are actually rotation-powered pulsars with unusually strong magnetic fields, which would be able to account for their large rates of spin-down, far in excess of those expected from usual rotation-powered pulsars with the above periods and canonical magnetic field strengths ∼ 1012 G. We return to AXPs later, summarizing the our current understanding of these pulsars. In the pulse-frequency histories shown above, which span ∼ 20 − 25 years, and which had been gathered from several generations of X-ray satellites starting with the pioneering Uhuru (see Chapter 1), the reader will note that the data points are very closely spaced over the last ∼ 5 years, appearing as a continuous line for several of the pulsars. These are the contributions from the BATSE (Burst and Transient Source Experiment; see Horack 1991 for details) detector aboard the Compton Gamma Ray Ob-
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Fig. 9.8 Secular pulse-period changes in accretion-powered pulsars (continued). Reproduced with permission by the Astronomical Society of Japan from Nagase (1989): see Bibliography.
servatory (CGRO), launched in 1991. Originally designed as a continuous, all-sky monitor to be used for the detection, location and study of γ-ray bursts (see Fishman et al. 1992) that was one of CGRO’s main objectives, BATSE’s continuous monitoring capability proved instrumental for retrieving data4 on accretion-powered pulsars in the hard 20–50 keV X-ray energy range, which is the region of overlap between the “tail” of the X-ray spectra of canonical accretion-powered pulsars and the energy-range of significant response of this detector. The interpretation of the above secular variations is, of course, in terms of our detailed understanding of accretion torques, which is a subject by 4 Essentially
from the background data of the large NaI(Tl) scintillation γ-ray detectors, through period-folding or Fourier techniques. Occasionally, when counts from a well-known pulsar like Vela X-1 in its high state dominated the data in a particular satellite orbit, pulses at the known period of the pulsar could be seen directly in the raw data, “riding” on top of the background; see Bildsten et al. 1997.
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itself, and which we shall discuss in Chapter 12. Secular variations in the pulse-frequency ν normally mean those on timescales of months to years, since short-term effects in the accretion torque, and in the response of the neutron star to this torque, will have averaged out, or reached a quasisteady state, by then. Usual torque theories, as we shall see in Chapter 12, are steady-state theories, and so apply only to truly secular variations in the above sense. How do we interpret observed variations of ν on short time scales (∼ 1 − 10 days)? Such data were available even in the 1970s, e.g., in the classic 1972 observation of the spindown episode in Cen X-3 by the Uhuru satellite [Fabbiano & Schreier 1977], and in the 1980s, e.g., in the close coverage of the spin-frequency variation of Vela X-1 by the Hakucho and Tenma satellites [Deeter et al. 1987, 1989]. Now the BATSE detector has produced such data in abundance in the time-frame 1991– 2000. An example of this shown in Fig. 9.9 for the classic accretion-powered pulsar Cen X-3. How do we use these data to probe the above shortterm dynamical phenomena in the accretion torque, or in the neutron-star response?
Fig. 9.9 Pulse-frequency changes in Cen X-3. Reproduced with permission by the AAS from Bildsten et al. (1997): see Bibliography.
Consider the physical causes of these short-term effects, taking the accretion process first. For pulsars with extensive accretion disks, the longest timescale in the disk, viz., the viscous drift time from the outer edge of the disk to its inner edge (see Chapter 10), is ∼ hours to days. Thus, the
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inherent fluctuations in the mass-transfer process and so the accretion rate M˙ , due, e.g., to those in the mass-supply from the companion star, or to the formation of plasma blobs in the accretion disk by various instabilities, will be smoothed out only on timescales shorter than the above, leaving all others essentially unchanged. Further, fluctuations in the interaction between the accreting plasma and the stellar magnetic field at the interface between the magnetosphere and the inner edge of the disk will occur at the much shorter (> ∼ seconds to minutes) timescales characteristic of this interface, and these will, of course, not be smoothed out by the disk viscosity. Finally, as we shall show in Chapter 12, for pulsars which are fast rotators in the sense explained there, the steady-state accretion torque is very small, due to a near-balance between the material torque (which tends to spin the star up) and the magnetic torque (which tends to spin the star down), so that small fluctuations at the disk-magnetosphere interface can cause large fluctuations in the accretion torque about this balance point, including reversals in the torque’s direction. For pulsars accreting from the stellar wind of a supergiant companion, on the other hand, the small accretion disk that is expected to form is thought to undergo actual reversals in the direction of rotation on timescales ∼ hours to days, so that there should be reversals in the accretion torque on this timescale in such systems also, but for a different reason. Now consider the response of the neutron star to a given torque. As we explained in Chapter 8, the coupling time between the crustal lattice and the vortex lines of the neutron superfluid in the crust is ∼ days to tens of days, according to current understanding of pinning, so that variations in crustal rotation frequency (which is what we observe as ν) are expected on this timescale, if unpinning, vortex creep and repinning occur during the spin-up and spin-down processes, particularly in the fast-rotating pulsars. How are we to disentangle these various effects observationally? Note first that the ν-variations we observe are the response of the rotating, multicomponent neutron star to the accretion torque variations. A popular electronic-circuit analogy would be to regard the accretion torque variations as the “applied signal”, the internal structure of the neutron star as the “filter”, and the observed ν-variations as the “output” [Lamb et al. 1978]. Note next that we cannot really monitor this output continuously, despite the apparent impression the reader may have from the continuous lines in Fig. 9.9. Rather, we “sample” the output at discrete time intervals δt, to use another standard term from signal-processing terminology. When the typical duration of an individual short-term event T1 is much shorter than
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δt, so that there are many unresolved events within the sampling interval, and, furthermore, if the average rate R of the occurrence of these events is so large that there is a very large number of events within the total observational time-span T , i.e., RT 1 (this is almost always true in practice), we can give a statistical description of the above variations in terms of the standard methods developed for noise processes. This is what we do next.
9.4
Timing Noise
We have discussed timing noise in rotation-powered pulsars in Chapter 7, indicating Groth’s (1975) pioneering formulation of its analysis. The corresponding, first formulation for accretion-powered pulsars, which largely followed the same approach, came from Lamb et al. (1978). In terms of the above analogy with electronic circuits, these authors expressed the applied signal, i.e., the time-dependent accretion torque, as N (t) = PN (t) + δN (t), separating the best-fit polynomial trend PN (t) (see Chapter 7) from the fluctuating part δN (t). They then focused on two specific types of δN (t), viz., white noise in the torque N , or, white noise in the time-derivative of the torque N˙ , conveniently expressed in the forms: (9.11) White noise in N : δN1 (t) = i δJi δ(t − ti ) , White noise in N˙ : δN2 (t) = δNi θ(t − ti ) , (9.12) i
where δ(t) and θ(t) are respectively the delta function and the step function in time. The natures of these two types of noise are shown schematically in Fig. 9.10. White noise in N occurs when N makes excursions away from the above, deterministic polynomial trend such that N returns to this trend in a characteristic time T1 (see above) which is much less than the observation span T . However, such individual events are not visible, because the rate R1 of such events is so high that many new events occur before a given event has decayed away, i.e., R1 T1 1. This leads to the form of applied signal sketched in panel (a) of the figure. By contrast, white noise in N˙ occurs when N executes a random walk away from the polynomial trend, i.e., it retains no memory of this trend, at least over the observation span T . This leads to a sequence of step-changes in N , giving the applied signal shown in panel (b) of the figure. Formally, a white noise in N = J˙ corresponds to a sequence of step-changes in the angular momentum J, which we have called δJi in Eq. (9.11), and a white noise in N˙ corresponds to step-changes δNi in the torque N itself, as above.
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Fig. 9.10 Torque fluctuations in accretion-powered pulsars: schematic forms. See text for detail. Reproduced with permission by the AAS from Lamb et al. (1978): see Bibliography.
Among the various physical origins of short-term torque events we have detailed in the previous subsection, Lamb et al. (1978) cited some specific ones as possible causes for the above two kinds of noise. White torque noise, for example, could come from either (a) “clumpiness” in the matter crossing the inner edge of the accretion disk into the magnetosphere, or, (b) sudden unpinning of superfluid vortices in the neutron-star crust during its secular spin-up or spin-down (see Chapter 8). On the other hand, white N˙ noise could come from step-changes in the accretion rate M˙ , due, e.g., to fluctuations in the collection of individual, small “streams” of matter that leave the inner edge of the accretion disk to fall into the stellar magnetosphere, if this is, in fact, what actually happens in the accretion process. The strengths of the above two types of noise are easily estimated as S1 = R1 δJ 2 for the first type, and S2 = R2 δN 2 for the second, in terms of the event-rates R1 and R2 for the two types, and the mean-square values (or second moments) of the fluctuations δJi and δNi in the two types of noise. Physical inputs are then needed to evaluate these second moments. For the unpinning events suggested above, for example, Lamb et al. (1978) gave the following estimate. If a fraction γ of the pinned superfluid of total
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moment of inertia Is unpins suddenly, and spins up from its spin-frequency νs ≡ Ωs /2π to the observed spin frequency ν of the crust, the jump in the angular momentum is δJ = 2πγIs (ν − νs ) = 2πγIs ν. Here, we have expressed the jump in νs as a fraction of ν. So the strength Su of the unpinning noise is given in terms of the rate Ru of the unpinning events as Su = Ru δJ 2 = 4π 2 Ru Is2 ν 2 γ 2 2 . What does the “output” signal, i.e., the pulse frequency ν or angular velocity Ω = 2πν, look like after the above “applied” signal is “filtered” by the neutron star? Consider the case of a rigid star at first. This is very simple, because the whole star of total moment of inertia I responds together as a rigid body to any torque fluctuation δN , producing a fluctuation in the derivative of the angular velocity, i.e., the angular acceleration ˙ which is given by δ Ω ˙ = δN/I. The noise in Ω ˙ and ν˙ is then simply a Ω, copy of the torque noise δN , scaled by the stellar moment of inertia. This ˙ or ν˙ is often loosely referred to as torque noise in is why the noise in Ω the literature. A white torque noise, for example, produces a white noise ˙ equivalent to a red f −2 noise, or a random walk, in Ω (see above) in Ω, [Lamb et al. 1978], and so on. But what actually happens, or is expected to happen, in this filtering by a neutron star? In their 1978 work, Lamb et al. described this process with the aid of the two-component theory of the rotational dynamics of neutron-star interiors in vogue then, wherein a slow, frictional coupling was invoked between the solid crust and the superfluid core. We summarized this theory in Chapter 7, and mentioned that current understanding of the rotational dynamics of neutron-star interiors is entirely different. The crust is now believed to be tightly copuled to the core on quite short timescales, while any slower coupling, essential for a non-rigid response of the neutron star, is believed to have its origins in the behavior of the crustal superfluid, as its vortices unpin from crustal lattice during spinup/spindown, the vortex-creep rate slowly relaxes to another steady state, and the vortices pin again (see Chapter 8). Slow modes of oscillation of the neutron superfluid, e.g., the Tkachenko mode (see Chapter 8), originally considered by Lamb et al. (1978) in conjunction with the above two-component theory, may still play an indirect role in the neutron star’s response, but the details are likely to be different. Finally, note that the mean square deviations in the essential rotational variables due to the above noise processes and their associated random walks can be expressed in a straightforward manner in terms the strength S of the random walk [Deeter et al. 1989]. Consider, for example, a random walk in Ω, corresponding to a white torque noise (see above), which
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is characterized by an rms step size δΩ and a step rate R. The resultant mean square deviation in Ω in an observation span T is given by δΩ2 = RT δΩ2 = ST , where S ≡ RδΩ2 is the strength of the random walk. This has the characteristic ∼ T dependence expected for a random walk. The associated variables, namely, the rotational phase φ and the ˙ have the same ∝ S dependence on the strength of angular acceleration Ω this random walk, but different dependences on T , as expected. The phase, which is obtained by integrating Ω, has a mean square deviation which is basically obtained by multiplying with one factor of T 2 (see below), i.e., δφ2 = ST 3 /2. Similarly, the angular acceleration, which is obtained by differentiating Ω, has a mean square deviation which is obtained by dividing by one factor of T 2 (see below), i.e., δ Ω˙ 2 = ST −1 . 9.4.1
Power-Density Spectra
In the early 1980s, Deeter and Boynton undertook a pioneering study of the techniques of estimation of the noise power spectra from actual data on accretion-powered pulsars [Deeter & Boynton 1982, Deeter 1984, and references therein], analyzing and suggesting solutions for many of the practical difficulties that arise in this particular case. These authors emphasized the crucial role played by sampling, the concept of which we have introduced in the previous subsection, and which is generally quite non-uniform in real astrophysical data. They formulated the idea in terms of a sampling function, g(t), so that the information we extract from the available data on a time-varying quantity x(t) is actually from the integral x(t)g(t)dt, i.e., the convolution of the time-series we wish to study with the sampling function at our disposal, over the domain of observation. Note that, for pulsar observations, a discrete sampling is usually more relevant than a continuous one, which just amounts to writing the sampling function as a sum of delta functions: g(t) =
1,n
gj δ(t − tj ).
(9.13)
j
As Deeter and Boynton showed, what really occurs in the problem is: 2 (9.14) P ≡ x(t)g(t)dt , which these authors called the power-density estimator . The key question is: if we are dealing with an r-th order red noise in x(t), as above, how can
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we reliably extract the strength of this noise from the above power-density estimator? Several practical difficulties stand in the way, a major one being the “leakage” of power from the low frequencies (where most of the power is for a red noise) to higher ones. This occurs because any power-density estimator we can employ in practice will not have non-zero response only at a given frequency, or even in a narrow band of frequencies around a given one, but rather have a finite response at nearly all frequencies, so that the leakage from the enormously power-laden low frequencies of a steep red noise can be disastrous [Deeter & Boynton 1982]. The only viable way of handling this problem is by designing estimators that minimize this leakage. Deeter and Boynton devised a scheme for designing g(t) appropriate for handling an r-th order red noise of strength S(f ) ≡ S0 f −2r with the aid of their moment conditions, i.e., the requirement that the first r moments of g(t) vanish: 0 ≤ i < r. (9.15) g(t)ti dt = 0, Why does this work? We can look at it in two equivalent ways. Let us write the expectation value of the above power-density estimator for an r-th order red noise as:
2 2 2r g (−r) (t) dt, (9.16) P = |G(f )| S(f )df = S0 (2π) where G(f ) is the Fourier transform of g(t), and g (−r) (t) is the r-th repeated integral5 of g(t). We obtain the first form of the right-hand side of Eq. (9.16) from standard Fourier transform relations, and the second form by repeated integration by parts (see below) of Eq. (9.14), subject to the moment conditions given by Eq. (9.15). First look at the situation in the frequency domain: as Eq. (9.16) shows, the integrand will not have undesirable behavior at low frequencies if the fisrt r + 1 terms in a (Maclaurin) series expansion of G(f ) around f = 0, i.e., those containing f 0 , f 1 , . . . , f r , all vanish , since S(f ) ∝ f −2r . Now look at the time domain: the condition equivalent to the above frequency conditions is precisely Eq. (9.15), i.e., the moment conditions. This is so because, when evaluating the integral in Eq. (9.14), we can do repeated integration of by parts on g(t), thereby converting it to g (−1) (t), g (−2) (t), . . . , g (−r) (t), and simultaneously extracting the first, second, . . . , r-th derivative of x(t), provided the moment condi5 Subject to the endpoint conditions that g (−i) vanishes at both ends of the observational time domain for 0 < i ≤ r; see Deeter 1984, and Deeter and Boynton 1982.
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tions are satisfied, as the reader can easily show. But the noise in the r-th derivative of x(t) is a white noise, since that in x(t) itself is an r-th order red noise, as we have indicated earlier. For example, a second order red noise in the phase φ is a white noise in φ¨ or ν, ˙ i.e., a white torque noise, or a spindown noise, as we saw in Chapter 7. Thus, the noise in the r-th derivative of x(t) is a constant, and we end up with the second form of the right-hand side of Eq. (9.16), minimizing leakage by dealing, in effect, with a white noise. The beauty of the Deeter-Boynton approach is the immediate connection it can make with the method of polynomial trend removal discussed above and in Chapter 7 in connection with rotation-powered pulsars, and so with the use of variances discussed in Chapter 7. We summarize only the first connection here, referring the interested reader to the original papers for discussion on related topics. It may not be quite obvious at first, but imposing a moment condition of order r on the sampling function g(t) basically amounts to removing a polynomial trend of degree r − 1 in the data x(t) [Deeter 1984]. To see this, note first that a moment condition of order r removes a polynomial trend of degree r − 1 from g(t). This is obvious, because when we fit an orthonormal set of polynomials pj of order j = 0, 1, . . . , r −1 to g(t), the coefficients Cj of the polynomials are given by Cj = g(t)pj (t)dt, and the moment conditions (9.15) ensure that all these coefficients vanish. Consider now what happens when we sample x(t) with such a trend-removed g(t). The part of x(t) that contains the polynomial trend of degree r − 1 is expressible as a linear combination of the above polynomials pj . Since we have removed just these polynomials from g(t), what remains of g(t) must be orthogonal to this part of x(t). Thus, the convolution integral of x(t) and g(t) (see above) has no contribution from a polynomial trend of degree r − 1 in x(t), and all power-spectral results derived from it behave as if we had actually taken out a polynomial trend of degree r − 1 from the data x(t). 9.4.1.1
Observed power spectra
In the late 1980s, Deeter and co-authors applied the above methods to an extensive study of timing noise in Vela X-1 (4U 0900-40), using an extensive database obtained from observation with the X-ray satellites Hakucho and Tenma in the 1980s, and from those with the earlier satellites OSO 8, HEAO 1, and SAS 3 in the 1970s [Deeter et al. 1987, 1989, and references therein]. The amount of timing noise in accretion-powered pulsars is generally much
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higher than that in rotation-powered pulsars, and that in Vela X-1 is particularly high. It has been known for a long time that a considerable part of the noise in the pulse phase in accretion-powered pulsars is generally due to the short-term variations in their pulse profiles, in addition to the usual photon-counting noise, i.e., Poisson fluctuations, in our detection systems. These variations are obvious in the pulse-to-pulse fluctuations of the pulse profile in any given accretion-powered pulsar, as described earlier, similar to the situation described for rotation-powered pulsars in Chapter 7. Reference or “master” pulses are, of course, constructed [Deeter et al. 1987], just as for rotation-powered pulsars, but cross-correlation of a locally-averaged pulse with the master still shows considerable residual noise in the pulsephase due to the above cause, which we can call pulse-shape noise. This masks the signal we are looking for, i.e., the torque noise described above, for sufficiently high values of the frequency f (often called the analysis frequency; f ∼ 1/T , where T is the observation span) at which we wish to measure the power-density P (f ), as we shall see below. Thus, it would be very useful if we could reduce this pulse-shape noise. These authors devised the following scheme for reducing pulse-phase noise by appropriately filtering the pulses. They noticed that some harmonics obtained by a Fourier analysis of the pulse profile were the noisiest, and so empirically designed specific filters to attenuate these harmonics, guided by the actual measured spectrum of pulse-shape noise, and the harmonic content of the master pulse. This resulted in a ∼ 25 − 60 % reduction of the pulse-shape noise, as measured by the variance of the phase estimates [Deeter et al. 1987]. The power-density spectrum Pν˙ (f ) of the noise in the pulse-frequency derivative ν, ˙ which is sometimes loosely referred to as torque noise, and which is also the noise in the second derivative φ¨ of the pulse phase, is shown in Fig. 9.11 for Vela X-1, as given by Deeter et al. (1989). Below an analysis frequency of f ∼ [4 days]−1 , it is clear that torque noise dominates, and it is white noise within the uncertainties of measurement. This covers a logarithmic range ∼ 3 decades in f below the cross-over frequency given above. Above this frequency, pulse-shape noise takes over, and dominates at all higher frequencies: it is a blue noise in ν, ˙ rising with increasing > frequency. For analysis periods ∼ 4 days, then, the noise in this accretionpowered pulsar is basically a white noise in ν, ˙ which is equivalent to a firstorder f −2 red noise, or a random walk, in the pulse frequency ν, and also equivalent to a second-order f −4 red noise, or a so-called double random walk, in the pulse phase φ. The pulse-shape noise that dominates at shorter observation periods or higher analysis frequencies is understood to be a
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˙ in Vela X-1 vs. analFig. 9.11 Power density of fluctuations in angular acceleration Ω ysis (circular) frequency f . Reproduced with permission by the AAS from Deeter et al. (1989): see Bibliography.
white noise in the pulse phase φ, caused by these fluctuations in pulse ˙ by shape. Such a noise translates into a second-order f 4 blue noise in ν, a reversal of the above scheme. As the reader can see in the figure, the observed pulse-shape noise actually rises roughly like f 3 , and this may not seem quite right at first. However, as Deeter et al. (1989) have argued, this is due to the use of the hierarchical sampling procedure [Deeter 1984] on the data at these high analysis frequencies, wherein the sample duration remains roughly constant, but the sampling density increases as f ∼ 1/T with increasing frequency. If the pulse-phase noise is actually white, i.e., of uniform density, the apparent density will decrease as f −1 because of this more frequent sampling. Transformation by two factors of f 2 to the variable ν˙ will then give an f 3 spectrum.
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The remarkable similarity between the above power-density spectrum of noise in the accretion-powered pulsar Vela X-1, as shown above, and the spectrum of Taylor variance of noise in the rotation-powered millisecond pulsar B1937+21, as shown in Chapter 7, may not be obvious at first. This becomes clear when we note that (i) the latter plot is in terms of the observation span t, related to the analysis frequency f of the former one as f ∼ t−1 , and (ii) the Taylor variance σz is related to power S in the phase noise as σz ∼ S 1/2 /t3/2 , and S scales with the above power P in the torque (or ν) ˙ noise as S ∼ P f −4 . At large values of f or small values of t, then, the white phase noise, S = const., which we described above, and which corresponds to measurement errors (dominated by pulse-shape noise for Vela X-1), then translates into σz ∼ t−3/2 for the Taylor-variance spectrum in Chapter 7, and into a P ∼ f 4 power spectrum in the figure here, which is modified into an f 3 spectrum because of the sampling effect described above. At small values of f or large values of t, on the other hand, the white torque noise, P = const., described above and shown earlier, should translate into a σz ∼ t1/2 Taylor-variance spectrum of Chapter 7. As Kaspi et al. (1994) pointed out, the exponent of t in this r´egime is not precisely determined observationally, but lies in the range 0.5 – 2.0, which is consistent with the above arguments. As these authors also indicated, the behavior of usual rotation-powered pulsars with slower rotation is similar. Thus, if all accretion-powered pulsars behaved like Vela X-1, we could argue that the spectrum of timing noise had the same qualitative behavior for the entire collection of pulsars powered by accretion and rotation, viz., white torque noise at low analysis frequencies f , and white phase noise due to measurement errors at high values of f . The only difference would then be one of noise strength, accretion-powered pulsars generally having the highest noise, millisecond rotation-powered ones the least, and “usual” rotation-powered ones somewhere in between. But the actual situation appears to be more complicated. Whereas the well-known accretion-powered pulsar Her X-1 was also known from Deeter’s work in the early 1980s to have white torque noise at low f , recent work from the BATSE database indicate that at least two types of behavior may be present in accretion-powered pulsars [Bildsten et al. 1997]. The power spectra thus obtained have relatively low resolution in f , but they clearly suggest that there may be one class of accretion-powered pulsars like Vela X-1, Her X-1, and 4U 1538-52 which have white torque noise at low f , and another class of accretion-powered pulsars like Cen X-3 and GX 1+4 which have red torque noise at these frequencies, with P ∼ f −1 approximately.
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The reader will remember from Chapter 7 that a red f −2 torque noise corresponds to a white noise in N˙ , or, equivalently, a random walk in the torque N . The intrinsic noise in accretion-powered pulsars of the second class could thus be a combination of a white torque noise and a random walk in N , the two coming from distinct physical causes. This would have interesting implications for the natures of the accretion flow in the above two classes of accretion-powered pulsars, in view of the discussion given earlier. 9.4.2
Time-Domain Analysis
An alternative analysis of timing noise in pulsars is in the time domain, i.e., directly in terms of the time series {t1 , t2 , . . . , tn } at which the pulse frequency ν has been measured, instead of being in the frequency domain, i.e., the Fourier space, as is the case for the power-spectrum approach described above. A popular method used in this domain is autoregression analysis, the basic idea of which we indicate now. Consider the “input signal”, introduced earlier, as a sequence of fluctuations6 at the above times {δJ1 , δJ2 , . . . , δJn }, and the “output” as the resultant sequence of pulsefrequency fluctuations {δν1 , δν2 , . . . , δνn }. We shall assume at the outset that any secular variation in ν has already been taken out of the series, in the form of a secular derivative ν, ˙ which produces a linear trend in the output. In the autoregression picture, the i-th output δνi is thought to be determined not only by the corresponding applied signal δJi , but also by several previous values of δν itself, say δνi−j , with j = 1, 2, . . . , p. Hence the name autoregression. It is a linear theory, so that all the above dependences are linear. The simplest case is, of course, that with p = 1, i.e., a correlation with only the immediately previous value, which means that the system has a memory of only the immediately previous step in the random walk. This is called a first-order Markov process, and expressed mathematically as δνi = Aδνi−1 + BδJi ,
(9.17)
in terms of the coefficients A and B of linear dependence. The correlation coefficient A between the current and the last step is of crucial importance in deciding the character of the noise in ν. For an applied white-noise signal in J, for example, A = 1 (a very strong correlation) corresponds to 6 These are fluctuations in the time-integral of the torque, which has the dimensions of angular momentum, and so is denoted by J.
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a random walk or red noise (see above) in ν, i.e., a white noise in ν, ˙ and A = 0 (no correlation) corresponds to a white noise in ν. On the other hand, for a white noise in the applied torque N , which corresponds to a red noise or random walk in J, A = 1 corresponds to a second-order red noise or double random walk in ν (see Chapter 7), or, equivalently, a red noise in ν, ˙ while A = 0 corresponds to a red noise in ν, or a white noise in ν. ˙ Time-domain methods7 were applied to rotation-powered pulsars in the ¨ 1980s, and, in 1993, Baykal and Ogelman made the pioneering application to accretion-powered pulsars. These authors used the recorded pulsefrequency histories of 16 accretion-powered pulsars as the above time series, and obtained estimates of the noise strength and the correlation coefficient A. They calculated the autocovariance of the k-th order, Ck ≡ δνi δνi+k (the angle-brackets mean ensemble average), from the data. Then the variance of the applied noise, σ 2 ≡ δJ 2 , is related8 to the autocovariance of the zeroth order, C0 ≡ δν 2 as δν 2 = B 2 δJ 2 /(1 − A2 ), and similar relations hold for higher orders. ¨ Baykal and Ogelman (1993) found values of the above correlation parameter A to be in the range ∼ 0.8 − 1.0 for the pulsars Vela X-1, Her X-1, and Cen X-3, and, for the first two pulsars, values of the noise strength consistent with those found earlier (see above) by Deeter and co-authors in their frequency-domain analysis. Since these values were close to the limit A = 1, which correspond to a white noise in ν˙ for an assumed white-noise signal in J (see above), they then assumed this value of A for the rest of the pulsars, and estimated the noise strengths in them. From the ensemble of pulsars showing a secular spin-up, these authors found an empirical relation between the noise strength S and the secular spinup rate ν ˙ of the form S ∼ ν ˙ 2 , which the reader can compare with the corresponding relation for rotation-powered pulsars given in Chapter 7. As we described earlier, recent BATSE power spectra have indicated that Cen X-3 actually seems to show a red noise in ν, ˙ which raises the natural question as to whether there is a contradiction here with the above result [Bildsten et al. 1997]. There is not, of course, since the above result would imply a white spectrum in ν˙ only if we assume white-noise signal in J, which amounts to a blue noise in the torque N , for which there is no evidence in Cen X-3. On the other hand, a white torque noise would, in fact, give a red noise in ν˙ for A ≈ 1 (see above). The reader will notice that much of the internal rotational dynamics of neutron star — the so-called 7 See
8 The
the discussion by Scargle (1981). reader can consult standard references, e.g., Box and Jenkins (1970).
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filter that we described earlier — is embodied in the correlation parameter A of this simple Markov-chain approach. In practice, such a one-parameter description of the complexities of the above dynamics may be problematic.
9.5
Quasi-Periodic Oscillations (QPOs)
In our discussion of accretion-powered pulsars, we have so far confined ourselves to periodic phenomena like pulsing and binary orbital motion, and to timing noise. A related behavior in time- or frequency-domain are the quasi-periodic oscillations or QPOs, which manifest themselves most often as rather broad (usually Lorentzian) peaks in the power spectrum, in contrast to the very sharp — often line-like — peaks characteristic of the pulses of accretion-powered pulsars, as shown in earlier figures in this chapter. In the more quantitative terms of a centroid frequency ν and a width λ of the peak, we can define the usual quality factor Q ≡ ν/λ, and QPOs will have moderate values of Q as opposed to the very large values characteristic of periodic oscillations. We can adopt Q > ∼ 2, say, as a criterion by which a local maximum in a power spectrum qualifies as a QPO [van der Klis 2000, henceforth vdk00]. The physical idea is as the name implies: as strictly periodic oscillations indicate the presence of periodic physical phenomena like pulsing, orbital motion, etc, QPOs indicate the presence of physical phenomena that have clear, distinct timescales associated with them, and yet do not lead to precisely periodic repetitions because of a variety of other effects that intervene, e.g., those that are dissipative, or have other timescales associated with them, as we shall see below. As such phenomena are very likely associated with accretion flows in these systems, they are potentially valuable diagnostics of these flows. Further, the highest-frequency QPOs — the so-called kilohertz (henceforth kHz) QPOs — provide diagnostics of regions close to the neutron stars in the X-ray binaries which emit these signals, and therefore are excellent probes of possible effects of relativistic gravity in these regions. 9.5.1 9.5.1.1
QPOs in LMXBs Low-frequency QPOs
In the mid-1980s, QPOs were first detected in LMXBs with the satellite EXOSAT . As pointed out earlier, periodic pulsations from LMXBs have not been seen to this day except in the recently-discovered accretion-powered
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Fig. 9.12 Low-frequency (LF) QPOs in the bright LMXBs GX 5-1 and Sco X-1. Note the presence of the strong LF red noise to the left of the QPO peak in GX 5-1, and its absence in Sco X-1. Reproduced with permission by Annual Reviews from van der Klis (1989): see Bibliography.
millisecond pulsars (see Chapter 6), so that the power spectra of the bright LMXBs known in the 1980s were known to contain only broad components with gradual variations, e.g., power laws, indicating aperiodic behavior, usually “red” noise of various kinds. (We have already given above an account of analogous timing noise in accretion-powered pulsars in HMXBs.) The 1985 discovery of low-frequency (henceforth LF) QPOs in such LMXBs by van der Klis, van Paradijs, and co-authors was thus a breakthrough, being our first — if indirect — glimpse at possible periodic features of
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accretion flows in these systems. These LF QPOs were observed in the frequency range ∼ 5 − 60 Hz, with quality factors Q > ∼ 2 and amplitudes of oscillation ∼ 1 − 10 % [van der Klis 1989; henceforth vdk89]: typical examples of LF QPOs in two well-known LMXBs are shown in Fig. 9.12. It was soon clear that these LF QPOs came in different types, which were later named horizontal branch oscillations (HBOs), normal branch oscillations (NBOs), and flaring branch oscillations (FBOs): these branches are those seen in X-ray color-color diagrams (see below), so that there are clear correlations between QPOs and spectral states of LMXBs. The discovery of these systematics was among the most important clues to the nature of the accretion flow and radiation transfer in LMXBs, partcularly Comptonization effects. X-ray color-color diagrams were introduced in analogy with analogous optical diagrams as a rough spectral indicator [White & Marshall 1984]. In the simplest form, the whole energy-range of response of an X-ray detector is divided into three bands, e.g., 1–3 keV, 3–6 keV, and 6–20 keV for the above EXOSAT studies of LMXBs; then the “hard color” is defined as the ratio of the intensities in the top and middle energy bands (e.g., [6-20/3-6] keV ratio here) and the “soft color” as the ratio of the intensities in the middle and bottom energy bands (e.g., [3-6/1-3] keV ratio here). The remarkable discovery from these EXOSAT studies was that many bright LMXBs traced out chracteristic Z-shaped tracks in the X-ray color-color diagram [vdk89], as shown in Fig. 9.13.
Fig. 9.13 X-ray color-color diagrams of two Z-sources — bright LMXBs tracing out Z-shaped tracks in the diagram. Shown are the horizontal branch (HB), the normal branch (NB), and the flaring branch (FB). Only rudiments of HB are seen for GX 17+2. Reproduced with permission by Annual Reviews from van der Klis (1989): see Bibliography.
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The names “horizontal”, “normal”, and “flaring” are largely historical, as vdk89 pointed out, but not always the most accurate scientifically. Thus, HB was actually horizontal in many of the first-discovered sources but not always so, NB is no more normal (i.e., occur no more commonly) than the other branches, and FB was so named because Sco X-1 showed flares in this state, but some others actually show a decrease in flux there. Nevertheless, the names have stuck. What is important is the close relation between these spectral states and the corresponding power-spectral states, i.e., QPOs and the associated noise. Thus, HBOs come in a characteristic range of frequencies νQP O ∼ 15 − 55 Hz, with a strong positive correlation between νQP O and the X-ray intensity Ix , describable by a power-law relation νQP O ∼ Ixn , with an exponent n around unity. This latter correlation proved very useful for probing accretion flows in LMXBs in the framework of QPO models for HBOs considered in the 1980s (see below). NBOs come in a characteristic range of frequencies νQP O ∼ 5 − 7 Hz, with no obvious, strong trend of correlation with Ix . Finally, FBOs come in a characteristic range of frequencies νQP O ∼ 10 − 20 Hz, with positive correlation between νQP O and Ix . As these QPOs of Z-sources were studied further, it became clear that those belonging to different branches could sometimes occur simultaneously in the same source. Thus, HBOs and NBOs could be observed simultaneously, particularly near the HB/NB vertex (see Fig. 9.13), and NBOs turned into FBOs in sources like Sco X-1 as the source proceeded through NB to FB, the nomenclature being something of a matter of definition there. This led to the conclusions that (a) HBO was probably qualitatively different from NBO/FBO, and (b) NBO and FBO could be different manifestations of the same basic phenomenon in two different r´egimes. We return to this point below. A QPO property that proved to be of much diagnostic value for Comptonization processes at work in LMXBs was that of time lags between QPOs in different X-ray energy bands, 1–6 keV and 6–20 keV for the above EXOSAT work. Between the QPOs observed in these bands, the one in the harder X-ray band lags behind the one in the softer band, as may be expected if the harder photons are produced by Comptonization in a cloud of hot electrons. For further details of LF QPOs, including LMXBs which do not show the Z-tracks, and have been named “atoll” and “banana” sources because of their characteristic behavior on X-ray color-color diagrams, we refer the reader to vdk89, and return to LF QPO models below.
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9.5.1.2
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High-frequency QPOs
It had long been recognized that fast variability on timescales ∼ milliseconds, or, equivalently, at frequencies ∼ kilohertz, would be a signature of phenomena near compact objects like neutron stars and black holes, since this is the dynamical timescale for motion under gravity near them [van der Klis 2000, henceforth vdk00]. In fact, it was during earlier attempts to detect such variability that LF QPOs had been found in LMXBs. The wealth of millisecond phenomena, particularly the high-frequency (henceforth HF) QPOs, were finally discovered and studied in detail by detectors aboard RXTE in the mid- to late-1990s. “Kilohertz” (henceforth kHz) QPOs in bright LMXBs were the first of these to be discovered: this was done independently by Strohmayer and van der Klis in early 1996, shortly after the launch of RXTE. These appeared as two simultaneous QPO peaks — so-called “twin peaks” — in the frequency range ∼ 200 − 1200 Hz, usually a few hundred Hz apart in frequency, moving together up and down in frequency in correlation with source state [van der Klis 2004, henceforth vdk04]. Frequencies of both peaks usually increase with increasing X-ray flux, and the separation frequency in some sources is close to that of the ∼ millisecond-period oscillations detected during the thermonuclear bursts observed from some of these LMXBs (see below). The discovery of these “burst oscillations” was another major discovery of HF phenomena near compact objects by RXTE : for further details, we refer the reader to vdk00, and references therein. In panel (a) of Fig. 9.14, we show the twin KHz peaks in Sco X-1. Such kHz QPOs have now been detected in nearly all Z-sources and “atoll” sources (see above), as also in some accretionpowered millisecond pulsars (see Chapter 6). Weak sidebands have also been detected in some kHz QPO sources. More recently, “hectohertz” (henceforth hHz) QPOs have been detected in atoll sources. These hHz QPOs have frequencies ∼ 100 − 200 Hz, which do not vary greatly and are rather similar across sources [vdk06]. In panel (b) of Fig. 9.14, we show the hHz QPO in 4U 0614+09. The valuable insights that QPOs offer into accretion flows and general relativistic effects near compact objects are closely connected with our attempts to construct basic, viable models for these oscillations, and so to understand their dynamical origins. We now summarize some aspects of such diagnostics.
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Fig. 9.14 Panel (a): Twin peaks of kilohertz QPOs in Sco X-1. Panel (b): Hectohertz QPO in 4U 0614+09. Reproduced with permission by Cambridge University Press from van der Klis (2006): see Bibliography.
9.5.1.3
LF QPO diagnostics
The interpretation of HBOs as the beat frequency between the neutron-star spin frequency, νs ≡ 1/P ≡ Ωs /2π, and the Keplerian frequency at the inner edge of the accretion disk (see above, and also Chapters 10 and 12), 3 /2π, was very popular in the 1980s and early 1990s [Alpar νK0 ≡ GM/rin & Shaham 1985; Lamb et al. 1985; Ghosh & Lamb 1992 and references therein]. In this magnetospheric beat-frequency (henceforth MBF) model, then, νHBO = νB = νK0 − νs .
(9.18)
The physical picture envisaged in the MBF model is as follows. Interactions between the magnetosphere of the neutron star and the accretion disk (see Chapter 10) produce fluctuations in the plasma density and the magnetic field at the inner edge of the disk, which circulate at rin with the local Keplerian frequency νK0 . On the other hand, the stellar field lines rotate with neutron star at the frequency νs . Consequently, a given plasma/magnetic field fluctuation feature reappears at a given magnetic azimuth of the stellar field with a frequency that is the beat frequency νB given above. Thus, the mass flux from the disk-magnetosphere boundary layer (see Chapter 10) is expected to be modulated quasi-periodically at this frequency νB , and this is identified with the HBO in the MBF model.
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This model provides a possible probe, therefore, of the conditions in the disk-magnetosphere interface, which was used by various authors. In particular, Ghosh and Lamb (1992; henceforth GL92) explored its value as a diagnostic of accretion disk models known at the time. These models give characteristic dependence of rin on the essential parameters, i.e., accretion rate M˙ , stellar magnetic moment µ, and stellar mass M . As detailed in Chapter 12, the form is: rin ∝ M˙ a µb M c ,
(9.19)
where the exponents a, b, c carry the signature of the disk model. GL92 noticed that the MBF model given above leads to νK0 = (νHBO + νs ) ∝ M˙ α µβ M γ ,
(9.20)
where the exponents are given by 3 α ≡ − a, 2
3 β ≡ − b, 2
3 1 γ ≡− c− , 2 2
(9.21)
as the reader can easily show. Values of these are given in Table 12.2 for various well-known disk models. It still remained to relate the accretion rate M˙ to the observed X-ray intensity I, and GL92 expressed it as I ∝ M˙ λ , which yielded the final diagnostic as: (νHBO + νs ) ∝ M˙ α/λ µβ M γ ,
(9.22)
which these authors used on the HBO data available at the time. As remarked above, νHBO is strongly dependent on I, and GL92 used this dependence as the diagnostic probe. Note, however, that νs was still unknown, and had to be guessed. These authors showed that, for choices of νs ∼ 100−350 Hz, the data were consistent with some disk models (e.g., 1R and 1G models in the notation of Table 12.2), and inconsistent with some others (e.g., the 2T model). Subsequent discovery of accretion-powered millisecond pulsars (see Chapter 6), as also inferred rotation frequencies of neutron stars displaying burst oscillations (see above), have shown that the above choice was reasonable. However, the actual values of νs for most LMXBs (particularly the Z-sources, which exhibit HBOs) still remain unknown. For more detail, we refer the reader to GL92. Accretion-flow models of NBOs and FBOs were explored in the late 1980s and early 1990s, the key feature there being the formation of a
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hot corona around the accretion disk, which became extensive at nearEddington accretion rates. Comptonization in such coronae were studied as natural causes for the observed lags between photons in various X-ray energy bands, as also for the observed “rocking” of the X-ray spectrum around a pivotal energy of Ep ∼ 5 keV as a source moved along the NB. For understanding the dynamical origin of NBOs, it was suggested that there would be quasi-radial inflow of matter in these coronae due to the removal of angular momentum by radiation drag, and oscillations modes in such flows at high luminosities were explored. Characteristic oscillations at the flow timescale, fosc ∼ 2/tf ∼ 6 Hz, were indeed found, and identified with NBOs. For more detail, we refer the reader to the review by Lamb (1991). 9.5.1.4
HF QPO diagnostics
The key diagnostic point about kHz QPOs from the beginning has been the fact that the frequency scale for orbital motion around the neutron star, νorb
r −3/2 M 1/2 orb ≈ 1200 Hz, 15 km 1.4M
(9.23)
readily shows the dynamics of interest is occurring in a region where generalrelativistic (GR) effects must necessarily be significant, since the radius of the innermost stable circular orbit (henceforth ISCO) in GR (in the Schwarzschild metric9 ) is given by: 6GM M RISCO ≡ ≈ 12.5 km. (9.24) c2 1.4M Inside RISCO , no stable orbital motion is possible [see, e.g., Weinberg 1972]. Thus, these QPOs can serve as probes into GR effects in close vicinity of neutron stars, and, in building models for these, we must necessarily include GR effects. An dramatic use of this diagnostic would occur if we were to find, for example, that the upper kHz QPO frequency νu , which normally increases with increasing X-ray intensity I, levels off at some maximum value νmax 9 The corresponding radius can be readily calculated for rotating stars in the Kerr metric with the aid of the expressions originally found by Bardeen et al. (1972). The rotational modifications enter through the Kerr angular-momentum parameter a ≡ Jc/GM 2 , where J is the stellar angular momentum. For prograde orbits 0 < a < 1, and for retrograde orbits −1 < a < 0. The correction factor to RISCO to the first order in a is (1 − 0.54a).
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and does not increase any more as I increases, as this might be an indication that the ISCO has been reached. In that case, we would have a direct, if qualitative, confirmation of an essential strong-field feature of GR, and would also get an immediate measure of the neutron-star mass with the aid of Eq. (9.24), if we were to identify νu with orbital frequency, as is done in many models (see below). In the late 1990s, such a phenomenon seemed to have been observed in the very compact LMXB 4U 1820-30, with νmax ≈ 1060 Hz, corresponding to a neutron-star mass M ≈ 2.2M . Unfortunately, further observations have cast doubt on this [vdk00, vdk06].
Fig. 9.15 Constraints on neutron-star matter EOS from maximum kHz QPO frequency νmax . Excluded area shown shaded. Left panel: Non-rotating neutron star, with various values of νmax . For clarity, excluded area for only νmax = 1220 Hz is shaded. Right panel: Rotating star, with values of angular-momentum parameter a as indicated (see text); only the νmax = 1220 Hz case is shown in this panel. EOS coded as follows (see Chapters 4 and 5), showing progression from softer to harder EOS. A: Pandharipande (1971), FPS: as in the above chapters, UU: Wiringa et al. (1988), L and M: Pandharipande & Smith (1975). Reproduced with permission by the AAS from Miller et al. (1998): see Bibliography.
However, very similar approaches can lead to constraints on the EOS of neutron-star matter (see Chapters 3, 4, and 5), as shown by Miller and co-authors in 1998. The idea here is to use two constraints, namely, (1) that the stellar radius R must be smaller than the orbital radius, R < rorb , and, (2) that RISCO must also be smaller than the orbital radius, RISCO < rorb , since no stable orbits can occur inside RISCO . Next, to tighten the constraints, the idea is to use the smallest possible value of rorb , which, through Eq. (9.23), corresponds to the highest frequency νmax seen in the
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kHz QPOs. With the aid of Eqs. (9.23) and (9.24), the first constraint boils down to an M -dependent upper limit on R, and the second one to an upper limit on M . Together, they trace out an allowed region in an M − R plot, as shown in Fig. 9.15, where we show both rotating and non-rotating stars (see above). Finally, this region is to be compared with the M − R relation given by various EOS, displayed on the same plot, as shown. Note how the allowed area shrinks for higher values of νmax , as expected, and so constrains EOS better, possibly ruling out some of the hard EOS. Several types of models have been proposed for the dynamical origin of kHz QPOs. Consider first beat-frequency models, a variation of which we have already encountered above for HBOs. In the scenario for kHz QPOs, the preferred radius changes from the magnetospheric radius to the sonic radius rs , where the inflow velocity first becomes supersonic. This is thus the sonic-point beat-frequency (henceforth SBF) model. Generally, rs is in the vicinity of RISCO , but radiative stresses determine its precise value. In the SBF model proposed by Miller et al. (1998), the envisaged physical picture is as follows. Matter orbiting at rs becomes supersonic as pressure support fails, and clumps of matter follow a spiral path to accrete onto the neutron-star surface. In a frame of reference rotating with the Keplerian orbital frequency νKs at rs , the spiral path is fixed once we know the azuimuth at which a given clump starts its infall from rs , and so is the “footpoint” of the spiral on the stellar surface, where the clump hits it and produces a “hot spot” of emission. To a non-rotating observer, however, this hot spot travels around the stellar surface with a frequency νKs , which is then identified with upper kHz QPO frequency νu . What about the lower kHz QPO frequency νl ? These authors argued that a part of the accretion would still continue to the magnetic poles of the star, and generate a beam of radiation from the poles in the usual manner, as explained in Chapter 1. This beam rotates with the neutron star at a frequency νs , and so irradiates the clumps at rs periodically at the beat frequency νKs − νs , and therefore modulates at this frequency the rate at which clumps provide matter to the spiral inflow, and hence the intensity of emission from the hot spots. Thus, νKs − νs is identified with the lower kHz QPO frequency νl in the SBF model. Thus, the difference ∆ν = νu − νl between the two peaks would be constant at νs in the simplest version of this model, although later versions included the possibility of a changing ∆ν, due, for example, to slowly decreasing rs . Consider now the relativistic precession model proposed by Stella and co-authors [Stella & Vietri 1998], in which the physical picture is entirely
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different, based completely on the systematics of the precession of particle orbits that occur in general relativity. The basic idea is simple: inclined, eccentric particle orbits around a compact object in GR show two kinds of precession. First, nodal precession, which is that of the orbital plane around the stellar rotation axis, caused by Lense-Thirring frame dragging effect [see, e.g., Weinberg 1972]. Second is a relativistic periastron precession, entirely analogous to the well-known precession of Mercury’s perihelion, a cornerstone in the early work on general relativity (see Chapter 7 for details of the analogous effect in binary rotation-powered pulsars). In this model, the upper kHz QPO frequency νu is identified with the orbital frequency, the lower kHz QPO frequency νl is identified with that of the above periastron precession, and an appropriate LF-QPO frequency (HBO frequency, say) is identified with that of the above nodal precession. Since these GR effects are, in a sense, geometrical and so independent of many details of the accretion flow, this model gives straightforward mathematical forms for the above frequencies in terms of stellar and orbital parameters. Thus, if the nodal precession frequency is calculated to the lowest order, the relation between the LF-QPO frequency νLF and the upper kHz QPO frequency νu would be: νLF =
8π 2 Iνu2 νs , M c2
(9.25)
where I is the stellar moment of inertia [vdk00]. Similarly, the frequency difference ∆ν = νu − νl between the two peaks is related to νu by: RISCO , (9.26) ∆ν = νu 1 − r where RISCO is the Schwarzschild value of the ISCO radius given by Eq. (9.24), and r the orbital radius. The prediction of Eq. (9.26) is compared with kHz QPO data in Fig. 9.16. The form predicted by Eq. (9.25), namely, νLF ∝ νu2 has also been generally borne out by observation [vdk06], but a problem appears to be that the constant of proportionality, which is essentially the stellar I/M , does not agree with the values given by acceptable EOS for neutronstar matter. Indeed, this type of model requires more detailed physics to decide essential features like (a) what decides preferred orbital radii r, (b) what mechanisms might be relevant for sustaining tilted orbits in viscous accretion disks, and so on. These problems have been studied by Stella and co-authors, and others.
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Fig. 9.16 Predicted relation between the upper kHz QPO frequency νu and the frequency difference ∆ν from the relativistic precession model, compared with data on well-known LMXBs, as marked. The former is given for three neutron-star masses: (1.8, 2.0, 2.2)M , as labeled. After Stella & Vietri (1999). Reproduced with permission by Cambridge University Press from van der Klis (2006): see Bibliography.
Consider, finally, the relativistic resonance model proposed by Abramowicz and co-authors in recent years [Abramowicz & Klu´zniak 2001], which again makes use of “geometrical” properties of particle orbits in GR. In general relativity, such orbits are not closed, but the motion is still describable in terms of three frequencies: the GR orbital frequency νφ — the analogue of the usual Keplerian orbital frequency, and the radial and vertical epicyclic frequencies νr and νθ , which may looked upon as those of oscillations in these directions of an orbit whose azimuthal frequency is νφ . It is because of the latter motions that particle orbits “waltz” in space — to quote the picturesque language of vdk06 — with nodal precession at a frequency (νφ − νθ ), and perihelion precession at a frequency (νφ − νr ), as described above. These authors pointed out that, at some particular orbital radii, the above frequencies νφ , νr and νθ had simple integral ratios between pairs, e.g., νr /νφ = 1/2 or 1/3, or νr /νθ = 2/3, and so on, i.e., they were commensurate, and these were ideal sites for exciting resonances, which would produce the observed oscillations. Various physical mechanisms can be envisaged for such resonance excitation, a particular one studied in detail being parametric resonance, wherein a normal mode
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of the system is perturbed at a frequency which is commensurate with the frequency of the mode. The particular resonance sites explored by these authors in detail were, in fact, the ones listed above. The nice thing about this scenario is that GR effects themselves would pick out these sites in an accretion disk [vdk06]. Similar ideas have been explored about possible resonances between stellar spin frequency and orbit frequencies in the disk. For further detail, we refer the reader to vdk06 and the references to the original literature cited therein.
Fig. 9.17 Correlation between the HBO frequency νHBO , scaled by the inferred stellar spin frequency νs , and upper kHz QPO frequency νu , as observed in Z-sources (names as indicated). Also shown is a quadratic fit (solid line), as expected from relativistic precession models, and a best power-law fit (dashed line). Reproduced with permission by Cambridge University Press from van der Klis (2006): see Bibliography.
We close our discussion of QPOs in LMXBs with the observation that correlations between LF and HF QPOs have proved to be a most valuable diagnostic in the subject, since any viable set of models for these QPOs must be consistent with this diagnostic. For example, the well-known correlation between the HBO frequency and the upper kHz QPO frequency in Z-sources is shown in Fig. 9.17, with a quadratic fit suggested by Eq. (9.25). Other correlations are discussed in the excellent review by van der Klis (vdk06), to which we refer the reader.
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QPOs in HMXBs
Observing QPOs in accretion-powered pulsars in HMXBs has the merit that the pulse period and so the stellar spin frequency νs is known there, so that diagnostics may become easier there if the QPO mechanism is understood. LF QPOs were first detected in the Be-star HMXB EXO 2030+375 [Angelini et al. 1989], and have now been seen in ∼ 10 HMXBs, with frequencies in the range ∼ 10 − 400 mHz [Ghosh 1998]. As a typical example, we show in Fig. 9.18 the discovery of ∼ 35 mHz QPOs in Cen X-3 by Takeshima et al. (1991). The figure shows the essential features clearly: the sharp, spike-like peaks in the power-spectrum at the pulse frequency of this famous accretion-powered pulsar (P = 4.8 sec; see above and Chapter 1) and its harmonics (as referred to earlier), the broad QPO at ∼ 35 mHz, and the characteristic low-frequency noise.
Fig. 9.18 Low-frequency QPOs in Cen X-3. Shown is the power spectrum with the QPO at ∼ 35 mHz, the pulse frequency and its harmonics (sharp spikes), and the lowfrequency noise. Reproduced with permission by the Astronomical Society of Japan from Takeshima et al. (1991): see Bibliography.
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If these LF QPOs in HMXBs were the analogues of HBOs in LMXBs, and if the MBF model applied to them, we could make ready use of the ideas and arguments detailed in Sec. (9.5.1.3), and do much useful diagnostics of accretion disks and torques in HMXBs. In particular, the fastness parameter, ωs ≡ νs /νK0 , of the pulsar, which occupies a central place in our understanding of disk-magnetosphere interaction and accretion torques (see Chapters 10 and 12), would be immediately calculable from Eq. (9.18) as νs ωs = . (9.27) νs + νQP O Such studies were used by various authors, and the results are summarized in Ghosh (1998), to which we refer the reader for further detail, as also for general discussions of diagnostic possibilities of QPOs in HMXBs. The basic point here is that it is not clear if these LF QPOs in all HMXBs are, in fact, describable by the MBF model. We close our discussion on this topic by mentioning that the first (and so far the only) detection HF QPOs in HMXBs has been reported by Jernigan et al. (2000) for Cen X-3, at frequencies ∼ 330 and 760 Hz. 9.6
X-Ray Spectra
X-ray spectra of accretion-powered pulsars are determined with the aid of a variety of instruments aboard various X-ray satellites, which have different ranges of effective response for the X-ray energy, e.g., soft (< ∼ 1 keV, say) X-rays, canonical “medium” (2–10 keV) energy X-rays, most commonly used since the early, Uhuru days, and hard (> ∼ 20 keV) X-rays, and so can effectively observe only the appropriate part of the whole spectrum. In Fig. 9.19, we show a representative collection of accretion-powered pulsar spectra [Coburn et al. 2002], whose essential features we discuss below. Note first that these are the so-called phase-averaged spectra, i.e., they have been obtained by integration over many pulse periods. We shall see below how we can concentrate on a small phase-range around a given phase of the pulse, and obtain the X-ray spectrum specifically in that range, leading to interesting conclusions for the processes of emission and transfer of radiation near the magnetic poles of the neutron star. Note next that, in Fig. 9.19 (as also generally in the literature), three spectra are shown for each pulsar. The observed spectrum of photon counts in the detector (usually denoted by crosses) is fitted by assuming a source or incident spectrum
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(usually denoted by a solid line), and “folding” it through the instrumental response, as also including the effect of the photoelectric absorption of lowenergy X-rays by interstellar (and also, in some cases, circumstellar) matter, thus obtaining the fitting spectrum (usually denoted by a histogram). The response of our detector is, of course, known to us, and the low-energy absorption depends on the column density NH of the absorbing matter and its temperature T , which are fitted as parameters. This incident (or deconvolved, as it is sometimes called, since the above procedure amounts to a deconvolution of the observed spectrum) spectrum is where the pulsar properties come in, and the following discussion will be entirely on the incident spectrum, unless noted otherwise. 9.6.1
Continuum Emission
The continuous photon spectrum N (E) (i.e., number of photons, or number of counts in our detector, per second per keV) is often fitted by a functional form: 1, E ≤ Ecut −Γ (9.28) N (E) = N0 E × e(Ecut −E)/Ef , E > Ecut characterized by a power-law E −Γ with the photon index Γ in the range 1 – 2 [Coburn et al. 2002], upto a cutoff energy Ecut , above which it is cut off steeply by the above exponential roll-off, Ef being the e-folding (sometimes also called just “folding”) energy. Typical values of both Ecut and Ef lie in the range 10–20 keV [White et al. 1983], as shown in Table 9.2 for 10 accretion-powered pulsars studied by Coburn et al. (2002), using RXT E. Exceptions to this general picture are a few pulsars like X Per which have much softer spectra, i.e., steeper slopes, Γ ∼ 3: these generally have −2 37 much lower (by factors < ∼ 10 ) luminosities than the canonical ∼ 10 erg −1 s luminosity of accretion-powered pulsars. Apart from this, there seems to be no particular correlation with the luminosity Lx in either Γ or Ecut . By contrast, EF tends to increase with decreasing Lx , with the exception of a couple of very luminous sources like GX 1+4 [White et al. 1983]. Soft spectra are also observed for the so-called anomalous X-ray pulsars or AXPs. While the form (9.28) seems to have been used most widely, alternative empirical forms for continuous spectra are also available, such as Tanaka’s Fermi-function form for the cut-off, which replaces the above exponential form by [1 + exp{(E − Ecut )/Ef }]−1 , while keeping the power-law
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Fig. 9.19 X-ray spectra of accretion-powered pulsars from RXT E. For each pulsar, top panel gives counts spectrum (crosses), inferred photon spectrum (smooth curve), and model “folded” through the X-ray detector’s response (histogram). Bottom panel gives residuals after the fit. Reproduced with permission by the AAS from Coburn et al. (2002): see Bibliography.
as above, or Mihara’s double power-law form, which generalizes the above single power-law to E −Γ1 + aE Γ2 (note the positive exponent in the second power-law), while keeping the exponential cutoff the same as above. All of these functional forms represent little more than convenience, however,
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Table 9.1 SPECTRAL PARAMETERS OF ACCRETION POWERED PULSARS (after Coburn et al. 2002) System
Γ
Ecut (keV)
Ef (keV)
Her X-1 4U 0115+63 Cen X-3 4U 1626-67 XTE J1946+274 Vela X-1 4U 1907+09 4U 1538-52 GX 301-2 4U 0352+309
0.93 0.26 1.24 0.88 1.14 0.00 1.236 1.161 -0.02 1.82
22.0 10.0 21.3 6.8 22.0 17.9 13.5 13.6 17.3 57
10.8 9.3 6.67 38.8 8.3 8.8 9.8 11.9 6.7 50
and inferred parameters from different forms are hard to compare without physical input. We return to the question of our physical understanding of pulsar continuum emission in Chapter 11. We now briefly mention two additional features of the continuous spectra. First, an excess of low-energy (∼ 0.5−4 keV) X-rays over that expected from the above spectral form with photoelectric absorption (see above) due to cool, homogeneous matter has been reported by various authors in various classes of accretion-powered pulsars, e.g., LMXBs like Her X-1 and 4U 1626-67, HMXBs like Vela X-1 and SMC X-1, and more recently in transient Be-star binaries in SMC [see Nagase 1989, 2001, and references therein]. This is the so-called soft excess. In their detailed Tenma study of this phenomenon in the 1980s, Nagase and co-authors focused on the ∼ 1 − 3 keV region of the spectrum of the well-known pulsar Vela X-1 in highly-absorbed states, and argued that the origin of the excess was the reduced absorption by matter when it was “clumpy”, and so covered the source of emission only partially. However, spectral studies of the same source in the 1990s with the satellite ASCA, done by the same and other authors [Nagase et al. 1994 and references therein], showed a flat continuum with the emission lines of highly-ionized (He-like) atoms of Mg, Si, and S (see below) in the same ∼ 1 − 3 keV energy range, which supported the idea that the soft component was due to scattering by the stellar wind of the massive companion in this binary, rather than the above picture of a leaky, clumpy absorber. Work in recent years has focused more on the soft excess in the region < ∼ 1 keV. These are often fitted with blackbody
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emission models with kT ∼ 0.1 − 0.3 keV, although their physical origin is not quite clear. Second, a curious feature, variously described as a “wiggle” or “bump”, has been noticed for some time at ∼ 10 keV by various authors using satellites like Ginga and BeppoSAX, and is now becoming quite clear in the RXTE spectra, as emphasized by Coburn et al. (2002), and as seen in Fig. 9.19. It is not a simple absorption line (see below), nor a cyclotron feature (see below), and, as these authors argue, its origins need to be understood in order to help accurate modeling of pulsar spectra obtained with future generations of X-ray detectors. 9.6.2
Pulse-Phase Spectroscopy
The fact that pulse profiles change with the X-ray energy, as described earlier, implies that the spectrum of the X-rays emitted by an accretionpowered pulsar changes across the pulse, i.e., it is a function of pulse phase [White et al. 1983]. Hence, if we could determine spectra at specific pulse-phases, we could directly observe how the X-ray spectrum goes through periodic changes during each pulse. In the late 1970s, Pravdo and co-authors pioneered this study, which is now known as pulse-phase spectroscopy [Pravdo et al. 1977, 1979]. The technique is straightforward, as explained by Pravdo et al. (1978): we divide the pulse period P into N phase bins (as we always do for pulse-profile determination), and determine the TOA of each photon, after it has been duly analyzed in energy and put in the correct energy bin by our pulse-height analyzer (which processes the counts recorded by our proportional counter), to an accuracy of P/N . With sufficiently long observations, therefore, we can accumulate sufficient numbers of photons in each phase bin to obtain a good spectrum in that bin, and so obtain a set of N spectra, which shows the march of spectra across the pulse. We can, of course, fit each of these spectra then by models described above, and so determine the variation of the model parameters across the pulse: for the continuum, for example, these would be Γ, Ecut , and Ef . We show in Fig. 9.20 results obtained by Pravdo and co-authors (1979) for the pulsar 4U 1626-67, panel (a) showing the spectra at two pulsephases which differ by ∼ 0.5, and panel (b) showing the variation of the continuum parameters Γ, Ecut , and Ef with pulse-phase. The reader can compare panel (b) with the pulse-profile of this pulsar in different energy ranges, as given earlier. The exponent Γ is the largest, i.e., the spectrum
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Fig. 9.20 Pulse-phase spectroscopy of 4U 1626-67. Shown are the spectra (marked 3 and 8) at two pulse-phases differing by ∼ 0.5. Reproduced with permission by the AAS from Pravdo et al. (1979): see Bibliography.
is the softest, where the peak of the low-energy pulse (or the “soft” pulse, as it is sometimes called) occurs, as may have been expected. Similarly, where the high-energy cutoff Ecut has its trough, i.e., the spectrum is cut off at low energies and so has very little hard component, is also the place where the high-energy or “hard” pulse has its trough. If the energy-variation of the pulse-profile has the same information as the pulse-phase variation of the spectrum, why do we do pulse-phase spectroscopy? As Pravdo et al. (1979) argued, there is actually clearer information in the latter. This is so because, in a pulse profile, there are energy-independent intensity changes, in which the overall profile moves up or down, as well as energy-dependent changes because of spectral changes with pulse phase, and it is difficult to disentangle the two in practice. By accumulating spectra in the phase-bins to begin with, we eliminate the normalization problem in the latter case, keeping only the spectral-change information. The ultimate use of this information is in shaping our understanding of the mechanisms of X-ray emission and transport in the vicinity of the magnetic poles of accreting neutron stars, which we consider in Chapter 11.
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9.6.3
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Cyclotron Features
First discovered in Her X-1 by Tr¨ umper et al. (1978), cyclotron features in the spectra of accretion-powered pulsars at X-ray energies > ∼ 10 keV, commonly known as “cyclotron lines” and more formally as cyclotron resonance scattering features (CRSFs), are of fundamental importance, as they are our best observational handle on neutron-star magnetic fields to this day. This is so because the energy Ec of these features directly probes of the magnetic field B on the neutron-star surface through the cyclotronresonance frequency Ωc ≡ (eB/me c)(1 + z)−1 , me being the mass of the electron, so that Ec = Ωc ≈ 11.6B12 (1 + z)−1 keV.
(9.29)
Here, z is the gravitational redshift on the neutron-star surface, which appropriately reduces the frequency seen by a distant observer, and B12 is B in units of 1012 G, as usual. The physical process underlying these features is believed to be resonant scattering of the X-ray photons by electrons in the surface material of the neutron star, whose energy states are the quantized Landau levels (see Chapter 13) determined by the strong magnetic field at the neutron-star surface. Thus, features are expected at the resonance energy Ec , and also possibly at the higher harmonics 2Ec , 3Ec , . . . (see below). This is the physics through which the neutron-star magnetic field leaves its imprint on the X-ray spectrum: further details are given in Chapter 11. Historically, the cyclotron feature in Her X-1 (Tr¨ umper et al. 1978) was first interpreted as ∼ 58 keV cyclotron emission, leading to a magneticfield estimate of B12 ≈ 5, and a later interpretation of it as a ∼ 38 keV cyclotron absorption feature led a lower value, B12 ≈ 3. The next pulsar in which this feature was discovered was 4U 0115+63, which showed a higher harmonic for the first time, i.e., features at ∼ 11.5 and 23 keV, leading to an estimate B12 ≈ 1. The rapid increase in the number of pulsars with cyclotron features in the late 1980s and early 1990s was largely due to the satellite Ginga [see Nagase et al. 1992]. Currently, confirmed cyclotron features are known in 12 accretion-powered pulsars: these are listed in Table 9.3. Except for the unusually high value for A 0535+26, the inferred magnetic fields all lie in the (narrow) range B12 ≈ 1 − 5, confirming our general beliefs about magnetic fields of neutron stars (see Chapter 13).
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Table 9.2 CYCLOTRON FEATURES IN ACCRETION POWERED PULSARS (after Coburn et al. 2002 and White et al. 1995) System
Ec (keV)
Her X-1 4U 0115+63 Cen X-3 4U 1626-67 XTE J1946+274 Vela X-1 4U 1907+09 4U 1538-52 GX 301-2 4U 0352+309 X 0332+530 Cep X-4 A 0535+26
40.4 16.4 30.4 39.3 34.9 24.4 18.3 20.7 42.4 28.6 28.5 32.0 110
Several correlations have been observed over the last ∼ 10 years between the cyclotron-feature energy Ec , the cutoff energy Ecut , and two other parameters of the cyclotron feature, which we can define [Coburn et al. 2002] conveniently as the width σc and the strength, or optical depth, τc of a Gaussian-shaped function τ (E) with which we fit the cyclotron feature, so that it modifies the underlying continuum N (E), given by Eq. (9.28), in the following way:
N (E) = N (E)e−τ (E) ,
τ (E) = τc e−(E−Ec )
2
/2σc2
.
(9.30)
The oldest known correlation is that between Ec and Ecut . Found originally by Makishima and Mihara (1992), and refined by subsequent work, it is a positive correlation of the form Ecut ∝ Ec0.7 , the physical origin of which is still poorly understood. While there is clearly a functional relation between the cutoff energy and the magnetic field (which determines Ec ), this does not necessarily imply that the magnetic field directly controls Ecut . It is possible that Ecut depends causally on another physical variable, which is actually determined by the magnetic field. A recent version of the correlation is shown in Fig. 9.21 [Coburn et al. 2002], which suggests that the actual relation may be more complicated than the above power-law. For Ec > ∼ 35 keV, there appears to be a saturation in the value of Ecut , as compared to those given by this power-law. This would seem to be rather supportive of the above notion of an indirect relation with B.
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Fig. 9.21 Cutoff energy Ecut vs. CRSF centroid energy Ec for accretion-powered pulsars. Reproduced with permission by the AAS from Coburn et al. (2002): see Bibliography.
More recently found correlations between the feature-width σc and other parameters are also interesting. The positive correlation between σc and Ec [Heindl et al. 1999, Coburn et al. 2002, and references therein] is not surprising, since, as we indicate in Chapter 13, the electrons responsible for the CRSF reside in quantized states at the neutron-star surface (also see above), and so constitute an effectively one-dimensional gas, able to move freely only along the magnetic field lines. This causes a Doppler shift between the rest-frame of the electrons, where resonance scattering must occur at the energy Ec , and the neutron-star surface,from where the Xrays are emitted, the fractional shift δEc /Ec being ∼ kTe /me c2 cos θ for a Maxwell-Boltzmann distribution of electrons characterized by a temperature Te , where θ is the angle between the photon’s propagation direction √ and B. This would lead to a width of the form σc ∝ Ec Te cos θ, readily explaining the observed correlation: the problem, as Coburn et al. pointed is the observed goodness of the correlation, in face of the smearing effect expected from variation of Te and cos θ from pulsar to pulsar. A resolution of this may be possible if (a) the electron temperature is itself tied to Ec , as is the case in some theoretical models [Lamb et al. 1990], and (b) we are viewing the systems along some preferred direction θ, or a small range of
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directions around it [Coburn et al. 2002]. The second point implies a selection effect, which tends to enhance the probability of detecting systems in a preferred range of θ-values, since there is no conscious observational bias. There seems to be some indication of this, as Coburn et al. (2002) argue, in the fact that most of the systems in which cyclotron features have been observed so far are either eclipsing ones or thought to be viewed nearly edge-on (i.e., i close to 90◦ ), thus possibly implying a preferred range of inclinations. For an excellent discussion of related points, we refer the reader to Coburn et al. (2002). These authors also reported a correlation between σc /Ec and τc , which appeared to be tighter than the one described above. 9.6.4
Emission Lines: Fluorescence, Recombination, Resonance
First discovered10 in Her X-1 by Pravdo and co-authors [Pravdo et al. 1977] with the OSO-8 satellite, characteristic emission lines of iron around the energy E ∼ 6.4 keV began to be detected in the X-ray spectra of accretionpowered pulsars in the latter half of the 1970s by detectors on board OSO-8 and HEAO-1. In their 1983 review, White et al. listed 11 pulsars displaying such iron lines, and dozens of pulsars showing these lines, as well as those characteristic of other elements, e.g., Ne, Mg, Si, S, Ar, and Ca, are known today (see below), such detections being possible as spectral resolution of our detectors improved progressively with the passage of time. We show in Fig. 9.22 a collection of pulsar spectra showing the iron line, taken from Nagase’s 1989 review: these were obtained with the Gas Scintillation Proportional Counter (GSPC) on the Tenma satellite, and are typical of the era. We shall see below what remarkable enhancements of spectral resolution became possible with subsequent generations of detectors. 9.6.4.1
Fluorescence
The identification of the first-discovered X-ray line with the Kα fluorescence line of Fe was immediate: it is the astrophysical analogue of the classic laboratory method of X-ray production. A metal atom, or an ion with one or more L-shell electrons still present in it, yields the Kα X-ray line when a K-shell electron is removed by an appropriately powerful stimulus, and an L-shell electron makes a radiative 2p − 1s transition to the vacated 10 The absorption edge of Fe at ∼ 7.1 keV was discovered even earlier by Swank et al. (1976), in the pulsar GX 301-2.
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Fig. 9.22 T enma X-ray spectra, showing the iron line in acctretion-powered pulsars. Reproduced with permission by the Astronomical Society of Japan from Nagase (1989): see Bibliography.
state. In accretion-powered pulsars, the stimulus is an energetic photon from the pulsar’s X-ray emission spectrum described above, i.e., the process is photoionization. Apart from the correctness of the expected energy for the Fe Kα lines with E ∼ 6.4 keV, corresponding to Fe ions having five or more electrons, i.e., Fe I – Fe XXII [Hatchett & McCray 1977], there is other, excellent evidence that these lines are in fact produced through fluorescence of the continuous X-ray spectrum emitted by the neutron star, since there is a general proportionality between the line intensity and the continuum intensity as they vary by ∼ one order of magnitude [Nagase 1989].
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Two points need to be clarified at the outset. First, why iron, and not some other element, since Fe is not the most abundant among the heavier elements present in the stellar matter accreting from the companion to the neutron star? The answer, as summarized lucidly by Hatchett and Weaver (1977) long ago, comes from basic atomic physics, contained in the idea of fluorescence yield. Following a photoionization, there can be either a K-fluorescence photon emission or the emission of an Auger electron11 , and the fluorescence yield ω is the probability of the former. As ω scales with atomic number Z of the element roughly as Z 4 , the most abundant elements C, N, and O have only ω < ∼ 0.01, and even elements like Ne, Si, and S have ω ∼ 0.1, while Fe has ∼ 0.3. Thus, iron has the highest value for the product of fluorescence yield and abundance among all the above elements, and so produces the strongest fluorescence signal . No wonder, then, that it was the first to be discovered, particularly since its Kα-line energy ∼ 6.4 keV is in a part of the spectrum which is relatively unattenuated during subsequent propagation. The second, and most crucial, point is where this matter is which fluoresces in the above manner. This will occupy a major part of our attention here, since this decides what diagnostic use we can make of the observed line properties, which contain information about the properties of the ionized gases which emit them, and so are potentially useful for probing accretion flows in the binary systems in which they are. It has been realized long ago where this matter is not [White et al. 1983]. Fluorescing material has to be relatively cool, and so will be in a state of relatively low ionization, which rules out any place close to the magnetic poles of the neutron star, from where the primary X-ray spectrum is emitted, since the intense radiation field there will fully ionize the matter. Similarly, the neutron-star surface is ruled out because any line generated there would show a substantial gravitational redshift in the line-frequency, which we do not see. The atmosphere of the companion star (i.e., that part of it which is illuminated by the X-rays from the neutron star, at any given orbital phase) is a possibility, and, indeed, was the first one to be studied in the 1970s. However, as detailed calculations showed, this source is too weak to explain the observed line-strengths, because of the relatively small solid angle subtended by the companion at the neutron star. Whereas the observed equivalent widths12 are in the range ∼ 102 − 103 eV, X-ray-illuminated companion 11 Also called autoionization, wherein the energy released by the electron falling into the vacated K-state is used to eject a second electron from the ion. 12 It is customary to measure the strength of a line by its equivalent width, defined as the width of an imaginary rectangle so centered on the actual line profile that the rectangle has the same area as that under the line profile, in an intensity vs. wavelength plot.
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atmospheres can only provide fluorescent emission with equivalent widths ∼ a few tens of eV. Where, then, is the bulk of the fluorescent material? In systems like the well-known pulsar Her X-1, which has a relatively low-mass companion, and is fed by Roche lobe overflow through an accretion disk, the flourescent matter may be in a shell at the Alfv´en surface, such as was considered once for explaining the soft X-ray flux from Her X-1, or in an accretion disk. But it is widely accepted today that, in HMXBs, which, by far, constitute the majority of the accretion-powered pulsars and of the systems displaying fluorescent emission, the site of fluorescence is the stellar wind from the massive companion. This is the site, therefore, which has been studied extensively in the literature, particularly over the last ∼ 10 − 15 years, and whose essential properties we summarize below. For this, we must first consider the basic phenomena that occur when compact X-ray sources are placed in stellar winds. 9.6.4.2
X-ray ionization of stellar winds
The essential physics of the interaction of a point-like X-ray source (the accreting neutron star) with the stellar wind driven by a large, massive companion was clarified by the pioneering work of McCray and coauthors in the mid-1970s and early 1980s [Hatchett & McCray 1977; Kallman & McCray 1982]. The X-ray continuum ionizes the matter in the wind: if matter is optically thin13 to the ionizing radiation, and in local ionization and thermal balance everywhere, then the ionization state and temperature at any point is determined by a single parameter, viz., the ionization parameter , defined by Tarter et al. (1969) as: ξ≡
L nr2
(9.31)
Here, L is the luminosity of the ionizing X-ray source, and n(r) is the number density in the wind at a distance r from the source, and we have assumed for simplicity that the X-ray source is isotropic. Contours of constant ξ are of prime importance in this problem, because (a) they are isothermal surfaces, and (b) they delineate distinct ionization zones with characteristic line-emission properties14 . 13 The optical depth τ is defined to be the line-integral of the opacity κ, weighted by matter-density ρ, along the line of sight: τ ≡ κρdl. Thus, κ has units cm2 g−1 in CGS system, while τ is dimensionless. Optically thin means τ 1 and optically thick, τ 1. 14 Note that ξ is not dimensionless, but has units of erg cm s−1 in CGS system.
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How do we obtain these contours? Their shape depends on the density profile n(r) of the wind, which, in turn depends on the velocity profile v(R) of the wind through the continuity equation n(R) =
M˙ w −2 ≈ 2 × 1010 M˙ 6 v8−1 R12 cm−3 . 4πµmp R2 v(R)
(9.32)
Here, M˙ w is the wind mass-loss rate from the massive companion, µ is the mean atomic weight of the wind-material, and R is the distance from the center of the companion. In the second form of the right-hand side of the above equation, we have scaled the physical variables in terms of their typical values in HMXBs, i.e., M˙ w in units of 10−6 M yr−1 , v in units of 108 cm s−1 , and R in units of 1012 cm, and have used µ ≈ 1.4 [Liedahl et al. 2001]. Note well the difference between this co-ordinate R, which is centered on the companion and determines the density structure, and the one introduced above as r, which was centered on the X-ray source and determined the radiation-field structure, as given by Eq. (9.31). It is this interplay between the two centers which suggests that, irrespective of the details, constant-ξ contours should have a basic bispherical-like geometry around the centers of the neutron star and the companion. This is indeed the case, as we now show. To determine the actual shape of the contours, it is customary to use the velocity profiles appropriate for winds from isolated O/B supergiant stars, which is of the form β R∗ , v(R) = v∞ 1 − R
(9.33)
although this may not be quite correct, as we shall discuss below. In the above equation, v∞ is the terminal velocity of the wind, and R∗ is the stellar radius. The exponent β determines the rate at which the wind accelerates to its terminal velocity: β = 0 corresponds to a constant-velocity wind, β = 0.5 corresponds to the profile first suggested by Castor and co-authors in the mid-1970s, and β = 0.8 is a value advocated by more recent work. Combining Eqs. (9.31), (9.32), and (9.33), we obtain for the ionization parameter: 2 β R R∗ , 1− ξ = ξ0 r R
(9.34)
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where ξ0 ≡
4πµmp v∞ L ≈ 50L36 v8 M˙ 6−1 erg cm s−1 . M˙ w
(9.35)
In Eq. (9.35), we have again scaled the physical variables in terms of their typical values in HMXBs to obtain the numerical form, with M˙ w and v∞ as before, and L in units of 1036 erg s−1 [Liedahl et al. 2001]. To obtain explicit contour shapes, we need to connect the two coordinates r and R for a given point, which we do by writing the second one as (R, θ), the angle θ being measured with respect to the line joining the two √ centers. The distance r from the X-ray source is then given by r = R2 + a2 − 2Ra cos θ, where a is the orbital separation between the centers of the two stars, and this, when inserted in Eq. (9.34), gives the actual contour-shapes. For the simple case of the constant-velocity wind (β = 0), the reader can easily show, as Hatchett and McCray (1977) did, that the contours, as given by 2 R ξ0 R 1− −2 cos θ + 1 = 0, (9.36) a ξ a are two sets of non-concentric spheres, corresponding to ξ < ξ0 and ξ > ξ0 , one set surrounding the neutron star, and the other, the companion. The boundary, or separatrix , between the two sets is the plane ξ = ξ0 , which passes through the mid-point of the line joining the two centers, and is perpendicular to this line. This, in fact, defines the classic bispherical geometry, and the corresponding co-ordinate system (see, e.g., Morse & Feshbach 1958). In Fig. 9.23, we show the contours for a β = 0.5 wind [Liedahl et al. 2000], with parameters typical of a pulsar like Vela X-1. The ξ-contours are clearly bispherical-like, and contours of constant density n are also shown. The above scale ξ0 for the ionization parameter is useful, so we consider it briefly. Note first that ξ approaches the constant value ξ0 far away from the system, which is clear from Eq. (9.34), since r ≈ R then, and R R∗ . What is more interesting, however, is that, at the separatrix mid-plane between the two stars described above, ξ exactly equals ξ0 for β = 0, and is ∼ ξ0 even for other values of β. Since most of the X-ray line emission from the wind is expected to come from between the two stars, this implies that ξ0 is a good indicator of the value, ξf , at the site where the line actually forms — the formation-ξ, if we wish [Liedahl et al. 2001]. The expected range of numerical values (in CGS units, which
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Fig. 9.23 X-ray ionization of stellar winds. Solid lines: ionization contours (labeled by values of log ξ). Dotted lines: density contours (labeled by values of log n). Diagonal lines mark shadow cone. Reproduced with permission by Rev. Mex. Astron. Astrofis. from Liedahl et al. (2000): see Bibliography.
< 3, follows directly from we shall use throughout) of this scale, 2 < ∼ log ξ0 ∼ Eq. (9.35), and the corresponding values of the ionization parameter are in the range 1 < ∼ log ξ < ∼ 4. Detailed calculations [Kallman & McCray ] 1982 give a variety of ionization states for various elements over this entire range of ξ-values, some ionized all the way upto the K-shell, and a given ionization stage occurring over only a limited range of values of ξ. For example, the relatively low-ionization (Fe I – Fe XVI, say) stages of iron, which characterize the fluorescent material (see below), occur at relatively low values, log ξ < ∼ 2, of the ionization parameter. The behavior of the above photoionized wind material is in sharp contrast to that of thermally, or collisionally, ionized plasmas commonly found in other astrophysical situations, notably inside normal stars. Whereas the degree of ionization in the latter material depends crucially on the temperature characterizing the collisional equilibrium, that in the former is determined by the above ionization parameter, which characterizes the photoionizational equilibrium: this means that the crucial dependence here is on the luminosity of the ionizing source, given a density profile n(r). As a result, photoionized plasmas around relatively luminous sources, such as the HMXBs we are discussing here, are highly overionized for their temper-
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atures, or, alternatively, are much cooler than we would expect from their ionization states. This is a classic signature of X-ray ionized plasmas. For example, in the line-dominated spectrum of Vela X-1 at eclipse, observed by Nagase and co-authors in 1994 (see below), the highly-ionized, He–like, atoms of Mg, Si, S, . . . were inferred to be in ionization zones where the electron temperature was < ∼ 100 eV, whereas temperatures of several keVs would be required in collisionally-ionized plasmas to produce such He-like ions in abundance. As we indicate below, more recent work with Chandra has suggested kTe ∼ 10 eV from the radiative recombination continua in Vela X-1, and kTe ∼ 100 eV in Cen X-3. A question of principle here is whether this high ionization of the wind caused by the X-rays from the neutron star would not hinder the winddriving mechanism. The physics is transparent: as the atoms become highly ionized, their capability for absorbing the UV line-radiation from the companion star, which drives the wind, becomes reduced. This was pointed out in the original Hatchett-McCray (1977) work, and has been confirmed by various studies in the 1980s and ’90s [see Liedahl et al. 2001 and references therein]. This would lead us to think that winds from the massive companions in HMXBs might be somewhat different from those from isolated OB stars: for example, the mathematical description given above may still roughly apply, but the terminal wind velocity v∞ may be smaller, and the index β may be somewhat different. Despite this, parameters corresponding to isolated, massive stars have been traditionally used for describing HMXBs. However, as we shall see later, pioneering observations from the X-ray satellite Chandra are beginning to indicate that the terminal velocity in HMXBs may, indeed, be smaller by factors ∼ 2 − 3 compared to the canonical value ∼ 1000 km s−1 for single OB stars. 9.6.4.3
Observations
With the launch of the satellite ASCA in 1993, a new era of X-ray spectroscopy of accretion-powered pulsars began, because the energy resolution ∆E/E of the Solid-state Imaging Spectrometer (SIS) on it was ∼ 2%, a great improvement over the ∼ 8% resolution of GSPCs (see above), and a vast improvement over the ∼ 20% resolution of usual proportional counters. SIS spectra began to reveal rich complexes of emission lines in the ∼ 0.5−10 keV energy-range, making one aspect of the X-ray spectra of HMXBs readily clear, namely, the remarkable difference between the spectra seen when the neutron star is eclipsed by the massive companion, and those seen away
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from the eclipse. While the continuous, pulsed spectrum dominates away from the eclipse, near the eclipse the softer part (E < ∼ 3 keV) of the spectrum is largely unpulsed and shows prominent emission lines, particularly at the pre-eclipse phases. At eclipse, the pulsed, direct component disappears, and the spectrum becomes remarkably flat, containing a profusion of emission lines. This is shown in Fig. 9.24 for Vela X-1, from the pioneering work of Nagase and co-authors (1994). The interpretation was straightforward: when the neutron star’s pulsed emission with relatively hard spectrum is eclipsed, the scattered component from the stellar wind, which has a rich collection of emission lines from the various ionization zones described above, dominates. Study of eclipse spectra thus became the standard tool for probing HMXB stellar winds, which is now yielding rich dividends in the era of the satellites Chandra and XMM-Newton, as we shall see below. In addition to the fluorescent Fe K-α (6.42 keV) and K-β (7.07 keV) lines from neutral or relatively low-ionization (Fe I – Fe XVI) iron, Nagase et al. (1994) also found strong K-α lines from He-like ions of Ne (0.92 keV), Mg (1.35 keV), Si (1.86 keV), S (2.45 keV), Ar (3.13 keV), Ca (3.89 keV), and Fe (6.67 keV). He–like ions are, of course, highly-ionized atoms (Fe XXV for iron, for example) with only the last two electrons of the atom left in the K-shell: hence the name. (Similarly, after one more electron is removed, we get H –like ions, e.g., Fe XXVI.). There is no question of fluorescent emission from such atoms. How do they emit K-α lines? The answer takes us to another line-emission process of great importance in photoionized plasmas.
9.6.4.4
Recombination
Ions surrounded by electrons in photoionized plasmas can recombine with these electrons, and emit photons in the process. This process of radiative recombination, which is only of marginal significance in hot, collisionallyionized plasmas (see above), becomes extremely important in cooler, photoionized plasmas. Radiative recombination proceeds in the two following steps. First, the free electron is captured into a bound state (n, l) characterized by the usual quantum numbers n and l, decreasing the ionization state by one, and emitting a photon whose energy lies in a continuum above the recombination edge, i.e., the binding energy Enl of the above bound state. This is the called the radiative recombination continuum, RRC for short. RRC from photoionized plasmas actually appear as narrow, “line-like” fea-
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Fig. 9.24 ASCA spectra of Vela X-1 at post-eclipse (top panel), pre-eclipse (middle panel), and eclipse (bottom panel) phases. Reproduced with permission by the AAS from Nagase et al. (1994): see Bibliography.
tures in the spectra, because the electron temperature Te is low for the state of ionization in these plasmas, as explained above, so that the relative line width ∆E/E ∼ kTe /Enl is small [Liedahl et al. 2001]. Recombination usually leaves the captured electron in an excited state, which then decays into the ground state in a series of spontaneous radiative transitions, or a radiative cascade. It is this second step, the radiative cascade, which is responsible for the profusion of emission lines from highly-ionized atoms of the above elements in the X-ray spectra of HMXBs. There is, of course,
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no difficulty about K-α emission from He-like ions by this mechanism: a H-like ion, with only one electron left, acquires another electron by recombination, which cascades radiatively down to the K-shell, giving a He-like ion in its ground state, and the last, L- to K-shell, transition gives the K-α line of such a He-like ion. We can name such a line from the He-like ion of a heavy element as simply the He–α line for that element. Similarly, the K-α line from a H-like ion of a heavy element, emitted when a fully-ionized atom (or bare nucleus) of such an element recombines with a free electron, and the resultant excited H-like ion undergoes a radiative cascade to its ground state, can be named a Ly–α line of that element, by analogy with the well-known nomenclature for the hydrogen atom. Subsequent ASCA observations of the well-known HMXB Cen X-3 by Ebisawa et al. (1996) confirmed the presence of similar lines, although the details were different because of the much higher luminosity of this pulsar, which leads to higher values of ξ for similar values of other variables. In addition to the fluorescence lines of Fe (see above), Si (1.74 keV), and Mg (1.25 keV) from their low-ionization states, these authors found recombination lines from He-like ions (or He-α lines) of Fe and Si (see above), and a profusion of recombination lines from H-like ions (or Ly-α lines) of Ne (1.02 keV), Mg (1.47 keV), Si (2.01 keV), S (2.64 keV), and Fe (6.97 keV). The difference from the above Vela X-1 spectrum is readily understood in terms of the higher luminosity, and therefore higher ξ-values, in Cen X3. As Nagase et al. (1994) had estimated, the formation-ξ (see above) in Vela X-1 of He-like ions of Mg, Si, and S is log ξf ≈ 2, and that of Fe is log ξf ≈ 2.9, while that of the H-like ions and bare nuclei of Fe in Cen X-3 is 3.4 < ∼ 3.8 [Ebisawa et al. 1996]. The ASCA spectrum of Cen X-3 ∼ log ξf < at eclipse is given by Liedahl et al. (2000). Extending the above study of K-shell recombination lines, we could consider Fe L-shell lines from photoionized plasmas, for ionization stages Fe XVII – XXI, say, i.e., intermediate between the low-ionization fluorescent stage and the highly-ionized K-shell emission stage. The analogy is with bright Fe L-shell lines from collisionally-ionized coronal plasmas, such as have been long-known in X-ray spectra of the Sun, and appreciated as good diagnostics of the density of such plasmas. In the early 1990s, such studies were undertaken by Liedahl, Kahn, and co-authors, who clarified the potential diagnostic value of these lines [Liedahl et al. 1992]. However, it was shown shortly afterwards by Kallman and the above authors [Kallman et al. 1996] that, in the photoionized plasmas occurring in accretion-powered pulsars (as opposed to collisionally-ionized plasmas, where the above Fe L-
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shell ions have unusually large rate coefficients), such lines would be completely swamped by the above K-shell recombination lines of elements like O, Ne, and Mg, which occur in the same energy range. Indeed, as summarized above, ASCA observations of the eclipse spectra of Vela X-1 and Cen X-3 in the same time-frame bore this out, clearly showing the dominance of K-shell lines from He- and H-like ions. This has now been confirmed further by recent Chandra observations of Vela X-1 (see below), in which very weak Fe L-shell lines have been detected, which probably originate from collisionally-ionized plasmas in those parts of the wind which are very close to the hot surface of the massive companion [Schulz et al. 2002]. 9.6.4.5
Recent observations
We now indicate the great improvements in energy resolution that have been possible with X-ray spectrometers aboard the recent satellites Chandra and XMM-Newton, and the consequent advances in our knowledge of line emission from accretion-powered pulsars. The High-Energy Transmission Grating Spectrometer (HETGS) on Chandra has an energy resolution which varies between ∼ 0.07% at ∼ 1 keV to ∼ 0.55% at ∼ 7 keV, which the reader can compare with the best earlier resolutions given above. In Fig. 9.25, we show the exquisite detail in which HETGS maps out the eclipse spectrum of Vela X-1 [Schulz et al. 2002], clearly showing the multitude of fluorescence and recombination lines, and also RRCs (see above). Note that the spectrum is shown here in terms of the wavelength λ of the X-rays, which is related to the energy E of the X-ray photons by λ = hc/E, so that λ(˚ A) ≈
12.4 , E(keV)
(9.37)
and the ∼ 3 − 12 ˚ A wavelength range in Fig. 9.25 corresponds to an energy range of ∼ 1 − 4 keV. Fluorescence lines from a variety of low-ionization states of the elements described earlier were observed by these authors, as the figure shows: for mid-Z elements like S, Si, and Mg, some individual ionization states could be resolved, e.g., S(IX), Si(VI, VII, VIII, IX), but only a range of likely states could be inferred in other cases, e.g., S(IVVIII). For Fe, Ca, and Ar, only a range of states could be inferred, e.g., Fe(II-XI). A magnified version of the narrow wavelength region ∼ 6 − 7.5, ˚ A, which displays the details of the Si lines, is shown in Fig. 9.26 [Sako et al. 2003]. The distinct K-α fluorescent lines from various charge states, which are separated by ∼ 70 m˚ A per charge and are, therefore, well-resolved at the above resolution of Chandra HETGS, are clearly seen.
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Fig. 9.25 Chandra HETGS spectrum of Vela X-1. Shown are recombination lines (red), fluorescence lines (green), and highly ionized lines (blue). Reproduced with permission by the AAS from Schulz et al. (2002): see Bibliography.
Detection of RRCs is clear in Fig. 9.25, for example that of Ne X at 9.1 ˚ A. From the width of the feature, the electron temperature can be readily estimated as kTe ≈ 10 eV, i.e., Te ≈ 1.2 × 105 K [Schulz et al. 2002]. This low temperature is entirely consistent with the general behavior of photoionized plasmas indicated above. Similar results have been obtained from Chandra eclipse spectra of Cen X-3 [Wojdowski et al. 2003], where kTe ≈ 100 eV, i.e., a warm, photoionized plasma is indicated from the width of the RRCs. Strong recombination lines from H- and He-like ions of S, Si, Mg, and Ne are also observed, as can be seen in Figs. 9.25 and 9.2615 . For example, for 15 Note that Fig. 9.25 is the spectrum at eclipse, while Fig. 9.26 is that at phase φ = 0.5, i.e., 180◦ away from eclipse. However, as Sako et al. (2003) have demonstrated, the line spectra at these two phases are almost identical in shape, differing only by an overall normalization factor. We discuss this remarkable result later. Here, we only use this result to show Fig. 9.26 as a good representation of the magnified version of a small part of Fig. 9.25.
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Fig. 9.26 Chandra HETGS spectrum of Vela X-1: detail showing the “forest” of Si K lines. Kα lines from eight charge states are detected in this narrow waelength-range. Reproduced with permission by M. Sako & G. Branduardi-Raymont from Sako et al. (2003): see Bibliography.
Ne X, four lines of the Lyman series, Ly-α, β, γ, δ are detected, and the Ly-α line of Si XIV is prominent in Fig. 9.26. It is the He-α lines, however, which play the dominant role in plasma diagnostics. The He-α triplet of Si XIII is shown in detail in Fig. 9.26, in which the different lines arise from the following transitions: • The intercombination, or x and y, lines come from the two transitions 3 P2,1 → 1 S0 . These are at ≈ 1.85 keV (not resolved even by the Chandra spectrometer) for the above Si XIII He-α triplet. • The forbidden, or z, line comes from the transition 3 S1 → 1 S0 , and is at ≈ 1.84 keV for the above triplet. • The resonance, or w, line comes from the transition 1 P1 → 1 S0 , and is at ≈ 1.86 keV for the above triplet. The relative strengths of these x, y, z, and w lines are well-known diagnostics of the state of the plasmas that emit them. In collisionallyionized plasmas, w is the brightest line, while in photoionized plasmas, z is the brightest line in the low-density limit, and, at higher densities, the blend x+y can be brighter than z. Thus, the ratio G ≡ (x+y +z)/w should
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be a good diagnostic of whether collisional ionization dominates, in which case G ≈ 1, or photoionization does, in which case we expect G ∼ 3 − 4 [Liedahl et al. 2001]. It also follows from the above description that the ratio R ≡ z/(x + y) should be a good density diagnostic, since R decreases with increasing electron density ne , roughly as R ∝ 1/[1 + (ne /nc )], where nc is a critical density, whose value for He-like Fe XXV, for example, is ∼ 1017 cm−3 . However, the actual application of these diagnostics to accretionpowered pulsars may require more care: as the reader can see in Fig. 9.26, the G ratio appears to be close to unity, and yet, as argued above, we have every reason to believe that the wind in Vela X-1 or Cen X-3 consists largely of warm, photoionized plasma. The resolution of the problem comes from the realization that resonance scattering can enhance the strength of the resonance line by factors upto ∼ 4, and so bring G down appropriately [Schulz et al. 2002; Wojdowski et al. 2003]. What is resonance scattering? Transitions with large oscillator strengths have the property that they can be excited directly by photons, and the subsequent decay of such photoexcited ions then adds to the strength of the lines emitted by these transitions. For H- and He-like ions, the photoexcitation from the ground (n = 1) state to the n = 2 state is almost invariably followed by the exact inverse process for the n = 2 → 1 decay. This is called resonant scattering [Wojdowski et al. 2003]. Now, it turns out that, for He-like ions, the w line has a large oscillator strength (this is also true for the Ly-α line of H-like ions), but the x, y, and z lines do not. Hence, the w line in the He-α triplet is preferentially enhanced by resonant scattering of the X-ray photons from the neutron star by the photoionized wind plasma, reducing the G ratio to ≈ 1. This idea is supported by the observation that the above enhancement is seen in Si He-α triplet in Cen X-3 in eclipse, but not outside eclipse, as explained by Wojdowski et al. (2003). During the eclipse of the neutron star, only the wind is visible, and we get the enhancement described above. Outside eclipse, however, we see the direct radiation from the neutron star as well: this continuum radiation has a depletion of photons at the w-line frequency, due to the resonant scattering of these photons out of the direct beam. In the total spectrum, the enhancement and depletion at the w-line frequency roughly cancel each other, and we get a value of G roughly characteristic of pure recombination.
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9.6.4.6
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Wind diagnostics
We indicated above that diagnostics of the conditions in the stellar wind is a major purpose of the study of X-ray emission lines from HMXBs: we now summarize some major lessons learnt from these studies. One basic point has been established since the days of ASCA and Ginga. The cool, low-ionization, fluorescing material should have a smaller characteristic size than the warm, highly-ionized, bulk of the wind emitting through recombination, since the strength of the fluorescent emission drops (sometimes drastically) at eclipse, when the companion blocks from our view the much of the fluorescent region, but little of the extended photoionized wind [Ebisawa et al. 1996]. The size of the companion (∼ 10−15R) is thus an upper limit to the size of the fluorescent region. A stricter limit of ∼ 3 × 1010 cm was obtained for Cen X-3 by Nagase and co-authors from the Ginga observation that the intensity of the Fe K-α line dropped by a factor ∼ 20 in less than 10 minutes, as the system entered eclipse. If we assumed that this fluorescent region was disjoint from the stellar wind and surrounded 10 the neutron star at distances < ∼ 3 × 10 cm from it, we could ask what its physical nature was. An accretion wake, trailing the neutron star, a shell of matter at the Alfv´en surface (see Chapter 10), or the inner part of an optically-thick accretion disk could all be regarded as possibilities. Alternatively, we could consider the possibility that the stellar wind is inhomogeneous, consisting of cool, dense, low-ionization, fluorescing clumps of matter, embedded in a warm, highly-ionized outflow emitting recombination lines [Sako et al. 2003]. In this picture of Vela X-1, the fluorescing clumps may contain most (> ∼ 90%) of the total wind mass, while the highly ionized matter occupies most (> ∼ 95%) of the total wind volume. Of course, these dense clumps would be ionized to a greater extent in a more luminous pulsar like Cen X-3 than in Vela X-1, as discussed earlier. In the era of Chandra and XMM-Newton, other wind diagnostics are becoming possible because of the excellent energy resolution of the spectrometers on these satellites (see above). A major advance has been the ability to actually measure the motion of wind material, and it is easy to see why this is so. In order to resolve the (Doppler) line shifts and widths corresponding to the expected, typical wind velocities vw ∼ 1000 km s−1 , we need a spectral resolution of better than vw /c ∼ 0.3%, which became possible with only the above satellites [Sako et al. 2003]. However, these measurements have yielded a surprise (which often comes with the first possible direct observation in a field), viz., that the average line shifts are
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|vw | ∼ 300 − 400 km s−1 , considerably less than the above expected values. Attempts to understand this are under way: one obvious possibility is closely connected with the conceptual point discussed earlier, namely, that photoionization of the wind by the X-rays from the neutron star reduces the effectiveness of the UV radiation from the companion for driving the stellar wind, and so reduces the wind’s terminal velocity by the above factor. We close our account of line emission from accretion-powered pulsars with another recent, if unexpected, result from Chandra observations, which reveals a structural complexity of the stellar winds in HMXBs that we are just beginning to appreciate. In their study of Vela X-1 spectra, Sako et al. (2003) compared the line spectra at phases φ = 0 and 0.5 (i.e., at eclipse and at a phase 180◦ before or after it), and found that (when normalized by adjusting the peak value of a well-known line, e.g., the Si XIII z-line, to be the same in both cases) the two were nearly identical, to within ∼ 20%. This is in direct contradiction to our expectation from a spherically symmetric description of the stellar wind, which is that there would be much higher ionization at eclipse, due to the fact that much denser wind material, close to the companion, is photoionized at that phase by the radiation that reaches us. It can be argued that, in contrast to the situation for isolated massive stars, the wind profile in HMXBs is not really expected to be spherically symmetric, because of preferential photoionization of that part of the wind which faces the X-ray emitting neutron star at any given phase, and its effects on the wind-driving mechanism, as explained above. However, what the above observation tells us is that, apart from an overall normalization difference (i.e., much higher total emission at φ = 0.5), the spectra are nearly identical in shape at phases φ = 0 and 0.5, which means that the ionization structure of the above, preferentially illuminated part of the wind (which is roughly a cylinder joining the two stars, with radius comparable to that of the companion) is nearly identical to that of the rest of the wind. The reason for this identical behavior remains to be understood fully. Of course, it is this behavior which allowed us to consider Figs. 9.25 and 9.26 together, although, strictly speaking, the former is at φ = 0 [Schulz et al. 2002], while the latter is at φ = 0.5 [Sako et al. 2003].
9.7
Mutants: Anomalous X-Ray Pulsars (AXPs)
Continuing with the biological terminology we introduced in Chapter 6 for describing evolutionary origins of various types of pulsars — single or
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binary — we now give a brief account of a type of pulsars which has been somewhat of a mutant, not fitting easily into the categories described there. These are the anomalous X-ray pulsars or AXPs, which we mentioned in Chapter 1. Originally thought to be accretion-powered pulsars, either with very low-mass companions similar to the case of the system 4U 1626-67, or with more exotic scenarios for accretion (see, e.g., Ghosh et al. 1997; Woods & Thompson 2006), they are believed today to be single neutron stars with superstrong magnetic fields ∼ 1013 − 1015 G, spinning down by electromagnetic torques as rotation-powered pulsars do (see Chapters 1 and 7), and yet being “anomalous” in many ways, notably that (a) their emission is primarily in the X-rays, with pulse profiles very similar to those of canonical accretion-powered pulsars (which is one of the reasons why they were earlier thought to be accretion-powered), and very different from those of canonical rotation-powered pulsars, and, (b) their spin-down luminosities (see Chapters 1 and 7) are quite tiny compared to their observed X-ray luminosities of 1035 − 1036 erg s−1 . They are thus thought to be magnetars — a class of neutron stars with superstrong magnetic fields generated by unusual processes (we return to this point in Chapter 13) — and so rather similar to the sources called soft gamma repeaters or SGRs. It was realized by the mid-1990s that there was something unusual about this class of X-ray pulsars, which then had about half-a-dozen members including 4U 1626-67. They all had (a) pulse periods in the narrow range ∼ 5 − 9 sec, (b) fairly steady X-ray luminosities Lx ∼ 1035 − 1036 erg s−1 , and with the exception of 4U 1626-67, (c) relatively soft spectra, and (d) a steady spindown. There was no obvious evidence for the presence of a binary companion indicated by characteristic Doppler shifts. This led to the early conclusion that these were LMXBs with very low-mass companions, as summarized by Hellier (1994) and by Mereghetti and Stella (1995), the analogy with 4U 1626-67 being helpful, since the binary nature of the latter was strongly suggested by the Fourier characteristics of its optical pulsations. In the same time-frame, it was also suggested by van Paradijs et al. (1995) and Ghosh et al. (1997) that AXPs with the exception ˙ of 4U 1626-67 could be remnants of the Thorne-Zytkow objects described in Chapter 6, wherein an accretion disk forms around the central neutron star in this object as its envelope is driven away, with accretion powering the observed X-ray pulsar. (4U 1626-67 and similar objects could form in the same basic evolutionary scenario when complete spiral-in did not occur in the CE phase: see Chapter 6). A variation of this remnant (or “fossil”) accretion disk idea was explored a few years later, wherein such
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disks formed when part of the material ejected in the supernova explosion accompanying the birth of the neutron star “fell back” on the star later [Chatterjee et al. 2000]. About eight AXPs are known now, with pulse periods in the range ∼ 5 − 12 sec, as shown in Table E.2 in Appendix E. The emphasis has shifted in the last few years to the view that AXPs are single neutron stars with superstrong magnetic fields — so-called magnetars, as explained above — and their observed spindown is the usual electromagnetic braking of rotation-powered pulsars, as described in Chapters 1, 7 and 12. Interpreted this way with the aid of the expression for spindown rates given in these chapters, the observed spindown rates of these AXPs, P˙ ∼ (0.05 − 4.2) × 10−11 s s−1 lead to magnetic field strengths B ∼ (0.60 − 7.1) × 1014 G at the surfaces of the neutron stars in them [see Woods & Thompson 2006 for a review]. Such field strengths are rather similar to those inferred for SGRs (see above) — transient sources which show repeated outbursts in soft gamma-rays (and also at other wavebands), as opposed to the usual gamma-ray bursts, which do not repeat — which display oscillations during their outbursts with periods in a similar range ∼ 5 − 8 sec, and also have similar spindown rates. Naturally, a connection between these two classes of astrophysical objects has been suggested and studied in recent years, the notion receiving further support from the discovery of occasional bursts from some AXPs. There are many features of AXPs that are consistent with the magnetar scenario. The estimated AXP ages indicated that they seemed to be too young to be LMXBs (see Chapters 6 and 9), while a magnetar would naturally be a young neutron star. Indeed, some AXPs seem to be associated with supernova remnants, and their distance as a whole from the galactic plane is relatively small, indicative of a young population. Further, persistent, monotonic spindown over 10–20 years is reminiscent more of rotationpowered pulsars than of accretion-powered ones. The recent discovery of glitches in AXPs and SGRs is also reminiscent of the same connection. We refer the reader to the above review by Woods and Thompson (2006) for detail. However, there are still some “mutant” AXP features whose origin is unclear. The pulse shapes of AXPs show gradual variations over the pulse period, entirely like the those of accretion-powered pulsars and perhaps suggestive of thermal emission from a stellar surface heated by a suitable mechanism, but entirely unlike the sharp spike-like pulses characteristic of the non-thermal emission from rotation-powered pulsars. Of course, the energy budget here is exactly the opposite of those of rotation-powered
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pulsars. For the latter, only a tiny fraction of the total spindown luminosity appears in the pulsed radio emission, while for AXPs the pulsed X-ray emission carries far more than the spindown luminosity. Thus, the bulk of an AXP’s power must come from sources other than rotation, and this is believed to be magnetic energy in the magnetar model. We return to this point in Chapter 13. Thus, in this model, it is basically only their electromagnetic spindown (and possibly the associated glitch-behavior) which relates AXPs to rotation-powered pulsars. Clearly, the crucial proving ground for the fascinating magnetar concept would be an independent confirmation of superstrong magnetic fields of above strengths in neutron stars. Evolutionary scenarios for AXPs in the magnetar view are still in their infancy. These mutants appear at the top right-hand corner of the P − P˙ diagram (see Chapter 7) or P − B diagram (see Chapter 12) of pulsars, as expected, above and to the right of the “pulsar island”, but it is not clear yet if they form a separate cluster, or there is a continuous progression from them to the pulsar island. This is, of course, closely connected to the question of the character of magnetic field decay at such high field strengths. Also interesting is the related question of the possible existence of magnetars in binary systems, and their possible connection with accretionpowered pulsars like GX 1+4 with high (B ∼ 1014 G) inferred values of the neutron-star magnetic field. Finally, the narrow range of pulse-periods of AXPs (and also of SGRs) seems to have remained basically as much of an enigma as they were when these systems were first discovered. Neither the older accretion-related scenarios nor the recent magnetar scenarios appear to have yielded so far a compelling reason for this close bunching of pulse periods in this specific range.
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Chapter 10
Pulsar Magnetospheres
Neutron stars, whether observed as rotation or accretion powered pulsars, have a canonical magnetic field on the surface is which is very high, Bs ∼ 1012 G, although considerably lower and higher fields are also possible. It is natural, therefore, that there would be a region around such a magnetic neutron star where the magnetic field exerts the dominant force determining the motion of the plasma, whether it is accreting on the star, or flowing away from it. This region is called the magnetosphere: the term was coined by Gold in 1959 [Vasyliunas 1979], for describing a somewhat similar region that forms around the Earth, which, of course, has a significant magnetic field (although tiny compared to that of pulsars), and is immersed in the plasma streaming towards it from the Sun — the solar wind. This is called the geomagnetosphere, the study of which is a fascinating subject, particularly for understanding near-Earth charged-particle phenomena like aurorae, and for clarifying Earth-Sun connections. Studies of the planetary magnetospheres around other planets, e.g., Jupiter, are also interesting in their own rights. In this book, however, we shall focus entirely on pulsar magnetospheres, making only occasional references to planetary ones: indeed, as we shall see, the latter may be quite different from the former.
10.1
Magnetospheres of Accretion-Powered Pulsars
Matter accreting from the binary companion has a flow pattern far away from the neutron star on which the latter’s magnetic field has no effect at all. Yet, it is essential to know the pattern of this exterior flow , because it determines how the neutron-star magnetic field interacts with the acrreting plasma, and so decides the size and shape of the magnetosphere. 495
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The exterior flow depends on the nature of the companion, and on the binary separation. In LMXBs, this flow almost invariably results from Roche lobe overflow, which occurs when the size of the low-mass companion exceeds that of the limiting equipotential around it which can confine it (see Appendix B), because the companion expands in the course of its nuclear evolution, and/or because the binary system shrinks due, e.g., to magnetic braking or the emission of gravitational radiation. In HMXBs with O/B supergiant companions and a sufficiently large binary separation, the exterior flow is simply the strong stellar wind driven by the companion. For smaller binary separations, the atmosphere of the supergiant may begin to overflow the Roche lobe, and both of the above kinds of exterior flows may occur simultaneously, as may be the case in systems like Cen X-3. In HMXBs with Be-star companions, the companion is well inside its Roche lobe, but the outflow of matter from it is more complicated. Instead of driving spherically-symmetric winds, as isolated O/B stars are expected to do, and as similar stars in HMXBs are also expected to do except for modifications due to selective photoionization of those parts of the wind facing the X-ray emitting neutron stars, Be stars, which are rapidly rotating, shed mass in the form of rings of rotating matter, which spread through viscous forces (see below) to form outflowing disks of matter in the (rotational) equatorial plane of these stars, whether they are isolated or in accretionpowered binary X-ray pulsars. It is these disks which drive the exterior flow around the accreting neutron star with a Be-star companion. 10.1.1
Exterior Flow and Plasma Capture
The dynamical pattern of the exterior flow basically depends on its angular momentum content. A stellar wind emitted by a supergiant companion well within its Roche lobe has little angular momentum with respect to the companion, and so is essentially a radial outflow centered on the companion (see Fig. 10.1). Whether this flow can have, under certain conditions, an angular momentum with respect to the neutron star which is significant enough to have a dynamical effect on the accretion process is a point we discuss below. By contrast, a Roche lobe overflow, whether in an LMXB or in a sufficiently close HMXB, always produces a stream of matter which has a large angular momentum with respect to the neutron star. As a result, this stream goes into orbit around the neutron star, forming a ring, which then spreads through viscous forces, forming an accretion disk around the neutron star (see Fig. 10.1). By a similar argument, the rotating matter in
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Fig. 10.1 Schematic representations of HMXB and LMXB, showing the essential features. Reproduced with permission by Cambridge university Press from Tauris & van den Heuvel (2006): see Bibliography.
the outflow disk of a Be star has so much angular momentum that, when the neutron star passes through this disk during the course of its binary motion, the matter accreting on the neutron star has a large angular momentum with respect to the latter, and so forms an accretion disk. The process of accretion is, of course, the capture of plasma from the above two types of exterior flow, which we now consider in some detail.
10.1.1.1
Capture from stellar winds
We have already introduced earlier the essential physics of the capture of plasma from stellar winds: the effective capture radius is the accretion radius, ra ∼ GMns /v 2 , determined by the velocity v of the wind relative
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to the neutron star1 , and the neutron-star mass Mns , so that the effective capture cross-section is πra2 , and the mass-accretion rate (often called just accretion rate) is given by M˙ = πra2 ρ0 v,
(10.1)
where ρ0 is the plasma density in the vicinity of the accretion radius. More quantitative expressions for the accretion radius are written in the form ra =
2ξGMns ∼ 1010 mns v8−2 cm, v2
(10.2)
where mns is the neutron-star mass in solar units, as before, and the assumed dynamical details of the plasma-capture process determine the dimensionless parameter ξ ∼ 1. For example, if we assume that ra represents the radius inside which the escape velocity from the neutron star exceeds v, then ξ = 1, as the reader can easily show. Note first that the above description of ra assumes v to be supersonic, since it neglects the effects of the thermal-energy content of the wind plasma. This would not appear to be unjustifiable if vw ∼ 108 cm s−1 , as it was thought to be in massive X-ray binaries with OB companions by analogy with isolated OB stars, 7 −1 and vorb > ∼ 10 cm s . Recent inference of considerably lower values of vw from Chandra observations of massive X-ray binaries does not invalidate this assumption, as v would still be supersonic at these values2 , but it shows that, in sufficiently close binaries, vorb may well be comparable to vw . Of more concern is the fact, readily apparent from Eq. (10.2), that the accretion radius, which scales as v −2 , can be easily underestimated by an order of magnitude by an overestimate of v by a factor ∼ 3. This is even more serious for the accretion rate, which scales as v −3 , as the reader can show by combining Eqs. (10.2) and (10.1). A major question that arises is whether plasma captured from the above stellar wind has enough angular momentum to form an accretion disk around the neutron star. While a wind with a constant density and a constant velocity relative to the neutron star will necessarily lead to the accretion of exactly zero angular momentum by the neutron star, as must 1 This means that v = v − v w orb , where vw is the wind velocity far from the neutron star, and vorb is the orbital velocity of the neutron star. For a circular orbit, for example, 2 + v2 . v = vw orb 2 In any case, Bondi and Hoyle had suggested in 1944 that Eq. (10.2) could be extended to subsonic flows by replacing v2 in the first form of the right-hand side to v2 + c2s , cs being the sound speed in the wind material far from the neutron star.
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be intuitively obvious from symmetry considerations, and as we shall show below explicitly, such a stellar wind is impossible. This becomes clear when we look at the continuity relation that describes the flow of a sphericallysymmetric wind from a massive OB companion with a mass-loss rate M˙ w : M˙ w = 4πr2 ρvw ,
(10.3)
r being the distance from the center of the massive star. Even when vw is constant, as it appears to be far away from the surface of the winddriving star, ρ ∼ r−2 , and the density varies even more strongly in the “acceleration region” of the wind near the stellar surface, since vw increases there with increasing r. The arguments are qualitatively the same when the massive star drives its wind over a more restricted solid angle than 4π. Thus, even in a steady wind with no fluctuations, a density gradient is inevitable. In addition, a gradient in the relative velocity v of plasma with respect to the neutron star is also inevitable, even when the wind velocity vw is constant, due to variations in the orbital velocity, as we show below. These spatial gradients in ρ and v across the capture cross-section lead to an imbalance between the angular momenta of opposite signs accreted through different parts of the capture cross-section, and so to a net angular momentum accretion, breaking the above symmetry.
Fig. 10.2 Estimating angular momentum content of matter captured from stellar winds. Reproduced with permission by the AAS from Shapiro & Lightman (1976): see Bibliography.
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The basic calculations, pioneered in the 1970s by Shapiro and Lightman (1976) and others, are easy, and we give a flavor of them here. In Fig. 10.2, we consider the accretion cylinder of the neutron star in the Shapiro-Lightman (1976) picture: this cylinder is of radius ra and oriented along the relative velocity v, which is tilted at an angle α = tan−1 (vorb /vw ) with respect to the line joining the centers of the two stars forming the binary system, the distance between the two stellar centers being a. We set up a two-dimensional cartesian co-ordinate system (x, y) lying in the crosssectional plane of this cylinder, with the x-axis perpendicular to the orbital plane, as shown in Fig. 10.2, r being the radius vector from the center of the massive companion to the point (x, y). Then the center of the (circular) cylinder face corresponds to x = 0 = y and also to r = a, and the gradients in ρ and v across the cylinder face enter as follows through the standard Taylor expansion around the above cylinder-face center: dv dρ y sin α, v(x, y) = v(a) − y sin α, (10.4) ρ(x, y) = ρ(a) − dr a dr a assuming the wind to be spherically symmetric, as above. (If it is not, there will be additional gradient terms, of course.) The rate of accretion of ˙ through the total cross-sectional area is given by angular momentum, J, J˙ = ρ(x, y)v 2 (x, y)ydxdy, (10.5) which follows directly from the observations that (a) mass is accreted at the rate ρ(x, y)v(x, y)dxdy through an infinitesimal area dxdy at the position (x, y) on the cylinder face, and, (b) angular momentum per unit mass, or speicific angular momentum — a term we shall use constantly in this book, of this matter is v(x, y)y. It is now straightforward to use Eq. (10.3) to evaluate the density gra= −2ρ(a)/a. Next, we evaluate dient in Eq. (10.4). We get (dρ/dr)a 2 + v 2 (r) (see above), where the velocity gradient by using v(r) = vw orb 3 vorb (r) = Ωorb r . This yields (dv/dr)a = (v/a)(vorb /v)2 , the velocities on the right-hand side of this equation being evaluated at r = a. The first point to note is that the two gradients contribute with opposite signs to the net angular momentum flux [Shapiro & Lightman 1976], as expected, since ρ and v generally change in the opposite sense with increasing r, as discussed above. Note next that the latter contribution is smaller than the 3 A point on the cylinder face at a distance r from the massive companion’s center moves with a velocity Ωorb r with respect to this center [Shapiro & Lightman 1976].
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former by the factor (vorb /v)2 , and so was thought to be negligible when the Shapiro-Lightman work was done in 1976, since vw was then thought to be ∼ 108 cm s−1 , in analogy with typical values for isolated OB stars, as discussed above, which made vorb /v < ∼ 0.1. However, recent Chandra observations indicate vw ∼ 3 × 107 cm s−1 (see above) in massive X-ray binaries, implying that vorb /v could well be < ∼ 1 for close binaries, and the latter contribution significant. Nevertheless, it does not appear that the two contributions can be so close as to nearly cancel each other, giving a net value much smaller than either of them. Rather, the former contribution, i.e., that from the density gradient, is still a reasonable order-of-magnitude estimate of the net value, as was argued by Shapiro and Lightman (1976). ˙ we evaluate the right-hand side of Eq. (10.5) To obtain this estimate of J, by substituting the above value of the density gradient, neglecting the velocity gradient, transforming from the above cartesian (x, y) co-ordinates to polar (, θ) co-ordinates, and integrating over the accretion cylinder’s cross-section (0 ≤ ≤ ra , 0 ≤ θ ≤ 2π). The first, constant term (i.e., that which we have in absence of all gradients) vanishes identically by symmetry upon angular (θ) integration, confirming explicitly the above statement that gradients are essential for generating the asymmetry, and so the imbalance, that leads to non-zero accretion of angular momentum. The gradient term integrates to a value which can be written as J˙ = M˙ vorb ra2 /(2a) with the aid of Eq. (10.1), so that the specific angular momentum of the accreted matter is given by: ˙ M ˙ = 1 Ωorb r2 , w ≡ J/ a 2
(10.6)
which is the Shapiro-Lightman (1976) result. Is this angular momentum w large enough for an accretion disk to form? To answer this question, we use essential properties of such disks. In a Keplerian accretion disk formed around a neutron star of mass Mns , the angular velocity at a radius r from the neutron star’s center is given by ΩK = GMns /r3 , and, consequently, the specific angular momentum of the disk plasma at this radius is given by: K = ΩK r 2 =
GMns r.
(10.7)
Since K increases with increasing r, it follows that plasma captured from a wind with the above specific angular momentum w can form an accretion
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disk whose outer radius rout is given by the condition w = which gives rout =
2w . GMns
√ GMns rout ,
(10.8)
Inside rout , the rotating disk matter spirals inward due to the removal of angular momentum by effective viscous stresses that operate in the disk, as explained later in this chapter, and ultimately accretes onto the neutron star through processes described later in this chapter. The question then is: is an accretion disk with an outer radius as given above possible for an accreting neutron star? Why would it not be? An obvious reason is that, if there is a lower limit rin to the radius that a disk must have due to some other piece of essential physics involved, then rout ≥ rin for consistency, and this condition must be satisfied for an accretion disk to exist. For an unmagnetized neutron star, this lower limit is either the stellar radius Rns ≈ 106 cm or the radius of the innermost stable orbit rms around a mass Mns ≈ 1.4M as given by the general theory of relativity (i.e., rms = 6GMns /c2 ≈ 106 cm), whichever is larger. Whether Rns or rms is actually larger depends on the equation of state of the neutron-star matter (see Chapter 5). For a black hole, as considered in the pioneering work of Shapiro and Lightman (1976), the lower limit is, of course, rms corresponding to the mass of the black hole MBH ∼ 10M in the prototypical black-hole binary Cyg X-1 (see Chapter 1), i.e., rms = 6GMBH /c2 ∼ 9×106 cm. However, our main concern in this book is with magnetized neutron stars, i.e., accretion- and rotation-powered pulsars, for which this lower limit is the radius at which the disk is terminated by the magnetic stresses due to the neutron-star magnetic field, which, in turn, is roughly approxi4/7 mated by the Alfv´en radius rA ≈ 3 × 108 B12 M˙ −2/7 (M/M )−1/7 cm (see below). Thus, we need to compare rout with rA for accretion-powered pulsars, and with the other radii described above for unmagnetized neutron stars or black holes. We do this by first expressing rout in terms of the binary orbital separation a in the form −3 −4 Mtot rout v2 = 4ξ 4 , (10.9) 1 + 2w a Mns vorb where Mtot ≡ Mns + Mc is the total mass of the binary. Equation (10.9) is obtainedby combining Eqs. (10.8), (10.6), (10.2), and Kepler’s third law, 2 + v 2 (r) that holds Ωorb = GMtot /a3 , as also the relation v(r) = vw orb for circular orbits, as explained above. As indicated above, ξ ∼ 1, and the
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ratio Q ≡ Mtot /Mns of the total to the neutron-star mass is ∼ 10 − 30 for HMXBs with OB companions (since Mc ∼ (10 − 40)M in these systems and Mn s ∼ 1.4M ), so that Eq. (10.9) leads to the estimate rout /a ≈ 2 2 /vorb )−4 , where Q1 is Q in units of 10. Now, using 4 × 10−3 Q1 ξ 4 (1 + vw Kepler’s third law (see above) to relate a to the orbital period Porb again, we can write this estimate in the numerical form: 2/3 −4 2 Porb vw 4/3 rout ≈ 3.8 × 109 Q1 ξ 4 cm, (10.10) 1 + 2 5d vorb where we have scaled Porb in units of a rough average value of 5 days, in view of the fact that the orbital periods of known HMXBs with OB companions are in the range 1d − 10d [van den Heuvel 2001]. Immediately noteworthy in Eq. (10.10) is the extreme sensitivity of rout to the ratio vw /vorb of the wind velocity to the orbital one, which makes our conclusions qualitatively dependent on this ratio, and, as we shall see below, somewhat different from what was believed to be true in the 1970s. Using Kepler’s third law again to relate vorb to Porb and Mtot , the reader can easily show that 2/3 −2/3 2 Porb Mns vw −2/3 2 ≈ 11.1 Q1 v8 , (10.11) 2 vorb 5d 1.4M where v8 is v in units of 108 cm s−1 , as before. This shows that if v8 ≈ 1, 2 2 /vorb ∼ 10, while the as was earlier believed to be the case (see above), vw 2 2 ∼ 1. Thus, recent Chandra estimates v8 ≈ 0.3 (see above) lead to vw /vorb 4 the above sensitive factor changes by ∼ 10 between these two possibilities, and, not surprisingly, some conclusions undergo major revision. For the pioneering Shapiro-Lightman (1976) calculations, the former estimate was used, so that rout ∼ (4 − 8) × 105 cm, which is not only tiny compared to the Alfv´en radius rA for a magnetic neutron star, but also smaller than the radii Rns and rms relevant for an unmagnetized neutron star (see above). Naturally, the Shapiro-Lightman conclusion was that accretion disks cannot form around neutron stars accreting from stellar winds. By contrast, argued these authors, black holes accreting from winds of OB supergiant companions can have small accretion disks. This they showed by using the version of Eq. (10.9) for black holes, which is obtained simply by replacing Mns by MBH in that equation, and inserting reasonable parameters for the Cyg X-1 system, namely, MBH ∼ 10M , Mtot ∼ 40M , and 2 2 /vorb ≈ 6.25. With the orbital separation a appropriately recalculated vw for the above masses, this yields rout ≈ 3.1 × 107 cm, as the reader can
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verify, which is about 4 times the lower limiting radius rms for a 10M black hole given above. “A real but marginal accretion disk”, to use the words of Shapiro and Lightman (1976), may therefore form. With the new, Chandra estimate available now, as explained above, the conclusion about neutron stars changes completely, since we now have rout ∼ (4 − 8) × 109 cm. This is 10-20 times the Alfv´en radius rA for a magnetic neutron star (see above), and enormous compared to the radii Rns and rms relevant for an unmagnetized neutron star. So, a “real” accretion disk, which is not nearly as “marginal” as that given by the above ShapiroLightman estimate for black holes, but also not nearly as extensive as those expected to form when neutron stars accrete through Roche-lobe overflow of their companions (see below), may very well occur around neutron stars accreting from winds. Of course, the conclusion regarding wind-accreting black holes remains as above (and is, in fact, strengthened if the wind velocity in black-hole binaries is also appropriately lower, as above, although there appears to be no direct evidence as yet for this) so that it appears that all wind-accreting massive X-ray binaries — i.e., accretion-powered pulsars and black-hole binaries — can have accretion disks.
10.1.1.2
Fluctuations in stellar winds
An interesting consequence of the analysis presented above, as already appreciated in the 1976 Shapiro-Lightman work, is the small relative size of the wind-accreted angular momentum w , compared to what one expects for well-developed accretion disks forming in binaries where accretion proceeds through Roche-lobe overflow (see Appendix B) of the companion. Let us call the specific angular momentum in the latter case RL . As we show later, RL ∼ orb where orb ∼ Ωorb a2 is the specific angular momentum associated with the binary orbit of size a and angular velocity Ωorb , which sets the scale for the maximum value can have in an accreting binary. Then 2 2 −2 , the above relative size is w /orb ∼ (ra /a)2 ∼ 10−2 Q−2 1 (1 + vw /vorb ) using Eq. (10.2) and Kepler’s third law, as also the relation between v, vw , and vorb for circular orbits, as given earlier. The smallness of this relative size is clear when we look at Eq. (10.11) for the velocity ratio : even when 2 2 /vorb ∼ 1 for the slower winds suggested by Chandra observations (see vw 2 2 /vorb ∼ 10 (see above), w /orb ∼ 10−2 − 10−3 , and for faster winds with vw −4 −5 above), w /orb is much smaller, ∼ 10 − 10 . We have shown earlier that this w is the net average specific angular momentum corresponding to the average, steady gradients in the stellar wind, and its smallness would
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therefore make us think that fluctuations in the wind could compete with or dominate over this average net value: this is what Shapiro and Lightman (1976) suggested. As a result, argued these authors, the sign of the angular momentum, and so the sense of rotation in the accretion disk, could undergo reversals from time to time in wind-accreting binaries, in marked contrast to what happens in Roche-lobe overflow binaries. This implies quite different behavior for accretion torques and pulse-period changes of accretion-powered pulsars in these two kinds of X-ray binaries, as we shall see later. What would cause these fluctuations? A variety of causes are possible, e.g., random density- and velocity-fluctuations in the wind, turbulence in the wind, and fluctuations in any posssible asymmetry (e.g., lack of spherical symmetry) in the wind on the one hand, and the effects of fluctuations in the photosphere of the wind-driving companion due, e.g., to chaotic magnetic fields or “hot spots”, on the other. It is easy to estimate the required size of these fluctuations. For density fluctuations δρ, for example, we need only remember the size of the average density gradient (dρ/dr)a = −2ρ(a)/a from the discussion below Eq. (10.5) to realize that δρ/ρ ∼ ra /a across the face of the accretion cylinder, and so a densityfluctuation of this size would compete with the net average gradient. This size can be readily calculated by using relations from the previous para2 2 −1 . Thus, for fast winds with graph, namely, ra /a ∼ 0.1Q−1 1 (1 + vw /vorb ) 2 2 vw /vorb ∼ 10, the required fluctuation size is δρ/ρ ∼ 10−2 , as Shapiro and Lightman (1976) estimated: such fluctuations would appear to be quite 2 2 /vorb ∼ 1, the recommon. On the other hand, for slow winds with vw −1 quired fluctuation size is δρ/ρ ∼ 10 , which may be rather rare. The pioneering, analytic calculations of the 1970s by Shapiro-Lightman and others along the above lines have been subsequently elaborated on by a variety of authors, using analytic, semi-analytic, and numerical methods. We return later to a brief sketch of their results.
10.1.1.3
Capture from Roche-lobe overflow
As we explained in Chapter 6, a major method of mass transfer in accreting binaries is Roche lobe overflow. The companion star exceeds the size of its limiting gravitational equipotential (the geometry is detailed in Appendix B) due to one or more of the following reasons: expansion of the companion in the course of its nuclear evolution, and shrinking of the binary due to emission of gravitational radiation and/or magnetic braking. Matter from
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the outermost layers of the companion then “overflows” or “spills over” the inner Lagrangian point L2 , i.e., the saddle-point in the gravitational potential which lies between the two stars on the line joining their centers (see Appendix B), where the “figure-of-eight” shaped limiting equipotential crosses itself, as shown in Appendix B. This picturesque language, widely used in the literature, describes a flow of matter through L2 basically at the sound speed of matter at that point, the hydrodynamics of which was clarified by Lubow and Shu (1975). This overflow is slow compared to the fast stellar winds considered above, takes the form of a stream of matter flowing through L2 , and leads to immediate capture of plasma from this stream and the formation of an extensive, Keplerian accretion disk, due to the high angular momentum of the matter (see Fig. 10.1). The stages of formation of such an accretion disk are shown in Fig. 10.3 [Ghosh & Lamb 1991]. The stream enters the Roche lobe of the neutron star, is accelerated by the gravitational force of the latter into a ballistic trajectory at first, which goes around the neutron star, collides with itself producing shocks, and finally settles down through post-shock dissipation within a few orbital periods into a circular ring of matter. The radius r of this ring is determined by the specific angular momentum RL of the matter through an analogue of Eq. (10.8), namely,
r =
2RL , GMns
(10.12)
the argument being the same as before, i.e., that a Keplerian orbit around the neutron star at this radius must have a specific angular momentum RL . But what is the specific angular momentum of matter overflowing the Roche lobe at L2 ? The distance of L2 from the neutron star’s center is ∝ a, the actual value depending on the ratio q ≡ Mc /Mns of the companion mass to the neutron-star mass. For LMXBs, in which Roche lobe overflow is ubiquitous (although this is also possible in HMXBs, as argued earlier), and which, therefore, are of main interest to us when considering Roche lobe overflow in accretion-powered binary X-ray sources, Mc < ∼ Mns , and the distance of L2 from the neutron star’s center is ∼ a (see Appendix B). The velocity of the overflowing matter at L2 with respect to the neutron star is completely dominated by its orbital velocity ∼ Ωorb a, since its overflow velocity through L2 is small, as we argued above. Hence the angular momentum content of the Roche-lobe overflow matter with respect to the neutron star can be written as
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Fig. 10.3 Schematic representation of the stages of formation of an accretion disk around a neutron star. Panel (e) is a side view of panel (d). Reprinted with permission by Springer Science and Business Media from Ghosh & Lamb (1991) in Neutron c 1991 Kluwer Stars: Theory and Observation, eds. J. Ventura and D. Pines, p. 363. Academic Publishers.
RL = ηRL Ωorb a2 ,
(10.13)
where the dimensionless parameter ηRL ∼ 1 depends on the mass ratio and flow details. Equations (10.12) and (10.13), together with Kepler’s third law, as given earlier, yield: r 2 = ηRL Q, a
(10.14)
where Q ≡ Mtot /Mns is the ratio of the total binary mass to that of the neutron star, as introduced earlier. As Q ∼ 1 for LMXBs (see above), it follows that r ∼ a.
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Next, this ring spreads both inward and outward from r , as effective “viscous” stresses transport angular momentum: this forms an accretion disk on a slow, viscous timescale (see below), the inner radius rin of the disk reaching rA for a magnetized neutron star (or the larger of the two radii Rns and rms for an unmagnetized neutron star), as explained earlier, and the outer radius of the disk reaching rout > ∼ r . In a steady state, rout is usually determined by the tidal forces exerted on the outer disk by the companion. This is due to the requirement that, in such a state, matter entering into the accretion disk from the stream, spiralling inward through the disk, and ultimately accreting onto the neutron star must have its angular momentum removed at the appropriate rate, which is done first by the “viscous” shear stresses in the disk, which transport this angular momentum outward through the disk, and then by the tidal interaction between the outer edge of the disk and the companion star, which puts the angular momentum into the orbital motion of the binary system [Ghosh & Lamb 1991]. From the above estimates, it must be clear that rout ∼ a, i.e., that the accretion disk is well-developed and extensive, occupying most of the Roche lobe of the neutron star. As we discussed earlier, this is entirely because of the large, near-maximal angular-momentum content of matter accreted via Roche-lobe overflow, while wind-accretion disks with w /RL ∼ 10−2 − 10−3 are much smaller, although quite viable.
10.1.2
Formation of Magnetospheres
Plasma captured as above from the exterior flow has a negligible part of the magnetic flux of the neutron star threading it, as the capture radii are enormous compared to neutron-star radii. Yet, as argued at the beginning of this chapter, the high magnetic fields of canonical neutron stars make it imperative that there be a magnetosphere consisting of an appropriately restricted region surrounding the neutron star, whose linear dimensions we denote by rm ∼ rA (see above), inside which plasma flow is dominated by 8 the stellar magnetic field. (This is true even for relatively weak, B > ∼ 10 G, magnetic fields on the neutron-star surface, with a magnetospheric extent which is appropriately smaller.) The formation of a magnetosphere is the consequence of the transition of the plasma flow from the first to the second type above. Note that, since the accreting plasma is fully ionized and therefore highly conducting, we would certainly expect it to move along the magnetic field lines of the neutron star close enough to the stellar surface,
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and possibly expect it to do so over all or most of the magnetosphere. It is this high conductivity of the accreting plasma that complicates the plasmaflow dynamics, since it denies the plasma ready access to the magnetized neutron-star surface [Ghosh & Lamb 1991]. Since such plasma tends to be “frozen” to the magnetic flux, it tends to carry the flux along with its flow as soon as it begins to encounter stellar field lines. This “sweeps up” the stellar magnetic flux, building up magnetic stresses which would ultimately halt the flow, creating an empty magnetic cavity around the star, surrounded by plasma which is constantly piling up at its boundary, and driving this boundary ever inward as the plasma pressure there keeps on increasing. In reality, such transient behavior may possibly occur at the very beginning of the accretion flow to a magnetic neutron star, but a steady state is quickly reached because plasma starts entering the above magnetic cavity through a variety of processes described below, and flowing toward the stellar surface. In this steady state, the radius rm of the magnetosphere is determined by the stress-balance condition that applies to the steady plasma-flow through the magnetospheric boundary, which ultimately accretes onto the neutron-star surface and produces the observed X-rays. Details of the calculation of rm are given later in the chapter. Here, we make a few general comments. First, the rotation of the accreting plasma at rm influences the structure of the outer magnetosphere, and its dynamical importance is often measured by the dimensionless angular velocity ωp ≡
Ω(rm ) , ΩK (rm )
(10.15)
which is just the angular velocity Ω(rm ) of the plasma at the magnetospheric boundary in units of the Keplerian angular velocity ΩK (rm ) there [Elsner & Lamb 1977]. Plasma rotation is dynamically unimportant when ωp 1, as in the case of an approximately radial flow just outside the magnetospheric boundary, and important when ωp ∼ 1, as in the case of Kepler-like orbital flows just outside rm , an exactly Keplerian accretion disk corresponding to ωp = 1. Second, the magnetospheric radius rm (∼ 108 cm for canonical neutronstar magnetic fields and accretion rates, as above, and smaller for weaker magnetic fields) is generally much smaller than the radius at which plasma 10 is captured, i.e., either the accretion radius ra > ∼ 10 cm, or the Roche-lobe 1/3 radius rRL ∼ a ≈ 9.6 × 1011 Q1 (Porb /5d )2/3 cm (see above), whichever is appropriate. Hence, to a first approximation, the structure of the magne-
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tosphere and the capture process can be treated separately. Further, for canonical neutron-star magnetic fields ∼ 1012 G, as applicable to HMXBs and intermediate-mass X-ray binaries like Her X-1, rm as above is much larger than the stellar radius Rns ∼ 106 cm, so that, again to a first approximation, physics of the magnetospheric boundary can be separated from that of the accretion flow onto the stellar surface [Ghosh & Lamb 1991]. However, for much weaker neutron-star magnetic fields ∼ 108 G, as are thought to be applicable to LMXBs, rm can be comparable to Rns , and the above separation may no longer be possible, leading to more complicated situations. Third, in the stress-balance condition that determines rm , the plasma stresses (which must balance the magnetic stresses) are typically dominated by the material stresses rather than the thermal pressure, and these material stresses depend, in turn, on the accretion flow pattern outside the magnetosphere. Thus, the scale and structure of the magnetosphere do depend on the flow pattern, and we would expect them to be quite different for disk flows and radial flows. However, as far as the scale of the magnetosphere is concerned, a curious coincidence sometimes occurs, which makes the scales for radial flows and certain kinds of disk flows rather similar, as we discuss in detail later. Fourth, there may be more dissimilarity than similarity between the geomagnetosphere, as introduced at the beginning of this chapter, and the neutron-star magnetosphere being studied here in detail. The key point that introduces qualitative differences between these two types of magnetospheres is that, while almost all of the plasma in the solar wind that impinges on the geomagnetosphere flows on by it, almost all of the plasma impinging on the magnetosphere of an accreting neutron star penetrates it, rapidly accreting onto the neutron star surface. Thus, the geomagnetosphere behaves rather like the empty magnetic cavity described above for the initial, transient configuration for a neutron-star accretion magnetosphere, and its size (in particular, the distance of the tip, or “nose”, of the geomagnetosphere from the Earth’s center) is estimated by balancing the ram pressure of the solar wind against the static pressure of the geomagnetic field confined to the cavity. This static procedure is simply invalid for a neutron-star magnetosphere with ongoing accretion, since the material stresses associated with plasma flow into the magnetosphere are generally as important as the stresses outside the magnetosphere, and the dynamic balance between these and the magnetic stresses determines rm , as we shall presently see. There are many other points of difference between these two
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types of magnetospheres, of which we mention a few essential ones. First, the large gravitational field of the neutron star produces a converging accretion flow towards it, while the solar wind simply “blows” past the geomagnetosphere. Next, the geomagnetosphere is always immersed in a single type of flow environment, i.e., the solar wind, which may be somewhat variable. By contrast, a wide variety of flow environments are possible, ranging from radial flows to Keplerian accretion disks, for neutron-star accretion magnetospheres, and none of these is similar to the solar wind. Finally, the large plasma density near the stellar surface in accreting neutron-star magnetospheres tends to make the plasma there collisional, while that in the geomagnetosphere is generally collisionless. 10.1.2.1
Lengthscales of magnetospheres
What determines the lengthscale of an accreting neutron-star magnetosphere, i.e., what decides the scale of rm ? We can approach the problem from various directions. First consider a general upper limit to this scale, which comes from the basic idea of the magnetosphere, namely, that the magnetic field controls the motion of the plasma there. This implies that the magnetic field must be able to deflect the accreting plasma effectively, which is not possible to do on timescales shorter than that required by Alfv´en waves to travel from the plasma (situated on a stellar magnetic field line at a distance r from the stellar center, say) along this field line to the stellar surface and back. This is so because “kinks” and disturbances in the magnetic field lines travel along these field lines at the Alfv´en velocity B , vA ≡ √ 4πρ
(10.16)
B being the magnetic field strength and ρ the plasma density [see, e.g., Krall & Trivelpiece 1974]. Thus, the above condition means that the Alfv´en velocity must be high enough, and hence the magnetic field strong enough, that Alfv´en waves can travel the distance to the star faster than the plasma can, i.e., r/vA ≤ r/vp , which gives vA ≥ vp . Here, vp is the infall velocity of the accreting plasma. With the aid of Eq. (10.16), we can rewrite this condition as: ρvp2 ≤
Bp2 , 4π
(10.17)
which gives an implicit upper bound on the magnetospheric lengthscale [Ghosh & Lamb 1991], since ρ, vp , and Bp are all functions of r. Here, Bp
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and vp are the poloidal components of the magnetic field and the plasma velocity. It is an upper bound because the left-hand side of inequality (10.17) always falls off more slowly (∼ r−5/2 for spherically symmetric plasma infall at free-fall velocity, and ∼ r−7/2 for free-fall along the field lines of a magnetic dipole, as we shall see later) with increasing r than the right-hand one (∼ r−6 for a magnetic dipole), so that the inequality is violated outside a limiting radius. A local flow-stability condition, closely related to the above idea which dealt with a global aspect of the flow over the whole magnetosphere, leads (not surprisingly) to essentially the same upper bound. Let us assume that the plasma has been threaded by the magnetospheric field, and is falling toward the stellar surface along magnetic field lines. Will this flow be stable? Stability analyses [see, e.g., Williams 1975] indicate that such field-aligned flow must be sub-Alfv´enic to be stable, i.e., vp ≤ vA , where vp is the poloidal velocity of the plasma along the field line. Except for this slight difference in the definition of vp , this is the same condition as above, and so leads to the same inequality (10.17), which is an implicit upper bound on the magnetospheric lengthscale, as explained above. This is as expected, since the essential stability arguments are the same as those invoked earlier for magnetic control of the flow, namely that the flow cannot be faster than speed at which information about the flow can travel along the field lines, which is the Alfv´en speed. In fact, because of the crucial dependence of both of the above criteria on the Alfv´en speed, it is natural to define the radius at which vp = vA , i.e., the above upper bound on the magnetospheric radius rm , as the Alfv´en radius, denoted by rA . However, other definitions of rA are also found in the literature. We can now obtain an explicit value of the Alfv´en radius rA . Consider the first criterion above, taking the poloidal component of the stellar magnetic field to be that of a dipole — an approximation we shall use for estimates throughout this book, unless otherwise specified. Then the poloidal field Bp is given roughly by Bp (r) ≈
µ , r3
(10.18)
where µ is the dipole moment of the neutron star. Assume, for simplicity, that the poloidal velocity of the infalling plasma at radius r is roughly equal to the free-fall velocity, 2GMns , (10.19) vp (r) ≈ vf f (r) ≡ r
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at that radius, and that the inflow is (roughly) spherically symmetric, so that the accretion rate is given by M˙ = 4πr2 ρvp (r).
(10.20)
Combining Eqs. (10.19) and (10.20), we see that ρvp2 ∼ r−5/2 , as indicated earlier, while Eq. (10.110) gives Bp2 ∼ r−6 , and insertion of these into the equation obtained by taking the equality sign in (10.17) gives the Alfv´en radius as: −1/7 Mns µ4/7 8 4/7 ˙ −2/7 ≈ 3 × 10 µ30 M17 cm. (10.21) rA = M M˙ 2/7 (2GMns )1/7 This is the pioneering result obtained by Lamb et al. in 1973. A similar value of rA can be obtained from the second criterion, by using all of the above approximations except that of spherically symmetric inflow, which is not even roughly valid for field-aligned flow. Instead, the plasma follows the magnetic field lines, and so undergoes convergent, funnel-like flows towards the magnetic poles, which are appropriately called accretion columns or funnels (see Chapter 1). As we show later, the shape of such flows is given by the relation ρvp /Bp = const., which holds for each magnetic flux tube, and which replaces Eq. (10.20). For a dipole field, this leads to a scaling ρvp2 ∼ r−7/2 , as the reader can show, and as we have indicated earlier. The final value of rA is rather similar to that given above, and we refer the reader to the review by Ghosh and Lamb (1991) for details. Whereas the above upper limits could be obtained entirely from considerations of the flow inside the magnetosphere, without any reference to the flow outside it, actual calculations of rm for specific situations must take the outside flow into account in an essential way. We now give brief accounts of such calculations for radial and disk flows, stressing the basic similarities and differences between these cases. 10.1.3
Radial Flow
Formation of neutron-star magnetospheres from approximately radial flows, i.e., those with ωp 1 (see above), is a limiting situation which was studied in detail in 1970s and ’80s, when it was believed that this situation necessarily occurs in wind-accreting X-ray binaries. As explained earlier, recent observations lead to the conclusion that relatively small but distinct accretion disks may be quite common even in wind-accreting systems. Nevertheless, this limit is still conceptually useful, and we shall indicate later
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that it may find application to systems that were not really investigated until relatively recently. Accordingly, we summarize below some essential aspects of magnetospheres forming from radial flows. The transient situation as the plasma first starts flowing toward the neutron star (see above) is sketched in Fig. 10.4. The high-conductivity plasma “sweeps in” the magnetic field, and the magnetic pressure rises and decelerates the flow. A shock wave forms near the magnetic fieldplasma interface — often called the magnetopause using the terminology of the geomagnetosphere — and travels upstream through the accreting plasma, heating it and slowing down its infall. Ultimately, the magnetic field becomes “rigid” enough to completely halt the inflow at the boundary of an empty magnetic cavity, creating a static equilibrium, as indicated earlier. We can call this cavity the static magnetosphere, and denote its 0 [ Elsner & Lamb 1977]. Note that actual magnetospheric radius scale by rm rm will be a function of the angle θ with respect to the magnetic axis of the star (θ is often called the magnetic colatitude), or its complement
Fig. 10.4 Schematic picture of the stages of interaction between an initially radial inflow and the neutron-star magnetic field. See text for detail. Reprinted with permission by Springer Science and Business Media from Ghosh & Lamb (1991) in Neutron Stars: c 1991 Kluwer Academic Theory and Observation, eds. J. Ventura and D. Pines, p. 363. Publishers.
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λ ≡ π/2 − θ, which can be called the magnetic latitude, in all cases of interest, i.e., rm (θ) or rm (λ). This is so because the stellar magnetic field is not spherically symmetric, though the physical variables describing a radial flow are. Indeed, rm (θ) gives the shape of the static magnetosphere, which we consider below in detail. 10.1.3.1
Size of the magnetosphere
The boundary of the static magnetosphere is determined by the condition of static pressure balance between the inside and the outside of the magnetosphere, including both the plasma pressure P and the magnetic pressure B 2 /8π. This gives 2 2 Bt Bt + Pin = + Pout , (10.22) 8π in 8π out where Bt is the component of the magnetic field tangential to the magnetospheric boundary, and the subscripts in and out are self-explanatory [Elsner & Lamb 1977]. Here, we have used the continuity of the normal component of B across the boundary. In the absence of significant plasma inside the magnetosphere, Pin can be neglected, as can the normal components of both Bin and Bout , and (Bt2 /8π)out can be neglected if any (small-scale) magnetic field carried by the accreting plasma has insignificant pressure compared to that of the stellar magnetic field in the cavity. Equation (10.22) then simplifies into 2 B (10.23) = Pout , 8π where we have dropped the redundant subscript on B. As discussed earlier, this sort of static pressure balance has been the characteristic way of handling the geomagnetosphere since the pioneering work of Chapman and Ferraro in the 1930s, and the charasteristic size so obtained is called the Chapman-Ferraro radius in geophysics. What would be the analogous size for accreting neutron-star magnetospheres? Imagine that the plasma outside the magnetosphere is undergoing radial 0 (see below), free fall everywhere upto the shock radius, which is close to rm and forming an adiabatic atmosphere outside the magnetosphere, whose pressure is given by Pout ≈ ρvf2 f . The plasma velocity, then, is given by Eq. (10.19), and the continuity condition relating the plasma density and velocity to the accretion rate M˙ is again Eq. (10.20). If the stellar field is
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taken as dipolar, given by Eq. (10.110) again, the reader can readily show 0 obtained from Eq. (10.23) will be identical to that that the value of rm of rA given by Eq. (10.21) [Davidson & Ostriker 1973; Lamb et al. 1973; Elsner & Lamb 1977]. However, we must appreciate that the two concepts leading to these two calculations are completely different: this is a central point, whose importance cannot be over-emphasized. While the former calculation hinged on the concept of the Alfv´en surface, which referred only to the flow inside the magnetosphere (studying its stability or related matters), the calculation here hinges on the concept of pressure balance with a flow entirely outside the magnetosphere. The fact that they are identical owes itself to the identical scalings for ρvp2 and Pout (∼ r−5/2 ) in the two cases. As we shall see later, while scalings with accretion variables (M˙ , µ, and M ) different from those given by Eq. (10.21) can occur in other circumstances, the numerical value of rm comes out to be in the same range (for the same values of the accretion variables) in most situations. We discuss the origins of this below. What is the expected width of the magnetospheric boundary layer, which separates the interior magnetic field from the exterior plasma? Through this layer or “sheath”, the magnetic field drops from its full magnetospheric value to almost zero as we go outward, and the plasma density rises from the low magnetospheric value to the high exterior value. Plasma physics of such sheaths have been studied in much detail, a basic outcome being the notion that the high current densities required in them in order to sustain the very large magnetic-field gradient, or “screening”, would broaden the layer through various plasma instabilities. The resultant thickness of the boundary layer is expected to be of the order of the transverse screening length (or inertial length) of the ions, li ≡ c/ωpi , where ωpi ≡ 4πni e2 /mi is the ion plasma frequency. Here, ni is the number density of the ions, and mi their mass. For expected plasma densities at the magnetospheric boundary, which the reader can work out from the equations given earlier, li ∼ 1 cm, which is absolutely tiny compared to the size 0 ∼ 108 cm, as shown above4 . Thus, the magnetospheric boundary layer rm can be treated as a tangential discontinuity.
4 The
ion Larmor radius, ai ≡ mi cvi /eB, corresponding to the magnetospheric field also comes out, for typical accretion-powered pulsars, to be roughly of the same order as li .
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The shock
0 The radial free-fall outside rm being hypersonic, the infalling plasma can 0 adjust to the “wall” at rm only through the formation of a strong standoff shock, behind which the flow becomes subsonic. The shock is likely to be 0 is determined by a bit of physics collisionless, and its position outside rm which is interesting, and which we briefly sketch here. As mentioned in earlier, once the accretion flow to the neutron star starts and X-rays are generated, the dominant mechanism for cooling the electrons is Compton scattering, and this is the situation for which we shall give the shock position. The electrons undergo Compton cooling on a timescale [Arons & Lea 1976; Elsner & Lamb 1977] :
tc ≡
3me c ≈ 6 × 10−3 r82 L−1 37 s, 8σT u
(10.24)
i.e., their energy Ee = kTe decreases at a rate dEe /dt = −Ee /tc . Here, σT is the Thomson scattering cross-section, and u ≡ L/4πr2 c is the radiation energy-density at r, other symbols having their usual meanings. On the other hand, the electrons are heated behind the shock by energy exchange with the ions (which are heated by the shock to temperatures measured in units of the free-fall temperature Tf f , i.e., the highest temperature obtainable by release of gravitational energy, Ef f ≡ kTf f ≡ GM mp /r ≈ 1.4(M/M )r8−1 MeV, assuming the ions to be protons, i.e., the accreting plasma to consist of pure hydrogen). For Te /Ti me /mi , as is expected to be the case (see below), this energy exchange proceeds on a timescale: 3me mp tex ≈ √ (kTe /me )3/2 4 8 2πe ne ln Λ 1/2 3/2 15 kTe M 3/2 ≈ 7 × 10−2 r8 L−1 s, (10.25) 37 ln Λ M 10 keV where ne is the number density of electrons, ≡ GMns /Rc2 ≈ 0.15(Mns/ M )R6−1 is the energy-conversion efficiency for accretion onto a neutron star of mass Mns and radius R, ln Λ is the Coulomb logarithm, scaled in terms of its typical value ∼ 15, and other symbols have their usual meanings. The electron energy Ee = kTe then increases by this process at a rate given by dEe /dt = (Ei − Ee )/tex ≈ Ei /tex , since Ee Ei (see below). Note that both tc and tex are short compared to the free-fall timescale at r, which is the flow timescale there.
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The equilibrium electron energy Ee∗ (or temperature Te∗ ) is obtained by equating the above rates of increase and decrease of Ee , and it is, using the value for the ion energy Ei ≡ kTi ≈ 0.37kTf f ≈ 520 keV behind the shock, which comes from strong shock conditions [Arons & Lea 1976]: 1/5 2/5 M ln Λ −1/5 −2/5 ∗ ∗ −1 keV, (10.26) Ee ≡ kTe ≈ 26r8 M 15 showing that Ee is indeed Ei , as anticipated above. The distance of the shock from the magnetospheric boundary is ls ∼ lex , the energy-exchange length, given by lex ≡ (5/16)vf f t∗ex , where the superscript ∗ on the energyexchange time means that the equilibrium electron temperature Te∗ is to be used when evaluating it by Eq. (10.25). This shock distance ∼ ls turns out 0 , as indicated above, to be small compared to the magnetospheric radius rm [ ] the ratio being given by Arons & Lea 1976 : 47/35 Mns ls −6/35 −32/35 ∼ 0.1µ30 L37 , (10.27) 0 rm M 0 , and set and ln Λ to their where we have used Eq. (10.21) to evaluate rm typical values given above. A typical value for bright accretion-powered 0 /16 [Arons & Lea 1976], which shows that the shock pulsars is ls ∼ rm 0 0 . radius rs = rm + ls is approximated well by the magnetospheric radius rm
10.1.3.3
Shape of the magnetosphere
How do we calculate the shape of the magnetosphere, i.e., the way in which rm (θ) varies with θ about the mean value given above? Two main approaches were followed in the 1970s, each respectively adapting an earlier approach pioneered in the 1950s and ’60s during the study of the shape of the geomagnetosphere. The first approach is an analytic calculation of rm (θ) for the corresponding two-dimensional problem, used for the geomagnetosphere by Cole and Huth (1959), as well as others, and adapted for neutron-star magnetospheres by Elsner and Lamb (1977). The second method is an approximate, numerical calculation of rm (θ) for the actual three-dimensional problem using standard multipole-expansion techniques of electrodynamics, used for the geomagnetosphere by Midgley and Davis (1962), and adapted for neutron-star magnetospheres by Arons and Lea (1976), and by Michel (1977a). In the first method [Elsner & Lamb 1977], the idea is to first imagine a two-dimensional analogue of the actual three-dimensional magnetosphere,
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in which the (stellar or planetary) point-dipole is replaced by a line-dipole, and the spherically symmetric radial inflow outside the magnetosphere is replaced by a cylindrically symmetric radial inflow, the axis of the cylinder being the line-dipole. Since the system has translational invariance along this axis, we need only find the shape of the magnetospheric boundary on any plane perpendicular to this axis. Such two-dimensional problems are generally solved by the method of conformal mapping, which maps the curve rm (θ) in the (2-dimensional) co-ordinate system (x, y) onto a circle in a suitably modified (2-dimensional) co-ordinate system (u, v) which is related to the original one by a conformal transformation. Recall that, in the actual 3-dimensional problem, the dipolar magnetic field scales as r−6 , and that the external pressure as Pout ∼ r−n (n = 5/2, for example, in the specific case considered above). Now, a general feature of the shape of a 3-dimensional magnetosphere is that it depends only on the exponent ν ≡ 6 − n. (Indeed, the size of the magnetosphere also depends crucially on ν, as can be seen by noticing that the exponents in Eq. [10.21], which corresponds to n = 5/2 and so to ν = 7/2, are closely related to 1/ν = 2/7.) However, a 2-dimensional dipolar magnetic field scales as r−4 , so that ν ≡ 4 − n in this case. The idea, therefore, is to find an analytic solution for the 2-dimensional magnetospheric shape for the value of ν that one expects for the actual 3-dimensional case, and argue subsequently on general grounds that the 3-dimensional magnetospheric shape will be similar, differing only in small details. The analytic calculations are straightforward, and in Fig. 10.5 we reproduce from Elsner and Lamb (1977) the shapes of 2-dimensional magnetospheres for ν = 2 and ν = 4. In the latter case, which is not very different from the ν = 7/2 case discussed above, the shape is particularly simple [Cole & Huth 1959], and can be written in parametric form as: 3φ 3φ φ 1 φ 1 (2d) (2d) sin − sin cos − cos , y = rm , (10.28) x = rm 2 3 2 2 3 2 where (x, y) are the cartesian co-ordinates in two dimensions, as above. (2d) 0 given by Here, rm is the 2-dimensional analogue5 of the scale rm Eq. (10.21), and the angular parameter φ runs over the range 0 ≤ φ ≤ 4π for describing the full magnetosphere, the magnetic poles corresponding to φ = (0, 2π), and the equator to φ = (π, 3π). Thus, the polar radius of the (2d) (2d) magnetosphere is rp = 2rm /3, and its equatorial radius is re = 4rm /3. Generally, the larger n is (i.e., the steeper the fall of the external plasma 5 For
details, see Elsner and Lamb (1977).
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Fig. 10.5 Shapes of 2-dimensional magnetospheric boundaries. Only the first quadrant is shown, rest following by symmetry. See text for detail. Reproduced with permission by the AAS from Elsner & Lamb (1977): see Bibliography.
pressure is with increasing radius), which means that the smaller ν is, the more oblate is the magnetospheric shape [Elsner & Lamb 1977]. The reader will have noticed in Fig. 10.5 the formation of cusps in the magnetospheric boundary over the magnetic poles, i.e., at the intersections of this boundary with the line x = 0. A cusp is a point where the tangent to the boundary becomes singular: that such cusps must exist on the magnetospheric boundary along the magnetic axis has been argued for both the geomagnetosphere [Midgley & Davis 1962] and neutron-star magnetospheres [Lamb et al. 1973]. The basic argument is that the static magnetospheric boundary coincides with a limiting set of magnetic field lines, and the magnetic field must have a cusp on the magnetic axis. The latter result comes from basic electrodynamics, as follows [Arons & Lea 1976]: an axisymmetric magnetic field structure, derived from a non-singular vector potential, must have B entirely in the radial direction along the magnetic axis (i.e., θ = 0 in terms of the magnetic colatitude θ defined above), with Br nowhere equal to zero. This is, of course, unable to describe the present situation, where Br vanishes outside the magnetosphere, r > rp , but is nonzero inside it, r < rp . Here, rp is the polar radius of the magnetosphere introduced above. The situation is remedied by including one component
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of the vector potential which is singular on the magnetic axis for r > rp but is finite for r < rp , but the price paid for this is that the direction of the magnetic field at rp becomes undefined, or, equivalently the tangent to the magnetospheric boundary, (drm /dθ), becomes singular as θ → 0. This is the cusp. We discuss the possible significance of cusps for plasma entry into the magnetosphere later. Now consider the second method [Arons & Lea 1976], in which the following 3-dimensional potential problem is solved numerically. Screening currents circulating on the boundary of a static magnetosphere must generate a magnetic field outside this boundary which is exactly equal and opposite to that of the stellar dipole, so that the total magnetic field everywhere exterior to the boundary is zero, as it must be. Now, if we knew the shape of the magnetospheric boundary, this would be a simple, linear potential problem, rather similar to the ones routinely solved in standard electrodynamics courses. What makes this problem much more complicated, and quite non-linear [Arons & Lea 1976], is that we do not know this shape: indeed, the shape determines the screening magnetic field, and the (total internal) magnetic field determines the shape through Eq. (10.23), so that we have to solve for them simultaneously. This may sound prohibitive, but is actually quite practicable with a little numerical effort, as was demonstrated by Arons and Lea (1976) for the specific 3-dimensional case with n = 5/2, ν = 7/2 discussed above. We summarize only the basic principle here. The axisymmetric, poloidal field can be described in terms ˆ which is entirely in the azimuthal direction of a vector potential, A = φA [Jackson 1975], and which has two parts. The first comes from the dipolar stellar field: Astar =
µ cos λ , r2
(10.29)
in terms of the magnetic latitude λ introduced earlier, and the magnetic dipole moment µ of the star. The second part comes from the screening currents circulating on the magnetospheric boundary: outside this boundary, it can be described by the standard multipole expansion = Ascreen out
µ r2
re 0 rm
7/4 0,∞ k
αk
r 2k e
r
1 P2k+1 (sin λ).
(10.30)
Here, re is the equatorial radius of the magnetosphere, as introduced earlier, 0 1 is the scale length given by Eq. (10.21), and P2k+1 is an associated rm Legendre polynomial [Gradshteyn & Ryzhik 1980], whose occurrence is a
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characteristic of problems involving vector potentials (as opposed to the ubiquitous presence of ordinary Legendre polynomials in problems involving scalar potentials). Also, we have taken a factor of µ/r2 out in front to show the dimensions explicitly. It is the coefficients αk of this multipole expansion that hold the key to the problem, since, for complete screening of the magnetic field outside exthe magnetosphere, we must have (a) that the dipole part of Ascreen out star 0 −7/4 actly cancels A , which means α0 = −(re /rm ) , using the fact that P11 (sin λ) = − cos λ [Gradshteyn & Ryzhik 1980], and (b) that all higher must vanish, which means αk = 0 for all k ≥ 1. This multipoles of Ascreen out is what determines the size and shape of the magnetospheric boundary, rm (λ), since the coefficients αk depend on this shape through the integrals [Arons & Lea 1976]: 2k+7/4 π/2 1 rm (λ) αk ≡ 1 + u2 (k + 1)(2k + 1) 0 re 1 .P2k+1 (sin λ) cos λdλ,
(10.31)
where u ≡ (1/rm )(drm /dλ). To actually solve for the shape, Arons and Lea (1976) used a standard numerical technique, adapting the original Midgley-Davis (1962) method. They expanded rm (λ) as a finite power-series in λ, ⎡ ⎤ 2j 1,N λ ⎦, (10.32) cj rm (λ) = re ⎣1 + π/2 j and varied the parameters cj until the above coefficients αk for 1 ≤ k ≤ N vanished6 . (Note that the series (10.32) contains only even powers of λ, since the magnetosphere is symmetric under reflection in the equatorial plane.) Integrals in Eq. (10.31) have to be calculated numerically, of course, and the N non-linear equations for the N cj ’s have to be solved used standard numerical procedures like (generalized) Newton’s method. The Arons-Lea (1976) procedure converged reasonably fast for the magnetospheric shape in the sense that, when four multipoles in the above ex6 The
0 )−7/4 merely determines the equatorial size r of the condition α0 = −(re /rm e 0 given above, and and so may be looked upon as a magnetsphere in units of the scale rm sort of normalization condition. In practice, Arons and Lea (1976) first determined the cj ’s through the numerical method described in the text and so obtained the series within the square brackets in Eq. (10.32), and then substituted this series in the k = 0 version of 0 (−α )−4/7 in a straightforward Eq. (10.31) to obtain α0 . This gave the size as re = rm 0 manner.
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pansion were included, the resulting surface did not change by more than 1% as N was increased further. However, the magnetic-field configuration converged more slowly, the “bumps” in the field-lines not disappearing until N was increased to about seven.
Fig. 10.6 Shape of 3-dimensional magnetospheric boundary (solid line with crosses). Also shown is the shock (dot-dashed curve), its standoff distance from the above boundary being magnified seven times for clarity. See text for further detail. Reproduced with permission by the AAS from Arons & Lea (1976): see Bibliography.
In Fig. 10.6, we reproduce from Arons and Lea (1976) the shape of the 3-dimensional ν = 7/2 magnetosphere, with representative field lines. Its remarkable similarity with the 2-dimensional ν = 4 magnetosphere of 0 , Fig. 10.5 is clear: in fact, the equatorial radius in Fig. 10.6 is re ≈ 1.2rm (2d) as compared to re ≈ 1.3rm in Fig. 10.5, and the ratio rp /re ≈ 0.5 is essentially identical in the two cases. It is this post facto verification that gives us confidence that the 2-dimensional configurations, rapidly calculable by simple, analytic techniques, do represent the actual situation rather well. The position of the standoff shock outside the magnetospheric boundary is also shown in the figure, the standoff distance magnified for clarity. Cusps again play an interesting role in the above numerical method. Strictly speaking, the expansion (10.32) is not valid at the cusps, as drm /dλ must be singular there. The error made by ignoring this, as Midgley and Davis (1962) did, is not enormous, but Arons and Lea (1976) suggested
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improvements for taking better account of cusps. From general electrodynamic considerations, they argued that the surface-shape near the cusp at λ = π/2 or θ = 0 is roughly given by rm (θ) = rp + S
θ π/2
2/3 ,
(10.33)
where S is a constant. The tangent to this surface is manifestly singular at θ = 0. The improvement in the method was, then, to adopt the shape (10.32) well away from the cusps, and the shape (10.33) close to the cusps, joining the two at a suitable intermediate latitude λ0 through the conditions that rm , as well as its first two derivatives, must be continuous at λ0 .
10.1.3.4
Magnetospheric boundary with accretion flow
As indicated earlier, once plasma penetrates the magnetospheric boundary and flows inside the magnetosphere, the static pressure balance condition (10.22) is not really valid. How do we determine the magnetospheric boundary then? Note that the essential component missing from Eq. (10.22) is the dynamic or “ram” pressure of the plasma ρvn2 associated with its motion into the magnetosphere, vn being the plasma velocity normal to the magnetospheric boundary. In other words, if we add a term (ρvn2 )in to the left-hand side of this equation, and a term (ρvn2 )out to its right-hand side, it should become correct. Now, we can neglect (Bt2 /8π)out as before, and, inside the magnetosphere, the analogue of what we did earlier would be tot ≡ (P + ρvn2 )in there [Elsner & to neglect the total plasma pressure Pin Lamb 1977]. This approximation is now less secure, of course, with plasma flow across the magnetospheric boundary, but this is what has been done in the subject. In that case, the modified balance equation reduces to Eq. ( tot ≡ (P + ρvn2 )out , which 10.23), with the right-hand side replaced by Pout means that the static magnetospheric shape and size, calculated as above, tot , will still work. For exambut corresponding to an external pressure Pout tot −5/2 , the shapes described earlier for 3-dimensional ple, if Pout scales as r magnetospheres with ν = 7/2, or 2-dimensional magnetospheres with ν = 4 will roughly apply. The width of the magnetospheric boundary layer, discussed above, is expected to increase greatly upon penetration of plasma into the magnetosphere, due to a variety of processes which depend on the dominant mode of plasma entry (see below).
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Disk Flow
It became clear during the 1980s and 1990s that most accretion-powered X-ray pulsars were, in fact, largely or entirely disk-fed. This included not only LMXBs and Be-star binaries, but some of the “standard” HMXBs with O/B companions as well, since the companions in the latter were demonstrated to be Roche lobe-filling, as explained earlier. Thus, the nature of magnetospheres around disk-accreting neutron stars occupied a central position in the subject. Much of the pioneering work in the subject had already been done in the 1970s, since the occurrence of accretion disks in X-ray binaries had been appreciated by then. We now summarize the basic physics of the interaction between accretion disks and neutron-star magnetic fields, and the formation and properties of magnetospheres in such cases. Of particular interest is the nature of the transition zone between the accretion disk and the magnetosphere, and the formation of a boundary layer at the magnetospheric boundary at the inner edge of the disk. But we have to introduce the essential properties of accretion disks first. We have already decsribed the formation of accretion disks around neutron stars in binaries in Sec. (10.1.1.3), and mentioned the essential role of “viscous” shear stresses in the outward transport of angular momentum, which makes accretion possible by making the matter spiral inward through the disk. Although some features of disk flows had already appeared in numerical simulations of hydrodynamic flows in binary systems done since the late 1960s, construction of semi-analytic, comprehensive models of accretion disks began in the 1970s. We summarize here the most essential properties some main types of accretion disk models proposed in the literature.
10.1.5
Thin Keplerian Accretion Disks
Pioneering model-building started with thin, Keplerian accretion disks, which are still the best-understood disks, and which occur perhaps most frequently at the canonical accretion rates M˙ ∼ (10−10 − 10−9)M yr−1 of binary neutron-star X-ray sources. By thin, we mean that these disks are geometrically thin, their vertical semi-thickness h being r at all radii r. On the other hand, they are optically thick in almost all cases, as we shall presently see. The essential characteristic of such disks is that their thermal energy-density is small compared to their mechanical energy-density (dominated by that due to rotational motion in the disk) almost everywhere, from which it immediately follows that, in the radial direction, the gravita-
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tional force GM/r2 due to the central accreting star must be very closely balanced by the centrifugal force Ω2 r, since the thermal pressure gradient in the radial direction is negligible, and the radial drift due to angularmomentum removal by viscous stresses (see below) is extremely slow. This balance, leading to Ω = ΩK ≡ GM/r3 , shows why such disks must be Keplerian. On the other hand, this small thermal pressure does determine the (small) vertical thickness 2h of the disk (see below), which would be infinitely thin if it had no thermal energy. How do we build a model for such a disk in a steady state, which is characterized by a constant mass-accretion rate M˙ and a constant specific angular momentum 0 deposited on the central accreting star? Since the disk is axisymmetric, its structure is described by the r- and z-variations of the essential dynamical variables Ω (which we already know to be Keplerian, as above), the inward radial drift velocity vr , and the density ρ, as well as the essential thermal variables, viz., the pressure p, the temperature T , and the flux of energy F from a unit area in the disk. There is, in addition, the question of the stresses that remove angular momentum from the accreting matter and transport it outward, thus making accretion possible. We introduced them above as “viscous” stresses, but the reader must appreciate that they have nothing to do with our usual notion of ionic/atomic/molecular viscosity of gases or plasmas. In other words, if we write these stresses formally as ηr(dΩ/dr) in terms of a coefficient of dynamical viscosity η (as we shall do below), in analogy with what we do for normal terrestrial fluid flow, this η would be nothing like the kinetic-theory viscosity of the ions and electrons of the fully-ionized disk plasma. Indeed, the latter viscosity would be quite negligible compared to that which is implied by the observed accretion rates through accretion disks in X-ray binaries. This η is really an effective viscosity then, and the question is: what is its origin? Two origins have been identified and discussed since the early 1970s, namely, (a) turbulent motions in the disk plasma, and (b) magnetic stresses associated with either small-scale magnetic fields advected in by the disk plasma, and/or large-scale magnetic fields of the neutron star “invading” the disk plasma, the latter magnetic stresses having received particular attention in recent years in connection with the magneto-rotational — or Balbus-Hawley — instability. We return to these viscosity mechanisms below, but consider for a moment why we call these “viscosity”. For turbulent stresses, the name has been accepted for a long time now, as such stresses do transport momentum, and are dissipative in nature, just as viscous stresses are. The interesting point about the above magnetic stresses is that they also have
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the same nature, the dissipation in this case coming from reconnection of field lines twisted by either differential plasma motion in the disk or the different motions of the neutron-star surface and the disk plasma connected by a field line, as we shall see later. Both of the above types of stresses, then, dissipate gravitational energy energy into heat (see below), making the analogy with viscous stresses quite apt. 10.1.5.1
One temperature disks
The quantitative description is simple. Consider mechanical aspects first. In addition to the Keplerian rotation GM , (10.34) Ω = ΩK ≡ r3 we have the condition of hydrostatic equilibrium in the vertical or zdirection: ∂p GM z = −ρ 2 , ∂z r r
(10.35)
which follows from the fact the dominant force acting vertically downward is the z-component of the central star’s gravity gz at a height z from the disk’s equatorial plane. As the reader can easily show, gz ≈ −(GM/r2 )(z/r) for z r. The description is completed by the condition of mass conservation, h M˙ = 2πr ρvr dz, (10.36) −h
and that of angular momentum conservation h dΩK dz. M˙ 0 = M˙ r2 ΩK + 2π ηr3 dr −h
(10.37)
In Eq. (10.36), we are assuming that there is no mass loss from the disk. In Eq. (10.37), the viscous torque enters, of course, making up, together with the angular momentum advected by the radial inward drift, M˙ r2 ΩK , the total rate of transport of angular momentum to the accreting star. Now consider the thermal aspects. First, the condition of energy conservation tells us that the total vertical flux 2F of energy from the two surfaces — top and bottom — of the disk at any radius r must equal the rate at which energy is generated by viscous dissipation at that radius in the disk within a column of unit cross-sectional area and height 2h. (This is so because radial transport of energy is negligible in a thin, Keplerian
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disk, as is the hradial pressure gradient, as explained above.) Now, the latter rate is just −h ηr2 (dΩK /dr)2 dz from elementary fluid mechanics, so that, with the aid of Eq. (10.37), F can be written as: F =
˙ 3GM M L, 3 8πr
0 . L≡1− √ GM r
(10.38)
The factor L represents the effects of the inner boundary condition in 2 the disk: by defining a radius r0 ≡ 0 /GM , we can always write it in the form L = 1 − r0 /r. At large radii, this inner boundary condition has little effect, so that L ≈ 1, and F ≈ 3GM M˙ /8πr3 . This leads to the interesting result that the rate of release of energy between any two r radii r< and r> (the notation should be obvious here) is r<> 2F.2πrdr = (3GM M˙ /2)(1/r< −1/r>). When we recall that the binding energy per unit mass in a Keplerian orbit of radius r is GM/2r, we see that the release of gravitational binding energy between r< and r> by matter moving inward at a rate M˙ is (GM M˙ /2)(1/r< − 1/r> ), so that the actual energy release is three times the release of gravitational energy. Where is the rest of the energy coming from? The puzzle was solved by Thorne (see Novikov and Thorne 1972, and Shakura and Sunyaev 1973), who pointed out that the extra energy was, in fact, the gravitational binding energy released at radii smaller than r< , but carted outward by the above viscous stresses, and converted into thermal energy ultimately between radii r< and r> through the same viscous dissipation process as described above. The second part of the thermal description is that of vertical energy transport through the disk, which is through radiation and/or convection. We shall neglect convective transport here. Then the thermal flux F from the midplane of the disk (where most of the energy is dissipated) in either vertical direction — up or down — is related to the temperature gradient ∂T /∂z and the opacity κ of the disk matter by: 1 ∂ 4 − ∂z 3 acT . (10.39) F = κρ Here, a is the radiation constant. Note that, for fully ionized gases, such as are found in accretion disks around neutron-stars in X-ray binaries, there are basically two sources of opacity, namely, (a) free-free, or Kramers, opacity, given by κf f ≈ 6.4 × 1022 ρT −7/2 cm2 g−1 , and, (b) elctron-scattering opacity, given by κes ≈ 0.4 cm2 g−1 . Clearly, for a given density ρ, κes always dominates over κf f at sufficiently high temperatures, a point to which we return presently.
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The equation of state of the disk matter and radiation, which is p=
ρkT 1 + aT 4 µmp 3
(10.40)
completes the thermal description, the first term on the right-hand side of the above equation being the gas pressure pg and the second, the radiation pressure pr . Here, the number µ is so defined that ρ/µmp is the total number of particles in a unit volume of the fully ionized disk plasma. If this plasma consists entirely of an element with mass number A and atomic number Z, µ = A/(Z + 1). Most of the accreting matter is normally hydrogen, for which µ = 1/2. The reader will have noticed that we have used above a single temperature throughout to describe the disk plasma, thereby assuming a thermodynamic equilibrium in which ions and electrons in this fully-ionized plasma are at the same temperature at all times. This is an essential property of the pioneering thin disk models, which assumes that there is an efficient energy transfer between ions (which receive the bulk of the dissipated heat) and electrons (which radiate this heat away ultimately) which ensures that their temperatures are equal, and because of which they are often called one-temperature disk models. We shall consider below other, subsequent models, in which the one-temperature assumption can be relaxed, and ions can be hotter than electrons, as might happen when the energy-transfer rate is unable to keep pace with the ion-heating rate. 10.1.5.2
The α-model of disk viscosity
Equations (10.34), (10.35), (10.36), (10.37), (10.38), (10.39), (10.40) can be solved to determine the disk structure, but we must have one piece of vital information before we can do so: we need to know the disk viscosity, i.e., η. In 1973, Shakura and Sunyaev made a brilliant suggestion which had a deep and lasting influence on the subject, namely that the viscous stress would scale with the total pressure p, so that dΩK = αp, (10.41) ηr dr where α is a dimensionless proportionality factor. This is the famous αmodel of the viscous stress in accretion disks. Consider for a moment what this means. Stress and pressure have the same dimensions, so that we can always relate them in the above manner for any flow, but factor α would then be a function of position, and possibly of other properties
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of that particular flow. This α-description becomes meaningful, therefore, only when α is roughly independent of the above variables, i.e., a rough constant characterizing the overall strength of the effective viscous stresses in the whole flow. This, in effect, is what the above authors conjectured by proposing the α-model. Why should this be true? The original Shakura-Sunyaev (henceforth SS) arguments and justification went as follows. These authors noted first that the turbulence was likely in an accretion disk, as the Reynolds number of the flow was high (also see below), and the scale of the turbulent eddies in a thin disk of semi-thickness h was expected to be ∼ h. The turbulent viscosity would then be roughly ηt ∼ ρvt h, where vt is the turbulent velocity, and the turbulent — or Reynolds — stress would be ηt r(dΩK /dr) ∼ p(vt /cs ). Here, p ∼ ρc2s is the pressure in terms of the sound velocity cs , and we have used the well-known estimate of the thin-disk thickness, h ∼ cs /ΩK , which we derive below. So, α ∼ vt /cs from turbulence alone. Next, magnetic stresses from twisted loops of small-scale magnetic field in the disk would be roughly ∼ B 2 /4π, scaling with the mean square strength of the field. So, α ∼ B 2 /4πp from magnetic stresses alone. Combining the two contributions, we get the estimate: / α∼
B2 4π
0
vt + . cs ρc2s
(10.42)
Now, argued SS, both terms on the right-hand side of the above equation must be less than unity. This is so for the turbulence term because supersonic turbulence (vt > cs ) would produce shocks, which would heat the plasma until α dropped below unity. For the magnetic term, this is also true because a magnetic field with a suprathermal energy density would bulge out vertically from the thin disk, reconnect and escape, lowering α below unity again. Therefore, α < ∼ 1. However, this does not, by itself, imply that α would be a constant. Indeed, SS stated that α was expected to be a function of r, and some detailed models proposed in later years did, in fact, include some specific radial variations. But the crucial point came out only post facto: after finding the structure of disks with constant α (see below), SS found that many physical varaiables had only a very weak dependence on α. It became clear, then, that neither the actual numerical value of α, nor any radial or other dependence of it, would make a large difference in the final disk structure. In effect, a steady, thin, Keplerian disk behaves as if it can be described roughly by a constant α, irrespective of the
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details of mechanisms that provide the viscous stresses. It is ultimately this fortunate situation that led to the wide acceptance and use of the original constant-α prescription, and such disks (or moderate variations thereof) are called α-disks, Shakura-Sunyaev (SS) disks, or “standard” disks today. As expected, a “first-principles” calculation of α for either of the above mechanisms for effective viscous stresses proved much more elusive. This is partly because neither turbulence nor reconnection of small-scale magnetic fields is fully understood yet at the level of a universally-accepted, “first-principles” theory, although many important advances in these areas have been achieved over the years, and are still continuing. The rest of the difficulty has been that the partly-parameterized theories of the above phenomena have sometimes not yielded very much more than Eq. (10.42) would directly suggest, or, occasionally, a form that is not really like an α-model. 10.1.5.3
The nature of disk viscosity
Consider hydrodynamic turbulence first, the existence of which in accretion disks was expected, since the Reynolds number of the flow, defined as R ≡ ρvl/η is high. (Here, v is the flow velocity, and l is the lengthscale on which the v changes over the flow.) This can be seen by setting v ∼ vK and l ∼ r for an accretion disk, and noticing that R = ρvK r/η ∼ α−1 (h/r)−2 . The second expression on the right-hand side can be obtained by using h/r ∼ cs /vK , as given above and derived below, and Eq. (10.41). The estimates α ∼ 0.001 − 0.1 and h/r ∼ 0.01 given below then show that R ∼ 105 − 108 . Turbulence in accretion disks would therefore seem unavoidable, but there was a serious worry. It has been known since Rayleigh’s pioneering work that a rotating, inviscid fluid with its specific angular momentum increasing outward, i.e., d2 /dr > 0, such as a Kelperin accretion disk which has K = √ ΩK r2 = GM r, is stable against axisymmetric perturbations7 . Would it still become turbulent at high Reynolds numbers? In answer, SS had cited Taylor’s classic laboratory experiments in the 1930s, in which this author had shown that fluids rotating between two cylinders did break out into turbulence above a critical Reynolds number Rc , even when the above Rayleigh criterion for stability was satisfied, just as plane, or Couette, flows go turbulent above an appropriate Rc . The analogy was suggestive, but it 7 This
is called the Rayleigh criterion. The reader can show this by proving that, when the above condition on is satisfied, an angular-mmomentum conserving outward displacement of a fluid element causes an inward force on the element, and vice-versa.
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is, of course, a far cry from Taylor’s experiments, in which (a) the fluid is incompressible, (b) the rotation is very subsonic, and, (c) the cylinders are long and narrow with rout /rin ≈ 1 and a relatively small gap t < ∼ 0.1r for the fluid, to accretion disks, in which (a) the plasma is compressible, (b) the rotation is highly supersonic, and, (c) the disk is thin, with rout /rin 1. In 1981, Zeldovich made a beautiful connection between the Taylor experiments and accretion disks which still inspires us, and which we summarize briefly at this point, as a tribute to his scientific virtuosity and astute thinking. Analyzing Taylor’s (1936) results, Zeldovich pointed out that we could decsribe the onset of turbulence in such rotating flows in terms of an analogue of the familiar Richardson number in plane Couette flows (see, e.g., Drazin & Reid 1981). Zeldovich coined the name Taylor number for this parameter, defining it as Ty ≡ 4
d2 /dr . r5 (dΩ/dr)2
(10.43)
The idea is clear: the numerator of the ratio in the above equation measures the effect of the angular-momentum gradient, while the denominator is a measure of the shear in the flow. Clearly, T y > 0 for Rayleigh stability, as above. The expectation is, by analogy with what happens for plane flows, that turbulence would set in at a “universal”critical value of T y, the corresponding critical Reynolds number being a function of T y alone, i.e., Rc = Rc (T y). If the angular-velocity profile of the flow is Ω ∝ rn , the reader can easily show from Eq. (10.43) that Ty = 8
n+2 . n2
(10.44)
A Keplerian accretion disk with n = −3/2 will, therefore, have T y = 16/9 ≈ 1.8. In Taylor’s experiments, with t ≡ r2 − r1 , as above, and t r ≡ (r1 + r2 )/2, it can be shown from the same equation that t Ty = 8 . r
(10.45)
Therefore, if we were to make a formal connection between the above two equations, a Keplerian accretion disk would correspond to a Taylor experiment with t/r = 2/9 ≈ 0.22 But why should we make this connection? The answer Zeldovich (1981) gave was truly ingenious. He showed — and the reader can readily verify this from Eq. (10.43) — that an angular-velocity profile with an exponent
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n gave the same value of the Taylor number as one with an exponent n given by: n = −
2n . n+2
(10.46)
A Keplerian disk will therefore give the same T y as one with Ω ∝ r6 . It is this Zeldovich scaling which is at the heart of the connection, for it implies that, for those aspects of the flow which are decided by the Taylor number, the flow cannot distinguish between a Keplerian accretion disk and one with n = 6, and the two situations should behave identically in these aspects. The rest of the story then fell into place. From Taylor’s (1936) data, Zeldovich showed that an angular velocity profile with n ≈ 5.5 was, indeed, seen for the case t/r ≈ 0.23, very close to the values given above. In fact, Taylor had given Rc for the onset of turbulence as a function of t/r in his original work, which Zeldovich converted into a function Rc (T y) with the aid of Eq. (10.45), and gave an analytic approximation as Rc (T y) ≈ 2 × 103 exp(T y/0.3).
(10.47)
A Keplerian disk with T y = 16/9 would then be expected to become turbulent at a critical Reynolds number Rc ≈ 8 × 105 according to these remarkable, self-consistent arguments due to Zeldovich. Calculation of α for hydrodynamic turbulence in accretion disks proved more difficult, however. In the mid-1970s, Stewart (1975) pioneered such attempts, exploring the role of non-axisymmetric perturbations (∼ exp i(ωt − mφ)), which can go unstable even when the flow is stable according to the Rayleigh criterion. These perturbations would form “whorl tubes” parallel to the disk’s rotation axis around the mean circular Keplerian motion, and tap energy from the disk’s differential rotation. In attempting to calculate the associated Reynolds’ stresses, the first thing to notice is that they are not really local properties of motion in such flows, so that a local stress-strain relation — as envisaged in the α-model — may not work. Even in the simple model explored by this author, involving sinusoidal ripples on mean circular motion, the natural relation that seemed to emerge was one between the radial derivatives of stress and strain, with an effective viscosity η ∼ ρu2 h/u. Here, u2 is the autocorrelation function of the velocity fluctuations, and u is the mean flow velocity. In the same time-frame, Eardley and Lightman (1975) pioneered the exploration of stresses due to small-scale magnetic fields in the disk. A small loop of magnetic field of size ∼ l is elongated in the azimuthal direction
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by the differential rotation in the disk, until the long neck is pinched off by reconnection, forming two loops. On the other hand, two loops brought against each other by the flow can coalesce by reconnection, forming one loop. These authors had a purely two-dimensional picture in mind, envisaging a steady state in which the rate of shearing ∼ Ω was balanced by the rate of reconnection ξvA /l. Here, vA is the Alfv´en velocity, in terms of which the reconnection speed is expressed as ξvA , with ξ ∼ 0.1 usually. From this, α can be estimated as α ∼ B 2 /4πp ∼ (l/hξ)2 . However, the above process is essentially three-dimensional, as pointed out by Galeev et al. (1979): elongated magnetic-field loops are twisted by turbulent eddies in the vertical plane before they reconnect, and this slows the growth rate of the magnetic field. Further work was done by Pudritz in the 1980s on the level to which magnetic field-strengths can be built up in turbulent accretion disks. In the 1990s, attention shifted to the role of magneto-rotational, or Balbus-Hawley, instability in transporting angular momentum, wherein vertical magnetic field-lines of the neutron star penetrating through the disk were twisted and amplified by the disk’s circulation, ultimately limited by reconnection. The above theoretical works have generally given estimates of α in the range 0.01 − 0.1. Alternatively, we can try to infer values of α from observation. While the steady-state disk structure is rather insensitive to α, certain time-variations in the properties of accretion disks during their instabilities (such as are believed to occur when dwarf novae, soft X-ray transients, and other similar systems go into outbursts) depend on α sufficiently strongly that attempts have been made to estimate α from data on such outbursts. These estimates generally lie in the range 0.001 − 0.1, so that viscous stresses required to drive the accretion in these disks appear quite small compared to the pressure, as SS had conjectured originally.
10.1.5.4
The structure of α-disks
With the above α-model for the viscous stresses, Eqs. (10.34), (10.35), (10.36), (10.37), (10.38), (10.39), (10.40), and (10.41) can be solved to obtain the complete structure of α-disks. A further simplification in the procedure occurs naturally, as follows. In thin disks, the scales of space variation are remarkably different in the radial and vertical directions, namely, ∼ r in the former direction and ∼ h in the latter, so that radial and vertical structures can be thought of as being roughly decoupled from each other. Finding these structures then becomes much easier: for finding the radial
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structure, which varies slowly on a scale r, we can average in the vertical direction. This means that, in the above structure equations, we regard all physical variables as having their vertically-averaged value. In addition, we replace vertical derivatives ∂/∂z, where the occur, by the estimate 1/h, and replace z by the estimate h. As a result, the above equations turn into a series of algebraic equations involving these physical variables describing the disk, T , ρ (or, equivalently, the number density n = ρ/mp , assuming a composition of H for the accreting matter), h, vr (or, equivalently, the radial drift timescale, defined as td ≡ r/vr ), and so on, and the radius r. There are two simple results — in the form of estimates — that immediately follow from the above averaging procedure, and we discuss them before proceeding to the detailed solutions. We do so because these are of fundamental importance in our understanding of thin, Keplerian disks, and so are used constantly for various arguments. The first one follows readily from the vertically-averaged version of Eq. (10.35), and is h ∼ cs /ΩK = r
cs ∼ 0.01r, vK
(10.48)
setting the scale of “thinness” of a thin disk. It is thermal pressure that gives the disk any thickness at all, so that the result is entirely as expected, and the number follows from the detailed solutions given below, showing us the scale of the thermal energy in a thin disk relative to that of Keplerian motion, and making it clear how supersonic the Keplerian disk circulation is. The second one follows on combining the vertically-averaged versions of Eqs. (10.36) and (10.37), and neglecting the constant term M˙ 0 on the left-hand side of the latter equation, since it is important only near the inner edge of the disk. The result is 2 h vr ∼α ∼ 10−4 α, (10.49) vK r setting the scale for the “slowness” of the slow radial inward drift in a thin disk. The relevant ratio here is that of viscous stresses to Keplerian ones, which is quite tiny, in view of the above discussions. Consider now the detailed solutions of the algebraic equations described above, which give us the detailed radial disk structure, i.e., the above physical variables as functions of r. In doing this, a further simplification occurs, making the task quite simple. We introduced in Sec. (10.1.5.1) (a) the two components of the total pressure, namely, the gas pressure pg and the radiation pressure pr , and, (b) the two components of the total opacity, namely,
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the free-free, or Kramers, opacity κf f and the elctron-scattering opacity, κes . It turns out that there is a natural ordering of the relative importance of these components as we go radially inward in a disk and the temperature rises, creating three distinct regions. These are (SS) described below, and shown in Fig. 10.7: • The outer region, where pg pr and κf f κes , i.e., gas pressure and free-free opacity dominate. • The middle region, where pg pr and κes κf f , i.e., gas pressure and electron-scattering opacity dominate. • The inner region, where pr pg and κes κf f , i.e., radiation pressure and electron-scattering opacity dominate.
Fig. 10.7 Shakura-Sunyaev, or “standard”, accretion disk. Shown are three regions described in text. Reproduced with permission from Shakura & Sunyaev (1973): see Bibliography.
Within each such region, we need use only the dominant component of pressure and opacity, which makes the algebra quite simple, and the results are displayed in Table 10.1: we show the half-thickness h, the number density n, the temperature T , and the radial drift timescale td ≡ r/vr as functions of the radius r the essential accretion variables M˙ , M , the viscosity parameter α, and the factor L representing the effects of the disk’s inner boundary condition, as explained earlier. In each region, a physical variable u (which can be h, n, T , or td ) comes out in a power-law form B (M/M )C rD LE , u = u0 αA M˙ −9
(10.50)
where u0 is a constant, A, B, C, D, E are numerical exponents, and M˙ −9 is the accretion rate in units of 10−9 M yr−1 , as before. We list the values of u0 , A, B, C, D, E in Table 10.1. Note that, in this table, r is measured in units of 1010 cm, i.e., it is r10 , in the outer region in the standard notation. In the middle region it is r8 , and in the inner region it is r6 , in the same notation.
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h n T td
h n T td
h n T td
537
RADIAL STRUCTURE OF SHAKURA-SUNYAEV DISKS
Constant u0
1.3 × 108 1.2 × 107 2.9 × 104 4.8 × 104
cm cm−3 K sec
8.5 × 105 cm 4.5 × 1020 cm−3 1.2 × 106 K 1.3 × 103 sec
105
1.1 × cm 4.5 × 1020 cm−3 2.2 × 107 K 1.5 × 10−2 sec
A
B
OUTER
REGION
−1/10 −7/10 −1/5 −4/5
3/20 11/20 3/10 −3/10
MIDDLE
REGION
−1/10 −7/10 −1/5 −4/5
1/5 2/5 2/5 −2/5
INNER
REGION
0 −1 −1/4 −1
1 −2 0 −2
C
D
E
−3/8 5/8 1/4 1/4
9/8 −15/8 −3/4 5/4
3/20 11/20 3/10 7/10
−7/20 11/20 3/10 1/5
21/20 −33/20 −9/10 7/5
1/5 2/5 2/5 3/5
0 −1/2 1/8 −1/2
0 3/2 −3/8 7/2
1 −2 0 −1
The boundaries between these regions can be found from the defining conditions that (a) κes = κf f at the middle/outer boundary, and, (b) pr = pg at the inner/middle boundary. Alternatively, these can be estimated by taking the solutions for any physical variable (say T ) in two contiguous regions, and equating them. The middle/outer boundary is at a radius 2/3
rm/o ≈ 6 × 108 M˙ −9 (M/M )1/3 cm,
(10.51)
and the inner/middle boundary is at a radius 16/21 ri/m ≈ 107 α2/21 M˙ −9 .(M/M )2/7 cm,
(10.52)
Here, M˙ −9 is the accretion in units of 10−9 M yr−1 , as before. With these solutions, the reader can show that the disk his optically thick in the vertical direction, i.e., the optical depth τ ≡ 2 0 κρdz is 1. Note that the relevant opacity for this in the outer region is κf f , but in the middle and √ inner regions, the effective opacity for this purpose is κes κf f (SS). The radial optical depth is much higher, of course. The above radial structure is what is most often needed for further work with accretion disks, but the reader should not have the impression that
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the vertical structure cannot be determined because of our use of vertical averaging to simplify the calculation of radial structure, which varies on a scale ∼ r, i.e., much more slowly. The information about vertical structure is still contained in Eqs. (10.35) and (10.39), and can be extracted at any given radius. The calculations can be done by changing the vertical coordinate from z to the surface density Σ(z) upto the vertical height z, z defined by Σ(z) ≡ 2 0 ρdz. (In this notation, surface density of the whole disk at any radius is Σ0 = Σ(h).) We refer the reader to SS for detail of such calculations, giving examples of final results. In the middle and inner regions, where the electron-scattering opacity dominates, the temperature decreases away from the midplane value Tc as T 4 = Tc4 [1 − (Σ/Σ0 )2 ] for optically thick disks. The density is roughly independent of the vertical co-ordinate in the inner region, but in the middle region, it falls like a gaussian, ρ(z) ∼ exp(−z 2 /h2 ), for small excursions from the midplane. 10.1.5.5
Two temperature disks
One temperature α-disks are rather cool, as Table 10.1 shows, even the inner, hottest region reaching only upto ∼ 107 K, corresponding to X-ray photons of energy ∼ 1 keV. It became clear by the mid-1970s that the observed X-ray spectra from black-hole binary sources like Cyg X-1 could not be accounted for by such accretion disks, since these spectra showed clear hard “tails”, reaching to ∼ 10 − 100 keV. And yet, since there cannot be any direct X-ray emission from the central, accreting black hole in these systems (unlike the situation for accreting neutron stars), these hard photons had to come from the inner region of the accretion disk. Motivated by this, a different kind of model was proposed for the inner part of the accretion disk by Thorne and Price (1975) and by Shapiro et al. (1976), wherein the ions could be much hotter than the electrons, producing a twotemperature (henceforth 2T) disk, in which both electrons and ions would be much hotter (Te ∼ 109 K, for example) than their counterparts in 1T SS disks. Such disks would be optically thin, as opposed to the 1T ones, and it was believed that they merge into cooler, optically thick SS disks at larger radii. We discuss such disks here briefly, as there is no fundamental reason why they cannot occur in neutron-star X-ray binaries also, at least in principle. Why would electrons and ions have different temperatures? As explained earlier, the deposition of dissipated energy is into the ions, while
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the electrons provide the bulk of the energy loss, so that their temperatures are clamped together only if the energy transfer between them is efficient enough. When this is no longer true, ions will be maintained at a higher temperature Ti than that of the electrons, Te , the equilibrium temperatures being determined by the condition that rate of ion-electron energy transfer equals the rate of energy loss from electrons. By taking this possibility into account, self-consistent 2T disk models can be constructed. The reader should note that there is nothing “wrong” with 1T disks. By assuming Ti = Te for SS disks from the outset, we precluded ourselves from finding other viable disk models in which these temperatures are not equal. It turns out sometimes that, at the same accretion rate, several disk models are allowed from basic physical principles. It is then a question of which disk configuration is “chosen” by a given system, perhaps depending on its previous history. It is also quite possible for a disk to oscillate between two such configurations, under certain circumstances. We now summarize the basic features of the 2T disk model originally proposed by Shapiro et al. (1976, henceforth SLE), in which the dominant process for cooling the electrons is the inverse Compton scattering of soft photons by these hot electrons. This process, in which the soft photons gain energy and become harder, is referred to as “Comptonization”. Where do these soft photons come from? We come back to this question below. Note that the process is described in terms of the Comptonization parameter y, defined as y≡
4kTe 2 M ax[τes , τes ], m e c2
(10.53)
where the symbol “Max” denotesthe maximum of the two quantities within h the square brackets, and τes ≡ 2 0 κes ρdz is the electron-scattering optical depth, as above. Comptonization is described quantitatively through the Kompaneets equation: we refer the reader to SLE and the references therein for an account of the procedure, and give here only the bare essentials of the final results. As the photons gain energy (or, as the popular expression goes, they are “upscattered” in energy) by Comptonization, the energy-amplification factor A can be studied as a function of the above y-parameter, as shown in Fig. 10.8, and this serves as a major diagnostic of the process. Three regions can be identified in this plot, namely, (1) negligible Comptonization, y 1, A ≈ 1, at the lower left of the diagram, (2) unsaturated Comptonization, y ∼ 1, A 1, at the “knee” of the curve in the middle
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Fig. 10.8 Amplification factor A vs. Comptonization y-parameter. See text for detail. Reproduced with permission by the AAS from Shapiro et al. (1976): see Bibliography.
of the diagram, and, (3) saturated Comptonization, y 1, A 1, at the upper right of the diagram. SLE proposed that, with a copious supply of soft photons, 2T disks with Comptonization cooling would operate in the second region above, i.e., unsaturated Comptonization. In this r´egime, y changes little as A goes through a large change, as is clear from the figure. Therefore, for a given observed energy flux of hard photons, a large range of (unknown) energy flux of soft photons can be handled in this r´egime of operation, without any large changes in the disk’s properties. In fact, SLE used y = 1 in their calculations, and found electron temperatures Te ∼ 109 K, and ion temperatures Ti ∼ (3 − 300)Te in the hot, inner regions of the resulting disks. These disks are gas-pressure dominated, optically thin √ (as opposed to SS disks), i.e., have τ∗ ≡ τes τf f 1, and geometrically somewhat “bloated”, with h/r ∼ 0.2, as shown in Fig. 10.9 The structure of 2T SLE disks are calculated in a fashion rather similar to that described above for 1T SS disks. Indeed, many of the 1T SS equa-
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Fig. 10.9 The two-temperature disk of Shapiro et al. (1976). Reproduced with permission by the AAS from Shapiro et al. (1976): see Bibliography.
tions remain valid, namely, all those describing the mechanical aspects, as also Eq. (10.38). Equation (10.40) needs to be modified to: p ≈ pg =
ρk(Ti + Te ) , mp
(10.54)
assuming the accreting matter to be pure H. It is really the energy-transport description, Eq. (10.39), however, that needs a radical change, being replaced by two relations. The first is that the rate of viscous dissipation in Eq. (10.38) must equal that at which energy is transferred from ions to electrons by Coulomb collisions8 , which can be written as 3 ρk(Ti − Te ) F = νC , h 2 mp
νC ≈ 2.4 × 1021 ln ΛρTe−3/2 .
(10.55)
Here, νC is the Coulomb collision rate. The second relation is that describing Comptonization, which is simply the the SLE ansatz : y = 1.
(10.56)
This completes the mathematical description of 2T SLE disks, and the above equations are solved in a manner identical to that described above for SS disks. The radial structure of 2T SLE disks is displayed in Table 10.2, following the basic notation used for 1T SS disks in Table 10.1, except that r is measured here in the units of the gravitational radius GM/c2 , M in units of 3M (SLE were primarily interested in black-hole binaries), and M˙ in units of 1017 g s−1 . Thus, the analogous expression here is: C D M r A ˙ B LE , (10.57) u = u0 α M17 3M GM/c2 8 We
follow SLE here, who proposed Coulomb collisions as the dominant transfer mechanism. Other possible mechanisms have sometimes been mentioned in the literature.
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542
Table 10.2 RADIAL STRUCTURE OF TWO-TEMPERATURE DISKS (After Shapiro et al. 1976) Variable u
h/r ρ Te Ti
Constant u0
0.2 5 × 10−5 g cm−3 7 × 108 K 5 × 1011 K
A
B
INNER
REGION
−7/12 3/4 −1/6 −7/6
5/12 −1/4 −1/6 5/6
C
D
E
−5/12 −3/4 1/6 −5/6
−1/8 −9/8 1/4 −5/4
5/12 −1/4 −1/6 5/6
and the structure parameters are listed in Table 10.2. The variables given are relative thickness h/r, mass density ρ, electron temperature Te , and ion temperature Ti . Particularly noteworthy is the fact that the electron temperature Te depends very weakly on all the essential disk variables, and so is maintained at a near-constant value throughout the hot, inner region. The continuous emission spectra of such disks have, of course, a Comptonized hard “tail”, as we describe later. Finally, consider the possible sources of the soft photons postulated above. The most likely source of these photons is thought to be the outer, cool, 1T SS parts of the disk, which emit photons of typical energy ∼ 0.1 keV. The bloated inner SLE disk has a relatively large crosssection for intercepting such photons. An alternative proposed source has to do with the idea of disk instabilities (see below). The radiation-pressure dominated inner part of a 1T SS disk is known to be unstable, and the idea is that this area will break up into alternately hot and cold blobs. The latter may then supply soft photons, although their relevance to an inner disk which is already in the 2T SLE configuration is unclear. In the 1980s, 2T disk models with other cooling mecahnisms, e.g., Comptonized bremsstrahlung, were considered by several groups of authors, with particular reference the possible presence and role of electron-positron pair plasmas in very hot disks. We refer the interested reader to the original works, and shall return below to an occasional consideration of such disks. 10.1.6
The Disk-Magnetosphere Interaction
How would such accretion disks as described above interact with the magnetic fields of accreting neutron stars in accretion-powered pulsars? What would be the sizes and shapes of the magnetospheres that would form? These questions began to be answered in the 1970s, first qualitatively by
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Pringle and Rees (1972) and Lamb et al. (1973), and then more quantitatively by Ghosh and Lamb (1978, 1979a,b), and by Scharlemann (1978). At that time, the only known disk models known were the geometrically thin (h r) ones, so that all such studies considered only these disks. Since then, various kinds of slim and thick disk models have been studied, mostly in connection with accretion by black holes, either in X-ray binaries or in active galactic nuclei (AGN). It still appears that disks which are geometrically very thick (h ∼ r) have more relevance to accreting black holes than accreting neutron stars, and we shall confine our attention here to interactions between relatively thin disks (h/r < ∼ 0.1, say) and neutron-star magnetospheres. Indeed, subsequent studies of the problem in the 1980s also addressed only such disks, and so have, by and large, the more recent magnetohydrodynamic (MHD) simulations of the 1990s and 2000s. It was clear from the beginning that the geometry of this problem was radically different from that of the radial/quasi-radial flow discussed earlier in the chapter, and, consequently, many of the “intuitive” ideas given there about a magnetosphere confined completely by an external flow, a static balance at first, and so on, may not carry over to this situation. This indeed proved to be the case, and, in retrospect, it appears rather fortuitous that some essential parameters did turn out to be rather similar in the two cases, although the physics involved is quite different, as we shall see. The crucial point is that a thin accretion disk can never “confine” a stellar magnetosphere in any sense: the radial inward flow of the highly-ionized disk plasma does sweep the magnetic field lines inward at and near the disk’s mid-plane, but this merely gives the field lines a “pinched” shape, ballooning out in directions away from the disk plane, as shown in Fig. 10.10. Basically, such a system is much more “open” and, therefore, much more prone to mixing of magnetic fields and plasma than a system with radial external flow can ever be. This is the qualitative difference referred to above, which changes the character of the problem. 10.1.6.1
Basic electrodynamic processes
In understanding the basic electrodynamics of how plasma in a thin disk would interact with a neutron star’s magnetic field, consider first what would happen if we made an automatic analogy with the radial-flow case, i.e., assumed that the disk would be completely screened from the neutronstar magnetic field by currents circulating in thin layers at the top and bottom surfaces of the disk, as well as at its inner edge. This “diamag-
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Fig. 10.10 Schematic picture of disk accretion by magnetic neutron stars. Shown are the various distinct regions of flow. Reproduced with permission by the AAS from Ghosh & Lamb (1978): see Bibliography.
netic” disk picture is the analogue of the static force-balance magnetospheric boundary in the radial-flow case described earlier. The mathematical problem can be solved in a straightforward manner for an infinitesimally thin disk, and the field configuration obtained. We give the reader a flavor of the method and results, before explaining why such diamagnetic disks very likely have little to do with disk-fed accretion-powered pulsars. The calculational approaches are very close to those described above. The two-dimensional problem is solved, again, by the method of conformal mapping, using a complex potential (see below). The three-dimensional problem is also tractable analytically, through integral equations. Of course, there is no magnetospheric “boundary” rm (θ) to be found in this case, the only boundary being the disk — a mathematical plane in the limit of these calculations. So the only configuration to be determined is that of the magnetic field lines, subject to the boundary conditions that (a) on the disk plane, the component Bz perpendicular to this plane must vanish, and, (b) inside in the inner edge rin of the disk, the radial component B must vanish on the z = 0 plane by symmetry. Here, we are using a cylindrical co-ordinate system (, φ, z), centered on the neutron star, whose undistorted field is dipolar, with a moment µ, in general tilted to the rotation axis of the disk by an angle χ. The resulting field configuration in 2-dimensions can be described in an elegant way with the aid of a complex potential φ, as shown by Kundt and Robnik (1980). Using (x, y) as the cartesian co-ordinates in two dimensions
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Fig. 10.11 Magnetic-field structure obtained in the two-dimensional problem wherein an infinitesimally thin diamagnetic disk surrounds a magnetic dipole. Reproduced with permission from Kundt & Robnik (1980): see Bibliography.
(y-axis along the disk-rotation axis), defining z ≡ x + iy, and noting that that the complex potential φ0 of an undistorted, tilted line-dipole (appropriate for 2-dimensional calculations, as we saw earlier in this chapter) of complex dipole moment µ ≡ |µ|(sin χ + i cos χ) is just φ0 = µ/z, we can write the total potential of the dipole plus the disk currents as µ z2 sin χ + i cos χ 1 − 2 . φ= (10.58) z rin This form shows explicitly that, very close to the dipole (z → 0), the field approaches that of the undistorted dipole, as it must. The resulting field configuration is shown in Fig. 10.11. The corresponding 3-dimensional configuration is very similar. The calculation in three dimensions proceeds in terms of the usual vector potential in its standard Bessel-function representation, the resulting integral equations being solved with the aid of standard methods. For accounts of these, we refer the reader to the original works of Aly (1980), Kundt and Robnik(1980), and others. The final expressions for the field components, though analytic, are rather lengthy, and we shall not quote them here. Note the occurrence of two neutral points in these field configurations, as marked in the figure, for tilted dipoles (χ = 0), which recede to arbitrarily large distances as χ → 0.
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The problem with the above beautiful solutions to idealized electrodynamic problems, which undoubtedly provide good insights into possible field topologies in disk-magnetosphere interaction, is that this diamagnetic configuration does not appear to be a self-consistent description of the actual disk-accretion flows to magnetic neutron stars in X-ray binaries. This is so because the stellar magnetic field, excluded in the above manner from the disk, would rapidly penetrate into the disk on a timescale much shorter than the radial inward drift timescale td of the disk introduced above, as originally pointed out by Ghosh and Lamb (1978, 1979a,b, henecforth GL). These authors identified several processes that lead to this penetration, and specifically discussed the following. First, the velocity discontinuity between the disk plasma, which is rotating with an angular velocity ΩK (r) at a radius r, and low-density stellarmagnetic-field region above and below it, which is co-rotating with the star with an angular velocity Ωs , drives the Kelvin-Helmholtz instability at the disk-magnetic field interface, which mixes the magnetic field and the disk plasma. To see how fast this proceeds, consider first the growth rate γKH of the Kelvin-Helmholtz instability, which can be expressed in terms of the linear velocity discontinuity v0 = r(ΩK − Ωs ) and the wave vector, ˆ of the mode in the following way: k = krˆ r + kφ φ, γKH ≈ vA
kφ2 v02 c2
. − kr2 − kkg
(10.59)
√ Here, vA ≡ Br / 4πρ is an Alfv´en velocity defined in terms of the disk plasma density ρ and the radial component Br of the magnetic field just 2 is a characteristic wavenumber for this outside the disk, and kg ≡ gz /vA problem, defined in terms of this vA and the vertical component gz ≡ (GM/r2 )(h/r) of the stellar gravity at the disk-magnetic field interface at z = h. In the above equation, the first term within the square root is the instability-driving term, the second describes stabilization due to magnetic tension, and the third, that due to gravity. Generally, the wavenumber k is dominated by its azimuthal component, kφ kr , (i.e., the ripples on the disk surface are mainly in the φ-direction), so that the second term is relatively unimportant, and stabilization is dominated by the gravity gz . Accordingly, the critical wavenumber kc above which modes (ripples) become Kelvin-Helmholtz unstable, kc ≈ kg (c/v0 )2 is basically determined by kg . Note that this characteristic wavenumber kg for a “standard” SS
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disk can be written in the following way (GL) from its above definition, with the aid of the radial structure of such disks given in Table 10.1: 6/5 12/5 ˙ 3/5 r8 cm−1 , kg ≈ 1.1 × 10−7 α−4/5 µ−2 30 M17 (M/M )
(10.60)
where M˙ 17 is the accretion rate in units of 1017 g s−1 , as before. We can now −1 , of the Kelvin-Helmholtz unstable calculate the growth time, tKH ≡ γKH modes with k > kc , and compare it with the radial inward drift timescale td of an α-disk given in Table 10.1. The result is (GL): −1 1/2 k Ωs = 1.4 × , 1− tKH ΩK kc (10.61) showing immediately that the Kelvin-Helmholtz instability grows much faster than the rate at which the slow radial inward drift of the disk plasma can sweep the magnetic field inward. Consequently, we expect a thorough mixing of the disk plasma with the stellar magnetic field by the time a given blob of matter has drifted inward through the disk. The second process is turbulent diffusion, by which the stellar magnetic field would rapidly diffuse vertically into the disk, under the conditions of interest in this problem. We have discussed turbulence in accretion disks above, and have given estimates of turbulent viscosity. Turbulent diffusivity νt of magnetic fields is closely related to that, scaling also as νt ∼ vt h ∼ αcs h. GL used the Parker (1971) estimate νt ≈ 0.15vt h, and so calculated the vertical diffusion time as tt ∼ h2 /νt . Comparing this timescale with that for the radial inward drift again as above, these authors obtained 7/10 M td −2/5 −1/10 = 8.6 × 102 α1/5 M˙ 17 r8 , (10.62) tt M td
105 α−5/4 µ−1 30
M M
5/8
17/8 r8
so that, again, the magnetic field is expected to diffuse to the disk’s midplane by the time a given blob of matter has drifted inward through the disk. The third process that GL explored was the reconnection of the stellar magnetic field above and below the disk to small-scale magnetic fields advected with the disk plasma: we have already discussed the latter field above. The idea here is that the stellar magnetic field would reconnect to the disk field-loops first near the top and bottom surfaces of the disk, and these field configurations would then reconnect to field loops further inside the disk through shearing of field lines through differential rotation and
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convective overturn in the disk. Thus the stellar magnetic would penetrate into the disk. How fast would this process go? It is widely accepted that reconnection proceeds at an appropriate fraction ξ of the Alfv´en speed vA , a commonly-advocated value of the fraction from solar physics being ξ ∼ 0.1. So the reconnection timescale through the disk is tR ∼ 2h/ξvA . The question then is: what magnetic field strength should we use for calculating vA , since, at any given r, the strength of the stellar field will, in general, be different from that of the field in the disk? We can make the conservative assumption that, when two unequal fields reconnect, the overall reconnection speed is determined by the weaker one, which has the lower Alfv´en speed. Now, the reader can show with the aid of the radial structure of α-disks given above that if viscosity is dominated by the magnetic part in Eq. (10.42), then the small-scale magnetic field strength in the disk exceeds 8 that of the stellar magnetic field at radii > ∼ 10 cm, so that the reconnection speed is determined by the latter field. The above timescale tR can then be calculated and compared with the radial drift timescale td again, yielding (GL): td = 5 × 103 ξα−7/20 µ30 tR
M M
11/45
−73/40
r8
.
(10.63)
Once again, we see that the stellar magnetic field will penetrate the disk by this effect in a time much shorter than that which a blob of matter takes to drift inward through the disk. Thus, there are a variety of processes by which the stellar magnetic field will “invade” into the disk and mix with the disk plasma quite thoroughly during the residence time of accreting matter in the disk (= radial drift time of matter through disk), the region of such penetration and mixing being expected to extend well beyond the inner edge of the disk. The diamagnetic configuration is thus not self-consistent, and this realization made GL explore the steady-state configuration that was likely to emerge from the above picture of penetration and mixing. We summarize this below. 10.1.6.2
Steady flow models
The basic features of stellar magnetic field lines threading an accretion disk, and so modified by the motion of the high-conductivity disk plasma, follow from basic physics, are well-known today (GL), and displayed in Fig. 10.10. The disk plasma in Keplerian rotation at a radius r moves with an angular
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velocity ΩK (r), which is different from that of the neutron star’s rotation, Ωs , at all disk radii except at the co-rotation radius rc in the disk, defined by ΩK (rc ) = Ωs , so that rc is given by: rc =
GM Ω2s
1/3 .
(10.64)
Thus, all stellar field lines connecting the star and the disk will have their two “foot-points” on the star and the disk moving at different angular velocities, except for the solitary ones which pass through the disk at the corotation radius rc . As both foot-points are anchored in very high-conductivity material (the neutron-star crust and the disk plasma), this difference in angular velocities |ΩK − Ωs | will stretch the field lines in the azimuthal direction, generating a toroidal field9 Bφ from the poloidal field Bp . How far will this stretching go? As GL pointed out, if the usual (extremely high) Coulomb or Bohm conductivity of the disk plasma decided this, the steady-state value of Bφ — as determined by the condition that the above rate of stretching just balances the rate of diffusion of the magnetic field through the disk due to the finite disk conductivity — would be enormous. Long before this can happen, however, various plasma processes in the disk will allow the (large) toroidal magnetic fields with opposite directions in the upper and lower halves of the disk to reconnect. Ultimately, therefore, the steady-state value of Bφ will be determined by a balance between the above amplification by differential rotation between the star and the disk on the one hand, and the above reduction by reconnection on the other. In effect, the latter process then formally mimics a much lower “effective” conductivity σeff than the Coulomb or Bohm one, which maintains Bφ at a moderate value actually given by the above balance process. As GL argued, and as we shall presently see, the value of this effective conductivity σeff is completely determined by the condition of steady flow, and does not depend on the details of the above reconnection processes. The limiting value of Bφ — or the widely-used measure of winding, γφ ≡ Bφ /Bp , called the magnetic pitch — does, of course, depend on the overall speed of re9 This picture of stretching and generation of toroidal fields requires the presence of a small amount of plasma on those parts of the stellar field lines which lie between the star and the disk. This is so because if this region were a perfect vacuum, stellar field lines would continually reconnect there without dissipation of energy, maintaining the different angular velocities of the foot-points on the stellar surface and the accretion disk, so that no toroidal field would develop. However, as shown by GL (1979a), the required plasma density is small enough that such amounts are certain to be present in accreting binary systems.
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connection and related phenomena responsible for this limiting process. A similar picture holds for the stretching of the stellar field lines in the radial direction due to the radial inward motion of the disk plasma toward the star. This generates a radial component Br of the magnetic field from the dominant vertical component Bz in the disk plane, which is limited by the above processes, giving a “pinched” appearance to the stellar field lines, as shown in Fig. 10.10. In a simple theory, we can postulate, as GL did, that the “effective” conductivity σeff is a scalar, i.e., same in the azimuthal and radial directions. The overall GL disk-magnetosphere configuration is shown in Fig. 10.10. The basic feature of the disk-magnetosphere interaction is the occurrence of a transition zone between the unperturbed accretion disk far away from the star, and the magnetosphere close to the star. Over this transition zone, (a) screening currents flowing in the accreting plasma screened off the stellar magnetic field from the accreting plasma outside this zone, and, (b) dominant stresses responsible for removing the angular momentum of the accreting matter changed from viscous stresses to magnetic stresses of the stellar magnetic field as the matter moved inward. These authors found that this transition zone divided itself naturally into two different parts. First, there is a broad outer transition zone, where the accreting matter is still in Keplerian rotation like the matter in the unperturbed disk, but the stellar magnetic-field stresses have a significant role in transporting angular momentum, although the dominant mechanism for this are still the effective viscous stresses introduced earlier. Second, there is a narrow, inner transition region at the inner edge of the disk which has the character of a boundary layer , where the accreting matter’s rotation is non-Keplerian, and the stellar magnetic-field stresses dominate completely in transporting angular momentum, bringing the Keplerian-velocity matter into co-rotation with the star during its passage through this layer. The concept of boundary layers originated from terrestrial fluid flows near walls of pipes or channels, or near barriers in the flow, or near interfaces between two fluids: in such layers, the flow velocity and related properties undergo changes on lengthscales δ much shorter than the scale l on which the flow characteristics change in the main body of the fluid. This happens because these boundaries, e.g., walls, enforce a boundary condition on them which is completely different from the conditions in the main body of the fluid, and the flow adjusts from the one to the other over this boundary layer. The simple, classic example is that of a slightly viscous liquid in a plane flow parallel to a solid wall. Because of low viscosity, the laminar
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flow in the main body of the fluid can be considered as basically that of an ideal fluid with zero viscosity, but this does not hold on the wall, on which the fluid velocity v must vanish even at low√viscosities. This creates a plane boundary layer of small thickness δ ∼ l/ R at the wall, over which the velocity changes from v to zero. Here, R ≡ ρvl/η is the Reynolds number introduced in Sec. (10.1.5.3), η being the (low) coefficient of viscosity of the liquid, so that R is large, and δ l. In this simple hydrodynamic example, then, viscosity determines the properties of the boundary layer. The above concept proved useful, and was generalized to other situations in physics of fluids and electromagnetic fields where viscosity plays little or no role. Examples of these are magnetohydrodynamic (MHD) boundary layers, elctromagnetic boundary layers, and so on. The dissipative forces that determine the behavior of such boundary layers are naturally different: magnetic diffusivity, effective conductivity, and so on. In the GL transition zone described above, the innermost part of this zone was found to have the character of such a boundary layer, wherein the essential physical variables of the problem — flow velocity components and magnetic field — change on a lengthscale δ which is much shorter than than the scale r on which they change over the rest of the transition zone, and over the unperturbed disk flow outside the transition zone. In the usual nomenclature, then, we can call the broad, outer transition zone the main flow (as well as the unperturbed disk). Why does it change through a boundary layer near rin , the inner edge of the disk? The argument is, in spirit, the same as that given above for simple, viscous boundary layers: the co-rotating stellar magnetosphere enforces boundary conditions which are quite different from those which apply to the flow in the unperturbed disk or even in the broad, outer, Kelplerian transition zone, where electromagnetic effects are either absent or mild. It is to satisfy these new boundary conditions that plasma entering the magnetosphere has to adjust itself through the thin boundary layer at rin . Note that the stellar magnetospheric boundary behaves in this case not as a wall, but rather as an interface between two very different types of flows. What determines the essential properties of the GL boundary layer? We discuss this below, noting here only that the effective “viscous” stresses of accretion-disk flows are negligible in the boundary layer (see above), so that other dissipative phenomena must decide the nature of the boundary layer. As we shall see, the GL boundary layer is, roughly speaking, an electromagnetic one, but with some unique features of its own.
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Disk-Magnetosphere Boundary Layer
In the GL configuration (see Fig. 10.10), the inner edge of the disk is located at rin : we calculate this radius below. The boundary layer runs on a lengthscale δ from rin outward. Beyond the boundary layer, the outer transition zone runs upto a radius rs , beyond which the stellar magnetic field is screened off from the disk completely, so that the disk is an unperturbed SS one for r > rs . From the boundary layer, matter is lost from the disk and starts flowing toward the neutron star along the field lines threading the boundary layer. We summarize here the quantitative description of the boundary layer, which yields both δ and rin , as also the structure of this layer. The reader will notice in this description that its essential components appear rather similar to those of a description of α-disks, except that some basic mechanisms change character completely. In particular, (a) angular-momentum transport is entirely through stellar magnetic-field stresses, compared to which the viscous stresses are negligible, and also, therefore, (b) dissipation of energy is dominantly electromagnetic, rather than viscous, i.e., the dissipation of disk currents through the effective conductivity σeff introduced above. We give only the essentials here, referring the reader to the original work for more detail. Note that GL considered aligned rotators (wherein the dipole axis is aligned with the rotation axis), which are simpler than oblique ones, and yet highly non-trivial. These authors realized that almost all the essential physics of disk-magnetosphere interaction was contained in the aligned rotator. This is so because an oblique dipole can always be decomposed into an aligned one plus a perpendicular one (i.e., one whose dipole axis lies in the disk plane, and so is perpendicular to its rotation axis), and the latter produces little disk-star coupling because its field lines have little or no tendency to invade into the disk and thread it. (The reader can readily visualize this last point by drawing a perpendicular rotator and a thin disk.) Accordingly, we shall consider only aligned rotators henceforth. As for SS disks described earlier, consider mechanical aspects first. Angular-momentum conservation is expressed as: dM˙ d d(M˙ d r2 Ω) = r2 Ω + Bφ Bz r2 , dr dr
(10.65)
where M˙ d (r) ≡ 4πrhvr ρ is the radial mass-flow rate in the disk plane, and Ω is the angular velocity of the plasma. Remember that Ω is not Keplerian in the boundary layer, and that plasma continually lost from the disk plane
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accretes on the star along the field lines, so that M˙ d (r) decreases as plasma moves inward through the boundary layer. By contrast, Ω = ΩK and M˙ d has a constant value equal to the accretion rate M˙ throughout the outer transition zone and the unperturbed disk. Radial momentum conservation, unnecessary for explicit consideration in the outer transition zone and the SS disk because of the close balance between centrifugal and gravitational forces in these Keplerian regions and the consequent — very slow — radial inward drift, must be considered now, since vr attains much higher values in the boundary layer, where this close balance fails as Ω becomes subKeplerian. This reads as vr
GM r 1 dp [(∇ × B)×B].ˆ dvr = − 2 + Ω2 r − + . dr r ρ dr 4πρ
(10.66)
The condition of mass conservation, which relates the vertical mass loss from the boundary layer (see above) to the change in M˙ d (r) as plasma moves radially through the boundary layer, completes this part of the description. GL suggested a simple form for this: dM˙ d = 4πrρcs g(r). dr
(10.67)
The basic idea here is that the vertical flow velocity out of the disk should scale in terms of the local sound speed cs . GL introduced a “gate” function g(r) to model the radial profile of mass loss from the boundary layer. The latter profile depends on the detailed physics there, in particular on the poloidal shape of the field lines near the disk plane. To see this last point, recall the shape of undistorted dipole field lines from elementary magnetostatics to realize that the gravitational potential along such a field line has its maximum exactly at the disk midplane, so that plasma, when lost from the disk vertically, would have no difficulty in flowing toward the star along such a field line. But, as described above and shown in Fig. 10.10, dipole field lines threading through a thin disk are “pinched” inward, producing a local minimum in the gravitational potential at the midplane z = 0 and two local maxima above and below it, at z = ±zm , say. The mass-loss rate is then sensitive to the difference between the above maximum and minimum values, since the lost plasma, flowing roughly along the field line, must flow “over” the gravitational-potential maximum at zm . The analogous situation for diamagnetic disks was particularly stressed by Scharlemann in his pioneering 1978 study of near-diamagnetic disks. This author likened the situation with that in de Laval nozzles, and coined the name lip for the
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above potential maximum: for details, we refer the reader to the original work, and to Aly’s subsequent (1980) work on completely diamagnetic disks. In GL configuration, however, there is little difficulty in going through these lips, as the extent of radial pinching of field lines threading the boundary layer is quite small (since the magnetic field is already strong there and rapidly growing inward: see below), and, therefore, so is the above difference between the maximum and minimum values of the gravitational potential. Indeed, as long as the lips occur within vertical distances ∼ h from the midplane, there should be no significant hindrance to the flow, since the disk plasma extends vertically on a scale h (see Sec. (10.1.5.4)). Indeed, numerical simulations of the early 2000s, which confirm the GL configuration (see below), clearly show that the above radial pinching is quite small, and that the plasma flows unimpeded from the transition region to the star along an accretion funnel following the field lines: this is shown in Fig. 10.12 taken from Romanova et al. (2002). On the whole, the original GL gate function g(r) is expected to increase inward in the boundary layer, since the radial pinch of the field lines decreases inward. Consider thermal aspects next, as before. The equation of state remains unchanged from Eq. (10.40), of course. In the original GL work, only gas-
Fig. 10.12 Quasi-steady flow and magnetic-field configuration for disk accretion by magnetic neutron stars, according to recent MHD simulations. Color-coding of matter density, with color scale as given. White lines: magnetic field lines. Arrows: mass flux density ρv. Reproduced with permission by the AAS from Romanova et al. (2002): see Bibliography.
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pressure dominated (GPD) disks were considered, as it was clear that the inner edges of thin Keplerian disks around strongly magnetic neutron stars — such as are found in accretion-powered X-ray pulsars — would be at 8 radii rin > ∼ 10 cm (see below), so that the radiation-pressure dominated (RPD) innermost regions of these disks will never have a chance to occur. In subsequent work, GL also considered RPD disks because of their possible relevance to other X-ray binaries: we shall return to this point later. Now, what happens to Eq. (10.35) for hydrostatic equilibrium in the presence vertical flow as above? Surely it is not valid, and should be replaced by the equation for vertical momentum conservation? GL argued that, while this is true, the latter will still be of the form GM z ∂p = −Cp ρ 2 , ∂z r r
(10.68)
where Cp is a number ∼ 1. To see why this is so, note first that Cp = 1 when there is hydrostatic equilibrium in the vertical direction. When there is vertical flow with a flow velocity comparable to the sound velocity cs , as above, Cp = 1, but the correction term, ρcs (∂cs /∂z) ∼ (∂p/∂z), is comparable to the left-hand side of the above equation, so that the two sides of the equation can differ only by an amount comparable to each of them. Consequently, they are related to each other by the above equation. Finally, in the vertical energy transport condition, Eq. (10.39), we have to recognize the fact that the energy flux F is no longer given by the viscous dissipation as in α-disks, since such dissipation is negligible in the boundary layer, but by the resistive dissipation of the electric current J in the boundary layer, so that F =h
J2 , σeff
(10.69)
in terms of the effective conductivity σeff introduced above and elaborated on below. Consider, finally, the electromagnetic aspects, which had no analogue in the case of unperturbed α-disks, and which therefore contain the key to the problem. First, the above current J is related to the magnetic field B by the Maxwell equation ∇×B=
4πJ . c
(10.70)
Next, the generalized Ohm’s law for the plasma can be written down in the form
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1 J = σeff E + v × B . c
(10.71)
The value of the effective conductivity σeff then follows on combining the radial components of the above two equations, and can be written as follows in terms of the magnetic pitch, γφ ≡ Bφ /Bz , introduced above: σeff =
γφ c2 . 4π hr(Ω − Ωs )
(10.72)
In obtaining this equation, we have used the fact that Bφ reverses on a lengthscale h between the upper and lower halves of the disk, and that the radial component of the electric field is given by10 Er = −Ωs rBz /c. Thus, as mentioned above, the value of the effective conductivity σeff is completely determined by the condition of steady flow, without any direct reference to the details of the above reconnection and/or other processes which ultimately limit the flux-frozen stretching, and allow “slippage” of field lines, which σeff describes formally. An essential signature of these processes is still contained in a single parameter, viz., the magnetic pitch γφ : we shall return to this point later. As shown by GL, the boundary-layer equations given above can be studied in a way somewhat analogous to that described earlier for α-disks. First, we average in the vertical direction in a manner identical to that described in Sec. (10.1.5.4). The resulting description of radial structure contains information on the radial variation of the eight physical variables Ω, vr , Bz , M˙ d , h, T , p, and ρ, in the form of four differential equations (coming from Eqs. (10.65), (10.66), (10.67), and (10.70)), and four algebraic equations. We use the latter equations to eliminate the variables h, T , p, and ρ, which yields four differential equations for the four variables Ω, vr , Bz , and M˙ d . It is these equations that exhibit a boundary-layer behavior at rin , as we explain next. 10.1.7.1
Boundary-layer behavior
We summarize here only the physical essentials, as before, referring the reader to the original GL papers for detail. The basic idea is to show that, at a radius rin , there is a region of characteristic thickness δ rin such that the above four variables Ω, vr , Bz , and M˙ d change by large amounts on a lengthscale δ. This is usually accomplished by changing the 10 This follows from the argument that, when Ω = Ω , the magnetic field has no s toroidal component, and Jr = 0.
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independent radial variable in the above differential equations from r to the non-dimensional variable x≡
r − rin , δ
(10.73)
suitable for measurement within the boundary layer, whose thickness is ∼ δ. In addition, the four dependent variables also have natural units, which are used to non-dimensionalize them. Three of these are obvious: ω ≡ Ω/ΩK (rin ),
b ≡ Bz /Bz (rin ),
˙, F ≡ M˙ d /M
(10.74)
as the reader can readily understand. The fourth is a bit more subtle, namely, ur ≡ vr /( 2GM/rin δ/rin ).
(10.75)
Here, the natural unit of radial velocity is that velocity which matter would acquire if allowed to fall freely from rin +δ to rin in the gravitational field of the neutron star. As the centrifugal support is removed because Ω becomes sub-Keplerian in the boundary layer, this sets the scale for vr , as we shall see in the actual solutions given below. The above process then yields four differential equations for the four dependent variables ω, ur , b, and F in terms of the independent variable x. There is boundary-layer behavior if the coefficients in these equations are ∼ 1, since this shows that the variable-changes have been done consistently, and therefore that the variables are generally expected to undergo changes comparable to their original magnitudes as x changes by ∼ 1, i.e., the actual radial co-ordinate changes by ∼ δ. In reality, the situation is the reverse: since the above coefficients are not determined within the boundary-layer approximation, one assumes that they are ∼ 1, and this determines the boundary layer thickness δ and other parameters. In the current problem, it also determines rin , since it is an interface radius between two flows, and we have made no assumptions about its value. This has a rather deep significance, to which we return below. The point is, of course, that there was no assurance ´ a priori that this would lead to a consistent description in terms of a set of non-dimensional equations with coefficients ∼ 1 — either in this or in any other problem — so that, when this does happen, we say that there is a consistent boundary-layer solution to the problem. This is the case for the GL configuration.
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Boundary-layer: nature and structure
The actual non-dimensional equations are given in GL (1979a), and we shall not repeat them here. The boundary-layer thickness δ is given in terms of the disk’s inner-edge radius rin as δ ∼ 0.03rin .
(10.76)
Note that the approximation sign here includes the (relatively weak) dependence on the uncertain values of the boundary layer parameters, viz., the coefficients that occur in the above non-diemnsional equations for ω and b, the coefficient Cp that occurs in Eq. (10.68), and the magnetic pitch γφ in the boundary layer, all of which are expected to be ∼ 1. We return to the value of rin below. Consider next the physical nature of the above boundary layer. It is basically an electromagnetic boundary layer, in the sense that the dominant stresses in it are magnetic, and the dominant dissipation in it is through electromagnetic processes, mimicked by the effective conductivity σeff introduced above. However, plasma flow also plays an essential role in it, since it is the cross-field radial flow with velocity vr in the boundary layer which generates the toroidal electric currents in this layer, and these currents screen the stellar magnetic field. Accordingly, GL (1979a) gave the following estimate of δ by analogy with the usual form for electromagnetic screening length: δ∼
c2 , 4πσeff vr
(10.77)
which is close to the value which comes from the detailed calculations summarized above, and which brings out the essential physics transparently, as we can now see. Using the above for σeff and γφ ∼ 1, as applicable to the boundary layer, we get δ ∼ h(vK /vr )from the above equation. Now, we can use the scale of vr given by Eq. (10.75) in this to obtain the estimate11 : δ ∼ rin
h rin
2/3 .
(10.78)
This then sets the scale for δ, and the reader can quickly verify from the disk structure given earlier that this does indeed agree roughly with Eq. (10.78). 11 In
these estimates, we have neglected the effects of stellar rotation for simplicity, thus confining ourselves to values of ωs which are not close to unity. More complete expressions are given in GL.
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Fig. 10.13 Structure of the GL boundary layer for a neutron star rotating with a fastness parameter ωs = 0.3. Shown are the essential dimensionless variables introduced in the text: angular velocity ω, radial mass flux F , poloidal magnetic filed b, radial velocity ur , and gate function g, all as functions of the dimensionless distance x from the inner edge of the disk, suitably expanded to be a measure of distance within the boundary layer (see text). Reproduced with permission by the AAS from Ghosh & Lamb (1978): see Bibliography.
Consider now the structure of the boundary layer, which is obtained by numerically solving the above non-dimensional differential equations: representative results taken from GL are shown in Fig. 10.13. We give here only the general features of the structure, which are not affected qualitatively by the above boundary-layer constants, nor by the fastness parameter, ωs ≡ Ωs /ΩK (rin ), of the rotating neutron star introduced earlier. Further details may be found in the original work. The angular velocity of the plasma is reduced from the Keplerian value at the outer edge of
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the boundary layer to the stellar value at its inner edge by the magnetic stresses. Thus, ω decreases from 1 to ωs in passing inward through the boundary layer. The magnetic field is screened by the electric currents circulating in the boundary layer, and so is reduced by a factor ∼ 5 in going outward through this layer. Thus, b decreases from 1 to ∼ 0.2 in passing outward through this layer: this is incomplete screening, the rest of it being done by the weaker currents circulating in the outer transition zone (see above), which finally screen off the stellar field completely from the accretion disk by a radius rs ∼ (10 − 100)rin , as we describe later. Next, the radial velocity rises inward from the outer edge of the boundary layer (where it must equal, by continuity, the slow radial drift characteristic of the outer transition zone, which, in turn, is rather similar to the drift in the unperturbed SS disk) at first as centrifugal support fails (Ω having dropped from the Keplerian value, as above). But the rising magnetic-pressure gradient opposes this, so that the radial velocity passes through a maximum and is reduced to zero at the inner edge of the layer. Thus, ur rises inward through the boundary layer from very small (drift) values to a maximum value ∼ 0.1 − 0.3, and then falls to zero. Finally, the mass-flow rate through the disk decreases inward steeply through the boundary layer, as most of the mass is lost vertically, so that F drops sharply inward from 1. In modelling this loss, GL used various gate functions g(x)(see above) which, in keeping with the discussion given above, rose from zero inward through the boundary layer, saturating at unity somewhere in the middle of the layer. These authors found that the structure was completely insensitive to the details of g(x): the actual form of the gate function used, and related details, are given in GL (1979a). The structure is insensitive to the values of the boundary-layer constants, and the fastness parameter ωs only affects the maximum value reached by ur , as may be expected. 10.1.8
Inner Edge of the Disk
A remarkable feature about the GL approach is that it gives the radius rin of the inner edge of the disk automatically from the boundary-layer treatment. While this is not unexpected in principle, since rin is really an interface between two kinds of plasma flow, as explained above, and this interface must be at the correct place so that the boundary-layer description is valid and consistent, the implications turned out to be far-reaching for clarifying the qualitative differences between radial flow and disk flow in this context. Accordingly, we summarize them here briefly.
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Consider first the value of rin (GL), which can be written as 4/7 −2/7 rin ≈ 0.52rA ≈ 1.6 × 108 µ30 M˙ 17
M M
−1/7 cm,
(10.79)
in terms of the Alfv´en radius rA introduced earlier through Eq. (10.21). The numerical coefficient ≈ 0.52 in the above equation contains the dependence on the boundary-layer constants (as did that for the boundary-layer thickness δ in Eq. (10.78)), which is extremely weak in this case. Thus, for given values of the essential accretion variables µ, M˙ , and M , the disk’s inner edge is expected to form at a radius which is approximately half of what the Alfv´en radius would be if the external flow were radial with the same values of these variables. But what we must not forget is that the above way of writing is merely a convenient one, which is useful for making connections with lengthscales already defined in the subject in other contexts, but which does not imply any causal relation between the disk’s innermost radius and the Alfv´en radius. The physical arguments that determine them are completely different. Those that lead to rA have been summarized above; let us now consider those that lead to rin . A Keplerian disk is terminated when the stellar magnetic-field stresses become strong enough to remove the Keplerian angular momentum of the disk flow over a distance δ r (GL): viscous stresses of an α-disk are then completely negligible, centrifugal support fails, accreting plasma acquires considerable poloidal velocity, and goes into fluxfrozen magnetospheric flow. This conceptual condition may be quantified as (GL 1991) Bz Bφ 4πr2 δ ∼ M˙ rvK . 4π
(10.80)
Here, the left-hand side is the torque due to the magnetic stresses (Bz Bφ /4π) acting over the area 4πrδ of the boundary layer in the disk plane (including both upper and lower surfaces), and the right-hand side is the rate of arrival of Keplerian angular momentum — rvK per unit mass — into the boundary layer corresponding to an accretion rate M˙ . It is thus a question of balancing integrated magnetic and and material stresses, or, equivalently, one of equating a torque with a rate of change of angular momentum. This is, in effect, what the above boundary-layer equations do: a look at them will show the reader that we have dealt throughout with angular momentum transport in various forms.
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This would not be a problem, were it not for the fact that the similar numerical values and scalings of rin for disk flow on the one hand, and of rA or rm (the magnetospheric boundary radius; see above) for radial flow on the other, led to wide misconceptions in the subject that they come from the same physics and/or were equivalent. Worse than this, the static pressure balance condition discussed for the radial-flow case above, which is studied only as a simple, mathematical problem even in that case, and has to be modified into an appropriate momentum-balance condition for describing accretion flows, was indiscriminately used for determining rin , although it has no relevance to the disk-flow case at all. It is interesting that such pressure-balance condition, wherein the pressure of an undistorted dipolar magnetic field is usually balanced with the thermal pressure of a standard α-disk to determine rin , has often been used in the context of the diamagnetic disk configuration described earlier, the assumption being presumably that the plasma would somehow “fall” from the inner edge of the disk to the star. Actually, it is easy to see, with the aid of the following gedanken experiment (GL 1991), that such a diamagnetic disk would never be terminated. Imagine an initial-value problem, wherein a diamagnetic disk is forming around a magnetic neutron star, as in Fig. 10.14. As mentioned earlier, an accretion disk forms by spreading of a plasma ring through viscous stresses, and so the disk moves inward, as shown, maintaining static pressure balance with the magnetic field at its inner edge, and perfect diamagnetism in the disk, i.e., the stellar magnetic field is kept completely out of the disk at all times. Such a disk would never be terminated by stellar-magnetic stresses anywhere, since there is no coupling between the stellar magnetic field and a diamagnetic disk. What will happen is shown in Fig. 10.14: as more and more matter piles up in the disk, its pressure will increase, and inner edge will move to a smaller radius where the magnetic pressure is higher. As the disk is completely screened from the stellar magnetic field, angular momentum transport in it would be entirely by the effective viscosity of α-disks, which can only give a Keplerian disk, as explained earlier. So the disk will remain Keplerian at all times, and continue to drift inward, with no possible steady-flow state emerging. We have given arguments earlier why diamagnetic disks cannot occur in accreting X-ray pulsars, and we now see that a similar argument holds for the inapplicability of static pressure balance to disk-magnetopshere interaction. But why are the numerical values and the scalings with the essential accretion variables of rin , rA , rm , and the static pressure-balance radius
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Fig. 10.14 A gedanken experiment illustrating that the evolution of a diamagnetic disk would not lead to a steady accretion flow onto the neutron star. Reprinted with permission by Springer Science and Business Media from Ghosh & Lamb (1991) in Neutron c 1991 Kluwer Stars: Theory and Observation, eds. J. Ventura and D. Pines, p. 363. Academic Publishers.
so remarkably similar? The answer has been partly given above already : it lies in the extreme disparity between the radial dependence of magnetic stress, pressure, or energy density (all of which have the same dimension), and that of material stress or energy density, or thermal pressure (all of which have the same dimension) that occur in accretion problems. To see this disparity, note first that the above magnetic quantities scale as r−6 for a dipolar field, and even more steeply for higher multipoles. Note next that the above material or thermal quantities scale typically as r−5/2 : this is true (a) for the energy density ρvf2 f that occurs in the radial-flow case considered earlier, (b) roughly for the stress ρvr vK in a standard thin disk, and also (c) roughly for the thermal pressure in gas-pressure dominated regions of such a disk. (The reader can easily verify these scalings with the aid of the continuity condition, and the structure of thin Keplerian disks given above.) Therefore, if we determine a critical radius by equating a magnetic quantity in the above list to a material/thermal quantity
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in that list, the large difference between the exponents on the two sides, −5/2 + 6 = 7/2, would ensure that this critical radius would depend only weakly — like a ∼ 2/7 power — on the combination of numbers and the accretion variables µ, M˙ , and M (raised to different powers) that occur in the problem. Accordingly, the numerical values of the critical radius would come out rather similarly from these approaches, and the dependence on the accretion variables would be rather weak, and often rather similar. The reader can readily verify this by actual calculations. It appears now that it was the above mathematical coincidence, which has no physical content, that had led to the long-standing confusion in the literature, and the use of static pressure-balance condition for disk accretion by some authors for nearly two decades. Today, when numerical simulations (see below) have confirmed the absence of diamagnetic disks, the threading and mixing of disk plasma and stellar field lines, the termination of disk flow with simultaneous initiation of accretion-funnel flow to the star along field lines (see Figs. 10.12), we are left wondering about the reason for the above marathon incomprehension of the essential physics of disk-magnetosphere interaction, viz., angular momentum transport, which we have explained above. In hindsight, it seems to have been an automatic carry-over of the intuitive ideas of static pressure balance and confinement of a stellar magnetosphere by accreting matter — used in radial-flow situations and the geomagnetosphere — to a situation where they are irrelevant. It appears to have been forgotten that a thin accretion disk cannot “confine” a stellar magnetosphere in any sense (see the remarks at the beginning of Sec. (10.1.6), and Figs. 10.10 and 10.12) because of the open geometry of the situation. Stellar magnetic field lines are not like walls: when distorted radially by a thin disk at the magnetic equator, they simply readjust at other latitudes, and continue to interact with the disk over a wide region, as these numerical simulations have clearly shown. Indeed, the name magnetospheric boundary, so widely used in the subject and so useful in the radial-flow case (see above), seems a bit of a misnomer in the disk-flow case, since there is really no such boundary in this case: the disk is merely terminated at rin by magnetic stresses, and the open stellar magnetosphere extends all around it. 10.1.9
Outer Transition Zone
We now briefly consider the basic properties of the outer transition zone introduced above. Since this region is Keplerian, as already explained, most
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of the basic physics and mathematical descriptions used for standard αdisks, as detailed in Sec. (10.1.5.1), carry over, and we do not repeat them here. Rather, we mention and clarify only the features that need modification, or those which are new. In the former category are two aspects: one mechanical and one thermal. The first is a modification of the condition for angular-momentum conservation, Eq. (10.37), in order to include the magnetic stresses. The modified version can be written conveniently as: d dr
2 3 dΩK ˙ M r ΩK + 4πηhr = Bφ Bz r2 , dr
(10.81)
making the role of magnetic stresses in angular-momentum transport quite transparent. The second is a modification of the rate of energy generation in this region, wherein the usual viscous dissipation is augmented by resistive dissipation of screening currents flowing in the disk. The modified version of Eq. (10.38) reads: F = ηhr2
dΩK dr
2 +h
J2 . σeff
(10.82)
In the latter category, there is only one feature, namely, the electrodynamics that determines the magnetic pitch γφ . The GL prescription for this has been given above, wherein the amplification timescale for Bφ due to differential motion between star and disk, which is 1/|ΩK − Ωs |, is balanced against its reduction timescale ∼ 2h/ξvA , which represents the overall effect of reconnection through the disk and related complex processes. Here, √ vA = Bφ / 4πρ. This gives the magnetic pitch as γφ =
γ0 ξ
ΩK − Ωs 2h 4πρ, Bz
(10.83)
where γ0 is a numerical factor ∼ 1. This completes the description. The structure of the GL outer transition zone is very similar to that of a standard α-disk at the same radius, as far the usual disk variables h, T , ρ, etc, are concerned. The new feature is the screening of the stellar magnetic field by the disk currents, which makes the magnetic field negligible outside a screening radius rs ∼ (10 − 100)rin . We refer the reader to the original work for more detail. Since the magnetic pitch in the outer transition zone is crucial for determining the accretion torque, we return to this question in Chapter 12.
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Further Work
In the 1980s, several authors revisited GL-type configurations with general comments and ideas for modification/improvement. In a perceptive review of accretion magnetospheres published shortly after the original GL papers, Vasyliunas (1979) pointed out the uncertainties associated with the modeling of reconnection and related dissipative processes by a finite effective conductivity σeff , as above. The idea is that the reconnection process, by itself, conserves magnetic flux except in microscopically small regions around neutral points, whereas processes with finite resistivity are inherently flux non-conserving. This does not appear to be a serious problem, however, since such resistivity was postulated by GL in the transition region, where there is much cross-field flow, which is manifestly flux non-conserving. The issue would rather be whether reconnection, by itself, can lead to both the Bφ -limitation and the dissipation required for consistency: the answer is that this is neither expected nor required in disk-magnetosphere interaction, where turbulent overturn of eddies would twist the field lines stretched in a vertical plane (see above), a variety of current-driven instabilities would operate when Bφ becomes large, all with their attendant dissipation. Therefore, the idea of σeff must be to give a time-averaged description of a collection of complex, inherently-fluctuating, dissipative processes in the interaction region, the central question about reconnection being whether the limit to Bφ given from reconnection speed in the original GL prescription is viable or not: we return to this point below. Arons and co-authors (1984, 1987) considered the problem, calling the GL configuration “closed” for reasons which are not entirely clear today, since the inherently “open” nature of the magnetosphere for disk flow is well-known, and has been particularly stressed above. It is possible that these authors were referring to the fact that all stellar field lines were envisaged as threading the disk in the original GL picture, as in Fig. 10.10. This may well be too restrictive a view, and some field lines, which would thread the disk in the outer parts of the GL outer transition zone in this view (and whose foot-points on the star are therefore at high latitudes: see Fig. 10.10), may actually become open, as has been discussed in the literature subsequent to the original GL work. However, these authors suggested that, while the mixing of disk plasma and stellar magnetic field in the GL boundary-layer appeared to be viable in their view, such mixing in the GL outer transition zone did not appear to be so, since field lines coming into the disk would rise back due to magnetic buoyancy, go out of the disk, and
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become open. Accordingly, these authors suggested a centrifugally-driven outflow along these open field lines. Unfortunately, this magnetic buoyancy argument does not seem to stand up to scrutiny, as can be seen in the following way. The rise of such buoyant flux tubes through the disk would be 12 at velocities < ∼ cs , the local sound speed . The reader can compare the rise −1 timescale tr > ∼ h/cs ∼ ΩK with the several invasion and mixing timescales tKH , tt , and tR given in earlier, and show that tr tKH , tr < tt , and 8 tr > ∼ tR for representative values of α and r ∼ 10 cm. Hence the invasion processes together dominate over buoyancy, and the disk plasma stays mixed with the stellar field considerably beyond the inner edge of the disk. Recent mumerical simulations (see below) have amply confirmed this, the basic physics being as follows. The field interacting with the plasma in the outer transition zone is relatively weak, so that its energy-density is smaller everywhere than the matter energy-density in the disk. Hence, we are in the matter-dominated r´egime, where field lines do not open easily. Indeed, these simulations find that the field lines threading the outer transition zone upto radii ∼ 10rin keep on opening and closing, and there is little or no outflow along these. However, at larger radii, field lines do open up due to differential rotation, but even these carry little matter. Only by artificially increasing the magnetic field-strength and decreasing the matter-density to levels which have no relevance to disk-accreting neutron stars can the numerical simulations be pushed to magnetic-field dominated r´egime, when field lines do open dramatically, and magneto-centrifugal winds are indeed possible. In 1987, Wang pointed out that the magnetic pitch given by the original GL prescription, Eq. (10.83), was an overestimate, since it would give such large values of Bφ in the outer parts of the GL outer transition zone that the magnetic pressure associated with it would exceed the thermal pressure of the plasma there, which was inconsistent. This author suggested that such a situation would disrupt the disk, but it is perhaps more likely that such large fields would bulge out of the disk, reconnect, and escape before that happens, in effect reducing the field strength and limiting the magnetic pitch to a self-consistent saturation level. There is no doubt today that the original GL prescription for the magnetic pitch gives an untenable overestimate in the outer parts of the outer transition zone. But what is the correct prescription? Following the original GL work, several authors in the 1980s had suggested that, instead of equating the timescales for am12 From
elementary physics, the rise velocity is < ∼ the expression for gz given earlier.
√
2gz h, which becomes ∼ cs on using
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plification and reduction of Bφ , as above, a more accurate procedure might be to equate the rate of growth of Bφ , which is |ΩK − Ωs |Bz (this follows from the earlier statement that Bφ is generated from Bz , and also, formally, from the Maxwell equations) to its rate of reduction by reconnection, which is ∼ Bφ /(2h/ξvA ). Following the procedure for the original prescription, the reader can readily show that the limiting pitch so obtained is the square root of that given by Eq. (10.83). Wang (1987) also advocated this value, from rather similar arguments. Over the last decade, the general thought has been that even such a magnetic pitch is hard to sustain if it very much exceeds unity. A variety of current-limiting plasma instabilities and related processes has been invoked to argue that the large currents implied by such large values of Bφ are unlikely to be sustained in reality in the outer transition zone. This is generally supported by numerical simulations of recent years, which find γφ ∼ 1 in the transition zone. However, a convincing prescription for limiting magnetic pitch has not emerged yet. As this point has crucial significance for accretion torques, we return to it in Chapter 12. As we have mentioned already, attention in 1990s turned to the role of the magneto-rotational instability (MRI) — or Balbus-Hawley instability (BHI) — wherein magnetic field threading through differentially rotating fluids, such as occur in accretion disks, are amplified until limited by dissipative processes like reconnection. For details of MRI, we refer the reader to the excellent review by Balbus and Hawley (1998). The reader will appreciate, no doubt, that MRI has immediate consequences for not only for (a) the fate of small-scale magnetic field loops in accretion disks, since there is a clearly a possibility here of calculating the famous α-parameter of effective viscous stresses from “first principles” here (see above), if we can identify the bulk of these stresses as magnetic ones, but also (b) the interaction of stellar magnetic fields with accretion disks, which bears directly on the problem we are discussing here. Pioneering numerical simulations were done on the role of MRI/BHI in disk-magnetosphere interaction by Miller and Stone in 1997. These authors explored the evolution of diamagnetic disks, as well as those threaded everywhere by stellar field lines, their main aim being to study the initial, rapid evolution of such disks due to the onset of MRI. These simulations were 2-dimensional, and we refer the reader to the original work for more detail. In the early 2000s, Romanova and co-authors did a series of MHD simulations of disk-magnetosphere interactions which have shed light on some
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of the most important features of these interactions described above: see Fig. 10.12. First, there has been a rather dramatic confirmation of the overall GL configuration, as pointed out already, viz., the termination of the disk and the accretion-funnel flow along the field lines to the star’s polar caps, the threading of the disk by stellar field lines well beyond rin , and so on. Second, instead of the thin boundary layer where plasma rotation became sub-Keplerian in the original GL configuration, these simulations found a relatively broad region of width δ ∼ (0.3 − 0.8)rin over which this happened. This was expected, as it had been realized in the 1980s and ’90s that the original GL separation of the total, broad transition region into a thin, non-Keplerian inner boundary layer and a broad, Keplerian, outer zone was for conceptual convenience, enabling them to give a semi-analytic description of each region. In reality, one region would change smoothly into another in any numerical simulation. The fact that the outer transition zone in these simulations turns out to have an extent of ∼ (5 − 10)rin confirms the qualitative correctness of the GL scenario. Third, the magnetic pitch in the transition region is ∼ 1, so that the large pitches envisaged by GL in the outermost parts of the broad transition region were incorrect. The specific mechanism suggested by these authors for limiting γφ was that, if the pitch became 1, torsional Alfv´en waves would propagate along the field lines, and dissipate themselves in the upper, tenuous parts of the disk, also called the corona. Finally, while there was strong coupling between the star and the disk over the above transition region, field lines could and did become open further out in the disk, as described above, although the outflow did not appear to be significant for conditions relevant for disk-accreting neutron stars. 10.1.11
Plasma Entry into Magnetospheres: Radial Flow
While discussing the properties of static magnetospheres in the radial-flow case earlier, we reminded the reader that the actual magnetospheres of interest in connection with accretion-powered pulsars are those through which considerable plasma flow is going on, since this plasma then accretes onto the neutron star, and generates the observed X-rays. Accordingly, the plasma outside the magnetospheric boundary must be able to enter into the magnetosphere, and we now consider the various possible processes by which they can do so. That the polar cusps in the magnetosphere can be entry points for the exterior plasma has been discussed since the days of geomagnetospheric research. In the 1970s, pioneering works by Elsner and
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Lamb (1977) and Arons and Lea (1976) established the dominant role of the Rayleigh-Taylor instability for plasma entry into accreting neutron-star magnetospheres in the radial-flow case, and suggested that cusp entry was unlikely to be significant compared to this mode, and so were other possible modes, e.g., diffusion across the magnetospheric boundary, and magnetic field reconnection. We take up each process in turn. 10.1.11.1 Entry via Rayleigh-Taylor instability The boundary of a static magnetosphere, with accumulated plasma resting on top of a magnetic cavity completely devoid of plasma, is inherently unstable because the plasma-magnetic field interface is acted upon by the downward gravitational pull of the neutron star. The situation is analogous to the familiar, everyday situation of a heavy fluid resting on top of a light fluid in the Earth’s gravitational field, which goes unstable at the slightest perturbation, developing “fingers” and “blobs” of the heavier fluid which descend into the lighter fluid. The fluids thus interchange places and so minimize energy, in an attempt to reach stability. This is called the interchange instability, and also the Rayleigh-Taylor instability, in honor of Rayleigh’s (1883) pioneering analysis of it, and also in honor of Taylor’s (1950) pioneering recognition that an acceleration of the interface is all that is essential for the phenomenon [Drazin & Reid 1981], and that this acceleration need not be gravitational. (The latter concept proves crucial for studying this instability in many other situations, e.g., in supernova explosions.) Interchange at the interface between (magnetically-confined) plasma and vacuum was thoroughly studied in the controlled-fusion research of the 1950s, where the role of gravity is played by magnetic-field curvature, and the corresponding instability is sometimes called the flute instability, due to the shape of the plasma fingers that form [Krall & Trivelpiece 1973]. The situation for accretion magnetospheres is rather close to the original Rayleigh-Taylor instability, since the magnetic field, which supports the overlying plasma at rm against the stellar gravity, may be thought of as the “light” fluid, and the plasma as the “heavy” fluid. The quantitative analysis proceeds in terms of the energy principle [Bernstein et al. 1958], which states that the boundary is unstable to infinitesimal perturbations ξ if δW ≡
B2 n ˆ .∇ P − (ˆ n.ξ)2 dS > 0, 8π
(10.84)
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for all possible ξ, the integral being over the boundary surface, and n ˆ denoting a unit vector normal to the boundary and pointing into the magnetosphere. If the condition n ˆ .∇(P − B 2 /8π) > 0 holds, we can always find a ξ for which the condition (10.84) is satisfied. This, therefore, is the criterion for Rayleigh-Taylor instability [Elsner & Lamb 1977], which we can rewrite, using the hydrostatic equilibrium condition at the boundr is the neutron-star gravity), and the ary, ∇P = ρg (where g = −(GM/r2 )ˆ force-free condition on the magnetic field at the boundary, (∇ × B)×B = 0, in the suggestive form: αρg > 0,
α ≡ cos χ −
κB 2 , 4πρg
(10.85)
where χ is the angle between the radius vector and the outward normal to the boundary, so that cos χ = −ˆ r.ˆ n, and κ is the curvature of the boundary surface, and therefore of the magnetic field lines at the boundary (since the former are parallel to the latter there) at that point. This curvature can be written in terms of boundary-shape rm (λ) in the standard way: 1 du/dλ κrm = √ − , 1 + u2 1 + u2
(10.86)
where u ≡ (1/rm )(drm /dλ), as before. In obtaining Eq. (10.85), we have used the relation n ˆ.(B.∇)B = κB 2 [Elsner & Lamb 1977]. The virtue of the criterion in the form (10.85) is that it makes immediate contact with our understanding of the original Rayleigh problem in ordinary hydrodynamics: the analogue of α there is simply the density contrast α ≡ (ρ1 −ρ2 )/(ρ1 +ρ2 ), often called the Atwood number , where ρ1 and ρ2 are the densities of the heavy fluid and the light fluid, respectively. For ordinary Rayleigh instability to occur, α must obviously be > 0, and it is the same criterion here, as Eq. (10.85) shows, except that α is a bit more complicated now. We can call it the effective Atwood number [Arons & Lea 1976], and its meaning is transparent when we consider αg = g cos χ − κB 2 /4πρ: the first term on the right-hand side of this equation is the component of stellar gravity perpendicular to the boundary, which is de-stabilizing, as in ordinary Rayleigh instability, and the second term is due to the curvature of magnetic field lines, which is stabilizing, since the the field lines are convex toward the plasma. We now rewrite α as: Ti cos χ , (10.87) , A≡ α = 2κrm A − Tf f 2κrm
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Fig. 10.15 Shape parameter A (closely related to the effective Atwood number: see text) vs. colatitude θ for model two- and three-dimensional magnetospheres. Dashed line: 2-D Elsner-Lamb magnetosphere, and open circles: 3-D Arons-Lea magnetosphere, both discussed earlier. Other magnetospheres as indicated. Reproduced with permission by the AAS from Elsner & Lamb (1977): see Bibliography.
to bring out the interplay between the shape of the magnetosphere, as measured by number A (calculated and plotted by Elsner and Lamb13 [1977], as reproduced in Fig. 10.15), and the plasma’s thermal content, as measured by Ti /Tf f , where Tf f ≡ GM mp /kr is the free-fall temperature, as introduced earlier. In arriving at Eq. (10.87), we have used the balance condition P tot = B 2 /8π given earlier, and have assumed that, behind the shock, P tot ≡ P therm + ρvn2 (see above) is dominated by the thermal pressure P therm = nk(Ti + Te ) ≈ nkTi , since the kinetic energy of the infalling plasma is largely converted into thermal energy in the shock14 . The instability criterion α > 0 is, therefore, basically a requirement that the plasma must be cooled enough (by whatever the dominant cooling mechanism is, e.g., Compton cooling) so that it satisfies Ti /Tf f < A. The profile of A as a function of the colatitude θ in Fig. 10.15 shows that this requires Ti to be ≤ 0.2Tf f over most of the magnetospheric boundary, except for a region of angular extent ∼ 10◦ around the pole, where the plasma must be even cooler to drive the boundary unstable. We gave above the ion temperature Ti ≈ 0.37Tf f just behind the shock; model calculations show that the ions cool to Ti ≈ 0.15Tf f by the time they reach the magnetospheric boundary rm [Arons & Lea 1976]. This means that most of the magnetospheric boundary, from the equator to the above 13 Note
that these authors denoted our A by α. assume, of course, that accretion onto the neutron star is going on, and Compton cooling is maintaining Te far below Ti behind the shock, as explained earlier. 14 We
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region around the poles, will, indeed, be Rayleigh-Taylor unstable. In the latter region, A decreases as θ does, vanishing at the poles (θ = 0), so that, for a given value of Ti /Tf f , the instability criterion will be violated in a region sufficiently close to the polar cusps, and the boundary will be stable there. This is the well-known Rayleigh-Taylor stability of the cusp region: the angular extent of this region for the ion temperature quoted above (which would be typical of bright, accretion-powered pulsars) is ∼ 5◦ [Arons & Lea 1976]. Thus, most of the magnetospheric boundary is expected to go RayleighTaylor unstable. What happens then? An instability means that perturbations of the magnetospheric boundary grow with time, and for RayleighTaylor instability in its original form, the growth rate γ (which is the imaginary part of a complex frequency, so that perturbations grow as exp[γt]) is given roughly by γ 2 ∼ αgkmax , where kmax is the most unstable wavenumber [Drazin & Reid 1981]. The result for this problem is closely analogous: if we do a normal-mode analysis of the above perturbation ξ in the form ξ = ξm (r, λ, t) exp(mφ), the growth rate of the mth mode at a latitude λ is given by [Arons & Lea 1976]: tanh(km ls ) 2 (λ) ≈ αgkm √ ∼ αgkm tanh(km ls ). γm 1 + u2
(10.88)
Here, ls is distance of the shock from the magnetospheric boundary introduced earlier, km is the wavenumber of the mth mode in a direction normal to the boundary at this latitude, and u is as before. The second form of the right-hand side of Eq. (10.88) follows from the fact that u = 0 at the equator (λ = 0), and increases monotonically with λ to only about unity even at the end of the unstable region near the polar cusps described above: the plot of u as a function of λ is given by Arons and Lea (1976). Note that the normal modes cannot extend outside the magnetospheric boundary any more than upto the position of the shock, i.e., 2π/km < ∼ ls , since the flow is hypersonic at radii greater than the shock radius rs = rm + ls , and so cannot be influenced by pressure perturbations at rm . This implies m 1. Further, at any latitude λ, the only significant modes are confined in their latitudinal extent to the small value δλ ∼ cos λ/(km rm )2/3 [Arons & Lea 1976]. What do the growing modes look like? At first, they resemble the flute modes of plasma-confinement studies referred to above, forming small, thin plasma filaments of length ∼ rm cos λ/mp (p = 1/2, 2/3) and width ∼ rm cos λ/m, which dip slightly into the magnetosphere and interchange
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Fig. 10.16 Stages of development of the Rayleigh-Taylor instability at the magnetospheric boundary. See text for description of stages (i)–(v). Reproduced with permission by the AAS from Arons & Lea (1976): see Bibliography.
positions with nearby flux tubes, which rise slightly, as shown in Fig. 10.16, panel (i). The rise of these magnetic “bubbles” then continues into the nonlinear r´egime, as shown in panel (ii), in which process the buoyancy of the bubble pushes it further up into the post-shock plasma flowing down subsonically toward the magnetosperic boundary, working against the downward ram pressure of the plasma. The rise stops at the stagnation height h where these two forces balance. During this process, a preferred filament
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size, or, equivalently, a preferred mode number m0 is likely to emerge. This is so because, while smaller wavelengths grow faster in the linear r´egime, thinner flux-tubes have smaller buoyancy and so rise more slowly, thus losing in the competition for space in which to rise. Ultimately, the tubes are expected to bunch towards an optimum width ∼ rm cos λ/m0 , with an optimal mode number m0 > ∼ rm /ls ∼ 30 [Arons & Lea 1976]. The stagnation point is shown in panel (iii), by which time the “valleys” into which the plasma has descended (in between the “hills” of rising magnetic flux tubes) sharpen further into “spikes”, falling almost freely into the magnetosphere. What happens then? As the peak of a bubble reaches h and comes to a halt, the sides of the bubble are still buoyant, and continue to rise: the bubbles thus acquire flat tops and the open necks of the spikes are narrowed considerably by lateral magnetic pressure , as shown in panel (iv). Ultimately, these necks close off, and it is believed that Kelvin-Helmhotz instability may play a crucial role in this. This is the instability that occurs at the interface between two fluids moving with respect to each other, the familiar terrsetrial example being wind blowing over the sea surface and causing growing surface-waves. In the magnetospheric case, it is the the motion of the plasma with respect to the magnetic field: it causes growing ripples on the surface of the moving plasma necks, and when the growth rate of this instability equals the narrowing rate of the necks, the plasma is expected to mix thoroughly with the magnetic field, and close off the necks completely. This produces detached filaments, or plasma blobs, as shown in panel (v), which sink through the magnetosphere at almost free-fall velocity vf f . At the same time, the “recovery” phase starts, i.e., new flutes begin to form at the surface of the cooled plasma, and the whole process (i)-(v) described above then repeats itself. This is the overall scenario pioneered by Arons and Lea (1976). In the above picture, the magnetospheric boundary then clearly “flaps” about its equibrium position rm (λ), given above, with an amplitude h. This flapping is, in general, only local, which means that different parts of the boundary may pinch off their plasma blobs and shower them into the magnetosphere in the above manner at various instants of time. Thus, we should think of “steady” accretion only in sense of a time-averaged flow over many such cycles. Finally, what happens to these diamagnetic √ plasma blobs of length ∼ rm / m0 and width ∼ rm /m0 which fall into the magnetosphere, and which are initially not threaded by the stellar magnetic field? We shall come back to the question of their ultimate fate.
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10.1.11.2 Cusp entry Plasma above the polar cusps can enter the magnetosphere by the loss-cone mechanism, which is a generalization of a simple, well-known concept used for understanding the behavior of charged particles moving in spiral paths around the Earth’s magnetic field lines in the geomagnetosphere. The particle’s helical motion is quantified by the pitch angle, which is determined by the ratio of its velocity perpendicular to the field line to that parallel to the field line. As the particle moves towards a magnetic pole, the parallel velocity decreases and vanishes at some point, where the pitch angle becomes 90◦ . The particle “bounces” back from that point along the field line, and approaches the other pole until it bounces back from the corresponding point near that pole. These are called mirror points, and the particle is thus trapped between them. The exception comes when a mirror point is so far inside the Earth’s atmosphere that the particle is lost by collision with atmospheric particles. Now, how close to the pole the mirror point is, is determined by how small the initial pitch angle of the particle was on the magnetic equator, where the magnetic field is the smallest along a given field line. Thus, there is a small range of pitch angles around 0◦ in the equatorial plane, i.e., a velocity cone, from which come the particles which are lost from the above mirror-trap. Hence the name loss cone. The idea is easily generalized in the present problem to define the loss cone as that region of phase-space in the particle distribution above the polar cusps which makes it possible for the particle to enter the cusp hole, the latter being the small region around the cusp where collisions and energy losses can lead to particle entry into the magnetosphere. The cusp hole is expected to have an area ∼ a2 , where a ≡ mcv/eB is the particle’s Larmor radius, v ∼ kT /m being the particle’s thermal velocity, and B the cusp magnetic field. The mass entry rate through the cusp hole by this loss-cone mechanism scales roughly as [Elsner & Lamb 1984]: np mp kTp ccusp , M˙ cusp ∼ np mp acusp v ∼ eB
(10.89)
where np is the number density of protons at the cusp and Tp their temperature, and cusp is the cylindrical radius of the cusp at its maximum extent, as shown in Fig. 10.17. In arriving at Eq. (10.89), it has been assumed that the loss cone is replenished constantly by plasma as it is depleted, and that the plasma inside the magnetosphere is collisionless. The above entry rate is understood in terms of an effective entry area ∼ acusp which is huge compared
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Fig. 10.17 Co-ordinate system used in calculating cusp entry: see text. Reproduced with permission by the AAS from Elsner & Lamb (1984): see Bibliography.
to the geometrical area a2 (by the factor cusp /a): this happens because some of the particle orbits that thread the field lines at the magnetospheric boundary lie deep within the magnetosphere at the cusp hole, as noted by Elsner and Lamb (1984). Modifications to the simple entry rate given above due to other effects, e.g., the appearance of transverse electric fields in the cusp due to the fact that protons tend to penetrate further into the magnetosphere than the electrons, have been described by the these authors. The basic point, however, is that the above mass-entry rate is tiny compared to the inferred accretion rates for canonical accretion-powered pulsars, M˙ ∼ 1017 gm s−1 . The ratio fcusp ≡ M˙ cusp /M˙ obs goes as ≈ −3/7 with the accretion rate [Elsner & Lamb 1984]. Thus, 3 × 10−10 M˙ 17 it is clear that cusp entry by loss-cone mechanism is quite negligible in accreting neutron-star magnetospheres at all accretion rates of interest for the observed accretion-powered pulsars. Michel (1977b) called attention to another interesting, possible mode of cusp entry. He argued that, since the magnetospheric boundary is stable at and near the cusps, the plasma collecting, stagnating, and cooling there drives the cusp down due to the increasing weight of the stagnant plasma. Eventually, a detached cusp, shaped rather like a plasma “drop”, descends to the stellar surface, while the original cusp re-forms behind it, unable to descend until the plasma cools sufficiently. The cycle then repeats itself,
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rather like the cyclic showering of plasma drops from a Rayleigh-Taylor unstable boundary described above. Elsner and Lamb (1984) argued that this “cusp drip” mode was unlikely to be significant in practice, as the the rest of the boundary, from the equator to fairly high latitudes, goes Rayleigh-Taylor unstable faster than this process (due to the fast Compton cooling of the plasma) and so provides another mode of entry, which the cusp plasma can avail of by moving a short distance in latitude, rather than stagnating over the cusps. Michel (1991) counter-argued that such movement of the stagnant cusp plasma would have to be uphill in the gravitational potential of the neutron star (because of the shape of the cusps), and so was not possible. The relative contribution of this cusp-drip entry mode to the total accretion rate remains to be clarified.
10.1.11.3 Entry by other modes We briefly mention two other possible modes of plasma entry into accreting neutron-star magnetospheres, namely, (a) by diffusion across the magnetospheric boundary, and (b) by reconnection of “frozen-in” magnetic field carried by the accreting plasma with the magnetospheric field. Consider diffusion first. We have argued earlier that the width of the magnetospheric boundary layer or “sheath” is ∼ the proton Larmor radius ap ≡ mp cvp /eB. How fast would plasma diffuse across such a boundary due to scattering of particles with other particles or with plasma waves? This is described by a cross-field diffusion coefficient D ≡ P c2 /σeff B 2 , where P is the pressure, and σeff ≈ ne e2 /νeff me is the effective electrical conductivity of the plasma in terms of the electron density ne and the effective collision frequency νeff [Krall & Trivelpiece 1973]. This frequency depends on the dominant scattering process in the sheath plasma: for a quiescent sheath, for example, Coulomb collision frequency, νCoul = 8πne e4 ln Λ/m2e ve3 is appropriate, while for very strong plasma turbulence, the Bohm value νBohm = eB/16me c is relevant. The sheath thickness δ is ∼ ap in quiescence, as given above, and remains of this order even if the sheath broadens somewhat due to various plasma instabilities — principally electrostatic in nature [Krall & Trivelpiece 1973] — which are excited when the current density in the sheath becomes sufficiently large. For Bohm diffusion,however, the diffusive layer broadens much more, to a thickness vp /vf f ap rm /32. The corresponding diffusive mass-entry rates δ ∼ M˙ diff have been estimated by Elsner and Lamb (1984), and expressed conveniently as a fraction of the fiducial rate of plasma arrival at the
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2 magnetospheric boundary M˙ 0 ≡ 4πrm ρm vf f . These authors find that −8 ˙ ˙ < fdiff ≡ Mdiff /M0 is ∼ (δ/rm ) ∼ 10 for turbulent diffusion with the usual sheath thicknessδ, and is rather similar for a quiescent sheath, increasing only to fdiff ∼ ap vp /32rm vf f ∼ 10−5 for the strong-diffusion, or Bohm, case, which is an extreme upper bound for diffusive entry rate. Thus, diffusion is insignificant as a mode of plasma entry into magnetospheres of the observed accretion-powered pulsars. Consider next entry via magnetic field reconnection, the idea of which is as follows [Elsner & Lamb 1984]. Accreting plasma contains embedded magnetic field, which can reconnect with the magnetospheric field in the above boundary layer, in analogy with what is believed to happen in the geomagnetosphere [see, e.g., Sonnerup et al. 1981]. After this reconnection, plasma originally threaded by these loops of embedded field can flow along the stellar field lines, and so accrete onto the polar regions of the star. Note that, for reconnection to occur, the magnetic field orientations must be correct, i.e., the two reconnecting fields must be either be directed oppositely or have significant components which are oppositely directed. The speed of reconnection vR is measured in units of the Alfv´en speed vA in the weaker of the two fields. Thus, vR = αR vA , where αR ∼ 0.1 is indicated by studies of magnetic-field reconnection on the solar surface, whereas αR ∼ 1 may be possible under forced reconnection. What is the expected plasma entry rate due to this process? Under the most favorable circumstances, wherein reconnection would go on simultaneously over the entire magnetospheric boundary, a straightforward 2 ρvR . We can translate this into a fraction fR of the estimate is M˙ R ∼ 4πrm e /vf f ), fiducial accretion rate introduced above, which yields fR ∼ αR (vA e e 2 where vA ≡ (B ) /4πρ is the Alfv´en speed in the magnetic field strength B e just exterior to the magnetospheric boundary. In reality, the scale λR of the reconnecting magnetic loops is expected to be rm , and only ∼ 2 reconnection sites of this size are expected to be simultaneously effective at each azimuth, since plasma entering at equatorial sites will have its flow to the poles impeded by the sites forming at high latitudes [Elsner & Lamb 1984]. This would reduce fR by a factor ∼ λR /rm . Thus, even if αR ∼ 1, fR 1 unless the external field B e has both the same scale as the magnetospheric field, λR ∼ rm , and also the same strength, since this is what e = vf f or, equivalently, (B e )2 /4πρ = vf2 f implies. As this is essentially vA impossible, this mode of entry is not considered significant for accreting neutron-star magnetospheres.
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Plasma Entry into Magnetospheres: Disk Flow
As indicated earlier, there is no inherent difficulty in plasma entry into the magnetosphere in this case, since there is no static magnetospheric boundary to cross. Indeed, as we have explained above, there is really no magnetospheric boundary here, except possibly at the “boundary layer” at the inner edge of the disk. Thus, the picture envisaged originally in the 1970s (GL) and confirmed by numerical simulations in the 2000s is that of a disk flow turning smoothly into an accretion-funnel flow along the field lines of the neutron star. It is possible, of course, that the innermost parts of the flow in what was called the “boundary layer” in the original GL picture can display Rayleigh-Taylor instability, but this is incidental, not crucial, to the entry process. Magnetic-field reconnection was, as the reader may recall, a crucial part of the GL scenario for both (a) mixing the disk plasma and the stellar magnetic field, and (b) limiting the distortion of the stellar field lines. The main concerns of the radial-flow case do not apply here, therefore, but a point of long-standing interest in the subject has been that of the stability of the flow from the disk to the accretion funnel. We now consider this briefly. The key physical criterion has already been introduced earlier, namely that field-aligned flow must be sub-Alfv´enic to be stable. Indeed, this criterion was used earlier to define the extent of the stable magnetosphere. For ease of discussion, we can define, as GL did, a flow-alignment or “threading” (see below) radius rt on each field line, so that plasma motion interior to rt is entirely along the field line: we discuss such motion below, and show that it is, indeed, sub-Alfv´enic, as it must be for consistency. Our concern here is, within the GL picture, with the vertical flow out of the boundary layer on its way to rt . A full calculation would be complex, but we can estimate the Alfv´enic Mach number MA of the vertical flow in the boundary layer from the boundary-layer structure given above. . In terms of the variables defined there, the Alfv´enic Mach number was given by GL (1979a) as: MA ≈
h rin
1/2
δ rin
1/4
gF bur
1/2 ,
(10.90)
which is shown as a function of the boundary-layer co-ordinate x in Fig. 10.18 for a representative situation. Clearly, MA < 1, i.e., the flow is sub-Alfv´enic over the entire boundary layer. The real uncertainty, however, is in our knowledge of the correct stability criterion in the presence
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Fig. 10.18 Alfve´ n Mach number MA in the flow just above and below disk plane vs. the boundary-layer co-ordinate x introduced earlier: see text. Reproduced with permission by the AAS from Ghosh & Lamb (1979a): see Bibliography.
of cross-field motion and finite conductivity, as is the case in the boundary layer. To the extent that it is still MA < 1, the vertical flow out of the boundary layer is stable. Recent numerical simulations have confirmed the sub-Alfv´enic nature of this flow. 10.1.13
Accretion Flows Inside Magnetospheres
After the plasma enters the neutron-star magnetosphere, how does it flow to the stellar surface? We explained earlier that, once it is threaded by the stellar magnetic field, it flows along the field lines to the magnetic poles of the star. On the other hand, if it is not so threaded, it still falls into the magnetosphere under the action of stellar gravity, in a sense “between field lines”, although the magnetic field may yet have an influence on its motion, as we shall see. We now summarize the essential characteristics of each of the above kinds of accretion flow. In deciding which of these kinds of flow actually occurs in a given system, we have to determine where in the magnetosphere the plasma becomes threaded by the stellar magnetic field: let us call this threading radius rt . If rt ≈ rm , then basically the entire flow in the magnetosphere is field-aligned, and described by the first scenario. On the other hand, if rt is substantially less than rm but still the stellar
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radius R, then the flow is between field lines in the outer magnetosphere, but field-aligned at r < ∼ rt . For this situation, Arons and Lea (1980) coined the term plasmasphere for the former region, rt < r < rm : as we shall see below, even in this plasmasphere, plasma drops may be “stripped” off the blobs, and may become frozen to the magnetic field lines. Finally, if rt < ∼ R, the flow would be between field lines over essentially the entire magnetosphere and plasma blobs would rain down on the entire stellar surface: but this is, of course, somewhat contrary to the basic notion of a magnetosphere (since the stellar magnetic field is supposed to dominate the accretion flow there, as the reader will recall), and we can say that the magnetosphere is then basically replaced by the plasmasphere. 10.1.13.1 Field-aligned flow Field-aligned flow of fully ionized plasma is rather easy to describe quantitatively, because the magnetohydrodynamics is greatly simplified by the fact that the electrical conductivity of the plasma is effectively infinite. We shall consider only steady flow here, in which case the continuity equation for the plasma flow is ∇.(ρv) = 0,
(10.91)
ρ being the plasma density and v its velocity. The basic electrodynamics is simply the statement that the electric field in the frame of the moving plasma vanishes, i.e., 1 E + (v × B) = 0. c
(10.92)
The last equation comes from that version of generalized Ohm’s law [Krall & Trivelpiece 1973] which applies to neutral plasmas in steady state. This law is normally expressed as an equation whose left-hand side is identical to that of Eq. (10.92), but whose right-hand side contains several essential electrodynamic and hydrodynamic terms instead of the zero that (10.92) does. Foremost among these terms15 is the finite-resistivity one, J/σeff , where J is the current-density, and σeff the effective conductivity of the plasma. For the magnetospheric plasma, all estimates of σeff indicate such a large value that the resistive term J/σeff is negligible compared to 15 The other terms are: (a) the Hall term, J × B/n ec, (b) the nonlinear term, e (me /ne e2 )∇.(vJ + Jv), and (c) the pressure term, (me /ne e)(∇.Pi /mi − ∇.Pe /me ). All of these can be shown to be negligible for the magnetospheric plasma: for details, see Ghosh et al. 1977.
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either of the terms on the left-hand side of Eq. (10.92). Indeed, the ratio of the resistive term to the reference term (1/c)(v × B) on the left-hand side of (10.92) is roughly the reciprocal of RM , the magnetic Reynolds number of the plasma flow, which is a measure of the strength of magnetic diffusion in the flow due to finite resistivity. This number is so defined that the resistive effects are smaller for higher values of RM . As an upper limit to σeff , we can take the Coulomb conductivity [Krall & Trivelpiece 1973]: σCoul ≡
3me √ 16 πe2 ln Λ
2kTe me
3/2
3/2
≈ 1.4 × 108
Te s−1 , ln Λ
(10.93)
where Te is the electron temperature, ln Λ is the Coulomb logarithm introduced earlier, and other symbols have their usual meanings. With this −3/2 in the magnetosphere at a distance r from conductivity, RM ∼ 1018 r8 the neutron-star center (r8 is r in units of 108 cm, as before), for Te ∼ 108 K, as above, ln Λ ∼ 10, and typical accretion rates M˙ ∼ 1017 g s−1 [Ghosh et al. 1977]. Clearly, RM is so large that infinite conductivity is an excellent approximation in this limit. As a lower limit to σeff , we can take the Bohm conductivity, which corresponds to strong, turbulent diffusion, as indicated above, and so can be taken as an extreme upper limit to diffusive effects. It is given by σBohm ≡
nec , 16B
(10.94)
in terms of the magnetospheric field B. As shown by Ghosh et al. (1977), even this conductivity yields a magnetic Reynolds number RM ∼ 107 r8−1 . Thus, infinite conductivity is an excellent approximation in the magnetosphere. If the flow is stationary, as we shall assume throughout, Maxwell’s equations imply that ∇ × E = 0, so that Eq. (10.92) leads to ∇×(v × B) = 0.
(10.95)
Equations (10.91) and (10.95) describe the basic geometry of field-aligned flows: they have been studied in detail in astrophysics for understanding stellar winds, i.e., outflows, from rotating, magnetic stars by Mestel (1968) and co-authors, and for understanding accretion inflows in rotating neutron-star magnetospheres by Ghosh et al. (1977) and others. The former problem is essentially the inverse of the latter one. Note that, throughout
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these and related calculations (such as those of accretion torques: see Chapter 12) described in this book, we shall consider only the case where the magnetic axis of the neutron star is aligned with its rotation axis, unless specified otherwise. We can call this the aligned rotator, which is the simplest case, and the only one that has been given a complete, quantitative treatment so far in the subject16 . The geometrical character of the flow is best seen by decomposing the velocity and magnetic field vectors into poloidal and toroidal components: ˆ v = vp + Ωφ,
ˆ B = Bp + Bφ φ,
(10.96)
and substituting these in Eqs. (10.91) and (10.95). Here, is the cylindrical radius, as before, i.e., we are working here with a cylindrical co-ordinate system (, z, φ), the z-axis being along the common roation and magnetic axis of the aligned rotator. The integrals of the resulting equations are [Mestel 1968, Ghosh et al. 1977]: vp =
f Bp , ρ
Ω = Ωs +
f Bφ . , ρ Bp
(10.97)
where f is the inward mass flux along a unit magnetic flux tube, and Ωs is the stellar angular velocity, both of these being constants along a given field line. This is the geometry of field-aligned flows: the poloidal flow is, quite transparently, entirely along the poloidal field, and the rotation of the plasma can also be seen to follow the toroidal field, once we take into account the fact the stellar magnetic-field structure is rotating with the star with an angular velocity Ωs , so that what appears as motion along Bφ in a frame rotating with the star will acquire an additional part Ωs when viewed from an inertial frame. The description is completed by identifying the third constant of motion along a field line, which is the specific angular momentum transported along the line by material and magnetic stresses, and ultimately added to the neutron star. must be conserved in the magnetospheric flow, since there is no mechanism within it for exchanging angular momentum with the outside world. The quantitative description comes through the steady-state momentum equation [Krall & Trivelpiece 1973]: ρ(v.∇)v = −ρ∇Φ + 16 In
1 (∇ × B) × B − ∇.P, 4π
(10.98)
misaligned rotators, the concept of a steady state has inherent difficulties, since the stresses due to a misaligned, rotating magnetic field vary periodically at any given point. A time-averaged description may still be useful.
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which generally includes gravitational, electromagnetic, and pressuregradient forces, the last one being negligible in this case, as above. Here, Φ is the gravitational potential. The toroidal component of this equation yields the integral [Mestel 1968]: Ω2 −
Bφ = , 4πf
(10.99)
which is a statement of conservation of angular momentum. Plasma rotation and the toroidal magnetic field are obtained by combining Eqs. (10.97) and (10.99), which yields: Ω=
Ωs − M2A (/2 ) , 1 − M2A
Bφ = M2A .
Bp Ωs 2 − . , vp 1 − M2A
(10.100)
where MA ≡ vp /vA is the poloidal velocity vp in units of the Alfv´en velocity 2 ≡ Bp2 /4πρ. We can call MA the Alfv´enic vA in the poloidal field, i.e., vA Mach number, in analogy with the usual Mach number where the reference velocity is that of sound. Complete specification of plasma motion and magnetic-field structure is obtained when we know the poloidal velocity vp and magnetic field Bp . In the interior of the magnetosphere, vp ≈ vf f , and Bp is approximately the undistorted stellar magnetic field — taken to be a dipole field throughout this book unless otherwise specified. Near rA , there may be an acceleration region for the infalling plasma, and the magnetic field may be considerably distorted by the currents circulating in the boundary layer, but, as argued by Ghosh et al. (1977), such regions are likely to be small. We shall adopt the interior magnetosphere values, as above, for the results described here. Field-aligned flow, as described by Eq. (10.100), shows characteristic properties, some of which we anticipated above already. First, the point where MA = 1 is clearly of great importance, since our solutions fail there: this is, of course, the Alfv´en radius rA , as defined by the condition ρvp2 = Bp2 /4π, as we saw above. Physically, our solutions are invalid at r = rA as completely field-aligned flow is not possible there: we have discussed earlier the various mechanisms of cross-field motion in the vicinity of rA . As argued earlier, stable, field-aligned flow is possible only for r < rA , i.e., inside the magnetosphere, and such flow is necessarily subAlfv´enic, i.e., it has MA < 1. That this condition is satisfied for our fieldaligned flow can be easily seen by noting that M2A = 4πρvp2 /Bp2 ∝ vp /Bp , using Eq. (10.97), and then using vp ≈ vf f ∼ r−1/2 and Bp ≈ Bdipole ∼ r−3 to obtain MA ∼ r5/4 . Thus, the Alfv´enic Mach number decreses in this
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manner as we go into the magnetosphere, and since MA = 1 at rA , the flow must be sub-Alfv´enic throughout the magnetosphere. Next, consider the behavior of the pitch-angle ψ of the magnetic field, defined as: tan ψ ≡
Ωs − (/2 ) Bφ = M2A . . Bp vp 1 − M2A
(10.101)
ψ is a measure of the toroidal “winding” of stellar field due the fact that field-aligned flow generally rotates at an angular velocity different from that of the neutron star, thereby generating a toroidal component of the field even if the original stellar field had no such component. It is clear from the above equation that, while the pitch is small in the deep interior of the magnetosphere (where MA 1), the field-lines are wound into an increasing tight spiral as r increases and approaches rA . This is as expected, since the magnetic stresses completely dominate over material stresses of the plasma at r rA , but are comparable to the latter as r → rA . Finally, note a curious, slightly paradoxical, behavior of plasma rotation in field-aligned magnetospheric flow, as described by Ghosh et al. (1977). As Eq. (10.100) clearly shows, Ω can change sign depending on the relative sizes of the two terms in the numerator (the denominator is positive throughout the magnetosphere, as MA < 1 there). Consider the situation where Ωs and have the same sign, and the star is slowly rotating in the sense that 2 |. Then the sign of Ω is opposite to that of Ωs over essentially |Ωs | < |/rA the whole magnetosphere: to see this, first note in Eq. (10.100) for Ω that, in the numerator, the size of the second term M2A (/2 ) exceeds that of the first in the vicinity of rA because of the above condition, since MA ≈ 1 there. Now note further that, as we go into the magnetosphere and r decreases, this term rises roughly as r−1/2 : this can be seen by recalling the scaling M2A ∼ r5/2 given above, and noting that, on a given dipole field-line (whose equation is r ∝ sin2 θ, where θ is the angle with respect to the dipole’s axis), 2 ∝ r3 , since = r sin θ. Thus, the second term dominates even more over the first as we go deeper into the magnetosphere, and Ω has a sign opposite to that of Ωs throughout aligned free-fall in the magnetosphere, i.e., the accreting plasma rotates “backward”, as seen by an inertial observer. This is shown in Fig. 10.19. This may appear paradoxical, since Ωs and having the same sign means that the angular momentum deposited on the star in the accretion process makes it spin faster, and yet the accreting matter itself rotates backward in the magnetosphere. How can this be? The resolution comes
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Fig. 10.19 Schematic angular-velocity profile in various regions of flow: accretion disk, transition region, magnetosphere, and deceleration region above stellar surface. Reproduced with permission by the AAS from Ghosh et al. (1977): see Bibliography.
when realize that the torque on the star associated with deposition of has two sources: (a) the stresses of the stellar magnetic field, and, (b) the circulation of the accreting matter. The former, which dominates by far in the interior magnetosphere, is in the forward direction, since the spiral shape into which the magnetic field is drawn has a forward pitch. This follows directly from Eq. (10.101), when we remember that vp has a negative sign (inflow), and that, in the numerator Ωs − (/2 ), the second term dominates throughout the magnetosphere, since it dominates at rA by our above assumption of slow rotation, and rises as −2 ∝ r−3 as we go into the magnetosphere. But why does the matter rotate backward? The answer also comes from the shape of the spiral: since it has a forward pitch, matter falling inward along the field lines has a backward azimuthal velocity, or backward rotation, in a frame rotating with the star (often called the corotating frame — a term we use extensively in this book). For slow rotation, as above, the transformation from the corotating frame to the inertial frame, which involves Ωs , does not change the direction of rotation, so that the matter rotates backward even in this frame: this is what Eq. (10.100) shows quantitatively.
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10.1.13.2 Flow “between” field lines The notion of diamagnetic plasma blobs falling toward the star between field lines originated from the result described above, namely, that RayleighTaylor instability was likely to be the dominant mode of plasma entry into the magnetosphere for radial exterior flow. Since these blobs interact with the stellar magnetic field only in a thin sheath of microscopic width on their surfaces, there is no reason to think that the magnetic field would thread through them automatically. Rather, the blobs would, at first, go into an approximately radial free-fall with a blob velocity vb ∼ vf f towards the stellar surface under the action of the neutron star’s gravity, remaining in approximate pressure equilibrium with the surrounding magnetic field as they fall. The last condition determines the size Rb of the blobs at any instant. What happens then? In 1980, Arons and Lea suggetsed that, as a blob with little internal magnetic field moves past the surrounding magnetic field with little plasma frozen to the field lines, the interface between them (i.e., the blob’s surface) becomes unstable to the hydromagnetic Kelvin-Helmholtz instability: this produces a spray of small plasma drops, which diffuse onto the magnetic field lines and become attached to them, eventually going into field-aligned flow towards the magnetic poles of the neutron star. Let us look into the various steps of this process. We introduced hydromagnetic Kelvin-Helmholtz instability above in connection with the “pinching off” of the plasma spikes at the magnetospheric boundary into blobs, and considered it in more detail in connection with the interaction of accretion disks with magnetospheres. The growth rate of the Kelvin-Helmholtz instability (KHI) in this problem is obtained from that given by modifying Eq. (10.101) (for the latter problem discussed earlier) in the following ways. First, the KHI driving term within the square root will now be (k.vb )2 /c2 , since it is the motion of the blob with a velocity vb relative to the magnetospheric field17 that causes KHI here, instead of the azimuthal velocity of the disk plasma relative to the magnetospheric field in the earlier case. Second, there is only one stabilizing term within the square root now, viz., the magnetic tension term k 2 , corresponding to the component of the wavevector parallel to the local direction of the magnetospheric field, since the 17 Strictly speaking, the relative velocity should be v − Ω φ, ˆ since the stellar field s b is corotating with the star, but this correction is small within the magnetosphere. The reason is that Ωs < ΩK (rA ) (see Chapter 12) for accretion to be possible, and vb ∼ vf f rises as r −1/2 as we go into the magnetosphere.
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gz -term of the earlier disk problem is absent here. The growth rate is thus [Arons & Lea 1980]: (k.vb )2 γKH = vA − k 2 , (10.102) c2 where vA ≡ B 2 /4πρb is the Alfv´en velocity in terms of the magnetospheric field B and the blob density ρb , and, for simplicity, we have neglected the presence of any plasma outside the blobs. What do the growing Kelvin-Helmholtz modes look like? Note first from Eq. (10.102) that growth occurs only for wavenumbers k exceeding a critical −1 value kc ∼ (c/vb )k > ∼ (c/vb )Rb , where we have used the fact that, for a −1 blob of size Rb , k > ∼ Rb . Thus, the depth δ of penetration of these modes into the blob, given by δ ∼ kc−1 ∼ (vb /c)Rb , is quite small compared to the size of the blobs, since vb < ∼ vf f c. However, the growth is very rapid, −1 with the blob-fall time as seen by comparing the growth time tKH ∼ γKH tb ∼ r/vb , their ratio being tKH /tb ∼ (vb /vA )(Rb /r) and so quite small, since vb < ∼ vf f < vA in the magnetosphere, and Rb /r ∼ 1/10−1/30. We can call these high-wavenumber eddies of size ∼ δ in the turbulent, thin surface layers of the blobs cat’s eyes, a term coined by Kelvin to describe the shape of the eddies in his original 1910 investigation of purely hydrodynamic Kelvin-Helmholtz instability. The scale δv of the turbulent velocities in these cat’s eyes is given by δv ∼ 0.1γKH /k ∼ 0.1(vb /c)vA ∼ 0.1(cs /c)vb , where we have used pressure balance between the blob and its surrounding magnetic field to set vA ≈ cs in the last step. Once a cat’s eye is formed, entrained in a drop of size δ Rb , and surrounded by an external magnetic field, it is likely to be subject to further KHI, forming smaller cat’s eyes on its surface, and the cascade is likely to continue down to smaller sizes, the n-th order cat’s eyes and drops having a size δn ∼ (vb /c)n Rb . This cascading process is extremely rapid, as a result of which the surface of the blob would disintegrate into an arbitrarily fine spray on essentially the timescale tKH given above, were it not for a suitable limiting process that sets a lower bound to the size of the drops. For purely hydrodynamic KHI, this process is surface tension, and for hydromagnetic KHI relevant to the present case, Arons and Lea (1980) proposed that finite Larmor-radius effect is the size-limiting process, the lower bound to the drop-size being δmin ∼ (c/vb )ri , where ri is the ion Larmor radius. As these authors argued, there are familiar, everyday examples of the hydrodynamic version of this phenomenon, e.g., the disintegration of a large blob of water
