High Energy Density Laboratory Astrophysics

HIGH ENERGY DENSITY LABORATORY ASTROPHYSICS

Edited by: SERGEY V. LEBEDEV Imperial College London, United Kingdom

Reprinted from Astrophysics and Space Science Volume 307, Nos. 1–3, 2007

Library of Congress Cataloging-in-Publication Data is available

ISBN 978-1-4020-6054-0 (hardbook) ISBN 978-1-4020-6055-7 (eBook) Published by Springer, P.O. Box 17, 3300 AA Dordrecht, The Netherlands.

Credit cover figure, left: Credit: NASA, ESA, John Krist (STScI /JPL), Karl Stapelfeldt (JPL), Jeff Hester (Arizona State Univ.), Chris Burrows (ESA/STScI) Credit cover figures, right: Imperial College London, Andrea Ciardi (Paris Observatory/Imperial College) and Sergey Lebedev (Imperial College)

Printed on acid-free paper All Rights Reserved c Springer 2007
No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without written permission from the copyright owner. Printed in the Netherlands

TABLE OF CONTENTS

Preface

1

Astrophysical jets, high-Mach-number flows, magnetized radiative jets, magnetic reconnection P. Hartigan / An Introduction to Observations Relevant to Astrophysical Jets and Nebulae Eric G. Blackman / Distinguishing Propagation vs. Launch Physics of Astrophysical Jets and the Role of Experiments

3–6 7–10

Yoshiaki Kato / Magnetic-Tower Jet Solution for Launching Astrophysical Jets

11–15

A. Ciardi, S.V. Lebedev, A. Frank, E.G. Blackman, D.J. Ampleford, C.A. Jennings, J.P. Chittenden, T. Lery, S.N. Bland, S.C. Bott, G.N. Hall, J. Rapley, F.A. Suzuki Vidal and A. Marocchino / 3D MHD Simulations of Laboratory Plasma Jets

17–22

Akira Mizuta, Tatsuya Yamasaki, Shigehiro Nagataki, Shin Mineshige / Outflow Propagation in Collapsars: Collimated Jets and Expanding Outflows

23–27

David J. Ampleford, Andrea Ciardi, Sergey V. Lebedev, Simon N. Bland, Simon C. Bott, Jeremy P. Chittenden, Gareth N. Hall, Adam Frank and Eric Blackman / Jet Deflection by a Quasi-Steady-State Side Wind in the Laboratory

29–34

A. Frank / Hypersonic Swizzle Sticks: Protostellar Turbulence, Outflows and Fossil Outflow Cavities

35–39

U. Shumlak, B.A. Nelson and B. Balick / Plasma Jet Studies via the Flow Z-Pinch

41–45

S. Sublett, J.P. Knauer, I.V. Igumenshchev, A. Frank and D.D. Meyerhofer / Double-Pulse Laser-Driven Jets on OMEGA

47–50

D.J. Ampleford, S.V. Lebedev, A. Ciardi, S.N. Bland, S.C. Bott, G.N. Hall, N. Naz, C.A. Jennings, M. Sherlock, J.P. Chittenden, A. Frank and E. Blackman / Laboratory Modeling of Standing Shocks and Radiatively Cooled Jets with Angular Momentum

51–56

R.F. Coker, B.H. Wilde, J.M. Foster, B.E. Blue, P.A. Rosen, R.J.R. Williams, P. Hartigan, A. Frank and C.A. Back / Numerical Simulations and Astrophysical Applications of Laboratory Jets at Omega

57–62

Ikkoh Funaki, Hidenori Kojima, Hiroshi Yamakawa, Yoshinori Nakayama and Yukio Shimizu / Laboratory Experiment of Plasma Flow Around Magnetic Sail

63–68

Markus B¨ottcher / Astrophysical Jets of Blazars and Microquasars

69–75

Ian J. Parrish and James M. Stone / Simulation of the Magnetothermal Instability

77–82

Lucas F. Wanex and Erik Tendeland / Sheared Flow as a Stabilizing Mechanism in Astrophysical Jets

83–86

Ph. Nicola¨ı, V.T. Tikhonchuk, A. Kasperczuk, T. Pisarczyk, S. Borodziuk, K. Rohlena and J. Ullschmied / How to Produce a Plasma Jet Using a Single and Low Energy Laser Beam

87–91

Radu Presura, Stephan Neff and Lucas Wanex / Experimental Design for the Laboratory Simulation of Magnetized Astrophysical Jets

93–98

V.I. Sotnikov, R. Presura, V.V. Ivanov, T.E. Cowan, J.N. Leboeuf and B.V. Oliver / Excitation of Electromagnetic Flute Modes in the Process of Interaction of Plasma Flow with Inhomogeneous Magnetic Field

99–101

B. Loupias, E. Falize, M. Koenig, S. Bouquet, N. Ozaki, A. Benuzzi-Mounaix, C. Michaut, M. Rabec le Goahec, W. Nazarov, C. Courtois, Y. Aglitskiy, A. YA. Faenov and T. Pikuz / Plasma Jet Experiments Using LULI 2000 Laser Facility

103–107

D. Martinez, C. Plechaty and R. Presura / Magnetic Fields for the Laboratory Simulation of Astrophysical Objects

109–114

Supernova remnants, shock processing, radiative shocks C.C. Kuranz, R.P. Drake, T.L. Donajkowski, K.K. Dannenberg, M. Grosskopf, D.J. Kremer, C. Krauland, D.C. Marion, H.F. Robey, B.A. Remington, J.F. Hansen, B.E. Blue, J. Knauer, T. Plewa and N. Hearn / Assessing Mix Layer Amplitude in 3D Decelerating Interface Experiments

115–119

A.B. Reighard and R.P. Drake / The Formation of a Cooling Layer in a Partially Optically Thick Shock

121–125

A.D. Edens, R.G. Adams, P.K. Rambo, I.C. Smith, J.L. Porter and T. Ditmire / Measurement of the Growth of Perturbations on Blast Waves in a Mixed Gas

127–130

Roland A. Smith, James Lazarus, Matthias Hohenberger, Alastair S. Moore, Joseph S. Robinson, Edward T. Gumbrell and Mike Dunne / Colliding Blast Waves Driven by the Interaction of a Short-Pulse Laser with a Gas of Atomic Clusters

131–137

Alastair S. Moore, James Lazarus, Matthias Hohenberger, Joseph S. Robinson, Edward T. Gumbrell, Mike Dunne and Roland A. Smith / Investigating the Astrophysical Applicability of Radiative and Non-Radiative Blast wave Structure in Cluster Media

139–145

J.F. Hansen, H.F. Robey, R.I. Klein and A.R. Miles / Mass-Stripping Analysis of an Interstellar Cloud by a Supernova Shock

147–152

Vikram V. Dwarkadas / Hydrodynamics of Supernova Evolution in the Winds of Massive Stars

153–158

C. Michaut, T. Vinci, L. Boireau, M. Koenig, S. Bouquet, A. Benuzzi-Mounaix, N. Osaki, G. Herpe, E. Falize, B. Loupias and S. Atzeni / Theoretical and Experimental Studies of Radiative Shocks

159–164

Matthew G. Baring and Errol J. Summerlin / Electrostatic Potentials in Supernova Remnant Shocks

165–168

X. Ribeyre, L. Hallo, V.T. Tikhonchuk, S. Bouquet and J. Sanz / Non-Stationary Rayleigh-Taylor Instabilities in Pulsar Wind Interaction with a Supernova Shell

169–172

Compact object accretion disks, x-ray photoionized plasmas D.D. Ryutov, J.O. Kane, A. Mizuta, M.W. Pound and B.A. Remington / Phenomenological Theory of the Photoevaporation Front Instability

173–177

R.J.R. Williams / Photoionized Flows from Magnetized Globules

179–182

Akira Mizuta, Jave O. Kane, Marc W. Pound, Bruce A. Remington, Dmitri D. Ryutov and Hideaki Takabe / Nonlinear Dynamics of Ionization Fronts in HII Regions

183–186

Marc W. Pound, Jave O. Kane, Dmitri D. Ryutov, Bruce A. Remington and Akira Mizuta / Pillars of Heaven

187–190

T. Sano / The Evolution of Channel Flows in MHD Turbulence Driven by Magnetorotational Instability

191–195

David Alexander / Laboratory Exploration of Solar Energetic Phenomena

197–202

S. Nagataki / Explosion Mechanism of Core-Collapse Supernovae and Collapsars

203–206

Stellar evolution, stellar envelopes, opacities, radiation transport John I. Castor / Astrophysical Radiation Dynamics: The Prospects for Scaling

207–211

P.A. Rosen, J.M. Foster, M.J. Taylor, P.A. Keiter, C.C. Smith, J.R. Finke, M. Gunderson and T.S. Perry / Experiments to Study Radiation Transport in Clumpy Media

213–217

Supernovae, gamma-ray bursts, exploding systems, strong shocks, turbulent mixing J.F. Hansen, M.J. Edwards, D.H. Froula, A.D. Edens, G. Gregori and T. Ditmire / Laboratory Observation of Secondary Shock Formation Ahead of a Strongly Radiative Blast Wave

219–225

Nathan C. Hearn, Tomasz Plewa, R. Paul Drake and Carolyn Kuranz / FLASH Code Simulations of Rayleigh-Taylor and Richtmyer-Meshkov Instabilities in Laser-Driven Experiments

227–231

Markus B¨ottcher and Charles D. Dermer / Models of Very-High-Energy Gamma-Ray Emission from the Jets of Microquasars: Orbital Modulation

233–236

S. Gupta and M. B¨ottcher / Time-Dependent Synchrotron and Compton Spectra from Microquasar Jets

237–240

Sergei S. Orlov and Snezhana I. Abarzhi / New Experimental Platform for Studies of Turbulence and Turbulent Mixing in Accelerating and Rotating Fluids at High Reynolds Numbers

241–244

Mikhail V. Medvedev / Weibel Turbulence in Laboratory Experiments and GRB/SN Shocks

245–250

M. Herrmann and S. I. Abarzhi / Diagnostics of the Non-Linear Richtmyer-Meshkov Instability

251–255

Planetary Interiors, high-pressure EOS, dense plasma atomic physics M. Koenig, A. Ravasio, A. Benuzzi-Mounaix, B. Loupias, N. Ozaki, M. Borghesi, C. Cecchetti, D. Batani, R. Dezulian, S. Lepape, P. Patel, H.S. Park, D. Hicks, A. Mckinnon, T. Boehly, A. Schiavi, E. Henry, M. Notley, R. Clark and S. Bandyopadhyay / Density Measurements of Shock Compressed Matter Using Short Pulse Laser Diagnostics

257–261

G. Chabrier, D. Saumon and C. Winisdoerffer / Hydrogen and Helium at High Density and Astrophysical Implications

263–267

Raymond F. Smith, K. Thomas Lorenz, Darwin Ho, Bruce A. Remington, Alex Hamza, John Rogers, Stephen Pollaine, Seokwoo Jeon, Yun-Suk Nam and J. Kilkenny / Graded-Density Reservoirs for Accessing High Stress Low Temperature Material States

269–272

S. Mazevet, M. Challacombe, P. M. Kowalski and D. Saumon / He Conductivity in Cool White Dwarf Atmospheres

273–277

Jonathan J. Fortney / The Structure of Jupiter, Saturn, and Exoplanets: Key Questions for High-Pressure Experiments

279–283

J. Hawreliak, J. Colvin, J. Eggert, D.H. Kalantar, H.E. Lorenzana, S. Pollaine, K. Rosolankova, B.A. Remington, J. St¨olken and J.S. Wark / Modeling Planetary Interiors in Laser Based Experiments Using Shockless Compression

285–289

Ultrastrong fields, particle acceleration, collisionless shocks D.D. Ryutov and B.A. Remington / Scaling Laws for Collisionless Laser–Plasma Interactions of Relevance to Laboratory Astrophysics

291–296

Matthew G. Baring / Topical Issues for Particle Acceleration Mechanisms in Astrophysical Shocks

297–303

Koichi Noguchi and Edison Liang / Three-Dimensional Particle Acceleration in Electromagnetic Cylinder and Torus

305–308

Edison Liang / Simulating Poynting Flux Acceleration in the Laboratory with Colliding Laser Pulses

309–313

Koichi Noguchi and Edison Liang / Three-Dimensional Particle Acceleration in Electromagnetic Dominated Outflows with Background Plasma and Clump

315–318

K.-I. Nishikawa, C.B. Hededal, P.E. Hardee, G.J. Fishman, C. Kouveliotou and Y. Mizuno / 3-D RPIC Simulations of Relativistic Jets: Particle Acceleration, Magnetic Field Generation, and Emission

319–323

Justin D. Finke and Markus B¨ottcher / Spectral Features of Photon Bubble Models of Ultraluminous X-ray Sources

325–327

F´elicie Albert, Kim TaPhuoc, Rahul Shah, Frederic Burgy, Jean Philippe Rousseau and Antoine Rousse / Polychromatic X-ray Beam from the Acceleration of Energetic Electrons in Ultrafast Laser-Produced Plasmas

329–333

T. Baeva, S. Gordienko and A. Pukhov / Scalable Dynamics of High Energy Relativistic Electrons: Theory, Numerical Simulations and Experimental Results

335–340

Sebastien Le Pape, Daniel Hey, Pravesh Patel, Andrew Mackinnon, Richard Klein, Bruce Remington, Scott Wilks, Dmitri Ryutov, Steve Moon and Marc Foord / Proton Radiography of Megagauss Electromagnetic Fields Generated by the Irradiation of a Solid Target by an Ultraintense Laser Pulse

341–345

Astrophys Space Sci (2007) 307:1 DOI 10.1007/s10509-006-9276-8

Preface

C Springer Science + Business Media B.V. 2007


The 6th International Conference on High Energy Density Laboratory Astrophysics was held on March 11–14, 2006 at Rice University in Houston, Texas. This is a continuation of the very successful previous conferences, held in 1996 in Pleasanton, California, in 1998 at the University of Arizona, in 2000 at Rice University, in 2002 at the University of Michigan, and in 2004 at the University of Arizona (organized by the University of Rochester). During the past decade, research teams around the world have developed astrophysics-relevant research utilizing high energy-density facilities such as intense lasers and z-pinches. Research is underway in many areas, such as compressible hydrodynamic mixing, strong shock phenomena, radiation flow, radiative shocks and jets, complex opacities, equations of state, superstrong magnetic fields, and relativistic plasmas. Ongoing research is producing exciting results using the Omega laser at the University of Rochester, the Z machine at Sandia National Laboratories, and other facilities worldwide. Future astrophysics-related experiments are now being planned for the 2 MJ National Ignition Facility (NIF) laser at Lawrence Livermore National Laboratory, the 2 MJ Laser Megajoule (LMJ) in Bordeaux, France; petawatt-class lasers now under construction in several countries, and future Z pinches. The conference brought together scientists interested in this emerging research area with topics including:

Stellar evolution, stellar envelopes, opacities, radiation transport Planetary Interiors, high-pressure EOS, dense plasma atomic physics Supernovae, gamma-ray bursts, exploding systems, strong shocks, turbulent mixing Supernova remnants, shock processing, radiative shocks

Astrophysical jets, high-Mach-number flows, magnetized radiative jets, magnetic reconnection Compact object accretion disks, x-ray photoionized plasmas Ultrastrong fields, particle acceleration, collisionless shocks These proceedings cover many of the invited and contributed talks presented at the conference. Of over 100 papers that were presented at the conference, 62 are included in this publication. The conference was organized by: Edison Liang, Rice University, Houston, TX Paul Drake, University of Michigan, Ann Arbor, MI George Kyrala, LANL, Los Alamos, NM Sergey Lebedev, Imperial College London, UK Bruce Remington, LLNL, Livermore, CA Hideaki Takabe, Osaka University, Japan The organizers would like to thank Umbe Cantu for the conference administration, as well as the sponsor and endorsing organisations: Rice University, Houston, TX Los Alamos National Laboratory, Physics Division Lawrence Livermore National Laboratory, High Energy Density Program APS Division of Plasma Physics APS Topical Group for Plasma Astrophysics DOE–NNSA Finally, the editor would like to thank all the authors and the referees for their contribution, time and effort.

Sergey Lebedev Guest Editor London, 2006

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Astrophys Space Sci (2007) 307:3–6 DOI 10.1007/s10509-006-9228-3

ORIGINAL ARTICLE

An Introduction to Observations Relevant to Astrophysical Jets and Nebulae P. Hartigan

Received: 30 May 2006 / Accepted: 25 July 2006 C Springer Science + Business Media B.V. 2006


Abstract This article reviews the basic physics and jargon associated with astronomical observations of nebulae, with an emphasis on processes relevant to shock waves in astrophysical jets.

continuum processes, and to ‘Astrophysics of Gaseous Nebulae and Active Galactic Nuclei’ by Osterbrock (1989) for emission line spectra.

Keywords Astronomical observations · Shock waves · Nebulae

2 Images

1 Motivation The HEDLA meetings bring together laboratory experimentalists, numerical modeling experts and observational astrophysicists to study how plasmas and fluids behave in a wide variety of conditions in nature. While this synthesis presents unique opportunities for collaborative research, communication between the different disciplines can be problematic, as each field has jargon and conventions that are not immediately transparent to scientists in other fields. As the oldest science, it is perhaps not surprising that astronomical conventions can be particularly arcane, though with a bit of background they quickly become second nature. This contribution provides a brief overview of the physics, conventions, and nomenclature used when describing and interpreting astronomical images of nebulae, with the objective to make it easier for a non-specialist to understand an observational astronomical talk on this subject. Most astronomical observations are either images or spectra (though some are both!), and in what follows I treat each of these in turn. For more information about physical processes that influence spectra I refer the reader to the classic texts ‘Radiative Processes in Astrophysics’ by Rybicki and Lightman (1979) for P. Hartigan Department of Physics and Astronomy, Rice University, P.O. Box 1892, Houston, Texas 77251-1892, USA

Astronomical images can be spectacularly beautiful, but what they tell us physically about the objects is determined to a large degree by the answers to two questions: (i) What is the scale/resolution?, and (ii) What do the colors represent? 2.1 Distance scales and units Astronomers typically use cgs-Gaussian units, but for convenience we like to represent planetary and solar-system scale distances in Astronomical Units (1 AU = 1.495 × 1013 cm), stellar distances in parsecs (1 pc = 3.09 × 1018 cm = 3.26 light years), and masses and luminosities in solar units (1 MO = 1.99 × 1033 gm, 1 L O = 3.83 × 1033 erg s−1 ). The AU is the distance from the Earth to the Sun, and a parsec is a typical distance between stars in the solar neighborhood. For reference, the average distance from the Sun to Pluto is about 40 AU, to the nearest star is 1.3 pc, the nearest region of massive star formation (Orion) is 460 pc, the center of our galaxy 8.5 kpc, and the nearest large external galaxy (M31) 0.77 Mpc. Distances are determined from a variety of methods, but the most direct one, applicable for the closest objects, is to observe the angular shift, known as the parallax, of the object relative to distant background stars as the Earth moves around the Sun (Fig. 1). A parsec is defined by the distance an object would have to be in order to have a parallax of 1 arcsecond. Hence, the number of AU in a pc is 206265, the same as the number of arcseconds in a radian. Springer

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Fig. 1 Definition of a parsec

The theoretical spatial resolution limit of an image is roughly λ/D, where λ is the wavelength of the light and D the diameter of the telescope. However, atmospheric turbulence limits all ground-based optical images to ∼1 arcsecond; observations in the near-infrared fare somewhat better and routinely obtain 0.5 arcseconds at a good site. With the use of adaptive optics to correct for atmospheric distortion one can approach the theoretical limit of λ/D, but such images have very small fields of view at present. A good rule of thumb is that a ground-based image is 1 arcsecond, and a spacebased or radio image is λ/D (e.g. Hubble Space Telescope is ∼0.07 arcseconds). By combining the above considerations it is straightforward to quickly infer distance scales for any astronomical image, provided you know the distance to the source. For example, if you see a ground-based image of the Orion Nebula, then the spatial resolution is ∼1′′ , which at 460 pc corresponds to 460 AU, or about 6 times the diameter of our solar system. An HST image of a large region of ionized gas at 2 kpc has a resolution of 2000 × 0.07 = 140 AU, and so on. 2.2 Colors, magnitudes, photometry, broadband and narrowband images Modern multicolor astronomical images are created by loading individual images into red, blue, and green channels, each image consisting of continuum and emission lines transmitted by the filter. Often the longest wavelength is put into the red channel, and the shortest into the blue, but one can also put images of the lowest ionization states in red and the highest ionization states in blue, or simply choose whichever

channels make the result pleasing aesthetically. Hence, color composites can provide a great deal of information about the physics of the region, or none at all, depending on the composite. It is also possible to create a ‘false color’ image where the intensities of a single image are assigned a specific color. Such images give no more information than greyscale images or contour plots do, and in this case the colors are completely arbitrary. X-ray and radio continuum images often appear in false color. The Earth’s atmosphere transmits from about 0.35 µm to about 1 µm, and this range is typically broken up into five bandpasses roughly 0.1 µm in width labeled U, B, V, R, and I for ultraviolet, blue, visual, red, and infrared, respectively. The Earth’s atmosphere is also transparent in several ‘windows’ throughout the near- and mid-infrared, including the three near-IR bandpasses J, H, and K at 1.25, 1.65, and 2.2 µm, respectively, and several bandpasses at mid-IR wavelengths that range out to 20 µm. However, ground-based telescopes emit thermal radiation at mid-IR wavelengths, which is why a small space telescope like Spitzer, cooled to cryogenic temperatures, is much more sensitive than are larger telescopes on the ground at these wavelengths. To quantify brightnesses in these broad bandpass filters, astronomers define a magnitude at each wavelength as mag = 2.5 log10 (F2 /F1 ), where mag is the magnitude difference between objects with fluxes F1 and F2 . The scale is defined so brighter is smaller, with the Sun having an apparent magnitude of about −26.5 at V, while the stars in the Big Dipper are about +2. The zero point for each wavelength is approximately the brightness of the star Vega in the northern summer sky. Color indices are defined as the difference between two magnitudes, e.g. B−V, where by convention the bluer filter is first so that redder objects have more positive colors. For objects like stars that radiate nearly like blackbodies, bluer colors mean higher surface temperatures (Fig. 2). In stellar evolution studies one often plots an ‘H-R diagram’, of either magnitude vs. color, or log(L) vs. log(T), where L is the luminosity (erg/s) and T the surface temperature of the star. If the object is a nebula that emits primarily line radiation and not continuum, then the apparent brightness simply depends on how many emission lines fall within the bandpass

Fig. 2 Examples of three optical spectra of stars that have different surface temperatures. Blackbody fits to the spectra appear as dotted curves

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of the filter. Typically one uses a narrowband filters to isolate specific emission lines of interest. Narrowband images also minimize the relative brightness of stars, which, being continuum sources, emit at all wavelengths including those transmitted by narrow band filters. 3 Spectra 3.1 Emission line spectra, densities, temperatures, velocities The most direct way to determine physical conditions within a nebula is to obtain a spectrum at each point. Many spectrographs disperse the light perpendicular to a long slit, and generate a spectrum at each position along the slit. Mapping the slit across the entire nebula produces a ‘data cube’ for each emission line, where the counts at a fixed wavelength produce an image at the corresponding radial velocity. Adding all the velocities together gives an emission line image. Instruments such as image slicers and Fabry-Perots give data cubes, as do most molecular line maps obtained with radio telescopes. Data cubes are both images and spectra, and, when combined with proper motion measurements that show how the object moves in the plane of the sky, represent the most thorough observational description of the object possible. Data cubes are time-consuming to obtain and can be challenging to analyze. The Doppler shift of emission lines in the spectra relative to the velocity of the ambient medium (usually the parent molecular cloud or a protostar in the case of stellar jets) reveals the kinematics of the gas, and the observed emission line width clarifies the gas dynamics such as temperature and turbulence. Because nebulae are optically thin (transparent) to most photons with energies a few eV, the emission lines observed at one position are actually integrals over the entire line of sight through the nebula. Heating and cooling within nebulae are governed by nonLTE processes. Nebular densities are very low, so nearly all the atoms and ions are in their ground states. Typical nebular densities range 10–106 cm−3 (note that this number must be multiplied by ∼1.67 × 10−24 to obtain g cm−3 for pure H composition). Nebular gas cools primarily as free electrons collide with atoms and ions and excite them to upper states, which then decay back to the ground level. In this way, kinetic energy of the electrons is converted to light, which escapes, so the nebula cools. Heating is usually accomplished by photoionization or by shock waves. In an ‘H II’ region, ultraviolet photons from a hot star ionize nebular gas and deposit any energy in excess of the ionization threshold into kinetic energy of the freed electron. Alternatively, shock fronts suddenly decelerate gas and convert a fraction of the bulk kinetic energy into heat. The kinetic temperature is determined by a balance of heating and cooling, and for an H II region is typically 104 K.

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Temperatures immediately behind shock waves are proportional to the square of the shock velocity, and the temperature then declines in an extended ‘cooling zone’ as the gas radiates emission lines. Astronomers refer to shocks that cool by emitting photons as ‘radiative’, a definition which differs from that used in high energy density physics, where radiative shocks are those where radiation is an important component to the total energy content of the gas. In astronomical nomenclature, a nonradiative shock is one where the timescales for cooling are so long that the postshock gas cools by some other means, such as expansion. It is usually straightforward to measure the electron density and temperature in a nebula. Many of the most abundant ionization states (e.g. O I, O II, O III, N I, N II, C I, S II, S III) have electronic configurations which have 2, 3, or 4 electrons in an outer p-shell. As shown in Fig. 3, p3 configurations always have a close doublet at a few eV above ground, and transitions from these levels are particularly good for measuring densities. Let us denote level 1 as the ground state, and levels 2 and 3 as a closely spaced doublet excited state. Because levels 2 and 3 have nearly identical excitation energies, their relative density n2 /n3 is independent of temperature. However, the flux ratio F21 /F31 of doublet lines to the ground state depends strongly on the density. In the low density limit, every collisional excitation is followed by radiative decay. In this limit, F21 /F31 equals the ratio at which the two levels are populated from the ground, which is typically the ratio of statistical weights g2 /g3 (ν 21 /ν 31 ∼1 for a

Fig. 3 Left – an energy level diagram for O I, an ion with 4 electrons in an outer p-shell. Line ratios between states highly separated in energy (e.g. λ5577/λ6300) constrain the temperature. Right – the same diagram but for O II, which has 3 electrons in the p-shell. The emission line ratio of λ3727/λ3729 determines the electron density between the low density limit (∼50 cm−3 ) and the high density limit (∼2 × 104 cm−3 ). All the lines depicted in this figure are forbidden transitions (electric quadrupole or magnetic dipole), with wavelengths in Angstroms

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closely-spaced doublet). When the density exceeds some critical value where collisional deexcitation rates exceed those for spontaneous decay, n2 /n3 approaches the LTE value of g2 /g3 , and the flux ratio F21 /F31 = (n2 A21 ν 21 )/(n3 A31 ν 31 ) ∼ (g2 A21 )/(g3 A31 ), which differs by a factor of A21 /A31 from the value in the low density limit. Hence, measuring the line ratio from a closely-spaced doublet gives the electron density if the density lies between the low density limit and high density limit for the transition (see Osterbrock 1989, p134, for examples of diagnostic curves). In general, an observed flux ratio between two transitions from an ion or atom defines a curve in Ne–Te space. The temperature dependence is more pronounced when the levels are more separated in energy. Line ratios between different ions determine the ionization fraction, and between different elements measure the relative abundances. A good historical example of such an analysis applied to shocks in jets is the classic paper by Brugel et al. (1981). In shock waves, the gas gradually becomes more neutral as it recombines. A hot star that illuminates a dark cloud of gas and dust sets up a similar stratified ionization structure, where higher ionization states occur on the side of the nebula which faces the star, and more neutral species dominate at greater distances into the cloud where ultraviolet light from the star becomes attenuated. In photoionized regions there is a balance between the ionization rate, which is proportional to aν Fν N Xi , and the recombination rate αN Xi+1 Ne. Here, aν is the photoionization cross section, α is the recombination rate coefficient, Fν is the ionizing flux, Ne, N Xi , and N Xi+1 are, respectively, the density of electrons, atoms in ionization state i, and atoms in ionization state i + 1. Equating these rates, the ionization parameter N Xi+1 /N Xi is proportional to aν Fν /Ne, which is why lower density nebulae have higher ionization fractions for a given ionizing flux.

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3.2 Continuous spectra Nebulae also emit continuum radiation from a variety of processes. Although less diagnostic than emission lines, continuum radiation also provides information concerning the state of the plasma responsible for the emission. Continuum radiation is characterized by the spectral index α, where Fν ∼ ν α . For example, α = 2 on the Rayleigh-Jeans portion of a blackbody spectrum. A spectral index is convenient because as long as one observes the source at two different frequencies, it is possible to fit a power law through those two points and obtain a spectral index. Of course, the source is probably not emitting according to a power law, but one can always define α = d ln Fν /d ln ν. Spectral indices for continuum processes change depending on whether or not the optical depth τ of the source is thin (≪1 or thick ≫1) For example, optically thick thermal free-free radiation from a plasma is identical to that from a blackbody, while optically thin free-free has a nearly flat spectral index (α ∼ −0.1 at optical/IR wavelengths). For synchrotron radiation, which occurs as electrons spiral relativistically around magnetic field lines, optically thick emission at long wavelengths has a spectral index of 5/2, but α = –(p−1)/2 at short wavelengths, where p represents the energy spectrum of the electrons, n(E)dE ∼ E− p (cf. Rybicki and Lightman, 1979).

References 1. Brugel, E., Bohm, K.-H., Mannery, E.: ApJS 47, 117 (1981) 2. Osterbrock, D.: Astrophysics of Gaseous Nebulae and Active Galactic Nuclei. University Science Books, Sausalito (1989) 3. Rybicki, G., Lightman, A.: Radiative Processes in Astrophysics. Wiley, New York (1979)

Astrophys Space Sci (2007) 307:7–10 DOI 10.1007/s10509-006-9205-x

ORIGINAL ARTICLE

Distinguishing Propagation vs. Launch Physics of Astrophysical Jets and the Role of Experiments Eric G. Blackman

Received: 15 April 2006 / Accepted: 26 June 2006 C Springer Science + Business Media B.V. 2006


Abstract The absence of other viable momentum sources for collimated flows leads to the likelihood that magnetic fields play a fundamental role in jet launch and/or collimation in astrophysical jets. To best understand the physics of jets, it is useful to distinguish between the launch region where the jet is accelerated and the larger scales where the jet propagates as a collimated structure. Observations presently resolve jet propagation, but not the launch region. Simulations typically probe the launch and propagation regions separately, but not both together. Here, I identify some of the physics of jet launch vs. propagation and what laboratory jet experiments to date have probed. Reproducing an astrophysical jet in the lab is unrealistic, so maximizing the benefit of the experiments requires clarifying the astrophysical connection. Keywords Astrophysical jets . Experiments . Magnetic fields

1. Distinguishing jet launch vs. jet propagation physics Jets in astrophysics emanate from accretion disk engines. The available jet mechanical luminosity is inversely proportional to the radius from the central engine, so the jet power is drawn from the inner most regions of the disk. Material must be accelerated to outflow speeds comparable to the escape speeds at the launch point. Radiation pressure is typically incapable of providing the directed momentum and magnetic launch models are favored (see Livio, 2004; Pudritz, 2004; Lynden-Bell, 2003). E. G. Blackman Department of Physics and Astronomy, University of Rochester, Rochester, NY, 14627

Magnetic models take different forms. In steady-state “fling” models (e.g. Blandford and Payne, 1982), mass flux is sustained by centrifugal and toroidal magnetic pressure forces along the poloidal field. Explosive “spring” models (e.g. Wheeler et al., 2002; Matt et al., 2004, 2006; Moiseenko et al., 2006) also thrive on a gradient of magnetic field pressure, but are time dependent and do not require an initially imposed mass flux. Such “springs” may operate in gammaray bursts (GRB) and maybe supernovae. In both spring and fling models, the launch region is Poynting flux dominated but on scales < ∼50Rin , (where Rin is the scale of the inner engine) the jet becomes flow dominated. Springs and flings can be further distinguished from magnetic tower Poynting flux dominated outflow models (Lynden-Bell, 2003; Uzdensky and MacFadyen, 2006); for the latter, Poynting flux domination remains even in the propagation region (R > ∼ 50Rin ). Related models have been applied to GRB and active galactic nuclei (AGN) assuming the baryon loading is low. In the relativistic jets of AGN, microquasars, and GRB it is not certain how far in the propagation region the outflow remains PF dominated, In the non-relativistic jets of young stellar object (YSO), jets are baryon rich and likely flow dominated outside the launch region. Presently, observations do not resolve the launch region at R < ∼ 50Rin for any source, although best indirect evidence for MHD launch perhaps comes from rotation of YSO jets < ∼100 AU scales (Coffey et al., 2004; Woitas et al., 2005). That B-fields are important to jet launching (R < ∼ 50Rin ) is more widely agreed upon than the role of B-fields in the asymptotic propagation region (despite the dearth of resolved observations of the former.) For example, if, by ∼50Rin , a magnetically collimated supersonic launch accelerates material to its asymptotic directed supersonic speed, then the tangent of the opening angle is just the inverse Mach number Springer

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and the dynamical role of magnetic fields at larger radii may be inconsequential. In all standard magnetic jet models, the magnetic field is dominant in a corona above the rotator, and the magnetic field has a large scale, at least compared to the scale of the turbulence in the rotator into which it is anchored. In recent magnetic tower models (Lynden-Bell, 2003; Uzdensky and McFadyen, 2006), the tower has both signs of vertical magnetic flux since it is composed of loops anchored with both footpoints in the engine. Traditional MHD models which start with a large scale dipole field, produce a jet composed of one sign of magnetic flux and the return flux meanders at large distances, being dynamically unimportant. Instabilities in both geometries can disconnect blobs and produce knotty jets. In short, the physics of the launch region (not yet resolved by astrophysical telescopes) involves such issues as: (1) Origin of magnetic fields, field buoyancy to coronae, magnetic helicity injection and relaxation into larger coronal structures, (2) physics of centrifugal+magnetic acceleration of material from small to super-Alfv´enic speeds, or Poynting flux driven bursts of acceleration, (3) criteria for steady or bursty jets, and (4) assessment of the extent of Poynting flux domination. The physics of the propagation region (resolved by astrophysical telescopes) involves such issues as: (1) Propagation, instability formation, and sustenance of collimation in as a function of internal vs. external density and strength of magnetic fields, (2) bow shocks, cocoon physics, particle acceleration, (3) effect of cooling on morphology, and (4) interaction with ambient media, stars, or cross-winds.

2. Insights on launch from the sun and a two-stage paradigm for jet fields Coronal holes and the solar wind provide an analogy to the more extreme jet launching from accretion disks. The launch region of the solar wind IS resolved. The coronal magnetic carpet (e.g. Schrijver and Zwaan, 2000) is composed of large scale “open” field lines as well as smaller scale “closed” loops. Both reverse every 11 years, so we know that the field is not a residue of flux freezing and must be produced by a dynamo. There are three types of dynamos in astrophysics (e.g. Blackman and Ji, 2006): (1) Velocity driven small scale dynamos, for which magnetic energy amplification occurs without sustained large scale flux on spatial or temporal scales larger than the largest scale of the turbulence, (2) velocity driven large scale dynamos which can amplify field on scales larger than the largest turbulent scale, and (3) magnetically dominated large scale dynamos, also known as magnetic relaxation, whereby an already strong field, adjusts its Springer

geometry and such that any twists migrate to large scales. Both type (2) and type (3) involve magnetic helicity and an associated mean turbulent electromotive force aligned with the local mean magnetic field. Type 1 and type 2 operate in the interior of a rotator, but some version of type 2, followed by a type 3 dynamo, provides the observable coronal field of the sun: First, a velocity driven helical dynamo amplifies fields of large enough scale that they buoyantly rise to the corona without shredding from turbulent diffusion. Once in the magnetically dominated corona, continued footpoint motions twist the field and inject magnetic helicity. In response, the loops incur instabilities which open up them or make them rise. Fields that power jets from disks may arise similarly. The sun and disk are helicity injecting boundaries to their magnetically dominated corona, (analogous to spheromak helicity injection (Bellan, 2000)). The type 2 dynamo occurs beneath the launch region and type 3 occurs in the launch region. Neither occurs in the propagation region.

3. Insights on propagation and launch from jet experiments Astrophysical jet experiments are in their first incarnation, and presently involve non-relativistic jet motion. We cannot expect any experiment to reproduce any astrophysical source, but rather, address specific physics pieces. To gain insight on astrophysical problems, a careful assessment of how a given experiment specifically relates to the Section 1 distinction between jet formation and propagation is required. 3.1. Insights from coaxial gun helicity injection experiments: Launch Hsu and Bellan (2002) employ a coaxial plasma gun composed of two coaxial electrodes linked by an axisymmetric vacuum magnetic field. This is analogous to an accretion disk with a dense set of poloidal magnetic loops, axisymmetrically distributed with zero initial toroidal field. At eight azimuthal locations, plasma is injected onto to the field lines while an electric potential is driven across the anchoring electrodes. An E × B toroidal rotation of the plasma results which then twists the poloidal field, amplifying a toroidal component. Equivalently, magnetic helicity is injected along the field. Once the twist is injected and the toroidal field amplified, the loops rise and merge on the axis. (This is related to a type 3 dynamo, defined above.) A twisted unipolar core tower forms, rises, and remains collimated by hoop stress. The force free parameter αin j ≡ J · B/B 2 = I /ψ (where I is the current from the imposed voltage across the electrodes and ψ is the initial poloidal magnetic flux) measures the amount of twist injected. The measurements roughly

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agree with theoretical expectations of the Kruskal-Shafranov kink instability criterion. For αin j ≤ 4π/L, where L is the magnetic column length, the collimated structure is stable. When αin j > ∼ 4π/L, the magnetic tower forms exhibits a kink instability but the structure stays connected. For αin j ≫ 4π/L, the magnetic tower forms, a kink instability occurs, and a disconnected magnetic blob forms. The experiments show that a kink instability need not immediately destroy jet collimation, even when disconnected blobs are produced. Real jets might be a series of ejected magnetically blobs, rather than a continuous flow. In astrophysics, pressure confinement may play an important role of collimating any magnetic tower. The experiments probe jet formation in a plasma with β ∼ 0.02 − 0.1, T ∼ 5 − 20eV , fields of ∼1kG, and number density n = 1014 /cm3 . The Alfv´en Mach number <1 so this is a launch region experiment not a propagation region experiment. The value of αin j in real astrophysical system is determined by shear, resistivity, and coronal density. 3.2. Insights from pulsed wire array experiments: Launch and propagation For the pulsed power machine MAGPIE, Lebedev et al. (2005) set up radial array of tungsten wires arranged like spokes on a wheel (with a modest concavity) and applied a radial current. The current ablates the wires and produces a magnetic field around each. The mean magnetic field has a net toroidal component in the plane above the array and a net toroidal field of opposite sign below the array. The J × B force from the toroidal magnetic pressure accelerates material vertically. Toward the axis of the array, plasma is denser and an initial hydrodynamic precursor jet forms. As the wires ablate and the magnetic force accelerates the plasma, a magnetically dominated cavity forms, Hoop stress collimates the denser plasma along the axis of the tower. The axis becomes a β ∼ 1 plasma surrounded by β < 1 toroidal magnetic field dominated cavity. Outside of the cavity is a β > 1 ambient plasma supplied from the early stage of wire ablation. The dimensionless Reynolds and magnetic Reynolds numbers are larger than unity so the parameters are crudely OK for MHD an astro-comparison. Cooling is important as the cooling length is short compared to dynamical spatial scales. The experiment addresses principles of BOTH launch AND propagation physics. The very narrowly collimated β = 1 core jet has an internal Mach number of ∼4. The surrounding large toroidal magnetic pressure driven cavity proceeds at Mach 10 with respect to the weakly magnetized ambient medium while the radial expansion is only Mach 3 so the cavity is collimated. In the experiments, these Mach numbers are already reached even when the tower height is only of order ∼Rin , where Rin the array diameter (∼ 4 mm.). That the vertical expansion is

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supersonic with respect to the ambient medium implies that the jet head has evolved from its formation region into its propagation region. The structures produced in the experiments are analogous to pressure confined magnetic tower models (Lynden-Bell 2003; Uzdensky and MacFadyen 2006), however there is very little polodial field, and the net toroidal field is produced from poloidal loops oriented perpendicular to the radial direction. A magnetically dominated tower encircling a β = 1 highly collimated core may also apply to astrophysical jets. At later stages of the evolution, the magnetic tower becomes kink unstable and a magnetic blob is ejected. But, as in Hsu and Bellan (2002), here too the instability does not destroy the collimation of the tower. In this case, the ambient thermal pressure slows radial expansion. Blob formation again highlights the importance of time dependent dynamics, and that disconnected blobs may be the true nature of magnetized jets. Were more material available from the wires, the blob formation process in the experiment could repeat. Though not the main focus of Lebedev et al. (2005), it is important to emphasize the precursor jet which precedes the magnetic cavity and results from the initially ablated plasma from the inner region of the wire array. This jet is hydrodynamic and collimated by radiative cooling. In fact, the analogue of this precursor jet is a close cousin to the the main focus of earlier conical wire array experiments of Lebedev et al. (2002) and Lebedev (2004). In these experiments, the conical array was more nearly cylindrical (concave at angles of 30 deg. with respect to the array axis rather than 80 deg.). Once the current is driven, this lower inclination implies an increased density on the axis of the jet compared to Lebedev (2005), thereby increasing cooling enough to break the flux freezing. Lebedev et al. (2002, 2004) are thus supersonic hydrodynamic jet experiments. Given the discussion of Section 1, experiments for which the magnetic field is not important inside of jet are relevant at most to the propagation region, not the launch region. The particular hydrodynamic jet experiments do show that that collimated supersonic launch may obviate the need for asymptotic magnetic collimation of a given jet when cooling is important. The collimation is enhanced when the wire material has a larger ion charge, enhancing radiative losses. This is consistent with a model of asymptotic protostellar jet collimation discussed in Tenorio-Tagle et al. (1988). The Lebedev et al. (2002, 2004) experiments show Mach number ≥15 jets. Jet deflection and shock propagation are studied in Lebedev et al. (2004) experiments, where an additional cross-wind is introduced into across the propagating jet flow. Generally, the hydrodynamic cooling-collimated jets seem to be relatively stable to non-axisymmetric perturbations. Springer

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3.3. Insights from laser ablation experiments: Propagation Another class of hydrodynamic jet experiments have been performed in laser inertial confinement facilities (Blue et al., 2005; Foster et al., 2005). These probe aspects of the jet propagation regime only. The experiments involve laser illumination of a thin metal disk such as titanium or aluminum. The thin target is placed flush against a washer about 6 times thicker, often of the same material. The lasers ablate the thin target disk and the ablated plasma is driven through the washer hole, exiting the hole in the form of a supersonic jet. The jet then propagates into a foam. A variety of features can be studied from the jet propagation into the foam using X-ray radiography and X-ray back-lighting. Blue et al. (2005) report on experiments performed at NIF, They studied aspects of nozzle angle on jet structure by comparing axially symmetric (2-D) vs. titled (3-D) nozzles. The 3-D case leads to an earlier transition to turbulence than in the 2-D case. Code testing of 2-D vs 3-D effects and the efficacy of the 3-D radiative HD code HYDRA (Marinak et al., 96) was confirmed, although the Reynolds numbers of the experiment are R = 107 while only R = 102 –103 in the simulations. Similar experiments performed by Foster et al. (2005), on OMEGA obtain Mach numbers as high as 5. The images are somewhat clearer than in Blue et al. (2005). Turbulent flows, dense plasma jets, bow shock structures are seen. Modeling was done using 2-D hyrdo simulations with RAGE (Baltrusaitis et al., 1996). These experiments probe a jet and foam density ratio of ρ j /ρa ∼ 1. This is intermediate

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between YSO jets which have ρ j > ρa vs. AGN jets which may have ρ j < ρa . The latter however, are relativistic, and the experiments involve only non-relativistic flows.

References Baltrusaitis, R.M., Gittings, M.L., Weaver, R.P., Benjamin, R.F., Budzinski, J.M.: Phys. Fluids 8, 2471 (1996) Bellan, P.M.: Spheromaks, Imperial College Press, London (2000) Blackman, E.G., Ji, H.: in press, MNRAS, astro-ph/0604221 (2006) Blandford, R.D., Payne, D.G.: MNRAS 199, 883 (1982) Blue B.E., et al.: Physical Review Letters 94, 095005 (2005) Coffey, D., Bacciotti, F., Woitas, J., Ray, T.P., Eisl¨offel, J.: ApJ 604, 758 (2004) Foster J.M., et al.: ApJL 634, L77 (2005) Hsu, S.C., Bellan, P.M.: MNRAS 334, 257 (2002) Lebedev et al.: ApJ 563, 113 (2002) Lebedev et al.: ApJ 616, 988 (2004) Lebedev et al.: MNRAS (2005) Livio, M.: Baltic Astronomy 13, 273 (2004) Lynden-Bell, D.: MNRAS 341, 1360 (2003) Matt, S., Frank, A., Blackman, E.G.: ASP Conf. Ser. 313: Asymmetrical Planetary Nebulae III: Winds, Structure and the Thunderbird, 449 (2004) Matt, S., Frank, A., Blackman, E.G.: In press ApJ Lett. (2006) Moiseenko, S.G., Bisnovatyi-Kogan, G.S., Ardeljan, N.V.: Submitted to MNRAS, astro-ph/0603789 (2006) Pudritz, R.E.: Astrophys. Space Science 292, 471 (2004) Schrijver, C.J., Zwaan, C.: Solar and Stellar Magnetic Activity, Cambridge Univ. Press, Cambridge (2000) Tenorio-Tagle, G., Canto, J., Rozyczka, M.: A & A 202, 256 (1988) Uzdensky, D.A., MacFadyen, A.I.: Submitted to ApJ, astro-ph/0602419 (2006) Wheeler, J.C., Meier, D.L., Wilson, J.R.: ApJ 568, 807 (2002) Woitas, J., Bacciotti, F., Ray, T.P., Marconi, A., Coffey, D., Eisl¨offel, J.: A&A, 432, 149 (2005)

Astrophys Space Sci (2007) 307:11–15 DOI 10.1007/s10509-006-9220-y

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Magnetic-Tower Jet Solution for Launching Astrophysical Jets The formation of the first jets in the universe Yoshiaki Kato

Received: 23 April 2006 / Accepted: 17 July 2006 C Springer Science + Business Media B.V. 2006


Abstract In spite of the large number of global threedimensional (3-D) magnetohydrodynamic (MHD) simulations of accretion disks and astrophysical jets, which have been developed since 2000, the launching mechanisms of jets is somewhat controversial. Previous studies of jets have concentrated on the effect of the large-scale magnetic fields permeating accretion disks. However, the existence of such global magnetic fields is not evident in various astrophysical objects, and their origin is not well understood. Thus, we study the effect of small-scale magnetic fields confined within the accretion disk. We review our recent findings on the formation of jets in dynamo-active accretion disks by using 3-D MHD simulations. In our simulations, we found the emergence of accumulated azimuthal magnetic fields from the inner region of the disk (the so-called magnetic tower) and also the formation of a jet accelerated by the magnetic pressure of the tower. Our results indicate that the magnetic tower jet is one of the most promising mechanisms for launching jets from the magnetized accretion disk in various astrophysical objects. We will discuss the formation of cosmic jets in the context of the magnetic tower model.

Keywords Accretion . Accretion disks . Black hole physics . ISM: jets and outflows . MHD . Relativity

PACS: First, Second, More

Y. Kato University of Tsukuba, 1-1-1 Tennodai, Tsukuba, Ibaraki 305-8577, Japan e-mail: [email protected]

1. Introduction Jets in active galactic nuclei (AGNs) are one of the most largest single objects in the universe, and also they are ubiquitous in different systems such as X-ray binaries (XRBs) and young stellar objects (YSOs). The comprehensive understanding of astrophysical jets is that they are driven by the gravitational energy of material accreting towards central objects, such as stars and compact objects. In fact, many observations suggest that astrophysical jets are associated with disks/tori, that may feed inner accretion disks. Therefore, the launching mechanism of jets strongly depends on the underlying accretion flows. In this review, we mainly focus on the development of a theory of accretion disks and jets after the early 1990s. In the followings, we first remark on magnetohydrodynamical (MHD) studies of accretion flows and outflows. We then move on to studies of jets from accretion disks using global MHD simulations. In Section 2, we introduce our recent discovery of the formation of magnetic tower jets in dynamo-active accretion disks. Finally, in Section 3, we summarize the study of magnetic tower jets. Although an alpha-viscosity prescription provides an convenient tool for representing a steady structure of the accretion flows, the magnitude of viscosity is not a free parameter and may not be a constant in space and time. Since magnetic fields provide a source of disk viscosity, as a consequence of magneto-rotational instability (MRI; Balbus and Hawley, 1991), we expect that magnetic fields play crucial roles in the dynamics of accretion flows (see Stone and Pringle, 2001). That is, the magnetohydrodynamical (MHD) approach is indispensable. The first global 3-D MHD simulations of non-radiative accretion flows were performed by Matsumoto (1999). He calculated the evolution of magnetic fields and structural changes of a torus which is initially threaded by toroidal magnetic fields. Hawley (2000), on the other hand, Springer

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calculated the evolution of a magnetized torus which confines poloidal magnetic fields. After 2000, many global 3-D MHD disk simulations starting with locally confined fields were published (e.g., Machida et al., 2000; Hawley and Krolik, 2001; Hawley, 2001; Hawley et al., 2001). All of these global simulations revealed that MRI maintains turbulent flows in the magnetized accretion disk and provides spontaneous generation of subthermal magnetic fields, however, the connection between the disks and the outflows was not resolved by these simulations (see Balbus, 2003 for a review). The acceleration mechanism of the MHD jet has been studied extensively by many groups. Blandford and Payne (1982), for the first time, suggested a disk wind driven by interaction between disks and magnetic fields permeating the disk as the origin of the jets (see also Pudritz and Norman, 1983, 1986; Lovelace et al., 1987). They assumed that the poloidal magnetic field is much stronger than the toroidal magnetic field in the surface layer of the disk or in the disk corona, where plasma-β is low, and jets are accelerated by a magneto-centrifugal force along the magnetic field line. In this case, the plasma corotates with the magnetic field lines until the Alfv´en point, beyond which toroidal fields start to dominate and hence collimation begins via the magnetic pinch effect. The pioneering simulations of MHD jets from accretion disks were performed by Uchida and Shibata (1985; see also Shibata and Uchida, 1986). They calculated the evolution of a disk threaded with vertical fields extending to infinity and found the propagation of a torsional Alfv´en wave along the magnetic field lines, where the jet was accelerated by a twisted magnetic field (see also Shibata and Uchida, 1985; Meier et al., 2001: they named this process the “sweeping magnetic twist mechanism”). Accordingly, they proposed another kind of magnetically driven jet, in which the toroidal magnetic field is dominant everywhere (see also Shibata et al., 1990; Fukue, 1990; Fukue et al., 1991; Contopoulos, 1995; Kudoh and Shibata, 1995, 1997), where the jets are accelerated by the magnetic pressure. If this is the case, the Alfv´en point is embedded in the disk or there is no Alfv´en point, and the Blandford-Payne mechanism cannot be applied to such jets. Later, many 2-D MHD simulations of jets driven by large-scale magnetic fields permeating disks were performed (e.g., Matsumoto et al., 1996; Kudoh et al., 1998; Casse and Keppens, 2002, 2004). On the contrary to toroidal field dominated jets, in these simulations, the Alfv´en point is far from the disk surface indicating that the jet is primarily accelerated by the magneto-centrifugal force (see Kudoh et al., 1998). It has alos been argued that such toroidal field dominated jets are very unstable to kink instabilities in real three-dimensional space and cannot exist in actual situations (e.g., Spruit et al., 1997). In order to study the structure and the stability of outflows driven by large-scale magnetic fields beyond the Alfv´en point, some groups carried out MHD simulations of outflows from disks treated as boundary condiSpringer

tions (e.g., in 2-D: Todo et al., 1992; Ustyugova et al., 1995; Ouyed and Pudritz, 1997a,b, 1999; in 3-D: Ouyed et al., 2003; Ouyed, 2003). In relation to MHD disk simulations, these simulations were more concerned with the jet structure driven by vertical magnetic fields, where the disk only plays a passive role. Since angular momentum can be efficiently extracted from the surface of the accretion disks by the vertical fields, a surface avalanche produces anomalous mass accretion in those simulations. Thus, we need to be careful as to whether or not the launching mechanism of a jet depends on magnetic fields, which are provided externally or generated internally. This is the first stage in the research of astrophysical jets. Previous studies of jets concentrated on the effects of large-scale magnetic fields permeating accretion disks. One may ask what the origin of such a large scale field is? Unfortunately, the origin of such a magnetic field is poorly understood (see Kronberg, 1994 and references therein). In addition, large-scale jet models predict that the direction of the jets are expected to be aligned with that of the large-scale magnetic field lines. Recent observations, however, show that the direction of large-scale magnetic fields are not correlated with the direction of the jets in young stellar objects (M´enard and Duchˆene, 2004). Rather, we expect that the magnetic fields generated by the disk itself are the most promising sources of magnetic fields that drive outflows. In order to study the outflows from the magnetized disk, some groups carried out 2-D MHD simulations of outflows from dynamo-active disks treated as boundary conditions (e.g., Turner et al., 1999; von Rekowski et al., 2003). On the other hand, Kudoh et al. (2002) carried out 2-D axisymmetric MHD simulations of a thick torus involving poloidal magnetic fields and found a rising magnetic loop, which behaves like a jet, from the torus. This is the second stage in the research of astrophysical jets. Recently, outflows have also appeared in 3-D MHD simulations of accretion disks. Hawley and Balbus (2002; hereafter HB02) calculated the evolution of a torus with initial poloidal fields and found three well-defined dynamical components: a hot, thick, rotationally supported, high-β Keplerian disk; a surrounding hot, unconfined, low-β coronal envelope; and a magnetically confined unbound high-β jet along the centrifugal funnel wall (see also Igumenshchev et al., 2003) These studies are a key to developing the next stage of magnetic jet models; what we call a magnetic tower jet. Now, we have entered the third stage in the research of jets in magnetized accretion flows.

2. Magnetic tower jets Lynden-Bell and Boily (1994: hereafter LB94) studied the evolution of force-free magnetic loops anchored to the star

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and the disk. They obtained self-similar solutions for the evolution of magnetic loops. They found that the loop is unstable against twist injection from rotating disks and that the loop expands along a direction of 60 degrees from the rotation axis of the disk (see also Uzdensky et al., 2002a,b; Uzdensky, 2002). Lovelace et al. (1995) pointed out that the dipole magnetic field of the star deforms itself into an open magnetic field due to the differential rotation between the star and the disk. Hayashi et al. (1996: hereafter HSM96) carried out, for the first time, MHD simulations of the magnetic interaction between a protostar and its surrounding accretion disk. They discovered an outflow driven by expanding magnetic loops and a magnetic flare as a result of magnetic reconnection in the loop. Later, Goodson et al. (1997) carried out similar simulations and found the density collimation along the rotation axis of the disk, which looks like a jet. (see also Goodson et al., 1999; Goodson and Winglee, 1999; Fendt and Elstner, 1999; Keppens and Goedbloed, 2000; Matt et al., 2002). Although they found the expanding magnetic loops, such magnetic loops are not collimated. Subsequently, magnetostatic configuration of collimated magnetic loops (a so-called as magnetic tower) anchored between the star and the disk were studied by Lynden-Bell (1996, 2003: hereafter L96, L03, respectively). He showed a solution of a magnetic tower surrounding by external plasma with finite pressure (see also Li et al., 2001). However, the formation and the evolution of such a magnetic tower have not been resolved until 2004. In the followings, we review the published simulations of magnetic tower jets.

a disk by using 2-D axisymmetric MHD simulations. Initial models of their study are illustrated in Fig. 1a. They assume a rotating torus surrounding a weakly magnetized neutron star with a dipole magnetic field. Outside the torus, they assume an isothermal, hot, low-density hydrostatic corona. They found an expansion of the magnetic loops as a result of the twist injection from the disk, due to the differential rotation of the disk and the star. The magnetic loop ceases to splay out when the magnetic pressure balances with the ambient gas pressure. Afterwards, the expanding magnetic loop forms a cylindrical tower of helical magnetic fields whose height increases with time (Fig. 2a). A key discrimination from previous simulations is the ambient corona. In previous MHD simulations of disk-star magnetic interactions, the magnetic tower structure was not so prominent, because the ambient gas pressure was too low to confine the magnetic tower inside the computational box. It is interesting to note that expanding magnetic loops can also be collimated by large-scale vertical magnetic fields, if they are associated with accretion disks (see Matt et al., 2003). Lastly, KHM04 discovered, for the first time, the formation and evolution of a magnetic tower, which is consistent with that proposed in L96. Independently, Romanova et al. (2004) also found the formation of a magnetic tower in a magnetosphere of a protostar in the propeller regime (see also Romanova et al., 2005).

2.1. Formation of a magnetic tower in the magnetosphere of a neutron star

Kato et al. (2004b; hereafter KMS04) studied the structure of non-radiative MHD flows starting with a rotating torus with initially poloidal localized fields around a nonspinning black hole by using the pseudo-Newtonian potential (Paczy´nski and Wiita, 1980). Initial models of their study are

Kato et al. (2004a: hereafter KHM04) extended HSM96 and studied the magnetic interaction between a neutron star and

2.2. Formation of a magnetic tower in a black hole accretion flow

Fig. 1 Initial models of our simulations: (a) A rotating torus (light-blue region) is surrounded by a weakly magnetized neutron star (metallic-gray region). Solid lines indicate a dipole magnetic field threading the torus. (b) A rotating torus (light-blue region) is surrounded by a non-rotating black hole. Solid lines indicate subthermal poloidal magnetic fields confined within the torus

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Fig. 2 Snapshots of our simulations: (a) Formation of a magnetic tower jet in the magnetosphere of a neutron star. (b) Formation of a magnetic tower jet in the magnetized accretion disk around a black hole. In both figures, the blue region, light-blue region, and solid lines indicate the isovolume of the density of the accretion disk, the isovolume of the Poynting flux, and magnetic field lines, respectively. To see animations of our simulations, the interested reader is directed to the link at http://www. ccs.tsukuba.ac.jp/ people/ykato/researches/

illustrated in Fig. 1b. They found the emergence of a magnetic tower from the magnetized accretion flows, when the bulk of the torus material reaches the innermost region close to the central black hole (Fig. 2b). The fields are mostly toroidal in the rim regions of the jets, whereas poloidal (vertical) fields dominate in the inner core of the jet. The collimation width of the magnetic tower depends on the coronal pressure; the more enhanced the coronal pressure is, the more collimated the jet is. Non-negligible coronal pressure tends to suppress the emergence of MHD jets. In contrast to magnetic towers in the magnetoshere of neutron stars, which are generated by winding-up a dipole magnetic field, the magnetic tower in black hole accretion flows is generated by inflating toroidal magnetic fields accumulated inside the accretion disk. Our 3-D magnetic tower solution in black hole accretion flows is basically the same as LB96 proposed. A magnetic tower jet in KMS04 is consistent with the toroidal field dominated jet, since the magnetic tower is made of a toroidal field generated by dynamo action within the disk. KMS04 showed, for the first time, that such toroidal field dominated jets survive at least for a few orbital periods of the initial torus. The most striking feature of a magnetic tower jet in KMS04 is the natural emergence of magnetic fields from the disk, that can accelerate the jets, and hence a magnetic tower jet is a promising model for launching astrophysical jets from accretion disks in various astrophysical objects. Independently, De Villiers et al. (2003) carried out 3-D general relativistic MHD simulations of the magnetized accretion flows plunging into the spinning black hole and found the formation of magnetically dominated evacuated region near the poles where outflows exist (they called it a funnel: see also McKinney and Gammie, 2004; Hirose et al., 2004). In contrast to HB02, the funnel is magnetically dominated, indicating that the funnel is the main product of the emerSpringer

gence of a magnetic tower from the disk (see also De Villiers et al., 2005; Hawley and Krolik, 2006).

3. Conclusion In these proceedings, we have briefly reviewed the MHD study of accretion flows and jets and have discussed recent progress in the study of magnetic tower jets. We should remark on the definition of a magnetic tower jet, because the formation process of magnetic towers is different in the magnetosphere of a star as compared to that in a dynamo-active accretion flow. A magnetic tower is generated by a twisted magnetic loop, supported by an external force, anchored between differential rotation mediums (see L03), however, many MHD simulations of magnetized accretion flows indicate that a magnetic tower can also be produced via the emergence of toroidal magnetic fields generated inside a dynamoactive accretion disk. In other words, magnetic tower jets can extend more than the scale of pre-existing magnetic fields that drive the jet. Thus, jets that are accelerated by smallscale magnetic fields may be appropriate for the definition of magnetic tower jets. Magnetic tower jets could well be the first jets formed in the early universe, because the largescale structure of strong magnetic fields are yet to develop in the star forming regions and galaxies at high redshift. Finally, we expect that magnetic tower jets will give a standard framework for the next stage in the research of launching jets. Acknowledgements The author would like to thank organizers for inviting me to a wonderful meeting. It was a great opportunity for me to present a talk for the laboratory astrophysics community. Numerical Computations were carried out on VPP5000 at the Astronomical Data Analysis Center, ADAC, of the National Astronomical Observatory (ryk22a).

Astrophys Space Sci (2007) 307:11–15

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Astrophys Space Sci (2007) 307:17–22 DOI 10.1007/s10509-006-9215-8

O R I G I NA L A RT I C L E

3D MHD Simulations of Laboratory Plasma Jets A. Ciardi · S.V. Lebedev · A. Frank · E.G. Blackman · D.J. Ampleford · C.A. Jennings · J.P. Chittenden · T. Lery · S.N. Bland · S.C. Bott · G.N. Hall · J. Rapley · F.A. Suzuki Vidal · A. Marocchino

Received: 24 April 2006 / Accepted: 4 July 2006 C Springer Science + Business Media B.V. 2006


Abstract Jets and outflows are thought to be an integral part of accretion phenomena and are associated with a large variety of objects. In these systems, the interaction of magnetic fields with an accretion disk and/or a magnetized central object is thought to be responsible for the acceleration and collimation of plasma into jets and wider angle flows. In this paper we present three-dimensional MHD simulations of magnetically driven, radiatively cooled laboratory jets that are produced on the MAGPIE experimental facility. The general outflow structure comprises an expanding magnetic cavity which is collimated by the pressure of an extended plasma background medium, and a magnetically confined jet which develops within the magnetic cavity. Although this structure is intrinsically transient and instabilities in the jet and disruption of the magnetic cavity ultimately lead to its break-up, a well collimated, “knotty” jet still emerges from the system; A. Ciardi (
) LUTH, Observatoire de Paris et UMR 8102 du CNRS, 92195 Meudon, France A. Ciardi . S.V. Lebedev . J.P. Chittenden . S.N. Bland . S.C. Bott . G.N. Hall . J. Rapley . F.A.S. Vidal . A. Marocchino The Blackett Laboratory, Imperial College, London, SW7 2BW, UK A. Frank . E.G. Blackman Department of Physics and Astronomy, University of Rochester, Rochester, NY, USA A. Frank . E.G. Blackman Laboratory for Laser Energetics, University of Rochester, Rochester, NY, USA T. Lery Dublin Institute for Advanced Studies, Dublin, Ireland D.J. Ampleford . C.A. Jennings Sandia National Laboratory, Albuquerque, New Mexico, USA

such clumpy morphology is reminiscent of that observed in many astrophysical jets. The possible introduction in the experiments of angular momentum and axial magnetic field will also be discussed. Keywords MHD plasmas . Accretion . Accretion discs . Laboratory astrophysics . Winds . Jets and outflows

Introduction The formation and collimation of jets is a problem of great interest in astrophysics. Jets are observed in a diversity of often unrelated systems and range from the sub-parsec and parsec scale in the case of young stellar object jets (Reipurth and Bally, 2001) to the galactic scale jets, thought to be powered by super-massive black holes present in the centre of active galactic nuclei (see Begelman et al., 1984). Jets may also play a critical role in the formation of gamma-ray bursts (see Piran, 2005 for a review) and by association supernovae (Galama et al., 1998; Stanek et al., 2003; LeBlanc and Wilson, 1970; Khokhlov et al., 1999; MacFadyen and Woosley, 1999; Wheeler et al., 2002; Akiyama et al., 2003). Such diversity, in otherwise similar outflow structures may suggest the presence of a universal formation mechanism and over the last twenty years, magnetic fields and the occurrence of rotation have been identified as the principal agents for creating collimated outflows. A common feature of the many variations of the magneto-rotational scenario (Blandford and Payne, 1982; Pudritz and Norman, 1986; Pelletier and Pudritz, 1992; Wardle and Koenigl, 1993; Shu et al., 1994; Ustyugova et al., 1999, 2000; Uchida and Shibata, 1985; Contopoulos and Lovelace, 1994; Ouyed et al., 1997; Goodson et al., 1999; Goodson and Winglee, 1999; Kudoh et al., 2002), is that a magnetic field can extract the rotational Springer

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energy and launch the plasma from a gravitational potential well to escape velocities. In many of the models, the winding of an initially poloidal magnetic field results in a flow pattern dominated by a toroidal magnetic field. In this context the interaction of a dominant toroidal magnetic field with thermal ambient plasma has been investigated in the laboratory both experimentally and numerically (Lebedev et al., 2005b, Ciardi et al., 2005). One of the aims of the present work is to extend the numerical work to 3D MHD simulations and to investigate the late stages of the evolution of “magnetic tower” in the laboratory. Laboratory experiments, performed on a variety of high energy density facilities, are starting to address some important astrophysical issue (Remington, 2005). Conical wire arrays have been successfully used to produce radiatively cooled, hypersonic jets (Ciardi et al., 2002; Lebedev et al., 2002) and to study their interaction with an ambient medium (Lebedev et al., 2004; Ampleford et al., 2005; Lebedev et al., 2005a). In these experiments a conical cage of micron-sized metallic wires was driven on the MAGPIE pulsed-power facility, which delivers a current ∼1 MA over ∼240 ns. The jet formation mechanism relies on the combination of a high rate of radiative cooling together with the redirection of flow across a conical shock. Magnetic fields were not important in these jets and the formation process was purely hydrodynamic. The current experiments are modified in order to introduce a dynamically significant magnetic field in the system.

Numerical experiments A radial wire array consists of two concentric electrodes connected radially by 16 tungsten wires 13 µm in diameter and with the radius of the inner and outer electrodes 2 mm and 20 mm respectively. Figure 1 shows the initial set-up of a three-dimensional simulation of a radial wire array. Magnetic field lines are also shown and the regions where the toroidal global field dominates over the local field of the wires can be clearly distinguished. Due to limitations in resolving the micron-sized wire cores in these large scale simulations, the wires are initiated as relatively cold dense gas and not as solid metallic wires. Nevertheless, these artificial initial conditions reproduce correctly the ablation rate of the wires (Lebedev et al., 2001) and the rapid formation of a hot coronal plasma surrounding the wires. The electrodes are treated in the computations as highly conductive but thermally insulated regions. The code solves on an Eulerian grid the three-dimensional single fluid, two temperatures and resistive MHD equations. The evolution of the electromagnetic fields is followed through an explicit Runge-Kutta type timeintegration solver and corrected transport. The LTE ionization is calculated using a Thomas-Fermi average atom model, Springer

Fig. 1 A radial wire array consists of thin metallic wires connecting two concentric electrodes. Current flows along the wires and into the central electrode. The J×BGlobal force acting on the plasma ablated from the wires is accelerated in the axial direction. The “global” magnetic field, which dominates the system, is purely toroidal. The wires’ “private” field is also plotted for some of the wires.

and we also include optically thin radiation as a loss term in the electrons energy equation; the latter is coupled to the energy equation for the ions through an energy equilibration term; more details on code are in (Chittenden et al., 2004). A typical simulation of a radial wire array is shown in Figure 2. The ablation of the wires initially produces an ambient plasma cloud which expands above the plane containing the wires. This thermally dominated plasma provides the collimating environment for the magnetic cavity. When sections of the wire cores are fully ablated the proper magnetic tower jet begins to form, consisting of a magnetic cavity with a jet on its axis. Axial expansion of the cavity and instabilities disrupt the system, leaving a clumpy and collimated jet behind. The formation of ambient plasma is due to the steady ablation of the wires which produces hot plasma (∼10 eV) of relatively low resistivity (η) with respect to the cold (∼1 eV) wire cores. For a Spitzer like resistivity η ∼ T −3/2 , where T is the temperature of the plasma, a marked difference in the resistivity develops in this two-component structure, with currents preferentially flowing in the ablated plasma. The global magnetic field (see Figure 1) accelerates the ablated plasma in the axial direction, while the wire cores, which are virtually force-free, act as a continuous but stationary source of plasma. We note that resistive diffusion dominates over the advection of the magnetic field up to a height of l R ∼2–3 mm above the wires, this is approximately the length scale over which the ablated plasma is magnetically accelerated to characteristic velocities of vabl ∼130 km s−1 . Close to the wires the electron temperature is a few eV and the magnetic Reynolds number ReM = vlR /DM ∼ 0.1; DM = η/µ0 is the magnetic diffusivity and we used v ∼ 30 km s−1 . At axial

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Fig. 2 Time evolution of a radial wire arrays composed of 16 tungsten wires 7.5 µm in diameter. Mass density x-z slices from a 3D simulation are shown. The hatched areas indicate the electrodes. Regions with a density below 10−4 kg m−3 (white) are treated as vacuum.

positions above lR the magnetic Reynolds number increases as a consequence of increasing plasma velocity, temperature and diffusion length scales. Nevertheless, the magnetic field rapidly decays above the wires to ∼10% its value calculated in the vacuum region below the wires. The ambient plasma has β > 1 and the thermal pressure will act to confine the magnetic cavity that forms later. Near the wires the magnetic field pressure dominates and the plasma β < 1. Over the ablation time (tabl ∼ 250 ns) a region of height ∼30–40 mm

above the plane of the wires is filled with plasma, its density varies as ∼1/r where r is radial distance from the array’s axis. The axially peaked plasma distribution (ne ∼1018 cm−3 on axis) occurs as the shock heated radially converging plasma is cooled by radiation losses, resulting in a plasma “column” that is hydrodynamically confined. In the axial direction, because of the time dependent ablation rate (∼I2 ) the density decreases rapidly away from the plane of the wires. Because of the discrete nature of the wires, the plasma distribution is

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Fig. 3 Four iso-density contours are shown at the same time (235 ns). The densities are, from left to right, 5 × 10−4 , 5 × 10−3 , 5 × 10−2 , 5 × 10−1 kg m−3 , and can be thought of as being nested. The background plasma is visible in the leftmost panel, while in the two mid panels, the well developed magnetic cavity can be seen. The rightmost panel shows the jet that forms inside the magnetic cavity.

Fig. 4 Magnetic field (yellow) and current density (red) distribution inside the magnetic cavity at 225 ns (left) and 245 ns (right). To show the inside of the magnetic cavity the iso-density contours (same as in Figure 3) are sliced vertically.

highly modulated in the azimuthal direction (see Figure 3). Nevertheless the evolution of the magnetic cavity is highly symmetric and it is only at later times, as instabilities develop, that asymmetric features become apparent. Because the mass ablation rate decreases with the strength of the global magnetic field as ∼1/r (Lebedev et al., 2001), the highest ablation rate occurs in the proximity of the inner electrode. It is there that during the current discharge full ablation of millimetre-sized sections of the wire cores takes place. Because of the disappearance of the force-free wire cores and thus of the plasma source, the magnetic field pressure associated with the global toroidal magnetic field is now able to sweep the remaining plasma upwards and sideways. The magnetic field acts as a piston, snowploughing the surrounding plasma and forming a magnetic “bubble” inside the background plasma (Figure 3). In Figure 4 the magnetic field (yellow lines) and the current density (red lines) distribution inside the magnetic cavity are shown for two distinct times. Similarly to the

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experiments, astrophysical magnetic tower jets are dominated by a toroidal magnetic field which is confined by the pressure of an ambient plasma (Lynden-Bell, 1996, 2003; Kato et al., 2004a, b). With the appearance of the magnetic cavity, a current-carrying jet forms on axis and it is confined by the magnetic field hoop stress. The characteristic density and temperature in the jet are n i ∼ 3 × 1019 cm−3 and T ∼ 30 eV respectively. The characteristic velocity of the jet is ∼150–200 km s−1 , which is higher than the initial flow velocity present before the appearance of the magnetic cavity and indicating that the plasma is actually accelerated in the jet formation process. Initially the plasma beta of the jet is ∼1 and the magnetic Reynolds number is ∼5–10. With the exception of the jet, the magnetic cavity is mostly void of any plasma. The principal current path is thus along the walls of the magnetic cavity and through the jet (see Figure 4). As noted above, in the jet itself, acceleration of material occurs as plasma swept by the converging magnetic piston, is compressed and redirected axially. In the simulation of

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tungsten arrays, radiation losses are such that the plasma shell surrounding the magnetic cavity remains fairly thin. Variation of the ambient plasma distribution and the driving magnetic field strength can significantly alter the growth rate of the magnetic tower. The characteristic axial expansion velocity of the magnetic tower is ∼200–400 km/s, while its radial expansion occurs with a velocity of about 50 km/s. The higher velocities are observed for arrays made with the 7.5 µm wires, where the magnetic tower forms during the rise of the current pulse and propagates in an environment having a smaller axial extent. A dominant kink (m = 1) mode instability develops immediately after the jet formation and leads to its break-up. For typical jet parameter the growth time (∼2.5 ns) of the instability is significantly smaller than the evolution time (Lebedev et al., 2005b). Nevertheless the combination of the axial expansion of the magnetic tower and instabilities do not lead to the destruction of the jet; instead a collimated, clumpy jet is launched out of the cavity. During this transient phase the current and field distribution change significantly (Figure 4): the current begins to connect once again at the base of the magnetic cavity while the magnetic field develops a significant axial component and becomes highly tangled, thus promoting reconnection. Finally, the radiatively cooled, “knotty” jet emerging from the cavity has typical velocities of 200–300 km s−1 , Mach numbers of >10, plasma beta ∼1–10 and ReM ∼1–5. Because of the high Mach number the jet will remain collimated over long distances. In addition, the clumps that form the jet have generally different axial velocity and will interact with each through a series of internal shocks, reminiscent of the internal shocks observed in proto-stellar jets (Hartigan et al., 2001). The rapid development of instabilities in the jet may be partly suppressed by the presence of a poloidal field in the jet and we are currently developing a series of experiments to investigate its effects. A typical radial array set-up involves the presence of a solenoid-like electrode below the plane of the wires which introduces a longitudinal magnetic field of the order of ∼15% of the toroidal field. Field compression, resistive diffusion in the plasma and electrode geometry can all influence the actual topology of the field prior to the jet formation; also the presence of an axial field introduces angular momentum in the flow, further complicating the analysis. Although the exact role of such effects has not yet been clarified, preliminary numerical and experimental results indicate that the inclusion of axial fields and angular momentum can have a major effect on the overall evolution of magnetic towers and on the jet collimation. These results open up the prospect of significantly extending the range of jet studies that can be performed in the laboratory. Acknowledgements The present work was supported in part by the European Community’s Marie Curie Actions – Human Resource and Mobility within the JETSET network under contract MRTN-CT-2004

21 005592. The authors also wish to acknowledge the SFI/HEA Irish Centre for High-End Computing (ICHEC) and the London e-Science Centre (LESC) for the provision of computational facilities and support.

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Kudoh, T., Matsumoto, R., Shibata, K.: Are jets ejected from locally magnetized accretion disks? Publications Astron. Soc. Jpn. 54, 267–274 (2002) Lebedev, S.V., Ampleford, D., Ciardi, A., Bland, S.N., Chittenden, J.P., Haines, M.G., Frank, A., Blackman, E.G., Cunningham, A.: Jet deflection via crosswinds: Laboratory astrophysical studies. Astrophys. J. 616, 988–997 (2004) Lebedev, S.V., Beg, F.N., Bland, S.N., Chittenden, J.P., Dangor, A.E., Haines, M.G., Kwek, K.H., Pikuz, S.A., Shelkovenko, T.A.: Effect of discrete wires on the implosion dynamics of wire array Z pinches. Phy. Plasmas 8, 3734–3747 (2001) Lebedev, S.V., Chittenden, J.P., Beg, F.N., Bland, S.N., Ciardi, A., Ampleford, D., Hughes, S., Haines, M.G., Frank, A., Blackman, E.G.: Laboratory astrophysics and collimated stellar outflows: The production of radiatively cooled hypersonic plasma jets. Astrophys. J. 564, 113–119 (2002) Lebedev, S.V., Ciardi, A., Ampleford, D.J., Bland, S.N., Bott, S.C., Chittenden, J.P., Hall, G.N., Rapley, J., Jennings, C., Sherlock, M., Frank, A., Blackman, E.G.: Production of radiatively cooled hypersonic plasma jets and links to astrophysical jets. Plasma Phys. Contr. Fus. 47, 465–B479 (2005a) Lebedev, S.V., Ciardi, A., Ampleford, D.J., Bland, S.N., Bott, S.C., Chittenden, J.P., Hall, G.N., Rapley, J., Jennings, C.A., Frank, A., Blackman, E.G., Lery, T.: Magnetic tower outflows from a radial wire array Z-pinch. Mon. Not. Roy. Astron. Soc. 361, 97–108 (2005b) LeBlanc, J.M., Wilson, J.R.: A numerical example of the collapse of a rotating magnetized star. Astrophys. J. 161, 541 (1970) Lynden-Bell, D.: Magnetic collimation by accretion discs of quasars and stars. Monthly Notice—Roy. Astron. Soc. 279, 389–401 (1996) Lynden-Bell, D.: On why discs generate magnetic towers and collimate jets. Mon. Not. Roy. Astron. Soc. 341, 1360–1372 (2003) MacFadyen, A.I., Woosley, S.E.: Collapsars: Gamma-ray bursts and explosions in “failed supernovae”. Astrophys. J. 524, 262–289 (1999) Ouyed, R., Pudritz, R.E., Stone, J.M.: Episodic jets from black holes and protostars. Nature 409–414 (1997)

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Pelletier, G., Pudritz, R.E.: Hydromagnetic disk winds in young stellar objects and active galactic nuclei. Astrophys. J. 394, 117–138 (1992) Piran, T.: The physics of gamma-ray bursts. Rev. Modern Phys. 76, 1143–1210 (2005) Pudritz, R.E., Norman, C.A.: Bipolar hydromagnetic winds from disks around protostellar objects. Astrophys. J. 301, 571–586 (1986) Reipurth, B., Bally, J.: Herbig-Haro flows: Probes of early stellar evolution. Ann. Rev. Astron. Astrophys. 39 403–455 (2001) Remington, B.A.: High energy density laboratory astrophysics. Plasma Phys. Contr. Fus. 47, A191–A204 (2005) Shu, F., Najita, J., Ostriker, E., Witkin, F.: Magnetocentrifugally driven flows from young stars and disks. I. A generalized model. Astrophys. J. 429, 781 (1994) Stanek, K.Z., Matheson, T., Garnavich, P.M., Martini, P., Berlind, P., Caldwell, N., Challis, P., Brown, W.R., Schild, R., Krisciunas, K., Calkins, M.L., Lee, J.C., Hathi, N., Jansen, R.A., Windhorst, R., Echevarria, L., Eisenstein, D.J., Pindor, B., Olszewski, E.W., Harding, P., Holland, S.T., Bersier, D.: Spectroscopic Discovery of the Supernova 2003dh Associated with GRB 030329. Astrophys. J. 591, L17–L20 (2003) Uchida, Y., Shibata, K.: Magnetodynamical acceleration of CO and optical bipolar flows from the region of star formation. Publ. Astron. Soc. Jpn. 37, 515–535 (1985) Ustyugova, G.V., Koldoba, A.V., Romanova, M.M., Chechetkin, V.M., Lovelace, R.V.E.: Magnetocentrifugally driven winds: Comparison of MHD simulations with theory. Astrophys. J. 516, 221–235 (1999) Ustyugova, G.V., Lovelace, R.V.E., Romanova, M.M., Li, H., Colgate, S.A.: Poynting jets from accretion disks: Magnetohydrodynamic simulations. Astrophys. J. 541, L21–L24 (2000) Wardle, M., Koenigl, A.: The structure of protostellar accretion disks and the origin of bipolar flows. Astrophys. J. 410, 218 (1993) Wheeler, J.C., Meier, D.L., Wilson, J.R.: Asymmetric supernovae from magnetocentrifugal jets. Astrophys. J. 568, 807–819 (2002)

Astrophys Space Sci (2007) 307:23–27 DOI 10.1007/s10509-006-9256-z

ORIGINAL ARTICLE

Outflow Propagation in Collapsars: Collimated Jets and Expanding Outflows Akira Mizuta · Tatsuya Yamasaki · Shigehiro Nagataki · Shin Mineshige

Received: 24 April 2006 / Accepted: 20 September 2006 C Springer Science + Business Media B.V. 2006


Abstract We investigate the outflow propagation in the collapsar in the context of gamma-ray bursts (GRBs) with 2D relativistic hydrodynamic simulations. We vary the specific internal energy and bulk Lorentz factor of the injected outflow from non-relativistic regime to relativistic one, fixing the power of the outflow to be 1051 erg s−1 . We observed the collimated outflow, when the Lorentz factor of the injected outflow is roughly greater than 2. To the contrary, when the velocity of the injected outflow is slower, the expanding outflow is observed. The transition from collimated jet to expanding outflow continuously occurs by decreasing the injected velocity. Different features of the dynamics of the outflows would cause the difference between the GRBs and similar phenomena, such as, X-ray flashes. Keywords Hydrodynamics . Jet . GRBs . Supernovae . Shock . Relativity 1 Introduction The gamma-ray bursts (GRBs) are the most energetic phenomena in the sky. A collimated and relativistic jet is A. Mizuta (
) Max-Planck-Institute f¨ur Astrophysik, Karl-Schwarzschild-Str. 1, 85741 Garching, Germany e-mail: [email protected] T. Yamasaki · S. Mineshige Yukawa Institute for Theoretical Physics, Kyoto University, Oiwake-cho, Kitashirakawa, Sakyo-ku, Kyoto 606-8502, Japan S. Nagataki Yukawa Institute for Theoretical Physics, Kyoto University, Oiwake-cho, Kitashirakawa, Sakyo-ku, Kyoto 606-8502, Japan; KIPAC, Stanford University, P.O.Box 20450, MS 29, Stanford, CA, 94309, USA

necessary to explain the observational features of GRBs (Piran, 2000). The central engine of the GRBs is not fully understood yet. However, recent observations of the long duration GBRs associated with SNe, for example, GRB980425/SN1998bw (Galama et al., 1998; Iwamoto et al., 1998) and GRB030329/SN2003dh (Hjorth et al., 2003; Price et al., 2003; Stanek et al., 2003) link the GRBs and the death of massive stars. Note, both SN1998bw and SN2003dh are categorized to a sub-class of the SNe, such as, hypernovae whose explosion energy is ∼1052 ergs which is one order magnitude higher than that of normal supernova explosion. The similar phenomena called as X-ray flashes (XRFs) are also observed (Heise et al., 2001). XRFs have larger fluence in the X-ray band than in the gamma-ray. Since the event rate of XRFs is similar to GRBs, several hypotheses are proposed to link these events. Nakamura (2000) proposed an unified model that explains the different properties of GRBs and XRFs by the different viewing angle of the collimated outflow. Lamb et al. (2005) proposed a model that explains the different properties by the different opening angle of the outflow. Theoretically the relation between the death of the massive stars and GRBs was predicted by Woosley (1993). That is so called collapsar model. When an iron core of a rapidly rotating massive star collapses, a proto neutron star or black hole is formed in the center of the progenitor. Though the gas along the rotational axis can freefall quickly, the gas along the equatorial plane gradually falls into the center because of the large centrifugal force. As a result an accretion disk is formed. MacFadyen and Woosley (1999) performed hydrodynamic simulations of this model. They deposit thermal energy in the polar region around the core, assuming neutrino emission from the accretion disk, and neutrino and anti-neutrino annihilation there. Then the gas expands and forms an bipolar flow. Since the calculation was Newtonian Springer

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one, the relativistic effects which are important for GRBs are not included. Aloy et al. (2000) did relativistic hydrodynamic simulations of the same type of problems done by MacFadyen and Woosley (1999). They showed collimated and relativistic jet along the polar axis of the progenitor. The jet finally breaks out from the progenitor, making a highly Lorentz factor component up to Ŵ ∼ 40. Another type of relativistic hydrodynamic simulations have also done by Zhang et al. (2003, 2004), and Umeda et al. (2005). They inject not only a thermal energy but also a kinetic energy from the computational boundary, assuming an outflow formation around the center of the progenitor. They followed the outflow propagation in the progenitor and interstellar medium. All their model were initially outflows with a large thermal energy and showed successful eruption from the progenitor, i.e., relativistic jets. But there still remain some issues on the propagation of the outflows in the progenitor. Which type of the outflow can keep the collimated structure and how do they keep the good collimation. In this paper, we show the different types of outflows in the collapsar.

2 Model We study the outflow propagation in the progenitor, assuming an outflow formation after the core collapse. The radial mass profile of the progenitor developed by Hashimoto (1995) is used. The progenitor had a mass of about 40 solar masses in the main sequence and has 16 solar masses in the pre-supernovae stage. The radius of the progenitor is 3.7 × 1010 cm. We use non-uniform grid points, assuming the axisymmetric geometry (r − z). Logarithmically uniform 500 grid points are spaced for 2 × 108 cm < z < 6.6 × 1010 cm. We also set uniform 120 zones for 0 < r < 1.2 × 109 cm and logarithmically uniform 130 zones for 1.2 × 109 < r < 1.1 × 1010 cm. The inner boundary of the computational box is located at the distance of 2 × 108 cm from the center of the progenitor. In this study the origin of the coordinate corresponds to the center of the progenitor. The boundary conditions at the cylindrical axis (r = 0) and z = 2 × 108 cm, are reflective one except 0 < r < 7 × 107 cm at z = 2 × 108 cm where an outflow is injected. The boundary conditions at other boundaries are outflow boundary condition. The mass densities of the progenitor is ∼106 g cm−3 (around the inner boundary), ∼1g cm−3 (at the surface of the progenitor), and 10−6 g cm−3 (constant outside of the progenitor). The 2D special relativistic hydrodynamic equations are solved, using our relativistic hydrodynamic code based on Godunov-type scheme (Mizuta et al., 2004, 2006). An ideal equation of state p = (γ − 1)ρǫ is also solved to close the equations, where p is pressure, the constant γ (= 4/3) is specific heat ratio, ρ is rest mass density, and ǫ is specific internal energy. As our current numerical code can handle only Springer

constant specific heat ratio, we take precedence the state for the relativistic temperature γ (= 4/3) in this paper. Since the timescale for the outflows to cross the progenitor is much shorter than that of the freefall of the envelopes, we ignore the gravitational potential by the formed black hole or proto neutron star at the center of the progenitor. The initial gas temperature of the envelope and outside of the surface is set to be very low (ǫ/c2 = 10−9 and ǫ/c2 = 10−6 ). We assume an outflow formation from the center of the progenitor. It is also assumed that the outflow is parallel to the cylindrical axis. We inject this outflow from the boundary described above. Four parameters are necessary to define the outflow condition. In this paper, we fixed two of them. The first one is the power of the outflow which is fixed to be 1051 ergs s−1 . The total energy by ten seconds injection satisfies 1052 erg which is the energy of the hypernova explosion. The second one is the radius of the injected outflow which is fixed to be 7 × 107 cm. We vary other two parameters, such as, the specific internal energy ǫ0 and the bulk Lorentz factor Ŵ0 , where subscripts ‘0’ stand for the values of the injected outflows from the computational boundary. The bulk Lorentz factor is varied from Ŵ0 = 1.05 to Ŵ0 = 5, corresponding 3-velocity is from v0 = 0.3c to v0 = 0.98c, where c is speed of light. The specific internal energy is varied from ǫ0 /c2 = 0.1 to ǫ0 /c2 = 30. The outflow of the model (Ŵ0 , ǫ0 ) = (30, 5) is similar to the models used by Zhang et al. (2003, 2004) and Umeda et al. (2005). This is the most attractive model for GRBs, since the outflow contains a large amount of thermal energy. Such an outflow could be formed in the quickly rotating progenitor. The outflow of the model (Ŵ0 , ǫ0 /c2 ) = (1.05, 0.1) is the most slowest and coldest one. The mass density of the injected outflow in model (Ŵ0 , ǫ0 /c2 ) = (1.05, 0.1) is ∼104 g cm−3 and the highest one in all models. To the contrary, the mass density of the injected outflow in model (Ŵ0 , ǫ0 /c2 ) = (5, 30) is ∼1g cm−3 and the lowest one in all models. As the mass density of the progenitor (before an outflow comes) around the injection point is 106 g cm−3 , the outflows in the all models are so-called “light jet” whose mass density is lower than that of the ambient gas. Thus we can expect strong interaction between the outflow and progenitor gas.

3 Results and discussions Figure 1 shows the density (top) and Lorentz (bottom) contours of two models [left panel : (Ŵ0 , ǫ0 /c2 ) = (5, 30) and right panel : (Ŵ0 , ǫ0 /c2 ) = (1.05, 0.1)], when the outflow breaks out from the progenitor surface. The outflow of the former model keeps good collimation in the progenitor, since the high Lorentz factor is localized along the cylindrical axis. To the contrary, the outflow of the latter model shows expanding feature. In both cases, the bow shock which drives progenitor

Astrophys Space Sci (2007) 307:23–27

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bow shock

bow shock

2

(Γ0,ε0/c )=(5,30)

2

reverse shock

(Γ0,ε0/c )=(1.05,0.1) reverse shock

Fig. 1 The contours of rest mass density (top) and Lorentz factor (bottom) of two modes. Left panel shows collimated jet at t = 3.5 s for case [(Ŵ0 , ǫ0 /c2 ) = (5, 30)]. Right panel shows expanding out-

flow at t = 10 s for case [(Ŵ0 , ǫ0 /c2 ) = (1.05, 0.1)] Figures are taken from Mizuta et al. (2006) and reproduced by permission of the AAS.

Fig. 2 The results of a series of calculations in which ǫ0 /c2 is fixed to be 5. Models (Ŵ0 , ǫ) = (5, 5), (4, 5), (3, 5), (2, 5), (1.4, 5), (1.25, 5) and (1.15, 5) are shown The contours of the rest mass density and

Lorentz factor in each models are presented as same as in Fig. 1. Figures are taken from Mizuta et al. (2006) and reproduced by permission of the AAS.

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gas to high pressure and temperature can be seen. Since the bow shock is enough strong, the pressure driven by the bow shock can keep the outflow to be collimated structure in case of the collimated jet. The reverses shocks also appear in both models. In case of the collimated jet this shock is close to the bow shock, and located at the point where the bulk Lorentz factor decreases to unity. To the contrary, in case of expanding outflow, the distance between the bow shock and reverse shock increases as time goes on. A back flow which is an anti-parallel flow to the main jet is observed in case of the collimated jet. This back flow begins from the shock heated gas through the reverses shock. Internal oblique shocks appear in the collimated jet which helps the jets to keep the collimated structure during the propagation in the progenitor (Norman, 1982; Falle, 1991; Leahy, 1991). There are two possibilities to appear such internal structures. The first is the dynamical nonlinear effect of Kelvin-Helmholtz instability which occurs at the boundary of the jet and the back flow. The second is the shear flow instability which occurs in the jet itself (Urpin, 2002). We need to do higher resolution calculations to identify the reason of the internal structures. No back flow is observed in case of the expanding outflow. In model (Ŵ0 , ǫ0 /c2 ) = (5, 30), the Lorentz factor increases up to 34 during the propagation in the progenitor, and to more than 100 after the break. The narrow opening angle for high Lorentz factor cases is good agreement with theoretical estimate of the opening angle ∼1/ Ŵ. The appearance of such a high Lorentz factor component corresponds to the feature of the GRBs. This acceleration is caused by the energy conversion from the thermal energy to kinetic one. Since the outflow of model (Ŵ0 , ǫ0 /c2 ) = (1.05, 0.1) does not include so much thermal energy, no large acceleration is occurs. The flow is non-relativistic one. Figure 2 shows the results of a series of the calculations, fixing ǫ0 /c2 = 5 and various Ŵ0 . The feature of the outflow changes from the collimated jet to the expanding outflow by decreasing the Lorentz factor of the injected outflow, i.e., Ŵ0 . The maximum Lorentz factor seen in each model also decreases from relativistic regime to non-relativistic regime by decreasing the Ŵ0 . A same continuous transition by changing the Ŵ0 is observed in the series of the calculations in which ǫ0 /c2 is fixed to be 1 or 0.1 (Mizuta et al., 2006). We have observed different types of the outflow propagation in the progenitor. The outflows which can keep collimated structure and becomes high Lorentz factor would be observed as GRBs, since the properties correspond to those of GRBs. Even if the outflows keeps collimated structure, the Lorentz factor increases up to a few in some models. Such outflows could be observed as XRFs. The outflows which do not keep good collimation but are mildly relativistic flows also would be the candidate of XRFs. The outflows which have large opening angle and expanding features would be observed as aspherical SNe (no accompanied GRBs). Springer

Recently several types of laboratory experiments to produce jet like flows have been proposed and done by using laser produced plasmas (Farley et al., 1999; Shigemori et al., 2000; Mizuta et al., 2002; Foster et al.,2005) and Z-pinch plasmas (Lebedev et al., 2002). Those are usually dense outflows and suitable to study the dynamics of protostar jets. Wheres the all outflows presented in this paper are light jet which shows a variety of properties of morphology and dynamics. We hope that we can produce such light jets in the laboratory to study the different type of the morphology and dynamics shown in this paper in the near future. 4 Conclusion We investigate the outflow propagation in the collapsar in the context of gamma-ray bursts (GRBs) with 2D relativistic hydrodynamic simulations. We observed a variety of the outflow properties by changing the specific internal energy and bulk Lorentz factor of the injected outflow from nonrelativistic regime to relativistic one. The feature of the outflow changes from the collimated jets to expanding outflows by decreasing the Ŵ0 . The observed different features of the dynamics possibly explain the different features of the similar phenomena such as, GRBs and XRFs. The production of the light jet in the laboratory is expected to study the features observed in this study.

Acknowledgment This work was carried out on NEC SX5, Cybermedia Center and Institute of Laser Engineering, Osaka University, and Fujitsu VPP5000 of National Observatory of Japan. This work was supported in part by the Grants-in-Aid of the Ministry of Education, Science, Culture, and Sport (14079205, A.M., S.M.) and (14102004, 14079202, and 16740134, S.N.), This work was supported by the Grantin-Aid for the 21st Century COE “Center for Diversity and Universality in Physics” from the Ministry of Education, Culture, Sports, Science and Technology (MEXT) of Japan.

References Aloy, M.A., M¨uller, E., Ib´an˜ ez, J.M., Mart´ı, J.M., MacFadyen, A.: ApJ 531, L119 (2000) Falle, S.A.E.G.: MNRAS 250, 581 (1991) Farley, D.R. et al.: PRL 83, 1982 (1999) Foster, J.M., et al.: ApJ 634, L77 (2005) Galama, T.J. et al.: Nature 395, 670 (1998) Hashimoto, M.: Progress of Theoretical Physics 94, 663 (1995) Heise, J., in’t Zand, J., Kippen, R.M., Woods, P.M.: Gamma-ray Bursts in the Afterglow Era, 16 (2001) Hjorth, J. et al.: Nature 423, 847 (2003) Iwamoto, K. et al.: Nature 395, 672 (1998) Lamb, D.Q., Donaghy, T.Q., Granziani, C.: Il Nuovo Cimento (astroph/0505156) (2005) Leahy, J.P.: In: Hughes, P.A. (ed.), Beams and jets in astrophysics. Cambridge Astrophysics Series, No. 19. Cambridge, UK: Cambridge University Press, 100 (1991)

Astrophys Space Sci (2007) 307:23–27 Lebedev, S.V. et al.: ApJ 564, 113 (2002) MacFadyen, A.I., Woosley, S.E.: ApJ 524, 262 (1999) Mizuta, A., Yamada, S., Takabe, H.: ApJ 567, 635 (2002) Mizuta, A., Yamada, S., Takabe, H.: ApJ 606, 804 (2004) Mizuta, A. et al.: ApJ 651, 960 (2006) Nakamura, T.: ApJ 534, L159 (2000) Norman, M.L., Winkler, K.-H.A., Smarr, L., Smith, M.D.: A&A 113, 285 534, L159 (1982)

27 Piran, T.: Phys. Rep. 333, 529 (2000) Price, P.A. et al.: Nature 423, 844 (2003) Shigemori, K. et al.: PRE 62, 8838 (2000) Stanek, K.Z. et al.: ApJ 591, L17 (2003) Umeda, H., Tominaga, N., Maeda, K., Nomoto, K.: ApJL 633, L17 (2005) Urpin, V.: A&A 385, 14 (2002) Woosley, S.E.: ApJ 405, 273 (1993)

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Astrophys Space Sci (2007) 307:29–34 DOI 10.1007/s10509-006-9238-1

O R I G I NA L A RT I C L E

Jet Deflection by a Quasi-Steady-State Side Wind in the Laboratory David J. Ampleford · Andrea Ciardi · Sergey V. Lebedev · Simon N. Bland · Simon C. Bott · Jeremy P. Chittenden · Gareth N. Hall · Adam Frank · Eric Blackman

Received: 16 May 2006 / Accepted: 18 August 2006 C Springer Science + Business Media B.V. 2006


Abstract We present experimental data on the steady state deflection of a highly supersonic jet by a side-wind in the laboratory. The use of a long interaction region enables internal shocks to fully cross the jet, leading to the development of significantly more structure in the jet than in previous work with a similar setup (Lebedev et al., 2004). The ability to control the length of the interaction region in the laboratory allows the switch between a regime representing a clumpy jet or wind and a regime similar to a slowly varying mass loss rate. The results indicate that multiple internal oblique shocks develop in the jet and the possible formation of a second working surface as the jet attempts to tunnel through the ambient medium. Keywords Hydrodynamics . ISM . Herbig . Haro objects . Methods . Laboratory . Stars . Winds . Outflows 1 Introduction Astrophysical observations have shown that some jets produced by protostars are not straight, and instead exhibit a steady curvature over a significant fraction of their length D. J. Ampleford (
) Sandia National Laboratories, Albuquerque, NM 87123-1106, USA e-mail: [email protected] A. Ciardi Observatoire de Paris, LUTH, Meudon, 92195, France S. V. Lebedev · S. N. Bland · S. C. Bott · J. P. Chittenden · G. N. Hall Blackett Laboratory, Imperial College, London SW7 2BW, UK A. Frank · E. Blackman Department of Physics and Astronomy, Laboratory for Laser Energetics, University of Rochester, Rochester, NY 14627, US

(many jet radii). Deflected jets normally occur as a pair of counter-propagating jets from a common source. These deflected bipolar jets fall into two categories – those with Sshaped (Reipurth et al., 1997) and those with C-shaped symmetries (Bally and Reipurth, 2001). The mechanisms behind the deflection of the C-shaped jets has been the subject of various studies; these studies have indicated that the deflection of the many of these jets cannot be explained by an ambient magnetic field (Hurka et al., 1999), photo-ablation of the surface of the jet (Bally and Reipurth, 2001), or a pressure gradient in the ISM (Canto and Raga, 1996). It has emerged that the most likely explanation for the deflection of these jets is the effect of a ram pressure due to a side-wind as discussed by Balsara and Norman (1992) and Canto and Raga (1995). For protostellar jets such a wind may be produced by differential motion of the source star and the surrounding interstellar medium. This is substantiated by observations which show that within a nebula many C-shaped jet structures are present, each with the jets deflected back towards the central star forming region, hence the effective wind is produced by the motion of the stars outward through the ISM (Bally and Reipurth, 2001). In previous experiments we have studied the deflection of highly supersonic jets in the laboratory using conical wire array z-pinches and a photo-ablated CH foil (Ampleford et al., 2002; Lebedev et al., 2004; Frank et al., 2005). The previous work indicated that these experiments are in the correct parameter regime to study the propagation of astrophysical jets in a side-wind, similar to the mechanism for deflection of Cshaped jets (the experiments aim to model the propagation of one of the jets far from the source; the formation mechanism and other jet are neglected). An important feature observed in the previous experiments was the presence of shocks in the jet during the deflection (as also shown by simulations utilizing astrophysical codes (Frank et al., 2005; Lim and Springer

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Fig. 1 (a) The experimental setup used by Lebedev et al. (2004), (b) illustration of requirements for a shock to cross the jet and (c) the setup used in this paper. The target is long compared to the jet diameter, and angled to provide a uniform wind density on the jet

Raga, 1998)). In this paper we will use a modification of the experimental setup used by Lebedev et al. (2004) to study the deflection of a supersonic radiatively cooled jet by a side wind that is steady state on the typical time scale of the jet; shocks are allowed to fully evolve within the jet while the jet is still subjected to a constant side wind. 2 Criteria for producing a steady state deflection and experimental setup In order to determine whether the interaction of a jet in a side wind is steady state it is useful to consider an oblique shock in the jet. If the jet is still influenced by the wind for the full spatial scale required to allow a shock to fully cross the jet then the interaction can be considered steady state. A shock will cross the jet in a time tcross =

φj , vs

(1)

where φ j is the jet diameter and vs is the transverse velocity of the shock (see Fig. 1b for the setup and parameters discussed). The maximum time that the jet is influenced by the sidewind (of axial extent L) is vLj . Hence for a shock to be allowed to cross the jet (and potentially be reflected or break-out) the transit time of the shock should be less than the time that the jet is influenced by the wind: φj L > vj vs

(2)

φj cs

(3)




where it has been assumed that the transverse shock in the jet is weakly driven, so the shock velocity vs can be approximated as the sound speed cs . This can be reformulated to incorporate the definition of the internal Mach number of the v jet (the axial Mach number) M = csj : L
M φj Springer

(4)

Satisfying Equation (4) guarantees that the interaction is steady state (it should be noted that not satisfying Eq. (4) does not necessarily indicate that the interaction is not steady state). Depending on the clumpiness of the jet and wind, it is possible that C-shaped protostellar jets could fall into the steady-state and non-steady-state regimes. For the case discussed by Lebedev et al. (2004), assuming the jet remains in a constant wind density for the full length of the foil (L ∼ 5 mm), then the length of the interaction was ∼10 jet diameters, however the jet Mach number was 20 (the actual Mach number depending on heating of the jet during the interaction). This does not satisfy Equation (4), so shocks were unlikely to be able to cross the jet, and the experimental data suggests that they did not (Lebedev et al., 2004). To explore a steady state interaction a longer interaction region is required. The overall experimental setup used in this paper is broadly similar to that used by Lebedev et al. (2004). Current produced by the MAGPIE generator (1MA, 240ns described by Mitchell et al. (1996)) is passed through a conical arrangement of 16 fine tungsten wires (each 18 µm in diameter). The current and self-generated magnetic field of the array produce a J × B force that acts on the low density coronal plasma which surrounds each static wire producing a steady flow of plasma (Lebedev et al., 2002a). This Lorentz J × B force has components which are both radial and axial (Fig. 1a). The formation of a conical shock on the array axis thermalizes the kinetic energy associated with the radial component of the velocity, leaving the axial component unaffected (Canto et al., 1988). At the top of this conical shock a pressure gradient is present which accelerates the flow; strong radiative cooling enables the formation of a highly supersonic (Mach number M
30), well collimated outflow (Lebedev et al., 2002b). Data from two diagnostics will be discussed in this paper. A 532 nm, 0.4 ns Nd-YAG laser is used for laser shadowgraphy, with a schlieren cut-off of 1 × 1020 cm−3 . An XUV imaging system which is sensitive to photon energies hν > 30 eV and has an integration time of 3 ns (Bland et al., 2004) is also fielded. Following the previous discussion of the ability of shocks to cross the jet in a characteristic time-scale, we note that the

Astrophys Space Sci (2007) 307:29–34

jet production process continues for many shock crossing times (i.e. the jet itself can be considered steady-state if no interaction occurs). Previously jets produced by this method have been used to explore various aspects relevant to the understanding of protostellar jets, such as the effect of radiative cooling and the effect of symmetry of convergent flows on jet production (Lebedev et al., 2002b; Ciardi et al., 2002), the effect of angular momentum on the jet (Ampleford et al., 2006a), the effect of an ambient medium on jet propagation (Ampleford et al., 2005) and the effect of a side-wind on the jet (Lebedev et al., 2004). To impose such a side wind on the jet a CH foil is photo-ablated by soft X-ray emission from the wire array; the expansion of the foil causes the wind to impact on the jet, as discussed in more detail by Lebedev et al. (2004). In this paper we expand on our previous discussion of jet deflection experiments, with the aim of investigating the dynamics of jet deflection in a regime that is more suited to some astrophysical jets, namely in a configuration which allows shocks to propagate across the jet whilst the jet is still under the influence of the side-wind. To increase the axial extent of the wind the size of the foil is increased, however to ensure that the jet is propagating through a near-constant wind density it is necessary to angle the foil with respect to the initial jet axis (Fig. 1c). This alteration to the foil angle also changes the position of the stagnation point (the point where the velocities of the of jet and wind are perpendicular) so it can be better diagnosed. The jet and wind parameters are expected to be broadly similar to those discussed by Lebedev et al. (2004).

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et al., 2004), with numerous structures now present between the jet and foil. For clarity this image has been repeated in Fig. 2b, with the many different features that will be discussed drawn and labelled. The axial position of the tip of the curved portion of the jet (at the left of the interaction) corresponds to the expected axial position of the tip of a jet propagating in vacuum. At the base of the target we expect a downward component to the wind (due to the angle of the foil and divergence). On the schlieren image (Fig. 2a) two shocks are present where the expanding wind meets the upwards travelling halo plasma surrounding the jet as it exits the wire array (labelled Halo shocks in Fig. 2b). The lower of these two shocks is a shock in the halo and the upper is a reverse shock in the wind (they are marked Halo shock and Wind shock respectively in Fig. 2d). In the next three sections we will describe the other structures observed in the interaction. 3.1 Internal oblique shock formation On the high magnification image (Fig. 2d) we see that there is an internal shock in the centre of the jet (labelled OS1).

3 Dynamics of jet propagation in a side-wind Figure 2a shows a schlieren image of the deflection of a jet in this modified configuration. In the image the jet is seen propagating vertically from array, which is below the base of the image. The side-wind is produced by photo-ablation of the CH foil and propagates right to left (away from the foil), with a small downward component. As the jet is subjected to the side wind the jet is steadily deflected in the direction of the wind motion, as drawn on Fig. 2b (see Lebedev et al., 2004 for a more detailed discussion of the basic deflection). The plasma jet in these experiments is highly supersonic, hence any perturbation to it, such as the ram pressure due to the side wind should generate strong shocks in the flow (as was observed by (Lebedev et al., 2004)). The schlieren diagnostic used in Fig. 2a is sensitive to density gradients in the plasma, such as those produced by these strong shocks. Correlation of these structures with increased XUV emission (Fig. 2c) is consistent with the thermalization of kinetic energy in these shocks. The interaction of the jet is much more complex than was seen in the previous study using a shorter wind (Lebedev

Fig. 2 Shocks within the jet shown in both (a) low and (d) high magnification schlieren images (both at 343 ns). (b) is a repeat of (a) with labels on the image which are discussed in the text. (c) shows an XUV emission image (Bland et al., 2004) at 380 ns

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As with the earlier experiments with a shorter interaction this is likely to be the oblique internal shock responsible for the initial deflection. We see that this internal shock is not straight, but instead bends each way by a few degrees. It is interesting to note that the most pronounced of these bends coincides with a continuation of the shock in the halo flow. Thus it is likely that this bend in the internal shock is associated with a change in the wind density and hence ram pressure – further experiments would be required to investigate the effect of variations in the ambient density. Further along the jet-wind interaction on the low magnification image we see that more structure is present; one obvious shock is labelled OS2. In this image it is unclear what the significance of this shocks is, however we can understand this better if we look at XUV emission. Figure 2c shows a gated XUV emission image from the same experiment, however 40 ns after the schlieren image. This image was taken at 22.5◦ from the plane containing the laser probe beam and foil, hence some emission from the surface of the foil can be seen in the XUV image. The structure seen in the XUV image is broadly similar to that in the schlieren image, however these shock features have developed slightly. Again we see the shock previously labelled OS2; it appears that this

is static in time, and remains almost parallel to the jet, so is likely to be a second internal oblique shock in the jet (OS2), further deflecting the jet.

3.2 Formation of a new working surface The nature of the shock WS2 becomes clear if we look at simulations of a jet in a side wind. Figure 3 shows a 2D slice taken from a 3D HD simulation of a jet propagating in a sidewind. For simplicity this simulation has a constant mass flux in the jet, constant jet injection velocity and uniform wind density and velocity. In these simulations we see that as the jet propagates the upwind surface becomes unstable and a second (and in the last frame a third) working surface begins to form. This is similar to what is observed in Fig. 2a and c – the feature labelled WS2 is likely to be the formation of this secondary working surface (the first working surface being at the head of the jet, labelled WS1). The development of this structure with time can be seen experimentally in Fig. 4, which shows a series of gated XUV images (for a different experiment). The development of a second working surface has also been observed for a different setup using a conical wire array (Ampleford et al., 2005).

Fig. 3 Simulations of a jet in propagating in a side-wind. 2D slice from a 3D HD simulation (Chittenden et al., 2004) with uniform jet and wind (i.e. different from the experiments)

Fig. 4 Development of the jet-wind interaction with time is shown experimentally by time resolved XUV emission (hν > 30 eV)

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ent ambient configurations (here and Ampleford et al. (2005)) both lead to the formation of secondary working surfaces in the jet. A laboratory 3D HD code has recovered many of the features of the present experiments; the data should provide a useful testbed for astrophysical computer simulations of such a case. Future experiments will aim to follow the evolution of shocks more closely and attempt to evaluate the shock jump conditions.

Fig. 5 (a) Low and (b) high magnification schlieren images showing the interaction of the low density, un-collapsed tip of the jet (from a different experiment to all other images)

Acknowledgements This research was sponsored by the NNSA under DOE Cooperative Agreement DE-F03-02NA00057 and in part by the European Communitys Marie Curie Actions – Human resource and mobility within the JETSET (Jet Simulations, Experiments and Theory) network under contract MRTN-CT-2004 005592. Sandia is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Company, for the US DOE’s National Nuclear Security Administration under contract DE-AC04-94AL85000.

3.3 Interaction of a low density jet with a denser wind If we look above the tip of the jet on the earlier schlieren image (Fig. 2a) we see that more shocks have formed. The axial position of this material implies that it was ejected before the well formed jet that has previously been discussed (Lebedev et al., 2002b), and instead consists of material that reached the axis before the conical shock was well formed (Bott et al., 2006). On a different experiment we can see this interaction in more detail on a high magnification schlieren image (Fig. 5). This image shows the low density jet through shadowgraphy, and shocks through schlieren effect. We see that there are actually two shocks present. The shock furthest from the foil is an internal shock in the jet, producing yet another region of deflection. The closest shock to the foil is a reverse shock forming in the wind. It is believed that when this jet material passed through the lower area of wind the ambient material was of sufficiently low density that either there was not enough momentum in the wind at that time or the mean free path of the jet was too long to be deflected (i.e. a particle effect that cannot be modeled using a hydrodynamic simulation). Also on this experiment the low magnification schlieren image shows a well defined reverse shock in the wind near the first deflection of the jet. 4 Conclusions We have discussed experimental data for the deflection of highly supersonic jets by a cross wind where the cross wind is effectively continuous in relation to the typical spatial scales of the jet. Such a configuration could be of interest in modeling the propagation of a jet in a side-wind that is neither clumpy or gusty (experiments that reach the inverse regime were discussed in Lebedev et al. (2004)). The data has shown that many different shocks are formed in the interaction. It is interesting to note that experiments utilizing two very differ-

References Ampleford, D.J., Lebedev, S.V., Ciardi, A., Bland, S.N., Bott, S.C., Hall, G.N., et al.: Laboratory modeling of standing shocks and radiatively cooled jets with angular momentum. Astrophys. Space Sci. DOI 10.1007/s10509-006-9258-x (2006) Ampleford, D.J., Lebedev, S.V., Bland, S.N., Ciardi, A., Sherlock, M., Chittenden, J.P., et al.: Deflection of Supersonic Plasma Jets by Ionised Hydrocarbon Targets. In: Davis, J., Deeney, C., Pereira, N.R. (eds.) AIP Conf. Proc. 651: Dense Z-Pinches, pp. 321–324 (2002) Ampleford, D.J., Lebedev, S.V., Ciardi, A., Bland, S.N., Bott, S.C., Chittenden, J.P., et al.: Formation of working surfaces in radiatively cooled laboratory jets. Astrophys. Space Sci. 298, 241–246 (2005) Bally, J., Reipurth, B.: Irradiated Herbig-Haro jets in the Orion Nebula and near NGC 1333. Astrophys. J. 546, 299–323 (2001) Balsara, D.S., Norman, M.L.: Three-dimensional hydrodynamic simulations of narrow-angle-tail radio sources. I – The Begelman, Rees, and Blandford model. Astrophys. J. 393, 631–647 (1992) Bland, S.N., Ampleford, D.J., Bott, S.C., Lebedev, S.V., Palmer, J.B.A., Pikuz, S.A., et al.: Extreme ultraviolet imaging of wire array zpinch experiments. Rev. Sci. Instrum. 75, 3941–3943 (2004), doi: 10.1063/1.1787927. Bott, S.C., Lebedev, S.V., Beg, F., Bland, S.N., Chittenden, J.P., Ciardi, A., et al.: Dynamics of cylindrically converging precursor plasma flow in wire array Z-pinch experiments. Phys. Rev. E. DOI 10.1103/PhysRevE.74.046403 (2006) Canto, J., Raga, A.C.: The dynamics of a jet in a supersonic side wind. MNRAS 277, 1120–1124 (1995) Canto, J., Raga, A.C.: The steady structure of a jet/cloud interaction – I. The case of a plane-parallel stratification. MNRAS 280, 559–566 (1996) Canto, J., Tenorio-Tagle, G., Rozyczka, M.: The formation of interstellar jets by the convergence of supersonic conical flows. Astron. Astrophys. 192, 287–294 (1988) Chittenden, J.P., Lebedev, S.V., Jennings, C.A., Bland, S.N., Ciardi, A.: X-ray generation mechanisms in three-dimensional simulations of wire array Z-pinches. Plasma Phys. Control. Fusion 46, B457– B476 (2004) Ciardi, A., Lebedev, S.V., Chittenden, J.P., Bland, S.N.: Modeling of supersonic jet formation in conical wire array Z-pinches. Laser Part. Beams 20, 255–261 (2002)

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Frank, A., Blackman, E.G., Cunningham, A., Lebedev, S.V., Ampleford, D., Ciardi, A., et al.: A HED Laboratory Astrophysics Testbed Comes of Age: Jet Deflection via Cross Winds. Astrophys. Space Sci. 298, 107–114 (2005) Hurka, J.D., Schmid-Burgk, J., Hardee, P.E.: Deflection of stellar jets by ambient magnetic fields. Astron. Astrophys. 343, 558–570 (1999) Lebedev, S.V., Ampleford, D., Ciardi, A., Bland, S.N., Chittenden, J.P., Haines, M.G., et al.: Jet deflection via crosswinds: laboratory astrophysical studies. Astrophys. J. 616, 988–997 (2004) Lebedev, S.V., Beg, F.N., Bland, S.N., Chittenden, J.P., Dangor, A.E., Haines, M.G.: Snowplow-like behavior in the implosion phase of wire array Z pinches. Phys. Plasmas 9, 2293 (2002a)

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Lebedev, S.V., Chittenden, J.P., Beg, F.N., Bland, S.N., Ciardi, A., Ampleford, D., et al.: Laboratory astrophysics and collimated stellar outflows: the production of radiatively cooled hypersonic plasma jets. Astrophys. J. 564, 113–119 (2002b) Lim, A.J., Raga, A.C.: 3D numerical simulations of a radiative HerbigHaro jet in a supersonic side wind. MNRAS 298, 871–876 (1998) Mitchell, I.H., Bayley, J.M., Chittenden, J.P., Worley, J.F., Dangor, A.E., Haines, M.G., et al.: A high impedance mega-ampere generator for fiber z-pinch experiments. Rev. Sci. Instrum. 67, 1533–1541 (1996) Reipurth, B., Bally, J., Devine, D.: Giant Herbig-Haro flows. Astrophys. J. 114, 2708 (1997)

Astrophys Space Sci (2007) 307:35–39 DOI 10.1007/s10509-006-9283-9

O R I G I NA L A RT I C L E

Hypersonic Swizzle Sticks: Protostellar Turbulence, Outflows and Fossil Outflow Cavities A. Frank

Received: 17 July 2006 / Accepted: 20 November 2006 C Springer Science + Business Media B.V. 2007


Abstract The expected lifetimes for molecular clouds has become a topic of considerable debate as numerical simulations have shown that MHD turbulence, the nominal means of support for clouds against self-gravity, will decay on short timescales. Thus it appears that either molecular clouds are transient features or they are resupplied with turbulent energy through some means. Jets and molecular outflows are recognized as a ubiquitous phenomena associated with star formation. Stars however form not isolation but in clusters of different density and composion. The ubiquity and high density of outflows from young stars in clusters make them an intriguing candidate for the source of turbulence energy in molecular clouds. In this contribution we present new studies, both observational and theoretical, which address the issue of jet/outflow interactions and their abilityto drive turbulent flows in molecular clouds. Our studies focus on scales associated with young star forming clusters. In particular we first show that direct collisions between active outflows are not effective at stirring the ambient medium. We then show that fossil cavities from “extinct” outflows may provide the missing link in terms of transferring momentum and energy to the cloud. Keywords Hydrodynamics . Methods: Laboratory . ISM: Herbig-Haro objects . Stars: Winds . Outflows

1 Introduction Star formation occurs within Molecular Clouds (MCs), complex structures whose physical evolution is still not clearly A. Frank Department of Physics and Astronomy and Laboratory for Laser Energetics, University of Rochester, Rochester NY 14627–0171

understood (Ballesteros-Parades et al., 2006). MCs are hierarchical structures with smaller substructures known as clumps and cores. Star formation is believed to occur in cores with larger clusters forming from more massive cores. The expected lifetimes for molecular clouds has become a topic of considerable debate as numerical simulations have shown that MHD turbulence, the nominal means of support for clouds against self-gravity, decays on a crossing timescale (Goldreich and Kwan, 1974; Arons and Max, 1975; Stone et al., 1998; MacLow et al., 2004). In light of this result the traditional view that MCs are long-lived, quasi-static equilibrium structures has been challenged by a paradigm in which star formation occurs on a timescale comparable to the freefall time (Ballesteros-Paredes et al., 1999; Hartman, 2003). In the former case turbulence in the cloud is an important source of support and regulation of the Star Formation Efficiency (SFE) and it must be re-supplied over time. In the latter case turbulence is produced with the cloud (Yamaguchi et al., 2001) or only needs to be driven up to the point that a cloud is disrupted. Feedback from protostars forming within a MC has been cited by many authors as a principle means of either re-energizing turbulence or disrupting clouds (Bally and Reipurth, 2001). When massive stars form their ionization fronts, strong stellar winds and eventual supernova blastwaves are expected to be the major contributor to feedback (Krumholtz, 2005). In lower mass cluster environments and environments where the effects of massive stars have not been felt protostellar outflows will likely be the dominant form of feedback. In these cases, even if energy is re-supplied from supra-cluster scales, at some wavenumber the outflow injection may come to dominate global dynamics. In fact, Energetic outflows associated with low and moderate mass young stellar objects are known to exert a strong effect on their parent clouds (for a recent review see Bally et al., 2006). The idea Springer

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Fig. 1 Left: Volume rendered image of density from AMR colliding jet simulations. 90 degree collision of 2 outflows displaced by an impact parameter of b = 1r j . Right comparison of Log10 Mass vs. Log10 Velocity for 3 impact parameters: b = 0 (solid line), b = 1r j (dashed line), b = 8r j (dotted line). Note that the colliding jet is least effective at accelerating ambient material (Cunningham et al., 2006a)

that feedback from TT winds could lead to a self-regulating state of star formation dates back at least as far as Norman and Silk (1980). Since that time measurements have backed up the idea. Young stellar outflows in settings such as NGC 1333 have been shown to contain sufficient kinetic energy to excite a significant fraction of supersonic turbulence in their surroundings and/or unbind and disperse portions of their parent cloud (Bally and Reipurth, 2001; Knee and Sandell, 2000; Warin et al., 1996; Matzner, 2002). In spite of the importance of this process, the global dynamics of multiple outflow/cloud interactions has yet to be explored in detail. In particular the nature of the coupling between multiple outflows and the cloud and their efficacy for generating turbulence and/or disrupting their parent clouds, remains to be characterized. While invoking jets and outflows to drive turbulent motions appears attractive, the principle means of energy transfer from jet to cloud appears to come via shock waves, the so-called “prompt entrainment” mechanism (Chernin et al., 1994). Thus the effect of a single supersonic outflow is bounded by the shock wave which defines it. Only those regions of a cloud which have been swept over by an outflow will gain energy. Given such a localization of energy and momentum deposition, the action of multiple, overlapping, randomly oriented outflows may be required to drive the random motions associated with isotropic turbulence. Somehow the energy and momentum in the localized region engulfed by a jet or outflow must be randomized and distributed over many scales. This may occur via MHD waves but the linking of outflows to wave generation has not been shown in detail as yet (Koduh and Basu, 2006). To explore the role of outflows in generating random motions we first carried out a study of colliding active outflows. 2 Collisions between active outflows We first estimate the probability that two protostellar outflows interact as a function of protostellar density in the cloud. We Springer

consider a volume V that contains an average outflow density N and assume that each protostar emits a bipolar outflow. We approximate the volume of these bipolar outflows as that of a cylindrical column of length L and radius R. Assuming that the production frequency of outflows in the cloud is constant, we can cast the density of outflows active at any given instant in terms of the stellar density N∗ as N = N∗ (toutflow /tcloud ). The probability that two active outflows occupy the same region of space in the cloud at the same time is then P ≈ (Voutflow /Vcloud )2 . Solving for N∗ we have N∗ (P) =

√

P tcloud π R 2 L toutflow

(1)

We define Ncritical as the protostellar density that achieves a volume fill ratio of 10% bowshock overlap: Ncritical ∼ N (0.1). Above this intersection probability we expect the effect of collisions to become appreciable. Assuming typical values for the protostellar outflow size, bow shock radius, outflow lifetime and cloud lifetime we find a Ncritical = 500 pc−3 . This is comparable to the protostellar density of many star forming regions. Outflow interactions of some form are therefore statistically likely to occur in a typical star forming region. Based on this conclusion the efficacy of active outflow interactions in stirring the ambient medium was explored in a 3-D AMR study of jet collisions (Cunningham Frank and Blackman, 2006a). Our study focused on hydrodynamic simulations of the interaction of two orthogonal outflows. The simulations included the effect of radiative energy loss on the flow and we investigated the role of the impact parameter and degrees of collimation. The simulations were carried out in 3D using the AstroBEAR adaptive mesh refinement (AMR) code (Fig. 1). If the collision of outflow streams from adjacent YSO’s contribute to the turbulent energy budget of their parent cloud, it would do so by increasing the rate at which the flow

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Fig. 2 Bottom Left: Simulation of fossil cavity driven by a wide angle wind which is active for 104 years. Image taken at 105 years. Note that backfilling via rarefaction waves has occurred. Top Left: Cavity driven by active wide angle wind shown for comparison with frame taken at

t = 3 × 104 y. Right: fractional difference between analytical scaling relation shown in Equation (2) and results of simulation. Vertical line shows point when the wind shuts off. After this point the scaling relation and simulation differ by at most a factor of 2 due to geometrical effects

Fig. 3 2-D slices of 3-D simulations of jets with different decay times driven into turbulent media. Left Top: Control case in which a constant jet is driven into a quiescent media. Top Right: Constant jet driven into turbulent media. Lower Left: Slowly decaying jet driven into turbulent

media. Lower Right: Rapidly decaying jet driven into turbulent media. Note effect on bow shock in turbulent simulations. In rapidly decaying jet simulations the outflowing material has been completely subsumed by the turbulence.

imparts momentum into the surrounding molecular gas. This could occur if the redirected outflow has a volume greater than the individual outflows. Also if the redirected flow generates more “splatter” in the sense that a wider range of scales become energized though vortices generated during the collision then the increased rate of momentum deposition into the ambient molecular gas would result in an increased rate of generation of turbulent energy and could thereby provide support for the parent cloud against gravitational collapse and star formation. To test this idea we ran simulations of colliding outflows for impact parameters b = (0, 1.0r j , 5.33r j ) for fully collimated jets and b = (0, 1.0r j , 8.0r j ) for “wide angle jets” with an opening angle of θ = 15◦ . We then examined different measures of the acceleration of ambient material by the outflows including the outflows ability to initiate turbu-

lent motions. These included exploration of the mass velocity plots M(v) and measures of average vorticity in the grid as a function of time. Surprisingly, our results indicated that the high degrees of compression of outflow material, achieved through radiative shocks near the vertex of the interaction, prevent the redirected outflow from spraying over a large spatial region. Furthermore, the collision reduces the redirected outflow’s ability to entrain and impart momentum into the ambient cloud. Thus combining the results of our simulations with consideration of the probabilities of outflow collisions for direct collisons led us to conclude that individual low velocity fossil outflows, interacting only on long timescales, are the principle coupling between outflows and the cloud. Springer

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3 Observations and simulations of fossil protostellar outflows (NGC 1333) Many authors have used “0-D” estimates of the energy present in active outflows and compare these with cloud turbulent energy (Bally and Reipurth, 2001). These initial efforts were important but studies of individual objects express a more complicated picture. In particular the explicit time-dependent nature of the coupling between cloud material and outflows is not addressed in these estimates. Outflow power evolves rapidly in time in the strongest phases (Class 0 sources) lasting a fraction of the star formation timescale. Thus outflows will continually be turning on and fading across the history of an active star forming region. The pitfalls of ignoring the temporal domain was highlighted in a recent study of NGC1333 (Quillen et al., 2005). In this work it was found that velocity dispersions, measured in 13 CO, did not vary across the cloud. There was no link between active outflows and turbulence. Instead a new class of outflow signature was identified in the form of fossil cavities. These fossil cavities proved to be a smoking gun showing strong coupling between outflows and the molecular cloud. 20 cavities were identified with typical diameter of about 0.1–0.2 pc, and velocity widths 1–3 km/s. Cavities at a range of sizes and velocities were seen in the cloud. If these cavities were simply empty regions in the cloud, the timescale for them to fill in would be less than a million years implying that they were created relatively recently. To make an explicit link with outflow models, cavity properties were compared with scaling relations for momentum injection, Po , derived from similarity conditions. Po ≈ 0.8 Mo kms−1 ×




l 0.4pc







n 104 cm−3

t 2 × 105 yr




R 0.1pc

3 (2)

Using this relation one can estimate the total momentum injection required to excavate a given cavity. Quillen et al. (2005) compared the number of cavities and their momentum requirements with those measured from currently active outflows. From this it could be estimated that much of the total momentum flux from outflows is fed back into the molecular cloud via the fossil cavities. In a recent, more detailed study, AMR simulations of fossil cavity evolution using full H2 chemistry and cooling were performed (Fig. 2, Cunningham, 2006b). In this work jets and wide angle winds were simulated with an injected momentum flux that decreased in time. These simulations where compare with runs with constant momentum flux. The decaying flux models exhibited deceleration of the outflow head Springer

and backfilling via expansion off of the cavity walls. They also showed lower density contrasts and reached constant aspect ratios. Most importantly the simulations recover the basic properties of observed fossil cavities including verifying the scaling relations (Equation (2)). This work also provided synthetic observations in terms of P-V diagrams which demonstrate that fossil cavities from both jets and wide angle outflows are characterized by linear “Hubble-law” expansions patterns superimposed on “spur” patterns indicative of the head of a bow shock. These should prove useful in future observational work. 4 Conclusions and future work We have shown that fossil cavities, rather than active outflows, may be the direct link between stellar injection of mechanical energy and turbulence on scales of young clusters. We note the useful study by Li and Nakumura (2006) who explored outflow collisions within a collapsing turbulent cloud and concluded that outflow activity could re-energize turbulence. These studies do not contradict our results as we agree with their main conclusion however their simulations were of lower resolution (1283 for the entire cluster) and could not resolve the interactions of individual outflows or include the outflow power evolution (a topic we will explore in the next section). Thus while turbulent energy may be supplied at larger scales to the clouds as a whole (via supernova or gravitational collapse) which then cascades down, there is also a separate injection of energy at smaller scales which constitutes a feedback from the stars within the cluster. Future work will need to make the interplay between turbulence and energy injection from jets more explicit. Figure 3 shows initial work in this direction in the form of simulations of jets into fully turbulent media and the subsequent evolution of the jet driven cavity (Cunningham et al., 2007) Acknowledgment We acknowledge support for this work from the Jet Propulsion Laboratory Spitzer Space Telescope theory grant 051080– 001, Hubble Space Telescope theory grant 050292–001, National Science Foundation grants AST-0507519, AST-0406799, AST 00– 98442 & AST0406823, DOE grant DE-F03–02NA00057, the National Aeronautics and Space Administration grants ATP04–0000-0016 & NNG04GM12G issued through the Origins of Solar Systems Program, and the Laboratory for Laser Energetics.

References Arons, Max.: ApJ 196, L77 (1975) Ballesteros-Parades, H., Vazquez-Seandeni.: ApJ 527, 285 (1999) Ballesteros-Paredes, J., Klessen, R., Mac Low, M.-M., VazquezSemadeni, E.: Protostars and Planets V. in: Reipurth, B., Jewitt, D., Keil, K. (eds.) University of Arizona Press, Tucson, in press (2006) Bally, J., Reipurth, B.: ARAA 39, 403 (2001)

Astrophys Space Sci (2007) 307:35–39 Bally, J., Reipurth, B., Davis, C.: Protostars and Planets V. in: Reipurth, B., Jewitt, D., Keil, K. (eds.) University of Arizona Press, Tucson, in press (2006) Chernin, L., Masson, C., Gouveia dal Pino, E.M., Benz, W.: ApJ 426, 204 (1994) Cunningham, A., Frank, A., Blackman, E.: ApJ, in press (2006a) Cunningham, A., Frank, A., Blackman, E., Quillen, A.: ApJ, in press (2006b) Cunningham, A., Frank, A., Blackman, E., Quillen, A.: ApJ, in preparation (2007) Goldreich, P., Kwan, J.: ApJ 189, 441 (1974) Hartmann, L.: ApJ 585, 398 (2003) Krumholtz, M., McKee, K., Klein, R.: Nature 438, 333 (2005)

39 Knee, L.B.G., Sandell, G.: A&A 361, 671 (2000) Kodu, T., Basu, S.: ApJ 642, 270 (2006) Li, Z.-Y., Nakamura, F.: ApJL 640, L187 (2006) Matzner, C.D.: ApJ 566, 302 (2002) MacLow, M.-M., Klessen: Rev. Mod. Phys. 76, 125 (2004) Norman, C., Silk, J.: ApJ 238, 158 (1980) Quillen, A.C., Thorndike, S.L., Cunningham, A., Frank, A., Gutermuth, R.A., Blackman, E.G., Pipher, J.L., Ridge, N.: ApJ 632, 941 (2005) Stone, J., Ostriker, E., Gammie, C.: ApJ 508, L99 (1998) Yamaguchi, et al.: PASJ 53, 985 (2001) Warin, S., Castets, A., Langer, W.D., Wilson, R.W., Pagani, L.: AAP 306, 935 (1996)

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Astrophys Space Sci (2007) 307:41–45 DOI 10.1007/s10509-006-9218-5

O R I G I NA L A RT I C L E

Plasma Jet Studies via the Flow Z-Pinch U. Shumlak · B. A. Nelson · B. Balick

Received: 14 April 2006 / Accepted: 12 July 2006 C Springer Science + Business Media B.V. 2006


Abstract The ZaP sheared-flow Z-pinch produces high density Z-pinch plasmas that are stable for up to 2000 times the classical instability times. The presence of an embedded radial shear in the axial flow is correlated with the observed stability, and is in agreement with numerical predictions of the stability threshold. The case is made that using a higherZ working gas will produce supersonic plasma jets, consistent with dimensionless similarity constraints of astrophysical jets. This would allow laboratory testing of some regimes of astrophysical jet theory, computations, and observations. Keywords Z-pinch . Plasma jets . Herbig-Haro . Planetary nebulae

1. Introduction Astrophysical jets arise from many sources such as planetary nebulae (PNe), massive black holes, active galactic nuclei (AGN), Herbig-Haro (HH) objects, and other young stellar objects (YSO). PNe (Balick and Frank, 2002) and HH objects (Reipurth and Bally, 2001) both produce supersonic yet non-relativistic jets, v ∼ 100–300 km/s, but over differing durations. PNe produce episodic jets and shock fronts, typically less than 1000 years old. HH objects produce jet outflows lasting up to 105 years. Massive black holes and other objects produce highly relativistic collimated jets that can be megaparsecs in length. Most jets are a fraction of a few parsecs in length and 20–50 times their diameter. Internal flowing clumps are observed, and the flows are seen to interact with surrounding subsonic plasma producing bow shocks. Little U. Shumlak (
) . B. A. Nelson . B. Balick University of Washington, Washington, USA

is known about the production and stability of astrophysical jets, nor about details of interactions with their background. Opportunities to simulate (scaled) astrophysical flow conditions in a stable plasma jet are rare, and almost unparalleled. Astrophysical jets can be compared with laboratory plasma through scaling laws and by the use of a variety of gases for interaction with laboratory plasma jets. For example background gas entrainment and shock formation can be studied by using an argon jet impinging upon a helium background, and using filters to separately record spectra of the jets and shocks using Ar and He emission lines. Using working gases of different mass also allows, within the scaling laws, simulations of differing velocities and varying mass ratios between the jets and the background. Modification of operating conditions and monitoring the time evolution of jets is an obviously impossible task with actual astrophysical jets. The ZaP flow Z-pinch experiment at the University of Washington produces plasma jets with remarkable similarities to plasma jets from HH and PNe objects; stable longlived high aspect ratio jets, propagating “knots”, and shearedflow. By varying the jet and background working gases, ZaP can serve as a powerful testbed for background interaction, shock formation and evolution, and scaling studies. This paper explores the possibility of using ZaP to produce and study plasma jets of interest to astrophysics.

2. Astrophysical jets While magnetic fields are generally accepted to be involved in the origin of HH jets (Reipurth and Bally, 2001), and likely in PNe jets (Gardiner and Frank, 2001; Matt and Balick, 2004), they prove difficult to measure. Values in the range of 0.03 – 0.1 mG have been reported in the region of the bow shock of Springer

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Astrophys Space Sci (2007) 307:41–45

Fig. 1 Schematic of the ZaP experiment showing the relevant features. Green cross-hatched region represents the magnetically-confined and expansion plasma. The inner and outer electrodes are separated

by an insulator at z = −125 cm, the electrode end wall is located at z = 75 cm, and the neutral gas injection plane is located at z = −75 cm

HH111, (Morse et al., 1993), fields approaching a few Gauss have been cited (Ray et al., 1997) in an HH jet from T Tau S. Surface fields on the order of kG have been detected in the nuclei of a few PNe by Zeeman splitting in their atmospheric lines (Jordan et al., 2005), 1–10 G fields are implied in small regions associated with water and OH masers in PN winds through the circular polarization of the narrow maser lines (Miranda et al., 2001). These magnetic fields are often posited to be azimuthal (toroidal) relative to the jet flow. This requires an axial current, (with its return path through the surrounding plasma) not unlike a Z-pinch. Figure 1 of Chevalier and Luo’s (1994), model of a PN jet is indistinguishable from a Z-pinch equilibrium, and Contopoulos (1995) makes an analogy of jet formation with an “astrophysical plasma gun”. However, pure hydrodynamics may also play a role in shaping these jets. In both PNe and HH flows (and even in simulations of them), velocity shear is observed (Balick and Frank, 2002; Reipurth and Bally, 2001; Hardee, 2004), e.g. large flow velocity (approximately 300 km/s) is observed in the main portion of HH47 moving through a slower velocity (35– 120 km/s) background gas (Heathcote et al., 1996). PNe flows also exhibit a great range of velocities in the same object (Balick and Frank, 2002; Frank, 1999).

Table 1 ZaP machine and operating parameters (for hydrogen working gas)

3. The ZaP flow Z-pinch experiment The ZaP experiment (Shumlak et al., 2001, 2003), shown in Fig. 1, produces a hydrogen Z-pinch column (100 cm long and approximately 1 cm radius) with flow velocities on the order of 100–200 km/s (see Table 1 for more information). These pulses are seen to be stable up to 2000 times the ideal growth period (corresponding to several flowthrough times). This stability is correlated with the presence of high flow velocity, radial shear of the axial veSpringer

Parameter Inner electrode radius Outer electrode radius Assembly region length Capacitor bank energy Peak plasma current Injector voltage (sustainment) Average density Total temperature

Value rinner router z assembly Wbank Ip Vinj

n e  Te + Ti

5 cm 10 cm 100 cm 30–60 kJ 250 kA 2 kV 2 × 1016 cm−3 200 eV

locity, and continual flow of plasma from the acceleration region. 3.1. Sheared-flow plasma production and stability ZaP flow Z-pinches are produced as follows: Neutral gas is puffed in the coaxial electrode (“acceleration”) region where it is ionized by high voltage applied between the electrodes. J × B forces accelerate the annulus of plasma from the coaxial region to the “assembly” region where a Z-pinch is formed. The 1/r 2 magnetic pressure in the acceleration region assures a radial variation in the axial flow, which is embedded in the final Z-pinch assembly. The virulent m = 1 “kink” mode is stabilized by a sufficient radial shear of axial velocity, vz , (Shumlak and Hartman, 1995) dvz /dr
0.1 kv A

(1)

where k is the axial wavenumber of the instability, and v A is the Alfv´en velocity. Remnant plasma (not shown) in the acceleration region near the outer wall continuously flows into the assembly region, which aids maintenance of a shearedflow.

Astrophys Space Sci (2007) 307:41–45 250

1.0

Pulse 21014009

Ip

0.8

B1/

150

0.6

100

0.4

50

0.2

0 3

0.0

2

Vgap (kV)

Ip (kA)

200

B1/

Fig. 2 Typical ZaP hydrogen pulse: (a) Plasma current and normalized m = 1 azimuthal mode data at midplane, (b) gap voltage, (c) line-averaged electron density at z = −65 (earlier peak) and z = −25 cm, and (d) wall magnetic field at several axial locations

43

1 0

z = -65 cm z = -25 cm

-3

(10 m )

-1 8 21

6 4 2

Bwall (T)

0 0.6 0.5 0.4 0.3 0.2 0.1 0.0

z = -120 cm z = -65 cm z = -25 cm z = 0 cm

0

20

40

60

80

Time (µs)

Waveforms for a typical ZaP pulse are shown in Fig. 2. The m = 1 component of the azimuthal field at midplane (normalized to the azimuthally-averaged field, Fig. 2(a)) is typically seen to settle to a low value for a period of 20–40 (or more) µs. During this “quiescent” period, density remains in the coaxial region at z = −65 cm (Fig. 2(c)) and a radial current flows in the coaxial region, as seen by the variation in Bθ with z in Fig. 2(d). When the coaxial region density goes towards zero, the radial current (proportional to d Bθ /dz) decreases, and the m = 1 component at midplane increases. During peak plasma current, a Rowogski loop at the endwall shows up to 5 kA of net current (waveform not shown) leaving through the hole in the endwall. Fast camera images with a Balmer Hα filter are shown in Fig. 3 for a similar pulse. The circular aperture is a 5 cm hole in the outer electrode, viewed at an oblique angle. (The dark object on the other side of the plasma is a similar outer electrode hole on the opposite side.) The images show a stable, well-centered pinch until later in the pulse, after the quiescent period, where the light emission is seen to twist and break up. Many characteristics of the pulse are seen to change as the normalized m = 1 component approaches the approximate value of 0.2 (corresponding to a radial displacement of ∼1 cm): oscillations of the m = 1 magnitude are not only larger, but the phase is more erratic; the presence of flow and shear disappears; and Imacon images no longer show well-formed light emission.

Fig. 3 Simultaneous view through two ports, z = 0 (lower 8) and z = 17 cm, (upper 8); Z -axis vertical (pulse 10830012). Frames, 1 µs apart, advance in time from the far lower left pair to the pair above, then advance to the next lower right pair to above, as indicated by the arrows

4. Relevance of laboratory plasma jets to astrophysical jets Laboratory plasmas can be of great relevance for testing theory and observations of astrophysical jets, and for benchmarking computational studies. Many researchers have examined the relevant scaling parameters (Ryutov Springer

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Table 2 Comparison of ZaP operation with hydrogen and argon (estimated for Z = 4) with high and low B astrophysical jets (AJ). (Values are from Refs. (Reipurth and Bally, 2001; Heathcote et al., 1996; Ray et al., 1997)

Parameter

ZaP (H)

ZaP (Ar, est.)

AJ (High B)

AJ (Low B)

n (cm−3 ) B (G) Te (eV) Ti (eV) L/r vflow (km/s) vflow /cs √ B/ p

1016 25×103 100 100 100 80–120 0.75–≈ 1 8

5 × 1017 140×103 1 1 300 32 8 63

102 –104 5 0.1–1 0.1–1 10–>100’s 300 10–30 1.6 × 104 –5 × 105

102 –104 3 × 10−5 0.1–1 0.1–1 10–>100’s 300 10–30 0.09–3

and Remington, 2001, 2002; Kane et al., 1999) and have performed or proposed relevant experiments (Lebedev et al., 2005; You et al., 2005; Lebedev et al., 2004; Raga et al., 2001; Remington et al., 2000). To cover the physics of astrophysical jets, the ideal laboratory experiment would be able to produce supersonic collimated jets (both magnetized and unmagnetized) and associated shocks, with sufficient diagnostics for comparison with theory and computation. 4.1. Similarities of ZaP plasmas with astrophysical jets Light emission from ZaP at certain times looks remarkably like those from HH objects, viz., columns of high flow with propagating clumps. Note the motion of the clumps in Fig. 3. Although the clumps may be the result of local instabilities, the global jet structure remains. These ZaP plasmas are highly magnetized flows, while plasma flowing past the end wall is mostly unmagnetized. The addition of background gas will produce HH-like bow shocks and PNe-like bubbles, allowing detailed laboratory study of MHD and hydrodynamical formation and evolution of shocks. 4.2. Similarity constraints A very lucid derivation of the relevant “similarity criteria” is given by Ryutov and Ramington (2001) Using ideal MHD equations, it is shown that for similar spatially varying initial conditions, two systems will have similarity in their time evolution when the following terms are held invariant in both systems:  B ∗ / p∗

and

 v ∗ / p ∗ /ρ ∗ ,

(2)

where B is the magnetic flux density, v is the fluid flow, p is the pressure, and ρ is the mass density, evaluated at some characteristic point. The first constraint is proportional to  the inverse square root of β ≡ p/ B 2 /8π , and the second constraint is proportional to the Mach number M ≡ v/cs = √ v/ γ p/ρ, where γ = 5/3 for polytropic fluids. Springer

4.3. Plasma jet parameters in ZaP Table 2 shows parameters for ZaP experimental results with hydrogen and estimates for operation with argon and are compared with parameters for jet HH47 (similar values are assumed for PNe jets) using estimated high and low values of B. For ZaP, the first constraint of Equation (2) will remain √ around the value required by force balance, B ∗ / p ∗ ≈ 8– 63 (depending on Z ) for all conditions and is bracketed by √ estimates for HH jets. The higher value of B ∗ / p ∗ results from using a higher-Z working gas. Radiative losses cause the plasma radius to decrease while maintaining the plasma current. The magnetic field correspondingly increases to high values to maintain equilibrium. An applied axial field is not required for either equilibrium or stability, since the required flow shear does not change if v A is constant. The second constraint, high Mach number, requires a colder plasma, which can be achieved by radiative cooling of high mass working gases. “Radiative collapse” from bremsstrahlung is not required, nor is it predicted for ZaP current levels. Many experimental “knobs” on ZaP allow exploration of these constraints (e.g. working gas, gas puff timings, current waveforms, etc.), producing a variety of subsonic, supersonic, magnetized, and unmagnetized jets.

5. Summary The production and evolution of shocks produced in ZaP by plasma jets can be studied in a variety of ways with ZaP’s diagnostic suite. The spectrometer can be placed at angles of 0◦ , 35◦ , and 90◦ to the Z -axis and can discriminate between the jet and the shock through their distinct emission lines, to allow both spatial resolution of velocities and their time evolution. The bow shock itself can be imaged with Imacon framing or streak photographs (using appropriate line filters), ruby laser holography, or schlieren imaging. Internal jet and shock densities can be time-resolved with multi-chord interferometry, PDA tomography (with 2 of the PDAs at 90◦ to the Z -axis) and with Langmuir probes. Internal magnetics can be time-resolved with magnetic field probes.

Astrophys Space Sci (2007) 307:41–45

These diagnostics could provide detailed information of the bow shock and jet structure, and be compared with and used to refine computations. These results for the specific astrophysical jet configuration under study will be compared to the astrophysical jet observations, theory, and modeling, to identify further laboratory experiments and astrophysical jet observations to be made. This creates a “feedback loop” of astrophysical jet observations, theory, and modeling leading to laboratory experiments (and computational comparisons) which then suggest new observations to be made, modifications and refinement of the theory, and further modeling.

References Balick, B., Frank, A.: Shapes and shaping of planetary nebulae. Ann. Rev. Astron. Astrophys. 40, 439–4486 (2002) Chevalier, R.A., Luo, D.: Magnetic shaping of planetary nebulae and other stellar wind bubbles. Astrophys. J. 421(1, pt. 1), 225–35 (1994) Contopoulos, J.: A simple type of magnetically driven jets: An astrophysical plasma gun. Astrophys. Space Sci. 450, 616–627 (1995) Den Hartog, D.J., Golingo, R.P.: Telecentric viewing system for light collection from a Z-pinch plasma. Rev. Sci. Instrum. 72(4), 2224– 2225 (2001) Frank, A.: Bipolar outflows and the evolution of stars. New Astron. Rev. 43, 31–65 (1999) Gardiner, T.A., Frank, A.: Magnetic collimation in planetary nebulae. Astron. J. 557, 250–255 (2001) Golingo, R.P., Shumlak, U., Nelson, B.A.: Formation of a sheared flow Z pinch. Phys. Plasmas 12(6), 62505–1–9 (2005) Hardee, P.E.: The stability properties of astrophysical jets. Astrophys. Space Sci. 293, 117–129 (2004) Heathcote, S., Morse, J.A., Hartigan, P., Reipurth, B. et al.: Hubble space telescope observations of the HH 47 jet: narrowband images. Astronom. J. 112(3), 1141–1168 (1996) Jordan, S., Werner, K., O’Toole, S.J.: Discovery of magnetic fields in central stars of planetary nebulae. Astron. Astrophys. 432, 273– 279 (2005) Kane, J., Arnett, D., Remington, B.A., Glendinning, S.G. et al.: Scaling supernova hydrodynamics to the laboratory. Phys. Plasmas 6(5), 2065–2071 (1999)

45 Lebedev, S.V., Ampleford, D., Ciardi, A., Bland, S.N. et al.: Jet deflection via crosswinds: laboratory astrophysical studies. Astronom. J. 616, 988–997 (2004) Lebedev, S.V., Ciardi, A., Ampleford, D.J., Bland, S.N., et al.: Production of radiatively cooled hypersonic plasma jets and links to astrophysical jets. Plasma Phys. Control. Fus. 47(4), B465–B479 (2005) Matt, S., Balick, B.: Simultaneous production of disk and lobes: a singlewind MHD model for the η carinae nebula. Astrophys. Space Sci. 615, 921–933 (2004) Miranda, L.F., G´omez, Y., Anglada, G., Torrelles, J.M.: Water-maser emission from a planetary nebula with a magnetized torus. Nature 414(6861), 284–286 (2001) Morse, J.A., Heathcote, S., Cecil, G., Hartigan, P., et al.: The bow shock and Mach disk of HH 111V. Astrophys. J. 410, 764–776 (1993) Raga, A., Sobral, H., Villagr´an-Muniz, M., Navarro-Gonz´alez, R., et al.: A numerical and experimental study of the time-evolution of a low Mach number jet. Mon. Notices R. Astron. Soc. 324, 206–212 (2001) Ray, T.P., Muxlow, T.W.B., Axon, D.J., Brown, A. et al.: Evidence for magnetic fields in the outflow from T Tau S. In: IAU Symposium, vol. 182, pp. 475–480 (1997) Reipurth, B., Bally, J.: Herbig-haro flows: probes of early stellar evolution. Ann. Rev. Astron. Astrophys. 39, 403–455 (2001) Remington, B.A., Drake, R.P., Takabe, H., Arnett, D.: A review of astrophysics experiments on intense lasers. Physics of Plasmas 7(5), 1641–1652 (2000) Ryutov, D.D., Remington, B.A.: Magnetohydrodynamic scaling: from astrophysics to the laboratory. Phys. Plasmas 8(5), 1804–1816 (2001) Ryutov, D.D., Remington, B.A.: Scaling astrophysical phenomena to high-energy-density laboratory experiments. Plasma Phys. Control. Fus. 44, B407–B423 (2002) Shumlak, U., Golingo, R.P., Nelson, B.A., Den Hartog, D.J.: Evidence of stabilization in the Z-pinch. Phys. Rev. Lett. 87(20), 205005/1–4 (2001) Shumlak, U., Hartman, C.W.: Sheared flow stabilization of the m = 1 kink mode in Z pinches. Phys. Rev. Lett. 75(18), 3285–3288 (1995) Shumlak, U., Nelson, B.A., Golingo, R.P., Jackson, S.L. et al. Sheared flow stabilization experiments in the ZaP flow Z-pinch. Phys. Plasmas 10(5), 1683–1690 (2003) You, S., Yun, G.S., Bellan, P.M.: Dynamic and stagnating plasma flow leading to magnetic-flux-tube collimation. Phys. Rev. Lett. 95, 045002–1–4 (2005)

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Astrophys Space Sci (2007) 307:47–50 DOI 10.1007/s10509-006-9206-9

O R I G I NA L A RT I C L E

Double-Pulse Laser-Driven Jets on OMEGA S. Sublett · J. P. Knauer · I. V. Igumenshchev · A. Frank · D. D. Meyerhofer

Received: 14 April 2006 / Accepted: 29 June 2006 C Springer Science + Business Media B.V. 2006


Abstract A double-pulse laser drive is used to create episodic supersonic plasma jets that propagate into a low density ambient medium. These are among the first laser experiments to generate pulsed outflow. The temporal laserintensity profile consists of two 1-ns square pulses separated by 9.6 ns. The laser is focused on a truncated conical plug made of medium Z material inserted into a high-Z washer. Unloading material from the plug is collimated within the cylindrical washer hole, then propagates into the low-Z foam medium. The resulting jet is denser than the ambient medium. Double-pulse jet evolution is compared to that driven by a single laser pulse. The total drive energy is the same for both jets, as if a source with fixed energy generated a jet from either one or two bursts. Radiographs taken at 100 ns show that a single-pulse jet was broader than the double-pulse jet, as predicted by hydrodynamic simulations. Since the initial shock creating the jet is stronger when all the energy arrives in a single pulse, the jet material impacts the ambient medium with higher initial velocity. Detailed comparisons between S. Sublett (
) Laboratory for Laser Energetics, Rochester, NY 14623; Department of Physics and Astronomy, University of Rochester, 14627 e-mail: [email protected]

single- and double-pulsed jet rheology and shock structure are presented. 2-D hydrodynamic simulations are compared to the experimental radiographs. Keywords Laser experiments . Plasma jet . Episodic outflow PACS: 52.30.−q, 41.75.Jv, 42.62.−b, 42.68.Sq, 47.40.−x, 47.56.+r

1. Introduction Episodic outflows are nearly as ubiquitous as outflows themselves in astronomical observations. Comparing doublepulsed plasma jets with the single-pulsed plasma jets in the laboratory provides insight into these observations. The University of Rochester’s OMEGA laser (Boehly et al., 1997) produces plasma jets with much higher densities than young stellar objects (YSO) or planetary nebulae (PN), but the jetto-ambient density ratio (a dimensionless parameter) is 1 in experiments, as observed in some astronomical jets. The experimental Mach numbers are approximately 3, approaching a range relevant to astronomical jets. Table 1 shows some ranges of density and velocities, with dimensionless parameters in boldface.

J. P. Knauer · I. V. Igumenshchev Laboratory for Laser Energetics, Rochester, NY 14623 A. Frank Department of Physics and Astronomy, University of Rochester, 14627 D. D. Meyerhofer Laboratory for Laser Energetics, Rochester, NY 14623; Department of Physics and Astronomy, University of Rochester, 14627; Department of Mechanical Engineering, University of Rochester, 14627

2. Laser setup OMEGA laser beams are used to launch a strong shock into a medium-Z conical plug set inside a high-Z washer. Material unloads off the plug, flows through the washer, and forms a jet once it gathers enough force to penetrate a low-Z ambient medium. Figure 1 shows a schematic of the target. Another set of OMEGA laser beams hits a backlighter target to Springer

48

Astrophys Space Sci (2007) 307:47–50 Table 1 Hydrodynamic similarity

Jet density cm−3 Ambient density cm−3 Density ratio Jet velocity km/s Sound speed km/s Mach number

OMEGA

YSO

PN

1021 1021 1 6–12 3 2–4

108 107 –109 0.1–10 102 –103 10–100 10–40

103 –106 10−4 –104 1–1000 10–100 1–100 10

Fig. 2 Target rendering by VISRAD. The shield cone appears on gold mesh, the foam medium in turquoise, and the fiducial grids in green. Color images available online

Fig. 1 Schematic of target showing the conical plug in gray, the washer in magenta, and fiducial grids in green and the circular field of view in red. Color images available online

radiograph the jets 100 ns after the initial laser drive. A shield cone prevents x-rays from the region where the laser hits the conic plug from illuminating the CCD. Figure 2 shows a scaled 3-D image of the target. The same total energy is deposited on the plug in both the single- and double-pulse cases. Either 7 drive beams impact the plug at the same time, or the beams are split into 2 pulses: first 3 beams, then 4 beams 9.6 ns later. The pulse separation time is large compared to the pulse duration of 1 ns. Laser ablation pressure launches a shock wave into the plug. A rarefaction wave is created after the laser pulse is over, and the rarefaction overtakes the shock wave 1.8 ns later, when a single-pulse has traveled 38 percent of the way through the plug or the first of two pulses has traveled 29 percent of the way through the plug. Then the shock decays as it transits the rest of the plug. Material unloading from the plug travels through the remainder of the washer hole, and impacts the foam ambient medium, forming a jet. In the double-pulse case the first shock reaches the back of the plug and begins to unload in under 4 ns, well before the second shock enters the plug. A region Springer

of the ambient medium between fiducial grids is backlit with a point-projection x-ray source whereby the jet is imaged onto a non-gated time-integrated vacuum x-ray CCD detector inside a Spectral Instruments series 800 camera (Spectral Instruments, www.specinst.com). 3. Jet images Figure 3 shows radiographs of single- and double-pulsed jets created from the same type of target 100 ns after the first laser pulse. The 7-beam single laser pulse is more than twice as intense as the 3-beam initial laser pulse of the doublepulse jet. Since ablation pressure is proportional to the 2/3 power of intensity (Lindl, 1998), the initial ablation pressure of 43 Mbar is 1.5 times stronger for the single pulse jet. The strong shock propagated through the plug is therefore over 20 percent faster for the single pulse jet. The single-pulse jet is also broader than the double-pulse jet. Figure 4 shows another double-pulse jet, also at 100 ns, but with a lower energy backlighter that probes less deeply into the core of the jet. The entire length of the jet is not visible, but the radius, bow shock, and head of the jet is similar to the jet in Fig. 3(b). Figure 5 shows simulations for the single- and doublepulse jets produced in the experiments. The concave morphology of the single-pulse jet head is distinguishable in Figs. 3(a) and 5(a) from the more lenticular head in

Astrophys Space Sci (2007) 307:47–50

49

Fig. 3 Experimental radiographs of single-pulse jet (a) and double-pulse jet (b) at 100 ns

Flatfielding removes the spatially dependent CCD response. Taking the flatfield, dark image, and jet image all at the same exposure time means that the bias is built into the dark image, and no scaling has to take place before the dark image is subtracted from the flatfield or jet image. Each jet image has the known transmission through the foam removed so that only the jet areal density remains. Wiener filtering with a Lucy-Richardson algorithm removes large scale features and makes shock fronts more clear, as shown in Fig. 6.

Fig. 4 Double-pulse jet at 100 ns backlit with lower energy than the jet in Fig. 3(b)

Figs. 3(b), 4, and 5(b). The shocks inside the cocoons are not do differ in simulations and experiments. The relative jet lengths were not established since the full lengths of the experimental jets could not be seen past the lip of the shield cone, visible in Fig. 2. The single-pulse jet has the greatest density in its Mach disk in both experiments and simulations. The double-pulse jet has a region as dense as the Mach disk at the base of the jet in both simulations and experiments, evidence of the second pulse. 4. Image analysis Subtracting a dark image from each jet image removes exposure time dependent dark noise and DC offset pixel values.

5. Conclusions Episodic hydrodynamic outflows generated by the OMEGA laser can be used to refine simulations. Two laser drive conditions generated single- and double-pulse jets by unloading material from a moderate-Z plug in a high-Z washer into a low-Z foam ambient medium. The ratio of jet velocity to the speed of sound in the ambient foam gives a Mach number of approximately 3. 2-D axisymmetric DRACO simulations (Keller et al., 1999) predict the shape of the head of the jet well. The simulated deposition of laser energy will be refined to better reproduce experimental results, especially to reduce the overestimated radial expansion of the jets. Targets will be refined so that the full length of each jet can be observed and the relative lengths can be compared to predictions. The observable features of these jets will add to the parameter space Springer

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Astrophys Space Sci (2007) 307:47–50

Fig. 5 Simulated radiographs of single-pulse jet (a) and double-pulse jet (b) at 100 ns

can study the effects of 3-D structure on the stability of the jet. Acknowledgements This work was supported by the U.S. Department of Energy Office of Inertial Confinement Fusion under Cooperative Agreement No. DE-FC52-92SF19460, the University of Rochester, and the New York State Energy Research and Development Authority. The Support of the DOE does not constitute an endorsement by DOE of the views expressed in this article.

References

Fig. 6 Wiener-filtered image of the single-pulse jet showing enhancement of shocks compared to Fig. 3. Note the phase contrast visible on the outer shock

achieved by experimental jets and refine the scaling of experiments to observed astronomical objects. Future work

Springer

Boehly, T.R., Brown, D.L., Craxton, R.S., Keck, R.L., Knauer, J.P., Kelly, J.H., Kessler, T.J., Kumpan, S.A., Loucks, S.J., Letzring, S.A., Marshall, F.J., McCrory, R.L., Morse, S.F.B., Seka, W., Soures, J.M., Verdon, C.P.: Initial performance results of the OMEGA laser system. Opt. Comm. 133, 495–506 (1997) Keller, D., Collins T.J.B., Delettrez, J.A., McKenty, P.W., Radha, P.B., Town, R.P.J., Whitney, B., and Moses, G.A.: DRACO – A new multidimensional hydrocode. Bull. Am. Phys. Soc. 44, 37 (1999) Lindl, J.D.: Inertial confinement fusion. Springer-Verlag, New York (1998) Smalyuk, V.A., Yaakobi, B., Marshall, F.J., Meyerhofer, D.D.: X-ray spectroscopic measurements of areal density and modulations of compressed shells in implosion experiments on OMEGA. LLE Rev. Q. Rep. 83, 124–129 (2000) Spectral Instruments, Tuscon AZ 85745 (www.specinst.com)

Astrophys Space Sci (2007) 307:51–56 DOI 10.1007/s10509-006-9258-x

O R I G I NA L A RT I C L E

Laboratory Modeling of Standing Shocks and Radiatively Cooled Jets with Angular Momentum D. J. Ampleford · S. V. Lebedev · A. Ciardi · S. N. Bland · S. C. Bott · G. N. Hall · N. Naz · C. A. Jennings · M. Sherlock · J. P. Chittenden · A. Frank · E. Blackman

Received: 16 May 2006 / Accepted: 19 September 2006 C Springer Science + Business Media B.V. 2006


Abstract Collimated flows ejected from young stars are believed to play a vital role in the star formation process by extracting angular momentum from the accretion disk. We discuss the first experiments to simulate rotating radiatively cooled, hypersonic jets in the laboratory. A modification of the conical wire array z-pinch is used to introduce angular momentum into convergent flows of plasma, a jet-forming standing shock and into the jet itself. The rotation of the jet is evident in laser imaging through the presence of discrete filaments which trace the rotational history of the jet. The presence of angular momentum results in a hollow density profile in both the standing conical shock and the jet. Keywords Herbig-haro objects . Laboratory . Stars . Winds . Outflows 1 Introduction Convergent flows and plasma jets are ubiquitous in astrophysics; active galactic nuclei, protostars and planetary D. J. Ampleford (
) · C. A. Jennings Sandia National Laboratories, Albuquerque, NM 87123-1106, USA e-mail: [email protected] S. V. Lebedev · S. N. Bland · S. C. Bott · G. N. Hall · N. Naz · M. Sherlock · J. P. Chittenden Blackett Laboratory, Imperial College, London SW7 2BW, UK A. Ciardi Observatoire de Paris, LUTH, Meudon, 92195, France A. Frank · E. Blackman Department of Physics and Astronomy, University of Rochester, Rochester, NY, USA; Laboratory for Laser Energetics, University of Rochester, Rochester, NY, USA

nebulae each have associated outflows (Begelman et al., 1984; Reipurth and Bally, 2001; Balick and Frank, 2002). Many of these jets are likely to contain angular momentum – recent observations of the DG Tauri and other protostellar jets (Coffey et al., 2004; Bacciotti et al., 2002) have indicated azimuthal velocities ∼10 km/s (compared to an axial velocity vz ∼ 300 km/s). The presence of angular momentum in these jets is likely to be significant to the dynamics of the jets and the source object. Specifically for the case of protostars, the extraction of angular momentum by a jet is widely believed to be necessary to allow the accretion of material from the disk onto the central star. The ability to model these rotating jets in the laboratory can aid in understanding these jets provided that the laboratory experiment can reach the appropriate regime of scaled parameters to make astrophysical connections (Ryutov et al., 1999; Remington et al., 2005). In this paper we discuss a laboratory technique to produce rotating convergent plasma flows and we will discuss experimental evidence for rotation of the standing shocks and highly supersonic jets produced. We will also make a preliminary comparison between the results presented and protostellar jets.

2 Experimental configuration To produce rotating convergent flows, standing shocks and jets we use a modification of the conical wire array z-pinch (Lebedev et al., 2002; Ciardi et al., 2002; Lebedev et al., 2005a). The jets produced in these arrays have been shown to have certain characteristic dimensionless parameters in the regime needed for the scaling of protostellar jets. Specifically, they are highly supersonic and radiatively cooled (Lebedev et al., 2002), and experiments can be designed such that the Springer

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Fig. 1 Time gated XUV (hν > 30 eV) self emission images taken endon to the array (a–d, at ∼210 ns) and interferometry images (e–f, at 344 ns and 319 ns respectively). Conical arrays of 16, 18 µm W wires with 30◦ wire inclination angles are shown with (a, e) no twist present and with a twists of (b) 2π/64 (c) 2π/32 and (d,f) 2π/16 between the electrodes. The wires are seen in the outer portion of each image, the

precursor streams are seen between these wires and the central conical shock. The insets in images (a–d) are soft x-ray emission (hν > 220 eV) from the conical shock for the same twist angles, on the same scale. (g) shows the size of conical shock measured on (b–d), with error bars indicating the imaged thickness of the wall of the shock

jet propagates in an ambient medium (Lebedev et al., 2004; Frank et al., 2005; Ampleford et al., 2005a). The current pulse from the MAPGIE pulsed-power generator (1MA, 250 ns, Mitchell et al., 1996) drives a conical array of fine tungsten wires. The self-generated global azimuthal field and the presence of axial and radial components of current through the wires results in continuous ablation and acceleration of precursor plasma flows by the J × B force in the radial (inward) and axial (upward) directions (see Fig 1e). To introduce rotation into this system the two electrodes are twisted with respect to each other (see Fig. 1 and f). The azimuthal current that is now present introduces an axial magnetic field. This results in an additional component to the Lorentz force in the azimuthal direction (Fθ = Jr Bz ). The magnitude of angular momentum introduced into the system can be directly controlled by adjusting the rotation angle between the electrodes. The precursor plasma flows meet on the axis of the wire array producing a standing conical shock. This shock thermalizes the kinetic energy associated with radial motion, resulting in an essentially axial flow. A pressure gradient at the top of the shock accelerates the material; radiative cooling enables the production of a highly supersonic, highly collimated flow. There is no mechanism to remove the net angular momentum from the material as it enters the conical shock and jet; hence it is expected that the shock and jet will rotate.

3 Rotating standing shock

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Figure 1 shows experimental data for conical wire arrays without and with a twist present (i.e. where angular momentum is not present and where it is expected to be present respectively). The upper images (Fig. 1a–d) are XUV emission images taken end-on to the array (looking down the array axis) for no twist, and with twist angles between the electrodes  = 2π/64, 2π /32, 2π /16. The lower images (Fig. 1e–f) are shadowgraphy images taken side-on to the array, with no twist and with a twist of  = 2π/16. In each of the XUV images there is a bright area at the centre of the image corresponding to the thermalization of kinetic energy in the standing shock. The insets in these images are soft x-ray emission (hν > 220 eV), which show that only this central region is emitting at these higher energies. This shock is seen on the shadowgraphy images as a dark region on the array axis where the laser beam is refracted out of the imaging system by the electron density gradients (some beam penetrates the centre of the shock in the twisted case). All the diagnostics show that with increased twist angle, and hence angular momentum, the shock becomes hollow and the size of this shock increases (see Fig. 1g). This is consistent with an increase in the centrifugal force as the angular momentum is increased. It can be shown (Ampleford, 2006a) that the diameter of the hollow shock is consistent with the balance between the centrifugal force on the rotating

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Fig. 2 Schlieren images of jets from conical wire arrays. (a) shows filament like structures, which rotate around the jet for a twist angle of ∼2π/64. (b–d) are schlieren images of the base of jets with no twist (b) and with twists of (c) 2π/64 and (d) 2π/16. All images are at t = 315–340 ns after start of current

column and ram pressure from the continuing convergent flows. This mechanism is also consistent with hybrid code simulations performed by Sherlock (2003). Experiments using twisted cylindrical wire arrays, where no angular momentum is present, demonstrate that the pressure associated with the magnetic field present on the axis of the array has a negligible effect on the shock dynamics (Ampleford et al., 2006a). Analytic estimates of the shock parameters, MHD simulations and hybrid code simulations each indicate a rotation velocity in the setups shown in Fig. 1 & f of v > 40 km/s (simi-

lar to the axial velocity of the material in the shock). The maximum estimated temperature in the shock of 50 eV and charge state Z ∼ 13 indicates a sound speed of ∼20 km/s, hence the rotating flow within the standing shock is supersonic. The density of material in the shock is expected to increase with time as mass is accumulated from the incident flows. 4 Rotating jet The standing shock in a conical array produces a highly supersonic jet, which is surrounded by a lower density halo of Springer

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Fig. 3 Interferograms of the base of jets with no twist (a) and with twists of (b) ∼2π/64 and (c) 2π/16. All images are at t = 315–340 ns after start of current. (d) shows radial electron density profiles for twists of 2π/64 (dotted) and 2π/16 (solid), obtained by Abel inversion of the interferometer images (the error margin is large inside the sharp density jump due to performing the Abel inversion). Abel inversion of the untwisted case is not possibly due to the large density gradient present; on the scale of the other images this profile is effectively a δ-function

plasma (Lebedev, 2005a; Ampleford, 2005b). If the shock that produces this jet is rotating and there is no mechanism for the plasma to dissipate this angular momentum, then it is expected that the jet which emerges will contain angular momentum. Rotation within the central portion of such a jet can be seen experimentally in Fig. 2a, which is a shadowgraphy image of a jet produced by an array with a relatively small twist angle ( = 1/64, the same setup as Fig. 1b). In the image two filament-like structures are seen, which form a double-helix shape. This filamentary structure might be linked to the discrete nature of the convergent plasma streams, or an instability in the flow. When the jets from untwisted conical arrays exhibit similar filamentary structure no rotation is seen. If it is assumed that these filaments have a source point at the top of the conical shock which is static in time, the angle of the filaments can be used to estimate the rotation rate of the jet. Where these two streams cross at the centre of the image (marked point ‘A’) we measure the pitch angle to be θ = 10◦ deg. This implies a ratio Springer

vθ /vz = 0.18, or assuming that the axial flow velocity at this point is 200 ± 50 km/s (Lebedev et al., 2002; Ciardi et al., 2002; Ampleford, 2005b) we estimate a rotation velocity vθ ∼ 35 ± 9 km/s. Figures 2b–d shows jets produced by arrays with three different twist angles; the jet becomes wider as the angular momentum in the system is increased. Comparing the untwisted case and the largest twist angle (Fig. 2b and d) it is clear that the introduction of angular momentum has caused the main jet to become divergent. Results with this larger twist angle indicate that the halo around the jet is likely to be rotating. Fig. 3c shows an interferometer image of the jet for a setup with a twist angle of 2π /16. At the right of the jet near the base of the image a filament is again present illustrating the rotation of the central jet (also seen on the shadowgraphy image, Fig 2d); the angle of this filament indicates a rotation velocity of ∼75 km/s. From comparison to a pre-shot reference interferogram (i.e. without plasma present) the position of the

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outermost boundary of the plasma can be determined. The approximate position of this plasma-edge as the flow propagates upwards has been marked on the image (white line). As there is no material outside the halo which surrounds the jet, the halo should freely expand at its rotation velocity. From the image we find that the outermost plasma is at a radius of 3.3mm and the velocity of this expansion vexp ∼ 60 km/s (the thermal velocity cs ∼ 5 km/s is negligible in comparison). From the interferometer images in Fig. 3 it is possible to reconstruct the electron density profile of the jet. Figure 3d shows Abel inversions of the phase difference through the axial positions indicated on Fig. 3b and c. The density profile shows that the jet produced is in fact hollow. Furthermore, with the larger twist there is a very sharp outer edge to the dense portion of the flow. This shock-like structure is likely to be formed as the expanding central portion of the jet snowplows material as it expands. Additional work is required to investigate this further. 5 Conclusions and connections to astrophysical jets Angular momentum is believed to be an integral part of the dynamics of proto-stellar jets. We have demonstrated a new technique to introduce angular momentum into radiatively cooled, highly supersonic jets in the laboratory. The experimental data indicates that the presence of rotation significantly affects the dynamics of both the jet forming shock and the jet, most notably producing a hollow density profile. The ability to introduce angular momentum in laboratory jet experiments allows the investigation of a large parameter space; however the connection to astrophysical jets requires certain scaling criteria to be satisfied, such as the hydrodynamic scaling relations described by Ryutov et al. (1999). Providing that the portion of the jet of interest is far from the source star, the effects of the magnetic field can be neglected (Hartigan et al., 2003). For radiatively cooled, supersonic jets some of the criteria can be summarized by the Mach number, the cooling parameter and the density ratio between the jet and the ambient medium (Blondin et al., 1990). Angular momentum is represented in the scaling relations developed by Ryutov et al. (1999) within the velocity vector. If it is known that the axial velocity is scaling correctly between the laboratory and protostellar jet (as defined in the Mach number of the flow), then the ratio of the axial to azimuthal velocities is sufficient to define the angle of the flow, and hence the angular momentum. In the experiments described here the ratio of azimuthal and axial velocities measured is ∼0.1–0.4, which is of a similar magnitude to that observed in astrophysical jets (Coffey et al., 2004; Bacciotti et al., 2002). That being said, we emphasize that because the flows in our jets are supersonic and essentially hydrodynamic, the rotating jets of our experiments are most relevant for the propagation regime in astrophysical jets rather than the launch regime where the

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magnetic fields likely dominate the flows (Blackman, 2006). Varying the array parameters in the experiments (e.g. diameter, length, and twist angle) allows control of the ratio of the axial and azimuthal velocities. The laboratory data shows that different radial positions of the jet are rotating; it may be possible to modify the array configuration to allow some control of this radial angular momentum profile. Such control would be required to attempt a laboratory representation of the azimuthal velocity profile thought to be present in an astrophysical jet. The experiments described here have been performed with no ambient medium surrounding the jet. Previous experiments have demonstrated the ability to propagate jets from conical wire arrays (without angular momentum) in an ambient medium that is static (Ampleford et al., 2005a) or has transverse momentum (Lebedev et al., 2004, 2005a; Ampleford et al., 2006b). Interesting topics that could be studied in the future include the angular momentum exchange between a jet and an ambient medium, and the effect of angular momentum on the deflection of jets. Similarly, the capability to use an equivalent technique to introduce angular momentum into laboratory jets where magnetic fields are dynamically significant will allow an investigation of the effect of rotation on magnetically driven launch mechanisms (for example using either radial wire arrays (Lebedev et al., 2005b) or imploding conical wire arrays (Ampleford et al., 2006c). Future experiments would greatly benefit from directly measuring the radial distribution of the azimuthal velocity (along with the mass density) in the shock and jet, for example using spectroscopic techniques, with the aim of developing a fuller understanding of the angular momentum distribution within the system. Acknowledgements The authors would like to thank Dr D.D. Ryutov for useful discussions on the scaling of rotating jets, Prof M.G. Haines for useful discussions on the nature of the filaments observed in the jet and Dr G.S. Sarkisov for assistance with one of the Abel inversions shown in Fig. 3d. This research was sponsored by the NNSA under DOE Cooperative Agreement DE-F03-02NA00057 and in part by the European Community’s Marie Curie Actions – Human resource and mobility within the JETSET (Jet Simulations, Experiments and Theory) network under contract MRTN-CT-2004 005592. Sandia is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Company, for the US DOE’s National Nuclear Security Administration under contract DE-AC04-94AL85000.

References Ampleford, D.J., et al.: Astrophys. Space Sci. 298, 241 (2005a) Ampleford, D.J.: Experimental study of plasma jets produced by conical wire array z-pinches. PhD thesis, University of London ( 2005b) Ampleford, D.J., et al.: Introduction of angular momentum into convergent plasma flows and radiatively cooled jets, in preparation (2006a)

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Ampleford, D.J., et al.: Jet deflection by a quasi-steady-state side wind in the laboratory. Astrophys. Space Sci. DOI 10.1007/s10509-0069238-1 (2006b) Ampleford, D.J., et al.: AIP Conf. Proc. 808, 33 (2006c) Bacciotti, F., et al.: Astrophys. J. 576, 222 (2002) Balick, B., Frank, A.: Ann. Rev. Astr. Astrophys. 40, 439 (2002) Begelman, M.C., et al.: Rev. Mod. Phys. 56, 255 (1984) Blackman, E.G., et al.: ‘Distinguishing Propagation vs. Launch Physics of Astrophysical Jets and the Role of Experiments’ in this issue (2006) Blondin, J.M., et al.: Astrophys. J. 360, 370 (1990) Ciardi, A., et al.: Laser Part. Beams 20, 255 (2002) Coffey, D., et al.: Astrophys. J. 604, 758 (2004) Frank, A., et al.: Astrophys. Space Sci. 298, 107 (2005)

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Hartigan, P.: Astrophys. Space Sci. 287, 111 (2003) Kato, Y., et al.: Astrophys. J. 605, 307 (2004) Lebedev, S.V., et al.: Astrophys. J. 564, 113 (2002) Lebedev, S.V., et al.: Astrophys. J. 616, 988 (2004) Lebedev, S.V., et al.: Plas. Phys. Contr. Fus. 47, B465 (2005a) Lebedev, S.V., et al.: MNRAS 361, 97 (2005b) Lynden-Bell, D.: MNRAS 341, 13 (2003) Mitchell, I.H., et al.: Rev. Sci. Instruments 67, 1533 (1996) Reipurth, B., Bally, J.: Ann. Rev. Astron. Astrophys. 39, 403 (2001) Remington, B.A., et al.: Plas. Phys. Contr. Fus. 47, A191 (2005) Ryutov, D.D., et al.: Astrophys. J. 518, 821 (1999) Sherlock, M.: Ion Collisional Effects in Z-Pinch Precursor Plasma and Laboratory Astrophysical Jets, PhD thesis, University of London (2003)

Astrophys Space Sci (2007) 307:57–62 DOI 10.1007/s10509-006-9268-8

ORIGINAL ARTICLE

Numerical Simulations and Astrophysical Applications of Laboratory Jets at Omega R. F. Coker · B. H. Wilde · J. M. Foster · B. E. Blue · P. A. Rosen · R. J. R. Williams · P. Hartigan · A. Frank · C. A. Back

Received: 28 April 2006 / Accepted: 17 October 2006 C Springer Science + Business Media B.V. 2006


Abstract We have conducted experiments on the Omega laser at the University of Rochester that have produced jets of supersonic Ti impacting and being deflected by a ball of high density plastic. These mm-sized jets of dense plasma are highly complex, have large Reynolds numbers, and, given sufficient time and shear, should produce a fully turbulent flow. The experiments are diagnosed with a point-projection backlighter, resulting in a single image per shot. Simulations of the 3D hydrodynamics capture the large-scale features of the experimental data fairly well while missing some of the smaller scale turbulent-like phenomena. This is to be expected given the limited characterization of the targets as well as the finite resolution of the 3D simulations. If Euler scaling holds, these experiments should model larger astrophysical jets in objects such as HH 110 where an outflow can be seen colliding with a molecular cloud. However, Euler scaling demands that not only the isothermal internal Mach numbers of the two systems be similar but also that any dissipative mechJ. M. Foster · P. A. Rosen · R. J. R. Williams AWE, Aldermaston, Reading, UK R. F. Coker (
) · B. H. Wilde Los Alamos National Laboratory, Los Alamos, NM B. E. Blue Lawrence Livermore National Laboratory, Livermore, CA P. Hartigan Department of Physics and Astronomy, Rice University, Houston, TX A. Frank Department of Physics and Astronomy, University of Rochester, Rochester, NY C. A. Back General Atomic, Inertial Fusion Group, San Diego, CA

anisms, such as radiative cooling or viscous dissipation, be of equal importance relative to each other. Similar equations of state are required as well. We discuss such issues in the context of these experiments and simulations. Keywords Hydrodynamics . ISM: Herbig-Haro objects . ISM: Jets and outflows . Methods: Laboratory

1 Background HH 110 (the left image in Fig. 1) is an astrophysical jet roughly a parsec in size. Our experiments on Omega (the right image in Fig. 1) are a few mm in size. What can we learn about the former from the latter? Strict scaling arguments (see below) show that under certain conditions the two systems behave the same way although they are nearly 20 orders of magnitude different in size. Thus, behavior seen in one should be applicable to the other. In addition, validation of codes used to model one system successfully should result in codes that can be applied to the other equally well; the better the scaling, the more confidence one has in applying a code to the other problem. As such, these experiments are part of a large validation program spanning many national laboratories and the astrophysical community. In this work, we focus on simulations of the LANL/SAIC code RAGE, a radiative hydrodynamics Eulerian code with continuous adaptive grid refinement that uses a Godunov scheme with implicit 2T hydrodynamics. The experiments have been very successful in that they have shown clearly what the codes can and cannot do. For example, the experiments pointed out a temporary code issue with shocks converging at r = 0 in RZ co-ordinates. Without detailed knowledge of initial conditions or a turbulence model, the codes do not reproduce Springer

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Fig. 1 Images of deflected jets. The left image is an HST observation of HH110 (Reipurth et al., 1996) while the right image is a Zn radiograph from our experiments on Omega

Fig. 2 The experimental setup. Lasers enter the hohlraum from below, resulting in a pressure drive on the Ti disc. A jet of Ti forms and enters the RF foam. A backlighter (not shown) perpendicular to the target is used to illuminate the foam. A transmission radiograph is then captured on film

some of the small-scale details of the experiments. However, the larger scale features are captured quite well (see below). Figure 2 shows a schematic of the experimental setup. We use indirect drive, where a number of laser beams (here, 12 beams at 450 J each) are directed into a gold hohlraum. The laser beams radiatively ablate the gold and plasma accumulates in the hohlraum. The resulting pressure build-up in the hohlraum drives the target, a cap and washer of TiAlV alloy. A shock enters the Ti cap, breaks out into the ‘free-run’ region (the hole in the Ti washer), and produces a jet of material that enters a foam. A few additional laser beams are directed towards a backlighter pinhole target made of Zn (or Fe in some cases). The pinhole focuses the resulting 4 to 7 keV photons so that they illuminate the cylindrical 100 mg/cc RF target foam. Finally, a transmission radiograph image is captured

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on film by a camera. Each experiment or ‘shot’ results in a single image. A gold grid with a plastic coating is included on each shot for target registration and resolution modeling. A gold shield is used to minimize the number of photons from the drive side of the target that reach the film while a gold washer or ‘cookie cutter’ is included to minimize direct shocking of the Ti washer. In some shots, a normal density plastic ball is placed as in obstacle in the foam. Codes that include laser drive physics (e.g. NYM at AWE) were used to match DANTE time-resolved observations of the apparent brightness temperature of the hohlraum wall from previous experiments with a very similar drive profile (Foster et al., 2002). The profile used by RAGE in the models presented here is shown in the left plot in Fig. 3. The profile corresponds to the modeled “air” temperature (although the targets are

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Fig. 3 A plot of the temperature profile used to drive the target (left). The nominal laser pulse is a square 1 ns pulse with a peak drive temperature of ∼180 ev. Note that the temperature source (used in the RAGE simulations) is not quite the same as the profile seen by DANTE for reasons such as hole closure (Foster et al., 2002). The right figure shows

the KH instability growth rate using typical values for the experiments. Both linear (dotted) and nonlinear (solid) growth rates are shown. The wavelengths of perturbations in the targets correspond to wavenumbers (k) that are larger than the maximum unstable wavenumber so KH growth is not likely to occur

shot in a vacuum, the simulations use low-density air since RAGE requires non-zero starting densities) in the middle of the hohlraum that is required by RAGE to get the observed hohlraum wall brightness temperature.

3 Scaling These experiments will scale to arbitrary dimensions if the polytropic Euler equations encompass all the relevant physics:

2 Instabilities Kelvin-Helmholtz (KH) instabilities do not appear in these experiments. If one assumes a finite velocity gradient across an interface such that over a scale δ there is a velocity change V, one gets a KH growth rate like that shown in the right figure in Fig. 3. The figure shows the growth rate, in revolutions of growth per unit time, for δ = 5 µm and V = 5 × 106 cm/s, typical values for these experiments. Also shown is the linear growth rate that corresponds to an infinite velocity gradient (Chandrasekhar, 1961). The minimum wavelength (λ = 2π/k) required for KH growth is λmin = 50 µm while the most rapid growth occurs at λpeak = 80 µm. Thus, with λinitial ∼1 µm, KH is not initially important for these experiments. Note that the wavenumber corresponding to the most rapid KH growth (∼32 revolutions of growth over the 100 ns of the experiment) is kpeak = 0.64/ δ. However, the rate of growth for Rayleigh-Taylor (RT) √ instabilities goes as (kg) ∼ 1 rev/ns for λ = 1 µm. Since the target surface roughness is ∼1 µm, RT may be important over the hundreds of ns duration of the experiments; RT is quite possibly the cause of the ‘smoke’ seen in the right hand image in Figs. 1 and 5. We have run simulations that included target perturbations to investigate instability growth. To simulate burrs on the target of roughly 1 µm, since we need ∼8 cells across a feature, we require sub-micron resolution. Other 3D features, such as the offset of the Au washer relative to the free-run region, need to be included as well. Such high-resolution 3D models are not yet feasible. However, coarser 3D simulations with large (> 10µm) perturbations show that, as expected, the macroscopic features of the flow are sensitive to the initial conditions.

ρ




 ∂v + v · ∇v = −∇ p ∂t

∂ρ + ∇ · (ρv) = 0 ∂t p ∂ρ p ∂p −γ + v · ∇ p − γ v · ∇ρ = 0 ∂t ρ ∂t ρ For a given adiabatic index, γ , the Euler equations are invariant under transformations that preserve the Euler num√ ber, Eu = v (ρ/P), where v is a velocity, ρ is density, and P is pressure. Thus, if one assumes dissipative mechanisms (thermal diffusion, viscosity, and radiation) are negligible, the Euler equations contain all the relevant physics and the experiment can be scaled to arbitrary dimensions (Ryutov et al., 1999). The conditions for these three mechanisms to be unimportant are vL/κ = Pe ≫ 1, ρ Lv/η = Re ≫ 1, and vτ/L = χ ≫ 1, where κ is the thermal diffusivity, η is the dynamic viscosity, τ is the radiative cooling time, and L and v are some length and velocity scale, respectively. Strictly speaking, the experiment can be shown to scale even if radiation is non-negligible (Ryutov et al., 2001). Table 1 shows some characteristic values for the experiments as well as for HH 110, a Herbig-Haro object with a jet extending from a young star system and being deflected by a large molecular cloud (Riera et al., 2003). For all regions of the flow that are important, the local Reynolds number, Re, is more than 105 , so turbulence may develop at late times if there is enough shear (Robey et al., 2003); it is not clear how fully developed turbulence affects scaling. In the experiment, which does not have a fully ionized plasma, viscosity is determined by Springer

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Astrophys Space Sci (2007) 307:57–62 Table 1 List of characteristic values for the jet ‘flute’ and bow-shock regions of the Omega experiments and for HH 110. The local isothermal Mach number, known as the Euler number, is spatially and temporally variable in the experiments, making firm scaling

to HH 110 difficult. If the functional shape of the Euler number, as a function of timescale, is the same between the experiments and HH 110, scaling will persist; otherwise, the experiments scale only for a finite time

Quantity

Symbol

Omega (core sheath)

Omega (bow-shock head)

HH 110

Temperature Density Pressure Fluid velocity Lengthscale Timescale Jet sound speed Local Mach number Kinematic viscosity Reynolds number Peclet number

T ρ P u L √ t = L (ρ/P) cs  M = u cs v Re Pe

0.1 eV 3 g/cc 2 × 1010 dyn/cm2 10 km/s 20 µm 25 ns 2 km/s 5 ∼0.1 cm2 /s ∼105 ∼109

0.06 eV 2 g/cc 2 × 1011 dyn/cm2 4 km/s 200 µm 60 ns 4 km/s 1 ∼0.05 cm2 /s ∼105 ∼1010

8000 K 1 × 10−21 g/cc 10−9 dyn/cm2 300 km/s 500 AU 50 yrs 10 km/s 30 ∼1019 cm2 /s ∼105 ∼105

Fig. 4 Comparison of simulation (PETRA in blue and RAGE in yellow) results to the experimental data. In the left figure is displacement of the bow-shock and the ‘pedestal’ (the large rounded feature at the

bottom of the data shown in the right image of Fig. 1). The right figure shows the comparison for the diameter of the bow-shock. Scatter in the data reflects uncertainty of initial and drive conditions

the degree of coupling between ions (Clerouin et al., 1998). In the simulations, viscosity is dominated by shock treatments so one can use the sound speed, cs , to estimate Re by (Landau and Lifshitz, 1987) ρcs sh /4. In RAGE the 5 to 95% shock width (sh ) is ∼7 cells, regardless of the details of the problem, so Re = 7/4ρcs x. In that viscosity is unimportant in both cases, one must have a numerical Re which is also ≫1 in order for the simulations to correctly represent the experiments. To get astrophysical scaling, Re for the experiment, the simulation, and the astrophysical object all need to be ≫1. This holds true for these Omega experiments and for HH 110 but not for coarse resolution (10 s of µm) simulations. To get ReRAGE >∼10 in all regions of interest, we require ∼1 µm resolution. Such high-resolution simulations do start to capture much of the observed 3D be-

havior (Foster et al., 2005). Of course, such conclusions depend on the choice of length scale; here, we are interested in the jet ‘flute’ and so we choose the width of that feature (∼200 µm) as our scale. Re in such a case is large enough in the simulations to capture the relevant features of the experi√ ments. Note that Euler scaling transforms time as L (ρ/P), so from Table 1 it can be seen that ∼25 ns in the experiments is equivalent to ∼50 yrs in HH 110. Table 1 also shows that thermal diffusion is not important in these experiments.

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4 Code comparisons Comparison of simulations by both RAGE and PETRA to data with no plastic ball is shown in Fig. 4. RAGE does slightly better than PETRA on the pedestal formation,

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Fig. 5 Images of simulated (left) and experimental (right) radiographs. The circle shows the initial location of the plastic ball. Scale is in units of microns. The stalk holding the ball (not included in this simulation) and

the Au grid can be seen. Due to the coarse 12 µm resolution, the ‘grass’ on the pedestal is not clearly seen in the simulations. The Y -shaped ‘flute’ (see Section 4) is labeled in the simulated radiograph

Fig. 6 Density contours and velocity vectors for simulations of HH 110 (lower left) from de Gouveia Dal Pino (1999), velocity vectors imposed over a scaled color image of density for a simulation of the Omega

experiments (top), and simulated Zn radiographs at 100 (lower middle) and 150 ns (lower right)

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but both codes over-predict the displacement of the bowshock, implying the experiment has more late-time dissipation than the codes. The scatter in the experiment data, reflects shot-to-shot variability; both codes are within the scatter for the bow-shock diameter but not for the displacement of the bow-shock and the pedestal. For RAGE, new 2D to 3D linking permits better numerical resolution at early time (during ablation) to capture more of the resulting late time 3D features (such as the smoke and the grass seen in Fig. 5). Figure 5 shows a comparison of a simulated RAGE radiograph to the data. The simulation matches macroscopic features such as bow-shock location quite well. It of course does not reproduce small-scale features that are related to (unknown) details of the initial conditions. We are starting to use other codes such as FLASH (Calder et al., 2002) to model the Omega experiments. Initial hydrodynamics-only FLASH results, using a material energy source in the Ti plug, with coarse (30 µm) resolution show no fine details at the jet head but there is apparent KH behavior along the jet that is not seen in the experiments or RAGE or PETRA simulations. Figure 6 shows a comparison of a RAGE simulation of the experiments to a simulation of HH 110 (de Gouveia Dal Pino, 1999). The images show that the coarse structure of the flow is similar for the two systems; this is expected since both are jets colliding with and being deflected by a large obstacle. However, there has been as yet no attempt to match the fine details. Figure 6 also shows images of a 3D RAGE simulation at 100 and 150 ns, illustrating how the jet evolves. Note the fairly poor resolution (higher resolution simulations show more 3D ‘smoky’ behavior). Observe the bow-shock in the plastic ball, the deflection of part of the jet, and the formation of the pedestal. The ‘flute’ formation and breakup is particularly hard to model. There is also ‘grass’ on the pedestal behind (or at – one cannot tell the difference in the 2D transmission image) the edge of the Mach ring. The

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grass is most likely debris being kicked up by the Mach ring shock traveling perpendicular to the backside of the Ti washer surface. Such features help point the way to where better physics models are required.

5 Summary We have developed a test bed for experiments on Omega that can be scaled to astrophysical objects. Our present Omega jet experiments can be well modeled by simulations using a variety of codes. These particular experiments scale roughly to HH 110. These types of experiments help validate codes so they can then be used directly on astrophysical problems. In the future, we hope to move to a higher Mach number, a higher aspect ratio, and perhaps a radiative jet (where scaling will still apply). These experiments continue to drive code improvements as well as quantitative image analysis.

References Calder, A.C., et al.: ApJS 143, 201 (2002) Chandrasekhar, S.: Hydrodynamic and Hydromagnetic Stability. Oxford Univ. Press, New York (1961) Clerouin, J.G., Cherfi, M.H., Zerah, G.: Europhys. Lett. 42, 37 (1998) de Gouveia Dal Pino, E.M.: ApJ 526, 862 (1999) Foster, J.M., et al.: ApJ 634, L77 (2005) Foster, J.M., et al.: Phys. Plasmas 9, 2251 (2002) Landau, L.D., Lifshitz, E.M.: Fluid Mechanics, Vol. 6, Course of Theoretical Physics, 2nd ed. (1987) Robey, H.F., et al.: Phys. Plasmas 10, 614 (2003) Reipurth, B., Raga, A.C., Heathcote, S.: A&A 311, 989 (1996) Riera, A., et al.: AJ 126, 327 (2003) Ryutov, D.D., et al.: ApJ 518, 821 (1999) Ryutov, D.D., Remington, B.A., Robey, H.F., Drake, R.P.: Phys. Plasmas 8, 1804 (2001)

Astrophys Space Sci (2007) 307:63–68 DOI 10.1007/s10509-006-9251-4

O R I G I NA L A RT I C L E

Laboratory Experiment of Plasma Flow Around Magnetic Sail Ikkoh Funaki · Hidenori Kojima · Hiroshi Yamakawa · Yoshinori Nakayama · Yukio Shimizu

Received: 14 April 2006 / Accepted: 12 September 2006 C Springer Science + Business Media B.V. 2006


Abstract To propel a spacecraft in the direction leaving the Sun, a magnetic sail (MagSail) blocks the hypersonic solar wind plasma flow by an artificial magnetic field. In order to simulate the interaction between the solar wind and the artificially deployed magnetic field produced around a magnetic sail spacecraft, a laboratory simulator was designed and constructed inside a space chamber. As a solar wind simulator, a high-power magnetoplasmadynamic arcjet is operated in a quasisteady mode of 0.8 ms duration. It can generate a simulated solar wind that is a high-speed (above 20 km/s), high-density (1018 m−3 ) hydrogen plasma plume of ∼0.7 m in diameter. A small coil (2 cm in diameter), which is to simulate a magnetic sail spacecraft and can obtain 1.9-T magnetic field strength at its center, was immersed inside the simulated solar wind. Using these devices, the formation of a magnetic cavity (∼8 cm in radius) was observed around the coil, which indicates successful simulation of the plasma flow of a MagSail in the laboratory. Keywords Spacecraft propulsion . Magnetic sail . M2P2 . Laboratory simulation . Magnetoplasmadynamic arcjet I. Funaki (
) · Y. Shimizu Japan Aerospace Exploration Agency, Sagamihara, Kanagawa, 229-8510, Japan e-mail: [email protected] H. Kojima University of Tsukuba, Tsukuba, Ibaraki, 305-8573, Japan (currently, Ishikawajima-Harima Heavy Industries Co. Ltd.) H. Yamakawa Kyoto University, Gokasho, Uji, Kyoto, 611-0011, Japan Y. Nakayama National Defense Academy, Yokosuka, Kanagawa, 239-8686, Japan

1 Introduction In 2005, after a cruising of 28 years, it was reported that Voyager 1 spacecraft had entered the solar system’s final frontier where the Sun’s influence ends. The spacecraft is now passing the termination shock where the solar wind starts to slow down and reverse due to its first encounters with pressure from interstellar space. Although the Voyager successfully unveiled our solar system during its very long travel, future exploration to the outer planets, or even beyond the heliosphere should be conducted within several years to make such explorations attractive. To drastically shorten the mission trip time to deep space, some new in-space propulsion systems are proposed. High priority candidates are: (1) next generation ion thruster which features high exhaust velocity of more than 50 km/s, (2) sail propulsion utilizing the energy of the Sun, and (3) aerocapturing/breaking systems, which are expected to be used in combination with high-performance ion thrusters or the sails if you want to put an orbiter to the outer planets with atmosphere, because the aerocapture system will help the orbiter to decelerate without fuel consumption. Among the sail propulsion systems, solar sails are intensively studied by NASA and other space agencies targeting at future deep space missions (Montgomery and Johnson, 2004; Kawaguchi, 2004). Unfortunately, acceleration of the solar sails is usually small due to heavy materials used for the sail, hence it is difficult to shorten the mission trip time in particular for the missions within our solar system. To overcome this difficulty, a magnetic sail (usually abbreviated as MagSail) is proposed by Zubrin because it is expected to achieve high thrust per weight by capturing the momentum of the solar wind (Zubrin and Andrews, 1991). When the MagSail is in operation, as shown in Fig. 1,charged particles approaching the current loop (coil) are blocked according to the B-field they experience, forming Springer

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Fig. 1 Schematics of magnetic sail. (a) Three-dimensional image of the plasma flow and magnetosphere of MagSail. (b) A schematic illustration of the trajectories of charged particles at the magnetopause; ions and electrons incident on a boundary layer when the polarization electric field due to charge separation is present

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a magnetosphere (or a magnetic cavity) around the coil current. The solar wind plasma flow and the magnetic field are separated by the magnetopause, at which ions entering the magnetic field are reflected except near the polar cusp region where the ions can enter deep into the magnetic cavity. Due to the presence of the magnetosphere, the supersonic solar wind flow is blocked and decelerated being accompanied by a bow shock; a wave-drag force is then transferred to the coil

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Magnetic Cavity (B 0)

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current through electromagnetic processes (Nishida et al., 2006). Thus the spacecraft is accelerated in the direction of the solar wind. The idea of the MagSail, however, did not draw attention so far because the original MagSail by Zubrin was unrealistically large spacecraft design with a large hoop coil of 30 km in radius to form 100 km-radius blocking area (corresponding to 20-N-class thrust). For such a large

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magnetosphere (or equivalently, a large stand-off distance, L), fluid-like interactions depicted in Fig. 1 in analogy to the geomagnetic field are expected. In contrast, ion or electron kinetic movement becomes important for a smaller L below the ion Larmor radius (r Li , defined in Fig. 1b). For the MagSail with a large r Li /L value, coupling between the ion flow and the magnetic field becomes loose, leading to only negligible thrust production. Although many theoretical works are going to study the effect of magnetospheric size (L) on the thrust of the MagSail (Fujita, 2004; Akita and Suzuki, 2005) and its derivatives (Winglee et al., 2000; Asahi et al., 2004; Khazanov et al., 2005), such a complicated plasma flow in a transitional regime between the MHD scale and the ion kinetic scale will be more confidently treated by a scale-model experiment. This paper firstly discusses the most important things, the scaling considerations that characterize the plasma flow of the MagSail. Our newly developed experimental simulator of the magnetic sail is then introduced with some initial test results, which help understanding the plasma flow structure of the MagSail.

2 Thrust and scaling parameters of MagSail

nm i u 2sw =

(2Bmp )2 2µ0

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where n is the plasma number density, m i ion mass, u sw the velocity of the solar wind, 2Bmp the magnetic flux density at the boundary, µ0 the permeability in vacuum. The magnetic flux density Bmp at a distance L from the dipole center is expressed as, Bmp =

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hence the detachment distance of the boundary from the dipole center, L, is derived as follows. L=




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The force on the current loop depends on the area that blocks the solar wind. By increasing this blocking area, a larger thrust is obtained. Therefore, the force exerting on the coil of the MagSail, F, can be formulated as (Funaki and Nakayama, 2004),

(1)

where Cd is thrust coefficient, 1/2ρu 2sw the dynamic pressure of the solar wind, and S = π L 2 the characteristic area of the magnetosphere.

2.2 Definition of magnetospheric size, ion Larmor radius, and skin depth Because the density of the solar wind plasma flow around a Magsail is very small, the charged particles are collision-less and their movement separates the plasma region outside the magnetic cavity and the region inside the magnetic cavity. Simplified picture of this boundary is depicted in Fig. 1b. When a magnetic dipole Md is located at the center, there is a balance between the total internal (magnetic) and the

(4)

This boundary is usually called a magnetopause, on which the charged particles, ions and electrons, impinge. The external space is considered as magnetic field-free. In the idealized situation, one finds the thickness of the magnetopause is the order of the plasma skin depth δ as δ = c/ω p

2.1 Thrust exerting on MagSail

1 F = Cd ρu 2sw S 2

external (plasma) pressures at the boundary:

(5)

where c is the light velocity, and ω p the plasma frequency. The thickness of the magnetopause, however, is considered larger than δ; it is about the ion gyration radius at the magnetopause (Willis, 1971): r Li =

m i u sw e2Bmp

(6)

Because of their heavier mass, the ions tend to penetrate more deeply into the magnetic field than electrons. This sets up a charge separation, thus the outward pointing polarization field restrains the ions. Before the ions can be deflected by the magnetic field, they are returned by this polarization field. The electrons, however, experience the Lorentz force and gain energy in the polarization field. The transverse velocity component of the electrons accounts essentially for the electric current in the interface, which in case of the magnetopause is usually referred to as Chapman-Ferraro current. From Equations (5) and (6), δ ∼ 2 km and r Li = 71 km for the solar wind flow in Table 1. 2.3 Non-dimensional parameters The solar wind is a super sonic plasma flow which consists of collisionless particles. These features are described by the Mach number, M > 1 as well as the magnetic Reynolds number, Rm = σ µ0 u sw L ≫ 1. Incorporating typical plasma Springer

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Astrophys Space Sci (2007) 307:63–68 Table 1 Parameters of plasma flow around MagSail Laboratory Parameter Solar wind parameters Velocity, u sw Plasma number density, n Electron temperature, Te Ion temperature, Ti Plasma duration MagSail (coil) parameters Magnetic moment, Md Size of magnetic cavity (stand-off distance), L B-field at magnetopause, Bmp Expected thrust level, F Duration of coil exciting current Dimensionless parameters Mach number, M Ratio of ion Larmor radius to L , r Li /L Ratio of electron skin depth to L , δ/L Magnetic Reynolds number, Rm = σ µ0 u sw L

velocity and temperatures of the solar wind, M ∼ 8. In addition to these two scaling parameters, we defined r Li/ /L, and δ/L, hence four non-dimensional parameters in total are introduced. Among them, the parameters Rm, r Li /L, and δ/L, are dominated by the size of the magnetosphere, which was selected as L < 70 km in our preliminary study (see Table 1). Corresponding non-dimensional parameters are 1 < r Li /L (the ion gyration radius is comparable to or larger than L, which is in contrast to the MHD scale requireing r Li /L ≪ 1 in the case of geophysics (Bachynski and Osborne, 1967; Yur et al., 1995)) and δ/L < 1 (the skin depth is comparable to or smaller than L). If the thickness of the magnetopause is small enough in comparison to L, almost all of the incident ions are reflected at the magnetospheric boundary, hence large thrust on the coil of the MagSail is expected. Vice versa, if the thickness of the magnetopause is much larger than L, no interaction between the plasma flow and the magnetic field is anticipated. We treat a transitional region between the MHD scale (thin magnetopause mode) and the ion kinetic scale (thick magnetopause mode) in this experiment.

3 Experimental simulation of MagSail Our simulator consists of a high-power magnetoplasmadynamic (MPD) solar wind simulator and a coil simulating a MagSail spacecraft, both of which are operated in a pulse mode. As shown in Fig. 2,the MPD arcjet is mounted on a flange of a space chamber (2.5-m in diameter), providing up to 0.7-m-diameter plasma plume at the center of the space chamber. A coil of 20 mm in diameter was located in the plume of the MPD arcjet. Into the artificially produced Springer

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<60 km/s 1018 m−3 1 eV (not available yet) 0.8 ms

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<3×10−5 Tm3 <0.1 m <1.9 mT <0.2 N 0.9 ms

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magnetic field by the coil, a plasma jet from the MPD arcjet was introduced to observe possible interactions.

3.1 Solar wind and MagSail simulator The MPD arcjet is the key device which provides a highdensity and high-speed plasma jet necessary for our MagSail experiment. The discharge chamber of the MPD arcjet is 50 mm in inner diameter and 100 mm in length, which consists of eight anode electrode rods (made of molybdenum) that are azimuthally equally spaced, and a short 20-mmdiameter cathode rod (thoriated tungsten); both electrodes are insulated by boron nitride (Shimizu et al., 2005). Since the MPD arcjet is operated in a pulse mode, a fast-acting valve (FAV) is used to feed hydrogen gas. When the FAV is opened, the gas in the reservoir flows through choked orifices of 1.2 mm in diameter. The mass flow rate of hydrogen gas was controlled by adjusting the reservoir pressure, obtaining a gas pulse of about 11 ms duration in a rectangular waveform. After the gas pulse reaches its quasi-steady state, the ignitron of a pulse-forming network (PFN1, capacitor bank for the MPD arcjet) is triggered. PFN1 has an L-C ladder circuit that supplies the discharge current up to 20 kA with a 0.8 ms flat-topped waveform in a quasi-steady mode. In contrast, PFN2 for the MagSail simulator (coil) supplies rather small current (below 3 kA), and a 20-turn-coil is required to produce up to 1.9 T magnetic field strength at the center of the 2-cm-diameter coil. These pulsed devices are synchronized; after the gas feed and the magnetic field by the coil (MagSail) reach their steady states, the discharge of the MPD arcjet is initiated.

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Fig. 2 Experimental setup

Vacuum Chamber Coil (φ20mm)

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Operational parameters of these devices are summarized in Table 1, in which the temperatures and density of a plasma jet obtained by Langmuir probes as well as the ion velocity by the time of flight method are provided. One can see that the MPD arcjet obtains smaller velocity and temperature in comparison with the solar wind, but its density is quite large; such large density is necessary to simultaneously meet the conditions of r Li /L ∼ 1, δ/L < 1, and L ∼ 0.1 m. The last condition (L ∼ 0.1 m) severely limits the range of the magnetic Reynolds number, Rm; so far we obtained Rm ∼ 15. Rm may be increased by further increasing u sw , L, or Te . Such improvement is also required for the Mach number, because until now we did not achieve the Mach number of the solar wind.

MSS Capacitor Bank(PFN2) 3kA/0.9ms

a bow shock, the dark region between the two boundaries may be the magnetosheath flow hence the magnetic cavity is bounded by boundary 2. Before identifying the structure of the flow around the scale-mode, however, the field and flow structure should be checked using ether optical or probe measurement. Anyway, it is confirmed that the radius of the magnetic cavity observed in Fig. 3c is consistent with L calculated from the plasma data using Equation (4), which was about 8 cm in this case; hence it is expected that the plasma flow around a MagSail spacecraft was successfully scaled down and was demonstrated in our new experimental simulator.

4 Summary 3.2 Plasma flow around the coil Figure 3 shows close-up photos of simultaneous operation of both the solar wind simulator (MPD arcjet) and the MagSail (coil). In Fig. 3a (the case without coil current, hence no magnetic field is present), it was found that the plasma jet directly impinges on the coil surface and the light emission is limited to the region only near the coil, whereas the light emission pattern is away from the coil when the coil current, hence the B-field, is applied (Figs. 3b and c). Expected field structure in the case of Fig. 3c is depicted in Fig. 3d. Two light emission regions can be identified; one is the upstream region where the field is expected to be magnetic field free; another is the region near the coil, in which the plasma being trapped in the deformed magnetic field is observed. The dark region between the two light emission regions may correspond to the magnetospheric boundary, where the plasma flow cannot enter but being reflected. If we assume that boundary 1 is

An experimental simulator of a plasma flow around MagSail was designed and fabricated. A high-density plasma jet above 1018 m−3 is supplied by a hydrogen MPD arcjet, from which a high-speed plasma jet (20–60 km/s) is ejected. In case a 20-mm-radius coil with 1.9 T magnetic field strength at the center of the coil was inserted into the plasma plume, a magnetic cavity was observed around the coil, which indicated the plasma flow around the properly scaled MagSail was experimentally simulated. The original MagSail by Zubrin required a spacecraft with a large coil of 30 km in radius to form 100-km-radius blocking area to produce 20-N-class thrust. Because the spacecraft proposed by Zubrin will operate in the MHD scale, the ion Larmor radius (r Li ) is smaller than the magnetospheric size (L); such a MagSail spacecraft in the MHD scale requires unrealistically large magnetosphere, hence huge coil, that is impossible to build in space. In this experiment, however, we have attempted the scale-model experiment of a smaller (ion Springer

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the coil is off course inevitable to demonstrate the MagSail, which is the next step of our model experiment. Acknowledgments We would like to thank the members of the magnetoplasma sail research group (in particular, Dr. H. Ogawa, Dr. K. Fujita, Dr. I. Shinohara, and Mr. H. Nishida in Institute of Space and Astronautical Science (ISAS), Prof. K. Toki of Tokyo University of Agriculture and Technology, Dr. H. Otsu of Shizuoka University, and Dr. Y. Kajimura of Kyushu University) for their hearty support and valuable advices. This research is supported by the Grant-in Aid for Young Scientist (B) (No. 15760594) by Japan Society for Promotion of Science, and by the Inamori Grant Program of the Inamori Foundation. The supports by the engineering committee as well as the space plasma committee of ISAS are also appreciated.

References

Fig. 3 Close-up view of a plasma flow around the Coil; MPD arcjet is operated at a discharge current of 20 kA and a mass flow rate of 0.4 g/s; also, the coil produces (a) no magnetic field, (b) 0.2-T, and (c) 1.8-T magnetic field strength at the coil center. The magnetic dipole axis is normal to the plasma flow as is expected in (d)

Lamor scale) MagSail, which is at most L = 70 km in space, hence r Li ∼ L. In this case of moderately sized MagSail, our preliminary probe diagnostics of the scale-model shows that thrust coefficient Cd is approximately 0.5, which also indicates that thrust production is possible by the MagSail in the ion Larmor scale. Direct measurement of thrust exerting on

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Akita, D., Suzuki, K.: Kinetic analysis on plasma flow of solar wind around magnetic sail. 36th AIAA Plasmadynamics and Lasers Conference, AIAA 2005-4791, Toronto, June (2005) Asahi, R., Funaki, I., Fujita, K., Yamakawa, H., Ogawa, H., Nonaka, S., Sawai, S., Nishida, H., Nakayama, Y., Otsu, H.: Numerical study on thrust production mechanism of a magneto plasma sail. AIAA, 40th AIAA/ASME/SAE/ASEE Joint Propulsion Conference, AIAA 2004-3502, Florida, July (2004) Bachynski, M.P., Osborne, F.J.F.: Laboratory geophysics and astrophysics. In: Anderson, T.P., Springer R.W., (eds.) Advances in Plasma Dynamics. Northwestern University Press (1967) Fujita, K.: Particle simulation of moderately-sized magnetic sails. J. Space Technol. Sci. 20(2), 26–31 (2004) Funaki, I., Nakayama, Y.: Sail propulsion using the solar wind. J. Space Technol. Sci. 20(2), 1–16 (2004) Kawaguchi, J.: A solar power sail mission for a Jovian Orbiter and Trojan asteroid flybys. 55th International Astronautical Congress, IAC-04-Q.2.A.03, Vancouver, October (2004) Khazanov, G., Delamere, P., Kabin, K., Linde, T.J.: Fundamentals of the plasma sail concept: Magnetohydrodynamic and kinetic studies. J. Propulsion Power 21(5), 853–861 (2005) Montgomery, E.E., Johnson, L.: The development of solar sail propulsion for NASA science missions to the inner solar system. AIAA-2004-1506, 45th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics & Materials Conference, Palm Springs, April (2004) Nishida, H., Ogawa, H., Funaki, I., Fujita, K., Yamakawa, H., Nakayama, Y.: Two-dimensional magnetohydrodynamic simulation of a magnetic sail. J. Spacecraft Rockets 43(3), 667–672 (2006) Shimizu, Y., Toki, K., Funaki, I., Kojima, H., Yamakawa, H.: Development of magnetoplasmadynamic solar wind simulator for Magsail experiment. 29th International Electric Propulsion Conference, IEPC-2005-201, Princeton, October–November (2005) Willis, D.M.: Structure of the magnetopause. Rev. Geophys. Space Phys. 9(4), 953–985 (1971) Winglee, R.M., Slough, J., Ziemba, T., Goodson, A.: Minimagnetospheric plasma propulsion: tapping the energy of the solar wind for spacecraft propulsion. J. Geophys. Res. 105(A9), 21067– 21077 (2000) Yur, G., Rahman, H.U., Birn, J., Wessel, F.J., Minami, S.: Laboratory facility for magnetospheric simulation. J. Geophys. Res. 100(A12), 23727–23736 (1995) Zubrin, R.M., Andrews, D.G. : Magnetic sails and interplanetary travel. J. Spacecraft Rockets 28(2), 197–203 (1991)

Astrophys Space Sci (2007) 307:69–75 DOI 10.1007/s10509-006-9213-x

O R I G I NA L A RT I C L E

Astrophysical Jets of Blazars and Microquasars Markus B¨ottcher

Received: 14 April 2006 / Accepted: 7 July 2006 C Springer Science + Business Media B.V. 2006


Abstract Some recent developments in the study of relativistic jets in active galactic nuclei and microquasars are reviewed. While it has been well established for some time that extragalactic jets found in radio galaxies, quasars, and BL Lac objects are the site of ultrarelativistic particle acceleration, the recent identification of the Galactic jet source and microquasar LS 5039 as a source of very-high-energy gamma-ray emission has underlined the striking similarity between the two types of astrophysical jet sources. In this paper, I will present an overview of the dominant radiation and particle acceleration processes and observational tests to distinguish between such processes. The wide-ranging analogies between Galactic and extragalactic jets, but also their distinct differences, in particular those caused by the presence of the companion star in Galactic microquasar systems, will be exposed. Keywords Gamma-rays: theory . Radiation mechanisms: non-thermal . X-rays: binaries . Stars: winds . Outflows 1. Introduction High-velocity, collimated outflows are a frequent phenomenon, commonly associated with the accretion of material onto central objects. For example, they occur as nonrelativistic jets in young stellar objects, as mildly relativistic outflows from Galactic X-ray binaries, and as highly relativistic outflows from active Galactic nuclei (AGN) and gamma-ray bursts. In this article, I will focus on analogies and similarities, but also distinct, characteristic differences

M. B¨ottcher Astrophysical Institute, Department of Physics and Astronomy, Ohio University, Athens, OH 45701, USA

between the highly relativistic jets of AGN, in particular the class of AGN termed “blazars”, and X-ray binaries.

2. Phenomenology of blazars Blazars (BL Lac objects and γ -ray loud flat spectrum radio quasars [FSRQs]) are the most extreme class of active galaxies known. They have been observed at all wavelengths, from radio through very-high energy (VHE) γ rays. 46 blazars have been identified with high confidence as sources of >100 MeV emission detected by the EGRET telescope on board the Compton Gamma-Ray Observatory (Hartman et al., 1999; Mattox et al., 2001), and 11 blazars (Mrk 421, Mrk 501, PKS 2155−314, 1ES 2344+514, 1H 1426+428, 1ES 1959+650, PKS 2005−489, 1H 2356−309, 1ES 1101−232, 1ES 1218+304, and PG 1553+113) have now been detected at VHE γ -rays (>350 GeV) by groundˇ based air Cerenkov telescopes. Many of the EGRET-detected γ -ray blazars appear to emit the bulk of their bolometric luminosity at γ -ray energies. Blazars exhibit variability at all wavelengths on various time scales. Radio interferometry often reveals one-sided kpc-scale jets with apparent superluminal motion. The broadband continuum spectra of blazars are dominated by non-thermal emission and consist of two distinct, broad components. A sequence of blazar sub-classes, from FSRQs to low-frequency peaked BL Lac objects (LBLs) to high-frequency peaked BL Lacs (HBLs) can be defined through the peak frequencies and relative ν Fν peak fluxes, which also seem to be correlated with the bolometric luminosity (Fossati et al., 1998, see also Fig. 1). Figure 1 already illustrates that in particular the highenergy emission from blazars can easily vary by more than an order of magnitude between different EGRET observing Springer

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Fig. 1 SEDs of 3C 279 (Hartman et al., 2001), BL Lacertae (B¨ottcher and Bloom, 2000; B¨ottcher and Reimer, 2004), and Mrk 501 (Petry et al., 2000). For each object, two simultaneous broadband spectra at

two different epochs are shown. The curves show model fits, using a leptonic jet model

epochs (see also von Montigny et al., 1995; Mukherjee et al., 1997, 1999). However, high-energy variability has been observed on much shorter time scales, in some extreme cases less than an hour (Gaidos et al., 1996). BL Lac objects occasionally exhibit particularly interesting X-ray variability patterns, which can be characterized as spectral hysteresis in hardness-intensity diagrams (e.g. Takahashi et al., 1996; Kataoka et al., 2000). This has been interpreted as the synchrotron signature of gradual injection and/or acceleration of ultrarelativistic electrons into the emitting region, and subsequent radiative cooling (e.g. Kirk et al., 1998; Georganopoulos and Marscher, 1998; Kataoka et al., 2000; Kusunose et al., 2000; Li and Kusunose, 2000; B¨ottcher and Chiang, 2002). Figure 2d shows the results of a recent BeppoSAX observation of BL Lacertae (Ravasio et al., 2003; B¨ottcher et al., 2003). Rapid flux and spectral variability of blazars is also commonly observed in the optical regime, often characterized by a spectral hardening during flares (see, e.g., Fig. 2c, or Lainela et al., 1999; Villata et al., 2002).

of material in the jet (Spada et al., 2001), stochastic particle acceleration in shear boundary layers of relativistic jets (e.g., Ostrowski and Bednarz, 2002; Rieger and Duffy, 2004), magnetic reconnection in Poynting-flux dominated jets (e.g., Sikora et al., 2005), or hadronically initiated pair avalanches (Kazanas and Mastichiadis, 1999). While the electron-synchrotron origin of the lowfrequency emission is well established, there are two fundamentally different approaches concerning the high-energy emission. If protons are not accelerated to sufficiently high energies to reach the threshold for pγ pion production on synchrotron and/or external photons, the high-energy emission will be dominated by ultrarelativistic electrons and/or pairs (leptonic models). In the opposite case, the high-energy emission will be dominated by cascades initiated by pγ pair and pion production as well as proton, π ± , and µ± synchrotron radiation (hadronic models). These two approaches have so far been mostly discussed separately. However, the recent observation of isolated TeV flares without simultaneous X-ray flares in 1ES 1959+650 (Krawczynski et al., 2004) and Mrk 421 (Bla˙zejowski et al., 2005) may provide rather strong support for scenarios in which elements of both classes of jet models might be relevant (hybrid models).

3. Models of blazar emission The high inferred bolometric luminosities, rapid variability, and apparent superluminal motions provide compelling evidence that the nonthermal continuum emission of blazars is produced in  light day sized emission regions, propagating relativistically along a jet directed at a small angle with respect to our line of sight. It is generally agreed that the lowfrequency component of blazar SEDs might be synchrotron radiation from nonthermal, ultrarelativistic electrons. Several electron injection/acceleration scenarios have been proposed, e.g. impulsive injection near the base of the jet (Dermer and Schlickeiser, 1993; Dermer et al., 1997), isolated shocks propagating along the jet (e.g., Marscher and Gear, 1985; Kirk et al., 1998; Sikora et al., 2001; Sokolov et al., 2004), internal shocks from the collisions of multiple shells Springer

3.1. Leptonic blazar models In leptonic models, the high-energy emission is produced via Compton scattering off the same ultrarelativistic electrons which are producing the synchrotron emission at lower frequencies. Possible target photon fields are the synchrotron photons produced within the jet (the SSC process: Marscher and Gear, 1985; Maraschi et al., 1992; Bloom and Marscher, 1996) or external photons (the EC process). Possible sources of external seed photons include accretion-disk photons entering the emission region directly (Dermer et al., 1992; Dermer and Schlickeiser, 1993) or after reprocessing in

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the optical color-magnitude correlation; (d) comparison of the fits from panel (a) to the X-ray hardness-intensity correlation during a short-term flare observed by BeppoSAX on Nov. 1, 2000. Based on count rates in the three BeppoSAX NFI energy channels LECS [0.5–2 keV], MECS [2–4 keV], and MECS [4–10 keV], and X-ray hardness ratios: HR1 = MECS[2–4]/LECS[0.5–2], HR2 = MECS[4–10]/MECS[2–4]

surrounding material like the broad-line regions of a quasar (Sikora et al., 1994; Dermer et al., 1997), jet synchrotron emission reprocessed by circumnuclear material (Ghisellini and Madau, 1996), infrared emission from circumnuclear dust (Bla˙zejowski et al., 2000), or synchrotron radiation from other (earlier/later, faster/slower) emission regions along the jet (Georganopoulos and Kazanas, 2003). Radiation spectra may be modified by γ γ absorption internal and external to the source, with the former leading to the injection of additional relativistic electron-positron pairs, and synchrotron self absorption must be taken into account in a self-consistent leptonic blazar model. As the emission region is propagating relativistically along the jet, continuous particle injection and/or acceleration and subsequent radiative and adiabatic cooling, as well as particle escape have to be considered. Also the deceleration of the jets, in particular in HBLs, may have a significant impact on the observable properties of blazar

emission (Georganopoulos and Kazanas, 2003; Ghisellini et al., 2005). Detailed modeling of both spectra and spectral variability of blazar spectra with time-dependent leptonic jet models has been done very successfully for a variety of blazars. Several examples for spectral fits to contemporaneous spectral energy distributions of various blazars are shown in Fig. 1. However, it has also been demonstrated that spectral fitting alone is generally insufficient to constrain the multitude of parameters in leptonic jet models, in particular in cases in which external photon sources turn out to be non-negligible (see, e.g. B¨ottcher et al., 2002). Invaluable additional constraints can be obtained when variability information is included in time-dependent modeling efforts. Figure 2 illustrates recent results of spectral variability modeling of the broadband emission from BL Lacertae during a broadband observing campaign in 2000 (Ravasio et al., 2003; B¨ottcher

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et al., 2003). One of the key results of the leptonic modeling part of that work (B¨ottcher and Reimer, 2004) was that leptons would need to be accelerated out to energies of several hundred GeV, into a power-law distribution with number index q = 2.3, consistent with first-order Fermi acceleration at relativistic, parallel shocks. 3.2. Hadronic blazar models If a significant fraction of the kinetic luminosity in the jet is converted into the acceleration of relativistic protons and those protons reach the threshold for pγ pion production, synchrotron-supported pair cascades will develop (Mannheim and Biermann, 1992; Mannheim, 1993). The acceleration of protons to the necessary ultrarelativistic energies requires high magnetic fields of at least several tens of Gauss. In the presence of such high magnetic fields, also the synchrotron radiation of the primary protons (Aharonian, 2000; M¨ucke and Protheroe, 2000) and of secondary muons and mesons (Rachen and M´esz´aros, 1998; M¨ucke and Protheroe, 2000, 2001; M¨ucke et al., 2003) must be taken into account in order to construct a self-consistent synchrotronproton blazar (SPB) model. Electromagnetic cascades can be initiated by photons from π 0 -decay (“π 0 cascade”), electrons from the π ± → µ± → e± decay (“π ± cascade”), p-synchrotron photons (“ p-synchrotron cascade”), and µ-, π - and K -synchrotron photons (“µ± -synchrotron cascade”). M¨ucke and Protheroe (2001) and M¨ucke et al. (2003) have shown that the “π 0 cascades” and “π ± cascades” generate featureless γ -ray spectra, in contrast to “ p-synchrotron cascades” and “µ± -synchrotron cascades” that produce a double-humped γ -ray spectrum. In general, direct proton and µ± synchrotron radiation is mainly responsible for the high energy bump in blazars, whereas the low energy bump is dominanted by synchrotron radiation from the primary e− , with a contribution from secondary electrons. Figure 2b shows a fit to the broadband SED of BL Lacertae in 2000, using the hadronic SPB model. 3.3. Hybrid blazar models While standard leptonic SSC models predict a close temporal flux correlation between the synchrotron and Compton components, recent monitoring observations of the TeV-blazars 1ES 1959+650 (Krawczynski et al., 2004) and Mrk 421 (Bla˙zejowski et al., 2005) at X-ray and TeV-energies revealed TeV flares without accompanying X-ray flares, a behaviour sometimes referred to as ”orphan TeV flares”. In the case of 1ES 1959+650, this phenomenon was preceded by an ordinary, correlated X-ray and TeV-flare, which can be generally well understood in the context of leptonic SSC models. This finding strongly suggests the need for models that explain Springer

flares dominated by leptonic interactions as well as flares where non-leptonic components must play an important role within the same system. The recent orphan TeV flare of 1ES 1959+650 led to the development of the hadronic synchrotron mirror model (B¨ottcher, 2005; Reimer et al., 2005). In this model, the primary, correlated X-ray and TeV flare is explained by a standard SSC model while the secondary orphan TeV-flare is explained by π 0 -decay γ -rays as a result of photomeson production in the Delta(1232)-resonance from relativistic protons interacting with the primary synchrotron flare photons that have been reflected off clouds located at pc-scale distances from the central engine. Some earlier developments along the lines of hadronic processes in the context of models with leptonically dominated blazar emission include the following: (a) The “supercritical pile” model of (Kazanas and Mastichiadis, 1999) suggests a runaway pair production avalanche initiated by mildly relativistic protons on reflected synchrotron photons via pγ pair production as the primary pair injection mechanism in blazar jets. (b) Atoyan and Dermer (2003) suggest the conversion of ultrarelativistic protons into neutrons via pγ pion production on external soft photons as a possible mechanism to overcome synchrotron losses of protons near the base of blazar jets and, thus, to allow blazar jets to remain collimated and transport their kinetic energy out to kpc scales.

4. The jets of microquasars In the standard picture, the high-energy (X-ray – γ -ray) spectra of X-ray binaries generally consist of two major components: A soft disk blackbody with a typical temperature of kT ∼ 1 keV, and a power-law at higher energies. Neutronstar and black-hole X-ray binaries exhibit at least two main classes of spectral states, generally referred to as the high/soft state, and the low/hard state (for a review see, e.g. Liang, 1998; McClintock Remillard, 2004). The high-energy spectra of X-ray binaries in the soft state are characterized by a thermal blackbody component, believed to be associated with thermal emission from an optically thick, geometrically thin accretion disk (Shakura and Sunyaev, 1973), and a powerlaw tail with a photon index Ŵ ≥ 2. Generally, no high-energy cutoff of the high-energy power-law is detected. In the hard state, the spectrum is dominated by a power-law, with a slope Ŵ < 2 and a cut-off at ∼ a few hundred keV. Two additional states, namely the Very High State (VHS) and the Intermediate State (IS) share common features of both the high and the low state, namely a hard powerlaw and a prominent disk blackbody. In addition to prominent X-ray emission, many X-ray binaries are also known to be associated with extended radio jet structures (for a recent review see, e.g., Fender et al., 2004). In analogy to their extragalactic cousins, those sources

Astrophys Space Sci (2007) 307:69–75

are generally termed “microquasars”. When the sources are in the high/soft or low-hard states, one observes quite often a positive correlation between the radio and hard X-ray emission, apparently anti-correlated with the soft X-ray emission (e.g. Corbel et al., 2000, 2001), indicating that the production of a steady, relatively slow (bulk Lorentz factor Ŵ j  2) radio jet is suppressed in the high/soft state. In the IS and VHS, occasionally intermittent, faster (Ŵ j
2) jets are observed (e.g., Fender et al., 2004). While, in the conventional view of X-ray binaries (including microquasars), the X-ray and γ -ray emission is attributed to Comptonized emission (Sunyaev and Titarchuk, 1980; Titarchuk, 1994) arising from hot thermal (kT ≫ 1keV) or relativistic, non-thermal electrons close to the black hole (Liang and Price, 1977; Bisnovatyi-Kogan and Blinnikov, 1977; Shapiro et al., 1976; Narayan and Yi, 1994; Chen et al., 1995, e.g.), the tentative EGRET detections of at least two Galactic microquasars at MeV–GeV γ -ray energies, namely LS 5039 (Paredes et al., 2000) and LSI+61◦ 303 (Gregory and Taylor, 1978; Taylor et al., 1992; Kniffen et al., 1997), the detection of X-ray jet structures in several microquasars using Chandra and XMM-Newton (Corbel et al., 2002; Tomsick, 2002, e.g.,), and, most recently, the detection of veryhigh-energy (VHE) γ -ray emission from LS 5039 (Aharonian et al., 2005) have re-ignited interest in jet models for the high-energy emission from microquasars, analogous to the commonly favored models for blazars. A jet origin of the Xray emission of microquasars has been suggested by several authors, e.g., Markoff et al. (2001, 2003a,b), who discussed the possibility of synchrotron emission from relativistic electrons in the jet extending from the radio all the way into the X-ray regime.

5. Models of high-energy emission from microquasar jets Microquasars now join blazar AGNs as a firmly established class of very-high energy γ -ray sources. Because of their apparent similarity with their supermassive AGN cousins, it has been suggested that Galactic microquasars may be promising sites of VHE γ -ray production (e.g., BoschRamon et al., 2005a). High-energy γ -rays of microquasars can be produced via hadronic (e.g. Romero et al., 2003) or leptonic processes. In the latter case, the most plausible site would be close to the base of the mildly relativistic jets, where ultrarelativistic electrons can Compton upscatter soft photons. Possible sources of soft photons are the synchrotron radiation produced in the jet by the same ultrarelativistic electron population (SSC = synchrotron selfCompton; Aharonian and Atoyan, 1999), or external photon fields (Bosch-Ramon and Paredes, 2004; Bosch-Ramon et al., 2005a). The X-ray – soft γ -ray spectral variability

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features expected in such a microquasar jet model have recently been investigated in a detailed parameter study by Gupta et al. (2006), and a brief summary of the main results of that work can also be found in these proceedings (Gupta and B¨ottcher, 2006). Up to this point, radiation physics widely analogous to blazar jets may be considered. However, both LS 5039 and LSI+61◦ 303 are high-mass X-ray binaries which are rather faint in X-rays, with characteristic 1–10 keV luminosities of ∼1034 ergs s−1 . This is much lower than the characteristic bolometric luminosity of the high-mass companions of these objects, at L ∗
1038 erg/s. Consequently, the dominant source of external photons in LS 5039 and LSI+61◦ 303 is the companion’s optical/UV photon field. The intense radiation field of the high-mass companion will also lead to γ γ absorption of VHE γ -rays in the ∼ 100 GeV–TeV photon energy range if VHE photons are produced close to the base of the jet (Aharonian et al., 2005; B¨ottcher and Dermer, 2005). It is primarily the presence of the prominent azimuthally asymmetric photon source provided by the companion star, which leads to substantial differences in the expected spectra and variability patterns of microquasars, compared to blazars. The most obvious of these consequences would be a temporal modulation of the high-energy emission on the time scale of the orbital period. Both effects of the orbital-phase dependent Compton scattering and γ γ absorption will lead to characteristic spectral variability patterns. The orbital modulation patterns due to γ γ absorption patterns have been investigated in detail in B¨ottcher and Dermer (2005), while the isolated orbital-modulation effects on Compton scattered starlight photons have been worked out in detail by Dermer and B¨ottcher (2006). A brief summary of the orbital modulation patterns due to those two effects, with particular focus on model parameter choices appropriate to LS 5039 (Casares et al., 2005) can also be found in these proceedings (B¨ottcher and Dermer, 2006). In addition to the orbital modulation effects from γ γ absorption and starlight Compton scattering, there are at least two other effects that could potentially lead to a quasiperiodic temporal variation in the high-energy emission from microquasar jets: (a)

If the binary orbit has a substantial eccentricity (e.g., e = 0.35 for the case of LS 5039, Casares et al., 2005), the rate of mass transfer from the stellar companion to the compact object, which is believed to be dominated by wind accretion, is likely to be periodically modulated. This modulation would also be expected to appear at radio and X-ray energies and would be expected to lead to an overall hardening of the γ -ray spectrum at all energies with increasing γ ray flux. In contrast, the orbital-phase dependent γ γ Springer

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(b)

absorption trough due to interactions of VHE photons with companion starlight would lead to a spectral hardening at E  100 GeV, but a spectral softening at E
100 GeV at increasing γ -ray flux levels, and the starlight Compton-scattering signature would provide an overall softening of the VHE γ -ray flux with increasing flux level because of the more severe effect of the Klein-Nishina cutoff at a more favorable orientation for starlight Compton scattering (Dermer and B¨ottcher, 2006; B¨ottcher and Dermer, 2006). The orientation of the jet may also be mis-aligned with respect to the normal of the orbital plane (Maccarone, 2002; Butt, 2003) and possibly precessing about the normal (Larwood, 1998; Torres et al., 2005), leading to additional modulations, including a changing Doppler boosting factor. A stationary misalignment of the jet could lead to a slight enhancement of the orbital modulation (if the jet makes a smaller angle with the line of sight than the orbitalmotion axis) or reduce it (in the opposite case). A γ -ray flux modulation due to jet precession can easily be disentangled from the orbital modulation since the precession period is generally different from the orbital period, so that its effect would average out when folding observational data with the orbital period.

Acknowledgements This work was partially supported by NASA through XMM-Newton GO grant no. NNG 04GI50G and NASA INGEGRAL Theory grant no. NNG 05GK59G.

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Astrophys Space Sci (2007) 307:77–82 DOI 10.1007/s10509-006-9243-4

ORIGINAL ARTICLE

Simulation of the Magnetothermal Instability Ian J. Parrish · James M. Stone

Received: 14 April 2006 / Accepted: 22 August 2006 C Springer Science + Business Media B.V. 2006


Abstract In many magnetized, dilute astrophysical plasmas, thermal conduction occurs almost exclusively parallel to magnetic field lines. In this case, the usual stability criterion for convective stability, the Schwarzschild criterion, which depends on entropy gradients, is modified. In the magnetized long mean free path regime, instability occurs for small wavenumbers when (∂ P/∂z)(∂ ln T /∂z) > 0, which we refer to as the Balbus criterion. We refer to the convective-type instability that results as the magnetothermal instability (MTI). We use the equations of MHD with anisotropic electron heat conduction to numerically simulate the linear growth and nonlinear saturation of the MTI in plane-parallel atmospheres that are unstable according to the Balbus criterion. The linear growth rates measured from the simulations are in excellent agreement with the weak field dispersion relation. The addition of isotropic conduction, e.g. radiation, or strong magnetic fields can damp the growth of the MTI and affect the nonlinear regime. The instability saturates when the atmosphere becomes isothermal as the source of free energy is exhausted. By maintaining a fixed temperature difference between the top and bottom boundaries of the simulation domain, sustained convective turbulence can be driven. MTI-stable layers introduced by isotropic conduction are used to prevent the formation of unresolved, thermal boundary layers. We find that the largest component of the time-averaged heat flux is due to advective motions as opposed to the actual thermal conduction itself. Finally, we explore the implications of this instability for a J. M. Stone: Program in Applied and Computational Mathematics, Princeton University, Princeton, NJ 08544 I. J. Parrish (
) · J. M. Stone Department of Astrophysical Sciences, Princeton University, Princeton, NJ 08544 e-mail: [email protected]

variety of astrophysical systems, such as neutron stars, the hot intracluster medium of galaxy clusters, and the structure of radiatively inefficient accretion flows. Keywords Accretion . Accretion disks . Convection . Hydrodynamics . Instabilities . MHD . Stars: neutron . Turbulence . Galaxy clusters

1 Introduction In many dilute, magnetized astrophysical plasmas, the electron mean free path between collisions can be many orders of magnitude larger than the ion gyroradius. In this regime, the equations of ideal magnetohydrodynamics (MHD) that describe the fluid plasma must be supplemented with anisotropic transport terms for energy and momentum due to the near free-streaming motions of particles along magnetic field lines (Braginskii, 1965). Thermal conduction is dominated by the electrons compared to ions by a factor of the square root of the mass ratio. In a non-rotating system, it is sufficient to neglect the ion viscosity (Balbus, 2004). A fully collisionless treatment may be done using more complex closures such as Hammett and Perkins (1990). The implications of anisotropic transport terms on the overall dynamics of dilute astrophysical plasmas is only beginning to be explored (Balbus, 2001; Quataert et al., 2002; Sharma et al., 2003). One of the most remarkable results obtained thus far is that the convective stability criterion for a weakly magnetized dilute plasma in which anisotropic electron heat conduction occurs is drastically modified from the usual Schwarzschild criteria (Balbus, 2000). In particular, stratified atmospheres are unstable if they contain a temperature (as opposed to entropy) profile which is decreasing upward. There are intriguing analogies between the stability Springer

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properties of rotationally supported flows (where a weak magnetic field changes the stability criterion from a gradient of specific entropy to a gradient of angular velocity), and the convective stability of stratified atmospheres (where a weak magnetic field changes the stability criterion from a gradient of entropy to a gradient of temperature). The former is a result of the magnetorotational instability (MRI; Balbus and Hawley, 1998). The latter is a result of anisotropic heat conduction. To emphasize the analogy, we will refer to this new form of convective instability as the magnetothermal instability (MTI). The MTI may have profound implications for the structure and dynamics of many astrophysical systems. In this paper, we use numerical methods to explore the nonlinear evolution and saturation of the MTI in twodimensions. We adopt an arbitrary vertical profile for a stratified atmosphere in which the entropy increases upward (and therefore is stable according to the Schwarzschild criterion), but in which the temperature is decreasing upwards (and therefore is unstable according to the Balbus criterion, (∂ P/∂z)(∂ ln T /∂z) < 0). We confirm the linear growth rates predicted by Balbus (2000) for dynamically weak magnetic fields and numerically measure the growth rates for stronger fields. We find that in the nonlinear regime vigorous convective turbulence results in efficient heat transport. Full details are published in Parrish and Stone (2005), hereafter PS. These results may have implications for stratified atmospheres where anisotropic transport may be present. The most exciting potential application is to the hot X-ray emitting gas in the intracluster medium of galaxy clusters (Peterson and Fabian, 2005; Markevitch et al., 1998). The hot plasmas are magnetized and have very long mean free paths along the magnetic field lines; thus, they are a prime candidate for this instability. For example, the Hydra A cluster has T ≈ 4.5 keV and a density of n ≈ 10−3 − 10−4 cm−3 giving a mean free path that’s almost one-tenth of the virial radius. Other applications are to the atmospheres of neutron stars with moderate magnetic fields and radiatively inefficient accretion flows.

2.1 Equations of MHD and linear stability The physics of the MTI is described by the usual equations of ideal MHD with the addition of a heat flux, Q, and a vertical gravitational acceleration, g.

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∂B + ∇ × (v × B) = 0, ∂t 
  B2 ∂E B(B · v) +∇ · v E + p+ − ∂t 8π 4π +∇ · Q + ρg · v = 0,

(1)

(3)

(4)

where the symbols have their usual meaning with E the total energy. The heat flux contains contributions from electron motions (which are constrained to move primarily along field lines) and isotropic transport, typically due to radiative processes. Thus, Q = QC + Q R , where QC = −χC bˆ bˆ · ∇T,

(5)

Q R = −χ R ∇T,

(6)

where χC is the Spitzer Coulombic conductivity (Spitzer, 1962), bˆ is a unit vector in the direction of the magnetic field, and χ R is the coefficient of isotropic conductivity. Some progress can be made analytically in the linear regime of the instability. I introduce two useful quantities, χC′ =

γ −1 χC P

and

χ′ =

γ −1 (χC + χ R ) . P

(7)

With WKB theory, one is able to obtain a dispersion relation. The details of this process are to be found in Section 4 of Balbus (2000). The most important result of the linear analysis is the instability criterion, k 2 v 2A −

χC′ ∂ P ∂ ln T < 0, ρχ ′ ∂z ∂z

(8)

where v 2A = B 2 /4πρ is the Alfv´en speed. In the limit of infinitesimal wavenumber, the instability criterion shows that any atmosphere with the temperature and pressure gradients in the same direction is unconditionally unstable, i.e. ∂ P ∂ ln T > 0. ∂z ∂z

2 Physics of the MTI

∂ρ + ∇ · (ρv) = 0, ∂t


  B2 BB ∂(ρv) + ∇ · ρvv + p + I− + ρg = 0, (2) ∂t 8π 4π

(9)

We refer to the instability criterion Eq. (9) as the Balbus criterion. The instability criterion of the magnetorotational instability (Balbus and Hawley, 1998) can be written k 2 v 2A +

d > 0, d ln R

(10)

where  is the angular velocity. The similarity between the MRI and the MTI is self-evident. Strong magnetic fields are

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capable of stabilizing short-wavelenth perturbations in both instabilities through magnetic tension. 2.2 Computational method We use the 3D MHD code ATHENA (Gardiner and Stone, 2005) with the addition of an operator-split anisotropic thermal conduction module for our simulations. Our initial state is always a convectively stable state (d S/dz > 0) in hydrodynamic equilibrium. We implement two different boundary conditions for exploring the nonlinear regime. The first boundary condition is that of an adiabatic boundary condition at the upper and lower boundaries, i.e. a Neumann boundary condition on temperature. This situation is ideal for single-mode studies of linear growth rates. In the Neumann boundary condition the magnetic field is reflected at the upper and lower boundaries, consistent with the adiabatic condition on heat flow. The second boundary condition fixes the the temperature at the upper and lower boundaries of the atmosphere, i.e. a Dirichlet boundary condition on temperature. This setup is useful for driven simulations where we wish to study the effects of turbulence. In this boundary condition, the magnetic field is again reflected at the boundaries of the box, but heat is permitted to flow across the boundary from the constant temperature ghost zones. With these choices of boundary conditions, the net magnetic flux penetrating the box is constant in time as there is zero Maxwell stress at the boundary. If the net magnetic flux penetrating the box is initially zero, then in two dimensions the magnitude of the magnetic field must decay in time as a result of Cowling’s anti-dynamo theorem, but otherwise the saturated state is not affected.

Fig. 1 Snapshots of the magnetic field lines for a single-mode perturbation with adiabatic boundary conditions at various times during the evolution of the instability. (upper left) Inititial condition; (upper right) Linear phase; (lower left) Non-linear phase; (lower right) Saturated state

t=0 t = 320 t = 640

t = 1280 t = 2880

3 Results 3.1 Single-mode perturbation and qualitative understanding of the MTI By examining a single mode-perturbation to the background state, we can gain an intuitive understanding of the physical mechanism of the instability. We begin by perturbing a convectively-stable, but MTI-unstable atmosphere with a very weak sub-sonic and sub-Alf´enic sinusoidal velocity perturbation in a box with adiabatic boundary conditions. The evolution of the magnetic field lines is shown in Fig. 1 for several different times. Notice in the upper right plot that at x ≈ 0.03 that a parcel of fluid (as traced by the frozen-in field line) has been displaced upward in the atmosphere. As this parcel of fluid comes to mechanical equilibirum with the background state, it adiabatically cools. The magnetic field line now is partially aligned with the background temperature state, thus thermally connecting this parcel of fluid with a

z

Fig. 2 Vertical profile of the horizontally-averaged temperature profile in the single mode case at various times. The initial state is a monotonically decreasing temperature profile with respect to height, and the final state is isothermal

hotter parcel deeper in the atmosphere. As a result, heat flows along the field, causing the higher parcel to become buoyant. This buoyant motion causes the field line to be more aligned with the background temperature, increasing the heat flux, and generating a runaway instability. It is instructive to examine the behavior of the temperature profile of the atmosphere. Figure 2 shows the horizontally Springer

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3.2 Linear growth rates By following Section 4.3 of Balbus (2000) and using the Fourier convention, exp(σ t + ikx), one can derive a weakfield dispersion relation for the MTI as


Fig. 3 Solutions of the MTI dispersion relation in the weak field limit for an atmosphere with d ln T /d ln S = −3. The axes are normalized to the local Brunt-V¨ais¨al¨a frequency, N . The crosses are growth rates measured from simulations

averaged temperature profile of this run at various times (normalized to the sound crossing time). The initial state is a linear temperature profile decreasing with height. As the instability progresses, the temperature profile becomes more and more isothermal. Saturation occurs, as one would expect, when the temperature profile is almost completely isothermal, since the source of free energy has been depleted. More details are to be found in Section 4 of PS.




 1 σ 2 χ ′T k2 σ + γ N N N
′ 2 d ln T χc T k + = 0, d ln S N

σ N

3

+

(11)

where N is the Brunt-V¨ais¨al¨a frequency, the natural frequency of adiabatic oscillations for an atmosphere. With the single-mode perturbation simulations we are able to measure the growth rate for a variety of situations. Figure 3 plots the nondimensionalized growth rate versus wavenumber for theory (solid line) and the measured values from simulations (crosses). As can be seen these are in very good agreement. There are essentially two ways to suppress the growth of the MTI. First, strong magnetic fields can exert tension that limits the growth and saturation of the instability. Second, isotropic conduction, as would result from radiative transport, can effectively short-circuit the thermal driving along field lines necessary for this instability to occur. For more detailed analysis, we refer the reader to Section 3 of PS. 3.3 Nonlinear regime and efficiency of heat transport In order to assess the efficiency of heat transport in the magnetothermal instability, one needs to examine multimode simulations seeded with Gaussian white noise perturbations and conducting boundary conditions at the top and bottom of the

Fig. 4 Snapshots of the magnetic field in the run with stable layers. (far left) Early linear phase; (middle left) early non-linear phase. (middle right) The MTI drives penetrative convection into the stable layers, and at late times (far right) magnetic flux is pumped into the stable layers

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Fig. 5 Time evolution of the horizontally-averaged heat flux at the midplane and 80% height of the simulation domain in the run with stable layers. The total heat flux (thick solid line) is subdivided into Coulombic (thin solid line), radiative (dashed lined), and advective (dotted line) components. The instantaneous total heat flux is dominated by advective motions

domain. The simplest such set-up results in narrow, unresolved boundary layers at the upper and lower boundaries, thus, making the heat flux difficult to measure accurately. As an alternative, we utilize a more physically relevant simulation. This setup involves an atmosphere that is convectively stable throughout, but MTI unstable only in the central region. The surrounding regions are stabilized to the MTI through the addition of isotropic conducitivity. As a result, the central unstable region is well-resolved. Figure 4 shows the evolution of magnetic field lines as this instability progresses. The magnetic field lines shown essentially track the central unstable region; however, the third panel clearly shows a plume of fluid that is penetrating into the stable layer as a result of convective overshoot. This phenomenon is well-known in the solar magnetoconvection literature (Tobias et al., 2001; Brummell et al., 2002). In three dimensions this type of behavior greatly amplifies the magnetic field in a local magnetic dynamo. More quantitative measurements can be made by comparing the time- and horizontally-averaged vertical heat fluxes and breaking it down into Coulombic, radiative, and isotropic components. Figure 5 shows these quantities plotted as a function of time. At the midplane, the oscillatory advective flux is clearly dominant at any given instant in time; however, averaged in time the advective heat flux contributes roughly 23 of the total heat flux. The Coulombic flux, which is relatively constant in time, contributes the remaining 31 . To determine the heat conduction efficiency of this instability, we compare it to the expected vertical heat flux across the simulation domain for pure uniform isotropic conductivity, namely, Q 0 ≈ 3.33 × 10 −5 . The time-averaged heat conduction at the midplane is Q tot,50% ≈ 3.54 × 10−5 , which indicates that the instability transports the entire applied heat flux efficiently.

4 Conclusions and application The most important conclusion of this work is that atmospheres with d S/d Z < 0 are not necessarily stable to convection. In fact, dilute atmospheres with weak to moderate magnetic fields can be convectively unstable by the Balbus criterion resulting in an instability that we call the magnetothermal instability. We have verified using MHD simulations that the measured linear growth rates agree with analytic WKB theory as predicted by Balbus (2000). For adiabatic boundary conditions, we find the saturated state is an isothermal temperature profile, corresponding to the exhaustion of the free energy in the system. For a driven instability with conducting boundary conditions, we find that the MTI efficiently transports heat, primarily by advective motions of the plasma in the vigorous convection that results. The most promising application of this instability is to clusters of galaxies. Structure formation calculations assuming CDM cosmologies predict monotonically decreasing temperature profiles of the intracluster gas (Loken et al., 2002). Observations of clusters with Chandra, such as Hydra A (DeGrandi and Modlendi, 2002), however, indicate essentially flat temperature profiles. The intracluster medium is dilute, magnetized, and has a mean free path that could be as high as one-tenth of the cluster virial radius. It may be that these temperature profiles are representative of the saturated state of the MTI. This possibility will be explored in future work.

References Balbus, S.A., Hawley, J.F.: Rev. Mod. Phys. 70, 1 (1998) Balbus, S.A.: ApJ 534 420 (2000)

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Balbus, S.A.: ApJ 562, 909 (2001) Balbus, S.A.: ApJ 616, 857 (2004) Braginskii, S.I.: in: Leontovich, M.A. (ed.), Reviews of Plasma Physics, Consultants Bureau, New York, Vol. 1, p. 205 (1965) Brummell, N.H., Clune, T.L., Toomre, J.: ApJ 570, 825 (2002) DeGrandi, S., Molendi, S.: ApJ 567, 163 (2002) Gardiner, T., Stone, J.: J. Comp. Phys. 205, 509 (2005) Hammett, G.W., Perkins, F.W.: Phys. Rev. Lett. 64, 3019 (1990)

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Loken, C., et al.: ApJ 579, 571 (2002) Markevitch, M., et al.: ApJ 377, 392 (1998) Parrish, I.J., Stone, J.M.: ApJ 633, 334 (2005) Peterson, J.R., Fabian, A.C.: astro-ph0512549 (2005) Quataert, E., Dorland, W., Hammett, G.W.: ApJ 577, 524 (2002) Sharma, P., Hammett, G., Quataert, E.: ApJ 596, 1121 (2003) Spitzer, L.: Physics of Fully Ionized Gases. Wiley, New York (1962) Tobias, S.M., et al.: ApJ 549, 1183 (2001)

Astrophys Space Sci (2007) 307:83–86 DOI 10.1007/s10509-006-9250-5

O R I G I NA L A RT I C L E

Sheared Flow as a Stabilizing Mechanism in Astrophysical Jets Lucas F. Wanex · Erik Tendeland

Received: 12 April 2006 / Accepted: 12 September 2006 C Springer Science + Business Media B.V. 2006


Abstract It has been hypothesized that the sustained narrowness observed in the asymptotic cylindrical region of bipolar outflows from Young Stellar Objects (YSO) indicates that these jets are magnetically collimated. The jz × Bφ force observed in z-pinch plasmas is a possible explanation for these observations. However, z-pinch plasmas are subject to current driven instabilities (CDI). The interest in using z-pinches for controlled nuclear fusion has lead to an extensive theory of the stability of magnetically confined plasmas. Analytical, numerical, and experimental evidence from this field suggest that sheared flow in magnetized plasmas can reduce the growth rates of the sausage and kink instabilities. Here we propose the hypothesis that sheared helical flow can exert a similar stabilizing influence on CDI in YSO jets. Keywords Astrophysical jets . Linear analysis . Sheared flow

1 Collimation of astrophysical jets The sustained narrowness of the asymptotic cylindrical region of many YSO bipolar outflows spawns the hypothesis that an intrinsic collimating mechanism is present in the jet plasma. Here we use the term “asymptotic cylindrical region” to mean the narrow cone between the disk-wind initial compression region and the jet termination region in the ambient medium. A typical example of this comes from Hubble Space Telescope Observations of HH30 (Burrows et al., 1996). The apparent opening angle of this jet becomes narrower in the asymptotic cylindrical region of the outflow when compared L. F. Wanex (
) · E. Tendeland University of Nevada, Reno, NV, USA

to the opening angle near the jet source (Mundt et al., 1990). Similar examples of recollimation in other jets have also been observed (K¨onigle and Pudritz, 2000). Observations of this nature have lead to the concept that the jet narrowness is maintained by self-collimation (Shu et al., 2000). Selfcollimation can be caused by hoop stresses from a toroidal magnetic field in the jet plasma. This collimating mechanism is similar to the jz × Bφ force observed in z-pinch plasmas. If this mechanism is to be considered as the cause of the sustained narrowness of some YSO jets the possibility that these jets carry current far from the accretion disk must be admitted. It is well known that a cylindrical current-carrying plasma column with a helical magnetic configuration is subject to MHD instabilities (Bateman, 1978; Chandrasekhar, 1961). The key instabilities are the interchange (sausage) and kink modes (Freidberg, 1987). The sausage m = 0 and kink m = 1 instabilities have been invoked to explain the observation of knots, wiggles, and filamentary structures in astrophysical jets (Nakamura and Meier, 2004; Reipurth and Heathcote, 1997). It has also been suggested that the disruption caused by the kink instability discredits the magnetic collimation model (Spruit et al., 1997). However, it is an open question whether MHD instabilities will disrupt YSO jet collimation (K¨onigle and Pudritz, 2000). Two mechanisms for reducing z-pinch plasma instabilities may explain why this is so. Z-pinch plasmas will be stable if the ratio of axial to azimuthal magnetic field strength is greater than the Kruskal-Shafranov limit (Kruskal and Schwarzschild, 1956; Shafranov, 1954). Some stellar evolution models predict that low-mass protostars form in an interstellar medium that is supported by a magnetic field (Ray, 2004; K¨onigl and Pudritz, 2000). This interstellar field could supply the instability reducing mechanism in these jets; however the efficiency of the pinch effect is reduced because the Springer

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axial magnetic field must be compressed as well as the jet plasma itself (Shumlak and Hartman, 1995). Analytical, experimental and numerical results show that sheared flow in z-pinch plasmas can reduce the growth of MHD instabilities (Arber and Howell, 1995; Bateman, 1978; DeSouzaMachado et al., 2000; Golingo et al., 2005; Shumlak and Hartman, 1995; Sotnikov et al., 2002, 2004; Ruden, 2002; Winterberg, 1985, 1999; Wanex et al., 2004; Wanex, 2005a). Velocity shear stabilization is primarily a phase mixing process that disrupts the growth of unstable modes (DeSouzaMachado et al., 2000; Wanex et al., 2005b). There are sound theoretical reasons why Keplerian shear could be present in jets originating from an accretion disk (Bacciotti et al., 2004; Bally et al., 2002; V¨olker et al., 1999). The disk-wind model of jet formation in protostars postulates that the source of the jet comes from a wide radial band of the accretion disk (K¨onigl and Pudritz, 2000). Since the rotational motion in the disk varies with distance from the central object the disk-wind driven jet velocity profile may also vary with distance from the central object. Axial and rotational sheared flows have been observed in astrophysical jets (Bally et al., 2002; Bacciotti et al., 2002, 2003, 2004; Coffey et al., 2004). In this paper we will focus on the stabilizing effects of sheared helical flow. Due to space limitations a brief summary of the evidence that sheared flow reduces instabilities in z-pinch plasmas will be presented and the possibility that sheared flow could be responsible for the sustained narrowness in YSO jets will be considered.

2 MHD model of an astrophysical jet

(in cylindrical coordinates) where v0φ and v0z are constants of proportionality and a is a small number to prevent the velocity from going to infinity at r = 0. The functional form of both velocity components is approximately Keplerian. In a frame that moves with the axial velocity at the center of the jet the axial plasma flow appears to be increasingly swept back with increasing radius. At the origin the axial velocity is zero and increases to a maximum at the edge of the jet. Both the azimuthal and axial components of velocity have Keplerian shear because the disk-wind is modeled as the source of plasma in the jet. This model is intended to simulate the asymptotic jet cylinder far from the surface of the accretion disk and termination region.

3 Stability criteria 3.1 Kelvin–Helmholtz instability The form of the Kelvin–Helmholtz instability (KHI) considered here occurs at the tangential boundary between the edge of the jet and the ambient medium. The KHI can be excited by the velocity discontinuity that exists at this boundary (Keppens et al., 2005). For astrophysical jets this instability will be excited if the velocity discontinuity is greater than the Alfv´en velocity (Nakamura and Meier, 2004). For the Keplerian velocity profile considered here the velocity at the outer edge of the jet is lower than the velocity at the center of the jet (relative to the ambient medium). Sub-Alfv´enic discontinuities are in general not subject to KHI so our analysis will focus on velocity profiles that meet this stability criterion. 3.2 Sausage instability

The MHD jet is modeled as perfectly conducting cylindrical plasma. The jet is then considered as a region in space where supersonic plasma and an electric current flow. The electric current I is balanced by a return current Ir of equal size. This return current can be modeled as a diffuse flow in the ambient medium or as a sheet on the jet surface (Lery and Frank, 2000). Jet models with return current have been referred to as “cocoon jets” (Appl and Camenzind, 1992; Lesch et al., 1989; Nakamura and Meier, 2004). MHD instabilities grow, but do not propagate in stationary z-pinch plasmas (Appl et al., 2000). The kink mode in a plasma column moving with uniform velocity would simply move with the flow and thus grow at the same rate as a stationary kink (Shumlak and Hartman, 1995). By transforming to a frame that moves with the jet, the jet plasma becomes analogous to a stationary cylindrical z-pinch plasma. The velocity of the jet in this frame is modeled with

where γ is the growth rate and R is a dimensionless parameter analogous to the Reynolds number (DeSouza-Machado et al., 2000). In this case γ ∼ vT /r0 where vT is the average thermal velocity and r0 is the radius of the plasma. If we let V ′ ∼ ∇V /r0 , where ∇V is the difference in the axial velocity between the center of the jet and the edge of the jet, the stability requirement is

√ √ √ v = ⌊0, v0φ / a + r , v0z (1/ a − 1/ a + r )⌋

∇V >

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(1)

The sausage instability is an interchange mode that can cause axisymmetric pinches or bulges to grow exponentially in the jet plasma. It has been shown analytically that the sausage instability can be stabilized in z-pinch plasmas with sheared axial flow V ′ meeting the following criteria  V ′ > γ ln (R),

(2)



(3)

ln (R) vT .

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Thus we see that ∇V must be above a threshold to prevent the sausage instability. 3.3 Kink instability A global stability analysis of the kink mode using nonrelativistic compressible ideal MHD with gravity neglected will demonstrate the stabilizing influence of sheared flow in current-carrying jets. The MHD equations are made dimensionless by normalizing the scales to the average radius of the jet in the asymptotic cylindrical region, the average thermal velocity of the jet plasma, the ambient pressure, and the ambient density. The linearized equations are solved numerically with a generic two-step predictor-corrector, second-order accurate space and time-centered advancement scheme (see Wanex et al., 2005b for details). The problem is treated by introducing perturbations into the plasma equilibrium state and following their linear development in time. All perturbed plasma variables (magnetic field, density, pressure, and velocity) have the form ξ (r )ei(kz z+mφ−ωt) . Initially the growth rates of the perturbed plasma variables are uncorrelated, however, after several growth times the solution converges to the fastest growing unstable mode. For this analysis the jet is considered to be in equilibrium with the surrounding medium across its boundary at r = r0 . The use of fixed boundary conditions allows a global stability analysis of internal unstable modes (Arber and Howell, 1995; Appl et al., 2000). It has been shown that sheared azimuthal flow is effective at reducing the growth rate of the kink instability in z-pinch plasmas but has little effect on axisymmetric modes (see Wanex et al., 2005b for a detailed explanation). Sheared axial flow is effective for stabilizing the sausage mode (DeSouzaMachado et al., 2000). This suggests that the growth of both the sausage and kink instabilities can be reduced by combining axial and azimuthal velocity components to produce helical sheared flow. For this reason sheared helical flow will be used in this analysis of the kink instability. Here we present the results of this analysis for two equilibrium profiles. The parabolic profile is obtained by using the magnetic field produced by a constant current density in the jet. We also present results for the constant electron velocity (Bennett) equilibrium profile with field maximum at 2r0 /3.

4 Results Figure 1 shows the results of the linear analysis on the kink instability for the √ parabolic equilibrium profile with azimuthal velocity 0.3/ 0.1 + r . The instability growth rates are reduced to zero for v0z > 1. Figure 2 shows the results for the Bennett equilibrium profile with the same velocity as in Fig. 1. The instability growth rates are reduced to zero for

Fig. 1 This is a 3D plot of the kink instability growth rates for the constant current density equilibrium profile with v0φ = 0.3 and a = 0.1. The growth rate is shown on the z-axis (in units of vT /r0 ), the wave number is shown of the y-axis (in units of 1/r0 ) and the value of the coefficient v0z on the x-axis (in units of vT ). As an example of how to interpret the plot consider the kink instability growth rate for v0z = 0.3, the growth rate for axial wave numbers 2 and 3 are zero, the growth rate then increases with increasing wave number to a maximum of ∼0.25 for wave numbers 8 and 9 and then decreases to zero for wave numbers above ∼14. Instability growth rates are zero for wave numbers 2–20 for v0z > 1

Fig. 2 This is a 3D plot of the kink instability growth rates for the constant electron velocity equilibrium profile with v0φ = 0.3 and a = 0.1. The growth rate is shown on the z-axis (in units of vT /r0 ), the wave number is shown on the y-axis (in units of 1/r0 ) and the value of the coefficient v0z on the x-axis (in units of vT ). As an example of how to interpret the plot consider the kink instability growth rate for v0z = 0.9, the growth rate for axial wave numbers 1 to 3 are zero, the growth rate then increases with increasing wave number to a maximum of ∼0.1 for wave number 5 and then decreases to zero for wave numbers above ∼7. Instability growth rates are zero for wave numbers 1–10 for v0z > 1.2

v0z > 1.2. This can be interpreted to mean that sheared helical flow can stabilize the kink instability for the parameters and profiles considered here if v0z > 1.2.

(4) Springer

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Using (1) and (3) one finds that √ ln (R) vT v0z > 2.2

(5)

is the stability criterion for the sausage instability (r = 1 and a = 0.1). Thus if (4) and (5) are satisfied the growth rates for both the sausage and kink instability can be reduced to zero for both of these examples. The Kelvin-Helmholtz stability condition can also be satisfied if the velocity at the edge of the jet is sub-Alfv´enic in a frame at rest with respect to the ambient medium. 5 Conclusion The results of this analysis suggest that the KelvinHelmholtz, sausage and kink instabilities in current carrying jets can be suppressed by Keplerian helical sheared flow for some equilibrium profiles. These results are sufficiently positive to motivate further analysis of the hypothesis that sheared helical flow can stabilize YSO jets. More work is required to extend the investigation to a larger range of parameters and equilibrium profiles. References Appl, S., Camenzind, M.: Astron. Astrophys. 256, 354 (1992) Appl, S., Lery, T., Baty, H.: Astron. Astrophys. 355, 818 (2000) Arber, T.D., Howell, D.F.: Phys. Plasmas 3, 554 (1995) Bacciotti, F., Ray, T.P., Mundt, R., Eisl¨offel, J., Solf, J.: ApJ 576, 222 (2002) Bacciotti, F., Ray, T.P., Eisl¨offel, J., Woitas, J., Solf, J., Mundt, R., Davis, C.J.: Astrophys. Space Sci. 287, 3 (2003) Bacciotti, F., Ray, T.P., Coffey, D., Eisl¨offel, J., Woitas, J.: Astrophys. Space Sci. 292, 651 (2004) Bally, J., Heathcote, S., Reipurth, B., Morse, J., Hartigan, P., Schwartz, R.: Astron. J. 123, 2627 (2002) Bateman, G., Schneider, W., Grossman, W.: Nucl. Fusion 14, 669 (1974) Bateman, G.: MHD Instabilities. The MIT Press, Cambridge (1978) Burrows, C.J., Stapelefeldt, K.R., Watson, A.M., et al.: ApJ 473, 437 (1996) Chandrasekhar, S.: Hydrodynamic and Hydromagnetic Stability. Clarendon Press, Oxford (1961)

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Coffey, D., Bacciotti, F., Woitas, J., Ray, T.P., Eisl¨offel, J.: Ap&SS 292, 553 (2004) Coker, R., Wilde, B., Keiter, P., et al.: Bull. Am. Phys. Soc. 49(33), 255 (2004) Coppins, M., Bond, D.J., Haines, M.G.: Phys. Fluids 27, 2886 (1984) DeSouza-Machado, S., Hassam, A.B., Sina, R.: Phys. Plasmas 7, 4632 (2000) ˇ Fendt, C., Cemelji´ c, M.: A&A 395, 1045 (2002) Freidberg, J.P.: Ideal Magnetohydrodynamics. Plenum, New York (1987) Golingo, R.P., Shumlak, U., Nelson, B.A.: Phys. Plasmas 12, 062505 (2005) Keppens, R., Baty, H., Casse, F.: Space Sci. Rev. 121, 65 (2005) K¨onigl, A., Pudritz, R.E.: In: Mannings V., Boss A.P., Russell S.S. (eds.) Protostars and Planets IV. The University of Arizona Press, Tucson (2000) Kruskal, M.D., Schwarzschild, M.: Proc. R. Soc. London Ser. A 223, 348 (1956) Lery, T., Frank, A.: ApJ 533, 897 (2000) Lesch, H., Appl, S., Camenzind, M.: A&A 225, 341 (1989) Mundt, R., Ray, T.P., Buhrke, T., Raga, Z.C., Solf, J.: A&A 232, 37 (1990) Nakamura, M., Meier, D.L.: AIP Conf. Proc. 703, 308 (2004) Ray, T.Muxlow, T.W.B., Axon, D.J., Brown, A., Corcoran, D., Dyson, J., Mundt, R.: Nature 385, 415 (1997) Ray, T.: Bull. Amer. Phys. Soc. 49, 70 (2004) Reipurth, B., Heathcote, S.: In: Reipurth B., Bertout C. (eds.) HerbigHaro Flows and the Birth of Low Mass Stars, p. 3. IAU Symposium 182 (1997) Ruden, E.: IEEE Trans. Plasma Sci. 30, 611 (2002) Shafranov, V.D.: At. Energ. 5, 38 (1954) Shu, F.H., Najita, J.R., Shang, H., Li, Z.: In: Mannings, V., Boss, A.P., Russell, S.S. (eds.), Protostars and Planets IV. The University of Arizona Press, Tucson (2000) Shumlak, U., Hartman, C.W.: Phys. Rev. Lett. 75, 3285 (1995) Sotnikov, V.I., Paraschiv, I., Makhin, V., Bauer, B.S., Leboeuf, J.N., Dawson, J.M.: Phys. Plasmas 9, 913 (2002) Sotnikov, V.I., Bauer, B.S., Leboeuf, J.N., Helinger, P., Tr´avniˇcek, P., Fiala, V.: Phys. Plasmas 11, 1897 (2004) Spruit, H.C., Foglizzo, T., Stehle, R.: Mon. Not. R. Astron. Soc. 288, 333 (1997) Vlemmings, W.H.T., Diamond, P.J., Imai, J.: Nature 440, 58 (2006) V¨olker, R., Smith, M.D., Suttner, G., Yorke, H.W.: A&A 343, 953 (1999) Wanex, L.F., Sotnikov, V.I., Bauer, B.S., Leboeuf, J.N.: Phys. Plasmas. 11, 1372 (2004) Wanex, L.F.: Astrophys. Space Sci. 298, 337 (2005a) Wanex, L.F., Sotnikov, V.I., Leboeuf, J.N.: Phys. Plasmas. 12, 042101 (2005b) Winterberg, F.: Beitr. Plasmaphys. 25, 117 (1985) Winterberg, F.: Z. Naturforsch. 54a, 459 (1999)

Astrophys Space Sci (2007) 307:87–91 DOI 10.1007/s10509-006-9222-9

O R I G I NA L A RT I C L E

How to Produce a Plasma Jet Using a Single and Low Energy Laser Beam Ph. Nicola¨ı · V. T. Tikhonchuk · A. Kasperczuk · T. Pisarczyk · S. Borodziuk · K. Rohlena · J. Ullschmied

Received: 10 April 2006 / Accepted: 18 July 2006 C Springer Science + Business Media B.V. 2006


Abstract Under suitable conditions on laser intensity, focal spot radius and atomic number a radiative jet was launched from a planar target. This jet was produced using a relatively low energy laser pulse, below 500 J and it presents similarities with astrophysical protostellar jets. It lasts more than 10 ns, extends over several millimeters, has velocity more than 500 km/s, the Mach number more than 10 and the density above 1018 cm−3 . The mechanism of jet formation was inferred from the dimensional analysis and hydrodynamic two-dimensional simulations. It is related to the radiative cooling while the magnetic fields play a minor role. Keywords Jets . Outflows . Laser-plasma PACS numbers: 98.38.Fs, 52.50.Jm, 95.30.Qd

1. Introduction The jets are ubiquitous in the Universe, from active galactic nuclei (Cecil et al., 1992; Bride and Perley, 1984) to Young

P. Nicola¨ı (
) · V. T. Tikhonchuk Centre Lasers Intenses et Applications, UMR 5107 CEA – CNRS – Universit´e Bordeaux 1, 33405 Talence cedex, France e-mail: [email protected] A. Kasperczuk · T. Pisarczyk · S. Borodziuk Institute of Plasma Physics and Laser Microfusion, ul. Hery 23, 00-908 Warsaw 49, Poland K. Rohlena Institute of Physics AS CR, Na Slovance 2, 182 21 Prague 8, Czech Republic J. Ullschmied Institute of Plasma Physics AS CR, Za Slovankou 3, 182 00 Prague 8, Czech Republic

Stellar Objects (Zinnecker et al., 1998; Reipurth et al., 1986) (YSO). The physics involved in jets formation is complicated and covers a large range of subjects. Consequently the numerical simulations require multidimensional codes accounting for hydrodynamics, ionization, radiation transport, equations of state and magnetic fields. The complexity of the phenomenon makes it challenging to devise laboratory experiments which are needed to benchmark the codes and to model certain aspects of large scale astrophysical phenomena. Recent experiments, carried out with Z-pinches (Lebedev et al., 2002; Ampleford et al., 2005) or with high energy lasers (Farley et al., 1999; Shigemori et al., 2000; Foster et al., 2002; Rosen et al., 2005) showed the interest and the relevance of the laboratory jets to some astrophysical jets. In these experiments, jets were produced from a radiative collapse of a convergent plasma flow or a shock wave, at a stagnation point. In terms of the experiment geometry, this method of jet production requires a high energy and/or a multiple laser beams. The present study addresses the problem of jet formation using a single and low energy laser beam. The experiments were carried out at the Prague Asterix Laser System (PALS) iodine laser facility (Jungwirth et al., 2001). The experimental images showed a jet formation under certain conditions (Borodziuk et al., 2004; Kasperczuk et al., 2006). Although the radiative effects are important, the jet creation is neither induced by plasma collision nor by shock convergence. The numerical simulations carried out with a multi-physics, two-dimensional (2D) radiative magneto-hydrodynamic code (Buresi et al., 1986; Drevet, 1997; Nicola¨i et al., 2000). It was found out that the main mechanism which transforms the ablated plasma in a plasma jet is the radiative cooling of expanding plasma. Under appropriate conditions on the laser energy and the focal spot radius, the jet is formed having a Mach number larger than Springer

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10, temperatures around 50 eV and electron densities above 1018 cm−3 . Moreover, these jets verify the scaling criteria, detailed by Ryutov et al. (1999), of the laser experiment to astrophysical conditions. 2. Experiment The experiment was performed at the PALS iodine laser facility (Jungwirth et al., 2001; Borodziuk et al., 2004; Kasperczuk et al., 2006). A laser beam was focused on a flat massive target by an aspherical lens at the first (1.315 µm) and the third (0.438 µm) harmonic. The essential part of experiments was carried out with the laser energy on target E L = 100 J for both harmonics. The laser pulse duration of ∼ 0.4 ns was kept constant in all shots. To study the expansion of ablated plasma, a three-frame interferometric system with an automatic image processing was employed. The diagnostic system was illuminated by the third harmonic of the iodine laser. Each of the interferometric channels was equipped with its own independent interferometer of the folding wave type with a 250 mm focal length and a wedge angle of 3 degrees. The images were recorded by high resolution 10 bits dynamics Pulnix TM-1300 CCD cameras, with the matrix of 1300 × 1300 pixels. The temporal resolution of the interferometer was determined by the probe pulse duration, which, in the case of the third harmonic was approximately 0.3 ns full width at half maximum (FWHM). The spatial resolution of 20 µm was deduced from the target edge washers. The delay between subsequent frames was set to 3 ns. A typical sequence of electron density distributions is presented in Fig. 1. The first three lines correspond to three materials: aluminium, copper and silver. In all cases, a plasma jet is present and it remains visible during a long time. Distinctly, the jet becomes narrower as the atomic number increases. This observation indicates that radiative processes play an important role in the experiment. The last sequence shows the effect of a laser focal spot radius change and a laser wave length decrease. It was shown in Ref (Borodziuk et al., 2004; Kasperczuk et al., 2006) that a transition to shorter wavelength improves the jet parameters. Also the jet formation depends strongly on the focal spot size. As the spot radius decreases, but the laser energy is kept constant, the ablated plasma ejection becomes quasi-isotropic, whereas, as the spot radius increases, the plasma extends along the laser axis. Last, for this radius, 100 µm, a complex multi-bubble structure can be seen in the figures (pointed out with arrows). It corresponds to an annular plasma ejection at angles ∼40– 45◦ . This plasma ejection also appears in the simulation. 3. Interpretation of the jet formation The simulations of the plasma dynamics were performed with the laser-plasma interaction hydrodynamic code Springer

Fig. 1 Experimental sequence of the electron density isolines at instants of 2, 5, 8 ns for R L = 300 µm, E L = 100 J and the wavelength of 0.438 µm for three target materials: Al (a), Cu (b) and Ag (c). For R L = 100 µm and Al target (d). The densities are in 1018 cm−3

FCI2 (Buresi et al., 1986; Drevet, 1997). This code is currently used to simulate laser experiments. It has been compared with other codes (Lindl et al., 2004) and successfully reproduced experiments (Nicola¨i et al., 2000). The code includes two-dimensional hydrodynamics, ion and classical or nonlocal electron conduction, thermal coupling and detailed radiation transport. Ionization, equations of state and opacity data are tabulated, assuming a local thermodynamic equilibrium (LTE) or a non-LTE depending on the plasma parameters. The laser propagation, refraction and collisional absorption are treated by a ray tracing algorithm. A resistive MHD package accounting for the azimuthal magnetic fields generated by the thermal sources (crossed gradients of the density and temperature) was also included. Our numerical study started with the following parameters: the pulse duration was τ L = 0.4 ns with the temporal shape I (t) ∼ sin2 (π t/2τ L ) and the laser energy E L = 100 J, which leads to the maximum laser power PL = 0.2 TW. For a given spot radius R L = 300 µm, the maximum laser intensity was I ≃ 8.8 × 1013 W/cm2 . The laser beam was focused on the surface of a massive copper solid target. The intensity profile on the target was not well-characterized and we approximate it by a super-Gaussian function I (r ) ∼ exp[−(r/R L )n ] with n = 8.

Astrophys Space Sci (2007) 307:87–91

In order to identify the physical origin of the jet formation, we performed several runs where certain physical effects were artificially switched off. The first simulation was carried under the following conditions: (i) without magnetic fields, (ii) without the radiation transport and (iii) using a flux limiter equal to 8% for the electron heat conduction. Figure 2a presents iso-densities, at 5 ns, well after the end of the laser pulse. This simulation obviously does not reproduce the experimental result. The plasma flow is too broad. This result has to be compared with those shown in Fig. 2b. In this simulation, the radiation transport is turned on. The results are much closer to the experiment. Initially spherical, the plasma plume is elongated after a few ns. Its radius is twothree times smaller than in the previous case and the height is about 3 mm is also in agreement with the experiment. Due to the radiative cooling, the plasma temperature decreases 2.5 times to the value around 50 eV. The electron heat flux, which scales as T 7/2 , induces, if the radiation transport is switched off, a more homogeneous density distribution. The intensity of the X-ray emission is proportional to the square of the plasma density, it cools more efficiently the dense plasma and reduces the internal plasma pressure. Comparing both simulations, we can see that the radiation cooling reduces the pressure by a factor 30. Consequently, with the radiation transport, the characteristic expansion velocities are smaller, specifically in the radial direction. This latter simulation is considered as the reference. All the following simulations were performed with the radiation transport switched on. The magnetic field generation and the nonlocal heat conduction are the competing processes that may operate under the present conditions and could be partly responsible of the jet formation. Indeed, the density gradients are rather steep and non-collinear with the temperature gradients. Therefore the nonlocal electron heat conduction (Luciani et al., 1983; Schurtz et al., 2000) and self-generated magnetic fields (Nicola¨i et al., 2000; Glenzer et al., 1999) could be operational. To check on these effects, we analyzed the temporal behavior of the magnetic field. It reaches a maximum, about 1 MG, during the pulse duration and then it decreases quickly after the pulse end. Its maximum at t = 5 ns is only 8.6 kG and its effect on the plasma motion is completely negligible (Fig. 2c). In order to address the effect of a tighter laser focusing, for the same energy and pulse duration, the focal spot size was reduced three times (R L = 100 µm), so the laser intensity was nine times higher. The smaller energy deposition volume leads to a higher temperature, a smaller absorption and a higher plasma pressure. The expansion from a smaller focal spot size is faster and the plasma expansion becomes more spherical (Fig. 2d). In addition, the density falls below the resolution limit of a few 1017 cm−3 at the distance less than 1 mm. Note that one observes as in the experiment, a ringlike structure in the direction of about 45 degrees (marked

89

(cm)

0.1

(a)

(b)

0.0 –0.1

43 2 1

654 3 21

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1 : 0.90 2 : 1.49 3 : 2.46 4 : 4.06 5 : 6.71 6 : 11.1 7 : 18.3 8 : 30.3

0.1 0.2 0.3 0.4 (cm)

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(f)

(cm)

0.1 65

0.0 4

3 2 1

–0.1

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Fig. 2 Electron density distributions at t = 5 ns, for E L = 100 J, τ L ∼ 400 ps, λ L ∼ 0.438 µm and R L = 300 µm. Without the radiation transport (a), the reference simulation (b), with the magnetic fields (c), with a smaller laser spot radius (d), with a different energy distribution in the the focal spot (e) and with an aluminium target (f). See text for details. The densities are in 1018 cm−3

by an arrow). This is due to collision of a hot central plasma with a cold material at the crater border. The laser energy distribution in the focal spot has an impact on the jet formation. For comparison, we present in Fig. 2e, the density profile obtained using a gaussian shape (n = 2). The jet still appears, but an energy distribution change modifies the form of the jet. Last, in order to test the target atomic number (Z) dependence, we replaced the copper by aluminium in the simulation (Fig. 2f). As expected, the jet becomes wider and more isotropic, which is consistent with a less efficient radiation cooling. The computational results can be confirmed from a dimensional analysis. The characteristic hydrodynamic time depends on the focal spot radius R L and on the ion acoustic velocity cs = ((Z + 1)T /m i )1/2 . th = R L /cs = 0.1 R L (A/Z )1/2 T −1/2 ns,

(1)

where A is the atomic mass in units of proton mass, R L is in µm and T in eV. The radiative cooling time is the ratio of the plasma thermal energy and radiated power. For simplicity here we neglect the line emission which is not too important in light materials. Then the power of the bremsstrahlung emission is given by (Book, 1980): Pbr = 1.7 × 10−32 Z n 2e T 1/2 W/cm3 , where the electron plasma Springer

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density is in cm−3 . The plasma energy density is defined as E p = 1.5 n e T = 2.4 × 10−19 n e T J/cm3 . Consequently, the radiation cooling time reads tr = E p /Pbr = 1.4 × 1022 T 1/2 /Z n e ns.

(2)

For the typical parameters of this experiment: R L = 300 µm, A = 63, Z = 15, T = 100 eV, and n e = 1022 cm−3 (between the critical density and the ablation front, that is, around the launching zone), we find the hydrodynamic time ∼ 6 ns and the radiation time ∼ 0.9 ns. The radiative cooling makes an important effect in agreement with the observations and the simulations. In the contrary, for the tighter focusing R L = 100 µm, the temperature is higher T ∼ 300 eV (Z = 20) and consequently the hydrodynamic time is shorter ∼ 1 ns, while the radiation time becomes longer ∼ 1.2 ns. In that case the radiative losses are less important and the plasma expansion is more symmetric. However, one should keep in mind that these estimates are rather qualitative. From one hand, we certainly underestimate the radiative losses by neglecting the line emission. From the other hand, the hydrodynamic motion leading to the jet formation is in reality two-dimensional. Equations (1) and (2) just point out the main qualitative dependencies of the jet formation mechanism. The jet is better formed if the the spot radius and the atomic number increase, and if the temperature and the laser wavelength decrease. In order to present interest in the astrophysical context, the laboratory plasma jet needs to have the same dimensionless parameters as the astrophysical jets. In addition, independently of the objet size and its time evolution, the hydrodynamics is scalable if the dissipative processes are negligible. These processes may be expressed in terms of dimensionless parameters: the Peclet and the Reynolds numbers (Ryutov et al., 1999). The first one mesures the convective transport relative to the conduction: Pe = Ujet Rjet /κ, where Ujet and Rjet are the jet characteristic velocity and radius. The thermal diffusivity reads 21

κ = 2 × 10 T

5/2

2

/(Z + 1)n e cm /s,

(3)

where  is the Coulomb logarithm. The Reynolds number defines the ratio of inertial force to viscous force: Re = Ujet Rjet /ν, where the kinematic viscosity reads: ν = 2 × 1019 T 5/2 A−1/2 Z −3 /n e cm2 /s.

(4)

Experiments in which Pe ≫ 1 and Re ≫ 1 could be scaled to the astrophysical system. In laser experiment, the jet typical parameters are T ∼ 50 eV, n e ∼ 5 × 1018 cm−3 , Ujet ∼ 500 km/s, Rjet ∼ 0.5 mm, Z ∼ 10 and  ∼ 5. Under these conditions one obtains κ ∼ 105 cm2 /s, ν ∼ 2 cm2 /s Springer

and so Pe ∼ 20 and Re ∼ 106 . Therefore, our experiment satisfies the conditions for the hydrodynamic scaling with nevertheless a particular attention for the thermal conduction.

4. Conclusions We have studied the formation of a plasma jet using a single laser beam and a simple planar massive target. The experiment and the simulations indicate that this jet may be launched using a relatively low laser energy. Under these conditions, the dissipative processes can be neglected and the laser produced jet can be scaled to astrophysical conditions. Moreover, such a jet is rather flexible and could be used for modeling of interaction between the astrophysical jet and the ambient clouds. For that, one can place in front of the jet, a solid foil, a foam or a gas jet (Lebedev et al., 2002; Ampleford et al., 2005; Foster et al., 2002; Rosen et al., 2005), by changing the angle of the laser beam incidence by a few tens of degrees. In addition, by modifying the pulse duration and intensity, one can modify the velocity and the density of the jet. Always using a single laser beam, one could build a series of pulses with an increasing intensity. Each pulse induces a jet faster than the previous one. A correct timing of such a pulse sequence should create a series of plasma jets interacting one with another. Acknowledgements This work is partly supported by the Aquitaine Region Council, by the Association EURATOM-IPPLM (contract No FU06-CT-2004-00081), by the Ministry of Scientific Research and Information Technology in Poland (project No 3 T10B 024 273), and by the Ministry of Schools, Youth and Sports of the Czech Republic (project No LC528).

References Ampleford, D., Lebedev, S.V., Ciardi, A., Bland, S.N., et al.: Astrophys. Space Sci. 298, 241 (2005) Book, D.L.: NRL Plasma Formulary, (Washington D.C., Naval Research Laboratory) p. 52 (1980) Borodziuk, S., Kasperczuk, A., Pisarczyk, T., Gus’kov, S. et al.: Opt. Appl. 34, 31 (2004) Bride, A.H., Perley, R.: Annu. Rev. Astron. Astrophys. 22, 319 (1984) Buresi, E., Coutant, J., Dautray, R., et al.: Laser Part. Beams 4, 531 (1986) Cecil, G., Wilson, A.S., Tully, R.B.: Astrophys. J. 390, 365 (1992) Drevet, C.: In: Proceedings of the 16th International Conference Fusion Energy, Montreal 1996, IAEA, Vienna, vol.3, p. 57 (1997) Farley, D., Estabrook, K.G., Glendinning, S.G. Glenzer, S.H., et al.: Phys. Rev. Lett. 83, 1982 (1999) Foster, J.M., Wilde, B.H., Rosen, P.A., Perry, T.S.: Phys. Plasmas 9, 2251 (2002) Glenzer, S., Alley, W., Estabrook, K., DeGroot, J. et al.: Phys. Plasmas 6, 2117 (1999)

Astrophys Space Sci (2007) 307:87–91 Jungwirth, K., Cejnarova, A., Juha, L., Kralikova, B. et al.: Phys. Plasmas 8, 2495 (2001) Kasperczuk, A., Pisarczyk, T., Borodziuk, S., Ullschmied, J. et al.: Phys. Plasmas 13, 062704 (2006) Lebedev, S.V., Chittenden, J.P., Beg, F.N., et al.: Astrophys. J. 564, 113 (2002) Lindl, J.D., Amendt, P., Berger, R.L., Gendeinning, S.G. et al.: Phys. Plasmas 11, 339 (2004) Luciani, J.-F., Mora, P., Virmont, J.: Phys. Rev. Lett. 51, 1664 (1983) Nicola¨ı, Ph., Vandenboomgaerde, M., Canaud, B., Chaigneau, F.: Plys. Plasmas 7, 4250 (2000)

91 Reipurth, B., Bally, J., Graham, J., Lane, A., et al.: Astron. Astrophys. 164, 51 (1986) Rosen, P.A., Wilde, B.H., Williams, R.J., et al.: Astrophys. Space Sci. 298, 121 (2005) Ryutov, D., Drake, R.P., Kane, J., Liang, E. et al.: Astron. Astrophys. 518, 821 (1999). Schurtz, G., Nicola¨i, Ph., Busquet, M.: Phys. Plasmas 7, 4238 (2000) and reference therein Shigemori, K., Kodama, R., Farley, D.R., et al., Phys. Rev. E.: 62, 8838 (2000) Zinnecker, H., McCaughrean, M., Rayner, J.: Nature 394, 862 (1998)

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Astrophys Space Sci (2007) 307:93–98 DOI 10.1007/s10509-006-9286-6

O R I G I NA L A RT I C L E

Experimental Design for the Laboratory Simulation of Magnetized Astrophysical Jets Radu Presura · Stephan Neff · Lucas Wanex

Received: 22 April 2006 / Accepted: 28 November 2006 C Springer Science + Business Media B.V. 2006


Abstract Concepts of several experimental configurations for the investigation of magnetized jets and their interaction with magnetized environments are presented. In the planned experiments, the plasma jets will be created by laser ablation of shaped targets, while magnetic and electric fields with the required configurations will be produced independently by a pulsed power generator. In particular, the recently coupled Terawatt laser Tomcat and Terawatt pulsed power generator Zebra will be used for experiments. Keywords Laboratory astrophysics . Astrophysical jets . Ablation plasma

1 Introduction Highly-collimated, oppositely directed jets are observed to originate from a variety of astrophysical systems at scales differing by many orders of magnitude, from young stellar objects (YSO; e.g. Reipurth and Bally, 2001; de Gouveia Dal Pino, 2005), to neutron stars and black holes (e.g. Mirabel and Rodriguez, 1998, 1999), to active galactic nuclei (AGN; e.g. de Gouveia Dal Pino, 2005; Livio, 2002; Massaglia, 2003). Despite similarities in their appearance, the observed and inferred properties of these jets are very different (see the first two lines in Table 1). For example, the AGN jets are relativistic, less dense than the environment, and emit nonthermal continuum radiation in a broad spectrum. The YSO jets are denser than the ambient medium and the line radiation behind the shock regions produces strong cooling that influences the morphology and dynamics of the jets (Blondin R. Presura (
) · S. Neff · L. Wanex Nevada Terawatt Facility, University of Nevada, Reno e-mail: [email protected]

et al., 1990). However, both types of jets form when material from an accretion disk falls onto a central object in the equatorial plane and is ejected at supersonic speeds along the rotation axis. Most likely, both are accelerated and collimated by mechanisms in which magnetic fields play dominant roles (Blandford and Payne, 1982; Shu et al., 1994; Lovelace et al., 1987). Such fields have been measured or inferred from recent observations for jets with different origins (Ray et al., 1997; Zavala and Taylor, 2005). Nevertheless, open questions remain, mainly regarding the formation of jets; their propagation in different environments with the related effects on stability, matter entrainment, and turbulence; and their termination to form Herbig-Haro objects, hot-spots, and radio lobes. To help address such issues, the laboratory simulation of astrophysical jets emerged as a very promising complementary investigation tool. Well-controlled laboratory experiments with scaled-down plasma jets can provide additional information, valuable for identifying the essential physical mechanisms, deciding between different models, and benchmarking simulation codes. The focus of this paper is on the design of experiments to investigate the properties of magnetized plasma jets and their interaction with the environment. Such experiments at the Nevada Terawatt Facility (NTF), using two independent Terawatt (TW) sources of plasma and magnetic field, can explore parameter ranges that become accessible through the independent control of the plasma and field parameters. Considering the vast difference in parameters between astrophysical and laboratory plasma jets (Table 1), it is obviously impossible to achieve a complete scaling down, that is to design an experiment which is identical in all essential dimensionless parameters with the natural system. Instead, the jet dynamics and the role of dissipative processes are adequately simulated if the dimensionless plasma parameters of the order of unity in the astrophysical system are also of Springer

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Table 1 Typical parameters of plasma jets originating from active galactic nuclei (AGN) and young stellar objects (YSO), and possible to produce in the laboratory: from the front and back of a laser irradiated target and from a previous experiment (Presura et al., 2005). Physical parameters of the jets are compared in the first five columns: size, velocity, electron density, electron temperature, magnetic field. In Flow AGN YSO Front Back Presura et al. (2005)

L (cm) 18

v (cm/s) 24

10 –10 1016 – 1019 0.2 0.2 0.02

10

10 1–5×107 2–6×107 ∼107 3×107

n e (cm−3 ) NA 10 – 105 1018 –1019 ∼1020 1018

Te (eV) NA 10−3 – 1 1–3×102 ∼50 150

the order of unity in the laboratory, and those that are much larger (smaller) than one in nature are likewise much larger (smaller) than one in the experiment (Ryutov et al., 2001, 1999; Baranov, 1969). For example, a magnetic Reynolds number of the order of 10 in the laboratory may have the same physical significance as a magnetic Reynolds number many orders of magnitude larger has for an astrophysical jet, if the magnetic field diffusion is negligible over the time scale of interest, which is many orders of magnitude shorter in the laboratory. Once such “scaling” can be justified, the small scale laboratory simulations of astrophysical phenomena become relevant. For magnetized plasma jets, one additional layer of difficulty results from the fact that when the plasma temperature T increases, the hydrodynamic Reynolds number and the Peclet number decrease (∝ T −1 for magnetized electrons) while the magnetic Reynolds number increases (∝ T 3/2 ). In this case, the particular physical aspect investigated determines which dimensionless parameters have to be maximized. Overcoming scaling difficulties, several well controlled experiments produced valuable results that boosted the confidence of the astrophysical community in laboratory-based astrophysics. These experiments investigated a variety of topics including: the effect of radiative cooling on the collimated propagation of supersonic plasma jets produced with intense lasers from shaped targets (Farley et al., 1999; Shigemori et al., 2000) and with conical-array z-pinch experiments (Lebedev et al., 2002); the hydrodynamic stability and the termination shock of a laser-produced supersonic plasma jet launched in a dense ambient medium (Foster et al., 2002); the supersonic jet deflection and internal shocks produced by ambient plasma winds (Lebedev et al., 2004); the stabilizing effect of an axial magnetic field upon the kink instability of a plasma jet produced in a coaxial gun (Hsu and Bellan, 2003); the generation of rotating jets with twisted wire arrays (Ampleford et al., this issue; Lebedev et al., 2005a); the formation of magnetic tower outflows from a radial wire array z-pinch (Lebedev et al., 2005b).

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the next six columns, the comparison includes dimensionless parameters: the Mach number, the Reynolds number, the Peclet number, the density contrast, the plasma β, and the magnetic Reynolds number. The numbers in parentheses are inferred from modeling. The Mach number for the laboratory jet was estimated without taking into account the effect of radiative cooling B (G) −3

M −5

10 –10 10−4 ∼5×105 – 8

(10–100) 10 – 50 ∼10 ∼4 10

Re – 1013 ∼1000 1500 40

Pe – 1011 ∼10 1000 0.1

η −2

(<10 ) 1–20 10 100 10

β

Re M

(100) 1–10 1 – 1

– 1015 ∼100 – 30

At the NTF, the role played by magnetic fields in the acceleration, collimation, and interaction of jets with the ambient medium is of primary interest. Two prominent jet theories, the magnetocentrifugal model (Blandford and Payne, 1982; Shu et al., 1994), and the Poynting-flux-dominated model (Lovelace et al., 1987) require the astrophysical jets to carry current and to be collimated by helical magnetic fields. This configuration is subject to magnetohydrodynamic (MHD) instabilities. In fact, the sausage m = 0 and kink m = 1 instabilities have been invoked to explain the observation of knots, wiggles, and filamentary structures in astrophysical jets (Pearson, 1996; Reipurth and Heathcote, 1997). If such instabilities develop indeed, they should eventually disrupt the jet collimation. On the other hand, axial magnetic fields or radial velocity shears could prevent the disruption (Hsu and Bellan, 2003; Wanex et al., 2005). Jets from young stellar objects are most likely threaded by axial magnetic field (K¨onigl and Pudritz, 2000). Both axial and rotational sheared flows have been observed in astrophysical jets (Bacciotti et al., 2004; Bally et al., 2002; Coffey et al., 2004). In addition, stellar evolution models predict that low-mass protostars form in an interstellar medium that is supported by a magnetic field (Ray et al., 1997; K¨onigl and Pudritz, 2000). This magnetized medium could supply the instability reducing mechanism in current carrying jets. However, the velocity shear layer formed at the interface between the jet and the ambient creates conditions favorable for the growth of the KelvinHelmholtz instability. Three-dimensional MHD simulations (Ryu et al., 2000) showed that the magnetic fields embedded in the plasma flow affect the instability differently depending on their strength, indicated by the Alfv´en-Mach number. When M A ≥ 50, the instability remains hydrodynamic in nature, but the dissipation is enhanced through magnetic reconnection; when 4 ≤ M A ≤ 50, the magnetic field is amplified in the Cat’s Eye, leading to the reorganization of the flow into a stable configuration; when 2 ≤ M A ≤ 4, the magnetic field tension, enhanced during the linear growth, prevents the transition to the nonlinear phase, so the flow is nonlinearly

Astrophys Space Sci (2007) 307:93–98

95

Table 2 Facilities operational and in different stages of development at the Nevada Terawatt Facility. For Zebra as magnetic field generator, the values given are the magnetic energy in the experimental region and

the duration over which the field varies with less than 5%, around the current maximum

Device

Implementation

E (J)

τ (ps)

I (W/cm2 )

B (T)

Operational

Zebra pulsed power generator Tomcat laser

Wire array B (Presura et al., 2005) B (Martinez et al., this issue) Q-switch (Presura et al., 2005) Long pulse Short pulse Long pulse Short pulse

5×104 103 103 5 15 10 200 35

105 6×104 6×104 5×103 8×102 1 103 0.35

1011

>250 20 200

1999 2004 2005 2004 2005 2006 2007 2007

Leopard laser

stable; when M A ≤ 2, the instability is not initiated, so the MHD flow is linearly stable.

2 Experimental facilities at the NTF Magnetized plasma jet experiments will be performed at the Nevada Terawatt Facility (NTF) using the facilities operational and in various stages of development listed in Table 2 together with their main parameters. The synergistic combination of these devices allows independent control of the plasma flow and magnetic field parameters and thus offers access to parameter ranges and experimental configurations unattainable otherwise. A previous experiment (Presura et al., 2005) based on the Tomcat Q-switch laser (Table 2) coupled with the Zebra generator (Table 2, line 2) focused on the interaction of an explosive ablation plasma plume with ambient magnetic field, with relevance to active magnetospheric experiments and to the expansion of supernova remnants in magnetized interstellar medium. The typical values of the parameters in the plasma-field interaction region are included in the last line of Table 1. The 10 TW short pulse laser Tomcat (Wiewior, to be published) and the 2 TW z-pinch generator Zebra (Bauer et al., 1997) were recently coupled. Examples of plasma jet parameters achievable with the coupled devices are given in Table 1. The plasma flow parameters are based on simulations with the 1-dimensional hydrodynamic code MULTI (Ramis et al., 1988) for the front/back irradiation of CH thick/thin targets. The effects of a magnetic field B ≈ 20T were estimated analytically. The values of the dimensionless parameters indicate that relevant regimes can be attained. These facilities are complemented by a mature suite of laser diagnostics: shadowgraphy, schlieren imaging, interferometry, and Faraday rotation. They provide information regarding the spatial distribution of plasma density and magnetic field. Spectroscopy in several spectral ranges and Thomson scattering are under development for plasma temperature estimates and additional density and field measurements.

1014 3×1015 1018 3×1016 1019

3 Plasma jet experiments Although previous experiments demonstrated that both pulsed power generators and lasers can generate plasma jet conditions relevant to astrophysical situations, the focus here will be on laser produced jets and their interaction with magnetic and electric fields or magnetized ambient plasma produced by current carrying conductors. Conical targets are especially suited to create jets by laser ablation (Farley et al., 1999; Shigemori et al., 2000). The ablated plasma initially expands supersonically perpendicular to the concave surface of the target. Due to the symmetry of the target, this creates a conical shock which refracts the converging flow to form a supersonic jet propagating along the symmetry axis (Tenorio-Tagle et al., 1988). The velocity shear layer formed at the interface between protostellar jets and the ambient creates conditions favorable for the growth of the Kelvin-Helmholtz instability. However, an axial magnetic field embedded in the plasma flow can provide stabilization mechanisms (Ryu et al., 2000). An experimental set-up for the study of the effect of a magnetic field upon the Kelvin-Helmholtz instability is illustrated in Fig. 2. A plasma jet created by ablation of a conical target is directed towards a region where ambient plasma and magnetic field can be generated independently. For typical jet density n e ≥ 1018 cm−3 and ambient density n e ≥ 1017 cm−3 , the density contrast η = ρjet /ρambient ≈ 10, similar to that observed for protostellar jets. Magnetized ambient plasma can be created by ablation or desorption from a current carrying conductor. Non-magnetized ambient plasma can be generated by laser ablation. The magnetic fields can be generated with coils driven by the Zebra pulsed power generator. Conductor ablation was observed for helical coils, at fields around 200 T (Fig. 1a,b). Magnetic flux densities up to 100 T were obtained with horse shoe coils with no conductor ablation (Fig. 1c) (Martinez et al., this issue). In these experiments the stability of the plasma jet will be monitored with time gated laser shadow imaging. The jet density and magnetic field in the boundary layer will be determined with time gated

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Fig. 1 Laser shadow images: (a) helical coil (reference); (b) helical coil (at 150 T); (c) horse shoe coil (at 100 T). All coils are made of 1 mm thick stainless steel wire

Fig. 2 Plasma jet produced by laser ablation of a conical target, propagating along a uniform external magnetic field. An ambient plasma, magnetized or not can be created at the location of the coil

interferometry and Faraday rotation. Faraday rotation in glass probes (Martinez et al., this issue) will be used to monitor the magnetic field produced by the coil. Using independent plasma and field sources (Fig. 2) allows the injection of the laser generated jet at an arbitrary angle with respect to the ambient magnetic field. This general configuration allows the investigation of termination shocks and of plasma flow penetration across the magnetic field. In such experiments, the plasma density and temperature, and the magnetic field distributions are the relevant parameters. They can be determined with time gated laser imaging, short wavelength imaging, and Faraday rotation, respectively. Astrophysical jet models (Blandford and Payne, 1982; Shu et al., 1994; Lovelace et al., 1987) assume that the jets carry current. As a result, they are expected to be MHD unstable and, in fact, the sausage m = 0 and kink m = 1 instabilities might explain the structures observed in astrophysical jets (Pearson, 1996; Reipurth and Heathcote, 1997). To investigate in the laboratory the effect of possible stabilizing mechanisms such as axial magnetic fields and axial or azimuthal Springer

sheared flows, additional experimental configurations are considered. These rely on plasma flows created by laser ablation in external electric or magnetic fields produced with the pulsed power generator. In this case, instead of a shaped target, a shaping environment is used for the jet formation. For example, a ring focus on an insulating target surrounding the high voltage electrode will produce a plasma plume threaded by a radial electric field Er (Fig. 3a). This will induce into the axial plasma flow with velocity vz an azimuthal magnetic field Bϕ = Er /vz . For example, embedding a 25 T magnetic field in a plasma flow with velocity around 400 km/s requires an electric field around 100 kV/cm, which is below the threshold for field electron emission, about 300 kV/cm. Applying an additional axial magnetic field Bz will produce jet rotation with vϕ = Er /Bz . In a variation of this set-up (Fig. 3b), the plasma is produced by laser ablation of a pin tip, so the expansion, guided by the radial electric field, bridges the inter-electrode gap. When a radial current is established through the plasma, the system behaves like a plasma focus or magnetic tower (Lebedev et al., 2005b) under the action of a force with density f z = jr Bϕ , producing a dense current-carrying column and a large radius cocoon-like diffuse current return plasma. An additional axial magnetic field Bz in this set-up, such that Bz ≥ K µ0 Iz /2πr , is expected to have a stabilizing effect upon the current-carrying jet. Here K is a constant dependent on the instability mode, and the rest of the right hand side is the azimuthal magnetic field. A configuration of wires thick enough to delay ablation can be used in z-pinch configuration to create an ambient magnetic field. For instance, the Joule heating of 8 wires with 1 mm diameter each is insufficient to reach the melting temperature before the Zebra current peak. Such a twisted conical array (Fig. 3c) will produce an axial magnetic field, similar to that inferred in previous experiments (Ampleford et al., this issue; Lebedev et al., 2005a), parallel to the current flowing in the plasma jet. Other quasi-force-free configurations

Astrophys Space Sci (2007) 307:93–98

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Fig. 3 Possible experimental arrangements for the generation of magnetized and current carrying jets using laser ablation in strong external fields. The coaxial inner (brown) and outer (grey) cylinders are the electrodes of the pulsed power generator. The plume on the top of the center electrode represents the laser produced plasma. The straight lines represent electric field lines in (a) and (b), and conductors in (c) and (d)

(Furth et al., 1988), including twisted cylindrical wire arrays and helical wire arrays, in which the current and the magnetic field vector are parallel, can be used to investigate the stabilizing effect of the magnetic field. The set-up presented in Fig. 3d, with a regular conical wire array, can be used for null results, without axial magnetic field. In actual experiments, to diagnose the jet formation phase, the outer electrode, represented as a cylinder in Fig. 3c and 3d, consists of individual current return rods, to allow diagnostics access. The jet stability and interaction during its propagation is investigated above the electrode structure. The jet stability will be monitored with laser and short-wavelength imaging and the magnetic field configuration with Faraday rotation. The radiative cooling of jets by optically thin radiation emission has a significant effect on the dynamics and morphology of supersonic flows such as the protostellar jets. This effect was investigated through simulations (Blondin et al., 1990) and was evidenced in laboratory experiments with lasers (Farley et al., 1999; Shigemori et al., 2000) and z-pinch (Lebedev et al., 2002). These papers show that the radiative cooling is stronger for higher atomic numbers. This dependence can be exploited to vary the Mach number and the density contrast in the experiment. The effect can be evidenced by measuring the density distribution with laser imaging and by imaging the self-emission in soft x-rays with multi-frame instruments.

4 Conclusions The stability and collimation of magnetized jets and their interaction with magnetized ambient plasma will be investigated experimentally at the NTF. The experiments will take advantage of the existing coupled TW laser and TW pulsed power generator, which allow the independent variation of the plasma flow and magnetic field parameters. A variety of experimental configurations were identified that address significant aspects of the physics of jets and are complementary to other current efforts. Acknowledgements The authors thank V. Ivanov and P. Laca for the help with the laser diagnostics. The authors are thankful for very useful discussions of some of these concepts with A. Frank, D. Ampleford, S. Lebedev, B. Remington, and R. P. Drake. This work was supported by DOE/NNSA under the UNR grant DE-FC52–01NV14050.

References Ampleford, D.J. et al.: Astrophys. Space Sci. (this issue) Bacciotti, F., Ray, T.P., Coffey, D. et al.: Astrophys. Space Sci. 292, 651 (2004) Bally, J., Heathcote, S., Reipurth, B. et al.: Astron. J. 123, 2627 (2002) Baranov, V.B.: Cosmic Res. 7, 98 (1969) Bauer, B.S., Kantsyrev, V.L., Winterberg, F., et al.: AIP Conf. Proc. 409, 153 (1997) Blandford, R.D., Payne, D.G.: MNRAS 199, 883 (1982)

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Blondin, J.M., Fryxell, B.A., Konigl, A.: Astrophys. J. 360, 370 (1990) Ciardi, A., Lebedev, S.V., Chittenden, J.P., et al.: Laser Part. Beams 20, 255 (2002) Coffey, D., Bacciotti, F., Woitas, J., et al.: Astrophys. J. 604, 758 (2004) de Gouveia Dal Pino, E M.: Adv. Space Res. 35, 908 (2005) Farley, D.R., Estabrook, K.G., Glendinning, S.G., et al.: Phys. Rev. Lett. 83, 1982 (1999) Foster, J.M., Wilde, B.H., Rosen, P.A., et al.: Phys. Plasmas 9, 2251 (2002) Furth, H.P., Jardin, S.C., Montgomery, D.B.: IEEE Trans. Magn. 30, 1467 (1988) Hsu, S.C., Bellan, P.M.: Phys. Rev. Lett. 90, 215002 (2003) K¨onigl, A., Pudritz, R.E.: in Protostars and Planets IV, edited by Mannings, V. et al., The University of Arizona Press, Tucson, 2000, p. 759 Lebedev, S.V., Ampleford, D., Ciardi, A., et al.: Astrophys. J. 616, 988 (2004) Lebedev, S.V., Chittenden, J.P., Beg, F.N., et al. : Astrophys. J. 564, 113 (2002) Lebedev, S.V., Ciardi, A., Ampleford, D.J., et al.: Plasma Phys. Control. Fusion 47, B465 (2005a) Lebedev, S.V., Ciardi, A., Ampleford, D.J., et al.: MNRAS 361, 97 (2005b) Livio, M.: Nature 417, 125 (2002) Lovelace, R.V.E., Wang, J.C.L., Sulkanen, M.E.: Astrophys. J. 315, 504 (1987) Martinez, D., et al.: Astrophys. Space Sci., doi: 10.1007/s10509-0069275-9 (2006)

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Massaglia, S.: Astrophys. Space Sci. 287, 223 (2003) Mirabel, I.F., Rodriguez, L.F.: Nature 392, 673 (1998) Mirabel, I.F., Rodriguez, L.F.: Ann. Rev. Astron. Astrophys. 37, 409 (1999) Pearson, T.J.: in Energy Transport in Radio Galaxies and Quasars, edited by Hardee, P.E., et al. (ASP Conf. Series Vol. 100, NRAO, Greenbank, 1996), p. 97 Presura, R., Ivanov, V., Sentoku, Y., et al.: Astrophys. Space Sci. 298, 299 (2005) Ramis, R., Meyer-ter-Vehn, J., Schmaltz, R.: Comput. Phys. Commun. 49, 475 (1988) Ray, T.P., Muxlow, T.W.B., Axon, D.J., et.al.: Nature 385, 415 (1997) Reipurth, B., Bally J.: Ann. Rev. Astron. Astrophys. 39, 403 (2001) Reipurth, B., Heathcote, S.: in Herbig-Haro Flows and the Birth of Low Mass Stars, edited by Reipurth, B., Bertout, C., IAU Symposium 182, 3 (1997) Ryu, D., Jones, T.W., Frank, A.: Astrophys. J. 545, 475 (2000) Ryutov, D., Drake, R.P., Kane, J.: Astrophys. J. 518, 821 (1999) Ryutov, D.D., Remington, B.A., Robey, H.F., Drake, R.P.: Phys. Plasmas 8, 1804 (2001) Shigemori, K., Kodama, R., Farley, D.R.: Phys. Rev. E 62, 8838 (2000) Shu, F., Najita, J., Ostriker, E., Wilkin, F.: Astrophys. J. 429, 781 (1994) Tenorio-Tagle, G., Canto, J., Rozyczka, M.: Astron. Astrophys. 202, 256 (1988) Wanex, L.F., Sotnikov, V.I., Leboeuf, J.N.: Phys. Plasmas 12, 042101 (2005) Wiewior, P., et al.: to be published. Zavala, R.T., Taylor, G.B.: Astrophys. J. 626, L73 (2005)

Astrophys Space Sci (2007) 307:99–101 DOI 10.1007/s10509-006-9244-3

O R I G I NA L A RT I C L E

Excitation of Electromagnetic Flute Modes in the Process of Interaction of Plasma Flow with Inhomogeneous Magnetic Field V. I. Sotnikov · R. Presura · V. V. Ivanov · T. E. Cowan · J. N. Leboeuf · B. V. Oliver

Received: 17 May 2006 / Accepted: 25 August 2006 C Springer Science + Business Media B.V. 2006


Abstract Laboratory experiments on the interaction of a plasma flow, produced by laser ablation of a solid target with the inhomogeneous magnetic field from the Zebra pulsed power generator demonstrated the presence of strong wave activity in the region of the flow deceleration. The deceleration of the plasma flow can be interpreted as the appearance of a gravity-like force. The drift due to this force can lead to the excitation of flute modes. In this paper a linear dispersion equation for the excitation of electromagnetic flute-type modes with plasma and magnetic field parameters, corresponding to the ongoing experiments is examined. Results indicate that the wavelength of the excited flute modes strongly depends on the strength of the external magnetic field. For magnetic field strengths ∼0.1 MG the excited wavelengths are larger than the width of the laser ablated plasma plume and cannot be observed during the experiment. But for magnetic field strengths ∼1 MG the excited wavelengths are much smaller and can then be detected. Keywords Flute instability . Laser plasma ablation . Laboratory astrophysics experiments

1 Introduction Interaction of plasma flows with magnetic fields plays an important role in astrophysics and space physics, ranging V. I. Sotnikov (
) · R. Presura · V. V. Ivanov · T. E. Cowan University of Nevada at Reno, NV 89557 J. N. Leboeuf JNL Scientific, Casa Grande, AZ 85222 B. V. Oliver Sandia National Laboratories, NM 87123

from supernova explosions and interaction of the solar wind with the magnetopause to the barium release experiments in the ionosphere. During recent experiments on interaction of laser ablated plasma flows with the magnetic field created by the Zebra pulse power generator strong wave activity was detected in the region of plasma flow deceleration by the magnetic field (Presura et al., 2006). Similar phenomena can take place during Novae explosions (Zakharov, 2003) and artificial magnetospheric releases, similar to the AMPTE magnetotail release (Bernhardt et al., 1987; Ripin et al., 1993). To study its linear excitation and nonlinear evolution, a nonlinear set of equations for electrostatic potential, magnetic field, and density has been derived in the low frequency limit (ω ≪ ci ,where ω is the frequency of the excited mode and ci = Z eB0z /Mi c is the cyclotron frequency of the ion with charge Z and mass Mi = µMp , Mp being the proton mass) from two-fluid macroscopic equations which include gyroviscosity (Sotnikov et al., accepted by IEEE TPS, 2006). The experimental set-up is illustrated in Fig. 1.The plasma flow was created by laser ablation of a massive solid CH2 target using the “Tomcat” laser (5 J, at 1054 nm, 6 ns). At best focus, the laser irradiance on target was ≈1014 W/cm2 . An azimuthal magnetic field Bϕ (r ) = µ0 I /2πr was produced by an axial current I (0.6 MA) flowing in a 14 mm diameter rod used as z-pinch load of the Zebra pulsed power generator with 200 ns rise time (see Fig. 1). The magnetic field generated at the peak of the pulse was 8 T at the laser target surface and 17 T at the rod surface, measured with magnetic probes. The laser was synchronized with the z-pinch so that the ablation plasma was produced and evolved during a ≈30 ns period of constant magnetic field at the current peak. The background pressure was <10−5 Torr to assure that collisions with neutrals had a negligible influence on the plasma evolution. No plasma was generated at the rod surface. The plasma was probed with a laser operating at 532 nm with a Springer

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Astrophys Space Sci (2007) 307:99–101

Fig. 1 Experimental set-up

0.2 ns pulse width. We fielded two-frame schlieren and twoframe interferometry diagnostics with 7 ns delay between the two frames. Schlieren images were recorded either in the plane x–z containing the magnetic field lines or in the plane x-y perpendicular to B (Figure 1). Interferometric measurements were performed in the x–y plane, simultaneously with x–z Schlieren images.

which drives the instability of the flute modes. In the finite beta plasma in the region where plasma flow kinetic energy is equal to magnetic field energy, the electric field in the flute oscillations is not irrotational (i.e. ∇ × E = 0), in contrast  − 1 ∂ A . with the low beta case, and is written as E = −∇ c ∂t The plasma density N can be expressed as the sum of a slowly varying component with x, the equilibrium plasma density n0 (x), and a perturbed component due to flute oscillations δn(x, y, t). Likewise the magnetic field is written as Bz = B0z (x) + δ Bz (x, y). We also assume the quasineutrality condition Ne = Z Ni . The equilibrium condition in this case is written as:
 Z Te βi βi κB = 1 + κ N + 2 g, Ti 2 VT i where κN = −

2 Electromagnetic flute mode instability Two-fluid macroscopic equations will be used to describe low frequency flute modes (ω ≪ ci ) in a weakly inhomogeneous plasma with external magnetic field B0z (x) (Sotnikov et al., accepted to IEEE TPS, 2006 and references there in). The following notations were chosen: direction of the plasma flow along the x-axis, and external magnetic field along the z-axis. As is customary in flute mode turbulence, oscillations are taken to be uniform in the direction of the magnetic field, i.e. the wave vector along the magnetic field k|| = 0. We consider an inhomogeneous high beta plasma in slab geometry with density n0 (x) in the presence of an inhomogeneous magnetic field B0z (x), where x is the direction of inhomogeneity. In Fig. 2 the experimental plasma density profile and the adjusted magnetic field profile are presented. Enhanced wave activity was observed in the (x, y) plane along the ydirection (along the plasma-magnetic field interface) inside the plasma–magnetic field interface (region A in Fig. 2). Deceleration of plasma flow in the process of interaction with the magnetic field causes the gravity-like term g = gex ,

Bθ

n

1 d B0z 8π n 0i Ti 1 dn 0 > 0, κ B = > 0, βi = n0 d x B0z d x B02

As will be shown later, the characteristic wave lengths of the flute modes which can be excited for the plasma parameters observed in the experiment (Fig. 1) are much larger than the typical size of the region where interaction takes place, when the strength of the magnetic field, produced by the pulsed power generator is ∼0.1 MG. But for much stronger magnetic field strength (∼1 MG) typical wavelengths of the excited flute modes are much smaller than the size of the region where interaction occurs. The following relation connects δn and δ Bz : δ Bz 1 δn =− β , B0z 2 n0

where β = βi + βe .

+

1 (0)   k 2y Vi(0) y Vi Dy 1 4 1 + 2β

1 + 12 βe k 2y gκ N = 0. 2 Z 1 + 12 β k⊥

In this equation: x Fig. 2 Experimental density and adjusted magnetic field radial profiles

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(2)

From (2) it follows that in a low beta case we can neglect the electromagnetic component in the flute mode and consider it as electrostatic. The dispersion equations for the frequency and the growth rate of the electromagnetic flute oscillations, valid for both low and high beta plasma, was derived in Sotnikov et al., 2006. In the system where ions are at rest the dispersion equation is written as: ω2 + k y Vtot ω −

A

(1)

Vtot = Vi(0) y −

1 + 14 β 1+

1 β 2

Vi(0) Dy −

1 Vey(0) ; 1 + 21 β

(3)

Astrophys Space Sci (2007) 307:99–101

101

Fig. 3 Dependence of growth rate γ (in (gκ n )1/2 units) of the electromagnetic flute-type modes on the wave vector k y (in units of the inverse ion skin depths ω pi /c). In (a) magnetic field B0 = 0.1 MG and in (b) magnetic field B0 = 1 MG

Vi(0) y = −

g + Vi(0) Dy ; 0i

Vi(0) Dy = −κ N

Vey(0) = κ N

VT2e ; 0e

VT2e =

Te ; me

VT2i 2 Ti 2 ; VT i = ; β = βi + βe ; k⊥ = k x2 + k 2y . 0i Mi

We have solved numerically the dispersion relation in Eq. (3) using plasma parameters typical of the experiments: n 0e = 1.0 × 1018 cm−3 , n 0i = 3.0 × 1017 cm−3 , Ti ∼ Te = 150 eV, B0 = (0.1 − 1.0) MG and g ∼ 5 × 1014 cm/s2 . Only modes, propagating along the plasma-magnetic field interface region (along the y-direction) were considered since during the experiment only perturbations along the y-axis were observed. The solution is displayed in Fig. 3, where the growth rate of the flute-like electromagnetic perturbations is plotted as a function of wave vector k y . As follows from Fig. 3, in the case when the external magnetic field strength B0 = 0.1 MG, the smallest possible wavelength of the excited flute modes along the y-direction is λ∼2 cm. These waves cannot be excited in the system, because their wavelengths are larger then the characteristic size of the laser ablated plasma plume in this direction. But when the magnetic field strength is increased to B0 = 1 MG, the part of the excited wave spectrum with large k y corresponds to the wavelengths λ∼1 mm and these waves can be excited along the y-direction inside the plasma-magnetic field interface region.. The growth rate yields a characteristic time for the instability to develop of ∼10 ns. 3 Conclusion In support of the planned experiments in the NTF laboratory to investigate generation of the flute modes in the region of plasma flow deceleration, we investigated the solutions

of the dispersion relation which describes excitation of the electromagnetic flute modes in a finite beta plasma. Obtained growth rates for the plasma and magnetic field parameters corresponding to the experimental setup show that the width of the excited wave spectrum of the flute modes strongly depends from the strength of the external magnetic field B0 . For the smaller values of the magnetic field (B0 ∼ 0.1 MG) the characteristic scale of the flute mode perturbations is of the order of ∼2 cm and it exceeds the characteristic width of the region where excitation of the flute modes is expected to take place. But with increase of the magnetic field strength (B0 ∼ 1 MG) it is possible to excite much shorter wavelengths ∼1 mm. This allows to excite flute modes inside the deceleration region along the plasma-magnetic field interface. Acknowledgements This work was supported by the United States Department of Energy under the following grants: Grant No. DE-FC5201NV14050 at the University of Nevada at Reno, Grant No. DE-AC0494AL85000 at Sandia National Laboratories.

References Bernhardt, P.A., et al.: J. Geophys. Res. A 92, 5777–5794 (1992) Presura, R., Ivanov, V.V., Esaulov, A., Sentoku, Y., Sotnikov, V.I., Laca, P., Astanovitskiy, A.L., Chiu, C., Ditmire, T., Horton, W., Keely, S., Le Galloudec, B., Renard-Le Galloudec, N., Cowan, T.E.: Anisotropic expansion of a laser-produced-plasma in a strong magnetic field. Submitted to PRL (2006) Ripin, B.H., Huba, J.D., McLean, E.A., Manka, C.K., Peyser, T., Burris, H. R., Grun, J.: Phys. Fluids B 5, 3491 (1993) Sotnikov, V.I., Ivanov, V.V., Cowan, T.E., Leboeuf, J.N., Oliver, B.V., Coverdale, C.A., Jones, B.M., Deeney, C., Mehlhorn, T.A., Sarkisov, G.S., LePell, P.D.: Investigation of electromagnetic flute mode instability in a high beta Z-pinch plasma. IEEE TPS, accepted for publication in (2006) Zakharov, Y.P.: Plasma Sci. 31, 1243–125 (2003)

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Astrophys Space Sci (2007) 307:103–107 DOI 10.1007/s10509-006-9249-y

ORIGINAL ARTICLE

Plasma Jet Experiments Using LULI 2000 Laser Facility B. Loupias · E. Falize · M. Koenig · S. Bouquet · N. Ozaki · A. Benuzzi-Mounaix · C. Michaut · M. Rabec le Goahec · W. Nazarov · C. Courtois · Y. Aglitskiy · A. YA. Faenov · T. Pikuz

Received: 21 April 2006 / Accepted: 29 August 2006 C Springer Science + Business Media B.V. 2006


Abstract We present experiments performed with the LULI2000 nanosecond laser facility. We generated plasma jets by using specific designed target. The main measured quantities related to the jet such as its propagation velocity, temperature and emissive radius evolution are presented. We also performed analytical work, which explains the jet evolution in some cases. Keywords Astrophysical jet . Laboratory astrophysics . Self-similar solutions . Laser B. Loupias (
) · M. Koenig · N. Ozaki · A. Benuzzi-Mounaix · M. Rabec le Goahec Laboratoire pour l’Utilisation des Lasers Intenses, UMR7605, CNRS – CEA – Universit´e Paris VI – Ecole Polytechnique, 91128 Palaiseau Cedex, France e-mail: E. Falize · S. Bouquet CEA/DIF/D´epartement de Physique Th´eorique et Appliqu´ee, BP 12, 91680 Bruy`eres-le-Chˆatel, France C. Michaut Laboratoire de l’Univers et de ses Th´eories, UMR8102, Observatoire de Paris, 92195 Meudon, France W. Nazarov University of St Andrews, School of Chemistry, Purdie Building, North Haugh, St Andrews, United Kingdom C. Courtois CEA/DIF/DCR, BP 12, 91680 Bruy`eres-le-Chˆatel, France Y. Aglitskiy Science Applications International Corporation, McLean, Virginia 22102, USA A. YA. Faenov · T. Pikuz Multicharged Ions Spectra Data Center of VNIIFRTI, Mendeleevo, Moscow Region, 141570, Russia

1 Introduction Astrophysical jets take place in astronomical systems exhibiting accretion disk such as proto-stars, supernovae, pulsars, active galactic nuclei and so forth (de Gouveia Dal Pino, 2005). Whereas from these systems the launching could be different, the jets share some common characteristics, as the collimation. Furthermore, more or less spaced emission knots, which move at high speed away from the central source, have been observed. Finally the jets are ended with bow shocks, which can be understood like a “working surface” with the ambient surrounding medium. To understand these phenomena, astrophysical jets have been the subject of elaborate studies in both theory (Raga et al., 1990) and observations (Reipurth et al., 2002). Nevertheless, they still raise problems such as the “jet collimation”. Jet experiments can offer an additional approach to improve our understanding of the physical processes, which occur during the jet propagation (Shigemori et al., 2000; Blue et al., 2005; Lebedev et al., 2005; Farley et al., 1999). Using for the target a cone filled with foam (Koenig et al., 1999) or high-Z material doped foam, we generate plasma jets and we observe its propagation in vacuum. We present jet characteristic measurements such as its velocity, Vjet , the jet emissive radius evolution, R(t), and its corresponding temperature, Te (t). For the first time all of these parameters were obtained simultaneously on a single shot basis, allowing a complete characterization of the jet. We also derive a purely hydrodynamic analytical solution for the evolution of the jet radius, which is in good agreement with the experimental measurements. This new analytical result well describes the whole evolution of the expanding fluid.

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2 Experiments The target is composed by a three layer pusher (0.16 µm Al/20 µmCH/3 µm Ti) from laser side and a cone filled with foam at two different densities, 50 mg/cc and 100 mg/cc. We also used high-Z material doped foam in order to create a more radiative jet and to use X-ray shadowgraphy. For some of the targets, we added a washer at the cone exit to increase the plasma jet collimation. The two LULI2000 beams (wavelength: 527 nm – total energy: 500 J – pulse duration: 1.5 ns – 500 µm spot diameter – I L ∼ 2 × 1014 W/cm2 ) are focused onto the pusher to generate by rocket effect a shock that propagates through the foam. The cone geometry allows us to drive the plasma flow along the cone axis to create the jet. Our main goal, regarding the experimental diagnostics, is measuring the jet characteristics to determine the best target design to have the most collimated plasma jet. In order to fulfill this goal, we implemented several diagnostics as shown in Fig. 1. 2.1 Transverse diagnostics We use transverse VISAR (Celliers et al., 2004) to measure the jet velocity. The front edge of the jet is higher or equal to the critical density for our probe beam (YAG 2ω). Therefore, the fringe break slope (Fig. 2(a)) provides Vjet . We can also notice a shift at the end of the fringes. From this shift, we can get the jet density gradient evolution ahead the jet but the data analysis is in progress. The measurements provide Vjet ≈ 160 km/s for 50 mg/cc foam density and Vjet ≈ 120 km/cc for 100 mg/cc. We produce jets with a mach number M ≈ 8 and we obtain the shape of the plasma jet by 2D shadowgraphy at two different times for each shot. We use two Gate Optical Imagers (GOI) with 120ps time resolution (Fig. 2(b)). We have a common shape that is more or less a plasma jet with a higher aspect ratio (jet radius/jet length) for cones without washer than for cones with washer. It confirms the guiding role of the washer. This result has to

Fig. 1 Experimental set-up for visible transverse diagnostics

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Fig. 2 Transverse diagnostic results for cone target at 50 mg/cc with washer: (a) VISAR: the line slope provides the front jet edge velocity which is around 176 km/s. (b) GOI: Shadowgraphy of the jet at 5.5 ns, aspect ratio ∼0,72

be compared with the emissive radius evolution where it is smaller. 2.2 Rear side diagnostics Thanks to the self-emission diagnostic, we measure the emissive radius evolution (Fig. 3(a)). We notice that both for 50 mg/cc and 100 mg/cc foam density, two phases on the emission radius variation arise. For the target without washer, we have a second phase where the radial expansion speeds up (Fig. 3(b): upper curve) whereas with washer the radial expansion slows down (lower curve). Finally, we also performed a 2D X-ray monochromatic shadowgraphy of the jet with the He-like Vanadium line at 5.4 keV using a spherical bent crystal (Experimental set-up in Fig. 4(a)). Figure 4(b) shows a result of high-Z material foam target. We can observe a dense jet core, radius around 40 µm, smaller than the 50 µm cone rear side radius hole. This result lets us thinking of a jet structure where a dense jet

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Fig. 3 Self-emission results: (a) Streaked image for 100 mg/cc with washer, (b) Radius evolution for cone target at 50 mg/cc foam density without washer (upper curve) and 100 mg/cc with washer (lower curve)

core (shape from x-ray shadowgraphy) is surrounded with an envelope by comparing GOI and self-emission radius. Furthermore, with an absolute calibration of the selfemission diagnostic we measure an equivalent black body temperature of the jet equal to 4 eV for 50 mg/cc foam density and 2.5 eV for 100 mg/cc.

Fig. 4 X-ray monochromatic shadowgraphy: (a) Experimental set-up, (b) X-ray monochromatic shadowgraphy of the jet

where ρ(r, t), P(r, t), v(r, t), γ and κ are respectively the density, the pressure, the velocity, the polytropic coefficient and the polytropic constant in the flow. In order to solve these equations, we use the rescaling theory, a method based on group transformations (Munier and Feix, 1983; Bouquet et al., 1985; Ribeyre et al., 2005). Following Ribeyre et al. (2005), we introduce the four scaling functions A(t), B(t), C(t) and D(t), such as: r = C(t)ˆr ;

3 Analytical model

dt = A2 (t)d tˆ;

ρ(r, t) = D(t)ρ(ˆ ˆ r , tˆ);

We consider the jet as a cylinder with an infinite length. The cylinder has an initial radius R0 and a central density ρ c and we try to derive the time-dependent radius, R(t), of the jet. We consider that the plasma can be described by the hydrodynamics equation and the jet evolution is then given by the Euler’s equations: 1 ∂ ∂ρ + (rρv) = 0, ∂t r ∂r ∂v 1 ∂P ∂v +v =− , ∂t ∂r ρ ∂r γ

P = κρ ,

(1)

v=

ˆ r , tˆ); P(r, t) = B(t) P(ˆ

C vˆ + C˙ rˆ ; A2

(2)

  where ‘∧’ denotes the quantities in the new space rˆ , tˆ . We impose the following initial conditions: C(0) = A(0) = ˆ B(0) = D(0) = 1. At t = t = 0, the two spaces (r, t) and rˆ , tˆ coincide and in the new space, the system (1) becomes:
˙  ∂ ρˆ 1 ∂ C˙ 2 D + (ˆr ρˆ v) ˆ +A +2 ρˆ = 0, rˆ ∂ rˆ ∂ tˆ D C
˙  ∂ vˆ ∂ vˆ A˙ 2 C + vˆ + 2A − vˆ ∂ tˆ ∂ rˆ C A +

1 B A4 ∂ Pˆ C¨ A4 , rˆ = − C ρˆ C 2 D ∂ rˆ

(3)

(4)

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Dγ γ κ ρˆ , Pˆ = B

(5)

where the upper dot stands for d/dt in (3) and (4). We now search a static solution, i.e., vˆ ≡ d rˆ /d tˆ = 0, in the new space. In order to remove the explicit time-dependence in Equations (3)–(5) we must impose different constraints on the rescaling functions, namely: D = C −2 ,

B A4 = C 2 D,

B = Dγ ,

C¨ A4 = ̟ 2 , (6) C

where the dimension of the free parameter ̟ is the inverse of a time. After some simple manipulations, we have to solve the ¨ 2γ −1 = ̟ 2 and we obtain ordinary differential equation CC its implicit solution t(C), which is written: ̟ (t − t0 ) ∝ C(t)
 1 1 3 − 2γ × 2 F1 , ; ; [C(t)]2(1−γ ) , 2 2 − 2γ 2 − 2γ

(7)

where t0 is an arbitrary constant and where 2 F1 is the hypergeometric function. The scaling function C(t) provides the time evolution of the radius of the jet since from Equations (2) we have, in particular, R(t) = R0 C(t). This law is a wide extension of the usual self-similar solutions (SSS) for which the radius (and the various physical quantities) evolve as a power law of t. Once the scaling function C(t) is obtained, the additional functions A(t), B(t), and D(t) can be computed from relations (2) and the density, pressure and velocity profiles ρ(r, t), P(r, t) and v(r, t) respectively can be obtained from (2) also. It should be noticed that this approach is not the classical one to derive SSS’s since in our case (Munier and Feix, 1983; Bouquet et al., 1985; Ribeyre et al., 2005), we have a new space including a new time tˆ. This transformation is called “the zooming coordinates method” in astrophysics (Blottiau et al., 1988; Hanawa and Matsumoto, 2000; Hennebelle, 2001; Shadmehri and Ghanbari, 2001). It is clear that, provided we lookfor stationary or static solutions (∂/∂ tˆ = 0) in  the new space rˆ , tˆ , this technique reduce to SSS’s; however, they are much more general than the t n -classical SSS’s. Introducing conditions (6) into Equation (4), we get the density profile in the (real) physical space (r, t): 1

2  2−γ 1 r 1 ρ(r, t) = 2 1− 2 . ρc C C R0

(8)

Fig. 5 Analytical (full line) and experimental radius comparison for 50 mg/cc target with washer

agreement. To plot the analytical radius evolution, a sound velocity in the jet core around 19 km/s has been assumed. We estimate from the experimental result a sound velocity about 15 km/s. This difference is due to the measurement of the temperature and the estimation of the ionization rate. Thanks to the self-emission diagnostic we measure an equivalent black body temperature of the jet around 4.5 eV. The temperature measurement does not take into account possible absorption in the visible band and it corresponds merely to the jet surface temperature. In order that the analytical model matches the experimental values of the radius, we have to consider the core jet temperature. Therefore, when we take 4.5 eV, we under-evaluate the temperature. Furthermore, up to now, we could only evaluate an ionization rate from 1D MULTI simulations (Ramis et al., 1988). For 100 mg/cc foam density (experimental radius evolution Fig. 3(a)), the polytropic model fits the experimental data over 5 ns roughly. This can be understood as the apparition of a different dynamic regime. To conclude, we have measured: – the jet shape by visible shadowgraphy, – its propagation velocity, – the jet radius evolution from the self-emission diagnostic. We have a new analytical work which is in quite good agreement (for 50 mg/cc foam density) with the experiment. We also tested the faisability of x-ray 2D radiography and high-Z material doped foam (work in progress). References

4 Discussion The analytical evolution of R(t) presented previously is compared with the experiments (Fig. 5) and it is in good Springer

Blue, B.E., et al.: Phys. Rev. Lett. 94, 095005 (2005) Blottiau, P., et al.: Astron. Astrophys. 207, 24–36 (1988) Bouquet, S., et al.: Astrophys. J. 293, 494–503 (1985)

Astrophys Space Sci (2007) 307:103–107 Celliers, P.M., et al.: Rev. Sci. Instrum. 75, 4916–4929 (2004) de Gouveia Dal Pino, E.M.: astro-ph/0505521 Farley, D.R., et al.: Phys. Rev. Lett. 83, 1982–1985 (1999) Hanawa, T., Matsumoto, T.: Astrophys. J. 540, 962–968 (2000) Hennebelle, P.: Astron. Astrophys. 378, 214–227 (2001) Koenig, M., et al.: Phys. Plasmas 6, 3296–3301 (1999) Lebedev, S.V., et al.: Plasma Phys. Control. Fusion 47, B465–B479 (2005)

107 Munier, A., Feix, M.R.: Astrophys. J. 267, 344–357 (1983) Raga, A.C., et al.: Astrophys. J. 364, 601–610 (1990) Ramis, R., et al.: Comput. Phys. Comm. 49, 475 (1988) Reipurth, B., et al.: Astron. J. 123, 362–381 (2002) Ribeyre, X., et al.: Astrophys. Sp. Sc. 298, 75–80 (2005) Shadmehri, M., Ghanbari, J.: Astrophys. Sp. Sc. 278, 347–355 (2001) Shigemori, K., et al.: Phys. Rev. E 63, 8838–8841 (2000)

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Astrophys Space Sci (2007) 307:109–114 DOI 10.1007/s10509-006-9275-9

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Magnetic Fields for the Laboratory Simulation of Astrophysical Objects D. Martinez · C. Plechaty · R. Presura

Received: 14 April 2006 / Accepted: 8 November 2006 C Springer Science + Business Media B.V. 2006


Abstract Strong magnetic fields were generated using a fast pulsed power generator, to investigate the interaction of plasma flows with magnetic fields and magnetized background plasmas. The inductive loads used in these experiments were designed using a filament and a finite element modeling approaches. Magnetic fields up to 2 MG (200 T) were measured by using the Faraday rotation technique. Keywords Faraday rotation . Laboratory astrophysics . Finite element method . Filament model . Magnetic field

Introduction The interaction of intense laser produced plasmas with strong, externally generated magnetic fields is a robust solution for creating in the laboratory matter with parameters relevant to astrophysical phenomena. Depending on the fact studied, the magnetic field can determine the plasma dynamics or can provide a small-scale dissipation mechanism. Experiments to study these effects require a wide range of plasma and field parameters. Such experiments are being developed at the Nevada Terawatt Facility (NTF) taking advantage of the 10 TW laser Tomcat coupled with the 2 TW pulsed power generator Zebra. With this set-up, the parameters of the plasma and its environment can be varied independently. Conceptual designs of experiments for the laboratory simulation of the interaction of astrophysical jets with the magnetized interstellar medium are presented in a companion paper (Presura et al., this volume). These experiments focus on the stability of magnetized plasma jets and on their D. Martinez · C. Plechaty (
) · R. Presura Nevada Terawatt Facility, University of Nevada, Reno e-mail: [email protected]

interactions with magnetized ambient medium. In another type of experiment, the isochoric heating of a solid target is pursued by attempting to control the heat transport by the hot electrons, produced by a high intensity laser, with a strong external magnetic field (Sentoku, 2006). The expected result is solid density matter uniformly heated to several hundred eV in volumes large enough (∼105 µm3 ) and for durations long enough (several ps) to investigate radiation transport. This would be directly relevant to the energy transport in stellar interiors. Experiments to generate the required magnetic fields were performed at the NTF at the University of Nevada, Reno using Zebra, a pulsed power generator which can produce a 0.6 MA current with a fast rise time (200 ns). Two types of coil designs were used, namely the two-turn helical coil (TTHC) and the horseshoe coil (HSC). Two different methods of modeling were used for designing the coil loads. The first method, used to determine the magnetic field spatial distribution, was based on solving the Biot-Savart equation and implemented in MatLab. The second method utilized Comsol Multiphysics, a commercial solver that uses the Finite Element Method (FEM), to solve for the magnetic and electric potentials. To illustrate the results obtained, two TTHC shots and one HSC shot are presented. The magnetic fields generated by these loads were measured using the Faraday effect. Fields up to 2 MG were measured.

Coil designs The filament model solves for the magnetic field of current carrying filaments using the Biot-Savart law. Equations describing the load geometry were derived for the TTHC and the HSC and a MATLAB program was written to solve and display the magnetic fields generated by various coil designs. Springer

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Fig. 1 The smooth curve is the magnetic flux density calculated along the axis of the HSC with the filament model. The second curve is the result from COMSOL. The resistivity of tantalum, a loop radius of 3.2 mm and a coil height of 8.1 mm were considered

The FEM models were solved in COMSOL Multiphysics and setup to solve for the magnetic and electrical potentials using a time harmonic quasi-static approximation. In this way, the skin depth and inductive currents are taken into account. Using the current waveform information from Zebra and the geometry of the coil, COMSOL, taking into consideration the resistivity of the coil material, was able to solve the potentials. The spatial magnetic field distributions calculated far from the wire with the two different models were similar. For example, the HSC models show very close matches to the magnetic field with 90 T in the center of the coil with a current of 0.7 MA (Fig. 1). The TTHC models, on the other hand, predict different magnetic fields. Figure 2 shows that the distributions are similar until the very center of the coil. The FEM solution has a magnetic field strength of 220 T at the center while the filament solution predicts 255 T, both with a current of 0.7 MA through the coil. To determine the origin of this difference, we solved in COMSOL a filament model similar to that treated in MATLAB. The magnetic field was practically the same with that predicted by the filament model (250 T) as shown in Fig. 2. We have determined that the difference observed in the FEM and filament model have a physical origin, namely the preferential distribution of the current on the lateral surfaces of the coil. This is supported by the accuracy of the results for the HSC. To assess the importance of the magnetic pressure on the coil, the Maxwell stress was calculated. For the examples Springer

shown here we calculated that the Maxwell stress for the TTHC at its highest point is 3.3E10 Pa (0.33 Mbar) and the HSC has 1.7E10 Pa (0.17 Mbar). The stress distribution will be used to calculate the motion and deformation of the coil on the experiment time scale in the future. Present estimates, that ignore the heating of the coil, indicate that this effect is negligible on the time scale of 100 ns.

Faraday diagnostics The Faraday effect was used to measure the magnetic fields. To obtain localized, non-perturbative, and time resolved measurements of the magnitude of the magnetic field in vacuum, the Faraday-active probes used were F2 (flint glass) disks, 3 mm in diameter and 1.75 mm thick. The glass probes were calibrated with a known magnetic field that was produced by an electromagnet. The measured Verdet constant was VF2 = 16.2 rad/(T·m), which is comparable with the value obtained by others, 14.2 rad/(T·m) (Lide, 2003). The Faraday diagnostic setup used on Zebra is shown in Fig. 3. A 532 nm, 250 mW diode pumped solid state laser beam is first collimated and then sampled by a photodiode that monitors the laser stability. After sampling, the beam is polarized by a Glan-Thompson prism and then is passed through a λ/2 plate, used to control the polarization plane. The beam is then focused into the Faraday probe located in the vicinity of the coil. After passing through the probe,

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Fig. 2 This figure compares the axial distribution of the magnetic flux density along the axis of a TTHC, with 2.7 mm radius and 2.2 mm pitch, calculated with the filament model (smooth curve), the COMSOL filament model (dashed curve), and the complete COMSOL model (stepwise curve)

Fig. 3 Faraday diagnostic setup on Zebra

the beam is sampled again, by the “emission” photodiode, which monitors for light generated from electrical breakdown – pinpointing the time in which the Faraday measurement is no longer valid. During a shot, the polarization plane of the beam undergoes a rotation, and its new polarization state is analyzed with a Glan-Taylor prism. This allows the rotation to be determined from an intensity measurement, according to Malus’

Law. For the shots presented later in this paper, the Faraday diagnostic set up was used to measure rays with polarization perpendicular and parallel with that of the original beam. With this differential method, the signal-to-noise ratio is increased. The plasma formation and evolution in the vicinity of the Faraday probe were monitored with laser shadow and Schlieren imaging.

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Fig. 4 Faraday diagnostic data for a TTHC shot showing the measured differential Faraday rotation signal, the calculated Faraday rotation signal, the current waveform, the self-emission signal and shadow images

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taken at the stated times. (A) Point in which the first maximum in Malus’ law is obtained. (B) Point at which the Faraday measurement is compromised. (C) Parameters of the TTHC used in the Zebra shots

Fig. 5 Faraday diagnostic data for a TTHC shot showing the same quantities as Fig. 4 and negative Schlieren images at the times stated. (A) Point at which the Faraday signal becomes no longer valid

Experimental results Two shots with TTHC loads made of 1 mm diameter 316 L stainless steel wire (Fig. 4C) are presented here. The F2 glass probe was placed in different positions in the two cases anSpringer

alyzed. In one case, the back face of the Faraday probe was placed about 0.6 mm in front of the coil. The results for this shot are shown in Fig. 4. As one can see, the calculated Faraday signal, based on measurements of the current, and the measured Faraday signal closely follow each other until

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Fig. 6 The Faraday diagnostic data for a HSC shot showing the calculated Faraday signal, the components of the differential Faraday measurement, and the current waveform. (A) The HSC setup. (B) Point at which the measured signal diverges from the calculated one. (C) Point in which the components of the Faraday signal diverge

t = 50 ns (point B) at which point they diverge, making the Faraday measurement invalid. It is theorized that at t = 50 ns the Faraday beam path becomes compromised (the emission photodiode starts to record a signal). In addition, the last shadow image taken during the shot shows plasma on the surface of the glass probe. Since the current peaks at t = 150 ns, the maximum possible magnetic field was not reached before the Faraday signal was compromised. At this time, the magnetic field through the volume of the glass was estimated to be 65 T, which leads to an estimate of 160 T in the center of the coil and 290 T near the conductor surface. In another TTHC shot, the front face of the glass was placed flush with the surface of the coil. Instead of shadow images, Schlieren images were taken (Fig. 5). In this shot, the Faraday beam path is compromised at t = 0 ns (point A), and practically at the same time, the emission photodiode detects electrical breakdown. The current peaks approximately 100 ns later. Near the current peak, as seen by the Schlieren images, plasma had formed on both the glass and the coil. At the latest time when the Faraday measurement is valid, the magnetic field is estimated to be 75 T at the location of the probe, which leads to an estimated 100 T at the center of the coil and 160 T near the conductor surface. The two TTHC shots show that if the probe is placed too close to the coil, the probe may flashover or the load may electrically break down, causing the Faraday measurement to become compromised earlier in the current waveform, as shown by changing the probe placement. Attempting to produce strong magnetic fields without creating a measurable amount of plasma, a four-wire HSC was shot (Fig. 6). In this case, two Faraday probes were used to increase the rotation effect and were placed roughly in the middle of the HSC with the front face of the first probe about

0.25 mm inside. For this shot, the differential Faraday signal diverges from the calculated Faraday signal when t = 300 ns (point B). However, the components of the differential setup do not diverge from each other until t = 800 ns (point C). It is theorized at t = 300 ns, the current stops following along the horseshoe path and arcs through plasma that forms in the gap between the leads of the horseshoe. The current in the horseshoe structure decays exponentially due to resistive effects until t = 800 ns when the Faraday beam path is compromised. At the current peak, the estimated magnetic field was 50 T at the coil center. In another experiment with a smaller HSC, magnetic fields of the order of 100 T were measured, without any plasma formation. Since the Faraday probe has a finite volume, the measurement taken is not a simple line integral. Each point in the volume of the glass contributes to the overall measurement. In order to estimate the effect of the probe volume on the field measurement, a MatLab program was written, based on the filament model. The magnitude of this effect was estimated as a function of the probe location.

Conclusion To date, magnetic fields up to 2 MG were produced. Fields of the order of 1 MG were obtained without generating any measurable plasma. The experiments show that the magnetic field strength predicted in the simulations closely matches the field strength actually produced. The simulations have also been useful in interpreting the results from the Faraday measurements. In future models, a more comprehensive study will be preformed to obtain a self-consistent treatment of magnetic Springer

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and electric fields, coil heating, and coil deformation. In addition, in future Faraday measurements, a smaller probe will be used to minimize the effect of electrical breakdown as seen in the two TTHC shots and to better localize the measurement. Acknowledgements We would like to thank V. Ivanov and P. Laca for laser diagnostics. This work was supported by DOE/NNSA under the UNR grant DE-FC52-01NV14050 and by UNR undergraduate awards granted to D. Martinez and C. Plechaty.

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References Presura, R., Neff, S., Wanex, L.: Experimental design for the laboratory simulation of magnetized astrophysical jets. Astrophys. Space Sci., DOI 10.1007/s10509-006-9286-6 (2006) Sentoku, Y., Kemp, A., Bakeman, M., Presura, R., Cowan, T.E.: Isochoric heating of hot dense matter by magnetization of fast electrons produced by ultra-intense short pulse irradiation. J. Physics IV 133, 521 (2006) Lide, D.R. (ed.): CRC Handbook of Chemistry and Physics, 84th edition. CRC Press LLC (2003)

Astrophys Space Sci (2007) 307:115–119 DOI 10.1007/s10509-006-9255-0

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Assessing Mix Layer Amplitude in 3D Decelerating Interface Experiments C. C. Kuranz · R. P. Drake · T. L. Donajkowski · K. K. Dannenberg · M. Grosskopf · D. J. Kremer · C. Krauland · D. C. Marion · H. F. Robey · B. A. Remington · J. F. Hansen · B. E. Blue · J. Knauer · T. Plewa · N. Hearn

Received: 21 April 2006 / Accepted: 20 September 2006 C Springer Science + Business Media B.V. 2006


Abstract We present data from recent high-energy-density laboratory experiments designed to explore the Rayleigh– Taylor instability under conditions relevant to supernovae. The Omega laser is used to create a blast wave structure that is similar to that of the explosion phase of a core-collapse supernova. An unstable interface is shocked and then decelerated by the planar blast wave, producing Rayleigh–Taylor growth. Recent experiments were performed using dual, sideon, x-ray radiography to observe a 3D “egg crate” mode and an imposed, longer-wavelength, sinusoidal mode as a seed perturbation. This paper explores the method of data analysis and accurately estimating the position of important features in the data. Keywords Rayleigh-Taylor instability . Supernova . Laboratory astrophysics Introduction In 1987, a core-collapse supernova (SN) occurred ∼160000 light years away, making it the closest SN in modern times. The proximity of SN1987A made it possible to use contemporary astronomical instruments to collect data from the C. C. Kuranz (
) · R. P. Drake · T. L. Donajkowski · K. K. Dannenberg · M. Grosskopf · D. J. Kremer · C. Krauland · D. C. Marion University of Michigan, Ann Arbor, MI, USA H. F. Robey · B. A. Remington · J. F. Hansen · B. E. Blue Lawrence Livermore National Laboratory, Livermore, CA, USA

SN. At the time, existing models did not agree with the data collected, specifically, the high velocities and early x-ray emission of dense core elements. These discrepancies motivated improvement of the understanding of core-collapse SNe. Current models have started to explain the mysteries of SN1987A, but many questions remain unanswered. Of particular interest is the effect of hydrodynamic instabilities on the transport of the heavy core elements. High Energy Density (HED) facilities make it possible to study specific, well-scaled areas of astrophysical phenomena, in our case, the blast-wave-driven interface of a corecollapse supernova. Intense lasers can create the extremely large energies in mm-scale targets previously seen only in astrophysical systems. Experiments of this type have been done or are planned at numerous laser facilities (Drake et al., 2004; Kane et al., 2000; Robey et al., 2001; Remington et al., 2000). It is possible to compare the SN and the experiment because the targets can be well-scaled to the SN explosion phase so that the two will have similar hydrodynamic evolution (Ryutov et al.,1999). The Rayleigh–Taylor (Rayleigh, 1900; Taylor, 1950) instability occurs when a system has a density gradient and effective pressure gradient in opposing directions. This is the case both the SN, where a blast wave propagates from the dense core through less dense, outer layers of the star, and in the laboratory experiment, where a planar blast wave moves through a dense plastic layer into a less dense foam layer. The resulting evolution is the flow of dense elements “sinking” outward in the form of fingers or spikes. Also, the less dense material “floats” inward and is referred to as bubbles.

J. Knauer University of Rochester, Rochester, NY, USA

Experiments

T. Plewa · N. Hearn University of Chicago, Chicago, IL, USA

During the experiment ten Omega (Boehly et al., 1995) laser beams irradiate a 150 µm layer of polyimide of a density Springer

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1.41 g/cc. The total energy of the beams is ∼5 kJ and the irradiance is ∼1015 W/cm2 , producing an ablation pressure of ∼50 Mbars, which creates a strong shock in the plastic layer of the target. After 1 ns, the laser pulse ends, causing the material to rarify. When the rarefaction wave overtakes the shock wave, a planar blast wave is formed. After about 2 ns, the blast wave crosses an interface between the plastic and carbonized resorcinol formaldehyde (CRF) foam. The foam has a density of 50 mg/cc, making the density drop between the plastic and foam similar to that expected in the case of the H/He interface in SN1987A. The interface is initially accelerated by the blast wave and then decelerated over a long period of time by the foam layer. The interface is unstable to both Richtmeyer-Meshkov (Richtmyer, 1960; Meshkov, 1969) and Rayleigh–Taylor instabilities. However, interface growth due to Rayleigh–Taylor dominates after the first few nanoseconds. Diagnostics This experiment uses an ungated Static Pinhole Camera Array (SPCA) loaded with Direct Exposure Film (DEF) behind Be, plastic and Ti or Sc light shields. To protect the ungated diagnostic from laser beams and hot plasma created during the experiment, a large gold shield is part of the target structure. The polyimide and foam components are placed inside a polyimide tube and attached to the gold shield. The target package and gold shield as well as the placement of pinhole backlighters and the two SPCAs can be seen in Fig. 1. The target package is placed at the center of the Omega chamber and each pinhole backlighter is perpendicular to the polyimide tube. The two Fig. 1 Image of target with positions of backlighters and ungated detectors. Inside the polyimide tube attached to the gold shield contains a 150 µm plastic layer followed by a 2–3 mm CRF foam layer

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backlighters are orthogonal to each other. The diagnostics are on the opposite side of the target from each pinhole backlighter. The main diagnostic is dual, orthogonal, x-ray radiography. There are two pinhole backlighters each having a 5 mm square Ta foil with a stepped pinhole in the center. The step refers to a large hole on one side of the Ta and a smaller hole on the other. The pinhole backlighters are very sensitive to rotational alignment. Therefore, a stepped structure increases the size of the source while maintaining high resolution. The large opening is about 50 µm stepped to 20 µm. About 500 µm behind the pinhole is a 50 µm thick plastic square; attached to the rear of the plastic is a 500 µm square foil of either Ti or Sc. These foils are irradiated with 4 omega laser beams that have 200–400 J/beam, 1000–1200 µm spot size and a 1 ns square pulse. These beams overfill the metal foil, irradiating the plastic under the foil so that the expanding plastic provides radial tamping of the expanding metal plasma. The Sc and Ti create 4.09 and 4.51 keV x-rays, respectively. These x-rays pass through the pinhole in the Ta then pass through the target to the ungated DEF on the opposite side of the target. On the rear surface of the polyimide piece, a 200 µm wide, 50–75 µm deep slot has been machined out of the plastic. A “tracer” strip of 4.3 at.% bromine doped plastic, C500 H457 Br43 (CHBr), is glued into that slot. The CHBr has a density 1.42 g/cc. Since the CHBr and the polyimide have similar densities and are both predominately low Z materials, they will have similar evolutions in response to extreme pressures. The tracer strip is used because the bromine component of the CHBr more readily absorbs x-rays than the CH or polyimide; therefore, it provides contrast on the x-ray radiographs

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Fig. 2 (a) Single-mode perturbation, a = 2.5 µm, k = 2π/(71 µm). (b) 2-mode perturbation, a = 2.5 µm, additional mode is k = 2π/(212 µm)

obtained by the primary diagnostics of the experiment. Also, since the strip is in the center of the target it allows the diagnostic to “look through” the polyimide since it is nearly transparent to the He-alpha x-rays used to diagnose the experiment. This allows the radiograph diagnose primarily the center of the target where the experiment is the least affected by target walls and sound waves created during the experiment. After the tracer strip is in place a seed perturbation is machined onto the rear surface of the plastic component. This paper will discuss two types of perturbations. The basic pattern for both perturbations is two orthogonal sine waves with a0 = 2.5 µm and k = 2π /(71 µm). The result is an “egg crate” pattern as seen in Fig. 2a, which will be referred to as a single-mode perturbation. The second type of perturbation has an additional mode whose wave vector is parallel to the long edge of the tracer strip. In this case, the additional mode has a0 = 2.5 µm and a k = 2bπ /(424 µm). This perturbation is referred as a 2-mode perturbation and can be seen in Fig. 2b. The reason to add additional modes is to explore enhanced spike penetration that these modes may produce. This has been seen in past experiments (Drake et al., 2004) and in simulations (Miles et al., 2003). This experiment uses dual, orthogonal radiography with one diagnostic line of sight down the tracer strip and the other across the tracer strip. The view across the strip allows one to see about 13 spikes on the radiograph and view down the strip allows one to see 3 or 4 spikes. Results and discussion Radiographs from recent experiments taken at 17 ns after the laser beams have fired can be seen in Fig. 3. Figure 3a is a radiograph of a single-mode target with the view across the tracer strip. Figure 3b is also a view across the tracer strip, but of a 2-mode target. The shock and interface are moving to the right in both images. Also, the tube walls are seen around

Y = ±470 µm and a gold grid is seen in each figure for calibration of magnification and position. In Fig. 3a there are several very bright lines due to scratches on the film. Notice that Fig. 3a has less contrast and more noise than Fig. 3b. This is because the radiograph in Fig. 3a is from a second layer DEF, where the first layer was overexposed and acted as a filter in this case. The resulting lineouts of this radiograph have been adjusted so that it is possible to compare relative positions between the two radiographs for the purposes of this paper. The positions of notable features are more clearly seen in the horizontal lineouts taken from each radiograph seen in Fig. 4a and 4b for the single-mode and 2-mode cases, respectively. Two lineouts were taken for each radiograph in order to estimate the distance from the spike tip to bubble tip. One lineout was taken across a Rayleigh–Taylor spike, shown by the dark grey line, and the other across a bubble structure, shown by the black line. The location of each lineout is shown on the corresponding radiograph by a black rectangle. On each lineout the position of the shock, spike tip and bubble head are shown. The lineouts across the spike and bubble have a sharp decrease in intensity across the shock. Notice the sharp differences in the lineout across the spike as compared to the one across the bubble. The lineout across the spike then has a gradual decrease in intensity and then another abrupt decrease at the tip of the spike. The lineout across the bubble also has a gradual decrease in intensity after the shock followed by an abrupt decrease and then a gradual increase in intensity from the remaining plastic layer. However, the bubble has a higher intensity than the spike since it appears lighter in the radiograph. The sharp increased in intensity on the left portion of the single-mode lineout are from the scratches in the film mentioned earlier. The positions of the shock, spike tip, and bubble head are shown by abrupt transitions in intensity, although the lineout shows them spread out over some horizontal distance. This is due to the finite resolution in experiment, the curvature of Springer

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Fig. 3 Radiograph at 17 ns of (a) single-mode perturbation. (b) 2-mode perturbation Fig. 4 (a) Lineouts at 17 ns across a spike and bubble in radiograph from single-mode perturbation and a (b) 2-mode perturbation

these features, and perhaps larger-scale variations in density and spike shape. In order to accurately and consistently define the locations of these features, a systematic method of analysis must be developed. The present, preliminary method involves finding the midpoint of the sharp decreases in intensity from the lineout in the portion determined to be the shock, spike tip or bubble head. The position of the bubble head is subtracted from the position of the spike tip to estimate the amplitude of the interface. At 17 ns, the amplitude of the single-mode perturbation is 143 µm and that of the 2mode perturbation is 168 µm. The larger interface amplitude of the 2-mode perturbation is consistent with the results of simulations indicating that additional modes may cause increased growth (Miles et al., 2003). We are now working on the analysis required to subtract out the effects of refraction, which will be necessary to determine a meaningful growth rate for comparison with theory. Springer

An additional radiograph can be seen in Fig. 5. This image is of a single-mode perturbation, but it is the view is down the tracer strip. Therefore, it is looking down 3 or 4 rows of about 13 spikes each. Note that on either side of the high contrast tracer strip are fainter spikes. These are spikes of the polyimide material surrounding the tracer strip. While polyimide is nearly transparent to the He-alpha x-rays used in this experiment, these spikes are a result of seeing an entire row of spikes aligned to be seen as one. For the same reason it is very difficult to see the location of the bubble head within the tracer strip. Looking through a row of ∼13 aligned spikes and bubbles causes a blurring of features in the radiograph. In contrast, the view across the tracer strip is only looking through ∼3 rows of aligned spikes and bubbles. The inaccuracy of the bubble head position in the view down the tracer strip can be seen from a lineout across the spike and bubble from the radiograph in Fig. 6. There is only a slight

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Conclusions

Fig. 5 Radiograph at 17 ns from single-mode perturbation down the strip view

Recent experiments have been successful in obtaining data from targets well scaled to the expected conditions at the H/He interface in the explosion phase of SN1987A. Initial findings have shown that adding additional modes over the basic “egg crate” perturbation create larger amplitudes of the mixed layer. Dual, orthogonal radiography has allowed two views of the interface and the resulting growth of the mix layer from the Rayleigh–Taylor instability. Estimating this amplitude can be done by finding the positions of the spike tip and bubble head. This paper has shown that it is challenging to estimate the position of the bubble head in the view down the strip because many spikes form a line blurring its position. However, the bubble head position can be found more easily and more accurately from the view across the tracer strip. The view down the strip is still very useful for confirming the spike tip and shock position as well as diagnosing target abnormalities.

Acknowledgements The author would like to acknowledge Kai Ravariere, Aaron Miles, Dave Arnett, and Casey Meakin for their useful technical discussions. Financial support for this work included funding from the Stewardship Science Academic Alliances program through DOE Research Grant DE-FG03-99DP00284, and through DE-FG03– 00SF22021 and other grants and contracts. This work is also supported in part by the U.S. Department of Energy under Grant No. B523820 to the Center for Astrophysical Thermonuclear Flashes at the University of Chicago.

References

Fig. 6 (a) Lineout from radiograph at 17 ns of single-mode data across bubble and spike from a view down the strip as compared to (b) the across the strip view

difference in the lineouts of the bubble and spike, making it almost impossible to locate the bubble head. Compare these lineouts to the lineouts from a view across the strip for the same perturbation at the same time in Fig. 4a. Notice that the bubble position in the across the strip view is much more distinct than in the down the tracer strip view.

Boehly, T.R., Craxton, R.S., et al.: Rev. Sci. Instr. 66(1), 508 (1995) Drake, R.P., Leibrandt, D.R., et al.: Phys. Plasmas 11(5), 2829 (2004) Kane, J., Arnett, D., et al.: ApJ 528, 989 (2000) Meshkov, E.E.: Fluid Dyn. 4, 101 (1969) Miles, A.R., Edwards, M.J., et al.: The Effect of a Short-wavelength Mode on the Nonlinear Evolution of a Long-wavelength Perturbation Driven by a Strong Blast Wave. Inertial Fusion and Science Applications, Monterey, CA (2003) Rayleigh, L.: Scientific Papers II. Cambridge, England, Cambridge (1900) Remington, B.A., Drake, R.P., et al.: Phys. Plasmas 7(May), 1641 (2000) Richtmyer, D.H.: Commun. Pure Appl. Math. 13, 297 (1960) Robey, H.F., Kane, J.O., et al.: Phys. Plasmas 8, 2446 (2001) Ryutov, D.D., Drake, R.P., et al.: ApJ 518(2), 821 (1999) Taylor, S.G.: Proc. R. Soc. A201, 192 (1950)

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Astrophys Space Sci (2007) 307:121–125 DOI 10.1007/s10509-006-9237-2

ORIGINAL ARTICLE

The Formation of a Cooling Layer in a Partially Optically Thick Shock A. B. Reighard · R. P. Drake

Received: 13 April 2006 / Accepted: 21 August 2006 C Springer Science + Business Media B.V. 2006


Abstract The mechanics of a radiative shock which has “collapsed,” or been compressed to high density, via radiative cooling is discussed. This process is relevant to an experiment in xenon gas that produced a driven, radiatively collapsed shock, and also to a simulation of the supernova 1987A shock wave passing through the outer layers of the star and into the low-density circumstellar material. Keywords Radiation . Radiation hydrodynamics . Optically thick shock . Driven shock . Collapsed shock . Laboratory astrophysics Introduction To reach a regime where the transport of radiation through a system can affect the hydrodynamic properties of the material, two conditions must be met. Firstly, one must create a system with sufficient “optical depth,” a measure of the attenuation of radiation. Said another way, there must be enough material that the radiation is affected by its presence. This system must also be relatively hot, on the order of at least tens of eV. This can be difficult in laboratory systems, requiring energy sources capable of delivering high laser power or X-ray flux to a small target, driving shock waves into materials of relatively low density. In astrophysics, energetic explosions into diffuse material, like a supernova explosion, produce similar effects. We have performed experiments at the Omega laser that have created a planar, driven, radiatively collapsed shock wave in xenon gas. This system is optically thick (highly absorptive of radiation) downstream in the shocked gas, and A. B. Reighard (
) · R. P. Drake Department of Atmospheric, Oceanic, and Space Sciences, University of Michigan, Ann Arbor, MI 48109, USA

after a brief transition period where the upstream gas is ionized by precursor radiation, optically thin upstream in the unshocked gas. This experiment (described further in Reighard et al. (2006)) has produced a shock wave that radiated away enough energy to cool both the electrons and the ions in the shocked material, causing the gas to become highly compressed in response. A metrology image of a target used in this experiment is shown in Fig. 1a. The beryllium drive surface is in the lower right-hand corner of the image, mounted on a polyimide tube 5 mm in length and 0.6 mm OD. Shown in Fig. 1b is a radiograph of collapsed xenon gas in this experiment, taken via side-on backlit pinhole radiography. The shock is moving to the right in this figure, where the x coordinate is the distance along the target axis from the initial position of the drive disk, while the y coordinate is the radial coordinate of the target. The dense xenon shows as a dark absorption feature. The shock front is just to the right of the dark absorption feature. The shape of the dense collapsed layer is affected by line of sight of the diagnostic and drag along the walls of the target, as well as any non-uniformity in the production of the layer from instabilities. It is then natural to seek astrophysical shocks with the same optical depth structure for comparison to this system. In an article by Ensman and Burrows, (1992), 1D hydrodynamic simulations of shock breakout in SN 1987A show a highly compressed, cooled shock as the disturbance passes through the outer layers of the star and into the lower density circumstellar material. The temperature and density profiles show a remarkable similarity to temperature and density profiles calculated in the experiment described above, as shown in Fig. 2. The density and temperature profiles for the experimental system in Fig. 2b. were generated using Hyades (Larsen and Lane, 1994), a 1D lagrangian code with a three-fluid treatment of the material and a multigroup treatment of the radiation with flux-limited diffusion. Springer

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Fig. 1 (a) Metrology image from experiments described in Reighard et al. Drive beams hit a Be disk (lower right corner of image), and accelerate it into the xenon filled plastic tube. The experiment is approximately 5 mm long. (b) Data image from Reighard et al. at 8.2 ns after drive beams turn on. The thin layer of collapsed xenon shown by X-ray radiography is approximately 65 µm thick, and has an average velocity of 140 km/sec. The gold grid used as a spatial and magnification indicator is in the top right of the image, while the edges of the target tube are visible as horizontal lines near the bottom and top of the image

These systems will produce this kind of collapsed shock structure in fundamentally the same way. This begs a causal explanation of the physics producing this structure in both systems. Here, we offer a step-by-step explanation of the physics behind the formation of a collapsed radiative shock. For each step in the discussion, parameters from the experiment described above (and in more complete detail in Reighard et al. (2006)) are used as an example.

Radiatively collapsed shock formation As in any shocked system, a shock wave heats ions as it passes through cold material. The shock accelerates everything to a higher velocity, but most of the energy of the shock goes into heating the heavier ions. This happens quickly, over the span Springer

Astrophys Space Sci (2007) 307:121–125

Fig. 2 1D Lagrangian simulations of the two systems discussed. (a) 1D VISPHOT calculation adapted from Ensman and Burrows (1992). (b) 1D Hyades simulation of discussed experimental system. Both systems show a layer of highly compressed material accompanied by quick cooling through a region that is thin compared to the distance the shock has traveled

of just a few ion-ion mean free paths. Through conservation equations, the matter is also compressed by a certain amount, depending on the polytropic index of the material, and the pressure increases. In an ionized system that can exchange energy by collisions, the ions begin to transfer energy to the cooler electrons. The rate at which this happens is dictated by the ion-electron collision rate, ν ie , given by

νie = 3.2 × 10−9

n i Z 3 ln  , ATe1.5

(1)

where ni is number density in cm−3 , Z is the average ionization state, ln  is the coulomb logarithm, A is the atomic weight of the material, and Te is electron temperature in eV (Drake, 2006). The rate of ion cooling depends on the difference between the ion temperature and the electron temperature; ∂ Tion = −νie (Tion − Te ). ∂t

(2)

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for an optically thin material, where σ is the StefanBoltzmann constant. On the approximation that T = Z Te + Ti = (Z + 1)Te if Te ≈ Ti , where Z is the average ionization state, this rate simplifies to

Temperature (eV)

(a) Ion temperature Electron temperature

∗ νrad = 2.2

Temperature (eV)

(b)

Ion temperature Electron temperature

Fig. 3 Ion and electron temperatures behind the shock front for the parameters from the experimental system, assuming a shock moving at 150 km/sec. The shock front is located at 0 cm. (a) Temperature profiles with only collisional heat exchange. (b) Temperature profiles with both collisional heat exchange and radiative cooling of the electrons. In this system, radiative cooling becomes important before collisional heat exchange can equilibrate the electron and ion temperatures

This process slows as the temperatures equilibrate. In xenon gas, with n i = 1.5 × 1019 , A = 131.3, with initially Te negligible, and Ti approximately 500 eV, this equilibration process takes on the order of a few hundred picoseconds. The ion and electron temperature profiles from collisional heat exchange in xenon gas are shown in Fig. 3a. The initial ion temperature is dictated by the strong shock equations for a shock wave moving at 150 km/sec, for which the equilibration length would be approximately 30 µm. This calculation is relevant to the experimental system described above, where a driven shock moves through xenon gas at velocities in excess of 100 km/sec. 1D Hyades simulations give a similar result, showing a equilibration length of approximately 25 µm at an instantaneous velocity of 160 km/sec. In some systems, the electron temperature may become large enough to cause the free electrons in the system to radiate. As the electrons in the system get warmer, they begin to radiate more. Though electrons in the system may continue to gain energy from collisions with ions, now they may also cool through radiative losses, at a fractional energy radiation rate given by the ratio of the radiative flux from two surfaces of a planar slab, 2Frad to the energy content of the shocked material of density ρ, slab thickness d, specific opacity κ, and specific heat (per unit Te ) cv . This ratio is

νrad =

2Frad 2κdρσ Te4 = ρdcv T ρdcv T

(3)

A κT 3 (Z + 1) e

(4)

in sec−1 for Te in eV and κ in cm2 /g. For the model calculation presented, we evaluated κ to be 2500 cm2 /g using a SEASAME table value for xenon at 100 eV, averaged over values for relevant densities (Leibrandt et al., 2005). If the collisional heating happens quickly, the heating and cooling of electrons may be treated as a two-step process. For the experimental system, significant radiative cooling begins before the ion and electron temperatures equilibrate, so the twostep approximation does not hold. Figure 3b. shows the temperature profiles produced by the combination of electronion heat exchange and radiative losses evaluated as just described. If the shocked gas can quickly radiate away a significant fraction of its energy, the system will form a collapsed layer in response to this loss. The cooling layer, where radiation cools the system, must be optically thin for energy to escape. As the electron temperature rises in the system not only will the radiative rate become large, the opacity of the system κ will decrease, making the hot material optically thin, allowing the energy to escape via radiation. As the shocked material loses energy, the system must respond to keep certain parameters constant across the shock. The conservation equations for mass and momentum must still hold, given by ρ1 u 1 = ρ2 u 2

(5a)

p1 (ρ, T ) + ρ1 u 1 = p2 (ρ, T ) + ρ2 u 2 .

(5b)

Pressure, density, and temperature are linked through these equations, so as the temperature drops another quantity must respond to keep the system in balance. If the pressure were to drop, there would be a void in the system, and the material would compress in response to energy loss. Pressure can be expressed as a function of temperature and density using the ideal gas law, p = ρ RTi ,

(6a)

where R is the gas constant, R=

(Z + 1)k B , Am p

(6b)

and k B is Boltzmann’s constant. Springer

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Fig. 4 Density profile when affected by radiation losses. In this simple analytic model, the radiative transfer equation was not fully employed, nor was the influence of increasing opacity as electron temperature decreased. Both these factors would affect the final density of the cooling layer. Here, the final density is dictated by the loss of all electron energy via radiation

Estimating the density profile from this kind of relationship requires some understanding of how the “gas constant” R changes with ionization. Using an estimate of how the average ionization Z changes with temperature allows an estimate of R. Using the Saha equation to calculate the average ionization state (only strictly valid in a equilibrium distribution, where ionization balances recombination exactly), Zbal is 
3/2  1 Te Z bal = 19.7 Te (1 + 0.19) ln − , (7) n 24 2 where Te is in eV and n24 is number density in units of 1024 cm−3 . Given these substitutions, pressure can be expressed solely as a function of temperature and density. The density profile can then be calculated from the temperature profile and the momentum conservation equation. While the profile generated in this way should be qualitatively correct, one would need to solve the radiative transfer equation to actually predict the final density value, taking into account the amount of radiation emitted and absorbed in each differential slab. In Fig. 4 we show the qualitative profile of radiative collapse, here without a full treatment of the radiative transfer. The layer continues to collapse until the system has radiated away all of the thermal energy of the system, where the electron temperature approaches zero, and the ion temperature is very low. Therefore in this incomplete model, the final value of the compression is not indicative of the value produced in the experimental systems. As the system becomes more compressed, the optical depth of the dense, collapsed layer increases, and the opacity can change as the system cools. As the optical depth of the material increases, it becomes more difficult to remove energy from within the dense material, in response to which the rate of density increase eventually reaches zero. If the system relaxes to a quasi-steady state, where the shocked Springer

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system maintains the same general shape, the ultimate thickness of the layer will depend on the balance of energy flowing through the layer. The optically thick downstream material will radiate at its blackbody temperature. This radiation will pass through the optically thin cooling layer but must be balanced by the radiation from the cooling layer to have steady state. The cooling layer will radiate equally both upstream and downstream. Because it is optically thin, the upstream, unshocked gas will radiate negligibly, and energy will escape from the system through it. Flux at the boundaries of the layer coupled with the hydrodynamic equations then give the final, post-shock temperature and the spatial extent of the cooling layer, as discussed in more detail in Drake (2006). In a driven system like the experiment described, the amount of momentum in the system is fixed. As the piston driving the shock amasses collapsed xenon, the system will decelerate. It is possible that the system will eventually slow to the point where the driving forces no longer heat the system to the point where radiation cooling can be effective. At this point, the evolution of the shock will become hydrodynamic in nature, and radiative collapse will cease.

Conclusion Radiative cooling in a system that is optically thick downstream (behind the shock) and optically thin upstream (in the unshocked material) can lead to dramatic effects in the overall structure of the shocked material. Collapse via radiative cooling of the shocked gas can lead to compression of material to much higher densities than those in a strong shock with no radiative cooling. Future work includes analysis of radiative effects at different initial driving velocities. This is achieved by varying the thickness of the Be layer illuminated by the laser. Computational efforts include work to better understand the effects of opacity on xenon at temperatures between 50 and 200 eV. In addition, by watching the long-term evolution of the shocked layer, such experiments might observe the onset of hydrodynamic instabilities like those discussed by Vishniac and Ryu (1989). Beyond such work, this type of system could be developed as a radiation source for experiments to examine other issues such as radiation transport.

Acknowledgements The authors acknowledge the vital contributions of the Omega technical staff and the target fabrication group at the University of Michigan. This work is supported by the National Nuclear Security Agency under DOE grants DE-FG0399DP00284 and DE-FG03-00SF22021, and by other grants and contracts.

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References Drake, R.P.: High Energy Density Physics: Foundations of Inertial Fusion and Experimental Astrophysics. (Springer, New York, 2006) Ensman, L., Burrows, A.: ApJ 393, 742 (1992)

125 Larsen, J.T., Lane, S.M.: J. Quant. Spectrosc. Radiat. Transfer 51(1), 179 (1994) Leibrandt, D.R., Drake, R.P., Reighard, A.B., et al.: ApJ 626, 616 (2005) Reighard, A.B., Drake, R.P., Dannenberg, K.K., et al.: Phys. Plasmas 13, 082901 (2006) Vishniac, E.T., Ryu, D.: ApJ 337, 917 (1989)

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Astrophys Space Sci (2007) 307:127–130 DOI 10.1007/s10509-006-9278-6

O R I G I NA L A RT I C L E

Measurement of the Growth of Perturbations on Blast Waves in a Mixed Gas A. D. Edens · R. G. Adams · P. K. Rambo · I. C. Smith · J. L. Porter · T. Ditmire

Received: 13 April 2006 / Accepted: 21 November 2006 C Springer Science + Business Media B.V. 2007


Abstract We have performed a series of experiments examining the properties of high Mach number blast waves. Experiments were conducted on the Z-Beamlet laser at Sandia National Laboratories. We created blast waves in the laboratory by using ∼1000 J laser pulses to illuminate millimeter scale solid targets immersed in gas. Our experiments studied the validity of theories forwarded by Ryu and Vishniac (1987, 1991) and Vishniac (1983) to explain the dynamics of perturbations on astrophysical blast waves. These experiments consisted of a systematic scan of the decay rates of perturbations of known primary mode number induced on the surface of blast waves by means of a regularly spaced wire array. The amplitude of the induced perturbations relative to the radius of the blast wave was tracked and fit to a power law in time. Measurements were taken for a number of different mode numbers in a mixed gas consisting of 7.5 Torr xenon and 2.5 Torr nitrogen and the results are compared to theoretical predictions. It is found that two of the three mode numbers imply one polytropic index while the third case, which is the most complicated for several reasons, implies a higher polytropic index. Keywords Vishniac overstability . Z-beamlet . KiloJoule laser . Laboratory astrophysics . Blast wave . Radiating shock Supernovae are some of the most energetic and impressive phenomena in the universe. There are a number of

A. D. Edens (
) · R. G. Adams · P. K. Rambo · I. C. Smith · J. L. Porter Sandia National Laboratories, Albuquerque, NM 87123 T. Ditmire University of Texas at Austin, Austin, TX 78712

instabilities associated with different times in the supernova process, from the collapse of the star to the merging of the supernova remnant (SNR) into the background medium. The particular instability we are interested in was theorized by Vishniac (1983), and is therefore known as the Vishniac overstability. In this overstability, illustrated in Fig. 1, there is a mismatch between the ram and thermal pressures at the blast wave surface. This mismatch can create an oscillating ripple in the blast wave surface. Depending on the thickness of the blast wave and the wavelength of the perturbation in its surface, this oscillating ripple can grow. After the initial paper Vishniac and Ryu wrote several papers generalizing their theory for more realistic blast waves. Vishniac and Ryu determined that the amplitude of a perturbation on a blast wave varies as a power law in time, A ∝ Ct s . For a given wavelength, C is a constant that depends on the spherical harmonic for the mode number in question, A is the amplitude of the perturbation, and s is the growth rate. The growth rate should depend on two variables: the mode number of the perturbation and the thickness of the blast wave, measured by its polytropic index. The polytropic index is a measure of the number of degrees of freedom for a gas, and decreases (along with the blast wave thickness) with an increasing number of degrees of freedom (Grun et al., 1991). Radiation provides an effective degree of freedom for a gas, lowering its polytropic index. There have been a number of experiments looking at laser produced blast waves. The first we could find in the literature was performed in 1972 by Basov et al. (1972), who used multi-sided illumination of a spherical target to create a blast wave in 15 Torr of residual air in order to gauge the amount of laser energy absorbed by that target. Later, Grun et al. (1991) looked at blast waves produced by single-sided illumination of plastic foils in 5 Torr of nitrogen or xenon gas in order to look at the Vishniac overstability. They showed that blast Springer

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Fig. 2 Illustration of half-cylinder array used in blast wave experiments. There are 1 mm spaced grooves surrounding the open area

Fig. 1 Illustration of the physics behind the Vishniac overstability. A mismatch in the direction of the two main forces on a blast wave surface, the thermal pressure directed normal to the surface and the ram pressure directed opposite the direction of motion, creates an oscillating ripple that can grow if conditions are right

waves in xenon were unstable to the Vishniac overstability, while those in nitrogen remained stable. Unfortunately, there was interference on the growth rate measurement due to the effects of the drive laser (Edens et al., 2004). More recently there have been a number of experiments looking at blast waves in planar (Keiter et al., 2002), cylindrical (Ditmire et al., 2000; Edwards et al., 2001; Shigemori et al., 2000), and spherical (Edens et al., 2004, 2005a, b; Hansen et al., 2006) geometries at a number of facilities. The present authors have been involved in experiments (Edens et al., 2005a, b) measuring the evolution of perturbations induced on blast wave and comparing those evolution rates to the theoretical predictions of Vishniac and Ryu. In the present work we present the results of those experiments for blast waves traveling in a mixture of nitrogen and xenon gas. We performed our experiments on the Z-Beamlet laser at Sandia National Laboratories (Rambo et al., 2005), and the general experimental setup is described in a previous publication (Edens et al., 2005b). The only changes from the setup described in that publication are the design of the wire array and the choice of gas. In order to reduce the modal noise caused by the use of a planar wire array, we used a halfcylinder array for this experiment. The array was designed to be one half of a 1 cm diameter cylinder with extended sides, and an illustration of the design can be seen in Fig. 2. With this setup it is possible to change both of the primary variables involved in the growth rate of perturbations on blast waves. The polytropic index of the gas will depend on the amount of radiation emitted by the gas, which will vary with the gas type. The primary mode number of the induced perturbation will vary with the spacing of wires in the array. The experiments reported here were performed in one gas (our mixed gas) and with three different wire array spacings.

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The gas used was 7.5 Torr xenon and 2.5 Torr nitrogen and will be referred to as the mixed gas from this point on. This mixture was chosen to maximize the radiative properties of the gas while still providing useable data. When experiments were attempted in pure xenon, the radiation from the main blast wave irradiated the wires in our array, creating small blast waves off the surface of the blast wave. These additional blast waves were large enough to obscure the main blast wave and make obtaining useable data impossible. To combat this, we decided to add in some nitrogen gas, which is less radiative, to reduce the overall radiation level. We first tried a mixture of 5 Torr xenon and 5 Torr nitrogen, and the additional blast waves with this mixture were small enough to see the main blast wave clearly. We then moved on to 7.5 Torr xenon and 2.5 Torr nitrogen and found the data were marginally useable depending on the choice of wires in the array. As can be seen in Fig. 3 when plastic coated copper wires were used in the array, the additional blast waves obscured the main blast wave. However, when uncoated copper wires were used, the main blast wave could be seen and measured. One can barely discern in the coated wire image a main blast wave similar to that seen in the uncoated wire image, but it is very faint. This dependence of the image quality on the wire type indicated to us we were at a radiation level that was at the edge of our tolerance and therefore we used this gas mixture for the experiments. Images of blast waves traveling in our mixed gas are seen in Fig. 4. The edge of the main blast wave was traced out in each image and the resulting plot transformed into polar coordinates. This transformed data was then interpolated to give constant spacing between the points and the mean radius of the blast wave was subtracted out. The angular coordinate was plotted in fractions of a circle and the graph was Fourier transformed so that we could isolate the frequency of interest. The amplitude of this frequency was then examined as a fraction of the mean radius and plotted versus time. We then fit a power law in time to the normalized amplitude versus time plot in order to compare the fit exponent to the theoretical predictions and an example of this can be seen in Fig. 5. The comparison of our experimentally determined evolution rates for perturbations on blast waves in mixed gas

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Fig. 3 Two Schlieren images of blast waves traveling in mixtures of 7.5 Torr xenon gas and 2.5 Torr nitrogen gas past a 3 mm spaced wire array. The wires in the array in the left image are coated in plastic, while those in the right hand image are bare copper wires. You can clearly make out a blast wave in the right hand image, while the blast wave in

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the left hand image is obscured by additional blast waves off the wires in the array. This difference based on the wires in the array indicated that we had chosen a mixture of gas that provided the maximum radiation while maintaining discernable data

Fig. 5 Plot of the normalized amplitude versus time for perturbations with a mode number corresponding to that induced by traveling past a 4 mm spaced wire array. The results are fit to a power law in time so that they can be compared to theoretical predictions

Fig. 4 Images of blast waves in 10 Torr of our mixed gas traveling past a 3 mm spaced wire array

to theoretical predictions for several different values of the polytropic index are shown in Fig. 6. We can see that the growth rates for the two lower mode number perturbations imply a polytropic index between 1.1 and 1.2, but that the decay rate for the highest mode number perturbation is more consistent with a polytropic index of 1.3. There are several possible explanations as to why the highest mode number data point does not agree with the other two. The first is that the oscillating nature of the overstability is

not taken into account when analyzing the data. The theory says the oscillation rate should be highest at the higher mode numbers. It may be that the effect of the oscillation during the time period studied was to reduce the amplitude of the perturbation and increase the apparent decay rate, thus raising the implied polytropic index. The next explanation for the behavior of the high mode number perturbation is that the smaller wavelength perturbation was simply more difficult to resolve from the data. The smaller length scale of the perturbations may increase the error in the raw data. Finally, the wavelength of the perturbation may be similar to the thickness of the blast wave during the time period studied and if so, the physics becomes more complicated and this may account for the behavior of the array. Unfortunately, simulations like those performed in support of our earlier work (Edens et al., 2005b) are not possible, due to the fact that there is no available equation of state for such a gas mixture. Springer

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discrepancy, including oscillatory behavior of the perturbations being more pronounced in the higher mode number data. Future experiments should look at a gas that is more radiative that nitrogen, but less than xenon (perhaps krypton). Perturbations on a blast wave in such a gas may show similar behavior while being possible to simulate.

References

Fig. 6 Comparison of experimentally determined growth rates for perturbations on blast waves traveling in mixed gas to the theoretical predictions of Vishniac and Ryu. Note that the growth rates for the two lower mode number perturbations correspond to a polytropic index (gamma) of 1.1–1.2 while the highest mode number perturbation has a growth rate that implies a polytropic index of 1.3

In conclusion, we have looked at the evolution of perturbations on blast waves traveling in a gas consisting of 7.5 Torr xenon and 2.5 Torr nitrogen. For low mode number perturbations, growth is observed and the implied polytropic index is between 1.1 and 1.2, but at a higher mode number the observed decay rate is more consistent with a higher polytropic index of 1.3. There areseveral possible explanations for this

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Basov, N.G., Shikanov, A.S., Sklizkov, G.V., et al.: Sov. Phys. Jetp-Ussr 35, 109 (1972) Ditmire, T., Shigemori, K., Remington, B.A., et al.: Astrophys. J. Suppl. Ser. 127, 299 (2000) Edens, A.D., Ditmire, T., Hansen, J.F., et al.: Phys. Plasmas 11, 4968 (2004) Edens, A.D., Ditmire, T., Hansen, J.F., et al.: Astrophys. Space Sci. 298, 39 (2005a) Edens, A.D., Ditmire, T., Hansen, J.F., et al.: Phys. Rev. Lett. 95, 244503 (2005b) Hansen, J.F., Edwards, M.J., Froula, D.H., et al.: Phys. Plasmas 13, 022105 (2006) Edwards, M.J., MacKinnon, A.J., Zweiback, J., et al.: Phys. Rev. Lett. 8708, 085004 (2001) Grun, J., Stamper, J., Manka, C., et al.: Phys. Rev. Lett. 66, 2738 (1991) Keiter, P.A., Drake, R.P., Perry, T.S., et al.: Phys. Rev. Lett. 89, 165003 (2002) Rambo, P.K., Smith, I.C., Porter, J.L., et al.: Appl. Opt. 44, 2421 (2005) Ryu, D., Vishniac, E.T.: Astrophys. J. 313, 820 (1987) Ryu, D., Vishniac, E.T.: Astrophys. J. 368, 411 (1991) Shigemori, K., Ditmire, T., Remington, B.A., et al.: Astrophys. J. 533, L159 (2000) Vishniac, E.T.: Astrophys. J. 274, 152 (1983)

Astrophys Space Sci (2007) 307:131–137 DOI 10.1007/s10509-006-9260-3

ORIGINAL ARTICLE

Colliding Blast Waves Driven by the Interaction of a Short-Pulse Laser with a Gas of Atomic Clusters Roland A. Smith · James Lazarus · Matthias Hohenberger · Alastair S. Moore · Joseph S. Robinson · Edward T. Gumbrell · Mike Dunne

Received: 18 May 2006 / Accepted: 22 September 2006 C Springer Science + Business Media B.V. 2006


Abstract Collisions between shocks are commonly found in many astrophysical objects, however robust numerical models or laboratory analogues of these complex systems remain challenging to implement. We report on the development of scaled laboratory experiments which employ new techniques for launching and diagnosing colliding shocks and high Mach number blast waves, scalable to a limited subset of astrophysically-relevant regimes. Use of an extended medium of atomic clusters enables efficient (>80%) coupling of 700 fs, 1 J, 1054 nm laser pulses to a “cluster” gas with an average density of ≈1019 particles cm−3 , producing an initial energy density >105 J cm−3 , equivalent to ≈5 × 109 J/g. Multiple laser foci are used to tailor the spatial profile of energy deposition, or to launch pairs of counter-propagating cylindrical shocks which then collide. By probing the collision interferometrically at multiple view angles in 5◦ increments and applying an inverse Radon transform to the resulting phase projections we have been able to tomographically reconstruct the full three-dimensional, time-framed electron density profile of the system. Keywords Hydrodynamics . Instabilities . Plasmas . Radiative transfer . Shock waves . Supernova remnants . Atomic clusters . Lasers . Laboratory astrophysics . Blast wave collision . Tomography . Radon transform R. A. Smith (
) · J. Lazarus · M. Hohenberger · J. S. Robinson The Blackett Laboratory, Imperial College, London, UK e-mail: [email protected] A. S. Moore · E. T. Gumbrell AWE Aldermaston, Reading, Berkshire, UK M. Dunne The Rutherford Appleton Laboratory, Chilton, Didcot, Oxfordshire, UK

1 Introduction The dynamics of shocks and blast waves and the behavior of spatial instabilities are of key importance in many areas of plasma physics, including fields as diverse as laser-matter interaction experiments and the evolution of structures within supernova remnants (SNR’s). Many systems are further complicated by interactions between shocks, and shock collisions can commonly be observed in astrophysical objects such as nebulae and supernova remnants over an extraordinary range of spatial and temporal scales. They can also occur in laboratory-scale experiments, including high-energy-density plasmas driven by Z-pinches (Lebedev et al., 2002) and energetic laser systems (Woolsey et al., 2001). Suitably-scaled laboratory experiments may thus provide an important tool for both improving physical insight and for code benchmarking. The understanding of the complex dynamics of many high-energy-density astrophysical systems is underpinned by numerical simulation (Stone and Norman, 1992; Casanova et al., 1991). However, modeling of shock propagation, and in particular shock collisions, remains challenging despite several decades of effort. Both fluid and particle interactions need to be tracked simultaneously over a large computational grid and the situation can be further complicated by the interplay between plasma radiation processes and macroscopic hydrodynamics. For example, radiation produced by the shock may pre-ionize surrounding material, thus modifying the properties of the matter it propagates into. Addressing this interplay numerically may necessitate the computationally expensive use of linked atomic physics and hydro routines. Woodward and Colella (1984) have also highlighted shock collisions as an area that most challenges the abilities of numerical models designed for the simulation of astrophysical objects. There consequently exists a clear demand Springer

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for well-characterized laboratory experiments in this area to help explain the complexities of the physics involved (Report for the US NRC, 2003). The absolute length, time and density scales in laboratory experiments are clearly very different from those characteristic of objects such as supernova remnants. However, the underlying dynamics can be invariant under an Euler transformation linking time, space and velocity (Ryutov et al., 1991). This invariance can be maintained by matching the dimensionless parameters which describe dissipative processes such as heat conduction, convection and viscous hydrodynamic flow, and by careful control of initial conditions in an experiment (Moore et al., 2006). This enables the use of well-characterized laboratory experiments to test both sophisticated numerical codes, and the theories describing the development of rich spatial structures seen for example in SNR’s (Weiler et al., 1998). Recent developments of both high-power laser systems and atomic cluster target sources have enabled small-scale laboratory experiments to be conducted in regimes of relevance to a limited range of astrophysical systems. Here we describe new techniques for both creating and diagnosing collisions between shocks which allow us to acquire a full three-dimensional, time-framed map of the electron density profile of the shock collision.

2 Atomic clusters as a target medium Atomic clusters are cold, weakly bound aggregates of a few to many millions of cold atoms. These fragile entities can be produced in low-temperature, expanding gas streams (Smith et al., 1998) and, rather counter intuitively, a mm scale extended medium of such clusters can exhibit extremely efficient absorption of high intensity laser light. In contrast to the <1% absorption typically seen in monatomic gases, sub-picosecond laser pulses with energies of order 1 J can deposit >80% of their energy in such a medium (Ditmire et al., 1997a). This has resulted in their use to drive a diverse range of processes including x-ray generation (Ditmire et al., 1998b), the production of highly-charged ions (Smith et al., 1999) and “tabletop” thermonuclear fusion (Zweiback et al., 2000a,b). More recently it has been shown that laser energies up to ≈450 J (Moore et al., 2005a) can be deposited in a Xe cluster medium, further increasing the available energy density to ≈108 J cm−3 , or equivalently ≈5×1010 J/g. The resulting increase in blast wave velocity may enable access to new regimes where strong growth of plasma instabilities such as the Vishniac overstability are likely. The coupling of single atomic clusters with sub-picosecond lasers has been studied extensively and can be well explained in terms of a transient Mie-like resonance (Ditmire, 1998) which drives an energetic Coulomb or hydrodynamic explosion (Ditmire et al., 1997b). Springer

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Heating of an extended cluster medium by a single highintensity laser beam focused into the plume produced by a pulsed gas jet (Smith et al., 1998) creates a plasma filament with a diameter comparable to that of the laser focus – typically a few 10 s of microns. This hot plasma cylinder is created on a timescale comparable to the laser pulse duration (typically sub-picosecond) and is initially surrounded by cold gas at an average atomic density of ≈1019 cm−3 . On the spatio-temporal scales of the subsequent energy transport, this unique type of interaction can be accurately viewed as delta function heat deposition. After this abrupt energy input the system initially exhibits non-local effects and, over time, thermalises, forming a hydrodynamic blast wave (Ditmire et al., 2000). Energy flow in the system can be driven by electron transport (both non-local and diffusive), radiation transport or hydrodynamic flow. Typically several of these mechanisms will dominate the dynamics at various times during the evolution. A more complete discussion of these processes is given by Moore et al. (2006) in this issue. By selecting high Z target species it is also possible to form radiative blast waves (Moore et al., 2006; Ditmire et al., 1998a). This can lead to the pre-heating of un-shocked material, significantly altering the propagation dynamics. The fragility of atomic clusters also provides a unique route for creating “tailored” shock and blast waves in a range of interesting geometries. Clusters can be broken apart by a lower intensity (<1014 W cm−2 ) laser pulse in selected regions of a target, leaving a cold medium with constant atomic density but strongly modulated absorption properties. A system for producing and optically probing such a tailored cluster medium is shown in Fig. 1. Heating of this pre-prepared medium with a second, high-intensity (≈1017 W cm−2 ) laser pulse then launches a tailored ionization wave (Fig. 2) (Symes et al., 2002; Moore et al., 2005b) which could, for example, be used to study growth of instabilities seeded by a range of spatial frequencies.

3 Creation of colliding blast waves The creation of blast waves in the laboratory using highpower lasers is an area of long standing interest (Grun et al., 1991; Dunne et al., 1994; Drake et al., 2002). Sub-picosecond lasers now allow energy deposition to be decoupled from the subsequent 100 ps–100 ns hydrodynamic motion of a plasma, greatly simplifying the understanding of these experiments. As a result, laboratory studies investigating the radial expansion of spherical and cylindrical blast waves in gas media have made significant progress in the last few years (Edens et al., 2004; Ditmire et al., 1998a). In contrast, only a limited number of counter-propagating plasma experiments to investigate shock collisions have been attempted (Elton et al., 1994; Bosch et al., 1992). These (and older unpublished studies which utilized electromagnetically-driven

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Fig. 1 Experimental layout for producing and imaging tailored blast waves. A cluster medium produced in a gas jet expansion is “machined” by a low-power 1054 nm beam at ≈1014 W cm−2 . A strong shock is then launched using a high-power 1054 nm picosecond pulse focused to ≈1017 W cm−2 and imaged using a time delayed 527 nm interferometric probe

Fig. 2 Electron density map of a strongly modulated blast wave in an extended hydrogen cluster medium at 6 ns after the heating pulse. A thin-shelled blast wave is present but no evidence of instability growth is seen at this time

shock tubes) were not able to access the strong shock or blast wave regime. The creation of a well-characterized, high energy density colliding shock experiment in the laboratory is rather challenging and very limited data on such systems has appeared in the literature to date. Pairs of laser-driven foil targets have, for example, been used to create counter-propagating plasma flows with collisions diagnosed through X-ray spectroscopy (Elton et al., 1994), while only a few attempts have been made to recover the electron density profile of pairs of colliding shocks (Bosch et al., 1992; Gregory et al., 2005). Our data contrasts significantly with these previous studies carried out using long pulse (ns) lasers, in which the collisions occurred between steady state shocks, not blast waves. Moreover, we have been able to provide significantly improved spatial resolution for imaging of collisions. This is pre-requisite to studying thin-shell shocks (blast waves).

The absorption characteristics of a cluster target medium enables high-Mach-number shocks (Ditmire et al., 1998a) to be created efficiently by delta function heating using a small scale, high-repetition-rate laser system. This also enabled us to quickly acquire large (>100 shot) data sets in order to minimize statistical fluctuations arising from shot-to-shot variations in laser energy, and to investigate shock collisions over a broad range of initial particle and energy densities. In order to launch pairs of colliding cylindrical blast waves we employed a split focus system (Fig. 3). A 700 fs, 800 mJ, 1054 nm heating beam from an Nd:Glass CPA laser system was spatially split into two halves using a shallow angle 15◦ Fresnel bi-prism. These two beamlets propagated at opposite angles to the original laser axis and were recombined using a second identical prism placed ∼0.25 m downstream. By tilting the second prism about an axis normal to its triangular cross-section, phase can be added to the two recombined beamlets asymmetrically, resulting in a slight angular deviation from parallel. Subsequent focusing with an F/10 plano-convex lens produced two focal spots of peak intensity Imax ≈ 1017 W cm−2 with variable separation in space. These foci were used to heat two near-parallel, cylindrical regions of a 2 mm-scale cluster medium produced by a cryogenically cooled, pulsed gas jet (Smith et al., 1998). The subsequent blast wave evolution was probed with a number of diagnostics including high-spatial resolution (≈3 µm) interferometric imaging perpendicular to the heating beam axis. To create the probe a small amount of light split from the main heating beam was frequency doubled to produce a 527 nm, 500 fs pulse which was passed through a variable time delay (0–150 ns) and used to backlight the plasma channel (in a manner similar to Fig. 1). Transmitted light was imaged through a Michelson interferometer onto a CCD image capture system. Interferograms captured in this way were unwrapped to retrieve a two-dimensional phase map Springer

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Fig. 3 A bi-prism arrangement for launching two parallel, cylindrical blast waves in a cluster gas medium. The system can be rotated around the laser axis to allow optical probing at a range of angles. Here the viewing angle is defined as 90◦ Fig. 4 Experimental line-integrated phase image of a pair of colliding cylindrical blast waves in an H2 cluster gas probed at 15◦ . The blast waves have evolved 5 ns after deposition of ≈105 J cm−3 (5 × 109 J/g) into pre-shocked material at an average atomic density of ≈1019 cm−3 . There is an apparent enhancement of phase shift in the collision region. However, the lack of cylindrical symmetry precludes recovery of the electron density profile via Abel inversion

and the three-dimensional electron density profile of a single cylindrically-symmetric blast wave (as per Fig. 2) recovered via an Abel inversion of the phase profile. However, the presence of two shocks or a shock collision region (Fig. 4) breaks the cylindrical symmetry necessary for retrieval of the electron density by this technique. 4 Tomographic reconstruction of 3D electron density profiles Due to the strong departure from cylindrical symmetry in the case of colliding blast waves, Abel inversion schemes are no longer appropriate for retrieval of electron density. To overcome this limitation we have implemented a tomoSpringer

graphic technique, which acquires multiple two-dimensional phase maps of colliding blast waves over a range of view angles. By rotating the prism system used to generate pairs of blast waves about the laser axis while keeping the optical probe system fixed we are able to image the collision region interferometrically in order to probe at different angles on successive laser shots. A ±7% energy bin (together with the modest E1/4 scaling of blast wave radius in cylindrical geometry with deposited energy) ensures that successive view angles sample plasmas derived from very similar initial conditions. We expect an E1/4 scaling of an adiabatic blast wave radius here as a result of our cylindrical geometry, rather than the more usual E1/5 case for spherical geometry. A projection can then be built up from the two-dimensional phase maps

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Fig. 5 A 2D section taken perpendicularly through the tomographically reconstructed 3D electron density profile of two colliding, cylindrical, thin shelled blast waves in an H2 cluster gas. 18 individual interferograms taken at 5◦ intervals at t = 9.75 ns in a 350 ± 25 mJ energy bin were used in the reconstruction. Electron density spikes are seen at the two collision ‘cusps’ where the thin shells intersect

obtained through standard unwrapping routines, and an electron density cross section can be calculated using the inverse Radon transform (Kak and Slaney, 2001). Due to symmetry considerations we were able to obtain a reconstruction with 5◦ resolution using 18 individual energy binned laser shots. In this work we have assumed a single plane of symmetry perpendicular to the shock collision to reduce the number of angles required by a factor of two. However, we emphasize that more generally this technique can recover the full three-dimensional electron density profile of the system for a single time interval, without recourse to any assumptions about the symmetry of the system, provided a sufficiently large number of view angles can be sampled. Figure 5 shows a two-dimensional slice through the reconstructed three-dimensional electron density profile of two colliding cylindrical blast waves driven in a hydrogen cluster gas at 9.75 ns after initiation and 1.75 ns after the blast waves first begin to collide. Successive blast waves were launched using 350 ± 25 mJ of laser energy approximately equally split between the two individual blast waves and images were recorded for a series of view angles taken at 5◦ intervals. The faint striations visible in the image are the result of reconstruction artifacts caused by the finite number of view angles used in the tomographic process, however the structure of the two thin-shelled blast waves and the collision boundary between them is clearly well resolved. A slight curvature of the boundary between the two blast waves is seen, which we attribute to a small asymmetry in the energy split between the heating beamlets resulting in a difference in internal energy density of the two waves. The thin-shell structure of the regions of the individual blast waves expanding away from the collision zone is also clearly visible, as is an enhancement of the electron density by a factor of ≈1.5× where the blast waves have collided. At the apex of the collision region

and the uncollided edges of the two cylinders two localized spikes in electron density ≈2.5× the peak shell density can also be seen. Figure 6 shows plots of electron density for sections taken perpendicular to and parallel to the collision region for clarity. The mechanism by which this additional increase in density occurs is currently unclear and will be investigated in future work. One possibility is that this is the result of the early stages of a Mach stem being formed. 5 Discussion While extremely powerful, the tomographic technique we report here has a number of important limitations – and possible enhancements – which are worth noting. The methodology we have described above requires multiple laser shots and is thus limited to situations where good shot-to-shot reproducibility of the plasma dynamics can be assured. This necessitates energy binning or a high degree of laser pulse reproducibility, together with a sufficiently large data set. More importantly, it also demands that the underlying physical processes should be robust under small-scale fluctuations in deposited energy and gas density etc. For the gross dynamics of the thin-shelled hydrogen blast wave that we have used as an example case, these conditions are well met (Moore et al., 2006). However, there are important situations in which such a tomographic technique would be invaluable, which are interesting precisely because shot-to-shot reproducibility is likely to be poor. One key example would be the study of instability growth where strongly non-linear processes can result in significant variations in both small and large scale spatial structures on near identical shots, for example the Vishniac overstability (Vishniac, 1983; Grun et al., 1991). In such a case a multi-shot technique is likely to blur out important details. However, we believe that the tomographic Springer

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of colliding blast waves can also be created using a modest table-top-scale laser system. By combining a split prism system able to vary the orientation of two colliding cylindrical blast waves with high resolution optical interferometry we have been able to apply tomographic reconstruction techniques to a blast wave collision for the first time. This has allowed us to reconstruct the fully three-dimensional, time-framed electron density profile of a blast wave collision for the first time. Studies are now underway to investigate collision dynamics in a range of high and low Z gases with the aims of improving physical understanding, and to provide high quality experimental data which can be used for benchmarking astrophysical codes. Acknowledgements This work was supported by grants from the EPSRC and AWE Plc. We are pleased to acknowledge the technical support of P Ruthven, B Ratnasekara and M. Dowman.

References

Fig. 6 Electron density profiles of (a) a section perpendicular to the collision region and (b) along the collision region for the pair of blast waves described in Fig. 5. Enhancements in electron density compared to the thin shell can be seen in both the collision region itself, and at the apex of the two thin shells and the collision region. The quantities x ∼ 0.1 mm and y ∼ 0.5 mm are the width and length of the collision region respectively

technique reported here could be implemented on a single shot basis for cases where the increased complexity of the optical system required is warranted by the payoff in terms of the physical insight gained. 6 Conclusion We have developed new techniques based on an extended medium of atomic clusters which allow us to study the propagation dynamics and collisions of high Mach number cylindrical shocks with unprecedented levels of detail. The combination of high absorption efficiency of a cluster medium and the fragility of individual clusters allows tailored blast waves to be produced for instability growth studies. Pairs Springer

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Astrophys Space Sci (2007) 307:139–145 DOI 10.1007/s10509-006-9266-x

O R I G I NA L A RT I C L E

Investigating the Astrophysical Applicability of Radiative and Non-Radiative Blast wave Structure in Cluster Media Alastair S. Moore · James Lazarus · Matthias Hohenberger · Joseph S. Robinson · Edward T. Gumbrell · Mike Dunne · Roland A. Smith

Received: 15 May 2006 / Accepted: 17 October 2006 C Springer Science + Business Media B.V. 2006


Abstract We describe experiments that investigate the capability of an experimental platform, based on laser-driven blast waves created in a medium of atomic clusters, to produce results that can be scaled to astrophysical situations. Quantitative electron density profiles were obtained for blast waves produced in hydrogen, argon, krypton and xenon through the interaction of a high intensity (I ≈ 1017 Wcm−2 ), sub-ps laser pulse. From this we estimate the local post-shock temperature, compressibility, shock strength and adiabatic index for each gas. Direct comparisons between blast wave structures for consistent relative gas densities were achieved through careful gas jet parameter control. From these we investigate the applicability of different radiative and Sedov-Taylor self-similar solutions, and therefore the (ρ, T ) phase space that we can currently access. Keywords Laser-cluster interactions . Laser-driven shocks and discontinuities . Hydrodynamic and radiative plasma instabilities . Laboratory astrophysics . Radiative blast waves . Blast waves 1 Introduction Shocks are ubiquitous throughout the observed universe and are thought to play a crucial role in the transport of energy A. S. Moore (
) · E. T. Gumbrell AWE Aldermaston, Reading, Berkshire, RG 74 PR J. Lazarus · M. Hohenberger · J. S. Robinson · R. A. Smith Laser Consortium, The Blackett Laboratory, Imperial College, London, SW7 2BZ, United Kingdom M. Dunne Rutherford Appleton Laboratory, Chilton, Didcot, Oxfordshire, OX11 0QX, United Kingdom

into the interstellar medium, setting in motion processes observed in nebulae that eventually could lead to the creation of new stars. Radiation is key to this since it plays a significant role in energy transport over the vast distances encountered between stellar objects, and can significantly modify the dynamics of a shock or blast wave. Consequently the study of radiative shocks has been an active area of numerical and theoretical research over many years, but recently has been highlighted as an area requiring further experimental investigation (US National Research Council, 2003). A radiative shock can broadly be defined as a shock in which radiative cooling occurs over a shorter time than the hydrodynamic timescale of the shock, τh ≈ shock thickness (h)/shock speed. Astrophysical shocks, structures commonly found in supernova remnants (SNRs), often lie within this classification and have been seen to exhibit many different physical forms and modes of instability and overstability (e.g. Cassiopeia A SNR (Hwang et al., 2000)). Very much related to this is the study of radiative blast waves. A blast wave, in which the rarefaction has caught up with the decelerating shock front, is affected in a similar way by the presence of radiation. Such radiation can transport energy ahead of the shock front and preheat the upstream unshocked material, under certain conditions causing small ripples or wrinkles in the blast wave surface to grow non-linearly. Extensive astrophysical research has focused on these problems (Kimoto and Chernoff, 1997; Ostriker and McKee, 1988; Vishniac, 1983; Ryu and Vishniac, 1991, 1987), and there is clear evidence from the structure seen in astrophysical observations that instabilities must be driven in certain phases of the expansion of supernova remnants. However, current code capabilities do not enable adequate understanding of the many areas of physics that must be included in a full description due to the differing length-scales involved (Drake et al., 2000). To aid the understanding of the physics of Springer

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some of these observations laboratory experiments have been conducted with the aim to reproduce the unstable blast wave structure which has been analytically predicted (Vishniac, 1983; Ryu and Vishniac, 1991, 1987; Grun et al., 1991; Schappert et al., 1994). As part of an ongoing campaign to experimentally investigate these instabilities, we report on new results that focus on the blast wave parameters – namely the strength and structure – that can be created in the laboratory. In this investigation we use a well-established platform that exploits the efficient absorption of short pulse (sub-ps) laser energy in an atomic cluster medium to produce strong blast waves (Ditmire et al., 2000; Shigemori et al., 2000; Edwards et al., 2001). An intense laser pulse (∼1017 Wcm−2 ) heats a plasma filament a few mm long, which expands to form a shock and later a blast wave. We study the interaction of a 1 J, 700 fs laser system with large H2 , Ar, Kr, and Xe clusters. Limiting our investigation to studying the structure of the blast wave front, in this paper we aim to fully identify the current range of hydrodynamic parameters that we are able to access and the most appropriate physical regime to scale to astrophysical blast waves. Through high-resolution interferometry using a sub-ps, second-harmonic optical probe beam, we were able to extract time-framed, two-dimensional electron density information, and measure the shock structure with ±3µm accuracy. 2 Creating hydrodynamically-similar laboratory and astrophysical systems Astrophysical phenomena and laboratory experiments differ vastly in both scale and temperature. However, this scale difference can be overcome by fulfilling four criteria for hydrodynamic similarity (Ryutov et al., 1999): (i) localisation, (ii) convective heat flow, (iii) negligible radiation flux, (iv) viscosity. Quantitatively these must be satisfied to approximately the same degree in both the laboratory and astrophysical case for the hydrodynamic results to be comparable. In order to study radiative effects, criterion (iii) is relaxed and the opacity of the upstream and shocked material and the radiative cooling rate then become critical to the flow characteristics. The differing peak temperatures of the plasmas in each case also results in different temporal characteristics in a laboratory experiment from those seen in an astrophysical object. For the case of a cooling SNR, which has been extensively studied in the astrophysical community, the temperature is initially much higher than in the laboratory (Tastro ∼1–100 keV) and the system must evolve for a long time (∼104 yrs) before radiative cooling becomes important (Blondin et al., 1998). Conversely, in the laboratory, once local thermodynamic equilibrium (LTE) is achieved, the typical plasma temperature is 0.1–100 eV and so causes Springer

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radiative effects to become apparent almost immediately that the system becomes hydrodynamic. 3 Blast wave self-similarity The commonly-used self-similar solution for blast wave propagation, first formulated by Sedov (1959), is found through consideration of the total energy in the system and results in the reduction in the number of non-dimensional parameters to just two. For a simple spherical or cylindrical system this analysis yields five dimensional variables – density (ρ), pressure (p), energy (E), radius (r) and time (t) – and one dimensionless variable – the adiabatic index, γ . In the case of a strong shock (Mach number, M > 1.3) the ambient pressure can be neglected and the list of dimensional parameters is further reduced to four, all of which are dependent on three dimensions: mass, length and time. Since the system is thus reduced to four equations and three dimensions, it is straightforward to re-write the equations for a dimensionless parameter, ξ , confirming that the system can be described as self-similar, reproducing the common parameters in spherical (sph) and cylindrical (cyl) coordinates (Sedov, 1959; Zel’dovich and Raizer, 1966, 1967). ⎧ 1 ⎪ ⎪ β
βsph = ⎨ ρ0 5 ... (1) ξ =r ⎪ Et 2 1 ⎪ ⎩ βcyl = 4 Although this Sedov-Taylor solution has proven invaluable in describing blast wave evolution, it is reliant on some critical assumptions: namely that for robust self-similarity no other length-scales are involved in the system. Consequently, this derivation can only be applied during the energy-conserving phase of blast wave evolution when all dissipative processes via radiation or conduction are negligible. This limits the validity of the Sedov-Taylor solution to a stage in the evolution of the system when the mass set in motion by the shock is far greater than that initially heated and the energy can be considered as deposited in an instantaneous, point-source. Consequently, the Sedov-Taylor description is appropriate during the adiabatic phase and holds until the blast wave shell density becomes comparable to that of the ambient background or until radiation losses become large. Importantly, the concept of self-similarity can be further extended provided that the fraction of energy radiated is constant throughout the duration of the ‘problem’. As discussed by Barenblatt, this extends the use of solutions of the form r ∼ t α beyond the adiabatic case and Liang and Keilty extended this to show that an analytic description exists for any constant energy loss fraction, ε, where 0 > ε > 1 (Barenblatt, 1979; Liang and Keilty, 2000). Therefore, while either the Sedov-Taylor or this modified self-similarity holds, for a blast wave radius

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Table 1 Experimental gas jet parameters for each of the four gases, calculated to ensure that an approximately equal ambient density is produced. (a) denotes the ideal case and (b) the experimentally-realised case

(a) Ideal H2 Ar Kr Xe (b) Real H2 Ar Kr Xe

P (bar)

T (K)

×10−4 (g cm−3 )

282.5 42.5 20.2 12.9

100 300 300 300

3.40 3.40 3.40 3.40

– – – –

49.9 50.5 21.0 20.3

100 300 300 300

0.60 4.04 3.52 5.35

85% 0% 13% 32%

at a particular time tcomp , the structure in different gases can be compared to determine the effect of increased radiation flux. Then, provided that the deposited energy, background density and radius are approximately equal at tcomp , the structure of the front can be directly compared to learn about the energy loss through radiation. Astrophysical blast waves from SNRs are rarely observed to evolve through different phases owing to the short ‘human’ timescale over which observations can take place. Conversely, in laboratory experiments all such phases are readily observed, and so it becomes crucial to study the formation, structure and evolution of laboratory blast waves if they are to be comparable and applicable to astrophysical problems. It is therefore important to acknowledge that, depending on the conditions in the experiment, there is only a specific length of time during which the blast wave exists, and possibly a reduced duration for which a self-similar description (and hence any hydrodynamic comparison with astrophysics) is appropriate. When constructing experiments to study the change in blast wave structure due to different clustered gas media, it is crucial that resulting plasmas, of the same deposited energy, are compared at a time when the blast waves have unambiguously formed and have swept up equal amounts of mass. The second point is important in order to compare the structure, since if self-similarity does not hold (for example due to substantial radiative flux in higher Z gases) then the flow variables are no longer ‘frozen’ and a comparison can only be made if the blast waves are expected to be in a similar phase of evolution. Consequently, by comparing the blast wave structure at the same radius with the density held constant, these factors are satisfied. Interestingly, from the simple analytical theory leading to Equation (1), the only variables in ξ are deposited energy (E), density (ρ0 ), radius (r), and time (t), so if the first three of these factors are kept constant, any deviations from the expected power-law in time are an indication of the extent to which the self-similarity does or does not hold. Finally it is important to note that the

∗

Nc

Rc (nm)

130000 35000 42000 64000

3000000 130000 220000 580000

30 11 13 20

48000 35000 26000 48000

300000 140000 66000 300000

14 11 9 16

analytical blast wave theory we have described here is purely fluid-based, and assumes a single fluid remaining instantaneously in thermodynamic equilibrium. In reality, the timescales of different aspects of the flow such as ionization, recombination or electron conduction may be shorter than thermodynamic equilibrization processes. 3.1 Calculating experimental parameters We present high spatial-resolution data comparing blast wave structure in hydrogen, argon, krypton and xenon, and so it is necessary to produce equivalent mass densities in all four gases.   +1  γ A
2  γγ −1  m˙ = P0 a (2) RT0 γ + 1

˙ through a nozzle is related to the The rate of mass flow, m, pressure and temperature of the gas pre-expansion (P0 and T0 respectively), nozzle diameter, a, and ratio of specific heats, γ = C p /Cv , of the gas (atomic mass A) according to Equation (2) (Miller, 1988). The laser-interaction and subsequent dynamics are instantaneous on the mass-flow timescale, and the gas jet forms a steady flow prior to the laser interaction. Consequently, since the volume profile of the gas jet is independent of gas species (Smith et al., 1998), provided the mass flow is the same then the neutral gas density profile in the jet will be equivalent, and thus the ambient density, ρ 0 , from Equation (1) will be the same for each gas, provided P0 and T0 are scaled appropriately. The ratios of specific heats in Equation (2) are estimated using the temperature dependence of C p in ref Forsythe (1954, 2003). Table 1 shows the equivalent pressures and temperatures required to achieve this comparison together with the resulting individual cluster parameters: Hagena parameter, Ŵ ∗ (Hagena and Obert, 1972), which governs cluster formation, and average cluster size, Nc, and radius, Rc. Due to the large difference in mass between, for example, hydrogen and xenon, it is technically difficult to achieve the Springer

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Fig. 1 A schematic, based on experimental data, of the timescales involved in the formation of a blast wave from a laser-heated cluster medium. Initially non-local effects occur due to the fast, free-streaming electron component (green). The electrons thermalise after several picoseconds, and form an electron thermal wave (red) that follows a t0.2 trajectory according to Kimoto and Chernoff (1997) Finally, after a few nanoseconds, the plasma becomes LTE, and a shock forms. As the rarefaction wave catches up with the shock and sufficient mass is shock-heated, a blast wave forms, the deceleration of which identifies it as radiative (purple) or adiabatic (blue)

necessary pressures in the laboratory to directly compare blast waves produced in hydrogen with heavier cluster gases – an issue we will address in future work. While it is clear that the individual cluster parameters differ significantly between the gases, which will have an effect on the ps-timescale laser interaction with each cluster, the large Hagena parameter implies that clustering has saturated, and energy absorption studies (Ditmire et al., 1997; Moore et al., 2004, 2005) show that when clusters are present to this degree a high fraction of the laser energy (>80%) is coupled into the plasma, so on the timescales we investigate the detailed physics of the individual laser-cluster interaction can be neglected. 3.2 Blast wave formation As we have highlighted, to be confident of the blast wave formation it is important to have an insight into the processes that occur prior to this. Figure 1 shows a schematic representation of data obtained at the different stages of expansion occurring after the initial nanoplasma explosions of individual clusters have merged into a uniform plasma. Initially from timescales ∼10 ps, non-local electron transport, as observed in Ditmire et al. (1998), is expected to occur, resulting in a precursor or ‘foot’ to the electron density profile due to a small fraction of free-streaming electrons transporting energy ahead of the diffusively-driven thermalized electron wave. As more matter is ionised by the propagating electrons, Springer

Fig. 2 Electron density profiles of blast waves formed in H2 , Ar, Kr, and Xe, at equivalent times in their self-similar trajectories. Shell-thinning (increased compression), should occur in higher atomic number gases due to the lower ‘effective’ adiabatic index caused by the increased radiation drive. However, we observe that the shell thickness increases with increasing atomic number (Z), probably due to increased electron conduction in Kr and Xe

the ionization front slows. The electrons quickly thermalize on a timescale of a few ps, but since tii ∼ (m i /m e )1/2 tee , the ions thermalize considerably more slowly on the 100 ps timescale. Equilibration between electrons and ions will take longer still: tei ∼ (m i /m e )tee and so thermal equilibrium in the plasma is only reached on a timescale of many nanoseconds. By this time the shock has already begun to form and after sufficient time (∼10 ns), the swept-up mass is greater than that initially heated and the rarefaction wave will have caught up with the shock front and a blast wave forms. From this point on it is possible to make the self-similar approximations described since we assume the plasma components are in local thermodynamic equilibrium (LTE).

4 Achievable blast wave parameters In Fig. 2 we show comparative electron density profiles for the blast waves formed in each of the four gases studied.

Astrophys Space Sci (2007) 307:139–145

The gas jet backing pressures were as indicated in Table 1(b). The profile in hydrogen was taken at a backing pressure of 50 bar, somewhat lower than that required by the mass flow calculations since this was not achievable with current equipment. Although this difference between hydrogen density and those of the other gases are quite large, this should only marginally affect the blast wave radius according to the all self-similar derivation in Equation (1). The comparative blast wave structures in Fig. 2 show several key differences. By assuming that the initially-heated plasma filament extends over an ≈80 µm focal spot, the mass swept up by the blast wave at r ≈ 400 µm that we measure here is approximately 100 times greater, allowing complete confidence that a blast wave has formed. A clear distinction can be seen between the blast wave structure in each case. In all four gases a fairly clean, sharp shock is observed, however krypton and xenon are subject to considerably more preheating (and thus ionization) of the upstream gas and the shock is less steep. In both Kr and Xe there is significant ionization up to 200 µm ahead of the shock, providing confirmation that the shock is radiating energy upstream, since the electron mean free path is estimated to be ≈50 µm at most at this point in time. Such radiative blast waves are typically classified into three regions: pre-shock or pre-cursor, transition, and cavity regions (Keilty et al., 2000; Mihalas and Weibel-Mihalas, 1999). For a monotonic gas, in the absence of radiation and ionization, it is appropriate to use an ideal gas description with a constant value for γ . However, radiative blast waves of the type we produce can not be described in this simplified way and a more accurate description utilises a separate γ in each region. In the regime of our experiments, the degree of ionization on either side of the shock will alter the equation of state. For simplicity we use two approximations both utilising an ‘effective polytropic index’. First we simply calculate a single-valued γid−eff equivalent to the analysis for a monatomic ideal gas. However we also compare this to the calculated γeff that takes account of different levels of ionization either side of the shock, by redefining the equation of state: γeff − 1 = p/ρǫ(Z ∗ + 1) (Drake, 2006), so that γ eff reflects more accurately the affect of just radiation. Here, ε is the specific internal energy, Z ∗ is the level of ionisation, and ρ is the molar gas density. In this slightly more complex ∗ approximation, while Z is estimated on either side of the shock, a single γeff is assumed, having the same value in the pre-shock and post-shock gas (Liang and Keilty, 2000). Blast waves that strongly radiate should undergo shellthinning, since the polytropic index of the post-shock gas is reduced through radiation and the material becomes more compressible. By measuring the shell thickness, h, as a function of the blast wave radius, R, in all four gases we calculate the compressibility, C = ρshell /ρ0 , through the relation C = R 2 /(2Rh − h 2 ). Provided that the blast wave is selfsimilar and that all the mass can be assumed to be in the thin

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shell, then the compression can be estimated to better than 10%, based on the resolution of the interferometric data. The comparison in Fig. 2 shows much thicker shells in Kr and Xe than in Ar and H2 for the same blast wave radius. This implies, counter intuitively, that H2 and Ar are more compressible. We measure a compression factor C ≈ 4 in both H2 and Ar, demonstrating that we access the strong shock limit for an ideal gas. Surprisingly, the compression we measure for Kr and Xe is C ≈ 2–3, whereas a radiating gas with a lowered γeff should lead to a compression exceeding 4. This contradiction indicates that either the blast waves we generate are not in the strong shock limit or that another of the assumptions implicit in the calculations is invalid. A consequence of preheating gas ahead of the shock is the considerable increase in the upstream sound speed, causing a reduction in the Mach number. In Krypton, the shock speed is ≈17 ± 2 km s−1 . Without preheat this would correspond to a Mach number in excess of 60. In fact it is more likely that in the precursor region material is heated to several eV ahead of the main shock, so that in the pre-shocked medium the sound speed could be as high as 3 km/s. This drastically reduces the Mach number to ∼6. However, we find that even with precursor temperatures as high as 5 eV, as shown in Table 2, the Mach numbers for both Kr and Xe do not drop sufficiently to preclude being in the strong shock limit. For an ideal gas, ignoring ionisation, compression is related to the adiabatic index and Mach number: C = (γid-eff + 1)M 2 /[(γid-eff − 1)M 2 + 2] (Shigemori et al., 2000). Using this ideal case, the range of predicted adiabatic indices for each gas are shown in Table 2. In H2 and Ar these are not unreasonable agreeing, within experimental limitations, with that of an ideal gas. However, the large error prevents any real insight into whether or not the value has fallen due to any increase in the effective number of degrees of freedom, f, in the plasma. In the higher-Z, more-radiative cases of Kr and Xe reduced compression leads to an increase in the polytropic index. The cylindrical blast waves that we create are quasi2D, which can be argued to decrease f, resulting in a maximum γid-eff = 2. However, the larger-still calculated γid-eff for Xe demonstrates the limitations of the ideal gas approximation of a constant polytropic index across the shock. Recalculating C to take account of differing levels of ionization ahead of and behind the shock produces much more realistic values of γeff . In H2 γeff = 1.48 ± 0.13 and Ar, Kr and Xe γeff is between 1 and 1.3. Interestingly, we achieve the lowest effective index in Krypton, where radiation is significant but not so strong that the shock strength is reduced like in Xenon. The large uncertainty in the precursor temperature leads to error in the Mach number. The shock velocity, which can be measured directly over a number of different laser shots or estimated from a single shot assuming a self-similar blast wave trajectory and associated deceleration parameter, typically Springer

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Astrophys Space Sci (2007) 307:139–145 Table 2 Table of calculated plasma parameters that we access in H2 , Ar, Kr and Xe, for the experimentally-realised blast wave profiles in Fig. 2 Parameters Experimental Time (ns) Shell thickness h (cm) Shock velocity Vs × 106 (cm s−1 ) Upstream Mach number M Compression Density ρ (g cm−3 ) Pressure P0 (Pa) Postshock temperature T2 (eV) Ideal gas polytropic index γid-eff Effective polytropic index γeff Localised Mean free path lii (cm) Collisionality lii / h Heat conduction/viscosity Peclet number Pe Reynolds number Re Radiation l R (cm) lc (cm) τthin (ns) Hydro τhydro (ns) e-i equilibration distance (µm) Euler Number

H2

Ar

Kr

Xe

6.8 54 ± 4 3.0 ± 0.1 56.2 ± 2.3 4.0 ± 0.3 5 × 10−5† 1.5 × 108† 1.3 ± 0.2 1.5–1.7 1.48 ± 0.13

11.5 58 ± 5 1.7 ± 0.1 33.1 ± 29.3 3.7 ± 0.3 4 × 10−4† 9 × 109† 7.3 ± 3.3 1.4–1.7 1.18 ± 0.17

12.0 70 ± 5 1.7 ± 0.2 6.7 ± 4.3 3.1 ± 0.2 6 × 10−4† 2 × 1010† 15.4 ± 3.3 1.5–1.9 1.07 ± 0.14

15.4 100 ± 5 1.3 ± 0.1 5.7 ± 4.3 2.3 ± 0.1 3 × 10−4† 9 × 1010† 16.2 ± 3.5 2.0–2.4 1.12 ± 0.14

2.7 × 10−5† 4 × 10−4†

7.2 × 10−4† 0.12†

1.5 × 10−3† 0.2†

5.3 × 10−3† 0.5†

300† 1000†

1.2† 10000†

0.5† 30000†

0.5† 40000†

0.3† 43000† 5200†

120† 30000† 570†

860† 38000† 310†

900† 51000† 400†

2† 3.2† 0.8†

3† 440† 0.1†

4† 2020† 0.1†

8† 5300† 0.1†

Scaling parameters are estimated following the analysis of Hagena and Obert (1972) and indicate that in hydrogen the system has become hydrodynamic and can be compared to the self-similar models. However, in the higher atomic number gases electron conduction remains significant, causing the low Peclet numbers we calculate † – order of magnitude estimate

introduces an error of ∼10%. 
2 T2 2 M 2γeff (γeff − 1) − (γeff − 1)2 = + T1 (γeff + 1)2 M 2 (γeff + 1)
∗  Z1 + 1 × (3) Z 2∗ + 1 Assuming an equation of state that includes ionization: p = (Z ∗ + 1)ρk B T /(Am p ), the post-shock temperature can be calculated using Equation (3), of which only the first term remains in the absence of ionization and in the strong shock limit. However, the dependence of the post-shock temperature on T1 is relatively small, since M 2 α1/T1 . Consequently, for large M, T2 is independent of T1 . In the general case this leads to the largest obstacle to accurately predicting T2 being the effective polytropic index, leading to the large error (∼20%) in the estimated post-shock temperature. In the ideal gas derivations of Zel’dovich and Raizer (1966, 1967), Mihalas and Weibel-Mihalas (1999), this is assumed constant across the shock, which is clearly not the case here. With the exception of hydrogen, where γeff = 5/3 can be used with relative confidence, the preheating ahead of the shock will Springer

undoubtedly change the adiabatic index of the upstream gas, preventing an accurate estimation of the post-shock temperature and Mach number. The order-of-magnitude estimates of the scaling parameters provide an indication of the hydrodynamic parameter space we are able to access. The limiting parameter is the Peclet number, Pe, which must be ≫1 in order for heat conduction to be negligible. In hydrogen this is not a problem, since the thermal diffusivity is relatively low. However, due to the higher temperature and lower number density in Ar, Kr and Xe, the electron transport is sufficient to limit the hydrodynamic scalability of the plasmas we create. In agreement with the conclusions of Edwards et al. (2001), the relatively thick shocks that we currently observe in the high Z gases are due to electron conduction that transports energy over 10’s µm in the case of Kr and Xe. Finally, the e-i equilibration distances further demonstrate the need to account for ionization. While there is probably LTE within the electron and ion fluids, only in hydrogen is it likely that a single fluid description is accurate, since for Ar, Kr and Xe the equilibration distances are on the order of or larger than the scale of the hydrodynamic system.

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5 Conclusions

References

These results provide convincing evidence that the blast waves we produce in both Kr and Xe are strongly radiative and that, because the gas medium we use is optically thin, radiation transports energy ∼200 µm ahead of the blast wave front, preheating the upstream material. We are able to produce high-Mach number (M > 20) blast waves in less radiative gases, Ar and H2 , and find that measurements of the shock compression match well with the expected values for an ideal gas. However, in the more radiative Kr and Xe, the assumptions of a single-valued effective polytropic index throughout the shock are less appropriate due to the degree of ionization. This is indicative that a simple monatomic ideal-gas theory invoking a single-value for γid−eff is not sufficient to predict our post-shock conditions. When Z ∗ is included in the equation of state more accurate estimates of the post-shock temperature and γeff can be made, however numerical modeling that can manage generalized jump conditions in an ionized, radiating gas is required to fully explain the behavior we observe. The radiative self-similar solutions in part address this problem, but without accurate knowledge of the pre-shock conditions, owing to the very strong preheating, it is not possible to calculate meaningful parameters. The scaling parameters necessary to link this work to an astrophysical situation are closest in hydrogen, but the additional electron conduction in Ar, Kr and Xe prevent realistic scaling being made without further modeling.

Barenblatt, G.I.: Similarity, Self-Similarity and Intermediate Asymptotics. Consultants Bureau, New York (1979) Blondin, J.M., et al.: Astrophys. J. 500, 342 (1998) Ditmire, T., et al.: Phys. Rev. Lett. 78, 3121 (1997) Ditmire, T., et al.: Phys. Rev. Lett. 80, 720 (1998) Ditmire, T., et al.: Astrophys. J. Supp. Series 127, 299 (2000) Drake, R.P.: High-Energy-Density Physics. Springer Publications (2006) Drake, R.P., et al.: Phys. Plasmas 7, 2142 (2000) Edwards, M.J., MacKinnon, A.J., Zweiback, J., et al.: Phys. Rev. Lett. 87, 8708 (2001) Forsythe, W.E.: Smithsonian Physical Tables, 9th edition. Table 157, Knovel (1954, 2003) Grun, J., et al.: Phys. Rev. Lett. 66, 2738 (1991) Hagena, O.F., Obert, W.: J. Chem. Phys. 56, 1793 (1972) Hwang, U., et al.: Astrophys. J. Lett. 537, 119 (2000) Keilty, K.A., et al.: Astrophys. J. 538, 645 (2000) Kimoto, P., Chernoff, D.: Astrophys. J. 485, 274 (1997) Liang, E.P., Keilty, K.A.: Astrophys. J. 533, 890 (2000) Mihalas, D., Weibel-Mihalas, B.: Foundations of Radiation Hydrodynamics. Dover Publications Inc. (1999) Miller, D.R.: in Scholes, G. (ed.) Free Jet Sources Vol. 1 of Atomic and Molecular Beams. Oxford Univ. Press (1988) Moore, A.S., et al.: CLF Annual Report 2004–2005, p. 34 Ostriker, J., McKee, C.: Rev. Mod. Phys. 60, 1 (1988) Ryu, D., Vishniac, E.T.: Astrophys. J. 313, 820 (1987) Ryu, D., Vishniac, E.T.: Astrophys. J. 368, 411 (1991) Ryutov, D., et al.: Astrophys. J. 518, 821 (1999) Schappert, G.T., et al.: AGEX II, LANL Quarterly Report (1994) Sedov, L.I.: Similarity and Dimensional Methods. Academic Press, New York (1959) Shigemori, K., et al.: Astrophys. J. Lett. 533, L159 (2000) Smith, R.A., et al.: Rev. Sci. Inst. 69, 3798 (1998) US National Research Council: Frontiers in High Energy Density Physics. National Acadamies Press, Washington, D.C. (2003) Vishniac, E.: Astrophys. J. 274, 152 (1983) Zel’dovich, Raizer: Physics of Shock Waves and High Temperature Phenomena. Dover Publications Inc. (1966, 1967)

Acknowledgements This work was supported by grants from the EPSRC and AWE. We are pleased to acknowledge useful conversations with D. R. Symes and the technical support of P. Ruthven and B. Ratnasekara.

Springer

Astrophys Space Sci (2007) 307:147–152 DOI 10.1007/s10509-006-9269-7

O R I G I NA L A RT I C L E

Mass-Stripping Analysis of an Interstellar Cloud by a Supernova Shock J. F. Hansen · H. F. Robey · R. I. Klein · A. R. Miles

Received: 24 April 2006 / Accepted: 19 October 2006 C Springer Science + Business Media B.V. 2006


Abstract The interaction of supernova shocks and interstellar clouds is an important astrophysical phenomenon since it can result in stellar and planetary formation. Our experiments attempt to simulate this mass-loading as it occurs when a shock passes through interstellar clouds. We drive a strong shock using the Omega laser (∼5 kJ) into a foam-filled cylinder with an embedded Al sphere (diameter D = 120 µm) simulating an interstellar cloud. The density ratio between Al and foam is ∼9. We have previously reported on the interaction between shock and cloud, the ensuing Kelvin-Helmholtz and Widnall instabilities, and the rapid stripping of all mass from the cloud. We now present a theory that explains the rapid mass-stripping. The theory combines (1) the integral momentum equations for a viscous boundary layer, (2) the equations for a potential flow past a sphere, (3) Spalding’s law of the wall for turbulent boundary layers, and (4) the skin friction coefficient for a turbulent boundary layer on a flat plate. The theory gives as its final result the mass stripped from a sphere in a turbulent high Reynolds number flow, and it agrees very well with our experimental observations.

1 Introduction

Keywords Shock . Interstellar . Cloud . Turbulent . Turbulence . Mass-loading . Star formation

2 Experimental setup and diagnostics

J. F. Hansen (
) · H. F. Robey · A. R. Miles Lawrence Livermore National Laboratory, Livermore, CA 94550, USA e-mail: [email protected] R. I. Klein Lawrence Livermore National Laboratory, Livermore, CA 94550, USA; Department of Astronomy, University of California, Berkeley, CA 94720, USA

We report here the results from experiments designed to study the interaction between a supernova shock and interstellar accumulations of matter (ISM) or “molecular clouds.” The experiments were carried out at the Omega laser at the Laboratory of Laser Energetics in Rochester, New York (Boehly et al., 1997). We observe rapid stripping of all mass from a cloud, and we present a model based on turbulent boundary layers where the time scale of the mass-stripping agrees with the experiment. This may be important for understanding mass-loading in the astrophysical case, and for comparison to numerical simulations of the interaction between shocks and interstellar clouds which have shown that turbulence can generate structure which acts as seeds for star formation (Clark and Bonnell, 2005). We note that aided by the forerunner of this experiment (Klein et al., 2000; Robey et al., 2002), shock-cloud interaction evolved to a late time has recently been observed in supernova remnant Puppis A (Hwang et al., 2005).

The strong shock of a supernova explosion is simulated at experiments at Omega in the following manner: a small Be shock tube (2.25 mm long; 0.8 mm inner diameter; 1.1 mm outer diameter) is filled with a low density (300 mg/cm3 ) carbonized resorcinol formaldehyde foam (CRF). The CRF at one end of the shock tube is then ablated by laser beams, causing the ejection of ablated material in one direction to launch a planar shock in the opposite direction. Good planarity of the shock is ensured by using multiple (ten), superimposed beams, each with a super-gaussian beam profile created by a phase plate in the focusing optics; the super-gaussian is of Springer

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Fig. 1 HYADES calculation of the free-stream velocity U∞ = U∞ (t) , the Mach number M, the compression ρ 1 (t) /ρ 10 (where the initial density ρ 10 = 300 mg/cm3 ), and the temperature T = T (t)

order eight and has a flat top matching the diameter of the shock tube. Each laser beam has an energy of ∼500 J with a pulse duration of 1.0 ns. The scaled ISM cloud is simulated by an Al sphere (radius R0 = 60 µm) embedded in the CRF a short distance into the shock tube (on the shock tube axis 500 µm from the ablated CRF surface). The density ratio between the Al (density 2.7 g/cm3 ) and the surrounding CRF is chosen to match the density ratio for an actual ISM cloud and other experimental parameters are also scaled to preserve the physics regime of the astrophysical case by following the guidelines set out by Ryutov et al. (1999). Physical quantities in the CRF (without an embedded Al sphere) can be accurately calculated using the 1D radiative hydrodynamic code HYADES (Larsen and Lane, 1994). We have used this code to calculate the free-stream velocity U∞ (t) , the temperature T (t) , and the density ρ 1 (t) for the CRF and these quantities are plotted in Fig. 1. (These values were used in our Euler scaling estimates above.) The simulation used an in-line quotidian equation of state (QEOS) model with a bulk modulus of 3 × 109 Pa and 112 zones to represent the 2.25 mm long CRF, with the first 46 zones feathered for ablation with a zone-to-zone scaling ratio of 1.15, the final 46 zones feathered for shock release with a scaling ratio of 0.87. The cloud is imaged using a gated x-ray framing camera (Budil et al., 1996). X rays for the image are generated by a second set of time-delayed laser beams (backlighter beams) pointed at a metal foil, typically Ti, located on the opposite side of the shock tube from the camera. He-α radiation from the Ti (at 4.7 keV) moves through the shock tube and is imaged by either a 10 µm pinhole located at the front end of the camera (“area radiography”) or by a 20 µm pinhole next to Springer

Astrophys Space Sci (2007) 307:147–152

the Ti foil (“point projection radiography”). The point projection radiography technique results in a higher photon flux through the pinhole, and thus a better signal-to-noise ratio. The imaging element of the camera is either a microchannel plate (MCP) + film or MCP + charge coupled device (CCD), and in both cases has a size of ∼35 mm. The exposure of the MCP was kept in the linear regime (Landen et al., 1994). The distance from shock tube to Ti foil is 4.0 mm for area radiography, 6.5 mm for point projection radiography. The time-delay for the backlighter beams is chosen to obtain an image at a desired time t after the initial, ablative laser pulse has started the shock in the shock tube. The camera MCP is triggered to coincide with the backlighter beams. The MCP pulse length was set to 500 ps in a trade-off between maximizing x-ray exposure on the MCP while minimizing motion blurring (e.g., when the plasma moves 20 km/s the motion blurring is 10 µm, comparable to the pinhole diameter). The experiment is repeated with different time-delays to generate an image sequence.

3 Results Results from the experiment can be seen in Fig. 2. As the shock runs over the cloud (t = 5 ns), its speed inside the cloud is greatly reduced, leading to a Kelvin-Helmholtz instability and its characteristic roll-up (t = 12 ns). Soon thereafter, a Widnall-type instability (Widnall et al., 1974) occurs, creating a low mode number azimuthal perturbation of order five when viewed from a point on the extended shock tube axis (Robey et al., 2002). Here we see the Widnall instability as four “fingers” at the trailing edge of the cloud at t = 30 ns, indicating a mode number of four to eight (depending on if each finger is or is not overlapping another finger along the line of sight). Material is constantly being stripped away from the Al plasma cloud and is visible in the images as a cone of diffuse material behind the cloud (t ≥ 19 ns). By t = 40 ns this cone extends outside our diagnostic field of view. By t = 60 ns so much material has been stripped away that the remaining cloud is quite diffuse (we are showing the 60 ns image at a higher contrast than the earlier point projection radiography images in Fig. 2). By t = 100 ns the cloud has been completely stripped away and can no longer be identified in the point projection radiography images. We also obtained an image at 80 ns in which the cloud is completely gone, but this target unfortunately had its Au spatial reference grid mounted outside the view of the x-ray framing camera and was therefore not included in Fig. 2. Because the point projection radiography technique illuminates the MCP in a very uniform fashion, we can use the point projection radiography images to estimate the cloud mass using the formula I = I0 exp(−m/µA) + Ib where I is the measured pixel intensity, I0 is the x-ray source intensity

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Fig. 2 A time sequence of images showing how the cloud evolves after the passage of a shock. In each image, the direction of motion of the shock is approximately from left to right, and is perpedicular to the imaged shock at time t = 19 ns, and the imaged Au grids at times t = 30 ns and 40 ns . In the first image, at t = 5 ns, the shock is intersecting the cloud and the left hand side of the cloud is compressed by a factor of 4, the strong shock limit for a polytropic gas with an adiabatic index γ = 5/3. The cloud undergoes a classical Kelvin-Helmholtz roll-up, as seen at t = 12 ns and later. Cloud material is stripped away from the

cloud. Stripped material is clearly evident trailing the cloud at t ≥ 19 ns and is shaped as a cone that extends all the way to the shock (or extends outside the field of view at t = 40 ns). A rarefaction changes the direction of the surrounding flow at approximately t = 40 ns, and by t = 60 ns the reverse flow has caused the right hand side of the cloud to become fairly round. By t = 100 ns the cloud has disappeared. The first four images were obtained with area backlighters, the last four with point projection radiography

(i.e., the intensity we would have expected to measure had there not been mass attenuation), m is the integrated line of sight mass, µ is the x-ray attenuation coefficient in units of mass per area, A is the pixel area in the image, and Ib is background intensity from all sources of non-directional exposure, such as film fogging, non-directional x rays, energetic particles, etc. We estimate the cloud mass to be 0.67 ± 0.11 µg at t = 30 ns and 0.54 ± 0.11 µg at t = 40 ns (it is coincidental that the error is 0.11 µg in both of these images – the error is normally different from image to image). This can be compared to the original sphere mass of 2.44 µg.

We begin with the integral momentum equations for a stationary, viscous boundary layer:

4 Analysis We present in this section a new mathematical model that describes mass stripping from a cloud under turbulent, high Reynolds number conditions. We compare this model to our experimental data and to an existing model (Taylor, 1963; Ranger and Nicholls, 1969) for laminar mass stripping. Our model combines four separate concepts of fluid mechanics: (1) the integral momentum equations for a viscous boundary layer, (2) the equations for a potential flow past a sphere, (3) Spalding’s law of the wall for turbulent boundary layers (Spalding, 1961), and (4) the skin friction coefficient for a turbulent boundary layer on a flat plate.

∂ ∂x



+

δ1

u 1 (U − u 1 ) dy1 +

0

1 dr r dx

δ1

0



δ1 0

(U − u 1 ) dy1

u 1 (U − u 1 ) dy1 = ν 1

 ∂u 1  ∂ y1  y1 =0

 1 dr δ2 2 1 dp δ2 + u 2 dy2 + r dx ρ 0 0 2 dx  ∂u 2  = −ν 2 ∂ y2  y2 =0   ∂u 1  ∂u 2  ρ1ν1 = −ρ ν 2 2 ∂ y1  y1 =0 ∂ y2  y2 =0 ∂ ∂x



δ2



dU dx

u 22 dy2

(1)

(2) (3)

where x is a coordinate along the surface of the cloud (we will approximate the cloud with a sphere at all times so that x = 0 at the flow stagnation point and x = π2 R at the equator), y is a coordinate perpendicular to the cloud surface, r is the distance from the cloud surface to the cloud axis of symmetry, U = U (x) is the free stream flow velocity behind the shock, u = u (x, y) is the flow velocity inside the boundary layer, ν is the kinematic viscosity, ρ is the density, p = p (x) is the pressure, and δ = δ (x) is the thickness of the boundary layer. The flow properties U, u, ν, ρ, p, and δ are also functions of time t. Subscript 2 denotes plasma in the cloud and subscript 1 the surrounding flow, e.g., δ 2 is the boundary Springer

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defined through v ∗ = ν (du/dy)| y=0 and Coles (1995) gives the coefficients κ = 0.41 and B = 5.0. This is as far as we can go without saying something about the wall-friction velocity v ∗ , or equivalently the skin-friction coefficient C f , as the two are related through 2

v∗ =

Fig. 3 The flow geometry around the cloud is modelled with the potential flow around a sphere with the boundary layer flow calculated using a local cartesian coordinate system where the coordinate x is along the flow (i.e., along the surface of the sphere) and y is the distance into the boundary layer from the sphere surface. The velocity in the boundary layer is u 1 outside the sphere radius R and u 2 inside the sphere radius. At the outside edge of the boundary layer y1 = δ 1 the velocity u 1 = U, i.e., matches the potential flow velocity. At the sphere radius u 1 = u 2 = AU, where A is a constant. At the inside edge of the boundary layer y2 = δ 2 the velocity u 2 = 0. The cylindrical coordinate r is the distance from the axis of symmetry to the sphere surface

layer thickness inside the cloud. The geometry is sketched out in Fig. 3. We will assume that the free stream velocity around the cloud follows the potential flow of a sphere, x 3 U (x) = U∞ sin , 2 R

(4)

where U∞ is the flow velocity far from the sphere. This has the advantage that dU/dx = 0 at the equator x = π R/2, which will simplify the algebra substantially. Also, at the equator d p/dx = −ρ g U dU/dx = 0 and trivially dr/dx = 0. To express the boundary layer velocity u as a function of the distance y from the sphere surface, we will use Spalding’s law of the wall for turbulent boundary layers (Spalding, 1961):

y+ = u+


(κu + )3 (κu + )2 −κ B κu + + − (5) +e e − 1 − κu − 2 6

where the dimensionless coordinate y + ≡ yv ∗ /ν and the dimensionless velocity u + ≡ u/v ∗ , or in the appropriate coordinate frame of reference u 1 = v1∗ u + 1 + AU and u 2 = AU − v2∗ u + , where A is a constant such that AU is the flow 2 velocity on the sphere surface. The wall-friction velocity v ∗ is Springer

1 C f (x) U 2 (x) . 2

(6)

For calculations of skin-friction drag, many renowned researchers, beginning with Dryden and Kuethe (1930) and Millikan (1932), have used velocity distributions for a flat plate in non-flat geometries and found that the results do not differ seriously from measured values (Goldstein, 1965). We will do the same and use the skin friction coefficient for a turbulent boundary layer on a flat plate (White, 1974): C f (x) ≈ 0.0592 Re−1/5 x

(7)

where the Reynold’s number Rex = U x/ν, but with a modification; if we use Equation (7) as is, the problem is overdetermined. We replace the coefficient 0.0592 with a coefficient that will be determined by our system of equations. We set: C f (x) =

2 (U x/ν)−1/5 . α2

(8)

We can now rewrite Equation (1) as ξ 2 1200κ 2 eκ B K 19 (1 − A) α −1 1 = 120e [(1 + 2A) ξ

− (3 + 2A) ξ + 4] − 3 (3 + 10A) ξ 5 +

−20 (1 + A) ξ 4 + 20(κ −1 eκ B − 1) (1 + 2A) ξ 3

−120 (1 − 2A) ξ − 480

(9)

where ξ = κ K 1 (1 − A) α 1 . 
3πU∞ R 1/10 . K1 = 4ν 1

(10) (11)

We expect A to be a fairly small quantity (it will certainly be smaller than unity) so one might be tempted to linearize Equation (9) w.r.t. A, but this only simplifies terms where A (or ξ ) does not appear in the exponents and does not lead to an analytical solution for A (as it does in the laminar model). Consequently some form of simple numerical scheme must be employed to calculate A, and we have therefore chosen to not linearize Equation (9) w.r.t. A, but to keep the exact form.

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Similarly in the Al cloud defining K2 =




3πU∞ R 4ν 2

1/10

(12)

allows us to rewrite Equation (2) as 5 4 η 600κ 2 eκ B K 29 α −1 2 = 120e (η − 2) − 3η − 10η 
1 κB e − 1 η3 + 120η + 20 κ

+ 240

(13)

where η = κ K 2 Aα 2 .

(14)

Next relate α 1 to α 2 by rewriting Equation (3) as α1 = K 3α2

(15)

where K3 =




ρ1 ρ2

1/2


ν1 ν2

1/10

.

(16)

Eliminate α 1 by substituting Equation (15) in Equations (9)– (10), leaving us with two equations, Equations (9) and (13), for two unknown coefficients A and α 2 . This equation pair can easily be solved numerically, e.g., Equation (9) can be solved for A by simple iteration as it converges nicely, and a simple regula falsi (secant) method can be used for Equation (13), but other numerical schemes will work, too, and we used a globally convergent Newton’s method. With A and α 2 at hand, one easily calculates the mass stripped from the cloud by integrating the cloud material flowing through the boundary layer at the equator (Ranger and Nicholls, 1969): dm = 2π Rρ 2 dt



0

δ2

u 2 dy2 = 2π Rρ 2 ν 2 ψ (η)

(17)

where we have defined a mass-strip coefficient ψ (η) =

Fig. 4 Cloud mass remaining as a function of time calculated using a laminar model (Taylor, 1963; Ranger and Nicholls, 1969) (dashed line) and the turbulent model presented in this manuscript (solid line), compared to experimentally measured values of the cloud mass (two squares). The turbulent model agrees with the measured values and also predicts that the cloud is completely stripped by ∼90 ns, which compares well with the experimental observation of the cloud being stripped by 80 ns–100 ns . In the laminar model (assuming unchanged condition from 80 ns) the cloud is not stripped until ∼1 µs

1 2 1 η + 2 2κ κ 
1 4 1 2 1 3 η −κ B ×e 1 − η − η − η − η + e . (18) 2 6 24

It should be noted that dm/dt is not proportional to R, ρ 2 , or ν 2 because η = η(R, U∞ , ρ 1 , ρ 2 , ν 1 , ν 2 ) from the numerical solution above. Using the specific physical quantities for our experiment, we can now calculate the mass stripped as a function

of time and see how the calculation compares to our experimental data. For the cloud radius R (t) we use measured values from the experiment images, and interpolate to other times. For values of the free stream flow velocity U∞ (t), the density ρ 1 (t), and the temperature T (t) , we use values from HYADES. The density ρ 2 (t) is obtained by applying the same compression as for ρ 1 (t). All of these values are in full agreement with values from CALE. Additionally, the peak compressions are independently verified from the experiment at t = 5 ns where the left side of the sphere is compressed to an ellipsoid shape with minor radius ∼30 µm, corresponding to a compression of ∼4 (which is the strong shock limit for a polytropic gas with adiabatic index γ = 5/3). With our given physical quantities, the coefficients K 1 ≈ 5, K 2 ≈ 3, and K 3 ≈ 41 at all times. From solving Equations (9), (13) and (15) we calculate the coefficients A ≈ 1/5, α 1 ≈ 6, and α 2 ≈ 27 at all times, and we find that the compound quantity η varies between 4  η  8 (except very briefly when the rarefaction changes the direction of the flow) so that the mass-strip coefficient ψ is in the range 4 × 102  ψ  4 × 103 . The mass of the cloud as a function of time is plotted in Fig. 4 and reaches m = 0 (fully stripped) by t ≈ 90 ns . This agrees well with the experiment where the cloud can no longer be observed by 80 ns– 100 ns . By comparison, the equivalent mass-strip coefficient  −1 dm/dt in the laminar model is  4 × 102 ψ = 2π Rρ l ν l for all times of interest in the experiment, which is too low to Springer

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achieve the cloud being completely stripped by t ≈ 80 ns; if the mass-stripping was done by laminar flow and continued under the same conditions past t = 80 ns (ignoring experimental limitations) the laminar mass-stripping time would be ∼1 µs. As a final note, to illustrate the non-linearity between dm/dt and the various physical quantities one can arbitrarily double, say, the value of the viscosity ν 2 and see that this leads to only a 12% increase in dm/dt. 5 Summary We observe the rapid stripping of all mass from a simulated interstellar cloud in a laser experiment. We present a model that agrees very well with our experimental observations. The model combines (1) the integral momentum equations for a viscous boundary layer, (2) the equations for a potential flow past a sphere, (3) Spalding’s law of the wall for turbulent boundary layers, and (4) the skin friction coefficient for a turbulent boundary layer on a flat plate. By comparison, a laminar model overestimates the stripping time by an order of magnitude. This suggests that mass-stripping in the experiment must be of a turbulent nature, and with its even higher Reynolds numbers, this must hold also in the astrophysical case. Acknowledgements We would like to thank C. F. McKee, Departments of physics and Astronomy, University of California, Berkeley for his support of this project. This work was performed under the auspices of the U. S. Department of Energy by the University of California, Lawrence Livermore National Laboratory under Contract No. W-7405Eng-48.

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References Boehly, T.R., Brown, D.L., Craxton, R.S., Keck, R.L., Knauer, J.P., Kelly, J.H., et al.: Opt. Commun. 133, 495 (1997) Budil, K.S., Perry, T.S., Bell, P.M., Hares, J.D., Miller, P.L., Peyser, T.A., et al.: Rev. Sci. Inst. 67, 485 (1996) Clark, P.C., Bonnell, I.A.: MNRAS 361, 2 (2005) Coles, E.: In: Tollmien, W., G¨ortler, W. (eds.) Fifty Years of Boundary Layer Research. Vieweg, Brunswick (1955) Dryden, H.L., Kuethe, A.M.: N.A.C.A. Report 342, 12 (1930) Goldstein, S. (ed.): Modern Developments in Fluid Dynamics. Dover, New York (1965) Hwang, U., Flanagan, K.A., Petre, R.: Astrophys. J. 635, 355 (2005) Klein, R.I., Budil, K.S., Perry, T.S., Bach, D.R.: Astrophys. J. Suppl. S. 127, 379 (2000) Landen, O.L., Bell, P.M., Oertel, J.A., Satariano, J.J., Bradley, D.K.: UCRL-JC-112384 Gain Uniformity, Linearity, Saturation and Depletion in Gated Microchannel-Plate X-Ray Framing Cameras, Lawrence Livermore National Laboratory, Livermore (1994). Copies may be obtained from the National Technical Information Service, Springfield, VA 22161 Larsen, J.T., Lane, S.M.: J. Quant. Spectrosc. Radiat. Transfer 51, 179 (1994) Millikan, R.A.: Trans. Amer. Soc. Mech. Eng. 54, 29 (1932) Ranger, A.A., Nicholls, J.A.: AIAA J. 7, 285 (1969) Robey, H.F., Perry, T.S., Klein, R.I., Kane, J.O., Greenough, J.A., Boehly, T.R.: Phys. Rev. Lett. 89, 085001 (2002) Ryutov, D., Drake, R.P., Kane, J., Liang, E., Remington, B.A. WoodVasey, W. M.: Astrophys. J. 518, 821 (1999) Spalding, D.B.: J. Appl. Mech. 28, 455 (1961) Taylor, G.I.: The shape and acceleration of a drop in a high speed air stream. In: Batchelor, G.K. (ed.) The Scientific Papers of G.I. Taylor. University Press, Cambridge (1963) White, F.M.: Viscous Fluid Flow. McGraw-Hill, New York (1974) Widnall, S.E., Bliss, D.B., Tsai, C.: J. Fluid Mech. 66, 35 (1974)

Astrophys Space Sci (2007) 307:153–158 DOI 10.1007/s10509-006-9227-4

ORIGINAL ARTICLE

Hydrodynamics of Supernova Evolution in the Winds of Massive Stars Vikram V. Dwarkadas

Received: 20 April 2006 / Accepted: 27 July 2006 C Springer Science + Business Media B.V. 2006


Abstract Core-Collapse supernovae arise from stars greater than 8 M⊙ . These stars lose a considerable amount of mass during their lifetime, which accumulates around the star forming wind-blown bubbles. Upon the death of the star in a spectacular explosion, the resulting SN shock wave will interact with this modified medium. We study the evolution of the shock wave, and investigate the properties of this interaction. We concentrate on the evolution of the SN shock wave in the medium around a 35 solar mass star. We discuss the hydrodynamics of the resulting interaction, the formation and growth of instabilities, and deviations from sphericity.

star’s death. The rich and complex dynamics of the various interactions leads to the formation and growth of a variety of hydrodynamic instabilities, which we will focus on in this paper. It may be possible to simulate some of these hydrodynamic situations with available laboratory apparatus, and we hope that this work will further stimulate laboratory experiments of realistic astrophysical phenomena, particularly those involving radiative shocks.

Keywords Supernova remnants . Hydrodynamics . Instabilities . Stellar winds . Massive stars . Wind-blown bubbles . Shock waves

It has been realized over the years that the medium around a core-collapse SN is continually being sculpted during the progenitor star’s lifetime, by the action of winds and outbursts. Chevalier and Liang (1989) discussed the interaction between a SN shock wave and the surrounding wind-blown bubble formed by the pre-SN star. However analytic arguments can only be extended so far, and numerical simulations are required to study the subsequent non-linear behavior. A series of papers in the early 90’s (Tenorio-Tagle et al., 1990, 1991; Rozyczka et al., 1993) explored some aspects of this. Since then our observational knowledge of this phenomena has multiplied exponentially, thanks to the availability of space based data in the optical, X-ray and infrared bands, and the stream of data pouring in from observations of SN 1987A. The latter has become the poster-child for SN evolution in wind-blown bubbles, having shaped and confirmed many of our views. The basic details of SN interaction with wind-blown bubbles were outlined in the papers listed above, and further elaborated on by Dwarkadas (2005). The mass loss results in the formation of a circumstellar (CS) wind blown cavity surrounding the star, bordered by a thin, dense, cold shell. The typical structure of this wind-blown bubble for constant wind

1 Introduction Mass-loss from stars is a ubiquitous process. Massive stars (>8M⊙ ) lose a considerable amount of mass before they explode. This material collects around the star, forming a circumstellar (CS) wind-blown bubble. At the end of its life, the star will explode in a cataclysmic supernova (SN) explosion, and the resulting shock wave will interact with this medium. The further evolution of the resulting supernova remnant will depend on the properties of this medium. In this paper we discuss the evolution of the surrounding medium around massive stars, and the subsequent interaction of the SN shock wave with this medium following the V. V. Dwarkadas Astronomy and Astrophysics, University of Chicago, 5640 S Ellis Ave. AAC 010c, Chicago, IL 60637 e-mail: [email protected]

2 SN-Circumstellar interaction

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Fig. 1 Density and pressure profiles for a circumstellar wind-blown bubble

properties (Weaver et al., 1977) is shown in Fig. 1, and consists of an outwards expanding shock wave (Ro ), and a wind termination shock (Rt ) that expands inwards in a Lagrangian sense, separated by a contact discontinuity (Rcd ). In general most of the volume between Rt and Rcd is occupied by a low-density, high pressure shocked wind bubble, surrounded by the extremely dense shell. Most of the mass is contained in the dense shell. When the SN shock wave interacts with this bubble, it quickly finds itself in a medium with density much lower than that of the ISM. Consequently, the emission from the remnant, which arises mainly from CS interaction (Chevalier and Fransson, 1994), will be considerably reduced compared to evolution within the ISM. It comes as no surprise then that the subsequent evolution depends primarily on a single parameter , the ratio of the mass of the dense shell to that of the ejected material. For very small values  ≪ 1 the effect of the shell is negligible as expected. For values of   1 interaction with the shell results in considerable deceleration of the SN shock wave. The X-ray luminosity can increase by orders of magnitude upon shock-shell collision. A transmitted shock wave enters the shell, while a reflected shock wave moves back into the ejecta. If X-ray images were taken just after the interaction, they would show the presence of a double-shelled structure as the reflected shock begins to move inwards. In about 10 doubling times of the radius the SN begins to ‘forget’ about the existence of the shell. The remnant density profile changes to reflect this, and consequently the X-ray emission from the remnant, which depends on the density structure, will also change. The reflected shock will move to the center and presumably be reflected back, while the transmitted shock will slowly exit the shell and eventually separate from it. As the ratio  increases, more of the kinetic energy from the remnant is converted to thermal energy of the shell. The transmitted shock is considerably slowed down, and in extreme cases ( ≫ 1) may even be trapped in the shell. The high pressure behind the reflected shock will impart a large velocity to the shock, and therefore thermalization of the ejecta is achieved in a much shorter time as compared to

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thermalization by the SN reverse shock. Upon reaching the center the reflected shock bounces back, sending a weaker shock wave that will collide with the shell. In time a series of shock waves and rarefaction waves are seen traversing the ejecta. Each time a shock wave collides with the dense shell a corresponding (but successively weaker) rise in the X-ray emission from the remnant is seen. We have outlined the basics of SN interaction with CS wind-blown shells. One-dimensional models are fully described in Dwarkadas (2005). We wish to present herein results from multi-dimensional models.

3 CS medium around a 35 M⊙ star The above description considered an idealized wind-blown bubble formed by the interaction of a fast wind with the surrounding medium, where the properties of both are constant in time. In reality, as a massive star evolves, the wind properties change with time. In particular after a star leaves the main sequence, its mass-loss properties change considerably. This will give rise to a much more complicated bubble structure than is shown in Fig. 1. In order to explore more realistically the medium surrounding a core-collapse SN, we have taken stellar evolution calculations from several groups, and investigated the evolution of the surrounding medium as the star evolves. In this paper we discuss the evolution of the medium around a 35 M⊙ star, from an evolutionary model provided to us by Norbert Langer. The star begins its life on the main sequence as an O star, then expands to become a Red Supergiant (RSG), and finally ends its life as a Wolf-Rayet (WR) star. The massloss rate and wind velocity over the evolution are shown in Fig. 2. During the main-sequence stage, the mass-loss rate is a few times 10−7 M⊙ /yr, and the wind velocity is about 3000– 4000 km/s. Once the star swells to become a red supergiant, the wind velocity reduces by more than 2 orders of magnitude, and the mass-loss rate increases to almost 10−4 M⊙ /yr. The WR wind shows a slight drop in mass-loss rates by a

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Fig. 2 Evolution of the mass-loss rate (left) and wind velocity (right) for the 35 M⊙ star during its lifetime

factor of a few from the RSG phase, but a steep increase in wind velocity by two orders of magnitude. We use these values as input boundary values at each timestep to our code, which then computes the structure of the nebula over time. Unlike a previous computation (Garcia-Segura et al., 1996) our computation is fully two-dimensional right from the start. We use a grid consisting of 600 zones in both the radial and azimuthal directions. The code used is the VH-1 numerical hydrodynamics code, a multi-dimensional code that solves the equations of fluid dynamics on a Lagrangian grid, and then remaps them onto an Eulerian grid. Radiative cooling is included via a cooling function, but we have not included the effects of ionization. A grid that expands outwards with the outgoing shock wave is used, although no new zones are added, i.e. the grid is not adaptive. Initially the wind occupies about 20 zones on the grid, depending on the grid resolution. The mass-loss rate and velocity of the wind are used to compute the density and velocity of the inflow, which are used as the input boundary conditions at each timestep. The initial setup is uniform and no perturbations are applied to the system. Perturbations that arise are due to effects such as non-spherical shocks on the spherical grid. The evolution of the medium is shown in Fig. 3. In the main sequence stage (3a,b), although the wind properties are changing continuously, the nebular structure is not very different from that expected from the idealized, two-wind case of a fast wind interacting with a slower wind, both of which have constant wind properties. A thin shell of swept-up material is formed, and the volume of the nebula is mostly occupied by a hot, low density bubble. The shell is on the whole mostly stable, although it shows some wrinkles. These arise mainly from shearing, due to flow of gas along the contact discontinuity. However these instabilities are not highly pronounced, and do not appear to grow to any significant extent. In lower-resolution simulations presented earlier

(Dwarkadas, 2001, 2004) we had suggested that the shell is unstable to some type of thin-shell instability (Vishniac 1983). We have seen the same instability in simulations of the medium around a 40 M⊙ star. The higher resolution calculations presented here do not show a strong presence of such an instability however. This is a topic still under investigation. The interior of the nebula shows significant fluctuations in density and pressure, and vortices are visible in the velocity flow. Since the mass-loss rate and wind velocity are changing at every timestep, the position of the reverse shock is not fixed on the grid, but moves slightly every timestep with respect to the outer shock. The changing position of the reverse shock from one timestep to another results in the deposition of vorticity into the shocked wind, which is then carried out with the shocked flow. This results in an interior that is quite inhomogeneous, with significant density fluctuations. When the star leaves the main sequence and becomes a RSG star, its radius increases considerably, the wind velocity (Vwind ) drops by two orders of magnitude, while the mass˙ increases appropriately. Thus the wind density, loss rate ( M) ˙ wind , goes up by several orders of magniproportional to M/V tude. A new pressure equilibrium is established, and a shock front is formed in between the RSG wind and main sequence bubble as the RSG wind is decelerated by the bubble pressure. The RSG wind piles up against this shock, forming a thin dense shell of RSG material. No hot, low-density cavity is present. The shell decelerates as it expands outwards, satisfying the classic case of Rayleigh-Taylor (R-T) instability, and Rayleigh-Taylor fingers are seen expanding outwards from the high-density shell into the low-density ambient medium (3c). Some of the filaments show the presence of sub-filaments growing from the main one, and the expanded heads of many of the filaments are a sign of Kelvin-Helmholtz instabilities resulting from the shear flow in between the

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Fig. 3 Density evolution of the medium a round a 35 M⊙ star with time. The wind properties at each stage are given in the top right hand corner of each panel. The velocity is in km/s, the mass loss rate in M⊙ /yr, and

the time in years. The color bar shows the logarithm of the gas density in units of g cc−1

filaments and the surrounding medium. Unfortunately, since we need to resolve the entire bubble, the resolution is not large enough to study the growth of the fingers in detail. The star leaves the RSG phase and loses its outer hydrogen envelope, becoming a WR star in the process. The compact star now gives off a very fast wind, not unlike in the O star stage, but with a mass-loss rate that is much higher than in the main sequence, and just a few times lower than the RSG stage. The supersonic WR wind creates a wind-blown bubble in the RSG wind. The dense W-R shell is accelerated by the high pressure, low density interior as it expands outwards, leading to the triggering of the Rayleigh-Taylor instability (3d). In this case R-T fingers are seen expanding inwards from the dense shell into the low-density cavity. The large momentum of the WR shell causes the RSG shell to fragment, and carries the material outwards (3e), speeding up as it enters the low-density bubble. Due to the fact that the WR wind is carrying fragments of the RSG material, and that it travels through a medium with considerable fluctuations in density and pressure, its expansion is not completely spherical. The collision of this slightly aspherical wind with the main

sequence shell gives rise to a reflected shock that moves back into the bubble. The asphericity is accentuated in the reflected shock, which moves inwards, before finally coming to rest in a wind-termination shock where the ram pressure of the freely expanding wind is equal to the thermal pressure within the bubble. The wind-termination shock when it forms is consequently also not spherical but slightly elongated along the equator (3f). As we shall show later this has important consequences for the expansion of the SN shock wave.

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4 SN-CSM interaction in the case of the 35 M⊙ star At the end of the WR stage, the stellar mass remaining is 9.1 M⊙ . We assume that the star then explodes in a SN explosion. A remnant of 1.4 M⊙ is left behind, and the remaining mass is ejected in the explosion. We use the prescription of Chevalier and Fransson (1994) to describe the ejecta structure as a power-law with density, with power-law index of 7. We compute the evolution of the SN described by this density profile expanding into the unshocked wind, and then

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Fig. 4 Pressure evolution of the SN shock wave within the WR bubble

map it onto the grid containing the bubble simulation. This calculation was also carried out using 600 × 600 zones. The evolution of the SN shock wave is shown in Fig. 4. It starts out as expected, with the formation of a forward and reverse shock structure (Fig. 4a). The interaction of the spherical forward shock with the aspherical wind termination shock, susceptible to the Richtmeyer-Meshkov instability, reveals quite interesting dynamics. Since the SN shock is spherical while the wind termination shock is slightly more elongated towards the equator, the interaction first takes place close to the symmetry axis. A transmitted shock moves out into the shocked bubble, while a reflected shock moves back. Different parts of the SN shock collide with the windtermination shock at different times, leading to transmitted shocks with a small but non-negligible velocity spread (Fig. 4b). The composite transmitted shock then expands in the inhomogeneous medium, interacting with several large density fluctuations on the way. The net result is a very corrugated shock wave that expands outwards towards the main sequence shell (Fig. 4c). The wrinkles are similarly prevalent in the reverse shock also. The wrinkled shock collides

in a piecemeal fashion with the main sequence shell, with some parts of the shock colliding before others (Fig. 4d,e). Each collision with the shell will give rise to an increase in the optical and X-ray emission at that point. Therefore some parts of the shell will brighten before others. It is interesting to note that a similar phenomenon has been observed in SN 1987A, where bright spots appear successively around different parts of the equatorial ring (Sugerman et al., 2002) As each portion of the forward shock wave collides with the shell, a reflected and transmitted shock pair is formed. The shell is dense enough that the transmitted shock does not emerge from the shell for a long period. The reflected shock meanwhile travels back towards the origin. However, as seen in Fig. 4, the velocity of each piece differs considerably from the next, both in magnitude as well as direction. The shape of the reflected shock therefore deviates significantly from spherical, and some parts of the reflected shock reach the symmetry axis before the rest has traveled far into the interior (Fig. 4f). This gives the remnant a very asymmetric shape, and results in some portions of the ejecta being much hotter than others.

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5 Conclusions The surroundings of massive stars are shaped by the massloss from the progenitor star. This can lead to a complicated density structure for the surrounding medium, the formation and growth of various hydrodynamical instabilities, deposition of vorticity and onset of turbulence. When the star explodes as a SN remnant, the SN shock wave will interact with this ambient medium. The inhomogeneous structure of the ambient medium can cause distortions in the SN shock wave as it expands outwards, which are magnified by the turbulence in the wind-blown structure. In this paper we have shown that the end result can be a wrinkled shock wave whose impact with the surrounding shell occurs in a piecemeal fashion. As each part of the shock wave hits the shell, it will brighten up in the optical and X-ray regime, a phenomenon that is observable in SN 1987A. We do caution that this comparison is illustrative only. Our numerical models are not meant to simulate SN 1987A, whose progenitor star was a much lower mass B3Ia star. Our simulations show that the complicated structure of the medium may result in deviations from spherical symmetry for the SN shock wave. Even though the expansion starts out as spherical, the final shape of the remnant may deviate considerably from sphericity. Most of the emission from the remnant arises from the high pressure region in between the forward and reverse shocks. The distorted shape of this emitting region is clearly visible in Fig. 4, and this will be reflected in observations of the remnant. Herein we have summarized the features of multidimensional models of SN evolution in the environments

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shaped by massive stars. Further details are available from Dwarkadas (2006). Acknowledgements Vikram Dwarkadas is supported by award # AST0319261 from the National Science Foundation, and by NASA through grant # HST-AR-10649 from STScI. We thank the anonymous referee for suggestions that helped to improve this paper. We acknowledge useful discussions with Roger Chevalier which were particularly helpful in identifying the various instabilities that were observed. We are grateful for comments from John Blondin and Thierry Foglizzo. This research was supported in part by the National Science Foundation under Grant No. PHY99-07949 to the KITP.

References Chevalier, R.A., Fransson, C.: ApJ 420, 268 (1994) Chevalier, R.A., Liang, E.P.: ApJ 344, 332 (1989) Dwarkadas, V.V.: to be submitted to ApJ (2006) Dwarkadas, V.V.: ApJ 630, 892 (2005) Dwarkadas, V.V.: In: Hoflich, P., Kumar, P., Wheeler, J. C., (eds). Cosmic explosions in three dimensions : asymmetries in supernovae and gamma-ray bursts. Cambridge: CUP, pp. 74 (2004) Dwarkadas, V.V.: JKAS 34, 243 (2001) Garcia-Segura, G., Langer, N., MacLow, M.-M.: A&A 316, 133 (1996) Rozyczka, M., Tenorio-Tagle, G., Franco, J., Bodenheimer, P.: MNRAS 261, 674 (1993) Sugerman, B.E.K., et al.: ApJ 572, 209 (2002) Tenorio-Tagle, G., Rozyczka, M., Franco, J., Bodenheimer, P.: MNRAS 251, 318 (1991) Tenorio-Tagle, G., Bodenheimer, P., Franco, J., Rozyczka, M.: MNRAS 244, 563 (1990) Vishniac, E.T.: ApJ 428, 186 (1994) Vishniac, E.T.: ApJ 274, 152 (1983) Weaver, R., McCray, R., Castor, J., Shapiro, P., Moore, R.: ApJ 218, 377 (1977)

Astrophys Space Sci (2007) 307:159–164 DOI 10.1007/s10509-006-9203-z

ORIGINAL ARTICLE

Theoretical and Experimental Studies of Radiative Shocks C. Michaut · T. Vinci · L. Boireau · M. Koenig · S. Bouquet · A. Benuzzi-Mounaix · N. Osaki · G. Herpe · E. Falize · B. Loupias · S. Atzeni

Received: 20 April 2006 / Accepted: 19 June 2006 C Springer Science + Business Media B.V. 2006


Abstract This paper deals with the radiative shock from both theoretical and numerical points of view. It is based on the whole experimental results obtained at Laboratoire ´ d’Utilisation des Lasers Intenses (LULI, Ecole Polytechnique). Radiative shocks are high-Mach number shocks with a strong coupling between radiation and hydrodynamics which leads to a structure governed by a radiative precursor. These shocks are involved in various astrophysical systems: stellar accretion shocks, pulsating stars, interaction between supernovae and the interstellar medium. In laboratory, these radiative shocks are generated using high power lasers. New diagnostics have been implemented to study the geometrical shape of the shock and the front shock density. Data were obtained varying initial conditions for different laser intensities and temperature. The modeling of these phenomena is mainly performed through numerical

C. Michaut (
)· G. Herpe LUTH UMR 8102, Observatoire de Paris, 92195 Meudon cedex, France e-mail: [email protected] L. Boireau LUTH UMR 8102, Observatoire de Paris, 92195 Meudon cedex, France, CEA/DIF, D´epartement de Physique Th´eorique et Appliqu´ee, 91128 Bruy`eres-le-Chˆatel, France T. Vinci · M. Koenig · A. Benuzzi-Mounaix · N. Osaki · B. Loupias LULI, Ecole Polytechnique, 91128 Palaiseau cedex, France S. Bouquet · E. Falize CEA/DIF, D´epartement de Physique Th´eorique et Appliqu´ee, 91128 Bruy`eres-le-Chˆatel, France S. Atzeni Dipartemento di energetica, Universit´a di Roma La Sapienza and INFM, Italy

simulations (1D and 2D) and analytical studies. We exhibit results obtained from several radiative hydrodynamics codes. As a result, it is possible to discuss about the influence of the geometry and physical parameters introduced in the 1D and 2D models. Keywords Radiative shock . High-power laser . Hydrodynamics . Radiation transfer . Plasma

1. Introduction In astrophysics, radiative shocks are very common phenomena. A radiative shock is a shock sufficiently strong that radiation transport alters the hydrodynamic structure. The ionized gas emits a radiation flux, and a part of this radiation goes ahead the shock and heats up the gas before the compression wave arrives. This heating is called the radiative precursor. In this case, the temperature is high during the compression like for a classical shock. But the temperature profile presents a slow decreasing curve ahead the shock. As an example, in the supernova remnant Puppis-A, one can shows details of the strong shock wave disrupting an interstellar cloud (Hwang et al., 2005), or in the Rotten Egg Nebula (also known as the Calabash Nebula and OH231.8 + 4.2,), the central regions are contracted into a white dwarf. The gas strikes the surrounding medium, with a velocity around 300 km/s. A supersonic gas front forms ionizing hydrogen and nitrogen (Bujarrabal et al., 2002). In order to better understand these phenomena, astrophysicists need radiative hydrodynamics models which have to be validated by confrontation with experimental results. Since the hydrodynamics is structured by radiation effects, the relevant models are sensitive to the treatment of radiation transport and its coupling with hydrodynamics. Code Springer

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benchmarking is one of the motivations for the development of high-energy density studies. High-energy density laboratory astrophysics (HEDLA) experiments are mostly driven on large-scale lasers. But the connection of experimental results to astrophysical situation is not straightforward. In this work, our step is to check theoretical assumptions or computational models through radiative shock experiments performed with LULI 2000 laser. 2. Theoretical considerations on radiative shocks In laboratory, the shock velocity is determined by the laser energy and the pulse duration as a technology limitation. At LULI, the laser intensity is typically about 1014 W/cm2 , and the pulse duration is around 1 ns. Koenig et al., have shown (Koenig et al., 1999) that, under these conditions, the maximum achievable shock velocity at the interface pusher/gas cannot exceed 60 km/s. In order to produce radiative regime, Bouquet et al., have shown (Bouquet et al., 2000) from a simple analytical model in an ideal gas, that the shock velocity Us should be larger than the following threshold: ρ 1/3 . (1) A2/3 This threshold value depends on the gas density ρ and the atomic weight A. Since Us is determined from laser specifications and target design, this ratio should be as low as possible. Consequently, we need a low gas density ρ and a high atomic weight A. For these reasons, the heavy element xenon has been chosen for experiments. Equation (1) has been derived assuming that the magnitude of the radiative pressure and the thermal pressure is equivalent. Other authors (Keiter et al., 2002; Reighard et al., 2006) consider that the radiative regime is reached provided a radiation flux exists. They obtain a condition similar to Equation 1 but with less constraining exponents. In more details, one can discuss the kind of regime which is achieved according to the importance of radiative flux and pressure. Once the appropriate gas is chosen, we can define various types of shocks from a simple analytical model considering

Us ≥ Cste

Table 1 Threshold value Us (km/s) for the radiative shock calculated at three densities ρ0 (g/cm3 ) and required to reach two regimes respectively where the radiative flux equal the thermal flux and the radiative pressure equals the thermal pressure. The corresponding temperature T (eV) is also evaluated for these two cases Fr = Fth

the diffusion approximation (Mihalas et al., 1999; Zel’dovich et al., 2001; Drake, 2005). In Table 1, the shock velocity Us is estimated for two cases. The first one arises when the radiative flux Fr = σ T 4 (σ Stefan-Boltzman constant, T temperature) is equal to the thermal flux Fth = ρ0 Cv T Us (ρ0 initial density, Cv heat capacity at constant volume). The second case occurs when the radiative pressure Pr = (4σ/3c) × T 4 (c light velocity) is equal to the thermal pressure Pth = ρ0 k T /m (k Boltzman constant, m average particle mass). As expected, on the one hand, the velocity Us increases for increasing densities and, on the other hand, the condition Pr = Pth requires a higher value for Us than for the case Fr = Fth . Obviously, the same behavior is observed for the shock temperature. The results are summarized in Fig. 1 where we plotted the two frontiers Fr = Fth and Pr = Pth giving rise to three distinct regimes. Below the lower line we have the pure hydrodynamical regime and above the upper line one find the fully radiative regime dominated by the radiative pressure. In contrast to our approach developed initially (Bouquet et al., 2000) where the radiative shock was expected only in the upper zone, we proved experimentally that in between these two limits, the shock is also radiative (its structure is governed by a radiative precursor (Bouquet et al., 2004)). In the following, this intermediate region will be called therefore “hybrid radiative domain”. Our experiments at LULI are located in this area (see the grey patch in Fig. 1). If we want to perform experiments in the fully radiative regime, we need higher energy/power lasers.

Pr = Pth

ρ0

Us

T

Us

T

10−4 10−3 10−2

20 47 97

7.5 23 74

165 271 420

110 256 595

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Fig. 1 Representation in the xenon (ρ, T ) diagram of the three shock regimes. The working LULI domain is also drawn in grey oval

3. Experimental set-up In the laboratory the radiative shock is generated using the LULI 2000 laser which interacts with the target. This target

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161 Table 2 Diameter and curvature depth of the radiative shock recorded by GOI from 5 to 10 ns Time (ns)

5

6

7

8

9

10

Diam. (µm) Depth (µm)

730 104

783 157

835 209

887 261

1043 365

1148 470

Fig. 2 Experimental set-up used to generate radiative shock driven by high-power laser and diagnostics arrangement

is composed by a three-layers pusher and a gas cell. The cell filled with xenon is 7 mm long. When the laser energy is deposited on the first layer, it produces an ablation of the CH plastic. The laser energy is converted into mechanical energy, in the direction of the laser interaction by rocket effect during this ablation phase. As X-rays are emitted, a second layer in Titanium screens these X-rays so they do not reach the xenon cell. Titanium is heavier than plastic, therefore a third (CH) plastic layer is added to re-accelerate the shock wave. Then the shock emerges in the xenon cell. From then on the xenon compression is strong, the gas is ionized and a radiation flux goes faster than the shock. Therefore a matter flux is produced combined with a faster radiation flux towards the gas cell end. Figure 2 shows the experimental set-up which has been previously described (Bouquet et al., 2004; Fleury et al., 2002; Vinci et al., 2006) in details. The cell is placed in the center of the vacuum chamber. Two laser beams, at 2ω, deliver 1 kJ maximum on the target with a square pulse ranging from 1 to 5 ns. Several diagnostics are implemented and they are shown in Fig. 2 each one with a small picture of the type of data recorded. On the transverse side of the target, a VISAR measures the precursor electron density and records both the shock front and the precursor velocities. Two Gated Optical Imager (GOI) take a picture of the shock front and precursor by imaging the probe laser absorption, at two different times per shot. We measure the shock curvature and follow its propagation. As this diagnostic is the more recently implemented we exhibit in this issue raw experimental data. Figure 3 represents a series of GOI snapshots at 6 different times from 5 to 10 ns after the high-power laser interaction. For each image the exposure time is 100 ps, the shock takes place in xenon with an initial pressure of 0.1 bar. The sequence of images in Fig. 3 is obtained by performing several experiments in the same conditions, and changing the time delay for each picture. Thus indirectly we point out that

Fig. 3 Series of GOI snapshots at 6 different times imaging a transverse view of the shock front and its precursor

our way to generate radiative shocks and to measure its physical parameters are reproducible. This snapshot series allow to follow the shock propagation and its curvature. One can clearly see in Fig. 3 that the shape of the shock becomes more and more curved with the time while its lateral extension increases. At the beginning the shock front is around 400 µm in diameter, equalling that of the focal spot laser. Table 2 summarizes the shock front diameter and its curvature depth (penetration depth in the unshocked xenon) corresponding to each image in Fig. 3. From values in Table 2, it is quite obvious that the shock front is expanding and curving. This phenomena seems relatively linear from 5 to 8 ns and it undergoes an acceleration from 8 ns. In Section 4.4, we discuss the radial shock expansion with numerical support. On the rear side, two VISAR measure the third layerXenon interface velocity. Therefore this CH rear side is used as pusher and as witness to determine the shock velocity and the temperature at the beginning. In addition since we have also an emissivity diagnostic on the rear side, we know the shock front temperature during all the experiment. The typical results obtained have been widely discussed elsewhere (Bouquet et al., 2004; Vinci et al., 2002, 2005a; Koenig et al., 2005, 2006). We produce a shock with a velocity typically in the range [50–100] km/s. The shock front temperature is measured between 10 to 20 eV. The precursor electron density is typically in the range [1017 –1020 ] cm−3 . Springer

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And the precursor velocity is always around twice the shock velocity. Of course the precursor velocity depends on the chosen electron density. Even if the broad lines of experimental data can be recovered from rough but simple analytical models, numerical simulations (1D and 2D) are quite necessary to understand the fine structure and evolution of the radiative shock. 4. Numerical simulations and experimental results Numerical simulations require specific atomic data. But actually, xenon opacity calculations are very difficult especially at low temperature. However we have the opportunity to use three opacity tables from LULI (Eidmann, 1994), LUTH (Michaut et al., 2004) and CEA (in FCI code (Dautray and Watteau, 1993)).

Fig. 4 Non-stationary 1D calculation with the code HADES for a radiative shock in hydrogen (ρ0 = 1.2 × 10−3 g/cm3 , T0 = 1 eV). The shock front velocity is 150 km/s. All shock front positions are aligned to easily compare precursor lengths. The code LASTAR gives the reference for the stationary shock profile (grey curve)

4.1. A non-stationary code for astrophysics: HADES For astrophysical purposes we developed a new numerical code, namely, HADES. It deals with radiative hydrodynamics in gas, without laser consideration. However, it can also describe in details the physics of radiative shocks generated in the gas-cell target during experiments. Therefore, it can be also validated by comparing experimental data with numerical results in the same way it is done with laser/matter interaction codes. In addition, improved numerical schemes and radiative transfer allow HADES to especially model various specific astrophysical situations. HADES is based on the public package CLAWPACK (Leveque, 2002) coupled with a moment method for radiation transport (Dubroca and Feugeas, 1999; Gonzales and Audit, 2005), for a 1D-geometry. Figure 4 shows the way a nonstationary shock wave converges to a stationary structure (provided by the code LASTAR (Michaut et al., 2004a,b)) as the time elapses. Here Hydrogen is chosen for astrophysical application. In Fig. 4 the precursor length increases with time until reaching its final stable length is about 1.2 mm. Based on this simulation, a laser radiative shock wave in H gas might become steady in 1D at about 100 ns. This suggests that a more direct connection between experiments and astrophysics can be achieved in the near future with higher energy/power lasers. At the foot of the radiative precursor (z = 1.2 mm), the discrepancy between HADES (t = 100 ns) and LASTAR profiles comes from the fact that the stationary shock is calculated in the diffusion approximation with a Local Thermodynamics Equilibrium (LTE). P. Drake suggested (Drake, 2005) we switch this approximation by a Non-LTE diffusion model to recover the right shape of the precursor. As a matter of fact, the photon mean free path is larger than the gradient temperature length in the precursor foot. This property is in contradiction with the LTE assumption. Springer

At the moment, the coupling between CLAWPACK and a radiation transport is in progress in cylindrical geometry to obtain the first steps for the further HADES 2D. 4.2. Precursor length 1D simulations containing laser interaction are used to design targets, especially the thickness of each pusher layer. Figure 5 compares the precursor length between a 1D simulation obtained with MULTI (Ramis et al., 1998) and a 2D simulation performed with DUED (Atzeni, 1987) with the same initial conditions.The shock starts in xenon at 2 ns. One clearly see in Fig. 5 that the 1D-precursor length is growing during at least 10 ns, until around 350 µm. In contrast, the 2D length becomes constant from 6 ns and its length saturates about 190 µm. Thus the 1D or 2D behaviors of the precursor length evolution are completely different. The same difference between 1D and 2D simulations was yet encountered in many other confrontations between experimental data and simulations (Koenig et al., 2005; Vinci et al., 2005a; Michaut et al.,

Fig. 5 Evolution of the precursor length for a 1D simulation (MULTI) and a 2D simulation (DUED) under same conditions

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Fig. 6 Experimental data of the evolution of the shock front temperature compared with a 1D simulation (MULTI) and a 2D simulation (DUED) under same conditions. The bump in the experimental data for t < 0 is due to emission from the shocked CH

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Fig. 7 2D simulations with DUED reproducing the emissivity diagnostic comparing a full calculation (left side) and a pure hydrodynamics calculation (without radiation, right side)

2006), or theoretical approach (Leygnac et al., 2006), which allows to affirm that 1D simulation always overestimates the precursor. This fact is due to the intrinsic 1D-geometry which does not take into account the lateral radiation losses. In the same way, the evolution of the shock front temperature can be studied with 1D and 2D simulations. 4.3. Temperature The shock front temperature is measured by an emissivity diagnostic which records the self light emitted by the shock front during all the experiment (Vinci et al., 2006). Same previous 1D and 2D simulations have been compared with experimental results (Vinci et al., 2005b) and same previous conclusions were drawn. Figure 6 shows experimental temperature recorded during 7 ns in xenon and two numerical profiles. The experimental shock front temperature suddenly increases to 16 eV during less than 2 ns and adopts a slow decreasing to reach 13.5 eV at 7 ns. The 2D-DUED simulation gives a better agreement with experimental result, excepted during the first ns. In opposition, the temperature keeps growing up in 1D-MULTI simulation. Another point which can be examined with 2D simulation is the time-dependent radial expansion of the shock and its precursor. 4.4. Radial expansion In addition using DUED, one can reproduce the emissivity diagnostic and like in experiments, the shock undergoes a radial expansion with the time. Therefore in order to understand the radial expansion of the shock, we perform a simulation experiment with DUED. Figure 7 shows two simulations reproducing the emissivity diagnostic: on the left side, the

Fig. 8 2D simulation performed with FCI showing the shock front curvature and its precursor in xenon at 5 ns. Local velocities along the shock front and in the shock frame are indicated with black arrows

calculation is complete with hydrodynamics and radiation, but on the right side radiation is turned off. In Fig. 7 the laser spot is 400 µm in diameter and the shock is almost twice at 8 ns. One can notice that the lateral expansion is almost the same in these two cases until 6 ns. Until at least this time the radial heating is due only to hydrodynamic effects. However, the radial temperature gradient is modified because the shock, in the left case, penetrates material previously heated by the radiation flux. This very likely explains the larger extent of the heating after 6 ns in the case with the radiation flux. Figure 8 is still a 2D simulation performed with the FCI code, the radiative shock is in xenon at 5 ns. On the propagation axis, the shock velocity is 60 km/s. In black arrows, the local velocity vectors are drawn, in the shock frame. Therefore we have subtracted 60 km/s in the propagation direction (z-axis). The result is that along the shock front radius there are large radial velocities. It explains why the shock is expanding Springer

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radially. From different simulations in which we turned off radiation or electronic conduction, we understand that this expansion is due to the non uniformity of the laser spot.

5. Conclusions In conclusion, we performed radiative shock experiments with long pulse laser. From all analyzed experiments, one can conclude that our experimental results are reproducible. Special targets are designed according analytical model and 1D simulation. These 1D-codes, taking into account laser/matter interaction, are essential to prepare and to design experiments. Because of their running times are compatible with this type of work consisting to optimization instead of 2Dcodes. But 1D simulations always overestimate precursor lentghs. In our experiments, we measured the shock front and precursor velocities, the electron density in the precursor and the shock front temperature. In addition, the time-dependant shock curvature and radial expansion were recorded. Thus 2D behavior of the radiative shock was clearly identified. With simulations we obtain good agreement for all parameters excepted the precursor length which depends on the dimensionality of the code. We have shown that the 1D evolution of both the precursor length and the temperature is continuously increasing with time. And yet, same parameters reach almost their asymptotic value after few ns in 2Dsimulations like in experiments. The shock front and its radiative precursor expand in the lateral direction as the time elapses. This expansion is measured using GOI and is recovered by 2D simulations. We have shown this effect is mainly due to the non uniformity of the initial energy deposition i.e. the laser spot is not spatially homogenous. The precursor length remains difficult to predict, it probably depends on opacities. Acknowledgements Authors would like acknowledge Ravasio, A., Rabec le Gloahec, M., Barroso, P., Bauduin, D., for their contribution to the experiments and for the target manufacturing.

References Atzeni, S.: Plasma Phys. Contr. Fusion 29, 1535 (1987)

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Astrophys Space Sci (2007) 307:159–164 Bujarrabal, V., Alcolea, J., S´anchez Contreras, C., et al.: Astron. Astrophys. 389, 271 (2002) Bouquet, S., Teyssier, R., Chi`eze, J.-P.: Astrophys. J. Suppl. 127, 245– 252 (2000) Bouquet, S., Stehl´e, C., Koenig, M., et al.: Phys. Rev. Lett. 92, 225001 (2004) Drake, R.P.: Astrophys. Space Sci. 298, 49 (2005) Drake, R.P.: Private Communication (2005) Dautray, R., Watteau, J.-P.: La Fusion Thermonucl´eaire par laser Eyrolles, Paris (1993) Dubroca, B., Feugeas, J.L.: C.R. Acad. Sci. 329, 915 (1999) Eidmann, K.: Laser Part. Beams 12, 223 (1994) Fleury, X., Bouquet, S., Stehl´e, C., et al.: Lasers Part. Beams 20, 263 (2002) Gonz´alez, M., Audit, E.: Astrophys. Space Sci. 298, 357 (2005) Hwang, U., Flanagan, K.A., Petre, R.: Astrophys. J. 635, 355 (2005) Koenig, M., Benuzzi-Mounaix, A., Philippe, F., et al.: Appl. Phys. Lett. 75, 3026 (1999) Keiter, P.A., Drake, R.P., Perry, T.S., et al.: Phys. Rev. Lett. 89, 165003 (2002) Koenig, M., Vinci, T., Benuzzi-Mounaix, A., et al.: Phys. Plasmas, accepted for publication (2006) Koenig, M., Vinci, T., Benuzzi-Mounaix, A., et al.: Astrophys. Space Sci. 298, 69 (2005) Leveque, R.J.: Finite volume methods for hyperbolic problems, Cambridge Texts in applied math (2002) Leygnac, S., Boireau, L., Michaut, C., et al.: J. Phys. IV, accepted for publication (2006) Mihalas, D., Weibel-Mihalas, B.: Foundations of radiation hydrodynamics. Dover Pub. Inc., Mineola, New York, (1999) Michaut, C. Stehl´e, C., Leygnac, S., et al.: Eur. Phys. J. D 28, 381 (2004a) Michaut, C., Stehl´e, C., Boireau, L., et al.: Microscopic aspects in radiative shock structures. In: Hammel, B.A., Meyerhofer, D.D., Meyer-ter-Vehn, J., Azech, H. (eds), Proc. of the Conf. IFSA 2003, Monterey, CA, USA, Sept. 2003, Inertial Fusion Sciences and Applications 2003, State of the art, p. 954 (2004b) Michaut, C., Boireau, L., Vinci, T., et al.: J. Phys. IV, accepted for publication (2006) Ramis, R., Schmalz, R., Meyer-Ter-Vehn, J.: Comp. Phys. Comm. 49, 475 (1988) Reighard, A.B., Drake, R.P., Dannenberg, K.K., et al.: Submitted to Phys. of Plasmas (2006) Vinci, T., Koenig, M., Benuzzi-Mounaix, A., et al.: Phys. Plasmas Lett. 13, 010702 (2006) Vinci, T., Koenig, M., Benuzzi-Mounaix, A., et al.: Astrophys. Space Sci. 298, 333 (2005a) Vinci, T., Koenig, M., Benuzzi-Mounaix, A., et al.: Radiative shocks: new experiments for laboratory astrophysics. In: Hidalgo, C., van Milligen, B. Ph, (eds), Proc. of the 32nd EPS plasma physics conference, Tarragona, Spain, 27 June–1 July 2005, ECA Vol. 29C, O-2.021 (2005b) Zel’dovich, Ya.B., Raizer, Yu.P.: Physics of shock waves and hightemperature hydrodynamic phenomena. In: Hayes, W.D., Probstein, R.F. (eds), Dover Pub. Inc., Mineola, New York, (2001)

Astrophys Space Sci (2007) 307:165–168 DOI 10.1007/s10509-006-9254-1

O R I G I NA L A RT I C L E

Electrostatic Potentials in Supernova Remnant Shocks Matthew G. Baring · Errol J. Summerlin

Received: 14 April 2006 / Accepted: 12 September 2006 C Springer Science + Business Media B.V. 2006


Abstract Recent advances in the understanding of the properties of supernova remnant shocks have been precipitated by the Chandra and XMM X-ray Observatories, and the HESS ˇ Atmospheric Cerenkov Telescope in the TeV band. A critical problem for this field is the understanding of the relative degree of dissipative heating/energization of electrons and ions in the shock layer. This impacts the interpretation of X-ray observations, and moreover influences the efficiency of injection into the acceleration process, which in turn feeds back into the thermal shock layer energetics and dynamics. This paper outlines the first stages of our exploration of the role of charge separation potentials in non-relativistic electron-ion shocks where the inertial gyro-scales are widely disparate, using results from a Monte Carlo simulation. Charge density spatial profiles were obtained in the linear regime, sampling the inertial scales for both ions and electrons, for different magnetic field obliquities. These were readily integrated to acquire electric field profiles in the absence of self-consistent, spatial readjustments between the electrons and the ions. It was found that while diffusion plays little role in modulating the linear field structure in highly oblique and perpendicular shocks, in quasi-parallel shocks, where charge separations induced by gyrations are small, and shock-layer electric fields are predominantly generated on diffusive scales.

Keywords Shock acceleration . Cosmic rays . Supernova remnants . Electrostatics

M. G. Baring (
) · E. J. Summerlin Department of Physics and Astronomy MS-108, Rice University, P.O. Box 1892, Houston, TX 77251, U.S.A. e-mail: [email protected]

1 Introduction The understanding of the character of shells and interiors of supernova remnants (SNRs) has been advanced considerably by groundbreaking observations by the Chandra X-ray Observatory. These have been enabled by its spectral resolution coupled with its impressive angular resolution. Of particular interest to the shock acceleration and cosmic ray physics communities is the observation of extremely narrow non-thermal X-ray spatial profiles in selected remnants (see Long et al., 2003 for SN1006; Vink and Laming, 2003 for Cas A; and Ellison and Cassam-Chena¨ı, 2005, for theoretical modeling), which define strong brightness contrasts between the shell, and the outer, upstream zones. If the synchrotron mechanism is responsible for this non-thermal emission, the flux ratios from shock to upstream indicate strong magnetic field enhancement near the shock. These ratios considerably exceed values expected for hydrodynamic compression at the shocked shell, so proposals of magnetic field amplification (e.g. Lucek and Bell, 2000) in the upstream shock precursor have emerged. Another striking determination by Chandra concerns electron heating by ions in the shock layer. Dynamical inferences of proton temperatures in remnant shocks can be made using proper motion studies of changes in a remnant’s angular size, or more direct spectroscopic methods (e.g. Ghavamian et al., 2003). In the case of remnant 1E 0101.2-7129, Hughes et al. (2000) used a combination of ROSAT and Chandra data spanning a decade to deduce an expansion speed. Electron temperatures Te are determined using ion line diagnostics (assuming the equilibration Te = T p ), via both the widths and relative strengths for different ionized species. From these two ingredients, Hughes et al. (2000) observed that deduced proton temperatures were considerably cooler, i.e. 3kT p /2 ≪ m p (3u 1x /4)2 /2, than would correspond to Springer

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standard heating for a strong hydrodynamic shock with an upstream flow speed of u 1x . The same inference was made by Decourchelle et al. (2000) for Kepler’s remnant, and by Hwang et al. (2002) for Tycho’s SNR. This property is naturally expected in the nonlinear shock acceleration scenario that is widely used in describing cosmic ray and relativistic electron generation in SNRs: the highest energy particles tap significant fractions of the total available energy, leading to a reduction in the gas temperatures. This nonlinear hydrodynamic modification has been widely discussed in the cosmic ray acceleration literature (e.g. see Jones and Ellison, 1991; Berezhko and Ellison, 1999, and references therein), and has been extensively applied to multiwavelength SNR spectral models (e.g. see Baring et al., 1999; Berezhko et al., 2002; Ellison and Cassam-Chena¨ı, 2005; Baring et al., 2005). The extent of equilibration between electrons and ions in SNR shell shocks needs to be understood, and can potentially be investigated by laboratory plasma experiments. A critical ingredient is the electrostatic coupling between electrons and protons in the shock layer, which offers the potential for considerable heating of e− , coupled with cooling of protons, setting m p (3u 1x /4)2 /2 ≫ 3kTe /2 ≫ m e (3u 1x /4)2 /2 with Te = T p . Probing this coupling is the subject of this paper. Here we describe preliminary results from our program to explore electrostatic energy exchange between these two species in SNR shocks, using a Monte Carlo simulation of charged particle transport, their spatial distribution and associated electric field generation. The goal is to eventually obtain a simulation with self-consistent feedback between the charge separation potentials, and the Lorentz force they impose on the charges. The research progress outlined here indicates that the role of diffusion in quasi-parallel shocks is very important, and can readily influence charge separation potentials.

2 Shock layer electrostatics in supernova remnants Cross-shock electrostatic potentials arise in the shock layer because of the different masses of electrons and ions: upstream thermal ions gyrate on larger scales than do their electron counterparts when they transit downstream of the shock for the first time. In shocks where the field is oblique to the shock normal by some angle Bn1 upstream (and therefore a greater angle downstream), on average, protons will be located further downstream of the shock than electrons. This naturally establishes an electric field E, to which the plasma responds by accelerating electrons and slowing down ions to short out the induced E. A feedback loop ensues, mediated by fields and currents that vary spatially on the order of, or less than, the ion inertial scale, which is typically shorter than the ion gyroradius in astrophysical shocks such as those associated with supernova remnant shells. Springer

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Particle-in-cell (PIC) simulations are a natural technique (e.g. see Forslund and Friedberg, 1971 for an early implementation) for exploring signatures of such electrostatics in shock layers. These trace particle motion and field fluctuations, obtained as self-consistent solutions of the NewtonLorentz and Maxwell’s equations, in structured zones or cells in spatially-constrained boxes. Such simulations have been used recently to probe the Weibel instability at weaklymagnetized, perpendicular, relativistic pair plasma shocks (see Silva et al., 2003; Hededal et al., 2004; Nishikawa et al., 2005). They have also been used by Shimada and Hoshino (2000) to treat electrostatic instabilities at non-relativistic, quasi-parallel electron-ion shocks. Rich in their turbulence information, due to their intensive CPU requirements, such simulations have difficulty in modeling realistic m p /m e mass ratios, and fully exploring 3D shock physics such as diffusive transport. Moreover, they cannot presently address the wide range of particle momenta and spatial/temporal scales encountered in the acceleration process; they often do not obtain time-asymptotic states for the particle distributions. Monte Carlo techniques provide an alternative method that can easily resolve electron and proton inertial scales, treat fully 3D transport and large dynamical ranges in spatial and momentum scales, all at modest computation cost. While they parameterize the effects of turbulence via diffusive mean free paths (e.g. see Jones and Ellison, 1991), they can accurately describe the microphysics of cross-shock electrostatic potentials. This simulational approach has been well-documented in the literature (e.g. Jones and Ellison, 1991; Ellison et al., 1996), with definitive contributions to the study of heliospheric shock systems, cosmic ray production, SNR applications and gamma-ray bursts. It models the convection and diffusion of gyrating particles in spatiallystructured flows and fields, with transport back and forth across the shock effecting diffusive Fermi-type acceleration directly from the thermal population. The mean free path λ is usually prescribed as some increasing function of particle momentum p or gyroradius r g . Here we use this approach, with λ ∝ p adopted as a broadly representative situation: see Baring et al. (1997) for a discussion of evidence from observations and plasma simulations in support of such a specialization. Here, λ/r g = 5 is chosen for illustrative purposes, to begin to investigate electrostatic influences on low-energy particles in non-relativistic electron-ion shocks of arbitrary Bn1 . Clearly then, the diffusive scales for protons and electrons are disparate by their mass ratio. In the Monte Carlo simulation, the shock is defined magnetohydrodynamically, consisting of laminar, uniform flows and fields upstream and downstream of a sharp, planar discontinuity. The magnetic fields and flow velocities either side of the shock are related uniquely through the standard Rankine-Hugoniot solutions for energy and momentum flux conservation (e.g. see Boyd and Sanderson, 1969). These

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Fig. 1 Electrostatic profiles for an electron-proton plasma shock of speed u 1x = 9000km/sec and sonic Mach number MS = 10, with upstream field obliquities Bn1 = 0◦ (green), Bn1 = 30◦ (blue) and Bn1 = 60◦ (red), as labelled. Left panel: scaled charge density distribution ρ(x); Right Panel: resulting “linear” electric field profile E ≡ E x (x) computed by solving Gauss’ Law (the Bn1 = 0◦ case

displays E x (x)/2). The profiles are exhibited on proton gyroscales r g, p = m p u 1x c/eB1 , so that fluctuations on electron gyroscales are collapsed into the shock layer (dashed vertical line). The panels display both oscillations associated with proton gyrations in oblique cases, and diffusive upstream “precursors,” which are most prominent when Bn1 = 0◦ (a parallel shock).

solutions include essential elements of Maxwell’s equations, such as the divergenceless nature of the B field. In the reference frames of the local upstream and downstream fluids, the mean electric field is assumed to be zero, a consequence of very effective Debye screening, so that the only electric fields present in shock rest frames are u×B drift fields. The charged electrons and protons (more massive ions are omitted in this paper to simplify the identification of the principal effects) are treated as test particles, convecting into and through the shock, initially with the prescribed upstream fluid velocity u1 . This neutral beam is entirely thermal, and moreover is in equipartition, so that it has an input temperature Te = T p . The charges constantly diffuse in space to mimic collisions with magnetic turbulence that is putatively present in the shock environs, and in so doing, can be accelerated. These non-thermal particles form a minority of the total population, and provide only a minor contribution to the fields illustrated. The charges transiting the shock distribute their downstream density in a manner that couples directly to their gyrational motion (e.g. see Baring, 2006), and the local densities of electrons and protons can easily be tracked in the Monte Carlo technique by accumulating “detection” data at various distances from the shock. Monte Carlo simulation runs clearly exhibit non-zero charge excursions within a proton gyroradius of the shock, an effect similar to those found in PIC codes. For example, a cold, neutral e − p upstream beam develops an electron concentration near the shock in the downstream region, with protons distributed on their larger inertial scales. The resulting charge distributions

ρ(x) depend on both the upstream field obliquity  Bn1 , and also on the sonic Mach number MS ≈ u 1x / 5kT p /(3m p ) in situations where the upstream beam is warm. Due to the steady-state, planar nature of the simulation, these distributions depend only on the coordinate x along the shock normal. It is straightforward using Gauss’ law for electrostatics, ∇ · E = 4πρ(x), to integrate the charge distribution profile to obtain E x (x) = −∂/∂ x. Eventually, such “linear” fields will then be used to compute the energy exchange between electrons and ions as they cross the non-monotonic charge separation potential (x). Linear determinations of electrostatic spatial profiles are shown in Fig. 1 to illustrate the key features; these did not self-consistently include the acceleration of electrons and protons in the produced E field. The left panel depicts large charge density fluctuations that trace ion gyration in the downstream magnetic field. Similar fluctuations of opposite sign are created by electrons, but on much smaller scales that are not resolved in the Figure. Accordingly, “striped” zones of positive and negative charge density result, and this electrostatic analog of a plasma oscillation integrates to produce the E fields in the right panel that can accelerate or decelerate electrons and protons. The outcome depends ◦ on the shock obliquity Bn1 when Bn1 < ∼ 60 , whereas ◦ quasi-perpendicular shocks with Bn1 > ∼ 60 possess profiles fairly close to the Bn1 = 60◦ case depicted in Fig. 1, since they all have field obliquities Bn2 ≈ 80–90◦ downstream. Note also, that while the gyrational contributions are prominent, there is also a diffusive contribution, manifested as an upstream precursor modification to ρ(x) and E. This is Springer

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particularly marked in the parallel shock (Bn1 = 0◦ ) case, where the diffusive scale along the field achieves a maximal component orthogonal to the shock plane. This diffusive influence originates in accelerated particles returning to the upstream side of the shock (x < 0), enhancing the density there before convecting downstream again: protons effect this on larger scales, and so control the precursors seen in the figure (i.e. ρ(x) > 0 for x < 0). Since the fields are established on the scale of a proton gyroradius, their magnitude scales as E x ∼ 4πρr g, p = 4π en p (m p u 1x c/eB1 ), so that |E x /B1 | ∼ 4π n p m p u 1x c/B12 ≡ M2A (c/u 1x ) ≫ 1 for Alfv´enic Mach numbers MA > 1. The competition between gyrational and diffusive influences on electrostatics is a principal conclusion of this paper, defining a dichotomy delineating quasi-parallel and quasiperpendicular shocks. The Monte Carlo technique can accurately trace both influences, while comfortably resolving the disparate scales for the e − p shock problem. Since the “linear” results illustrated need to be upgraded to account for the E-field’s influence on e− and p motions, it is presently unclear whether ions can energize electrons overall (the right panel of the Figure suggests they may even decelerate them), and how the net work done depends on field obliquity. A noticeable feature of the electric field profiles ◦ in Fig. 1 is that for Bn1 < ∼ 60 , these linear field calculations do not establish |E| → 0 asymptotically as |x| → ∞, as required by net charge neutrality. The next step of this program will be to solve the Newton-Lorentz equation of motion dp/dt = q(E + v × B/c) to determine both drift and accelerative contributions to the charges’ motions. These will necessitate a recomputation of the E field profiles, and a feedback loop will result, with shock layer currents generating magnetic field excursions via Amp`ere’s law, ∇ × B = 4πJ/c. This iterative process will continue to convergence (establishing |E| → 0 as |x| → ∞), with relaxation to equilibriumoccuring on the spatial response scale u 1x /ω p , where ω p = 4π e2 n p /m p is the proton plasma frequency. Since u 1x /(ω p r g, p ) ∼ u 1x /(cMA ) ≪ 1), this response scale is far less than a proton gyroradius for typical SNR environmental parameters, and indeed for any strong, non-relativistic astrophysical shock. The degree of electron energization in the cross shock potential may offer significant insights into the well-known electron injection problem at non-relativistic shocks. Electrons do not resonantly interact with Alfv´en waves until they become relativistic. Levinson (1992) suggested that e− interaction with a presumably abundant supply of whistler waves could effect pre-injection into diffusive acceleration processes, if electrons could achieve energies in excess of around 10 keV to access the whistler resonance branch. The planned self-consistent extension of the developments outlined here will help determine whether this channel of access to continued acceleration is opened up by shock layer elecSpringer

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trostatics. Moreover, crafted laboratory plasma experiments may cast light on this aspect of shock physics. 3 Conclusion In this paper, charge density and associated cross-shock electric field spatial profiles are presented for different magnetic field obliquities. It was found that in highly oblique and perpendicular shocks diffusion plays little role in modulating the field structure, which is controlled by the magnetic kinking and compression on the downstream side of the shock. In contrast, in quasi-parallel shocks, where the gyrational charge separation is small, diffusion scales upstream and downstream of the shock dominate the generation of shock-layer electric fields. This is an interesting twist, suggesting that observationally, thermal X-ray emission could be distinctly different in portions of an SNR rim that establish quasi-parallel and quasi-perpendicular shocks. The work discussed here paves the way for self-consistent determination of the acceleration/deceleration of electrons and protons, their spatial distributions, and the electric fields normal to non-relativistic shocks. This development will impact the understanding of electron injection and acceleration in shocks of all obliquities.

References Baring, M.G.: On-line proceedings of the 2006 KITP/UCSB conference “Supernova and Gamma-Ray Burst Remnants” (2006) [http://online.kitp.ucsb.edu/online/grb c06/ baring/] Baring, M.G., Ellison, D.C., Reynolds, S.P., Grenier, I.A., Goret, P.: ApJ 513, 311 (1999) Baring, M.G., Ellison, D.C., Slane, P.O.: Adv. Space Res. 35, 1041 (2005) Baring, M.G., Ogilvie, K.W., Ellison, D.C., Forsyth, R.J.: ApJ 476, 889 (1997) Berezhko, E.G., Ellison, D.C.: ApJ 526, 385 (1999) Berezhko, E.G., Ksenofontov, L.T., V¨olk, H.J.: ApJ 395, 943 (2002) Boyd, T.J.M., Sanderson, J.J.: Plasma Dynamics. Nelson & Sons, London (1969) Decourchelle, A., Ellison, D.C., Ballet, J.: ApJ 543, L57 (2000) Ellison, D.C., Baring, M.G., Jones, F.C.: ApJ 473, 1029 (1996) Ellison, D.C., Cassam-Chena¨ı, G.: ApJ 632, 920 (2005) Forslund, D.W., Freidberg, J.P.: Phys. Rev. Lett. 27, 1189 (1971) Ghavamian, P., Rakowski, C.E., Hughes, J.P., Williams, T.B.: ApJ 590, 833 (2003) Hededal, C.B., Haugbolle, T., Frederiksen, J.T., Nordlund, A.: ApJ 617, L107 (2004) Hughes, J.P., Rakowski, C.E., Decourchelle, A.: ApJ 543, L61 (2000) Hwang, U., et. al.: ApJ 581, 110 (2002) Jones, F.C., Ellison, D.C.: Space Science Rev. 58, 259 (1991) Levinson, A.: ApJ 401, 73 (1992) Long, K.S., et al.: ApJ 586, 1162 (2003) Lucek, S.G., Bell, A.R.: MNRAS 314, 65 (2000) Nishikawa, K.-I., et al.: ApJ 622, 927 (2005) Shimada, N., Hoshino, M.: ApJ 543, L67 (2000) Silva, L.O., et al.: ApJ 596, L121 (2003) Vink, J., Laming, J.M.: ApJ 584, 758 (2003)

Astrophys Space Sci (2007) 307:169–172 DOI 10.1007/s10509-006-9263-0

O R I G I NA L A RT I C L E

Non-Stationary Rayleigh-Taylor Instabilities in Pulsar Wind Interaction with a Supernova Shell X. Ribeyre · L. Hallo · V. T. Tikhonchuk · S. Bouquet · J. Sanz

Received: 14 April 2006 / Accepted: 6 October 2006 C Springer Science + Business Media B.V. 2006


Abstract The Rayleigh-Taylor instability (RTI) plays an important role in the dynamics of several astronomical objects, in particular, in the supernovae (SN) evolution. In the present paper we examine the dynamics of a shell (representing a type II SN remnant) blown-up by a wind emitted by a central pulsar. The shell is accelerated by the pulsar wind and its inner surface experiences the RTI. We develop an analytical approach by using a specific transformation into the coordinate frame co-moving with the SN ejecta. We first derive a non-stationary spherically symmetric solution describing an expansion of a gas shell under the pressure of a central source (pulsar). Then, we analyze its 3D stability with respect to a small perturbation on the inner shell surface. The dispersion relation is derived in the co-moving reference frame. The growth rate of the perturbation is found and its temporal evolution is discussed. We compare our result with the previous published studies and apply it to the Crab nebula evolution. Keywords Rayleigh-Taylor instabilities . Pulsar wind nebulae . Supernova remnants

X. Ribeyre (
) · L. Hallo · V. T. Tikhonchuk Centre Lasers Intenses et Applications, UMR 5107, CNRS, Universit´e Bordeaux 1, CEA, Universit´e Bordeaux 1, 351, Cours de la lib´eration, 33405 Talence, France e-mail: [email protected] S. Bouquet Commissariat a` l’Energie Atomique, DIF/D´epartement de Physique Th´eorique et Appliqu´ee, BP 12, 91680, Bruy`eres-le-Chˆatel, France J. Sanz E.T.S.I., Aeron´auticos, Universidad Polit´ecnica de Madrid, Madrid 28040, Spain

1 Introduction The Crab nebula observations lead us to conclude that the expansion of the supernova remnant (SNR) is non stationary (Trimble, 1968). Indeed, the supernova (SN) ejecta are accelerated by the central pulsar wind and a initially homogeneous shell is decomposed in a complex filamentary structure. For a typical filament position about r ≃ 1 pc from the central pulsar, with an age about t ∼ 930 years (Davidson and Fesen, 1985), the ratio r/t gives the average filament velocity about 1050 km s−1 ; but the current velocity is 10% greater (∼1150 km s−1 ). This result shows clearly that the nebula was accelerated (Trimble, 1968; Davidson and Fesen, 1985). Moreover the Hubble Space Telescope high resolution observations of the Crab nebula show that the filaments are arranged in a structure that morphologically is similar to the non linear stage generated by the RTI. It develops at the interface between the pulsar-driven synchrotron radiation and a shell of swept-up ejecta (Hester et al., 1996). The Crab nebula belongs to the so-called family of pulsar wind nebula (PWN). Hereafter we use this more general expression. A hydrodynamic model of PWN has been developed by Blondin et al. (2001) and Chevalier (2005). Several analytical and numerical studies are devoted to the interaction between the pulsar wind and the SN ejecta (Jun, 1998). Reynolds et al. (1984) studied a self-similar homologue spherical expansion model where the radial velocity is proportional to the radius. The fact that the synchrotron luminosity of the pulsar decreases in time plays an important role in the PWN evolution. However, in his simulations (Jun, 1998) considers a constant pulsar wind pressure in interaction with expanding ejecta. He carried out 2D simulations of the RTI development in a thin shell expanding during several thousand years, while the pulsar luminosity decreases in a shorter time scale. Blondin et al. (2001) studied the evolution phase beyond 104 years, Springer

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3

Fig. 1 Density profile for a PWN, the density is in g/cm and the radius in cm. There are an expanding pulsar wind, a region of shock pulsar wind at radius r0 (pulsar bubble), a swept shell of ejecta is bounded by a shock at r1 (or R p ). We consider stability of the shell of ejecta of thickness r0 -r1 . From Blondin et al. (2001)

when the reverse shock in the ejecta (denoted R2 in Fig. 1), due to the interaction between the ejecta and the interstellar medium interacts with the pulsar wind. More recently, Bucciantini et al. (2004) carried out a MHD simulation to take into account the magnetic field effect on the RTI development over a time scale of 2000 years. However, the relation between SN fragments characteristics and the wind pressure remains unclear. In this paper, our study is based on the analytical model developed by Ribeyre et al. (2005), which is similar to that of Bernstein and Book (1978). In this model, the pulsar wind composed by high energetic particles and photons, is modelled by a pressure law that varies during the shell expansion. The evolution phase that we consider in this paper is the same that Jun (1998) studied. It concerns the period about 1000 years after the SN explosion and we are taking into account the non-stationarity of the shell evolution, and the acceleration phase is followed by the ballistic movement. The following assumptions have been made: – The mass brought into the expanding shell is negligible and consequently the mass of the shell is constant. – The pulsar wind density is small compared to the density at the inner shell interface. – The outer shell interface is in contact with vacuum. The first assumption is valid if one considers a period of evolution of the PWN not too close to the explosion time, i.e., when the shell was already formed and consequently, its mass does not evolve much more (Chevalier, 2005). The second assumption is valid because the pulsar wind density is low and it brings little matter to the shell. In our study, we do not consider the effect of the ablation process on the instability development driven by the wind (Atzeni and Meyer-Ter-Vehn, 2004; Sanz and Betti, 2005). Although ablation is possible, it should not have a strong effect since the radiation of the pulsar is trapped inside the shell volume and, Springer

Fig. 2 Simplified model of the PWN. A shell of ejecta is blown by the pulsar wind and the RTI develops at the inner shell interface. We assume that the pulsar wind have a negligible density and that the outer shell is in contact with vacuum

consequently, the ablation and the deposition of the wind matter on the inner shell surface balance. Moreover, it is known that ablation stabilizes the small wavelengths, whereas we are interested in large wavelengths compared to the shell thickness. The third assumption is valid for a shell expanding in an interstellar medium of a very low density or for a young supernova remnant. Moreover, the derivation below shows that the external boundary condition of the shell has a little influence on the RTI growth rate, because only the inner interface of the shell is unstable. In our paper we consider a perturbed amplitude smaller than the shell thickness. This configuration is appropriate for studies of the shell fragmentation. The case of thin shell, where the perturbed shell amplitude is greater than the shell thickness, was considered in Ref. (Kull, 1991).

2 Shell space-time evolution Our analysis of expansion of a shell blown by a pulsar wind (see Fig. 2) is based on the model developed by Ribeyre et al. (2005). We consider a family of solutions of the Euler equations, for a polytropic gas, such as p = Kρ γ . The effect of the pulsar wind is treated by a pressure law acting on the inner interface of the ejecta. A non-stationary spherically-symmetric solution for the unperturbed radial shell flow is described by [see in Ribeyre et al., 2005]: 1/(γ −1)
 ρ0  r2 2 −1/(γ −1) , (1) 1 − R 1 − 0 C3 C 2 r12
 γ  ρ0  r 2 γ /(γ −1) 2 −γ /(γ −1) p(r, t) = K 3γ 1 − R0 1− 2 2 , (2) C C r1   t r 2 vr (r, t) = 2 β + (β + 1) , (3) C τ τ ρ(r, t) =

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where ρ(r, t), p(r, t), vr (r, t) are respectively the density, the pressure and the radial velocity, r0 and r1 are the inner and outer initial shell radii respectively. The parameter β definines the magnitude of the velocity profile at t = 0 and τ is the characteristic time for the dynamics of the shell. Moreover, R0 ≡ r0 /r1 and ρ0 ≡ ρ(r0 , 0) is the initial density at the inner border of the shell. A relation exists between the inner density, the constant K and r1 :    1/(γ −1) ρ0 = r12 (γ − 1) 1 − R02 /2K γ τ 2 .

(4)

Finally, the scale function C(t) is given by the solution of the following differential equation: τ −2 = C¨ C 3γ −2 ,

(5)

where the upper dot stands for the time derivative of C. For the polytropic constant γ = 5/3, i.e., for a monoatomic ideal gas, an analytical expression for C(t) reads as:

(Bouquet et al., 1985; Ribeyre et al., 2005). More precisely, for γ = 5/3, one can express the angular perturbed displacement evolution as: δ ∼ eωtˆ/τ Ylm (θ, φ),

(9)

where the relation between the co-moving time tˆ and the time in the physical space t is given by: tˆ ≡ τ g(t) = τ {arctan[β + (β 2 + 1) t/τ ] − arctan β}.

(10)

Moreover, although Bernstein and Book (1978) do not succeed to exhibit a dispersion relation, the one we obtain is very simple. The parameter ω is given by the roots of the following equation : ω4 − ω2 − l(l + 1) = 0.

(11)

Therefore, there are four linearly independent eigenmodes: C(t) =




 βt 2 t 2 1+ + 2. τ τ

(6)

As explained earlier, it is clear that the parameter τ is the characteristic time of shell expansion and the constant ˙ β ≡ τ C(0) characterizes its initial velocity, v0 (r ) = βr/τ . Moreover, the temporal evolution of the internal face is given by r0 (t) = C(t)r0 and for the outer interface r1 (t) = C(t)r1 . The scale function C(t) is a linear function of t for t ≫ τ , i.e., when the shell is in ballistic motion. The function C(t) characterizes also the acceleration of the shell. Indeed, the temporal derivative of v(t), is: v˙ =

r τ 2 C 3γ −1

.

(12)

One mode ω1 is unstable and three others are stable. One obtains the same expression (8) that Bernstein and Book (1978) by considering a linear superposition of the modes ω1 and ω2 for β = 0. However this particular solution does not describe a general perturbation of the inner shell surface. By taking account of all four modes, one can describe various initial conditions for the shell perturbations and study their stability as well as the interaction between the inner and outer shell interfaces.

(7) 4 Qualitative RTI analysis

3 Rayleigh-Taylor instability Bernstein and Book (1978) obtained an exact expression for the incompressible perturbation evolution for a given radial flow. They consider a radially symmetric shell flow described by Equations (1)–(3) and a 3D perturbation with an angular dependence corresponding to the spherical harmonic: Ylm (θ, φ). Applying their expression to the case of an ideal gas, γ = 5/3, the time evolution of the perturbation δ, is given by: √ δ ∝ C(t) cosh [ l + 1 arctan (t/τ )],

√ ω3,4 = ±i l.

√ ω1,2 = ± l + 1,

(8)

for β = 0, i.e., without initial shell velocity. A similar expression can be obtained by solving the perturbed Euler equations in the expanding co-moving frame

The dispersion relation defined by the Equation (11) and the solutions (12) are surprisingly simple and they correspond to the incompressible perturbations. In √ particular, if one considers only the unstable mode ω1 = l + 1, then, the perturbation evolves as exp [ω1 tˆ/τ ] [see Equations (9) and (10)]. This behavior can be compared with a simple model, by supposing that at every moment the √ growth rate of the RTI is given by the formula ω = τ w0 k0 (Rayleigh, 1883), describing the instability of a plane surface in acceleration w0 . In our case the acceleration of the inner interface is defined by Equation (7). Consequently, the temporal evolution of the perturbation δ(t) in this model is given by: δ(t) ∝ exp

 t  0

k0

(t ′ )w

0

(t ′ ) dt ′



,

(13)

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Astrophys Space Sci (2007) 307:169–172

Table 1 Comparison of the growth rate and the amplification of the perturbation for t → ∞, given by Eq. (15) with β = 0 and the solution of Bernstein and Book (1978) [see the Eq. (51) in Bernstein and Book (1978)], for γ = 4/3, 5/3, 2 and for l = 100. One gives for each value of γ respectively, the growth rate and also the amplification which corresponds to exponential of the product growth rate multiplied by time γ 4/3 5/3 2

Our approximate model

Bernstein and Book (1978) solution

21.63–2.48 × 109 15.09–3.60 × 106 12.19–1.98 × 105

20.06–5.20 × 108 15.09–3.60 × 106 12.84–3.78 × 105

5 Conclusion

where k0 (t) is the wave number of the perturbation. In spherical geometry, the wavelength of the perturbation λ evolves in time as, 2π/λ(t) = (l + 1)/[r0 C(t)] (where l is the mode number of the perturbation). Then using expression (7) in Equation (13), one obtains: √   l + 1 t dt ′ δ(t) ∝ exp . 3γ −1 τ 0 C 2

(14)

In particular for γ = 5/3, one finds the exact solution (8) if β = 0 or a more general solution if β = 0. One can go further and calculate the growth rate for an arbitrary γ . Indeed, the expression of k0 (t) does not change and it is enough to calculate the new acceleration w0 (t, γ ). Then, the perturbation growth (14) can be presented in the following way: ⎡  √ δ(t) ∼ exp ⎣ l + 1

0

C(t)

C (1−3γ )/2 dC   β 2 + 3(γ2−1) 1 −

1 C 3γ −3

⎤

 ⎦.

We examined the dynamics of a shell (representing a type II SN remnant) blown by the strong wind emitted by a central pulsar. Assuming that the shell mass is constant and evolves in vacuum, we developed a simplified model describing a non-stationary shell evolution. Moreover, we derive a simple dispersion relation for the RTI growth rate for an ideal polytropic gas (γ = 5/3). We compare our approach with the work achieved by Bernstein and Book (1978) and show that the RTI growth and the growth rate of perturbation are comparable even for over values of γ . Finally, this model can be applied to the Crab nebula and results are in pretty agreement with others studies (Jun, 1998; Bucciantini et al., 2004) and with observations as well (Hester et al., 1996). Acknowledgements The authors thank Dr T. Foglizzo for the fruitful discussion and useful comments. This work is partly supported by the Aquitaine Region Council.

References

(15)

Starting from this expression, one can find the asymptotic growth rate [C(t) → t for t → +∞] with β = 0 which could be compared with the expression of the asymptotic growth rate provided by Bernstein and Book (1978) for an arbitrary γ [see their expression (51)]. In this case, for l = 100 and for three values of γ given by Bernstein and Book, γ = 5/3, 4/3 and 2, the variations between the two growth rates do not exceed 8% (see Table √ 1). Therefore, the instantaneous expression, ω = τ w0 k0 gives a good approximation of the growth rate of RTI, even for γ = 5/3. On the other hand, there is a significant uncertainty, up to a factor five, on the value of the amplification of the perturbation, since one takes the exponential variation. This simple analytical model can be used successfully to compute the early evolution of the Crab nebula (Ribeyre

Springer

et al., 2006). Starting with Jun’s (Jun, 1998) input data, the main properties of the Crab nebula are recovered (mass of the filament, size of PWN and time corresponding to the early fragmentation of the shell ∼400 years). In addition, it is found that the most probable mode, ldisup , leading to the disruption of the shell is ldisrup = 60. This result is in agreement with others studies (Bernstein and Book, 1978; Bucciantini et al., 2004).

Atzeni, S., Meyer-Ter-Vehn, J.: The Physics of inertial fusion. Oxford Unversity Press (2004) Bernstein, I.B., Book, D.L.: ApJ 225, 633 (1978) Blondin, J.M., Chevalier, R.A., Frierson, D.M.: ApJ 563, 806 (2001) Bouquet, S., Feix, M., Munier, A.: ApJ 393, 494 (1985) Bucciantini, N., Amato, E., Bandiera, R., Blondin, J.M., Del Zanna, L.: A&A 423, 253 (2004) Chevalier, R.A.: ApJ 619, 839 (2005) Davidson, K., Fesen, R.A.: ARA&A 23, 119 (1985) Hester, J.J., Stone, J.M., Scowen, P.A., Jun, B.-I., Gallacher III, J.S., Norman, M.L., Ballester, G.E., Burrows, C.J., Casertano, S., Clarke, J.T., Crisp, D., Griffiths, R.E., Hoessel, J.G., Holtzman, J.A., Krist, J., Mould, J.R., Sankrit, R., Stapelfeldt, K.R., Trauger, J.T., Watson, A., Westphal, J.A.: ApJ 456, 225 (1996) Jun, B.-I.: ApJ 499, 282 (1998) Kull, H.J.: Physics Reports 206, 5 (1991) Rayleigh, Lord: Proc. Lond. Math. Soc. 14, 170 (1883) Reynolds, S.P., Chevalier, R.A.: ApJ 278, 630 (1984) Ribeyre, X., Tikhonchuk, V.T., Bouquet, S.: Astrophys. Space Sci. 298, 75 (2005) Ribeyre, X., Hallo L., Tikhonchuk, V.T., Bouquet, S., Sanz, J.: Publication to be submitted to Astron. Astrophysics (2006) Sanz, J., Betti, R.: Phys. Plasmas 12, 042704 (2005) Trimble, V.: Astron. J. 73(7), 535 (1968)

Astrophys Space Sci (2007) 307:173–177 DOI 10.1007/s10509-006-9233-6

O R I G I NA L A RT I C L E

Phenomenological Theory of the Photoevaporation Front Instability D. D. Ryutov · J. O. Kane · A. Mizuta · M. W. Pound · B. A. Remington

Received: 11 April 2006 / Accepted: 11 August 2006 C Springer Science + Business Media B.V. 2006


Abstract The dynamics of photoevaporated molecular clouds is determined by the ablative pressure acting on the ionization front. An important step in the understanding of the ensuing motion is to develop the linear stability theory for an initially flat front. Despite the simplifications introduced by linearization, the problem remains quite complex and still draws a lot of attention. The complexity is related to the large number of effects that have to be included in the analysis: acceleration of the front, possible temporal variation of the intensity of the ionizing radiation, the tilt of the radiation flux with respect to the normal to the surface, and partial absorption of the incident radiation in the ablated material. In this paper, we describe a model where all these effects can be taken into account simultaneously, and a relatively simple and universal dispersion relation can be obtained. The proposed phenomenological model may prove to be a helpful tool in assessing the feasibility of the laboratory experiments directed towards scaled modeling of astrophysical phenomena. Keywords HII regions . Ablation front instability . Eagle nebula . Laboratory astrophysics PACS Numbers: 98.38.Dq, 98.38.Hv, 52.38.Mf, 5257.FG, 52.72.+v

D. D. Ryutov (
) · J. O. Kane · B. A. Remington Lawrence Livermore National Laboratory, Livermore, CA 94551 A. Mizuta Max-Planck-Institut f¨ur Astrophysik, Garching 85741, Germany M. W. Pound Astronomy Department, University of Maryland, College Park, MD 20742, USA

1 Introduction The shape of photoevaporated molecular clouds (e.g., Hester et al., 1996; Pound et al., 2003) is most probably caused by a variety of hydrodynamical processes occurring under the action of the ablation force. Some of the models relate the observed structures to the existence of large initial density perturbations (see, e.g., Bertoldi, 1989; Bertoldi and McKee, 1990; Williams et al., 2001). The others (see below) attribute the shape to the development of instabilities that grow from small perturbations at the ablation (photoevaporation) front. In the present paper, we consider several aspects of this second approach, concentrating on the linear stage of instability. We present a simple phenomenological model that allows one to describe, in a unified way, all the stabilizing and destabilizing factors which have been studied thus far in a piecemeal fashion. The linear analyses of the ablation front instability can be traced back to the papers by Spitzer (1954) and Frieman (1954) where the instability was identified as the RayleighTaylor (RT) instability of an accelerating interface. Kahn (1958) has argued that the partial absorption of the ionizing radiation in the ablated material should lead to a stabilization of the RT instability. Vandervoort (1962) developed a detailed theory of the ionization front instability, with radiation tilt included, at zero acceleration (i.e., this was an instability different from the RT instability). In the limit of negligible density in the ablated plasma, η ≡ ρa /ρ → 0 (where ρ a and ρ are the density in the ablation flow and in the molecular cloud, respectively), the instability is present only for nonzero tilt; it can be called the “tilted radiation” (TR) instability. Axford (1964) and Sysoev (1997) included the effect of absorption into the stability analysis of a non-accelerating front and radiation at normal incidence and have found a generally stabilizing effect. Williams (2002) included both the Springer

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radiation tilt and absorption (but no acceleration) and concluded that radiation tilt makes the system more unstable at all wavelengths. Ryutov et al. (2003) considered the TR instability in the presence of acceleration (but without absorption in the ablation flow). In numerical simulations by Kane et al. (2005) and Mizuta et al. (2005a, b), which contained both the linear and nonlinear stages of evolution, acceleration and absorption were present, but no radiation tilt. It was found that, in such a situation, the absorption has a strong stabilizing effect on the linear RT instability but non-linear perturbations would grow (Mizuta et al., 2005a). In the present paper, which is limited entirely to the linear theory, we include in the analysis all three factors: acceleration, radiation tilt, and absorption in the ablation flow. We discuss also the “impulsive acceleration” instability. By the latter we mean the situation where the intensity of the photoionizing radiation comes as a short pulse, with the time-scale shorter than the dynamical time of the system. This scenario is of some interest because the light curves of the young OB stars may indeed have a substantial spike early in time (Iben and Talbot, 1966; Cohen and Kuhi, 1979). We call our model “phenomenological” because we use a simplified description of several processes affecting the instability: the absorption of the incident radiation in the blow-off plasma is described just by a constant absorption coefficient κ; the ablation pressure is assumed to depend only on the intensity of radiation at the ablation front, and the specific shape of this dependence is not considered. Therefore, some parameters that enter the final dispersion relation have to be found either experimentally, or from more detailed analysis. This is an obvious drawback of the phenomenological approach. On the other hand, the benefit of this approach is related to the possibility of describing, in a unified fashion, a broad variety of factors affecting the instability both in astrophysics and in possible laboratory experiments (of the type described in Remington et al., 1993), and identify effects that have the strongest influence on the instability.

2 Basic assumptions We assume that the radiation comes from a direction that forms an angle θ with the normal to the unperturbed planar surface (Figure 1). The absorption coefficient along the ray is assumed to be a constant κ, so that the intensity along the ray varies according to the equation d I ∗ /ds = −κ I ∗ , where s is a coordinate along the ray. We use an asterisk to designate the energy flux at a plane normal to the direction of the rays. We denote this intensity at the unperturbed surface of the cloud as I0∗ . When radiation reaches the molecular cloud, the absorption is assumed to occur in a zero-thickness layer. In this last respect, our model is identical to that used in Vandervoort (1962) and Ryutov et al. (2003). Springer

Astrophys Space Sci (2007) 307:173–177

z y

g k x

h Fig. 1 The geometry of the problem

In astrophysics, where absorption in the ablated lowdensity material is caused by the presence of the neutral hydrogen produced by the recombination process, the absorption coefficient is proportional to the square of the density of this material. In the laboratory experiments of the type (Remington et al., 1993), the absorption can be caused by a variety of factors and, in particular, by the presence of the higher-Z admixtures. In our phenomenological model we have to adjust the coefficient κ so as to fit absorption properties of a particular system. Following the model used in Kahn (1958), we assume that the ablation pressure is some growing function of I, the energy flux through the surface of the cloud: pa = pa (I )

(1)

In the unperturbed state this energy flux is I0 = I0∗ cos θ. In what follows, we do not need the specific dependence of pa vs I, just that this is a smooth dependence with (I / pa )(d pa /d I ) ∼ 1. This allows us to cover in a unified fashion not only an instability of the photoevaporated clouds, but also the instability of ablation fronts in possible laboratory experiments with intense lasers, where details of formation of the ablation flow are different from astrophysical setting (e.g., Ryutov et al., 2003). In this brief communication we discuss only the simplest model of the cloud, within which the cloud material is considered an incompressible fluid. As was shown in Ryutov et al. (2003), the model of an incompressible fluid yields the results that are very close to a more sophisticated model treating the cloud as a compressible ideal gas. The ablation pressure accelerates the cloud in the negative direction of the axis z (Fig. 1). The absolute value g of the acceleration is equal to g=

pa , ρh

(2)

where h is the cloud thickness. The effective gravity force in the frame of the unperturbed ablation front is directed towards z>0.

Astrophys Space Sci (2007) 307:173–177

175

We assume that the density ρ a in the ablation flow is much smaller than ρ, and present results corresponding to the limit η ≡ ρa /ρ → 0. We work in the frame moving with the unperturbed ablation front. In this frame, the cloud material flows through the surface of the ablation front with velocity.  v ∼ η pa /ρa

(3)

As shown in Lindl (1995) and Takabe et al. (1999), the flow through the interface does not have a significant effect on the instability if the condition |k|v < Ŵ holds, with k and Ŵ being the wave number and the growth rate of unstable perturbations, respectively. For the typical RT growth rate Ŵ ∼ (kg)1/2 , this means that the flow through the interface is unimportant if kh < 1/η. In the limit η ≪ 1 this condition is not very restrictive and we neglect the flow through the interface in the rest of the paper.

The linearity constraint includes not only the smallness of the amplitude compared to 1/k, but also the constraint that no shadowing effects are present. The latter constraint reads as kξ ≪ cos θ and becomes dominant at grazing incidence (θ close to π /2). By considering the dynamics of perturbations inside the slab, one can relate the pressure perturbation at the inner (molecular cloud) side of the perturbed interface to the displacement of the interface. This can be done in a standard way (in particular, see the corresponding derivation in Ryutov et al., 2003). As the pressure perturbation at the inner side of the perturbed surface has to be equal to δpa , we obtain that (cf. Eq. (10) in Ryutov et al., 2003) ξ =

kδpa ρ   1 1 × − . (1 − e2kh )(ω2 − kg) (1 − e−2kh )(ω2 + kg)

(6)

3 Equations for perturbations Perturbation of the interface between the cloud and the ablation flow leads to the perturbation of the energy flux I through the perturbed surface. There are two sources for this perturbation. First, if the surface gets tilted with respect to its original orientation, the angle between the rays and the surface changes, thereby leading to a change of I. If the plane of incidence of the incoming radiation is the xz plane, as shown in Fig. 1, then the corresponding change of I is δ I = I0∗ sin θ∂ξ/∂ x, where ξ (x, y) is the displacement of the surface in the z direction. Second, if a certain element of the surface is displaced, the intensity changes because of the change of the absorption along the ray. This contribution is, obviously, δ I = I0∗ κξ , so that the total perturbation of intensity is δ I = I0∗ (sin θ∂ξ/∂ x + κξ ). This leads to the perturbation of the ablation pressure, 
∂ξ + κξ , δpa = C pa sin θ ∂x

(4)

where C>0 is a coefficient of order of unity: C = (I0∗ / pa )[∂ pa (I0 )/∂ I0 ]. At this point, it is convenient to perform a Fourier transform in the xy plane, and separate the spatial and temporal variables. In other words, the perturbation will have the following dependence on x, y, and t: exp(−iωt + ik x x + ik y y). An instability would correspond to Ŵ ≡ Imω > 0. We use also the notation α for the angle between the two-dimensional wave vector k and the axis x (Figure 1), so that k x = kcosα. For such perturbations, according to Eq. (4), δpa = C pa (ik cos α sin θ + κ)ξ.

(5)

Then, from Eqs. (5) and (6), one obtains the following dispersion relation, that contains the effects of radiation tilt, radiation absorption, and acceleration: ω4 − ω2 kghC (ikh sin θ cos α + κh)

coth kh − k 2 g 2 [1 − C(ikh sin θ cos α + κh)] = 0.

(7)

4 The analysis of the dispersion relation It is instructive to see what this dispersion relation predicts in the limiting cases that have been analyzed in the past. To consider a situation of a semi-infinite cloud with no acceleration (as it was done in Vandervoort (1962), Axford (1964), Sysoev (1997), and Williams (2002)), one has to replace g in Eq. (7) by its expression (2) and take the limit of large h. One then obtains ω2 − k( pa /ρ)C (ik sin θ cos α + κ) = 0.

(8)

In the limit of no absorption (κ = 0), we essentially recover the results by Vandervoort (for a low-density ablation flow, η → 0): no instability for radiation at normal incidence (θ = 0), and instability in the presence of radiation tilt, with the growth rate proportional to the wave number,  C pa sin θ| cos α| Im ω = ±k . (9) 2ρ If we include absorption, then, for normal incidence, one obtains non-damped oscillations, whereas in the presence of tilt, the instability is present at arbitrarily large absorption coefficient. The latter result corresponds to that obtained in Springer

Astrophys Space Sci (2007) 307:173–177

1.0

176

0.75

Im /(kg)1/2 0.5 0.75

h=2, kh=10

Re /2(g/h)1/

0.5

kh=1

kh=0.1

2

Im /(g/h)1/

0.25

0.25 0

0.25

2

0.5

0.75

1

0

/4

/2

3 /4

/2

h Fig. 2 The normalized growth rate at a zero tilt vs. the normalized absorption coefficient. At κh > 1 the linear RT instability ceases to exist

Fig. 3 The growth rate (solid line) and real frequency (dashed line) for κh = 2, C = 1, and kh = 1 vs. the tilt angle θ. Note the different normalization of the real and imaginary parts

the linear analysis by Williams (Williams, 2002). In the limit of large absorption, κ ≫ k, the growth rate is equal to:

For large-enough absorption coefficients such that Cκh > 1, the system becomes stable. This agrees with the results of Mizuta et al. (2005). Dependence of the growth rate on the absorption coefficient in the unstable domain (Cκh < 1) is illustrated in Fig. 2 (for C = 1). The real part of the frequency of the unstable modes is equal to zero, i.e., in this regard, they behave as standard RT perurbations. Finally, if we include all the ingredients, absorption, tilt, and gravity (i.e., return to the general equation (7)), we find an instability that exists at any absorption coefficient (for nonzero tilt). This is illustrated in Fig. 3, where the normalized growth rate is presented for the case of κh = 2, where the system would be stable at a normal incidence (θ = 0). Unlike the “standard” RT instability, perurbations here have a finite real frequency (i.e., a finite phase velocity along the surface)– a feature that can be exploited to experimentally identify this mode in possible laboratory experiment (Ryutov et al., 2003).

ary time of the typical O-type stars, the ones that produce the ionizing radiation. This circumstance points at a possibility that the stars are still in a transient stage of their formation, and their luminosity may have varied significantly during the past years. Such variations, including non-monotonous variations, with the luminosity passing through a maximum, is a common phenomenon in the evolution of very young stars (e.g., Iben and Talbot 1966; Cohen and Kuhi, 1979)). To get some insights into the possible implications of this effect, we consider the following simple model: that the ablative pressure “turns on” at t = 0, reaches the maximum and “turns off” at some t = t0 , which is much shorter that the growth time of perturbations. This model corresponds to the model of “impulsive acceleration,” which is sometimes used to imitate the Richtmyer-Meshkov instability. In order for our model of absorption to work, the time t0 should, on the other hand, exceed the transit time of the ablated gas over the distance of the order of 1/k. We will assume that this condition is satisfied, i.e., our results would not be applicable to very short bursts of radiation. For simplification, we consider only perturbations with kh > 2–3, so that we can neglect the feed-through to the back surface of the cloud and concentrate on what is going on at the front surface. For the time-dependent ablative pressure, one can no longer consider the exp(−iωt + ik x x + ik y y) dependence of perturbations on time. We have to seek perturbations of the form f (t)exp(ik x x + ik y y). Quite analogously to Eq. (8) but with the acceleration effects included, one then obtains for the function ξ (t):

5 Impulsive irradiation

∂ 2ξ pa (t) −1 =k (h − ikC sin θ cos α − κC)ξ. 2 ∂t ρ

Im ω = ±k sin θ cos α



C pa k . ρκ

(10)

Development of perturbations in the presence of acceleration and absorption, was studied numerically in Mizuta et al. (2005) for normal incidence. In this case, our Eq. (7) yields: ω4 − ω2 kκgh 2 C coth kh − k 2 g 2 (1 − Cκh) = 0.

(11)

It was noted by Pound (1998) that the dynamical time of evolution of the Eagle Nebula is much shorter than the evolutionSpringer

(12)

Assuming that the initial conditions are ξ (t = 0) = ξ0 , ξ˙ (t = 0) = 0, one readily finds that for a very short pulse

Astrophys Space Sci (2007) 307:173–177

ξ (t) = ξ0 kvt (1 − ikhC sin θ cos α − κhC) + ξ0 ,

177

(13)

t

where v = (1/ρh) 0 0 pa dt. With no tilt and no absorption, one finds a standard result for the impulsive acceleration. For normal incidence, large absorption (κhC > 1) causes the front inversion. If a substantial (θ ∼ 1) tilt is present, the second term in parentheses in Eq. (13) becomes dominant. It causes a 90◦ phase shift in the x direction. The impulsive acceleration just after the “lighting up” of the OB-type stars may be an additional mechanism for launching a subsequent evolution of molecular clouds. Acknowledgements Work performed under the auspices of the U.S. DoE by UC LLNL under contract No. W-7405-Eng-48; M.W. Pound is supported by NSF Grant No. AST-0228974.

References Axford, W.I.: ApJ 140, 112 (1964) Bertoldi, F.: ApJ 346, 735 (1989) Bertoldi, F., McKee, C.F.: ApJ 354, 529 (1990)

Cohen M., Kuhi L.V.: ApJ. Suppl. 41, 743 (1979) Frieman, E.: ApJ 120, 18 (1954) Hester, J.J., Scowen, P.A., Sankrit, R., et al.: Astron. J. 111, 2349 (1996) Iben, I., Jr., Talbot, R.J.: ApJ 144, 968 (1966) Kahn, F.D.: Rev. Mod. Phys. 30, 1058 (1958) Kane, J.O., Mizuta, A., Pound, M.W., et al.: Astrophys. Space Sci. 298, 261 (2005) Lindl, J.D.: Phys. Plasmas 2, 3933 (1995) Mizuta, A., Kane, J.O., Pound, M.W., et al.: ApJ 621, 803 (2005) Mizuta, A., Takabe, H., Kane, J.O., et al.: Astrophys. Space Sci. 298, 197 (2005) Pound, M.W., Reipurth, B., Bally, J.: Astron J. 125, 2108 (2003) Pound, M.W.: ApJ 493, L113 (1998) Remington, B.A., Weber, S.V., Haan, S.W., et al.: Phys. Fluids B5, 2589 (1993) Ryutov, D.D., Kane, J.O., Pound, M.W., Remington, B.A.: Plasma Phys. Contr. Fusion 45, 769 (2003) Spitzer, L.: ApJ 120, 1 (1954) Sysoev, N.E.: Astr. Lett. 23, 409 (1997) Takabe, H., Nagamoto, H., Sunahara, A., et al.: Plasma Phys. Contr. Fusion 41, A75 (1999) Vandervoort, P.O.: ApJ 135, 212 (1962) Williams, R.J.R:. MNRAS 331, 693 (2002) Williams, R.J.R., Ward-Thompson, D., Whitworth, A.P.: MNRAS 327, 788 (2001)

Springer

Astrophys Space Sci (2007) 307:179–182 DOI 10.1007/s10509-006-9257-y

O R I G I NA L A RT I C L E

Photoionized Flows from Magnetized Globules R. J. R. Williams

Received: 10 May 2006 / Accepted: 20 September 2006 C Springer Science + Business Media B.V. 2006


Abstract Low mass star formation may be triggered by the dynamical effects of radiation fields and winds from massive stars on nearby molecular material. The columns of neutral material observed at the edges of many HII regions may be the tracers of this process. Magnetic fields are dynamically important in the molecular clouds from which new stars form, but their effect on the development of molecular columns has not been studied in detail. In this paper, I present initial MHD simulations of this process. Keywords MHD . Shock waves . ISM: Clouds 1 Introduction In the region around a young massive star, intense ultraviolet fields ionize the molecular gas. Columns of neutral gas are frequently found at the boundaries of these regions, for example in the Eagle nebula. As well as the overall morphology of the columns, observations of optical and molecular line emission give information about flow velocities in the ionized and neutral gas, respectively (Pound, 1998). Hydrodynamic models for the development of these columns have been presented by Williams et al. (2001), while recent three-dimensional modelling has demonstrated the formation of columns in turbulent flows (Dale et al., 2005; Mellema et al., 2005). The radiation-driven collapse of the molecular material at the tips of these columns is a mechanism by which high-mass star formation may catalyse further low-mass star formation. The British Crown reserves the right to retain a non-exclusive, royality free licence in and to any copyright. c British Crown Copyright 2006/MOD.
R. J. R. Williams AWE plc, Aldermaston, RG7 4PR, UK

These models, however, neglect the influence of magnetic fields on the flow. Strong magnetic fields are a common feature of star forming regions (Crutcher, 1999): the magnetic energy density is typically ten times greater than the thermal energy of the gas, and comparable with the kinetic energy in unresolved velocity fields. Bertoldi (1989) discussed the effect of magnetic fields on the evaporation of dense globules with an approximate analytic approach. He describes the structures expected in the initial collapse, but notes that numerical calculations will be necessary to follow the evolution to late time. Ryutov et al. (2005) have discussed the influence of magnetic fields on the development of ionized columns. They suggest that magnetostatic turbulence may explain the support of the columns which appear to have internal pressures significantly less than those in the surrounding ionized nebula. Note, however, that molecular line widths suggest the gas is in fact moving with sufficient energy density to provide support (Williams et al., 2001), although the rapid decay of turbulence in MHD simulations (Mac Low, 1999) means it is unclear how these high velocities can be maintained. Ryutov et al. also argue that magnetic tension in the swept-back field may significantly suppress the development of columns, unless strong reconnection processes are present. While magnetic field support may suppress the initial formation of a cool thin shell between the ionization front and leading shock (Ryutov et al., 2005), one-dimensional analysis suggests that the subsequent emission of a slow-mode shock from the interface can allow the material to collapse to high density (Williams et al., 2000). While it has been argued that plasmas with high magnetic pressures are intrinsically unstable (Falle and Hartquist, 2002), it is not, however, clear that this organized loss of magnetic field support will be possible in multiple dimensions. In this paper, we model this photoionization process using a two-dimensional magnetohydro-dynamic code. As a Springer

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Fig. 1 Structure of flow at late time, with a weak initial magnetic field perpendicular to the impinging radiation. Grey scale is log density, and the contours are magnetic field lines. The location of the ionization front is at x ≃ 2, where the density changes rapidly

simple initial condition, we assume that a uniform density field contains one or more clumps of dense, cool gas. These clumps have an initial density of n H = 2 × 105 cm−3 , ten times higher than the surrounding diffuse material, and are initially in hydrostatic equilibrium. The pressure in the material corresponds to an initial temperature of 10 K in the clumps and 100 K in the gas which surrounds it. A simple γ = 5/3 equation of state is used, as the material is predominantly either ionized or cool enough to suppress rotational and vibrational modes. Once ionized, the material is assumed to relax to a temperature between 100 K and 104 K as a function of ionization fraction on a timescale given by the ionization rate. This artificial prescription avoids numerical problems which can result from the explicit treatment of thermal source terms in regions with unresolved heating fronts, as the emphasis here is on dynamical processes rather than detailed modelling. The numerical grid covers 2 pc × 4 pc, with a uniform numerical resolution of 800 × 1600 cells. All of the boundaries are set to allow free flow of material out of the grid, but no inflow. The use of these boundary conditions maintains the initial conditions in equilibrium, but allows material to expand freely from the ionized regions of the flow, to emulate the effect of large-scale divergent geometry characteristic of blister H II regions. The region as a whole is threaded by a uniform magnetic field. There is evidence for that the magnetic fields are ordered on large scales in molecular clouds (Ward-Thompson et al., 2000). An alternative would be to use the results of a turbulent flow simulation to initialize the flow. However, due to the uncertainty about how the observed flow speeds are maintained, the results here would remain dependent on uncertain assumptions. For the present work, a quiet start is appropriate. 2 Numerical methods We have modelled the photoevaporated flows from magnetized globules using the two-dimensional MHD code ATHENA (Gardiner and Stone, 2005). Springer

We assume that the flow is governed by the equations of MHD, with additional ionization balance terms. The coupling distances between ions and neutral species are far smaller than those which characterize the dynamics (Williams, 2006), which is generally taken to be a sufficient condition for the validity of these equations. Note, however, that Tytarenko et al. (2002) found that in these circumstances, accelerated flows are subject to strong instabilities which initially segregate the ionized and neutral components of the gas and eventually result in the generation of highly turbulent flows. The direct component of the ionizing continuum, of intensity 1012 cm−2 s−1 , is plane parallel and incident parallel to a co-ordinate axis. Diffuse radiation is treated using the case B assumption, as is usual. Transfer of the direct continuum, and its coupling to the ionization balance of the gas, is treated using an implicit scheme similar to that detailed by Williams (2002). In the present work, we have also used an improved model of the thermal balance. Instead of assuming the material heats (or cools) to an equilibrium temperature corresponding to its ionization fraction immediately, as assumed in Williams (2002), we limit the rate of heating to that provided by the absorption of ionizing radiation.

3 Simulation results 3.1 Weak field We first consider the flow from a single large clump where the initial magnetic pressure is ∼10× the thermal pressure, rather than ∼10× as observed. The clump is initially at (0.5, 0.5) pc and has radius 0.15 pc. In Fig. 1, we show the flow structure which develops. Compression and shear flow driven by the increased pressure in the photoevaporating gas soon increase the magnetic field strength in parts of the flow. The field in the ionized wind is dragged perpendicular to the surface of the column, with the magnetic field concentrated into a tulip-shaped region surrounding the barrel of the column. The flow from the tip

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Fig. 2 Initial field at 45◦ to the impinging radiation, and multiple initial clumps

Fig. 3 Structure of flow at late time, with a strong initial magnetic field parallel to the impinging radiation

is supersonic, while around the barrel, significant variations in flow speed occur along streamers following field lines from the surface. The velocity field within the column is structured, with several 2–4 km s−1 shocks. The main surface of the ionization front has a more irregular structure, with numerous small clumps pointing into the ionized region. Behind the clump, the magnetic field is swept back into a concentrated core, with its strength limited by reconnection (as inferred by Ryutov et al., 2005). No explicit resistivity is included in these simulations – instead, the reconnection is caused entirely by the effects of numerical resisitivity. However, the reconnection occurs in regions of the flow where we would expect that, for smaller but finite resistivity, even higher small-scale field line curvature will result in the largescale effects of reconnection being similar to those found here. The results are strikingly similar for other initial angles for the field. 3.2 Multiple clumps Figure 2 shows the structure of the flow found for a model in which the dense gas was initially in a regular array of small clumps with radius 0.02 pc with a centre–centre separation of 0.1 pc. The large-scale form of the flow is similar to that at

the head of one of the simulations seeded by a single dense clump, as might be expected. On smaller scales, the tension of the field swept back by the movement of the shocked diffuse field tends to aggregate the dense clumps. As the shock progresses, the initial compression by the shock tends to make the clumps longer in the direction of the radiation field, unlike the hydrodynamical case where the initial compression produces an oblate structure. While the velocity field in the neutral gas is highly variable, the influence of the magnetic field means that it is organized rather than turbulent. 3.3 Strong field With an initial field with magnetic pressure 10 times the gas pressure, comparable to the situation in observed molecular clouds, magnetic effects dominate the dynamics, and the increased pressure in the region heated by photoionization is far less significant for the flow. For a magnetic field parallel to the surface of the molecular cloud, the field pressure cushions the gas, and the dense clump can be accelerated with little disruption. The surface of the molecular cloud is only weakly perturbed by the presence of the condensation. More interesting is the case where the initial field is perpendicular to the ionization front, shown in Fig. 3. In this Springer

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case, the gas is constrained to move close to the initial field direction. The overall development of the flow is similar to that which would be expected for a set of one-dimensional simulations. A plug of low-density gas is trapped between the pressure of the ionized gas and the inertia of the dense core, with higher density than the surrounding material flowing from the unperturbed front. Looking in more detail at the simulations, however, deep fingers of ionized gas intrude into the neutral material. These propagate across the surface of the main ionization front away from the dense core, and consist of slow-mode shocks followed by oblique ionization fronts. The leading shock has an overall switch-on/switch-off internal structure. The second, slow-mode, sub-shock also has a corrugated surface, suggestive of the instability mechanism discussed by Stone and Edelman (1995). For an oblique initial magnetic field, the early-stage evolution is similar to that for parallel field. However, in this gas the trapped plug of neutral gas is offset, and does not entirely shield the dense clump from the radiation. When eventually the ionization front reaches the dense material, a weakly-magnetized transonic wind forms at the surface. 4 Conclusions Magnetic fields are seen to have a significant influence on the development of the columns around H II regions, for strengths comparable with those observed. With weak initial fields, shocks and shear flows amplify the field until it becomes dynamically important. For single clumps, the flows are remarkably similar, whatever the initial orientation of the magnetic field. Once an ablation flow has established, the magnetic field is oriented perpendicular to surface of the neutral gas and is concentrated into intermittent tubes. These tubes correlate with velocity and density modulations in the ionized wind from the clump. The wider surface of the cool material has a turbulent structure, with the magnetic field concentrated into sheets.

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For multiple clumps, the flows are more complex, with highly structured ablation flows. In the neutral region, the interaction between the leading shock and the globules again leads to the formation of a highly turbulent flow threaded by intense sheets of magnetic flux. For stronger initial fields, the pressure increase due to ionization becomes a less significant influence, but the development of the flows is more strongly influenced by the initial magnetic field. Ionization fronts and shocks are subject to strong instabilities, as might be expected for such low flows (Stone and Edelman, 1995; Falle and Hartquist, 2002). Important questions remain, such as the form of the flows in fully three-dimensional situations, and the processes which control reconnection in the multicomponent molecular material. References Bertoldi, P.: ApJ 346, 735 (1989) Crutcher, R.M.: ApJ 520, 706 (1999) Dale, J.E., Bonnell, LA., Clarke, C.J., Bate, M.R.: MNRAS 358, 291 (2005) Falle, S.A.E.G., Hartquist, T.W.: MNRAS 329, 195 (2002) Gardiner, T.A., Stone, J.M.: in: Magnetic Fields in the Universe. AIP Conf. Proc. 794, 475 (2005) Mac Low, M.-M.: ApJ 524, 169 (1999) Mellema, G., Arthur, S.J., Henney, W.J., Iliev, I.T., Shapiro, P.R.: Submitted to ApJ (astro-ph/0512554) Pound, M.W.: ApJ 493, L113 (1998) Ryutov, D., et al.: Astrophys. Space Sci. 298, 183 (2005) Stone, J.M., Edelman, M.: ApJ 454, 182 (1995) Tytarenko, P.V., Williams, R.J.R., Falle, S.A.E.G.: MNRAS 337, 117 (2002) Ward-Thompson, D., Kirk, J.M., Crutcher, R.M., Greaves, J.S., Holland, W.S., Andr´e, P.: ApJ 537, L135 (2000) Williams, R.J.R., Dyson, J.E., Hartquist, T.W.: MNRAS 314, 315 (2000) Williams, R.J.R., Ward-Thompson, D., Whitworth, A.P.: MNRAS 327, 788 (2001) Williams, R.J.R.: MNRAS 331, 693 (2002) Williams, R.J.R.: in: T.W. Hartquist, J.M. Pittard, and S.A.E.G. Falle (eds.), Diffuse Matter from Star Forming Regions to Active Galaxies. Springer, in press (2006)

Astrophys Space Sci (2007) 307:183–186 DOI 10.1007/s10509-006-9252-3

O R I G I NA L A RT I C L E

Nonlinear Dynamics of Ionization Fronts in HII Regions Akira Mizuta · Jave O. Kane · Marc W. Pound · Bruce A. Remington · Dmitri D. Ryutov · Hideaki Takabe

Received: 18 April 2006 / Accepted: 12 September 2006 C Springer Science + Business Media B.V. 2006


Abstract Hydrodynamic instability of an accelerating ionization front (IF) is investigated with 2D hydrodynamic simulations, including absorption of incident photoionizing photons, recombination in the HII region, and radiative molecular cooling. When the amplitude of the perturbation is large enough, nonlinear dynamics of the IF triggered by the separation of the IF from the cloud surface is observed. This causes the second harmonic of the imposed perturbation to appear on the cloud surfaces, whereas the perturbation in density of ablated gas in the HII region remains largely single mode. This mismatch of modes between the IF and the density perturbation in the HII region prevents the strong stabilization effect seen in the linear regime. Large growth of the perturbation caused by Rayleigh-Taylor-like instability is observed late in time. Keywords HII regions . ISM: molecules . ISM: kinematics and dynamics . Hydrodynamics . Instabilities . Methods: numerical . ISM individual object: M16

A. Mizuta (
) Max-Planck-Institute f¨ur Astrophysik, Karl-Schwarzschild-Str. 1, 85741 Garching, Germany e-mail: [email protected] J. O. Kane · B. A. Remington · D. D. Ryutov University of California, Lawrence Livermore National Laboratory, 7000 East Ave., Livermore, CA 94551, USA M. W. Pound Department of Astronomy, University of Maryland, College Park, MD 20742 USA H. Takabe Institute of Laser Engineering, Osaka University, 2-6 Yamada Oka, Suita, Osaka, 565-0871, Japan

1 Introduction Columns or pillars are common structures seen in HII regions. One of the most popular examples of this structure is the Eagle Nebula, which has three famous pillars beside some O stars (Hester et al., 1996). The boundary between the cloud and HII region is an ionization front (IF) where photoevaporation occurs, resulting in photoevaporated flow. The HII region consists of almost fully ionized hydrogen and is isothermal at T ∼ 104 K. The inside of the pillars consists of dense and cold molecular hydrogen. The hydrogen number density in the pillar in the Eagle Nebula is ∼104 –105 cm−3 and the temperature is about a few tens of Kelvins (Pound, 1998). Pound found the velocity gradient along pillar from the head to the bottom (see also Pound et al., 2006; Kane et al., 2006 in this volume). Dense clumps, some of which will become young stars, are observed in the pillar (McCaughrean and Andersen, 2002). Since this cloud is optically thick for incident photons, the thickness of the IF is very thin. Because of the similarity of this phenomena with laser ablation, scaled laboratory experiments using laser ablation are proposed to study IF dynamics (Kane et al., 2005). It is still not fully understood how the pillars form, though several hypotheses have been proposed. For example, some models are based on hydrodynamic instability of the IF. Spitzer (1954) proposed a model that the shape of pillars are at due to nonlinear phase of the Rayleigh-Taylor instability which occurs when a light fluid accelerates a denser fluid. Vandervoort (1962) theoretically found unstable modes at an IF without acceleration, but the important role recombination plays in the HII region was not included. Axford (1964) extended Vandervoort’s work, including recombination and found that recombination in the HII region works to stabilize the perturbation, as suggested by Kahn (1958). Sysoev (1997) did more complete analysis and found the growth Springer

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of long-wavelength instabilities for normally incident radiation. Williams (2002) confirmed this with 2D simulations. He also included effects of the radiation tilt in the analysis. The effect of tilted rays of the incident radiation to the IF with acceleration was theoretically studied by Ryutov et al. (2003, 2006). Recently (Mizuta et al., 2005) have numerically studied an accelerating IF and concluded that stabilization due to recombination works even with acceleration in the linear regime (i.e. small amplitude of the perturbations). This paper discusses cases with larger initial imposed perturbation.

2 Model We studied the dynamics of an accelerating IF by varying the initial amplitude of the imposed perturbations. The 2D hydrodynamic equations are solved using the code described in Mizuta et al. (2005). The energy equation includes sources of cooling and heating due to absorption of the incident photons, recombination in the HII region, and radiative molecular cooling. The transport equation for the incident photoionizing photons is also solved, considering the photon absorption by neutral hydrogen and recombination of ionized hydrogen. On the other hand, we do not consider far ultraviolet photons which will act to heat gas in the photo dissociation region behind the IF. Recombination to the ground state is ignored, assuming that the diffusive photon is locally absorbed (on the spot approximation). See Mizuta et al. (2005) for more details. A 0.46 pc × 3 pc computational box with 184 × 1200 uniform grid points is used. Periodic boundary conditions are employed at x = 0 and x = 0.46 pc. Outflow boundary condition is imposed at y = 0 and y = 3 pc. A quarter pc thickness finite cloud is located at a distance of 0.5 pc from the boundary at y = 3 pc, where the incident photon flux comes in. The hydrogen number density n(H) of the cloud and other regions are 105 and 10 cm−3 , respectively. After the compression through a shock, the density of the cloud is a few times 105 cm−3 which is comparable to current observed number density at the pillar in the Eagle Nebula. The region y < 2.5 pc is isothermal with T = 40 K and the region y > 2.25 pc is pressure matched. A constant incident photon number flux of 5 × 1011 cm−2 s−1 is taken to be parallel to the y axis. When a neutral hydrogen atom absorbs an incident photon and becomes ionized, an internal energy increment of 1.73 × 10−12 erg is locally deposited into the gas. An isothermal state whose temperature is about 104 Kelvin is achieved in the HII region as a result of the balance between this heating by photon absorption and cooling by the recombination of the ionized hydrogen. The molecular cloud is very cold (∼40 Kelvin) due to the strong radiative molecular cooling, even if shocks cross in the cloud. Springer

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A sinusoidal surface perturbation is initially imposed. Here we have studied four different initial amplitude perturbations: 3.8 × 10−3 pc (case S038), 9.0 × 10−3 pc (case S090), 1.4 × 10−2 pc (case S140), and 1.9 × 10−2 pc (case S190), respectively. Figure 1(a) shows the number density contour of the initial condition, where the initial amplitude is 9.0 × 10−3 pc. 3 Results and discussions Figure 2 shows the amplitude of the perturbation as a function of time, where the amplitude is defined as half the peak-tovalley amplitude of the contour corresponding to an ionization fraction of f = 0.5. Since the IF sometimes separates from the cloud surface, this amplitude shown in Fig. 2 does not always correspond to the amplitude of the perturbation at the cloud surface. Case S038 shows a small amplitude which oscillates with time, and does not grow, as shown in Mizuta et al. (2005). When the incident photon flux comes in, the flow of the photoionized plasma begins from cloud surface. Since this flow is normal to the cloud surface, the density of the photoevaporated flow around the bubble region (surface concavities at x = 0, 0.46 pc) becomes higher than around spike region (x = 0.23 pc), due to focusing effect. Higher number density results in stronger absorption of the incident photon flux in the HII region. As a result, the ablation pressure around the bubble region becomes lower than that around the spike region. The local difference of the ablation pressure works to strongly stabilize the perturbation in linear regime. The amplitude of the other cases, however, increase, with time, in striking contrast to the strong stabilization observed in the linear regime. The reason is due to the ‘separation of the IF’ from the cloud surface. Such cases were theoretically studied by Newman and Axford (1968) and Beltrametti et al. (1982) in other contexts, assuming spherical symmetry. The strong absorption of incident photon around the bubble occurs, reducing the ablation pressure there, also in the larger initial amplitude case. The surface perturbation inverts phase completely during the shock propagates in the dense cloud (Fig. 1(b) and (c)). The number density around the bubble region becomes higher as the IF becomes strongly concave. The increasing density around the bubble in the HII region causes absorption of all incident photons before the cloud surface as shown in Fig. 1(d), since the number of incident photons is finite. This causes the separation of the IF from the cloud surface, which has not been considered before. When the separation of the IF from the cloud surface occurs, a tiny and warm HI region (neutral hydrogen) appears between the cloud surface and the IF, since the gas was once ionized and recombined to neutral hydrogen. The cloud surface locally does not feel any ablation pressure and expands in the y direction, when the separation of the IF appears.

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Fig. 1 Number density contours (color) and incident photon number flux contours (solid white) at intervals of 1 × 1011 cm−2 s−1 , starting from the IF where ionization fraction (f) goes to zero. The times shown correspond to (a) 0 kyr, (b) 30 kyr, (c) 60 kyr, (d) 90 kyr, (e) 120 kyr, (f) 200 kyr, (g) 340 kyr and (h) 480 kyr, respectively. As an example, the IF is indicated with an arrow in (f), a case where there is clear separation with the cloud surface (ablation front). The separation of the IF can be seen in (d), resulting the appearance of the tiny and warm HI region (neutral hydrogen atom). Figures are taken from Mizuta et al. (2006) and reproduced by permission of the AAS

S190 S140 S090 S038

0.1 Amplitude (pc)

Fig. 2 Time evolution of the “amplitude” of the perturbation for cases S038, S090, S140, and S190. Figure is taken from Mizuta et al. (2006) and reproduced by permission of the AAS

0.01

0.001 0

100

200

300

400

500

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After a few tens of kyr, the IF corresponds to the cloud surface again. But the perturbation on the cloud surface is not longer single mode; a second harmonic of the imposed perturbation has appeared (Fig. 1(e)). To the contrary the density perturbations in the HII regions above the cloud surface are still single mode (Fig. 1(e)). This mismatch between the perturbation modes on the cloud surface and those in the density of the ablated plasma mean that the stabilization observed in the linear regime will not occur. (Recall, this stabilization requires that these two perturbations be “mode locked” 180 degrees out of phase.) As a result, a Rayleigh-Taylor-like instability evolves and large growth of the second harmonic of the imposed perturbation appears (Fig. 1(f)–(h)). There still remains another possibility for the growth of the perturbations. That is thin-shell instability for the ionization shock front which caused by the unbalanced forces between the ram pressure to the shock front and thermal pressure by the HII region (Garcia-Segura and Franco, 1996). We can measure the velocity gradient along the y axis as Pound observed in the pillars in the Eagle Nebula. We find the velocity gradient to be about 12 km s−1 pc−1 in the central column at t = 480 kyr. This is good agreement with observed one (an average magnitude of 8.3 km s−1 pc−1 ).1 Although we fixed the wavelength of the initially imposed perturbations in this paper, the essentially nonlinear dynamics is observed from the beginning of the simulation in cases which show the evolution of the columns. The case of multi mode perturbation should be studied in the near future. The dynamics described in this paper is good target to model the scaled laboratory experiments, since theoretical analysis is difficult to do because of the nonlinear dynamics. We will demonstrate some numerical simulations to seek the possible laboratory experiments to study the instability presented in this paper.

4 Conclusion We present a new type of the instability for the accelerating IF triggered by the separation of the IF from the cloud surface. When the initial amplitude of the perturbation is small enough, the strong stabilization is observed. When

1

This value does not include the effect of the inclination angle.

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the surface becomes concave, the number density increases. Strong absorption through this region locally reduces the ablation pressure as theoretically discussed by Axford (1964) for the non-accelerating IF. To the contrary, when the initial amplitude of the perturbation exceeds a critical value, roughly when the ratio of the initial amplitude to wavelength is greater than 0.02, the nonlinear dynamics is triggered by the separation of the IF. The separation of the IF causes the appearance of the second harmonic of the imposed perturbation. The perturbation in density in the HII region, however, remains largely single mode. This mismatched of the modes between the perturbations on the cloud surface and in the density of the ablated flow prevents the stabilization effect seen in the linear regime. A kind of Rayleigh-Taylor instability takes over. The large growth of the second harmonic of the imposed perturbation is observed in the later phase. Acknowledgments Work performed under the auspices of the U.S. Department of Energy by the Lawrence Livermore National Laboratory under Contract No. W-7405-ENG-48 and with support from NASA Grant NRA 00-01-ATP-059 and from National Science Foundation under Grant No. AST-0228974. MWP supported by NSF Grant No. AST0228974.

References Axford, W.I.: ApJ 140, 112 (1964) Beltrametti, M., Tenorio-Tagle, G., Yorke, H.W.: A&A 112, 1 (1982) Garcia-Segura, G., Franco, J.: ApJ 469, 171 (1996) Hester, J.J., et al.: AJ 111, 2349 (1996) Kane, J.O., et al.: Ap&SS 298, 261 (2005) Kane, J.O., et al.: Ap&SS, this issue (2006) Kahn, F.D.: Rev. Mod. Phys. 30, 1058 (1958) McCaughrean, M.J., Andersen, M.: A&A 389, 513 (2002) Mizuta, A., et al.: ApJ 621, 803 (2005) Mizuta, A., et al.: ApJ 647, 1151 (2006) Newman, R.C., Axford, W.I.: ApJ 151, 1145 (1968) Pound, M.W.: ApJ 493, L113 (1998) Pound, M.W., et al.: Ap&SS, this volume (2006) Sysoev, N.E.: Astronomy Letters 23, 409 (1997) Ryutov, D.D., et al.: Plasma Physics and Controlled Fusion 45, 769 (2003) Ryutov, D.D., et al.: Ap&SS, this volume (2006) Spitzer, L.J.: ApJ 120, 1 (1954) Vandervoort, P.O.: ApJ 135, 212 (1962) Williams, R.J.R.: MNRAS 331, 693 (2002)

Astrophys Space Sci (2007) 307:187–190 DOI 10.1007/s10509-006-9214-9

ORIGINAL ARTICLE

Pillars of Heaven Marc W. Pound · Jave O. Kane · Dmitri D. Ryutov · Bruce A. Remington · Akira Mizuta

Received: 14 April 2006 / Accepted: 7 July 2006 C Springer Science + Business Media B.V. 2006


Abstract Sometimes the most beautiful things are the hardest to understand. Pillars like those of the Eagle Nebula form at the boundary between some of the hottest (10000 K) and coldest (10 K) gas in the Galaxy. Many physical processes come into play in the birth and growth of such gaseous pillars: hydrodynamic instability, photoionization, ablation, recombination, molecular heating and cooling, and probably magnetic fields. High-quality astronomical observations, quantitative numerical simulations, and scaled laser experiments provide a powerful combination for understanding their formation and evolution. We put our most recent hydrodynamic model to the test, by creating simulated observations from it and comparing them directly to the actual radioastronomical observations. Successfully reproducing major characteristics of the observations in this manner is an important step in designing appropriate laser experiments. Keywords Eagle Nebula . Radio astronomy . Hydrodynamic models . Aperture synthesis

1. Introduction The pillars of the Eagle nebula are the most spectacular example of a phenomenon that is commonly seen wherever molecM.W. Pound (
) Astronomy Department, University of Maryland, College Park, MD 20742 J.O. Kane . B.A. Remington . D.D. Ryutov Lawrence Livermore National Laboratory, Livermore, CA 94551 A. Mizuta Max-Planck-Institut fur Astrophysik, Garching, 85741, Germany

ular clouds are situated near O stars. Proposed formation mechanisms for such pillars generally fall into two broad categories: (i) instabilities at the boundary between the cloud and the ionized region which grow with time (e.g. Spitzer, 1954; Frieman, 1954; Williams et al., 2001; Mizuta et al., 2005a,b) and (ii) pre-existing density enhancements (i.e., clumps) which locally retard the ionization front creating “cometary globules” (Reipurth, 1983; Bertoldi and McKee, 1990). We have developed a comprehensive, 2-D hydrodynamic, cometary globule model of pillar formation (Mizuta et al., 2005; Kane et al., in this volume) that includes energy deposition and release due to the absorption of UV radiation, recombination of hydrogen, radiative molecular cooling, magnetostatic pressure (Ryutov et al., 2002), and geometry/initial conditions based on Eagle observations. Pillar formation by both ionization-front instability (Mizuta et al., 2005b and dense cores (Kane et al.) have been examined; both methods can grow a pillar a few tenths of a parsec long in a few hundred thousand years. This timescale is comparable to the dynamic time measured for the Eagle Pillars (Pound, 1998). The CO(J = 1−0) observations, taken with the BerkeleyIllinois-Maryland interferometer are those of Pound (1998), with the addition of more recent higher spatial resolution data (see Figure 1b of Pound et al., 2005). To facilitate comparison between model and observations, we create “synthetic observations” from the model by filtering it through the known telescope response function and processing the resultant data using identical methods as for the observations to produce maps. The details of this technique are described in Pound et al. (2005). In that paper, we compared synthetic integrated intensity maps of the Mizuta et al. (2005) instability- model pillars to the observed maps. Here, we take the Kane et al. cometary-model pillar (Figure 1) and perform a similar analysis but with a significant improvement: the new model allows us to create full synthetic data cubes (position, Springer

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Fig. 1 Phases of pillar growth from Kane et al. model; (left) initial condition of a dense core embedded in a molecular cloud, (middle) after 125,000 years, (right) after 250,000 years with total length of about 2 pc.

Astrophys Space Sci (2007) 307:187–190

The model at 250,000 years is taken as the basis for creating synthetic observations

Fig. 2 (left) The synthetic integrated intensity map derived from processing the model. (right) The actual integrated intensity map from Pound (1998). Maps are to same physical scale and use identical intensity range. Lines indicate locations of position-velocity cuts shown in Figures 3 and 4

position, velocity). The addition of velocity information provides a powerful test of the model, as it allows detailed comparison of the model and observed gas dynamics.

2. Results Figure 2 shows the synthetic and actual integrated intensity maps. Integrated intensity is a measure of the amount of gas along the line of sight. Brighter intensity means higher molecular hydrogen column density and, since the pillars are assumed to be roughly cylindrically symmetric, also indicates higher particle density. The size and shape of the model pillar reasonably match that of Eagle Pillar II (the middle of the 3 Eagle pillars). Furthermore, the final average particle density of the model volume density is close the that inferred from the observations However, there does not appear to be enough material in the model “tail”; it is underdense compared with Pillar II. Figure 3 shows the position-velocity (p-v) diagrams along the length of the synthetic pillar and Eagle Pillar II. Both show a velocity gradient from “head” to “tail”, with the observed gradient being slightly larger. However, since any measured velocity gradient (synthetic or observed) is a function of inclination angle, decreasing the inclination angle of the model

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pillar can make the gradients match. The extent of the emission on the velocity axis is a measure of the internal velocity dispersion of the gas. One can see that the observed velocity dispersion is about 2.5 times greater than the synthetic one. This should not be surprising since no turbulent support was put into the model; this is an area where the model could be improved. Figure 4 shows the p-v diagrams across the head of the synthetic and observed pillars. There are two features of the synthetic p-v diagram (upper panel) worth noting. The first is overall shape: rounded contours on the top and flattened contours on the bottom. This is what would be expected of “inside-out” velocity shear—that is, material interior to the pillar is flowing more slowly than its surface (see inset). The second feature is the two symmetrically-placed bright spots indicative of limb-brightening. Both these features make sense in light of the dynamics of the model: the massive core in the head resists motion and the lighter material gets pushed around it, sweeping back a cometary shape with a dense outer shell. Material directly behind the head is less affected and flows more slowly. The observed p-v diagram (lower panel of Figure 4) shows neither of these features. There are two bright spots, but they are not symmetric and are likely two individual cores. There is no evidence of inside-out velocity shear.

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Fig. 3 (top) The synthetic position-velocity diagram along the length of the model pillar. The “head” of the pillar is on the left, the “tail” on the right. (bottom) Actual length-wise position-velocity diagram of Eagle Pillar II

Fig. 4 (top) The synthetic position-velocity diagram across the head of the model pillar. Double-peak is indicative of limb-brightening and overall shape (curved outer contours on top, flat contours on bottom) indicates velocity shear. Inset at top right shows velocity flow pattern which would produce such features in a p-v diagram (bottom) Actual cross-wise position-velocity diagram of Eagle Pillar II

3. Conclusions The cometary model can produce a large pillar with roughly the correct size and shape within the measured dynamical timescale. The model starts with an isolated 30 M⊙ core in a lower density envelope. This produces a pillar head with properties like the Eagle’s but without enough material in the tail. However, the structure of the Eagle pillars is clearly not as simple as a single core, and that suggests the next step to enhance the simulation. Models with multiple cores of different size and mass should increase both the amount of tail material as well as the internal velocity dispersion.

Acknowledgements Work performed under the auspices of the U.S. DoE by UC LLNL No W-7405-Eng-48. MWP supported by NSF Grant No. AST-0228974.

References Bertoldi, F., McKee, C.F.: ApJ 354, 529 (1990) Frieman, E.A.: ApJ 120, 18 (1954) Mizuta, A., Kane, J.O., Pound, M.W., Remington, B.A., Ryutov, D.D., Takabe, H.: ApJ 621, 803 (2005) Mizuta, A., Takabe, H., Kane, J.O., Pound, M.W., Remington, B.A. Ryutov, D.D.: Astrophysics & Space Science 298, 197 (2005)

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190 Pound, M.W.: ApJ 493, L113–L116 (1998) Pound, M.W., Reipurth, B., Bally, J.: AJ 125, 2108 (2003) Pound, M.W., Kane, J.O., Remington, B.A., Ryutov, D.D., Mizuta, A., Takabe, H.: Astrophysics & Space Science 298, 177 (2005) Reipurth, B.: A&A 117, 183 (1983)

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Astrophys Space Sci (2007) 307:187–190 Ryutov, D., Kane, J., Mizuta, A., Pound, M., Remington, B.: APS Meeting Abstracts, 1004P (2002) Spitzer, L.: ApJ 120, 1 (1954) Williams, R.J.R., Ward-Thompson, D., Whitworth, A.P.: MNRAS 327, 788 (2001)

Astrophys Space Sci (2007) 307:191–195 DOI 10.1007/s10509-006-9182-0

ORIGINAL ARTICLE

The Evolution of Channel Flows in MHD Turbulence Driven by Magnetorotational Instability T. Sano

Received: 14 April 2006 / Accepted: 22 May 2006 C Springer Science + Business Media B.V. 2006


Abstract MHD turbulence driven by magnetorotational instability (MRI) in accretion disks is investigated using the local shearing box calculations. The growth of many shortwavelength MRI modes, which are called “channel flows”, is found in the spatial distribution of the current density. These small channel flows can be regarded as a unit structure of MRI driven turbulence. Nonlinear evolution of the channel flow affects the saturation amplitude and time variability of the Maxwell stress. Exponential growth of a channel mode is stopped by the Kelvin-Helmholtz type instability which triggers the subsequent magnetic reconnection. The characteristics of the magnetic reconnection are consistent with the Sweet-Parker model. These studies of the nonlinear evolution of the channel flow are required to understand the saturation mechanism of the MRI. Keywords Accretion disks . MHD . Turbulence 1 Introduction Magnetorotational instability (MRI) is the most promising source of angular momentum transport in accretion disks (e.g., Balbus & Hawley, 1998). Local and global simulations of magnetized accretion disks have revealed that the Maxwell stress in MHD turbulence driven by the MRI can transport angular momentum significantly (e.g., Hawley et al., 1995; Hawley, 2000). However the nonlinear saturation mechanism of the MRI have not been understood yet, so that what determines the saturation amplitude of the stress, or the size of the α parameter of Shakura & Sunyaev (1973), is still unclear. T. Sano (
) Institute of Laser Engineering, Osaka University, Suita, Osaka 565-0871, Japan e-mail: [email protected]

It is important to investigate throughly the nature of MHD turbulence in accretion disks to understand the saturation processes of the MRI. In this paper, we focus on the nonlinear evolution of channel flow in MRI driven turbulence. The channel flow is an unstable mode of axisymmetric MRI whose wavevector is parallel to the rotation axis. The channel mode is important because it is the equal fastest-growing of all linear MRI modes, even including those with non-zero radial and azimuthal wavenumbers. The eigenfunctions of this mode satisfy not only the linearized MHD equations but also the nonlinear equations in the incompressible limit (Goodman & Xu, 1994). MRI driven turbulence can be treated as incompressible because the turbulent velocity is about the Alfv´en speed and much smaller than the sound speed. Thus the amplitude of channel flow can grow exponentially even in the nonlinear regime. The nonlinear growth of channel modes is the most efficient mechanism of field amplification in the disks. In two-dimensional axisymmetric simulations under the ideal MHD approximation, a two-channel flow appears at the nonlinear stage and continues to grow without saturation (Hawley & Balbus, 1992). When the vertical wavelength of a channel flow is fitted to the height of computational domain or the disk thickness, it is called a two-channel flow. The vertical length scale of MRI modes increases as the magnetic field is amplified, and finally becomes comparable to the box size. A two-channel flow consists of two streams moving radially inward with a sub-Keplerian rotational velocity and radially outward with a super-Keplerian rotational velocity. As the amplitude of two-channel flow increases, strong vertical shear is developed which can generate and amplify the oppositely directed horizontal fields. Thus the growth of the two-channel mode is associated with both the development of the vertical shear in the horizontal flow and the formation of a pair of current sheets. Springer

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Fig. 1 Spatial distributions of (a) the current density |J| and (b) the azimuthal component of the current Jy in MRI driven turbulence drawn on a radial-vertical slice. Snapshot data are obtained from a local shearing box simulation including the ohmic dissipation. The initial field geometry is a zero net flux Bz and the plasma beta is β0 = 104 . The magnetic diffusivity is assumed to be uniform and the initial Lundquist number

is SMRI = 20. Model parameters are identical to those of model S52r in Sano et al. (2004) except for the grid resolution (128 × 512 × 128). The data are taken at t/trot = 100. The gray colors are linearly scaled (a) from 0 (black) to 814 J0 (white) and (b) from −250J0 (black) to 250J0 (white) where J0 ≡ cB0 /4π L z . The growth of many channel modes with short wavelengths can be recognized in this figure

If the ohmic dissipation is taken into account, magnetic reconnection could break up the structure of channel flow. On the other hand, the strong shear brings the Kelvin-Helmholtz type instability (Goodman & Xu, 1994) which could also interrupt the growth of a channel mode. The saturated turbulent state is achieved by a balance between the growth of the MRI (channel modes) and the field dissipation through magnetic reconnection. Then the breakup of the channel flow must be relevant to the saturation mechanism. In the following sections, we demonstrate the importance of the channel flow in MRI driven turbulence and examine the nonlinear evolution of a channel mode. The local shearing box calculations including the ohmic dissipation are most appropriate for this purpose. Low Lundquist number simulations are inevitable to resolve the dissipation scale. Such simulations would be quite useful to compare the properties of turbulence with various theories of incompressible MHD turbulence (e.g., Schekochihin et al., 2004) and applicable directly to the dynamics of protoplanetary disks (Stone et al., 2000; Sano et al., 2000; Inutsuka & Sano, 2005).

exponential decay of the magnetic field by reconnection are the origin of spike-shaped variations in the magnetic energy. The joule heating contributes almost all the increase of the thermal energy. Thus the nonlinear evolution of the MRI is characterized by the two-channel flow for this case. On the other hand, if the net magnetic flux of the shearing box is zero, the two-channel flow never appears in the nonlinear regime. Instead, many small-scale structures can be seen in the spatial distribution of the magnetic field. We find that these small fluctuations have the similar field geometries to the channel modes. Figure 1shows snapshots of the current densities |J| and Jy during the turbulent phase. Model parameters are identical to those of model S52r in Sano et al. (2004) except that the grid resolution is 128 × 512 × 128 instead of 32 × 128 × 32. The initial Lundquist number for the MRI is SMRI ≡ v 2A /η = 20, where v A = Bz /(4πρ)1/2 is the Alfv´en speed, η is the magnetic diffusivity, and  is the angular velocity. Numerical resolution of this case is sufficient to capture most of the heating by the ohmic dissipation term. Note that the number SMRI is called the magnetic Reynolds number in our previous papers (Sano & Miyama, 1999; Sano et al., 2004). We would emphasize that this number SMRI is specified for the MRI, because the most unstable wavelength of the MRI v A /  is adopted as a typical length scale. In fact, both the linear and nonlinear evolution of the MRI can be well-characterized by this number SMRI . As seen from Figure 1, many pairs of current sheets fill the domain. The current directions of each pair are opposite. These structures are stretched in the azimuthal direction as a result of the Keplerian shear motions. At each pair of

2 Presence of Channel Flows When accretion disks are penetrated by a uniform vertical field Bz , a two-channel flow appears quasi-periodically during the turbulent phase in three-dimensional simulations (Sano & Inutsuka, 2001). The channel flow is related to large time variability of the magnetic energy and the Maxwell stress. Exponential growth of the two-channel mode and Springer

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small fluctuations impede the growth of a large-scale channel flow and reduce the efficiency of field amplification. Here we define the typical thickness of current sheets as λ J = 2π/k J where k 2J = (∇ × B)2 / B 2 . The thickness of the current sheets is about λ J /L z ≈ 9.3 × 10−2 for this model, but we find the grid size dependence of λ J at least within the limit of our calculations. The current sheets are thinner in the higher resolution cases, so that the timescale of magnetic reconnection can be shorter. This could also cause the smaller saturation amplitude of the field. Further results on the resolution dependence will be reported elsewhere. Fig. 2 Correlation between the radial and azimuthal components of the magnetic field. We use snapshot data taken at 100 orbits for the same model as in Figure 1. The solid line denotes B y /Bx = 2.7, which is the typical ratio of their rms values in MRI driven turbulence

current sheets, both the radial and azimuthal components of the field are sinusoidal with respect to z but their signs are opposite. These properties are similar to the channel mode. The correlation between Bx and B y in the saturated turbulence is shown by Figure 2.Anti-correlation of the horizontal fields is one of the most important features of MRI driven turbulence, because this ensures outward transport of angular momentum. The ratio of the horizontal components is about

B y2 1/2 /

Bx2 1/2 ≈ 2.7, which is interestingly independent of any initial conditions (Sano et al., 2004). The kurtosis of the magnetic field is B 4 / B 2 2 ≈ 4.0, which denotes the intermittency of the field. The origin of these small structures may be the growth of the channel modes, although the directions of the current sheets are not always horizontal owing to the turbulent motions. Therefore each small channel flow can be regarded as a characteristic unit of the structure in MRI driven turbulence for the zero net flux cases. Both the saturation level and time variability of the Maxwell stress for zero net flux cases are much smaller than those in the uniform vertical field runs (Sano et al., 2004). The difference can be explained by the characteristics of the channel mode. When the shearing box is dominated by a two-channel flow, the magnetic field is amplified effectively and a spike-shaped excursion is expected in the time evolution. However if there are many channel modes with small amplitude in the box and each mode develops independently, the time variability in the volume average of the stress must be small. Note that the saturation level of the Maxwell stress is slightly lower as the grid resolution is higher for the zero net flux cases. The time- and volume-averaged Maxwell stress in the model shown by Figure 1 is

−Bx B y /4π /

P ≈ 3.2 × 10−4 which is about half of that for the lower resolution run in Sano et al. (2004). The channel mode can be an exact solution of the nonlinear MHD equations only when the amplitudes of all the other modes are negligible. Thus,

3 Nonlinear Evolution of a Channel Mode Next we perform two-dimensional simulations of the nonlinear evolution of a single channel mode. Consider a small box in the disk threaded by a uniform vertical field B0 . We ignore the vertical gravity so that the initial density ρ = ρ0 and pressure P = P0 are spatially uniform. The box height is set to be the most unstable wavelength of the MRI, L z = 2π v A0 / , where v A0 = B0 /(4πρ0 )1/2 . The initial pressure and field strength are assumed to be P0 = 5 × 10−3 and v A0 = 10−3 . We use ρ0 = 1 and  = 10−3 as the normalizations, and thus the initial plasma beta is β0 = 104 and the box height corresponds to L z ≈ 0.63H where H is the scale height of the disk. The ohmic dissipation terms are included in the energy and induction equations. The magnetic diffusivity is assumed to be uniform η = 10−3 which means the initial Lundquist number for the MRI SMRI ≡ v 2A0 /η = 1. We use a rectangular box with a wide width L x = 2L z and the grid resolution of 256 × 128. As for the initial perturbations, small random fluctuations are added to the velocity with the maximum amplitude |δv|max = v A0 . 3.1 Parasitic Instability At the beginning, the most unstable mode evolves dominantly and then the wavelength of the channel flow is equal to the box height, λch = L z . The evolution of this mode is represented by a Fourier mode amplitude of the radial velocity Vx (k) whose radial wavenumber is zero, k˜x ≡ k x L x /2π = 0, and vertical wavenumber is unity, k˜z ≡ k z L z /2π = 1. This is shown by the solid curve in Figure 3as a function of time normalized by the rotation time trot = 2π/. The growth rate obtained from our simulation is ωMRI ≈ 0.20 and this is exactly the same as the prediction by the linear analysis for the case of SMRI = 1 (Sano & Miyama, 1999). However, the exponential growth is stopped and turned to decrease when the amplitude of the channel flow exceeds the strength of the vertical field Vx /v A0 > 1. Goodman & Xu ∼ (1994) have shown that the channel mode is unstable for the Kelvin-Helmholtz type instability (parasitic instability). The Springer

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Fig. 3 Time evolution of Fourier mode amplitudes of the velocities normalized by the initial Alfv´en speed. The growth of a two-channel flow is represented by a mode amplitude of the radial velocity with the radial wavenumber k˜x ≡ k x L x /2π = 0 and the vertical wavenumber k˜z ≡ k z L z /2π = 1 (solid curve). A mode amplitude of the vertical velocity with k˜x = 1 and k˜z = 0 is shown by the dashed curve, which indicates the growth of the parasitic instability. The growth rates of the MRI and the parasitic instability expected from the linear analysis (ωMRI /  = 0.2 and ωPI /  = 0.6) are also plotted by thin lines in this figure

maximum growth rate is given by ωPI ∼ 0.2b where b ≡ Bh /B0 is the amplitude of the channel flow and Bh = (Bx2 + B y2 )1/2 is the horizontal component of the field. Thus the growth rate of the parasitic instability increases exponentially with time. The most unstable growth is expected when the vertical wavenumber is zero and the radial wavelength is about twice the length of the channel flow. Note that the unstable growth of the parasitic instability requires the longer wavelength of disturbances than λch . The evolution of the parasitic mode is given by a Fourier mode amplitude of the vertical velocity Vz (k) with k˜x = 1 and k˜z = 0. When the mode amplitude b becomes unity, the growth rate of the parasitic mode is of the order of the angular velocity  and then catches up with the channel mode. The amplitude of the channel mode can be estimated approximately by using a mode amplitude of the radial velocity, b ∼ Vx (k˜x = 0, k˜z = 1)/v A0 . The amplitude is b ≈ 3.0 at the peak (t ≈ 6.4trot ), and thus the maximum growth rate of the parasitic instability is ωPI ∼ 0.6. The growth rate obtained numerically around the peak is consistent with this theoretical prediction (see Fig. 3). The magnetic energy begins to decrease just after the fast growth of the parasitic instability, so that this instability breaks up the channel flow and triggers magnetic reconnection. 3.2 Sweet-Parker Reconnection After the breakup of the channel flow, magnetic reconnection takes place and the magnetic energy decreases exponentially. The gas pressure increases rapidly by the joule heating. Figure 4shows the time history of the volumeaveraged magnetic energy. This simple calculation of a chanSpringer

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Fig. 4 Time history of the volume-averaged magnetic energy in a simple simulation of a channel mode. The nonlinear evolution of the channel flow reproduces a spike-shaped time variation. The dashed line denotes the decay rate of the Sweet-Parker reconnection model; B 2 ∝ exp(−ωSP t) = exp(−2πt/tSP )

nel mode can reproduce a spike-shaped variation successfully. We find that the decay timescale and the reconnection rate are consistent with the Sweet-Parker reconnection model (Sweet, 1958; Parker, 1957), although the reconnection in our calculation is unsteady. The decay timescale of the magnetic energy obtained from this simulation is about τdec ≈ 3.9trot , while the model prediction is τSP = (τ A τη )1/2 ≈ 4.1trot . The Alfv´en timescale is τ A = L/v A ≈ 0.67trot and the diffusion timescale τη = L 2 /η ≈ 25trot , where we assume the typical length scale as L ∼ 2λch and the Alfv´en speed is v A ∼ bv A0 . The reconnection rate is given by Mi = vi /v A where vi is the inflow velocity to the diffusion region. We can extract the inflow velocity from the simulation data by means of a mode amplitude of the vertical velocity with k˜x = k˜z = 1. The inflow velocity obtained numerically is vi /v A0 ≈ 0.43 at the peak of the channel mode. Then the reconnection rate becomes Mi ≈ 0.15 which is consistent with the Sweet-Parker model Mi = (η/Lv A )1/2 ∼ 0.16. When we use a square box L x = L z , the magnetic dissipation proceeds gradually without topological change of the magnetic field. For this case, the parasitic instability cannot grow because the radial width is smaller than the critical wavelength. This suggests the importance of the parasitic mode as a triggering mechanism of magnetic reconnection.

4 Discussion The Maxwell stress in MRI driven turbulence is proportional to the magnetic pressure (e.g., Hawley et al., 1995; Sano et al., 2004). However the relation between the gas and magnetic pressure in the saturated turbulence has still many uncertainties. The local shearing box calculation would be a useful tool to investigate MHD turbulence in accretion disks. In this paper, we have shown that the evolution of the channel

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flow determines the characteristics of MRI driven turbulence. The channel flow like structures can be seen also in the global disk simulations (Machida & Matsumoto, 2003). The nonlinear saturation level of the MRI has dependences on some physical quantities, such as the gas pressure and initial field strength (Sano et al., 2004). Using our simple simulations of a channel flow, we find that the reconnection rate is unaffected by the gas pressure. We change the initial gas pressure over 3 orders of magnitude fixing the other parameters, but the reconnection rate Mi is always around 0.1. We also perform the same calculations of a channel flow in three-dimension. The reconnection rate and timescale are almost the same as those in two-dimensional simulations. The growth rate of the parasitic instability is comparable to the MRI when the amplitude of the channel mode b is about 5. The ratio of the horizontal field to the vertical one in MRI driven turbulence is about

Bh2 1/2 /

Bz2 1/2 ≈ 5.2 (Sano et al., 2004). This is consistent with a picture that the parasitic instability constrains the field amplification in MRI driven turbulence. For the zero net flux cases, the typical wavelength of channel modes is small because of the weak vertical field. The saturated stress is much smaller than that in the uniform Bz runs. Therefore, the net flux of the vertical field may be essential for the efficient transport of angular momentum in accretion disks.

195 Acknowledgements Numerical computations were carried out on VPP5000 at the National Astronomical Observatory of Japan and on SX-6 and SX-8/6A at the Institute of Laser Engineering, Osaka University. This work was also supported by the Grant-in-Aid (16740111, 17039005) from the Ministry of Education, Culture, Sports, Science, and Technology of Japan.

References Balbus, S.A., Hawley, J.F.: Rev. Mod. Phys. 70, 1 (1998) Goodman, J., Xu, G.: ApJ 432, 213 (1994) Hawley, J.F.: ApJ 528, 462 (2000) Hawley, J.F., Balbus, S.A.: ApJ 400, 595 (1992) Hawley, J.F., Gammie, C.F., Balbus, S.A.: ApJ 440, 742 (1995) Inutsuka, S., Sano, T.: ApJ 628, L155 (2005) Machida, M., Matsumoto, R.: ApJ 585, 429 (2003) Parker, E.N.: J. Geophys. Res. 62, 509 (1957) Sano, T., Inutsuka, S.: ApJ 561, L179 (2001) Sano, T., Inutsuka, S., Turner, N.J., Stone, J.M.: ApJ 605, 321 (2004) Sano, T., Miyama, S.M.: ApJ 515, 776 (1999) Sano, T., Miyama, S.M., Umebayashi, T., Nakano, T.: ApJ 543, 486 (2000) Schekochihin, A.A., Cowley, S.C., Taylor, S.F., Maron, J.L., McWilliams, J.C.: ApJ 612, 276 (2004) Shakura, N.I., Sunyaev, R.A.: A&A 24, 337 (1973) Stone, J.M., Gammie, C.F., Balbus, S.A., Hawley, J.F.: in Protostars & Planets IV, ed. V. Mannings, A.P. Boss, S.S. Russell (Tuscon: Univ. of Arizona Press), 589 (2000) Sweet, P.A.: in IAU Symp. 6: Electromagnetic Phenomena in Cosmical Physics, ed. B. Lehnert (Cambridge Univ. Press), 123 (1958)

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Astrophys Space Sci (2007) 307:197–202 DOI 10.1007/s10509-006-9225-6

ORIGINAL ARTICLE

Laboratory Exploration of Solar Energetic Phenomena David Alexander

Received: 14 April 2006 / Accepted: 25 July 2006 C Springer Science + Business Media B.V. 2006


Abstract The solar atmosphere displays a wide variety of dynamic phenomena driven by the interaction of magnetic fields and plasma. In particular, plasma jets in the solar chromosphere and corona, coronal heating, solar flares and coronal mass ejections all point to the presence of magnetic phenomena such as reconnection, flux cancellation, the formation of magnetic islands, and plasmoids. While we can observe the signatures and gross features of such phenomena we cannot probe the essential physics driving them, given the spatial resolution of current instrumentation. Flexible and well-controlled laboratory experiments, scaled to solar parameters, open unique opportunities to reproduce the relevant unsteady phenomena under various simulated solar conditions. The ability to carefully control these parameters in the laboratory allows one to diagnose the dynamical processes which occur and to apply the knowledge gained to the understanding of similar processes on the Sun, in addition directing future solar observations and models. This talk introduces the solar phenomena and reviews the contributions made by laboratory experimentation. Keywords Laboratory astrophysics . Solar physics . Dynamic phenomena . Magnetic reconnection . Plasma jets

mosphere and provide a unique perspective into the physics governing astrophysical processes. Observations of the Sun from ground- and space-based observatories provide a wealth of data, often continuous, with high resolution in space, time and energy (or wavelength or frequency). Yet, despite its proximity and the array of instrumentation available it is still 150 million km away and is a ‘laboratory’ where the experiments are uncontrolled and where the physics occurs on spatial scales far smaller than we can resolve. To fully understand the observed solar phenomena and the physics driving them, carefully designed terrestrial laboratory experiments, scalable to solar conditions, are necessary. In this paper, I introduce some of the solar phenomena of particular interest and illustrate some of the relevant laboratory investigations currently underway. By the very nature of the subject, this cannot be considered a comprehensive survey but it should give a flavor of some of the issues involved. Some key energetic solar phenomena are discussed in Section 2 although we consider, in a little more depth, the specific phenomena of plasma jets on the Sun in Section 3. Scaling issues, associated with the connection between the laboratory experiments and the Sun, are discussed in Section 4, and we conclude in Section 5.

2 Solar phenomena 1 Introduction The Sun is often quoted as being a “Laboratory for Astrophysics” and in many ways it lives up to this sobriquet. A wide range of physical phenomena involving the interaction of magnetic fields and plasma occur throughout the solar atD. Alexander Department of Physics and Astronomy, Rice University, 6100 Main St, Houston, TX 77005

The Sun exhibits a wide array of energetic phenomena, often resulting from the interaction between plasma and magnetic field. The very generation of the Sun’s magnetic field is under intense study with observational and theoretical advances in heliosiesmology and solar dynamo physics being one of most exciting areas of solar physics research over the last decade. Solar transient phenomena and their role in driving space weather is another area of intense interest involving a synergistic approach between observation, theory, and modeling. Springer

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˚ channel of the Fig. 1 (left) Post-flare loop arcade seen in the 195 A TRACE telescope for an X281 class flare (soft X-ray intensity of 2.8 mW/m2 ) which occurred at the west limb of the Sun on 2003 Nov 4. The plasma in the magnetic loops shown has a temperature of order 1.5 MK, having cooled from 20–40 MK. The image size represents 175,000 km × 233,000 km at the Sun. (right) Associated CME

seen in Thomson scattered white light by the SOHO/LASCO telescope. The white circle in the center of the LASCO image marks the location of the Sun, the grey disk is the occulter, and the scale of the image is ∼30 solar radii or 21 million km. The CME is clearly evident, erupting from the south-west limb of the Sun (bottom right on the figure)

In this paper, we focus on the transient phenomena and leave the interested reader to explore the physics of helioseismology and solar dynamo physics from better and more comprehensive treatments (see Christensen-Dalsgaard, 2004 for an excellent review).

fast CME, and a proton storm at 1 AU. The coronal EUV im˚ channel of age on the left of the figure was taken in the 195 A the Transition Region and Coronal Explorer (TRACE), while the CME white light image was taken by the Large Angle and Spectrometric Coronagraph (LASCO) on the Solar and Heliospheric Observatory mission (SOHO). The initiation of these events is an ongoing problem in the solar physics community with distinct efforts focusing on flares and CMEs separately, although the commonality of some of the physics is understood. Several models have been put forward to explain how the solar corona builds up and then releases magnetic energy in the form or large-scale eruptions, localized heating, and the ejection of mass and magnetic flux. Loss-of equilibrium (Lin and Forbes, 2000), magnetic breakout (Antiochos, DeVore, and Klimchuk, 1999) and tether-cutting (Moore et al., 2001) models have been the most prevalent in recent years, although see also Chen and Krall (2003). In many of these cases, magnetic reconnection is critical to the initiation or evolution of the energy release. The breakout model requires reconnection in the overlying arcade field to occur in order for the energized sheared field to erupt (‘break-out’), whereas the loss-of-equilibrium model requires fast reconnection in a current sheet formed below a rising magnetic fluxrope to enable the fluxrope to erupt as a CME: for a review of all of these models see Lin, Soon, and Baliunas (2003). While there is no direct evidence that magnetic reconnection occurs in the solar atmosphere (gyro-radii are on the cm scale while observations are on the hundreds of km scale), many of the expected signatures of the reconnection process (counter-flowing jets, particle acceleration, topological changes in the magnetic field, etc) have been observed. A better understanding of the consequences of magnetic reconnection under conditions relevant (or scaleable)

2.1 Solar flares and coronal mass ejections Some of the most energetic phenomena in the Universe occur on the Sun. A solar flare can generate power as high as 1022 W from as little as 1014 kg of mass (10−16 solar masses), or approximately 108 W/kg. Similar numbers for blazars, with an energy output from 109 solar masses and 1042 W, and gamma-ray bursts, ∼1 solar mass and 1045 W, are 103 W/kg and 1015 W/kg, respectively. A gamma-ray burst is clearly more powerful than a solar flare but it is interesting to note that while gamma-ray bursts accelerate particles to the order of 1 MeV, solar flares can generate ions and electrons with energies in excess of 100 MeV. The largest solar flares tend to be associated with a quite distinct but no less impressive phenomenon known as a Coronal Mass Ejection (CME). CMEs are large expulsions of mass and magnetic field from the Sun. Their velocities can exceed 2000 km/s, some 3–5 times faster than the ambient solar wind. The CME drives a shock in interplanetary space which subsequently accelerates ions to 10–100 MeV/nucleon. Figure 1 shows an event from 2003 Nov 4 (one of the famous Hallowe’en storms: Gopalswamy et al., 2005) which included a very large flare (GOES class X28), a 1

The X28 designation was based on saturated X-ray detectors. it is now thought this flare, the largest on record, may have been as high as an X45 (4.5m W/m2 ), see Brodrick et al., (2005).

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to the solar corona would significantly enhance our understanding of CME initiation. The largest flares/CMEs are often associated with coronal destabilization triggered by the eruption of a solar filament.

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whether the eruption is triggered by a global destabilization of the corona or by a local change in magnetic field via the emergence of new flux or the cancellation of old flux. Models exist which explore all of these issues. For a comprehensive review see Lin et al. (2003).

2.2 Solar filaments and prominences 2.3 Magnetic fluxtubes on the Sun Solar filaments (or prominences) are structures in which cool, dense chromospheric plasma is embedded in the hotter solar corona (see Gilbert et al., 2001). The typical length scale of a solar prominence is 104 –105 km and the magnetic geometry is thought to contain regions of upward concavity in the local magnetic field, either helical structures such as fluxropes (e.g. Rust and Kumar, 1994) or in a series of dips on individual fieldlines (e.g. Kuperus and Raadu, 1974) where the substantial mass of the filaments plasma can collect (although see Karpen et al., 2001). Despite the presence of large amounts of dense plasma, the dynamics and evolution of the filament are governed by the dominant magnetic forces: the plasma beta in filaments is on the order 10−3 to 10−1 . Eruptions of these filaments are commonly associated with energy release in the corona resulting in a range of phenomena including solar flares and CMEs. Determining how these eruptions are driven is a primary goal in the physics of CME initiation. Factors which influence the eruption process include; the role of magnetic reconnection (where it occurs, whether it is a driver or a consequence of the eruption, how fast it proceeds), magnetic topology (whether the filament is a helical fluxrope prior to eruption or develops into a fluxrope as a consequence of the eruption), whether an MHD instability occurs (Fig. 2 shows an example where the kink instability is thought to be responsible for the observed kinking of the filament structure),

A fundamental component of magnetic fields in astrophysics and in the laboratory is the magnetic fluxtube. In the Sun, fluxtubes are generated in the solar interior by solar dynamo processes occurring at the base of the convective zone, the strong magnetic fluxtubes then buoyantly rise through the convective zone to emerge through the solar surface to generate the coronal loops which characterize the key building blocks of the solar corona. It is the coronal magnetic structures which, when stressed, generate the array of energetic transient phenomena discussed above. Understanding the physical processes which govern the creation, transport, and evolution of magnetic fluxtubes throughout the various regimes of the solar atmosphere and interior is necessary to understand solar variability, both on the long- and the shortterm. As an example of theoretical progress in this area, Linton et al. (2001) have recently performed a series of numerical simulations focused on understanding the interaction of fluxtubes. They find a wide range of behavior from the merging of fluxtubes, fluxtubes ricocheting off each other, reconnection and slingshot-type dynamics, and even the tunneling of one fluxtube through another. The various behaviors result from the degree of twist in the interacting fluxtubes, their angle of approach, and the value for the Lundquist number adopted.

Fig. 2 A dynamic solar filament occurring on 2002 May ˚ channel of 27 seen in the 195 A TRACE. The maximum height attained by the filament (see last frame) is ∼80,000 km. The contours show locations of hard X-ray emission associated with the filament activation (see Alexander et al., 2006 for details)

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Laboratory experimentation is the only means by which we can study fluxtube behavior in detail. However, to understand the full range of consequences of fluxtube interaction in regimes pertinent to astrophysical systems closer collaboration between the laboratory astrophysics, solar theoretical, and solar observational communities is essential.

3 Laboratory simulation of solar plasma jets There are many examples of laboratory simulations of solar phenomena. I would like to call out for particular attention, given the focus of solar transients in this paper, the work of the Princeton MRX team (Yamada, 2004; Ji et al., 1998) on driven reconnection, magnetic helicity, wave generation and ion heating, all of which are important in the solar atmosphere, and that of Bellan and Hansen (1998), Hansen and Bellan (2001), and Hansen, Tripathi, and Bellan (2004) on simulating solar prominences and their interaction in the laboratory. Much of the astrophysical focus of the HEDLA 2006 conference centered on jet phenomena and I would like to bring a solar perspective to those discussions by focusing on jet phenomena in the Sun and what we might learn from laboratory simulations. Figure 3 shows an example of a solar plasma jet studied by Alexander and Fletcher (1999). Such jets are commonly observed and take a number of forms depending on the nature of the magnetic field interactions driving them. The basic characteristics of these jets include: • plasma velocities of order, 150–200 km/s Fig. 3 Plasma jet in the solar ˚ corona. Left panel: The 171 A image from the TRACE telescope at 02:55:20 UT on 1998 Aug 19 with the jet clearly visible extending out of an ‘anemone’ like kernel. Right panel: A difference image from 7 minutes later (03:02:20–03:01:20 UT) showing that the jet bifurcated with evidence of twisting motions from the alternating black and white striations: white/black implies positive/negative change in emission (from Alexander and Fletcher, 1999)

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• the presence of adjacent hot (T∼100 eV) and cool (T∼1 eV) jets • evidence for twisting and bifurcation of the jet with time The observed jets are mostly associated with the emergence or cancellation of magnetic flux with modeling efforts suggesting that they result from the coalescence of magnetic islands and require enhanced resistivity to generate the necessary velocities and energy release rates (Shibata, Yokoyama, and Shimojo, 1996; Karpen et al., 1995). Multi-thermal solar plasma jets have also been detected out to 3 solar radii from the Sun (Fig. 4) and traveling as fast as 500 km/s (Ko et al., 2005). Some preliminary work has been performed at the Lawrence Livermore National Laboratories using a Spheromak-like Compact Torus (SCT) formed by the Compact Torus Injection Experiment (CTIX; Hwang et al., 2000). The internal field of the SCT is of order 1 kG with a temperature of 50–75 eV and a plasma beta around 10−3 . The idea here is that the interaction of two SCTs focused through an appropriately shaped exit aperture results in the creation of oblique shocks which interact to accelerate the plasma (Ryutova and Tarbell, 2000) and thereby mimic chromospheric and transition region shocks observed at the Sun (Tarbell et al., 2000). 4 Scaling issues One of the major issues in the applicability of laboratory experiments to astrophysical phenomena is the markedly different regimes in which the various physical interactions occur.

Astrophys Space Sci (2007) 307:197–202 Table 1 Physical parameters in laboratory and solar regimes (SCT parameters from Howard, 2006)

201

Parameter

Solar photosphere

Solar corona

CTIX/SCT

Plasma density (cm−3 ) Plasma temperature (eV) Magnetic field strength (G) Characteristic length (cm) Alfven timescale (s) Resistive timescale (s) Lundquist number Spitzer resistivity (s) Plasma frequency (GHz) Debye length (cm)

1017 0.5 500 4 × 106 14.3 8.5 × 105 5.9 × 104 5.3 × 10−13 2.8 × 103 1.5 × 10−4

109 100 10 109 12 2 × 1012 1.7 × 1011 4.8 × 10−16 0.284 23.5

1015 10 1000 20 5.8 × 10−7 1.54 × 10−3 2.6 × 103 4.2 × 10−15 2.8 × 102 4 × 10−4

Fig. 4 Solar plasma jet occurring on 1999 Aug 26 found to extend out to 3 solar radii (from Ko et al., 2005). The arc starting in the top left corner of each frame is the solar limb. The scale of each frame is 175,000 km × 124,000 km

Typical scales in astrophysics can be 10–20 orders of magnitude greater than achievable in the laboratory. Consequently, the issue of the scalability of laboratory phenomena is crucial to their application to astrophysics (Ryutov and Remington, 2002). Table 1 details the various parameters in the solar photosphere and solar corona (two very different regimes in the Sun separated by only 2000 km). For comparison the equivalent parameters for the CTIX SCT are also listed. The key difference is in the relative strengths of the plasma and the magnetic field and the consequences this has for dynamical scale-heights and transient phenomena. Clearly, the physical scales are widely disparate, particularly in density, characteristic length scales, and the relevant

timescales. However, more important is how the various dimensionless parameters of the system compare (see Ryutov, this volume). We see that for the case of the SCT the Lundquist number is within an order of magnitude of that in the solar photosphere while several orders of magnitude smaller than in the corona.

5 Conclusions We can learn a lot about the physics behind energetic solar phenomena from well-tailored laboratory experiments. However, the inclusion of magnetic field is crucial. The range of dynamic phenomena occurring across a wide array of Springer

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physical regimes on the Sun can act as a “laboratory for laboratory astrophysics”, especially for providing guidance on the role of magnetic fields. Consequently, the comparison of laboratory experiments and simulations with detailed solar observations may provide insight into the physical processes at work in astrophysical plasmas. Scaling is still a major issue but the range of solar phenomena and wealth of data increase the chances that scalability can achieved. References Alexander, D., Fletcher, L.: Sol. Phys. 190, 167 (1999) Alexander, D., Liu, R., Gilbert, H. R.: ApJ 653, 720 (2006) Antiochos, S.K., DeVore, C.R., Klimchuk, J.A.: ApJ 510, 485 (1999) Bellan, P.M., Hansen, J.F.: Phys. of Plasmas, 5, 1991 (1998) Brodrick, D., Tingay, S., Wieringa, M.: JGR 110, A09S36 (2005) Chen, J., Krall, J.: J. Geophys. Res. 108(A11), 1410 (2003) Christensen-Dalsgaard, J.: Equation-of-State and Phase-Transition in Models of Ordinary Astrophysical Matter, Celebonovic, V., Gough, D., D¨appen W. (eds). New York, 18 (2004) Gilbert, H.R., Holzer, T.E., Low, B.C., Burkepile, J.T.: ApJ 549, 1221 (2001)

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Astrophys Space Sci (2007) 307:197–202 Gopalswamy, N., et al.: J. Geophys. Res. 110, A09S00 (2005) Hansen J.F., Bellan, P.M.: ApJ 563, 183 (2001) Hansen, J.F., Tripathi, S.K.P., Bellan, P.M.: Phys. Plasmas 11, 3177 (2004) Howard, S.J.: PhD Thesis, UC Davis (2006) Hwang, D.Q. et al.: Nuc. Fusion 40, 897 (2000) Ji, H., Yamada, M., Hsu, S., Kulsrud, R.: Phys. Rev. Lett. 80, 3256 (1998) Lin, J., Forbes, T.H.: J. Geophys. Res. 105, 2375 (2000) Lin, J., Soon, W., Baliunas, S. L.: New Astron. Rev. 47, 53 (2003) Linton, M.G., Dahlburg, R.B., Antiochos, S.K.: ApJ 553, 905 (2001) Karpen, J.T., Antiochos, S.K., Devore, C.R.: ApJ 450, 422 (1995) Karpen, J.T., Antiochos, S.K., Hohensee, M., Klimchuk, J.A., MacNeice, P.J.: ApJ 553, 85 (2001) Ko, Y.-K. et al.: ApJ 623, 519 (2005) Kuperus, M., Raadu, M.A.: A&A 31, 189 (1974) Moore, R.L., Sterling, A.C., Hudson, H. S., Lemen, J. R.: ApJ 552, 833 (2001) Rust, D.M., Kumar, A.: Solar Phys. 155, 69 (1994) Shibata, K., Yokoyama, T., Shimojo, M.: Adv. Sp. Res. 17, 197 (1996) Yamada, M.: Proc. 35th COSPAR Scientific Assembly, p. 4411 ( 2004) Tarbell, T.D., Ryutova, M., Shine, R.A.: Sol. Phys. 193, 195 (2000) Ryutov, D.D., Remington, B.A.: Plasma Phys. Control. Fusion 44, 407 (2002) Ryutova, M., Tarbell, T.D.: ApJ 541, 29 (2000)

Astrophys Space Sci (2007) 307:203–206 DOI 10.1007/s10509-006-9230-9

O R I G I NA L A RT I C L E

Explosion Mechanism of Core-Collapse Supernovae and Collapsars S. Nagataki

Received: 12 April 2006 / Accepted: 2 August 2006 C Springer Science + Business Media B.V. 2006


Abstract We have performed 2-dimensional MHD simulations of collapsars with magnetic fields and neutrino cooling/heating processes. It is found that explosion energy of a hypernova is not obtained from the neutrino heating process. However, strong jet is found when magnetic fields are included, and total energy of the jet component can be of the order of 1052 erg, which is comparable to the one of a hypernova. Keywords Supernova . Collapsar . GRB

1. Introduction There has been growing evidence linking long gamma-ray bursts (GRBs; in this study, we consider only long GRBs, so we call long GRBs as GRBs hereafter for simplicity) to the death of massive stars. The host galaxies of GRBs are star-forming galaxies and the positions of GRBs appear to trace the blue light of young stars. Also, ‘bumps’ observed in some afterglows can be naturally explained as contribution of bright supernovae. Moreover, direct evidences of some GRBs accompanied by supernovae have been reported such as the association of GRB 980425 with SN 1998bw and that of GRB 030329 with SN 2003dh. It should be noted that these supernovae are categorized as a new type of supernovae with large kinetic energy (∼1052 ergs), nickel mass (∼0.5 M⊙ ), and luminosity, so these supernovae are sometimes called as hypernovae. Also, since GRBs are considered to be jet-like phenomena, S. Nagataki Yukawa Institute for Theoretical Physics, Kyoto University, Oiwake-cho Kitashirakawa Sakyo-ku, Kyoto 606-8502, Japan e-mail: [email protected]

it is natural to consider the accompanying supernovae to be jet-induced explosions. The central engine of GRBs accompanied by hypernovae is not known well. But it is generally considered that normal core-collapse supernovae can not cause an energetic explosion of the order of 1052 erg. So another scenario has to be considered to explain the system of GRBs associated with hypernovae. One of the most promising scenario is the collapsar scenario (MacFadyen and Woosley, 1999). In the collapsar scenario, a black hole is formed as a result of gravitational collapse. Also, rotation of the progenitor plays an essential role. Due to the rotation, an accretion disk is formed around the equatorial plane. On the other hand, the matter around the rotation axis falls into the black hole. It was pointed out that the jet-induced explosion along to the rotation axis occurs due to the heating through neutrino anti-neutrino pair annihilation that are emitted from the accretion disk. MacFadyen and Woosley (1999) demonstrated the numerical simulations of the collapsar, showing that the jet is launched ∼7 s after the gravitational collapse and the duration of the jet is about 10 s, which is comparable to the typical observed duration of GRBs. However, detailed neutrino heating process is not included in MacFadyen and Woosley (1999). Also, it is pointed out that effects of magnetic fields may be so important (Proga et al., 2003; Fujimoto et al., 2005). So in this study, we solved the dynamics of collapsars with neutrino cooling/heating processes and magnetic fields.

2. Models and numerical methods Our models and numerical methods of simulations in this study are shown in this section. First we present equations of ideal MHD, then initial and boundary conditions are explained. Micro physics included in this study, equation of Springer

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state (EOS), nuclear reactions, and neutrino processes are also explained. 2.1. Magnetohydrodynamics We have done two-dimensional MHD simulations taking account of self-gravity and gravitational potential of the central point mass. The calculated region corresponds to a quarter of the meridian plane under the assumption of axisymmetry and equatorial symmetry. The spherical mesh with 150(r ) × 30(θ) grid points is used for all the computations. The radial grid is nonuniform, extending from 5.0 × 106 cm to 1.0 × 1010 cm with finer grids near the center, while the polar grid is uniform. The location of the inner most radius is same with MacFadyen and Woosley (1999). The basic equations in the following form are finite differenced on the spherical coordinates: Dρ = −ρ∇ · v Dt 1 Dv = −∇ p − ρ∇ + (∇ × B) × B ρ Dt 4π
 D e + ρ = − p∇ · v − L− ν + Lν + Lnucl Dt ρ ∂B = ∇ × (v × B), ∂t

(1) (2) (3) (4)

where ρ, v, P, , e, L ± ν , L nucl , and B are density, velocity, pressure, gravitational potential, internal energy density, heating/cooling rates due to neutrino processes, energy gain (loss) rate due to nuclear reaction, and magnetic field. The Lagrangian derivative is denoted as D/Dt. The gravitational potential of the central point mass is modified to account for some of the effects of general relativity, φ = −G M/(r − rs ) where rs = 2G M/c2 is the Schwartzshild radius, G is the Gravity constant, and M is the mass of the black hole. The initial mass of the black hole is set to be 1.69 M⊙ and M becomes larger along with time since mass accretes from the inner boundary. The ZEUS-2D code developed by Stone and Norman (1992) has been used to solve the MHD equations with second order accurate interpolation in space. Heating/cooling rates due to neutrino processes and energy gain (loss) rate due to nuclear reaction are described in Subsections 2.3.2 and 2.3.3.

be suitable to make a baryon poor fireball. The mass of iron core is 1.69 M⊙ that is covered with Si layer whose mass is 0.55 M⊙ . So we assume that the iron core has collapsed and formed a black hole at the center. Angular momentum was distributed so as to provide a constant ratio of 0.04 of centrifugal force to the component of gravitational force perpendicular to the rotation axis at all angles and radii, except where that prescription resulted in j16 greater than a prescribed maximum value, 10. This treatment is exactly same with MacFadyen and Woosley (1999). Total initial rotation energy is 5.7 × 1048 erg that corresponds to initial ratio of the rotation energy to the gravitational energy, T /W = 8.3 × 10−3 . Configuration and amplitude of the magnetic fields in a progenitor prior to collapse are still uncertain. So in this study we choose a simple form of the initial configuration and the amplitude is changed parametrically. Initial configuration of the magnetic fields is chosen as follows:
3 1 r0  B( r ) = B0 (2 cos θ er + sin θ eθ ) for r ≥ r0 3 r

(5)

2 B0 (cos θ er − sin θ eθ ) for r < r0 . 3

(6)

=

This configuration represents that the magnetic fields are uniform in a sphere (r < r0 ), while dipole at outside of the sphere. We set r0 to be the boundary between CO core/Si layer. B0 corresponds to the strength of the magnetic field in the sphere. We have chosen B0 to be 0, 108 G, 109 G, and 1010 G. Initial ratios of the magnetic energy relative to the gravitational energy are 0, 1.1 × 10−8 , 1.1 × 10−6 , 1.1 × 10−4 , respectively. The initial lowest plasma beta, which are realized at the outer boundary, are ∞, 5.8 × 105 , 5.8 × 103 , and 5.8 × 101 , respectively. As for the boundary condition in the radial direction, we adopt the outflow boundary condition for the inner and outer boundaries. That is, the flow from the central black hole is prohibited at the inner boundary and the inflow from the surface of the progenitor is prohibited at the outer boundary. Of course, the mass of the central black hole becomes larger due to the mass accretion from the inner boundary. As for the boundary condition in the zenith angle direction, axis of symmetry boundary condition is adopted for the rotation axis, while reflecting boundary condition is adopted for the equatorial plane.

2.2. Initial and boundary conditions 2.3. Micro physics We adopt the Model E25 in Heger et al. (2000). The star in this model has 25 M⊙ initially with solar initial metallicity, but lose its mass and becomes to be 5.45 M⊙ as a Wolf-Rayet star at the final stage. This model seems to be a good candidate as a progenitor of a GRB since losing their envelope will Springer

2.3.1. Equation of state The equation of state (EOS) used in this study is the one developed by Blinnikov et al. (1996). This EOS contains an

Astrophys Space Sci (2007) 307:203–206

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Fig. 1 Density contour (from 1 – 1012 g cm−3 ) with velocity fields at t = 2.5 s after the collapse for the region r ≤ 108 cm. The case without magnetic fields is shown in the left panel, while the case with magnetic fields (109 G) is shown in the right panel

electron-positron gas with arbitrary degeneracy, which is in thermal equilibrium with blackbody radiation and ideal gas of nuclei. 2.3.2. Nuclear reactions Although the contribution of ideal gas of nuclei to the total pressure is negligible, effects of energy gain/loss due to nuclear reactions are important. In this study, nuclear statistical equilibrium (NSE) was assumed for the region where T ≥ 5 × 109 [K] is satisfied as Nagataki et al. (2003), while no nuclear reaction occurs for the region where T < 5 × 109 [K]. This treatment is based on the assumption that the timescale to reach and maintain NSE is much shorter than the hydrodynamical time. Note that complete Si-burning occurs in explosive nucleosynthesis of core-collapse supernovae for the region T ≥ 5 × 109 [K]. The hydrodynamical time in this study, ∼ sec, is comparable to the explosive nucleosynthesis in core-collapse supernovae, so the assumption adopted in this study seems to be farely well. 5 nuclei, n, p,4 He,16 O,56 Ni was used to estimate the binding energy of ideal gas of nuclei in NSE for given (ρ, T , Ye ). Ye is electron fraction that is obtained from the calculations of neutrino process in Section 2.3.3.

pair annihilation process, the formulation of Goodman et al. (1987) is adopted. We assume that the matter is optically thin against neutrinos to obtain the neutrino heating rate as mentioned above. 3. Results In Fig. 1, density contour with velocity fields at t = 2.5 s after the collapse. The case without magnetic fields is shown in the left panel, while the case with magnetic fields (109 G) is shown in the right panel. It is clearly shown that a jet propagates along to the rotation axis for the case with magnetic fields. The total energy of the jet component can be of the order of 1052 erg, which is comparable to the one of a hypernova. We found that the jet is launched by the magnetic pressure of Bφ , which is amplified by the winding-up effect. As for the amplitude of Br , Bθ , they are much smaller than the amplitude of Bφ . In Fig. 2, total emitted energy by neutrino processes as a function of time (solid line), total absorbed energy by

2.3.3. Neutrino processes Neutrino cooling processes due to pair capture on free nucleons, pair annihilation, and plasmon decay are included in this study. Since photoneutrino and bremsstrahlung processes are less important ones at 109 < T < 1011 [K] and ρ < 1010 [g cm−3 ] where effects of neutrino cooling are important in our calculations, we do not include these processes. Neutrino heating processes due to νe and ν¯ e captures on free nucleons and neutrino pair annihilation with blocking factors of electrons and positrons are included in this study. The νe and ν¯ e captures on free nucleons are inverse processes of electron/positron captures. As for the neutrino

Fig. 2 Solid line: total emitted energy by neutrino processes as a function of time. Dashed line: total absorbed energy by neutrino antineutrino pair annihilation. Dotted line: total absorbed energy by neutrino capture on nucleons

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neutrino anti-neutrino pair annihilation (dashed line), and total absorbed energy by neutrino capture on nucleons (dotted line) are shown. It is noted that total absorbed energy is so little that this effect can not explain the explosion energy of a hypernova. As for the dependence of the strength of the initial magnetic fields, we found that strong jet is launched when strong initial magnetic fields are assumed initially.

4. Summary and discussion We have done 2-dimensional MHD simulation of collapsars with magnetic fields and neutrino cooling/heating processes. It is found that explosion energy of a hypernova is not obtained from the neutrino heating process. However, strong jet is found when magnetic fields are included, and total energy of the jet component can be of the order of 1052 erg, which is comparable to the one of a hypernova. To tell the truth, we consider that neutrino heating process can be a possible key process to drive a GRB jet when effects of general relativity are taken into consideration. Since

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our code is Newtonian at present. So we are developing a GRMHD code now. We hope to present new results in the very near future. Acknowledgement The author is grateful to M. Watanabe and S. Yamada for useful discussion. The computation was partly carried out on NEC SX-5 and SX-8, SGI Altix3700 BX2, and Compaq AlphaServer ES40 at Yukawa Institute for Theoretical Physics, and Fujitsu VPP5000 at National Astronomical Observatory of Japan. This work is partially supported by Grants-in-Aid for Scientific Research from the Ministry of Education, Culture, Sports, Science and Technology of Japan through No. 14102004, 14079202, and 16740134.

References Blinnikov, S.I., Dunina-Barkovskaya, N.V., Nadyozhin, D.K.: ApJ 106, 171 (1996) Fujimoto, S., et al.: ApJ 644, 1040 (2006) Heger, A., Langer, N., Woosley, S.E.: ApJ 528, 368 (2000) MacFadyen, A.I., Woosley, S.E.: ApJ 524, 262 (1999) Nagataki, S., Mizuta, A., Yamada, S., Takabe, H., Sato, K.: ApJ 596, 401 (2003) Proga, D., MacFadyen, A.I., Armitage, P.J., Begelman, M.C.: ApJ 599, L5 (2003)

Astrophys Space Sci (2007) 307:207–211 DOI 10.1007/s10509-006-9236-3

O R I G I NA L A RT I C L E

Astrophysical Radiation Dynamics: The Prospects for Scaling John I. Castor

Received: 14 June 2006 / Accepted: 10 August 2006 C Springer Science + Business Media B.V. 2006


Abstract The general principles of scaling are discussed, followed by a survey of the important dimensionless parameters of fluid dynamics including radiation and magnetic fields, and of non-LTE spectroscopy. The values of the parameters are reviewed for a variety of astronomical and laboratory environments. It is found that parameters involving transport coefficients – the fluid and magnetic Reynolds numbers – have enormous values for the astronomical problems that are not reached in the lab. The parameters that measure the importance of radiation are also scarcely reached in the lab. This also means that the lab environments are much closer to LTE than the majority of astronomical examples. Some of the astronomical environments are more magnetically dominated than anything in the lab. The conclusion is that a good astronomical environment for simulation in a given lab experiment can be found, but that the reverse is much more difficult. Keywords Hydrodynamics . Radiation . Scaling PACS Nos: 95.30.Jx, 95.30.Lz, 97.10.Ex, 97.10.Gz, 98.62.Mw

1 Introduction Radiation hydrodynamics is the discipline in which not only the material fluid but also the radiation (photons or neutrinos) The U.S. Government’s right to retain a non-exclusive, royalty-free license in and to any copyright is acknowledged. J. I. Castor Lawrence Livermore National Laboratory, L-16, Livermore, CA 94550, USA e-mail: [email protected]

must be treated dynamically. Since the speed of light is so large, it is tempting and often successful to neglect the fluid velocity in the dynamical equation for the radiation. Accounting for the effects thereby ignored is the business of radiation hydrodynamics. Chief among these are the advective flux of radiation energy and the subtraction of momentum and energy from the radiation when it exerts a force on the material. From a computational point of view, the proper accounting for the velocity effects is one of two main challenges in radiation hydrodynamics; the other challenge is meeting the requirement of a full transport solution with all the spectral and angular detail that the radiation field possesses. The latter challenge is faced even when the fluid velocity is negligible. The computational complexity of radiation hydrodynamics is the motivation for seeking laboratory analogues of astronomical environments for which, owing to the radiation hydrodynamic effects, numerical simulations are very difficult; the analogue experimental results can provide benchmarks for the simulations. In this paper I will give a quick review of the principle of scaling for physical systems described by a small set of partial differential equations. The central point is the non-dimensionalization of the equations, which leads to a minimum set of non-dimensional parameters the values of which must all match for two physical systems in order for one system to be the scaled version of the other. Next I provide a list of possibly relevant dimensionless parameters that arise in describing various astronomical environments. After a brief explanation of the parameters I provide a table of the parameter values for several astronomical environments and also several laboratory environments that may be proposed as scaling candidates for the astronomical ones. The discussion of this table is the main point of this paper, and after the discussion I offer a short conclusion. Springer

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2 The principle of scaling The scaling concept is described as follows: It is assumed that our physical system is described fully and with sufficient accuracy by providing the values of a few fields, such as the density, fluid velocity, perhaps magnetic induction, etc., as functions of a few independent variables, such as coordinates x, y, z and time t. It is also assumed that to sufficient accuracy the fields obey a certain set of partial differential equations over these coordinates. These equations may be put into non-dimensional form by expressing each field or coordinate as the product of a representative value and a dimensionless function. When the transformed equations are simplified, the dimensional representative values can be grouped together into dimensionless combinations, which are the fundamental parameters of the problem. The two systems can be scaled versions of each other if the non-dimensional partial differential equations that describe them are identical, and, in particular, if the dimensionless parameters are identical. This means that the same set of physical processes is an accurate description of both systems, and that the relative magnitudes of the different processes that are included are also identical between the two systems. The test for scaling is therefore this: identify the relevant dimensionless parameters and test each of them for equality. Each dimensionless parameter can be expressed as the ratio of two physical quantities that appear in the governing equations and which have the same dimensions. When the dimensionless ratio is either extremely large or extremely small, it means that one of the physical quantities is negligible compared with the other. In this case the equations could be simplified by discarding the negligible term(s). So when we test for equality of the dimensionless parameters for two systems, we can ignore a parameter that is different for the two systems if it happens to be extremely large or extremely small in both; that parameter involves physics that is not actually relevant for these systems.

3 The astronomical environments Astronomical bodies have characteristic length and time scales that are huge compared with terrestrial laboratories, of course, but there is also a great dynamic range among them. But just to pick one example, consider the interstellar medium. The typical length scale is roughly one parsec,1 and the typical time scale is very roughly 1000 years. Each of these numbers is 3 × 1019 larger than laser experiment scales of 1 mm and 1 nanosecond. The density may be 1–1000 par-

1

1 parsec (pc) is 3.08568 × 1018 cm Springer

Astrophys Space Sci (2007) 307:207–211 Table 1 Selected astronomical environments Environ

Length

Velocity

# density

Temp

B

Warm ISM Dense cld Stellar atm Stellar env AGN disk XRB disk NS acc col Stellar wnd

3 × 1018 3 × 1018 109 1010 2 × 1013 106 4 × 104 1012

107 5 × 105 107 107 3 × 108 3 × 107 3 × 109 108

1 103 1015 1018 1012 3 × 1021 1023 1011

104 102 104 106 107 107 108 105

10−5 10−4 102 102 106 106 1012 5 × 101

Table 2 Selected laboratory environments Environ Burn thru  hohlraum NIF hohl Z expt Short pulse

Length −3

10 10−2 3 × 10−2 10−1 10−3

Velocity 6

10 107 2 × 107 107 108

# density 24

10 1022 1022 1022 1024

Temp

B 5

6 × 10 106 3 × 106 106 107

106 106 106 106 108

ticles per cubic centimeter, which is at least 1019 times less than the typical laser target density 1022 cm−3 . It is a stringent test of scaling to span nineteen orders of magnitude! For the present discussion I have selected eight astronomical environments as candidates for scaling to the lab: (1) warm interstellar medium; (2) a dense interstellar cloud; (3) a stellar photosphere; (4) an interior point in a stellar envelope; (5) the accretion disk around an active galactic nucleus; (6) an x-ray binary accretion disk; (7) in a neutron star accretion column; and (8) a point in the wind of a hot star. Table 1 shows the characteristic properties and dimensions of the environments. The units are cgs, kelvins and Gauss, and the particle density is atoms per cubic centimeter.

4 Some laboratory environments I will consider here a few selected laboratory environments that have been employed or proposed for laboratory astrophysics experiments. These are: a burn-through foil that might be inserted in an Omega hohlraum wall; plasma at critical density in a modest-temperature hohlraum; the same thing but sized for a NIF hohlraum; the same thing on the Z pulsedpower machine; conditions produced by a short-pulse laser. Table 2 lists the characteristics chosen to represent the different experiments. As above, the units are cgs-kelvin-Gauss.

5 Relevant dimensionless parameters The physical processes that dominate the behavior in an astronomical environment that can be scaled to the laboratory are necessarily simple: ideal gas dynamics with radiation

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209

flow and perhaps MHD. The gas dynamics by itself introduces one parameter, the Mach number M = u/cs , in which cs = (γ p/ρ)1/2 is the adiabatic sound speed. Viscosity might be significant, in which case the Reynolds number Re = ρu L/µ is a relevant parameter. We suppose that molecular heat conduction is negligible compared with the radiative heat flux, so we need not be concerned with the Prandtl number. The radiative flux is described with the Boltzmann number, which is the ratio of the convective heat flux to the 1-way radiative flux σ B T 4 : Bo = ρuC p /(σ B T 3 ). In some of the environments the radiation mean free path λ p is short and in some it is long; the optical depth parameter τ = L/λ p is the measure of it. The mean free path is λ p = 1/(κρ L) in terms of the opacity κ, which may be the Rosseland mean or some other fiducial value. When the optical depth is large, the radiation is said to be in the diffusion limit, and the net radiative flux can be computed from a heat-conduction-like formula. The ratio of the convective flux to the diffusive radiative flux is the P´eclet number Pe = (3/4)τρuC p /(σ B T 3 ) = (3/4)τ Bo. If the characteristic optical depth of the plasma, κρ L, is small, the optical depth may in fact not be a useful concept. In such plasmas, e.g., a galactic nebula, the opacity and therefore the optical depth varies dramatically between the cores of strong lines and the continuum regions. The total cooling rate per unit mass by radiative emission is given formally by the expression C = 4π κ P B in LTE, in which κ P is the Planck-weighted opacity and B = σ B T 4 /π is the Planck function. A similar expression applies in non-LTE, but it can only be evaluated by treating in detail the cooling processes of collisional excitation, radiative recombination, etc. The cooling function has been studied by Raymond et al. (1976) and Dalgarno and McCray (1972), among others. We may regard κ P as formally accounting for all those processes. The optically-thin cooling time based on C is tcool = C p T / C, and the relevant dimensionless number is the ratio of tcool to the characteristic flow time, L/u. This is

Cool = utcool /L = uC p T /(L C ). For those cases in which it is appropriate to identify C with 4π κ P B, the parameter becomes utcool /L = uC p T /(4π κ P L B). In this last form Cool is the Boltzmann number defined above divided by 4κ P ρ L; the last has the form of a characteristic optical depth, which is expected to be very small in the case being discussed. With the qualification that C = 4π κ P B is not always appropriate, the manipulation to express the cooling time parameter in terms of the Boltzmann number is justified by the parallel between this relation in the optically thin case and that between the P´eclet number and the Boltzmann number in the diffusive case. It goes without saying that quantitative studies must account for all the detailed physical processes rather than using the crude expressions presented here. When the magnetic field is significant the equations of MHD replace the Euler equations. An additional parameter appears for ideal MHD, the plasma beta, β = 8π p/B 2 . If the electrical conductivity is not effectively infinite then the equations of resistive MHD must be used, and an additional parameter is the magnetic Reynolds number, Rm = µ0 u L/η = 4π σ u L/c2 , in which η is the electrical resistivity in SI units, and σ is the conductivity in Gaussian units (s−1 ). These are the relevant parameters in a collisional plasma. In the weak-collision regime there are additional parameters, such as the Larmor radius divided by L and the collision frequency times L/u, as discussed in these proceedings by Ryutov (2006).

6 Scaling parameters for astronomical and lab environments I have evaluated the eight parameters discussed earlier, τ , M, Re, Rm, Bo, Pe, Cool and β, for the various astronomical and laboratory environments. These are shown in Table 3. For the warm ISM, dense cloud and stellar wind environments the effective value of the opacity was based on the cooling

Table 3 Scaling parameters for astronomical and laboratory environments Environ

τ

M

Re

Rm

Bo

Pe

Cool

β

Warm ISM Dense cld Stellar atm Stellar env AGN disk XRB disk NS acc col Stellar wnd Burn thru Omega hohl NIF hohl Z expt Short pls

10−4 2 × 10−5 101 3 × 103 5 103 103 2 × 10−2 7 × 101 9 × 10−3 10−3 9 × 10−2 10−2

101 6 101 1 101 1 4 × 101 4 × 101 2 × 10−1 1 1 1 4

107 7 × 1013 5 × 1012 5 × 1011 7 × 107 109 6 × 108 1010 2 × 104 3 × 103 103 3 × 104 103

1019 6 × 1014 4 × 109 4 × 1013 6 × 1019 4 × 1011 5 × 1013 1015 2 × 10−1 5 × 101 2 × 103 5 × 102 103

2 × 10−3 10−9 6 × 10−2 3 × 10−5 2 × 10−12 5 × 10−4 8 × 10−2 3 × 10−6 3 × 101 4 × 10−1 10−2 2 × 10−1 3 × 10−1

2 × 10−7 2 × 10−14 6 × 10−1 7 × 10−2 6 × 10−12 4 × 10−1 8 × 101 5 × 10−8 2 × 103 2 × 10−3 10−5 10−2 3 × 10−3

5 2 × 10−5 10−3 2 × 10−9 8 × 10−14 10−7 2 × 10−5 3 × 10−5 10−1 101 3 5 × 10−1 5

3 × 10−1 3 × 10−2 3 3 × 105 3 × 10−8 8 × 101 3 × 10−8 10−2 2 × 103 3 × 101 9 × 101 3 × 101 3

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function (T ) given by Dalgarno and McCray (1972), and for the warm ISM, stellar wind and accretion column environments a dilution factor for the radiation field was included in the definition of the Boltzmann number. The way in which we would like to use these tables is to look up the astronomical environment we want to simulate in Table 3, and then find a laboratory environment in the table that has similar values of the scaling parameters. But we see at a glance that the scaling parameters for the astronomical environments have a huge dynamic range while the laboratory parameters do not. Some of the notable differences between the astronomical and laboratory environments that are seen in Table 3 are these: The transport parameters Re and Rm, the ordinary and magnetic Reynolds numbers, are much larger in all the astronomical environments. In view of the comments earlier about very large parameters, it seems that viscosity and resistivity can quite generally be neglected in the astronomical environments, while this may not be true in the laboratory. The Boltzmann number is generally very small for the astronomical environments; this means that they are radiationdominated. The derived values of the astronomical P´eclet and cooling numbers are also small, with one exception in each case. In the laboratory, because the density is so much higher, radiation is generally not dominant. This is a major impediment to simulating astronomical radiation hydrodynamics problems in laboratory experiments. Also because of the high density, the lab environments have a great difficulty achieving a low β. Some, but not all, astronomical environments are very highly magnetized, which is not true of the laser and pulsed-power experiments considered here. Leaving aside the radiation and magnetic field effects, in other words just looking at the Mach number, we see a better overlap between astronomical and laboratory parameter values. So pure gas dynamics looks promising for scaling astronomical problems to the laboratory.

Astrophys Space Sci (2007) 307:207–211

The competition between collisional excitation and deexcitation processes and their radiative counterparts is the heart of non-LTE excitation and ionization equilibrium. The scaling parameters that express this competition are the ε values defined by ε=

in which u → ℓ is an atomic transition forming a spectral line, Cuℓ is the rate coefficient for collisional de-excitation and Auℓ is the spontaneous radiative decay rate. In order for the emitted spectrum to match, all the values of ǫ should match. This expression for ε can also be written as ε = Ne /(Ne )crit , where (Ne )crit ≡ Auℓ /Cuℓ is called the critical density for this transition. Many of the lines in the spectra of nebulae are electric-dipole forbidden, and the values of Auℓ are of order 1 s−1 or less, which makes the critical density of order 102 –106 cm−3 . Obviously these lines are not candidates for laboratory studies. For permitted lines Van Regemorter’s (1962) semi-empirical formula may used to approximate the collisional rate, then the dependence on Auℓ cancels out and the result is ε ∝ Ne λ3ph , where λph is the photon wavelength. This explicit dependence on Ne means that the line formation process is not scalable. In nebulae in our galaxy, and in emission-line regions of active galactic nuclei and elsewhere, we have conditions very far from LTE in which the plasma is strongly photoionized by a diluted but energetic radiation field. The plasma temperature comes to an equilibrium in which photoionization heating balances radiative cooling, mostly in line emission. The ionization balance and the temperature then depend, for a given shape of the ionizing spectrum, on the ionization parameter  defined by Krolik et al. (1981), Tarter et al. (1969) =

7 Non-LTE and astronomical spectra Another area in which we would hope to simulate an astronomical problem in the lab is in plasma spectroscopy: Can we create a radiation source in the lab of which the spectrum would be a good match to that of the astronomical object? Scaling spectroscopy is harder than scaling hydrodynamics. Replacing one element by another does not work very well, since complex spectra are unique; hydrogenic spectra are the exception. So we suppose that the ions of interest are not hydrogen-like, and that the same element will be used in the simulation that occurs in the astronomical problem. Since atomic excitation and ionization depend on the ratio of the ionization potential to kT , we conclude that T will also not be scaled. Springer

Ne Cuℓ , Auℓ

U . ρC p T

Here U is the diluted radiation energy density for photon energies above the photoionization threshold; U is related to the radiative flux F by U = F/c. (Different authors have defined the ionization parameter in different ways; the citations above give two; for a third definition see (Netzer, 1990). The definitions are simply related to each other, and are ways of parameterizing the ionization ratio in a photoionized plasma, namely, the ratio of the photoionization rate to the radiative recombination rate. The ionization ratio depends on atomic properties in addition to the environmental characteristics.) We see that  is roughly the same as u/(cBo), except that here the Boltzmann number must be defined using the actual radiative flux F rather than the thermal flux σ B T 4 ; the relation is approximately F = W σ B T 4 , in which W is the geometrical dilution factor. The Boltzmann number is again the

Astrophys Space Sci (2007) 307:207–211

important scaling parameter for non-LTE. The typical values of  in nebulae and active galactic nuclei are 102 –103 , while the values in the lab environments are closer to 10−1 . 8 Prospects We have seen that the dynamic range of the scaling parameters for the astronomical environments is very large indeed, much larger than the range among the available laboratory experiments. A large part of the range covered by the astronomical environments is inaccessible in the laboratory. This means that the the odds that a given astronomical environment can be simulated in the laboratory are not good. However, the odds that a given laboratory environment has an analogue in astronomy are much better. Some processes do not scale very well – the viscosity and resistivity effects are generally much smaller in the astronomical environments, and both radiation and magnetic fields tend to be stronger (small Bo and β) in the astronomical cases. Scaling appears to be most successful for pure gas

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dynamics. Within these limitations the prospects for scaling are good. Acknowledgements This work was performed under the auspices of the U.S. Department of Energy by the University of California Lawrence Livermore National Laboratory under Contract No. W-7405-Eng-48. I would like to thank B. A. Remington and R. P. Drake for an education in laboratory astrophysics on Nova, and D. Ryutov for his insightful comments on scaling. I am grateful to the organizers of HEDLA 2006 for the opportunity to present this work. The article was improved by the referee’s helpful comments.

References Dalgarno, A., McCray, R.A.: ARA&A 10, 375–426 (1972) Krolik, J.H., McKee, C.F., Tarter, C.B.: ApJ 249, 422–442 (1981) Netzer, H.: In: Courvoisier, T.J.-L., Mayer, M. (eds.), Swiss Society for Astrophysics and Astronomy, Springer, New York (1990) Raymond, J.C., Cox, D.P., Smith, B.W.: ApJ 204, 290–292 (1976) Ryutov, D.: In: Lebedev, S. (ed.), Proceedings of HEDLA (2006) Tarter, C.B., Tucker, W.H., Salpeter, E.E.: ApJ 156, 943–951 (1969) Van Regemorter, H.: ApJ 136, 906–915 (1962)

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Astrophys Space Sci (2007) 307:213–217 DOI 10.1007/s10509-006-9235-4

O R I G I NA L A RT I C L E

Experiments to Study Radiation Transport in Clumpy Media P. A. Rosen · J. M. Foster · M. J. Taylor · P. A. Keiter · C. C. Smith · J. R. Finke · M. Gunderson · T. S. Perry

Received: 12 May 2006 / Accepted: 11 August 2006 C Springer Science + Business Media B.V. 2006


Abstract Clumpiness of the interstellar medium may play an important role in the transfer of infrared continuum radiation in star forming regions (Boisse, 1990). For example, in homogeneous models, C II emission should be confined to the cloud edge (Viala, 1986). However, in star formation regions (such as M17SW, M17 and W51), it is observed to extend deep into the molecular cloud (Stutzki et al., 1988; Keene et al., 1985). One plausible interpretation of these observations is that, due to their clumpiness, the clouds are penetrated by UV radiation far deeper than expected from simple homogeneous models. The interaction of H II regions around young massive stars with a clumpy medium is another area of interest. Molecular clouds are well established to be clumpy on length scales down to the limits of observational resolution. Clumps can act as localized reservoirs of gas which can be injected into the surroundings by photoionization and/or hydrodynamic ablation (Dyson et al., 1995; Mathis et al., 1998). The calculation of radiation transport in hot, clumpy materials is a challenging problem. Approximate, statistical treatments of this problem have been developed by several workers, but their application has not been tested in detail. We describe laboratory experiments, using the Omega laser to test modelling of radiation transport through clumpy media in the form of inhomogeneous plasmas.

Keywords Radiation transfer . Radiation flow . Clumpy media . Inhomogeneous plasma

P. A. Rosen (
) · J. M. Foster · M. J. Taylor · C. C. Smith AWE Aldermaston, Reading, RG7 4PR, UK e-mail: [email protected]

2 Experiment design

P. A. Keiter · J. R. Finke · M. Gunderson Los Alamos National Laboratory, Los Alamos, NM 87545, USA T. S. Perry Lawrence Livermore National Laboratory, Livermore, CA 94550, USA

1 Introduction Calculations of radiation or particle transport are greatly complicated by the presence of regions in which two (or more) materials are randomly and inhomogeneously mixed (that is, where discrete chunks of random size are randomly dispersed in a host material). A variety of statistical methods has been developed to treat this problem, and these are discussed extensively in the literature (Henke et al., 1984; Vanderhagen, 1986; Pomraning, 1988, 1991; Haran et al., 2000; Smith, 2003). In brief summary, it is assumed that at any point in the inhomogeneously mixed material, the probability of occurrence of each of the constituents is known, and the goal is to calculate the mean particle or radiation intensity, averaged over all possible configurations of the ensemble, and thus the effective opacity of the mixture. Theoretical treatments differ according to the methods they employ and the statistical distribution assumed for the components of the mixture. Our aim is to carry out laboratory investigations of radiation transport through inhomogeneously mixed plasmas, that can be analysed using these statistical methods.

We have developed a laboratory experiment to study radiation transport through inhomogeneous media. Figure 1 shows the experimental concept. A sample of inhomogeneous, gold-loaded hydrocarbon foam is contained within a gold tube, driven from one end by thermal radiation from a hohlraum target heated by the Omega laser (Soures et al., Springer

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Fig. 1 Experimental configuration showing the laser-heated hohlraum and the experimental package. The gold tube is 1 mm length and 0.8 mm diam

1996). Radiation burnthrough of the foam is diagnosed using an x-ray framing camera viewing through a longitudinal slit in the wall of the gold tube. We compare the radiation burnthrough times of samples of pure foam, and foam containing either uniformly distributed gold particles, or an equal mass of atomically mixed gold. Differences of opacity result in differences of burnthrough time, and the experimental data are compared with calculational models for the inhomogeneous and homogeneous mixtures. In our present experiments, a peak radiation drive temperature of 200 eV is obtained in a 1.6 mm diam., 1.2 mm length hohlraum that is heated by 15 beams of the Omega laser, with a total energy of ∼6.5 kJ in a 1 ns duration laser pulse of 0.35 µm wavelength. The experimental package is mounted over a 0.8-mm-diam. hole in the end wall of the hohlraum; it comprises a 0.8-mm-diam., 1-mm-length gold tube containing the inhomogeneous gold/foam mixture. In the design of the experiment, we consider various candidate gold/foam mixtures under different conditions of temperature and density. We assume that the gold particles are uniformly heated, and that they expand until pressure equilibrium with the surrounding host material is attained. For particles of micron size appropriate to our experiments at the Omega laser, the timescale for this expansion is approx. 0.5 ns. We calculate separately the (spectrally dependent) opacities of the heated hydrocarbon foam and gold particles, using the IMP opacity code (Rose, 1992). We then apply the inhomogeneous-mixture opacity models discussed above, to obtain the effective opacity of the mixture, and from this calculate the mean opacity. Figure 2 shows calculations of Rosseland mean opacity, as a function of initial particle size and temperature, for a triacrylate (C15 H20 O6 ) foam of 0.05 g cm−3 density containing 15% by weight of admixed gold. Particle size following pressure equilibrium is approx. a factor of three larger than initial particle size. We note that Springer

Astrophys Space Sci (2007) 307:213–217

Fig. 2 Rosseland mean opacity of an inhomogeneous, pressureequilibrated gold/foam mixture (0.05 g cm−3 total density, 15% by weight gold, various initial particle sizes)

the mean opacities of pure foam, and foam containing 15% by weight admixed gold, differ by approx. a factor of five. The mean opacity in the particulate mix case falls midway between these two limits, for particles of 2-µm initial size. For our present experiments, triacrylate foams have been manufactured at the University of St. Andrews, UK (Falconer et al., 1995), and have been successfully loaded with particles with a narrow distribution of diameters (1.5–3 µm), or with an organo-metallic gold compound. Pre-shot characterisation of these materials is accomplished using a variety of techniques including weighing (to determine bulk density), x-ray radiography (to determine uniformity of bulk density and gold loading), and examination by scanning electron microscopy (SEM) (to determine particle size distribution and to detect agglomeration of particles). X-ray fluorescence and neutron activation measurements are also carried out in samples contained in polymer cylinders, to establish gold content. Figure 3 shows data from a multi-channel, time-gated xray pinhole camera viewing the outside of the foam-filled tube. This camera employs absorption-edge x-ray filters to limit its response to narrow spectral regions close to 300 and 450 eV. The multiple images shown in Figure 3 (obtained using a pure foam sample) were recorded at 0.5, 1.5, 2.5 and 3.5 ns after the onset of radiation drive. These data clearly show the progression of the radiation-driven heat wave as well as the closure (at late time in the 450-eV images) of the diagnostic slit. Comparison of the burnthrough times of the different foams (pure, particle-loaded, and atomically mixed) is the basis of our experiment. 3 Modelling Our modelling of the experiment has adopted two approaches: design studies using a radiation-temperature drive

Astrophys Space Sci (2007) 307:213–217

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Fig. 3 X-ray framing camera data from the 300-eV channel (top) and 450-eV channel (bottom) showing progression of the radiation-driven heat front along the foam-filled tube. The (white) shadow of a vertical fiducial wire is visible in some images

prescription; and a more detailed, fully integrated calculational model of a limited number of specific experiments. Both originate from a detailed model of the laser-heated hohlraum, but the radiation-temperature drive prescription uses an equivalent Planckian drive spectrum for simplicity in subsequent simulations, whereas the detailed model includes the hohlraum in each and every simulation and thus provides a better approximation of multi-frequency effects in the radiation transport. In all cases, we use the Lagrangian radiation hydrocode NYM (Roberts et al., 1980), multi-group implicit Monte-Carlo radiation transport, and opacities generated using the IMP code (Rose, 1992). In the design simulations, we use a drive prescription identical to that described by Foster et al. (2002), and apply this (by means of appropriate boundary conditions) to a simulation of the experimental package (foam-filled tube) alone. The inhomogeneous gold-loaded foam mixture is simulated approximately by means of opacity multipliers, using as a basis opacity data for both pure foam (required opacity multiplier >1) and atomically mixed, gold-loaded foam (opacity multiplier <1). The appropriate opacity multipliers are obtained from calculations of the effective opacity of the mixture, using the methods of Pomraning et al. (1988, 1991) and Vanderhaegen (1986). The radiation hydrocode simulation is post-processed by ray tracing to generate spatially resolved x-ray emission intensity for comparison with the x-ray framing camera data. Figure 4 summarises the position of the emission front (in the 300-eV channel of the x-ray framing camera), from simulations of both pure and atomically mixed foam, together with these same calculations in which the radiative opacity has been scaled by factors of 2 and 5 (pure foam case) and 0.5 and 0.2 (atomically mixed case). It is evident that these simulations are consistent with the factor-of-5 difference of opacity between the pure-foam and atomic-mix cases (Figure 2), and that the use of an opacity multiplier thus provides a good first approximation to the inhomogeneous-mixture case.

Fig. 4 Emission (in the range 200–300 eV) front position versus time from post-processed NYM simulations of pure foam and atomicallyloaded foams using nominal and scaled radiative opacities

The fully detailed calculations of specific experimental shots include both the hohlraum and experimental package in each simulation, and so better represent the detailed drive spectrum (including the long-range preheat originating in the gold M-band emission from the hohlraum). Nevertheless, their results are not greatly different from simpler, pre-shot models of the experiment. Again, ray-tracing post-processing is used to generate synthetic diagnostic signals for comparison with experiment. In the fully detailed calculations, we have considered only the case of pure foam and atomically mixed gold-loaded foam; we have not yet carried out opacity multiplier simulations as a representation of the particulatemix case. 4 Comparison of experimental data with simulation Figure 5 compares the measured and calculated positions of the x-ray emission front (300-eV x-ray images) with pre-shot Springer

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Fig. 5 Emission front position as a function of time for experimental data and simulation of pure foam. A multiplier to the radiative opacity has been applied

simulations (including ray-tracing post-processing) in which the particulate- and atomic-mix cases are approximately represented by opacity multiplier of 2 and 5 (applied to simulations of the pure-foam case). Although the density and gold mass fraction of the foam samples differ somewhat from their nominal values (50 mg cm−3 , and 15% by weight, respectively), it is clear that the experimental data demonstrate the trends evident in simulation: data from experiments in 2002 and 2005 are self consistent (within the timing uncertainty of the x-ray framing camera), and the radiation burnthrough time of the particle-mix case falls between that of the pure foam and atomic-mix cases. Figure 6 compares these experimental data with both preshot and detailed simulations of the experiment. In both cases, simulations are for foams of nominal density (50 mg cm−3 ), and gold mass fraction (15% by weight, in the case of the atomic mixture). The pre-shot and detailed simulations show differences of emission front position that arise from differences of detail in the drive prescription and spectrum (overall energetics, and multi-group radiation transport effects), but these differences are comparable to the uncertainty inherent in the experimental measurement. An arbitrary (but plausible) offset of 300 ps of the 2002 data results in very close agreement of the 2002 and 2005 data sets, and a detailed simulation of the pure foam experiment for 60–65 mg cm−3 foam density (not shown) lies very close (within experimental error) to the experimental data. Experimental data from the particulate-mix foams appears to lie close to simulations assuming atomic mix, and we speculate that this may be as a result of break up of the gold particles following radiation (M-band) preheating. This speculation can only be resolved in further experiments using larger diameter gold particles. Data from the single shot using atomically mixed gold difSpringer

Astrophys Space Sci (2007) 307:213–217

Fig. 6 Measured and calculated positions of the x-ray emission front, for the 300-eV x-ray images. The experimental data (points) are for pure, particle-loaded and atomically loaded foams. All foams were of 0.05 g cm−3 nominal density, although the densities apparent in pre-shot characterisation measurements (figures in parentheses) differ somewhat from this value. The simulations (lines) are for foams of 0.05 g cm−3 density and 15% by weight gold (atomic mix case)

fers from both the pure-foam and particulate-mix cases. This difference implies a foam of nominal density, although preshot characterisation of this single case implied a rather lower (than nominal) foam density.

5 Summary We have reported the results of experiments carried out at the Omega laser in 2002 and 2005, to study radiation transport through an inhomogeneous mixture. Good experimental data have been obtained, and it has been possible to measure radiation-burnthrough times that are correlated with the composition of the pure-foam, and inhomogenous or atomic gold-foam mixture, that was used. Modelling successfully reproduces both the burnthrough time of the pure foam samples, and the trend of burnthrough time with material composition. Differences between the measured and modelled burnthrough times of the particulate and atomic mixtures may point to pre-heat induced break-up of the gold particles (in which case, the particulate mix behaves as an atomically dispersed mixture), or to uncertainties in characterisation of the material samples. Further experiments using better characterised materials, and a wider range of particle sizes, are planned. Acknowledgements It is our pleasure to thank the target fabrication teams at AWE and LANL, Chris Bentley (AWE) for his work on foam characterisation, and the laser-operations team at LLE.

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References Boisse, P.: Astron. Astrophys. 228, 483 (1990) Dyson, J.E., Williams, R.J.R., Redman, M.P.: Mon. Not. R. Astron. Soc. 277, 700 (1995) Falconer, J.W., Nazarov, W., Horsfield, C.: J. Vac. Sci. Technol. A 13, 1941 (1995) Foster, J.M., Wilde, B.H., Rosen, P.A., Perry, T.S., Fell, M., Edwards, M.J., Lasinski, B.F., Turner, R.E., Gittings, M.L.: Phys. Plasmas, 9, 2251 (2002) Haran, O., Shvarts, D., Thieberger, R.: Phys. Rev. E 61, 6183 (2000) Henke, B.L., et al.: J. Opt. Soc. Am. B 1, 818 (1984) Keene, J., Blake, J.A., Phillips, T.G., Huggins, P.J., Beichman, C.A.: Astrophys. J. 299, 967 (1985)

217 Mathis, J.S., Torres-Peimbert, S., Peimbert, M.: Astrophys. J. 495, 328 (1998) Pomraning, G.C.: J. Quant. Spectros. Radiat. Transfer 40, 479 (1988) Pomraning, G.C.: Linear Kinetic Theory and Particle Transport in Stochastic Mixtures. World Scientific, Singapore (1991) Roberts, P.D., Rose, S.J., Thompson, P.C., Wright, R.J.: J. Phys. D 13, 1957 (1980) Rose, S.J.: J. Phys. B 25, 1667 (1992) Smith, C.C.: J. Quant. Spectros. Radiat. Transfer 81, 451 (2003) Soures, J., McCrory, R.L., Verdon, C.P., et al.: Phys. Plasmas 3, 2108 (1996) Stutzki, J., Stacey, G.J., Genzel, R., Harris, A.I., Jaffe, D.T., Lugten, J.B.: Astrophys. J. 332, 379 (1988) Vanderhagen, D.: J. Quant. Spectros. Radiat. Transfer 36, 557 (1986) Viala, Y.: Astron. Astrophys. Suppl. 64, 391 (1986)

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Astrophys Space Sci (2007) 307:219–225 DOI 10.1007/s10509-007-9292-3

ORIGINAL ARTICLE

Laboratory Observation of Secondary Shock Formation Ahead of a Strongly Radiative Blast Wave J. F. Hansen · M. J. Edwards · D. H. Froula · A. D. Edens · G. Gregori · T. Ditmire

Received: 24 April 2006 / Accepted: 18 December 2006 C Springer Science + Business Media B.V. 2007


Abstract We have previously reported the experimental discovery of a second shock forming ahead of a radiative shock propagating in Xe. The initial shock is spherical, radiative, with a high Mach number, and it sends a supersonic radiative heat wave far ahead of itself. The heat wave rapidly slows to a transonic regime and when its Mach number drops to two with respect to the downstream plasma, the heat wave drives a second shock ahead of itself to satisfy mass and momentum conservation in the heat wave reference frame. We now show experimental data from a range of mixtures of Xe and N2 , gradually changing the properties of the initial shock and the environment into which the shock moves and radiates (the radiative conductivity and the heat capacity). We have successfully observed second shock formation over the entire range from 100% Xe mass fraction to 100% N2 . The formation radius of the second shock as a function of Xe mass fraction is consistent with an analytical estimate.

Keywords Radiative . Shock . Heat wave . Supernova

J. F. Hansen (
) · M. J. Edwards · D. H. Froula Lawrence Livermore National Laboratory, Livermore, CA 94550, USA e-mail: [email protected] A. D. Edens Sandia National Labs, Albuquerque, NM 87185, USA G. Gregori CCLRC – Rutherford Appleton Laboratory, Chilton, Didcot OX11 0QX, UK T. Ditmire University of Texas at Austin, Austin, TX 78712, USA

1 Introduction We previously reported on the experimental discovery of secondary shock formation ahead of strongly radiative blast waves in Xe (Hansen et al., 2005, 2006). The process can be summarized as follows (c.f. Fig. 1): a shock (which we will refer to as S1) is initially fast enough to radiate very strongly, with an inverse Boltzmann number Bo−1 = σ T 4 /vs ρ0 cv T ∼ 50 (where T is the shock temperature, vs is the shock speed, and ρ0 is the density of the ambient gas). The radiation mean free path in the cold ambient gas ahead of S1 is relatively short, resulting in the formation of a supersonic, radiative heat wave (RHW), which propagates in advance of S1. As S1 continually slows down, it radiates less and less, and the radiated power soon drops below the rate at which S1 sweeps up energy from gas heated by RHW, i.e., the energy loss rate becomes negative. At this time, most of the energy that was originally in S1 has been radiated to RHW, and RWH is far ahead of S1. However, the velocity of RHW has also been diminishing rapidly because of expansion and a rapidly weakening driving source. Eventually RHW becomes transonic and gives birth to a second shock wave (S2). RHW then falls behind S2, which itself is too slow to be radiative. S1 continues to weaken as it propagates in the downstream material of S2 and soon dissipates. After S2 has roughly doubled its radius, it is no longer influenced by the details of how it was formed, and the shock trajectory closely assumes that of a self-similar Sedov-Taylor blast wave (Taylor, 1950; Sedov, 1959; Zeldovich and Raizer, 1966; Liang and Keilty, 2000). The motivation behind our experiments [and behind many other experiments in laboratories around the world (Bozier et al., 1986; Remington et al., 1999; Ryutov et al., 1999; Shigemori et al., 2000; Robey et al., 2001; Keiter et al., 2002; Fleury et al., 2002; Bouquet et al., 2004)] is an interest in astrophysical shocks which have high Mach numbers and Springer

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Fig. 1 Sequence of events (time increasing from left to right): (a) A spherical shock, S1, radiatively drives a heat wave far ahead of itself. (b) As S1 expands it slows down and it radiates less and less. The radiative heat wave, RHW, also slows down and its expansion rate can be estimated using Barenblatt’s solution. (c) S1 gains on RHW, but before it catches up, RHW becomes transonic and gives birth to a second shock wave, S2. RHW immediately falls behind S2, which itself is too slow to be radiative. (d) S1 continues to weaken as it propagates in the downstream material of S2 and soon dissipates. In the experiment, shocks that are drawn here with solid curves are visible in the schlieren images, features drawn with dashed curves are not

which may be radiative (Blondin et al., 1998), e.g., shocks originating in supernova (SN) explosions (M¨uller et al., 1991; Reed et al., 1995; Sonneborn et al., 1998; Remington et al., 1999; Bartel et al., 2000). The nature of these shocks is important to understand as the shocks mix up interstellar matter and thus affect mass-loading, stellar formation (McKee and Draine, 1991; Allen and Burton, 1993; Klein and Woods, 1998), and the history of the Milky Way and other galaxies. Although the motivation for our experiments was an interest in astrophysical shocks, we should point out that the character of these laboratory shocks may be different from any of astrophysical importance, and that secondary shock formation has not been observed in astrophysical shocks to date. 2 Background The radiative nature of a shock, coupled with the optical opacity and heat capacity of its surroundings, largely determines the evolution of the shock and its rate of expansion. The energy loss rate can be quantified (in units of how much energy the expanding shock sweeps up) by the dimensionless number (Cohen et al., 1998; Liang and Keilty , 2000) ε = − (dE/dt) (2πρ0 )−1 rs−2 (drs /dt)−3 , where E is the total energy content and rs the shock radius. In a fully radiative case, in which radiation escapes to infinity, the incoming kinetic energy swept up by a shock is entirely radiated away (ε → 1) and the shocked material collapses to a thin shell directly behind the shock. For an adiabatic case (ε = 0) once the shock has swept up more mass than what was initially present, the shock could be regarded as without characteristic length or time scales, and one would expect the well-known self-similar motion of a Sedov-Taylor blast wave (Taylor, 1950; Sedov, 1959; Zeldovich and Raizer, 1966; Liang and Keilty , 2000), rs ∝ t α , where the exponent Springer

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α = 2/5. In a case where radiation removes energy from the shock in an optically thin environment, analytical and numerical studies predict a slower shock expansion, such as α = 2/7 (the “pressure-driven snowplow”), α = 1/4 (the “momentum-driven snowplow”; the shock is simply coasting) (McKee and Ostriker, 1977; Blondin et al., 1998), and 2/7 < α < 2/5 (the thermal energy of the shocked gas is not completely radiated away) (Cohen et al., 1998; Liang and Keilty , 2000). In a case where the environment is not optically thin – which is the case for many experiments including ours – radiation is reabsorbed in the upstream material and if the shock is travelling fast enough a supersonic, radiative heat wave (RHW) breaks away from the shock in a situation analogous to a supercritical shock wave (Zeldovich and Raizer, 1966). It has been shown that the shock and RHW will coexist and eventually propagate as r ∝ t α where α is larger for the shock (Reinicke and Meyer-ter-Vehn, 1991). This means that the shock would eventually catch RHW, after which a second state is obtained in which RHW is of the ablative type and the shock moves in a classical Sedov-Taylor trajectory with α = 2/5. In our earlier experiments (Hansen et al., 2006) we came across an additional possibility for optically thin environments, namely that prior to the shock catching RHW, the latter enters a transonic regime, stalls, and generates a second shock (S2). We showed that an analytical estimate for the formation radius of S2 can be obtained from a standard result of heat front physics (Zeldovich and Raizer, 1966; Mihalas and Weibel Mihalas, 1984; Hatchett, 1991), using the 1D fluid equations for conservation of mass ρ1 u 1 = ρ2 u 2 and momentum p1 + ρ1 u 21 = p2 + ρ2 u 22 in the frame of RHW, where subscript 1 denotes the region ahead of RHW, and subscript 2 denotes the region behind RHW. Assuming an ideal gas (so that p = ρc2 where c is the isothermal sound speed (Mihalas and Weibel Mihalas, 1984)) we combine these to obtain ρ2 = ρ1

c12 + u 21 ±

 2 c12 + u 21 − 4c22 u 21 2c22

.

(1)

A supersonic (u 1 > c1 ) RHW and a real compression η ≡ ρ2 /ρ1 requires u 1 ≥ c2 +

 c22 − c12 ≈ 2c2

(2)

(where the approximation is valid because the temperature behind RHW is much higher than the temperature before it), i.e., requires the mixed Mach number M≡

u1 ≥ 2. c2

(3)

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Once the Mach number drops to 2, RHW can no longer fulfill Equation (1), and S2 forms at RHW. S2 immediately moves ahead of RHW and acts to slow down u 1 so that RHW is now subsonic, satisfying  c2 u 1 ≤ c2 − c22 − c12 ≈ 1 . 2c2

(4)

In the limit of transition from super- to subsonic, ρ2 /ρ1 = u 1 /c2 and u 2 = c2 . To estimate the radius rh of RHW at the moment when its Mach number is 2, and thus the formation radius of S2, we can assume a radiative conductivity of the ambient gas of the form a

χ = χ0 ρ T

b

(5)

and use Barenblatt’s solution for an instantaneous point release of energy (Barenblatt, 1979; Reinicke and Meyer-terVehn, 1991): rh (t) = r0 t δ

(6)

where  δ r0 = K 1b K 2

1 3b + 2 E γ −1
 K1 = 3 1 ρ0 2π B ,1 + 2 b δ=

K2 =

2χ0 (γ − 1) 1 , Ŵ b+1 ρ01−a bδ

(7) (8) (9)

(10)

where γ is the adiabatic index, m 0 is the molecular weight, B (x, y) is the beta function, and Ŵ is the gas constant from the ideal gas equation of state (γ − 1) e = ŴT,

(11)

where e is the internal energy [i.e., the heat capacity cv = Ŵ/(γ − 1)]. The RHW Mach number is obtained from u1 =

drh dt

(12)

and δ/2  −3δ/2 (rh /r0 )−3/2 . c2 (r = 0) = K 12 K 2−3 t −3 = K 1δ K 2

(13)

Fig. 2 Measured shock radius versus time in experimental images of shocks in xenon (El ≈ 100 − 200 J). Note the step in radius around 12 mm when both shocks are visible

Using the sound speed at the RHW center r = 0 is a reasonable approximation as the temperature profile throughout RHW is quite flat (Hansen et al., 2006; Barenblatt, 1979; Reinicke and Meyer-ter-Vehn, 1991). We should also point out that using Barenblatt’s solution to estimate the RHW radius assumes that radiation can be treated in the diffusion approximation. In the earliest expansion phase this is not the case, but Barenblatt’s solution turns out to be reasonable approximation when the wave has cooled somewhat and has a large enough optical depth. For pure Xe this would be at t ≈ 20 ns; at this time rh ≈ 4 mm, compared to rs ≈ 1.4 mm and only about one eight the total energy still residing in the initial shock (the rest is in the RHW). For pure Xe with ρ0 ∼ 10−5 g/cm3 and E = 5 J, this analytical estimate says that the RHW Mach number drops to Mach 2 when rh ≈ 10 mm, in reasonable agreement with the experiment where we first observed S2 with r ≈ 12 mm (Hansen et al., 2006), see Fig. 2. We report here on a new series of experiments using mixtures of Xe and N2 . The goal of these experiments was to further study the previously unreported second shock, to see what the effect is of changing the radiative conductivity (or equivalently the opacity) and the heat capacity, and to see if the above analytical estimate holds under a range of conditions.

3 Experiment set-up and diagnostics We create spherically expanding blast waves in the following fashion: a high-energy infrared pulsed laser (1064 nm wavelength) is focused onto the tip of a solid (stainless steel) pin surrounded by a Xe/N2 mixture with a density of ρ0 = 3.6 × 10−5 g/cm3 . The laser pulse is 5 ns in duration with energy El ≈ 5 J (the exact energy fluctuates slightly from shot to shot but is measured and recorded). The laser Springer

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energy is deposited in pin material which then becomes very hot and expands rapidly, pushing at the surrounding gas, setting up a strong, radiative initial shock (S1). At the end of the laser pulse (t = 5 ns), S1 is traveling in excess of 60 km/s and is (at least for mixtures with a significant fraction of Xe) strongly radiative. When the radius rs ∼ 0.4 mm, S1 has swept up enough material that the details of its initial conditions are unimportant. Radiation from S1 heats the surrounding gas. The (inverse) Boltzmann number Bo−1
50 initially, i.e., S1 is supercritical and drives a supersonic RHW that travels rapidly outward, leaving a large separation between S1 and RHW. With time, S1 slows and its ability to radiate efficiently quickly decreases. Also, S1 is traveling into the counter pressure of hot RHW plasma, which is becoming comparable to the ram pressure; the Mach number of S1 drops rapidly, and the post-shock compression reduces correspondingly. The Mach number for RHW is also decreasing, and when it reaches ∼ 2, RHW stalls and creates S2, with a radius measured in earlier experiments in pure Xe of ∼ 12 mm. After this time, S1 continues to weaken until it dissipates, while S2 is essentially non-radiative and once it has swept up enough mass (doubled its initial radius), it propagates like rs ∝ t 2/5 , see Fig. 2. To image S1 and S2 on spatial scales up to ∼ 5 cm, we used two lenses in a telescope configuration and a gated, singleframe, high-speed CCD camera (2 ns gate), along with a low energy, green laser pulse (λ = 532 nm wavelength, 15 ns duration) as a backlighter. We employed a schlieren technique with a vertical knife edge at the telescope focal point to remove light which had not been deflected by the plasma. With this method, image brightness corresponds to the spatial derivative of plasma electron density in the horizontal direction, so that vertical structures in the plasma are readily seen. A monochromatic filter was placed in front of the camera to prevent damage to the CCD (damage occured in our previous experiments), with the disadvantage that glow from the heated plasma could not be seen in this experiment. A spectrometer was used to obtain spectral line intensity as a function of position (ahead of and behind the blast wave), which was then Abel inverted to get spectral line intensity as a function of radius. An estimate of temperature as a function of radius was then calculated from pair-wise line ratios. This was done in pure N2 using two NII lines (399.5 nm and 444.70 nm) and two NIII lines (451.485 nm and 463.413 nm) and in pure Xe using three XeII lines, 441.48 nm, 446.22 nm, and 460.3 nm.

4 Results Keeping the density constant (by keeping the partial pressures p X e m X e + p N2 m N2 = C where C is a constant), we varied the composition of the ambient gas from 100% Xe (by mass) to 100% N2 and tracked the formation of the Springer

Astrophys Space Sci (2007) 307:219–225

second shock. We found that the higher the fraction of N2 , the smaller the formation radius became (and the sooner the second shock forms). This trend is shown in Fig. 3, where each column represent a certain mixture of Xe and N2 , and time runs toward the bottom of each column. In the top row of images, we only see the initial shock. As we follow each column down, the second shock forms, and the initial shock dissipates; this should be particularly obvious around the middle of each column. The bottom row shows images where only the second shock can be seen. It is worthwhile noting the experimental difficulty in observing the second shock. Previous experiments have not observed secondary shock formation, and this may be because of any of the following: (a) the experiment was studying radiative shocks, so images were obtained only relatively early in time, while S1 is still radiative, (b) the experiment was studying blast waves, so images were only obtained relatively late in time, to ensure that a stable Sedov-Taylor blastwave had formed, (c) images were too sparse in time, i.e., the sequence of images shows S1 in the first few images, then switches to S2 without capturing the moments when both exist simultaneously, and the experimenter believed – quite naturally – that the same shock was observed in all the images, (d) the schlieren technique was not sensitive enough. The latter condition is one that we struggled with. When the knife-edge position was not carefully calibrated, we obtained images for the intermediate times (when both shocks exist) that simply show no shock at all (both shocks are too weak to perturb the plasma enough to overcome the crudely positioned knife-edge). Figure 3 can be represented in a bar plot, where each bar represents an uncertainty in the formation radius of the second shock; the lower end of each bar is the radius of S1 in an image where the second shock cannot yet be seen, while the upper end is the radius of S2 in the earliest image for each mixture in which we can see both S1 and S2. We have measured all shock radii and are showing this data in Fig. 4. Also shown in Fig. 4 are second shock formation radii from our previous work in pure Xe (Hansen et al., 2006).

5 Comparison to analytical estimate Barenblatt’s solution assumes a radiative heat conductivity which is a function of temperature (and density), but the heat capacity is implicitly assumed to be constant. This means that, at best, Barenblatt’s solution will only be an approximation to real gases with non-constant heat capacities. To estimate when the RHW becomes transonic, we have used this solution taking the heat capacity for the conditions around where we expect the transition to occur, namely T ∼ 3.8 eV based on emission spectroscopy data in the near-ultraviolet

Astrophys Space Sci (2007) 307:219–225

223

Fig. 3 Images of shocks in Xe/N2 mixtures with a density ρ0 = 3.6 × 10−5 g/cm3 . created by a laser focused on a pin (visible in most images). The mixture is the same within each column of images, and the mass-fraction of each gas is written at the top of the column. The number at the top left of each image is the experimental time in nanoseconds (after the laser pulse)

range in pure Xe and pure N2 . The following steps illustrate our method in this regard: 1. The adiabatic index γ for each gas mixture is assumed unknown and is determined from Sedov-Taylor’s formula for a self-similar blast wave:

rs =




75 (γ − 1) (γ + 1)2 16π 3γ − 1

 51


E ρ0

 51

2

t5,

(14)

where we use the measured blast-wave radius rs from an image obtained at a very late time t so that the second shock is well-developed past the point where its initial conditions matter, in addition to the recorded laser energy E and measured density ρ0 . We typically find γ ≈ 1.05, see Table 1. 2. The gas constant Ŵ is calculated from Equation (11) using the adiabatic index γ from Step 1, an initial estimate of a representative temperature T ∼ 3.8 eV, and an internal Springer

224

Astrophys Space Sci (2007) 307:219–225 Table 1 Radiative conductivity and heat capacity parameters for mixtures of nitrogen and xenon χ0 5% Xe, 95% N2 20% Xe, 80% N2 40% Xe, 60% N2 60% Xe, 40% N2 80% Xe, 20% N2 100% Xe

Fig. 4 Calculated and experimentally observed second shock formation radii as a function of Xe mass fraction in Xe/N2 mixtures. The lower/upper end of each bar represents an image where the second shock cannot/can be seen

e = e0 ρ f T g

(15)

with parameters e0 , f , and g from Table 1. 3. The various parameters appearing in Barenblatt’s solution [in Equations (7–10)] are calculated using the adiabatic index γ from Step 1, the gas constant Ŵ from Step 2, and a radiative conductivity from Equation 5 using the parameters χ0 , a, and b from Table 1. 4. The RHW radius rh can be solved for analytically using Equations (12–13) above (but the expression is too complicated to cast any light on the physics and is not included here). We then set the mixed mach number M = u 1 /c2 = 2 to obtain the radius r2 = rh | M=2 when the second shock forms. [Note that if the parameter b ≫ 1 (which is the case here) the simple estimate


K 1b K 2 6b

1/3b

f

g

γ

−2.2 −2.2 −2.2 −2.2 −2.2 −2.2

7 7 7 7 7 10

160 50 40 30 7 2.6

0.1 0.1 0.1 0.1 0.1 0.1

1.4 1.5 1.5 1.5 1.6 1.65

1.03 1.05 1.06 1.05 1.05 1.05

spectrometer estimate. If we go back and use the temperature from Equation 17 in Step 1 we get less than a 4% difference in our final answer for r2 . The calculated values for r2 in six different mixes of Xe and N2 are shown alongside the experimental data in Fig. 4. The trend of smaller formation radii for higher fractions of N2 is reproduced by analytical estimates, and generally the agreement between analytical estimate and the experimental data is quite good.

We have varied in a systematic way the opacity and heat capacity of the gas into which a spherical shock wave expands (by mixing Xe with N2 keeping the mass density constant) and measured when a second shock forms ahead of the initial shock. The formation radius of the second shock as a function of Xe mass fraction is consistent with an analytical estimate where the expansion of a radiatively driven heat wave is estimated using Barenblatt’s solution and the heat wave then stalls as its Mach number drops to ∼ 2. Acknowledgements We thank Dwight Price and the staff at the Janus facility (where the experiments were conducted) for their valuable assistance. This work was performed under the auspices of the U. S. Department of Energy by the University of California, Lawrence Livermore National Laboratory under Contract No. W-7405-Eng-48.

(17)

Allen, D.A., Burton, M.G.: Nature 363, 54 (1993) Barenblatt, G.I.: Similarity, Self-Similarity and Intermediate Asymptotics. Consultants Bureau, New York (1979) Bartel, N., Bietenholz, M.F., Rupen, M.P., Beasley, A.J., Graham, D.A., Altunin, V.I., Venturi, T., Umana, G., Cannon, W.H., Conway, J.E.: Science 287, 112 (2000) Blondin, J.M., Wright, E.B., Borkowski, K.J., Reynolds, S.P.: Astrophys. J. 500, 342 (1998) Bouquet, S., Stehle, C., Koenig, M., Chieze, J.-P., Benuzzi-Mounaix, A., Batani, D., Leygnac, S., Fleury, X., Merdji, H., Michaut, C., Thais, F., Grandjouan, N., Hall, T., Henry, E., Malka, V., Lafon, J.-P.J.: Phys. Rev. Lett. 92, 225001-1 (2004)

inside RHW and compare to the initial estimate of T ∼ 3.8 eV obtained from our spectrometer data. We find that a temperature calculated from Equation 17 agrees quite well with the Springer

e0

References

As a final check one could calculate the temperature c22 . Ŵ

b

(16)

can be used to 15% accuracy.]

T =

1 × 10 3 × 10−32 5 × 10−32 8 × 10−32 1 × 10−31 1 × 10−44

a

6 Summary

energy

r2 ≈

−32

Astrophys Space Sci (2007) 307:219–225 Bozier, J.C., Thiell, G., LeBreton, J.P., Azra, S., Decroisette, M., Schirmann, D.: Phys. Rev. Lett. 57, 1304 (1986) Cohen, E., Piran, T., Sari, R.: Astrophys. J. 509, 717 (1998) Fleury, X., Bouquet, S., Stehle, C., Koenig, M., Batani, D., Benuzzi-Mounaix, A., Chieze, J.-P., Grandjouan, N., Grenier, J., Hall, T., Henry, E., Lafon, J.-P., Leygnac, S., Malka, V., Marchet, B., Merdji, H., Michaut, C., Thais, F.: Laser Part. Beams 20, 263 (2002) Hansen, J.F., Edwards, M.J., Froula, D., Gregori, G., Edens, A., Ditmire, T.: Astrophys. Space Sci. 298, 61 (2005) Hansen, J.F., Edwards, M.J., Froula, D.H., Gregori, G., Edens, A.D., Ditmire, T.: Phys. Plasmas 13, 1 (2006) Hatchett, S.P.: UCRL-JC-108348 Ablation Gas Dynamics of Low-Z Materials Illuminated by Soft X-Rays. Lawrence Livermore National Laboratory, California. Copies may be obtained from the National Technical Information Service, Springfield, VA 22161 (1991) Keiter, P.A., Drake, R.P., Perry, T.S., Robey, H.F., Remington, B.A., Iglesias, C.A., Wallace, R.J., Knauer, J.: Phys. Rev. Lett. 89, 165003-1 (2002) Klein, R.I., Woods, D.T.: Astrophys. J. 497, 777 (1998) Liang, E., Keilty, K.: Astrophys. J. 533, 890 (2000) McKee, C.F., Draine, B.T.: Science 252, 397 (1991) McKee, C.F., Ostriker, J.P.: Astrophys. J. 218, 148 (1977) Mihalas, D., Weibel Mihalas, B.: Foundations of Radiation Hydrodynamics. Oxford University Press, Oxford (1984)

225 M¨uller, E., Fryxell, B., Arnett, D.: Astron. Astrophys. 251, 505 (1991) Reed, J.I., Hester, J.J., Fabian, A.C., Winkler, P.F.: Astrophys. J. 440, 706 (1995) Reinicke, P., Meyer-ter-Vehn, J.: Phys. Fluids A 3, 1807 (1991) Remington, B.A., Arnett, D., Drake, R.P., Takabe, H.: Science 284, 1488 (1999) Robey, H.F., Kane, J.O., Remington, B.A., Drake, R.P., Hurricane, O.A., Louis, H., Wallace, R.J., Knauer, J., Keiter, P., Arnett, D., Ryutov, D.D.: Phys. Plasmas 8, 2446 (2001) Ryutov, D., Drake, R.P., Kane, J., Liang, E., Remington, B.A., Wood-Vasey, W.M.: Astrophys. J. 518, 821 (1999) Sedov, L.I.: Similarity and Dimensional Methods in Mechanics. Academic, New York (1959) Shigemori, K., Ditmire, T., Remington, B.A., Yanovksy, V., Ryutov, D., Estabrook, K.G., Edwards, M.J., MacKinnon, A.J., Rubenchik, A.M., Keilty, K.A., Liang, E.: Astrophys. J. 533, 159 (2000) Sonneborn, G., Pun, C.S.J., Kimble, R.A., Gull, T.R., Lundqvist, P., McCray, R., Plait, P., Boggess, A., Bowers, C.W., Danks, A.C., Grady, J., Heap, S.R., Kraemer, S., Lindler, D., Loiacono, J., Maran, S.P., Moos, H.W., Woodgate, B.E.: Astrophys. J. Lett. 492, L139 (1998) Taylor, G.I.: Proc. R. Soc. London A 201, 159 (1950) Zeldovich, Y.B., Raizer, Y.P.: Physics of Shock Waves and HighTemperature Hydrodynamic Phenomena. Academic, New York (1966)

Springer

Astrophys Space Sci (2007) 307:227–231 DOI 10.1007/s10509-006-9226-5

ORIGINAL ARTICLE

FLASH Code Simulations of Rayleigh-Taylor and Richtmyer-Meshkov Instabilities in Laser-Driven Experiments Nathan C. Hearn · Tomasz Plewa · R. Paul Drake · Carolyn Kuranz

Received: 15 April 2006 / Accepted: 25 July 2006 C Springer Science + Business Media B.V. 2006


Abstract We present two- and three-dimensional simulations involving Richtmyer–Meshkov and Rayleigh-Taylor instabilities run with the adaptive mesh refinement code, FLASH. Variations in the rate of mixing layer growth due to dimensionality, perturbation modes, and simulation resolution are explored. These simulations are designed for detailed comparisons with experiments run on the Omega laser to gain understanding of the mixing processes and to prepare for validation of the FLASH code. Keywords Adaptive mesh refinement . Rayleigh-Taylor . Richtmyer-Meshkov . Supernova . Code validation . Omega laser 1 Introduction Rayleigh-Taylor and Richtmyer-Meshkov instabilities are thought to play a critical role in the final distribution of elements, the spectra, and the light curve evolution of a supernova explosion (Kifonidis et al., 2006; Haas et al., 1990; Blinnikov, 1999). Therefore, the supernova calculations currently being run at the ASC Flash Center with its adaptive mesh refinement code, FLASH (Fryxell et al., 2000), must be able to model these processes accurately. To gain an understanding of the processes, and confidence in the models, we are constructing a set of FLASH simulations that can be directly compared with experiments of shock-induced mixing on the Omega laser. N. C. Hearn (
) · T. Plewa ASC Flash Center, University of Chicago e-mail: [email protected] R. P. Drake · C. Kuranz Space Physics Research Laboratory, University of Michigan

Such detailed comparisons are a critical part of FLASH validation process (see (Calder et al., 2002; Weirs et al., 2005)). As the Rayleigh-Taylor instability plays an important role in these experiments, this work may help to resolve discrepancies between experimentally measured and numerically estimated growth rates (see Dimonte et al. (2004)). Simulation studies of similar experiments have been performed using other codes (see, for instance, Miles et al. (2004)). The aim of these experiments is to study these mixing processes in a two-fluid medium (see the article by C. Kuranz et al., this issue; (Drake et al., 2004; Kane et al., 1997; Miles et al., 2004; Robey et al., 2001)). Here, a polyimide plastic and a lower-density foam are formed into a cylinder, such that the interface between the two media is planar (transverse to the cylinder axis) with sinusoidal perturbations, where one or more perturbation modes are present. A thin layer of polyimide (herein referred to as the “tube wall”) was also placed around the perimeter of the cylinder. In the experiment, a planar blast wave propagates across the interface from the polyimide into the foam. Here, the Richtmyer-Meshkov instability initiates the mixing, and the Rayleigh-Taylor instability continues it.

2 Methods and simulation setup The simulations presented here have dimensions compatible with those of the Omega laser experiments. Not including the ∼25 µm tube wall, the diameter of the cylinder is 900 µm, and the amplitude of each interface perturbation mode is 2.5 µm. The primary perturbation mode has a wavelength of 71 µm, applied along both transverse axes, with an additional 213 µm wavelength perturbation applied along one axis in the double-mode cases. These perturbations are applied in either two or three dimensions as described below. Springer

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Astrophys Space Sci (2007) 307:227–231

Fig. 1 A section of the mixing layer isosurface from a 3D double-mode simulation. The time relative to the start of the simulation is shown in the corner of each panel. The initial perturbation is shown in the upper left panel, and the subsequent panels show the shape of the mixing layer after the passage of the shock wave

The shape and magnitude of the initial pressure pulse was computed with a one-dimensional HYADES (Larsen and Lane, 1994) simulation that models the initial laser-driven shock propagation in the cylinder. The equation of state for these simulations is approximated by a ideal gas with an adiabatic index of 1.4. The effective adiabatic index remains below 5/3 because the polyimide is not fully ionized. The simulations discussed in this paper were run with adaptive mesh refinement at three different resolutions. With the lowest resolution (labeled L20), there are 256 elements across the diameter of the tube, or roughly 20 elements per wavelength of the primary perturbation. The intermediate resolution (L40) has 40 elements per wavelength, and the highest resolution (L80) has 80 elements per wavelength. Reflective boundary conditions are used for the sides of the cylinder, except for the case where the polyimide tube wall enclosing the cylinder is present (Fig. 3), in which an outflow boundary is used. This set of simulations was run with a uniform polyimide density distribution that does not include the enhanced density at the location of the pressure pulse. However, newer simulations that include the correct density profile do not show any qualitative differences in their evolution.

3 Dynamics overview Comparing the dynamics of 2D and 3D simulations is important for gaining an understanding of the growth of the instaSpringer

bilities. Simulations in three dimensions allow us to explore situations that are more like those found in the actual experiments and in supernovae. By working in a three-dimensional space, the fluid has an additional degree of freedom for shaping its flow. Abundance isosurfaces denoting equal polyimide and foam composition from a double-mode, 3D simulation are shown in Fig. 1 at four moments in time. The upper-left panel shows the initial perturbation before the shock has passed. Here, the interface has a 71 µm wavelength perturbation running in both directions, of the form cos y cos z, and a 213 µm wavelength perturbation along the y-axis only, of the form cos y. (In all cases, the x-axis is parallel to the axis of the cylinder.) After the shock crosses the interface, the spike-andbubble structures are produced, which are illustrated in the subsequent panels. The final time of this simulation (30 ns) will be the reference time used in all of the simulations presented in this paper.

4 Self-convergence in 2D The convergence properties of the simulations (with respect to resolution) are depicted in Fig. 2. Here, the mixing layer of a 2D, double-mode simulation is shown at the final time. The plotted variable is the mass fraction of the polyimide material, with the color-to-mass-fraction correspondence shown in the inset legend. The convergence of the mixing layer thickness with respect to resolution is illustrated in Fig. 6.

Astrophys Space Sci (2007) 307:227–231

229

Fig. 2 Polyimide mass fractions for the final times (30 ns) of three 2D simulations with the same perturbation (double mode), but different resolutions. From left to right are resolutions L20, L40, and L80. (See the inset legend for colors corresponding to polyimide mass fractions.) Note the increase of structure with resolution

Fig. 3 Evolution of a flat interface with and without a tube wall present. The top row of figures shows two times (10 and 30 ns) from a simulation with a flat (perturbation-free) interface, without a tube wall, but with reflective boundary conditions along the sides of the cylinder. The second row shows the same times from a simulation with a tube wall. In the presence of the tube wall, the interface shows some significant deformation near the edges, but remains reasonably flat in the center

5 Adding the tube wall We have implemented a simple model of the tube wall by expanding the computational domain and filling the additional volume with polyimide. In this trial, an additional 100 µm of polyimide material was added around the cylinder, with outflow boundary conditions applied to the outside of the tube wall. This tube is somewhat thicker than the real tube in the experiments, but this arrangement allows us to study the tube wall’s interaction with the interior media while avoiding the need to model the external vacuum. The initial pressure pulse was set in the tube wall in the same fashion as the rest of the material, resulting in a shock wave that is planar prior to crossing the interface. A comparison of the dynamics of a flat (unperturbed) interface in 2D with and without a tube wall at resolution L80 is shown in Fig. 3. Once the shock wave crosses the material interface, boundary effects are seen due to the different shock speeds in the foam and the polyimide tube walls. A number of important phenomena result from the presence of the tube:

Fig. 4 Comparison of polyimide mass fractions between single and double perturbation modes. The final time of a 2D simulation with the single-mode perturbation is shown in the left panel, and an image from the double-mode simulation at the same time is displayed to the right. The position of the mixing layer is the same, but the thickness of the mixing layer is 8% smaller in the single-mode case

1. 2.

A curved shock front (not visible here, but seen in density plots), Deposition of foam along the inner tube walls behind the interface,

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Astrophys Space Sci (2007) 307:227–231

Fig. 5 Polyimide mass fractions from two-dimensional slices through the three-dimensional simulations. The left and middle panels show the final times for 3D simulations with single and double mode perturbations, respectively. For reference, the image from a 2D double-mode

simulation at the same resolution is displayed at the right. The 3D simulations have a much thicker interface compared to their 2D counterparts (by roughly 40%), with spikes that are narrower

Fig. 6 The growth of the mixing layer for the various perturbation modes (single and double) and resolutions (L20, L40, and L80) in 2D. A trend towards faster growth with higher resolution is clear. Here we see evidence for convergence in the double-mode simulations, but more analysis is required for the single-mode simulations. For comparison, the preliminary growth curves for single and double mode experiments are shown

3. 4.

A transverse displacement of the tube walls behind the interface, and Deformation of the interface due to lateral motion and wave propagation along the interface.

At late times, the central regions of the interface tend to flatten out, which agrees with simulations of the perturbations and tube walls combined that have also been run. Only a small deviation in the position of the interface along the cylinder axis is seen. The other simulations shown in this paper do not include the tube walls, so their analysis can only focus on the properties of the mixing layer near the axis of the cylinder. 6 Single- versus double-modes The polyimide mass fractions for the final time of a singlemode 2D simulation is shown in the left panel of Fig. 4. For comparison, the final state of the 2D double-mode run of the Springer

same resolution (L80) is displayed to the right. The thickness of the mixing layer is about 9% larger in the double-mode simulation.

7 3D Models Two 3D simulations are shown in Fig. 5, one with a single perturbation mode (left panel), and one with the double mode (middle panel). For reference, a double mode 2D simulation at the same resolution (L20) is shown in the right panel. The double mode 3D simulation is the same as shown in Fig. 1, with the longer wavelength perturbation varying only along the y-axis. The slices shown in Fig. 5 are parallel to the x-z plane, at a y-coordinate where the amplitude for the short wavelength perturbation is at a maximum. Owing to the greater coordinate freedom, the mixing layer in the 3D

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Fig. 7 The growth of the mixing layer for the various perturbation modes (single and double) at resolution L20 in 3D. The faster growth with double-mode perturbations is seen here, as well as reasonably good agreement with the experimental data (see Fig. 6). The non-monotonic behavior at select points is likely due to inaccuracies in estimating the mixing layer thickness

runs is thicker, and features narrower, more-elongated structures compared with the 2D counterparts. These results are consistent with previous work on this subject (see Kane et al. (2000)). 8 Discussion We have explored the effects of resolution, dimensionality, and perturbation modes on the growth of the mixing layer for these simulations. We note three specific factors that can increase the thickness of the mixing layer: 1.

2. 3.

Increasing resolution, with some signs of convergence at higher resolutions (as seen in comparisons between L40 and L80; even higher resolutions may be needed for convergence in 3D), Increasing the number of perturbation modes, and Greater dimensionality (2D versus 3D).

Of these factors, it would appear that the last (dimensionality) has the greatest effect in this sample, with a roughly 40% increase in mixing layer thickness between the 2D and 3D L20 runs. As the resolution is increased in the 2D runs (from L20 to L80), the difference in mixing layer thickness between the single and double-mode runs decreases from about 25% to just under 10%.

Acknowledgements This work is supported in part by the U.S. Department of Energy under grant no. B523820.

References Blinnikov, S.I.: Astron. Lett. 25, 359 (1999) Calder, A.C., Fryxell, B., Plewa, T., et al.: ApJS 143, 201 (2002) Dimonte, G., Youngs, D.L., Dimits, A., et al.: Phys. Fluids 16, 1668 (2004) Drake, R.P., Leibrandt, D.R., Harding, D.R., et al.: Phys. Plasmas 11, 2829 (2004) Fryxell, B., Olson, K., Ricker, P., et al.: ApJS, 131, 273 (2000) Haas, M.R., Erickson, E.F., Lord, S.D., et al.: ApJ 360, 257 (1990) Kifonidis, K., Plewa, T., Scheck, L., Janka, H.-Th., M¨uller, E.: A&A 453, 661 (2006) Kane, J., Arnett, W.D., Remington, B.A., et al.: ApJ 478, L75 (1997) Kane, J., Arnett, W.D., Remington, B.A., et al.: ApJ 528, 989 (2000) Larsen, J.T., Lane, S.M.: J. Quant. Spectrosc. Radiat. Transfer 51, 179 (1994) Miles, A.R., Braun, D.G., Edwards, M.J., et al.: Phys. Plasmas 11, 3631 (2004) Miles, A.R., Edwards, M.J., Blue, B., et al.: Phys. Plasmas 11, 5507 (2004) Robey, H.F., Kane, J.O., Remington, B.A., et al.: Phys. Plasmas 8, 2446 (2001) Weirs, G., Dwarkadas, V., Plewa, T., Tomkins, C., Marr-Lyon, Mark.: Ap&SS 298, 341 (2005)

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Astrophys Space Sci (2007) 307:233–236 DOI 10.1007/s10509-006-9216-7

O R I G I NA L A RT I C L E

Models of Very-High-Energy Gamma-Ray Emission from the Jets of Microquasars: Orbital Modulation Markus B¨ottcher · Charles D. Dermer

Received: 14 April 2006 / Accepted: 12 July 2006 C Springer Science + Business Media B.V. 2006


Abstract The recent detection of very-high-energy (GeV – TeV) γ -ray emission from the Galactic black-hole candidate and microquasar LS 5039 has sparked renewed interest in jet models for the high-energy emission in those objects. In this work, we have focused on models in which the high-energy emission results from synchrotron and Compton emission by relativistic electrons in the jet (leptonic jet models). Particular attention has been paid to a possible orbital modulation of the high-energy emission due to azimuthal asymmetries caused by the presence of the companion star. Both orbitalphase dependent γ γ absorption and Compton scattering of optical/UV photons from the companion star may lead to an orbital modulation of the gamma-ray emission. We make specific predictions which should be testable with refined data from HESS and the upcoming GLAST mission. Keywords Gamma-rays: theory . Radiation mechanisms: non-thermal . X-rays: binaries . Stars: winds, outflows

1. Introduction Recent observations (Aharonian et al., 2005) of
250 GeV γ -rays with the High Energy Stereoscopic System (HESS) from the X-ray binary jet source LS 5039 establish that microquasars are a new class of γ -ray emitting sources. These results confirm the earlier tentative identification of LS 5039 with the EGRET source 3EG J1824-1514 (Paredes et al., M. B¨ottcher (
) Astrophysical Institute, Department of Physics and Astronomy, Ohio University, Athens, OH 45701, USA C. D. Dermer E. O. Hulburt Center for Space Research, Code 7653 Naval Research Laboratory, Washington, D.C. 20375-5352

2000). In addition to LS 5039, the high-mass X-ray binary LSI 61◦ 303 (V615 Cas) also has a possible γ -ray counterpart in the MeV – GeV energy range (Gregory and Taylor, 1978; Taylor et al., 1992; Kniffen et al., 1997). Microquasars now join blazar AGNs as a firmly established class of very-high energy γ -ray sources. The nonthermal continuum emission of blazars is believed to be produced in a relativistic plasma jet oriented at a small angle with respect to our line of sight. Their radio through UV/X-ray emission is most likely due to synchrotron emission by relativistic electrons in the jet, while the high energy emission can be produced by Compton upscattering of lower-energy photons off relativistic electrons (for a recent review, see, e.g. B¨ottcher, 2002), or through hadronic processes (Mannheim and Biermann, 1992; Atoyan and Dermer, 2001; M¨ucke et al., 2003). Because of their apparent similarity with their supermassive AGN cousins, it has been suggested that the same processes may operate in the jets of Galactic microquasars, which may thus also be promising sites of VHE γ -ray production (e.g., Romero et al., 2003; Bosch-Ramon et al., 2005). In the case of leptonic microquasar jet models, possible sources of soft seed photons for Compton upscattering are the synchrotron radiation produced in the jet by the same ultrarelativistic electron population (SSC = synchrotron selfCompton; Aharonian and Atoyan, 1999), or external photon fields (Bosch-Ramon and Paredes, 2004; Bosch-Ramon et al., 2005). In high-mass X-ray binaries like LS 5039, the 1– 10 keV luminosities from the accretion disk is typically much lower than the characteristic bolometric luminosity of the high-mass companions. Consequently, the dominant source of external photons is the companion’s optical/UV photon field. In addition to providing an orbital-period dependent seed photon field for Compton scattering, the intense radiation field of the high-mass companion will also lead to γ γ Springer

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absorption of VHE γ -rays in the ∼100 GeV – TeV photon energy range if VHE photons are produced close to the base of the jet. We (B¨ottcher and Dermer, 2005; Dermer and B¨ottcher, 2006) have recently presented detailed analyses of the expected high-energy spectrum and its orbital modulation due to the orbital-period dependence of both the soft photon field and the γ γ absorption. Here, we briefly summarize and discuss those results.

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2. Model description We choose a generic model set-up in which the orbital plane of the binary system defines the (x1 , x2 ) plane. The jet, assumed to be perpendicular to this plane, defines the x3 axis, and is inclined with respect to our line of sight by an inclination angle i. An azimuthal (phase) angle φ is defined such that φ = 0 in the direction of the x2 axis. The line of sight lies in the (x2 , x3 ) plane. The γ -ray production site is located at a height z 0 along the jet. Model parameters of the system have been motivated by recent observational results of Casares et al. (2005) for LS 5039: Mass of the compact object M X = 3.7+1.3 −1.0 M⊙ , orbital period P = 3.91 d, luminosity of the companion star L ∗ = 105.3 L ⊙ , effective temperature of the companion star Teff = 39, 000 ◦ K, inclination angle i = 25◦ , and an orbital separation of s ≈ 2.5 × 1012 cm. 3. Results VHE γ -ray photons in the range ∼100 GeV – 1 TeV will be efficiently absorbed by the intense photon field of the companion star. The inset to Fig. 1 illustrates the shape of the resulting absorption trough and its dependence on the orbital phase. Here we assume that the intrinsic γ -ray spectrum is a power-law with photon index αph = 2.5, and z 0 = 1012 cm. The various curves illustrate the orbital modulation of the absorption trough, with the lowest (most heavily absorbed) curve corresponding to φ = 0 and the highest (least absorbed) curve corresponding to φ = π . The modulation is a combined consequence of two effects: for phase angles closer to π , (a) the average distance of the star to any point on the line of sight is longer and (b) the angle of incidence θ is smaller, causing the threshold for γ γ pair production to increase as ǫthr = 2/(ǫ∗ [1 − µ]). This leads to a decreasing overall depth of the absorption trough, and a shift of the minimum of the absorption trough towards higher photon energies. The main frame of Fig. 1 shows the dependence of the absorption feature on the location z 0 of the VHE γ -ray production site. The γ γ opacity is plotted for two photon energies, E = 250 GeV, and E = 1 TeV at φ = 0. GeV – TeV Springer

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Fig. 1 γ γ opacity at 250 GeV and 1 TeV as a function of the distance of the photon production region from the central compact object at phase φ = 0. The figure illustrates that (1) VHE photons produced within a few ×1012 cm (i.e., of the order of the orbital separation of the binary system) would be subject to substantial γ γ absorption; (2) the minimum of the absorption trough (maximum of τγ γ as a function of photon energy) is shifting towards higher energies for larger distances from the central source. Inset: Orbital modulation of the expected γ γ absorption trough, assuming a power-law spectrum with photon index αph = 2.5 and a photon production site at z 0 = 1012 cm. The different curves represent the escaping photon spectrum at various orbital phases, from φ = 0 (lowest curve) to φ = π (highest curve) in steps of π/10

photons produced within z 0 ∼ s from the compact object will be heavily attenuated for this phase angle. For photons produced at z 0 ≫ s, γ γ attenuation becomes negligible. Our results concerning the effects of γ γ attenuation can briefly be summarized as follows: (1)

(2)

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VHE γ -rays produced closer to the central engine than z 0 of the order of the binary separation s, would be subject to very strong γ γ absorption due to the stellar radiation field at orbital phases close to φ = 0. For VHE photon production sites at z 0  s, the γ γ opacity – and, thus, the VHE γ -ray flux – would be strongly modulated on the orbital period of the binary system. At orbital phases close to φ = π , the intrinsic VHE γ -ray flux would still be virtually unabsorbed even for z 0 ∼ 1012 cm. The orbital modulation of the VHE γ -ray flux would be characterized by a spectral hardening in the ∼300 GeV – 1 TeV range during flux dips. At lower energies, the spectrum softens with decreasing flux, while the opposite trend would be observed at lower photon energies, E  100 GeV.

Besides γ γ opacity effects, an orbital modulation of the γ -ray flux is also expected from azimuthally asymmetric Compton scattering of companion-star photons in the

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E(eV) Fig. 2 The ν Fν spectral energy distribution from Compton scattering of starlight photons of a high-mas companion star in a microquasar jet. Standard parameters of the system, appropriate to LS 5039, have been chosen. In particular, the height of the emission region above the orbital plane is z 0 = 2.5 × 1012 cm, the jet plasma outflow Lorentz factor Ŵ = 2, and the number index of the electron distribution is p = 3. Broken lines indicate the spectra at various orbital phases, while the solid line shows the spectrum averaged over all phases

microquasar jet. This effect is illustrated in Fig. 2. Analogous to the phase-dependent modulation of the incidence angle for γ γ absorption, this geometrical effect would also yield a more favorable angle for Compton scattering of starlight photons into the γ -ray regime at phases φ ≈ 0. This effect leads to an overall softening throughout the GeV – TeV photon energy range with increasing flux because the Klein-Nishina cutoff becomes noticeable at lower observed photon energies for scattering events happening closer to head-on. This is in contrast to the trend caused by γ γ absorption, where a higher flux will be accompanied by a harder spectrum at photon energies ∼100 GeV. Figures 3 and 4 illustrate the dependence of the orbital modulation of the starlight Compton-scattering spectrum on the height z 0 of the emission region. Both figures indicate a peculiarity in that the orbital modulation is particularly strong under a specific orientation of the system, in which at phase φ = π the companion star would appear directly behind the emission region as seen by the observer. In that case, Compton scattering events producing γ -rays in the direction of the observer, would have to happen almost tail-on – a geometry which is highly unfavorable for the efficiency of Compton scattering. Consequently, under this special orientation, the starlight-Compton-scattering spectrum in the direction towards the observer would be severely suppressed, and the orbital modulation is maximized. Acknowledgement This work was partially supported by NASA through XMM-Newton GO grant no. NNG 04GI50G, NASA INGE-

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References Aharonian, F., et al.: Sci. 309, 746 (2005) Aharonian, F., Aharonian, A.: MNRAS 302, 253 (1999) Atoyan, A., Dermer, C.D.: Phys. Rev. Lett. 87, 221102 (2001) B¨ottcher, M.: In: Goldwurm, A., Neumann. D.N., Vˆan, J.T.T. (eds.) The Gamma-Ray Universe, Proceedings of the XXII Moriond Astrophysics Meeting, p. 151 (2002) B¨ottcher, M., Dermer, C.D.: A&A 634, L81 (2005)

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236 Bosch-Ramon, V., Paredes, J.M.: A&A 417, 1075 (2004) Bosch-Ramon, V., Romero, G.E., Paredes, J.M.: A&A 429, 267 (2005a) Casares, J., Rib´o, M., Ribas, I., Paredes, J.M., et al.: MNRAS 364, 899 (2005) Dermer, C.D., B¨ottcher, M.: A&A 643, in press (2006) Gregory, P.C., Taylor, A.R.: Nature 272, 704 (1978) Kniffen, D.A., et al.: ApJ 486, 126 (1997)

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Astrophys Space Sci (2007) 307:233–236 Mannheim, K., Biermann, P.L., A&A 253, L21 (1992) M¨ucke, A., et al.: Astropart. Phys. 18, 593 (2003) Paredes, J.M., Mart´ı, J., Rib´o, M., Massi, M.: Sci. 288, 2340 (2000) Romero, G.E., Torres, D.F., Kaufman Bernad´o, M.M., Mirabel, I.F.: A&A 410, L1 (2003) Taylor, A.R., Kenny, H.T., Spencer, R.E., Tzioumis, A.: ApJ 395, 268 (1992)

Astrophys Space Sci (2007) 307:237–240 DOI 10.1007/s10509-006-9241-6

ORIGINAL ARTICLE

Time-Dependent Synchrotron and Compton Spectra from Microquasar Jets S. Gupta · M. B¨ottcher

Received: 13 April 2006 / Accepted: 22 August 2006 C Springer Science + Business Media B.V. 2006


Abstract Jet models for the high-energy emission of Galactic X-ray binary sources have regained significant interest with detailed spectral and timing studies of the X-ray emission from microquasars, the recent detection by the HESS collaboration of very-high-energy γ -rays from the microquasar LS 5039, and the earlier suggestion of jet models for ultraluminous X-ray sources observed in many nearby galaxies. Here we study the synchrotron and Compton signatures of time-dependent electron injection and acceleration, and adiabatic and radiative cooling in the jets of Galactic microquasars. Keywords Gamma-rays: theory . Radiation mechanisms: non-thermal . X-rays: binaries

1 Introduction In many sources, the accretion process associated with high energy emission from X-ray binaries (XRBs) is coupled with the expulsion of collimated, mildly relativistic bipolar outflows (jets) most likely perpendicular to the accretion disk. In the standard picture, the high-energy (X-ray – γ -ray) spectra of X-ray binaries generally consist of a soft disk blackbody with a typical temperature of kT ∼ 1 keV, and a power-law at higher energies. Neutron-star and black-hole X-ray binaries exhibit at least two main classes of spectral states, generally referred to as the high/soft state (H/S), and the low/hard state (L/H) (for a review see, e.g. Liang, 1998; S. Gupta (
) · M. B¨ottcher Astrophysical Institute, Department of Physics and Astronomy, Clippinger Hall 251B, Ohio University, Athens, OH 45701–2979, USA e-mail: [email protected]

McClintock and Remillard, 2004). Additionally, the Very High (VHS) (Miyamoto et al., 1991) and Intermediate (IS) (M´endez and van der Klis, 1997) states have been identified. In the L/H state, microquasars exhibit a continuously generated, partially self absorbed compact jet, with its radio luminosity showing strong, non-linear correlation with X-ray luminosity. No radio emitting outflow is associated with the H/S state, whereas discrete, often mutiple ejections attributed to unstable disk radius, are associated with the VHS and IS. Jet models of microquasars have recently attracted great interest, especially after the detection of VHE γ -ray emission from the high-mass X-ray binary and microquasar LS 5039, in combination with the tentative identification of several microquasars with unidentified EGRET sources. These detections have confirmed the idea that microquasars are a distinctive class of high and very high energy γ -ray sources. In a recent paper (Gupta et al., 2006), we presented a detailed study of various plausible scenarios of electron injection and acceleration into a relativistically moving emission region in a microquasar jet, and subsequent adiabatic and radiative cooling, where we paid particular attention to the X-ray spectral variability, as motivated above. Here we briefly summarise the key results from our detailed parameter study presented in (Gupta et al., 2006).

2 Model setup The accretion flow onto the central compact object is ejecting a twin pair of jets, directed at an angle θ with respect to the line of sight. Two intrinsically identical disturbances, containing non-thermal plasma (blobs) originate from the central source at the same time, traveling in opposite directions along the jet at a constant speed v j = β j c. Let d be the distance to the source, and µ ≡ cos θ. The time at which Springer

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any radio component is observed, is denoted by tobs . We assume that over a limited range in distance x0 ≤ x ≤ x1 , relativistic electrons are accelerated and injected in the emission region. The blob’s (transverse) radius, R⊥ , scales with distance 0 from the central engine as R⊥ = R⊥ (x/x0 )a , i.e., a = 0 corresponds to perfect collimation, and a = 1 describes a conical jet. In the following, we will consider values of 0 ≤ a ≤ 1. If the magnetic field is dominated by the parallel component B , magnetic flux conservation leads to a magneticfield dependence on distance from the central black hole of 0 −2 B(x) = B0 (R⊥ /R⊥ ) = B0 (x/x0 )−2a . Each electron injected into the emission region at relativistic energies, will be subject to adiabatic and radiative cooling, described by −

dγ u 1 d VB γ 4 = + c σT γ2 dt VB dt 3 3 m e c2

(1)

in the co-moving frame of the blob, where the first term on the r.h.s. describes the adiabatic losses. The second term describes synchrotron and Compton losses, with u = u B + u rad , where u B = B 2 /8π is the magnetic-field energy density, and u rad is the seed photon energy density for Compton scattering in the Thomson regime. The term u rad consists of contributions from the X-ray emission from an optically thick accretion disk, from the intrinsic synchrotron radiation, and from external photons from the companion star. The standard parameter choices for our “baseline” model, summarized in Table 1, are broadly representative of GRS 1915+105 in the low/hard state, which also give equipartition between the energy densities of the relativistic electrons and the magnetic field in the ejecta. Table 1 Parameter choices for our baseline model

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Once a solution γi (ti ; γ , t) to Eq. (1) is found, the electron distribution at any point in time (and thus at any point along the jet) can be calculated through the expression

Ne (γ ; t) =



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Radiation mechanisms included in our simulations are synchrotron emission, Compton upscattering of synchrotron photons, namely synchrotron self-Compton (SSC) emission, and Compton upscattering of external photons. With the time-dependent (and thus x-dependent) non-thermal electron spectra in Eq. (2), we then use a δ-function approximation sy (to estimate the ν Fν spectral output f ǫ at a dimensionless photon energy ǫ = hν/m e c2 (in the observer’s frame). The δ function approximation here indicates that electrons at any particular energy γ emit only at a certain energy ǫ: u B σT c n˙ syn (ǫs , s ) = 3π ǫcomov



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(4)

Dermer et al. (1997), where D = [Ŵ j (1 − β j µ)]−1 is the Doppler boosting factor, ǫ B ≡ B/Bcr with Bcr = m 2 c3 / he = 4.414 × 1013 G, defined as the field at which the cyclotron quantum number equals the rest mass energy of the electron, ∗ and tobs is the observer time, so that tobs = t/D = ta,r /DŴ j .

Parameter

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Black-hole mass Distance Jet inclination angle Bulk Lorentz factor Binary separation Luminosity of companion star: Surface temperature of the companion star Initial blob radius Jet collimation parameter Accretion Fraction Accretion disk luminosity Electron injection spectrum, low-energy cutoff Electron injection spectrum, high-energy cutoff Electron injection spectrum, spectral index Beginning of electron injection zone End of electron injection zone Magnetic field at x0 Injection luminosity

M d θjet Ŵj s∗ L∗ T∗ R0 a ˙ M˙ Edd m˙ = M/ LD γmin γmax q x0 x1 B0 L inj

15 M⊙ 3.75 × 1022 cm 70◦ 2.5 1012 cm 8 × 1037 ergs s−1 3 × 104 K 103 Rg 0.3 0.01 1.9 × 1037 ergs s−1 10 104 2.4 103 Rg 105 Rg 5 × 103 G 4.4 × 10−5 L Edd

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approximations used for some of our spectral calculations. The respective magnetic fields are indicated in the legend; the other parameters are the baseline model values discussed in the text and listed in Table 1. The vertical lines in panel (a) indicate the photon energies at which the light curves in panel (b) were extracted

For more details, see Gupta et al. (2006). For the timedependent ν Fν spectral output f ǫEC due to Compton upscattering of external photons from the star and the disk, we use the Thomson approximation, represent the star as a monochromatic point source, the disk as a thin annulus at the radius Rmax where the differential energy output is maximized, and use the optically thick, geometrically thin, gas-pressure dominated accretion disk solution of Shakura and Sunyaev (1973).

As an example, we show in Fig.1, the results for the case of varying the initial magnetic field B0 , where we focus on the time-averaged photon spectra, light curves, and X-ray HIDs, and explore the effect of variations of B0 on these aspects. The detailed study of the whole set of parameters can be found in (Gupta et al., 2006). From this parameter study, one could draw the following general conclusions:

r A sudden increase of a light curve slope at a fixed observing 3 Results A large number of simulations have been performed to study the effects of the various model parameters on the resulting broadband spectra, light curves, and X-ray hardness intensity diagrams (HIDs). We started our parameter study with a baseline model for which we used the standard model parameters broadly representative of GRS 1915+105. Subsequently, we investigated the departure from this standard set-up by varying (1) the initial magnetic field B0 , (2) the luminosity of the companion star L ∗ , (3) the injection electron spectral index q, (4) the low-energy cutoff γ1 of the electron injection spectrum, (5) the high-energy cutoff γ2 of the electron injection spectrum, (6) the injection luminosity L inj , (7) and the observing angle θobs and thus the Doppler boosting factor.

r

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frequency, not accompanied by significant flaring activity at other wavelengths, usually indicates the passing of a new spectral component through the fixed observing frequency range. Most notably, this diagnostic can be used to investigate the presence of one or more external-Compton component(s) in the X-ray/soft γ -ray regime. Clockwise spectral hysteresis in the hardness-intensity diagrams indicates the dominance of synchrotron emission (in particular, before the end of the injection period in our generic model setup). In this case, the frequency-dependent light curve decay will be a useful diagnostic of the magnetic field strength in the jet. Counterclockwise spectral hysteresis in the hardnessintensity diagrams indicates the dominance of Compton emission (similar to the case of blazars, see, e.g., B¨ottcher and Chiang, 2002).

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r In our study, we found, quite often, a co-existence of clockwise and counterclockwise X-ray hysteresis loops, which would provide a particularly powerful diagnostic, as it would allow to probe the characteristic transition energy between synchrotron and Compton emission, and its time dependence. Various spectral components (synchrotron, SSC, externalCompton) could be easily distinguished if detailed snapshot SEDs could be measured for microquasars, on the (often subsecond) time scales of their X-ray variability. Unfortunately, such detailed snapshot broadband spectra are currently not available, and might not be available in the near future. Therefore, we have exposed several other features pertinent to the transition between different spectral components which will be more easily observable in realistic observational data of microquasars. We conclude therefore that the X-ray variability as predicted by our model can be used as a diagnostic to gain insight into the nature of the high energy emission in microquasar jets. In particular, a transition between clockwise and counter-clockwise spectral hysteresis would allow not

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only the distinction between different emission components, but also parameters such as the magnetic field, the Doppler boosting factor, and the characteristic electron injection / acceleration time. Acknowledgements The work of S.G. and M.B. was supported by NASA through XMM-Newton GO grant no. NNG04GI50G and INTEGRAL theory grant NNG05GK59G.

References B¨ottcher, M., Chiang, J.: ApJ 581, 127 (2002) Dermer, C.D., Sturner, S.J., Schlickeiser, R.: ApJS 109, 103 (1997) Gupta, S., B¨ottcher, M., Dermer, C.D.: astro-ph/0602439 (2006) Liang, E.P.: Phys. Rep. 302, 67 (1998) McClintock, J.E., Remillard, R.A.: astro-ph/0306213 (2004) M´endez, M., van der Klis, M.: ApJ 479, 926 (1997) Miyamoto, S., Kimura, K., Kitamoto, S., Dotani, T., Ebisawa, K.: ApJ 383, 784M (1991) Shakura, N.I., Sunyaev, R.A.: A&A 24, 337 (1973) Tavecchio, F., Maraschi, L., Ghisellini, G.: ApJ 509, 608 (1998) Mirabel, I.F., Rodr´ıguez, L.F.: Nature 371, 46 (1994)

Astrophys Space Sci (2007) 307:241–244 DOI 10.1007/s10509-006-9289-3

O R I G I NA L A RT I C L E

New Experimental Platform for Studies of Turbulence and Turbulent Mixing in Accelerating and Rotating Fluids at High Reynolds Numbers Sergei S. Orlov · Snezhana I. Abarzhi

Received: 25 July 2006 / Accepted: 8 December 2006 C Springer Science + Business Media B.V. 2007


Abstract We present a new experimental platform for studies of turbulence and turbulent mixing in accelerating and rotating fluids. The technology is based on the ultra-high performance optical holographic digital data storage. The state-ofthe-art electro-mechanical, electronic, and laser components allow for realization of turbulent flows with high Reynolds number (>107 ) in a relatively small form-factor, and quantification of their properties with extremely high spatio-temporal resolutions and high data acquisition rates. The technology can be applied for investigation of a large variety of hydrodynamic problems including the fundamental properties of nonKolmogorov turbulence and turbulent mixing in accelerating, rotating and multiphase flows, magneto-hydrodynamics, and laboratory astrophysics. Unique experimental and metrological capabilities enable the studies of spatial and temporal properties of the transports of momentum, angular momentum, and energy and the identification of scalings, invariants, and statistical properties of these complex turbulent flows.

Keywords Turbulent mixing and turbulence . Accelerating and rotating fluids . Multiphase and reactive flows . Shocks . Supernova . Accretion disc

S. S. Orlov (
) Stanford University, Stanford, CA, USA e-mail: [email protected] S. I. Abarzhi The University of Chicago, Chicago, IL, USA e-mail: [email protected]

1 Introduction Turbulence and turbulent mixing in accelerating and rotating fluids play a key role in a wide variety of astrophysical phenomena. Stellar interiors and solar non-Boussinesq convection, Rayleigh-Taylor (RT) and Richtmyer-Meshkov instabilities in explosions of supernova type Ia and II, accretion and proto-stellar disks, magneto-hydrodynamic and dynamo, formation of planets and stars are to list a few. In many of these cases the acceleration is spatially non-uniform and time-varying, the rotation is spatially varying, and the flow is highly anisotropic and multiphase. Its statistical, spectral, and scaling properties differ substantially from those of isotropic Kolmogorov turbulence (Abarzhi et al., 2005; Baroud et al., 2002; Frisch, 1995). Theoretical description of the turbulent flows subjected to acceleration and rotation remains one of the most challenging problems in hydrodynamic theory, whose solution requires innovative ideas and approaches. The computational treatments of the problem are met with tremendous difficulties as the numerical solutions appear to be very sensitive to the initial and boundary conditions as well as to the influence of small-scale structures on the turbulent dynamics and anomalous character of energy transport (Calder et al., 2002). On experimental side, flows with high rates of acceleration and rotation and high Reynolds numbers (e. g., >107 ) are not only difficult to implement in a well-controlled laboratory environment, but also very hard to quantify and measure with sufficient accuracy and spatio-temporal resolution due to the limitations of diagnostics and metrological tools currently available (Adrian, 2005; Frisch, 1995). Recent theoretical studies (Abarzhi et al., 2005) have suggested that the rate of momentum loss can be a better indicator of accelerated turbulent flows than the rate of energy dissipation, which is the basic quantity of isotropic Kolmogorov Springer

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turbulence (Frisch, 1995). Energy is complimentary to time, momentum is complimentary to space, and spatial distributions of the flow quantities are important to monitor for a reliable description of non-Kolmogorov turbulent processes, with the analysis involving the statistical properties of the transports of mass, momentum (angular momentum, if applicable) and energy. These statistical characteristics are quite difficult to obtain from the experimental diagnostics currently used, as the most of the existing fluid dynamics experiments perform measurements at one (or, a few) points and focus on the temporal dependencies of the velocity fluctuations and energy transport (Adrian, 2005; Frisch, 1995). To quantify and better understand the non-Kolmogorov turbulence and multi-phase turbulent mixing in accelerating and rotating fluids, new experimental approaches, capable of providing higher accuracy spatial and temporal measurements and adequate statistics, are needed.

Astrophys Space Sci (2007) 307:241–244

The goal of our experiments in flow visualization is to perform the fully resolved measurements with high accuracy and good statistics of the spatial and temporal distributions of the turbulent flow quantities in a well controlled laboratory environment. An advanced technology, which can be leveraged to further the experimental capabilities, is the digital holographic data storage (HDS). Coufal et al. (2000) provides a detailed review of the technology, its historical developments, and the state-of-the-art. One of the unique features of the HDS is the capability to image and to capture with high-resolution (106 pixels or more) and extremely low distortions the optical images at a very high transfer rate and speed (up to 104 frames-per-second). In data images used in HDS (see Fig. 1), each pixel is treated as a unique independent data channel, and the spatial position of each pixel

is controlled with submicron accuracy throughout the entire image, whereas the digitized pixels values are measured with high (8 bits or higher) precision, providing extremely high spatial resolution as well as signal-to-noise ratio over the entire spatial extent of the imaged domain. A state-of-the-art example of digital holographic data storage technology is an experimental facility built at Stanford University for the DARPA Holographic Data Storage Systems consortium (Orlov et al., 2004). This platform incorporates the advanced mechanical, optical, imaging, and programmable electronic components including the highprecision high-speed air-bearing spindle device, pulsed frequency doubled (λ = 532 nm) Nd:YAG lasers, dedicated synchronization electronics for precision timing and signals generation, 1000 frames-per-second 1 Mpixel digital camera, high resolution optical imaging system, and advanced automation. The experimental and metrological capabilities of the HDS technology can be employed to investigate the turbulent flows (including the reactive, compressible, and multiphase flows) subjected to accelerations and rotation. For the flow visualization, the Particle Imaging Velocimetry (PIV) approach (Adrian, 2005) can be employed and significantly improved by leveraging the high-resolution optical imaging capabilities of the HDS (see Fig. 2). Spatial (∼1 µm over 10 cm area) and temporal (<5 ns) resolutions of the HDS system allow for mapping the flow velocity fields and capturing the fast events with extreme accuracy and high data rate acquisition (>1000 flow images per second) providing thus improved statistical evaluation of the turbulent flow quantities (see Table 1 for the characteristic values of the system parameters). In order to provide simultaneous mapping of the velocity field and density fluctuations in unsteady turbulent flows, the PIV setup can be combined with an imaging interferometer (compressible flows, Fig. 2) and PLIF diagnostics (for multiphase flows). The high rotation rate (up to 250 Hz) can result

Fig. 1 (Left) A high-resolution (1024 × 1024 pixels; 13.1 × 13.1 mm) holographic image captured at 1000 frames-per-second (Orlov et al., 2004); the octagonal shape and “voids” were imprinted during the signal

encoding; (Middle) same image after digital thresholding; (Right) enlarged fragment. Bright area looks like a complicated “maze” of white pixels due to the spatial information coding used in HDS

2 New technology for advanced diagnostics

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Table 1 Experimental and metrological capabilities of HDS platforms Rotation rate Achievable Reynolds number Acceleration Radial Tangential Angular acceleration Radial stability Optical spatial resolution Temporal resolution Chamber size Height Diameter CMOS camera resolutions Frame rate Camera imaging data Interferometric density fluctuation measurements

Upto 15,000 RPM >107

Stability: ±0.001%

Upto 5 × 105 m/s2 (50,000 g) (∼2 g) 0–1000 RPM/s <10 nm ∼1 micron <5 ns

At 15,000 RPM At 1000 RPM/s Digitally controlled Radial air-bearing Over 10 cm field Jitter <100 ps

Up to 30 cm Up to 40 cm 103 ×103 pixels or more 103 (upto 104 ) fps Up to 24 GB per experiment Sensitivity: n L/λ ∼ 0.1 ∼100 × 100 points per frame

10 bit grayscale ∼1 Gpixel bandwidth ∼2 × 104 of 1 Mpixel images

Fig. 2 Schematic of the PIV setup for flow visualization combined with an imaging interferometer for the simultaneous, correlated, measurements of the velocity field and the density fluctuations in compressible fluids

in the radial accelerations of up to 50,000 g for sustained periods of time (tens of seconds), which, in combination with high mechanical stability of the air-bearing suspension (<10 nm spatial run-out), enables the high-accuracy experimental studies of Rayleigh-Taylor instability. Compared to linear accelerations obtained using, for instance, Linear Electric Motor (Dimonte et al., 2000), rotational arrangement allows one to produce high accelerations in a relatively small form-factor for a prolonged period of time and to study very late stages of the mixing process. Our experimental approach can provide the repeatability of the experimental conditions, detailed diagnostics, and the ability either to control the stabilizing/destabilizing factors or to measure their influences

with high accuracy. For instance, in experiments on RTI with high acceleration, one of the mixing fluids can be a jelly-like medium. As the value of acceleration induced by centrifugal force exceeds the visco-elastic stress, the jelly acts as a liquid-like medium (Meshkov et al., 1999). The initial perturbation in this case can be embossed mechanically on the jelly surface using pre-designed plastic forms, whereas the time-history of the rotation and acceleration can be controlled via programmable electronic components, ensuring thus the repeatability of the experimental conditions. In addition to tracking of the interface and its dynamics, the simultaneous PLIF and PIV measurements of the velocity fields in the bulk of the mixing fluids will allow to quantify the basic statistics

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of the RT turbulent mixing (Abarzhi et al., 2005). The conditions of high accelerations and the requirements for the high spatial resolution (∼1 µm) both demand the optimization of buoyancy and size of the imaging particle (for accurate flow tracking) as well as the matching of refractive indexes of the mixing fluids (McDougall, 1979) to minimize optical distortions in order to produce reliable high-spatial resolution data. The new diagnostics, high spatial and temporal resolutions, as well as the capability to collect and store large imaging data sets of tens of GBytes in a single experiment will enable studies of many of the fluid dynamics phenomena with accuracy and precision which has not been achieved or demonstrated so far. As a future outlook, using precision mechanical capabilities, in addition to RTI and multiphase turbulent mixing, the magneto-hydrodynamics (MHD) and turbulent dynamo experiments could be conducted under the conditions of very stable (or time varying with precise control) and accurate (< ± 0.001% variation) high-speed rotation by utilizing conducting and magnetic liquids (e. g., liquid sodium or liquid gallium) placed in a spherical or cylindrical chamber. 3 Phenomenology and validation Many of astrophysical objects (e.g., accretion disks, stellar and planetary interiors) undergo turbulence, accelerated turbulence mixing and rotation. Capturing scaling properties of rotating and accelerating turbulent flows is important for better understanding of the astrophysical phenomena. Recent studies (Abarzhi et al., 2005) suggest that the basic invariant properties of the accelerated turbulent mixing differ substantially from those of isotropic Kolmogorov turbulence. In particular, the rate of momentum loss is the fundamental quantity of accelerated turbulent mixing, similarly to the energy dissipation rate in isotropic Kolmogorov turbulence. The invariance of the rate of momentum loss can be employed further in studies of scaling and spectral properties. One can show, for instance, that in accelerating turbulent flow, the velocity scales as square root of length scale and the spectrum of kinetic energy (velocity fluctuations) is proportional to ∼k −2 , compared to the power 1/3 for velocity scale and ∼k −5/3 for velocity spectrum in classical Kolmogorov turbulence. These quantitative differences are sufficiently large to be distinguished in the high-precision experiments discussed in the foregoing. Another important application of our laboratory experiments is the validation of hydrodynamic simulations. Advanced astrophysical modeling platforms (such as the FLASH code developed at the University of Chicago, see

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Astrophys Space Sci (2007) 307:241–244

Calder et al., 2002) produce numerical data, which are highly resolved in space. These data are however rather difficult to compare directly to existing experiments, which provide mostly the temporal statistics at one (or just a few) spatial position(s). Higher accuracy spatial and temporal experimental quantification of the model hydrodynamic problems (such as multiphase turbulent mixing induced by Rayleigh-Taylor instability, interface evolution in shock-driven RichtmyerMeshkov instability, or decaying turbulence in rotating fluids) may significantly advance the code validation and, hence, reliability of the hydrodynamic modeling of the complex turbulent flows in astrophysical applications.

4 Summary The unique capabilities developed in digital holographic storage technology can be applied for advanced metrology and high-resolution laboratory studies of a broad variety of hydrodynamic phenomena relevant to astrophysical problems. High resolution measurements and quantitative characterizations with good statistics are important for understanding the basic invariant, scaling, spectral, and statistical properties of complex turbulent flows, including rotating, accelerating and multi-phase flows, and, particularly, turbulent mixing induced by Rayleigh-Taylor instability. High resolution measurements of these phenomena performed in a wellcontrolled laboratory environment can be applied for validation of hydrodynamic simulation platforms and for modeling of astrophysical phenomena.

References Abarzhi, S.I., Gorobets, A., Sreenivasan, K.R.: Phys. Fluids 17, 081705 (2005) Adrian, R.J.: Exp. Fluids 39, 159 (2005) Baroud, C.N., Plapp, B.B., She, Z.-S., Swinney, H.L.: Phys. Rev. Lett. 88, 114501 (2002) Calder, A.C., Fryxell, B., Plewa, T., Rosner, R., et al.: ApJS 143, 201 (2002) Coufal, H.J., Psaltis, D., Sincerbox, G.T. (eds.): Holographic Data Storage. Springer-Verlag, Springer Series in Optical Sciences, Berlin (2000) Dimonte, G., Schneider, M.: Phys. Fluids 12, 304 (2000) Frisch, U.: Turbulence, the Legacy of A. N. Kolmogorov. Cambridge University Press, Cambridge, UK (1995) McDougall, T.J.: J. Fluid Mech. 93, 83 (1979) Meshkov, E.E., Nevmerzhitsky, N.V., Sotskov, E.A.: Comparative study of evolution of periodical perturbations in liquid and solids using jellies. In: Meshkov, E.E., Yanilkin, Yu., Zhmailo, V. (eds.), Proceedings of the International Workshop Phys. Compressible Turbulent Mixing (1999) Orlov, S.S., Phillips, W., Bjornson, E., et al.: Appl. Opt. 43, 4902 (2004)

Astrophys Space Sci (2007) 307:245–250 DOI 10.1007/s10509-006-9288-4

O R I G I NA L A RT I C L E

Weibel Turbulence in Laboratory Experiments and GRB/SN Shocks Mikhail V. Medvedev

Received: 14 April 2006 / Accepted: 8 December 2006 C Springer Science + Business Media B.V. 2007


Abstract It has recently been realized that the Weibel instability plays a major role in the formation and dynamics of astrophysical shocks of gamma-ray bursts and supernovae. Thanks to technological advances in the recent years, experimental studies of the Weibel instability are now possible in laser-plasma interaction devices. We, thus, have a unique opportunity to model and study astrophysical conditions in laboratory experiments – a key goal of the Laboratory Astrophysics program. Here we briefly review the theory of strong non-magnetized collisionless GRB and SN shocks, emphasizing the crucial role of the Weibel instability and discuss the properties of radiation emitted by (isotropic) electrons moving through the Weibel-generated magnetic fields, which is referred to as the jitter radiation. We demonstrate that the jitter radiation field is anisotropic with respect to the direction of the Weibel current filaments and that its spectral and polarization characteristics are determined by microphysical plasma parameters. We stress that the spectral analysis can be used for accurate diagnostics of the plasma conditions in laboratory experiments and in astrophysical GRB and SN shocks.

and in active galactic nuclei jets, shocks in supernovae (SN) remnants, merger shocks in galaxy clusters and large scale structure (LSS), – all of them represent a single class of strong collisionless shocks. The theoretical prediction that the Weibel instability operating at the shock produces strong magnetic fields (Medvedev and Loeb, 1999) has recently been confirmed in a number of state-of-the-art numerical sumulations (Silva et al., 2003; Frederiksen et al., 2004; Nishikawa et al., 2003; Saito and Sakai, 2004; Kazimura et al., 1998). It has also been predicted that radiation of relativisitc electrons from Weilbel-generated magnetic fields, referred to as the “jitter radiation” has distinct spectral properties (Medvedev, 2000). The Weibel instability, its dynamics and properties, can now be studied in laboratory experiments od laser-plasma interactions (and even relevant to the fast ignition concept). We can, thus, directly probe conditions of distant astrophysical environments in the lab.

Keywords Weibel instability . Plasma physics . Radiation from plasma . Plasma diagnostics . Plasma astrophysics . Collisionless shocks

In general, shocks with the Mach number greater than three, must be highly turbulent. The source and the mechanism of the turbulence is thought to be of kinetic nature, in order to prevent multi-stream motion of plasma particles. It has been shown that the Weibel instability operates at the shock front (Moiseev and Sagdeev, 1963; Medvedev and Loeb, 1999). This instability is driven by the anisotropy of the particle distribution function (PDF) associated with a large number of particles reflected from the shock potential.

1 Introduction Internal and external shocks in gamma-ray bursters (GRBs), internal shocks produced in jets of micro- and normal quasars M. V. Medvedev Department of Physics and Astronomy, University of Kansas, KS 66045, USA e-mail: [email protected]

2 The Weibel mechanism

2.1 Linear regime of field growth and its saturation The instability under consideration was first predicted by Weibel (1959) for a non-relativistic plasma with an Springer

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and deflects their trajectories, as is shown in Fig. 1. As a result, the protons moving upward and those moving downward will concentrate in spatially separated current filaments. The magnetic field of these filaments appears to increase the initial magnetic field fluctuation. The growth rate and the wavenumber of the fastest growing mode (which, in fact, sets the spatial correlation scale of the produced field) are γ B = A ω p,s (vz /c), Fig. 1 Illustration of various stages of the Weibel instability. Color coding of particles: blue – the incoming particles from the IGM, red – the particles scattered from the shock

anisotropic distribution function. The simple physical interpretation provided later by Fried (1959) treated the PDF anisotropy more generally as a two-stream configuration of cold plasma. Below we give a brief, qualitative description of this two-stream magnetic instability. Let us consider, for simplicity, the dynamics of one species only (e.g., protons), whereas the other is assumed to provide global charge neutrality.1 The electrons are assumed to move along the z-axis with the velocities v = +ˆz vz and v = −ˆz vz , thus forming equal particle fluxes in opposite directions (so that the net current is zero). Such a particle distribution occurs naturally near the front of a shock (moving along z-direction), where the “incoming” (in the shock frame) ambient gas particles meet the “outgoing” particles reflected from the shock potential (loosely speaking, the low energy cosmic rays). Thus, the particle velocities vz are of order the shock velocity, vz ∼ vsh . The counter-streaming particles may also have some thermal spread. Since for high-Mach number shocks, vth ≪ vsh in the upstream region, we may neglect the parallel velocity spread in our consideration. The thermal spread in the transverse direction cannot be neglected, however. We parameterize the PDF anisotropy as follows: A=

ǫ − ǫ⊥ M2 − 1 ≃ 2 , ǫtot M +1

(1)

2 is the energy of particle along zwhere ǫ ∝ vz2 ≃ vsh 2 2 direction, ǫ⊥ ∝ (vx + v 2y ) ∝ vthermal ≃ cs2 is the thermal energy in the transverse direction, ǫtot = ǫ + ǫ⊥ is the total energy, cs is the sound speed upstream and the Mach number of the shock is M = vsh /cs . Clearly, for strong shocks M ≫ 1, the anisotropy parameter is close to unity, A ∼ 1. Next, according to the linear stability analysis technique, we add an infinitesimal magnetic field fluctuation, B = xˆ Bx cos(ky). The Lorentz force, e(v × B)/c, acts on the charged particles 1

In reality, the role of protons is more complicated. In particular, they can play a crucial role in the electron acceleration in the downstream region. We do not consider such effects in this paper; they will be studied elsewhere.

Springer

k B = A ω p,s /c,

(2)

where ω p,s =




4π e2 n s Ŵshock m s

1/2

≈ 1.32 × 103




ns m p n p ms

1/2

−1/2

Ŵshock s−1

(3)

is the plasma frequency defined for species s (electrons, protons, etc.), n p and m p are the number density and the mass of the protons, respectively; Ŵshock is the Lorentz factor of a relativisitc shock (for non-relativistic shocks, Ŵshock ∼ 1). We use cgs units throughout the paper, unless stated otherwise. Thus, the instability is indeed driven by the PDF anisotropy and should quench for the isotropic case. To put these facts in the context of cosmological shocks, we give estimates of the characteristic temporal and spatial scales. The order of magnitude estimates of the magnetic field e-folding time and the field correlation length at strong shocks (M ≫ 1) are readily obtained as −1/2

τ B ∼ 1/γ B ≃ 2 × 10−1 v8−1 n 0 λ B ∼ 2π/k B ≃ 10

8

−1/2 n0

cm

s,

(4) (5)

for a typical ISM proton density of n ∼ 1 cm−3 and a typical non-relativistic shock velocity v ∼ 108 cm s−1 ; as usual, we denote n 0 = n/(100 cm−3 ) and v8 = v/(108 cm s−1 ). The Lorentz force deflection of particle orbits increases as the magnetic field perturbation grows in amplitude. The amplified magnetic field is random in the plane perpendicular to the particle motion, since it is generated from a random seed field. Thus, the Lorentz deflections result in a pitch angle scattering, which makes the bulk of the PDF isotropic. If one starts from a strong anisotropy, so that the thermal spread is much smaller than the particle bulk velocity, most of the particles will eventually isotropize and the thermal energy associated with their random motions will be equal to their initial directed kinetic energy. This final state will bring the instability to saturation. The saturation level of the magnetic field may readily be estimated as follows. First of all, note that the instability is due to the free streaming of particles. As the magnitude of the magnetic field grows, the transverse deflection of particles

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gets stronger, and their free streaming across the field lines is suppressed. The typical curvature scale for the deflections is the Larmor radius, ρ L = v⊥B /ωc,s ,

(6)

where v⊥B is the particle velocity transverse to the direction of the local magnetic field and ωc,s = eB/m s c is the cyclotron (Larmor) frequency of species s. On scales larger than ρ L , particles can only move along field lines. Hence, when the growing magnetic fields become such that k B ρ L ∼ 1, the particles are magnetically trapped and can no longer amplify the field. Assuming an isotropic particle distribution at saturation (v⊥B ∼ vsh ), this condition can be re-written as ǫB =

B 2 /8π ≃ A2 . 2 m s n s vsh /2

Fig. 2 The temporal evolution of the magnetic field energy density normalized by the initial kinetic energy of the particles

(7)

For strong shocks (M ≫ 1, A ∼ 1), this corresponds to the magnetic energy density close to equipartition with the thermal energy of particles downstream the shock. Here again, we evaluated the field in a non-relativistic shock. Weibel instability has been modeled in numerical PIC 2D and 3D simulations by our group Silva et al. (2003) as well as by several other research groups (Frederiksen et al., 2004; Nishikawa et al., 2003; Saito and Sakai, 2004; Kazimura et al., 1998). We examined the instability, which occurs in a collision of two inter-penetrating unmagnetized electronpositron clouds with zero net charge. This is the simplest model for the formation region of a shock front, as well as a classic scenario unstable to electromagnetic and/or electrostatic plasma instabilities. The relativistic electromagnetic 3D PIC code OSIRIS (Fonseca et al., 2002) was used. The simulations were performed in a simulation cube of size 256 × 256 × 100 grid points, ten grid points correspond to one plasma skin depth c/ω p,e . We had more than 108 particles in the simulation box. Periodic boundary conditions were imposed. The initial state is spatially homogeneous with two identical groups of particles moving with some velocity ±vz . The particles in both groups have a small thermal velocity vth ≃ vz /6. The system has no net charge and no net current, and initially the electric and magnetic fields are set to zero. The results of the simulations are shown in Figs. 2 and 3. Figure 2 shows the temporal evolution of the magnetic equipartition parameter ǫ B . In Fig. 3 the three-dimensional structure of the magnetic fields and currents are shown at two different times: (a) during linear regime, at t ≃ 13ω−1 p,e , . The left and (b) just after the saturation, at t ≃ 20ω−1 p,e panels show the structure of magnetic field lines and the right panels show the number density of particles (blue are moving downward, red are moving upward). We see that during the linear stage of the instability (ω p,e t  15)

Fig. 3 The 3D structure of the magnetic fields and currents from the simulations at two different times: (a) during linear regime, at −1 t ≃ 13ω−1 p,e , and (b) just after the saturation, at t ≃ 20ω p,e . The left panels show the structure of magnetic field lines and the right panels show the number density of particles (blue are moving downward, red are moving upward). The units of axes x1, x2, x3 are c/ω pe

there is exponential generation of a magnetic field, which predominantly lies in the plane of the shock (x − y-plane), i.e., perpendicular to the direction of motion of the plasma clouds. The produced magnetic field is highly inhomogeneous, with the characteristic correlation scale comparable to the plasma skin depth length c/ω p,e . It is also seen that the magnetic field generation is associated with the separation of the particle streams in spatially distinct regions and the formation of straight current filaments. Saturation of the instability occurs at time t ∼ 15ω−1 p,e , which is indicated by the peak of ǫ B in Fig. 2. At this moment, most of the particles are randomized over the pitch angle by the Lorentz deflections. Thus the PDF anisotropy, which is Springer

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the free energy source for the instability is removed. At the time of saturation, the magnetic field energy density reaches its maximum ∼20%. 2.2 Nonlinear regime: Long-term field evolution At longer times, one sees the substantial decrease in ǫ B , during the initial stage of the nonlinear regime in which current filaments begin to interact with each other, forcing like currents to approach each other and merge. During this phase, initially randomly oriented filaments cross each other to form a more organized, large-scale quasi-regular pattern, hence much current and B-field is annihilated. At later times, t
35ω−1 p,e , the filament coalescence continues, as is indicated by the increase of the filament sizes. Note that the spatial distribution of currents is now quite regular, so that filaments with opposite polarity no longer cross each other, but simply interchange, staying always far away. The total magnetic field energy is ǫ B ∼ 0.25% and does not change any more. Note also that the residual magnetic field is highly inhomogeneous, seen as a collection of magnetic field filaments or “bubbles”. The amplitude of the field in the bubbles is close to equipartition. Therefore, the overall decrease of the magnetic field energy is mostly associated with the decreasing filling factor of the field. The filament coalescence is a hierarchical and self-similar process (Medvedev et al., 2005). Initially the system consists of filaments of diameter D0 , carrying current I0 and separated by distance d0 ≃ 2D0 , which are randomly located in space. This configuration is unstable because opposite currents repel each other, whereas like currents are attracted to each other and tend to coalesce and form larger current filaments. At the k-th merger level, i.e., after k pairwise mergers: Ik = 2k I0 , Dk = 2k/2 D0 , dk ∼ Dk /2. Expressing k interms of tite t, for the non-relativistic and relativistic filaments respectively, we obtain for the magnetic field correlation length λ B (t) = D0 2t/(2τ0,N R ) ,

λ B (t) ≃ ct.

at the time t∗ = 2 log2 (c/v0m ) τ0,N R ∼ 10τ0,N R .

(10)

We now compare our theoretical predictions with the results of particle-in-cell numerical simulations, performed using code OSIRIS. We have performed 2D simulations (1280 × 1280 cells, 128.0 × 128.0 (c/ω p,e )2 , 9 particles/(cell × species), 4 species) of the collision of electrically neutral clouds (electron-positron – e− e+ , and electron-proton – e− p) moving in the zˆ direction, across the x y simulation plane, with parameters similar to those in Silva et al. (2003). In order to save on computation time and trace a substantial period of field evolution, we set the particles bulk velocity to vsh ≃ v ≃ 0.6c. Relativistic effects do not play any significant role because the corresponding Lorentz factor is γsh ∼ 1.17. The change of the field correlation length with time is clearly seen in these runs (not shown). The growth of this length is substantially slower and the magnetic field filling factor is respectively larger in the electron-proton run. The temporal evolution of λ B as measured in the simulations is shown in Fig. 4. Both a non-power-law non-relativistic regime (until t ≈ 10 − 20/ω p,e ) and a power-law regime are clearly seen. The power-law fits yield λ B (t) ∝ t α with α ≈ 0.8. Note also that the second power-law segment with the same index is present at t
100/ω p,e in e− p run, indicating proton filament coalescences. A similar behavior was also observed in 3D simulations, but the significantly larger simulation planes presented here allow for improved statistics. At late times t
100/ω p,e , the evolution of λ B rolls off and is slower when the number of filaments in the simulation box becomes relatively small. We also note that in some respect, the field scale growth is analogous to the inverse cascade in two-dimensional magnetohydrodynamic (MHD) turbulence. The crucial difference is, however, the entirely kinetic nature of the process; at such

(8)

Here, the typical non-relativistic time-scale is determined by Equation (8). The coalescence time may be written as η (c/vsh ) ∼ 105 ω−1 τ0,N R ∼= √ p,s , ǫ B ω p,s

(9)

Here we assumed the typical values: ǫ B ∼ 10−3 and η ∼ 1. Also, the parameter η accounts for the fact that the field correlation scale at the onset of non-linear evolution (t ∼ 15ω−1 p,s ) is somewhat larger than λs . We include this uncertainty via the parameter η > 1 as D0 ∼ η (c/ω p,s ). The transition from the non-relativistic to relativistic coalescence regime occurs Springer

Fig. 4 Temporal evolution of the field correlation length in electronpositron (solid line) and electron-proton (dashed line) runs. In the range 7/ω p,e < t < 50/ω p,e , λ B ∝ t α with α ≃ 0.8

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small scales ∼c/ω p the MHD approximation is completely inapplicable. Numerous runs were performed to isolate the effect of the finite box size and make sure that the results are not affected. We found that the periodic boundary conditions affect the dynamics when the filament sizes are comparable or, at most, a factor of three-four smaller than the box size. Initially, the filaments are created on the skin-depth scale, c/ω pe , which is about 1/25 of the box size. Hence, the linear instability, saturation and early nonlinear dynamics are accurately simulated.

is continuously generated at the shock front. Thus, the field is also random in the parallel direction, i.e., the z-direction. Thus, Weibel turbulence at the shocks is highly anisotropic. Both the theoretical considerations and realistic 3D simulations of relativistic shocks indicate that the dynamics of of the Weibel magnetic fields in the shock plane and along the normal to it are decoupled. Hence, the Fourier spectra of the field in the x − y plane and in z direction are independent. Thus, for the Weibel fields at shocks, the correlation tensor has the form

3 Theory of jitter radiation in 3D

K αβ (k) = C(δαβ − n α n β ) f z (k ) f x y (k⊥ ),

The angle-averaged spectral power emitted by a relativistic particle moving through small-scale random magnetic fields, under the assumption that the deflection angle is negligible and the particle trajectory is a straight line, has been derived elsewhere (Rybicki and Lightman, 1979; Landau and Lifshitz, 1971; Medvedev, 2000) and it reads

where n is the unit vector normal to the shock front, C is the normalization constant proportional to B 2 , f z and f x y are the magnetic field spectra along n and in the shock plane, respectively, k⊥ = (k x2 + k 2y )1/2 and k = k z , and finally, the tensor (δαβ − n α n β ) is symmetric and its product with n is zero, implying orthogonality of n and B. Numerical simulations (Frederiksen et al., 2004) also indicate that the field transverse spectrum, f x y , is well described by a broken power-law with the break scale comparable to the skin depth, c/ω p , where ω p = (4π e2 n/ Ŵm)1/2 is the relativistic plasma frequency and Ŵ is the shock Lorentz factor. We expect that the spectrum f z , has similar properties. Therefore, we use the following models:

dW e2 ω = dω 2π c3



∞

ω/2γ 2

|wω′ |2 ω′2




ω2 ω 1− ′ 2 + ωγ 2ω′2 γ 4



dω′ . (11)

Here γ is the Lorentz factor of a radiating particle and ′ wω′ = weiω t dt is the Fourier component of the transverse particle’s acceleration due to the Lorentz force. We need to express the temporal Fourier component of the Lorentz acceleration, w = (e/γ mc)v × B, taken along the particle trajectory in terms of the Fourier component of the field in the spatial and temporal domains Medvedev (2006). In the static case, i.e., when the magnetic field is independent of time, the ensemble-averages acceleration spectrum reads:

|wω′ |2  = (2π V )−1



|wk |2 δ(ω′ + k · v) dk,

|wk |2 = (ev/γ mc)2 (δαβ − v −2 vα vβ ) V K αβ (k),

(12)

f z (k ) = 

k2α1

 , 2 β1

κ2 + k

f x y (k⊥ ) = 

(14)

2α2 k⊥

2 κ⊥2 + k⊥

(15)

β2 ,

where κ and κ⊥ are parameters (being, in general, a function of the distance from the front, Medvedev et al., 2005) determining the location of the peaks in the spectra, α1 , α2 , β1 , β2 are power-law exponents below and above a spectral peak (β1 > α2 + 1/2 and β2 > α2 + 1, for convergence at high-k).

(13)

K αβ (r, t) = T −1 V −1 Bα (r′ , t ′ )Bβ (r′ + r, t ′ + t) dr′ dt is the second-order correlation tensor of the magnetic field. 3.1 The magnetic field spectrum We adopt the following geometry: a shock is located in the x-y-plane and is propagating along z-direction. As it has initially been demonstrated by Medvedev and Loeb (1999) and later confirmed via 3D PIC simulations (Silva et al., 2003; Nishikawa et al., 2003; Frederiksen et al., 2004), the magnetic field at relativistic shocks is described by a random vector field in the shock plane, i.e., the x – y plane. As the shock is propagating through a medium, the produced field is transported downstream (in the shock frame) whereas new field

3.2 Radiation spectra from a shock viewed at different angles We now evaluate Equations (12, 13). The scalar product of the two tensors is (δαβ − vα vβ /v 2 )(δαβ − n α n β )

= 1 + (n α vα )2 /v 2 = 1 + cos2 ,

(16)

where we used that δαα = 3. Here  is the angle between the normal to the shock and the particle velocity (in an observer’s frame), which is approximately the direction toward an observer, that is vk for an ultra-relativistic particle (because of relativistic beaming, the emitted radiation is localized within Springer

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Fig. 5 The log − log plots of |wk |2 vs k (thin lines) and of d W/dω vs ω (thick lines), for three viewing angles  = 0, π/10, π/2. The axes units are arbitrary. In this calculation we used f z = f x y with α = 2, β = 20, κ = 10, v = 1. The exponent ζ = ζ (α, β) is model dependent. We also chose γ = 1 in order to align the peaks of |wk |2 and d W/dω. Note that the actual peaks are at values k, ω lower than 10 by a factor two or three. Note also that the spectrum d W/dω levels off at oblique angles at frequencies much smaller than κvγ 2 sin , whereas |wk |2 indeed starts to flatten at k ∼ κv sin 

C (1 + cos2 ) 2π  × f z (k ) f x y (k⊥ )δ(ω′ + k · v) dk d 2 k⊥ .

(17)

Equations (11, 17) fully determine the spectrum of jitter radiation from a GRB shock. We now consider special cases. A spectrum from a shock viewed at an arbitrary angle, 0 ≤  ≤ π/2, is illustrated in Fig. 5, which represents full numerical solutions of Equations (11), (15), (17) for three different viewing angles. In calculation of d W/dω, the emitting electrons were assumed monoenergetic, for simplicity. An important fact to note is that the jitter radiation spectrum varies with the viewing angle. When a shock velocity is along the line of sight, the low-energy spectrum is hard Fν ∝ ν 1 , harder than the “synchrotron line of death” (Fν ∝ ν 1/3 ). As the viewing angle increases, the spectrum softens, and when the shock velocity is orthogonal to the line of sight, it becomes

Springer

Acknowledgments This work has been supported by DOE grant DEFG02-04ER54790.

References

a narrow cone of angle ∼1/γ ). Equation (12) becomes

|wω′ |2  =

Fν ∝ ν 0 . Another interesting feature is that at oblique angles, the spectrum does not soften simultaneously at all frequencies. Instead, there appears a smooth spectral break, which position depends on . The spectrum approaches ∼ν 0 below the break and is harder above it. The jitter spectra can deliver much information on the structure of the Weibel magnetic fields. As one can see, the spectrum depends on the spatial spectra of the magnetic fields modeled by Equations (15). In particular, when viewing angles are 0 and 90 degrees, the contributions of the parallel and transverse magnetic field spectra are decoupled. For instance, for θ = 0, the peak of the jitter radiation spectrum and its high-energy asymptotic slope are uniquely determined by the parallel correlation length κ and the large-k magnetic field spectrum slope k s with s = 2α1 − 2β1 . Similarly, the transverse jitter spectrum (at θ = π/2) allows one to deduce these parameters for the transverse magnetic field spectrum, f x y . At intermediate angles, one can determine the relative orientation of the current (and magnetic) filaments in the target and the radiation detector.

˚ ApJ 608, Frederiksen, J.T., Hededal, C.B., Haugbølle, T., Nordlund, A.: L13 (2004) Fonseca, R.A., et al.: Lecture Notes in Computer Science 2329, III-342, Springer-Verlag, Heidelberg (2002) Fried, B.D.: Phys. Fluids 2, 337 (1959) Kazimura, Y., Sakai, J.I., Neubert, T., Bulanov, S.V.: ApJ 498, L183 (1998) Landau, L.D., Lifshitz, E.M.: The Classical Theory of Fields. Pergamon Press, Oxford (1971) Medvedev, M.V., Loeb, A.: ApJ 526, 697 (1999) Medvedev, M.V.: ApJ 540, 704 (2000) Medvedev, M.V., Fiore, M., Fonseca, R.A., Silva, L.O., Mori, W.B.: ApJ 618, L75 (2005) Medvedev, M.V.: ApJ 637, 869 (2006) Moiseev, S.S., Sagdeev, R.Z.: J. Nucl. Energy C 5, 43 (1963) Nishikawa, K.-I., Hardee, P., Richardson, G., Preece, R., Sol, H., Fishman, G.J.: ApJ 595, 555 (2003) Rybicki, G.B., Lightman, A.P.: Radiative Processes in Astrophysics. Wiley, New York (1979) Saito, S., Sakai, J.I.: ApJ 604, L133 (2004) Silva, L.O., Fonseca, R.A., Tonge, J.W., Dawson, J.M., Mori, W.B., Medvedev, M.V.: ApJ 596, L121 (2003) Weibel, E.S.: Phys. Rev. Lett. 2, 83 (1959)

Astrophys Space Sci (2007) 307:251–255 DOI 10.1007/s10509-006-9290-x

O R I G I NA L A RT I C L E

Diagnostics of the Non-Linear Richtmyer-Meshkov Instability M. Herrmann · S. I. Abarzhi

Received: 27 July 2006 / Accepted: 13 December 2006 C Springer Science + Business Media B.V. 2007


Abstract We study analytically and numerically the evolution of the two-dimensional coherent structure of bubbles and spikes in the Richtmyer-Meshkov instability (RMI) for fluids with a finite density ratio. New diagnostics and scalings are suggested for accurate quantification of RMI dynamics. New similarity features of the late-time instability evolution are observed. The results obtained can serve as benchmarks for high energy density laboratory experiments. Keywords Richtmyer-Meshkov . Supernovae . Diagnostics . Similarity . Multi-scale dynamics

Introduction The Richtmyer-Meshkov instability develops when a shock wave refracts through the interface between two fluids with different values of the acoustic impedance (Richtmyer, 1960; Meshkov, 1969). The instability results in a growth of the interface perturbations and produces with time extensive interfacial mixing of the fluids (Dimonte, 2000). The shock-driven turbulent mixing plays an important role in many astrophysical phenomena. In particular, in supernova type II, RMI is considered as a plausible mechanism of the extensive mixing of the inner and outer layers of the progenitor star (Chevalier, 1992). Laboratory experiments and simulations suggest the following evolution of RMI. Initially, the light fluid accelerates “implusively” the heavy fluid, with the acceleration value determined by the shock-interface interaction. With time, a coherent structure of bubbles and spikes appear: the M. Herrmann (
) · S. I. Abarzhi Center for Turbulence Research, Stanford University, Stanford, CA, USA e-mails: [email protected]; [email protected]

light (heavy) fluid penetrates the heavy (light) fluid in bubbles (spikes). Small-scale structures are produced by shear, and roll-up of vortices causes a mushroom-typed shape of the spike. Eventually a mixing zone develops, and in the chaotic regime the spikes and bubbles decelerate as power-law with time (Dimonte, 2000). The dynamics of RMI is governed by two, in general independent, length scales: the spatial period of the structure λ, set by the initial perturbation or by the mode of fastest growth, and the amplitude h, which is the bubble (spike) position (Abarzhi and Herrmann, 2003; Abarzhi et al., 2003). To quantify the evolution of RMI, experiments and simulations were focused on diagnostics of the length scale h, readily available for measurements. The observation results have been interpreted with several empirical models, which presumed a single-scale character of the interface dynamics, governed solely by the spatial period λ, and extensively used adjustable parameters for data comparison. Despite the efforts made, the primary issue for observations still remains ”How to quantify these flows reliably?” Here we suggest new diagnostics and scaling, which allow for a more accurate description of RMI dynamics, and discuss new similarity features exhibited at its late stages.

Diagnostics and evolution of the nonlinear unstable interface To study RMI dynamics, we find numerical and analytical solutions for the system of compressible two-dimensional Navier-Stokes equations with the initial and boundary conditions at the fluid interface. The density of the heavy (light) fluid is ρh(l) , and the Atwood number is A = (ρh − ρl )/(ρh + ρl ). The normal component of velocity and pressure are continuous at the fluid interface. The flow has no mass Springer

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sources, and the initial co-sinusoidal perturbation has small amplitude. In our numerical simulations, we solve the compressible Navier-Stokes equations using a finite volume hybrid capturing-tracking scheme (Smiljanovski et al., 1997; Schmidt and Klein, 2003). The contact discontinuity at the interface is tracked by a level set scalar, and the boundary conditions at the interface are used to reconstruct the exact states on each side of the interface, so that the interface remains an immiscible discontinuity with out unphysical fluxes across (Abarzhi and Herrmann, 2003). Several test runs are performed for code validation. Figures 1 and 2 show a comparison of the amplitude evolution obtained in our simulations with the experiments of Jones and Jacobs (1997) and with the linear theory of Wouchuk (2001), and the agreement with the experiments and the theory is excellent. Figure 2 shows that the growth-rate of the amplitude exhibits oscillations. These oscillations are caused by the reverberations of sound waves, and, predicted first by the linear

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theory of Wouchuk (2001), they are accurately reproduced in our simulations. The amplitude of the oscillations is very small compared to the speed of sound, so the oscillations do not induce significant pressure fluctuations. In Fig. 1 the velocity oscillations are also present and can be derived from our numerical solution as the time-derivative of the amplitude. The experiments of Jones and Jacobs (1997) do not capture the oscillations, as the data set in Fig. 1 is a combination of single shot measurements of a series of different experiments. To quantify the evolution of RMI, we define first the length scale and the time scale of the flow. The length scale is the period of the coherent structure λ. The proper choice of the time-scale is a non-trivial issue. To perform a comparative study of various stages of RMI (linear, weakly and highlynonlinear), most of exisiting observations use the time-scale set by the initial growth-rate v0 , as λ/v0 (Cheng et al., 2000; Robey et al., 2003; Glendinning et al., 2003; Miles et al., 2004; Jacobs and Krivets, 2005). However, as is seen from Fig. 2, the value of the growth-rate oscillates and the amplitude of the oscillations is ∼10% − 20% of v0 . Hence, for more accurate quantification of RMI evolution, we choose the time-scale set by the velocity v∞ , at which an ideally planar interface would move after shock passage. The velocity v∞ can be obtained straightforwardly from 1D calculations (Meshkov, 1969). For studies of the nonlinear RMI, four different Atwood numbers are considered in our simulations with A = 0.55, 0.663, 0.78, and 0.9. In all runs, the dynamic viscosity µ is the same in the heavy and light fluids, with Re = ρl v∞ λ/µref = 13042 (A = 0.55), Re = 11572 (A = 0.663), Re = 9700 (A = 0.78), and Re = 6968 (A = 0.9). Periodic boundary conditions are used in the transverse direction, and the effect of viscosity on the large-scale dynamics is negligible. The initial perturbation has spatial period λ = 2π/k = 3.75 cm and amplitude a0 = 0.064λ and is located at z(x, t = 0) = a0 cos(kx) inside a [−40.667λ, 1.333λ] × [−0.5λ, 0.5λ] box resolved by 5376 × 128 equidistant cartesian grid cells. The initial shock is weak, Ma = 1.2, and propagates from the light to the heavy fluid. Our simulations stop as the outlet reflected shock hits the interface. Still, as the domain size is large, they run much longer compared to other observations (Cheng et al., 2000; Robey et al., 2003; Glendinning et al., 2003; Miles et al., 2004; Jacobs and Krivets, 2005). To quantify the nonlinear evolution of RMI, we use two diagnostic parameters, the bubble velocity and curvature. The bubble velocity is a traditional diagnostic parameter. In many existing observations, the velocity of the bubble front in RMI is determined relative to a ”middle line” – half of the distance between the tips of the bubble and spike, i.e. half the amplitude. To calculate the bubble velocity more accurately, we account for the fact that RMI develops relative to a background

Astrophys Space Sci (2007) 307:251–255

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motion with a constant velocity v∞ . Therefore, in the laboratory frame of reference the bubble velocity is (v + v∞ ), whereas in the frame of references moving with velocity v∞ , the bubble velocity is v. We compare our numerical solutions with the results of the nonlinear theory of Abarzhi et al. (2003), which suggests the following evolution of the bubble front in RMI. After a short stage of the shock-interface interaction, the bubble curvature ζ and velocity v change linearly with time t; then, in the weakly non-linear regime, the curvature reaches an extreme value, dependent on the initial conditions and the Atwood number; asymptotically, the bubble flattens, ζ → 0, and decelerates, v → 0. For fluids with similar densities, A ≪ 1, the bubbles move faster than those with contrasting densities, A → 1. The flattening of the bubble front is a distinct feature of RMI universal for all A, which indicates that the nonlinear evolution of RMI is governed by two length scales, the amplitude h and the wavelength λ, and has therefore a multi-scale character (Abarzhi et al., 2003). The evolution of the bubble front is shown in Fig. 3, representing the phase diagram of the bubble velocity v(t) versus bubble curvature ζ (t) with time t being a parameter. Initially the bubble exhibits an abrupt acceleration caused by

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the shock- interface interaction (Richtmyer, 1960; Meshkov, 1969). Then the bubble starts to decelerate, while the absolute value of the bubble curvature increases and reaches an extreme value. We emphasize that the bubble velocity and curvature reach their extreme values at two distinct moments of time. As the instability evolves, the bubble continues to decelerate and its curvature approaches zero, as found by the theory of Abarzhi et al. (2003). Our results are in qualitative agreement with the experiments of Jacobs and Krivets (2005), who observed that in the nonlinear regime the front of the RM bubble flattens at the tip and its velocity decreases, whereas for fluids with similar densities the bubbles move faster than those for fluids with highly contrasting densities. Our simulation results indicate that it may be hard to estimate accurately the time-dependence of the asymptotic velocity of the bubble front in compressible RMI. Figure 4 presents the log-log plot of the bubble velocity versus time in the nonlinear regime of instability for A = 0.55. This density ratio is close to that in the experiments of Jacobs and Krivets (2005), while the time considered is longer. We see that the evolution of the bubble velocity is accompanied by oscillations. These oscillations appear in the linear regime of the instability and are induced by the revereberations of

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Summary

0.01 A = 0.55

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sound waves (see Fig. 2). In the nonlinear regime of RMI, the bubble velocity decreases, whereas the amplitude of the velocity oscilllations is damped only slightly, and an estimate of the time-dependence of the bubble velocity from the simulation data is thus a challenging problem. We emphasize that in most of exisiting experiments, the data sampling do not have high temporal resolution and do not capture the oscillations. Therefore, the accuracy and resolution of available experimental data may be insufficient to make a quantitative statement on the asymptotic time-dependencies of the nonlinear RMI. Further improvement of the interface diagnostics in experiments is required, in particular, in the temporal resolution and data rate acquisition. Our simulations confirm the results of the theory of Abarzhi et al. (2003) and do not confirm the solution given by drag models (Alon et al., 1995; Oron et al., 2001), which presume a single-scale character of the interface evolution and predict that for all values of the Atwood number, the RM bubble remains curved asymptotically with ζ D = −π/3λ. As we see from Fig. 3, the shape of the bubble front in RMI is not determined solely by the spatial period λ and is sensitive to the length scale h which plays the role of the integral scale for energy dissipation in small-scale structures. For fluids with contrasting densities, A = 0.663, 0.78, and 0.9, the decay in the curvature value is obvious, Fig. 3. In the case of A = 0.55 the flattening process is slower. At a time when the reflected shock hits the interface and our simulations stop, the absolute value of the bubble curvature is still finite |ζ | = 0.516/λ. Yet this is already about 50% smaller than |ζ D | (Oron et al., 2001). Figure 3 illustrates that the nonlinear dynamics in RMI is a multi-scale process, governed by two macroscopic scales: the spatial period of the structure λ and the amplitude h, which is the bubble displacement. In the nonlinear regime of RMI the velocity of the bubble front v = (dh/dt) and its curvature ζ mutually depend on one another: dh/dt = v∞ f (|ζ λ|), where f is an algebraic function. The processes of deceleration, d 2 h/dt 2 , and flattening, d(ζ λ)/dt, are therefore inter-related, and exhibit new features of universality and similarity, see Figure 3, which can be used in future models of RM turbulent mixing.

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We report analytical and numerical solutions describing the dynamics of the two-dimensional coherent structure of bubbles and spikes in the Richtmyer-Meshkov instability for fluids with a finite density ratio. Good agreement between the analytical and numerical solutions is achieved. To quantify accurately the interface evolution in the observations, new diagnostics and scalings are suggested. The velocity, at which the interface would move if it would be ideally planar, is used to set the flow time-scale as well as the reference point for the bubble (spike) position. Our data sampling has high temporal resolution and captures the velocity oscillations caused by sound waves. The bubble velocity and curvature are both monitored. The obtained results indicate that the evolution of the Richtmyer-Meshkov instability is a multi-scale process, governed by two length scales, the spatial period and the amplitude of the coherent structure, and exhibits new similarity features in the late-time evolution. Our results can serve as benchmarsk for high energy density laboratory experiments (Robey et al., 2003; Glendinning et al., 2003; Miles et al., 2004). References Abarzhi, S.I., Herrmann, M.: New type of the interface evolution in the Richtmyer-Meshkov instability. In: Annual Research Briefs2003, Center for Turbulence Research, pp. 173–183. Stanford, CA (2003) Abarzhi, S.I., Nishihara, K., Glimm, J.: Rayleigh-Taylor and Richtmyer-Meshkov instabilities for fluids with a finite density ratio. Phys. Lett. A 317, 470–476 (2003) Alon, U., Hecht, J., Offer, D., Shvarts, D.: Power laws and similarity of Rayleigh-Taylor and Richtmyer-Meshkov mixing fronts at all density ratios. Phys. Rev. Lett. 74, 534–537 (1995) Cheng, B.L., Glimm, J., Sharp, D.H.: Density dependence of RayleighTaylor and Richtmyer-Meshkov mixing fronts. Phys. Lett. A 268(4–6), 366–374 (2000) Chevalier, R.A.: A model for the radio brightness of the supernova remnant 1987a. Nature 355(6361), 617–618 (1992) Dimonte, G.: Spanwise homogeneous buoyancy-drag model for Rayleigh-Taylor mixing and experimental evaluation. Phys. Plasmas 7, 2255–2269 (2000) Glendinning, S.G., Bolstad, J., Braun, D.G., Edwards, M.J., Hsing, W.W., Lasinski, B.F., Louis, H., Miles, A., Moreno, J., Peyser, T.A., Remington, B.A., Robey, H.F., Turano, E.J., Verdon, C.P., Zhou, Y.: Effect of shock proximity on Richtmyer-Meshkov growth. Phys. Plasmas 10(5), 1931–1936 (2003) Jacobs, J.W., Krivets, V.V.: Experiments on the late-time development of single-mode Richtmyer-Meshkov instability. Phys. Fluids 17(034105), 1–10 (2005) Jones, M.A., Jacobs, J.W.: A membraneless experiment for the study of Richtmyer-Meshkov instability of a shock-accelerated gas interface. Phys. Fluids 9, 3078–3085 (1997) Meshkov, E.: Sov. Fluid Dyn. 4, 101 (1969) Miles, A.R., Edwards, M.J., Blue, B., Hansen, J.F., Robey, H.F., Drake, R.P., Kuranz, C., Leibrandt, D.R.: The effect of a short-wavelength mode on the evolution of a long-wavelength perturbation driven by a strong blast wave. Phys. Plasmas 11(12), 5507–5519 (2004)

Astrophys Space Sci (2007) 307:251–255 Oron, D., Alon, U., Offer, D., Shvarts, D.: Dimensionality dependence of the Rayleigh-Taylor and Richtmyer-Meshkov instability latetime scaling laws. Phys. Plasmas 8, 2883–2889 (2001) Richtmyer, R.: Taylor instability in shock acceleration of compressible fluids. Commun. Pure Appl. Math. 13(2), 297 (1960) Robey, H.F., Zhou, Y., Buckingham, A.C., Keiter, P., Remington, B.A., Drake, R.P.: The time scale for the transition to turbulence in a high Reynolds number, accelerated flow. Phys. Plasmas 10(3), 614–622 (2003)

255 Schmidt, H., Klein, R.: A generalized level-set/in-cell-reconstruction approach for accelerating turbulent premixed flames. Combust. Theory Modelling 7, 243–267 (2003) Smiljanovski, V., Moser, V., Klein, R.: A capturing-tracking hybrid scheme for deflagration discontinuities. Combust. Theory Modelling 1, 183–215 (1997) Wouchuk, J.: Growth rate of the linear Richtmyer-Meshkov instability when a shock is reflected. Phys. Rev. E 63(56303), 1–13 (2001)

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Astrophys Space Sci (2007) 307:257–261 DOI 10.1007/s10509-006-9184-y

ORIGINAL ARTICLE

Density Measurements of Shock Compressed Matter Using Short Pulse Laser Diagnostics M. Koenig · A. Ravasio · A. Benuzzi-Mounaix · B. Loupias · N. Ozaki · M. Borghesi · C. Cecchetti · D. Batani · R. Dezulian · S. Lepape · P. Patel · H. S. Park · D. Hicks · A. Mckinnon · T. Boehly · A. Schiavi · E. Henry · M. Notley · R. Clark · S. Bandyopadhyay Received: 21 April 2006 / Accepted: 31 May 2006 C Springer Science + Business Media B.V. 2006


Abstract In this paper, experimental results on X-ray and proton radiography of shock compressed matter are presented. It has been performed at the Rutherford Appleton Laboratory (RAL) using three long pulse beams to generate a shock wave in a multi-layer foil and a short pulse beam to create either an X-ray or protons source for a transverse radiography. Depending on the probe material (aluminium M. Koenig (
)· A. Ravasio· A. Benuzzi-Mounaix· B. Loupias· N. Ozaki Laboratoire pour l’Utilisation des Lasers Intenses, CNRS-CEA-Universit´e Paris VI-Ecole Polytechnique, 91128 Palaiseau, France M. Borghesi· C. Cecchetti Department of Physics and Astronomy, The Queen’s University of Belfast, Belfast, BT7 1NN, UK D. Batani· R. Dezulian Dipartimento di Fisica ‘G. Occhialini’, Universit`a di Milano-Bicocca and INFM, Piazza Della Scienze 3, 20126 Milano, Italy S. Lepape· P. Patel· H. S. Park· D. Hicks· A. Mckinnon Lawrence Livermore National Laboratory, Livermore, CA 94550, USA T. Boehly Laboratory for Laser Energetics, University of Rochester, USA A. Schiavi Department of Energetics, University of Rome “La Sapienza”, Rome, Italy E. Henry D´epartement de conception et r´ealisation des exp´eriences (DECRE), CEA-DIF, BP 12, 91680 Bruy`eres-le-Chˆatel, France M. Notley· R. Clark· S. Bandyopadhyay Central Laser Facility, Rutherford Appleton Laboratory, Chilton, Oxfordshire, OX11 0QX, UK

or carbon foam) a Molybdenum Kα source or a proton beam are used. Density data of the shocked aluminium, in the multimagabar regime are presented. Keywords Shocks . Laser plasmas . Radiography The knowledge of Equation Of State (EOS) and related parameters of dense matter is important in several fields of physics. For instance, in astrophysics the star evolution is mainly governed by the thermodynamic properties of matter. EOS is also fundamental for the knowledge of internal structure of giant or telluric planets (Stevenson, 1981). Inertial Confinement Fusion (ICF) success depends directly from the understanding of shell pellet implosion and the final core compression. Both of these processes implies a precise knowledge of the microballoon material (Koenig et al., 1998) and the fuel (deuterium) EOS at very high pressures (Collins et al., 1998; Knudson et al., 2001) (>100 Gpa). Since several years, a large effort has been done in laser driven shock wave experiment to develop relevant diagnostics to perform high precision EOS measurements (Cauble et al., 1997; Koenig et al., 1995) (Benuzzi-Mounaix et al., 2002; Hicks et al., 2005). Shock-wave-EOS experiments require that two parameters, usually the shock and fluid velocities, be measured to infer the thermodynamic properties of the material. While a few experiments have used X-ray radiography on low-Z materials to determine both velocities(Cauble et al., 1997; Collins et al., 1998), most of them rely on the shock velocities measurement via optical interferometry in transparent media or by observation of shock breakout times on steps of known thickness in optically opaque materials (Hicks et al., 2005; Koenig et al., 1999; Koenig et al., 1995). The latter technique result in indirect EOS determinations through a method known as impedance matching. In opaque high/mean Z materials, it is not possible to get information Springer

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about the shock characteristics inside the sample and then to measure fluid velocities directly unless the free surface velocity can be determined (Benuzzi-Mounaix et al., 2002). Therefore, impedance matching is the only possibility to infer EOS data. Moreover, in measurements based on velocity determination, it is impossible to have a good precision on such an important physical quantity as density. Indeed this crucial parameter, as seen in the recent deuterium various experiments (Collins et al., 1998; Knudson et al., 2001), due to error amplification going through Rankine-Hugoniot relations remains uncertain . The development of direct probing techniques to obtain information on another shock parameter, such as density, would allow precise absolute EOS determinations and would represent a real breakthrough in the field. Such attempt has been made on plastic several years ago (Hammel et al., 1993) and more recently (Hicks et al., 2004; Boehly et al., 2005) with “conventional” (long pulse laser beam) X-ray sources. Moreover, density measurements have been conducted using high explosives to drive a shock in a metal and a high energy proton beam (800 MeV) produced by a conventional accelerator to probe it (Holtkamp et al., 2003). These experiments, dedicated to the study of spallation, i.e., in a pressure regime much lower than the one we are interested in, provided a density measurement due to protons collisional stopping power. In a recent experiment, high-energy protons, produced by the interaction of an intense laser pulse with a thin solid target, in a point projection imaging scheme, were used to characterize in situ the spatial and temporal evolution of a laser-driven shock propagating through a low-Z material. The particular properties of laser produced protons beam (small source, high degree of collimation, short duration) make them of great interest for radiographic applications (Borghesi et al., 2001). However due to high level scattering of the proton beam, inferring density implies development of specific propagation code for the proton in the shock compressed material. In this paper we present density measurement of shocked aluminium using hard X-ray radiography (17.5 keV) and shock propagation in a carbon foam probed by a proton beam. This experiment was performed at the RAL (Rutherford Appleton Laboratory) in the TAW target area. Three long pulse beam converted at 2ω with a 2 ns pulse duration generated a shock wave into a multilayer target (Fig. 1) delivering a total maximum energy on target E2ω ≈ 300 J. To generate the shock wave, we used Random Phase Plates in order to eliminate large scale spatial intensity modulations and obtain a Gaussian type profile in the focal spot (FWHM ≈ 300 µm), corresponding to a maximum laser intensity I L ≤81013 W/cm2 . A schematic view of the experimental set-up and the target design is shown in Figure 1. The three laser beams, were Springer

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Fig. 1 Experimental set-up and diagnostics

focused on an ablator-pusher foil to generate a shock into a sliver which was aluminium or carbon foam for the X-ray or proton source respectively. The pusher design was optimized using 1D radiative hydrodynamic simulations according to the laser characteristics. We had a three layer target (10 µm CH-10 µm Al-10 µm CH), the last CH layer acting as a “witness” plate. Two main sets of diagnostics were implemented: on the rear side of the target, a streak camera (“Self Emission”) collected photons emitted by the target giving the shock mean shock velocity of the rear side CH layer. On the transverse axis, we had specific detectors for the probe beams either Image Plates (IP) for the hard X-ray source or Radiochromic Films (RCF) for protons. Magnification was adjusted according to the detector resolution (80 microns for IP or few microns for the RCF) and was 40 and 13 respectively.

X-ray radiography The relativistic electrons generated by the interaction of an intense short laser pulse induce Kα radiation. The energy of this radiation can be adapted to the material to be probed by choosing the appropriate backlit target material. In our case, we used a molybdenum foil that generated a 17.5 keV Kα line. According to previous electron transport experiments, the Kα X-ray source has a minimum size of the order of 50 µm (Martinolli et al., 2006; Stephens et al., 2004) which is too big for the desired spatial resolution. This is the reason why we used a scheme developed recently (Park et al., 2006) which consists to use the foil thickness (5 µm in our case) as the way to limit the source size (Fig. 1). In order to check the final resolution, we did a test with a 100 lpi tungsten mesh. From the data, we could determine a 20 µm resolution along the shock propagation (foil thickness), 60 µm in the other direction due the Kα source minimum size. We then performed several shots compressing the aluminum with the long pulse beams and varying the probe beam delay, between 5 and 10 ns

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Fig. 2 X-ray radiograph of compressed aluminum

Fig. 3 Deduced density profile of shock compressed aluminum. Dots are the Abel inverted data, plain curve is a fit to those data

Proton radiography

after the beginning of the shock drive beams. This ensures that the shock has propagate a suffisant length into the sliver in order to be able to deduce the density (Fig. 2). To determine the density, in fact, we must consider that the observed signal on the detector is a line integral of the local absorption. The use of RPP generated a symmetrical shock along the propagation axis making the situation treatable with typical Abel inversion techniques. Taking azimuthal symmetry into account, the areal density in the radiography z direction can be written as: ρz = 2

a y

Due to high level of scattering of protons generated by an ultra-intense laser pulse in a high Z solid target as pointed out recently (LePape et al., 2006), we deliberately tried to probe a low density material such as carbon foam (CRF). According to simple calculations (geometrical+scattering

 ρ(r )r dr  + ρ0 (L − 2 a 2 − y 2 ), r 2 − y2

where a is the y coordinate of the shock front and L the transverse thickness of the target. The X-ray intensity I, after propagation into the sliver is a function of the areal density, I = I0 exp(−µm ρz). The measured data transmission F(y) is then given by 1 F(y) = − ln ρ 0 µm




I I0



−L=2



a y

Fig. 4 Resolution test for the proton beam through CRF foam and a 1000 lpi grid. A line out is taken where protons goes through the foam

β(r )r dr  , r 2 − y2

where β(r ) = ρρ0 − 1 is inferred from Abel inversion. From the data shown in Fig. 2, we deduced the shock compressed density (Fig. 3). The density is maximum in the center, the compression ratio ρ/ρ0=2.2 in good agreement with expected value given, for example, by simulations. The error bars, taking into account the total incident spectrum uncertainty, Abel inversion evaluation, are at least of the order of ±10%.

Fig. 5 Line out of Fig. 4 on the foam+grid region

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Fig. 6 Proton radiograph of compressed carbon foam taken 7 ns (a) & 11 ns (b) after the main pulses

angles given by the SRIM code), the expected resolution for the proton beam going through a 500 µm wide 0.1 g/cm3 CRF foam sliver must be typically 25 µm. This was probed experimentally having a 1000 lpi grid mesh behind the CRF sliver shooting only the short pulse beams, i.e., just with the proton beam (Fig. 4). As we can observe in the zone where there is no foam, the grid is very contrasted and well defined. When the protons propagate through the CRF sliver, there is a loss of contrast and signal due to scattering and slight absorption. However by taking a lineout in the foam region (Fig. 4), we obtain the modulations (Fig. 5) due to the grid spacing (25 µm). As expected the deduced resolution is <15 µm, not as good as for direct transmission through the grid (with no foam) but high enough to give high contrast for the shock compressed case. As for the hard x-ray, we did several shots delaying the probe beam with respect to the main long pulses. Figure 6 shows a clear shock front in the middle of the foam sliver (a) and just at the end of the CRF target (b) for the 7 ns and 11 ns cases respectively. One can also remark (Fig. 6b) the position of the pusher behind the shock front. The shock between the two these shots traveled through 120 µm of foam, giving an estimate of the velocity Us ≈ 30 km/s which is in good agreement with expected values from hydrodynamic simulations. To deduce the density from the experimental results, it cannot be done as for the x-ray case because we are not in a pure absorption situation, moreover not in cold matter either; Therefore one has to treat the proton propagation in a 2D density and temperature profiles. To this end we developed a specific Monte-Carlo code to deal with the proton propagation in warm dense matter coupled to density and temperature profiles given by 2D hydrodynamic simulations (DUED code (Atzeni, 1986)). We then need to iterate between the MC code and density profile to retrieve the experimental data. This final stage will be completed in the next few months so final density from the compressed foam will be determined.

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Conclusions In this paper, we present new diagnostics for direct density measurements using hard X-ray or proton beam source generated by ultra intense short lasers pulses. Hard X ray radiography (E > 10 keV) allowed, for the first time, to infer shock characteristics inside the high-Z dense target impossible to obtain with standard thermal x-ray sources. Concerning the proton radiography, we have shown the possibility to diagnose shock propagation in a low density medium (carbon foam) with high spatial resolution (<25 µm). We are on the way to infer the compressed material density by coupling a Monte-Carlo code, for the proton propagation in a warm dense plasma, with 2D hydrodynamic simulations. Higher proton energy beams, using for example the newt PW laser at LULI (Pico, 2000) will probably allow to detect shock front in standard density materials such as plastic or diamond. Acknowledgements The authors would like to thank the RAL people who contributed to the success of this work. This experiment has been supported by the Laserlab EU program FP6 contract RII3-CT-2003506350.

References Atzeni, S.: Comput. Phys. Comm. 43, 107 (1986) Benuzzi-Mounaix, A., Koenig, M., Huser, G., Faral, B., Batani, D., Henry, E., Tomasini, M., Marchet, B., Hall, T.A., Boustie, M., Ress´eguier, T.D., Hallouin, M., Guyot, F., Andrault, D., Charpin, T.: Phys. Plasmas 9, 2466–469 (2001) Boehly, T.R., Vianello, E., Miller, J.E., Meyerhofer, D.D., Hicks, D.G., Eggert, J.H., Hansen, J.F., Celliers, P.M., Collins, G.W.: Bull. Am. Phys. Soc. 50, 73 (2005) Borghesi, M., Schiavi, A., Campbell, D.H., Haines, M.G., Willi, O., MacKinnon, A.J., Gizzi, L.A., Galimberti, M., Clarke, R. J., Ruhl, H.: Plasma Phys. Control. Fusion 43, 267–276 (2001) Cauble, R., Silva, L. B.D., Perry, T.S., Bach, D.R., Budil, K.S., Celliers, P., Collins, G.W., Ng, A., Barbee, T.W., Hammel, B.A., Holmes,

Astrophys Space Sci (2007) 307:257–261 N.C., Kilkenny, J.D., Wallace, R.J., Chiu, G., Woolsey, N.C.: Phys. Plasmas 4 (1997) Collins, G., Silva, L.D., Celliers, P., Gold, D., Foord, M., Wallace, R., Ng, A., Weber, S., Budil, K., Cauble, R.: Science 281, 1178 (1998) Hammel, B.A., Griswold, D., Landen, L., Perry, T.S., B, A.R., Mille, P.L., Peyser, T.A., Kilkenny, J.D.: Phys. Fluids B 5, 2259–2264 (1993) Hicks, D., Hansen, F., Bradley, D., Celliers, P., Eggert, J., Collins, G., Boehly, T., Meyerhofer, D.: Bull. Am. Phys. Soc. 49, 234 (2004) Hicks, D.G., Boehly, T.R., Celliers, P.M., Eggert, J.H., Vianello, E., Meyerhofer, D.D., Collins, G.W.: Phys. Plasmas 12, 082702 (2005) Holtkamp, D., Clark, D., Ferm, E., Gallegos, R., Hammon, D., Hemsing, W., Hogan, G., Holmes, V. King, N.: in: Furnish, M.D., Gupta, Y.M., Forbes, J. W. (eds.), A survey of high explosive-induced damage and spall in selected metals using proton radiography, AIP Press, Portland, Or, USA, pp. 477–482 (2003) Knudson, M.D., Hanson, D.L., Bailey, J.E., Hall, C.A., Asay, J. R., Anderson, W. W., Phys. Rev. Lett. 8722 (2001) Koenig, M., Benuzzi, A., Faral, B., Krishnan, J., Boudenne, J. M., Jalinaud, T., R´emond, C., A. Decoster, Batani, D., Beretta, D., Hall, T.A.: Appl. Phys. Lett. 72, 1033 (1998) Koenig, M., Benuzzi, A., Philippe, F., Batani, D., Hall, T., Grandjouan, N., Nazarov, W.: Phys. Plasmas 6, 3296–3301 (1999)

261 Koenig, M., Faral, B., Batani, D., Benuzzi, A., Bossi, S., R´emond, C., Perrine, J., Temporal, M., Atzeni, S. Boudenne, J. M.: Phys. Rev. Lett. 74, 2260 (1995) LePape, S., Koenig, M., Vinci, T., Martinolli, E., Bennuzzi-Mounaix, A., Hicks, D., Patel, P., Mackinnon, A., Romagnani, L., Borghesi, M., Kar, S., Boehly, T.: HEDP (2006) Martinolli, E., Koenig, M., Baton, S.D., Santos, J.J., Amiranoff, F., Batani, D., Perelli-Cippo, E., Scianitti, F., Gremillet, L., M´elizzi, R., Decoster, A., Rousseaux, C., Hall, T.A., Key, M.H., Snavely, R., MacKinnon, A., Freeman, R.R., King, J.A., Stephens, R., Neely, D., Clark, R.J.: Phys. Rev. E, to be published (2006) Park, H.-S., Chambers, D.M., Chung, H.-K., Clarke, R.J., Eagleton, R., Giraldez, E., Goldsack, T., Heathcote, R., Izumi, N., Key, M.H., King, J.A., Koch, J.A., Landen, O.L., Nikroo, A., Patel, P.K., Price, D.F., Remington, B.A., Robey, H. F., Snavely, R.A., Steinman4, D.A., Stephens, R.B., Stoeckl, C., Storm, M., Tabak, M., Theobald, W., Town, R.P.J., Wickersham, J.E., Zhang, B.B.: Phys. Plasmas, 13, 056309 (2006) Stephens, R.B., Snavely, R.A., Aglitskiy, Y., Amiranoff, F., Andersen, C., Batani, D., Baton, S.D., Cowan, T., Freeman, R.R., Hall, T., Hatchett, S.P., Hill, J.M., Key, M.H., King, J.A., Koch, J.A., Koenig, M., MacKinnon, A.J., Lancaster, K.L., Martinolli, E., Norreys, P., Perelli-Cippo, E., Gloahec, M.R.L., Rousseaux, C., Santos, J.J., Scianitti, F.: Phys. Rev. E 69, 066414 (2004) Stevenson, D.J.: Science 214, 611–619 (1981)

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Astrophys Space Sci (2007) 307:263–267 DOI 10.1007/s10509-006-9285-7

O R I G I NA L A RT I C L E

Hydrogen and Helium at High Density and Astrophysical Implications G. Chabrier · D. Saumon · C. Winisdoerffer

Received: 18 April 2006 / Accepted: 24 July 2006 C Springer Science + Business Media B.V. 2007


Abstract We briefly summarize the present status of theoretical and experimental investigations aimed at describing the thermodynamic properties of hydrogen and helium at high density. We confront various theoretical models to presently available experiments and we consider the astrophysical implications for giant planet interiors. Keywords Plasma physics . Equation of state

ture and cooling of these astrophysical bodies thus requires the knowledge of the equation of state (EOS) and the transport properties of such dense plasmas, including a correct description of pressure ionization and of the mixture’s thermodynamic stability. In this short contribution, we focus on the range of astrophysical objects characteristic of the socalled substellar objects, i.e. objects with masses below the hydrogen-burning limit (m ≤ 0.075 M⊙ ). This encompasses brown dwarfs and gaseous planets.

1 Introduction An accurate determination of the thermodynamic properties of matter under extreme conditions of temperature and density, in particular hydrogen and helium, is required for a correct description of the mechanical and thermal properties of various cool and dense astrophysical bodies, including giant planets, brown dwarfs and low-mass stars (i.e., stars less massive than the Sun). These objects are composed dominantly of ion-electron plasmas, where ions are strongly correlated and electrons are strongly or partially degenerate. Pressure ionization of both hydrogen and helium occurs along the internal profile of these bodies, which bears major consequences for their structural, cooling and magnetic properties. Furthermore, complex phenomena such as phase transition or phase separation may take place in their interior, presenting challenging problems. The characterization of the strucG. Chabrier (
) · C. Winisdoerffer Ecole Normale Sup´erieure de Lyon, C.R.A.L.(UMR 5574 CNRS), Lyon, France e-mail: [email protected] D. Saumon Los Alamos National Laboratory, Mail Stop P365, Los Alamos, NM 87545, USA

2 The equation of state of dense fluid hydrogen and helium 2.1 Hydrogen: pressure dissociation and ionization The description of hydrogen pressure-ionization, or metallization, has remained a challenging problem since the pioneering work of Wigner and Huntington (1935). Much experimental work has been devoted to this problem, but no definitive result has been reached yet. Several highpressure shock wave experiments have been conducted in order to probe the EOS of deuterium in the regime of pressure ionization. Gas gun shock compression experiments are generally limited to pressures below 1 Mbar (Nellis et al., 1983), probing only the domain of molecular hydrogen. New techniques include laser-driven shock-wave experiments (Collins et al., 1998, 2001; Mostovych et al., 2000; Boehly et al., 2004), pulse-power compression experiments (Knudson et al., 2004) and convergent spherical shock wave experiments (Belov et al., 2002; Boriskov et al., 2003) that can achieve pressures up to 5 Mbar in fluid deuterium at high temperature, exploring for the first time the regime of pressure-dissociation and ionization. These recent experiments give different results at P
1 Mbar, however, and this Springer

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controversy needs to be settled unambiguously before a robust comparison between experiment and theory can be made in the very domain of hydrogen pressure ionization. On the theoretical front, a lot of efforts has been devoted to describing the pressure ionization of hydrogen. The EOS commonly used for modeling Jovian planet interiors is the Saumon-Chabrier-Van Horn (SCVH) EOS (Saumon and Chabrier, 1991, 1992; Saumon et al., 1995) which includes a detailed description of the partial ionization regime. This EOS reproduces the Hugoniot data of Nellis et al. (1983) but yields temperatures about 30% higher than the gas reshock data, indicating insufficient D2 dissociation (Holmes et al., 1995). A slightly revised version (Saumon et al., 2000) recovers the gas gun reshock temperature data as well as the laser-driven shock wave results (Collins et al., 1998), with a maximum compression factor of ρ/ρ0 ≃ 6, where ρ0 = 0.17 g cm−3 is the initial density of liquid deuterium at 20 K. On the other hand, the earlier SESAME EOS (Kerley, 1972), based on a similar formalism, predicts a smaller compression factor, with ρ/ρ0 ≃ 4, in general agreement with all the other recent shock wave experiments. Ab initio approaches for the description of dense hydrogen include path integral Monte Carlo (PIMC) (Militzer and Ceperley, 2000; Militzer et al., 2001; Bezkrovniy et al., 2004) and Quantum Molecular Dynamics (QMD) simulations. The latter combine molecular dynamics (MD) and Density Functional Theory (DFT) to take into account the quantum nature of the electrons (Lenosky et al., 2000; Bagnier et al., 2001; Desjarlais, 2003; Bonev et al., 2004). The relevance of earlier MD-DFT calculations was questioned on the basis that these simulations were unable to reproduce data from gas-gun experiments (Lenosky et al., 2000). This problem has been solved with more accurate simulations (Bagnier et al., 2001; Desjarlais, 2003; Bonev et al., 2004). Although an ab initio approach is more satisfactory than the phenomenological approach based on effective potentials, in practice these simulations also rely on approximations, such as the handling of the so-called sign problem for the antisymmetrization of the fermion wave functions, or the calculation of the electronic density functional itself (in particular the exchange and correlation effects), or the use of effective pseudo-potentials of restricted validity, not mentioning finite size effects. Moreover, these simulations are too computationally intensive for the calculation of an EOS covering several orders of magnitude in density and temperature, as necessary for the description of the structure and evolution of astrophysical bodies. Figures 4 and 5 of (Saumon and Guillot, 2004) and Figure 1 of (Chabrier et al., 2006) compare experimental and theoretical Hugoniots in the P-ρ and P-T planes. Whereas the SCVH EOS achieves a maximum compression similar to the laser-driven data, most of the other models predict maximum compression factors in the P-ρ plane in agreement with the more recent data. The MD-DFT results, however, preSpringer

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dict temperatures for the second shock significantly larger than the experimental results (Holmes et al., 1995). Even though the experimental double-shock temperature may be underestimated due to unquantified thermal conduction into the window upon shock reflection, and thus may represent a lower limit on the reshock temperatures, the disagreement in the T-V plane is significant. As mentioned above, the degree of molecular dissociation has a significant influence on the thermodynamic properties of the fluid and insufficient dissociation in the simulations may result in overestimates of the temperature. It has been suggested that the LDA/GGA approximations used in MD-DFT underestimate the dissociation energy of D2 (St¨adele and Martin, 2000). A correct calculation would thus lead to even less dissociation. The fact that compression along the experimental Hugoniot remains small thus suggests compensating effects in the case of hydrogen. More recent, improved QMD simulations (Bonev et al., 2004), however, seem to partly solve this discrepancy and to produce reshock temperatures in better agreement with the experimental results. Peak compression in the modern MDDFT simulations occurs in the ∼0.2–0.5 Mbar range around a dissociation fraction of ∼50%. The differences in the behaviour of hydrogen at high density and temperature illustrated by these various results bear important consequences for the structure and evolution of our Jovian planets. These differences must be correctly understood before the description of hydrogen pressure dissociation and ionization stands on firm grounds. As noted by Boriskov et al. (2005), all the recent experiments agree quite well in terms of the shock speed us versus the particle velocity u p , almost within their respective error bars. When solving the Rankine-Hugoniot equations, error bars and differences in (us , u p ) are amplified in a P-ρ diagram by a factor of (ρ/ρ0 − 1). These are challenging experiments as the differences seen in the P-ρ diagram arise from differences in us and u p of less than 3%. Ongoing high-pressure isentropic compression experiments (Knudson, this volume) are promising techniques to help address this challenge. 2.2 Helium: equation of state and phase diagram A model EOS for helium at high density, covering the regime of pressure ionization, has been developped recently by Winisdoerffer and Chabrier (WC05) (Winisdoerffer and Chabrier, 2005). This EOS, based on effective interaction potentials between He, He+ , He++ and e− species, reproduces adequately experimental Hugoniot and sound speed measurements up to ∼1Mbar. Figure 12 of WC05 illustrates the expected phase diagram of fluid helium at high temperatures and densities, for conditions characteristic of the interior of substellar objects. In this model, pressure ionization is predicted to occur directly from He to He++ for T  105 K. Because of the

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uncertainties in the treatment of the interactions at high density, however, the predicted ionization density ranges from a few to ∼10 g cm−3 , i.e. P ∼ 9–20 Mbar, depending on the temperature. This is significantly larger than the ρ ≈ 1 g cm−3 density above which available measurements of electrical conductivity of helium (Ternovoi et al., 2001; Fortov et al., 2003) predict that the plasma is substantially ionized. These measurements, however, conflict with more recent conductivity measurements (Cellier, this volume) and MDDFT calculations (Mazevet, this volume). It must be kept in mind that the reported measurements are model dependent and that conductivity determinations imply some underlying EOS model.

at T = 0 is predicted to occur at a pressure P
4.0 Mbar, from DFT calculations based on exact exchange calculations (St¨adele and Martin, 2000). There is so far no published experimental evidence for the PPT but it cannot yet be ruled out. Given the difficulty of modeling this region of the phase diagram of hydrogen, only experiments can ultimately establish whether a PPT exists or not. The existence of a phase separation between hydrogen and helium under conditions characteristic of Jupiter and Saturn interiors was first suggested by Smoluchowski (1967) and Salpeter (1973) and the first detailed calculations were done by Stevenson and Salpeter (1977). A phase separation is a first order transition which implies a concentration and thus a density discontinuity below a critical temperature between two phases in equilibrium, as given by the Gibbs phase rule:

3 Plasma phase transition. Hydrogen-helium phase separation

µiI = µiI I ⇒ xiI = xiI I e− kT ,

The pressure ionization and metallization of hydrogen have been predicted to occur through a first order phase transition, the so-called plasma phase transition (PPT) (Wigner and Huntington, 1935; Saumon and Chabrier, 1992; Norman and Starostin, 1968; Ebeling and Richert, 1985; Saumon and Chabrier, 1989; Kitamura and Ichimaru, 1998). Nearly all of these PPT calculations are based on chemical EOS models. Such models are based on a model Helmholtz free energy that includes contributions from (1) neutral particles (atoms and molecules), (2) a fully ionized plasma, and (3) usually a coupling between the two. It is well-known that realistic fully ionized plasma models become thermodynamically unstable at low temperatures and moderate densities. This is analogous to the behavior of expanded metals at T = 0 that display a region where d P/dρ < 0 and even P < 0 (Pines and Nozi`eres, 1958). This behavior of the fully ionized plasma model is formally a first order phase transition and reflects the formation of bound states in the real system. In other words, the chemical models have a first order phase transition built in from the onset, and this phase transition coincides, not surprisingly, with the regime of pressure ionization. This represents a common flaw in this type of models and it follows that their prediction of a PPT in hydrogen is not credible. On the other hand, recent ab initio simulations find a sharp (6 ± 2%) volume discontinuity at constant pressure (Bonev et al., 2004; Scandalo, 2003) or d P/dT < 0 at constant volume (Magro et al., 1996), a feature consistent with the existence of a first order phase transition. At the same time, the pair correlation function exhibits a drastic change from a molecular to an atomic state with a metallic character (finite density of electronic states at the Fermi level). These transitions are found to occur in the ∼0.5–1.25 Mbar and ∼1500–3000 K temperature range. While these results are suggestive, a systematic exploration of this part of the phase diagram remains to be done. Note that a first order structural transition for H2

G

(1)

where µi , xi denote respectively the equilibrium chemical potential and number concentration of the species i in phase I and II and G is the excess (non-ideal) mixing enthalpy between the two phases. Under the influence of a planet’s gravity field, a density discontinuity yields an extra source of gravitational energy as the dense phase dropplets (namely helium-rich ones in the present context) sink towards the center of the planet. This, in turn, translates into a delayed cooling time and thus a longer time scale (age) t to reach a given (observed) luminosity L: Mg ρ R¯ E ρ t ≈ ≈ , L L

(2)

where M is the mass fraction experiencing phase separation, ρ is the density difference between the two phases, g is the planet gravity and R¯ is the planet’s radius. In Saturn’s case, such an additional source of energy is required to explain the observed luminosity at the correct age, i.e. the age of the solar sytem, ∼4.5 × 109 yr (Fortney and Hubbard, 2004). There are few studies of the H/He phase diagrams. Stevenson’s (1982) calculations are based on the socalled binary ionic mixture (BIM) model, where the electrons are considered as a rigid background. These calculations were extended by Guillot and Chabrier (unpublished) by including the electron polarizability. Although yielding different values for the critical temperature, these calculations yield similar qualitative results, i.e. a critical temperature increasing with decreasing pressure. Both calculations, however, assume that phase separation takes place in the fully ionized part of the planet interior, i.e. in the H+ /He++ domain. Electronic structure calculations for the T = 0 H/He solid alloy, with no assumption on the degree of ionization of the plasma, were first conducted by Kleipeis et al. (1991). Finite temperature results were obtained by applying an estimated entropy correction. The following, finite temperature calcuSpringer

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lations were performed by Pfaffenzeller et al. (1995) within the framework of the MD-DFT. A striking result of these latter calculations is the prediction of an increasing critical temperature with increasing pressure, a result qualitatively different from the fully ionized calculations, and which has major consequences for jovian planet evolution. Although the most detailed calculations of the H/He phase separation diagram so far, these calculations, however, suffer from some limitations. First of all, these calculations were performed in the canonical ensemble and a thermostat must thus be used to calculate the forces acting on the ions when solving Newton’s equation of motion. The effects due to the thermostat remain unquantified. Although expected to be small, the possibility that these effects yield uncertainties comparable to the calculated excess (non-ideal) enthalpy, itself a very small (1%) contribution to the total mixing enthalpy, can not be excluded. Second, the excess enthalpy in Pfaffenzeller et al. (1995) is calculated directly from the difference between the final and the initial states, yielding an incorrect evaluation of the contraction work thus of the pression along the changes of equilibrium states (see Winisdoerffer et al., 2004). No accurate H/He phase diagram is thus available yet. A new MD-DFT method, allowing the calculation of the excess chemical potential of a fraction of helium atoms immersed in a hydrogen fluid in the microcanonical ensemble has been derived recently (Winisdoerffer et al., 2004). The microcanonical ensemble is the most natural ensemble for MD calculations, preventing the use of thermostats. Derivation of the complete phase diagram with an accurate functional for the electrons, however, is prohibitively time consuming. Comparisons for a few test cases between these calculations and calculations conducted in a different ensemble, however, might enable us to quantify the effects due to external thermostats. The problem of the phase diagram of dense H/He mixtures is perhaps the most important remaining problem regarding the interior structure and evolution of Jupiter and Saturn and needs to be addressed both experimentally and theoretically. It is also one of the few such problems that is well constrained by astrophysical observations. Indeed, some of the aforementioned H/He critical point or critical line determinations can be excluded from astrophysical constraints arising from Jupiter and Saturn cooling history (Guillot et al., 1995).

4 Implications for Jupiter and Saturn interiors A detailed study of the influence of the EOS of hydrogen and, to a lesser extent, of helium on the structure and evolution of Jupiter and Saturn has been conducted recently (Saumon and Guillot, 2004). Fortunately, some shock wave experiments overlap the adiabats of Jupiter’s and Saturn. As demonstrated by Saumon and Guillot (2004), the small (≤5%) difference on Springer

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the (P, ρ) relation along the adiabat between the SCVH and SESAME EOS, representative of the two sets of experimental results, is large enough to affect appreciably the interior structure of the models. A slightly modified version of the SESAME EOS, which recovers the H2 entropy at low temperature and density, yields Jupiter models with a very small core mass, Mcore ∼ 1 M⊕ (M⊕ is the mass of the Earth) and a mass M Z ∼ 33 M⊕ of heavy elements (Z > 2) mixed in the H/He envelope. The SCVH EOS yields models with Mcore ∼0–6 M⊕ and M Z ∼ 15–26 M⊕ . Models of Saturn are less sensitive to the EOS differences, since only ∼70% of its mass lies at P > 1 Mbar, compared to 91% for Jupiter. Models computed with the SCVH and the modified SESAME EOS have Mcore = 10–21 M⊕ and M Z = 20–27 M⊕ and 16–29 M⊕ , respectively. As shown by Saumon and Guillot (2004), the temperature along the adiabat is quite sensitive to the choice of the EOS. This affects the thermal energy content of the planet and thus its cooling rate and evolution. Equations of state which are adjusted to fit the deuterium reshock temperature measurements (Ross, 1998) lead to models that take ∼3 Gyr for Jupiter to cool to its present state. Even when considering uncertainties in the models, or considering the possibility of a H/He phase separation, such a short cooling age is unlikely to be reconciled with the age of the solar system. This astrophysical constraint suggests that the reshock temperature data are too low. 5 Conclusions In this brief review, we have examined the present status of the description of the thermodynamic properties of dense hydrogen and helium. The description of the pressure ionization and more generally the EOS of these elements at high density determine the mechanical and thermal properties, thus the structure and the evolution of substellar objects, brown dwarfs and jovian planets. Modern high-pressure experiments and/or observations remain for now too uncertain to enable us to discriminate between most EOS models in planet interiors. Upcoming experiments like pre-compressed targets or isentropic compression experiments, however, will lead eventually to a better determination of these EOS and, eventually, of the characterization of phase separation and phase transition in the interior of these objects. Acknowledgments The work of DS was supported in part by the United States Department of Energy under contract W-7405-ENG-36.

References Bagnier, S., Blottiau, P., Cl´erouin, J.: Phys. Rev. E 63, 015301 (2001) Boehly, T., Hicks, D., Celliers, P., et al.: Phys. of Plasmas 11, L49 (2004) Belov S.I., Boriskov G.V., Bykov A.I., et al.: JETP Lett. 76, 433 (2002)

Astrophys Space Sci (2007) 307:263–267 Bezkrovniy, V., Filinov V.S., Kremp, D., Bonitz, M., Schlanges, M., Kraeft W.D., et al.: Phys. Rev. E 70, 057401 (2004) Bonev S.A., Militzer, B., Galli, G.: Phys. Rev. B 69, 014101 (2004) Boriskov G.V., Bykov A.I., Ilkaev R.I., Selemir V.D., Simakov G.V., Trunin R.F., et al.: Dokl. Phys. 48, 553 (2003) Boriskov G.V., Bykov A.I., Ilkaev R.I., Selemir V.D., Simakov G.V., Trunin R.F., et al.: Phys. Rev. B 71, 092104 (2005) Chabrier, G., Saumon, D., Potekhin, A.: J. Phys. A 39, 4411 (2006) Collins G.W., Celliers P.M., Da Silva L.B., et al.: Phys. Rev. Lett. 87, 165504 (2001) Collins G.W., Da Silva L.B., Celliers P.M., et al.: Science 281, 1178 (1998) Desjarlais M.P.: Phys. Rev. B 68, 064204 (2003) Ebeling, W., Richert, W.: Phys. Lett. 108A, 80 (1985) Filinov V.S., Bonitz, M., Fortov V.E., Ebeling, W., Levashov, P., Schlanges, M.: Contrib. Plasma Phys. 44, 388 (2004) Fortney, J., Hubbard, W.: Astrophys. J. 608, 1039 (2004) Fortov V.E., Ternovoi, V.Ya., Zhernokletov M.V., et al.: JETP 97, 259 (2003) Guillot, T., Chabrier, G.: (unpublished) Guillot, T., Chabrier, G., Gautier, D., Morel, P.: Astrophys. J. 450, 463 (1995) Holmes N.C., Ross, M., Nellis W.J.: Phys. Rev. B 52, 15835 (1995) Kerley, G.: Los Alamos Lab. Rep. LA-4476 (1972) Kitamura, H., Ichimaru, S.: J. Phys. Soc. Japan 67, 950 (1998) Kleipeis, J., Schafer, K., Barbee, T., Ross, M.: Science 254, 986 (1991) Knudson M.D., Hanson D.L., Bailey J.E., Hall C.A., Asay J.R., Deeney, C.: Phys. Rev. B 69, 144209 (2004) Lenosky, T., Bickham S.R., Kress J.D., Collins L.A.: Phys. Rev. B 61, 1 (2000) Magro W.R., Ceperley D.M., Pierleoni, C., Bernu, B.: Phys. Rev. Lett. 76, 1240 (1996)

267 Militzer, B., Ceperley D.M., Kress J.D., Johnson J.D., Collins L.A., Mazevet, S.: Phys. Rev. Lett. 87, 275502 (2001) Militzer, B., Ceperley D.M.: Phys. Rev. Lett. 85, 1890 (2000) Mostovych A.N., Chan, Y., Lehecha, T., Schmitt, A., Sethan J.D.: Phys. Rev. Lett. 85, 3870 (2000) Nellis W.J., Mitchell A.C., van Thiel, M., Devine G.J., Trainor R.J., Brown, N.: J. Chem. Phys. 79, 1480 (1983) Norman G.E., Starostin A.N.: High Temp. 6, 394 (1968) Pfaffenzeller, O., Hohn, D., Ballone, P.: 74, 2599 (1995) Pines, D., Nozi`eres, P.: Theory of Quantum Fluids (1958) Ross, M.: Phys. Rev. B 58, 669 (1998); erratum: Phys. Rev. B 60, 6923 (1999) Salpeter, E.: Astrophys. J. 181, L83 (1973) Saumon, D., Chabrier, G.: Phys. Rev. Lett. 62, 2397 (1989) Saumon, D., Chabrier, G.: Phys. Rev. A 44, 5122 (1991) Saumon, D., Chabrier, G.: Phys. Rev. A 46, 2084 (1992) Saumon, D., Chabrier, G., Van Horn H.M.: Astrophys. J. Suppl. Ser. 99, 713 (1995) Saumon, D., Chabrier, G., Xu and Wagner: High Press. Res. 16, 331 (2000) Saumon, D., Guillot, T.: Astrophys. J. 609, 1170 (2004) Scandalo, S.: Proc. Nat. Acad. Sci. 100, 3051 (2003) Smoluchowski, R.: Nature 215, 69 (1967) St¨adele, M., Martin R.M.: Phys. Rev. Lett. 84, 6070 (2000) Stevenson, D.: (1982) Stevenson, D., Salpeter, E.: Astrophys. J. Suppl. Ser. 35, 221 (1977) Ternovoi, V.Ya., et al.: In: Furnish, M., Thadhani, N., Horie, Y. (eds.), Shock Compression of Condensed Matter, p. 107 (2001) Wigner, E., Huntington H.B.: J. Chem. Phys. 3, 764 (1935) Winisdoerffer, C., Chabrier, G., Zerah, G.: Phys. Rev. E 70, 6403 (2004) Winisdoerffer, C., Chabrier, G.: Phys. Rev. E 71, 026402 (2005)

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Astrophys Space Sci (2007) 307:269–272 DOI 10.1007/s10509-006-9231-8

O R I G I NA L A RT I C L E

Graded-Density Reservoirs for Accessing High Stress Low Temperature Material States Raymond F. Smith · K. Thomas Lorenz · Darwin Ho · Bruce A. Remington · Alex Hamza · John Rogers · Stephen Pollaine · Seokwoo Jeon · Yun-Suk Nam · J. Kilkenny

Received: 1 May 2006 / Accepted: 7 August 2006 C Springer Science + Business Media B.V. 2006


Abstract In recently developed laser-driven shockless compression experiments an ablatively driven shock in a primary target is transformed into a ramp compression wave in a secondary target via unloading followed by stagnation across an intermediate vacuum gap. Current limitations on the achievable peak longitudinal stresses are limited by the ability of shaping the temporal profile of the ramp compression pulse. We report on new techniques using graded density reservoirs for shaping the loading profile and extending these techniques to high peak pressures. Keywords Quasi-isentropic compression . High compression rates Understanding high stress-low temperature compressive states is relevant to the study of planetary interiors (Remington et al., 2005). Traditionally multi-Mbar longitudinal stress (Px ) states have been accessed by launching a near instantaneous compressive shock into the material. Shock waves are associated with large jumps in temperature which greatly increases the thermal contribution to Px , and can cause melting of the material under study. In aluminum, shock stresses above 1.4 Mbar produce temperatures above 4000 K and melting of the sample (Chijioke et al., 2005).

R. F. Smith (
) · K. T. Lorenz · D. Ho · B. A. Remington · A. Hamza · S. Pollaine Lawrence Livermore National Laboratory, P.O. Box 808, CA 94550 e-mail: [email protected] J. Rogers · S. Jeon · Y.-S. Nam University of Illinois at Urbana-Champaign J. Kilkenny General Atomics, P.O. Box 85608, CA 92186–5608

Recent laser-driven shockless compression techniques (Smith et al., 2006; Edwards et al., 2004; Lorenz et al., 2006; Swift and Johnson, 2005) have demonstrated quasiisentropic compression (ICE) in an aluminum sample to peak stresses over 1 Mbar (Smith et al., 2006) and at estimated temperatures of 500 K. The isentrope generally lies to the compressive side of the Hugoniot in pressure-volume (P-V) space. Since the Al melt temperature is measured to increase with compression, isentropic loading from room temperature to multi-Mbar stresses will stay below the melt line. The smaller amount of internal energy imparted into the material within the ICE platform allows for greater compression for comparable Px on shock experiments. The technique has been demonstrated with several drivers such as the magnetic pulse loading of the Sandia Z-machine (Asay, 1999; Reisman et al., 2001; Hall et al., 2001), pillow impactors in gas guns facilities (Chhabildas and Barker, 1997; Asay, 1997) and the chemical energy of high explosives (Barnes et al., 1974; Tasker et al., 2004). The time scales for these experimental platforms range from 100’s of ns to several microseconds. In the case of the gas-gun driven ICE platform the mm thick impactor is constructed using a graded density layered-plate approach that initially produces a series of small steps in the loading, which subsequently transition to smooth compression as a result of wave interactions in the layer plates (Chhabildas and Barker, 1997). In laser-driven ICE experiments the loading time is over tens of nanoseconds. Laser-driven ramp compression experiments have recently been used to measure material strength (Lorenz et al., 2005) and the kinetics of polymorphic phase transformations (Smith et al., 2006). Currently the highest pressure achieved on laser-driven ICE targets is 2 Mbar (Lorenz et al., 2006). With current laser ICE target designs the ramp compression rise time scales inversely with peak Px , which for high Px results in hydrodynamic steepening Springer

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of the ramp compression wave into a shock over short distances. For a given target thickness this places a limit on the maximum Px that can be applied to a sample while still ensuring shockless compression. Within this paper we describe new techniques which incorporate graded densities into the standard laser ICE target design for increasing the rise time within laser-driven ramp compression experiments. The developments of these techniques are important for realizing the potential of shockless compression to a peak Px over 10 Mbar on the National Ignition Facility (NIF) (Remington et al., 2005; Edwards et al., 2004). The target design for the laser-driven shockless compression consists of a low-Z reservoir foil followed by a vacuum gap and the target to be shocklessly compressed (as shown in Fig. 1(a)). In the experiments described here one beam of the Janus laser at 527 nm delivered a maximum of 380 J in a 4 ns square pulse onto the front surface of the reservoir material. A kinoform phase plate (KPP), inserted into the beamline to spatially smooth and shape the laser focal spot, generated a ∼1 mm square planar (I/I ∼ 5%) region at the focal plane which contained an estimated 80% of the total drive energy, giving a maximum on-target intensity of ∼5 × 1012 W/cm2 . The focused laser beam launches an ablatively driven shock through the reservoir. Reservoir materials typically consist of a plastic foil (<300 µm thick) doped with a higher Z material (e.g. Br) in order to absorb x-rays generated in the laser corona. After breakout from the rear surface shock heating and momentum cause the reservoir material to dissociate and unload across a vacuum gap. Transit across the vacuum gap causes the mass density gradients along the target axis to relax as a function of distance from the original reservoir/vacuum gap interface. The unloading reservoir material monotonically loads up against the sample and the imparted momentum launches a ramp longitudinal stress wave through the material. The temporal profile of the compression wave may be shaped to a limited extent by varying the size of the vacuum gap, the density of the reservoir or by controlling the laser energy (Lorenz et al., 2006). In the experiments described here we apply the ramp compression wave to an Al/LiF target. As the ramp stress wave reaches the back surface of the Al, the sample begins to accelerate into a 500 µm thick LiF window material. The time history of the Al/LiF interface acceleration is recorded with a line imaging velocity interferometer (VISAR) with two channels set at different sensitivities (Celliers et al., 2004) (Fig. 1(b)). The output of the VISAR is recorded by a fast optical streak camera. Fringe movement is linearly proportional to the velocity of the Al/LiF interface. This allows for accurate measurement of the interface velocity (after taking into account the refractive index of the window (Wise and Chhabildas, 1986)) as a function of time. As Al and LiF are well impedence matched the VISAR effectively records the Al particle velocity history. This information allows for the Springer

Astrophys Space Sci (2007) 307:269–272

Fig. 1 (a) Schematic of Laser ICE target. (b) The output of the VISAR as recorded with a streak camera gives a temporal history of Al particle veocity and a spatial record of the applied longitudinal stress wave. Fringe movement is linearly proportional to the Al/LiF interface velocity

compression source to be determined via a back-integration technique where the time dependent particle velocity at the rear surface is used as an input (Hayes, 2001; Hayes et al., 2003, 2004). Due to increase of sound speed with increasing Px the ramp compression wave will eventually steepen up into a shock within the Al sample. This would result in a near instantaneous jump in entropy and off-isentropic compression (and possible target melting) would ensue. The maximum Al thickness is therefore designed to be less than the calculated shock-up thickness. For all reported Laser ICE experiments the laser pulse duration is designed to be less than the shock transit time through the reservoir. Therefore by the time the shock reaches the reservoir-vacuum interface the initial steady shock has transformed into a blast wave which contains no information about the temporal history of the laser drive. For a fixed target design higher ramp pressures may be achieved in the sample by increasing the input laser energy. This results in increased peak stresses in the blast wave exiting the reservoir and increased velocities of the material unloading across the vacuum gap. Increased peak stresses are then launched into the sample material but over increasingly shorter timescales. Using this technique peak longitudinal stresses of 0.1 and 2 Mbar have ramp compression rise times of ∼30 ns (Smith et al., 2006) and 5 ns (Lorenz et al., 2006), respectively. The dynamics of the unloading solid reservoir produces a shaped ramp compression profile which tends to shock up

Astrophys Space Sci (2007) 307:269–272

at low Px . There is a need therefore to develop techniques for shaping the time history of the compression wave which will firstly increase the loading up time for a given peak Px and secondly soften the gradient in the ramp wave such that eventual steeping into a shock will tend to occur at the top of the loading profile. The net effect in both cases, for a given peak Px , is to increase the shock up distance and hence for a fixed target thickness higher levels of shockless compression may be obtained. In the experiments reported here we use two different techniques for applying a density gradient onto the gap side of the reservoir material. The expectation is that the density gradient slows the rate at which momentum is imparted into the target and by customizing this gradient we can ultimately customize the shape of the pressure profile. The first reservoir design used consists of a 120 µm thick 1% Br/CH laser ablator with 60 µm of SU8 (CH6 O4 N5 ) photopolymer glued onto the vacuum side. Recently developed phase contrast lithographic techniques (Jeon et al., 2004) are used to produce 3D nanostructures which reduce in size with increasing depth within an 60 µm thick SU8 photopolymer (see Fig. 2(a) insert). The resultant density gradient is characterized with transmission x-ray radiography, with a spatial resolution of 0.3 µm, to go from full density to 19% density over 60 µm (Fig. 2(a)). Using the target conditions described in Fig. 1(a) data is taken for solid density and graded density SU8. The time history of the ramp pressure profiles are shown in Fig. 2(b). It is observed that for the same peak stress the rise time of the ramp compression wave is ∼30% longer for the graded density reservoir case. Importantly the slope of the ramp compression wave associated with the graded density reservoir is noticeable reduced in the stress range over which shocks typically develop within solid density reservoirs. Approximately 30% more laser energy was required in the graded density target to reach in the same peak stresses as for the solid density reservoir. The peak longitudinal stress is related to the total amount of momentum imparted by the impacting reservoir material into the Al sample. The higher amount of laser energy is therefore needed for the graded density material to match the momentum associated with the solid density reservoir. Also shown are the calculated pressure profiles from the LASNEX plasma physics code (Zimmerman and Kruer, 1975) which as an approximation used a linear density gradient from full density to 19% density over the 60 µm SU8 layer. The simulations show some disagreement in the gradient of the pressure profiles but qualitatively agree with the experimental observations of increased compression rise time with the use of a density gradient. Further improvements are expected in the compression rise time through customization of the density gradient profile. Another approach for incorporating an effective integrated density gradient into a solid density reservoir is by direct micro-machining three dimensional features into the gap

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Fig. 2 (a) Novel phase contrast lithographic techniques (Jeon et al., 2004) produce 3D nanostructures which reduce in size with increasing depth within an 60 µm thick SU8 photopolymer (see insert). The resultant density gradient is characterized with x-ray radiography to go from full density to 19% density over 60 µm. (b) Using the target conditions described in Fig. 1(a) the time history of the ramp pressure profile with and without a density gradient in the SU8 is shown. Also shown are the calculated pressure profiles from the LASNEX plasma physics code (Zimmerman and Kruer, 1975). The pressure and time axes have been normalized to make comparisons easier

side of the reservoir material. In these experiments, a sawtooth feature was diamond turned into one side of a 225 µm Polyimide [C22 H10 N2 O5 ] foil. This feature was machined in one dimension and was characterized with high resolution imaging to have a period of 10 µm and a depth of 8 µm (insert of Fig. 3). Longitudinal stress profiles measured from identical target and irradiation conditions for polyimide reservoirs with and without the sawtooth feature are shown in Fig. 3. The target without the micro-machined feature exhibits a smooth ramp up to a peak pressure of ∼0.19 Mbar. The rise time of the graded density reservoir has increased over the solid density case. The target with the density gradient shows a more structured rise with two mini-plateaus which is due to local softening followed by steepening of the ramp gradients when compared to the solid density case. Locally there is expected to be a lot of turbulence as the stress wave breaks out of the sawtooth reservoir but experimentally this is observed to be annealed out at the distance of the vacuum gap. Future experiments will concentrate on varying the structure of the Springer

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Fig. 3 An average graded density is produced within a 225 µm thick polyimide foil by machining of a sawtooth feature into the vacuum gap side. A scanning electron microscope (SEM) image (insert) reveals the diamond turned feature to have a depth of 8 µm and a period of 10 µm. The resultant ramp compression profiles with and without the sawtooth feature are shown

machined feature in order to extend the rise time and smooth out the gradients within the ramp profile. Increasing the rise time for a given peak stress in laser driven quasi-isentropic compression experiments is necessary to drive these technique into the multi-Mbar stress regime where low temperature compressive material states relevant to planetary interiors may be accessed. Using two separate techniques to introduce a graded density in the gap side of the reservoir it has been shown that the rise time of the compression wave in laser-driven ICE is increased. Further improvements are expected by customizing the shape of the density gradient which will facilitate shockless compression experiments on NIF to peak pressures over 10 Mbar. Acknowledgements This work was performed under the auspices of the U.S. Dept. of Energy by the University of California, Lawrence Livermore National Laboratory under contract No. W-7405-Eng-48.

References Asay, J.R.: Int. J. Impact Engng. 20, 27 (1997)

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Astrophys Space Sci (2007) 307:269–272 Asay, J.R.: Isentropic compression experiments on the Z Accelerator. In: Furnish, M.D., Chhabildas, L.C., Hixson, R.S. (eds) Shock Compression of Condensed Matter–1999, AIP Conf. Proceedings, NY Barnes, J.F., Blewett, P.J., McQueen, R.G., Meyer, K.A., Venable, D.: J. Appl. Phys. 45, 727 (1974) Celliers, P.M., Bradley, D.K., Collins, G.W., Hicks, D.G., Boehly, T.R., Armstrong, W.J.: Rev. Sci. Instrum. 75, 4916 (2004) Chhabildas, L.C., Barker, L.M.: Sandia Report SAND86–1888 (1986) Chijioke, A.D., Nellis, W.J., Silvera, I.F.: J. App. Phys. 98, 073526 (2005) Edwards, M.J., Lorenz, K.T., Remington, B.A., Pollaine, S., Colvin, J., Braun, D., Lasinski, B. F., Reisman, D., McNaney, J.M., Greenough, J.A., Wallace, R., Louis, H., Kalantar, D.: Phys. Rev. Lett. 92(7), 075002/1–4 (2004) Hall, C.A., Asay, J.R., Knudson, M.D., Stygar, W.A., Spielman, R., Pointon, T.D., Reisman, D.B., Toor, A., Cauble, R. C.: Rev. Sci. Instrum. 72, 3587 (2001) Hayes, D.B., Backward Integration of Equations of Motion to Correct for Free Surface Perturbations, Sandia Report SAND2001–1440 (2001) Hayes, D.B., Hall, C.A., Asay, J.R., Knudson, M.D.: J. Appl. Phys. 94(4), (2003) Hayes, D.B., Hall, C.A., Asay, J.R., Knudson, M.D.: J. Appl. Phys. 96(10), (2004) Jeon, S., Park, J.-U., Cirelli, R., Yang, S., Heitzman, C.E., Braun, P.V., Kenis, P.J.A., Rogers, J.A.: PNAS 101(34) 12428–12433 (2004) Lorenz, K.T., Edwards, M.J., Glendinning, S.G., Jankowski, A.F., McNaney, J., Pollaine, S.M., Remington, B.A.: Phys. Plasmas 12, 056309 (2005) Lorenz, K.T., Edwards, M.J., Jankowski, A.F., Pollaine, S.M., Smith, R.F., Remington, B.A.: Submitted to J. High Energy Den. Phys. (2006) Reisman, D.B., Toor, A., Cauble, R.C., Hall, C.A., Asay, J.R., Knudson, M.D., Furnish, M.D.: J. Appl. Phys. 89(3), 1625–1633 (2001) Remington, B.A., Cavallo, R.M., Edwards, M.J., Ho, D.D.-M., Lorenz, K.T., Lorenzana, H.E., Lasinski, B.F., McNaney, J.M., Pollaine, S.M., Smith, R.F.: Astrophys. Space Sci. 298(1–2) (2005) Smith, R.F., et al.: Submitted to Phys. Rev. Letts. (2006) Smith, R.F., et al.: Submitted to Phys. Rev. Letts (2006) Swift, D.C., Johnson, R.P.: Phys. Rev. E 71, 066401 (2005) Tasker, D.G., Goforth, J.H., Oona, H., Fowler, C.M., King, J.C., Herrera, D., Torres, D.: Shock Compression of condensed Matter- AIP Conference Proceedings, 706, pp. 1239–1242 (2004) Wise, J.L., Chhabildas, L.C.: pp. 441–454, Plenum, New York (1986) Zimmerman, G.B., Kruer, W.L.: Comments on Plasma Physics for Controlled Thermonuclear Fusion 2, 51–61 (1975)

Astrophys Space Sci (2007) 307:273–277 DOI 10.1007/s10509-006-9185-x

ORIGINAL ARTICLE

He Conductivity in Cool White Dwarf Atmospheres S. Mazevet · M. Challacombe · P. M. Kowalski · D. Saumon

Received: 14 April 2006 / Accepted: 13 June 2006 C Springer Science + Business Media B.V. 2006


Abstract We investigate the conductivity of warm dense helium under conditions found in the atmospheres of cool white dwarfs using ab initio simulations. The calculations performed consist of quantum molecular dynamics simulations where the electronic wavefunction at each time step is obtained using density functional theory, while the ion trajectories are calculated using the resulting quantum mechanical forces. We use both conventional DFT (PW91) and hybrid (PBE0) functionals to calculate the conductivities that provide an estimate of the ionization fraction. While the calculations are in good agreement with the measurements for the equation of state, a significant discrepancy exists with the recently measured conductivity. Keywords Atomic processes . Dense matter . Equation of state . Plasmas . Radiation mechanisms:general Introduction The possibility of using white dwarfs for dating stellar populations has generated a renewed interest in modeling their cooling rate (Fontaine et al., 2001). The coolest white dwarfs are the most important for this task but are rather poorly unS. Mazevet (
) · M. Challacombe Theoretical Division, Los Alamos National Laboratory, Los Alamos NM 87545, USA D. Saumon Theoretical Division, Los Alamos National Laboratory, Los Alamos NM 87545, USA; Applied Physics Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA P. M. Kowalski Applied Physics Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA

derstood. Modeling their outer layers requires the calculation of the physical properties of He and H-He mixtures at high densities (few g/cm3 ) and low temperatures (T < 10000 K) where conventional methods to calculate opacities usually fail (Iglesias et al., 2002). For very cool white dwarfs, the opacity of the outer layers controls the cooling rate of the star. In pure He atmospheres, the opacity is dominated by the free-free absorption process which depends directly on the ionization fraction. At the conditions of interest here, the ionization fraction is very low and uncertain as witnessed by the broad range of predictions for the density at which helium pressure ionizes (from 0.3 to 14 g/cm3 (Young et al., 1981; Fontaine et al., 1977)). To eliminate this uncertainty, we performed He conductivity calculations at conditions relevant to white dwarfs atmospheres using ab initio simulations. We performed quantum molecular dynamics (QMD) simulations where the electrons receive a full quantum mechanical treatment using finite temperature density functional theory (DFT) Mermin (1965), while the ions are propagated in time classically using the resulting quantum mechanical forces. We estimate the ionization fraction by performing electrical conductivity calculations within linear response theory (Mazevet et al., 2004). We find that the ab initio simulations are in good agreement with the experimental data for the equation of state. In contrast, the value of conductivity recently measured at densities of about 1 g/cm3 as well as its density dependence can not be reproduced by the method.

Equation of state For the calculations presented here, we used the VASP ab initio simulation code (Kresse and Hafner, 1996). The simulations were ran for typically 2 ps with time steps ranging from 2 fs for the lowest densities, to 0.5 fs at the highest, and using Springer

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1 (U0 − U1 ) + (V0 − V1 )(P0 + P1 ) = 0, 2

(1)

describes the shock adiabat through a relation between the initial and final volume, internal energy, and pressure, respectively, (V0 , U0 , P0 ) and (V1 , U1 , P1 ). As described in (Kress et al., 2001), the DFT-MD P and U for a fixed V1 are leastsquares fitted to a quadratic function in T to calculate the Hugoniot point for a given V1 and a set of T1 ’s. P1 and T1 are determined by substituting these functions and solving Equation (1). While the high pressure experimental data are very sparse, it is reassuring to find an overall good agreement with the QMD calculations as these four experimental points are extensively used to adjust a large majority of helium EOS at high pressures. For the principal Hugoniot calculation, the initial condition used is P0 = 0GPa, and ρ0 = 0.1245 g/cm3 , in agreement with the experimental data. The reference energy point for the QMD calculations was taken as U0 = 0 eV/atom and was obtained by calculating the in˚ For the ternal energy of an isolated atom in a box of 12 A. principal Hugoniot, the agreement at the highest pressure measured is rather good and within the experimental error bars. For the reshocked points, we used the QMD Hugoniot point obtained at the highest pressure as an initial condition, ρ1 = 0.4 g/cm3 , U1 = 1.7 eV/atom and P1 = 14.7 GPa. The highest QMD reshocked point calculated also agrees, within the error bars, with the experimental measurement. The QMD calculations indicate, however, a slightly softer reshock Hugoniot, in agreement with the EOS obtained using the chemical model. The chemical EOS is inspired from the He EOS of Saumon et al. (Saumon et al., 1995) which is widely used in the astrophysics community. While both EOS are calibrated to the Springer

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Exp. principal Hugoniot Exp. second shock QMD principal Hugoniot QMD second shock Chem. principal Hugoniot Chem. second shock

50 P (GPa)

54 and 128 atoms in the simulation cell. We used a Projected Augmented Wave (PAW) pseudo-potential in the PerdewWang 91 (PW91) (Perdew and Wang, 1992) parametrization of the Generalized Gradient approximation (GGA) (Blochl, 1994). We paid particular attention to the convergence of the various properties of interest as a function of the plane wave energy cutoff. This issue was recently raised for the case of hydrogen (Desjarlais, 2003). For helium, we find that it is necessary to use a cutoff energy of 800 eV in order to obtain an accurate description of the experimental Hugoniot. In contrast to the hydrogen findings, we find that the real space projection used in the QMD calculations does not introduce significant error for the pressure or the internal energy. In Fig. 1, we compare the QMD Hugoniot with the experimental data (Nellis, 1984) and with the results obtained from a free energy minimization method based on a chemical model we have developed. The Rankine-Hugoniot Equation (10)

40 30 20 10 0

0.3

0.4

0.6

0.5

0.7

0.8

3

ρ (g/cm )

Fig. 1 Helium principal and second shock Hugoniot as a function of density

first and second-shock gas gun data shown in Fig.1, they differ in several significant ways. The chemical EOS presented here was developed specifically for the calculation of opacities of He in the warm dense matter regime. Because opacities are dominated by processes involving trace species (especially free electrons), particular attention was devoted to the numerical accuracy of the chemical equilibrium. The trace species He+ 2 has been added as it affects the ionization at low temperatures and high densities. On the other hand, only the He-He interactions are included and the free electrons are treated classically. The latter approximation is justified as long as the degree of ionization remains small, but Fermi-Dirac statistics need to be introduced once pressure ionization becomes significant (at densities above 2 g/cm3 in this model). The agreement between the QMD and chemical EOS is excellent below 1 g/cm3 .

Electrical properties We now turn to the principal motivation of the current study which is the calculation of the electrical properties of helium at high pressures and for conditions relevant to white dwarf atmospheres. From the QMD trajectories produced, we calculate the electrical properties on a set of configurations using the Kubo-Greenwood formulation of the optical conductivity where the real part is given as(Harrison, 1970; Callaway, 1974). σ (ω) R =

"  2π e2 1 " W (k) f nk − f mk 3ω  k n,m,α  k   k 2  k  × ψn ∇α ψm  δ E m − E nk −
ω ,

(2)

In Equation (2), ω is the frequency, e is the electronic charge, ψnk and E nk are the electronic eigenstates and

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20

3.6

18 Hybrid Gap correction (eV)

16

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14 12 10 8 T=0K fcc MD T=0.52eV MD T=1.5eV MD T=3eV

6 4

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0.1

1 3 density (g/cm )

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Fig. 2 GGA gap variation as a function of density and for several temperatures

Fig. 3 Variation of the hybrid band gap correction as a function of density and temperature

eigenvalues for the electronic band n at a given k-point in the Brillouin zone, W (k) is the k-point weight in the Brillouin zone using the Monkhorst-Pack scheme and f nk is the Fermi distribution function. ∇α is the velocity operator along each direction (α = x, y, z) and  the volume of the simulation cell. Finally, the Kubo-Greenwood formulation is applied using the all-electron PAW potential, which does not require the correction term related to the non-locality of the pseudo-potential that would be needed if, for example, an ultrasoft pseudo-potential was used. Calculations were performed at the Ŵ point. Additional details on this type of calculation can be found in (Mazevet, 2004). At normal conditions, helium is an insulator with a rather large band gap of around 20 eV. As can be seen from Equation (2), the conductivity, and hence the ionization fraction, increases as the population of the conduction band becomes large. This occurs when either the band gap diminishes or if the temperature increases or both. Figure 2 shows the variation of the band gap as a function of density and temperature as obtained using DFT. For a regular Fcc lattice at T = 0, the method predicts that the He band gap closes at a density of 13.5 g/cm3 in agreement with previous LMTO calculations [3]. At low temperatures, 0.5 eV (1 eV = 11604 K), the band gap obtained from the simulations is almost constant in the density range investigated here. At finite temperature, the band gap shown in Fig. 2 is obtained by averaging the energy difference of the orbitals on either side of the Fermi energy along the trajectory. This is equivalent to the HOMO-LUMO gap which is defined, at zero temperature, as the energy difference between the highest occupied orbitals and the lowest unoccupied one. We show, in Fig. 2, the band gap obtained at finite temperature as the latter is the quantity entering in Equation (2). At higher temperatures, we further see that the band gap diminishes as a function of both density and temperature with a closure of the band gap predicted at around

10 g/cm3 . At the two highest temperatures, the band gap reduces faster than when an Fcc structure is used for the ions as a result of increased disorder. At low densities, the gap tends to a value of about 15 eV. For the isolated atom, calculation of the first excited state gives a value of 15.8 eV using either the PW91 (Perdew and Wang, 1992) or PBE (Perdew et al., 1996) functionals. The experimental value for the position of He first excited state is 19.82 eV. The DFT method, which is a ground state approach, is well known to systematicaly underestimate band gaps as well as excited state energies. In effect, the excited state energies used in Equation (2), E mk , and obtained from the diagonalization of the Kohn-Sham Hamiltonian are quasi excited states only and neglect, for example, the electronhole interaction which appears when an electron is promoted to a higher level. For semi-conductors and insulators, the eigenenergies need to be corrected using either time dependent DFT or Greens function approaches such as GW to allow for an accurate calculation of the electrical properties using Equation (2) (Onida et al., 2002). These methods are rather expensive computationally and can not, at present, be applied to dense plasmas where the simulation cell contains from fifty to a few hundreds atoms. To estimate the band gap correction needed, we performed DFT calculations using a hybrid functional, PBE0, with the chemistry code MondoSCF, a program suite for 0(N) SCF theory and ab initio MD (Challacombe et al., 1996). Hybrid functionals combine the orbital-dependent Hartree-Fock exchange and an explicit density functional, in the present case the GGA PBE functional (Perdew et al., 1996). Among other properties, hybrid functionals allow for an improved calculation of the band gap and at an accuracy comparable to GW calculations (Martin, 2004). Figure 3 shows the band gap correction obtained using hybrid calculations that were performed on a single configuration randomly selected from the Springer

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-1

σDC[10 (Ωm) ]

10

T=2000K T=6000K T=17406K exp. [ ] T=30240K

-2

-4

-6

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trajectory of interest. We further note that the hybrid calculations are performed at zero temperature. The temperature dependence indicated in Fig. 3 results from the use of ionic configurations obtained from a trajectory corresponding to the thermodynamics conditions indicated. We see in Fig. 3 that the correction is approximately constant in both density and temperature at about 3 eV. We notice a small increase of the correction of about 0.7 eV as the density increases between 0.1 and 4 g/cm3 . While this variation could be physical, we further note that it is within the uncertanties of the calculation performed here. The MondoSCF electronic structure code used to perform the hybrid DFT calculations uses Gaussian basis sets. In extended systems, care must be taken when the density is varied as numerical linear dependencies arise when the basis contains orbitals that are too diffuse. We find that the situation is exacerbated for helium which requires rather diffuse orbitals to converge the band gap. To account for the result obtained using the hybrid functional, we apply a uniform E = 3 eV correction to the eigenenergies above the Fermi energy when calculating the conductivities using Equation (2). We note that the occupation number f nk appearing in Equation (2) also needs to reflect the correction applied to the eigenenergies. To first order, the matrix elements appearing in Equation (2) do not need to be corrected (Hedin, 1965). We show, in Fig. 4, the variation of the DC conductivity along the 2000 K, 6000 K as well as the 17406 K (1.5 eV) isotherms. The DC conductivity is obtained by taking the zero frequency limit of Equation (2). For all the conditions studied here, the optical conductivity has a Drude like form for photon energies below 10eV. This confirms that the opacity is dominated by the free-free contribution which in turn is directly related to the electrical conductivity and the ionization fraction. For the three isotherms shown in Fig. 4, we find that pressure ionization remains moderate (weak) at these densities. While the DC conductivity varies by about 2 orders of magnitude as the density is increased from 0.5 to 6 g/cm3 , the fluid stays mostly neutral along the two lowest isotherms and up to the highest densities investigated. The temperature dependence of the DC conductivity is, in contrast, more drastic with an increase over ten orders of magnitude as the temperature is raised from 2000 K to 17406 K. Figure 4 also shows that the DC conductivity is reduced by a factor of about three when the hybrid band gap correction is introduced. We further note that at a temperature of 2000K, the calculated conductivity reaches the limit of accuracy of the method and are given as an order of magnitude estimate only. As such, we do not apply the hybrid correction at this particular temperature. Finally, Fig. 4, shows a direct comparison between the calculated conductivities and recent experimental measurements performed by (Fortov et al., 2003). The measurements strongly suggest that helium starts becoming a conductor at densities around 1 g/cm3 . It is further important to notice that

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10

10 10

-8

-10

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2

3

4

5

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density (g/cm )

Fig. 4 Comparison between the calculated and measured. He conductivities. Dotted lines: GGA calculations. (Solid lines) using the hybrid correction

while the conductivities are measured for temperatures between 15000 and 30000 K. there is no correlation between the temperature and the increase in conductivity shown in Fig. 4. In effect, the rise in the experimental conductivity around 1 g/cm3 was attributed to the effect of pressure ionization and to a rapidly closing band gap (Fortov et al., 2003). This result is in sharp contradiction with the QMD calculations which indicate conductivity values lower by about a factor of three, a weak density dependence, and a wide band gap persisting to densities well above 1 g/cm3 . The hybrid calculations which improve on the conductivity obtained using DFT properties further exacerbate the comparison with the experimental data by leading to lower conductivity values in this density range. Overall, while the experimental conductivities are of the same magnitude as those calculated, we can not reconcile the ab initio results with the experimental measurements. While the calculations performed here provide a solid benchmark for physical models to describe He at conditions found in white dwarf atmospheres, the significant disagreement with the currently available conductivity data calls for additional measurements to bring new light to this discrepancy. Measurements of the He reflectivity under similar conditions are currently performed at the OMEGA laser facility and preliminary results are reported in these proceedings (Cellier, Private communication). Conclusion Using Quantum Molecular Dynamics, we calculate the dynamical and electrical properties of helium in a regime relevant to white dwarf atmospheres. We find very good agreement with the experimental principal and second-shock Hugoniot below 1 g/cm3 . As the experimental measurements at high pressures are sparse, the current study first provides a useful benchmark for EOS modeling above 1 g/cm3 . The

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conductivity calculated using QMD and linear response theory is, however, in significant disagreement with recent experimental measurements. Acknowledgements Work supported under the auspices of the U.S. Department of Energy at Los Alamos National Laboratory under Contract W-7405-ENG-36.

References Blochl, P.E.: Phys. Rev. B 50, 17953 (1994); Kresse, G., Joubert, J.: Phys. Rev. B 59, 1758 (1999) Cellier, P., Loubeyre, P.: Private Communication Challacombe, M., Tymczak, C.J., Nemeth, K., Weber, V., Gan, C.K., Schwegler, E., Henkelman, G., Niklasson, A.: Los Alamos National Laboratory, LA-CC-04-086 Callaway, J.: Quantum theory of the solid state. Academic Press New York (1974) Desjarlais, M.P.: Phys. Rev. B 68, 064204 (2003) Fontaine, G., Brassard, P., Bergeron, P.: PASP 113, 409 (2001) Fontaine, G., Graboske, H.C., Van Horn, H.M.: ApJS 35 (1977) Fortov, V. et al.: JETP 97, 259 (2003)

277 Harrison, W.A.: Solid state theory. Mc Graw-Hill (1970) Hedin, L.: Phys. Rev. 139, A796 (1965) Iglesias, C.A., Rogers, F.J., Saumon, D.: Astrophysical Journal Letter, 569, L111 (2002) Kresse, G., Hafner, J.: Phys. Rev. B 47, RC558 (1993); Kresse, G., Furthm¨uller, J.: Comput. Mat. Sci. 6, 15–50 (1996); Kresse, G., Furthm¨uller, J.: Phys. Rev. B 54, 111 Kress, J.D., Mazevet, S., Collins, L.A., Wood, W.W.: Phys. Rev. B 63, 024203 (2001) Mermin, N.D.: Phys. Rev. 137A, 1441 (1965) Mazevet, S., Kress, J., Collins, L.A.: Atomic Processes in Plasmas, AIP 730, 139 (2004) Martin,R.M.: Electronic structure. Cambridge University Press, (2004) Nellis, W.J.: et al., Phys. Rev. Lett. 53, 1248 (1984) Onida, G., Reining, L., Rubio, A.: Rev. Mod. Phys. 74, 601 (2002) Perdew, J.P., Wang, Y.: Phys. Rev. B 46, 12947 (1992) Perdew, J.P., Burke, K., Ernzerhof, M.: Phys. Rev. Lett. 77, 3865 (1996) Perdew, J.P., Ernzerhof, M., Burke, K.: J. Chem. Phys. 105, 9982 (1996) Saumon, D., Chabrier, G., Van Horn, H.M.: Astrophysical Journal Supplement Series, 99, 713–41 (1995) Young, D.A., McMahan, A.K., Ross, M.: Phys. Rev. B 24, 5119 (1981) Zeldovich, Ya., Raizer, Yu.: Physics of shock waves and hightemperature hydrodynamic phenomena (Academic Press, New York, 1966)

Springer

Astrophys Space Sci (2007) 307:279–283 DOI 10.1007/s10509-006-9224-7

ORIGINAL ARTICLE

The Structure of Jupiter, Saturn, and Exoplanets: Key Questions for High-Pressure Experiments Jonathan J. Fortney

Received: 13 April 2006 / Accepted: 25 July 2006 C Springer Science + Business Media B.V. 2006


Abstract We give an overview of our current understanding of the structure of gas giant planets, from Jupiter and Saturn to extrasolar giant planets. We focus on addressing what highpressure laboratory experiments on hydrogen and helium can help to elucidate about the structure of these planets. Keywords Planetary systems . Jupiter . Saturn

1 Introduction In order to understand the formation of giant planets, and hence, the formation of planetary systems, we must be able to determine the interior structure and composition of giant planets. Jupiter and Saturn, our solar system’s gas giants, combine to make up 92% of the planetary mass of our solar system. Interestingly, knowledge of only a few key quantities allows us to gain important insight into their interior structure. The equation of state of hydrogen, together with measurements of the mass and radius of Jupiter and Saturn is sufficient to show that these planets are hydrogen-helium rich objects with a composition similar to that of the Sun (Demarcus, 1958). Furthermore, estimates of the transport coefficients of dense metallic hydrogen and the observation that Jupiter emits more infrared radiation than it absorbs from the Sun (Low, 1966), is sufficient to show that gas giant planet interiors are warm, fluid, and convective, not cold and solid (Hubbard, 1968). It has also been clear for some time that the composition of Jupiter and Saturn is not exactly like that of the Sun – these planets are enhanced in “heavy elements” J. J. Fortney Space Science and Astrobiology Division, NASA Ames Research Center, MS 245-3, Moffett Field, CA 94035 e-mail: [email protected]

(atoms more massive than helium), compared to the Sun (Podolak and Cameron, 1974). An understanding of how these planets attained these heavy elements, and their relative ratios, can give us a wealth of information on planetary formation and the state of the solar nebula. Looking beyond Jupiter and Saturn, we now have 200 extrasolar giant planets (EGPs) that have been found to orbit other stars. A subclass of these planets are the “hot Jupiters” that orbit their parent stars at around 0.05 AU. To date, ten planets (with masses from 0.36 to 1.5 MJ ) have been seen to transit their parents stars. All of these objects are hot Jupiters, with orbital periods of only a few days (see Charbonneau et al., 2006). These transiting planets are important because we can measure their masses and radii, thereby allowing us access to information on their interior structure (Guillot, 2005). While our understanding of the interiors of these planets will never be as detailed as that for Jupiter and Saturn, we will eventually have a very large sample of these transiting objects at various masses, compositions, and orbital distances, which will allow for an understanding of the massradius relation for giant planets under a variety of conditions. By far the most important physical input into giant planet structural models is the equation of state (EOS) of hydrogen. The decade of pressure that is most important for understanding the interiors of giant planets is 1–10 Mbar (100– 1000 Gpa) (Saumon and Guillot, 2004). In the past decade experiments have been able to probe into the lower end of this pressure range (Weir et al., 1996; Collins et al., 1998; Knudson et al., 2001; Boriskov et al., 2005). In this paper, instead of focusing on equation of state physics we will focus on key questions for understanding the structure and composition of giant planets. As we discuss giant planet interiors we will investigate how high pressure laboratory experiments have and will continue to allow us to better answer these questions.

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2 Key questions 2.1 Are planetary atmospheric abundances representative of the entire H/He envelope? This question is directly related to whether hydrogen’s molecular-to-metallic transition is continuous or first-order. Whether or not hydrogen’s transition to a metal in the fluid state is first order has always been an open issue. The importance of this question to giant planets cannot be overstated. If the transition is first order (a so-called “plasma phase transition”, or PPT) then there will be an impenetrable barrier to convection within the planet and there must also be several discontinuities at this transition. One is a discontinuity in entropy (Stevenson and Salpeter, 1977; Saumon and Chabrier, 1992). In the 1970s, W. B. Hubbard discussed that, for a fully convective and adiabatic giant planet, a measurement of the specific entropy in the convective atmosphere would essentially allow us to understand the run of temperature vs. pressure for the entire planet, as all regions would share this specific entropy (see Hubbard, 1973). However, if a PPT exists, this will not be true (Chabrier et al., 1992). Another discontinuity at the PPT would be in chemical composition, due to the Gibbs phase rule. Modern structural models of Jupiter and Saturn aim to constrain the bulk abundance and distribution of heavy elements in the interiors of these planets. We would like to understand what fraction of the heavy elements are distributed throughout the H/He envelope, and what fraction are in a central core. See Guillot (1999) and Saumon and Guillot (2004, 2005) for recent computations of the interior structure of Jupiter and Saturn. The main constraints on these models are planetary mass, radius, rotation period, and gravity field. Additional constraints would be most welcome. One potentially important constraint would be atmospheric abundances derived from entry probes or spectra. If it could be clearly shown that the molecular-to-metallic transition is indeed continuous, then mixing ratios of chemical species in the atmosphere should be representative of the entire H/He envelope, as the entire envelope should be well-mixed due to efficient convection. This could constrain the amount of heavy elements in the H/He envelope and allow for a much more precise determination of the core mass and bulk heavy element abundance. For Jupiter, the Galileo Entry Probe has measured the abundances of the important species methane and ammonia (Atreya et al., 2003). However, the abundance of water, presumably the most abundant species after helium, is still highly uncertain. Perhaps the clearest indication of the physical state of hydrogen in the molecular-to-metallic transition region (∼l–5 Mbar) would be a measurement of the hydrogen’s conductivity. To date, Weir et al. (1996) and Nellis et al. (1999) have measured the conductivity of hydrogen using a reverberation Springer

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shock technique up to 1.8 Mbar (180 Gpa). They found a four order of magnitude increase in conductivity from 0.93 to 1.4 Mbar that plateaued between 1.4 and 1.8 Mbar at a conductivity consistent with that of the minimum conductivity of a metal. These measurements appear to indicate that hydrogen’s transition to a metallic state is indeed continuous (at least at their measured temperature of 2600 K). However, the measured conductivity is still over an order of magnitude less than that expected for a fully ionized hydrogen plasma (Hubbard et al., 2002), so these measurements cannot be considered a definitive refutation of a PPT. Another open question is how the presence of neutral atomic helium (10% by number in a solar composition mixture) may affect this transition. 2.2 Heavy Elements: How much and where are they? The pressure-density relation of hydrogen is the single most important input in giant planet structural models. All things being equal, the more compressible hydrogen is, the smaller a planet will be at a given mass and composition. This has a direct bearing on model-derived constraints on the amount of heavy elements within a planet’s interior. Saumon and Guillot (2004) computed detailed interior models for Jupiter and Saturn that were consistent with all available observational constraints. They found that Jupiter models that used EOSs consistent with the 6-fold limiting deuterium compression data of Collins et al. (1998) lead to core sizes of 0–10 M⊕ , with total heavy element abundances (envelope plus core) of 10–25 M⊕ . Models computed using EOSs consistent with the harder 4.3-fold limiting compression of Knudson et al. (2001, 2004) and Boriskov et al. (2005) led to smaller cores sizes (0–3 M⊕ ) but larger heavy elements abundances (25–35 M⊕ ). Since other experiments have not been able to replicate the soft Collins et al. (1998) data, and the data of Knudson et al. and Boriskov et al. agree quite well while using different experimental setups, these harder EOS data sets are currently viewed by many as the most reliable. (For recent reviews, see Nellis, 2005, 2006). Tests of the hydrogen or deuterium EOS off of the single-shock Hugoniot, perhaps at pressures of up to a few Mbar, but temperatures below 104 K, would be most valuable. For helium, our second most important constituent, new EOS data are sorely needed. No helium EOS data have been published since Nellis et al. (1984), and this data set only reached a maximum pressure of 560 kbar (56 Gpa). In Fig. 1 we show schematic interior structures of Jupiter and Saturn. We show pressures and temperatures at three locations: the visible atmosphere (1 bar), near the molecular-to-metallic transition of hydrogen (2 Mbar), and at the top of the heavy element core of each planet. Atmospheric elemental abundances, as determined by the Galileo Entry Probe for Jupiter and by spectroscopy for Saturn, are shown within a grey box (Atreya et al., 2003). These abundances

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Fig. 1 Schematic interior structure of Jupiter and Saturn. Pressures and temperature are marked at 1 bar (100 kpa, visible atmosphere), 2 Mbar (200 Gpa, near the molecular-to-metallic transition of hydrogen), and at the top of the heavy element core. Temperatures are especially uncertain, and are taken from Guillot (2005). Approximate atmospheric abundances for “metals” (relative to solar) are shown within the grey box, in the molecular H2 region. Possible core masses, in M⊕ (labeled as “M E ”) are shown as well (Saumon and Guillot, 2004)

should at least be representative of the entire molecular H2 region. If a PPT does not exist, these abundances should be representative of the entire H/He envelope. In both planets, the molecular H2 region is depleted in helium relative to protosolar abundances (von Zahn et al., 1998; Conrath and Gautier, 2000) indicating sedimentation of helium into metallic H layers. Recent evolutionary models for Saturn indicate this helium may rain through the metallic H region and form a layer on top of the core (Fortney and Hubbard, 2003). 2.3 What are the temperatures in the deep interiors of Jupiter and Saturn? While the interior pressure-density relation sets the structure of the planet, it is the pressure-temperature relation that determines the thermal evolution. The temperature of the deep interior sets the heat content of the planet. The higher the temperatures in the planet’s interior, the longer it will take to cool to a given luminosity. This has been investigated recently by Saumon and Guillot (2004) for Jupiter. They computed evolution models of Jupiter using several different hydrogen EOSs that span the range of data obtained from LLNL laser (Collins et al., 1998) and Sandia Z (Knudson et al., 2004) data. These different EOSs predict temperatures than can differ by as much as 30% at 1 Mbar. They find that Jupiter models cool to the planet’s known luminosity in ∼3 to 5.5 Gyr using these various EOSs. This 2.5 Gyr uncertainty is rather significant. The atmospheres of Jupiter and Saturn are both depleted in helium relative to protosolar composition (Atreya et al., 2003). This observation, together with theoretical work indicating that helium has a limited solubility in metallic hydrogen at planetary interior temperatures of ∼104 K (Stevenson, 1975; Hubbard and Dewitt, 1985; Pfaffenzeller et al., 1995), indicates helium is phase separating from hydrogen and be-

ing lost to deeper layers in each planet. The evolution of Saturn, and perhaps Jupiter, must be able to accommodate the substantial additional energy source due to differentiation within the planet. This “helium rain”, if present, has been shown to be the dominant energy source for several-Gyrold giant planets (Stevenson and Salpeter, 1977; Fortney and Hubbard, 2003, 2004). In order to understand to what degree helium phase separation has progressed in Jupiter and Saturn, and how far down into the planet the helium has rained to, we must understand the deep interior temperature of these planets. To date, temperature measurements have been published by Holmes et al. (1995) and Collins et al. (2001). These experiments were performed using gas gun and laser apparatuses, respectively. Both found temperatures generally lower than most calculated hydrogen EOSs, which if indeed correct, would lead to shorter cooling timescales for giant planets. This faster cooling would more easily accommodate the additional energy source due to helium rain. Additional data, especially at the high pressures and “cool” temperatures of planetary interest (off of the single-shock Hugoniot) would be of great interest. 2.4 Do all giant planets possess heavy element enrichments? If we are to understand giant planets as a class of astronomical objects, we must understand how similar other giant planets are to Jupiter and Saturn. The mass-radius relation of exoplanets allows us, in principle, to understand if these planets have heavy element enrichments that are similar to Jupiter and Saturn. Figure 2 shows the mass and radius of Jupiter, Saturn, and the 10 known transiting hot Jupiters. It is interesting to note while Jupiter and Saturn differ in mass by a factor of 3.3, their radii only differ by 18%. However, while the hot Jupiters differ in mass by a similar factor (of 4) they differ in Springer

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Fig. 2 Radius and mass of Jupiter, Saturn, and the 10 known transiting hot Jupiters, as of April 2006. See Charbonneau et al. (2006) and references therein. One RJ is 71492 km, Jupiter’s equatorial radius at P = l bar. Curves of constant density (in g cm−3 ) are over-plotted with a dotted line. Data are taken from Charbonneau et al. (2006) and McCullough et al. (2006)

radius by a factor of 2. This large spread is presumably due to large difference in the interior heavy element abundances of these planets (Fortney et al., 2006; Guillot, 2005; Guillot et al., 2006). Giant planets under intense stellar irradiation cool and contract more slowly that those far from their parent stars, so radii larger than 1 RJ are expected (Guillot et al., 1996). Planet HD 149026b, with a radius of only 0.73 RJ , must be on the order of 2/3 heavy elements by mass to explain its small radius (Sato et al., 2005; Fortney et al., 2006). Its parent star has a metallicity 2.3 × that of the Sun, so this may point to a connection between stellar and planetary abundances. However, the determination of planetary core sizes appears to be complicated by the need for an additional interior energy source (yet to be definitely identified) for planet HD 209458b, and perhaps also OGLE-Tr-l0b (Bodenheimer et al., 2001; Guillot and Showman, 2002; Winn and Holman, 2005). These planets have radii that are too large to be explained by conventional cooling/contraction models (Chabrier et al., 2004; Laughlin et al., 2005). Therefore, the spread in Fig. 2 is likely due to a combination of differing magnitudes of this interior energy source and heavy element abundances, which adds significant complications to this picture. Guillot et al. (2006) have recently proposed a correlation between the heavy element abundances in transiting planets and the metallicity of the planets’ host stars, assuming an additional energy source that scales linearly with the incident stellar flux absorbed by the planets. In Fig. 3 we show a first look at comparative interior structure of the core-dominated planet HD 149026b, Saturn, and Neptune. The figure shows the current interior density distribution as a function of normalized radius for two HD 149026b models from Fortney et al. (2006) compared to inteSpringer

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Fig. 3 Interior density as a function of normalized radius for two possible models for HD 149026b compared with Neptune and Saturn. All planet models have been normalized to the radius at which P = l bar. The Neptune profile is from Podolak et al. (1995) and the Saturn profile is from Guillot (1999). The Saturn and Neptune models have a two-layer core of ice overlying rock. The two profiles of HD 149026b assume a metallicity of 3 times solar in the H/He envelope and a core made entirely of either ice or rock

rior models of Saturn (Guillot, 1999) and Neptune (Podolak et al., 1995). The Saturn and Neptune models both have twolayer cores of rock overlain by ice. The ratio of ice to rock in these cores is based more on cosmogonical arguments than on physical evidence. The interior structure of HD 149026b may be a hybrid of the ice giants and gas giants. Uranus and Neptune are ∼90% heavy elements, while Saturn is ∼25% and Jupiter 10% (Saumon and Guillot, 2004). Although HD 149026b is more massive than Saturn, it has a bulk mass fraction of heavy elements (50–80%) more similar to that of the solar system’s ice giants. Clearly, the field of exoplanets is allowing us to study and understand planets unlike any we have in our solar system.

3 The future The path towards a better understanding of the structure of giant planets seems clear. Along with additional laboratory work at high irradiance laser, Z-pinch, and other facilities, space missions will also allow us better insight into giant planets. For Saturn, NASA’s Cassini spacecraft will allow us to place better constraints on Saturn’s gravity field. For Jupiter, NASA’s Juno mission, still scheduled to launch in 2010, will map the planet’s gravity field at high precision and to high order, and will derive the abundances of water vapor and ammonia in the planet’s atmosphere below their respective cloud layers. For extrasolar planets, the European COROT and NASA Kepler missions will allow us to detect potentially hundreds of additional transiting planets. The scientific gain from all of these missions is directly dependent on our understanding of hydrogen and

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helium at high pressure. Experiments in the future should focus on the following issues:

r Is the fluid molecular-to-metallic transition of hydrogen a

r r

continuous transition? Does the presence of a 10% mixture of helium effect this transition? What are the EOSs of hydrogen and helium along the internal adiabats of Jupiter and Saturn? What is the temperature of hydrogen along the relatively “cool” adiabats of giant planets?

Acknowledgements JJF acknowledges the support of an NASA Postdoctoral Program (NPP) fellowship and a travel grant from the HEDLA conference organizers.

References Atreya, S.K., Mahaffy, P.R., Niemann, H.B., Wong, M.H., Owen, T.C.: Planet. Space Sci. 51, 105 (2003) Bodenheimer, P., Lin, D.N.C., Mardling, R.A.: ApJ 548, 466 (2001) Boriskov, G.V., Bykov, A.L., Il’Kaev, R.I., Selemir, V.D., Simakov, G.V., Trunin, R.F., Urlin, V.D., Shuikin, A.N., Nellis, W.J.: Phys. Rev. B 71, 092104 (2005) Chabrier, G., Barman, T., Baraffe, I., Allard, F., Hauschildt, P.H.: ApJ 603, L53 (2004) Chabrier, G., Saumon, D., Hubbard, W.B., Lunine, J.I.: ApJ 391, 817 (1992) Charbonneau, D., Brown, T.M., Burrows, A., Laughlin, G.: astroph/0603376, Protostars and Planets V, in press (2006) Collins, G.W., Celliers, P.M., da Silva, L.B., Cauble, R., Gold, D.M., Foord, M.E., Holmes, N. C., Hammel, B.A., Wallace, R.J., Ng, A.: Physical Review Letters 87, 165504 (2001) Collins, G.W., da Silva, L.B., Celliers, P., Gold, D.M., Foord, M.E., Wallace, R.J., Ng, A., Weber, S.V., Budil, K.S., Cauble, R.: Science 21, 1178 (1998) Conrath, B.J., Gautier, D.: Icarus 144, 124 (2000) Demarcus, W.C.: AJ 63, 2 (1958) Fortney, J.J., Hubbard, W.B.: Icarus 164, 228 (2003) Fortney, J.J., Hubbard, W.B.: ApJ 608, 1039 (2004) Fortney, J.J., Saumon, D., Marley, M.S., Lodders, K., Freedman, R.S.: ApJ 642, 495 (2006) Guillot, T.: Planet. Space Sci. 47, 1183 (1999) Guillot, T.: Annual Review of Earth and Planetary Sciences 33, 493 (2005)

283 Guillot, T., Burrows, A., Hubbard, W.B., Lunine, J.L., Saumon, D.: ApJ 459, L35 (1996) Guillot, T., Santos, N.C., Pont, F., Iro, N., Melo, C., Ribas, L.: A&A, 453, L21 (2006) Guillot, T., Showman, A.P.: A&A 385, 156 (2002) Holmes, N. C., Ross, M., Nellis, W. J.: Phys. Rev. B 52, 15835 (1995) Hubbard, W. B.: ApJ 152, 745 (1968) Hubbard, W. B.: Space Science Reviews 14, 424 (1973) Hubbard, W. B., Burrows, A., Lunine, J. I.: ARA&A, 40, 103 (2002) Hubbard, W. B. Dewitt, H. E.: ApJ 290, 388 (1985) Knudson, M. D., Hanson, D. L., Bailey, J. E., Hall, C. A., Asay, J. R., Anderson, W. W.: Physical Review Letters 87, 225501 (2001) Knudson, M. D., Hanson, D. L., Bailey, J. E., Hall, C. A., Asay, J. R., Deeney, C.: Phys. Rev. B 69, 144209 (2004) Laughlin, G., Wolf, A., Vanmunster, T., Bodenheimer, P., Fischer, D., Marcy, G., Butler, P., Vogt, S.: ApJ 621, 1072 (2005) Low, F.J.: AJ 71, 391 (1966) McCullough, P.R., Stys, J.E., Valenti, J.A., Johns-Krull, C.M., Janes, K.A., Heasley, J.N., Bye, B.A., Dodd, C., Fleming, S.W., Pinnick, A., Bissinger, R., Gary, B.L., Howell, P.J., Vanmunster, T.: ApJ in press, astro-ph/0605414 (2006) Nellis, W.J.: Ap&SS 298, 141 (2005) Nellis, W.J.: Reports of Progress in Physics 69, 1479 (2006) Nellis, W.J., Holmes, N.C., Mitchell, A.C., Governo, G.K., Ross, M., Young, D.A., Trainor, R.J.: Physical Review Letters 53, 1248 (1984) Nellis, W.J., Weir, S.T., Mitchell, A.C.: Phys. Rev. B 59, 3434 (1999) Pfaffenzeller, O., Hohl, D., Ballone, P.: Physical Review Letters 74, 2599 (1995) Podolak, M., Cameron, A.G.W.: Icarus 22, 123 Podolak, M., Weizman, A., Marley, M.: Planet. Space Sci. 43, 1517 (1995) Sato, B., Fischer, D.A., Henry, G.W., Laughlin, G., Butler, R.P., Marcy, G.W., Vogt, S.S., Bodenheimer, P., Ida, S., Toyota, E., Wolf, A., Valenti, J.A., Boyd, L.J., Johnson, J.A., Wright, J.T., Ammons, M., Robinson, S., Strader, J., McCarthy, C., Tah, K.L., Minniti, D.: ApJ 633, 465 (2005) Saumon, D., Chabrier, G.: Phys. Rev. A 46, 2084 (1992) Saumon, D., Guillot, T.: ApJ 609, 1170 (2004) Saumon, D., Guillot, T.: Ap&SS 298, 135 (2005) Stevenson, D.J.: Phys. Rev. B 12, 3999 (1975) Stevenson, D.J., Salpeter, E.E.: ApJS 35, 239 (1977) von Zahn, U., Hunten, D.M., Lehmacher, G.: J. Geophys. Res. 103, 22815 (1998) Weir, S.T., Mitchell, A.C., Nellis, W.J.: Physical Review Letters 76, 1860 (1996) Winn, J.N., Holman, M.J.: ApJ 628, L159 (2005)

Springer

Astrophys Space Sci (2007) 307:285–289 DOI 10.1007/s10509-007-9385-z

ORIGINAL ARTICLE

Modeling Planetary Interiors in Laser Based Experiments Using Shockless Compression J. Hawreliak · J. Colvin · J. Eggert · D. H. Kalantar · H. E. Lorenzana · S. Pollaine · K. Rosolankova · B. A. Remington · J. St¨olken · J. S. Wark

Received: 8 May 2006 / Accepted: 22 January 2007 C Springer Science + Business Media B.V. 2007


Abstract X-ray diffraction is a widely used technique for measuring the crystal structure of a compressed material. Recently, short pulse x-ray sources have been used to measure the crystal structure in-situ while a sample is being dynamically loaded. To reach the ultra high pressures that are unattainable in static experiments at temperatures lower than using shock techniques, shockless quasi-isentropic compression is required. Shockless compression has been demonstrated as a successful means of accessing high pressures. The National Ignition Facility (NIF), which will begin doing high pressure material science in 2010, it should be possible to reach over 2 TPa quasi-isentropically. This paper outlines how x-ray diffraction could be used to study the crystal structure in laser driven, shocklessly compressed targets the same way it has been used in shock compressed samples. A simulation of a shockless laser driven iron is used to generate simulated diffraction signals, and recent experimental results are presented. Keywords Laser shock . Shockless compression . Diffraction . Dynamic diffraction . Simulation 1 Introduction Models of planetary interiors rely on accurate laboratory measurements of equations of state (Guillot, 2005). High pressure measurements are generally limited by the experimental apparatus. Diamond anvil experiments can be used to study statically compressed samples up to only a few hundred J. Hawreliak (
) · J. Colvin · J. Eggert · D. H. Kalantar · H. E. Lorenzana · S. Pollaine · K. Rosolankova · B. A. Remington · J. St¨olken · J. S. Wark Lawrence Livermore National Laboratory, Livermore, CA, USA e-mail: [email protected]

GPa (Clark et al., 1995). Dynamic techniques are needed to reach pressures exceeding 4 TPa that exist at the cores of the gas giants (Guillot, 1999). These pressure should be obtainable in the next generation of high energy density experimental facilities such as the National Ignition Facility (NIF) (Remington et al., 2005). This paper discusses how the crystal structure can be measured in dynamically compressed samples using in-situ x-ray diffraction (Kalantar et al., 2003a,b; Loveridge et al., 2001). First, a brief discussion of shock and shockless compression is presented as a means of obtaining high pressures. Then we describe how the x-ray diffraction techniques would differ between the shock and shockless drive. Finally a comparison is made between data from a recent experiment and simulation.

2 Dynamic compression techniques Shock compression has been used over the last half century to provide a wealth of information about materials at high pressures (Meyers, 1994). Traditionally experiments have been done using a gas gun to launch a projectile at a stationary target, or using high explosives (HE) to apply a sharp pressure pulse (Asay, 1997). More recently, shock wave research has extended to large laser and z-pinch facilities. Even though the technology to create a shock is different in each method, the sample will always transform to a single temperature and pressure described by the shock Hugoniot, the curve in phase space that represents the shock response of a material. To study samples under the ultra high pressure conditions relevant to planetary interiors in the laboratory, dynamic compression techniques which do not produce a shock are required. These techniques are labeled as “quasi-isentropic” because they produce high pressures off the shock Hugoniot Springer

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3 X-ray diffraction from dynamically compressed materials With the development of x-ray diffraction of dynamic systems, (Kalantar et al., 2003; Loveridge et al., 2001; Kalantar Laser Drive Beams

Diffracted X-rays To Film Detector

X-ray Source

250 µm Plastic Reservoir 250 µm Gap 4 µm Berylium Heat Shield

8 µm Iron Crystal 250 µm Plastic Backing

Fig. 1 A schematic diagram of a shockless compression target. The drive beams are incident on the plastic reservoir, which causes the blow off from the back of the plastic to cross the gap and pile up on the iron sample. A thin layer of beryllium is used to prevent thermal contact between the iron sample and the plasma

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Pressure (GPa / 100) Compression (1-ρo/ρ)

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400 Thermal conduction wave

X-ray preheat

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Reservoir driven

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0 0

20

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in materials while maintaining low temperatures using a sequence of small shocks. Quasi-isentropic compression has been performed using pillow impactors on gas guns (Asay, 1997), temporal pulse shaping on z-pinch facilities (Hall, 2000), HE expanding across a vacuum gap (Barnes et al., 1974) and a two component target on laser facilities (Edwards et al., 2004; Lorenz et al., 2005). In a similar fashion to the HE case, the laser target uses a plastic reservoir which is shocked with a highpowered laser and the plasma blow off from the ensuing blast wave crosses a vacuum gap and pressurizes the sample. The parameters of the target can be tailored to suit a particular compression regime. Unlike shock compression, the sample does not end up on a single point on the shock Hugoniot, but samples a continuum of states along the compression ‘quasiisentrope’. Where measurements made at different times look at different pressure and temperatures. An example of a laser driven shockless compression target is shown in Fig. 1, and simulated pressure, temperature, and density profiles are shown in Fig. 2 for an on target drive laser intensity of 1 × 1013 W/cm2 . This shows that over 5 to 10 ns period the pressure increases in the sample until it reaches a peak pressure, which is maintained for approximately 10 ns. The beryllium layer serves as a heat shield to prevent the sample from melting on initial contact with the plasma, but ultimately a thermal wave causes the sample to heat up, setting up a large temperature gradient in the sample. The long time-scales and spatial gradients in the isentropically compressed case change the way the sample can be diagnosed using in-situ x-ray diffraction.

60

Time (ns) Fig. 2 A plot of the simulated temperature, pressure and compression at the center of the 8 µm iron foil from a 1D LASNEX simulation. The simulation shows slight preheating of the sample from the x-rays generated at the drive surface of the reservoir, causing a slight expansion before the pressure wave arrives at the center of the sample. The three lines indicate the time at which the simulated diffraction images are shown in Fig. 5

et al., 1999; Gupta et al., 1999) it is possible to make a determination of crystal structure in-situ during the compression process. These in-situ measurements are critical to understanding how materials react under presssure as the crystallographic structure is fundamental in determining material properties. A change in crystal structure, known as a phase change, is usually accompanied by a dramatic change in bulk properties, i.e. volume collapse, change in electrical conductivity, change in magnetism, etc. We will discuss modifications to the existing wide angle x-ray diffraction techniques used on shock compression experiments (Kalantar et al., 2003) to make crystallographic measurements on quasi-isentropically compressed samples. Figure 3 shows how the x-ray probes will differ between the two techniques. The gray shaded region denotes the time and the finite penetration depth of the x-ray probes on the crystal. In the shock case, shown in Fig. 3a, there is a discontinuity at the shock front between the uncompressed and shock compressed material. For x-ray diffraction this discontinuity means we can record a diffraction signal from both the unshocked sample and the shock compressed sample with a single pulse of x-rays which is time synchronous with the shock front reaching the material. The measurement of the unshocked signal is critical to making quantitative measurements because it provides a method for calibrating the experimental geometry of the detector and the initial crystal state. In the isentropically compressed case, the compression happens over a longer timescale, as illustrated in Fig. 3b. This means that it will require two separate x-ray pulses. One x-ray pulse is required to record a zero pressure signal. A second x-ray pulse is required to interrogate the sample at pressure. Unlike the shock case, where the shock front provides a discontinuity between compressed material and the zero pressure material , the isentropically compressed targets will have a continuous compression with large gradients into the sample.

Astrophys Space Sci (2007) 307:285–289

4 Diffraction from iron [100] crystals The properties of iron under high pressure and temperature have always been of interest owing to iron being a major component of the earth’s inner core and its technological relevance to modern society. The α − ǫ phase transition at 13 GPa is of particular importance to the shock physics community as it was first proposed by Bancroft et al. (1956) to explain the discrepancy between Walsh’s measurements (Walsh, 1954) and Bridgman’s original work up to 10 GPa (Bridgman, 1964). The crystallographic nature of the ǫ phase was not determined until later with static compression measurements using x-ray diffraction by Jamison and Lawson (1962). It was not until recent in-situ x-ray diffraction experiments of laser shocked samples that this was confirmed (Kalantar et al., 2005). With the recent success of in-situ xray diffraction in determining the crystal structure of shocked iron, we will use iron as the example to discuss the x-ray diffraction technique for isentropically compressed samples. We employ a simulation to show how the gradients in the sample will affect the width of the diffraction lines that will be recorded on the film. Figure 4 shows how the compression of the iron varies in the sample at different times in the simulation. During the pressure ramp, 30 ns after the drive beams are incident on the reservoir surface, the density gradient is large going from ambient density to a compression of nearly 15%. Since the x-rays can penetrate the entire sample, the diffracted peak will contain information from the full range of compressions at this time, and it will be significantly broadened. Close to the peak of the pressure pulse, 33.5 ns, maximum compression is reached at the front surface, but there is still a large gradient in compression which will broaden the diffracted signal and make it difficult to extract detailed information. Figure 5 illustrates line broadening effects due to the strain gradient in the iron sample. Figure 5a shows a line out from an experimental image from iron sample shock compressed with a laser using an iron K-shell backlighter (Kalantar et al.,

as two separate pulses are required to probe in the isentropically compressed case. d0 is the initial plane spacing, ds is the plane spacing in the shocked sample and ∇d is the direction of increasing plane spacing in the ramp compressed sample

0.2 30.0ns 33.5ns 41.5 ns

Compression (1-ρο/ρ)

Fig. 3 Schematic diagram showing a time-distance plot of representative lattice planes in iron that is compressed by (a) shock loading and (b) isentropic compression. The shaded region represents the x-ray probe time and depth in the crystal. Both the uncompressed and compressed lattice are probed by a single x-ray pulse in the shocked case, where

287

0.1

0 0µm

4µm Position from iron surface

8µm

Fig. 4 This plot shows the compression at different times through the sample. Both 30 and 33.5 ns show steep gradients in the compression. At 41.5 ns the compression is more uniform across the sample

2005; Hawreliak et al., 2006). As stated before in the shock case we can probe several states of the material with a single pulse due to the penetration depth of the x-rays. In this case we get signal from pristine material (deepest in the sample), a compressed BCC lattice (intermediate phase) and the transformed HCP phase (closest to the surface). For comparison, Fig. 5b and c show calculated signals at 3 times for iron and copper backlighters respectively. Two different examples are show for each time. The solid line includes some instrument broadening of the diffraction peaks. Figure 5b shows that the diffraction signals recorded at 30 and 33.5 ns will be broaden significantly by the gradient in compression of the sample. The diffraction signal at 41.5 ns forms a tighter peak than the diffraction peak at the other two times, which can be attributed to the smaller gradient in the compression. The broadening in the diffraction signal due to the strain gradient can be reduced by the choice of backlighter. When using K -shell laser generated x-rays from an iron backlighter at 6 keV the 1/e depth is about 7 µm for the (002) plane, generating diffraction signal from nearly the entire sample. Alternatively the K -shell radiation from a laser based copper backlighter at 8 keV the depth is reduced to less then 2 µm. Springer

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Scattering (Arb. Units)

a)

30 ns - Cu BL w Broadening 33.5 ns - Cu BL w Broadening 41.5 ns - Cu BL w Broadening

30 ns - Fe BL w Broadening 33.5 ns - Fe BL w Broadening 41.5 ns - Fe BL w Broadening

Unshocked Material (002) BCC

Unshocked Material recorded using early x-ray pulse

b)

Unshocked Material recorded using early x-ray pulse

c)

Compressed Material recorded using late x-ray pulse

Compressed Material recorded using late x-ray pulse

New Phase HCP

Compressed 0

200

400

600

2

Pixel

2.2

2.2

2.4

Scattering Vector [001]

The shape of the simulated peaks in (b) and (c) reflect the range of compressions in the sample. The solid curves includes the broadening observed due to the α − ǫ phase transition 12

Signal

Diffraction from uncompressed material

Compression

10 8

Probe time

6 4 2

0%

2%

Compression (1-a0 /a )100%

Simulations of the diffraction peaks using a copper backlighter are shown in Fig. 5c. The copper backlighter, with its shallower penetration depth, generates sharper peaks than an iron backlighter but the scattering signal levels will be reduced due to the higher absorption in the iron at the copper k-shell emission line. Figure 6 shows recent experimental data driven at a laser drive intensity of 1 × 1013 W/cm2 . The timing of the late x-ray pulse was set to probe the lattice as the compression wave reached the sample. In Fig. 6 the data is obtained by a single late x-ray pulse. The two distinct peaks is consistent with the initial small shock that is generated by the ramped compression profile steepening up in the plastic heat shield layer. While a thick heat shield over the sample is desired to keep the plasma heat from the sample, too thick a sample will lead to an initial shock or potentially shock compression of the sample.

5 Conclusion Quasi-isentropic compression gives us the ability to explore material states off the shock Hugoniot. This approach also provides us with one key and truly unique capability, namely access to compressions in solids that approach the terapascal regime. Typically, single shock techniques melt solids at pressures exceeding 200–400 gigapascals, and so properties Springer

2

Scattering Vector [001]

Fig. 5 A comparison of line outs from (a) raw experimental x-ray diffraction data from a shocked, single crystal sample and simulated diffraction using (b) an iron backlighter and (c) a copper backlighter. Fig. 6 Experimental data and simulated temporal profile of the compression in the sample both at a laser drive intensity of 1 × 1013 W/cm2 . The sample was Vanadium coated with a 5 µm plastic heat shield. The experimental data was generated using only the late time pulse, so the distinct peaks is consistent with a shocking of the sample

2.4

Compression (%)

0

4%

0 20

40

60 80 Time (ns)

100

120

of solids beyond these pressures remain completely unexplored. Investigations in the terapascal regime enabled by quasi-isentropic compression certainly will provide an unprecedented window into the extreme states found in planetary interiors. Time-resolved X-ray diffraction methodologies will play an unique and pivotal role in investigating the crystal lattice and other fundamental properties of the solid during these extreme conditions. For shocked solids, our group and others throughout the community have already demonstrated that x-ray diffraction can successfully probe atomistic phenomena such as phase transformations with nanosecond temporal resolution. However, diffraction measurements in a quasiisentropically compressed solid have never been demonstrated and pose important challenges; in this work, we have discussed a novel approach to enabling such investigations. We have described key experimental aspects of a technique to perform time resolved diffraction measurements, including a new geometrical target design as well as relevant backlighter and timing issues. Additionally, we have explored the need for a detailed hydrodynamic understanding of the compression process, either through bulk measurements or detailed simulations, that will permit quantitative analyses of the quasi-isentropically compressed lattice. Finally, we have presented preliminary experimental results showing in situ, real-time lattice measurements during quasi-isentropic compression and modelling to quantify the contributions from

Astrophys Space Sci (2007) 307:285–289

spatial and temporal gradients. Development of this technique will continue on the Omega laser at the University of Rochester, the Jupiter Laser Facility at LLNL, and the Vulcan laser at the Rutherford Appleton Laboratory in the UK. This work was conducted under the auspices of the U.S. DOE by the UC LLNL and LANL under Contract No. W-7405-Eng-48. The quasi-isentropic compression experiments were conducted at the University of Rochester Laboratory for Laser Energetics. Additional support was provided by LDRD program Project No. 06-SI-004 at LLNL and by the DOE under Grants No. DEFG0398DP00212 and No. DEFG0300SF2202, by the U.K. EPSRC under Grant No. GR/R25699/01. References Asay, J.R.: The use of shock-structure methods for evaluating highpressure material properties. Int. J. Impact Eng. 20(1–5), 27 (1997) Bancroft, D., Peterson, E.L., Minshall, S.: Polymorphism of iron at high pressure. J. Appl. Phys. 27(3), 291–298 (1956) Barnes, J.F., Blewett, P.J., Mcqueen, R.G., Meyer, K.A., Venable, D.: Taylor instability in solids. J. Appl. Phys. 45(2), 727–732 (1974) Bridgman, P.W.: Collected Experimental Papers. Harvard University Press, Cambridge (1964). [by] P.W. Bridgman. illus. 25 cm. Some papers in German. “In most cases the original reprints have been reproduced photographically, although in a few instances it proved necessary to reset a paper in type.” Clark, S.J., Ackland, G.J., Crain, J.: Theoretical stability limit of diamond at ultrahigh pressure. Phys. Rev. B 52(21), 15035–15038 (1995) Edwards, J., Lorenz, K.T., Remington, B.A., Pollaine, S., Colvin, J., Braun, D., Lasinski, B.F., Reisman, D., McNaney, J.M., Greenough, J.A., Wallace, R., Louis, H., Kalantar, D.: Laser-driven plasma loader for shockless compression and acceleration of samples in the solid state. Phys. Rev. Lett. 92(7), 075002 (2004) Guillot, T.: Interiors of giant planets inside end outside the solar system. Science 286(5437), 72–77 (1999) Guillot, T.: The interiors of giant planets: models and outstanding questions. Annu. Rev. Earth Planet. Sci. 33(1), 493–530 (2005) Gupta, Y.M., Zimmerman, K.A., Rigg, P.A., Zaretsky, E.B., Savage, D.M., Bellamy, P.M.: Experimental developments to obtain realtime x-ray diffraction measurements in plate impact experiments. Rev. Sci. Instrum. 70(10), 4008–4014 (1999)

289 Hall, C.A.: Isentropic compression experiments on the sandia z accelerator. Phys. Plasmas 7(5), 2069–2075 (2000) Hawreliak, J., Colvin, J.D., Eggert, J.H., Kalantar, D.H., Lorenzana, H.E., Stolken, J.S., Davies, H.M., Germann, T.C., Holian, B.L., Kadau, K., Lomdahl, P.S., Higginbotham, A., Rosolankova, K., Sheppard, J., Wark, J.S.: Analysis of the x-ray diffraction signal for the alpha-epsilon transition in shock-compressed iron: simulation and experiment. Phys. Rev. B 74(18), 184107 (2006) Jamieson, J.C., Lawson, A.W.: X-ray diffraction studies in the 100 kilobar pressure range. J. Appl. Phys. 33(3), 776–780 (1962) Kalantar, D.H., Belak, J., Bringa, E., Budil, K., Caturla, M., Colvin, J., Kumar, M., Lorenz, K.T., Rudd, R.E., Stolken, J., Allen, A.M., Rosolankova, K., Wark, J.S., Meyers, M.A., Schneider, M.: Highpressure, high-strain-rate lattice response of shocked materials. Phys. Plasmas 10(5), 1569–1576 (2003b) Kalantar, D.H., Belak, J.F., Collins, G.W., Colvin, J.D., Davies, H.M., Eggert, J.H., Germann, T.C., Hawreliak, J., Holian, B.L., Kadau, K., Lomdahl, P.S., Lorenzana, H.E., Meyers, M.A., Rosolankova, K., Schneider, M.S., Sheppard, J., Stolken, J.S., Wark, J.S.: Direct observation of the alpha-epsilon transition in shock-compressed iron via nanosecond x-ray diffraction. Phys. Rev. Lett. 95(7), 075502 (2005) Kalantar, D.H., Bringa, E., Caturla, M., Colvin, J., Lorenz, K.T., Kumar, M., Stolken, J., Allen, A.M., Rosolankova, K., Wark, J.S., Meyers, M.A., Schneider, M., Boehly, T.R.: Multiple film plane diagnostic for shocked lattice measurements (invited). Rev. Sci. Instrum. 74, 1929–1934 (2003a) Kalantar, D.H., Chandler, E.A., Colvin, J.D., Lee, R., Remington, B.A., Weber, S.V., Wiley, L.G., Hauer, A., Wark, J.S., Loveridge, A., Failor, B.H., Meyers, M.A., Ravichandran, G.: Transient x-ray diffraction used to diagnose shock compressed Si crystals on the nova laser. Rev. Sci. Instrum. 70, 629–632 (1999) Lorenz, K.T., Edwards, M.J., Glendinning, S.G., Jankowski, A.F., McNaney, J., Pollaine, S.M., Remington, B.A.: Accessing ultrahighpressure, quasi-isentropic states of matter. Phys. Plasmas 12(5), 056309 (2005) Loveridge-Smith, A., Allen, A., Belak, J., Boehly, T., Hauer, A., Holian, B., Kalantar, D., Kyrala, G., Lee, R.W., Lomdahl, P., Meyers, M.A., Paisley, D., Pollaine, S., Remington, B., Swift, D.C., Weber, S., Wark, J.S.: Anomalous elastic response of silicon to uniaxial shock compression on nanosecond time scales. Phys. Rev. Lett. 86(11), 2349–2352 (2001) Meyers, M.A.: Dynamic Behavior of Materials. Wiley, New York (1994) Remington, B.A., Cavallo, R.M., Edwards, M.J., Ho, D.D.M., Lasinski, B.F., Lorenz, K.T., Lorenzana, H.E., McNaney, J.M., Pollaine, S.M., Smith, R.F.: Accessing high pressure states relevant to core conditions in the giant planets. Astrophys. Space Sci. 298(1–2), 235–240 (2005) Walsh, J.M.: Bull. Am. Phys. Soc. 29, 28 (1954)

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Astrophys Space Sci (2007) 307:291–296 DOI 10.1007/s10509-006-9247-0

ORIGINAL ARTICLE

Scaling Laws for Collisionless Laser–Plasma Interactions of Relevance to Laboratory Astrophysics D. D. Ryutov · B. A. Remington

Received: 11 April 2006 / Accepted: 29 August 2006 C Springer Science + Business Media B.V. 2006


Abstract Scaling laws for interaction of ultra-intense laser beams with a collisionless plasmas are discussed. Special attention is paid to the problem of the collective ion acceleration. Symmetry arguments in application to the generation of the poloidal magnetic field are presented. A heuristic model for evaluating the magnetic field strength is proposed. Keywords Laser-plasma interaction . Particle acceleration . Magnetic field generation . Laboratory astrophysics PACS Numbers: 52.38Kd, 52.38.Fz, 41.75.Jv

1 Introduction Interaction of ultra-intense laser radiation with a plasma is of significant potential interest for laboratory astrophysics. In particular, it opens up a possibility of creating a platform for studying such effects as generation of dynamicallysignificant magnetic fields (Norreys et al., 2004; Moon et al., 2005), development of various collective instabilities, e.g., the filamentation instability (Silva et al., 2003; Wei et al., 2004), particle acceleration via collective effects (Katsouleas, 2004; Snavely et al., 2000; Mackinnon et al., 2001), and dynamics of relativistic plasmas (Liang, 2005; Wilks et al., 2005). Many of these effects have been studied numerically, beginning with Wilks et al. (1992). At these very high intensities, the Coulomb collisions often become sub-dominant, and the plasma behavior can be reasonably well described in the collisionless approximation. Recently, first steps have been made in developing scaling D. D. Ryutov (
) · B. A. Remington Lawrence Livermore National Laboratory, Livermore, CA 94551, USA e-mail: [email protected]

relations that would govern such situations (Pukhov et al., 2004; Gordienko and Pukhov, 2005; Ryutov and Remington, 2006a,b). The amplitude E0 of the incident wave defines the so called dimensionless vector potential, a0 ≡

eE 0 . mωc

(1)

Here we use the CGS-Gaussian system of units, with e and m being the charge and the mass of the electron, c being the speed of light, and E 0 and ω being the electric field amplitude and the wave frequency. The parameter a0 is the ratio of the electron quiver momentum normalized to mc. In the case a0 > 1 the oscillating electrons are relativistic, with γ ∼ a0 . An intensity I ≡ cE 02 /8π is related to a0 by I (W/cm2 ) ≈ a02

1.3 × 1018 . [λ(µm)]2

(2)

For completeness, we also present the amplitude of the oscillating magnetic field in vacuum in terms of a0 : B0 (MG) = a0

105 . λ(µm)

(3)

Note that, for a typical wavelength of λ ∼ 1 µm and a mildly relativistic incident wave (a0 ∼ 1), this oscillating field is already quite high, ∼100 MG. One of the most efficient applications of the scaling laws is their use for testing the applicability limits of various physical models. The main assumption of Ryutov and Remington (2006a,b), in addition to the absence of collisions, is that the initial plasma temperature is negligible compared to the energies that electrons and ions (if the ion dynamics is esSpringer

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Fig. 1 The temporal dependence of the electric field in an incident wave. The envelope function is of the form F(t/τ ), with τ being a characteristic temporal width. In the scaling exercise, the function F must remain the same function of its argument, although the parameter τ may vary from system to system

sential) acquire early in the laser pulse. Following this line, in Section 2 we consider possible experiments that would verify our scaling laws in the problem of the ion acceleration in the setting of Snavely et al. (2000) and Mackinnon et al. (2001). If the model is shown to be valid, then scaling arguments can be used as a predictive tool. We present an example of that, again using the problem of the ion acceleration, in Section 2. The presence of scaling laws reveals continuous symmetries of the problem. Of a comparable importance in a number of cases are discrete symmetries, allowing one to make very general conclusions regarding the geometrical properties of the system. An example of using such symmetries in the problem of generation of the quasi-static poloidal magnetic field was given in Ryutov and Remington (2006a), where the geometrical structure of such a field was established. Now, in Section 3, we extend the analysis of Ryutov and Remington (2006a) to add a heuristic estimate of the magnitude of the poloidal magnetic field. Throughout this paper we assume that the duration of the ultra-intense pulse τ is much greater than the wave period 2π /ω (Fig. 1). The shape of the envelope function F has to remain the same throughout the scaling exercise, although normalization can vary.

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potential well. On the rear surface, the hydrogen-containing impurities liberate hydrogen ions, which are accelerated by this ambipolar field, roughly speaking, in the normal direction to the foil. For a thin-enough foil, the fast electrons make many bounces inside the well, before they lose their energy. The ion acceleration by the ambipolar electric field is not affected by collisions. In other words, the ions can be treated as collisionless. We note that the foil in this discussion plays a passive role and is not involved in the process of the energy transfer from the oscillating electrons to the ions. In order for this to be correct, the foil has to be thin enough so as to make energy losses of “oscillating” relativistic electrons negligible. One can also note in passing that a very similar setting has been studied in great detail in the problem of ion acceleration by electrons generated in a high-current diode and injected into vacuum (e.g., Antonsen and Ott, 1976; Ryutov and Stupakov, 1976; Arzhannikov et al., 1976; see also survey Humphries, 1980 of these early studies). The full set of the Maxwell–Vlasov equations describing collisionless plasmas, with relativistic electrons and nonrelativistic ions was reduced to the dimensionless form in Ryutov and Remington (2006a,b). It was shown that, under the conditions described above, the system is fully characterized by the following six parameters: (4)

n, L , τ, ω, E 0 , M/Z ,

which are: the density at a characteristic point of the blowoff plasma; length-scale (e.g., spot size) of the incident beam and the blow-off plasma; the pulse duration τ of the main pulse; the frequency ω of the incident radiation; the maximum amplitude E 0 of the electric field of the incident wave (or, equivalently, the maximum intensity I), and the mass-tocharge ratio for the accelerated ions. [The latter parameter may be of interest in the context of comparing the acceleration of hydrogen vs. deuterium.] The dimensionless parameters that determine the scalability between any two (or more) systems are [Ryutov and Remington (2006a,b)]: T ≡ ωτ ; !

R ≡ Lω/c;

Z eE 0 . Mωc

S≡

4π n 0 ec ; E0ω

2 Scaling for the ion acceleration experiment

U ≡

In this section, we consider a standard setting for the experiment on the ion acceleration with a pulsed laser. The ultraintense pulse hits the surface of a thin foil, where some blowoff-plasma already created by a pre-pulse is usually present. The incident beam generates ultra-relativistic electrons which cannot leave the system because of a quasineutrality constraint. These electrons start oscillating in an ambipolar

They must be held constant in order that the dimensionless equations remain unchanged between the two systems such that the evolution of these systems is similar. We emphasize that we consider a system with relativistic electrons. There exists also a similarity transformation that covers both the non-relativistic and relativistic regimes (Ryutov and Remington, 2006b), but it requires a constancy of one more

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(5)

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collisional), or the two experiments are not perfectly similar in terms of their geometry (including the irradiation geometry), or the temporal dependence of the incident radiation. We have six input parameters (Equation (4)) subject to four constraints S = const, R = const, T = const, and U = const (Equation (5)). To check the validity of scaling laws, we can arbitrarily choose any two of six input parameters in the primed system, then adjust the remaining four so as to keep the dimensionless parameters (Equation (5)) constant. Consider, for example, that we increase the ion mass by a factor of 2 (switching from hydrogen to deuterium), and increase the intensity by a factor of 4 (columns 2 and 3 in Table 1). Then, the rest of the input parameters would have to be changed as shown in columns 4–7 of Table 1. This ensures the constancy of the dimensionless parameters (Equation (5)). The other parameters of the two systems (e.g., the average energy of the accelerated ions) changes according to the scalings formulated in Ryutov and Remington (2006b). Specifically, in our example, the average energy increases by a factor of 2, and the total number of accelerated ions also increases by a factor of 2 (columns 9, 11). One can also reverse these arguments and, if there is a good reason to believe in the validity of the underlying assumptions, use the scalings as a predictive tool. For example, if an experimentalist is interested in generating fast protons with a laser with a frequency two times less than in the earlier successful experiment, he/she can do it in a scaled fashion, thereby being able to predict all the details of a new experiment. This is illustrated by Table 2 which shows that twice as many fast ions with the same energy can be generated if the intensity is reduced by a factor of 4, and the other “input” parameters are changed according to the columns 4–6 of Table 2.

dimensionless parameter, a0 , Equation (1), and is in this regard more restrictive. In the case of relativistic electrons, the constancy of the parameter S means the constancy of the ratio of the frequency of the incident wave and the relativistic cut-off plasma frequency. This is the reason why the parameter a0 is absorbed into the parameter S. In addition to holding the dimensionless parameters (5) constant, in order that the two systems behave in a scaled fashion, the geometric similarities must also be observed, e.g., if the characteristic length-scale L of the plasma density distribution is increased by a factor of 2, so too must the focal spot radius be increased by the same factor. The geometrical characteristics of the incident radiation have to be identical between the two systems (up to the lengthscale change): the polarization must remain the same, as well as the direction and the convergence of the incident beam. The shape of the temporal dependence of the laser pulse must also remain unchanged (although its duration may change). Under such conditions, any systems for which the dimensionless parameters (5) are kept the same, behave identically, up to scale transformations identified in Ryutov and Remington (2006a,b). Here we discuss, at a conceptual level, the possible experimental verification of the underlying physics assumptions, of which the most important are the absence of collisions and smallness of the initial “temperature.” Within these two assumptions, the similarity covers all the processes involved, in all their complexity: distribution functions, the spatio-temporal characteristics of the reflected waves, possible presence of the filamentation and other instabilities, magnetic field generation, and so on. Any observed differences may signify that either the basic assumptions are wrong (e.g., the system is actually

Table 1 Switching to accelerating deuterium at an increased intensity 1

2

3

4

5

6

7

8

9

10

11

Quantity

Ion mass

Intensity

Frequency

M 2M

I 4I

ω ω

Spatial scale L L

Density

Original system “Primed” system

Pulse duration τ τ

Electron energy We 2We

Ion energy Wi 2Wi

Quasistatic m.f. B 2B

Number of fastions N 2N

n 2n

The relative amplitude of the harmonics remains unchanged

Table 2 A scaled experiment at a reduced frequency 1

2

3

4

5

6

7

8

9

10

Quantity

Frequency

Intensity

ω ω/2

I I/4

Spatial scale L 2L

Density

Original system “Primed” system

Pulse duration τ 2τ

Electron energy We We

Ion energy Wi Wi

Quasistatic m.f. B B/2

Number of fastions N 2N

n n/4

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3 Discrete symmetries and generation of the poloidal magnetic field A variety of mechanisms for generation of a quasi-static magnetic field have been discussed in the past, including the thermo-electric dynamo (Stamper, 1971) and a ponderomotive force (Sudan, 1993). These mechanisms easily explain the appearance of the toroidal magnetic field, as shown in

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Fig. 2 for a circular beam imprint. The experiments on the magnetic field generation carried out thus far have been interpreted in terms of the toroidal field (Norreys et al., 2004; Tatarakis et al., 2002), without much consideration given to the possible presence of the poloidal field. However, as was pointed out in Ryutov and Remington (2006a), the symmetry arguments lead to the prediction that a poloidal magnetic field can also be generated, if the incident radiation is linearly polarized; a peculiar structure of the currents generating this field had been predicted. In this article, we provide a more detailed description of the geometrical structure and present a rough heuristic estimate of the poloidal field. We consider the following model: A linearly polarized laser beam falls normally onto the plasma slab (Fig. 2b). The beam imprint is assumed to be circular, with the characteristic radius r0 exceeding the wave-length of the incident light, r0 ≫ λ.

Fig. 2 Structure of the magnetic field generated by a laser beam at normal incidence: (a) Toroidal magnetic field; it is generated by the current that flows away from the viewer near the axis and toward the viewer at the periphery. Shades of gray show the distribution of the intensity of the incident light. (b) The current streamlines (arrows) generated in the case where the incident wave is linearly polarized, with the wave electric field being directed along the y axis. This current pattern leads to the generation of the poloidal magnetic field. (c) Distribution of the z-component of the magnetic field at some intermediate depth (dots – towards the observer; hatching – away from the observer). The axial magnetic field lines are closed by a canopy of field lines near the ends of “solenoids”. The direction of the closure at the nearest end is shown by arrows

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The plasma density varies in the z direction (normal to the slab) from zero to some constant value, which can be both lower and higher than the critical density. The intensity of the beam varies along z due to effects of absorption (we do not specify the mechanism) and varying density (including possible cut-off beyond the critical point). We assume that the intensity of the reflected wave is small (because of the absorption of the incident radiation, or because of a smoothness of the density variation in the case where the maximum density is sub-critical). We assume that the electric field of the incident wave is parallel to y. As the system has two symmetry planes (xz and yz), the quasi-static current (which is a polar vector) normal to these planes must vanish (there is no preferential direction). Accordingly, the current pattern will look as shown in Fig. 2b. The shape of the streamlines is identical in all four quadrants, whereas the directions of the currents are shown by arrows. In principle, finer structures (but possessing the same symmetry) can also be present. The current of this form has some finite extent along the z axis. One can think of the current pattern as that of four solenoids of a finite length along the z axis, with the current direction alternating from one solenoid to another as shown in Fig. 2b. The distribution of the axial (z) magnetic field intensity in the xy plane at some depth z is shown in Fig. 2c by shading in dots (direction towards the viewer), and hatching (away from the viewer). In the limiting case of a small axial extent, the magnetic field structure will be that shown in Fig. 3: this would be a field of four current rings, with alternating direction of the currents. The aforementioned “canopy” of the magnetic field lines shows up, reminiscent of the field lines of a group of sunspots. Thus far, we have been using only symmetry arguments, without discussing the mechanism of the poloidal magnetic

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that it may actually reach a saturation earlier, if the neglected Hall term comes into play). We limit ourselves to a qualitative, order-of-magnitude estimate of the absolute value of the ponderomotive force. As it is related to the radial gradient of the intensity, it can be evaluated as f ∼ γ mc2 /r0 . Fig. 3 A more detailed structure of the magnetic field in the case of a small axial extent of the current-carrying zone. The current loops are shown by thick thick curves (loops). The magnetic field lines are shown as the thin loops; the dashed portions of these lines correspond to the zone beneath the plane where current loops are situated

field generation. Below, we provide some initial heuristic assessment. In the case of a relativistic drive, the amplitude of the electron periodic excursions in the wave field is of the order of c/ω. For the focal spot size r0 ≫ λ, this displacement is much less than the spot size. The presence of the radial gradient of the intensity under such circumstances creates an average force f acting on the electrons and varying over a scale ∼ r0 ; for generation of the poloidal magnetic field the x and y components of the force are important, as they can create the electron flow pattern shown in Fig. 2b. One should note that, for a non-relativistic drive, a0 ≪ 1, this force is potential (a so-called Miller force, Gaponov, Miller, 1958). As we shall see, the potential force f cannot drive a quasi-static field, i.e., the condition of a0 > 1 (Equation (1)) is important for our model. By averaging the electron momentum equation over the wave period and a spatial scale of a few wave-lengths, one obtains an equation for the evolution of thus averaged quantities: d p e = f  − e E − v × B dt c

(7)

Here v represents the average velocity of the electrons, related by the equation j = −en v to the average current density j. Neglecting for a moment the electron inertia (the lhs) and the Hall effect (the last term in the rhs), and using the Maxwell equation ∇ × E = −(1/c)∂ B/∂t, one obtains: ∂ B c = − ∇ × f ∂t e

(8)

One sees that, indeed, the generation of the magnetic field requires the presence of a solenoidal component of the ponderomotive force f , which is absent in the case of a nonrelativistic drive. Equation (8) also shows that, after the laser drive turns on, the magnetic field increases and reaches a steady state by the time the drive ends. (We shall see shortly

(9)

This estimate is valid only in the relativistic domain, at γ − 1 > 1. The geometrical structure of the force is similar to the current pattern shown in Fig. 2b. In the approximation described by Equation (8), the maximum magnetic field will be reached at the end of the pulse and will be equal to | B| ∼

γ mc3 τ er02

(10)

This is a rough, order-of-magnitude estimate. Let us now evaluate the possible contribution of the neglected terms. The complete version of Equation (8) reads: # $ ∂ B c c = − ∇ × f  + ∇ × [ v × B] + p ˙ ∂t e e

(11)



Noting that p ˙ ∼ γ m ˙v ∼ γ m j˙ /en ∼ (γ mc/4π en) ˙ where we have used ∇ × B = (4π/c) j, and ∇ × B, comparing the last term in the rhs with ∂ B /∂t, one finds that the inertial term can be neglected if γ mc2 c2 ≪ 1, ≡ 4πne2r02 ω′2pe r02

(12)

where ω′pe is a relativistically-corrected plasma frequency (a cut-off frequency). As the wave frequency is typically comparable to the cut-off frequency, this condition is automatically satisfied provided condition (6) holds. Now we assess the role of the Hall term. By using the relation v = − j/en = −(c/4π en)∇ × B, one can rewrite Equation (11) in the following way:

B × ∇ × B ∂ B c e . = − ∇ × f − ∇× ∂t e mc ω2pe

(13)

As the last term is nonlinear in B, it would lead to the saturation of the magnetic field for a long enough pulse. By balancing the forcing term, with f as in Equation (9), and the Hall term, one finds the following rough estimate for the saturated field: | B|2 ∼ γ mc2 n. 4π

(14)

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One can combine the estimate (10) and (14) in one heuristic relation, which covers both cases of a long and short driving pulses: B=

B0 τ τ + (2πr0 /λ)(r0 /c)

(15)

with B0 as in Equation (3). Here we assumed that the incident wave has a frequency near a cut-off frequency. Not surprisingly, Equation (15) can be presented in terms of B0 and the scaling parameters R and T: B = B0 /[1 + R 2 /T ], with r0 playing the role of the length-scale. Thus far, in the analysis of the magnetic field, we didn’t take into account the ion motion. In reality, due to the quasineutrality constraint, they will experience a force comparable to f (although, generally speaking, of a different structure). This would cause the whole plasma in the focal spot to expand. The condition that the expansion is negligible during the time τ , reads as:

r0 τ< c



M γmZ

(16)

If the opposite condition is valid, the zone of a high magnetic field is disassembled before the end of the pulse. It goes without saying that the poloidal field may be generated alongside the toroidal field, which we do not discuss in this paper.

4 Discussion In this paper we applied scaling and symmetry arguments to study two problems in the area of the interaction of ultraintense light with a plasma. The first problem is that of a collective ion acceleration by the ambipolar field where we identified a possible experimental test of the physics model based on two key assumptions: (1) that the system can be described reasonably well as collisionless, for both fast electrons and ions; (2) that the initial thermal spread is negligibly small compared to the energies that they acquire early upon arrival of the ultra-intense pulse. We discuss possible ways of experimental verification of this model by performing properly scaled experiments conducted so as to satisfy the similarity rules established in Ryutov and Remington (2006a,b). What is very attractive with this approach is that, if the similarity indeed holds, there are many experimental signatures, starting from the spectrum and spatio-temporal behavior of accelerated ions, through the quasi-static magnetic field evolution, and ending up with the spatio-temporal dependence

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of the reflected radiation. Deviations from the predictions of the scaling laws, in a carefully performed experiment, would mean the violation of the initial basic assumptions and would allow one to circumscribe a parameter domain in which the model is applicable. On the other hand, if the validity of the model is established, one can use the scaling laws as a predictive tool. In the second part of the paper, based on symmetry consideration, we establish a spatial structure of the poloidal magnetic field which may be generated alongside the toroidal field. For a linearly-polarized wave, the axial component of the field changes sign from one quadrant to the other. This is a signature that can be used if an experimental attempt to detect this field is made. Acknowledgements The authors are grateful to B.I. Cohen, L.L. Lodestro, and T.D. Rognlien for helpful comments. Work performed under the auspices of the U.S. DoE by UC LLNL under contract No W-7405-Eng-48.

References Antonsen Jr., T.M., Ott, E.: Appl. Phys. Lett. 28, 424 (1976) Arzhannikov, A.V., Burdakov, A.V., Koidan, V.S., Ryutov, D.D.: ZHETF Lett. 24, 19 (1976) Humphries Jr., S.: Nuclear Fusion 20, 1549 (1980) Gaponov, A.V., Miller, A.: Soviet Phys. JETP 7, 168 (1958) Gordienko, S., Pukhov, A.: Phys. Plasmas 12, 043109 (2005) Katsouleas, T.: Plasma Phys. Contr. Fus. 46, B575–582 (2004) Liang, E.: Astrophys. Space Sci. 298, 211 (2005) Mackinnon, A.J., Borghesi, M., Hatchett, S., et al.: Phys. Rev. Lett. 86, 1769 (2001) Moon, S.J., Wilks, S.C., Klein, R.I., et al.: Astrophys. Space Sci. 298, 293 (2005) Norreys, P.A., Krushelnick, K.M., Zepf, M.: Plasma Phys. Contr. Fus. 46, B13–21 (2004) Pukhov, A., Gordienko, S., Kiselev, S., Kostyukov, I.: Plasma Phys. Contr. Fus. 46, B179 (2004) Ryutov, D.D., Remington, B.A.: in: Superstrong Fields in Plasmas, Proc. 3rd Intern. Conf. on Superstrong Fields in Plasmas, Varenna, Italy, Sept. 19–24 2005, AIP Conference Proceedings v. 287, p. 341, Melville, NY (2006a) Ryutov, D.D., Remington, B.A.: Plasma Phys. Contr. Fus. 48, L23–L31 (2006b) Ryutov, D.D., Stupakov, G.V.: Fizika Plazmy 2, 767 (1976) Silva, L.O., Fonseca, R.A., Tonge, J.W., et al.: Ap. J. Lett. 596, L121 (2003) Snavely, R.A., Key, M.H., Hatchett, S.P., et al.: Phys. Rev. Lett. 85, 2945 (2000) Stamper, J.A.: Phys. Rev. Lett. 26, 1012 (1971) Sudan, R.: Phys. Rev. Lett. 70, 3075 (1993) Tatarakis, M., Gopal, A., Watts, I., et al.: Phys. Plasmas 9, 2244 (2002) Wei, M.S., Beg, F.N., Clark, E.L., et al.: Phys. Rev. E 70, 56412 (2004) Wilks, S.C., Chen, H., Liang, E., et al.: Astrophys. Space Sci. 298, 347 (2005) Wilks, S.C., Kruer, W.L., Tabak, M., Langdon, A.B.: Phys. Rev. Lett. 69, 1383 (1992)

Astrophys Space Sci (2007) 307:297–303 DOI 10.1007/s10509-006-9274-x

ORIGINAL ARTICLE

Topical Issues for Particle Acceleration Mechanisms in Astrophysical Shocks Matthew G. Baring

Received: 27 July 2006 / Accepted: 31 October 2006 C Springer Science + Business Media B.V. 2006


Abstract Particle acceleration at plasma shocks appears to be ubiquitous in the universe, spanning systems in the heliosphere, supernova remnants, and relativistic jets in distant active galaxies and gamma-ray bursts. This review addresses some of the key issues for shock acceleration theory that require resolution in order to propel our understanding of particle energization in astrophysical environments. These include magnetic field amplification in shock ramps, the non-linear hydrodynamic interplay between thermal ions and their extremely energetic counterparts possessing ultrarelativistic energies, and the ability to inject and accelerate electrons in both non-relativistic and relativistic shocks. Recent observational developments that impact these issues are summarized. While these topics are currently being probed by astrophysicists using numerical simulations, they are also ripe for investigation in laboratory experiments, which potentially can provide valuable insights into the physics of cosmic shocks. Keywords Shock acceleration . Cosmic rays . Magnetohydrodynamics . Supernova remnants . Gamma-ray bursts . Plasma physics

1 Introduction Supersonic flows abound in the cosmos, as do emission regions exhibiting non-thermal radiation. The intimate connection between the two establishes that particle acceleration in astrophysical shocks is germane to many systems, M. G. Baring Department of Physics and Astronomy MS-108, Rice University, P.O. Box 1892, Houston, TX 77251, U.S.A. e-mail: [email protected]

ranging from the heliosphere, to stars of various sorts expelling winds, to supernova remnants, to extragalactic jets and gamma-ray bursts. In the case of heliospheric shocks such as travelling interplanetary discontinuities and planetary bow shock environs, we can immerse ourselves in the plasma experiment via in situ spacecraft measurements of non-thermal ions, electrons and turbulent magnetic fields. While localized and therefore sparse in terms of the spatial sampling, these observations do provide profound insights into the complexity of the shock acceleration phenomenon. In astrophysical sites beyond the solar system, our role is passive, as observers of signals from remote sites of acceleration. Moreover, the information on plasma properties is subject to a convolution with radiative processes, complicated source morphology within our spatial resolution scale, and propagational modification along the line of sight to sources. Observationally, radio, optical and X-ray telescopes have provided groundbreaking insights into the shock acceleration phenomenon, due to advances in the angular resolution and spectral sensitivity. In addition, the gamma-ray field is generating a greater understanding of source energetics as we find that many non-thermal astronomical sources emit most of their power in the gamma-rays. Progress on the theoretical front builds on the observational advances, and has turned more to computer simulations due to the dramatically enhanced speed of computers over the last two decades. Yet astrophysical code verification is an increasingly salient issue as their complexity and computational demands burgeon. This provides a niche for laboratory plasma experiments that are tailored for the problem of astrophysical particle acceleration. Supersonic flows can be generated in controlled environments, and work along these lines focuses naturally on using lasers to mimic blast waves and jets, supernovae and supernova remnants (e.g. Borovsky et al., 1984; Drake et al., 1998; Shigemori et al., 2000; Kang et al., 2001; Lebedev Springer

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et al., 2002; Woolsey et al., 2004), and probe hydrodynamic and magnetohydronamic aspects. The question of scalability of such laboratory findings to astrophysical systems (e.g. Ryutov et al., 2001; see also the review of Remington et al., 2006) is obviously of central importance. Here, an offering on some topical issues for particle acceleration at astrophysical shocks is made, to provide a basis for the community in high energy density plasma physics to help identify germane astrophysical problems that might be well suited for interdisciplinary investigation.

2 Magnetic field enhancements in shocks One of the key properties of shock structure that is germane to the acceleration of high energy cosmic rays is the strength of the magnetic field B near the shock. In astrophysical shocks this cannot really be measured directly, since there is generally a lack of viable spectral line diagnostics: the Zeeman effect and cyclotron emission/absorption features are generally broadened, small or non-existent in diffuse, turbulent shock environs. Normally, proximity of an emission region to a stellar surface, such as in white dwarfs and neutron stars is required to afford precise magnetic field measurements. In heliospheric shocks, magnetometer data discern the chaotic nature of pre- and post-shock fields, and a prominent property appears to be (e.g. Baring et al., 1997) a shock-induced compression of the field in the downstream region that is fairly close to magnetohydrodynamic (MHD) determinations that are derived from momentum and energy flux conservation across the shock, i.e. the so-called Rankine-Hugoniot conditions (e.g. Drury, 1983; Jones and Ellison, 1991). If the interaction of charged particles with shock-associated field turbulence is gyroresonant at the Doppler-shifted cyclotron frequency (e.g., see Melrose, 1980), as is expected for Alfv´en and whistler modes, then the acceleration timescale τ naturally scales as the gyroperiod (i.e., τ ∼ 1/νg ∝ 1/B ; see Forman et al., 1974; Drury, 1983) and the corresponding diffusive lengthscale is comparable to the Larmor radius r g ( ∝ 1/B ). Hence the magnitude B establishes the temporal, spatial and energy scales of acceleration at a shock, and so is a critical parameter for the energization process. Since the general paradigm of galactic cosmic rays (CRs) invokes supernova remnants (SNRs) as the sites for their production (see Drury, 1983 for a review), knowledge of the field strength in proximity of their shocks is vital. Directional information can be obtained on fairly large spatial scales via radio synchrotron polarization data (e.g. see Rosenberg, 1970; Downs and Thompson, 1972; Anderson et al., 1995, for Cassiopeia A), but values of B = |B| are not forthcoming. Estimates for the field strength can be inferred by modeling the continuum flux level in a given wavelength band, but these are subject to a number of assumptions about the Springer

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medium, for example the mean density n e and the Lorentz factors γe of the radiating electrons. The understanding of the character of shells and interiors of SNRs has recently been advanced by groundbreaking observations with the Chandra X-ray Observatory, enabled by its impressive angular resolution coupled with its spectral capabilities. Of particular interest is the observation of extremely narrow non-thermal (1.2–2.0 keV) X-ray spatial profiles in selected remnants (see Long et al., 2003; Bamba et al., 2003 for the northeast limb of SN1006; Vink and Laming, 2003 for Cas A; for theoretical modeling see Ellison and Cassam-Chena¨ı, 2005; V¨olk et al., 2005), typically less than 5–10 arcsec across. Upstream of these shell shocks, the X-ray emission, which is thought to be synchrotron in origin, drops to effectively zero. These strong brightness contrasts between the shell, and the outer, upstream zones correspond to flux ratios exceeding R > ∼50 . The narrowness of profiles along image scans argues for the shocks being aligned perpendicular to the sky, i.e., offering no projectional smearing in the images. Note also that the surface brightness angular profiles in SN1006 and Cas A are much broader for the thermal X-rays (0.5–0.8 keV) and the radio synchrotron than for the non-thermal X-rays. If the synchrotron mechanism is indeed responsible for non-thermal Chandra emission, the electrons contributing to the Chandra signal are probably in a strongly-cooling regime: see Baring et al. (1999) for a comprehensive discussion of SNR cooling parameter space. Since the synchrotron cooling rate for an electron scales as γe2 B 2 , then the flux ratio R is approximately a measure of the ratio of B 2 downstream (d) to upstream (u). The observed lower bounds to R considerably exceed values R < ∼ 16 expected for magnetohydrodynamic compression at the shocked shell; at a plane-parallel shock with B along the shock normal there is no field compression, while in a strong (i.e. high sonic Mach number) perpendicular shock with B in the shock plane, Bd /Bu ∼ 4 . Hence, the pronounced brightness contrast is taken as strong evidence of magnetic field amplification in the shock precursor/ramp upstream. Higher fields are obviously advantageous to cosmic ray production issues. Historically-accepted values of B ∼ 1 – 10 µ Gauss (i.e. 0.1–1 nanoTesla) are somewhat too small to permit acceleration in SNR shocks of ages around 103 – 104 years right up to the cosmic ray knee at ∼ 3 × 1015 eV (e.g. see Lagage and Cesarsky, 1983). This problem has spawned the suggestion (Jokipii, 1987) that relatively ineffective diffusive transport of particles orthogonal to the mean field direction in quasi-perpendicular regions of SNR shocks can speed up acceleration of ions to higher energies, helping access the knee. Yet, this enhanced rapidity is accompanied by reduced efficiency of cosmic ray injection from thermal energies (Ellison et al., 1995). Hence, truly larger fields provide a cleaner path for acceleration in remnants to reach the cosmic ray knee (e.g. Kirk and Dendy, 2001).

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The X-ray observational developments have been accompanied by theoretical proposals of magnetic field amplification in the upstream shock precursor. Most notable has been the work of Lucek and Bell (2000), and subsequent papers such as Bell (2004), Amati and Blasi (2006), and Vladimirov et al. (2006). The idea of Lucek and Bell is that high energy cosmic rays in strong shocks could amplify B when streaming upstream, adiabatically transferring energy to the turbulent field by pushing against it, simultaneously decelerating the upstream flow. If this process is efficient, the rate of work done on the upstream Alfv´en turbulence of energy density UA naturally scales roughly with the CR pressure gra√ dient: dUA /dt = vA |∇ PCR | . Here vA = B/ 4πρ is the Alfv´en speed, and PCR is the cosmic ray pressure. The associated field amplification should then scale as (δ B/B)2 ∼ MA PCR /ρu 2u in an upstream flow of speed u u and mass density ρ; this then becomes very effective for high Alfv´enic Mach number (i.e. MA ≡ u u /vA ≫ 1 ), strong shocks that generate large cosmic ray pressures. While this hypothesis is reasonable, demonstrating it is non-trivial. Various MHDtype simulations have been employed by Bell, such as in Bell (2004), where large-scale currents are used to drive instabilities that amplify the upstream field. The persistence of currents on large scales is unclear, particularly due to the action of Debye screening. Moreover, self-consistent physical connection between the cosmic rays of large Larmor radii and the field turbulence of much shorter wavelengths is extremely difficult to explore with MHD or plasma simulations, due to the wide disparity in spatial scales involved. This is an issue also for a growing number of particle-incell (PIC) simulations (e.g. Silva et al., 2003; Hededal et al., 2004; Nishikawa et al., 2005; see Section 4 below) used to explore field enhancement via the Weibel instability in relativistic shocks; such developments are not that salient for the problem of amplifying Alfv´en turbulence in non-relativistic shocks, and mostly probe the inertial scales of thermal ions and electrons defined by their plasma frequencies.

3 Non-linear feedback between the acceleration and the hydrodynamics Non-relativistic collisionless shocks can be highly efficient accelerators, placing 10–50% of the bulk flow kinetic energy into non-thermal particles. Evidence from theory, computer simulations, and spacecraft observations supports this conclusion; in particular, see Ellison et al. (1990), for a study of the Earth’s bow shock, and Drury (1983), Blandford and Eichler (1987), and Jones and Ellison (1991) for reviews. With such efficiencies, the accelerated particles acquire a sizable fraction of the total energy budget, influencing the shock hydrodynamics, and therefore also the fraction of energy going into accelerated particles, in a non-linear manner. The

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modified flow velocity spatial profile in the shock deviates from the familiar step-function form in test-particle acceleration scenarios, with the energetic particles pushing against the upstream flow and decelerating it far ahead of the shock discontinuity. Accordingly an upstream shock precursor forms, with declining flow velocity as the shock is approached. This structure alters the shape of the energetic particle distribution from a power-law in momentum (e.g., Eichler, 1984; Ellison and Eichler, 1984; Ellison et al., 1996; Berezhko et al., 1996; Malkov, 1997; Blasi, 2002), the canonical testparticle form where the diffusively-accelerated particle distribution samples no spatial or momentum scale. The index σ = (r + 2)/(r − 1) of this power-law dn/d p ∝ p −σ is purely a function of the compression ratio r = u u /u d of upstream ( u u ) to downstream ( u d ) flow speed components normal to the shock in the shock rest frame (e.g. see Drury, 1983; Jones and Ellison, 1991), and is independent of the magnetic field orientation or the nature and magnitude of the turbulence effecting diffusive transport in the shock neighborhood. The spatial variation of the upstream flow in strong shocks that are efficient accelerators eliminates the scale independence. Since the highest energy particles have greater diffusive mean free paths λ (generally true for gyroresonant interactions with MHD turbulence, and certainly so near the Bohm diffusion limit λ ∼ r g ), they diffuse farther into the upstream shock precursor against the convective power of the flow, and therefore sample greater effective velocity compression ratios r . Accordingly, they have a flatter distribution, yielding a distinctive concavity to the overall particle spectrum, i.e. σ is now a declining function of momentum p . These departures from power-law behavior amplify the energy placed in the particles with the greatest momenta, which in turn feeds back into the shock hydrodynamics that modify the spatial flow velocity profile. Traveling discontinuities possessing this complex feedback are termed non-linear shocks, the non-linear label being ascribed to the interplay between the macroscopic dynamics and the microscopic acceleration process. Clearly, the possible magnetic field amplification in the upstream precursor that was discussed in Section 2 contributes to the overall dynamics/energy budget of the magnetohydrodynamic flow, and so intimately influences this non-linear aspect of astrophysical shocks. The deviations from power-law distributions obviously impact the radiation signatures produced by these particles, with alterations in the fluxes expected in X-ray and TeV gamma-ray bands in remnants, differing by as much as factors of 3–10 from traditional test-particle predictions (e.g., see Baring et al., 1999; Ellison et al., 2001; Berezhko et al., 2002; Baring et al., 2005). Conclusively confirming the existence of this non-linear spectral concavity is a major goal that is inherently difficult, since it demands broad, multi-wavelength spectral coverage. There is a limited suggestion of concavity Springer

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in radio data for Tycho’s and Kepler’s SNRs (Reynolds and Ellison, 1992), and in a multi-wavelength modeling of SN 1006 (Allen et al., 2004; see also Jones et al., 2003 for inferences from radio and infra-red data from Cassiopeia A), but this task really looks ahead to the launch of the GLAST gamma-ray mission in late 2007, when, in conjunction with ˇ ground-based Atmospheric Cerenkov Telescopes probing the TeV band, it may prove possible to determine gamma-ray spectra from SNRs spanning over 3 decades in energy. In the meantime, an interesting astrophysical manifestation of these non-linear effects has been offered by SNR observations by the Chandra X-ray Observatory, looking instead at the thermal populations. Inferences of ion temperatures in remnant shocks can be made using proper motion studies, or more direct spectroscopic methods (e.g. Ghavamian et al., 2003). For the remnant 1E 0101.2-7129, Hughes et al. (2000) used a combination of ROSAT and Chandra data spanning a decade to deduce an expansion speed. Electron temperatures Te are determined by line diagnostics, via both the widths, and the relative strengths for different ionized species. From these two ingredients, Hughes et al. (2000) observed that, in selected portions of the SNR shell, 3kTe /2 ≪ 3kT p /2 ∼ m p (3u u /4)2 /2 . Therefore, the electrons were considerably cooler than would correspond to equipartition with thermal protons heated in a strong shock with an upstream flow speed of u u : the thermal heating is assumed comparable to the kinematic velocity differential u u − u d ≈ 3u u /4 . The same inference was made by Decourchelle et al. (2000) for Kepler’s remnant, and by Hwang et al. (2002) for Tycho’s SNR. This property of comparatively cooler electrons may be indicative of them radiating very efficiently. Or it may suggest that the protons are cooler (i.e. 3kT p /2 ≪ m p (3u u /4)2 /2 ) than is widely assumed in the test-particle theory, the conclusion drawn by Hughes et al. (2000) and Decourchelle et al. (2000). This effect is naturally expected in the non-linear shock acceleration scenario: as the highest energy particles tap significant fractions of the total available energy, they force a reduction in the thermal gas temperatures. Such feedback can profoundly influence shock layer thermalization, inducing significant interplay with electrostatic equilibration between low energy electrons and ions, an issue addressed in these proceedings by Baring and Summerlin (2006). Note that nonlinear modifications may vary strongly around the shocked shell of an SNR, since the obliquity angle Bn,u of B to the shock normal varies considerably between different rim locales.

4 The character of relativistic shocks Relativistic shocks, for which the upstream flow Lorentz fac tor γu = 1/ 1 − (u u /c)2 considerably exceeds unity, are Springer

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less well researched than their non-relativistic counterparts, not in small part due to their greater cosmic remoteness: they predominantly arise in extragalactic locales like jets in active galaxies, and gamma-ray bursts. Yet, because of such associations, they are now quite topical. Diffusive test-particle acceleration theory in parallel (i.e., Bn,u = 0◦ ) relativistic shocks identifies two notable properties in such systems: (i) particles receive a large energy kick E ∼ γu mc2 in their first shock crossing (e.g., Vietri, 1995), but receive much smaller energy boosts for subsequent crossing cycles (factors of around two: e.g., Gallant and Achterberg, 1999; Baring, 1999); (ii) a so-called ‘universal’ spectral index, σ ∼ 2.23 exists in the two limits of γu ≫ 1 and small angle scattering, i.e., δθ ≪ 1/γu (e.g., Kirk et al., 2000; see also Bednarz and Ostrowski, 1998; Baring, 1999; Ellison and Double, 2004). Here, δθ is the average angle a particle’s momentum vector deviates in a scattering event, i.e. an interaction with magnetic turbulence. These characteristics are modified in parallel, mildly relativistic shocks with γu ∼ 1 . In such shocks, the distribution dn/d p remains a power-law (scale-independence persists), but hardens ( σ decreases) as either γu drops, or the scattering angle, δθ, increases (e.g., Ellison et al., 1990; Baring, 1999; Ellison and Double, 2004; Baring, 2004), even if the compression ration r = u u /u d is held constant (it usually increases with declining γu due to a hardening of the J¨uttnerSynge equation of state). These effects are consequences of large kinematic energy kicks particles receive when scattered in the upstream region after transits from downstream of the shock. It is particularly interesting that when scattering conditions deviate from fine pitch-angle-scattering regimes with δθ ≪ 1/γu , the power-law index is dependent on δθ, with a continuum of spectral indices being possible (Ellison and Double, 2004; Baring, 2004). Then the nature of the turbulence is extremely influential on the acceleration outcome, so that understanding the turbulence is of paramount importance. This sensitivity of σ to the field fluctuations when δθ > ∼ 1/γu , a large angle scattering domain, contrasts the canonical nature of σ in non-relativistic shocks mentioned above. In jets and gamma-ray bursts, ultra-relativistic shocks are typically highly oblique due to the Lorentz transformation of ambient, upstream magnetic fields to the shock rest frame. This introduces an added dimension of variation, with increasing Bn,u dramatically steepening the power-law, i.e. increasing σ . This is naturally expected since such systems are highly superluminal, that is, there exists no de HoffmanTeller (1950) shock rest frame where the flow velocities are everywhere parallel to the mean magnetic field (which would correspond to large scale electric fields being zero everywhere). Therefore, relativistic shocks are much less efficient accelerators because particles convect more rapidly away downstream from the shock (e.g. Begelman and Kirk,

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1990). In oblique, relativistic shocks, σ , and indeed the efficiency of injection from the thermal particle population, also depend on the ability of turbulence to transport particles perpendicular to the mean downstream field direction (Ellison and Double, 2004; Niemiec and Ostrowski, 2004). This perpendicular transport couples directly to the magnitude (δ B/B)2 and power spectrum of field fluctuations, i.e. the strength of the scattering. Steep spectra (σ > ∼ 4) result unless the ratio of the diffusive mean free paths perpendicular to and parallel to B is comparable to unity, which defines the Bohm diffusion regime. In summation, for relativistic shocks, the spectral index is sensitive to the obliquity Bn,u of the shock, the nature of the scattering, and the strength of the turbulence or anisotropy of the diffusion. These properties are reviewed in Baring (2004). Observational vindication of these theoretical predictions is clearly mandated. This is not readily forthcoming, since the only accessible information involves a convolution of shock acceleration and radiation physics. Yet, it is clear, for example in gamma-ray bursts (GRBs), that data taken from the EGRET experiment on the Compton Gamma-Ray Observatory (CGRO) suggest a broad range of spectral indices (Dingus, 1995) for the half dozen or so bursts seen at high energies. This population characteristic is commensurate with the expected non-universality of σ just discussed. Yet it is important to emphasize that the power-law index is not the only acceleration characteristic germane to the GRB problem: the shapes of the particle distributions at thermal and slightly suprathermal energies are also pertinent. This energy domain samples particle injection or dissipational heating in the shock layer, and is readily probed for electrons by the spectrum of prompt GRB emission by the BATSE instrument on CGRO. Tavani (1996) obtained impressive spectral fits to several bright BATSE bursts using a phenomenological electron distribution and the synchrotron emission mechanism. While there are issues with fitting low energy (i.e. < ∼ 100 keV) spectra in about 1/3 of bursts (e.g. Preece et al., 1998) in the synchrotron model, this radiative mechanism still remains the most popular candidate today for prompt burst signals. Tavani’s work was extended recently by Baring and Braby (2004), who provided additional perspectives, using acceleration theory to underpin a program of spectral fitting of GRB emission using a sum of thermal and non-thermal electron populations. These fits demanded that the preponderance of electrons that are responsible for the prompt emission constitute an intrinsically non-thermal population. That is, the contribution to the overall electron distribution that comes from a Maxwell-Boltzmann distribution is completely dominated by a non-thermal component that, to first order, can be approximated by a power-law in energy truncated at some minimum electron Lorentz factor. This requirement of nonthermal dominance strongly contrasts particle distributions

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obtained from acceleration simulations, as is evident in a host of the references cited on acceleration theory above: the non-thermal particles are drawn directly from a thermal gas, a virtually ubiquitous phenomenon. This conflict poses a problem for acceleration scenarios unless (i) radiative efficiencies for electrons in GRBs only become significant at highly superthermal energies, or (ii) shock layer dissipation in relativistic systems can suppress thermalization of electrons. A potential resolution to this dilemma along the lines of option (i) is that strong radiative self-absorption could be acting, in which case the BATSE spectral probe is not actually sampling the thermal electrons. It is also possible that other radiation mechanisms such as Compton scattering, pitch-angle synchrotron, or jitter radiation may prove more germane. Discerning the radiation mechanism(s) operating in bursts is a foremost goal of future research, and will be facilitated by the GLAST mission, with its good sensitivity in the 5 keV–300 GeV band, in conjunction with NASA’s current GRB flagship venture, Swift. Option (ii) is a conjecture that has no definitive simulational evidence to support it at present. The most comprehensive way to study dissipation and wave generation in collisionless shocks is with PIC simulations, where particle motion and field fluctuations are obtained as solutions of the Newton-Lorentz and Maxwell’s equations. Rich in their turbulence information, these have been used extensively in non-relativistic, heliospheric shock applications, and more recently, relativistic PIC codes have blossomed to model shocks in various astrophysical systems. PIC simulation research has largely, but not exclusively, focused on perpendicular shocks, first with Gallant et al. (1992), Hoshino et al. (1992), and then Smolsky and Usov (1996), Shimada and Hoshino (2000), Silva et al. (2003), Nishikawa et al. (2003, 2005), Spitkovsky and Arons (2004), Hededal et al. (2004), Liang and Nishimura (2004), Medvedev et al. (2005) and Hededal and Nishikawa (2005). These works have explored pair shocks, ion-doped shocks, Poynting flux-dominated outflows, and low-field systems with dissipation driven by the Weibel instability, in applications such as GRBs and pulsar wind termination shocks. PIC simulations are dynamic in nature, and rarely achieve a time-asymptotic state. Even in the minority of cases where there is some evidence of acceleration beyond true thermalization, none of these works has demonstrated the establishment of an extended power-law that is required in modeling emission from GRBs and active galactic nuclei. This is perhaps due to the severely restricted spatial and temporal scales of the simulations, imposed by their intensive CPU and memory requirements; these limit the modeling of realistic electron-to-proton mass ratios, full exploration of three-dimensional shock physics such as diffusive transport, and addressing the wide range of particle momenta encountered in the shock acceleration process. In particular, it is difficult to establish a broad inertial range Springer

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for cascading MHD turbulence when the maximum spatial scales in the simulation are not orders of magnitude larger than the principal ion inertial scales. Definitively observing the injection of electrons from a thermal population to establish a truly non-thermal distribution remains a pressing goal of plasma simulations, both for relativistic shocks and, as has been understood for more than two decades, also their non-relativistic cousins. 5 Conclusion This review is by no means a complete presentation of the topical issues for the shock acceleration problem, but it does offer a fair sampling suitable for motivating interdisciplinary activity. It is clear that several issues could benefit substantially from input from laboratory experimentation on the high energy density physics/astrophysics interface. One key question is whether or not ambient magnetic fields are amplified by both non-relativistic and relativistic shocks beyond standard MHD expectations. If so, is the amplification electrostatic in origin, or is it connected to energetic particles accelerated by the shock? It would be important to discern whether there are differences between high and low Alfv´enic Mach number systems, i.e. what role the ambient magnetic field plays in controlling the outcome. Another question concerns whether or not suprathermal electrons and ions can actually be seen, and whether one can identify their origin. It is salient to ascertain if they are diffuse in nature, or if they form coherent beams, both of which are seen at traveling shocks embedded in the solar wind. Also, if acceleration is observed, then identifying the role the highest energy particles have in modifying the shock hydrodynamics and the thermal structure of the shock layer would help solve an outstanding problem that has long been a principal goal within the cosmic ray community. Finally, specifically concerning relativistic systems, it would be desirable to elucidate how the distributions of any accelerated particles seen depend on external quantities such as the field obliquity and speed of the shock, whether thermal electrons can be suppressed relative to accelerated ones, and if there is an identifiable connection with the field turbulence near the shock. These are demanding goals, yet terrestrial experiments are very useful for probing global aspects of shock problems, and in particular for extracting insights into hydrodynamic and MHD behavior. In order to make progress, it is essential to prepare an experimental setup that is as tenuous as possible, to mimic the collisionless (in the Spitzer sense) shock environments offered throughout the cosmos. At this juncture, exciting prospects are on the horizon for this interdisciplinary forum, with contributions to be found in the laboratory, in computer simulations, and in astronomical observations, all of which can benefit from cross-fertilization with each other.

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Astrophys Space Sci (2007) 307:305–308 DOI 10.1007/s10509-006-9242-5

O R I G I NA L A RT I C L E

Three-Dimensional Particle Acceleration in Electromagnetic Cylinder and Torus Koichi Noguchi · Edison Liang

Received: 14 April 2006 / Accepted: 23 August 2006 C Springer Science + Business Media B.V. 2006


Abstract Particle acceleration via Poynting vector with toroidal magnetic field is studied in 3D PIC simulation of electron-positron plasma. We choose two different initial magnetic field configurations to compare how the particle acceleration is affected by the expansion of electromagnetic wave. In the cylindrical case, the electromagnetic field strength decays as (ct)−2 , and particles are accelerated in the radial direction as well as the axial direction. RayleighTaylor instability is also observed at the center of the cylinder. In the torus case, the field strength decays as (ct)−3 , making the acceleration less efficient. Particles accelerated in the axial direction by E × B force creates strong charge separation. Keywords Gamma ray bursts . Collapsars . PIC . Numerical . Relativity

1 Introduction One of the most famous models to produce a GRB and large explosion energy from the death of a massive star is the collapsar model (Woosley, 1993; MacFadyen and Woosley, 1999). When a massive star is collapsing to a black hole, it accretes envelope matter at a very high rate. Woosley pointed out that neutrinos should be emitted from the innermost region of the accretion disk through the electron-positron capture and electron-positron pair annihilation (Berezinskii and Prilutskii, 1987; MacFadyen and Woosley, 1999). A strong jet with a large explosion energy on the order of ∼1052 ergs, is generated from the polar region by the pair-annihilation K. Noguchi (
) · E. Liang Rice University, Houston, TX 77005-1892 e-mai: [email protected].

of neutrinos, and the shock wave propagates through a channel created inside of the stellar envelope. Two-dimensional relativistic hydrodynamic simulations with nucleosynthesis (MacFadyen and Woosley, 1999; Nagataki, 2000, 2001; Nagataki et al., 2003) confirms the formation and propagation of jets inside of the stellar envelope. Poynting flux acceleration (PFA) may be launched when the magnetic jet head or the magnetar stripe wind emerge from the stellar envelope surface due to the sudden deconfinement of the magnetic field and embedded electronpositron plasma. When the electron-positron plasma jet is emerged out from the envelope, electromagnetic wave expands into ambient medium with particles accelerated by the PFA (Liang and Nishimura, 2004; Noguchi et al., 2005). Since magnetic fields are connected to the accretion disk associated with the collapsar, the emerged jet is collimated and creates a magnetic tower (Lynden-Bell, 2003). Three-dimensional magnetohydrodynamic global simulations of a central black hole associated with low magnetized accretion disk (Kato et al., 2004) shows that vertically inflating toroidal fields supports the magnetic jet collimation. In this article we present 3D PIC simulations of particle acceleration driven by PFA with two different initial magnetic field configurations. In the first model, initial toroidal magnetic field and particle distributions are a decreasing function of radius, and have infinite length in the axial direction. In the second model, magnetic field and particles are distributed as a radially decreasing function as the same as the first model, but has a finite length in the axial direction. The first model represents the particle acceleration in the magnetic tower, and the second represents the magnetic jet head emerged from the stellar envelope. Hereafter we call the first model as the cylindrical case and the second as the torus case, respectively. Springer

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Fig. 1 3D contour plot of magnetic field strength (left) and the magnetic field profile of the plane y = 0 (right) at tce = 520

2 Initial setup of the simulation We use the 3D explicit PIC simulation scheme based on the Yee algorithm (Yee, 1966). Spatial grids for the fields are uniform in all directions, x = y = z = c/ω pe , where ω pe is the electron plasma frequency. We employ 1283 cells with triply periodic boundary conditions. In the simulation, the origin is located at the center, and x direction corresponds to the axis of the magnetic cylinder and torus. For both the cylindrical and torus cases, the radial profiles of initial magnetic field and particle density are the same. In order to make the magnetic field satisfy ∇ · B = 0, the initial magnetic field profile has toroidal component only, which is given as

Bφ =

%

B0r/(2x), 0 ≤ r < 2x 2B0 x/r,

2x ≤ r ≤ 8x

.

(1)

We note that the radial profile of Bφ corresponds to the profile around cylindrical infinite current column with j = cB0 /4πr (r < 2π x). For simplicity and self-consistency, we assume no initial electric field and current. Electric field and current are self-induced by magnetic field and particle motion associated with it. In magnetic-tower scenario, magnetic field near the tip of a jet is almost purely toroidal, and plasma motion is driven by magnetic field expansion. The density profile is given as

ρ=

%

ρ0r/(2x), 0 ≤ r < 2x 2xρ0 /r,

2x ≤ r ≤ 5x

.

3 Results First, we study the cylindrical case. Three-dimensional contour plot of magnetic field strength and the magnetic field profile of the plane y = 0 at tce = 520 are shown in Fig. 1.Only a quarter of the whole simulation box is shown in Fig. 1a. Electric field E z is automatically generated by the expansion of magnetic field, and the direction of the Poynting vector is always in the radial direction. EM wave expands to surrounding vacuum region with the speed ∼ c, carrying particles within its ponderomotive well. Electrons and positrons are also accelerated in the negative and positive z direction respectively, due to E × B drift, as we expected. Due to the periodic boundary condition, no charge separation occurs in the z direction. Rayleigh-Taylor instability occurs in the central region (r < 10x). The critical wave number k x for electronpositron plasma is given by Chen (1984)

(2)

Here, ρ0 is calculated by assuming ω pe / ce = 0.1, where ce is the electron cyclotron frequency. The initial temperature of plasma is a spatially uniform Maxwellian k B Te = k B T p = 100eV, where Te and T p are the Springer

electron and positron temperature, respectively. We note that we use c/ω pe instead of electron Debye length as a unit scale in the simulations, because EM field expands with the speed of light rather than the thermal speed. In the cylindrical case, Bφ and ρ are uniform in the z direction. In the torus case, however, both Bφ and ρ vanishes for z < −2c/ω pe and z > 2c/ω pe in order to simulate the magnetic jet head. Initial temperature is assumed to be uniform Maxwellian with k B Te = k B T p,i = 100eV.

k x2 v02 < −g

ρ0′ , ρ0

(3)

where v0 = −g/ ce is the positron centrifugal drift velocity in the x direction, ρ0′ = dρ0 /dr , and g = v02 /r is the centrifugal field strength in the radial direction by the magnetic field

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Fig. 2 Phase plots of electrons (blue) and positrons (red) in the cylindrical case at tce = 560

Fig. 3 3D contour plot of magnetic field strength (a) and the spatial distribution of electrons (blue) and positrons (red) (b) at tce = 520

curvature. The minimum wavelength λ = 2πr is ∼ 30x on the surface of the initial plasma column, which corresponds to the wavelength in magnetic field profile Fig. 1b. Only the center of the column becomes unstable, since ρ0′ rapidly goes to zero toward the edge of magnetic field column. Figure 2shows phase plots at tc e = 560. Fig. 2a shows that troidal magnetic field Bφ create a current in the x direction. Figures 2b and 2c indicates that the most energetic particles are accelerated with γ ∼ 10 in both x and y directions, and slow particles are not accelerated by PFA in y and z. Figures 2d and 2e shows that strong bifurcation occurs in the y direction. The first ponderomotive force well in the front of EM pulse (y ∼ ±50x) is too weak to hold all the energetic particles, and following ponderomotive wells (y ∼ ± 40x and ± 30x) capture such particles slipped out from the first well. Finally, Fig. 2f shows the expansion is uniform in the y − z plane.

Next, we show the results of the torus case. Figure 3a shows the three-dimensional contour plot of magnetic field at tce = 520. In this case, EM field expands almost spherically. Different from the cylindrical case, the magnetic torus has finite length in the x direction, resulting the charge separation between electrons and positrons. Figure 3b shows the spatial distribution of sample electrons (blue) and positrons (red). Positrons and electrons are tend to move in the positive and negative x direction, respectively, creating electric field in the x direction. Only few particles are captured and accelerated by PFA, since the magnetic field strength drops as (ct)−3 in the torus case, whereas (ct)−2 in the cylindrical case. Figure 4 shows phase plots at tc e = 560. Figure 4a indicates strong acceleration occurs on the edge of torus, creating charge separation. The highest γ factor for the most energetic particles is ∼10, which is the same order as the cylindrical Springer

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Fig. 4 Phase plots of electrons (blue) and positrons (red) in the cylindrical case at tce = 560

case. Charge separation may prevent particles from accelerating for long time, but simulation box is too small to see if the acceleration will stop. Figures 4b and 4 c shows that the acceleration in the x − y plane is highly non-uniform, and the highest γ in the y direction is almost half of γ in the x direction. Figures 4d and 4e shows that the acceleration of particles captured in the front of EM pulse (|y| > 25x) is not efficient, and more effective acceleration occurs in the second well (10x < |y| < 25x). As we mentioned, magnetic field strength drops as r −3 , and the first well is too shallow to capture particles in it. The location of the second well corresponds to the propagation of the initial field peak (r = 2x). Finally, Fig. 4 f shows the expansion is uniform in the y − z plane. 4 Summary We studied 3D PIC simulations of particle acceleration driven by PFA with two different initial magnetic field configurations. Acceleration by PFA is robust in both cases, without showing any instability. The efficiency, however, strongly depends on how strong magnetic field is and how it expands. In the cylindrical case, we observe acceleration of particles in the radial direction as well as the axial direction. Acceleration in the axial direction is due to E × B force, whereas in the radial direction is by PFA. Without support from external ambient pressure, EM wave expands indefinitely with decaying the field strength proportional to (ct)−2 . Bifurcation in the phase space occurs because the ponderomotive potential well becomes too shallow to hold particles in it. RayleighTaylor instability occurs at the center, which does not affect the particle acceleration by PFA. Springer

In the torus case, the expansion of magnetic field is spherical rather than cylindrical, and the magnetic field strength decays with (ct)−3 . As a result, the front potential well is too weak to hold particles, and radial acceleration by PFA is not as efficient as the cylindrical case. Another important difference is the charge separation between electrons and positrons, which may terminate the acceleration in the axial direction. Our simulation results show that hydrodynamical and/or MHD simulations are not sufficient to understand the acceleration process in collapsar jets. We are planning to run more realistic model to simulate the acceleration and radiation of particles by jets. Acknowledgements This research is partially supported by NASA Grant No. NAG5-9223, NSF Grant No. AST0406882, and LLNL contract nos. B528326 and B541027. The authors wish to thank ILSA, LANL, B. Remington and S. Wilks for useful discussions.

References Chen, F.F.: Plasma Physics and Controlled Fusion, Vol. 1, Plenum Press, New York (1984) Berezinskii, V.S., Prilutskii, O.F.: A&A 175, 309 (1987) Kato, Y., Mineshige, S., Shibata, K.: ApJ 605, 307 (2004) Liang, E., Nishimura, K.: Phys. Rev. Lett. 92, 175005 (2004) Lynden-Bell, D.: MNRAS 341, 1360 (2003) MacFadyen, A.I., Woosley, S.E.: ApJ 524, 262 (1999) Nagataki, S.: ApJS 127, 141 (2000) Nagataki, S.: ApJS 551, 429 (2001) Nagataki, S., Kohri, K., Ando, S., Sato, K.: Astropart. Phys. 18, 551 (2003) Noguchi, K., Liang, E., Nishimura, K.: Nuovo Ciment C 028, 381 (2005) Woosley, S.E.: ApJ 405, 273 (1993) Yee, K.S.: IEEE Trans. Antennas Propag. 14, 302 (1966)

Astrophys Space Sci (2007) 307:309–313 DOI 10.1007/s10509-006-9281-y

O R I G I NA L A RT I C L E

Simulating Poynting Flux Acceleration in the Laboratory with Colliding Laser Pulses Edison Liang

Received: 9 May 2006 / Accepted: 21 November 2006 C Springer Science + Business Media B.V. 2007


Abstract We review recent PIC simulation results which show that double-sided irradiation of a thin over-dense plasma slab with ultra-intense laser pulses from both sides can lead to sustained comoving Poynting flux acceleration of electrons to energies much higher than the conventional ponderomotive limit. The result is a robust power-law electron momentum spectrum similar to astrophysical sources. We discuss future ultra-intense laser experiments that may be used to simulate astrophysical particle acceleration. Keywords Electron acceleration . Laser-plasma interaction . Laboratory astrophysics 1 Introduction Most high energy astrophysical sources (pulsars, blazars, gamma-ray bursts, supernova remnants) emit a simple powerlaw spectrum in the X-gamma-ray range. The most common observed photon index lies in the range 2–3, which translates into an electron momentum index of 3–5 for optically thin radiation (Rybicki and Lightman, 1979). The most popular current models for astrophysical particle acceleration are shock acceleration (first-order Fermi), diffusive wave acceleration, and Poynting flux acceleration by large-scale electromagnetic fields. In earlier work (Liang et al., 2003) we demonstrated that Poynting flux acceleration driven by electromagnetic-dominated outflows (Liang et al., 2003) naturally produces robust power-law relativistic electron spectra. Poynting flux acceleration of e+e− plasmas is especially relevant to gamma-ray bursts and pulsar winds. It is therefore highly desirable to study particle acceleration in the labora-

tory that may mimic or at least shed new light on Poynting flux acceleration in astrophysics. Recent advances in ultra-intense short-pulse lasers (ULs) (Mourou et al., 1998; Ditmire, 2003) open up new frontiers on particle acceleration by ultra-strong electromagnetic (EM) fields in plasmas (Lontano et al., 2002). However, most conventional laser acceleration schemes (e.g. laser wakefield accelerator, plasma wakefield accelerator, plasma beat-wave accelerator, free wave accelerator, see Tajima and Dawson, 1979; Sprangle et al., 1990; Kawata et al., 1991; Hussein et al., 1992; Esarey et al., 1996; Woodworth et al., 1996; Pukhov et al., 1997; Malka, 2002) involve the propagation of lasers in an underdense plasma (ωpe = (4π ne2 /m e )1/2 < ωo = 2π c/λ, λ = laser wavelength, n = electron density). In such schemes the acceleration gradient (energy gain/distance) (Esarey et al., 1996; Malka, 2002) and energetic particle beam intensity are limited by the underdense requirement. They also do not produce a power-law electron spectrum. Here we review PIC simulation results of a radically different concept: comoving acceleration of overdense (ωpe > ωo ) plasmas using colliding UL pulses. In this case the acceleration gradient and particle beam intensity are not limited by the underdensity condition. This colliding laser pulses accelerator (CLPA) concept may have important applications to laboratory astrophysics since CLPA naturally produces a power-law electron spectrum, similar to the high energy spectra of observed astrophysical sources. Most other laser acceleration schemes produce either exponential or quasimonoenergetic electron momentum distributions.

2 Colliding laser pulses accelerator E. Liang Rice University, Houston, TX 77005-1892 USA e-mail: [email protected]

Figure 1 shows the basic idea of CLPA. Two linear polarized intense laser pulses with aligned B vectors irradiate a Springer

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B

2 opposite

B thin slab of e+eplasma

EM pulses

Fig. 1 Schematic diagram showing the CLPA concept

thin overdense plasma slab from opposite sides. They compress the slab until it becomes thinner than 2 relativistic

Fig. 2 Evolution of two linearly polarized plane EM pulses (I (λ/µm)2 = 1021 W/cm2 , cτ = λ/2) irradiating an overdense e+e− plasma (n o = 15 ncr , thickness = λ/2, kT = 2.6 keV) from opposite sides. We plot magnetic field B y (medium), electric field Ez (light), current density Jz (dark) and px /mc vs. x (inset) at tωo /2π = (a)1.25, Springer

skin depths. At that point the laser pulses “tunnel through” and capture the surface electrons as they reemerge at the far side of the slab. Due to plasma loading the laser pulses slow down and stay in phase with the fastest particles, and accelerate them continuously with self-induced comoving J × B forces. Figure 2 shows the PIC simulation of two linearly polarized plane half-cycle EM pulses with parallel B, irradiating a thin e+e− slab from opposite sides (thickness = λ/2, initial density n o = 15n cr (critical density)). Cases with nonparallel B are more complex and are still under investigation. Each incident pulse compresses and accelerates the plasma inward (Fig. 1a), reaching a terminal Lorentz factor of γmax ∼ (e /ωpe )2 ∼ 40. Only ∼10% of the incident EM amplitudes is reflected because the laser reflection front is propagating inward relativistically (Kruer et al.,

(b)1.5, (c) 1.75; (d) Snapshots of px /m e c vs. x (dots) for the rightmoving pulse at tωo /2π = 2.5 (black), 5 (red), 10 (blue), 22.5 (green) showing power law growth of γmax ∼ t 0.45 . We also show the profiles of B y (medium), Ez (light) at tωo /2π = 22.5 (from Liang, 2006)

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Fig. 3 Results of two Gaussian pulse trains (λ = 1µm, I = 1021 W/cm2 , cτ = 85 fs) irradiating a e+e− plasma (n o = 9n cr , thickness = 2λ/π, kT = 2.6 keV). (a) early B y and n o /n cr (B) profiles at tωo = 0; (b) time-lapse evolution of log (px /me c) vs. logx for the rightmoving pulse at tωo = (left to right) 180, 400, 800, 1600, 2400, 4000, 4800 showing power-law growth of γmax ∼ t 0.8 ; (c) evolution of elec-

tron energy distribution f (γ ) vs. γ showing the build-up of power-law below γmax with slope ∼ −1: tωo = (left to right) 180, 400, 800, 2400, 4800. (Slope = −1 means equal number of particles per decade of energy), (d) plot of γ vs. θ(= | pz |/| px |) in degrees at tωo = 4800, showing strong energy-angle selectivity and narrow beaming of the most energetic particles (from Liang, 2006)

1975). As the relativistic skin depths from both sides start to merge (Fig. 1b), the two UL pulses interpenetrate and tunnel through the plasma, despite ωpe > γ 1/2 ωo . Such transmission of EM waves through an overdense plasma could not be achieved using a single UL pulse, because there the upstream plasma is snowplowed by the laser pressure indefinitely. As the transmitted UL pulses reemerge from the plasma, they induce new drift currents J at the trailing edge of the pulses (Fig. 1c), with opposite signs to the initial currents (Fig. 1b), so that the new J × B forces pull the surface plasmas outward. We emphasize that the plasma loading which slows the transmitted UL pulses plays a crucial role in sustaining this

comoving acceleration. For a given e /ωpe the higher the plasma density, the more sustained the comoving acceleration, and a larger fraction of the plasma slab is accelerated. This unique feature distinguishes this overdense acceleration scheme from other underdense schemes. As slower particles gradually fall behind the UL pulses, the plasma loading of the UL pulses decreases with time. This leads to continuous acceleration of both the UL pulses and the dwindling population of trapped fast particles. The phase space evolution (Fig. 1d) of this colliding laser pulses accelerator (CLPA) resembles that of the DRPA discovered earlier (Liang et al., 2003, 2004; Nishimura et al., 2004).

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Fig. 4 Conceptual experimental setup for the demonstration of the CLPA mechanism using three PW lasers

3 Acceleration by colliding Gaussian laser pulse trains Figure 3 shows the results of irradiating an overdense e+eslab using more realistic Gaussian pulse trains (λ = 1 µm, pulse length τ = 85fs, Ipeak = 1021 Wcm−2 ). We see that γmax increases rapidly to 2200 by 1.28 ps and 3500 by 2.56 ps, far exceeding the ponderomotive limit ao2 /2 (∼ 360). The maximum Lorentz factor increases with time according to γmax (t) ∼ e E(t) dt/mc. E(t) is the UL electric field comoving with the highest energy particles. E(t) decreases with time due to EM energy transfer to the particles, plus slow dephasing of particles from the UL pulse peak. This leads to γmax growth slower than linear and γmax ∼ t 0.8 (Fig. 2b). In practice, γmax will be limited by the diameter D of the laser focal spot, since particles drift transversely out of the laser field after t ∼ D/c. The maximum energy of any comoving acceleration is thus < eEo D = 6 GeV(I/1021 Wcm−2 )1/2 (D/100 µm). The asymptotic momentum distribution forms a power-law with slope ∼ −1 (Fig. 2d) below γmax , distinct from the exponential distribution of ponderomotive heating (Kruer et al., 1985; Wilks et al., 1992; Gahn et al., 1999; Wang et al., 2001; Sheng et al., 2004). A quasi-power-law momentum distribution is formed below γmax since there is no other preferred energy scale below γmax , and the particles have random phases with respect to the EM field profile.

fluence. Such a high density e+e− jet can be slit-collimated to produce a ∼ micron thick e+e− slab, followed by 2-sided irradiation with opposite UL pulses. As an example, consider UL pulses with τ = 80 fs and intensity = 1019 Wcm−2 . We need focal spot diameter D > 600 µm for the pairs to remain inside the beam for >1 ps. This translates into ∼1 KJ energy per UL pulse. Such high-energy UL’s are currently under construction at many sites (Ditmire, 2003). Figure 3 shows the artist conception of such an experimental setup. We have also performed simulations of CLPA using electron-ion plasmas. Results so far suggest that as long the e-ion slab is sufficiently thin and laser pulses sufficiently intense, so that the electrons can be compressed to less than two relativistic skin depths before the lasers are reflected, the electrons are accelerated by the reemerging pulses similar to the e+e− case. However the ions lag behind the electrons due to their inertia and are accelerated only by the charge-separation electric field. The late-time partition between electron and ion energies depends on the plasma density and laser intensities. Note that CLPA is insensitive to the relative phases of the two pulses. If one pulse arrives first it simply pushes the plasma toward the other pulse until it hits. Then both pulses compress the slab together with the same final results. Acknowledgements EL was partially supported by NASA NAG59223, LLNL B537641 and NSF AST-0406882. He thanks Scott Wilks for help with running ZOHAR and the graphics of Fig. 4, and Bruce Langdon for providing the ZOHAR code.

4 Proposed laser experiment An experimental demonstration of the CLPA will require a dense and intense e+e− source. (Cowan et al., 1999, 2000) demonstrated that such an e+e− source can be achieved by using a PW laser striking a gold foil. Theoretical works (Liang et al., 1998; Shen et al., 2001) suggest that e+e− densities >1022 cm−3 may be achievable with sufficient laser Springer

References Cowan, T.E. et al.: Laser Part. Beams 17, 773 (1999) Cowan, T.E. et al.: Phys. Rev. Lett. 84, 903 (2000) Ditmire, T. (ed.): SAUUL Report, UT Austin (2003) Esarey, E., Sprangle, P., Krall, J., Ting, A.: IEEE Trans. Plasma Sci. 24, 252 (1996)

Astrophys Space Sci (2007) 307:309–313 Gahn, C. et al.: Phys. Rev. Lett. 83, 4772 (1999) Hussein, M.S., Pato, M.P., Kerman, A.K.: Phys. Rev. A 46, 3562 (1992) Hussein, M.S., Pato, M.P.: Phys. Rev. Lett. 68, 1992 (1991) Kawata, S., Maruyama, T., Watanabe, H., Takahashi, I.: Phys. Rev. Lett. 66, 2072 (1991) Kruer, W.L., Valeo, E.J., Estabrook, K.G.: Phys. Rev. Lett. 35, 1076 (1975) Kruer, W.L., Estabrook, K.G.: Phys. Fluids 28, 430 (1985) Liang, E., Nishimura, K., Li, H., Gary, S.P.: Phys. Rev. Lett. 90, 085001 (2003) Liang, E., Nishimura, K.: Phys. Rev. Lett. 92, 175005 (2004) Liang, E. Phys. Plasmas in press (2006) Liang, E.P., Wilks, S.C., Tabak, M.: Phys. Rev. Lett. 81, 4887 (1998) Lontano, L., et al. (eds.): Superstrong Fields in Plasmas, AIP Conf. Proc. No. 611 AIP, NY (2002) Malka, V.: AIP Conf. Proc. No. 611, p. 303, In: Lontano, M. et al. (eds.) AIP, NY (2002)

313 Mourou, G.A., Barty, C.P.J., Perry, M.D.: Phys. Today 51(1), 22 (1998) Nishimura, K., Liang, E.: Phys. Plasmas 11 (10) (2004) Pukhov, A., Meyer-ter-Vehn, J.: Phys. Rev. Lett. 79, 2686 (1997) Rybicki, G., Lightman, A.P.: Radiative Processes in Astrophysics, Wiley, NY (1979) Shen, B., Meyer-ter-Vehn, J.: Phys. Rev. E 65, 016405 (2001) Sheng, Z.M., Mima, K., Zhang, J., Meyer-ter-Vehn, J.: Phys. Rev. E 69, 016407 (2004) Sprangle, P., Esary, E., Ting, A.: Phys. Rev. Lett. 64, 2011 (1990) Tajima, T., Dawson, J.M.: Phys. Rev. Lett. 43, 267 (1979) Wang, P.X., et al.: App. Phys. Lett. 78, 2253 (2001) Wilks, S.C., Kruer, W.L., Tabak, M., Langdon, A.B.: Phys. Rev. Lett. 69, 1383 (1992) Woodworth, J.G., Kreisler, M.N., Kerman, A.K.: in: Tajima, T. (ed.), The Future of Accelerator Physics. p. 378, AIP, NY (1996)

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Astrophys Space Sci (2007) 307:315–318 DOI 10.1007/s10509-006-9291-9

O R I G I NA L A RT I C L E

Three-Dimensional Particle Acceleration in Electromagnetic Dominated Outflows with Background Plasma and Clump Koichi Noguchi · Edison Liang

Received: 14 April 2006 / Accepted: 18 December 2006 C Springer Science + Business Media B.V. 2007


Abstract The effect of background plasma on particle acceleration via Poynting fluxes is studied in 3D PIC simulation of electron-positron and electron-ion plasmas. When a strongly magnetized ejecta at the center expands to lowtemperature electron-positron ambient plasma background and a low-density clump, electromagnetic wave front accelerates particles in the background and clump, and captures them in the Ponderomotive potential well. We do not observe any instability, and the momentum distributions of background and clump form a power law of slope close to −1.5 with a sharp peak in the middle. When an ejecta expands to the ion-electron interstellar medium (ISM), the acceleration via Poynting flux is severely damped due to the charge separation. Keywords Gamma ray bursts . PIC . Numerical . Relativity 1 Introduction Gamma-ray bursts (GRBs) are the most luminous physical phenomenon in the universe, whose mechanism is still unknown. There are two competing paradigms for the origin of the prompt GRB emissions: hydrodynamic internal shocks (M´esz´aros, 2002; Piran, 2000) versus Poynting fluxes (Lyutikov and Blackman, 2001). Both pictures require the rapid and efficient acceleration of nonthermal electrons to high Lorentz factors in moderate magnetic fields to radiate gamma-rays. In the hydrodynamic internal shock scenario, shock wave energy is mainly transferred to ions, resulting emission with low energy peak, whereas in the Poynting flux K. Noguchi (
) · E. Liang Rice University, Houston, TX 77005-1892, USA e-mail: [email protected]

scenario, long-wavelength electromagnetic (EM) energy can be directly converted into gamma-rays using the electrons or electron-positron pairs as radiating agents. Recent large-scale 3D PIC simulations (Hededal and Nishikawa, 2005; Nishikawa et al., 2006) shows that relativistic jets propagating through a weakly or nonmagnetized ambient plasma exite the Weibel instability, and that accelerated electron jet in the electron-ion jet has a significant hump above a thermal distribution, whereas electron-positron jet does not. However, the maximum γ is around 10 in both cases, and acceleration mainly occurs in the perpendicular direction relative to the shock wave propagation. The recent 2D PIC simulations (Liang et al., 2003; Liang and Nishimura, 2004), which is particularly relevant to the Poynting flux scenario of GRBs, shows that intense EM pulses imbedded in an overdense plasma (EM wavelength λ ≫ plasma skin depth c/ω pe ) capture and accelerate particles via sustained in-phase Lorentz forces when the EM pulses try to escape from the plasma. Such Poynting flux may originate as hoop-stress-supported magnetic jets driven by strongly magnetized accretion onto a nascent blackhole, or as transient millisecond magnetar winds, in a collapsar event (Zhang et al., 2003) or in the merger of two compact objects (Ruffert and Janka, 2003). Liang and Nishimura (2004) showed that the Poynting flux acceleration (PFA) reproduces from first-principles many of the unique features of GRB pulse profiles, spectra and spectral evolution, and Noguchi et al. (2005) recently showed that the mechanism is robust even with the radiation damping force. In this article we report 3D PIC simulations of particle acceleration driven by Poynting flux with low-temperature background ambient medium and low-density clump with newly developed 3D PIC code, and we show the power spectrum and radiation power strength from each particle. Springer

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2 Initial setup of the simulation We use the 3D explicit PIC simulation scheme based on the Yee algorithm (Yee, 1966). Spatial grids for the fields are uniform in all directions, x = y = z = c/ω pe , where ω pe is the electron plasma frequency. The simulation domain is −600x ≤ x ≤ 600x, −5y ≤ y ≤ 5y and −5z ≤ z ≤ 5z with triply periodic boundary conditions. Following Noguchi et al. (2005), the background magnetic field B0 = [0, B y , 0] is applied at the center of the simulation box, −6x < x < 6x, −5y ≤ y ≤ 5y and −5z ≤ z ≤ 5z, so that the magnetic field freely expands toward the ambient plasma regions. The magnetic field strength B y is given by ⎧ |x| < 4x ⎪ ⎨ B0 , B y = B0 [−|x|/(2x) + 3], 4x < |x| < 6x . ⎪ ⎩ 0, otherwise

(1)

We note that B y has finite gradient at edges to avoid unphysical particle acceleration. In order to study the particle acceleration only from PFA, initial electric field and current are assumed to be zero, which seems rather too simple and artificial. However, our study shows (Noguchi et al., 2005) that the acceleration is insensitive to the initial field configuration, and the existence of ordered Poynting vector is the key to accelerate particles via PFA. The group velocity of EM wave front is very close to the speed of light, and any hydrodynamical instability due to the electric field or current non-uniformity can be ignored. The most energetic particles are concentrated in the wave

Fig. 1 The spatial distribution of particles in the ejecta (top), ambient medium (middle) and clump (bottom) at tce = 650, and the phase plot of particles at tce = 12000 with Px − x (top), Py − x (middle) and Pz − Px (bottom). The color of each particle in the left panel repreSpringer

front, and instabilities such as Weibel instability which may occur in downstream does not affect the PFA mechanism. The number density distribution of initial electronpositron ejecta ρej is proportional to B y in order to keep the ratio ω pe / ce = 0.1, where ce is the electron cyclotron frequency. The clump and the ambient plasma consist of either electron-positron or electron-ion. The clump is a 100x × 6y × 6z cuboid with density ρcl = 0.1ρej , whose center is located at (−60x, 0, 0) so that the distance between the front of the ejecta and the edge of the column is 4x. The remaining of the simulation box is filled by the ambient plasma with density ρam = 0.01ρej . The initial temperature of ejecta is assumed to be a spatially uniform relativistic Maxwellian, k B Te = k B T p = 1 MeV, where the subscripts e and p refer to electrons and positrons. The temperature of the clump and the ambient plasma is also uniform Maxwellian with k B Te = k B T p,i = 100 eV.

3 Results First, we study the electron-positron background case. Figure 1 shows the spatial distribution of particles at tce = 650 and the phase plot at tce = 12000. The color of each particle in Fig. 1a represents the magnitude of estimated radiation damping force using the relativistic dipole formula (Rybicki and Lightman, 1979)

P =

2e2 (F 2 + γ 2 F⊥2 ), 3m 2 c3 

(2)

sents the magnitude of estimated radiation damping power P. In the right panel, blue dots represent ejecta particles, green clump, and red ambient, respectively

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due to the ponderomotive force. At tce = 12000, acceleration by PFA still continues, and the clump is compressed into a thin layer co-moving with the wave front. Figure 1b shows that the highest γ in the ejecta is around 250, whereas γ ≃ 100 in the background and the clump. As we mentioned, there is no acceleration in the y direction, and the momentum distribution in the y direction does not change. There is no

where F and F⊥ are the parallel and perpendicular components of the force with respect to the particle’s velocity. As the ejecta expands, electric field is automatically generated in the z direction, expanding the clump in the z direction. Particles are also accelerated in the direction of the Poynting vector (positive x direction for x > 0 and negative for x < 0),

Fig. 2 The phase plot of electrons with ISM plasma (x > 0) and with ion-electron clump and ISM (x < 0) at tce = 10000 with Px − x (top), Py − x (middle) and Pz − Px (bottom). The meaning of colors is the same as Fig. 1 10

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charge separation in the x direction, and no instability occurs in front of the EM pulse. Figure 2 shows the phase plot of electrons in the ionelectron clump and ambient plasma case at tce = 3000. Different from the electron-positron case, the acceleration by the PFA is strongly reduced by the charge separation. Especially with the clump (x < 0), the initial EM field energy is too weak to accelerate ions, and electrons are bounced back to the center. However, electrons in the wavefront (x ≃ ±300) are still captured by the pontderomotive force well, and accelerated by PFA. Next, we compare the power spectrum of electrons in both runs at tce = 3000. Figure 3a shows the electron-positron case and Fig. 3b shows the electron-ion case with the clump (C) and without the clump (NC). In the electron-positron case, the existence of clump does not affect the acceleration of the ejecta. Particles in both clump and ambient plasma get acceleration, making a sharp peak at E = 1.2mc2 , and form a power law of slope close to −1.5. The ion-electron case shows that the energy peak of the ambient plasma shifts to the lower, and the acceleration of the ejecta is severely reduced because of the charge separation between ions and electrons even though the ejecta consists of electrons and positrons. The power spectrum for the clump and background plasma sharply drops around E = 1.3mc2 , indicating that the electric field by the charge separation slows down the accelerated particles.

4 Summary We studied the effect of electron-positron and ion-electron background ambient plasma on particle acceleration via Poynting fluxes. With electron-positron ambient plasma, the acceleration mechanism is still robust in the electron-positron background case, and particles in the background and clump are also accelerated. With the ion-electron case, however, the acceleration is severely suppressed due to the charge separation in the background and clump plasma. If the density of interstellar medium near the ejecta is more than 10%, higher

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initial magnetic field energy ω pe / ce ≫ 1 is required to create high energy tail by PFA. The advantage of PFA compared with the internal shock acceleration scenario is the efficient energy transfer to high energy particles, and less bulk heating. High energy tail of GRBs requires gamma > 100 or more, which can be explained by the energetic particles accelerated by PFA, but not by the internal shock. When electron-positron and electronion plasma coexist, charge separation between ions and electrons decelerates the acceleration. Internal shock acceleration may take place if the initial plasma temparature is so low that ion density is much higher than positron density, or the initial plasma is weakly or not magnetized. We are currently working on simulations of the electron-positron with low-density electron-ion plasma in the ejecta and background. Longer timescale simulations are required to show the final power distribution of particles and resulting radiation spectrum, which remains as a future problem. Acknowledgements This research is partially supported by NASA Grant No. NAG5-9223, NSF Grant No. AST0406882, and LLNL contract nos. B528326 and B541027. The authors wish to thank ILSA, LANL, B. Remington and S. Wilks for useful discussions.

References Birdsall, C.K., Langdon, A.B.: Plasma Physics via Computer Simulation. McGraw-Hill (1985) Hededal, C.B., Nishikawa, K.-I.: ApJ 623, L89 (2005) Liang, E., Nishimura, K., Li, H., Gary, S.P.: Phys. Rev. Lett. 90, 085001 (2003) Liang, E., Nishimura, K.: Phys. Rev. Lett. 92, 175005 (2004) Lyutikov, M., Blackman, E.G.: MNRAS 321, 177 (2001) M´esz´aros, P.: Ann. Rev. Astron. Astrophys. 40, 137 (2002) Nishikawa, K.-I., Hardee, P. E., Hededal, C. B., Fishman, G. J.: ApJ 642, 1267 (2006) Noguchi, K., Liang, E., Nishimura, K.: Nuovo Ciment C 028, 381 (2005) Piran, T.: Phys. Rep. 33, 529 (2000) Ruffert, M., Janka, H-Th.: Gamma-Ray Burst and Aftergrow Astronomy. In: AIP Conference Proceedings, vol. 662, p. 193 (2003) Rybicki, G.B., Lightman, A.P.: Raidiative Processes in Astrophysics. Wiley-Interscience, New York (1979) Yee, K.S.: IEEE Trans. Antennas Propag. 14, 302 (1966) Zhang, W., Woosley, S.E., MacFadyen, A.I.: ApJ 586, 356 (2003)

Astrophys Space Sci (2007) 307:319–323 DOI 10.1007/s10509-006-9234-5

O R I G I NA L A RT I C L E

3-D RPIC Simulations of Relativistic Jets: Particle Acceleration, Magnetic Field Generation, and Emission K.-I. Nishikawa · C. B. Hededal · P. E. Hardee · G. J. Fishman · C. Kouveliotou · Y. Mizuno

Received: 5 May 2006 / Accepted: 10 August 2006 C Springer Science + Business Media B.V. 2006


Abstract We have applied numerical simulations and modeling to the particle acceleration, magnetic field generation, and emission from relativistic shocks. We investigate the nonlinear stage of the Weibel instability and compare our simulations with the observed gamma-ray burst emission. In collisionless shocks, plasma waves and their associated instabilities (e.g., the Weibel, Buneman and other two-stream instabilities) are responsible for particle (electron, positron, and ion) acceleration and magnetic field generation. 3-D relativistic electromagnetic particle (REMP) simulations with three different electron-positron jet velocity distributions and also with an electron-ion plasma have been performed and show shock processes including spatial and temporal evolution of shocks in unmagnetized ambient plasmas. The growth time and nonlinear saturation levels depend on the initial jet parallel velocity distributions. Simulations show that the

K.-I. Nishikawa (
) National Space Science and Technology Center, Huntsville, AL 35805; Department of Physics and Astronomy, The University of Alabama, Tuscaloosa, AL 35487 e-mail: [email protected] C. B. Hededal Dark Cosmology Center, Niels Bohr Institute, Juliane Maries Vej 30, 2100 Copenhagen Ø, Denmark P. E. Hardee Department of Physics and Astronomy, The University of Alabama, Tuscaloosa, AL 35487 G. J. Fishman · C. Kouveliotou NASA-Marshall Space Flight Center, National Space Science and Technology Center, Huntsville, AL 35805 Y. Mizuno National Space Science and Technology Center/MSFC, Huntsville, AL 35805

Weibel instability created in the collisionless shocks accelerates jet and ambient particles both perpendicular and parallel to the jet propagation direction. The nonlinear fluctuation amplitude of densities, currents, electric, and magnetic fields in the electron-positron shocks are larger for smaller jet Lorentz factor. This comes from the fact that the growth time of the Weibel instability is proportional to the square of the jet Lorentz factor. We have performed simulations with broad Lorentz factor distribution of jet electrons and positrons, which is assumed to be created by photon annihilation. Simulation results with this broad distribution show that the Weibel instability is excited continuously by the wide-range of jet Lorentz factor from lower to higher values. In all simulations the Weibel instability is responsible for generating and amplifying magnetic fields perpendicular to the jet propagation direction, and contributes to the electron’s (positron’s) transverse deflection behind the jet head. This small scale magnetic field structure contributes to the generation of “jitter” radiation from deflected electrons (positrons), which is different from synchrotron radiation in uniform magnetic fields. The jitter radiation resulting from small scale magnetic field structures may be important for understanding the complex time structure and spectral evolution observed in gamma-ray bursts or other astrophysical sources containing relativistic jets and relativistic collisionless shocks. The detailed studies of shock microscopic process evolution may provide some insights into early and later GRB afterglows.

Keywords Gamma-ray bursts . Relativistic jets . Weibel instability . Particle acceleration . Magnetic field generation . Methods: numerical . Particle-in-cell simulation Springer

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1 Introduction This report presents the study of collisionless relativistic shocks associated with prompt gamma-ray bursts and their afterglows. Using a 3-D relativistic particle-in-cell code we have investigated the dynamics of relativistic shocks which play an essential role for afterglows. There is now general agreement among theorists that the prompt emission from a gamma-ray burst (regardless of the central engine) requires the formation of a highly relativistic, highly collimated jet of out-flowing material that emits the observed prompt gammaray emission. Observational evidence suggests that GRBs are produced by Doppler beamed and boosted emission from shocks associated with a jet like flow. Within these general scenarios it is proposed that synchrotron radiation from shock accelerated particles in a shock-generated magnetic field produces the gamma-ray burst and produces the associated afterglows (e.g., Piran, 2005). The Swift satellite is a multi-wavelength observatory designed to detect GRBs and their X-ray and UV/optical afterglows. Thanks to its fast pointing capabilities, Swift is revealing the early afterglow phase. The Swift X-Ray Telescope (XRT) found that most X-ray afterglows fall off rapidly for the first few hundred seconds, followed by a less rapid decline (Tagliaferri et al., 2005). In the early afterglows of GRB 050406 and GRB 050502b, XRT detected strong X-ray flares: rapid brightening of the X-ray afterglow after a few hundred seconds post-burst (Burrows et al., 2005). These results suggest the existence of additional emission components in the early afterglow phase besides the conventional forward shock (blast wave) emission (Kobayashi et al., 2005). Zhang et al. (2003) have discussed a clean recipe for constraining the initial Lorentz factor γ 0 of GRB fireballs by making use of the early optical afterglow data alone. The input parameters are ratios of observed emission quantities, so that poorly known model parameters related to the shock microphysics (e.g. ǫe , ǫB , etc.) largely canceled out. This approach is readily applicable in the Swift era when many early optical afterglows are expected to be regularly caught. This data has been combined with other information such f as ℜB ≡ (ǫBr /ǫB )1/2 where superscripts r and f represent the reverse and forward shock region and σ , the ratio between the electromagnetic energy flux and the particle energy flux. σ is closely related to the initial magnetization of the outflow (Zhang and Kobayashi, 2005). The result is that this method has provided, for the first time, information about the magnetic content of the ejecta. Such information about the initial Lorentz factor of the fireball and whether the central engine is strongly magnetized are helpful for the identification of the GRB prompt emission site and mechanism, which are currently uncertain (e.g. Zhang and Meszaros, 2002). In collisionless shocks, plasma waves and their associated instabilities (e.g., the Weibel, Buneman and other twostream Springer

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instabilities) are responsible for particle (electron, positron, and ion) acceleration and magnetic field generation. Threedimensional relativistic particle-in-cell (PIC) simulations have been used to study the microphysical processes in relativistic shocks. Recent PIC simulations using counterstreaming relativistic jets show that rapid acceleration is provided in situ in the downstream jet, rather than by the scattering of particles back and forth across the shock as in Fermi acceleration (Silva et al., 2003; Frederiksen et al., 2004; Hededal et al., 2004; Hededal and Nishikawa, 2005; Nishikawa et al., 2003, 2005, 2006a; Medvedev et al., 2005). Three recent independent simulation studies have now confirmed that the relativistic counter-streaming jets excite the Weibel instability (Weibel 1959). The Weibel instability generates current filaments and associated magnetic fields (Medvedev and Loeb, 1999; Brainerd, 2000; Pruet et al., 2001; Gruzinov, 2001), and accelerates electrons (Silva et al., 2003; Frederiksen et al., 2004; Hededal et al., 2004; Hededal and Nishikawa, 2005; Jaroschek et al., 2005; Spitkovsky, 2006; Nishikawa et al., 2003, 2005, 2006a). The current filaments and associated magnetic fields produced by the Weibel instability form the dominant structures in a relativistic collisionless shock. The growing current filaments generate highly nonuniform small-scale transverse magnetic fields around the current filaments. The “jitter” radiation to be expected from deflected electrons has different properties than synchrotron radiation (Medvedev, 2000; Medvedev, 2006; Fleishman, 2006), and may explain the complex time evolution and/or spectral structure in gamma-ray bursts (Preece et al., 1998; Preece et al., 2002). Rapid particle acceleration perpendicular and parallel to the jet propagation direction accompanied by the nonlinear development of the filamentary structures cannot be characterized as Fermi acceleration. 2 Simulations with 3-D remp code Four simulations were performed using an 85×85×640 grid with a total of 380 million particles (27 particles/cell/species for the ambient plasma) and an electron skin depth, λce = c/ωpe = 9.6, where ωpe = (4π e2 n e /m e )1/2 is the electron plasma frequency and  is the grid size. In all simulations, jets are injected at z = 25 in the positive z direction. Radiating boundary conditions were used on the planes at z = 0, zmax . Periodic boundary conditions were used on all other boundaries (Buneman, 1993). The ambient and jet electronpositron plasma has mass ratio m e /m p ≡ m e− /m e+ = 1. The ion-electron mass ration is m i = m e = 20. The electron thermal velocity in the ambient plasma is vth = 0.1c where c is the speed of light. The electron number density of the jet is 0.741nb where nb is the ambient electron number density. The jet makes contact with the ambient plasma at a 2D interface spanning the computational domain. Here the dynamics of the propagating

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jet head and shock region is studied. Effectively, we study a small portion of a much larger shock. We have simulated three different initial distributions of jet electrons (positrons). Relativistic jets are injected into an unmagnetized ambient electron-positron plasma (initially σ = 0). Two cases have narrow distributions of jet electrons (positron) with (γ V|| = 5, and 15). The other case mimics cold jet electrons and positrons created by photon annihilation (4 < γ V|| < 100) (Meszaros et al., 2001; Ramirez-Ruiz et al., 2006). For all three cases the temperature of jet particles is very cold (0:01c in the rest frame) (Nishikawa et al., 2006b). In order to examine the dependence of theWeibel instability on the jet particle species, one case with electron-ion jet (γ v|| = 5) has been simulated (Nishikawa et al. 2006a). Current filaments (Jz ) resulting from development of the Weibel instability behind the jet front are shown in Fig. 1 at time t = 59.8/ωpe . If the density of the ambient plasma (ISM) is 1 cm−3 , the electron plasma frequency ωpe /2π is about 9 × 103 Hz. Therefore this simulation time corresponds to 6.64 ms. The estimated radiation loss time t1/2 ∼ 6πm 3 c (Eq. 2.19 in Hededal, 2005) is at least 102 times µ0 e4 B 2 γ0 larger than the simulation time. Electrons are accelerated by

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“radial” electric fields accompanying the current filaments. The electrons are deflected by transverse magnetic fields (Bx , B y ) via the Lorentz force: −e(v × B), generated by current filaments (Jz ), which in turn enhance the transverse magnetic fields (Weibel, 1959; Medvedev and Loeb, 1999). The complicated filamented structures resulting from the Weibel instability have diameters on the order of the electron skin depth λce = 9.6. This is in good agreement with the predic1/2 tion of λ ≈ 21/4 cγth /ωpe ≈ 1.188λce = 10 (Medvedev and Loeb, 1999; Ramirez-Ruiz, Nishikawa, and Hededal, 2006). Here, γth ∼ 1 is a thermal Lorentz factor. The x-component of magnetic field is plotted as a function of z (along the jets) for the four different cases in Fig. 2.

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Fig. 1 2D images in the x–z plane at y = 43 for the electron-ion (a) electron-positron (b, c, and d) jet injected into an unmagnetized ambient electron-positron plasma at t = 59.8/ωpe . The colors indicate the zcomponent of current density with Jz,x represented by arrows for γ v|| = 5(a, b) and 15(c), and widely distributed pair injection 4<γ v|| < 100 (d)

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Fig. 2 The x-component of the magnetic field in the x–y plane along the z-direction for the electron-ion injected into an unmagnetized ambient electron-ion plasma (a), electron-positron (b, c, d), jet injected into an unmagnetized ambient electron-positron plasma at t = 59.8/ωpe γ v|| = 5 (a, b) and 15 (c), and widely distributed pair injection 4 < γ v|| < 100 (d). Onedimensional cuts for (a) and (c) are at x/ = 38 and y/ = 33 (blue), 43 (red), and 53 ( green), and cuts are separated by about an electron skin depth

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One-dimensional cuts for (a) and (c) are at x/ = 38 and y/ = 33 (blue), 43 (red), and 53 ( green), and cuts are separated by about an electron skin depth. TheWeibel instability generates transverse magnetic fields due to the current filaments shown in Fig. 1. It should be noted that Fig. 2a shows the evolution of the ion Weibel instability. However, in order to investigate its dynamics it requires a larger system with a longer simulation time (Spitkovsky, 2006). Figures 2b, c, and d show similar structures: near the jet head a narrow region has a large magnetic field energy. These regions with higher magnetic field may correspond to the preshock magnetic field discussed in Li and Waxman (2006). Behind that region the magnetic fields become smaller where the current channels created by the Weibel instability merge nonlinearly. The particles are strongly accelerated in the perpendicular direction and particles are accelerated and decelerated in the flow direction where the magnetic field energy is largest, and nonlinearly saturated at the some distance behind the jet front. In the case with γ v|| = 5, the saturation level is the largest (Nishikawa et al., 2006a). However, the magnetic fields dissipate strongly in the nonlinear region. The region with higher magnetic fields is nearly 10 electron skin depth (100 ≈ 10λce ) (Gruzinov, 2001). Since the case with 4 < γ v|| < 100 contains jet particles with higher Lorentz factors, the saturation level is higher in the nonlinear dissipation region behind the jet front of magnetic field energy (Fig. 2d) (Ramirez-Ruiz et al., 2006). This is due to sequential excitement of the Weibel instability by the lower to higher Lorentz factor. The growth time of the weibel instability is proportional to the Lorentz factor and γsh /ωpe = 17.2 for γsh = 5 (Medvedev and Loeb, 1999). The growth time for γsh = 30 is around 42/ωpe , which is within the simulation time tωpe = 58.9. It should be noted that in the jet front (570 < Z /  600) where the larger magnetic field exists as shown in Fig. 2b, c, and d, jet electrons are less accelerated in the perpendicular direction. The spatial and temporal structure of the magnetic field associated with accelerated electrons (positrons) produces different lightcurves and spectra; this will be investigated in the near future.

3 Discussion We have performed self-consistent, three-dimensional relativistic particle simulations of relativistic electron-ion and electron-positron jets propagating into unmagnetized electron-positron ambient plasmas (initially σ = 0) for a longer time and a larger simulation system than our previous simulations (Nishikawa et al., 2003, 2005, 2006a) in order to investigate the nonlinear stage of theWeibel instability. The main acceleration of electrons takes place in the downstream region. Processes in the relativistic collisionless shock are dominated by current filament structures produced Springer

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by the Weibel instability. This instability is excited in the downstream region behind the jet head, where electron density perturbations lead to the formation of current filaments. On average the nonuniform electric field and magnetic field structures associated with these current filaments decelerate the jet electrons and positrons, while accelerating the ambient electrons and positrons, and accelerate the jet and ambient electrons and positrons in the transverse direction. The nonlinear region current channels generated by the Weibel instability dissipate. Dissipation levels depend on the initial jet electron parallel velocity distributions. A key issue is the microscopic dynamics of the reverse shocks which play a crucial role for the generation of early afterglows (Zhang et al., 2003; Zhang and Kobayashi, 2005; Kobayashi et al., 2005; Fan et al., 2005). We have examined the microscopic dynamics of relativistic shocks with three different velocity distributions in order to understand Swift observations and models of GRB emission. The important parameters are the initial Lorentz factors of relativistic jets (γ ) and the ratio between the electromagnetic energy flux and the particle energy flux (σ ). In this report we used σ0 = 0 and have investigated the magnetization of the GRB outflows f (ℜB ≡ (ǫBr /ǫB )1/2 ). Figure 2 shows the x-component of the magnetic field along the jets. We found that the values of Bx and the spatial and temporal evolution depend on the initial jet Lorentz factors. They will provide different lightcurves and spectra. A self-consistent emission calculation based on the motion of electrons (positrons) in the simulation system has been developed. The calculation has reproduced the spectral structure of one GRB afterglow (Hededal, 2005; Hededal and Nordlund, 2005). In order to calculate the radiation (jitterlike) from the particles in the electromagnetic fields generated by the Weibel instability, the retarded electric field from a single particle is Fourier-transformed and gives the individual particle spectrum. The individual particle spectra are added together to produce a total spectrum over a particular simulation time span (Hededal, 2005; Hededal and Nordlund, 2005). In order to obtain lightcurves we can calculate a spectrum over short time spans, ts , relative to the longer simulation time span. The change in power in an energy band can then be followed from one short time span to the next giving a light curve in the energy band. It should be noted that in this calculation very long (large) simulations are required using small time steps in order to increase the upper frequency limit to the spectrum. Frequency resolution is limited by the short time span (ω=1/ts ) (Hededal, 2005). We expect that lightcurves obtained in this way will provide reasonable evolution of the higher energy bands. Future investigation with this newly-developed method (Hededal, 2005; Hededal and Nordlund, 2005) will provide self-consistent lightcurves, spectra of synchrotron/jitter emission, spectral evolution, polarization as functions of the

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viewing angle and allow their systematic comparison with early and later afterglows observed by Swift and GLAST. This research (K.N.) is partially supported by the NSF awards ATM-0100997, INT-9981508, and AST-0506719, and the National Aeronautic and Space Administration contract NASA-NNG05GK73G. P. Hardee acknowledges partial support by a National Space Science & Technology (NSSTC/NASA) cooperative agreement NCC8-256 and NSF award AST-0506666. The simulations have been performed on IBM p690 at the National Center for Supercomputing Applications (NCSA) which is supported by the NSF. References Brainerd, J.J.: ApJ 538, 628 (2000) Buneman, O.: Tristan, In: Matsumoto, H., Omura, Y. (eds.) Computer Space Plasma Physics: Simulation Techniques and Software, Terra Scientific Publishing Company, Tokyo (1993) p. 76. Burrows1, D.N., Romano, P., Falcone1, A., et al.: Sci. 309, 1833 (2005) Fan, Y.Z., Zhang, B., Wei, D.M.: ApJ 628, L25 (2005) Fleishman, G.D.: ApJ 638, 349 (2006) ˚ ApJ 608, Frederiksen, J.T., Hededal, C.B., Haugbølle, T., Nordlund, A.: L13 (2004) Gruzinov, A.: ApJ 563, L15 (2001) ˚ ApJ 617, Hededal, C.B., Haugbølle, T., Frederiksen, J.T., Nordlund, A.: L107 (2004) Hededal, C.B., Nishikawa, K.-I.: ApJ 623, L89 (2005) Hededal, C.B.: Gamma-Ray Bursts, Collisionless Shocks and Synthetic Spectra, PhD Thesis, 2005 (astro-ph/0506559) ˚ ApJ submitted (2005) (astro-ph/0511662) Hededal, C.B., Nordlund, A.: Jaroschek, C.H., Lesch, H., Treumann, R.A.: ApJ 618, 822 (2005)

323 Kobayashi, S., Zhang, B., Meszaros, P., Burrows, D.N.: ApJL submitted (2005) (astro-ph/0506157) Li, Z., Waxman, E.: ApJ submitted (2006) (astro-ph/0603427) Medvedev, M.V., Loeb, A.: ApJ 526, 697 (1999) Medvedev, M.V., Fiore, M., Fonseca, R.A., Silva, L.O., Mori, W.B.: ApJ 618, L75 (2005) Medvedev, M.V.: ApJ 540, 704 (2000) Medvedev, M.V.: ApJ 637, 869 (2006) Meszaros, P., Ramirez-Ruiz, E., Rees, M.J.: ApJ 554, 660 (2001) Nishikawa, K.-I., Hardee, P., Richardson, G., Preece, R., Sol, H., Fishman, G.J.: ApJ 595, 555 (2003) Nishikawa, K.-I., Hardee, P., Richardson, G., Preece, R., Sol, H., Fishman, G.J.: ApJ 622, 927 (2005) Nishikawa, K.-I., Hardee, P., Hededal, C.B., Fishman, G.J.: ApJ 642, 1267, (2006a) Nishikawa, K.-I., Ramirez-Ruiz, E., Hardee, P., Hededal, C.B., Fishman, G.J., Kouveliotou, C., Mizuno, Y.: ApJ, in preparation (2006b) Piran, T.: Rev. Mod. Phys. 76, 1143 (2005) Preece, R.D., Briggs, M.S., Mallozzi, R.S., Pendleton, G.N., Paciesas, W.S., Band, D.L.: ApJ 506, L23 (1998) Preece, R.D., Briggs, M.S., Giblin, T.W., Mallozzi, R.S, Pendleton, G.N., Paciesas, W.S., Band, D.L.: ApJ 581, 1248 (2002) Pruet, J., Abazajian, K., Fuller, G.M.: Phys. Rev. D. 64, 063002-1 (2001) Ramirez-Ruiz, E., Nishikawa, K-I., Hededal, C.B.: ApJ, submitted (2006) Silva, L.O., Fonseca, R.A., Tonge, J.W., Dawson, J.M., Mori, W.B., Medvedev, M.V.: ApJ 596, L121 (2003) Spitkosky, A.: (2006) (astro-ph/0603211) Tagliaferri, G., Goad, M., Chincarini, G., et al.: Nature 436/18, 985 (2005) Weibel, E.S.: Phys. Rev. Lett. 2, 83 (1959) Zhang, B., Kobayashi, S., Meszaros, P.: ApJ 595, 950 (2003) Zhang, B., Kobayashi, S.: ApJ 628, 315 (2005) Zhang, B., Meszaros, P.: ApJ 572, 876 (2002)

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Astrophys Space Sci (2007) 307:325–327 DOI 10.1007/s10509-006-9204-y

ORIGINAL ARTICLE

Spectral Features of Photon Bubble Models of Ultraluminous X-ray Sources Justin D. Finke · Markus B¨ottcher

Received: 14 April 2006 / Accepted: 26 June 2006 C Springer Science + Business Media B.V. 2006


Abstract The nature of Ultraluminous X-ray Sources – Xray sources which exceed the Eddington luminosity for a ∼ 10M⊙ black hole – remains a mystery. One possible explanation is an inhomogeneous accretion disk around a solar mass black hole where photon transport through radiationpressure dominated “photon bubbles” can lead to superEddington accretion. While previous studies of this model have focused primarily on its radiation-hydrodynamics aspects, here we explore some observational implications of such a model with a Monte Carlo–Fokker Planck radiation transport code. Keywords Accretion . Accretion disks . Methods: numerical . Radiative transfer . X-rays: binaries

et al., 2006). Formation of such objects, however, clearly presents theoretical difficulties. Several models for super-Eddington luminosities have been proposed, including anisotropic X-ray emission from relativistic beaming in the jets of microquasars (microblazars; Georganopoulos et al., 2002; King et al., 2001) and the “photon bubble” model which involves accretion disks with low density, optically thin cavities through which radiation can travel more freely, leading to superEddington accretion rates – and hence, luminosities – that do not blow away the disk (Begelman, 2002). Strong beaming is now considered unlikely based on recent observations of X-ray ionization of optical nebulae associated with some ULXs (Pakull and Mirioni, 2003; Gutierrez, 2006); however, the photon bubble remains a viable model for accretion– related ULXs.

1. Introduction In recent years, many off-nuclear X-ray sources which exceed the Eddington luminosity for a ∼10 M⊙ black hole have been discovered in nearby galaxies (Fabbiano, 1989; Colbert and Mushotzky, 1999; Liu and Mirabel, 2005). Dubbed Ultraluminous X-ray Sources (ULXs), their nature is one of the outstanding mysteries of modern astrophysics. While some can be identified as supernova remnants or background Active Galactic Nuclei, most seem to be the result of accretion from a high-mass star onto a compact object. If the accretion is Eddington-limited, the compact object would have a mass in the range 50–103 M⊙ – so-called intermediate-mass black holes (IMBHs; Makishima et al., 2000; Madhusudhan J. D. Finke (
) · M. B¨ottcher Astrophysical Institute, Department of Physics and Astronomy, Ohio University, Athens, OH 45701 e-mail: [email protected]

2. ULXs’ X-ray spectra Can one distinguish between IMBH and photon bubble models with ULXs’ X-ray spectra? From an accreting IMBH one would expect a disk blackbody spectrum with a temperature kT ∼ 0.1–0.3 keV, and possibly a high energy component from Comptonization in a hot, thermal coronal gas, or some other mechanism. Radiation in photon bubble cavities could be repeatedly Compton-reflected leading to strong fluorescence lines and ionization edges (Ballantyne et al., 2004) that may be observable, assuming it is not overwhelmed by other radiation sources such as the disk’s blackbody, which should be near that which is observed for sub-Eddington X-ray binaries, i.e., ∼1–3 keV, or smeared out by the multiple Compton scattering within the bubble. Distinguishing models by spectral modeling is nearly impossible for all but the highest quality ULX data sets, and Springer

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spectral fitting to these ULX spectra gives contradictory results. Roberts et al. (2005) found that 6 of the best-observed ULXs were fit approximately equally as well with a cool blackbody and hard power-law as with a soft power-law and warm blackbody. Even more surprisingly, they found that most spectra were fit best by a cool, kT ∼ .25 keV and a warmer kT ∼ 2 keV blackbody, a fit which exhibits considerable challenges for any physical model. One prevalent interpretation is currently that of a cool disk blackbody around an IMBH with an optically thick (τ ∼ 10) corona producing the hard feature; such a model has considerable problems, such as how one could observe a disk blackbody through an optically thick plasma. The observed spectra could also be explained by a hot, inner ADAF and a cool outer disk, in which case the cool blackbody temperature would not have any obvious relation to the black hole mass. Quasi-periodic oscillations (QPOs) have been observed for two ULXs, M82 X-1 (Strohmayer and Mushotzky, 2003) and Holmberg IX X-1 (Dewangan et al., 2006) indicating ULXs originate from compact objects similar to those in galactic black hole candidates. An Fe K line has been observed in M82 X-1 (Strohmayer and Mushotzky, 2003; Agrawal and Misra, (2006)); absorption edges have been observed in the 0.1–1 keV range in M101 ULX–1 (Kong et al., 2004); and possible edges at ∼ 0.7 keV and ∼8 keV, and an emission feature at ∼6 keV have been seen in Holmberg IX X-1 (Dewangan et al., 2006). It is hoped that with better predictions of the expected X-ray spectra from photon bubbles, particularly emission lines and ionization edges characteristic for these types of models, the nature of ULXs will be revealed. 3. Photon bubble simulations So far, investigations of photon bubbles in ULXs have focused on radiation–hydrodynamics (see, e.g., Ruszkowski and Begelman, 2003; Turner et al., 2005) leaving the expected spectral features essentially unexplored. We have examined the observational signatures of these objects with the two–

dimensional Monte Carlo/Fokker–Planck code described in B¨ottcher et al. (2003) and B¨ottcher and Liang (2001), which uses the Monte Carlo method of Pozdnyakov et al. (1983) for Compton scattering and the implicit Fokker–Planck method of Nayakshin and Melia (1998) for evolution of the electron distribution in a thermal two-temperature plasma. With this code, we set up a low density region sandwiched by two high density regions in a plane parallel geometry, representing an inhomogeneous disk at 10R S (R S = 2G M/c2 ≈ 3 · 106 cm for a 10M⊙ black hole). The low density region was divided into 24 homogeneous zones, 4 radial and 6 vertical, in which the Fokker–Planck equation was implicitly and independently solved. The high density regions were represented by blackbody (and later, blackbody + Compton reflection) spectra inserted at the upper and lower boundaries, accounting for Compton reflection of these surfaces. Photons escaping the simulation boundaries were added to an event file for later spectral extraction. The density of the low density region was chosen so that τT ∼ 0.1, for a length scale of ∼0.1R S . The MC/FP simulation was run and the spectrum incident on the upper and lower boundaries, representing the high density regions, was calculated. To calculate the expected Compton reflection, this spectrum was used as an input into the latest version of XSTAR (Kallman and Bautista, 2001). XSTAR was run in constant pressure mode; the pressure was calculated from the specified density by the ideal gas law. Due to XSTAR’s limitations, the density in the high density region was chosen to be lower than the value resulting from our standard model specifications: n e = 1 · 1017 cm−3 ; however, the column density was chosen to be quite large, (N = 1 · 1024 cm−2 ), so that, although reflected emission will not originate from a realistic depth, the reflected spectrum should still be consistent with a high density (n e ∼1020 cm−3 ), considering the spectral features will be substantially smeared out in the course of multiple Compton scatterings in the low density regions. The area of the high density region’s surface was 5.6 · 1014 cm2 ; from this and the total luminosity, the

Table 1 Ionization edges: Equivalent widths and fits Gaussian fit parameters

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Element’s K Edge

EW [eV]

N [1033 erg s−1 keV−1 ]

σ [eV]

E 0 [keV]

Be B C N O Ne Mg Si S

15 9.5 31 14 99 34 40 95 140

12 7.0 24 18 110 40 21 23 11

20 30 50 40 60 70 130 150 180

0.24 0.38 0.55 0.72 0.93 1.45 2.08 2.82 3.67

Astrophys Space Sci (2007) 307:325–327

10

Ne Mg

C N

35

Be

B

Si S

Acknowledgements This work was partially supported by NASA through XMM-Newton GO grant no. NNG04GI50G and INTEGRAL theory grant NNG05GK59G.

34

−1

−1

LE [erg s keV ]

10

observations. Results of a parameter study will be published in a forthcoming paper (Finke and B¨ottcher, 2006).

36

O

10

327

10

10

10

33

References

32

31

10

−1

0

10 Photon Energy [keV]

1

10

Fig. 1 Resulting X-ray spectrum of the bubble simulation with a 0.5 keV blackbody representing the high density regions (solid line). It is plotted with a 0.5 keV blackbody (dotted line), and 0.5 keV blackbody + power law fit for the soft excess + Gaussian fits to ionization edges (dashed line). See the electronic edition for a color version of this figure.

ionization parameter (ξ = L/(n R 2 )) was calculated. Solar metalicities were assumed. An example output of such a simulation with a high density temperature of kT = 0.5 keV can be seen in Fig. 1. The spectrum was fitted with an 0.5 keV blackbody continuum, a power law representing the soft-excess, and Gaussian fits for the spectral features, which are believed to be K shell ionization edges for various elements. Equivalent Widths and the fit parameters are listed in Table 1. The soft-excess is produced by Compton scattering of photons from the low density region by the high density region. This model successfully reproduces the major spectral features of ULXs: the soft-excess and the high energy blackbody. It also predicts features (i.e., the ionizationedges) that may be seen in future

Agrawal, V.K., Misra, R.: Astrophys. J. 638, L83 (2006) Ballantyne, D.R., Turner, N.J., Blaes, O.M.: Astrophys. J. 603, 436 (2004) Begelman, M.C.: Astrophys. J. 568, L97 (2002) B¨ottcher, M., Jackson, D.R., Liang, E.P.: Astrophys. J. 586, 339 (2003) B¨ottcher, M., Liang, E.P.: Astrophys. J. 552, 248 (2001) Colbert, E.J.M., Mushotzky, R.F.: Astrophys. J. 519, 89 (1999) Dewangan, G.C., Griffiths, R.E., Rao, A.R.: in submission, astroph/0602472 (2006) Fabbiano, G.: ARA&A, 27, 87 (1989) Finke, J.D., B¨ottcher, M.: in preparation Georganopoulos, M., et al.: A&A 388, L25 (2002) Gutierrez, C.M.: Astrophys. J. 640, L17 (2006) Kallman, T., Bautista, M.: Astrophys. J. Suppl. 133, 221 (2001) King, A.R., et al.: Astrophys. J. 552, L109 (2001) Kong, A.K.H., et al.: The XXII Texas Symposium on Relativistic Astrophysics, Stanford University (2004); Kallman, T., Bautista, M.: Astrophys. J. Suppl. 133, 221 (2001) Liu, Q.Z., Mirabel, I.F.: A&A 429, 1125 (2005) Madhusudhan, N., et al.: Astrophys. J. submitted (astro-ph/0511393) (2006) Makishima, K., et al.: Astrophys. J. 535, 632 (2000) Nayakshin, S., Melia, F.: Astrophys. J. Suppl. 114, 269 (1998) Pakull, M.W., Mirioni, L.: RevMexAA 15, 197 (2003) Pozdnyakov, L.A., et al.: Astrophys. Space Phys. Rev. 2, 18 (1983) Roberts, T.P. et al.: Proceedings of “The X-ray Universe 2005”, San Lorenzo de El Escorial, Spain, (2005) Ruszkowski, M., Begelman, M.C.: Astrophys. J. 586, 384 (2003) Strohmayer, T.E., Mushotzky, R.F.: Astrophys. J. 586, L61 (2003) Turner, N.J., et al.: Astrophys. J. 624, 267 (2005)

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Astrophys Space Sci (2007) 307:329–333 DOI 10.1007/s10509-006-9280-z

O R I G I NA L A RT I C L E

Polychromatic X-ray Beam from the Acceleration of Energetic Electrons in Ultrafast Laser-Produced Plasmas F´elicie Albert · Kim TaPhuoc · Rahul Shah · Frederic Burgy · Jean Philippe Rousseau · Antoine Rousse

Received: 14 April 2006 / Accepted: 23 November 2006 C Springer Science + Business Media B.V. 2006


Abstract Polychromatic beams of hard X-rays from ultrafast laser plasma interaction are studied. Just as in a conventional synchrotron, electrons are accelerated and wiggled, but on a much shorter scale of a few millimeters. By focusing a 50 TW CPA laser system (30 fs duration) onto a helium gas jet, we obtained a polychromatic collimated beam (50 mrad) of X-ray radiation in the keV range. In addition, its perfect synchronization with the laser system, its ultrafast duration (≃30 fs) and its brightness (up to 108 photons/shot/solid angle at 0.1% BW) will make it applicable to both X-ray science and backlighting to address laboratory astrophysics research issues. Keywords Ultrafast X-ray science . Laser-Plasma source . Electron acceleration PACS 52.59.Px, 52.50.Dg, 52.38.Ph, 52.25.Os

1 Introduction Laser produced X-ray sources have been widely used for a decade to probe dense fusion plasmas, jets and shocks resulting from laser plasma interactions (Hammel et al., 1993). Such X-ray sources, mainly used for backlighting in laboratory astrophysics experiments (Whitlock et al., 1984; Cole et al., 1982), are generated by focusing an intense laser onto a solid target, resulting in line and continuum emission (Rousse and Rischel, 2001; Workman and Kyrala, 2001). However, those X-ray sources radiate isotropically and it makes the collection of photons harder to realize in backF. Albert (
) · K. T. Phuoc · R. Shah · F. Burgy · J. P. Rousseau A. Rousse LOA-ENSTA-Ecole Polytechnique, Chemin de la huniere, 91761 Palaiseau, France e-mail: [email protected]

lighting experiments. In comparison, synchrotrons produce beams of keV X-rays with a broad and continuous spectrum, because the rays originate from relativistic electrons oscillating in periodic magnetic structures (undulators and wigglers) (Attwood, 1999). Here, by implementing the concepts of synchrotron mechanisms and using a plasma wiggler in a fully laser based scheme, we show that it is possible to produce a polychromatic (1–10 keV), low divergence (50 mrad) and ultrafast X-ray beam (TaPhuoc et al., 2005; Rousse et al., 2004). When an ultraintense laser (I > 1018 W/cm2 ) is focused onto a gaz, the ponderomotive force, proportional to the gradient of light intensity, plows the electrons away from the strong field regions, leaving a column free of electrons in the wake of the laser pulse (Pukhov and Meyer-ter-vehn, 2002; Whittum, 1992). Thus, due to longitudinal space charge separation, electrons trapped in the back of the ion channel will be accelerated up to relativistic energies of more than 100 MeV (Malka et al., 2002; Faure et al., 2004). Because of the transverse restoring force due to the ion bubble, the accelerated electrons displaced from the cavity axis undergo oscillations, called betatron oscillations (Kiselev et al., 2004; Esarey et al., 2002; Kostyukov et al., 2003). Just as in a synchrotron, a collimated beam of keV X-ray radiation will result from the relativistic motion of the electrons.

2 Theory In the case of laser based synchrotron radiation, the ion channel that serves as a plasma wiggler is created by the ponderomotive force of the laser pulse, which expels the electrons toward the low light intensity regions, leaving an ion column in the wake of the light pulse. This occurs for a normalized vector potential a0 greater than or on the order of unity, when relativistic effects can no longer be neglected. The laser Springer

Astrophys Space Sci (2007) 307:329–333

produced ion channel creates a restoring force, due to space charge separation, that can be calculated from Gauss’ law F = −m e ωp2r0 /2

 K = γ kβ r0 = 1.33 × 10−10 γ n e [cm−3 ]r0 [µm]

(2)

that is similar to the role of the wiggler strength parameter. Here r0 is the amplitude of the betatron oscillations and kβ = 2π/λβ . If K becomes high so that K ≫ 1, the radiation will be emitted in many harmonics within a narrow cone of divergence θ ≃ K /γ , and the emission frequency is a function of K (Kiselev et al., 2004; Esarey et al., 2002; Kostyukov et al., 2003). In that case, the spectrum becomes broadband and quasi continuous. It can be described by the synchrotron radiation spectrum function (Jackson, 2001) 

∞ ω ωc

K 5/3 (x)d x

(3)

Here ωc represents the critical frequency beyond which there is negligible radiation at any angle. For frequencies below ωc and up to ω ≃ 0.29ωc , the spectrum function increases as ω1/3 and then drops exponentially to zero. For a relativistic electron wiggled in an ion core, the critical frequency is (Kiselev et al., 2004; Esarey et al., 2002; Kostyukov et al., 2003)
ωc =

3 3 γ
cr0 kβ2 ≃ 5 × 10−24 γ 2 n e [cm−3 ]r0 [µm]keV (4) 2

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(a.u)

b

(a.u)

2

factor of the electron defined by γ = 1/ 1 − vc2 . This electron undergoing betatron oscillations in the ion channel will emit synchrotron radiation (Kiselev et al., 2004; Esarey et al., 2002; Kostyukov et al., 2003). In the case of an electron slightly displaced from the axis, the fundamental wavelength of the radiation can be approximated by λ = λβ /2γ 2 . In the case of high amplitude oscillations, high harmonics will be radiated, and just as in a synchrotron, this can be described by a dimensionless parameter

ω ωc

a

(1)

Here, m is the electron rest mass, r0 the radius of the ion column and ωp the plasma frequency, defined by ωp =  n e e2 /mǫ0 , with n e the electronic plasma density, e the elementary charge, and ǫ0 the vacuum permittivity. Since the restoring force scales with the electronic density, and that the electronic densities used in our experiments are on the order of n e = 1019 cm−3 , this yields a very high oscillation frequency in the plasma wiggler and reduces the constraint on the electron beam energy compared to conventional synchrotrons. For small amplitude oscillations, the electron will produce an harmonic motion at the fundamental betatron fre√ quency ωβ = ωp / 2γ , where γ is the  relativistic Lorentz

S(ω/ωc ) =

(a.u)

330

c

Fig. 1 Simulated on axis X-ray spectrum for (a) K = 0.18 (γ0 = 20, r0 = 0.1 µm), (b) K = 0.94 (γ0 = 20, r0 = 0.5 µm) and (c) K = 5.6 (γ0 = 20, r0 = 3 µm). If K is smaller than unity, radiation is emitted at the fundamental frequency, if K becomes high so that K ≫ 1, the spectrum becomes broadband and quasi continuous

We have simulated the theoretical on axis X-ray spectrum (Fig. 1) using a code that calculates the electron trajectories using a Runge–Kutta algorithm to integrate the equation of motion. In this calculation the electron plasma density, the electron energy and the initial transverse position r0 are the initial conditions. The electron spectrum can be either arbitrary or taken from experimental data. The ion cavity is centered on the laser propagation axis (where r0 = 0). Then the Synchrotron function from Equation (3) is used to calculate the radiation emitted by each particle along the oscillation path.

3 Experimental setup The experiment (Fig. 2) was performed in the “Salle Jaune” of the Laboratoire d’Optique Appliqu´ee, using a 50 TW Sapphire doped with titanium (Ti:Al2 O3 ) laser system based on Chirped Pulse Amplification technology (CPA). It has a 40 nm broadband spectrum centered on λ0 = 820 nm and a 30 fs Full Width Half Maximum (FWHM) pulse duration. The

Astrophys Space Sci (2007) 307:329–333

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simultaneous measurements of electron and X-ray spectra (Fig. 2).

4 Results and discussion

Fig. 2 Experimental setup for electron and X-ray beam characterization (top) and theoretical filters (with phosphor screen) transmissions for energies up to 20 keV (bottom)

laser can deliver energies up to 1.5 J on target with a linear horizontal polarization. We used a 1 m focal length parabolic mirror to focus the 55 mm diameter laser beam onto the edge of a supersonic Helium gaz jet, which has been fully characterized using a Mach-Zender Interferometer. It showed a uniform and sharp edged density profile all along the jet. Imaging of the focal plane shows a Gaussian intensity profile of the laser with a beam waist w0 of 18 µm which contains 50% of the total laser energy. Therefore, this system produces a vacuum focused intensity on the order of 3 × 1018 W/cm2 , which corresponds to a normalised vector potential a0 of 1.2. The electronic plasma density can be tuned from 1018 cm−3 to 5 × 1019 cm−3 by varying the backing pressure of the gaz jet with a regulator. During the experiment, we characterized the electrons accelerated above 40 MeV, by deviating them onto a phosphor screen imaged with a visible CCD, with permanent 1 T magnets placed in the path of the accelerated particles. The magnetic spectrometer resolution is limited by the dispersing power of the magnet and also by the electron beam spatial quality. The resolution is therefore respectively 17 MeV and 6 MeV for 200 MeV and 100 MeV energies. The X rays were observed in the forward direction on a phosphor screen imaged with another visible CCD. Different sets of filters (Be, Al, Nb, and Sn) were along the propagation axis to select different spectral bands (above 0.8 keV, 2 keV, 5 keV and 10 keV respectively). This setup allowed us to make

The X-ray radiation was measured on axis with an average divergence of 50 mrad at an electronic density n e = 1 × 1019 cm−3 . The 500 µm Be window in front of the CCD selected all X-ray photons above 3 keV. We also observed that the Xray divergence depends on the electronic density. It is a possible consequence of the variation in the betatron strength pa√ rameter K ∝ n e as the beam divergence varies with K /γ . Another feature of the betatron X-ray source is its size. We have measured the radiation source size by using the shadow of a razor blade acting as a knife edge (Fig. 3). The blade and the X-ray CCD were placed respectively 13 cm and 2 m away from the nozzle. A geometrical relation gives then a transverse source size of 13 µm, which is the same order of magnitude than the laser focal spot. The spectral distribution of the radiation was measured from 1 keV to 10 keV by placing a first set of Be, Al, Sn, and Nb filters in front of the detector. The spectral resolution was limited by the bandwidths of the filters. The spectrum decreases exponentially from 1 to 10 keV. The total number of photons (integrated over the bandwidths of the filters and over the divergence of the x-ray beam) is found to be more than 108 photons (per shot/solid angle at 0.1% BW). Another unique feature of the x-ray beam is its intensity as a function of the electron density of the plasma, also shown on Fig. 4. It is found to be sharply peaked at n e = 1.1 × 1019 cm−3 . Below this critical density, the x-ray signal rapidly vanishes mainly because the number of trapped electrons is too low. This is confirmed in the experiment for which no electrons were detected by the spectrometer. At larger densities

Fig. 3 Experimental setup and results for the source size characterization. A razor blade was placed 13 cm away from the source, in the path of the X-ray beam. The detector recorded the images 2 m away from the source. Geometrical relations gave a source size of 13 µm

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requires only a contrast ratio of 1:107 for our experimental conditions. 5 Conclusion and perspectives

Fig. 4 Experimental (circles) and PIC simulation (squares) variation of x-ray intensity with electronic density. The input parameters used in the simulation are the laser intensity (3 × 1018 W/cm2 ), the focal spot size (18 µm) and the electronic plasma density (varied from 1018 cm−3 to 6 × 1019 cm−3 )

the x-ray signal drops down and a plateau is reached. For these experimental conditions, the resulting amplitude of the plasma wave becomes too weak. The pulse must first be modulated and additional laser energy would be needed. As a result, the temperature of the electron beam decreases and its divergence increases. In order to obtain a better description of the X-ray properties, the laser-plasma interaction has been simulated in this high laser intensity regime with a 3D PIC Code (Pukhov, 2003) which has been modified to properly model the synchrotron emission. The PIC simulations clearly reproduce this experimental behaviour: a sharp increase of the x-ray intensity followed by a smoother decrease of the signal. Other laser plasma based mechanisms could potentially produce polychromatic X-ray radiation. Here, unlike in the case of non-linear Thomson scattering, the electrons do not overlap with the light field, and experiments already performed on the subject (TaPhuoc et al., 2003) necessitated a normalized laser vector potential a0 = 6 to produce radiation in the 100 eV range, hence a much higher laser intensity would be needed to produce a keV beam. Relativistic Bremsstrahlung could also produce a narrow divergence polychromatic X-ray beam, but it would be much weaker than what we observe at our electronic densities (Seltzer and Berger, 1986). High Harmonic Generation (HHG) in gaseous media can also coherently produce a bright beam of quasi continuous light but only in the soft X-ray region (down to 10 nm) (Tarasevitch et al., 2000; Brabec and Krausz, 2000). Recent work (Dromey et al., 2006) showed HHG from solid targets where, unlike in gases, there is no theoretical prediction for a sharp cutoff in harmonic generation. Wavelengths as short as 1.2 nm (850th order of 1,054 µm 600 fs fundamental laser light) have been obtained with this process. However, a high intrinsic laser-pulse contrast ratio (>1:1011 ) is required to observe the shortest wavelength harmonics in this experiment while the betatron X-ray source Springer

Laser based synchrotron radiation generates a broad-band hard X-ray beam. It reproduces the concept of synchrotron radiation in a plasma wiggler along a few milimeters, size of the gas jet in which the electrons are produced, accelerated and wiggled. It ensures a much more compact device than conventional synchrotrons. Based on numerical simulations, the X-ray pulse duration should be on the order of the 30 fs laser pulse duration. Nevertheless, this must be measured experimentally. Moreover, the source is perfectly synchronized with the laser system, which opens the way toward pump-probe experiments at a femtosecond timescale. This type of experiment was already performed with laser based K α sources, and has been a valuable improvement to understand the mechanisms of non thermal melting and phase transitions (Rishel et al., 1997; Siders et al., 1999). The Betatron X-ray source provides up to 108 photons/pulse/solid angle/0.1%BW, which can be collected and focused onto a sample by grazing incidence optics or X-ray lenses. Moreover, recent work showed that harder and brighter X-rays can be expected from the betatron mechanism as it scales with the electron energy that can be increased with higher laser intensities and larger electron acceleration length (TaPhuoc et al., 2005; Rousse et al., 2004). The main astrophysical research issues that could potentially benefit from this source are X-ray radiography and X-ray absorption to probe dense plasmas studied in laboratory astrophysics. Hard X-rays can indeed be used to transversely radiograph laser-driven shocks in solid materials for the determination of equation of state (EOS) (Koenig et al., 2005). Also, the use of short pulse X-ray blacklighter sources is a powerful diagnostic to investigate dense plasmas in transient local thermodynamic equilibrium (LTE) (Audebert et al., 2005). This type of plasma can be found in various fields of fundamental and applied research. The Betatron source has the advantage to offer a broad white light continuum in the hard X-ray region (above 1 keV) without emission lines to probe dense plasmas studied in laboratory astrophysics. Acknowledgements Work supported by the European Community under Contract Nos. HPRI-CT-1999-00086, HPRI-CT-2000-40016, and HPRI-CT-1999-50004 (FAMTO project). F.A. also acknowledges the support from the HEDLA organizing committee.

References Attwood, D.: Soft X-rays and Extreme Ultraviolet Radiation. Cambridge University Press, Cambridge (1999) Audebert, P., et al.: Phys. Rev. Lett. 94, 025004 (2005)

Astrophys Space Sci (2007) 307:329–333 Brabec, T., Krausz, F.: Rev. Mod. Phys. 72, 545 (2000) Cole, A.J., et al.: Nature 299, 329 (1982) Dromey, B., Zepf, M., et al.: Nature Phys. 2, 456 (2006) Esarey, E., Shadwick, B.A., Catravas, P., Leemans, W.P.: Phys. Rev. E 65, 056505 (2002) Faure, J., Glinec, Y., Pukhov, A., Kiselev, S., Gordienko, S., Lefebvre, E., Rousseau, J.-P., Burgy, F., Malka, V.: Nature 431, 541 (2004) Hammel, B.A., Griswold, D., Landen, O.L., Perry, T.S., Remington, B.A., Miller, P.L., Peyser, T.A., Kilkenny, J.D.: Phys. Fluids B: Plasma Phys. 5, 2259 (1993) Jackson, J.D.: Classical Electrodynamics, 3rd edn. Wiley, New York (2001) Kiselev, S., Pukhov, A., Kostyukov, I.: Phys. Rev. Lett. 93(13), 135004 (2004) Koenig, M., Benuzzi-Mounaix, A., Ravasio, A., Vinci, T., Ozaki, N., Lepape, S., Batani, D., Huser, G., Hall, T., Hicks, D., MacKinnon, A., Patel, P., Park, H.S., Boehly, T., Borghesi, M., Kar, S., Romagnani, L.: Plasma Phys. Controlled Fusion 47, B441 (2005) Kostyukov, I., Kiselev, S., Pukhov, A.: Phys. Plasmas 10, 4818 (2003) Lee, T., et al.: Chem. Phys. 299, 233 (2004)

333 Malka, V., Fritzler, S., Lefebvre, E., Aleonard, M.M., Burgy, F., Chambaret, J.P., Chemin, J.F., Krushelnick, K., Malka, G., Mangles, S.P.D., Najmudin, Z., Pittman, M., Rousseau, J.P., Scheurer, J.N., Walton, B., Dangor, A.E.: Science 298, 1596 (2002) Pukhov, A.: Rep. Prog. Phys. 66, 47 (2003) Pukhov, A., Meyer-ter-vehn, J.: Appl. Phys. B: Lasers Opt. 74, 355 (2002) Rishel, C., et al.: Nature 390, 490 (1997) Rousse, A., et al.: Phys. Rev. Lett. 93, 135005 (2004) Rousse, A., Rischel, C.: Rev. Mod. Phys. 73, 17 (2001) Seltzer, S., Berger, M.: At. Data Nucl. Data Tables 35, 354 (1986) Siders, C., et al.: Science 286, 1340 (1999) TaPhuoc, K., et al.: Phys. Plasmas 12, 023101 (2005) TaPhuoc, K., Rousse, A., Pittman, M., Rousseau, J.P., Malka, V., Fritzler, S., Umstadter, D., Hulin, D.: Phys. Rev. Lett. 91(19), 195001 (2003) Tarasevitch, A., Orisch, A., Von der Linde, D.: Phys. Rev. A 62, 023816 (2000) Whitlock, R.R., et al.: Phys. Rev. Lett. 52, 819 (1984) Whittum, D.H.: Phys. Fluids B 4, 730 (1992) Workman, J., Kyrala, G.A.: Proc. SPIE 4504, 168 (2001)

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Astrophys Space Sci (2007) 307:335–340 DOI 10.1007/s10509-006-9279-5

O R I G I NA L A RT I C L E

Scalable Dynamics of High Energy Relativistic Electrons: Theory, Numerical Simulations and Experimental Results T. Baeva · S. Gordienko · A. Pukhov

Received: 14 April 2006 / Accepted: 23 November 2006 C Springer Science + Business Media B.V. 2006


Abstract Similarity theory, which is necessary in order to apply the results of laboratory astrophysics experiments to relativistic astrophysical plasmas, is presented. The analytical predictions of the similarity theory are compared with PIC numerical simulations and the most recent experimental data on monoenergetic electron acceleration in diluted plasmas and high harmonic generation at overdense plasma boundaries. We demonstrate that similarity theory is a reliable tool for explaining a surprisingly wide variety of laboratory plasma phenomena the predictions of which can be scaled up to astrophysical dimensions. Keywords Similarity theory . Laser-plasma interaction . Particle acceleration . X-ray generation

1 Introduction The advent of chirped-pulse amplification (CPA) has opened a new era of laser technology and laser applications (Strickland and Mourou, 1985). The CPA method has allowed the construction of table-top amplifiers which can generate pulses with millijoule energies and femtosecond durations, leading to peak powers of several terrawatts (1 TW = 1012 W, corresponding to the electric output of 1000 large nuclear power stations). With appropriate focusing, the pulses, T. Baeva (
) · A. Pukhov Institut f¨ur Theoretische Physik I, Heinrich-Heine-Universit¨at D¨usseldorf, D-40225, Germany e-mail: [email protected] S. Gordienko Institut f¨ur Theoretische Physik I, Heinrich-Heine-Universit¨at D¨usseldorf, D-40225, Germany; L. D. Landau Institute for Theoretical Physics, Moscow, Russia

produced by these laser systems can create exotic conditions, never before achieved in a laboratory: intensity 1020 W/cm2 , electric field 1011 V/cm, temperature 106 eV. Matter that is exposed to these extreme conditions behaves in such a way that gives a new insight into fundamental phenomena from atomic, molecular and condensed matter physics (with characteristic energies in the eV range), to nuclear physics, high energy physics, astrophysics, and cosmology (with characteristic energies of MeV and GeV) (Remington et al., 2000; Takaba et al., 1999). Cosmological and astrophysical applications of the new laser technology (the so called “laboratory astrophysics”) have inspired great interest mainly because of the following reason. It is well known that the traditional astrophysical research is based on either observation or theoretical modeling. Yet this approach lacks the ability to quantitatively test models under experimental conditions where the initial and final states are well characterized (Remington et al., 1999). Therefore laboratory astrophysics is a new impetus for both astrophysical and laser-plasma studies, since it allows preparing and controlling the initial states rather than make us rely on uncontrollable observation results. However the new laser technology not only opens new opportunities for astrophysical research but also brings new scientific challenges and problems. Indeed, astrophysical and laboratory dimensions differ significantly. Consequently, a key question is how one can re-scale laboratory-size experimental results up to dimensions of astrophysical importance. This scientific difficulty is neither new nor unique. Quite analogous problems encounters for example the classical (magneto-)hydrodynamics. One of the most powerful theoretical tools in such situations is the so called similarity theory (Birkhoff, 1960; Sedov, 1993). The similarity allows engineers to rescale the behavior of a physical system from a laboratory acceptable size to a size of practical use. Springer

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Yet not only laboratory astrophysics but also laser-plasma interaction research is interested in developing effective similarity theory. As a matter of fact such topical phenomena as quasi-monochromatic electron acceleration in diluted plasmas and high harmonic generation at overdense plasma boundaries can be fully understood on the basis of the ultrarelativistic similarity theory. The aim of the present article is to present similarity theory for laser-plasma interactions in the ultra-relativistic regime and explain its application to the physical phenomena just mentioned. Our choice of effects to discuss fell on the quasimonochromatic electron acceleration in diluted plasmas and the high harmonic generation at overdense plasma boundaries not only because these spectacular phenomena have found impressive numerical confirmation, but mostly because they have been recently observed experimentally and, as a result, provide a powerful support for the applicability of the ultra-relativistic similarity theory.

Let us consider collisionless laser-plasma dynamics and neglect the ion motion. The evolution of the electron distribution function f (t, r, p) is described by the Vlasov equation (∂t + v · ∂r − e(E + v × B/c) · ∂p ) f (t, p, r) = 0,

(1)

where p = m e γ v and the self-consistent fields E and B satisfy the Maxwell equations. We suppose that the laser pulse vector potential at the time t = 0 short before entering the plasma is A(t=0) = a((y 2 + z 2 )/R 2 , x/cτ ) cos(k0 x), where k0 = ω0 /c is the wavenumber, R is the focal spot radius and τ is the pulse duration. If one fixes the laser envelope a(r⊥ , x), then the laser-plasma dynamics depends on four dimensionless parameters: the laser amplitude a0 = max |ea/m e c2 |, the focal spot radius k0 R, the pulse duration ω0 τ , and the plasma density ratio n e /n c , where n c = m e ω02 /4π e2 is the critical density. Now we are going to show that in the ultra-relativistic regime when a02 ≫ 1, the number of independent dimensionless parameters reduces to three: k0 R, ω0 τ and S, where the similarity parameter S is ne . a0 n c

(2)

Let us introduce the new dimensionless variables tˆ = S 1/2 ω0 t, rˆ = S 1/2 k0 r, pˆ = p/m e ca0 , ˆ = A

−1/2

−1/2

S eE ˆ S eB eA , Eˆ = , B= , 2 m e c a0 m e cω0 a0 m e cω0 a0

Springer

f =

ne ˆ τˆ ), ˆ rˆ , a0 , S, R, fˆ(tˆ, p, (m e ca0 )3

(3)

(4)

where Rˆ = S 1/2 k0 R and τˆ = S 1/2 ω0 τ . The normalized distribution function fˆ is a universal one describing the interaction of the given laser pulse with a fixed initial plasma profile. It satisfies the equations ˆ + (ˆv × B)) ˆ · ∂pˆ ] fˆ = 0, [∂tˆ + vˆ · ∂rˆ − (E

∇rˆ · Eˆ = 4π (1 + ρ), ˆ ∇rˆ · Bˆ = 0,

(5) (6)

ˆ ∇rˆ × E ˆ ˆ = 4π ˆj + ∂tˆE, ˆ = −∂tˆB, ∇rˆ × B  ˆ pˆ 2 + a0−2 is the electron velocity, ρˆ = where vˆ = p/ ˆ ˆj = − vˆ fˆ d pˆ and the initial condition for the − fˆ d p, vector potential is ˆ tˆ = 0) = aˆ (( yˆ 2 + zˆ 2 )/ R, ˆ x/ ˆ ˆ τˆ ) cos(S −1/2 x), A(

2 Ultra-relativistic similarity theory

S=

and the new distribution function fˆ defined as

(7)

with the slow envelope aˆ such that max |ˆa| = 1. Equations (5, 6) together with the initial condition (7) still ˆ τˆ , S and a0 . depend on the four dimensionless parameters R, However, the parameter a0 appears only in the expression for the electron velocity. In the limit a02 ≫ 1 one can write  ˆ pˆ 2 + a0−2 ≈ vˆ = p/ ˆ ˆ |p| vˆ = p/

(8)

Consequently, for the ultra-relativistic amplitude a02 ≫ 1, the laser-plasma dynamics does not depend separately on a0 and n e /n c . Rather, they converge into the single similarity parameter S. The ultra-relativistic similarity means that for different interaction cases with the same value of S, plasma electrons move along similar trajectories. The number of these electrons Ne , their momenta p, and the plasma fields scale as p ∝ a0 ; N e ∝ a0 ;

(9)

φ, A, E, B ∝ a0

(10)

for ω0 τ = const, k0 R = const and S = const. The ultra-relativistic similarity is valid for arbitrary Svalues. The S parameter appears only in the initial condition (7) so that S −1/2 plays the role of a frequency parameter since it separates relativistically overdense plasmas with S ≫ 1 from underdense ones with S ≪ 1. Notice that the similarity theory developed in this section neglects the ion motion, therefore it is applicable only for short laser pulses (compare with Ruytov and Remington, 2006)

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337

3 Scalings for laser wake-field acceleration(LWFA) In this section we apply the ultra-relativistic similarity theory developed in Section 2 to the special case of underdense plasma and consider laser wake-field acceleration, particularly in the bubble acceleration regime (Pukhov and Meyer-ter-Vehn, 2002). Here we present only the main consequences of the similarity theory and their numerical confirmation. Mathematically rigorous derivation of these results and their heuristic consideration can be found in Gordienko and Pukhov (2005), Pukhov and Gordienko (2006). Before starting with the scalings for the bubble acceleration regime let us first recall its main characteristics. When a short relativistically intense laser pulse enters underdense plasma, it is able to expel all electrons behind itself. Thus a cavity (bubble), free of electrons but containing positively charged ions, is generated and propagates in the plasma just behind the pulse. The electrons from the rear side of the bubble penetrate the cavity, get trapped and accelerated up to high energies. A distinctive feature of the bubble acceleration regime is that the spectrum of the high energy electrons is quasi-monochromatic what was first predicted numerically (Pukhov and Meyer-terVehn, 2002) and then observed experimentally (Faure et al., 2004). On the basis of the ultra-relativistic theory two families of scalings describing the bubble acceleration regime have been obtained. The first family describes trapped and accelerated electrons in the bubble regime with the same values of all of the ultra-relativistic similarity theory dimensionless parameters S, ω0 τ , k0 R. From the similarity theory presented in Section 2 (see for details Gordienko and Pukhov, 2005; Pukhov and Gordienko, 2006) follows that if one keeps S = const, then both the number of trapped monoenergetic electrons Nmono and the largest energy they can reach E mono grow as the dimensionless vector potential a0 of the laser pulse, i.e. Nmono ∝ a0 , E mono ∝ a0 .

(11)

In order to describe the second family we use the fact that for the bubble acceleration regime S ≪ 1. In this case, S can be considered as a small parameter and quite general scalings for laser-plasma interactions can be obtained. It follows from the theory that in the optimal configuration the laser pulse √ has the focal spot radius k p R ≈ a0 and the duration τ ≤ R/c. Here, k p = ω p /c is the plasma wavenumber and ω2p = 4π n e e2 /m e is the plasma frequency. The central result of the ultra-relativistic similarity theory is that the bubble regime of electron acceleration is stable, scalable and the scaling for the maximum energy

E mono of the monoenergetic peak in the electron spectrum is  P cτ 2 E mono ≈ 0.65 m e c . (12) Prel λ Here, P is the laser pulse power, Prel = m 2e c5 /e2 ≈ 8.5 GW is the natural relativistic power unit, and λ = 2π c/ω0 is the laser wavelength. The scaling (12) assumes that the laser pulse duration satisfies the condition cτ < R. The scaling for the number of accelerated electrons Nmono in the monoenergetic peak is

Nmono

1.8 ≈ k0 re



P , Prel

(13)

where re = e2 /m e c2 is the classical electron radius, and k0 = 2π/λ. The acceleration length L acc scales as L acc ≈ 0.7

cτ Z R, λ

(14)

where Z R = π R 2 /λ ≈ a0 λ2p /4π λ is the Rayleigh length. The parametric dependencies in the scalings (12)–(14) follow from the analytical theory. The numerical pre-factors are taken from 3D PIC simulations. These pre-factors may change depending on the particular shape of the pulse envelope. However, as soon as the envelope of the incident laser pulse is defined, the pre-factors are fixed. The parametric dependencies, on the other hand, are universal and do not depend on the particular pulse shape. To check the analytical scalings, we performed 3D particle-in-cell simulations with the code VLPL (Pukhov, 1999). In the simulations, we use a circularly polarized laser pulse with the envelope a(t, r⊥ ) = a0 cos(π t/2τ ) exp(−r⊥ 2 /R 2 ), which is incident on a plasma with uniform density n e . We used grid steps h x = 0.07λ, h y = h z = 0.5 λ, and 4 particles/cell. First, we check the basic ultra-relativistic similarity with S = const. We choose the laser pulse duration τ = 8 · 2π/ω0 . The laser radius is R = 8 λ, where λ = 2π c/ω0 is the laser wavelength. The laser pulse aspect ratio is cτ/R = 1 in this case. We fix the basic similarity parameter to the value S i = 10−3 and perform a series of four simulations with (i) a0i = iii 10, n ei = 0.01n c ; (ii) a0ii = 20, n ii e = 0.02n c ; (iii) a0 = 40, n eiii = 0.04n c ; (iv) a0iv = 80, n iv e = 0.08n c . Taking the laser wavelength λ = 800 nm, one can calculate the laser pulse energies in these four cases: W i = 6 J; W ii = 24 J; W iii = 96 J; W iv = 384 J. These simulation parameters correspond to the bubble regime of electron acceleration (Pukhov and Meyer-ter-Vehn, 2002), because the laser pulse duration τ is Springer

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Let us consider a short laser pulse of ultra-relativistic intensity, interacting with the sharp surface of an overdense plasma slab. Under the assumption that the incident laser pulse is short we can neglect the slow ion dynamics and consider only the electron motion. Since the laser pulse is both ulra-short and ultra-intense we can neglect the Coulomb collisions and related to them resistivity. Indeed, for T ≤ m e c2 the collision frequency is ν ∝ ω pe

Fig. 1 Electron energy spectra obtained in the simulations (i)-(iv) (see text). The control points 1–5 were taken after the propagation distances L 1 = 200λ, L 2 = 400λ, L 3 = 600λ, L 4 = 800λ, L 5 = 1000λ. The spectra evolve similarly. The monoenergetic peak positions scale ∝ a0 and the number of electrons in a 1% energy range also scales ∝ a0 in agreement with the analytic scalings (9)

√ shorter than the relativistic plasma period a0 ω−1 p . We let the laser pulses propagate the distance L ib = 1000 λ through the plasma in the all four cases. At this distance, the laser pulses are depleted, the acceleration ceases and the wave breaks. Figure 1 (i)–(iv) shows the evolution of the electron energy spectra for these four cases. One sees that the energy spectra evolve quite similarly. Several common features can be identified. First, a monoenergetic peak appears after the acceleration distance L ≈ 200 λ. Later, after a propagation distance of L ≈ 600 λ, the single monoenergetic peak splits into two peaks. One peak continues the acceleration towards higher energies, while the other one decelerates and finally disappears. Comparing the axes scales in Fig. 1, we conclude that the scalings (9) hold with good accuracy. At the end we want to emphasize that theoretical and numerical results just presented are in agreement with the experimental data (Faure et al., 2004).

4 Similarity and high harmonic generation at plasma boundaries In this section we explain the physical picture of high harmonic generation (HHG) at the boundary of over-dense plasma with emphasis on its close relation to ultra-relativistic similarity theory and present its numerical confirmation. Detailed derivation of the analytical results can be found in Gordienko et al. (2004), Baeva et al. (2006) and Baeva (2005), here we mainly concentrate on the physical origin of high harmonic spectrum universality the roots of which lie in the similarity properties of collisionless plasma dynamics. Springer

&

1/3

e2 n e T

'3/2

ln ,

(15)

where ω pe is the plasma frequency, T is the electron temperature and ln  ≈ 15 is the Coulomb logarithm. Since the laser pulse duration τ ∝ 2π/ω0 , for T ∝ m e c2 Equation (15) gives rise to ' & 1/3 3/2 2π e2 n e τν ∝ ω pe ln  ≈ 10−5 ≪ 1 ω0 T

(16)

for a laser pulse of wavelength 800 nm and typical solid state 1/3 density (e2 n e ≈4 eV). Consequently on the time scale of the laser we can neglect the Coulomb collisions and use collisionless description by means of the relativistic Vlasov equation. Notice that for the case of interest the kinetic energy of electrons is much larger than m e c2 and the role of collisions is even less. Note that only the surface area of very hot plasma is relevant to high harmonics generation. In spite of the rather high density of this plasma layer, Coulomb collisions in it are negligible. The cold plasma below the hot plasma layer could also be important if the “return current”, generated to guarantee plasma quasineutrality, plays a significant role. However, the similarity theory used for the theory of high harmonic generation (Section 2) demonstrates that the typical electric field in the area where important physical processes take place scales as the laser field (S = const). Since this field is very large, even if collisions were taken into account, all electrons would be deep in the run-away regime, in which resistivity is described by kinetic effects in collisionless plasma. Therefore the high harmonic generation at the boundary of overdense plasma is treated by means of the Vlasov equation in what follows. The plasma electrons are driven by the laser light pressure and the restoring electrostatic force connected to the Coulomb attraction to the ions. As a consequence, the plasma surface oscillates and the electrons gain normal momentum components pn . Since the plasma is overdense, the incident electromagnetic wave is not able to penetrate it. In other words there is electric current along the plasma surface. For this reason, the momenta of the electrons in the skin layer have, apart from

Astrophys Space Sci (2007) 307:335–340

339

the component normal to the plasma surface, tangential components pτ . According to the relativistic similarity theory (see Equation (9)), both the normal and tangential components of electron momentum scale as the dimensionless electromagnetic potential: pn , pτ ∝ a0 if the similarity parameter S is fixed. Consequently, there is a finite angle between the full electron momenta and the plasma surface for most of the times. An important point is that the typical value of this angle does not decrease if a0 increases for S = const. Since we consider a laser pulse with ultra-relativistic intensity, the motion of the electrons is ultra-relativistic. In other words, their velocity is approximately c. Though the motion of the plasma surface is qualitatively different. Its velocity vs is not ultra-relativistic for most of the times but smoothly approaches c only when the tangential electron momentum vanishes. Using simple algebra one can demonstrate that the γ factor of the surface γs also shows specific behavior. It has sharp peaks at those times for which the velocity of the surface approaches c. Thus, while the velocity function vs is characterized by its smoothness, the hallmarks of γs are its quasi-singularities. When vs reaches its maximum and γs has a sharp peak, high harmonics of the incident wave are generated and can be seen in the reflected radiation. Physically this means that the high harmonics are due to the collective motion of bunches of fast electrons moving towards the laser pulse (Baeva, 2005). These harmonics have two very important properties. First, their spectrum is universal: the exact motion of the plasma surface can be very complicated, since it is affected by the shape of the laser pulse and can differ for different plasmas. Yet the qualitative behavior of vs and γs at the vicinity of the surface velocity maxima is universal and since high harmonics are generated around these moments of time the spectrum of the high harmonics does not depend on details of the particular surface motion. This observation leads to the conclusion (Gordienko et al., 2004; Baeva, 2005) that the high harmonic spectrum contains two qualitatively different parts: a power-law decay and an exponential decay. In its power-law part the spectrum decays as In ∝ 1/n 8/3 ,

(17)

3 , where In up to a critical harmonics number that scales as γmax is the intensity of the nth harmonic. Here γmax is the maximal γ -factor of the point, where the component of the electric field tangential to the surface vanishes. The second important feature of the high harmonics is that they are phase-locked. This observation is of particular value, since it allows for the generation of attosecond and even sub-attosecond pulses (Gordienko et al., 2004).

Fig. 2 Spectra of the reflected radiation for the laser amplitudes a0 = 5, 10, 20. The broken line marks the universal scaling I ∝ ω−8/3

Fig. 3 Electron distribution function. The helix represents the electron surface motion in the laser field. The reddish downward spikes stay for the surface relativistic motion towards the laser. These spikes are responsible for the zeptosecond pulse generation

In order to check our analytical results, we perform a number of 1d PIC simulations with the VLPL code (Pukhov, 1999). A Gaussian laser pulse with a = a0 exp(−t 2 /τ L2 ) was incident onto a plasma layer with a step density profile. Figure 2 shows spectra of the reflected radiation for laser amplitudes a0 = 5, 10, 20, duration ωτ L = 4π and plasma density Ne = 30Nc , which roughly corresponds to the solid hydrogen or liquid helium. The log-log scale of Fig. 2 reveals the power-law scaling of the spectral intensity In ∝ 1/n 8/3 . The critical harmonic number n c , where the power-law scaling changes into the exponential decay increases for higher laser amplitudes. Let us take a closer look at the particular case a0 = 20 (the red line a) in Fig. 2). In this case, the power-law spectrum extends above the harmonic number 2000, and zeptosecond pulses (1 zs = 10−21 s) can be generated. As one sees from the electron distribution function f (t, x, px ), Fig. 3, the maximum surface γ -factor γmax ≈ 25 is achieved at the time t ≈ 6. The temporal profile of the reflected radiation is shown in Fig. 4. When no spectral filter is applied, Fig. 4a, a train of attosecond pulses is observed. However, when we apply a spectral filter selecting harmonics above n = 300, a train of much shorter pulses is recovered, Fig. 4b. Figure 4c zooms to one of these pulses. Its full width at half maximum is about 300 zs. At the same time its intensity normalized to the laser frequency is huge (eE zs /mcω)2 ≈ 14 that would correspond to the intensity Izs ≈ 2 × 1019 W/cm2 . Spectrum of high harmonics generated at an overdense plasma boundary in the ultra-relativistic regime including Springer

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laboratory plasma phenomena and provides a way to scale them up to astrophysical dimensions. References

Fig. 4 Zeptosecond pulse train: (a) temporal structure of the reflected radiation; (b) zeptosecond pulse train seen after spectral filtering; (c) one of the zeptosecond pulses zoomed, its FWHM duration is about 300 zs

about 850 harmonics with In ∝ n − p , p = 2.5(+0.2, −0.3) has recently been observed experimentally (Dromey et al., 2006), thus confirming the predictions of the ultra-relativistic similarity theory. The scaling of the spectrum roll-over 3 was also predicted analytically as proportional to γmax experimentally confirmed in a series of experiments producing keV-photons (Zepf, 2006). In conclusion, the ultra-relativistic similarity theory presented in this article has proved to be efficient in explaining

Springer

Baeva, T., et al.: Phys. Rev. E 74, 1 (2006) Baeva, T.: Diploma thesis: “Attosecond phenomena in laser-condensed matter interaction”, D¨usseldorf University (2005) Birkhoff, G.: Hydrodynamics. University Press, Princeton, NJ (1960) Dromey, B., Zepf, M., Gopal, A., et al.: Nature Phys. 2, 456 (2006) Faure, J., et al.: Nature 431, 541 (2004) Gordienko, S., et al.: Phys. Rev. Lett. 93, 115003 (2004) Gordienko, S., Pukhov, A.: Phys. Plasmas 12, 043109 (2005) Pukhov, A.: J. Plasma Phys. 61, 425 (1999) Pukhov, A., Gordienko, S.: Phil. Trans. R. Soc. A 364, 623 (2006) Pukhov, A., Meyer-ter-Vehn, J.: Appl. Phys. B 74, 355 (2002) Remington, B., Drake, R.P., Takabe, H., Arnett, A.: Science 284, 1488 (1999) Remington, B., et al.: Phys. Plasma 7, 1641 (2000) Ryutov, D.D., Remington, B.: Plasma Phys. Control. Fussion 48, L23 (2006) Sedov, L.I.: Similarity and dimentional methods in mechanics, 10th edn. CRC Press, Boca Reton, FL (1993) Strickland, D., Mourou, G.: Opt. Comm. 56, 219 (1985) Takaba, H., et al.: Plasma Phys. Control. Fussion 41, A75 (1999) Zepf, M.: International Conference on the Interaction of Atoms, Molecules and Plasmas with Intense Ultrashort Laser Pulses 1– 5 Szeged, Hungary (2006)

Astrophys Space Sci (2007) 307:341–345 DOI 10.1007/s10509-007-9386-y

ORIGINAL ARTICLE

Proton Radiography of Megagauss Electromagnetic Fields Generated by the Irradiation of a Solid Target by an Ultraintense Laser Pulse Sebastien Le Pape · Daniel Hey · Pravesh Patel · Andrew Mackinnon · Richard Klein · Bruce Remington · Scott Wilks · Dmitri Ryutov · Steve Moon · Marc Foord

Received: 12 May 2006 / Accepted: 22 January 2007 C Springer Science + Business Media B.V. 2007


Abstract Laser generated protons have been used to probe the temporal and spatial evolution of megagauss magnetic fields. Grid deflectometry techniques have been applied to proton radiography to obtain precise measurements of proton beam angles caused by magnetic fields in laser produced plasmas. Data are presented in two different regimes of interactions at ultra high intensity (1020 W/cm2 ) where hots electrons are supposed to be responsible of the B field, and at lower intensity (1017 W/cm2 ) and later time where the gradients of temperature and density are responsible of the B field. Keywords Laser generated-proton beam . Magnetic field . Deflectometry

(Sudan, 1993) or (iii) the current of fast electrons generated during the interaction (Wilks et al., 1992; Pukhov and Meyer-ter-Vehn, 1996). The localization of these fields near the critical density makes them very challenging to probe. Optical probe used for Faraday rotation are refracted on steep density gradients, so that the optical beam can only probe the outer part of the plasma and the lower amplitude B fields (around 10 MG) (Borghesi et al., 1998). Recently magnetic fields of 700(±100) megagauss were inferred from polarization shifts of low order VUV harmonics induced by the Cotton–Mouton effect (Wagner et al., 2002, n.d.). However this technique does not provide any spatial information on the B fields structure. The use of laser driven proton deflectometry thus seems to be the ideal technique to probe fast evolving B fields localized near the critical density.

Introduction There are a number of outstanding issues induced by large magnetic fields in the evolution of complex physical phenomena, including the formation of black holes and neutron stars and the acceleration of cosmic rays (Shapiro and Teukolsky, 1983). Magnetic fields of the megagauss order are generated by the interaction of a high intensity laser with a solid target (Stamper, 1991; Tatarakis et al., n.d.). These fields are predicted to exist in a localized region near the critical density surface. Such spontaneous fields can be generated by several mechanisms including: (i) non parallel temperature and density gradients in the ablated plasma (Stamper, 1991), (ii) the ponderomotive force associated with the laser radiation S. Le Pape (
) . D. Hey . P. Patel . A. Mackinnon . R. Klein . B. Remington . S. Wilks . D. Ryutov . S. Moon . M. Foord Lawrence Livermore National Laboratory 7000 East Avenue Livermore 94550 California USA e-mail: [email protected]

Laser driven proton deflectometry This technique exploits the spatial and temporal characteristics of a laser driven proton source. The proton beam is generated by focusing an ultra-intense laser (Clark et al., 2000a; Maksimchuk et al., 2000; Snavely et al., 2000; Zepf et al., 2003) on a thin metallic foil (typically 4.5 microns gold foil). The protons that are present at the surface of the foil as part of the surface contaminants are accelerated via the space charge force induced by the hot electrons that are directly accelerated by the laser pulse. Experiments conducted previously on the same installation (Allen et al., 2004) have shown that most of the protons are emitted from the back surface of the foil. The protons are typically accelerated to energies of a few hundred of keV to 25 MeV for an intensity of 1019 W/cm2 . The proton beam generated is temporally short (in the order of a ps), highly laminar and hence equivalent to a virtual point. In proton imaging, a point projection of the probed Springer

342 Fig. 1 Experimental setup of the proton deflectometry experiments

Astrophys Space Sci (2007) 307:341–345

Interaction target

Proton target mesh

Radiochromic film

CPA CPA

Proton beam

l

region is obtained with a spatial resolution set by the virtual source size. The magnification is given by M = (L + 1)/1 (see Fig. 1). This technique is mainly sensitive to field gradients, which are detected via proton density modulations in the probe beam cross section. In proton deflectometry a mesh (MacKinnon et al., 2004) is additionally inserted between the proton target and the interaction target in order to preimprint a periodical pattern on the probe beam. From the mesh distortions, the field strength can then be calculated.

B field measurement from the proton target The experiment was realized in the new Titan facility at the Lawrence Livermore National Laboratory. The laser delivers around 130 J on target in 1 ps at λ = 1053 micron. It is focused by a F/3 off axis parabola on a focal spot of about 10 microns diameter, leading to an intensity of 1020 W/cm2 . The laser is focused on a thin gold target of 4.5 microns. A 1000 lpi mesh is placed between the target and the radiochromic film pack. Figure 2a and b, present 18 and 22.5 MeV protons for magnifications of M = 87 and M = 147 respectively. The magnification is modified by changing the distance between the proton source and the mesh (from 1000 microns to 500 microns). On these images, the meshes clearly present strong distortions that might be induced by the presence of a B field. Previous work (Clark et al., 2000b) has explained ring Fig. 2 (a) 18.5 MeV protons with a magnification of 87. The mesh looks straight on the outer part on the beam and strongly distorted at its center. (b) 22.5 MeV protons with a magnification of 147. When the magnification is increased most of the mesh elements look distorted

Springer

L

structures observed in the proton beam by the presence of a B field in the bulk of the target. Indeed during the interaction of the short ultra intense laser pulse with the thin foil, a large current of hot electrons is driven in the foil by the laser pulse. This current is then responsible for the sheath at the back of the foil that accelerates the protons. This flow of hot electrons also induces large B fields in the target bulk and at the back of the target. LSP (Welch et al., n.d.) simulations have been realized to model the acceleration of the protons from a 5 microns gold foil irradiated by intensity around 1019 W/cm2 . Figure 3 presents these simulations. A large magnetic field (10 MG) can be seen at the back of the foil and the B fields persist after the electron pulse and are maximum near the edge of the laser spot. The electron pulse lasts 100 fs whereas the B field lasts up to 375 fs. In this work the assumption was made that the protons were emitted from the front of the foil. However, further works have shown that most of the generated protons are emitted from the back on the foil (Allen et al., n.d.). Their trajectory could then be sensitive to the presence of B fields in that region. Figure 4a shows the respective amplitude of the fields (electric and magnetic) as a function of the distance to the target. It shows that the B fields are really high (10 MG) when close to the target (less than a micron) and that the electric fields are much weaker in amplitude but extend over a longer distance (around 10 microns). Figure 4b shows their respective influence on the proton trajectory. The presence of a strong B field at the back

Astrophys Space Sci (2007) 307:341–345

343

Fig. 3 B field as a function of time (T0 is the time where the 100 fs electron burst is launched in the gold target, i.e 125 fs is 25 fs after the end of the electron burst). (a) 125 fs, (b) 250 fs and (c) 375 fs. The laser Er and BΘ

Particle trajectory velocities

1 107

0.06

5 106

0.05

Particle velocity (γβ)

Fig. 4 (a) Particle velocities as a function of the distance to the target. Vz is the velocity in the direction normal to the target, Vr is the radial velocity. (b) Magnitude of the electric and magnetic field as a function of the distance to the target

comes from the left, the target is a gold foil of 5 microns. The image shows only half of the focal spot, it is symmetric around y = 0. The focal spot is 10 microns diameter; the laser pulse duration is 100 fs

0 -5 106 -1 10

7

-1.5 107 -2 107

Surface Ez acceleration <1 µm

Vz

0.04

Vr (x 10) 0.03

Er acceleration 0.02

0.01

Bθ bump

-2.5 107 -3 107 0.0005 0.0006 0.0007 0.0008 0.0009 0.001 0.0011 0.0012 0.0013

Distance Z (cm)

a of the target foil induces a bump of the radial velocity of the protons on the first microns and then a constant acceleration is induced by the electric field. The distortions observed on the mesh on Fig. 2a are not observed over the entire proton beam but are localized on a portion of the beam only. The B fields are localized near the focal spot (Fig. 3) as the electric field that generated the proton acceleration is localized at the back of the proton foil on a much larger area that corresponds to the source size. A simple model of proton trajectories as they pass trough the B fields has been used to estimate the B field strength. For 22.5 MeV protons, a 500 MG field is used to fit the mesh distortions. Two beams experiment At the same time a two beams experiment has been set up to probe the B fields generated on a second target. This experiment has been realized on the Calisto facility at the Lawrence Livermore National Laboratory. The laser delivers 10 J in 100 fs at 800 nm. The beam is then split in two, using a doughnut mirror with a one-inch hole in the center. The reflected beam is then focused by an F/4 off axis parabola on a thin foil of gold that will generate the proton beam. The

0 0.0005 0.0006 0.0007 0.0008 0.0009 0.001 0.0011 0.0012 0.0013

Distance Z (cm)

b second beam that goes through the mirror is focused by an F/6 off axis parabola on a thin foil of aluminum to create the probed plasma (Fig. 5). The plasma is probed face-one. A 1000 lines per inch mesh is placed at 1.5 mm from the proton target. LSP hybrid PIC code simulations (Town et al., 2005) show that, face-on, the protons are sensitive only to B field, while side-on, the protons are sensitive to E field. The timing of the laser has been adjusted so that the 3.5 MeV protons arrive to the foil 90 ps (± 25 ps) after the laser pulse. Figure 6 shows an image of the protons probing the magnetic field. The aluminum foil edges are clearly visible on the radiochromic film; the mesh is blurred at the position of the foil because of the scattering of the protons in the aluminum. The B field in the interaction plasma induces the ring pattern that is visible on the film. As the time delay between the two pulses is large (90 ps), the probed B fields are not created by the flow of hot electrons. Indeed those B fields last only about 10 ps after the laser pulse (Sandhu et al., n.d.). The B fields probed at this late time are induced by the temperature and density gradients that are present in the plasma. These B fields are then much weaker than those induced by the hot electrons current. A first estimation using the same simple model of proton raytrace gives a B field amplitude of 0.01 G. Springer

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Fig. 5 Setup of the two beams experiment. The incoming beam is splited in two using a doughnut mirror, the dashed beam creates the probed plasma and the other one creates the proton beam. A timing slide allows to adjust the delay time between the two beams

Astrophys Space Sci (2007) 307:341–345

density with a temporal resolution of a picosecond. The first experiment presents data on self-generated B fields during the proton acceleration processes. The preliminary results show the presence of large B fields at the back of the target. LSP simulations confirm that a large B field (10 MG) on a really small distance (less than a micron) has a strong influence on the radial velocity of the protons. From the mesh distortions, the B field amplitude is estimated at 500 MG. Further experiments changing the intensity on target or the target material will be conducted to confirm the first measurement. The second experiment presents data from a two beams setup. In this experiment the B fields are generated on a second target and are thus decoupled from those generated during the proton generation processes. Data are recorded at late time, i.e. when the B fields are induced by temperature and density gradients. The first results estimate B fields in the order of 0.01 MG. Further experiments will be carried out to measure the B fields at shorter time, i.e. when the B fields are generated by the hot electrons current.

Acknowledgement This work has been performed under the auspices of the U.S DoE LLNL under contract No W-7405-Eng-48.

References

Fig. 6 Typical imaging data at probing time t = 90 ps. This image shows the proton beam probing the plasma created in the Aluminum foil. The 1000 lpi mesh, placed between the proton target and the interaction target, is visible on the image. The presence of the aluminum foil causes the blurring of the mesh due to the scattering of the proton is the foil. The annular structure observed in the proton beam is induced by the presence of B field in the probed plasma

Conclusion In this paper we present an attempt to use laser driven proton beams to probe high magnitude B fields created by a short laser pulse. With a laser driven proton beam, high magnitude electric and magnetic fields can be probed near critical Springer

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