The Equation of State in Astrophysics: IAU Colloquium 147 (International Astronomical Uni)

What do we understand of the properties of the dense ionised matter found in the interiors of low-mass stars and giant planets? The 147th IAU Colloquium gathered together international experts to address this question, and their answers are provided in this opportune proceedings. In this volume, reviews by world experts in plasma and dense matter physics and in stellar astrophysics cover everything from the cooling theory of white dwarfs and their accretion-induced collapse through to the internal structure of low-mass stars and giant planets. They cover a wide range of topics related to the equation of state in dense matter, from the fundamental basis of the iV-body problem to astrophysical applications. Together these articles provide an essential review of the most recent achievements in the field and give direction for future research, for graduate students and researchers

The Equation of State in Astrophysics

IAU Astronomical Union Union Astronomique International The following Colloquia of the International Astronomical Union are published for the Union by Cambridge University Press.

82. Cepheids. Edited by Barry F. Madore. 0 521 30091 6. 1985 91. History of Oriental Astronomy. Edited by G. Swamp, A. K. Bag and K. S. Shukla. 0 521 34659 2. 1987 92. Physics of Be Stars. Edited by A. Slettebak and T. P. Snow. 0 521 33078 5. 1987 101. Supernova Remnants and the Interstellar Medium. Edited by R. S. Roger and T. L. Landecker. 0 521 35062 X. 1988 105. The Teaching of Astronomy. Edited by Jay M. Pasachoff and John R. Percy. 0 521 35331 9. 1990 106. Evolution of Peculiar Red Giant Stars. Edited by Hollis Johnson and Ben Zuckerman. 0 521 36617 8. 1989 111. The Use of Pulsating Stars in Fundamental Problems of Astronomy. Edited by Edward G. Schmidt. 0 521 37023 X. 1989 136. Stellar Photometry - Current Techniques and Future Developments. Edited by C. J. Butler and I. Elliott. 0 521 41866 6. 1993 139. Stellar Pulsation and Pulsating Variable Stars. Edited by James M. Nemec and Jaymie M. Matthews. 0 521 44382 2. 1993 143. The Sun as a Variable Star. Edited by J. M. Pap, C. Frohlich, H. S. Hudson and S. K. Solanki. 0 521 42006 7. 1994 145. Supernovae and Supernova Remnants. Edited by Richard McCray. 0 521 46080 8. 1994 147. The Equation of State in Astrophysics. Edited by Gilles Chabrier and Evry Schatzman. 0 521 47260 1. 1994

The Equation of State in Astrophysics Proceedings of IAU Colloquium No. 147 Saint-Malo, France 14-18 June 1993

Edited by Gilles Chabrier Ecole Normale Superieure de Lyon and Evry Schatzman Observatoire de Meudon

Hi CAMBRIDGE UNIVERSITY PRESS

Published by the Press Syndicate of the University of Cambridge The Pitt Building, Trumpington Street, Cambridge CB2 1RP 40 West 20th Street, New York, NY 10011^1211, USA 10 Stamford Road, Oakleigh, Melbourne 3166, Australia © Cambridge University Press 1994 First published 1994 Printed in Great Britain at the University Press, Cambridge A catalogue record for this book is available from the British Library Library of Congress cataloguing in publication data available ISBN 0 521 47260 1 hardback

Contents

Group photograph List of participants Preface

X

xi xvi

Reviews 1 Equations of state in stellar structure and evolution H.M. Van Horn

page 1

2 Equation of state of stellar plasmas F.J. Rogers

page 16

3 Statistical mechanics of quantum plasmas. Path integral formalism A. Alastuey

page 43

4 Onsager-molecule approach to screening potentials in strongly coupled plasmas Y. Rosenfeld

page 78

5 Astrophysical consequences of the screening of nuclear reactions J. hern and M. Hernanz

page 106

6 Crystallization of dense binary ionic mixtures. Application to white dwarf cooling theory R. Mochkovitch and L. Segretain page 126 7 Non crystallized regions of White dwarfs. Thermodynamics. Opacity. Turbulent convection /. Mazzilelli page 144 8 White dwarf crystallization E. Garcia-Berro and M. Hernanz

page 161

9 Gravitational collapse versus thermonuclear explosion of degenerate stellar cores J. Isern and R. Canal

page 186

10 Neutron star crusts with magnetic fields D.G. Yakovlev and A.D. Kaminker

page 214

11 High pressure experiments for astrophysics P. Loubeyre

page 239

12 Equation of state of dense hydrogen and the plasma phase transition; A microscopic calculational model for complex fluids F. Perrot and C. Dharma-wardana

page 272

13 The equation of state of fluid hydrogen at high density G. Chabrier

page 287

14 A comparative study of hydrogen equations of state D. Sanmon

page 306

15 Strongly coupled ionic mixtures and the H/He equation of state H.M. DeWitt

page 330

16 White dwarf seismology: Influence of the constitutive physics on the period spectra G. Fontaine and P. Brassard page 347 17 Helioseismology: the Sun as a strongly-constrained, weakly-coupled plasma W. Da'ppen

page 368

18 Transport processes in dense stellar plasmas N. Itoh

page 394

Vll

Vlll

Contents

19 Cataclysmic variables: structure and evolution J.-M. Hameury

page 420

20 Giant planet, brown dwarf, and low-mass star interiors W.B. Hubbard

page 443

21 Searches for brown dwarfs J. Liebert

page 463

22 Jovian seismology B. Mosser

page 481

Observational projects 23 EVRIS: first space experiment devoted to stellar seismology A. Baglin

page 512

24 The HIPPARCOS mission and tests for the equation of state A. Baglin and Joao Fernandes

page 517

25 Ground based heliosismology: IRIS and GONG F.-X. Schmider

page 525

26 The spatial GOLF project S. Turck-Chieze

page 532

27 The DENIS survey T. Forveille

page 537

28 PRISMA: A mission to study interior and surface of stars P. Lemaire

page 540

Posters 29 Towards a helioseismic calibration of the equation of state in the solar convective envelope S. V. Vorontsov, V. A. Baturin, D. 0 . Gough, W. Dappen page 545 30 Thermal cyclotron and annihilation radiation in strong magnetic fields V.G. Bezchastnov and A.D. Kaminker

page 550

31 Modified adiabatic approximation for a hydrogen atom moving in a magnetic field V.G. Bezchastnov and A.Y. Potekhin

page 555

32 Computations of static white dwarf models: A must for asteroseismological studies P. Brassard and G. Fontaine

page 560

33 The Chandrasekhar mass of a gravitating electron crystal D.Engelhardt and I. Bues

page 565

34 Coulomb corrections in the nuclear statistical equilibrium regime D. Garcia and E. Bravo

page 571

35 Molecular Opacities: Application to the Giant Planets T. Guillot, D. Gautier and G. Chabrier

page 576

36 On Radiative Transfer Near the Plasma Frequency at Strong Coupling Yu. K. Kurilenkov and H.M. Van Horn

page 581

37 Effects of Superfluidity on Spheroidal Oscillations of Neutron Stars Umin Lee, T.J.B. Collins, R.I. Epstein and H.M. Van Horn

page 586

38 Magnetic Field Decay in the Non-superfluid Regions of Neutron Star Cores A. G. Muslimov and H. M. Van Horn

page 591

Contents

ix

39 On the equation of state in Jovian seismology J. Provost, B. Mosser and G. Chabrier

page 596

40 Analysis of the screening formalisms in solar and stellar conditions H. Dzitko, S. Turck-Chieze, P. Delbourgo-Salvador and Ch. Lagrange

page 601

41 Theoretical Description of the Coulomb Interaction by Pade-Jacobi Approximants W. Stolzmann and T. B locker

page 606

42 New Model Sequences from the White Dwarf Evolution Code M. Wood

page 612

43 Low temperature opacities C. Neuforge

page 618

LIST OF PARTICIPANTS

Numbers in parentheses refer to the group photograph

A. ALASTUEY, (58)

J. M. APARICIO, (37)

W. APPEL, (14)

A. BAGLIN, (2)

I. BARAFFE, (31)

V. BEZCHASTNOV, (61)

T. BLOCKER,

H. BOFFIN, (22)

P. BRASSARD, (46)

Laboratoire de Physique, Ecole Normale Superieure de Lyon, 69364 Lyon Cedex 07 - FRANCE [email protected] C.E.A. de Blanes, 17300 Blanes - SPAIN aparicio@ceab. es Laboratoire de Physique, Ecole Normale Superieure de Lyon, 69364 Lyon Cedex 07 - FRANCE wappel@physique. ens-lyon.fr DASGAL, Observatoire de Meudon, 92195 Meudon Cedex - FRANCE [email protected] Laboratoire de Physique, Ecole Normale Superieure de Lyon, 69364 Lyon Cedex 07 - FRANCE ibaraffe@physique. ens-lyon.fr Ioffe Institute of Physics and Technology, 194021 St Petersburg - RUSSIA [email protected] Institiit fiir Theoretische Physik und Sternwarte, Universitat Kiel, 2300 Kiel - GERMANY pas96 @rz. uni-kiel. dbp. de Institut d'Astrophysique, Universite Libre de Bruxelles, 1050 Brussels - BELGIUM [email protected] Departement de Physique, Universite de Montreal, H3C 3J7 Montreal, Quebec - CANADA brassard@astro. umontreal. ca

XI

List of participants I. BUES, (56)

G. CHABRIER, (11)

C. CHARBONNEL, (13) F. D'ANTONA, (40)

W. DAPPEN, (27)

H. E. DE WITT, (20)

W. EBELING,

D. ENGELHARDT,(60)

J. FERNANDES, (55)

G. FONTAINE, (45)

A. FORSTER, (17)

E. GARCIA-BERRO, (8)

Remeis-Sternwarte Bamberg, D-8600 Bamberg - GERMANY [email protected] Laboratoire de Physique, Ecole Normale Superieure de Lyon, 69364 Lyon Cedex 07 - FRANCE [email protected] Observatoire de Geneve, CH-1290 Sauverny - SWITZERLAND Osservatorio Astronomico di Roma, 00040 Monte Porzio - ITALIA [email protected] and [email protected] Department of Physics and Astronomy, University of Southern California, 90089-1342 Los Angeles, California - USA [email protected] Lawrence Livermore Laboratory, 94550 Livermore, California - USA [email protected] Hiimboldt Universitdt, Sektion Physik, 1040 Berlin - GERMANY [email protected] Dr. Remeis Sternwarte, 8600 Bamberg - GERMANY [email protected] DASGAL, Observatoire de Meudon, 92195 Meudon Cedex - FRANCE [email protected] Departement de Physique, Universite de Montreal, H3C 3J7 Montreal, Quebec - CANADA [email protected] Institiit fur Theoretische Physik, Hiimboldt Universitat, 1040 Berlin - GERMANY [email protected] Universidad Politecnica de Barcelona, Dept Fisica, 08031 Barcelona - SPAIN [email protected]

Xll

List of participants Institut d'Estudis Catalans, Laboratori d'Astrofisica, Barcelona - SPAIN dgarcia@fen. upc. es Centre d'Etudes de Limeil-Valenton, D. GILLES, (36) 94195 Villeneuve-Saint-Georges - FRANCE gilles@limeil. cea.fr J. F. GONZALEZ, (12) Laboratoire de Physique, Ecole Normale Superieure de Lyon, 69364 Lyon Cedex 07 - FRANCE [email protected] DASGAL, Observatoire de Meudon, M. J. GOUPIL, (43) 92195 Meudon Cedex - FRANCE goupil@mesiob. obspm. circe.fr Institute Physics of the Earth, T. V. GUDKOVA, (6) Russian Academy of Sciences, 123810 Moscow - RUSSIA [email protected] T. GUILLOT, (51) Observatoire de Nice, 06304 Nice Cedex 4 - FRANCE [email protected] nice.fr J. GUTIERREZ-CABELLO, Facultad de Fisica, Dipartamento di Astronomia, 08028 Barcelona - SPAIN [email protected] Observatoire de Strasbourg, J. M. HAMEURY, (34) 67000 Strasbourg - FRANCE [email protected] L.P.L., University of Tucson, W. B. HUBBARD, (44) 85721 Tucson, Arizona - USA [email protected] C.S.I.C., cami de Santa Barbara, J. ISERN, (5) 17300 Blanes - SPAIN [email protected] N. ITOH, (4), Mrs ITOH, (3) Department of Physics, Sophia University, 102 Tokyo - JAPAN [email protected] D. GARCIA-SENZ,

Xlll

List of participants T. KAHLBAUM, (49)

A. KOVETZ, (41)

P. 0 . LAG AGE, U. LEE, (25)

P. LEMAIRE, J. LIEBERT, (23)

P. LOUBEYRE, (15)

I. LOPEZ, (35)

G. MASSACRIER, (32)

J. MATIAS, (42)

I. MAZZITELLI, (57)

R. MOCHKOVITCH,

WIP, Arbeitsgruppe Nieder Temperaturplasmaphysik, 2200 Greifswald - GERMANY Department of Geophysics and Planetary, Sciences, Tel Aviv University, 69978 Tel Aviv - ISRAEL attay@etoile. tau.ac. il CEA - CEN Saclay, 91191 Gif-sur-Yvette Cedex - FRANCE Laboratoire de Physique, Ecole Normale Superieure de Lyon, 69364 Lyon Cedex 07 - FRANCE [email protected] IAS, Universite Paris 11, 91405 Orsay Cedex - FRANCE Steward Observatory, University of Tucson, 85721 Tucson, Arizona - USA [email protected] Universite Paris 6 Laboratoire de Physique des Milieux Condenses 75252 Paris - FRANCE DAPNIA, CEN Saclay, 91191 Gif-sur-Yvette Cedex - FRANCE [email protected] Laboratoire de Physique, Ecole Normale Superieure de Lyon, 69364 Lyon Cedex 07 - FRANCE [email protected] Observatoire de Toulouse, 31400 Toulouse - FRANCE [email protected] CNR, Instituto di Astrofisica Spaziale, 00044 Frascati (Rm) - ITALIA italo@irmias. ias.fra. cnr. it LA.P., 75014 Paris - FRANCE [email protected]

XIV

List of participants B. MOSSER, (33)

N. MOWLAVI,

C. NEUFORGE, (21)

A. PEREZ, (59)

F. ROGERS, (50)

Y. ROSENFELD, (48)

D. SAUMON, (47)

E. SCHATZMAN, (10)

M. SCHLANGES, (19)

F. X. SCHMIDER,

L. SEGRETAIN, (16)

G. SHAVIV, (38)

I.A.P., 75014 Paris - FRANCE [email protected] Universite Libre de Bruxelles, 1050 Brussels - BELGIUM nmowlavi@astro. ulb. ac. be Universite de Liege, Institut d'Astrophysique de Liege, 4000 Liege - BELGIUM Laboratoire de Physique, Ecole Normale Superieure de Lyon, 69364 Lyon Cedex 07 - FRANCE [email protected] Lawrence Livermore Laboratory, 94550 Livermore, California - USA [email protected] Physics Department, Nuclear Research Center-Negev, 84190 Beer-Sheva - ISRAEL L.P.L., University of Tucson, 85721 Tucson, Arizona - USA [email protected] DASGAL, Observatoire de Meudon, 92195 Meudon Cedex - FRANCE [email protected] Universitat Greifswald, Fachrichtung Physik, 2200 Greifswald - GERMANY Universite de Nice, Departement d'Astrophysique, 06108 Nice Cedex 2 - FRANCE Laboratoire de Physique, Ecole Normale Superieure de Lyon, 69364 Lyon Cedex 07 - FRANCE [email protected] Technion-Israel, Institute of Technology 32000 Haifa - ISRAEL [email protected]

XV

List of participants C. STEHLE, (30)

W. STOLZMANN, (39)

J. TULLY, (1)

S. TURCK-CHIEZE,

H. M. VAN HORN, (53)

G. VAUCLAIR, (52)

S. VORONTSOV, (26)

A. WEISS, M. A. WOOD, (9)

G. WUCHTERL, (62)

D. YAKOVLEV, (54)

V. N. ZHARKOV, (7)

DAMAP, Observatoire de Meudon, 92195 Meudon Cedex - FRANCE [email protected] Instittit fiir Theoretische Physik und Sternwaxte, Universitat Kiel, 2300 Kiel - GERMANY pas96@rz. uni-kiel. dbp. de Observatoire de Nice, 06304 Nice Cedex 4 - FRANCE [email protected] CEA - CEN Saclay, 91191 Gif-sur-Yvette Cedex - FRANCE turck@frsacl 1 .bitnet Department of Physics and Astronomy, University of Rochester, 14627 Rochester, New York - USA hmvh@uordbv. bitnet Observatoire de Toulouse, 31400 Toulouse - FRANCE vauclair@fromp 51. bitnet Astronomy Unit, Queen Mary and Westfield College, E l 4NS London - ENGLAND [email protected] Max Planck Institute fiir Astrophysik, 8046 Garching - GERMANY Florida Institute of Technology, 32901-6988 Melbourne, Florida - USA [email protected]. fit. edu Universitat Heidelberg, Instittit fiir Theoretische Astrophysik, 6900 Heidelberg - GERMANY [email protected] Ioffe Institute of Physics and Technology, 194021 St Petersburg - RUSSIA [email protected] Institute Physics of the Earth, Russian Academy of Sciences, 123810 Moscow - RUSSIA [email protected] (att. Prof. Zharkov)

XVI

PREFACE In the Internal Constitution of the Stars, published in 1926, Eddington gave a central temperature of white dwarfs of several billions degrees. The Fermi-Dirac statistics appeared just one year later, in 1927, and the new equation of state for degenerate matter provided the explanation of white dwarfs. The solution of the problems we have to consider presently are probably not relevant of the same kind of intellectual jump. But who knows! Anyhow, the time of the perfect gas law is definitely over. It is possible, in many cases, to get astrophysical orders of magnitude, using simple or oversimple relations betwen physical quantities. However, modeling correctly observational results has become, nowaday, more and more difficult. Data are of a better precision and provide more information, would it be chemical abundances, evolutionary tracks or those wonderful helioseismological data. An elementary statement is that, in order to look Inside the Stars (the title of a recent colloquium), we need more accurate descriptions of basic physical laws: equation of state, opacities, thermonuclear reactions. We are still facing many difficulties in the field, and we can give a few examples: we do not have a theory which decribes consistently both the equation of state of a plasma and the level population of the atoms; we still have to improve the theory of screening effects in dense plasmas; we have now a description of cold, dense, weakly ionized matter of brown dwarfs, but it is still incomplete. And this is not the end of the list! There has not been many meetings devoted to the equation of state. The 1978 meeting on dense matter in Paris dealt with several aspects of the equation of state, but the present meeting represents an important new landmark. It provides the link between the astrophysical problems related to the equation of state and the underlying most recent physical theories. It is not necessary to give here the list of the reviews: they can be found in the table of contents. Let just say that most regions in the pressure-temperature diagram have been covered by the present review papers. From main sequence stars to white dwarfs, from brown dwarfs to giant planets, from supernovae to neutron stars (ultra high density being excluded), most aspects of the equation of state have been considered. The connections with new observational results (HIPPARCOS, GOLF, PRISMA) have also been presented. Our knowledge of the equation of state will certainly have to be improved within the next few years in order to obtain the best description of a large amount of new observational results. We hope the present conference will help reaching this goal. We are very endebted to the different speakers for their excellent reviews, as well in the oral as in the written presentations. We are grateful to the Commission of the European Communities, the 'Centre National de la Recherche Scientifique (CNRS)', the 'Ministire de la Recherche el de la Technologie' and the International Astronomical Union for granting financial support for the organization of the conference and the edition of the present proceedings. We would like to express our special thanks to Marie-Pierre Fuchs (who unfortunately could not come to St-Malo) for carrying efficiently the practical organization and to the whole staff of the Palais du Grand Large for their efficiency and their warm hospitality. Finally we express our warmest thanks to all the participants who made this a fruitful, stimulating and enjoyable meeting.

Evry Schatzman and Gilles Chabrier February

XV11

1994

PREFACE Dans Internal Constitution of the Stars, publid en 1926, Eddington donnait une temperature centrale pour les naines blanches de plusieurs milliards de degree. La statistique de Fermi-Dirac apparut un an plus tard, en 1927, et la nouvelle Equation d'dtat pour la matiere ddgdndrde fournit 1'explication des naines blanches. La solution des problemes que Ton considere aujourd'hui ne relive probablement pas du meme genre de revolution intellectuelle. Mais qui sait! II est possible, dans la plupart des cas, d'obtenir des ordres de grandeur astrophysiques en utilisant des relations simples entre grandeurs physiques. Une modelisation correcte des rdsultats observationnels, cependant, est devenue de nos jours de plus en plus difficile. Les donndes sont de plus en plus prdcises et fournissent de plus en plus d'informations, que ce soit les abondances chimiques, les trace's devolution ou ces superbes donndes d'hdliosismologie. II est trivial de constater que, afin de voir Inside the Stars (titre d'un congres recent), nous avons besoin de descriptions de plus en plus correctes des outils physiques de base : Equation d'dtat, opacity's, reactions thermonucldaires. II reste encore beaucoup de difficulty's dans le domaine, et nous pouvons en donner quelques exemples : il n'y a toujours pas de thdorie qui ddcrive de facon cohdrente a la fois l'dquation d'dtat d'un plasma et la population des niveaux atomiques; il nous faut encore amdliorer la thdorie des effets d'dcran dans un plasma; nous avons ddsormais une description de la matiere froide, dense, partiellement ionisde dans les naines brunes, mais elle est encore incomplete. Et ce n'est pas une liste exhaustive! II n'y a pas eu beaucoup de congres ayant pour theme l'dquation d'dtat. La conference de 1978 sur la mati£re dense a Paris traitait de plusieurs aspects de l'dquation d'dtat, mais la prdsente conference reprdsente une marque importante, car elle fournit le lien entre les problemes astrophysiques lids a l'dquation d'dtat et les theories physiques sous-jacentes les plus rdcentes. II n'est pas ndcessaire de donner la liste des revues: elles sont donndes dans la table des matieres. Disons simplement que la plupart des rdgions du diagramme densitd-tempdrature sont couvertes par les revues de ce recueil. Des dtoiles de la Sdquence Principale aux naines blanches, des naines brunes aux planetes gdantes, des supernovae aux dtoiles a neutrons (en excluant la matiere ultra-dense), la plupart des aspects de l'dquation d'dtat ont dtd considdrds. Les liaisons avec les nouveaus rdsultats observationnels (HIPPARCOS, GOLF, PRISMA) sont dgalement prdsentdes. Notre connaissance de l'dquation d'dtat devra certainement Stre amdliorde dans les prochaines anndes afin d'obtenir une description correcte des nombreux nouveaux rdsultats observationnels. Puisse le prdsent recueil etre utile a un tel but. Nous sommes extremement reconnaissants aux diffdrents confdrenciers pour leurs excellentes revues, tant dans la prdsentation orale qu'dcrite. Nous remercions la Commission des Communautis Europiennes, le Centre National de la Recherche Scientifique, VUnion Astronomique Internationale et le Ministere de la Recherche et de la Technologie pour leur soutien financier dans l'organisation de la confdrence et l'ddition du prdsent recueil. Nous voudrions exprimer plus particulierement nos remerciements a Marie-Pierre Fuchs (qui malheureusement n'a pas pu venir a St-Malo) pour son efficacitd dans ('organisation pratique de la confdrence, ainsi qu'a tout le personnel du Palais du Grand Large pour son efficacitd et sa cordiale hospitalitd. Enfin nous exprimons nos plus sinceres remerciements a tous les participants, qui ont fait de cette confdrence une rdunion fructueuse, stimulante et sympathique.

Evry Schatzman et Gilles Chabrier Fdvrier 1994

XV111

Equations of State in Stellar Structure and Evolution H. M. VAN HORN Department of Physics and Astronomy, C. E. Kenneth Mees Observatory, and Laboratory for Laser Energetics, University of Rochester, Rochester, NY 14627-0011. Current address: Division of Astronomical Sciences, Room 1045, National Science Foundation, 4201 Wilson Boulevard, Arlington, VA 22230

Abstract In this paper I summarize some of the recent advances in studies of dense matter. Research on phase separation in the binary ionic mixtures (BIMs) that constitute the matter in white dwarfs has been motivated by the need to obtain accurate estimates for the ages of the faintest white dwarfs and thus of the disk of our Galaxy. Substantial age increases appear possible, but it is not yet clear whether such large increases occur in real white dwarfs. A second advance is the prediction, based on state-of-the-art physical calculations, that ionization of H at low temperatures and increasing densities may occur via a first-order "plasma phase transition" (PPT). Astrophysical consequences of this result are still being explored in an effort to test this prediction. Related to these equation-of-state calculations are calculations of the enhancement of nuclear reaction rates at high densities. New thermonuclear rates have been computed for C+C reactions in BIMs, although there is currently some controversy about results at the highest densities. New pycnonuclear reaction rates have also been calculated for BIMs, and it has been suggested that He-burning at T — 0 may occur through a first-order phase transition. Finally, calculations of the equation of state of matter in strong magnetic fields and of radiative opacities at high densities have undergone very recent and substantial improvements, which are just beginning to be utilized in astrophysical calculations. Dans cette revue, je resumme les travaux les plus recents dans l'etude

2

Van Horn: Stellar structure and evolution

de la matiere dense. Les recherches sur la separation de phase dans les melanges ioniques binaires qui constituent l'interieur des naines blanches ont ete motivees par la necessite d'obtenir une estimation correcte de Page des naines blanches les moins brillantes et done du disque de notre Galaxie. II apparait possible d'accroitre l'age substantiellement, mais il n'est pas encore certain qu'un tel accroissement ait reellement lieu dsans les naines blanches. Une seconde percee est la prediction, basee sur des calculs trs detailles, que l'ionisation de H a basse temperature et haute pression ait lieu au travers d'une "transition de phase plasma (PPT) ". Les consequences astrophysiques de ce resultat sont actuellement encore en cours d'etude, afin de tester cette prediction. En relation avec ces equations d'etat, ont lieu des calculs d'accroissement de taux de reactions nucleaires a haute densite. De nouveaux taux termonucleaires ont ete calcules pour la reaction C+C dans les melanges binaires, bien au'il y ait actuellement une controverse quant a ces resultats. De nouveaux taux pycnonucleaires ont ete calcules pour les melanges, et il a ete suggere que le brulage de He a T = 0 ait lieu au travers d'une transition de phase du premier ordre. Enfin, des calculs de l'equation d'etat de la matiere dans les champs magnetiques forts et des opacites radiatives a haute densite ont ete effectues recemment.

1.1 Introduction The purpose of this paper is to summarize various problems related to the equations of state (EOS) used in stellar structure and evolution calculations. I shall be concerned with quantities such as the EOS itself, various transport coefficients, nuclear reaction rates, and so forth. I shall endeavor to summarize some of the most recent advances in these areas, but it is not possible here to cover all topics of interest. In particular, there have been some interesting recent calculations of the EOS of matter in strong magnetic fields which are mentioned only briefly at the end of this paper. Let me begin with a reminder of the parameters needed for stellar evolution calculations. The four equations describing the time-dependent structure of a star are (c/. Clayton 1968, pp. 436 fF.): dr GMr

dP dr

P r2 '

dT

3 up

dr

3

(2) Lr

4acT 4irr2'

Van Horn: Stellar structure and evolution

and , OS x

/. v

lr)

(4)

In order to solve these equations, we need to know the following properties of stellar matter: the pressure P = P(p,T), the entropy 5 = S(p,T), the opacity K = n(p,T), and the net nuclear energy generation rate e = e(p,T). The relations giving the dependence of the pressure and entropy upon the thermodynamic state variables p, the mass density, and T , the temperature, are collectively called the "equation of state." Quantities such as the radiative opacity K and the so-called "conductive opacity" /c,., proportional to the inverse of the thermal conductivity, are termed transport coefficients; they determine the rate of transport of some physical quantity, such as heat. For some purposes, we also need to know the viscosity (which controls the rate of transport of momentum) or the ambipolar diffusion coefficients (which govern the rate of gravitational separation of elements in a star); for the most part I shall neglect these quantities in this brief review. The thermonuclear and pycnonuclear (density-induced) reaction rates e and the neutrino loss rates ev round out the list of physical quantities needed. It is also useful to have some idea of the values of the dimensionless physical parameters that characterize the state of matter at different temperatures and densities. Van Horn (1991) has illustrated this for plasmas composed of H or of Fe. Here we briefly describe the changes in the physical conditions, beginning at low p and at thermal energies kT much less than the ionization potentials of the atoms. Under these conditions, matter consists of an ideal gas of neutral atoms or molecules. As the temperature increases, but still at low densities, the molecules first dissociate into neutral atoms, and the atoms subsequently ionize. At sufficiently high T, matter consists of a fully ionized plasma of bare atomic nuclei and electrons. Under these conditions, matter can still be described as a mixture of ideal gases (the ions and electrons). If the density is now increased along some isotherm, new, density-dependent effects begin to appear. As the electrons in the plasma become crowded increasingly close together, degeneracy first becomes important. This occurs when kT ~ ep, where €p is the Fermi energy. At still higher densities, the Coulomb interaction energy (Ze)2/a between the electrons and ions becomes comparable to kT. Here a is a measure of the average separation of electrons and ions; it is of the order of the size of the "Wigner-Seitz sphere" containing just enough electrons to neutralize the ionic charge. The quantity F = (Ze)2/akT is the dimension-

4

Van Horn: Stellar structure and evolution

less parameter generally used to characterize the strength of the Coulomb interaction. The condition T << 1 corresponds to a high-temperature, low-density, fully ionized plasma. If we continue our tour through the (p,T) plane along an isochore at very high density but decreasing temperature, the parameter F continually increases. This corresponds to the increasing importance of the Coulomb interactions. At sufficiently low T, coresponding to F « 180, the plasma actually freezes into a crystalline lattice structure. This was first discovered more than 25 years ago as a result of numerical simulations of the properties of the idealized "one-component plasma" (OCP), consisting of a single species of ions in a uniform neutralizing background. Advances in computer technology have now permitted Monte Carlo simulations extending over tens or hundreds of millions of configurations for thousands of particles of several different species and have greatly improved the precision of our knowledge of the thermodynamic properties of very dense plasmas. Much of the discussion in the remainder of this paper concerns results we have learned from such investigations. Indeed, for "binary ionic mixtures" (BIMs), consisting of two species of ions in a uniform background, much of the recent discussion has focused on calculations of the phase diagrams of the mixtures and the investigation of the possible astrophysical consequences of phase separation. In §2 below, I first discuss the implications of the phase diagrams of various mixtures for the ages of the coolest white dwarf stars. Next, in §3, I consider the so-called "plasma phase transition," which has been suggested as the process by which the transition from low-density, molecular H2 to high-density metallic hydrogen occurs. I describe some recent work on nuclear reaction rates at .high densities in §4, and in §5 I conclude with a list of some important problems that still need attention. 1.2 EOS I: Phase Separation in White Dwarfs and the Age of the Galaxy Much of the reason for the current interest in the problem of phase separation in white dwarfs stems from the prospect of using white dwarf cooling theory to measure the age of the disk of our Galaxy. This method is based on the observed shortfall in the white dwarf luminosity function at logX/i© « -4.5 (c/. Liebert, Dahn, and Monet 1988); the "luminosity function" is effectively the number density of stars per unit interval of log L/LQ. Using the best existing white dwarf cooling calculations, the ages of the faintest white dwarfs are found to be ~ 9 Gyr (Winget et al. 1987;

Van Horn: Stellar structure and evolution

5

Iben and Laughlin 1989). This result is surprising, because it is much less than the ages of the oldest globular cluster stars, which lie in the range from ~ 12 to 18 Gyr (c/. Norris and Green 1989). This disagreement has important astrophysical consequences, because it implies either (1) that the ages of the white dwarfs are wrong because of unanticipated effects that lengthen the white dwarf cooling times or (2) that we require a new paradigm for galaxy formation. The possibility that a new model for galaxy formation may be needed is the more interesting one, as it would have profound implications for many different areas of astrophysics. There are even some indications that this may be the case (Blitz 1993). For example, Larson's early theoretical calculations of pressure-supported collapse (Larson 1976, cited by Norris and Green 1989) indicate that the disk of our Milky Way Galaxy may have formed from the inside out, with a lapse of several billion years between the beginning of star formation in the central bulge and the onset of star formation out at the ~ 10 kpc distance of the Sun from the galactic center. In addition, some observations (Sarajedini and King 1989, Lee 1992, van den Bergh 1993) appear to indicate a real spread in the ages of the globular clusters. If this is the case, it supports a gradual collapse model for the formation of the Galaxy. However, before one can uncritically accept the idea that a new model for galaxy formation is demanded by the evidence, it is important to investigate the alternative. In particular, are there additional energy sources that can "turn on" late in the evolution of a white dwarf and prolong the cooling times? One possibility that has been suggested to do precisely this is phase separation. The idea is that as a white dwarf cools to the point at which its center begins to freeze, it may undergo phase separation into, e.g., a less dense fluid and a denser solid. If this occurs, the solid can "snow out," falling toward the center of the white dwarf and liberating gravitational potential energy. If the density contrast is sufficiently great, the energy liberated can be substantial and may prolong the white dwarf cooling times significantly. It should also be noted, however, that this additional energy release produces a bump in the white dwarf luminosity function at low luminosities (Garcia-Berro et al. 1988); the prolongation of the cooling times means that more white dwarfs "pile up" in the lowest luminosity intervals. This effect is not seen in the observational luminosity function. However, the error bars are so large at the critical, low-luminosity end of the range that this effect might not have been detected, even if if it is present. White dwarfs are thought to consist mainly of C and 0 , and Stevenson (1980) originally suggested that the phase diagram of the C/O mixture

6

Van Horn: Stellar structure and evolution

might have either a eutectic or a spindle form. If the phase diagram were of the eutectic type, the density contrast between the two phases would be substantial, and a considerable prolongation of the cooling times would occur. Conversely, if the phase diagram were of the spindle type, the density contrast would be much smaller, and a correspondingly smaller age change would result. The importance of this issue led two groups independently to investigate this question using the best modern calculations. Bar rat, Hansen, and Mochkovitch (1988) performed a density-functional calculation and found a spindle-shaped phase diagram. Ichimaru, Iyetomi, and Ogata (1988) carried out an analysis based on their very high-precision Monte Carlo calculations and found an azeotropic phase diagram (that is, a phase diagram that resembles a spindle shape except at low 0 abundances). In both cases, very little density contrast was produced upon phase separation, and the age-extension inferred for the oldest white dwarfs was found to be only At ~ 0.5 Gyr). Another possibility that has been considered is C/Ne phase separation (Garcia-Berro et al. 1988). Although the cosmic abundance of Ne is too small to produce a significant effect, the Ne abundance in white dwarfs is thought to be produced by nucleosynthesis from the primordial CNO elements during the prior stages of evolution and to be considerably larger than the cosmic abundance. The CNO elements incorporated into stars at birth are all converted primarily to 14N during H-burning on the main sequence. The 14N is subsequently converted into 22Ne during He-burning in the red-giant phase, with an abundance as large as X^e ~ 0.02. This can in principle lead to significant energy release associated with C/Ne phase separation late in the cooling of a white dwarf. Preliminary estimates by Garcia-Berro et al. gave varying age-extensions At ranging up to several Gyr, depending upon the C/O ratio. To investigate further the effects of phase separation involving trace elements, Segretain and Chabrier (1993) and Ogata et al. (1993) have recently completed independent calculations of the phase diagrams of binary ionic mixtures (BIMs) involving various trace species. Segretain and Chabrier employed density-functional-based calculations, while Ogata et al. employed high-precision Monte Carlo calculations. Ogata et al. (1993) found spindle-shaped and azeotropic phase diagrams for low values of the charge ratio Z2/Z1, such as the values appropriate for O/Ne or C/O BIMs, while the phase diagram assumes a eutectic form for larger values of Z2/Z1, as appropriate for C/Ne or C/Fe BIMs. Segretain and Chabrier (1993) obtained results that differ in detail but which are qualitatively very similar

Van Horn: Stellar structure and evolution

1

to those of Ogata et al. This similarity provides considerable confidence that the general features of both sets of results are reliable. Ogata (1993) has recently performed the first detailed stellar evolution calculations that include C/Ne phase separation self-consistently. His computations show that the age of a 0.6 M© C white dwarf with X^e = 0.02 at log L/LQ = —4.5 increases by 5.7 Gyr when C/Ne phase separation is included. If it is not washed out by other effects, this is large enough to resolve much of the discrepancy between the white dwarf ages and those of the globular cluster stars. However, the primordial CNO abundances are considerably less than X^e = 0.02, and since the age-extension At scales approximately linearly with these abundances it is not yet dear that the ages of the coolest white dwarfs actually are substantially increased. In addition, while Ogata and his collaborators have not yet computed the white dwarf luminosity function associated with their evolutionary calculations, from rather general considerations one expects a substantial bump. Averaging over the pre-white-dwarf mass distribution may flatten this bump somewhat, but these computations remain to be completed. In two recent papers, Segretain et al. (1994) and Hernanz et al. (1994) carefully examined the effect of the different crystallization processes on the cooling time and on the luminosity function of white dwarfs. Their calculation of the luminosity function takes into account the white dwarf mass distribution function and a model for the chemical evolution of the Galaxy. These authors found out that the crystallization of trace elements alter severely the cooling time of the star, but do not modify significantly the position of the cut-off in the luminosity function, a direct consequence of the strong dependence of metallicity upon galactic evolution. 1.3 EOS II: Dense Hydrogen in Brown Dwarfs and Giant Planets A second arena in which our understanding of astrophysical equations of state has recently undergone a considerable advance concerns the properties of dense H and in particular an improvement in our understanding of the manner in which the transition from neutral to ionized hydrogen occurs at low temperatures with increasing density ("pressure ionization"). There has been a long history of interest in the physics of this problem, and the astrophysical applications — especially to the interiors of the giant planets — have long been recognized. The problem of the equation of state of dense H attracted renewed interest in the early 1980s with (1) the advent of experimentally attainable conditions approaching those needed for

8

Van Horn: Stellar structure and evolution

the pressure ionization (or equivalently the "metalization") of H; (2) the theoretical predictions that pressure ionization may occur via a first-order "plasma phase transition" (PPT); and (3) the flurry of astrophysical interest in the properties of "brown dwarf stars. It is beyond the scope of this paper to summarize all of the recent work on this problem, and I hope I may be forgiven for limiting myself to a few cases with which I am best acquainted. Several groups have recently produced new equations of state for dense H (see later papers in these proceedings). Among these, the calculations by Saumon (1989) and Saumon and Chabrier (1989) stand out in my opinion in that they have extended realistic EOS calculations to sufficiently high densities to encounter pressure ionization. They found, as had also been suggested by earlier, less detailed calculations (Robnik and Kundt 1983; Kraeft et al. 1986), that this transition appears to occur through a firstorder "plasma phase transition." The theoretical prediction of the existence of the PPT by Saumon and Chabrier has not yet been confirmed experimentally, however. For this reason, there have been several recent efforts to find astrophysical tests of this result. In one effort, Chabrier, Saumon, and their colleagues have explored the consequences of the PPT for the structure and evolution of the giant planets and brown dwarfs (Chabrier et al. 1992; Saumon et al. 1992). While the effects they found were not large, they did discover that models including the first-order PPT were in slightly better agreement with the observational constraints than were models without the PPT. A somewhat more promising test may involve studies of the effects of the PPT upon the oscillation modes of the giant planets. In the past several years, various groups have found observational evidence for periodic or quasiperiodic phenomena on Jupiter (Magalhaes et al. 1989, 1990; Deming et al. 1989; Schmider and Mosser 1990; Schmider, Mosser, and Fossat 1991). Lee, Strohmayer, and Van Horn (1992) have recomputed the global oscillation modes of Jupiter taking account of the PPT. If it exists, the associated density discontinuity permits the existence of global oscillation modes trapped at the PPT; this may provide an observational test of the existence or non-existence of the PPT. In addition, Marley (1991) has also shown that if global oscillations are excited on Saturn, the ring system may be a very sensitive detector of these modes.

Van Horn: Stellar structure and evolution

9

1.4 Nuclear Reaction Rates at High Densities Calculations of nuclear reaction rates at high densities are necessary for applications to supernova explosions and to accreting neutron stars and white dwarfs. Here I shall confine myself to brief discussions of three illustrative cases: the effects of screening on thermonuclear C+C reactions in pre-supernova models, the extension of pycnonuclear (density-induced) reaction rate calculations to binary ionic mixtures (BIMs), and preliminary investigations of He-"burning" at zero temperature.

H.l

C-ignition

Calculations of the effects of electron-screening on the C+C thermonuclear reaction rate date back to the original work by Salpeter (1954). He showed that the reaction rate is enhanced increasingly strongly by plasma screening at high densities. These calculations were subsequently improved by Salpeter and Van Horn (1969) and further refined by DeWitt, Graboske, and Cooper (1973). The statistical physics foundation for nuclear reactionrate calculations in dense plasmas was substantially clarified by Jancovici (1977) and Alastuey and Jancovici (1978). Ichimaru (1993) has recently reviewed the calculation of nuclear reaction rates in very dense plasmas. Both thermonuclear and pycnonuclear rates at high densities depend upon evaluations of the screening potential at small separations of the reacting nuclei. While Monte Carlo calculations have now developed to the point where relatively accurate values can be obtained for the screening potential when two reacting nuclei are separated by distances comparable to the average distance between nuclei in the plasma, the extrapolation of these results to near-zero separation, where the nuclear reaction occurs, has recently become the subject of some controversy. Ogata, Iyetomi, and Ichimaru (1991) have employed their most recent, highly accurate Monte Carlo results, extrapolated to zero separation by a technique they developed, to obtain a parametrized form for the screened C+C thermonuclear reaction rate both in the high-density fluid phase and in the solid phase of a C/0 plasma. This is the first detailed calculation of the enhancement factor for nuclear reaction rates for a binary ionic mixture (BIM). Ogata et al. found (i) that quantum-statistical effects greatly enhance the reaction rate above the value predicted by the classical enhancement factor and (ii) that the presence of 0 in the solid phase acts to produce a blocking effect on the pycnonuclear reaction rates. The carbonignition curves are accordingly sensitive to the 0 abundance (c/. Ichimaru

10

Van Horn: Stellar structure and evolution

1993). The calculations by Ogata et al. have been criticized by Rosenfeld (1993), however, and the issue is currently unresolved. 1.4.2

Pycnonuclear reactions for BIMs

Ichimaru, Ogata, and Van Horn (1992) have developed an expression for the pycnonuclear reaction rate in a zero-temperature plasma which for the first time is valid for binary ionic mixtures (BIMs). Their result is: ^

1.34 xlO32XjXj(Aj + Aj) 2 2 iiP " 1 + 6iS ZiZjiAiAj) 1809 X A" exp (-2.460A~ 1/2 ) reactions cm" 3 s" 1 ,

(5)

where Z{, A%, and X{{i = 1,2) are, respectively, the atomic charges, masses, and mass fractions of the two reacting species. The quantities Sij are the cross-section factors (in MeV-barns). The dimensionless quantity Xij is defined by /Q\

« =

(j) \4/

r*.

1/2

-^-, r

(6)

i j

where the r*j = [{A{ + Aj)h2]/2AiAjHZ{Zje2 are the nuclear Bohr radii, aij = |(a, + af) are the mean ion-sphere radii, and Tm
Van Horn: Stellar structure and evolution

11

however, ignition may occur via pycnonuclear ractions and may be quite violent. Thus calculations of He-burning reactions at low temperatures are of considerable astrophysical interest. Under normal astrophysical conditions, He burning occurs via the coupled reactions 4He + 4He -> 8Be and 4He + 8Be -*• 12 C*, which together comprise the so-called "3a reaction." Both reactions are endoergic, and 8Be and 12 C* normally decay spontaneously into the original reactants. The ground state of 12 C is formed through a rare 7-decay of the excited state 12C*. In the past few years, several authors have investigated the modifications of the 3a reaction that occurs with increasing density. Fushiki and Lamb (1987) showed how to take account of screening in these reactions by performing an 5-matrix calculation for the 3a process. Their general calculation is valid for both the thermonuclear and pycnonuclear regimes. Among other things, they took into account the shift in the effective reaction energy produced by screening. More recently, Schramm, Langanke, and Koonin (1992) have done a QM calculation of the 3a reaction as a three-body problem, not as a succession of two-body problems. An interesting alternative possibility is that He-burning may occur through a spontaneous, first-order phase transition instead of proceeding through the usual 3a reactions. When the density of the accumulated He at a neutron star surface increases to the point at which the electrostatic Coulomb interaction energy difference between a pair of reacting He nuclei and the resulting Be nucleus exceeds the —92 keV mass-energy difference, He ceases to be the lowest energy state. This occurs for the reaction 4He + 4 He <-^8 Be when the condition for thermodynamic equilibrium, 2/iHe — /*Be = 0 is satisfied. Calculations show that this condition is achieved at a density p = 4.1 x 109 g cm" 3 (Van Horn, Ogata, and Ichimaru 1991; Schramm, Langanke, and Koonin 1992 and references therein). This may have interesting consequences for the process by which matter accreted onto the surfaces of neutron stars reaches nuclear equilibrium at high densities, and thus affects the computations of the He shell-flashes that produce the X- or 7-ray bursts.

1.5 Some Unsolved Problems In this final section, I conclude by merely listing a number of interesting problems that I do not have time to to discuss in detail but which I believe merit further study.

12

Van Horn: Stellar structure and evolution

1.5.1 Matter in strong magnetic fields Since Ruderman's (1974) seminal paper on " magnetic matter," it has been recognized that the structure of matter in such strong magnetic fields at the surfaces of neutron stars is radically different from that of, e.g., the matter inside a white dwarf. Fushiki et al. (1989) have computed the equation of state in the non-relativistic limit, near the surface of a neutron star. In contrast with earlier calculations, they find that the density of matter does not approach zero gradually as the pressure P falls to zero; instead they find a finite and large surface (P = 0) density, p ~ several xlO 3 g cm" 3 , for magnetic field strengths ~ 1012 G. More recently, Lai and Shapiro (1991) have extended these calculations to include relativistic electrons. They have confirmed that the surfaces of strongly magnetic neutron stars have very large densities, increasing with the magnetic field strength. They have also studied the effects of such strong fields on the processes of /?-decay and electron capture. Two natural extensions of these calculations are (i) to augment them with new calculations of the transport properties of strongly magnetic matter and (ii) to incorporate the results into new models for the structure, evolution, and global oscillations of neutron stars. Some of these calculations are now beginning to be done (c/., Yakovlev 1993). 1.5.2 Radiative opacities Another area in which substantial advances have occurred in recent years involves the calculation of radiative opacities at high densities. The OPAL project, led by F. J. Rogers and C. A. Iglesias at Lawrence Livermore National Laboratory (LLNL), showed that "non-hydrogenic" transitions (i.e., transitions in which the principal quantum number of the level remains unchanged) lead to a pronounced increase (more than a factor of two) in the opacity of typical astrophysical mixtures in the temperature range T ~ 105 to 106 K (c/. Rogers, Iglesias, and Wilson 1987; Rogers and Iglesias 1992). The new OPAL opacities significantly improve the agreement both with astrophysical observations and with experimental data (Rogers and Iglesias 1992, 1993). The second new initiative is the international Opacity Project, led by M. J. Seaton at University College London. This is a massive effort conducted by an international team of more than 25 scientists from at least seven nations to compute and compile the atomic data needed for astrophysical opacity calculations (c/. Seaton et al. 1992 and references therein). In contrast to the OPAL calculations, which use a single-configuration atomic

Van Horn: Stellar structure and evolution

13

model with configuration interactions to generate the atomic data needed for opacity calculations, the OP calculations focus instead on the computation of high-accuracy transition data, using sophisticated theoretical models for the physics of the atomic properties and processes. As a result of the massive database thus generated, and of the effort to validate the calculations against experimental data wherever possible, the OP effort is just beginning to produce astrophysically useful results. Where the OPAL and OP calculations can be compared, the results are now in very good agreement. Another interesting, new opacity calculation was published recently by Lenzuni, Chernoff, and Salpeter (1991) for a gas consisting solely of H and He, with no heavy elements. This calculation was motivated by an interest in the properties of the first generation of stars, which were formed prior to enrichment of the interstellar medium by the products of stellar nucleosynthesis. These calculations are also relevant to models for the atmospheres of very cool white dwarfs. One of the main contributions provided by these calculations is that the authors have fully included the effects of collisioninduced absorption by molecular hydrogen, which becomes the dominant process at high densities. Lenzuni et al. have incorporated the latest calculations of this process in their work. Lenzuni and Saumon (1992) have subsequently extended these calculations to still higher densities, using the number densities from the equationof-state calculations by Saumon and Chabrier (1989, 1991). In addition to confirming the importance of collision-induced absorption by molecular H2, they found that absorption by H~ plays an increasingly important role at the highest densities considered, as "pressure-dissociation and ionization" produces relative increases in the number densities of neutral H atoms and electrons. Only very approximate information is available concerning the properties of H~ at very high densities, however, as Lenzuni and Saumon point out. Thus, the accuracy of the opacities at the very high densities remains uncertain, pending verification of the properties of H~ in high-density plasmas. Finally, it has been known for some time that plasma dispersion affects the absorption of radiation in dense plasmas (c/., Cox and Giuli 1968; Aharony and Opher 1979). However, this has never been taken into account properly in computations of astrophysical opacities. Because of the current interest in accurate, new calculations of opacities, Van Horn (1992) and Kurilenkov and Van Horn (1993) have recently initiated an effort to extend calculations of the Rosseland mean absorption coefficient KR in a dense plasma to higher densities. They have obtained the following expression, which they are now

14

Van Horn: Stellar structure and evolution

working to extend to more realistic physical conditions: f°° 1 KR

K t («ac)

J

00

dBu du Q T U)

dBu

dT

(7)

du

where the index of refraction of the plasma is nw = {kc/u) = [1—(w2/u>2)]'/2, and the absorption coefficient in plasma has been taken to scale from the corresponding value in vacuum as KW = n" 1 *^ . It remains to incorporate this result into the current generation of astrophysical opacity calculations. Acknowledgements This work has been supported in part by NSF grant AST 91-15132 and NASA grant NAGW-2444 through the University of Rochester. In addition, I thank the organizers of this conference for providing me with the opportunity to atend, and I especially thank Gilles Chabrier for his patience in waiting for this much-delayed manuscript! References Aharony, U., and Opher, R., Astron. Ap., 79, (1979) Alastuey, A., and Jancovici, B., Ap. J., 226, 1034, (1978) Barrat, J. L., Hansen, J. P., and Mochkovitch, R., Astr. Astr., 199, L15, (1988) Blitz, L., Nature, 364, 757, (1993) Chabrier, G., Saumon, D., Hubbard, W. B., and Lunine, J. I., Ap. J., 391, 817, (1992) Clayton, D. D., Principles of Stellar Evolution and Nucleosynthesis (McGraw-Hill: New York), (1968) Cox, J. P., and Giuli, R. T., Principles of Stellar Structure (Gordon and Breach: New York), pp. 64 ff., 189ft",(1968) Deming, D., Mumma, M. J., Espenak, F., Jennings, D. E., Kostiuk, T., Wiedemann, G., Lowenstein, R., and Piscitelli, J., Ap. J., 343, 456, (1989) DeWitt, H. E., Graboske, H. C , Jr., and Cooper, M. C , Ap. J., 181, 439, (1973) Fushiki, I., Gudmundsson, E. H., and Pethick, C. J., Ap. J., 342, 958, (1989) Fushiki, I., and Lamb. D. Q., Ap. J., 317, 368, (1987) Garcia-Berro, J., Hernanz, M., Mochkovitch, R., and Isern, J., Astron. Ap., 193, 141,(1988) Hernanz, M., Garcia-Berro, E., Isern, J., Mochkovitch, R., Segretain, L. and Chabrier, G., Ap.J, in press, (1994) Iben, I., Jr., and Laughlin, G., Ap. J., 341, 312, (1989) Ichimaru, S., Revs. Mod. Phys., 65, 255, (1993) Ichimaru. S., Iyetomi, H., and Ogata, S., Ap. J., 334, L17, (1988) Ichimaru, S., Ogata, S., and Van Horn, H. M., Ap. J., 401, L35, (1992) Jancovici, B., J. Stat. Phys., 17, 357, (1977) Joss, P., Ap. J., 225, L123, (1978) Kraeft, W.-D., Kremp, D., Ebeling, W., and Ropke, G., Quantum Statistics of Charged Particle Systems (Plenum Press: New York), (1986)

Van Horn: Stellar structure and evolution

15

Kurilenkov, Yu. K., and Van Horn, H. M., these proceedings Lee, U. M., Strohmayer, T. E., and Van Horn, H. M., Ap. J., 397, 674, (1992) Lee, Y.-W., P. A. S. P., 104, 798, (1992) Lenzuni, P., Chernoff, D., and Salpeter, E. E., Ap. J. Suppi, 76, 759, (1991) Lenzuni, P., and Saumon, D., Rev. Mexicana Astron. Astrof, 23, 223, (1992) Liebert, J., Dahn, C. C , and Monet, D. G., Ap. J., 332, 891, (1988) Magalhaes, J. A., Weir, A. L., Conrath, B. J., Gierasch, P. J., and Leroy, S. S., Nature, 337, 444, (1989) Magalhaes, J. A., Weir, A. L., Conrath, B. J., Gierasch, P. J., and Leroy, S. S., Icarus, 88, 39, (1990) Marley, M. S., Icarus, 94, 420, (1991) Norris, J. E., and Green, E. M., Ap. J., 337, 272, (1989) Ogata, S., personal communication. Ogata, S., Iyetomi, H., and Ichimaru, S., Ap. J., 372, 259, (1991) Ogata, S., Iyetomi, H., Ichimaru, S., and Van Horn, H. M., Phys. Rev. E, 48, 1344,(1993) Robnik, M., and Kundt, W., Astron. Ap., 120, 227, (1983) Rogers, F. J., and Iglesias, C. A., Ap. J. Suppi, 79, 507, (1992) Rogers, F. J., and Iglesias, C. A., in Strongly Coupled Plasma Physics, ed. H. M. Van Horn and S. Ichimaru (U. Rochester Press:Rochester, NY), p. 243, (1993) Rogers, F. J., Iglesias, C. A., and Wilson, B. G., Ap. J., 322, L45, (1987) Rosenfeld, Y., these proceedings Ruderman, M. A., in IAU Symp. No. 53: Physics of Dense Matter, ed. C. J. Hansen (Dordrecht: Reidel), p. 117, (1974) Salpeter, E. E., Aust. J. Phys., 7, 373, (1954) Salpeter, E. E., and Van Horn, H. M., Ap. J., 155, 183, (1969) Sarajedini, A., and King, C. R., A. J., 98, 1624, (1989) Saumon, D., Ph.D. thesis, University of Rochester, (1989) Saumon, D., and Chabrier, G., Phys. Rev. Lett, 62, 2397, (1989) Saumon, D., and Chabrier, G., Phys. Rev. A, 44, 5122, (1991) Saumon, D., Hubbard, W. B., Chabrier, G., and Van Horn, H. M., Ap. J., 391, 827,(1992) Schmider, F. X., and Mosser, G., in Progress of Seismology of the Sun and Stars, ed. Y. Osaki and H. Shibahashi (Springer: Berlin), (1990) Schmider, F. X., Mosser, G., and Fossat, E., Astron. Ap., 248, 281, (1991) Schramm, S., Langanke, K., and Koonin, S. E., Ap. J., 397, 579, (1992) Seaton, M. J., Zeippen, C. J., Tully, J. A., Pradhan, A. K., Mendoza, C , Hibbert, A., and Berrington, K. A., Rev. Mexicana Astron. Astrof, 23, 19, (1992) Segretain, L., and Chabrier, G., Astron. Ap., 271, L13, (1993) Segretain, L., Chabrier, G., Hernanz, M., Garcia-Berro, E., Isern, J. and Mochkovitch, R., Ap.J, in press, (1994) Stevenson, D. J., J. de Physique, Coll. C2, 41, C2-61, (1980) Van Horn, H. M., Science, 252, 384, (1991) Van Horn, H. M., Bull. A. A. S., 24, 824, (1992) Van Horn, H. M., Ogata, S., and Ichimaru, S., Proc. US-Japan Workshop on Nuclear Fusion in Dense Plasmas, IFSR #519, p. 57, (1991) Winget, D. E., et ai, Ap. J., 315, L77, (1987) Yakovlev, D. G., in Strongly Coupled Plasma Physics, eds. H. M. Van Horn and S. Ichimaru (University of Rochester Press: Rochester, NY), (1993)

Equation of State of Stellar Plasmas Forrest J. Rogers Lawrence Livermore National Laboratory, Livermore, CA 94550

Abstract The equation of state (EOS) of astrophysical plasmas is, for a wide range of stars, nearly ideal; with only small non-ideal Coulomb corrections. Calculating the EOS of an ionizing plasma from a ground state ion, ideal gas model is easy, whereas, fundamental methods to include the small Coulomb corrections are difficult. Attempts to include excited bound states are also complicated by plasma screening and microfield phenomena that weaken and broaden these states. Nevertheless, the high quality of current observational data, particularly seismic, dictates that the best possible models should be used. The present article discusses these issues and describes how they are resolved by fundamental many-body quantum statistical methods. Particular emphasis is placed on the activity expansion method that is the basis of the OPAL opacity code. Some comparisons with standard methods are given. Abstract L'equation d'etat des plasmas astrophysiques est, pour un large domaine d'etoiles, pratiquement ideale; avec de petites corrections coulombiennes. Calculer l'equation d'etat d'un plasma ionise a partir d'un modele de gaz ideal d'ions dans leur etat fondamental est facile, alors que les methodes fondamentales pour inclure les petites corrections coulombiennes sont difficiles. Des tentatives pour inclure des etats lies excites sont aussi rendues difficiles par les effets d'ecran et le phenomene de microchamp qui affaiblissent et elargissent ces etats. Neanmoins, la haute qualite des observations actuelles, en particulier en sismologie, impose l'utilisation des tous meilleurs modeles. Dans cet article, nous discutons les possibilites et nous decrivons les methodes de physique statistique quantique du probleme a N-corps pour resoudre ce probleme. La methode de developpe opacite OPAL. Nous presentons des comparisons avec des methodes standards. 16

Rogers: Stellar plasmas

2.1

17

Introduction

The density temperature range found in stars is very large. Nevertheless, it is frequently true that the Coulomb correlational effects are small. For example, the density temperature profile of a main sequence star is given approximately by constant values of R=p/T^, where p is the density in g/cm3 and T$ is the temperature in millions of degrees Kelvin. Main sequence stars are about 98% hydrogen and helium by mass with the remainder being mostly Carbon, Nitrogen, and Oxygen. Since log R is less than -1 for most stars, these two conditions limit Coulomb corrections to the equation of state to generally less than a few percent. For stars in more advanced states of evolution, e.g. white dwarf and neutron stars, R can be much larger and the composition in extreme cases can be pure iron. The discussion in the current paper will be primarily aimed at understanding the EOS in stars where the Coulomb interactions are small to moderate. Saumon and Chabrier (1992; 1991) have been concerned with dense hydrogen and helium equations of state. The simple ionization equilibrium model introduced by Saha (1920) led to a revolution in stellar modeling and has proved to be adequate for many purposes. Sana's model assumes ideal gas conditions and uses only the ground state configuration in the ionization balance equations. For example, in the simple case of a non-degenerate hydrogen plasma it gives L

=^ ^ e x p ( ^ )

(1)

where pi= Nj/V is the number density for ions of type i={e, p, H}, (2) is the thermal de Broglie wavelength, and ge, gp, and gH are the statistical weights. The corresponding pressure is

Even though, as already noted, the Coulomb corrections to equation (1) are generally small for most stars, it is necessary to have a precise theory to account for them. There are two reasons for this. Firstly, a more complete model has to include all excited states including possible plasma screening effects, as well as, the Coulomb coupling corrections. These

18

Rogers: Stellar plasmas

effects are interrelated and, as a result, ad hoc approaches are prone to count the same term twice. Consequently, the corrections to the Sana equation obtained by ad hoc approaches can vary by substantial amounts (gauged by modeling sensitivity). Secondly, the modeling of currently available high precision observational data requires very accurate EOS data (Christensen-Dalsgaard and Dappen 1993). Modeling of helioseismological data, for example, requires derivatives of the EOS that are accurate to better than 1%. This level of accuracy can only be expected from fundamental procedures. Considerable effort has been devoted to the fundamental treatment of hydrogen plasmas. An in depth description of much of this work can be found in the books by Kraeft, Kremp, Ebeling, and Ropke 1986 and Ebeling, Kraeft, and Kremp 1976; see also DeWitt 1966^ Krasnikov 1971 The more general problem of multi-component plasmas lias received much less attention (Rogers 1991; Krasnikov and Kucherenko 1978; Ebeling 1974). A new approach for treating multi-component plasmas is described in the articles by Aluestuey and Perez (1992) and Aluestuey elsewhere this volume. Numerous papers have been devoted to phenomenological modeling of multi-component plasmas (McChesney 1964). 2.2 Commonly Used Methods Typical stellar model calculations require the EOS for a large, variable set of temperature-density points and for variable composition. It is thus very desirable to have available an efficient model that allows online computation, or failing that, a model that can be used to readily produce tables. All such models in current use are based on free energy minimization methods. These approaches work in the chemical picture and deal explicitly with ions and atoms. Eggleton, Faulkner, and Flannery (1973) developed an EOS that is computationally simple and suitable for online use (EFF). They introduced an ad hoc free energy term to produce pressure ionization at high density, i.e., when the interparticle separation is less than a bohr. This overcomes a well known shortcoming of the Sana equation, which predicts that hydrogen is 30% neutral in the solar center where temperatures are 100timesthe binding energy of H. Similar to Sana, EFF assume that ions and atoms are in their unperturbed ground states,. The EFF method accounts for Fermi-Dirac statistics for electrons, but ignores Coulomb interactions. It can produce unphysical phase transitions when used outside its range of validity.

Rogers: Stellar plasmas

19

In view of the need for high accuracy in the EOS, a number of attempts have been made to improve the EFF equation. ChristiansenDalsgaard and Dappen (1991) and Swensen, VandenBerg, Alexander, and Irwin (1994) have added a Debye-Hiickel free energy term (see equation (22)), called the CEFF equation to indicate that the Debye Coulomb correction has been added. Swenson and Rogers (1993) also adjusted parameters in the CEFF equation in order to obtain improved agreement with the activity expansion method. Going beyond die simple models to include the excited states and more correct Coulomb interaction terms is fraught with difficulties. For example, the internal partition function for an isolated atom is divergent, so that some method for cutting off the partition function must be introduced. Typically this is accomplished by assuming that the presence of other particles in the vicinity of a given atom (ion) confines the particle to a sphere of order of the ion sphere radius, a, or that it interacts with the plasma through a short-ranged screened potential, e.g., the Debye-Huckel potential. The resulting free energy is finite but discontinuous at plasma conditions such that a bound state is just entering the continuum. This behavior cannot be present in physically consistent models. Mihalas, Hummer, and Dappen (Hummer and Mihalas 1988; Mihalas, Dappen and Hummer 1988; Dappen, Anderson, and Mihalas 1987) have presented an occupation probability formalism that is physically consistent and produces continuous free energies. It is commonly referred to as the MHD equation of state. Based on experimental measurements of level shifts (Goldsmith, Griem and Cohen 1984), they assume that the bound states of atoms and ions are unshifted by the plasma environment. They note that, if a configurational free energy, f(V,T,{ni}), that depends explicitly on the occupation numbers of the individual states is added to the ideal free energy terms, then the ratio of the occupation of a state i of a given ion to the total occupation is given by ni/n = exp[-0(£,. + of I dnj] IZ*

(4)

where Zint = Z; exp[-j3(E,. + df I an,)]

(5)

plays the role of the partition function, (6)

20

Rogers: Stellar plasmas

is the occupation probability, nj is the occupation number for state i and n is the total occupation for a given species. The occupation probability is a measure of the number of bound states of type i that are available to be occupied. The quantity l-C0i is, thus, a measure of the fraction of total states that have been severely affected by plasma perturbations and no longer act like localized states. In order to make progress it is necessary to have either a good estimate for f or a good estimate for coj, whichever is easier. The MHD approach mixes the two possibilities. In the case of neutral particle interactions they use the free energy of a parameterized hard sphere gas to determine the occupation probability. For ion-ion interactions they use the electric microfield (Stark-ionization theory) to determine the occupation probability. For ion-neutral interactions they propose using a product of the two forms. The method is thus phenomenological, but uses experimental data to fit free parameters in the occupation probability function. The MHD method has been used in numerous stellar modeling sensitivity studies (Christensen-Dalsgaard and DSppen 1993; Dziembowski, Pamyatnykh, and Sienkiewicz 1992). A method closely related to the MHD approach has also been developed by Sevastyanenko (1985). 2.3 Consistent Treatment of Bound and Scattering States The source of the discontinuity in free energy minimization methods is easily shown to be the result of an inconsistent treatment of bound and scattering states. The second virial coefficient is the leading term in the density expansion of the free energy. The quantum mechanical second virial coefficient can be expressed in the form (Uhlenbeck and Beth 1936; Beth and Uhlenbeck 1939)

(7)

^tf? where

B? = -V^X^,

z^ = £(2£ +1*"*-

(8)

and

f

J>^*^

(9)

Rogers: Stellar plasmas

21

In equations (7, 9) i and j are the particle types, X^ = (h212^kT)1'2, n and I, are quantum numbers, the Eni are bound state energies, 5/ is the phase shift, p is the relative momentum, and |iy is the reduced mass. The connection between bound and scattering states can readily be established by integrating equation (9) by parts and using the fact that the phase shift at zero energy is just nn (Levinson 1949). Adding the zero energy term to equation 8 (Rogers, Graboske, and DeWitt 1971) gives B

- -1)

(10)

If now each exponential (Boltzmann) factor is expanded in powers of PE n / it is apparent that the first term in the expansion just removes the negative term in equation (10). In other words, an analytical term having the form of the most divergent term in Zjm is not actually present in By; at least at temperatures where the Boltzmann factors are near unity. The sum over the PE n / terms in the expansion of BJ£ is also divergent, but with a second integration by parts of equation (9) it too can be shown to be analytically missing from the total By at high temperature (Rogers 1979; 1977; Bolle 1989; 1987; Pisano and McKellar 1989). There are divergences in By, but they are all in the final form for B^s after the two integrations by parts. These two integrations by parts have effectively redefined the continuum such that it begins at -kT, rather than the usual zero of energy. The effective internal partition function is thus,

(ID It is worthwhile noting that the value of By has not changed in this series of manipulations, rather it has been shown that the analytic properties of the complete set of states dictates a specific separation of By into effective bound and scattering state parts. This separation will play an important role in the many-body statistical mechanical methods to be described in the next section. 2.4 Classical Density Expansion for Plasmas The natural way to treat reacting, multicomponent plasmas is in the grand canonical ensemble (Hill 1956). In this approach one views the system in terms of its fundamental constituents, so that bound complexes arise naturally from the theory. As a result, this approach is commonly referred to as the physical picture method. The standard procedure is to

22

Rogers: Stellar plasmas

expand the pressure in terms of two body, three body clusters, etc., i.e., a cluster expansion. The same is true for plasmas, but the long range of the Coulomb potential introduces substantial complications. In addition, the quantum nature of electrons introduces degeneracy and exchange corrections. The attractive electron-ion interaction leads to short distance divergences in classical cluster coefficients, so that the use of quantum mechanical methods is essential. Graphical resummation procedures are required to remove the long-range divergences occurring in all cluster coefficients of plasmas. The divergences in B^js, mentioned in the previous section, are only the simplest examples. A detailed description of the procedure is given elsewhere (Dashen, Ma, and Bernstein 1969; Kraeft, Kremp, Ebeling and R6pke 1986; Rogers 1981). For illustrative purposes it is much easier to consider the related procedure in the canonical ensemble, i.e., an expansion of the free energy in the density. For simplicity we also focus on classical procedures where possible. In early graphical analysis of the classical one-component plasma, ie., heavy ions emmersed in a continuous neutralizing background of electrons, Mayer (1950)referredto the non-ideal free energy (aside from a factor VkT) as -S, so that, (F-F0)/VkT = -S

(12)

where S = 2LBj-i^

(13)

Bj is the jth virial coefficient. The classical second virial coefficient for the one component plasma (OCP) is given by B2 = -—\d~rx'd~ri(e-pu{ra)-\) v 2VJ

= -2n\"{e-MT) -l)r2dr / J o

(14)

where rl2 = r = The divergence in B2 is easily seen by expanding the classical Boltzmann factors in powers of |5u. The leading term diverges as r2, the second term as r, and the third term logarithmically as r-»°o. The leading divergence posses no problem, since it is canceled by an opposing term provided by the neutralizing electron background; leaving the (Pu)2 as the leading

Rogers: Stellar plasmas

23

divergence. In general the lowest order term from each of the virial coefficients is given by \ \

i

) . . . u { r

j x

)

(15)

The sum over the most divergent terms from each of the higher B j ; i.e., those that have just one power of the potential turned on between each pair of particles, can be represented graphically as shown in Figure 1.

Fig. 1 Graphic illustration of the classical ring sum for the OCP. Solid circles represent ions. A diagram involving n ions comes from B n .

Due to their topology, they are known as ring diagrams. The result of the summation over ring diagrams is t

^

j

(16)

y

where A = Z2e2/kTXD

(17)

A D =(*774/rZVp) 1/2

(18)

and

The pressure for a v component plasma, corresponding to multicomponent generalization of equation (12), is (19)

24

Rogers: Stellar plasmas

Using equation (16) this gives in the Debye-Huckel approximation for the OCP

P/kT=p(l-±) o

(20)

The Debye-Hiickel correction is just the leading term in an infinite expansion analogous to the virial expansion for an ordinary gas. However due to the Coulomb modifications it involves p 3 # rather than p 2 (see equation 16). The next higher term in the expansion is obtained by summing over the next most divergent diagrams, the three rung ladder diagrams; so named by their ladder-like quantum mechanical form. Figure 2 shows (in the classical limit) how one rung of the three rung ladder diagram is screened by summing over chains, of ever increasing length, that have one power of the potential, Bu, turned on between each pair of ions. Similar summations produce screening in the remaining two rungs of the diagram (often called watermelon diagrams in classical mechanics).

0 0 <• Fig. 2 Graphical illustration of the classical ladder sum for the OCP that screens the right hand rung of the three rung ladder diagram. Solid circles represent ions.

Continuing in this way provides an expansion for S, known as the Abe nodal (cluster) expansion (Abe 1959), that is valid at all values of the coupling. The result for the general case of a multi-component plasma is

V* 3

(22) (23)

Rogers: Stellar plasmas

• S2 = pj>l-B,(T,XD)-27tjdrr2(q,

25

-

(24)

_2 where _

Z Z

a ft<

(26)

is the Debye-Huckel screened potential, A© is the multi-component Debye length, and /ak=e~*--l

(27)

is the Mayer function. Figure 3 shows the convergence of the Abe cluster expansion to Monte-Carlo simulations (Slattery, Doolen, and DeWitt 1982) of the one component plasma. The Debye-Huckel pressure term (corresponding to S D H ) is seen to have a very small range of validity. Significant discrepancies appear for A > 0.2. The S2 correction, analogous to the second virial coefficient for the Debye-Hiickel potential, is somewhat of an improvement The important point is that the Abe series offers a systematic way of calculating the EOS of the OCP. It was shown in Rogers (1981) that, if large numbers of Abe nodal terms are included, this convergence persists to large values of the coupling parameter, T=Z2e2/&. This shows that, contrary to some claims in the literature, that, when properly implemented, the Debye-Huckel potential is a valid plasma potential. Similar studies have been done for multicomponent mixtures of heavy ions (TCP) in a neutralizing background. Our interest here is limited mainly to weak to moderately coupled plasmas, but this example shows that the Abe procedure could be applied to plasmas occurring in white dwarf stars. Related examples of the convergence of the activity expansion method (see Section 7) to the OCP are given in Rogers and DeWitt (1973). They found that the convergence is somewhat faster in the

26

Rogers: Stellar plasmas

activity expansion. Similar results have also been reported for electrolytic solutions (Wood, Lilley, and Thompson 1978). Since electrons in real plasmas are always quantum mechanical, direct use of classical methods is not possible. Nevertheless, the main complications in real plasmas have to do with the highly classical long range correlations. The divergences associated with short range quantum interactions are easily handled by replacing the classical virial coefficients occurring in the Sj with their quantum-mechanical equivalents (Rogers 1991; 1986; 1981; 1974; Rogers and DeWitt 1973). Quantum mechanical calculations for real reacting plasmas follow similar steps and the convergence should be similar to the example in Figure 3. In the current implementation the ion-ion correlations are included in all orders, but the electron-electron and electron-ion correlations are included only through 5/2 order in the density (activity). Approximate methods for including electron-electron and electron-ion terms when even the electron-ion coupling is strong are given in Rogers 1979. 1

1.0

0.5

-

3

2 1

4 1

1

\ -

OCP

0 —

\ \

i

N . . .

8

12

16

A Fig. 3 Comparison of the Abe nodal expansion with Monte Carlo simulations of the OCP by Slattery, Doolen, and Dewitt (1982). Labels indicate the free energy terms used to calculate the pressure.

Elaborating on the remarks of the previous paragraph, we note that for a real plasma, involving electrons and ions, the diagrammatic sum that gives the Debye-Hiickel correction equivalent to equation 16, involves additional terms corresponding to electron-electron and electron-ion interactions.

27

Rogers: Stellar plasmas

o <

O electron-electron

electron-proton

proton-proton

Fig. 4 Leading classical ring diagrams for an electron-proton plasma. Open circles represent electrons; closed circles represent protons.

The fact that electrons are now being treated as real particles, rather than a neutralizing fluid, introduces explicit diagrams in which electrons interact with electrons and diagrams in which electrons interact with ions. Due to the Coulomb repulsion the electron-electron terms are not divergent at small distances. However, quantum diffraction effects modify the result and can be treated by semi-classical methods. The classical electron-ion terms are divergent at small distances and can only be treated quantum mechanically; the large distance divergence is of course still classical. When 7ij is « 1, the ring sum free energy is (28)

where IcT

(29)

Equations (28, 29) are exactly the same as given by the classical theory (see equations (22-23)). This supports the earlier statement that the Coulomb correlations are largely classical. In general however, due to the quantum modifications at small distances, SDH is appreciably reduced when yii = Xjjl XD>0.5 . For Boltzmann statistics the quantum corrected Debye-Huckel term can be written in the form (30)

28

Rogers: Stellar plasmas

where f(0,0)=l. There are also some additional corrections due to exchange effects. The ion-ion diffraction parameter, "fa, does not appear in equation (30), since 7 i i « l due to the mass of the ion. Figure 5 displays the variation of f with Yee for the simple case of a non-degenerate electron gas for the "exact" calculations of Graboske and DeWitt (1974) and the approximate effective potential results of Rogers (1979).

100

Fig. 5 Diffraction corrections to the electron-electron ring sum.

2 5 EOS of a Hydrogen Plasma We have chosen the simple example of ionization equilibrium in a nondegenerate hydrogen plasma to compare the free energy obtained from a many body quantum statistical approach with a commonly used free energy minimization model, i.eM

e+pjtH The typical free energy minimization method would have

where the first three terms on the right correspond to the translational free energies for electrons, protons, and hydrogen atoms, respectively, Zmt is the sum over states (internal partition function), and SoH is gi v e n by

Rogers: Stellar plasmas

29

equation (28). As described in Section 2, some model must be introduced to make the sum in Zjm finite. In practice this can be one of a number of possibilities such as the use of the energy levels of the Debye potential. Since Zjnt changes discontinuously at conditions where a state moves into the continuum and is no longer counted, equation (31) is not physically consistent The free energy that results from the many body diagrammatic approach is (Rogers 1991; 1989; 1986) (32) \PA.)

\PPA>'p)

KPH^'H

)

where (33) which is closely related to Z^ t of equation (11), is the so called PlanckLarkin partition function. The sum in Zg}t ranges over the states in a screened potential which approaches the Debye-Hiickel potential at very low density. As described in Rogers (1986; 1981) the energy levels appearing in 3m are unscreened except for high lying states near the plasma continuum. The states that are screened change with plasma conditions. As a result Zg{t is both finite and a continuous function of temperature and density; although the density dependence is very slight for normal stellar conditions. The MHD EOS displays a similar property through the use of the occupation probability formalism. Figure 6 is an attempt to explain the result in equation (32). It shows (schematically) the contribution to the partition function for hydrogen as a function of principal quantum number at temperatures of 1-2 eV. Due to the large value of the Boltzmann factor at these temperatures, the ground state contribution is large. Because of the wide energy separation between n=l and n=2, the contribution from n=2 has already dropped close to unity. For higher states the n2 degeneracy causes the contribution to again increase. For the pure Coulomb potential the area under the curve becomes infinite and leads to the well known divergence of the atomic partition function. The high lying states are classical (closely spaced), which means that sums over quantum numbers can be replaced with integrals; while the low lying states are quantum mechanical. This suggests that the partition function can be separated into two parts; one requiring the explicit use of quantum numbers, the other only involving

30

Rogers: Stellar plasmas

integrals as shown in the figure. The natural way to do this seems to be to use the analytic properties (see Section 3). The boundary between the two regions is of course fuzzy and not abrupt as actually shown. The discussion leading to equation (28) showed that, even though quantum mechanics is required to remove the divergence in the electron-ion terms, the ring sum is highly classical. This suggests that the electron-ion part of the ring summation leading to equation (28) has in some sense included the classical parts of Zint. As a result, using the full Zjnt in equation (31) is incorrect, since it would double count much of the excited state part.

8 Fig. 6 Contributions to the hydrogenic bound state partition function. Lower shaded region corresponds to the quantum mechanical part; upper shaded region to the (approximate) classical part. The total is shown as a smoothed curve to emphasize the classical nature of the high n contribution.

The pressure for a partially ionized hydrogen plasma corresponding to equation (32), but also including degeneracy and exchange corrections, has the form (34)

where the In/2 functions are the usual Fermi Functions, a e =Me/kT is the degeneracy parameter,

31

Rogers: Stellar plasmas

(35)

is the first order electron exchange, and gee is the electron-electron distribution function for an ideal Fermi gas, the Debye length corrected for electron degeneracy is 1/2

kT

(36)

and fp(Yee.Yei) is the diffraction correction to the pressure similar to Figure 5. There are some additional exchange corrections to equation (34), which are not explicitly shown. The first order exchange correction (equation (35)) is frequently omitted in astrophysical EOS calculations, but in view of the current need for high precision that is no longer acceptable. The origin of the exchange term is easily seen from the electronelectron distribution function for an ideal gas, as shown in Figure 7. 1.0

Fig. 7 Schematic representation of the ideal Fermi gas electron-electron distribution function.

For an ideal Boltzmann gas, gee is everywhere unity, as is gpp ang gep (not shown), so that, the first order contribution to the pressure, i.e., the average over ZiZje2/r, is exactly zero. However, when the electron

32

Rogers: Stellar plasmas

distribution is allowed to redistribute itself, due to the Paulirepulsion,gee is forced to have the value 1/2 at r=0 and gradually increases to unity as r > ao. This produces a small, negative, non-cancellation in the average of ZiZje2/r. In other words the quantum statistical effects that produce the degeneracy corrections are always partially reduced by the Coulomb interactions. The presence of the Coulomb interaction also affects the electronic charge distribution, which produces additional corrections to the first order average over ZiZje2/r. This leads to what are known as second order (or higher) exchange corrections. The Fermi function ratios appearing in equation (34) are shown schematically in Figure (8) as a function of the degeneracy parameter. The ratio of I3/2/I1/2 increases with increasing degeneracy, indicating that degeneracy increases the electron contribution to the pressure, while the ratio L1/2/I1/2 decreases, indicating that highly degenerate electrons do not contribute to the screening.

Fig. 8 Fermi function ratios appearing in equation (34).

2.6 The Activity Expansion Method A many body Quantum statistical procedure for calculating the EOS of reacting multi-component plasmas is described in Rogers (1991). A somewhat different procedure, specialized to hydrogen plasmas is given in Bartsch and Ebeling (1971). These approaches treat the plasma in terms of its electrons and nuclei and are thus "physical picture" methods.

Rogers: Stellar plasmas

33

Composite particles, i.e., ions, atoms, and molecules, arise naturally in the physical picture, such that, plasma screening effects on the bound states are determined from theory. This is a definite advantage over chemical picture methods in current use; all of which introduce models to obtain these effects. For a discussion of the activity expansion method, as developed for multi-component plasmas, the reader is referred to Rogers (1991). The main steps in the method can be summarized as follows I.

Start from the classical multicomponent canonical ensemble.

n.

Carry out the Abe reorganization in powers of |3u (see Section 4) to obtain a finite expression for S(T, {pi}, XD). The reason for this step is that the analogous summation in the grand-canonical ensemble gives S(T, {z{}, X), i.e., exactly the same analytic function except density is replaced by the activity; where (37)

is the activity, i={e, 0Cj}, otj is an atomic nucleus of charge Zj, and A. is a screening length that approaches XD when the Coulomb coupling is weak. HI.

Construct a generating function method for developing the corresponding grand canonical ensemble from functional derivatives of S(T, [z[), X). The advantage of this procedure is that, as a result of the resummation in step n, S(T, {zj}, X) is divergence free and therefore so is the grand canonical expression.

IV.

Recollect terms to introduce screened cluster coefficients; i.e. construct the grand canonical ensemble analogs to equations (2127). At this point all states are screened by the plasma.

V.

Replace all the classical Boltzmann factors in the classical grand canonical ensemble with Tr exp-PH.

VI.

Introduce an augmented set of activity variables to account for the formation of ions, atoms, and molecules; i.e., construct composite particle activities from products of ze and za. This renormalization removes the screening of low lying, bound states that were initially present after step IV.

34

VII.

Rogers: Stellar plasmas

Reorganize the resulting activity expressions to take advantage of the charge asymmetry of high Z ions. Since the ion-ion terms are classical, it is possible to include them in all orders. Limitations on the range of validity are set by the electron-ion diagrams, which require quantum mechanics and are only included to 5/2 order in the coupling parameter.

This procedure will only recover the linear correction in Yij in the expansion of f(Yee>Yei)- However, it is possible to introduce additional corrections using known results from other sources (Rogers 1981). Since the activity expansion approach is based on a systematic method, it is possible to determine its range of validity. This is shown in Figure 9 for an argon plasma which shows contours of constant A=< Z^>e^/kTA.D* As already shown in Figure 3 for the classical OCP, the contour A=0.2 is the approximate limit of validity of the Debye-Hiickel theory. The current version of the code includes all orders of ion-ion interactions, but only terms through 5/2 order in the activity for electron-electron and electronion interactions. Consequently, in the shaded region on the right side of the figure the errors in the pressure are estimated to exceed 5%. The limitation at the low temperature end of this region is due to the inclusion of only two particle neutral-neutral and ion-neutral interactions. 1000

10"3 1(T 2 Density (fl/cm3)

1CT1

Fig. 9 Contours of constant A= e2/kTXD for an argon plasma.

35

Rogers: Stellar plasmas

The short dashed line in the lower right hand part of Figure 9 is the approximate location of some recent shock wave EOS measurements by Erskine, Rosznyai, and Ross (1994). Figure 10 compares this experimental work with the Saha equation and the activity expansion method. The coupling parameter, T, is around unity and the temperature is 2.5 eV at the densest point. The activity expansion method is in good agreement with the experiment. Additional experimental work is in progress.

-

OPAL-H-

0.16 —

^0.12

£

/y> -.

a °"0.08 — ; lonlzatlon onset v 0.04 1

> ^^£*

• i

0

6.004

i

i

0.008 0.012 Density (g/cm3)

i i

i

i i

i

i

i

i~

0.016

Fig. 10 Comparison of the activity expansion method (curve labeled OPAL) with the shock wave experiments of Erskin, Rosznyai, and Ross (1994). To demonstrate the importance of Coulomb interations, the simple Saha equation is also shown.

2.7 Equation of State Comparisons The EOS properties frequently needed in stellar modeling are the first order thermodynamic quantities and the following second order quantities (Cox and Guili 1968) XT =

{dinp)T

(37)

36

Rogers: Stellar plasmas

(38)

(39) Comparisons of the several EOS methods described above have been carried out by Da'ppen (1992). A few selected examples are repeated here. Figure 11 compares results for XT. Figure lla shows that for stellar envelope conditions, that there is only a few percent difference between the EFF and MHD approaches. Figure lib shows that the differences between the MHD and OPAL equations of state are much smaller than their differences with EFF. Similar results were found for the other second order quantities. The seismic modes are determined only by the deviations of Fi from the ideal gas value 5/3. Consequently, differences similar to those shown in Figure lla for XT substantially affect the agreement with the p-mode data. The MHD and OPAL equations of state are in somewhat better agreement with observational data than is the simple EFF model (Christensen-Dalsgaard and Dappen 1993). At higher densities, where the Coulomb coupling is outside the range of validity of the Debye-Hiickel theory, differences between MHD and OPAL become significant. Figure 12 shows a comparison of Fi along an isochore having density 0.1 gm/cm^- The region around two million degrees corresponds to conditions near the bottom of the solar convection zone. Differences in the two EOS methods are fairly small in this important solar region. However, substantial differences occur at temperatures of a few hundred thousand degrees. These differences could effect the modeling of very low mass stars. In a recent paper Dziembowski, Pamyatnykh, and Sienkiewicz (1992) used helioseismological data to test the MHD equation of state. They found evidence that this approach is inadequate for conditions that exist in the fractional solar radius range r//?=0.85 to 0.95. We have used the activity expansion method (Section 7) to study two points in this region (Rogers and Iglesias 1993). One point is located at r//?=0.88, T=744,000 K and p=0.0348 g/cm3; the other is at r/R=0.90, T=612,000 K and p=0.0259 g/cm3. We were able to draw several conclusions (see Figure 1 of Rogers and Iglesias 1993). 1) The activity expansion result is appreciably different than the MHD result for Fi t on the fine sensitivity

37

Rogers: Stellar plasmas

I

1

1

1

1

1

2.0 1.8 -

/

— —

.1.6

\

i

1.4 1.2 1.0 "~

V

|

1

— — 1

1

1

1 I

0.1 _

/

n

U

CO

i —

V—— —

-0.1

-0.2 -

\

1

4.2

/

—

i 4.4

i i i 4.6 4.8 5.0 log T (Kelvin)

i 5.2

I

5.4

Fig. 11 Comparison of %p on an isochore with p=10*5-5. Part a compares the EFF and MHD equations of state; solid line EFF and the dashed line MHD. The mixture is 90% H and 10% He by number abundance. OPAL is indistinquishable from MHD on this scale. Part b compares the MHD and OPAL equations of state by looking at the relative differences with EFF: i. e., (X-rOpal . X j E F F ^ E F F ^ 4 line) ^ j (ftMHD _x T EFF)/x T EFF (dashed line). [From Dappen (1992) with permission].

38

Rogers: Stellar plasmas

I

U"1.60

I

I

I

I

I

6.0

6.2

-

1.55 -

1.50

5.4

5.6 5.8 log T (Kelvin)

Fig. 12 Ti for p=0.1 g/cm3 for the mixture of Fig. 11; solid line CEFF; dashed line MHD: dotted line OPAL equation of state. [From Dappen (1992) with permission ]

scale of the experimental data; 2) the Coulomb corrections are important; 3) the metallicity has a surprisingly large effect on I*i; and 4) the composition of Z is important. For example, we found that if we assume Z is composed entirely of Si, that a close match of the inversion data is obtained with just solar metallicity (0.0193). 2.8 OPAL Equation of State Tables The inability to resolve a number of long standing discrepancies between theory and observation through improved modeling led to the speculation that the widely used Los Alamos opacities were missing important sources of opacity in the lO^-lO** K temperature range (Simon 1982). Due to this speculation and the need for the opacity of low Z materials to model laser produced plasmas, the OPAL opacity effort was undertaken (Iglesias, Rogers, and Wilson 1987; Rogers and Iglesias 1992; Iglesias and Rogers 1993). While it was necessary to calculate the occupation numbers as part of this effort, the much greater accuracy and table density required to calculate derivatives of the equation of state,

39

Rogers: Stellar plasmas

made it necessary to defer these calculations. The success of the OPAL opacities in helping to improve theoretical models (Guenther 1992) has made it essential to also provide EOS data that is consistent with the opacity tables. We now have available EOS tables for the following conditions: Grevesse 1991 mixture (Rogers and Iglesias 1992) 0.005>T6<100 10"14> p < 107 g/cm3 X=0,0.2,0.4,0.6, and 0.8 Z=0,0.02, and 0.04 Y=l-X-Z Data has been tabulated for the quantities discussed in Section 8. Figure 13 shows Fi-5/3 vs. T6 at several values of log R for a simple mixture of H and He. Figure 13a shows results for the full range of the tables; while Figure 13b is limited to the high temperature range above 100,000 K. The large negative deviations from 5/3 at low temperature in Figure 13a is due to placing energy into the formation of ions and atoms. In the few hundred thousand degree range the negative deviation due to bound complexes is competing with the positive deviations caused by Coulomb interactions. The positive Coulomb deviations start to dominate with increasing values of log R.

0.20

0.00

-0.20

-0.40

-O.60

0.1

1.0

10.0

Fig. 13 (a) Fi-5/3 vs. T6 for hydrogen mass content X=0.4 and helium mass content Y=0.6. Dotted line log R=-5: solid line log R=-3; dot-dash log R=-l; dashed line log R=0.

Rogers: Stellar plasmas

40 0.02

0.01

0.00

-o.oi

-0.02

1.0

10.0

Fig. 13 (b) Same as (a) except for reduced temperature range.

Figure 13c is similar to Figure 13b, but for the Grevesse 1991 mixture. The small amount of high Z admixture has a noticeable effect on the results. It is apparent that the modeling results will be sensitive to uncertanties in the abundances of heavy elements, e.g. neon in the million degree range.

0.02

-0.02

10.0

Hg. 13 (c) Same as (b) with X=0.4, Y=0.58 and Z=0.02.

Rogers: Stellar plasmas

41

Work performed under the auspices of the U.S. Department of Energy by the Lawrence Livermore National Laboratory under contract number W-7405-ENG-48. References Abe, R., 1959, Theor. Phys. 22,213 Alustuey, A., 1994, this volume Alustuey, A. and Perez, A., 1992, Europhys. Lett. 1,13 Bartsch, G. P., and Ebeling, W., 1971, Beitr. Plasma Phys. 11, 393 Beth, E. and Uhlenbeck, G. E., 1939, Physica 4,915 Bolle, D., 1987, Phys. Rev. A36, 3259 , 1989, Phys. Rev. A39,2752 Cox, J. and Guili, R., 1968, Stellar Structure (Gordon and Breach, New York) Christensen-Dalsgaard, J. and Dappen, W., 1993, Astron. Astrophys. Rev. 4,267 Christensen-Dalsgaard, J. and Dappen, W., 1991, in Challenges to Theories of the Structure of Moderate-Mass Stars, eds. D. O. Gough and J. Toomre (Lecture Notes in Physics, 388, Springer, Heidelberg) Dappen, W., Anderson, L. S., and Mihalas, D., 1987, Ap. J. 319,195 Dappen, W., 1992, Rev. Mexicana Astron. & Astrof. 23,1144 Dashen, R., Ma, S. K., and Bernstein, H. J., 1969, Phys. Rev. 187,345 DeWitt, H., E., 1966, J. Math. Phys. 7,6169 Dziembowski, W. A., Pamyatnykh A. A., and Sienkiewicz, R., 1992, Acta Astronomica 14,5 Ebeling, W . , 1974, Physica 73,573 Ebeling, W., Kraeft, W. p . , and Kremp, D., 1977, Theory of Bound States and Ionization Equilibrium in Plasmas and Solids (Berlin, AkademieVerlag; New York, Plenum) Eggelton, P . , Faulkner, J. and Flannery, G. P., 1973, A&A 23,,261 Erskine, D., Rosznyai, B., and Ross, M., 1994, JQSRT 50 Goldsmith, S., Griem, H. E., and Cohen, L., 1984, Phys. Rev. A30,2775 Graboske, H. C , and DeWitt, H. E., 1974 (unpublished) Guenther, D. B, 1992, Nature 359,585 Hill, T. L., Statistical Mechanics (McGraw-Hill, New York, 1956), Chap. 5 Hummer, D. G., and Mihalas, D., 1988, Ap. J. 331,794 Iglesias, C. A., Rogers, F. J., and Wilson, B. G., 1987, Ap. J. (Letters), 322, L45 Iglesias, C. A. and Rogers, F. J., 1993, ApJ 412,752 Kraeft, W. D., Kremp, D., Ebeling, W., and Ropke, G., 1986, Quantum Statistics of Charged Particle Systems, (Plenum Press, New York) Krasnikov, Yu. G., and Kucherenko, V. I., 1978, Teplofiz. Vys. Temp. 16, 43 Krasnikov, Yu. G., 1977, Sov. Phys. JETP 46,271

42

Rogers: Stellar plasmas

Larkin, A. L, 1960, Sov. Phys. JETP 11,1363 Levinson, N., 1949, Danshe Mat. Fys. Medd, No. 25 Mayer, J. E., 1950, J. Chem. Phys. 18,1426 McChesney, M., 1964, Can. J. Phys. 42,2473 Pisano, C, and McKellar, B. H. J., 1989, Phys. Rev. A40,6597 Rogers, F. J., 1974, Phys. Rev. A10,2441 , 1977, Phys. Lett. 61A, 358 , 1979, Phys. Rev. A19, 375 , 1981, Phys. Rev. A24,1531 , 1984, Phys. Rev. A29, 868 , 1986, Ap. J. 310,723 , 1990, Ap. J 352, 689 Rogers, F. J., 1991, in High Pressure Equations Of State: Theory and Applications, ed. S. Eliezer & R. A. Ricci (North Holland, New York, 1991) Rogers, F. J. and DeWitt, H. E., 1973, Phys. Rev. 8,1061 Rogers, F. J., Graboske, H. C, and DeWitt, H. E., 1971, Phys. Lett. 34A, 127 Rogers, F. J. and Iglesias, C. A., 1991, ApJS 79,507 Rogers, F. J. and Iglesias, C. A., 1993, A.S.P. Conf. Ser., V42, Ed. Timothy M. Brown Saha, M., 1920, Phil. Mag. 40,472 Saumon, D., and Chabrier, G., 1991, Phys. Rev. A44,5122 , 1992, Phys. Rev. A46,2084 Sevastyanenko, V., 1985, Beitr Plasmaphys. 25,151 Simon, N. R., 1982, ApJ 269, L87 Slattery, W, L., Doolen, G. D., and DeWitt, H. E., 1982, Phys. Rev. A26, 225 Swenson, F. J., and Rogers, F. J., 1994, (in preparation) Swenson, F. J., VandenBerg, D. A., Alexander, D. R., and Irwin, A., 1994, (in preparation) Uhlenbeck, G. E., and Beth, E., 1936, Physica 3,729 Wood, R. H., Lilley, T. H., and Thompson, 1978, J. Chem. Soc. Far. Trans. 174,1301

Statistical Mechanics of Quantum Plasmas. Path Integral Formalism. A. ALASTUEY Laboratoire it Physique. Ecole Normalt Superieure de Lyon, 69364 Lyon Ctdtx 07. France

Abstract In this review, we consider a quantum Coulomb fluid made of charged point particles (typically electrons and nuclei). We describe various formalisms which start from the first principles of statistical mechanics. These methods allow systematic calculations of the equilibrium quantities in some particular limits. The effective-potential method is evocated first, as well as its application to the derivation of low-density expansions. We also sketch the basic outlines of the standard many-body perturbation theory. This approach is well suited for calculating expansions at high density (for Fermions) or at high temperature. Eventually, we present the Feynman-Kac path integral representation which leads to the introduction of an auxiliary classical system made of extended objects, i.e., filaments (also called "polymers"). The familiar Abe-Meeron diagrammatic series are then generalized in the framework of this representation. The truncations of the corresponding virial-like expansions provide equations of state which are asymptotically exact in the low-density limit at fixed temperature. The usefulness of such equations for describing the inner regions of the sun is briefly illustrated.

Abstract Dans cette revue, nous considerons un fluide coulombien quantique con43

44

Alastuey: Statistical mechanics of quantum plasmas

stitue de charges ponctuelles (typiquement des electrons et des noyaux). Nous decrivons differents formalismes s'appuyant sur les premiers principes de la mecanique statistique. Ces met nodes permettent de calculer les quantites d'equilibre de maniere systematique dans des limites particulieres. La methode des potentiels effectifs est d'abord evoquee, ainsi que son application aux developpements a basse densite. Nous resumons aussi les grandes lignes de la theorie de la perturbation a N corps. Cette approche est bien adaptee au calcul des developpements a haute densite (pour des fermions) ou bien a haute temperature. Finalement, nous exposons la representation de Feynman-Kac qui conduit a l'introduction d'un systeme classique auxiliaire constitue d'objets etendus, i.e., les filaments (aussi appeles "polymeres"). Les series diagrammatiques de Abe et Meeron habituelles sont alors generalisees dans le cadre de cette representation. Les developpements correspondants de type viriel, fournissent des equations d'etat qui deviennent asymptotiquement exactes dans la limite de basse densite a temperature donnee. L'utilite de telles equations pour decrire les regions internes du soleil est brievement illustree. 3.1 Introduction In many situations, dense stellar matter can be described in terms of quantum plasmas made of electrons and nuclei. Beside its own conceptual interest, the study of the equilibrium properties of such systems turns then to be quite useful for astrophysics. Here, we review first principle formalisms with main emphasis on the path integral representation. First, we define the model and we recall the theorems which guarantee its thermodynamics stability. In Section 2, we describe few standard methods. In section 3, we present the Feynman-Kac representation which leads to the introduction of an equivalent system made of filaments (also called "polymers" in the literature). The corresponding formalism is applied to the derivation of low-density expansions of the thermodynamic functions in Section 4. It is shown that these expansions can be systematically inferred from diagrammatic series where the usual long-range Coulomb divergencies are resummed by a technique similar to what Meeron (1958) and Abe (1959) did in the purely classical case. As exposed in Section 5, the truncations of the expansion of the pressure provide analytic equations of state (EOS) which should be rather accurate at relatively low densities and high temperatures. For instance, these conditions are met in the core of the Sun which can be viewed, as a first approximation, as a three-component plasma made of electrons, protons and Helium nuclei.

Alastuey: Statistical mechanics of quantum plasmas

45

This review is mostly dedicated to a qualitative description of the basic mechanisms and identities underlying the formalisms. A few results relative to exact expansions of the pressure are also presented. 3.1.1 The model We consider a multicomponent system S, with an arbitrary number of species made of point particles. Each particle of species a (an electron or a nuclei in practical applications) has a mass ma and carries a charge ea and a spin aa. Two charges ea and ep separated by a distance r interact instantaneously via the usual two-body Coulomb potential eaepvc(r) with vc(r) = 1/r. The corresponding Hamiltonian for N particles enclosed in a box with volume A is

(i)

where t = [*] is a double index, while k runs from 1 to the number Na of charges of species a and a runs from 1 to the number n 5 of species (N = ^2 Na). The boundary conditions which define HN are of the Diricha

let type, i.e. the eigenwavefunctions of HN vanish at the surface of the box. This non-relativistic Coulomb Hamiltonian is well-suited for practical applications where the mean velocity of the particles is small compared to the speed of light. Let the system be in thermal equilibrium at temperature T (/? = 1 / kBT). The grand-partition function of the finite system reads EA = TrA exv[ where y,a is the chemical potential of species a. In the definition (2), the trace TT\ is taken over all the states satisfying the above boundary conditions and symmetrized according to the statistics of each species. Note that the total charge ^ ea Na carried by each of these states may be different a

from zero. 3.1.2 The Thermodynamic

Limit

Lieb and Lebowitz (1972) have shown that the thermodynamic limit (TL)

46

Alastuey: Statistical mechanics of quantum plasmas

of the present system exists if and only if at least one species obeys Fermi statistics. The TL is defined as the infinite volume limit (A —*• oo), while the chemical potential fia and the temperature T are kept fixed. The existence of the TL means that the thermodynamic quantities relative to the infinite system have the right extensive properties. In particular, the bulk pressure P given through UmlnSA

(3)

is a well-behaved function of the intensive parameters fia and /? which does not depend on the shapes of the finite boxes considered in the TL. If the fugacities za = exp (/?/ia) are small enough (at given temperature), the system surely is in a fluid phase. The local density of any species a then becomes uniform in the TL and reduces to

Furthermore, the infinite system is locally neutral, i.e. e

apa = 0

(5)

a

for any set of fugacities. This is due to the fact that all the excess charges (associated to non-neutral states ( ^ ea Na^0) go to the boundaries in a

order to minimize the electrostatic energy. Once the TL has been taken, these charges are rejected to infinity and the bulk region is neutral. The existence of the TL is guaranteed by the combination of several phenomena. First, the most probable microscopic configurations can be organized in ensembles of finite neutral clusters which are weakly coupled because of screening. This physical idea can be formulated in a precise mathematical way with the help of the cheese theorem and of the harmonicity of the Coulomb potential (see Lieb (1976) for a review). Of course, this screening mechanism requires the presence of charges with opposite sign (otherwise the system explodes). Second, the classical collapse between two opposite charges at short distances is avoided by the uncertainty principle. The latter ensures that the quantum density matrix remains finite, and consequently integrable, at zero separations in configuration space. Third, the H-stability prevents the macroscopic collapse of all the matter into a big molecule. This property stipulates that the hamiltonian Hjy in the infinite volume is bounded below by an extensive constant (i.e. the groundstate of is bounded below by cat x N). Dyson and Lenard (1967, 1968) have

Alastuey: Statistical mechanics of quantum plasmas

47

proved that the H-stability is enforced by Pauli's principle, which requires the presence of Fermions. The existence of the TL implies the stability of matter with respect to electrostatic Coulomb interactions. It explains the usual aspect of the physical systems which can be reasonably described by the present model. Moreover, it justifies the calculation of bulk quantities in the infinite volume without any explicit consideration of boundary effects. 3.2 First Principles Formalisms 3.2.1 Effective potential method This very general method, not specific to Coulomb systems, is due to Morita (1959). It consists in introducing an equivalent classical system made of point objects with many-body effective interactions. For the sake of simplicity, we expose this method in the framework of Maxwell-Boltzmann (MB) statistics and we just mention how the exchange effects due to Fermi or Bose statistics can be included in a straightforward way. In configuration space, the MB grand-partition function reads

r

EnfV«o7A,,II*

Na=0 a

iv

«'

•'A* ,•

(6)

x < RN\exp(-PHN)\RN > with | RN >= ®j | fi >. In (6), the contributions of the spins reduce to the trivial degeneracy factors (2aa + 1) a because Hjv does not depend on the spins. The diagonal part < Rpf | exp (—/Jifyv) \ Rpt > of the quantum density matrix can be always interpreted as a classical Boltzmann factor associated to an effective interaction
>= n ( 2 7 r A ? ) 3 / 1 exp(-/3v?N)

(7)

where Aj = (/?7i2/mt) is the de Broglie wavelength associated to the i"1 particle. Contrarily to the interaction part of Hpj,
48

Alastuey: Statistical mechanics of quantum plasmas

The total three-body effective interaction ^ ( F j , fj, fk) between three particles i, j and k is then decomposed as

with /[(2TTA2)3/2

Similar decompositions of 3) in terms of the two-, three-, ..., N-body u-interactions can be found recursively. The insertion of these decompositions in (7) and use of the configuration^ expression (6) for Hj lead to (2
°° *—^«

^

*

I 'IIP

1 *

N

f

I I J J 1 ra 1 *• IV

I

I A

Jtf

t=l

(ID

E -81+ • • • ] • i
>

This identity shows that the MB grand-partition function of the quantum system S is identical to the one of an auxiliary classical system made of point objects interacting via two-, three-, ... and N-body effective interactions. For Fermi or Bose statistics, the same equivalence holds with effective u-interactions defined in terms of the symmetrized matrix elements of exp (—(3HN). The introduction of the above equivalent system allows to study the pressure, and consequently all the thermodynamic functions of S, within the usual methods of classical statistical mechanics. However, we stress that the difficulties inherent to quantum mechanics rely in the deter(2)

mination of the many-body effective interactions. In the limit h —>• 0, u\j

Alastuey: Statistical mechanics of quantum plasmas

49

Fig. 1.1 Solid line : the effective-potential u+_(r) between two opposite charges (+e) and (—e) ; dashed line : the Coulomb potential — e 2 /r.

goes to et ej vc(\fi — fj\) while all the higher order effective interactions u vanish. The effective-potential expression (11) then reduces to the classical grand-partition function of S, as it should be. The effective potential method has been first applied to quantum plasmas by Ebeling (1967) with the purpose of deriving low-density expansions of the thermodynamic functions. This method is indeed well-suited for such calculations for the following reason. By construction, the N-body effective interaction u ^ goes to zero for any large separation of its arguments. So the configurations where vlN) contributes to HA, are made of finitesize clusters which all contain at least N particles. This suggests that the contributions of vf-N) to the pressure should be at least of order pN (p is a generic notation for the particle densities). According to this simple rough argument, the density-expansion of the pressure up to the order pN included can be calculated exactly by omitting all the effective interactions v^) with M > N. In particular, Ebeling and coworkers (see references in the book by Kraeft et al. (1986)) performed calculations up to the order p*l2 by keeping only the two-body effective interactions vf-2\ The potentials u*2) can be viewed as Coulomb interactions regularized at short distances. Indeed, at large distances u\j behaves as the Coulomb potential, i.e., e< ej / |n — ry|, while u\-' remains finite at zero separation because quantum diffraction smoothes out the 1/r-singularity at the origin. For fixing ideas, we have drawn in Figure 1 the effective interaction u!j._(r) between two oppposite charge (+e) and (—e) separated by a distance r. The auxiliary classical system with the two-body Coulomb-like interactions ir 2 ' can be studied by various standard methods. In particular, the

50

Alastuey: Statistical mechanics of quantum plasmas

density-expansion of the pressure may be readily obtained from Abe-Meeron diagrammatics. The corresponding graphs follow from systematic resummations of the convolution chains in the genuine Mayer graphs built with bonds (exp [—/3u^2'] — 1). This procedure eliminates the divergencies induced by the Coulomb long-range part of t^ 2 ), and make an integrable screened potential appear. The definition of the screened potential in terms of u^ is identical to the one of the familiar Debye-Hiickel potential in terms of the 1/r-Coulomb potential. As it will be discussed with more details in Section 5, and contrarily to what was expected, the density expansions calculated by the above authors are correct only up to the order p2 included. In fact, the contributions of the three-body effective interactions t^3) are of order ps^2 instead of p3. This is due to the fact that u^ is long ranged, i.e., not integrable at large distances. Of course the exact expression of the virial coefficient of order ph'2 might be obtained by the above method via a proper inclusion of the three-body effective interactions.

3.2.2 Many-Body Perturbation This method consists in a perturbative treatment of the interaction part of the Hamiltonian Hjy given by (1). It allows systematic expansions of the equilibrium quantities in powers of the charges which characterize the strength of Coulomb interactions. The formalism is carefully presented in the book by Fetter and Walecka (1971). Here, we just describe the main steps of the method. The perturbative expansion is carried out in the framework of the grandcanonical ensemble, at fixed temperature and fugacities (the corresponding expansion in the canonical ensemble is more cumbersome because of the so-called 1/N tails). The key starting identity is the Dyson series which represent the quantum Gibbs factor (i.e. the density matrix) as

exp (-0HN)

=

where Ho and V are the kinetic and interaction parts of HN respectively. Moreover, V® is the imaginary-time evoluted operator.

Alastuey: Statistical mechanics of quantum plasmas

V° = exp(TH0)Vexp(-TH0)

51

(13)

and TT is the familiar T-product which ranges all operators from left to right which decreasing "times" r. The insertion of the Dyson series (12) in the trace (2) leads to the required perturbative representation of any equilibrium function of S. Each term in these perturbative series obviously reduces to a statistical equilibrium average < ... >o calculated with the freeparticle Gibbs measure exp [—(3(Ho — ^^fia Na)] / E\to. For instance, the a

V-expansion of the pressure given by Eq. (3) involves

< TT[VT\ .. .Vr°n} >0

(14)

Since V is diagonal in configuration space and reduces to a sum of two body terms, one easily guesses that the above averages may be expressed as multiple spatial integrals over products of two-body interactions weighted by equilibrium configuration distributions relative to the reference system SQ. Moreover these distributions reduce to products of one-body terms because the particles of So are not coupled together. The precise structure of the perturbative terms, like (14) for instance, might be determined via suitable insertions of the familiar configurational Slater sums. However, this leads, at high orders in F , to complicated counting problems of permutations. In order to circumvent these difficulties linked to statistics, it is particularly convenient to introduce the second quantization representation defined in terms of the operators ^a(raz) or ^a(foz) which respectively annihilates or creates one particle of species a at position Fwith spin az along a given z-axis. In the framework of the grand-canonical ensemble, this representation is in fact quite appealing since the total number of particles is not fixed. Furthermore, the bosonic or fermionic nature of each species is automatically taken into account via the commutation or anticommutation relations,

V " ) ± ¥a(f,*")*i(f,(r') = «»•,*" X 6(r- f) *«(* O t a ( f > " ) ± »a(f>")*a(*?. "') = 0 while two operators associated to different species always commute. The second-quantized expressions of the operators of interest read

52

Alastuey: Statistical mechanics of quantum plasmas

*«(->*)

(15)

(16) x e a e,t> c (|f(17) (now JVa is the number-operator which counts the particles of species a). The averages like (14 ) appearing in the V-expansions then involve traces over the Fock space of the products of ( ^ ) T . and ($^) r . weighted by the free-particles measure

(18)

Since the measure (18) is Gaussian in the $'s, and since all the \P's commute or anticommute except for c-numbers, the above traces can be calculated with help of Wick's theorem. Similarly to the formula which relates any moment for a Gaussian distribution of real variables to the covariances, the free-particle average of any product of $'s is identically equal to the sum over the products of all the possible corresponding two-operators contractions. The sole non-vanishing contractions are those which conserve the number of particles of each species. They reduce to the finite-temperature one-particle Green's functions of So, i.e.

The second-quantization analysis of the V-expansions provides well-defined rules for interpreting each term as a Feynman graph similar to those which appear in field theory. Roughly speaking, the graphs are made of oneparticle fermionic or bosonic oriented loops connected by two-body interaction lines. The two points linked by a given interaction line are affected of the same "time" r,-. They may belong either to the same loop or to different

Alastuey: Statistical mechanics of quantum plasmas

53

loops. Each loop contains an arbitrary number nj_, of points f,- at time r<. Its statistical weight is given by the product (20) t=i

of the free-particle Green's function connecting two successive points on the loop (rnL+l = fi and rnL+1 = ri). The weight of a loop containing only one point reduces to

(21) which is nothing but />a,o/(2<xa + 1) with paf> the particle density of So (note that pa 3 (an example of such ring graphs is shown in Figure 3). The sum of all these ring contributions is finite. For the particle correlations, one must resum chain structures instead of rings (see

54

Alastuey: Statistical mechanics of quantum plasmas

CXO (W

Fig. 1.2 a, 1.2b Two Feynman graphs which contribute to the e2-expansion of the pressure. Big oriented circles : fermionic or bosonic loops ; dashed lines : two-body interactions connecting pairs of points (small black circles) on the loops.

o-b Fig. 1.3 A typical ring graph.

e.g. Cornu and Martin (1991)). All these mathematical recipes reflect the screening of the bare Coulomb interaction via many-body collective effects. The expansion parameter in the above perturbative series is the charge e, where e is a generic notation for the charges of the particles. The lowest order correction to the pressure PQ of the free gas So is given by the exchange graph (2a) which is of order c2 (the direct contribution of order e2 vanishes because of overall neutrality). The next correction is not of order ^* as a consequence of the long-range divergencies which prevent the pressure to be an analytic function of e2. This correction is given by the sum of all the ring graphs analogous to (2b) and (3) (physically, this contribution corresponds to the RPA mean-field approximation). The analytic evaluation

Alastuey: Statistical mechanics of quantum plasmas

55

of the ring sum for any temperature and fugacities is quite cumbersome. To our knowledge, it has been done explicitely in two cases. First, for an electron gas (one-component plasma of fermions) at zerotemperature, Montroll and Ward (1958) have shown that the ring contribution to the internal energy per particle E/N is of order e 4 lne 2 . In fact, they recover the e2-expansion.

E

N

35/37T4/3ft2

10m

34/3 H

/^

first derived by Gell-Mann and Brueckner (1957) in the framework of the adiabatic perturbation-switching formalism (the graphs introduced in this perturbative treatment of the ground state turn out to be quite similar to those described above). The expansion (22) can be rewritten in terms of the single dimensionless parameter rs defined as the ratio of the mean interparticle distance a = (3/4irp)1/3 by the Bohr radius 03 = h2 /me2. The appearance of rs is quite natural since ra is proportional to the ratio of the mean Coulomb energy (~ p1^3) divided by the Fermi kinetic energy (~ p 2 / 3 ). The expansion (22) should converge for small values of r,, i.e., at high densities. It can be also viewed as an asymptotic expansion in inverse powers of the density. The second analytic evaluation of the ring contribution has been performed in the high-temperature limit. This contribution then becomes of order e 3 and leads to the well-known classical Debye-term — K% / (24?r) with KD = (4TT (3 Yla e a Pa) • The e2-expansion should be quite appropriate at high temperatures since the mean Coulomb energy e 2 /a then becomes small compared to the thermal kinetic energy kpT. Consequently, a natural by-product of these perturbative series is the derivation of high-temperature expansions. However, the reorganization of the e2-series in /J-series is not straightforward because both expansions involve several independent dimensionless parameters. Indeed, in addition to the coupling parameter /3e 2 /a(~ (3) which measures the strength of Coulomb interactions, they depend on the quanticity parameter A/a(~ /31/2) which controls diffraction and degeneracy effects. The /^-expansion of f3P up to the order /?5/2 has been calculated by De Witt (1966) in the framework of Maxwell-Boltzmann

56

Alastuey: Statistical mechanics of quantum plasmas

statistics. Keeping only the free term /3PQ and the ring sum, he found

(23)

Notice that the exchange effects will give terms of orders /? 3 / 2 (the usual Fermi or Bose ideal term), (32 (contribution from the exchange graph (2a)), /3s/2 and so on. Moreover, the /?s/2-term arising from the /^-expansion of the ring sum is proportional to h. In fact, the higher order interaction terms in (23) involve any integer power of h, even or odd, positive or negative. This general singular structure with respect to h of the /^-expansions hold in particular for the OCP. In that case, the pressure (and the other thermodynamic functions) may be also represented by the familiar Wigner-Kirkwood (WK) expansion in powers of h around the classical limit (the WK expansion cannot be used for a multicomponent system because the collapse of opposite charges makes the classical limit singular). One then concludes that the /?- and WK-expansions do not coincide. As argued by De Witt (1962), this a priori surprising result can be interpreted by noting that, in the high-temperature limit, the de Broglie wavelength A = (/3h2 / m) becomes much larger than the Landau length I = /3e2 which is a classical average range of the Coulomb interactions. The validity of the WK-expansion requires, roughly speaking, A
Alastuey: Statistical mechanics of quantum plasmas

57

Fig. 1.4 A typical Ladder graph.

3.3 The Feynman-Kac Representation 3.3.1 The case of one particle For the sake of pedagogy, we first illustrate the Feynman-Kac representation for one particle with mass m submitted to an external potential V(f). According to the original path integral formulation introduced by Feynman (1965), the diagonal matrix element of exp[—/?(—^A + V)] reads

f) all paths

V

(24) /

where S(f(t)) is the classical action in the potential — V, (25) for a path r (t) going from r to r in a "time" /3h. The summation in (24) is taken over all such paths. The variable changes t = &(3h and f(t) = f+X { (s) with A = (/?/i 2 / m ) i n ( 24 ) and (25) lead to the so-called Feynman-Kac (FK) representation (see e.g. B. Simon (1979))

-/?(-^A + V)]\f >=

(2TTA2)3/2

with

I) = f1 dsV(r + Af(«))

(27)

58

Alastuey: Statistical mechanics of quantum plasmas

The factor exp[—(3V*(r,£)] obviously arises from the potential part of the action S. The corresponding kinetic factor, i.e. exp[—^/0 ds £*(«)], is absorbed in the normalized Gaussian measure V( f) which defines the functional integration over all the dimensionless Brownian bridges f (s) subjected to the constraint f (0) = £(1) = 0. This measure is intrinsic, i.e. independent of all the physical parameters, and its covariance is given by

It is very natural to interpret exp[—pV*(r,£)] as the Boltzmann factor associated to a classical closed filament located at f and with shape parametrized by f(s). The potential V* seen by this filament is the average of the genuine potential seen by the particle when it runs over the line f+ A £($). The FK representation (26) stipulates that the Gibbs factor for the quantum point particle exactly reduces to the shape-average of this Boltzmann factor with the Gaussian measure P ( f ). The de Broglie wavelength A controls the typical size of a filament: roughly speaking, the statistical weight of a filament with size R behaves as exp(—R2/X2). Note that the classical limit of the density matrix is immediately obtained from (26) by replacing V* by V(r) : in this limit, A goes to zero and the spatial extension of the filaments can be neglected in the calculation of V*( r,£). 3.3.2 The system S with Maxwell-Boltzmann statistics First, we only consider MB statistics. The corresponding grand-partition function Hj^B of S is given by the space-configurational expression (6). Similarly to the expression (26), the FK representation of the diagonal matrix elements of exp(—f3Hflf) reads

= (29) e

e

x exp " I E < J I

dsv

c{\Ti + A;6(s) - Tj - \j(j(s)\)

where each Brownian bridge &($) parametrizes the trajectory of the ith particle in the genuine Feynman path integral, and is distributed according to the intrinsic Gaussian measure defined above. This representation suggests to introduce the following auxiliary classical system S* made of closed

Alastuey: Statistical mechanics of quantum plasmas

59

Fig. 1.5 A few closed filaments which belong to S*. The small black circles represent the positions of the filaments, while the closed curves attached to each of them represent their shapes parametrized by

filaments interacting via two-body forces. Each filament is characterized by its spatial position r and two internal degrees of freedom, the dimensionless path f (s) associated to its shape and the species index a which specifies its spatial extension Xa and the strength ea of its coupling with the other filaments. We note £ = ( a, f, £ ) the state of such a filament. Two filaments in states £ and £' interact via the two-body potential eae0/v(€,£') with

v(£, £') = [X dsve(\f+ \J(s) -?-

\a.?(s)\)

(30)

Jo This potential is different from the electrostatic interaction energy between two uniformly charged filaments, because the average of vc is taken over positions at the same "time" s. However, it reduces to the Coulomb potential at large distances, i.e. v{£, £') ~ l/|r - f*|

when

\r - f*| -> oo

(31)

since the filaments can then be replaced by points. A few filaments are drawn in Figure 5. The insertion of the FK representation (29) in the space-configurational expression (6) of Hj^fl leads to a sum over the states of S* weighted by Boltzmann factors associated to the filaments interactions. In this sum, it is quite natural to define the phase-space measure d£ for a filament as d£="da"drD(£) and to set z{£) = (2
60

Alastuey: Statistical mechanics of quantum plasmas

JV=O

^ fc=l

where /(£*,£/) is the Mayer-bond associated to v(£k,£t),

f(£k,£i) = exp[-^eakea,v(£k,Si)]

- 1.

(33)

The identity (32) exemplifies the equivalence between the quantum system SMB and the classical system S* for studying equilibrium properties. In «S*, the quantum mechanical aspect of SMB is hidden in the complex nature of the filaments. In fact, these extended objects describe quantum fluctuations of the point particles. In the effective-potential method, the auxiliary classical system is still made of point objects while the quantum effects are taken into account in the two-body and higher order effective interactions. Here, we stress that the interactions between the filaments are strictly of the two-body type.

3.3.3 Inclusion of Fermi or Bose statistics In order to take into account the exchange effects due to Fermi or Bose statistics, we express the trace (2) over the properly symmetrized states {RjvOTtf > 3 in configuration and spin spaces. These states are defined via the usual Slater sums

>S=

n (jy a ;)i/a

£ n *"{Va) ®i |r-V>a(0<4Q(l) >

(34)

In (33), Va is a permutation of (l...Na), Va(i) = (Va(k),a), and ea(Pa) is either 1 if the particles of species a are bosons (oa integer) or the signature (±1) of Va in the fermionic case (aa half-integer). Furthermore, (8) means a tensioral product over the one-body states \faz > describing a particle localized at f with the projection of its spin along a given 2-axis equal to

Alastuey: Statistical mechanics of quantum plasmas

61

az (az may take (2 aa + 1) values). Using (34), we obtain

n

zNa ^(0>

(35)

Since the hamiltonian .fl^ (equation (1)) does not depend on the spins, the spin-part of the matrix elements contributes the trivial degeneracy factor Yiff* Hi < av' {i)\ava(i) > w n i c ^ o n ly depends on the pairs of permutations The Slater-sum representation (35) of HA alows a natural identification of the exchange effects. Indeed, the "square" terms (Va = V'a for any a), where the diagonal matrix elements of exp(—(3H]v) in configuration space appear, obviously correspond to MB statistics. A "rectangle" term (Va / "P'a f° r a * least one species) involves the exchange of n particles (n > 2). The corresponding matrix elements of exp(—(3HN) are off-diagonal with respect to the positions of the exchanged particles. The structure of the FK representation of these off-diagonal matrix elements can be interpreted in two ways which lead to different treatments of the exchange contributions. A first possible interpretation consists in introducing opened filaments J-^i associated to the exchange of a particle a from position rjt to position fi. The shape of J7^ is parametrized by

w&(*) = (1 - s)fk + sft + Xa({s)

(36)

which describes a path of the exchanged particle in the genuine Feynman path integral (uki (0) = Ffc and Uki (1) = ft)- The above closed filaments £ are again associated to the non-exchanged particles. For instance, if one considers the off-diagonal matrix element fN\ exp(-PHN)\rir2r3

...fN>

(37)

which corresponds to the exchange of two particles, there appear two opened filaments T"2 and T2\ a n ^ (N-2) closed filaments S3,..., £N- This situation is illustrated in Figure 6. By collecting together all the contributions with the same finite number n of exchanged particles, we are then left with a problem of impurities, the n opened filaments, immersed in the bath S* of closed

62

Alastuey: Statistical mechanics of quantum plasmas

Fig. 1.6 Two opened filaments F°2 and T%x surrounded by closed filaments of S*.

filaments. This inhomogeneous situation can be dealt with along standard perturbative techniques where the reference system is the homogeneous bath S* described in section 3.2. The second interpretation of the exchange contributions is due to Brydges (private communication). Any permutation of n objects, which characterizes the exchange of n particles, is the product of p cycles with p < n. Therefore, the corresponding n opened filaments may be always viewed as a set of p closed filaments. Each of these new closed filaments is made of q opened filaments, q > 1, and will be noted £(*). It can be associated to a closed path described in a "time" qflh. For instance, in the above example the union of T^ and T%± gives raise to £^2K Therefore, the whole Slater representation (35) of 3\ is identified as the grand-partition function of a mixture of classical closed filaments £^ with q = 1,2,3...oo (the S^'s are the closed filaments £ introduced in section 3.2). The typical size of £^ depends on the "time" q(3h. Its activity incorporates a self-energy term arising from two-body interactions of the type (30) between the opened filaments which constitute £^q\ In the present approach, the MB and exchange effects are treated on an equal footing, while the previous interpretation leads to a perturbative treatment of the exchange contributions.

3.3.4 Possible

applications

Within the above equivalences which follow from the FK representation, the equilibrium properties of the quantum system S can be studied by applying the usual methods of classical statistical mechanics to S*. Indeed, the system of closed filaments is isomorphic to an ordinary classical system

Alastuey: Statistical mechanics of quantum plasmas

63

of point objects with two-body interactions. In «S*, the position Fis replaced by the generalized coordinate £. The familiar calculation rules remain unchanged, apart from this simple substitution, because all the quantities of S* behave as commuting c-numbers (the operatorial structure of quantum mechanics "disappears" in the FK representation). The familiar Mayer series can be extended to S*. For systems with shortrange forces, Ginibre (1971) proved the convergence of the activity expansions by exploiting the classical structure of the Mayer-like graphs. For the present Coulomb systems, the Mayer-like series for S* constitute a powerful tool in the systematic derivation of density-expansions (see Section 4). Aside from these exact calculations, one might introduce approximate methods by extending well-known integral equations (likeHNC) to the correlations of S* (see e.g. Chandler (1981) for a review relative to systems with short-range forces). Although such extensions do not cause any trouble at a formal level, we stress that the explicit calculations might be rather difficult because of the functional integrations over the shapes of the filaments.

3.4 Virial-Like Expansions Like in the classical case, all the above Mayer-like graphs diverge because of the long-range Coulombic nature of the filament-filament potential. Alastuey, Cornu and Perez (1993) have shown that the corresponding series can be reorganized in series of finite resummed graphs. In this section, we just sketch the main steps of their method. First of all, they consider only MB statistics and the resummation procedure is applied to the Ursell function h(Sa, £(>) of S*. The density expansions of the MB thermodynamic functions of SMB are then evaluated via standard identities. The exchange effects are included perturbatively within the impurities approach exposed in Section 3.3.

3.4-1 Diagrammatic resummations The two-point Ursell function h(£a, £b) of S* is defined as usual by

p(£a)p(£b)h(£a,£b) = z(£a)z(£b)]im

f^fA

(38)

It can be represented by series of Mayer graphs F in terms of the filament density

64

Alastuey: Statistical mechanics of quantum plasmas

Fig. 1.7 A typical graph T which contributes to the Mayer-like densityexpansion of h(£a,£b). The closed filaments are drawn as in Figure 5, with the sole difference that the positions of the root filaments £a and £\, are representated by white circles. The tubes connecting the filaments are the Mayer bonds /(£<,,£i) and f(£i,£t).

The F's are defined via the familiar topological prescriptions, where the usual points are now replaced by filaments. Each graph is built with the two root filaments £a and £\, and n field filaments £\,...,£n which are integrated over. Each field filament £{ has a statistical weight p(£i). Two filaments are linked by at most one f-bond (33). Each F is connected and does not contain articulation filaments. A typical graph F is drawn in Figure 7 and its contribution reads

(40)

Note that the filament density p{£\) cannot be factored outside the integral in (40) because it depends on the shape of the filament. The contribution of each graph F is divergent because of the non-integrable 1/r-decay of the Mayer bond f associated to the Coulombic behaviour (31) of the potential v. For instance, the spatial integral over ri involved in (40) is not convergent at large distances. The resummation procedure starts with the following decomposition

Alastuey: Statistical mechanics of quantum plasmas

65

,n =fc(\r- fD + bldr- f\) ,1

+ / ds[\J(s).V + \a,?(s).V']fc(\r-r'\) + fT(f:,£')

(41)

Jo

where fe is the Coulomb shape-independent bond fc(\r - 7*1) = -(3eQea,vc(\r - f |)

(42)

This decomposition defines in fact the truncated bond fa. By construction fa decays as 1 / \r — r '| 3 when \r— f '| —* oo. Inserting (41) in each graph F, we obtain a new representation oih{£a, C\,) in terms of graphs f built with bonds / which maybe either fc, f% / 2, Af -V/c o r faSince fT is almost integrable, the divergencies in the F-diagrams are induced by the other non-integrable bonds / . They are of the same type as those encountered in the purely classical case, so they can be eliminated via the same mathematical recipe (first introduced by Mayer (1950) and Salpeter (1957)), i.e., the resummation of all the convolution chains built with the Coulomb bond fc. This procedure transforms the whole set of f-diagrams into a set of new graphs II built with resummed bonds. For obtaining the graphs II, one first distributes all the diagrams F in resummation classes characterized by given chain-structures. For instance in Figures 8a, 8b and 8c we show three diagrams f belonging to the same class. The summation of all the F-diagrams in a given class, then amounts to suppress all the intermediate filaments C in the Coulomb convolution chains, while the remaining filaments V are linked by resummed bonds F which reduce to the sums of these chains. The graph II generated by the f -class illustrated in Figure 8, is drawn in Figure 9. The topological structure of the genuine graphs is conserved through the resummation process. As a remarkable consequence of a factorization property of the symmetry counting factors, the resummed bonds F are generic in the sense that they do not depend on the global structure of the graphs II. In fact, F(Vi,Vj) only depends on the chain structures inserted between V% and Vj in the genuine F-diagrams. Only four kinds of F-bonds appear. The single chains with ending bonds which are either fc or Ajf,-.Vt/c (^jfi-Vj/e)> l ea d to three resummed bonds FD, (Af. V-^b) a n d Fdip. All the other structures, involving several chains and/or the bonds fa and /c/2, give raise to the fourth bond FR. All these cases are illustrated in Figures 8 and 9. The resummation procedure automatically excludes the convolutions FD *

66

Alastuey: Statistical mechanics of quantum plasmas

(a)

-o

Fig. 1.8 a, 1.8b, 1.8c : Three f-diagrams which belong to the same resummation class. For clarity, the shapes of the filaments are not represented. The big black circles are filaments V which remain fixed through the resummation process. The small black circles are filaments C which are "eaten" by the resummation "machinery". Solid lines : bonds fe ; lines with one arrow : bonds A£.v/ C (the arrow indicates the point with respect to which acts the gradient) ; double solid lines : bonds / c 2 /2 ; dashed lines : bonds

Fig. 1.9 The H-graph generated by the f-class illustrated in Figures 8a, 8b and 8c. Strings : bonds FQ ; strings with one arrow : bonds X^.^FD ; strings with two opposite arrows : bonds Fdip ; hatched bubbles : bonds FR.

*i ft- Vi^b * FD, FD* \j(j. S7jFD and A,-£. between j Ijtwo filaments (Vi,Vj) in any graph II. The bonds F can be calculated explicitely in terms of the MB particle densities. Indeed, since the /c-bonds are shape-independent, the functional

Alastuey: Statistical mechanics of quantum plasmas

67

integrations over the shapes of the intermediate filaments C in the chains lead to the replacement of each p(£) = pa( £) by the MB particle density

(43)

Thus the summation of all the convolution chains can be performed in terms of the familiar Debye potential 0o( r ) = exp (—KT)/T with K = (47r/?2_,e^p£ffl) . In particular, one finds a

FD(Vi,Vj) = -PeaieajD(\?i ~ ?j\)

(44)

We stress that , contrarily to FD and A £. V^D which decay exponentially fast (like 4>D essentially), the bonds Fap and FR are found to decay algebraically as 1/r3 when r —• oo. These behaviours are related to the efficiency of the screening of the multipole-like interactions, which appear in the expansion of the bare filament-filament potential (33) in powers of f and £'. The charge-charge and dipole-charge interactions are perfectly screened via the usual classical process while the screening of the higher order multipole interactions is inhibited by quantum fluctuations. The above slow decays should ultimately pollute the correlations with algebraic tails, in accord with the absence of exponential clustering predicted by Brydges-Seiler (1986), Alastuey-Martin (1988-1989) and Cornu-Martin (1991). Although the screening mechanisms are less efficient than in the classical case, they do eliminate the long-range Coulomb divergencies, i.e., each graph II does converge, as expected. The resummed diagrammatic expansion of the two-point correlations of SMB is immediately obtained by inserting the previous Il-representation of h(£a,£b) i n the identity

P%B(<*a7a;<xbrb) = J V((a)V{ib)p{£a)p{^)h{Sa,eh)

(45)

In (45), each graph II is multiplied by p{£a) p(£b) a nd integrated over the shapes £a and £b of the root filaments £a and £b. Note that if we set h —• 0, the bonds A ^.V^D a n d Fdip disappear, while the bond FR reduces to (exp (FD) — 1 — FD) and the statistical weights p(£) are replaced by the particle densities. The so-called nodal expansion of the classical correlations first derived by Meeron (1958) is then recovered.

68

Alastuey: Statistical mechanics of quantum plasmas

3.4-2 The MB thermodynamic functions For evaluating the density expansions of the MB thermodynamic functions, it is convenient to start from the identity a-pMB

a-pMB

a

i-X

d

f

~— = ^ - + £ £ JO/ 9 JI £>0, fifyaepvc{r) (46) A a,0

which expresses the free-energy per unit of volume in terms of the two-point correlations. In (46), Pxg 1S calculated for a fictitious system Sg where all the interactions are multiplied by the dimensionless coupling parameter g. Moreover the temperature and the MB particle densities of Sg are identical to those of S. The insertion of the II-representation of p"* in (46) provides a diagrammatic expansion of the free-energy. At this level, the Il-series for (0FMB/A) do not constitute an explicit expansion with respect to the particle densities, because of the presence of the shape-dependent statistical weights pg(£). In fact, the functional pg{€) can be itself expanded in powers of the p%B 's as follows. One starts from the familiar Mayer expansion of pg{€) in terms of the fugacities z{£) = z*. The corresponding set of divergent Mayer graphs G is then transformed into a set of convergent graphs P via a resummation process similar to the one described in Section 4.1. Now there appear five resummed bonds which can be expressed in terms of functions entirely scaled by KDfZ = (47r/?£ a e£z a ) and of / r scaled by the Landau and de Broglie lengths (these lengths depend only on the temperature). A scaling analysis with respect to KQ,Z of the spatial integrals in the graphs P allows to express pg(£) as a double integer series in z*1/2 and Inz*. The half-integer powers come from KD,« ~ z*1?2 while the logarithms arise from the l/r 3 -tail in / j . Eventually, the fugacities are eliminated in favor of the particle densities by and using (43), and pg(£) is rewritten as a double integer series in (pMB) MB \np , the first terms of which read

/ dsvc(\f + A7fi(s) - A«f()|)) Jo

'

(47)

- JX>(|))exp(-/3eae7 J dsvc(\r + X^(i(s) - Xa(t(s)\))\

After the replacement of the statistical weights pg(£) by their particle-

Alastuey: Statistical mechanics of quantum plasmas

69

densities expansions, it remains to determine the density dependence of the spatial integrals in the II-graphs. This analysis is carried out via a scaling method similar to the one used for the P-graphs with KD in place of KD,ZIndeed, all the resummed bonds F can be expressed in terms of KD-scaled functions and fa. The final form of the density expansion of (/?F Mfl /A) is a double integer series in (pMB)1/2 and lnpMB. The corresponding expansion of the pressure @PMB derived from the identity

has the same structure. We stress that the macroscopic instability of SMB does not cause any divergency in the virial coefficients of the above series. However, it should prevent their convergence.

3.4.3 Exchange contributions The exchange effects are taken into account by inserting the Slater expansion (35) of EA in the grand-canonical expression (3) of /?P. This gives

= (3PMB + }2En

(49)

n=2

where En is the contribution of n exchanged particles. According to the impurities point of view, En may be expressed in terms of the density p{£a\Fki) of S* in presence of n opened filaments J-ki- For instance, Ei reads

J x exp [ - / ? £ dg J dSaP(Sa\g; J? 2 , Jft)(t»(f., Jfi) + v(€a, In (50), the first exponential represents the exchange contribution in the vacuum while the second one describes the many-body effects on the twoparticle exchange. The structure of all the other En's is similar to the one displayed in (50). The diagrammatical method exposed in Section 4.1 can be extended to

70

Alastuey: Statistical mechanics of quantum plasmas

the inhomogeneous system S* in presence of the .F's. This provides a representation of p (€a\^Fki) in terms of graphs made of the n opened filaments Tki and closed filaments. The statistical weight of a closed filament is the density p(£) of the homogeneous system. Two closed filaments are linked by at most one resummed bond F. A closed filament £ and an opened filament T are linked by at most one "external" Mayer bond (exp [—f3v(£,Jr)] — 1). The density expansion of En follows from use of the previous diagrammatic representation of p{£a\Fki)- The integrals over the positions of the opened and closed filaments can be evaluated via a scaling analysis with respect to nD. The resulting expression for En takes the general form zn multiplied by a double integer series in (pMB) and lnpMB. 3-4-4 General structure of the density

expansions

The present method gives access first to the pressure. Use of the expansions of f)PMB and En in (52), allows to rewrite f3P as a triple integer and \npMB. The fugacities and MB densities are series in z, (pMB) then eliminated in favor of the real densities pa by combining the identities p%B = zad(/3PMB)/dza and pa = zad((}P) / dza (p%B differs from pa at given fugacities). The resulting virial expansion of the pressure is a double integer series in p1/2 and Inp. The density expansions of the other thermodynamic functions, obtained via the usual identities, have the same structure. According to the above scaling analysis, the contributions of a fully resummed graph of the II-type to a given virial coefficient, reduce to subgraphs built with bonds / j , FD, fc Af. V^b> A£. v/c- The physical nature of these contributions can then be identified by inspection of the bonds and filaments which appear in the subgraphs according to the correspondance rules, - long-range classical interactions : bonds fc and - quantum diffraction : bonds A^. v / c and Af. - bound and scattering states : bonds fa - exchange : opened filaments In general, all these physical effects are coupled at high orders in p. We stress that, in the present formalism, all the corresponding contributions are treated simultaneously in a systematic and coherent way. Of course, the exact density expansions can be also derived in the framework of the formalisms described in Section 2. However, the FK diagrammatic method is well-suited for this purpose. In particular, all the long-

Alastuey: Statistical mechanics of quantum plasmas

71

range Coulomb divergencies are automatically eliminated at any order in the density via the introduction of the generic resummed bonds F. In the effective-potential method, it remains to treat the long-range part of the many-body effective interactions in a systematic way. Furthermore, the density is the natural expansion-parameter in the FK diagrammatics. Indeed only a finite number of graphs II contributes to a given virial coefficient (roughly speaking the order in the density of II increases with the number of filaments and the number of bonds). On the contrary, the calculation of such a coefficient from many-body perturbation requires the collection of infinite sets of Feynman graphs with Ladder structures. Moreover, in this approach, the elimination of the fugacities in favor of the densities is more cumbersome than in the FK method where part of this transformation is automatically done. Although the FK representation is quite efficient for performing infinite resummations at a formal level, one has to keep in mind that, for practical calculations, the difficulty relies in the functional integration over the shapes of the filaments. The purely diffraction contributions, which only involve moments of the Gaussian measure T>(£), are evaluated from the covariance (28) by using Wick's theorem. The contributions of bound and scattering states, which involve one or more bonds /x, cannot be so easily computed in the space of filaments. In fact, it is more convenient to express them in terms of matrix elements of exp (-(3HN) by applying backwards the FK formula (26). The corresponding explicit calculations are then limited by the absence of exact solutions for the N-body quantum-mechanical problem, as soon as N is larger than two. Beside the complexity of the graphs, this problem intrinsic to quantum mechanics and which appear in any formalism, prevent a detailed knowledge of the virial coefficients at orders higher than

3.5 Low-Density Equations of State 3.5.1 the exact EOS at the order phl2 The truncation of the above virial expansion of the pressure at the order

72

Alastuey: Statistical mechanics of quantum plasmas

p5/2 gives (Alastuey and Perez (1992)) K3

a

a,fi

3

X)

l l l n («*«/

ID

C2 1/2

with Kx, = (47ry9 ^ e2apQ)

, / a/3 = ^CaC/3, map = mampl{ma+mp),\ap

=

a

(Ph2/mQl3)1/2, C\ = 15.205 ± .001, C2 = -14.733 ± .001 and EulerMascheroni's constant C = .577216... Moreover, Q{-y/2lQp/\ap) is the so-called quantum second-virial coefficient first introduced by Ebeling

(52)

while E(—y/2laa/Xaa)

is the exchange integral,

Alastuey: Statistical mechanics of quantum plasmas

l dr< -f\e-

73

ph

""\r>

(53)

In (52) and (53), hap is the one-body Hamiltonian of the relative particle with mass map submitted to the Coulomb potential e a e^/r. The functions Q and E only depend on the temperature via the single dimensionless parameter {—y/2l/X). All the physical effects mentioned is Subsection 4.4 give contributions to (51), the structures of which require the following comments. First, the classical contributions of the long-range part of the interactions are polynomials in the inverse temperature, the charges and the densities, which do not involve Planck's constant. The involved coefficients are evaluated analytically or, like C\ and C2, by numerical computations of dimensionless integrals. The lowest-order contribution is nothing but the familiar Debye-Hiickel term in p3?2, and constitutes the leading correction to the MB ideal pressure in (51). By specifying (51) to the case of the classical OCP, it can be checked that the classical terms found here, up to the order p 5 / 2 , do coincide with those calculated by Cohen and Murphy (1971), as it should be. The contribution of quantum diffraction at large distances appears only at the order p 5 / 2 and reduces to

The occurence of this term shows that the long-range part of the interactions cannot be entirely treated at a classical level. This diffraction term is missing in the expressions obtained by the effective-potential method. In this formalism, it arises from the three-body interactions w-3' which are longranged. Indeed, for large-triangular configurations, u\23 typically behaves as (f33 h e\ ei e$ / (12mi))ViVc( r i2)«Vi u e( r i3) ( a P ar * from a sum over the permutations of 1, 2, 3). These slow l/r 2 -decays lead to screened contributions which are of order phl2 instead of p3 ( a similar mechanism makes the Debye correction be of order p3!2 instead of p2). The diffraction correction (54) is merely proportional to h2, because it arises from large distances where the quantum effects can be treated perturbatively "a la WignerKirkwood". By the way, the presence of this term is crucial for recovering from (51) the well-known ^-correction (Hansen and Pollock (1973))

74

Alastuey: Statistical mechanics of quantum plasmas

(55) 6m to the classical pressure of the OCP calculated by the WK method. In fact, the sole contributions to (55) do arise only from the />2-term in (51), while (54) exactly cancels the 7i2-term which comes from p2KDQ. The term papp ^a/jQ(—\/2/ajg/Aajg) is the total contribution from both bound and scattering states of two charges ea and ep. The truncation of < f\ exp (—(3hap) | r > in the integral (52) defining Q ensures that this contribution is finite. This regularization in not an arbitrary mathematical artefact. It is directly related to the truncated stucture of the bond /y, and reflects the screening of the Coulomb interaction at large distances. For opposite charges such that eQep < 0, one may extract from Q a contribution of the bound states which reduces to the familiar Planck-Brillouin-Larkin (PBL) sum

f ) n2 [exp(-/fc?») - 1 + pef]

(56)

n = ll

where €%& = — e2 e% map/(2 h2n2) are the energy levels of the hydrogenoid atom with Hamiltonian hap. However, other definitions of the bound states contributions can be introduced from (52) by using the basic properties of the trace. For instance, as shown by Bolle (1987), there exists an infinite set of arbitrary decompositions in terms of bound and scattering contributions of the PBL sum itself. So, as far as thermodynamic quantities are concerned, only the total contribution of both bound and scattering states is an unambiguous quantity. The />5'2-contribution from bound and scattering states merely reduces to its p2 counterpart multiplied by /3eaepKD. This multiplicative factor arises from many-body effects which induce a constant shift —eaepKD on the energy levels of the two-particles states. The contribution pl,E(—y/2laa/Xaa) arises from the exchange of two charges ea in the vacuum. It is finite, independently of any screening effect, because the off-diagonal matrix elements < —f\ exp (—fihaa) \ r > are short-ranged. The magnitude of this contribution is smaller than the one relative to free particles because the repulsive potential e^/r inhibits the exchange. Similarly to what happens for the contributions of bound and scattering states, at the order p 5 ' 2 , the many-body effects on the twoparticles exchange amount to lower the repulsive barrier e2 / r by the constant —e\n D .

Alastuey: Statistical mechanics of quantum plasmas

75

Eventually, the high-temperature series can be recovered from (51) by expanding the virial-coefficients in powers of /?. Since //A is proportional to /3 1 / 2 , the /3-expansions of Q (-y/2l/X) and E (-V21/X) coincide with their Taylor series in powers of //A. In agreement with DeWitt's observations, such series do involve singular powers of h because //A is also proportional to 1/h. In fact, the expression (23) up to the order /3 5 / 2 is exactly recovered by inserting the /^-expansions of Q and E in (51). The calculation of the higher order terms from the virial expansion requires a suitable reorganization of the density series, similar to the one relative to the e2-series evocated in Section 2.2. 3.5.2 Application to the Sun For practical applications, the range of validity of the truncated EOS (51) is determined by the conditions I < a and X < a (with a ~ p" 1 / 3 ), which mean respectively weak coupling and weak degeneracy. It turns out that these conditions are met in the inner regions of the Sun which can be viewed, in a first approximation, as a three-component plasma made of electrons, protons and Helium nuclei. Perez (private communication) has evaluated the various physical corrections appearing in (51) within a typical temperature- and density-profile. Moreover, he used representations of Q (Kraeft et al (1986)) and E (Jancovici (1978)) which allow simple and accurate numerical computations. His results are sketched in Table 1. As expected, the inner regions almost behave as an ideal MB gas, since the relative magnitudes of the above corrections do not exceed a few percent. In the core, the correction to the ideal MB pressure is positive and mainly due to the electronic exchange. In the outer layers, this correction becomes negative because the classical Debye term then dominates. In the intermediate regions, the partial cancellation between the exchange and Debye terms increases the relative importance of the other contributions in (51), in particular those taking into account the recombinations between one nuclei and one electron in hydrogen atoms and He+ ions. The previous results illustrate the usefulness of the truncated EOS (51) for helioseismology. The actual observations lead to very accurate determinations of the speed of the sound and of adiabatic coefficients. For a proper interpretation of these data, one needs a reliable theoretical description of the small deviations from the ideal behaviour. The EOS (51) fullfills this requirement since it comes out from an exact expansion. Furthermore, its analytical character allows simple and consistent calculations of any thermodynamic coefficient via partial differentiations with respect to the density

76

Alastuey: Statistical mechanics of quantum plasmas

Table 1.1. Relative deviations from the MB ideal pressure for the inner regions of the Sun, calculated from the truncated EOS (51). R represents the distance to the center. The magnitudes of all the physical corrections arising in (51) are also indicated R

log/)

logT

PP/P-I

Class.

Bound/

Exch.

Diff.

[106] km

g/cc

(K)

[io-2]

[io-2]

Scatt.St. 2 2 [io-2] [io- ] [io- ]

0.62 0.15 0.00

-2.2 1.5 2.2

5.82 6.95 7.19

-0.698 0.110 1.446

-0.710 -1.000 -0.971

0.006 0.089 0.101

Int.

0.005 1.006 2.294

0.000 0.014 0.022

or the temperature. This EOS should be quite efficient for describing the regions close to the core, where all the chemical species are almost fully ionized (the inclusion of the contributions of heavier nuclei, like F%6+, is straightforward). Near the surface, the validity of the p5/2-truncated EOS is limited by the presence of atoms (like He) and ions which result from recombinations between one nuclei and two or more electrons. In the virial expansions, the contributions of these entities enter in terms of order f? (at least) which are not included in (51). Acknowledgment I am indebted to Marie-Pierre Fuchs for typing this manuscript, and to Asher Perez for communicating unpublished data.

References Abe R., Progr. Theor. Phys. 22, 213, (1959) Alastuey A., Cornu F. and Perez A., to be published in Phys. Rev. E, (1994) Alastuey A. and Martin Ph. A., Phys. Rev. A 40, 6485, (1989) Alastuey A. and Perez A., Europhys. Lett. 20, 19, (1992) Bolle D., Phys. Rev. A 36, 3259, (1987) Brydges D. and Seiler E., /. Stat. Phys. 42, 405, (1986) Chandler D., "Studies in Statistical Mechanics", edited by E. W. Montroll and J. L. Lebowitz (North Holland, Amsterdam, 1981) Cohen E. G. D. and Murphy T. J., Phys. Fluids 12, 1404, (1969) Cornu F. and Martin Ph. A., Phys. Rev. A 44, 4893, (1991) De Witt H. E., J. Math. Phys. 3, 1216, (1962) De Witt H. E., J. Math. Phys. 7, 616, (1966) Dyson F. J. and Lenard A., J. Math. Phys. 8, 423, (1967) Ebeling W., Ann. Phys. (Leipzig) 19, 104, (1967) Fetter A. and Walecka J. D., "Quantum Theory of Many-Particle Systems" (Me Graw-Hill, New York, 1971)

Alastuey: Statistical mechanics of quantum plasmas Feynman R. P. and Hibbs A. R., "Quantum Mechanics and Path Integrals" (Me Graw-Hill, New York, 1965) Gell-Mann M. and Brueckner K. A., Phys. Rev. 106, 364, (1957) Ginibre J., "Statistical Mechanics and Quantum Field Theory " (Les Houches Lectures, ed. by C. De Witt and R. Stora (Gordon and Breach, New York, 1971) Jancovici B., Physica 91 A, 152, (1978) Kraeft W. D., Kremp D., Ebeling W. and Ropke G., "Quantum Statistics of Charged Particle Systems" (Plenum Press, New York, 1986) Lenard A. and Dyson F. J., J. Math. Phys. 9, 698, (1968) Lieb E. H., Rev. Mod. Phys. 48, 553, (1976) Lieb E. H. and Lebowitz J. L.,Adv. Math. 9, 316, (1972) Mayer J. E., J. Chem. Phys. 18, 1426, (1950) Meeron E., J. Chem Phys. 28, 630, (1958) Montroll E. W. and Ward J. C., Phys. Fluids 1, 55, (1958) Morita T., Prog. Theor. Phys. (Japan) 22, 757, (1959) Pollock E. L. and Hansen J. P., Phys. Rev. A 8, 3110, (1973) Rogers F. J., Phys. Rev. A10, 2441, (1974) Salpeter E. E., Ann. Phys.(New York) 5, 183, (1958) Simon B., "Functional Integration and Quantum Physics" (Academic, New York, 1979)

11

Onsager-molecule approach to screening potentials in strongly coupled plasmas YAAKOV ROSENFELD Nuclear Research Center- Negev, P.O.Box 9001, Beer-Sheva, Israel

Abstract The solution of the exact integral equation for the liquid pair-structure in the asymptotic strong coupling limit for the plasma, as mapped on the Onsager charge-smearing optimization for the energy lower bound, features "Onsager atoms" and "Onsager molecules". The universal properties of this asymptotic limit make it a natural reference starting point for an asymptotic strong coupling expansion for the fluid structure and thermodynamics, playing the role of an "ideal liquid" state. In particular, the leading strong coupling terms for the potential energy, direct correlation functions, and screening potentials for the Coulomb and Yukawa mixtures (corresponding to classical plasmas and electron screened classical plasmas), with full thermodynamic consistency, are presented. These are in complete agreement with the Alastuey-Jancovici analysis of early simulations data by Hansen in strong coupling, and with recent highly accurate simulations data of Ogata, Iyetomi, and Ichimaru. Data analysis errors lead Ogata, Iyetomi, and Ichimaru to incorrect results for the short range screening potentials in strong coupling. Their calculations for the short range screening potentials, bridge functions, and enhancement factors for nuclear reaction rates in strongly coupled plasmas, should be revised. La solution de l'equation integrate exacte pour la structure du liquide dans la limite asymptotique de couplage fort pour le plasma, calquee sur 1'optimisation de charge d'Onsager pour la limite inferieure de l'energie, met en evidence des "atomes d'Onsager" et des "molecules d'Onsager". Les proprietes universelles de cette limite asymptotique en font un point de reference naturel pour un developpement asymptotique en couplage fort pour la structure et la thermodynamique du fluide. En particulier, les ter78

Rosenfeld: Onsager-molecule approach

79

mes dominants en couplage fort pour l'energie potentielle, les fonctions de correlation directes, et les potentiels d'ecran des melanges de Coulomb et de Yukawa (correspondant aux plasmas dassiques et aux plasmas dassiques ecrantes), sont pr'esentes dans cette revue. Ces resultats sont en parfait accord avec l'analyse de Alastuey et Jancovici des premieres simulations de Hansen en couplage fort, et avec les recentes simulations splus precises de Ogata, Iyetomi et Ichimaru. Des erreurs d'analyse de donnees ont conduit Ogata, Iyetomi et Ichimaru a publier des resultats incorrects pour la partie a courte portee des potentials d'ecran en couplage fort. Leurs calculs pour les potentiels d'ecran, les fonctions bridge, et les facteurs d'accroissement des taux de reactions nudeaires dans les plasmas fortement couples doivent etre revus. 4.1 Introduction : screening potentials as related to the inverse scattering problem for the fluid pair structure Classical plasmas composed of classical positive ions in a uniform neutralizing background charge density of degenerate electrons, are basic models for dense stellar materials and provide important reference systems in condensed matter physics (Rogers & DeWitt (eds.), 1987; Ichimaru (ed.), 1990). These systems are referred to as "one component plasma" (OCP), "binary ionic mixture" (BIM), or "multi ionic mixture", depending on the number of different ionic species. Considerable effort has been invested over the years (Hansen & Baus, 1980; Ichimaru 1982; Ichimaru et al., 1987; Isern, these proceedings) in increasing the accuracy of the calculations of the short range screening potentials in strongly coupled plasmas because of the following two main reasons: (1) The enhancement factors for the thermonuclear reaction rates which are important for stellar evolution, particularly for Carbon ignition in degenerate cores, are essentially controlled by the short range part of the screening potential, (the zero-separation value , H(0) , in particular). The ions inside dense stars (such as white dwarfs and neutron stars) are strongly coupled ,r ~(potential energy)/(kinetic energy) ~100, the enhancement factor is roughly proportional toexp[H(0)], and H(0) ~ T, so that an error of 2% in the value of H(0) may yield a reaction rate which is off by an order of magnitude. (2) The screening potentials play a key role in the study of the short range behavior of the bridge functions, notably their universal properties, which proved seminal for developing an accurate theory of liquid structure (Rosenfeld k Ashcroft, 1979; Rosenfeld 1980a,b). The zero separation theorem (Hoover & Poirer, 1962; Widom, 1963; Rowlinson & Widom, 1982) provides an important test for the accuracy of the

80

Rosenfeld: Onsager-molecule approach

equation of state of mixtures, and offers a consistency check (Rosenfeld, 1990a) for predictions based on closure approximations in integral equation theories for the fluid pair structure. The screening potentials H{J{T) of the classical plasma are defined in terms of the bare Coulomb interaction between the two ions of charges Z, and Zj , and the pair correlation function gij(r):

Hair) = ^

+ hfe(r)]

(1)

It is convenient to measure all distances in units of the Wigner-Seitz radius, a = ( j ^ ) 1 ^ 3 , where n is the total number density of the ions, and to define the Coulomb coupling parameter T = a£ y. The pair correlation function can be expressed through the free energy change upon fixing the positions of the pair of fluid particles in the appropriate configuration to form an interaction-site molecule (Hoover & Poirer, 1962; Widom, 1963; Jancovici, 1977; Rosenfeld, 1987a,b) H.(A_

WOO ~ FS

Here FQX is the configuration^ (excess over ideal gas) free energy of the Nparticle system (in a uniform neutralizing background) and F{X(T) is that of the same system but with the pair of particles kept at fixed separation , r , forming a two-site charge cluster. Ffx(r) does contain the intramolecular interaction ' r ; , so that Hij(r) is finite as r —• 0. The simulation data is limited to r < Rmin where Rmin ~ 1 for F ~ 160 and Rmin ~ 0.5 for T ~ 10. It is impossible to get H(r ~ 0) by simulations directly from eq.(l) because of the essentially zero probability for very close encounters. It is possible to try and extrapolate these results down to r = 0, but early attempts were not very successful (DeWitt, Graboske, and Cooper, 1973; Itoh, Totsiji, Ichimaru, and DeWitt,1978,1979) as pointed out by Rosenfeld (1980a, 1987a,b). On the basis of eq.(2), however, the calculation of H(0) can be carried out via the equation of state of the plasma mixture, if it is available. Salpeter (1954) and Salpeter & Van Horn (1969) pioneered the calculation of eq.(2) near r = 0 by means of the ion-sphere model. Using eq.(2) and assuming the validity of the linear mixing rule (Hansen et al., 1977; Brami et al., 1979) for the free energy of the BIM and by employing an accurate OCP equation of state (DeWitt, 1976), Jancovici (1977) obtained a good estimate of H(0) for the OCP. He also derived the exact leading r 2 term in the expansion around r = 0, which was used by

Rosenfeld: Onsager-molecule approach

81

Alastuey and Jancovici (1978) to extrapolate the simulation data for H(r) towards the origin r = 0. The first direct (structural, via eq.l) theoretical calculations of H{j{r) for the OCP and the BIM were performed by the assumption of universality of the bridge functions in the modified-hypernetted-chain (MHNC) theory (Rosenfeld, 1980a,b). These results are in agreement with Jancovici's calculations as well as with the ion-sphere scaling for mixtures. The classical direct (<£(r) —• S(k)) and inverse (S(k) —• (r)) problems for liquid pair structure, relating the structure factor, S(k) , to the pair potential, (f>(r), can be reduced by exact diagrammatic analysis to the solution of the hypernetted-chain (HNC) integral equation for an effective potential (denote: 0 = ^

*(r) =ttr)+ ^p-

(3)

The "exact" HNC equation is composed of the Ornstein-Zernike relation between the direct correlation function, C(T), and the radial distribution function g(r) = h(r) — 1, with the A;-space form (also defining the structure factor S(k)) h(k) = c(k) + nh(k)c(k) = S(k)c(k)

(4)

and the HNC-dosure for $(r):

H(r) - /ty(r) = lnfaf(r)] = -0#(r) + h(r) - c(r)

(5)

n — N/V is the number density . The heart of the problem is B(r) - the bridge function - which may be expanded in diagrams with the "dressed" /i(r)-bond. B(r) = 0, i.e. $(r) = (r), defines the HNC-approximation. The diagrammatic low density expansion represents a very slowly convergent route for obtaining meaningful results for a highly correlated system like a dense fluid, while, on the other hand , even the HNC-approximation provides an excellent point of departure for describing liquid pair structure. Along this alternative route, a first order improvement on the HNCapproximation , the ansatz of the universality (Rosenfeld & Ashcroft, 1979) of the repulsive short range structure of B(r) was found empirically to be very accurate. It provides the key (Rosenfeld, 1986a; Aers & Dharmawardana, 1984) to consistent solutions of the direct and inverse "scattering problems". Heuristic arguments were given in favor of "universality", but

82

Rosenfeld: Onsager-molecule approach

the computational intractability of the bridge diagrams prohibited its direct assessment. More recently, the "Onsager molecule" method (Rosenfeld, 1987a,b) was advanced for the evaluation of eq.(2). It is based on a general result in liquid state theory (Rosenfeld 1985,1986b), which holds under broad assumptions, that the leading term in the asymptotic strong coupling expansion of the configurational free energy is also its Onsager-type exact lower bound, e.g. the ion-sphere result for the classical plasmas. Using the Onsager-Molecule approach we can calculate the leading strong coupling expansion term of 77(r) (Rosenfeld, 1988, 1991a; Rosenfeld, Levesque, and Weis, 1989) and from it derive the asymptotic B(T), closing a fully self consistent cycle. The results using this method (Rosenfeld, 1992a) are in agreement with Jancovici's expression, with the MHNC calculations, and with the simulation results, as will be described below. Very recently, using an extrapolation of very accurate "extra long" computer simulations for strongly coupled classical plasmas, new results for the short range screening potential H(r) were obtained by Ogata, Iyetomi and Ichimaru (1991). These results are reported to be of 0.1% accuracy and to differ significantly from previous calculations (e.g. Jancovici's expression (1977) for H(0)). Careful analysis of the same data finds (Rosenfeld, 1992a), however, that these new simulations data are in excellent agreement with the previous calculations. Moreover, these high accuracy simulations provide an additional strong support to the asymptotic expansion ("Onsager molecule") analysis (Rosenfeld, 1987a,b) of short range screening potentials in multi-ionic plasmas. It is mainly their incorrect data analysis, which led Ogata, Iyetomi and Ichimaru (1991) to wrong conclusions. The analysis by Ogata, Iyetomi and Ichimaru (1991) is also used as the lounching platform for similar calculations of electronic-response effects by Ichimaru and Ogata (1991) and for general multi ionic mixtures by Ogata, Ichimaru and Van Horn (1991). The errors in these later calculations were also pointed out (Rosenfeld, 1992a). A very recent review by Ichimaru (1993), summarizes the many papers on screening potentials and enhancement factors for nuclear reaction rates by Ichimaru and coworkers, which perpetuate these errors. Although mentioning this criticism, the review by Ichimaru does not present any serious attempt to address it. In turn, the difference of results from the Ichimaru group vs the older Jancovici-Alastuey work could make a difference of a massive accreting white dwarf exploding into a Type I supernova or collapsing into a neutron star (Isern, these proceedings). The present review was invited in order to clarify the situation. This review describes the properties of strongly coupled plasmas as obtained analytically from the Onsager-molecule theory in complete agreement

Rosenfeld: Onsager-molecule approach

83

with the numerical simulations data. The Onsager molecule theory provides the asymptotic strong coupling properties of classical plasmas in a simple physically intuitive way, and is an essential tool for analysing numerical simulation results for strongly coupled plasmas. Ogata, Iyetomi and Ichimaru completely ignore the asymptotic analysis, which could have provided them an early worning about their wrong assumption. Along with revealing the errors in the analysis by Ogata, Iyetomi and Ichimaru, this review proposes a revised calculation of the short range screening potentials from their accurate simulations data. 4.2 Asymptotic strong coupling limit for plasmas (Onsager atoms and molecules) as a paradigm for the "ideal liquid" state Unlike the solid, on one hand, and the ideal gas, on the other hand, the dense fluid near freezing does not contain a "natural" small parameter. The solid features the crystal structure as the "ideal" reference state; the Lindemann melting rule identifies the ratio of particle's vibration to the nearest neighbor distance as a natural small parameter; the "elementary excitations" are phonons, providing the basis for the harmonic expansion. The low density fluid features the uniform distribution "ideal gas"as the reference state, with a natural small parameter given by the particle's size to the WignerSeitz radius. The "elementary excitations" are pair-collisions, and the virial power series in density is the natural expansion (for plasmas consider the Debye theory and the Abe-Meeron cluster expansion). There is, however, no natural "ideal liquid" available, but it can be theoretically constructed by using the asymptotic high density limit (bypassing the solid) of the exact diagramatic theory of liquids, to have all the needed properties in order to serve its purpose (Rosenfeld, 1985,1986b,1991a). This ideal liquid features universal structure-factor peaks, e.g. at tg(kia) = A;,o for D — 3. In parallel it features "Onsager atoms" with self energy that provides an exact energy lower bound. The small parameter is a generalized free-volume, the "elementary excitations" are "Onsager molecules", and the "basis functions" are the fundamental geometric measures of the sphere. The "ideal liquid" is defined through mapping of the asymptotic high density or strong coupling limit of the exact liquid state theory for pair-correlations onto the Onsager exact lower bound for the potential energy (the corresponding variational problems coincide). It is extended by recursive definitions for higher order correlations and provides basis functions for fluid structure and free energy functional (Rosenfeld, 1993a).

84

Rosenfeld: Onsager-molecule approach

Exact liquid state theory for pair correlations can be reduced to an HNC equation for some potential and thus general properties of the solution of this equation for different potentials are of central importance. For any non-singular potential $(r) with strong repulsion at short distances , the asymptotic T -*• oo solution of the HNC equation has the following universal features (Rosenfeld, 1988) ( use a (= (yf^) 1 / 3 in 3-D) as the unit of length, and /3$(r) = £ for the OCP in 3D as the specific example) : (1) Madelung behavior of the potential energy featuring: the Onsager exact lower bound for the potential energy = sum of self energy of individual dressed particles (Onsager atoms and molecules), e.g. for the OCP

5 £ = PUOA = -0.9I\

(6)

is the self-energy of an "Onsager atom" consisting of a point charge at the center of a neutralizing unit sphere having the background charge density. It is equal to the energy integral with a universal pair correlation function

J

D

[9D(r) -

(in our units n=ft^ 1 , where ftjrj is the volume of a unit D-dimensional sphere). The universal functions <jf£>(r) denote the limit r} —* \ of the solution of the PY equation for D-dimensional hard spheres of packing fraction t). (2) Saturation of the direct correlation functions: —^p- — ^ ( r ) = electrostatic interaction between the uniformly smeared charges (spheres with the ionsphere radius), for the OCP, given in ZD by

* ( r ) = | ^ 5 2 $(r) = -

r

+

16

r r 160

<

2 /a\

r >2

T

(3) Ewald identity and Mean Spherical Approximation boundary conditions:

N

\

J

= -{-nj

,

(9) D

[c(r) + P4>(r)]d r + c(r = 0)}

Rosenfeld: Onsager-molecule approach

85

(a) The direct correlation function is a short range Ewald function: c(r > 2) = — /3$(r > 2). (b) There exists a positive definite structure factor, S(k) > 0 : thus also c(k) < 0. (4) Pair exclusion: g(r < 2) = 0 , i.e. c(r < 2) + /3$(r < 2) > 0. The ideal liquid limit is associated with an effective packing fraction rj = 1. e = 1 — rj serves as the small parameter. (5) Universal structure factor peaks: Let ft(r) denote the overlap volume of two D-dimensional unit spferes at separation r (ft(0) = fto)» a n d define

u(r) = §£J. For D = 3 obtain « ( r ) = l - | r + ^

r<2

w(r) = 0

r >2

The zeroes c(fcj) = 0 are identical to the zeroes v(ki) = 0 , e.g. £,- = tan(ki) in 3Z). For the Coulomb potential c(k) ex. — T J T > ^ or the Yukawa potential c(fc) a —fcaVa^• (6) Correlation functions and screening potentials feature the Onsager molecules naturally and by recursive definition: In the limit

(11)

nx{r)

= (N-

2)UQA + UOM(T)

(12)

and thus from eq.(2)

HOM{T)

= - - PuOM{r) + 2(3UOA

(13)

Similarly to the Onsager atom, UOM{T) 1S * n e self-energy of an "Onsager molecule" consisting of a pair of ions separated by a distance r in a uniform neutralizing charge cloud of background charge density. The shape of this molecule is determined by the surface on which the electrostatic field vanishes. HOM{T) < 7 by virtue of the antibonding of the Thomas-Fermi molecules. Onsager molecules have the property to "dissociate" whenever the distance between the two point charges is larger than 2a, i.e. = 2UOA

T i.e.

HOM(T) = —

(14a) for

r

>2a

86

Rosenfeld: Onsager-molecule approach

Since the Onsager molecule is achieved by optimization, < »(r)

(146)

Since (14.) then the asymptotic functions obey H(r) = -c(r) - B(r)

(15)

(7) The asymptotic "energetics" is exact: recall that the energy integral is equal to that with a universal pair correlation function, and change in B(r) does not affect the asymptotic (Madelung) energy \HOM(T) ls * n e exact asymptotic limit for the screening potential1

4.3 Exact solution of the classical inverse scattering problem in the strong coupling limit, and asymptotic expansion for the correlations We now obtain (Rosenfeld, 1988) the solution of the inverse scattering problem in the ideal liquid limit in 3D. Consider the asymptotic HNC limit for $(r) = <j>(r) + kBTB(r), given the Onsager-molecule screening potential BoM(f)- Our purpose is to find Ac(r) = c(r) — co(r) and B(r) , where co(r) is the asymptotic HNC limit for ^(r). (1) Note that as part of the general "ideal liquid" properties: Ac(r > 2) = B(r > 2) = 0

(16a)

(2) Because c(r) is an Ewald function corresponding to the Onsager atom we obtain (recall eq.9) Ac(fc) < 0 3 f

, Ac(r = 0) = f- / Ac(r)d3 47T J

(3) There is only one solution: Ac(r) is proportional to the overlap volume function

Rosenfeld: Onsager-molecule approach

Ac(r) = -Au{r) < 0

87

(16c)

(4) The coefficient A > 0 is obtained by requiring thermodynamic consistency between the compressibility and energy equations of state, knowing that the latter is the exact asymptotic limit. For the OCP , A = 0.2F. (5) Finally, the bridge function is obtained from eq.(15): B(r) = - Ac(r) - co(r) - HOM{r)

(17)

(6) Because of (14a) the asymptotic bridge functions satisfy

BOM{T)

> -Ac(r)

(18a)

In the range relevant to the scattering problem of strongly coupled plasmas, r > ~ 1 ( as g(r < ~ 1) = 0, in strong coupling), the deviations between HOM(T) and \t(r) is less than 1%. However, the corresponding deviations between BOM(^) and — Ac(r) are much larger and may exceed 50%! The bridge function as obtained from the right hand side of (18a) is universal , i.e. it is of the same shape for all potentials, and equals

Buniver,ai(r) = -Ac(r) = ATu(r)

(186)

It thus represents the universal component of the strong coupling B(T). Buniversai(r) always provides an exact lower bound to the asymptotic bridge function. Here is a summary of the asymptotic strong coupling limit results for the OCP (Rosenfeld 1988, 1991a): (a) Energy:

^

-0.9r

(19)

(b) Structure: For the short range part (before "dissociation"), r < 2 , we have

(20)

Rosenfeld: Onsager-molecule approach

H(r,T) r

=

HOM(r,T) 9

r

1

(21)

^ Wr) = ^(2 5 / 3 - 2) - Y + h2(r)r* h2(r) ^ 0.038 - 0.0026r2 => h2{r = 2) * 0.0277

T

r

j

(22)

(23)

= 6(r) = ¥(r) + i * ( r ) - W(r) /i2(r) w a s obtained by fitting the Onsager molecule numerical data (Stein, Shalitin and Rosenfeld, 1988) ,while its value at r = 2 is determined by H{r = 2) = 0.5. In the longer range part (after "dissociation"),r > 2, we have

r

1 c(r,r) . ^(r,T) " r "" rr ' r ~ u

^

J

The Onsager molecule functions for mixtures, namely 1r»j(r) and w»y(r) are easily obtained analytically as generalizations of the OCP. HOM,%J{T < 2) is still needed to be calculated numerically (see, however, the next section). Finally, the ideal liquid Onsager limit and the liquid pair structure belong to the same basin of attraction with respect to the diagramatic iterative map as implied by eqs.3-5. This iterative map exhibits an instability of the pair function which correlates well with the simulation data for the freezing of simple fluids and plasmas. The availability of the Onsager limit result is seminal for these instability calculations (Rosenfeld, 1991b, 1992b).

4.4 Ion-sphere and Onsager molecule scaling properties for multi-ionic mixtures

The leading term in the asymptotic strong coupling expansion of the configurational free energy of the fluid is an exact lower bound (Rosenfeld, 1985, 1991a), e.g. the ion-sphere result for the classical plasmas (Lieb and Narnhofer, 1975). The two free energies in eq.2 are bounded from below by these leading terms. In particular, the excess free energy of the given mixture is bounded by the sum of Onsager-atom self energies:

Rosenfeld: Onsager-molecule approach

89

(25)

NkBT

i*OA,i = ~0-9~ft^ = —0.9F, is the self-energy of an "Onsager atom" consisting of a point charge Z{ at the center of a neutralizing sphere of radius Ri having the background charge density, R{ = (1/3 T where < Z > is the average charge per ion, then the linear-law reads (Hansen, Torrie, and Vieillefosse, 1979; Brami, Hansen, and Joly, 1979): pex

M^

= Z*>MT<)

(26)

where /o(r) is the OCP excess free energy per particle in temperature units.The screening potential is obtained as a difference between two exact lower bounds, as (Rosenfeld, 1987a,b)

Z'Z'Y HoM,ij(r) = —X-y uoM,ij{r) + uOA,i + UOAJ (27) 1S where u.oM,ij the self-energy of an "Onsager molecule" consisting of a pair of ions Z,-, Zj separated by a distance r in a uniform neutralizing charge cloud of the background charge density. The shape of this molecule is uniquely determined by the surface on which the electrostatic field vanishes. Onsager molecules as defined have the property to "dissociate" when the distance between the two point charges is larger than Ri + Rj , i.e. uoM,ij(r) = uOA,i + UOAJ

for

r < Rti + Rj

(28a)

The continuity of the function HoM,ij{T) and its first derivatives implies that near "dissociation" a few-term Taylor series expansion of (28b) around r = Ri + Rj will provide a good estimate of the function. This Taylor expansion has the following scaling form (dij = ' ^ ; ):

90

Rosenfeld: Onsager-molecule approach

HOM,ij{ r
^P-HOM(±-

< 2)

(28c)

< 2) is the corrresponding function for the OCP with Z, = 1, which is reasonably well represented in the region 1.3 < r < 2 by the electrostatic interaction of two uniformly charged spheres of unit radius and unit charge. The region of validity of this scaling approximation, which relates the screening potentials of the mixture to the OCP result, thus covers the region of availability of simulation data Rmm < r < 2 in strong coupling (Rosenfeld, 1987a,b). From elementary electrostatics we find that the small-r expansion takes the form HOM(T

—

= tiijto — /i»j,ir + ...

\^y)

where

hij,o = and (306) This coefficient of the r 2 term as obtained from the Onsager molecule is also the exact term for the plasma mixture, as obtained by Ogata, Iyetomi, and Ichimaru (1991). Comparison of (28) and (29) shows the relevance of two different lengths : rfy = ^ ; which is the "dissociation" radius, and 1 3 Ri+j = (R% + R^) / which is the ion-sphere radius of the molecule at small values of r. We need to consider separately the cases of "like" (t = j) and "unlike' (i / j) ions. For "like" (» = j) ions, da = R* by definition , and the Onsager-molecule screening potentials obey

HOM,ii(r) = §-HOM(^-)

(31)

where HOM{T) is the OCP result for Z< = 1. For "unlike' (i ^ j) ions there is a cross-over of the relevant scaling length ,as r changes from "dissociation" to "zero separation" , between d{j

Rosenfeld: Onsager-molecule approach

91

and Ri+j. In view of (28) and (29) then the following scaling approximation can be obtained for the " Onsager molecule" screening potentials of unlike ions:

HoM,ij(r) = G{T)^HOM{^-)

(32a)

where the prefactor G(r) varies slowly between G(r = d{j) = 1 and

G(r - 0) - Go =

4(22/3 _

where s = ^f > 1 . The Onsager molecule results suggest the following general scaling relation for plasma mixtures: nij(r) = G(r)H(Tijy-j-)

(33)

where H(T,r) is the OCP result for Z, = 1, Ty = Z f f i r , and where it is expected that G = 1 for like ions (i = j), and G = 1 for unlike ions in the region close to r = rftJ-.

4.5 Electron response corrections to the interionic screening potentials The starting point for analysing the electron response corrections to the interionic screening potentials is to consider classical mixtures of charged particles interacting through the repulsive Yukawa (screened Coulomb) potentials (Rowlinson, 1989). The Yukawa intermolecular potential, ^(r) = s-~, has the special property (Rosenfeld, 1993a,b) that it gives rise to the same functional form for the potential outside a spherically symmetric distribution of matter : Thus, if a point Yukawa charge Z is smeared out radially with distribution p(r) upto a finite radius R, and if at the same time the charge is appropriately renormalized, Zrenorm = ?(«, R)Z,

—,—^r = — / p(x)x smh(ax)dx (34) v v q(a,R) a J v ' ' ' then the potential outside the smearing radius R remains the same as for the

92

Rosenfeld: Onsager-molecule approach

original point charge. For a uniform distribution inside a sphere of radius R, p(r < R) = 2^55, obtain a, R)uniform = Q(aR)

(35o)

2_

3[e«(t - 1) + e~\t + 1)] t2 1

+

The renormalization property enables (Rosenfeld, 1993a,b) to follow the Onsager "smearing" procedure as developed for obtaining a lower bound for the potential energy of Coulomb systems (Onsager, 1939; Lieb & Narnhofer, 1975; Rosenfeld, 1982a, 1985; Rosenfeld & Gelbart, 1984; Rosenfeld & Blum, 1986) and apply it to general Yukawa systems. Consider classical mixtures consisting of iV,- positively charged, Z,e > 0 , point particles of type t , interacting through the Yukawa pair potentials : -ar

_ Z,Z,r obtain the following optimized Onsager type lower bound (denoted as

TNkBT

T

(37)

expressed as the sum of "self" (dependent only on each type t ) terms:

^

(38)

Hi

with the function uo(t) related to Q(t),

«o(o = %iQm+v*-'

( 39 )

and where the smearing radii , .ft, ,are obtained from the solution of the following set of non-linear coupled algebraic equations:

?

_

Rosenfeld: Onsager-molecule approach

93

The excess (over ideal gas) energy of the plasma as evaluated in the standard linear response approximation of the electrons screening effects is (Firey & Ashcroft, 1977; Chabrier & Ashcroft, 1990; Rosenfeld k Ashcroft, 1979)

2

ij

' r=olr

2 Pumix,lin.resp. ^i

(41«) v

"

^ \^ Zt, . > ^^i^T^O./tn.reap.l. 0 1 -^"/ t

/.in (.410)

*

corresponding to replacing uo(t) in (39) by

= Mt)-±-l = ~-£t>

+ ....

(42)

It should be noted that the functions Q(t) and tio,/m.rc»p.(') a^e even functions of their argument so that the Onsager bound for (41a) is an even function of a. Thus, for small values of a the Onsager bound predicts the following expansion for the linear-response excess free energy:

It is interesting to note here that the best fits (Hansen, 1975; Young, Corey, and DeWitt, 1991) to simulations data yield F(T,a)

8.992

18.165 ,

=

°

+

(AA,

(44)

-

for large values of F, in excellent agreement with the bound (43). The weak screening expansion for the Onsager bound for the mixture (eqs.(37)-(40)) is given by ttmi.,lin.raip. = -0.9 < Z 5 / 3 >< Z

+ ANon.Lin.<*2

>^3

+ O(aA)

where the first term is the standard linear-mixing (ion-sphere) result for unscreened plasmas, and

94

Rosenfeld: Onsager-molecule approach

< Z A

N o

n.Lin.-

<

> , 18 i +

< Z5/3 >2

3 l

Z5/3 >2

1

2/3

l

J

+

100 l < Z 7 / 3 This expansion is in agreement only with general trends among the coefficients in a fit (Hubbard & DeWitt, 1985) to simulations data which, as it stands, can not be compared in full detail with the present result. To obtain we solve eqs.(40) to leading order :

Zi

/3

The Onsager bound is the exact leading term in the strong coupling expansion for the fluid potential energy (Rosenfeld, 1985, 1986b, 1988). To the extent that its scaling properties apply also for arbitrary values of the coupling parameter, this strong coupling limit suggests the following general scaling approximation ("non-linear" mixing rule, because of eq.40) for repulsive-Yukawa fluids:

^

fmix =

(48a)

2 2 f ( T ) , a, a, = aRi

(486)

where u m , r and u refere to the mixture and to the one component energies (per particle, in temperature units), respectively, and fmix and / refere similarly to the corresponding excess free energies, and where eqs.(40) are used in (48b). The Onsager exact energy bound for charged Yukawa mixtures and the corresponding non-linear mixing rule have a simple physical meaning which is revealed by treating the "Yukawa" problem in "Coulomb" language. In the standard linear-response (Firey & Ashcroft, 1977; Chabrier & Ashcroft, 1990) treatment for point ions in (e.g.) a nearly degenerate Fermi sea of electrons, it is assumed that the electron number density varies in space according to (let e = 1)

Rosenfeld: Onsager-molecule approach

pe(?)

=< pe > +Ap c (7*) = n

95

a2

+^*(~H

(49)

where $(~r*) is the electrostatic Coulomb potential at the point r in space, and (Dharma-wardana & Taylor, 1981; Dandrea, Ashcroft, and Carlsson, 1986) a is related to the Fermi function I\fi. Let Up be the total electrostatic potential energy (relative to the infinite self energy of the point charges) of this electro-neutral system ,and let UK be the non-ideal kinetic energy of the electrons in the linear response leading order. Their sum,

(50) satisfies the following electrostatic inequality (Rosenfeld, 1991a,b, 1993b): UP

+ UK > £ NiUOA,i = UOTF

(51)

i

where

UOA,i =

^

(52)

The integral / ...dv{ is over the volumes ,«i, of individual, confined, isolated, spherical, and neutral "Onsager-atoms" composed of a central point charge Z{ and an electron cloud of number density (49) and radius R{. Using the definition of Q(t) in Eqs.35 the electron density at the surface can be written as

Pe{Ri)

~ [4«R>\

(53)

The optimized bound (51) subject to the condition of total charge neutrality, J2ixiR^ = 1> yields the bound (41b). Comparison of (40) with (53) reveals its true physical meaning: it is the condition of constant (independent of t) electron surface density, i.e. constant surface electrostatic potential. This is of course the expected result on physical grounds, and which is missing in the linear rule. In view of the dominance of the electrons contribution to the total pressure of the plasma, which is determined for Thomas-Fermi theory

96

Rosenfeld: Onsager-molecule approach

by their density at the surface of the confined atom, the condition (40) also corresponds to the well known "volume additivity rule" for Thomas -Fermi mixing of elements (More, Warren, Young, and Zimmerman, 1988),

t>(P,T) = J>t»,-(P,r)

(54)

i

namely combining specific-volumes , v=]^ , Vj=$- ,at same pressure and temperature. Using the non-linear mixing rule (48) obtain the following result for the zero separation screening potential of the Yukawa system ,-,a0 + f(Tj,aj) - f(Ti+i,ai+j)

(55)

rz?

a«+j = = aRi+j aRi+j ,, and where Z,+ - = Z{ + Zj . In the ,, a«+j 7 asymptotic strong coupling limit obtain

where Ti+j — ^iJf*

,ij(0) T

=

9 JZ}

Z?

t

10 \R{

Rj

ZlA Ri+j)

175 i.e.

>»/» (Z,5/3 + Z,5/3 - (Z,

~
+

3 < Z 5 / 3 > / 5/3

ioo*/3i

z<

+

v 5/3

,7

^ ' -(z* +

which for the one-component system takes the form (Rosenfeld, 1993b) HOM(0)

/

10 v

5/3x

/

= -1.05732 - a + 0.36860a2 + ... The present Eq. 58 corrects a small error in the original paper. Note however that in the physics context of the linear response for electron screening,

Rosenfeld: Onsager-molecule approach

97

the bare interaction between the ions is Coulomb, so that the "linear response" screening potential (Stein, shalitin, and Rosenfeld, 1988) , for a pair of ions in the electron screened plasma, is an even function of oc

HiS{0) = / ^ ( 0 ) + ( \

^ r

Mr)) / r=0

(59)

4.6 Relation of the Onsager molecule theory to the data obtained by simulations results and to its analysis 4-6.1 Concerning the simulation data Jancovici (1977) obtained his original expression

1.0531 + 2.293ir

I

(60)

by using the "linear law" approximation for the excess (over ideal gas contribution) free energy of plasma mixtures, F%fix. This expression was obtained with a fit (DeWitt, 1976) to the best available simulation data at the time. With a most accurate fit to the best presently available data (Young, COrey, and DeWitt, 1991), in the range ( r > 1) / 0 ( r ) = -0.8992r + 1.8322r03253 - 0.268 InT - 1.3693

(61)

the updated Jancovici expression is ( for T > 1)

= 1.05638 + 0.99643r-«-«-(l^i+0^1iD

(62)

It is interesting to note that the leading Madelung term for the energy, which is a free parameter in the fit to the simulation data, comes closer to our prediction of -0.9 as the accuracy of the simulations increases. Our asymptotic analysis predicts (Rosenfeld 1985, 1986b) an expansion of the type

+ ...

(63)

where 6 = 0.5 is the exact result for the HNC approximation. The linear and non-linear mixing rules for the thermodynamics of plasma mixtures without and with screening, respectively, are exact asymptotic

98

Rosenfeld: Onsager-molecule approach

scaling predictions of the Onsager-molecule approach which hold to a high accuracy also at relatively low values of F ~ 1. Concerning the accuracy of the linear law it should,be emphasized that it holds to about 0.1% in the strong coupling region for all available simulation and hypernetted-chain approximation results (DeWitt, Slattery, and Stringfellow, 1990; Iyetomi, Ogata, and Ichimaru, 1988,1991; Brami, Hansen, and Joly, 1979; Rosenfeld, 1982b). The same is true (Rosenfeld, 1993b) for the non-linear mixing rule for screened plasmas in the context of the hypernetted-chain approximation (extensive simulations data not yet available). We demonstrated (Rosenfeld 1988,1991a) that the OCP structure at finite F can be calculated very accurately by employing only the leading term in the strong coupling expansion of the bridge function around the "ideal" Onsager state:

5(r,r) = 6(r)r + &i(r)rA + ...

(64)

Indeed (Rosenfeld, Levesque, and Weis, 1989), by using B(r,T) = Tb(r) (see eq.23 ) in the structure equations (3)-(5) , the results for the structure functions and for the equation of state at large values of T (say, T > 40 ) are comparable to those obtained from the fits to the simulations presented by Iyetomi , Ogata, and Ichimaru (1992). The correction terms, probably led by a F* term with A ~ 1/3 , contain the long ranged contributions to the otherwise short ranged {BOM(J > 2) = 0) leading term. These corrections are of the order of the entropic contributions to the free energies involved in eq.(2), which are relatively small in comparison to the corresponding potential energy terms in strong coupling. The most sound procedure for extracting the bridge function from the simulation results for g{r) is to analyse the difference AB(r) = B(r) — F6(r), which is on the border of the statistical accuracy of the simulations. Eventhough this method was not followed, other numerical solutions (Poll, Ashcroft, and DeWitt, 1988; Iyetomi , Ogata, and Ichimaru, 1992) of the inverse scattering problem for the OCP structure, using the simulations data as input, compare very well with our asymptotic predictions (Rosenfeld, 1988; Rosenfeld, Levesque, and Weis, 1989). Note that the fit by Ietomi , Ogata, and Ichimaru (1992) leads to 2pl = —1.406 as the leading term, which compares with our prediction (eq.20) of -1.4. As already discussed (Rosenfeld, 1988), the Onsager molecule result for the bridge function and for its universal component is in accord with the demonstratively successful modified-HNC theory based on the empirical Percus-Yevick bridge functions for hard spheres (Rosenfeld & Ashcroft, 1979), BPYHS(T,V)I a n d an imposed thermodynamic consistency

Rosenfeld: Onsager-molecule approach

99

for determining the free parameter , n(T). The resulting BPYHS{T, T) features the long range oscilatory behavior, yet PY«^V< ) _• 0.2u>(r) for F -+ oo . Explicit expressions for the universal bridge functional for inhomogeneous fluids, and an updated view of the extent of universality of the bridge functions, was presented recently (Rosenfeld, 1993a). Finally, consider the expansion

^

= h0 - hxr2 + h2rA +

tor6...

(65)

where the coefficient hi is known exactly , hi = 0.25, and define the function H(r) - H(0) + 0.25IV2

j = h2 + h3r + ...

(66)

2

which according to the Onsager molecule asymptotic prediction (eq.22) is a slowly varying function in the range r < 2. Note that the Onsager molecule asymptotic result features the exact coefficient of the r 2 term in eq.(65), as well as the exact second moment of the electric microfield distribution (Rosenfeld, 1989). The fit by Alastuey and Jancovici (1978) to the Hansen data (Hansen & Baus, 1980) , yielding h2 = 0.039, and h3 = -0.0043, is in accord with our Onsager molecule prediction. We do not have at hand the raw simulation data of Ogata, Iyetomi, and Ichimaru (1991) for -^*^, but it was fitted by them in the range Rmin < r < 2 to an accuracy of 0.1% by the function / ( r ) f(r) = A-Br

+ exp(Cr1/2 - D)/r

(67)

where for 5 < T < 180 A= 1.356-0.0213 ln(r) B = 0.456- 0.0130 ln(r) C = 9.29 + 0.79 ln(r)

(68) V '

D = 14.83 +1.31 ln(r) We found out that the function f(r) , in the actual data range for r, can be fitted to 0.05% by the expansion (65) with 3 free coefficients (i.e. upto the r 6 term), for all values of T > ~ 70, with results in about 0.3% agreement with the original Alastuey-Jancovici extrapolation for H(Q). This also

100

Rosenfeld: Onsager-molecule approach

shows that the functional form of / ( r ) as chosen by Ogata, Iyetomi, and Ichimaru (1991) ,with 4 free coefficients, is inferior to the Alastuey-Jancovici form with 3 coefficients. Increasing the number of free coefficients to 5 almost does not affect the result of the extrapolation for these relatively large values of F. The resulting function /*o(r) decreases with F towords the asymptotic limit of 1.05732, while the function ^{T) gradually increases towards the predicted Onsager molecule result. For F = 10, however, only with 5 coefficients it is possible to fit the function / ( r ) to better than 0.1%, and the result features a negative coefficient h^. This last result, and the gradual change in the value of /12, from its asymptotic value of about 0.038 to zero (and even negative values) are in qualitative agreement with the direct simulation results of Ogata, Iyetomi, and Ichimaru (1991) for the coefficient /12, as also discussed below.

4.6.2 Concerning the data analysis of Ogata, Iyetomi, and Ichimaru Ogata, Iyetomi, and Ichimaru (1991) and Ichimaru & Ogata (1990) employ the following methodology: they estimated the coefficient /12 by the Monte Carlo method. They find that the computed values (which are smaller in magnitude than the extent of errors) are an order of magnitude smaller than h\. As a result they decide to truncate the expansion at the second term with hi = 0.25 , and obtain the extrapolated screening potential in the follwing form :

—±-L = ho — h\ r 2 for r < ro

p. = f(r)

for r>r0

The remaining two unknown parameters ho,ro are determined from the requirement that the function and its first derivative are continuous at TQ:

-0.5r 0 = . ho - 0.25rg = /(ro) Since / ( r ) is dominated by the linear part at ro , Ogata, Iyetomi, and Ichimaru (1991) obtain

Rosenfeld: Onsager-molecule approach

r 0 = IB = 0.912 - 0.026 ln(r) ho = A - B2 = 1.148 - 0.0094 ln(r) - 0.00017[ln(r)]2

101

(

'

Near freezing, for example at F = 160 , corresponding to their Figure lb, they obtain r 0 = 0.780 , h0 = 1.095. Jancovici's result (1977) , ho = 1.071, is 2.2% smaller. Ogata, Iyetomi, and Ichimaru (1991) estimate the error , in their H(Q) = ho so extrapolated, to stay on the order of 0.1%. This analysis by Ogata, Iyetomi, and Ichimaru (1991) is wrong. It is based on incorrect logic and incorrect methodology : (1) In strong coupling, e.g. F = 160 , the value of ro (e.g. TQ — 0.78) is much smaller than Rmin (e.g. Rmin ~ 1.15). Thus, the connection point for the extrapolation is beyond the actual data points, making the procedure dependent on the choice of fitting function for the data. The obvious next step is to check what happens with the addition of extra terms. Indeed, keeping the term /i2, and repeating the procedure it is found (Rosenfeld, 1992a) that the connection point is now inside the data range, and the results are in 0.3% agreement with Alastuey and Jancovici. (2) Even if the coefficient h2 is an order of magnitude smaller than h\ its effect on the extrapolated ho can be of order of 1% (e.g. with ro ~ 1 and hi ~ hi/25 ~ 0.01). More specifically, observe their equations A1,A2 and Table 4 . They write h2 in the general form Th2 = A — F 2 /32 , so that if Th2 is close to zero then A is close to F 2 /32. Since, for large F , A is a large number for which one needs high accuracy, then it is indeed a good idea to do, as they did, and monitor the quantity nu" in their eq.A2, which should provide the indication for the accuracy of the calculation for A. Now observe their Table 4 to find (e.g.) for T = 80 : u = 0.9910 ± 0.000089. in other words, the statistical accuracy is indeed very high, but the result for "«" is 1% off the exact value "u"= 1 in all the cases in Table 4. Thus, maybe the statistical error for A is high and indicates ±0.01 for /12, but on the basis of the "litmus test" via the quantity "u" (as indeed intended and presented by them), one expects at least a similar error of about 1% for A, which means an error of about T/3200 for h2 , i.e. 0.05 for F = 160 . Thus their claim h2 = 0.00 ± 0.01 is may be correct for F = 10 but not for F = 160. In addition to this expected systematic error, their data point for F = 160 is: h2 = 0.063 ± 0.133, which at its maximum is of the order expected by the Onsager molecule theory. Such a possible gradual transition in the behavior of h2 is in accord with our analysis ,above, of the simulation data as embodied by their function

f(r). Ogata, Iyetomi, Ichimaru, and Van Horn (1992,1993) find deviations from

102

Rosenfeld: Onsager-molecule approach

the linear mixing rule which seem to justify their results for H(0) based on /12 = 0. A closer look at their data reveals however that again, as above, they project F ~ 10 results to much stronger couplings. The significant data for the purpose of calculating H(0) is for very dilute ((12 ~ 0.01) mixture of larger charge (Z2 > 1)) in a solvent of unit charge. The effective coupling for these relevant mixtures in the simulations of Ogata, Iyetomi, Ichimaru, and Van Horn (1992,1993) are relatively weak (i.e. F ~ 10). Moreover, their results for a "negative" deviation from the linear mixing rule lead to unphysical crystalization diagrams (Mochkovitch, these Proceedings), and contradict more accurate simulations by DeWitt & Slattery (1993) which find a small positive deviation from the linear mixing rule. In their analysis of their simulation data for mixtures, Ogata, Iyetomi, and Ichimaru (1991) do not consider the prefactor G, and thus effectively use the scaling (32a) with G = 1. On the basis of the high accuracy of the linear law it is expected that scaling relations between the OCP and mixtures, which are similar to the linear law and are likewise based on the "Onsager molecule" (ion-sphere) limit, to be of similar high accuracy ( e.g. better than the 0.3% which is the uncertainty in the OCP results). The additional systematic deviation introduced for unlike ions (t ^ j) by the assumption that G = 1 , which is the assumption made by Ogata, Iyetomi, and Ichimaru can be estimated using (32b) by the deviations of Go from unity. Thus the results Go - 1 = 0.999,0.993 for s = 4/3,2 , respectively, justify the scaling relation used by them for Carbon/Oxygen and (somewhat less) Hydrogen/Helium mixtures. For larger charge ratios, however, significant deviations from the assumption G — 1 will occur as, e.g. Go — 1 ~ 0.03,0.06 for s = 5,10 , respectively. In later work Ogata, Ichimaru, and Van Horn (1991, 1993) and Ogata, Iyetomi, Ichimaru, and Van Horn (1992) they use for the mixture the same incorrect extrapolation method as for the OCP, assuming the ion-sphere scaling via d{j for the function / ( r ) , again without regard to the short range correcting function G(r). The raw simulations data of Ogata, Iyetomi, Ichimaru, and Van Horn agrees very well, however, with our Onsager molecule analysis. For the unscreened OCP at F = 44.9 Ichimaru and Ogata (1991) obtain H(0)/T = 1.110 in contrast to the Jancovici type extrapolation (Rosenfeld, 1992a) H(0)/T = 1.088. Considering the contribution of the electron response then the Ichimaru-Ogata result for a = 0.3125, namely 8 = 1.110 — 0.825 = 0.285 , is similar to the corresponding alternative extrapolation result (Rosenfeld, 1992a) 6 = 1.088 - 0.800 = 0.288, because the value of /12 changes little, for this small value of a = 0.3125 , from the

Rosenfeld: Onsager-molecule approach

103

a = 0 case. Ichimaru and Ogata compare their value of S with another estimate ,which they proposed earlier, given by (their eq.ll) 6 = 1.057(1 - e~Q) = 1.057a - 0.5285a 2 + ...

(72)

which gives 6 = 0.278 for a = 0.3125. This value also compares well with the Onsager molecule estimate (eq.58) 6 = a - 0.36860a2 + .... = 0.277. It is not clear what is the origin of eq.72, but it seems to be a fitting function. In any case, it should be noted that eq.(72) is analytically different (and for larger values of a also numerically different) from the Onsager molecule prediction eq.58 which is derived from basic physics.

4.7 Conclusion The present analysis finds excellent agreement between the simulations data for the equation of state and for the screening potentials of strongly coupled plasmas (F > ~ 70), in accordance with the "Onsager molecule" asymptotic limit. The Alastuey-Jancovici-type extrapolation of the highly accurate simulations data of Ogata, Iyetomi, and Ichimaru (1991) for the OCP screening potential, together with the correct "Onsager molecule" scaling (Eq.32) provide the short range screening potentials of strongly coupled plasmas to an accuracy of about 0.3%, in agreement with the linear law and equation of state data. The extended Jancovici formula (eq.55) for the Columb and Yukawa OCP agrees very well with the present extrapolation of the data of Ichimaru and Ogata (1991). The inevitable conclusion from the present analysis is that the calculations of the enhancement factors for nuclear reaction rates of Ogata, Iyetomi, and Ichimaru (1991) and the calculations of the short range bridge functions of Iyetomi, Ogata, and Ichimaru (1992), which are all based on incorrect analysis of the short range screening potential for strongly coupled plasmas, should be revised. In particular, the new data analysis should take into account the gradual change of the coefficient /^(F) from its Onsager F -+ oo limit value of about 0.038, to values near zero for F ~ 20 and to possibly negative values for F < ~ 10. Acknowledgements I thank Hugh DeWitt for bringing to my attention the papers by Ichimaru and coworkers, and for sending valueable material. Interesting discussions with Gilles Chabrier, Hugh DeWitt, Jordi Isern and Hugh Van Horn are acknowledged with gratitude. This work was supported by The Basic Re-

104

Rosenfeld: Onsager-molecule approach

search Foundation administered by The Israel Academy of Sciences and Humanities under Grant No.492/92.

References Aers G.C. and Dharma- wardana M.W.C., Phys.Rev.A29, 2734, (1984) Alastuey A. and Jancovici B., Ap.J. 226, 1034 , (1978) Brami B., Hansen J.P., and Joly F., Physica 905A, 505, (1979) Chabrier G. and Ashcroft N.W., PAya.iiew.i442 , 2284, (1990) Dandrea R.G., Ashcroft N.W., and Carlsson A.E., Phys.Rev.B 34 , 2097, (1986) Dharma-wardana M.W.C. and Taylor R. , J.Phys.Cl4, 629, (1981) DeWitt H.E., Graboske H.C., and Cooper M.S., Ap.J. 181 , 439, (1973) DeWitt H.E., Phy8.Rev.Al4,1290, (1976) DeWitt H.E., Slattery W.L., and Stringfellow G.S., in "Strongly Coupled Plasma Physics", edited by S.Ichimaru, North-Holland/Yamada Science Foundation, Amsterdam, 1990, page 635. DeWitt H.E. and Slattery W.L., privat communication, (1993) Firey B. and Ashcroft N.W., Phys.Rev.Al5 , 2072, (1977) Hansen J.P., J.Physique.36, L-133, (1975) Hansen J.P., Torrie G.M., and Vieillefosse P., PAys.iicw.yil6, 2153, (1977) Hansen J.P. and Baus M., Phys.Rep.59, 1, (1980) Hoover W.G. and Poirer J.C., J.Chem.Phys.37 ,1041, (1962) Hubbard W.B. and DeWitt H.E., Astrophys. J. 290, 388, (1985) Jancovici B., J.Stat.Phys.17, 357, (1977) Ichimaru S., Rev.Mod.Phys. 54, 1017, (1982) Ichimaru S. .Iyetomi H., and Tanaka S., Phys.Rep. 149, 91, (1987) Ichimaru S., Ed., Strongly Coupled Plasma Physics", North- Holland/Yamada Science Foundation, Amsterdam, 1990 Ichimaru S. and Ogata S., in "Strongly Coupled Plasma Physics", edited by S.Ichimaru, North-Holland/Yamada Science Foundation, Amsterdam, 1990, page 101. Ichimaru S. and Ogata S., Ap.7.374 ,647, (1991) Ichimaru S., Rev.Mod.Phys.65, 255, (1993) Itoh N., Totsuji H., Ichimaru S., Ap.J.21S, 477, (1978) Itoh N., Totsuji H., Ichimaru S., and DeWitt H.E., vlp.7.234, 1079, (1979) Iyetomi H., Ogata S., and Ichimaru S., Ap.J.334, L17, (1988) Iyetomi H., Ogata S., and Ichimaru S., in US-Japan Workshop on Nuclear Fusion in Dense Plasmas, Institute for Fusion Studies, The University of Texas, Austin. Preliminary Abstracts, Edited by S.Ichimaru and T.Tajima, Oct. 1991 Iyetomi H., Ogata S., and Ichimaru S., Phys.Rev.A46 ,1051, (1992) Lieb E.L. and Narnhofer H., J.Stat.Phys.12, 291, (1975) More R.M. , Warren K.H., Young D.A., and Zimmerman G.B., Phys.Fluids 31, 3059,(1988) Ogata S., Iyetomi H., and Ichimaru S., in "Strongly Coupled Plasma Physics", edited by S.Ichimaru, North-Holland/Yamada Science Foundation, Amsterdam, 1990, page 59 Ogata S., Iyetomi H., and Ichimaru S., Ap.7.372, 259, (1991)

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Ogata S., Ichimaru S., and Van Horn H.M., in US-Japan Workshop on Nuclear Fusion in Dense Plasmas, Institute for Fusion Studies, The University of Texas, Austin. Preliminary Abstracts, Edited by S.Ichimaru and T.Tajima, Oct. 1991 Ogata S., Iyetomi H., Ichimaru S., and Van Horn H.M. , in proceedings of the International Conference on the Physics of Strongly Coupled Plasmas, Rochester, New-York, USA, August 17 - 21,1992 (University of Rochester Press, in print, edited by H. VanHorn and S. Ichimaru); Ogata S., Ichimaru S., and Van Horn H.M. , preprint , 1993 Onsager L. , J.Phys.Chem. 43, 189, (1939) Poll P.D., Ashcroft N.W., and DeWitt H.E., Phys.Rev.A37 , 1672, (1988) Rogers F.J.and DeWitt H.E., Eds., Strongly Coupled Plasma Physics", Plenum, New York, 1987 Rosenfeld Y. and Ashcroft N.W., Phys.Rev.A20 ,1208, (1979) Rosenfeld Y., Phys. Rev. Letters, 44, 146, (1980a) Rosenfeld Y., /. Physique. 41 , C2 -77, (1980b) Rosenfeld Y. ( Phys.Rev.A25 ,1206(1982a) Rosenfeld Y., Phys.Rev.A26, 3622, (1982b) Rosenfeld Y. and Gelbart W.M., J. Chem. Phys., 81, 4574, (1984) Rosenfeld Y., Phys.Rev.A32, 1834(1985) Rosenfeld Y., J. Stat. Phys. 42, 437, (1986a) Rosenfeld Y., Phys.Rev.A 33, 2025, (1986b) Rosenfeld Y. and Blum L., J. Chem.Phys. 85 , 1556, (1986) Rosenfeld Y., Phys.Rev.A35, 938, (1987a) Rosenfeld Y., in "Strongly Coupled Plasma Physics", edited by F.J.Rogers and H.E.DeWitt, Plenum, New York, 1987b. Rosenfeld Y., Phys.Rev.A37, 3403, (1988) Rosenfeld Y., Levesque D., and Weis J.J., Phys.Rev.A 39 , 3079, (1989) Rosenfeld Y., Phys.Rev.440,1137, (1989) Rosenfeld Y., J.Chem.Phys. 93, 4305, (1990) Rosenfeld Y.,in High-Pressure Equations of State : Theory and Applications, edited by S. Eliezer and R. Rici, Italian Physical Society International School of Physics Enrico Fermi, Course CXIII ,1989, North- Holland, Amsterdam, (1991a) Rosenfeld Y., Phys.Rev.A43, 6526, (1991b) Rosenfeld Y., Phys.Rev.A46, 1059, (1992a) Rosenfeld Y., A46, 4922, (1992b) Rosenfeld Y., J.Chem.Phys.98,%126, (1993a) Rosenfeld Y., Phys.Rev.E47, 2676, (1993b) Rowlinson J.S. and Widom B., Molecular Theory of Capilarity (Clarendon, Oxford, 1982) Rowlinson J.S. , Physica 4 1 5 6 , 15, (1989) Salpeter E.E., Aust.J.Phys.7, 373, (1954) Salpeter E.E. and Van Horn EM.,Ap.J. 155, 183, (1969) Stein J., Shalitin D., and Rosenfeld Y., Phys.Rev.A37 , 4854, (1988) Widom B., J.Chem.Phys.39, 2808, (1963) Young D.A., Corey E.M., and DeWitt H.E., Phys.Rev.A44, 6508, (1991)

Astrophysical consequences of the screening of nuclear reactions J. ISERN AND M.HERNANZ Centre d'Estudis AvanqaU Blanes (CSIC), Camt dt Santa Barbara an, 17S00 Blancs, Spain.

Abstract The rate of nuclear reactions depends on the influence of the surrounding particles that compose the plasma. At high densities the situation is far from being satisfactory and the influence of electron polarization has not been completely elucidated. In particular, it is shown that the possibility of an accretion induced collapse of a carbon-oxygen white dwarf instead of a supernova explosion completely depends on the screening factors and pycnonuclear rates that are adopted. Similarly, the possibility of detecting isolated neutron stars that accrete matter from the interstellar medium depends on the adopted pycnonuclear rates. Low rates allow the formation of a metastable layer that can release energy explosively and produce a7-ray burst. Nevertheless, current rates seem to prevent such a situation. Le taux des reactions nucleaires depend de l'influence exercee par les particules voisines qui composent le plasma. A haute densite, la situation est loin d'etre satisfaisante et l'influence de la polarization electronique n'est pas sufisamment claire. En particulier, on montre que la possibility d'obtenir un collapse non explosif d'une naine blanche de carbone oxygene depend des facteurs d'ecrantage et des taux pycnonucleaires adoptes. Egalement, la possibility de detecter des etoiles a neutrons isolees depend des taux pycnonucleaires adoptes. Des petites valeurs favorisent la formation d'une couche metastable qui peut liberer de l'energie explosivement et produire 106

Isern & Hernanz: Screening of nuclear reactions

107

une eruption gamma. Quand- meme, les taux actuels semblent empecher cette situation

5.1 Introduction A nuclear reaction happens when two nuclei approach to a distance of the order of 10~13 cm after tunneling the Coulomb barrier. In the case of two isolated nuclei, this barrier is described by: Vu(r) = — - —

(5.1)

but in the case of a plasma it must be modified to account for the influence of the environment. It is possible to write: Vl2(r) = ^ ^ - + w(r)

(5.2)

where w(r) represents the average contribution of the surrounding particles to the interaction potential. Since the effect of this term is to lower the barrier, the transmission coefficient increases and the reaction rate can drastically differ from the case of isolated nuclei. For instance, in the case of a pure carbon plasma with a density p — 109 g/cm 3 and a temperature T = 109 K, the rate of the reaction 12 C+ 12 C increases by a factor of 1016 due to the influence of the surrounding particles. This effect was noticed for the first time by Schatzman (1948) and was later developed by several authors (Salpeter, 1954; Salpeter and Van Horn, 1969; DeWitt et al, 1973; Graboske et al, 1973; Jancovici, 1977; Alastuey and Jancovici, 1978; Itoh et al, 1978, 1979, 1990; Yakovlev and Shalybkov, 1988; Ogata et al 1991). Usually, the screening factor is written as

E=j-

(5.3)

where R and Ro are the rate of thermonuclear reactions with and without allowance for screening. If | E - 1 |<< 1, the screening is called weak. If | E — 1 |>> 1, it is called strong. In the first case, the theory developed by DeWitt et al (1973), Graboske et al (1973) or Yakovlev and Shalybkov (1988) is completely satisfactory from the astrophysical point of view since corrections are usually smaller than the current uncertainties in the astrophysical factors. This regime almost covers the evolution of stars in the main sequence (except the low luminosity end) but not the advanced stages of the evolution of intermediate stars. Once H is exhausted in the central regions, intermediate and low mass

108

Isern & Hernanz: Screening of nuclear reactions

stars develop an electron degenerate core. The typical densities of these cores do not allow to neglect the screening effects on nuclear reactions, which turn out to be dominant. This regime is not yet completely understood (see Rosenfeld , this volume), specially at high densities and improvements can change the chemical composition of degenerate cores or even the loci of the ignition curve in the cold, high density regions of the p-T plane. If densities are high enough, the barrier penetration is driven by the energy of the ground state of nuclei in a lattice instead of by the thermal energy and nuclear fusion happens in the so called pycnonuclear regime. This implies that, even at zero temperature, nuclear fusion can happen and that it is not possible to reach arbitrarily high densities without a nuclear rearrangement of matter. The transition from the strong screening to the pycnonuclear regime is not well known and all the work done up to now relies on interpolations that try to avoid discontinuities. A correct understanding of both regions, pycnonuclear and strong screening, is crucial in order to elucidate if CO white dwarfs can make a silent collapse to a neutron star or they will always explode as a Type la supernova when they accrete enough matter from a companion star. Both regions, specially the pycnonuclear one, are also relevant to elucidate if the long term evolution of slowly accreting neutron stars can result into a 7-ray burst, a thermonuclear explosion or just into a peaceful growth of their mass. Here in this paper, we will only address to the fate of accreting CO white dwarfs and accreting neutron stars. 5.2 The physical problem Dense plasmas can be characterized by several dimensionless parameters which facilitate the classification of physical properties in each problem. In order to simplify, it is possible to consider a plasma consisting of only one chemical species with atomic and mass numbers Z and A respectively, with a ion number density n< (or, quivalently, a density p = AHfii, where H is the atomic mass number) and a temperature T. The ion system can be described through the following parameters: The strength of the Coulomb coupling of ions is described by the so called Coulomb coupling constant, F,

where a is the ion-sphere or Wigner-Seitz radius, a = (3/47rn,)1/3. A weakly coupled plasma satisfies the relationship F << 1, while a strongly coupled

Isern & Hernanz: Screening of nuclear reactions

109

plasma satisfies F > 1. When T > 180, the ion plasma suffers a phase transition and solidifies into a bcc crystal (Slattery, Doolen, DeWitt, 1980, 1982; Heifer, McCrory, Van Horn 1984; Ogata and Ichimaru, 1987) The ratio between the thermal de Broglie wavelength of an ion, Ay, and the ionic spacing, a, describes the importance of the quantum effects in the description of the fluid properties. A = — = —.

(5.5)

A classical fluid satisfies A << 1. The importance of the quantum effects can also be outlined through the parameter r\ as:

where CJP is the ionic plasma frequency. Ze Values of r\ > 1 imply that the contribution of the quantum plasma effects to the reaction rates cannot be longer neglected. The electron component can be characterized through its Wigner-Seitz radius, usually expressed in units of the Bohr radius (Pines and Nozieres, 1966): r. = - ^

(5.8)

OB

where ae = (3/47rn c ) 1/3 and aB = H2/me2 = 5.292 10~9. The Fermi momentum of electrons in units of me is given by

X3A.) me

, ijOljg r,

while the Fermi energy, in units of me2, is given by EF = mc 2 (\/l-|-1.9610- 4 7-r 2 - 1)

(5.10)

The degree of degeneracy can be expressed through the well known parameter * kT~ kT where fi is the chemical potential of electrons.

110

Isern & Hernanz: Screening of nuclear reactions

The coupling strength between electrons and nuclei is given by the rate between the Thomas-Fermi screening length and the radius of the electronic sphere ae

For densities higher than 106 g/cm3, the electrons are weakly polarized and they can be treated as an ideal Fermi gas (Ajjr/a e >> 1). The temperature is usually referred to the Gamow energy, EQ. In the case of nuclei interacting via the bare Coulomb potential (case of a tenuous plasma), the rate, Ro, of the reaction is Ro oc exp(-3EG/kT)

(5.13)

and the parameter r, that only depends on the temperature, is denned as %jJ-/(J

r, £i % 7l

i £iaJL £J\ f^Q C

I F ~ ^~4~^ kTh

2

* 1 /O

J

*

(5

v

' ^

where y, is the reduced mass of the colliding particles. The radius, ro, of the classical turning point for colliding pairs with an energy equal to the Gamow energy is (Alastuey and Jancovici 1978) ro _ 3F _ a

T

TT

5.2.1 Strong screening regime

5.2.1.1 Nonresonant reactions In the case of nonresonant reactions, the reaction rate can be written as:

Rex jTS(E)P(E)exp(-^)dE

(5.16)

where P(E) is the Coulomb barrier-penetration factor, the exponential term is the Boltzmann factor and S(E) is the astrophysical factor. Since S(E) is a smooth function of the energy and P(E) exp(—E/kT) displays a pronounced peak around the Gamow energy, EG, it is possible to write: R oc S(EG) JQ°° P(E) exp (~)dE

(5.17)

and the integral can be interpreted as the pair correlation function at a distance of the order of the nuclear radius, g(r n ) (DeWitt, Graboske and

Isern & Hernanz: Screening of nuclear reactions

111

Cooper, 1973; Graboske et al, 1973; Alastuey and Jancovici, 1978; Yakovlev and Shalybkov, 1988): 9(r)

ex exp | -

^

|

(5.18)

Therefore, the enhancement factor can be written as (5.19)

)

where EG 0 and go are the Gamow energy and the pair correlation function for nuclei in an infinitely diluted plasma. Despite g(r n ) is a strongly varying function in the neighborhood of r n = 0 , the ratio g(r n )/go(rn) is a slowly varying function of r n and it is possible to write 9o(0) It has been shown (Jancovici, 1977; Yakovlev and Shalybkov, 1988) that AF

(T,n))

(5-21)

where the first term, which is dominant, corresponds to the purely classical contribution and is equal to the difference between the Coulomb excess of free energies after and before the interaction, and the second term contains the quantum corrections. This term, which is not negligible at all at high densities (Figure 1), has been computed by several authors using different methods: the WKB approximation (Itoh et al, 1977, 1979), the path integral (Alastuey and Jancovici, 1978) or directly (Ogata et al, 1991). A critical discussion of the consistency of the approximations involved in these calculations can be found in Yakovlev and Shalybkov (1989) and in Rosenfeld (this volume). Figure 2 displays the values of In E as a function of F for two values of 3F/r, and in section 3 we discuss the astrophysical consequences of using different approximations. 5.2.1.2 Resonant reactions In the case of resonant reactions, their rate can be written as: Ttot(Er)

6XP(

kT>

(5 22)

'

where E r is the resonant energy and Tin, Tout are the width of the direct and reverse reactions respectively, and Ttot — r,-n + Tout is the total width. In the case of a strongly coupled plasma, the resonant energy is shifted to £j. =

112

Isern & Hernanz: Screening of nuclear reactions

6.

8.5

9.5

10.5

log(RHO) Fig. 5.1 Ignition line for a pure carbon plasma. The continuous liie has been computed using the full expression of Alastuey and Jancovici (1978). The dashed line corresponds to the case where only the purely classical contribution is used. The dotted line, 3F/r = 1.6, outlines the limit of validity of the expression Er + AJP and the enhancement factor becomes (Mitler 1977, Mochkovitch and Nomoto, 1986):

AF Tin(Br)T(E'r)

(5.23)

5.2.1.3 Photodesintegration reactions The photodesintegration rate of a nucleus is given by

A oc< av > exv(-Q0/kT)

(5.24)

where < av > is the rate per pair of interacting particles of the forward reaction:

avxx. f

Jo

a(E)exp(-E/kT)dE

(5.25)

113

Isern & Hernanz: Screening of nuclear reactions

200

Fig. 5.2 Comparison of the screening factors obtained by Alastuey and Jancovici (1978), updated with the values proposed by Rosenfeld (this volume), H(RAIJ), and those obtained by Ogata et al (1991), H(OII). The continuous line corresponds to 3T/r = 1 and the dashed line to ZF/r = 0.5

E is the kinetic energy in the center of mass system, and increases by a factor E = ex P [-

, 17)] = e x p [ | £ + (T, r,)]

(5.26)

but since the energy threshold must include the contribution of the Coulomb interactions to the chemical,potential: Q = Qo + A/*, the photodesintegration rate is reduced by a factor exp(—Afi/kT) which cancels the dominant term of the screening factor (Mochkovitch, 1983; Mochkovitch and Nomoto, 1986). Therefore, the reaction rate becomes:

A = A(r =

(5.27)

and since (T, rf) is negative, the photodesintegration rate is reduced instead of enhanced.

114

Isern & Hernanz: Screening of nuclear reactions

5.2.1-4 Electron polarization In all the previous calculations it has been assumed that the degenerate electrons form a uniform, perfectly rigid background. This approximation is justified since ATf/a e , the ratio of the Thomas-Fermi screening length to the radius of the electronic sphere is larger than one in the regions of interest. Nevertheless, the polarization of electrons is not zero and this effect must be taken into account. If the polarization of electrons is small, the Coulomb excess of free energy per ion, or the chemical potential, can be written in the form of a term computed assuming a uniform electron background plus a small correction due to the electron polarization (Mochkovitch and Hernanz 1986; Yakovlev and Shalybkov 1988): inE = Ht + Hpot where Hpoi = -AFpoi/kT and AFPoi/kT is the difference of polarization energies of electrons. Notice that despite this term is usually called electron screening, it is not the screening due to electrons but the perturbation induced by the polarization of electrons to the screening computed assuming a uniform electron background. The contribution of electron polarization can be computed in the following way (Mochkovitch and Hernanz 1986): Hpoi = - 4 F r [ 2 4 / 3 / ( 2 5 / 3 r , x) - / ( r , *)]

(5.28)

where >S0(q)F(x,y)

-dq (5.29) q2 So(q) is the structure factor corresponding to the pure Coulomb unperturbed potential, F(x,y) is related to the dielectric function of relativistic electrons and x = hkF/mec, y = q/qF, 9 F = (97TZ/4)1/3, qrF = a/^TF, q = ak. In the case of a pure carbon plasma, Z=6, p = 109 g/cm3 and T= 108 K, H p0 | = 0.55 while H; = 37.2. Therefore, due to the uncertainties involved in the nuclear reactions and in the screening factor itself, it is usually possible to neglect the contribution of this term. These results, however, strongly differ from those of Ichimaru (1993) who found enhancements much more important, by one or two orders of magnitude. This issue is very important and must be clearly elucidated (see Rosenfeld, this volume) since these corrections can completely modify the results discussed below about the accretion induced collapse of white dwarfs. Recent independent calculations by Sahrling (1994) confirm the results of Mochkovitch and Hernanz, and Yakovlev and Shalybkov. The effect of the electron polarisability in the afore-mentioned example is found to be negligible (of the order of 1%).

Isern & Hernanz: Screening of nuclear reactions

115

5.2.2 Pycnonuclear reactions

The evaluation of the reaction rate in the pycnonuclear regime is quite complicated since the energy distribution of nuclei is no longer given by the Boltzmann distribution and it is necessary to solve the Schrodinger equation to obtain the wave function. The first discussion of this process was given by Cameron (1959) and was later elaborated by Wolf (1965), Salpeter and Van Horn (1969), Schramm and Koonin (1990) and Ogata et al (1991), in the case of one component plasma. The case of binary ionic mixtures has been recently treated by Ichimaru et al (1992), while the case of the 3a process has been treated by Salpeter and Van Horn (1969), Fushiki and Lamb (1987), Schramm et al (1992) and Langanke and Mtiller 1993). 5.2.2.1 Fusion rates in one component plasma In the pycnonuclear regime, no independent particle model provides an adequate description of the fusion process and there are not simple approximations to the potential of the interacting particles. It is possible, however, to find two limiting cases for the potential (Salpeter and Van Horn, 1969): The "static" and the "relaxed" approximations. The first one assumes that all the nuclei as well as the center of masses of the reacting pair are frozen at the equilibrium positions, while the second one assumes that the position of the center of masses is fixed and the remaining lattice points polarize into the positions determined by the separation of the two reacting nuclei. Wolf (1965), using a simplified study of the dynamics of the crystal, obtained an intermediate potential between these two limiting cases, but nearer to the static case. Lately, Ogata et al (1991) determined the interparticle potential using a MonteCarlo method and found a result that is near to the relaxed approximation. The result of Ogata et al (1991) can be easily understood in the following terms: Let At be the time spent by a reacting pair with a typical energy hup into the barrier and let u~x be the characteristic response time of the crystal. Since up At ~ 140Z5/2p^"1/4 (Schramm and Koonin 1991) is bigger than one for all the situations of physical interest, the lattice will respond adiabatically and will polarize under the influence of the reacting pair. Nevertheless, since the kinetic energy associated to the polarization must be taken into account during the calculation of the tunneling probability, it turns out that both effects of the polarization almost cancel and the final fusion rate is very near to the static one (Schramm and Koonin, 1991). The inclusion of dynamic terms has been contested by Ichimaru (1993) who argued that all the dynamic effects are already included in the

116

Isern & Hernanz: Screening of nuclear reactions

5.00

3.00 -

1.00 -

-1.00

RHO/1E9 Fig. 5.3 Comparison of different fusion rates normalized to the static rates,log(R/R,«o«,c), versus A, where A is the ratio of the nuclear Bohr radius and the lattice spacing (A a p 1 ' 3 ). Solid line is from Schramm and Koonin (1991). Dashed line is from Ogata et al (1993). Dotted line corresponds to the relaxed approximation of Salpeter and Van Horn (1969). pair correlation function through the dynamic structure factors since nuclear reactions are very rare events as compared with all the other collision processes. Figure 3 displays the comparison between the different fusion rates quoted here.

5.2.2.2 Fusion rates in a binary ionic mixture In the case of a plasma with several components the situation is rather complicated because of the uncertainties about the structure of the crystal. Usually, the fusion rate has been taken as c(x) = c(x = 1) x 2 , where x is the molar fraction. The validity of this expression relies on the assumption that the reacting nuclei are randomly distributed and that the lattice potential is uniform. The existence of lattice imperfections can be handled in the following way

Isern & Hernanz: Screening of nuclear reactions

117

(Salpeter and Van Horn 1969): The presence of a default produces a local increase of the density that translates into an increase of the local rate by a factor exp(1.3A~1/2 6X/X), where SX is the local excess of the parameter A. Because of the extreme sensitivity of the pycnonuclear reactions to the density, the nuclear fusion will start at these defaults and, if the relaxation of the reaction products produces new defaults, the reaction rate will be strongly enhanced. The presence of other chemical species can have two different consequences. If the reacting and non-reacting nuclei have similar charges (case of C O mixtures) the heavier species blocks the reaction and its rate is strongly reduced. If they have very different values, the heaviest one induces a local increase of the density and the reaction rate is strongly enhanced. The critical value that separates both behaviors is Z2/Z1 = 2.3, where Zi > Z\. Figure 4 shows the influence the presence of oxygen and iron on the pycnonuclear fusion of carbon. 5.2.2.3 The 3a fusion rate The 3a reaction in the pycnonuclear regime was considered for the first time by Cameron (1959) and later on by Salpeter and Van Horn (1969), Fushiki and Lamb (1987) and Schramm et al (1992). Besides the problem of its three-body nature, the pycnonuclear 3a reaction is characterized by the fact that temperatures are too low to allow the resonance of 8Be and by the fact that in the region of interest helium is a quantum fluid rather than a Coulomb lattice. The present results are quite satisfactory since different approaches give the same results. Nevertheless, the effect of the presence of impurities remains to be studied. It is interesting to notice that above p ~ 2 — 3 109 g/cm3 a lattice made of 8Be has a lower energy than a 4He fluid (Schramm et al 1992) and 8Be becomes stable. However, the pycnonuclear 3a reaction is able to convert helium into carbon before reaching this density and the possibility of the existence of 8Be matter seems to be prevented (Langanke and Miiller, 1993). 5.3 The astrophysical problem 5.3.1 Collapse or explosion of mass accreting white dwarfs More than fifteen years ago, Schatzman suggested that besides the standard mechanism for neutron star formation (collapse of the fuel exhausted core of a massive star), a second mechanism was possible: gravitational collapse of a white dwarf due to accretion of matter in a close binary system. In

118

Isern & Hernanz: Screening of nuclear reactions

2.00

IMII

0.00 -

-1.00

RHO/1E9 Fig. 5.4 Blocking and catalyzing effects of impurities in the pycnonuclear fusion of carbon. The vertical axis displays log[R(x £ 0)/R(x = 0)] for the 1 2 C+ 1 2 C reaction in the presence of oxygen (continuous line) and iron (dashed line). In both cases the molar abundance of the heavier species is x=0.5

both cases, collapse would be induced by growth above the effective Chandrasekhar's mass. The second mechanism would account for the presence of neutron stars in systems such as the low-mass X-ray binaries, where survival to a Type I supernova explosion appears unlikely. Accretion induced collapse of white dwarfs has later been incorporated in evolutionary scenarios for the origin of several kinds of binary systems containing neutron stars (Van den Heuvel 1983, 1989). Even if white dwarfs might grow in mass at any arbitrary rate, which is not the case (Isern et al 1983; Hernanz et al 1988; Canal et al 1990), accretion induced collapse would in any case encounter the difficulty that explosive ignition of the nuclear fuel in the electron degenerate cores always precedes reaching the Chandrasekhar's mass. For white dwarfs made of carbon-oxygen or oxygen-neon-magnesium, ignition can be delayed until losses due to electron captures overcome the energy released by ther-

Isern & Hernanz: Screening of nuclear reactions

119

monuclear reactions (Isern and Canal, this volume). CO white dwarfs were proposed by Canal and Schatzman (1976) while ONeMg white dwarfs were advocated by Miyaji et al (1980). Concerning CO white dwarfs, it was soon realized that central ignition in a partially solid core provided the most favorable conditions for accretion induced collapse (Canal and Schatzman 1976; Canal and Isern 1979). The behavior of a cool white dwarf interior during the accretion phase depends on the competition between physical processes that increase the temperature of the material (compression, nuclear reactions in the inner core and thermonuclear burning of the accreted matter) and those that cool down the star (photon and neutrino losses). Obviously, heat conduction plays an important role, as being the main mechanism for heat transport in the stellar interior. Conduction can be characterized by the time required for a thermal signal to travel from the center to the surface of the star. This time is given by Henyey and L'Ecuyer (1969) as: •

(5-30)

where / is the linear extent of the region considered and all the remaining symbols have their usual meanings. The time scale for increasing central density, when mass approaches the Chandrasekhar's limit, can be expressed as (Canal and Schatzman 1976):

where M is the mass accretion rate and yo = -Jl + x2F, where x/r is the dimensionless Fermi momentum. The effects of the compression induced by the accreted matter can be divided in two terms (Nomoto 1982). The first one is due to the increase in density at a fixed mass fraction as the stellar mass increases, and its effects are quite uniform throughout the whole star. The second term corresponds to compression as matter moves inwards in the mass-fraction space. It is negligible in the inner strongly degenerate regions of the star, but it is very large in the semi degenerate external layers. This means that a thermal wave is generated in those layers which diffuses inwards. A rough estimate of the compressional luminosity of the external layers is (Nomoto 1982): LNH -?— = 1.4 lO- 3 r 7 Mio I©

(5.32)

120

Isern & Hernanz: Screening of nuclear reactions

T7 being the temperature in units of 107K and M\Q the accretion rate in units of 1O"10 M©/yr. The effects of this thermal wave on the temperature profile, and thus on the physical state of the white dwarf interior, will depend on the time needed for the thermal wave to reach the center of the star as compared to the time required for the star to reach the Chandrasekhar's limit and also on the efficiency of thermal cooling of the white dwarf (Hernanz et al 1988). For low accretion rates (10~12M©/yr< M < 3 10"10M©/yr), the thermal wave generated by the accretion process in the external layers can reach the center of the star before any instability starts, but compression is so slow that radiative cooling through the star surface dominates. The white dwarf, in this case, evolves keeping an isothermal profile and the internal temperature is determined by the balance between compression and cooling through the envelope. For high accretion rates (M > 5 10~8 M©/yr), compressional luminosity is much higher. However, in this case, the thermal wave has no time to reach the center for high enough masses, and only the outermost solid layers are affected. In the central solid layers TTH » TP and they evolve quasiadiabatically. The slope for the evolutionary path in the log p — log T diagram is given by (Mochkovitch 1983) 0.815 + 0.215F1/4 (5 33) T3 1= ~ 0.945 + 0.6461^ ' F3 — 1 being the adiabatic index. For F values in the range 100 - 200, the adiabatic index is ~ 0.5, For intermediate accretion rates, (310- 10 M©/yr< M < 510~8M©/yr), the thermal wave has again enough time to reach the center of the star. Compressional heating dominates the normal cooling and matter heats up. The outcome, collapse or explosion, depends on the density at which the thermonuclear runaway starts and this critical density depends on the velocity at which the burning front propagates. The minimum density necessary to get a collapse is 8.5 109g/cm3 (Isern and Canal, this volume). Garcia and Bravo, also this volume, propose an even smaller value, 7109 g/cm3. However, if all the modes of propagation of the burning front are taken into account, this critical density adopts a value in the range 9.2 to 9.5 109 g/cm 3 , although these limits are rather uncertain. It is thus interesting to examine the dependence of the ignition density on the adopted value of the nuclear reaction rate. Table 1 displays the ignition density for different accretion rates. All the models have been constructed in the same way as in Hernanz et al (1988). Models labeled with a capital letter

Isern & Hernanz: Screening of nuclear reactions

121

Table 5.1. Ignition densities (in units of 109 g/cm3) for different pycnonuclear reaction rates M(M0/yr)

A

B

a

b

c

io- 6 io- 97

10.60 9.05 7.87 7.98 7.73

12.40 10.10 9.10 9.92 9.66

10.40 6.63 7.82 7.98 7.71

12.10 6.63 8.97 9.89 9.63

10.85 6.51 7.99 8.26

lO"

io- 10 io- 11

started at M^D ~ 1-2M®, those with a small letter started at 1.1. The initial temperature was To = 4 IO6 K for all of them. Pycnonuclear reaction rates from Ogata et al (1991) were used in models A and a. Models B and b were obtained using the rates of Schramm and Koonin (1990), while model c was obtained with the pycnonuclear rates of Ichimaru et al (1992), which include the blocking effects due to the presence of oxygen nuclei. Columns A and B show the importance of including the dynamics of the crystal in the reaction rates. In the first case, almost all of them collapse. Comparison of model c with a shows that the blocking effects of oxygen effectively delay the runaway but do not introduce any qualitative change in the above picture. The initial mass is also a critical factor. Below 1.05 M®, the accretion induced collapse is impossible. Since the outer thermal wave has less time to reach the centre, collapse is favored by increasing the initial mass. This implies that only the tail of most massive CO white dwarf can collapse to form a neutron star. The strong screening factors adopted and the interpolation algorithm between strong screening and pycnonuclear regimes are also critical. Table 2 displays the ignition density in the case of a white dwarf with an initial mass MWD = 1.2 M®, initial temperature T = 410 6 K, which accretes matter at a rate M = 10~ 7 M^/yr and is made of carbon and oxygen. The screening factors have been obtained from Ogata et al (1991), Oil model, and from Rosenfeld (1993) and Alastuey and Jancovici (1979), RAJ model. The pycnonuclear rate was obtained from Ichimaru et al (1993). The gap between pycnonuclear and strong screening reactions was covered with the interpolation algorithm described by Bravo et al (1983). In model A, the rates were obtained interpolating between the line 3F/r = 1 (strong screening regime) and 3 r / r = 8.1 (pycnonuclear region), Model B by interpolating between 3T/T = 1 and T = 180 and model C between ZT/T = 1.5 and T = 180. As it

122

Isern & Hernanz: Screening of nuclear reactions

Table 5.2. Ignition densities (in units of 109 g/cm3) for different strong screening factors B

OH RAJ

9.23 8.88

9.90 9.70

off 9.88

can be seen from table 2, the results are spectacular and even an off center ignition is obtained in case C model OIL 5.3.2 Slowly accreting neutron stars Neutron stars are rather abundant in the Galaxy. Their estimated density in the solar neighborhood is n ~ 104/(300pc)3. Few of them can be detected as pulsars, if they are young enough or/and they rotate fast enough, or as bright X-ray sources, if they are members of a close binary system (in this case, they accrete matter at a rate M ~ lO^Mo/yr and they emit Lx — 1038 erg/s in X-rays). The question is: what happens with the remaining, isolated and old, neutron stars that accrete at a typical rate of 1010 g/s?. The evolution of neutron stars that accrete mass at a very low rate, 10~ 16 < M < 10~llMQ/yr, has been studied many times (Hameury et al 1983; Blaes et al 1989, 1990, 1992; Miralda-Escude et al 1990; Zdunik et al 1992) In all of them the pycnonuclear rates have proved to play a critical role since the mass-accretion is so slow that the temperature is too low to allow thermonuclear rearrangement of nuclei. Their evolution can be summarized in the following way: a) Few years after the explosion, the neutron star captures the bound debris and burns them to 56 Fe. Because of the weight of the newly accreted matter, this material sinks and starts capturing electrons: 56Fe—>56Cr—^56Ti. b) After the pulsar phase, the neutron star can start accreting matter from the interstellar medium, M ~ 1010 V^QUISM g/s> where U40 is the speed of the star relative to the interstellar medium in units of 40 km/s and niSM is the hydrogen number density of the interstellar gas in cm" 3 (Hoyle and Lyttleton, 1939), which imply accretion rates of the order of 1010 g/s. The newly accreted matter compresses the crust and hydrogen is burnt to helium due to pycnonuclear reactions or electron captures, depending on the accretion rate. Since spallation reactions at the surface of the star destroy most of the metals (Bildstein et al 1991), burning is stable as far

Isern & Hernanz: Screening of nuclear reactions

123

as M < 2 1013 g/s and the crust can be considered isothermal, Ts = M^ (Blaes et al 1992, for such low rates. c) The helium newly formed in this way solidifies at a temperature T9 ~ 3.3 lOV 1 / 3 . For an isothermal crust of the kind mentioned in b), this will happen at a density pa = 2.8 107Mfflu (Blaes et al 1992). As He behaves like a quantum liquid for densities higher than 310 8 g/cm3, (Mochkovitch and Hansen 1979, Chabrier 1993), a layer of solid He surrounded by a classical liquid in the top and a quantum liquid in the bottom will form. When the density is high enough, helium ignites due to pycnonuclear reactions. The characteristic time scales for the a-captures are: r[ 12 C(a, 7 ) 16 O] ~ 10"9s r[ 16 0(a,7) 2O JVe]~lO 15 3 Thus, if the 3o>reaction were slow, the freshly synthesized oxygen would have time to capture an a-particle and approach to Si. Since the 3a rate is very high (Schramm, Langanke and Koonin, 1991) only 12C has time to capture an a-particle and an 16O-rich mixture forms. As oxygen crystallizes at higher temperatures than helium (pc a 1.3 105M10 ' g/cm 3 ), it immediately solidifies and sinks, accumulating at the bottom of the quantum He-ocean. d) As the accretion proceeds, the density at the bottom of the accreted material increases and finally at p = 1.921010 g/cm 3 , 1 6 0 undergoes a twostage electron capture to 16 C. If this 16C accumulates or fuses depends on the pycnonuclear rate of the reaction 16 C + 1 6 C. If it accumulates, it can undergo an elastic Rayleigh-Taylor instability with the old underlying crust resulting in a starquake which releases some 1040 ergs of gravitational and nuclear energy (Blaes et al 1990, 1992). The fusion of 16C was studied by Bravo et al (1983). It has the following characteristics: 16

C + 1 6 C -^ 28 Mg + An (90%) Q = l

i e c + i 6 c _^29 Mg

+ 3n

( 1Q% ) Q __ 1

with So = 5.5 108 MeV-barn. At the density at which it is produced, its characteristic fusion time scale is 1000 yrs, quantity that can be much shorter if some heavy elements coming from the original crust diffuse and mix with the 16C layers. Therefore, the composition will change to 28Mg. During all this process, the underlying original crust makes a transition to

124

Isern & Hernanz: Screening of nuclear reactions

56

Ti, 62Cr or 62Fe because of electron captures and, since its mean molecular weight per electron is equal or higher than that corresponding to 28Mg, the accreted layer will be stable. 5.4 Conclusions Carbon-oxygen white dwarfs can either collapse or explode depending on their initial properties. If they are massive and cool enough, they can nonexplosively collapse and form a neutron star. The frequency and relevance of this phenomenon, as compared with the accretion induced collapse of ONeMg white dwarfs, critically depends on the adopted rates and on the way as the transition between strong screening and pycnonudear regimes is handled. The effects due to the polarization of electrons can be critical. If they increase the reaction rates by one or two orders of magnitude as claimed by Ichimaru (1983), the collapse to a neutron star will be prevented in the majority of cases. The possibility of observing isolated neutron not detectable as pulsars also depends on the adopted pycnonudear rates. If the rates are high enough, due to the presence of defaults or heavy impurities, the slowly accreted material is gradually transformed and neutronized without forming a metastable layer that could release its energy in a burst. Therefore, due to the current uncertainties on the screening factors it is not possible to reach a definite conclusion in both cases. It should be of the highest interest to definitively elucidate the behavior of nuclear reaction rates at high densities (3F/r > 1) and in the pycnonudear region (F > 180). Acknowledgements This work has been partially financed by the DGICYT grant PB91-060, the Spanish-French Action "Physics of White Dwarfs and Brown dwarfs" and the CESCA grants "Structure and Evolution of Galaxies" and "Accretion onto White Dwarfs". References Alastuey A., Jancovici B.,Astrophys. J. 226, 1034 (1978) Bravo E., Isern J., Canal R., Labay J.,Astron. Astrophys.124, 39 (1983) Bildsten L., Salpeter E.E., Wasserman l.,Astroph. 7.384, 143 (1992) Blaes O., Blandford R., Goldreich P. and Madan P.,Astrophys. 7.343, 839 (1989) Blaes O., Blandford R., Madan P. and Koonin S.,Astrophys. 7363, 612 (1990) Blaes O., Blandford R., Madan P. and Yan L., Astrophys. 7.399, 634 (1992) Cameron A.G.W., Astrophys. J. 130, 916 (1959) Canal R., Isern., in White Dwarfs and Variable Degenerate Stars IAU Colloq N o l 5 3 , ed H.M. Van Horn, V. Weidemann, p.52, Rochester NY: Univ. Rochester Press (1979)

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Canal R., Isern J., Labay J., Ann. Rev. Astron. Astrophys. 28, 183 (1990) Canal R., Schatzman E., Astron. Astrophys. 46, 229 (1976) Chabrier G., Astrophys. J. 414, 695 (1993) DeWitt H.E., Graboske H.C., Cooper M.S.,Astrophys. J. 181, 439 (1973) Graboske H.C., DeWitt H.E., Grossman A.S., Cooper U.S.,Astrophys. 7. 181, 457 (1973) Fushiki I., Lamb D.Q., Astrophys. J. 317, 368 (1987) Hameury J.M., Heyvaerts J. and Bonazzola S.,Astron. Astrophys.121, 259 (1983) Heifer H.L., McCrory R., Van Horn H.M., 7. Stat. Phys 37, 577 (1984) Henyey L., L'Ecuyer J ., Astrophys. J. 156, 549 (1969) Hernanz M., Isern J., Canal R., Labay J., Mochkovitch R., Astrophys. 7. 324, 331 (1988) Hoyle F. and Lyttleton R.A.,Proc. Cambridge Phil. Soc.35, 592 (1939) Ichimaru S.,Rev. Mod. Phys. 65, 255 (1973) Ichimaru S., Ogata S., Van Horn H.M., Astrophys. J. Letters 401, L35 (1992) Isern J., Labay J., Hernanz M., Canal R., Astrophys. 7. 273, 320 (1983) Itoh N., Kuwashima F., Munakata E.,Astrophys. 7. 362, 620 (1990) Itoh N., Totsuji H., Ichimaru S.,Astrophys. 7. 220, 742 (1978) Itoh N., Totsuji H., Ichimaru S., DeWitt H.E.,Astrophys. J. 1079, (1979) Jancovici B.,7. Stat. Phys. 17, 357 (1977) Miralda-Escude J., Haensel P. and Paczynski B.,Astrophys. 7.362, 572 (1990) Miyaji S., Nomoto K., Yokoi K., Sugimoto D., Publ. Astron. soc. Japan 32, 303 (1980) Mochkovitch R., Astron. Astrophys. 122, 212 (1983) Mochkovitch R., Hansen J.?.,Phys. lett.73A, 35 (1979) Mochkovitch R., Hernanz, M., in Nucleosynthesis and its implications on nuclear and particle physics, eds. J. Audouze and J. Tran Thanh Van, p. 109, Reidel: Dordretch (1986) Nomoto K., Astrophys. 7. 257, 780 (1982) Ogata S., Ichimaru S., Phys. Rev A36, 5451 (1987) Ogata S., Iyetomi H., Ichimaru S.,Astrophys. 7. 372, 259 (1991) Pines D, Nozieres P., The theory of quantum liquids (Benjamin: New York) (1966) Sahrling M., to appear in A&A, (1994) Salpeter E.E., Aust. J. Phys. 7, 373 (1954) Salpeter E.E., Van Horn H.M., Astrophys. J 155, 183 (1969) Schatzman E., presented at InT. Sch. Cosmol. Gravit. Erice, Italy (1974) Schramm S., Koonin S.E., Astrophys. 7. 365, 296 (1990) Schramm S., Langanke K and Koonin S.E.,Astrophys. 7.397, 579 (1992) Slattery W.L., Doolen G.D., DeWitt H.E., Phys Rev A21, 2087 (1980) Slattery W.L., Doolen G.D., DeWitt H.E., Phys Rev A26, 2355 (1982) Van den Heuvel E.P.J., in Accretion-Driven Stellar X-Ray Sources Ed. W.H.G. Lewin, E.P.J. Van den Heuvel, p.303, Cambridge Univ. Press () Van den Heuvel E.P.J., in Timing Neutron Stars, ed H.O. Ogelman E.P.J. Van den Heuvel, Dordrecht: Kluwer, p.523 (1989) Yakovlev D.G., Shalybkov D.A., Adv. Space. Res. 8, (2)707 (1988) Zdunik J.L., Haensel P., Paczynski B. and Miralda-Escude J.,Astrophys. 7.384, 129 (1992)

6 Crystallization of dense binary ionic mixtures. Application to white dwarf cooling theory. R. MOCHKOVITCH Institut d'Astrophysique it Paris, 75014 Paris, France

L. SEGRETAIN Laboratoire de Physique. Ecole Normale Superieure de Lyon, 69364 Lyon Cedex 07. France

Abstract This paper is organized in two parts. First, phase diagrams for dense binary mixtures are computed with the density functional theory (DFT). The method of calculation is reviewed and the different approximations which are used are clearly stated. The DFT is then applied to several mixtures of astrophysical interest. A comparison is made between several existing phase diagrams and the origin of some discrepancies among them is discussed. In a second part, the consequences of these phase diagrams on the cooling of white dwarfs are presented in a pedagogical way starting from the simple Mestel theory. The importance of the partial separation of carbon and oxygen at crystallisation is emphasized and the possible effect of minor species such as 22Ne or Fe is also considered. The separation of carbon and oxygen adds 1 - 2 Gyr to age of the galactic disk estimated from the white dwarf luminosity function while the delay resulting from the presence of minor species is probably negligible when the chemical evolution of the Galaxy is properly taken into account. Cet article est organise en deux parties. Tout d'abord, les diagrammes de phase des melanges binaires denses sont calcules a l'aide de la theorie de la fonctionnelle de densite. La methode de calcul est detaillee et les differentes approximations utilisees sont clairement expliquees. Le theorie est ensuite appliquee a plusieurs melanges d'interet astrophysique. Une comparaison est faite entre plusieurs diagrammes de phase publies et l'origine de certains 126

Mochkovitch & Segretain: Crystallization of dense binary ionic mixtures

127

disaccords entre eux est discutee. Dans la seconde partie, les consequences de ces diagrammes de phase sur le refroidissement des naines blanches sont presentees de maniere pedagogique avec pour point de depart la theorie de Mestel. L'importance de la separation partielle du carbone et de l'oxygene durant la cristallisation est soulignee et l'effet possible des especes mineures, comme le 22Ne et le fer, est aussi considere. La separation du carbone et de l'oxygene ajoute 1 a 2 milliards d'annees a l'age du disque galactique obtenu a partir de la fonction de luminosite des naines blanches alors que le delai du a la presence des especes mineures est probablement negligeable quand revolution chimique de la Galaxie est convenablement prise en compte. 6.1 Calculation of the phase diagrams 6.1.1 Introduction The importance of Coulomb interactions in white dwarf interiors has been recognized long ago (Kirshnitz 1960; Abrikosov 1960; Salpeter 1961), and was shown to lead eventually to crystallization of the core of the star (Van Horn 1968; Lamb and Van Horn 1975). The importance of the alloying behavior of dense Coulomb lattices to astrophysics was first pointed out by Dyson (1971), and the importance of the crystallization diagram of C/O mixtures on the evolution of white dwarfs was first considered by Stevenson (1980). In order to study the crystallization of the white dwarf core, we consider this one as a multi-component plasma, an extension of the one component plasma model (OCP) (Brush, Sahlin, Teller 1966; Hansen 1973). The OCP is characterized universally by the dimensionless parameter T = Zs/3e2/kTae = Z 5 / 3 r c where Z is the charge of the ion and ae — (3/(4n-ne))1^3 is the electronic sphere radius with n e the number density of electrons. The present critical value of the paramater T at crystallization is Tc « 172 - 180 (Ogata, Ichimaru 1987; Strinfellow et al. 1990; Dubin 1990; Farouki and Hamaguchi 1993) and the release of the latent heat is about Ar^T'/'particle. However, besides carbon and oxygen, minor elements like neon and iron are present in the core of white dwarfs and the crystallization of this mixture is more difficult to study than for the one component system. In the next section we examine the different methods used to build a phase diagram and we present all existing results. 6.1.2 Methods of calculation The determination of a phase diagram for a multi-component plasma requires the determination of the free energy in the liquid (FL) and the solid

128

Mochkovitch & Segretain: Crystallization of dense binary ionic mixtures l.U

1.4 =

>

: p=cst

:

'

A

1.3 /

1 1 1 1

1.2

9

_

—

TI

—

/

. • '

|

1.1

n

1 0 . 2 0 4 0.6 0.8

—

1 0.8 0.6 0.4 0.2 0

Fig. 6.1 Construction of phase diagram (Fs) phase, as shown infig.1.1. We compute free energies in the two phases at a given temperature for all concentrations and we determine the equilibrium concentrations in the solid (is) and in the liquid (z£,) phase with a double-tangent method. We iterate this process for several temperatures to build the phase diagram. Two methods can be used to compute JFJ and Fa. The first one is the Monte-Carlo simulation. The advantage of this method is to be an "exact" calculation since the energy is calculated directly from the canonical average of the interaction , but a reliable result requires a large number of particles since the accuracy of the method is proportional to 1/N. This requirement becomes really restricting for a mixture when the concentration of one of the species is small ; in this case the total number of particle becomes prohibitive, and the simulation requires a tremendously large computation time. The second method is based on the density fonctional theory of freezing, and is described below. 6.1.3 The density functional theory of freezing The density functional theory (DFT) is based on the theorem (Honenberg and Kohn 1964) that any thermodynamic function can be written as a functional of the density. It has been applied to phase diagrams for the first time by Ramakrishnan and Yussouf (1979) for the crystallization of a hard

Mochkovitch & Segretain: Crystallization of dense binary ionic mixtures

129

sphere system. Since then, the DFT has been used for different short range repulsive potentials (V(r) oc r ~ n , n = oo,12,6,4) (Barrat 1987) and was shown to be in good agreement with the simulations. Within the framework of the DFT, we can calculate directly A F = Fa—F[ by considering the solid near melting as a non-uniform liquid and by calculating the related correlation functions from the Taylor expansion around the liquid correlation functions. In order to calculate AF, several approximations are necessary. i) The first one is the truncation of the Taylor expansion at a given order, usually the second order. That yields, for the OCP, to : A F = AFid + AFex

(1)

with

NkT ~ Jv

PL

l

PL

(2)

and

53^ represents the sum over all non zero reciproqual lattive vectors (RLV), C(G) denotes the Fourier transform of the direct correlation function (DCF) of the liquid and the one-particule density in the solid is written as :

p(r) = pL(l + Y1P(G) exp(ig.f))

(4)

However a problem appears for Coulomb systems if the Taylor expansion is truncated at the second order. In this case, Rovere and Tosi (1985) showed that the solid phase is never stable, its energy being overestimated because of the missing compensating negative contribution of the higher order terms. Two solutions can be used to solve this problem. The first one is to include higher orders terms (third order or more). This is tremendously complicated and has been done only for the OCP (Barrat 1987; Iyetomi and Ichimaru 1988; Likos and Aschroft 1992). Then a second, more phenomenological, approach is used. The effect of higher order terms is equivalent to a reduction of the contribution of the second RLV (i.e. the term C^p2^ ) (Likos and Aschroft 1992). Then the solution is to reduce arbitrarily this contribution in such a way that the crystallization of the OCP is recovered

130

Mochkovitch & Segretain: Crystallization of dense binary ionic mixtures

for Tc = 178 (Barrat et al. 1988; Ichimaru et al. 1988; Segretain and Chabrier 1993). ii) The second approximation concerns the description of the solid phase, namely the choice of a description of the solid density fluctuation (i.e. pg), of a crystal lattice (i.e. {G}), and of a type of alloy (for a BIM). For the description of the solid density fluctuation, a gaussian approximation is usually adopted (i.e pg = exp(—G2/a)). The accuracy of such an approximation has been assessed in all situations studied so far (Tarazona 1984; Baus and Colot 1985; Baus 1987; Barrat et al. 1987,1988; DeWitt et al. 1993). With respect to the choice of the crystal lattice, the BCC (body centered cubic) lattice has been shown to be the most stable configuration for the OCP (Stringfellow et al. 1990). Finally the type of alloy can be either ordered (like a CsCl crystal) or random, i.e. each species occupies a node of the lattice with a probability equals to xu so that :

XvM*)3'2 £ex P (-a,,(f- Rtf)

(5)

Although an ordered crystal is more stable at T = 0, a random alloy is favored at finite temperature because of the entropic contribution (Segretain and Chabrier 1993; Ogata et al. 1993). iii) The third and last approximation concerns the calculation of the correlation functions (Cg). Two N-body theories are used. The first one is the so-called mean spherical approximation (MSA) (Hansen and MacDonald 1976) which reads (for one component) : f g(r) = 0,

r
(6)

where V(r) is the potential, c(r) the direct correlation function and g(r) the pair distribution function. The great advantage of the MSA is that it can be solved analytically for a multi-component plasma (Parrinello and Tosi 1979). The second theory is the so-called (Improved) Hypernetted Chain approximation ((I)HNC) (Hansen and MacDonald 1976) where : g(r) = exp(-0V(r) + h(r) - c(r) + B(r))

(7)

The HNC solution corresponds to B(r) = 0 in eq. (7), where B(r) is the so-called bridge function, representing the elementary diagrams. These two approximations form a closed system of equations with the Ornstein-Zernicke relation :

h(r) = g(r)-l

= c(r) + pjc(\r-

P\)h(r')dr'

(8)

Mochkovitch & Segretain: Crystallization of dense binary ionic mixtures

131

The free energy of the liquid phase is calculated as : Fl({xi}tP,T)

= Fid + Fex

(9)

The ideal contribution Ftd is given by the standard well-known expression (Landau and Lifschitz 1958). The excess (non-ideal) free energy of the mixture Fex is given within an excellent approximation (< 1%) in term of the excess ionic free energies of the pure phases at the same pressure, i.e at the same electron density (and then constant F e ) under white dwarf interior conditions, by using the so-called linear mixing law, whose accuracy has been demonstrated by several authors (Brami, Hansen and Joly 1979; Ichimaru, Ogata and Iyetomi 1988; Chabrier and Ashcroft 1990; DeWitt and Slattery 1993): /C

*(P'T)

=

The free energy of the solid phase is then given by Fs = Fi + AF, where AF is given by eq. (1). It is important to stress that, within the DFT, the form of the phase diagram (i.e spindle, azeotrope, eutectic ...) is entirely determined by A F only, and does not depend on any approximation entering the calculation of FL6.I.4 Results In this section, we give a review of the existing calculations of phase diagrams of astrophysical interest. 6.1.4-1 Carbon-oxygen mixture The first calculation by Stevenson (1980) lead to an eutectic phase diagram for C/O mixtures. In his calculation, Stevenson assumed the solid BIM to be entirely random, so that the free energy was given in term of an effective coupling parameter (F c ^ M ({a; t },re) ~ - 0 . 9 r c / / with Teff =< Z > 5 / 3 Te, and < Z >= Q2iZiXi)), whereas the mixture retains in fact some short range order, through the direct correlation functions, and the free energy 0.9 < Z 5 / 3 > r e ) , is given by the linear mixing rule (F£ / A f ({*i},r e ) as demonstrated since then (see above). In Stevenson's scenario, the solid phase is less stable, since Fejj > F\m, which leads to an eutectic diagram. The calculations of Barrat et al. (1988) (fig. 1.2) are based on the DFT formalism, and use the MSA theory to calculate the correlation functions between the particles, but with the same diameter a for the two ionic species. Moreover, Barrat et al. used the old value F c = 168 for the cristallization of

132

Mochkovitch & Segretain: Crystallization of dense binary ionic mixtures

Fig. 6.2 Phase diagram for C/O mixture. Full line : Segretain and Chabrier; dotted line : Barrat et al.; dashed line : Ichimaru et al. . Note that for the diagram of Barrat et al. Tc = 168 instead of Tc = 178 for the other two. the pure phase. The diagram is found to be of a spindle form (dotted line). More recently, Segretain and Chabrier (1993) extended these calculations by considering different diameters for the two ionic species and by using Tc = 180 for the crystallization value for the OCP. The diagram (full line) confirms essentially the results of Barrat et al. The calculations of Ichimaru et al. (fig. 1.2) are also based on the DFT, but the correlation functions are calculated within the framework of the IHNC. They find an azeotropic diagram with an azeotropic point characterized by xc = 0.16, with XQ the concentration of carbon. As long as they are used for astrophysical applications, it is interesting to note that these diagrams are very similar, and lead essentially to the same results. 6.1.4-2 Arbitrary ionic mixtures Recently, Segretain and Chabrier (1993) (fig. 1.2 and 1.3) extended the former calculations to examine the evolution of the crystallization diagram of arbitrary ionic mixtures, as a function of the charge ratio. The calculations were performed within the framework of the DFT, and the correlation functions were calculated within the MSA theory, with different diameters for the two ionic species. The width of the gaussian, a,,, in the ion distribution function of the solid (eq. (5)), was found to agree perfectly (< 0.5%),

Mochkovitch & Segretain: Crystallization of dense binary ionic mixtures

133

with the value obtained by molecular dynamics simulations, for the OCP (Slattery and DeWitt 1993). They find that the phase diagram evolves from a spindle form for 0.72 < C = Zi/%2 < 10, to an azeotropic form for 0.58 < C < 0-72 and to an eutectic form for ( < 0.58. This bears important consequences for the cooling of white dwarfs, as will be detailed below. Some recent calculations, by Ogata et al. (1993) use MC simulations to calculate the free energies of the solid and the liquid ionic mixture. These authors claim a negative departure from the linear mixing rule (LMR) in the liquid phase for the mixture at small concentration of high charges. This negative departure in the free energy of the liquid of course favors the liquid phase, so that, when computing the phase diagrams, they obtain an azeotropic form for 0.71 < C < 1.0 and an eutectic form for ( < 0.58. For 0.58 < C < 0.71 they get the superposition of an eutectic and an azeotropic diagram. These calculations, however, are very dubious for two reasons. First, as we already pointed out in § 2, for the statistical error to be negligible in MC simulations, the number of particles must be large enough. In Ogata et al. calculations, the departure from the LMR is found to occur for X2 < 10%, i.e. JV2 < 10, where %i and N2 denote respectively the concentration and the number of particles of species 2. Second, recent similar MC simulations, involving a larger number of particles, by Slattery and DeWitt (1993), confirm a positive, though small, departure from the LMR in the liquid phase. Since the results by Ogata et al. rely entirely on their claim of a negative departure, this has to be confirmed with no ambiguity for their results to be trust.

6.2 Astrophysical consequences 6.2.1 Historical overview We now discuss the consequences on the cooling of white dwarfs of the phase diagrams computed above. We shall naturally focus on the delay introduced by the crystallization process which directly affects any estimate of the age of the galactic disk obtained from the white dwarf luminosity function. Before describing the results of detailed calculations we present an historical and very simple overview of the problem, starting from the original Mestel theory (1952). We first write down the energy equation of the cooling white dwarf L + Lu-

Lnuc = - / CvTdm T — Jo Jo \oT) = ~Eth — Egrav

p-^dm p

(11)

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Mochkovitch & Segretain: Crystallization of dense binary ionic mixtures

i—i

1—|—i—i—i

Z./Z2=0.75

Fig. 6.3 Phase diagram for an arbitrary mixture. (Segretain and Chabrier)

where L, Lv and Lnuc are respectively the photon and neutrino luminosities and the nuclear energy release in the envelope; Eth and Egrav are the thermal and gravitational energy sources of the white dwarf (all other symbols have their usual meanings). Mestel theory follows from the energy equation after a series of approximations, some of them being excellent and others more questionable: (i) The interior temperature is assumed to be uniform due to the high thermal conductivity of degenerate electrons. (ii) Only the photon luminosity is included, the radiation transfer through the envelope being computed with a Kramers opacity. (iii) The thermal energy source is dominant in the white dwarf interior. (iv) The heat capacity of the dense plasma inside the white dwarf is

Mochkovitch & Segretain: Crystallization of dense binary ionic mixtures

135

approximated by the ideal gas contribution of the ions. There is no phase transition at any stage of the cooling. All these simplifications being made eq. (11) reduces to 3 M , dT k

where fi and ma are the mean atomic weight and the atomic mass unit. Together with L oc T3S from (i) eq. (12) can be integrated to give L oc t~7^ and the luminosity function oc dt/dLog L oc L~hf7. Since Mestel's pioneering work the theory of cooling white dwarfs has received a continuous interest and has been improved in many different ways. While approximation (i) has proved to be valid after a few 105 years of cooling, major changes have been incorporated to (ii), (iii) and (iv). (ii) Nuclear burning at the end of the planetary nebula phase still affect the first ~ 104 years of white dwarf evolution (Iben and Tutukov, 1984). For approximately 107 years, neutrino losses dominates the cooling until the photon luminosity takes over. Naturally, the luminosity is now obtained from detailed atmosphere models instead of the simple result deduced from the Kramers law (Fontaine and Van Horn, 1976; D'Antona and Mazzitelli, 1979; Wood, 1992). However, in spite of the progress which have been made, the determination of an accurate relationship between the internal temperature and the luminosity remains one of the major challenge in white dwarf cooling theory. The source of the difficulties comes from uncertainties in the opacity which is still poorly known in some regions of the envelope where the material is partially degenerate (see D'Antona and Mazzitelli, 1990). (iii) The assumption that the gravitational energy source is negligible was first discussed by Mestel and Ruderman (1967). The relative importance of the different terms entering the cooling equation was then evaluated from the virial theorem in a very elegant and simple way by Lamb and Van Horn (1975). In a pure carbon white dwarf, where the separation of elements at crystallization discussed below does not occur, it can be shown that (13) where ep is the typical Fermi energy of the electrons in the white dwarf interior. (iv) The ions certainly do not behave like a perfect gas as can be seen from the large value of the plasma coupling constant T. The thermodynamics of the ions is controlled by Coulomb effects and is now known with

136

Mochkovitch & Segretain: Crystallization of dense binary ionic mixtures

a high degree of accuracy thanks to the work of a number of people since twenty years (see Baus and Hansen, 1980 and Ichimaru et al., 1987 for good reviews). The most spectacular consequence of the Coulomb effects is probably the existence of the fluid-solid transition at F c ~ 180. The related release of latent heat (of the order offcjgTper particle) introduces a delay in the cooling of white dwarfs which was first discussed by Van Horn (1968). At low temperature, when fcgT < hup {up being the plasma frequency), quantum effects become important and the heat capacity of the ions decreases according to the Debye law, leading to an acceleration of cooling in very faint white dwarfs, known as "Debye cooling". Since white dwarfs are made of carbon and oxygen (or of oxygen, neon and magnesium for the most massive ones) plus several other minor species such as 22Ne and Fe, it was clear that calculations ignoring a possible separation of elements at crystallization were uncomplete. The problem was considered by Stevenson (1980) who proposed an eutectic phase diagram for the C/O mixture. The implied total separation of carbon and oxygen had a drastic effect on the time needed to reach the cut-off of the luminosity function which was increased by about 5 109 years (Mochkovitch, 1983). The Stevenson phase diagram was the result of a very simple analytical model for the free energies in both liquid and solid phases. Much more reliable results have been obtained recently using the density functional approach described above. Instead of an eutectic all the new phase diagrams for carbon and oxygen have a spindle (or azeotropic) shape. The delays introduced in the cooling of white dwarfs are less spectacular but remain important.

6.2.2 The crystallization of carbon-oxygen white dwarfs The solid being more oxygen rich than the liquid, it is also slightly denser to maintain the continuity of total pressure at the phase transition. An estimate of the density change between the solid and liquid phases is given by ^

^

^

(14)

where P{ and Pe are respectively the ionic and electronic pressures, 7 is the adiabatic index for the electron gas and Ye is the number of electrons per nucleon. Inside a 0.6 M© white dwarf ^ « 10~4. The solid therefore settles down at the center while the lighter liquid left behind is redistributed with great efficiency by convective transport (Mochkovitch, 1983). The result of the crystallization process is then an enrichment in oxygen in the central

137

Mochkovitch & Segretain: Crystallization of dense binary ionic mixtures

1

1.60.8 _

0.6 \

: .

>:

0.4 -

\

:

0.2

0 0.2 0.4 0.6 0.8 1

n

........... r

'0 0.2 0.4 0.6 0.8 1

Fig. 6.4 Redistribution of carbon and oxygen in a crystallizing white dwarf; a spindle phase diagram leads to the formation of a composition gradient in a white dwarf where Xe = X0= 0.5 before crystallization. In the example shown a constant enrichment in oxygen, a = 0.3, has been adopted.

regions of the white dwarf and a depletion in the outer parts. This has been illustrated in fig. 1.4 in the case of a spindle phase diagram and for carbon and oxygen with initially equal mass fractions throughout the white dwarf. Making the rough approximation that the enrichment in oxygen is a constant a ~ 0.3 X'o = (1 + a)Xl0 ,

(15)

1

(X'' being respectively the oxygen mass fractions in solid and liquid phases) and assuming a perfect mixing outside the solid core it becomes possible to obtain the composition gradient analytically (Barrat et al., 1988). When the solid core has grown to a mass M» the remaining oxygen mass fraction in the liquid mantle is

' XI dm — MB

(16)

where Mwd is the total mass of the white dwarf. Taking the derivative of eq. (16) with respect to Ma and introducing (15) yields

dxi

_

,17)

138

Mochkovitch & Segretain: Crystallization of dense binary ionic mixtures

which after integration gives X'o= 0.5(1 + a ) ( l - - ^ ) « .

(18)

An order of magnitude of the release of gravitational energy produced by the redistribution of elements is simply e9rav ~—gRZ

1013 erg.g-1 ,

(19)

where we have adopted ^ „ i o " 4 , g - 1.2 108 cm.s" 2 and R = 8300 km, which are typical for the gravity and radius of a 0.6 M© white dwarf. The value of egrav is comparable to the latent heat and cannot be neglected in cooling calculations. The resulting time delay Atgrav

~

Mv

*

e

*™ Z 109 yr ,

(20)

where < Z/crya > ~ 10~4 L© is the average luminosity of the white dwarf during the crystallization process. After having shown from simple estimates the importance of the redistribution of carbon and oxygen at crystallization, we summarize the results from detailed calculations such as those discussed extensively by GarciaBerro et al. in this volume. An additional complication which has been included in these calculations comes from the possibility of an initial composition gradient in the white dwarf core. Mazzitelli and D'Anton a (1986) who followed the evolution of intermediate mass stars from the main sequence to the white dwarf stage found a high oxygen concentration at the center of 0.6 M© white dwarfs (Xo ~ 0.8 from the center to M/M©=0.4 and Xo ~ 0.4 in the outer parts). Only massive white dwarfs (M~ 1 M©) show a nearly uniform composition with Xc — Xo = 0.5. Table 1 gives the time needed to reach the cut-off of the luminosity function at Log(L j LQ) « —4.5 as a function of the white dwarf mass in models which either include or neglect the redistribution of carbon and oxygen and the initial stratification. The results listed in Table 1 have been obtained with the spindle phase diagram of Segretain and Chabrier (1993). We have checked that the azeotropic phase diagram of Ichimaru et al. (1988) gives more or less the same results. This is not surprising since the crystallization of the white dwarf occurs far from the azeotropic point. It can be seen from Table 1 that the typical delay introduced by the redistribution of carbon and oxygen is 1 - 2 Gyr. It is smaller when the initial stratification is taken into account. For massive white dwarfs however there is no stratification and cases 1-3 and 2-4 are respectively identical. The delay directly affects any age determination of

Mochkovitch & Segretain: Crystallization of dense binary ionic mixtures

139

Table 6.1. Values of the cooling time (in Gyr) at Log(L/Lq)) = —4.5 for different white dwarf masses, in the following four cases: (1) no initial stratification, no CO redistribution at crystallization; (2) no stratification, CO redistribution; (3) initial stratification, no redistribution; (4) initial stratification, CO redistribution. The delays due to CO redistribution are given for the cases 2-1 and 4-3. M/M0

0.5

(1)

8.73 9.24 9.39 9.27 8.96 8.53 8.10 8.07

(2)

10.78 11.45 11.53 11.27 10.78 10.13 9.45 9.13

A r ( 2 - 1)

2.05 2.21 2.14 2.00

(3)

8.10 8.76 9.24 9.28 8.97 8.53 8.10 8.07

(4)

9.18 9.96

10.76 11.12 10.73 10.13 9.45 9.13

A r ( 4 - 3)

1.08

1.52

0.6

1.20

0.7

0.8

1.84

0.9

1.82

1.76

1.0

1.60

1.60

1.1

1.35

1.35

1.2

1.06

1.06

the galactic disk coming from the white dwarf luminosity function. A calculation which would neglect the redistribution of carbon and oxygen would then yield an age underestimated by at least 1 Gyr. 6.2.3 The effect of minor species 6.2.3.1 -22Ne From the different minor species which can play a role in the cooling of white dwarfs 22Ne is certainly the most important. It is relatively abundant since it is produced from 14N during He burning by the chain of reactions 14 N(a,7) 18 O(a,7) 22 Ne. Its abundance is directly related to the initial concentration of CNO elements, so that one can expect a mass fraction X(22Ne) ~ 1 - 2% in population I white dwarfs. 22Ne is a neutron rich nucleus (Ve(22Ne)=10/22=0.45) which can produce a large release of gravitational energy if it accumulates at the center during crystallization. The physics of 22 Ne deposition should in principle be described by a 3-component (C-ONe) phase diagram which is unfortunately not presently available. One has to rely on N-Ne binary phase diagrams where nitrogen mimics the behavior of the CO mixture. As shown in Sect. 1.1.4, for a charge ratio less than 0.72

Mochkovitch & Segretain: Crystallization of dense binary ionic mixtures

140

0.98 Fig. 6.5 Neon poor side of the N-Ne phase diagram. The neon concentration is smaller in the solid. The liquid gets more and more neon rich until the azeotropic point is reached.

there is an azeotropic point. For the N-Ne mixture, the neon mass fraction of the azeotrope is Xa(Ne)=0.16 which means that with X(22Ne) ~ 1 - 2%, the crystallizing white dwarf is on the neon poor side of the phase diagram (see fig. 1.5). This has many interesting consequences. The solid which freezes out from the liquid has a smaller neon concentration and is therefore lighter due to a smaller Ye (eq. (14)). It will rise and melt in lower density regions so that the liquid at the center will become more and more neon rich until the azeotropic point is reached. Crystallization at the composition of the azeotrope then takes place until the whole 22Ne has been collected in a central sphere of mass

X(Ne)

< 0.1Mwd ,

(21)

This "neon distillery" releases a large amount of gravitational energy, three times more than CO redistribution for X(Ne)=0.0l! Translated into delays on the cooling times this implies that a 0.6 M© CO white dwarf with 1% of 22 Ne will take ~ 14 Gyr to reach the cut-off of the luminosity function.

Mochkovitch & Segretain: Crystallization of dense binary ionic mixtures

141

Does it mean that the age of the galactic disk is 14 Gyr? Probably not, for at least two independent reasons. First, the white dwarfs at the cutoff come from stars formed very early in the history of the disk and then have a low metallicity. Since the delay is proportional to the abundance of 22Ne, it follows that the faint end of the luminosity function and the estimated age of the disk are only weakly affected by neon deposition. A detailed calculation, where the luminosity function is obtained consistently with a model of galactic evolution is presented by Garcia-Berro et al. in this volume. Another more fundamental reason is simply that Ne may very well not deposit in the realistic case of a ternary mixture C-O-Ne. As mentioned above the phase diagram is not known for more than two components. Nevertheless, one can try to use the results for the CO and N-Ne cases to predict that the solid which freezes out from the ternary mixture will be oxygen rich and neon poor. The higher oxygen concentration and the depletion in neon have opposite effects on the density difference between the solid and the liquid. Since the "neon distillery" works only if the solid is lighter, this leads to a maximum allowed enrichment in oxygen for a given depletion in neon. Let us assume that in liquid phase Xj. = Xl0 = 0.495 and Xl(Ne) = 0.01. In the solid X*(Ne) = aXl(Ne) with a < 1. We have represented in fig. 1.6 the maximum oxygen mass fraction in the solid as a function of a which keeps Apti < 0. If we adopt the value a « 0.5 as indicated from the study of N-Ne mixtures, X™ax « 0.65, close to the oxygen mass fraction in the solid obtained for a CO mixture with Xj. = Xl0 = 0.5. The chance that the solid remains lighter are naturally increased if a w 0 as proposed by Ogata et al. (1993).

6.2.3.2 - Fe The possibility of iron deposition has been discussed by Xu and Van Horn (1992) and the problem appears very similar to that of neon. The phase diagram of the N-Fe mixture has an eutectic shape and the solid can then be expected to be iron free and lighter than the liquid. In principle an "iron distillery" could work and bring all the iron at the center. For an iron mass fraction of 10~3, the resulting additional delay would reach 0.8 Gyr. However, exactly as for neon, iron deposition is uncertain in a multicomponent plasma and even if it occurs, it does not affect the cooling of the old, low metallicity white dwarfs which make the faint end of the luminosity function.

Mochkovitch & Segretain: Crystallization of dense binary ionic mixtures

142

0.9

0.8

no Ne deposition

0.6

Ne deposition 0.5,

0.25

0.5 a

0.75

Fig. 6.6 Maximum oxygen mass fraction in the solid as a function of a (the neon depletion factor) to have Ap,i < 0 and neon deposition.

6.3 Conclusion During the past twenty years our understanding of the physics of the interior of white dwarfs has been transformed. The thermodynamics of dense plasmas is now known with a high degree of accuracy and the importance of Coulomb effects on the stucture and evolution of white dwarfs has been fully recognized. For one of the most difficult problem - the shape of the phase diagram for mixtures - the density functional theory provides for the first time a reliable solution (at least for binary mixtures) and cooling sequences of white dwarfs with the best input physics for the interior can now be calculated. The partial separation of carbon and oxygen at crystallization adds 1 - 2 Gyr to the time needed to reach Log(L/LQ) = -4.5 and hence to the age of the galactic disk deduced from the white dwarf luminosity function. The main source of error in white dwarf cooling theory now probably comes from the envelope, where the opacity is still very uncertain in some regions of the p — T plane.

References Abrikosov A.A., Zh. Eksp. i Teor. Fiz. 39, 1798, (Soviet Phys. JETP 12, 1254), (1960)

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Barrat J.L., Thesis, university of Paris VI (1987) Barrat J.L., Europhys. Lett. 3, 523, (1987) Barrat J.L., Baiis M., Hansen J.P.,, J. Phys. C 2 0 , 1413, (1987) Barrat J.L., Hansen J.P., Mochkovitch R., A&A 199, L15, (1988) Baus M., Colot J.L , Mol. Phys. 55, 653, (1985); Baus M., /. Stat. Phys., 48, 1129,(1987) Baus M., Hansen J.P., Phys. Rep. 59, 1, (1980) Brami F., Hansen J.P., Joly B., Physica A 95, 505, (1979) Brush S.G., Sahlin H.L., Teller E., J. Chem. Phys. 45, 2102, (1966) Chabrier G., Ashcroft N.W., Phys. Rev. A 42, 2284, (1990) D'Antona F., Mazzitelli I., A&A , 74, 161, (1979) D'Antona F., Mazzitelli I., Ann. Rev. Astron. Astrophys. , 28, 139, (1990) DeWitt H.E, Slaterry W.L., (1993) Private communication DeWitt H.E, Slattery W.L., Yang J., The international conference on the physics of strongly coupled plasmas, (1993), in press Dubin D.H.E., Phys. Rev. A 42, 4972, (1990) Dyson F., Ann. Phys. 63, 1, (1971) Farouki R.T., Hamaguchi S., Phys. Rev. E47, 4330, (1993) Fontaine G., Van Horn H.M., Ap.J. Suppl. , 31, 467, (1976) Hansen J.P., Phys. Rev. A 8, 3096, (1973) Hansen J.P., MacDonald I.R., Theory of simple liquids, (Academic Press), (1976,1989) Honenberg P., Kohn W., Phys. Rev. , 136, B 864, (1964) Iben I.Jr., Tutukov A., Ap.J. , 282, 615, (1984) Ichimaru S., Iyetomi H., Tanaka S., Phys. Rep. 149, 91, (1987) Ichimaru S., Iyetomi H., Ogata S., y i p / 3 3 4 , L17, (1988) Iyetomi H., Ichimaru S., Phys. Rev. B 38, 6761, (1988) Kirshnitz D.A., Zh. Eksp. i Teor. Fiz. 38, 503, (Soviet Phys. J E T P 11, 365), (1960) Lamb D.Q., Van Horn H.M., Ap.J. 200, 306, (1975) Landau L.D. , Lifshitz E.M., Statistical Physics, (London: Pergamon) (1958) Likos C.N., Aschroft N.W., Phys. Rev. Lett. 69, 316, (1992) Mestel L., M.N.R.A.S. , 112, 583, (1952) Mestel L., Ruderman M.A., M.N.R.A.S. , 136, 27, (1967) Mochkovitch R., A&A , 122 , 212, (1983) Mazzitelli I., D'Antona F., Ap.J. , 308, 706, (1986) Ogata S., Ichimaru S., Phys. Rev. A 36, 5451, (1987) Ogata S., Iyetomi H., Ichimaru S., Van Horn H.M. Phys. Rev. E 48, 1344, (1993) Parrinello M., Tosi M.P., Chem. Phys. Lett. 64, 579, (1979) Ramakrisnan T.V., Yussouf M., Phys. Rev. B 19, 2775, (1979) Rovere M., Tosi M.J.Phys. C, 18, 3345, (1985) Salpeter E.E., Ap.J. 134, 669, (1961) Segretain L., Chabrier G., A&A 271, L13, (1993) Singh Y., Phys. Rep. 207, 351, (1991) Stevenson D.J., J. Physique 41, C2-61, (1980) Stringfellow G.S., DeWitt H.E., Slaterry W.L., Phys. Rev. A 4 1 , 1105, (1990) Tarazona P., Mol. Phys. 52, 81, (1984) Van Horn H.M., Ap.J., 151, 227, (1968) Wood M., Ap.J. , 386, 539, (1992) Xu , Van Horn H.M., Ap.J. , 387, 662, (1992)

Non crystallized regions of White Dwarfs. Thermodynamics. Opacity. Turbulent convection. I. MAZZITELLI Istituto di Astrofisica Spaziak. Consiglio Nazionalc dtlk Ricerche, C.P. 67, 00044 Frascati. Italy

Abstract The evolution of White Dwarf stars along their cooling sequences is governed not only by their thermal content, but also by the rate at which heat flows through the external, partially degenerate and non-isothermal layers. In particular, cooling is found to be largely influenced both by the optical atmosphere, and by the convective envelope. The first one, in fact, determines the internal density stratification, down to the point at which electron degeneracy takes over, while the second one affects the temperature stratification in the same layers. The reliability of the present generation of models of White Dwarf envelopes is discussed, on the grounds of the main physical inputs (thermodynamics, opacity, convection theory), for both H-rich and He-rich surface chemical compositions. The conclusion is that, below LogL/LQ < —3, we can build little more than test models. L'evolution des naines blanches le long de leur sequence de refroidissement est gouvernee non seulement pas leur contenu thermique, mais aussi par la vitesse a laquelle la chaleur s'echappe a travers les couches externes, nonisothermes et partiellement degenerees. En particulier, le refroidissement est largement influence a la fois par l'atmosphere optique et par l'enveloppe convective. La premiere determine la stratification interne en densite jusqu'a ce que la degenerescence electronique prenne le dessus, alors que la seconde affecte la stratification en temperature dans les memes couches. Nous discutons la validite de la generation actuelle de modeles d'enveloppes de 144

Mazzitelli: Non crystallized regions of White dwarfs

145

naines blanches, sur la base des ingredients physiques (thermodynamique, opacite, theorie de la convection), pour a la fois des compositions chimiques de surface riches en hydrogene et en helium, respectivement. La conclusion est que, en dessous de LogL/L@ < —3, il n'est guere possible de proposer autre chose que des modeles test.

7.1 Introduction Observations seem to definitely show that the luminosity function of the nearby White Dwarfs (WDs) display an abrupt break-down in the luminosity range LogL/LQ= -4.0 4- -4.5 (Liebert et al. 1989). The obvious interpretation of this feature is the finiteness of the age of the galactic disk —in the vicinity of the Sun— such that even the first born among the WDs are not yet old enough to have cooled below this luminosity (D'Antona and Mazzitelli 1978). The alternative explanation of WDs being already in the fast Debye cooling phase (D'Antona and Mazzitelli 1989), should in fact lead to a milder decrease of the luminosity function, than apparently suggested by the observations. In this framework, it should be legitimate to assume WDs as powerful probes for evaluating the age of the galactic disk (Winget et al. 1987). However, if this is the goal, we have first to make sure about the reliability of the present generation of stellar models, also because the external layers of the WDs —which determine the cooling rate— are at such (relatively) high densities and (relatively) low temperatures, that matter is definitely far from ideal gas conditions. Also, in old (and cold) WDs, the largest fraction of the temperature difference between surface and centre is found in the convective region. Although the convective gradient is in any case very close to the adiabatic gradient —due to the large density— the Mixing Length Theory (MLT), tuned on the sun, is hardly expected to be the more realistic model for dealing with turbulent convection in dense, partially ionized and partially degenerate conditions. In the following, WDs models of different luminosities and surface chemical compositions will be discussed showing that, especially in the case of He-rich stars, the p-T region where our present physical understanding of the thermodynamics and of the heat transport properties of a partially ionized real gas is still rather poor, is met relatively soon during the evolution.

146

Mazzitelli: Non crystallized regions of White dwarfs

7.2 The main ingredients There is by now wide agreement about some general features of WDs. The masses of the most of WDs, for instance, seem to lie in the range 0.5<M<0.6 M©(Weidemann 1990). For this reason, all the following tests will be performed on a model having M = 0.55 M©. Also, gravitational settling has been established as a fast and powerful mechanism to achieve, at the surface of a WD, very low metal abundances, and complete separation between hydrogen —if any— and helium (Fontaine and Michaud 1979). Then, a very low metal abundance Z=10—5 has been chosen for the models discussed below, and both pure-H and pure-He surface compositions are tested. Finally, a more tricky point. The evolutionary models coming from the Asymptotic Giant Branch show that WDs are left with a thin hydrogen mantle, of the order of 5 10—5 M©(Iben and McDonald 1986). The pulsational properties, however, seem to suggest thinner H-layers, of the order of 10—8 Af©or even less (Winget et al. 1982). A compromise between the two alternatives has been chosen here, and the H-rich models tested below have an hydrogen mantle of thickness 10—7 M©. Since the basic evolutionary sequence has been evolved starting from the Horizontal Branch and through the main Thermal Pulses phase, an artificial metal settling and He-depletion algorithm has been applied at the beginning of the WD phase, in order to be consistent with the previously discussed chemical constraints. The adopted algorithm can perhaps lead to minor numerical disturbances at the beginning of the WD evolution which, however, do not give rise to any further noise during the main cooling phase, since the duration of the high luminosity WD phases is very short, and any perturbation of the external layers is fastly reset, the WD finally setting on its cooling track. It is worth noting that, due to the presence of breathing pulses (Caloi and Mazzitelli 1993) at the end of the HB phase, the most of the carbon in the core is transformed into oxygen (~ 90%, if the enhanced 12C + a reaction rates (Kettner et al. 1982) are assumed; about 70 As for the main physical inputs: (i) (ii) (iii) (iv)

thermodynamics is from Magni and Mazzitelli (1979) neutrinos are from Itoh et al. (1992, and references therein) thermal conduction from Itoh et al. (1984) atmosphere is grey, with a simple T(r) relation

and other inputs (radiative opacities, convection theory etc.) will be discussed in the following. The numerical structure is integrated in ~1000 mesh points, ant the evolution from the beginning of the WD phase down to final cooling (LogL/LQ< -4.5) takes about 3000 time steps.

Mazzitelli: Non crystallized regions of White dwarfs

147

Log p Fig. 7.1 The position on the p-T plane of the external, H-rich layers of a 0.55 MQwhite dwarf at several luminosities (solid lines). The dotted rectangle marks the region where pressure dissociation and ionization is approaching; the short-dashed line marks onset of electron degeneracy; the long-dashed line shows where the coulomb coupling parameter T reaches the value 10, and the dashed-dotted line marks the region where ions start behaving as quantum particles. 7.3 H y d r o g e n rich m o d e l s 7.3.1 Thermodynamics

The envelope structures at different luminosities of the test WD are plotted on the p-T plane in Fig. 1.1. It is clear that, at least as long as thermodynamics alone is concerned, the modeling of H-rich WD envelopes is relatively safe until down to low luminosities. In fact, only for LogL/L@< -4 a small fraction of the envelope is (marginally) close to the non-ideal gas region. Also, H-rich WDs interiors are always well far from the region where collective effect begins to be

148

Mazzitelli: Non crystallized regions of White dwarfs

present, since the Coulomb coupling parameter T is always relatively small (its value never overcomes F ~ 5). Finally, at least for WDs, we have not to worry about the quantum behavior of hydrogen. The tracks are shown down to the H/He interface. For deeper H-rich envelopes (up to the evolutionary remnant mass, which is three orders of magnitude larger than the one adopted in the present computations) the situation would not substantially change since, in the worst of the cases, the tracks would go ahead at larger temperatures and densities, eventually meeting the electron degeneracy conditions, for which a classical thermodynamical treatment is available. Also in the case of thinner envelope masses, down to 10—14 M©, as suggested by some spetroscopical features (Liebert et al. 1987), one should not expect substantial differences. In fact, as long as the optical atmosphere is hydrogen-dominated, the density at which the subatmospheric envelope begins is not affected by the thickness of the H-envelope and, since the slope of the tracks in the p-T diagram is mainly dictated by the adiabatic gradient of temperature, even after the transition to helium the tracks should not jump to p-T regions far different from the ones shown in Fig.1.1. One can then conclude that, from the point of view of thermodynamics alone, our present understanding of H-dissociation and ionization at high density is probably sufficient to give us reliable WD models down to very low luminosities. It is perhaps worth explicitly noting that it is not legitimate to extend the same conclusion also to Brown Dwarf stars, where the physical processes are quite different. 7.3.2 Opacity Most unfortunately, the optimistic conclusion of the proceeding section, cannot be extended to the other relevant physical input, that is: the radiative opacities. In fact, from Fig. 1.2, one can see that, for luminosities below LogLj LQ~ -3, the optical atmosphere lies in regions of low-T and high-p, where no extended opacity sets are presently available. The Los Alamos opacities (Huebner 1977) do not reach temperatures below Log T=4.2, where molecules can be still present, and the OPAL ones (Rogers and Iglesias 1992), even if they reach lower temperatures (T=6000oK), are provided only for relatively low densities. The lack of radiative opacities represent a serious limit to our abilities of modeling stellar structures, since the opacity in atmosphere determines, by orders of magnitude, the density at the base of the atmosphere, at which the envelope begins. In other terms, for H-rich WDs at LogL/Lo< -3, the location at

Mazzitelli: Non crystallized regions of White dwarfs

149

H-rich WD 6

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Log p Fig. 7.2 The same as Fig. 1.1 but for the opacities. Right of the shortdashed line, electron conduction takes over. Left of the long-dashed region, the Los Alamos radiative opacities are available; left of the dotted-dashed line, the OPAL opacities are available.

which the subatmosphere begins in Fig. 1.2, can actually lie on a horizontal segment (the temperature being fixed, according to luminosity and mass) of unknown amplitude. The coolest models shown, then, are very uncertain, since they have been modeled according to a given numerical recipe for extrapolating opacity tables, and not according to physically sound opacities. The conclusion is then that, for H-rich WDs, our present knowledge of radiative opacities confines our capability of correctly predicting the evolution of WDs to luminosities LogL/LQ> -3. Below this limit, we can certainly build test models but, basing upon these last, we can scarcely draw sound quantitative conclusions about galactic evolution.

Mazzitelli: Non crystallized regions of White dwarfs

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7.3.3 Turbulent convection Up today, the MLT has been widely applied to the modeling of WDs convective envelopes since, in any case, the rate of overadiabaticity in the dense external layers of these objects is expected to be negligible or quite so. This seems in fact to be the case as long as the thermal properties of WDs are concerned, even if other properties of WDs, as the pulsational ones, are instead much more sensitive to the treatment of convection (Pelletier et al. 1986). However, the problem is perhaps not so settled, as will be shown below. The MLT is usually tuned on the sun, and one can legitimately ask which can be its predictive power, when applied to conditions in which the density is up to six or seven orders of magnitude larger than in the solar subatmosphere. Let us examine this point more in detail.

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Log M/Mtot Fig. 7.4 The behavior of the Prandtl number as a function of depth, as in Fig. 1.3, at various luminosities, for a H-rich WD. For comparison, remember that the Prandtl number in the solar subatmosphere is a ~ 10~9. Much larger densities, could imply for instance larger viscosities than in the case of the almost inviscid sun. The coefficient of dynamical ion viscosity r) can be easily computed for a coulombic plasma, according to Wallenborn and Baus (1978). The results for the case of the subatmosphere of a H-rich WD, are shown in Fig. 1.3. Viscosity can be quite large, so large as to equal that of very viscous terrestrial fluids. Actually, the knowledge of viscosity alone is not sufficient, since also for a relatively large value of 77, matter in a star can still have room enough to generate a wide spectrum of viscous eddies, and full developed turbulence. The Prandtl number a = v/x, where v is the kinematical viscosity coefficient and x the thermometric conductivity, is in this respect, much more significant, since it is directly connected to the amplitude of the eddies spectrum.

152

Mazzitelli: Non crystallized regions of White dwarfs

As can be seen from Fig. 1.4, during the cooling of the WD, the value of a in the convective region steadily increases, reaching ~1 below LogLjZ©= -4. This leads to a consistent shrinking of the turbulent eddies spectrum with respect to an inviscid case, and the efficiency of the convective heat transfer decreases. In particular, from Canuto and Mazzitelli (1991) one can see that the integral of the energy spectrum of the turbulent eddies versus the vawenumber does not change significantly when a decreases below 10—3, but this is not the case for larger values of
Mazzitelli: Non crystallized regions of White dwarfs

153

Log k Fig. 7.5 The true energy spectrum of turbulent eddies as a function of the vawe number, for two different values of the Prandtl number a. The corresponding MLT spectrum would have been a delta function.

7.4 Helium rich models 7.4.1 Thermodynamics The case for He-rich WD models is far different from the H-rich ones. The main reason is that, at relatively low surface temperatures (T< 20000oA", helium gives an absolutely negligible contribution to opacity, which is due only to the trace metals. This implies, in turn, that the results presented in the following will have a significance only for the chosen metal abundance (Z=10-5). Any other metal abundance would have given completely different results, contrarily to the case for H-rich envelopes, where the above discussed results are indicative of a general low metal case. Due to the very low radiative opacities at low temperature, the He-rich optical atmospheres turn out to be very thick (in mass) and dense, so that, in the p-T diagram , the tracks of the subatmospheric envelopes are shifted

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towards much larger densities with respect to the H-rich case, and larger densities mean complicances due to real-gas effects, collective effects and quantum effects. It is worth explicitly noting here that, since the chances are that the metal abundances in the atmospheres of cold He-rich WDs could be even lower than Z=10-5, the following discussion elucidates perhaps the more favourable case, the atmospheres of real He-rich WDs being probably even thicker and denser. Figure 1.7 elucidates the thermodynamic problems met in the case of He-rich envelopes. As soon as the luminosity of the WD drops below LogL/L®= -2, part of the convective region begins to lie in a non-ideal gas zone. Since the thermal structure of the envelope is largely sensitive to the adiabatic temperature gradient, it is also clear that a very good equation of state for partly-ionized and partly-degenerate real gas is required.

Mazzitelli: Non crystallized regions of White dwarfs

155

-5

Log p Fig. 7.7 The analogous of Fig. 1.1, but for He-rich atmospheres. In this case, the non-ideal gas conditions are met very soon during the evolution. Unfortunately, due to the presence of two electrons, the case for helium is far more difficult to be dealt with than the case for hydrogen and, up today, no extensive numerical results from a realistic thermodynamic treatment of this situation are available. The conclusion is then that thermodynamics puts a severe constraint on our understanding of He-rich WDs already at relatively large luminosities. Another constraint, at much lower luminosities, comes from the fact that the convective region begins approaching the conditions where the quantum corrections take over, but this occurs only at LogL/Lo< 4 -j- -4.5. 7.4.2 Radiative opacity Even worse is the situation for He-rich WDs when we consider the radiative opacities. In fact, as can be seen in Fig. 1.8, we do not have reliable opacities

156

Mazzitelli: Non crystallized regions of White dwarfs

Log p Fig. 7.8 The same as Fig. 1.2, but for He-rich WDs. for He-rich atmospheres below LogL/L@= -2. Also the LAOL opacities at relatively high densities are probably not very realistic, since they have been computed in the ideal-gas approximation. However, in convective layers, this is perhaps a minor problem, since the thermal structure of WDs in these p-T regions is mainly dictated by thermodynamics. As soon as convection sets in, opacities become relevant only in the optical atmosphere. The situation is furtherly complicated by the fact that, even before trying to evaluate radiative opacities, a good knowledge of the real-gas thermodynamics is required. At present, it seems perhaps possible to make progresses on thermodynamics by means of Free Energy minimization schemes, or by means of the brute force of a Monte Carlo treatment but, unfortunately, none of these two alternatives is able to provide the detailed knowledge of the bound levels required for evaluating opacities. This is even more true since it is not only the structure of the bound levels of helium to be required, but mainly the structure of the (pressure perturbed) bound levels

Mazzitelli: Non crystallized regions of White dwarfs

157

of the trace metals, radiative opacity being due almost only to these last elements. To add further complicances to an already messy situation, let us focus on the already quoted dependence of opacity on metal abundance. The possible presence of small amounts of hydrogen or metals due to interstellar accretion, or to levitation from inside due to radiation pressure on the lines, makes models of He-rich WDs little more than a computer exercise, as soon as the surface temperature drops below the level at which the Hecontribution to opacity becomes negligible. In practice, from the point of view of radiative opacities, the same conclusions can be drawn as for the case of thermodynamics, even if somehow more severe: for luminosities below LogL/Lo~ -2, the models can be only test models, and they are likely to become absolutely meaningless below -3.

7.4.3 Turbulent convection In the case of He-rich WDs, viscosity in the convective envelopes turns out to be always negligible,the Prandtl number being about 3-4 orders of magnitude lower than for hydrogen. However, we are in the presence of another tricky feature. From an inspection of Fig. 1.7 it can be seen that, at low luminosities, the star seems to approaches the region where F = 180 —that is: crystallization— not only in the central layers, but also in the convective envelope, just below the optical atmosphere. This is more evident from Fig. 1.9, where the behavior of F is shown as a function of the depth, starting from the bottom of the optical atmosphere. The values of F in the more external region, below 10—10 M©, are meaningless, since matter very close to the surface is not completely ionized, but the peak around F=170 is indeed significant. The reason why F increases outwards up to a maximum is due to its power dependence F ~ f^/3/T, and to the fast drop of the temperature when approaching the surface, whereas density does not decrease so fast as in normal stars. Within the (admittedly narrow) limits of reliability of the above models, we then reach the conclusion that, at very low luminosities (LogL/L@< -4.3), He-rich WDs can start forming a solid crust close to the top of the convective region! The implications of this curious physical feature are far from being clear. Does this mean that convection is abruply stopped and, in the solid region, the temperature gradient suddenly jumps to the radiative

Mazzitelli: Non crystallized regions of White dwarfs

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one? Or does it mean that convection mixes downward matter, so that a dynamical equilibrium between crystallization and melting is reached? The only qualitative conclusion which can be drawn from these considerations is that we can in any case expect some sort of drag on convection, for very cold and faint He-rich WDs. However, since crust crystallization occurs well ahead of the onset of other physical processes which make our understanding of He-rich WDs uncertain, this feature is at present little more than a curiosity.

7.4-4 Mixing Just for the sake of completeness, it is worth recalling the reader's attention upon one final complicance, which arises if the hydrogen-rich envelope on H-rich WDs is thinner than 2 -j- 3 • 10-8 M©—a value which cannot be

Mazzitelli: Non crystallized regions of White dwarfs

159

dismissed on the basis of our present interpretation of astroseismological data. In that case, in fact, the sinking of the H-convection at low luminosity is such that, below LogL/L@~ -3.5, the H-convective and the He-convective layers join, and full mixing is achieved. Hydrogen at the surface is then diluted by orders of magnitude, and the problems with He-rich WDs are met again, adding one further degree of freedom to an already unmanageable problem. 7.5 Conclusions From the above discussion, it should now be clear that our understanding of the main physical inputs entering the modeling of very cool and old WDs, no matter if H-rich or He-rich at the surface, is still rather poor. Perhaps, radiative opacities in the optical atmospheres are by far the missing ingredient which makes so unsure our knowledge of the final cooling phases of these objects, even if the problems with thermodynamics and turbulent convection are still far from being settled. For these reasons, one derives the feeling that webetter try to be cautious in rising stringent constraints on the age of the galactic disk, from fits between observed luminosity functions and WDs evolutionary models; the technique is promising, but the error bar weighing on this kind of comparisons is still quite large and —even worse— its amplitude is presently unknown (maybe even in the range of billions of years?). References Caloi, V. and Mazzitelli, I. Astron. Asirophys. 271, 139, (1993) Canuto, V.M. and Mazzitelli, I. Ap.J. 370, 295, (1991) D'Antona, F. and Mazzitelli, I. Astron. Astrophys. 66, 453, (1978) D'Antona, F. and Mazzitelli, I. Ap.J. 347, 934, (1989) Fontaine, G. and Michaud, G. Ap.J. 231, 826, (1979) Huebner, W.F., Mertz, A.L., Magee, N.H.Jr., Argo, M.F., Astrophysical Opacity Library, L.A. 6760, M, (1977) Iben, I.Jr., McDonald, J. Ap.J. 301, 164, (1986) Itoh, N., Kohyama, Y., Matsumoto, J., Seki, M. Ap.J. 258, 758, (1984) Itoh, N., Mutoh, H., Hikita, A., Ap.J. 395, 622, (1992) Kettner, K.V., Becker, H.W., Buchman, L., et al., Z. Phys. 308, 73, (1982) Liebert. J., Fontaine, G., Wesemael, F., Mem. Soc. Astron. Ital.58, 17, (1987) Liebert, J., Dahn, C , Monet, D.G. Ap.J. 332, 891, (1989) Magni, G. and Mazzitelli, I. Astron. Astrophys. 72, 134, (1979) Pelletier, C , Fontaine, G., Wesemael, F., Michaud, G., Wegner, G., Ap.J. 307, 242,(1986) Rogers, F.J. and Iglesias, C.A., Ap.J. Suppl. 79, 507, (1992)

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Wallenborn, J. and Baus, M. Phys. Rev. A, 18, 1737, (1978) Weidemann, V. Ann. Rev. Astron. Astrophys. 28, 103, (1990) Winget, D.E., Van Horn, H.M., Tassoul, M., Hansen, C.J., Fontaine, G., Carrol, B.W. Ap.J. Lett 252, L65, (1982) Winget, D.E., Hansen, C.J., Liebert, J., Van Horn, H.M., Fontaine, G. Ap.J. Lett 315, L77, (1987)

8 White Dwarf Crystallization ENRIQUE GARCIA-BERRO Departamento de Fisica Aplicada. Universidad Politecnica de Cataluna, Jordi Girona Salgado 31, 08084 Barcelona, Spain

MARGARIDA HERNANZ Centre d'Estudis Avancats de Blanes, CS.I.C, Blanes, Spain

Camt de Santa Barbara, 11300

Abstract The inclusion of a detailed treatment of solidification processes in the cooling theory of carbon-oxygen white dwarfs is of crucial importance for the determination of their luminosity function. Carbon-oxygen separation at crystallization yields delays larger than 2 Gyr to cool down to luminosities corresponding to the observed cut-off. This leads to estimates of the age of the galactic disk 1.5 to 2 Gyr older than the ones obtained in previous studies (about 9 Gyr). Furthermore, the presence of minor chemical species, in particular 22Ne, alters significantly the crystallization process, and produces extra delays of 2 to 3 gigayears. However, the detailed computation of the theoretical white dwarf luminosity function, taking into account a reasonable model of galactic chemical evolution, and including the effect of these species, shows that the location of the cut-off, and then the estimated age of the disk, is not modified significantly. Le traitement detaille du processus de solidification revet une importance cruciale dans l'etude du refroidissement des naines blanches carbone-oxygene et la determination.de leur fonction de luminosite. La separation du carbone et de l'oxygene lors de la cristallisation introduit un retard de plus de 2 109 ans pour atteindre les valeurs de la luminosite correspondant au cut-off observe. Ceci conduit a une estimation de l'age du disque de 1.5 a 2 109 ans plus vieille que celles obtenues dans les etudes precedentes. De plus, la presence d'impuretes, en particulier le 22Ne, modifie sensiblement le 161

162

Garcia-Berro & Hernanz: White dwarf crystallization

processus de cristallisation et rajoute un delai de 2 a 3 109 ans. Cependant, un calcul detaille de la fonction de luminosite, prenant en compte un modele raisonnable d'evolution galactique, et incluant I'effet des impuretes, montre que la position du cut-off, et done la valeur estimee de l'age du disque ne sont pas beaucoup modifiees. 8.1 Introduction White dwarfs are remnants of stellar evolution. They are relatively well known objects and their evolutionary time scales are rather large. All these characteristics allow us to consider white dwarfs as fossil stars, so we can use them to do some sort of galactic archaeology. The fundamental tool for that purpose is the white dwarf luminosity function, that is the number of white dwarfs with bolometric magnitude M\,oi per cubic parsec per unit bolometric magnitude. The most reliable observational white dwarf luminosity function is that of Liebert et al (1988). The slope of the increasing part of the white dwarf luminosity function can be roughly explained in terms of a simple model the so-called Mestel law (Mestel, 1952). However, the most relevant feature of the observational white dwarf luminosity function is the pronounced cutoff at log(Z/Z©) w —4.5 (Liebert et al, 1988). There is strong evidence that this effect is not due to the incompleteness of the sample considered or to selection effects (Liebert et al, 1988). More evidence, although indirect, has come recently from the results of Monet et al (1992) who have been able to determine accurate parallaxes and visual magnitudes for a sample of faint red stars. Although this survey is not systematic, it seems quite clear from their results that the cut-off is of intrinsic nature and that there is a significant absence of white dwarfs with luminosities below that of the cut-off. On the other hand, and from a theoretical point of view, it seems clear enough that the drop-off in the white dwarf luminosity function is a consequence of the finite age of the galactic disk, or at least of the solar neighborhood, as pointed out by Winget et al (1987) and Gara'a-Berro et al (1988a, 1988b). However, the computation of theoretical white dwarf luminosity functions requires an accurate white dwarf cooling theory. In fact, the position of the cut-off depends essentially on the cooling time scales of white dwarfs, whereas some aspects of galactic evolution, such as the star formation rate or the initial mass function, affect only the shape of the luminosity function around its maximum (Yuan, 1989, 1992, Iben and Laughlin, 1989) and not the position of the cut-off, unless extremely unrealistic assumptions are

Garcia-Berro & Hernanz: White dwarf crystallization

163

made. Since the pioneering work of Mestel (1952) the cooling theory of white dwarfs has received continuous interest. ;From the theoretical point of view, once the complicated behaviour of the envelope is properly taken into account (see the review by Mazzitelli, this volume), the most important phenomenon occuring during white dwarf evolution is crystallization. The importance of Coulomb interactions in dense ionic plasmas which form the cores of white dwarfs was first realized by Kirshnitz (1960), Abrikosov (1960) and Salpeter (1961). The relevant parameter is the dimensionless Coulomb coupling constant, defined as: = 2.275 l O 5 ^ !;

(1)

where ae =

4irne

is the inter-electronic distance, n e is the free electron density and Ye is the molar fraction of electrons. When F c < 1 Coulomb interactions are negligible, and when F c « 10 crystallization sets in. Two essential physical processes are associated with crystallization. The first one is the release of latent heat and the second one is chemical fractionation in the mixture. Both provide extra energy sources and lengthen the cooling time scale of the star. Although the release of the latent heat has been properly included in the calculations of the energetic balance of cooling white dwarfs for long (Van Horn 1968; Lamb and Van Horn 1975; Iben and Tutukov, 1984, Wood, 1992), the effect of chemical fractionation, though energetically dominant, has been usually ignored. This effect depends strongly on the shape of the phase diagram. Mochkovitch (1983) first examined the consequences of an eutectic phase diagram for the carbon-oxygen mixture, as suggested by Stevenson (1980). The effect was shown to be drastic, because of the large release of gravitational energy associated with pure oxygen deposition at the center of the star. The resulting cooling times and luminosity functions (Garcfa-Berro et al, 1988a and b) led to age estimates for the galactic disk as large as 15 Gyr. This stimulated more accurate calculations of carbon-oxygen crystallization diagrams (Barrat, Hansen, Mochkovitch (1988); Ichimaru, Iyetomi and Ogata (1988)), which proved that this diagram is not of an eutectic shape. This implies more modest, though still significant, time delays (Isern et al, 1989b, Hernanz et al, 1990).

164

Garcia-Berro & Hernanz: White dwarf crystallization

Besides carbon and oxygen, minor chemical species are also present in white dwarf interiors, reflecting the initial metallicity of the parent star. The most important being 22Ne, a product of the helium burning of the 14N left by the CNO cycle. Its distribution throughout the star depends on the mass and metallicity of the progenitor of the white dwarf. The importance of 22Ne crystallization on white dwarf cooling was first pointed out by Isern et al (1991), who computed a preliminary phase diagram and estimated time delays as large as 3 Gyr. The potential importance of other minor species like iron (Xu and Van Horn, 1992, Chabrier et al, 1993), also stressed the need for further, more accurate calculations of the crystallization diagrams of these species. The most recent study of the properties of arbitrary binary ionic mixtures has been performed by Segretain and Chabrier (1993). Nevertheless, the computation of realistic theoretical white dwarf luminosity functions also requires the connection of a model of galactic evolution with the cooling sequences. This, in turn, requires a full treatment of the effects of the mass distribution of white dwarfs, the star formation rate and, specially, the evolution of the abundances of minor chemical species. Our aim is to present a complete, accurate treatment of the effect of the different crystallization processes on the white dwarf luminosity function, in order to extract reliable information about the galactic history. 8.2 Cooling sequences Recently Segretain and Chabrier (1993) calculated the phase diagrams of arbitrary binary ionic mixtures, under conditions encountered in dense stellar plasmas, within the framework of modern density-functional theory (DFT) of freezing. These authors characterized the dependence of the shape of the phase diagram on the charge ratio of the mixture Z\\Zi. The crystallization diagram was shown to evolve from a spindle form for 0.72 < Z1/Z2 < 1, into an azeotropic form for 0.58 < Z1/Z2 < 0.72, and finally into an eutectic form for Zi/Z2 < 0.58. Results of such calculations for C/O, CO/22Ne and CO/ 56 Fe mixtures under conditions of interest are shown on figure 1. Note that the diagrams are calculated at constant Pe, i.e constant F e for a given T, and are expressed in dimensionless units for the temperature, so that they apply to any thermodynamic conditions, as long as the pressure of the system is given by the pressure of a homogeneous, degenerate electron background. Given the tremendous complexity of calculating the true phase diagram of a three-component system, the true ternary mixtures have been approximated by effective binary mixtures, where the nitrogen ion mimics an effective, homogeneous C/O mixture with an average charge Z = 7.

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In the case of an azeotropic phase diagram, crystallization proceeds within a particular scheme, as described by Isern et al (1991). The density discontinuity at crystallization is given by (Mochkovitch 1983): A/? _ ps- PL

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166

Garcia-Berro & Hernanz: White dwarf crystallization

0.4

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This process will no longer occur above a maximum oxygen concentration in the solid, for which eqn. (2) yields Ap > 0, no matter the shape of the phase diagram. For a neon and an oxygen mass concentration in the liquid A"j2isje = 1% and [XO]L = 50%, the adopted phase diagram leads to an upper limit for the oxygen mass concentration in the solid ([Xo]s)hi»x = 62%. As shown on figure 1, 22Ne will crystallize first, in an effective CO mixture. All neon will deposit at the center of the star at the azeotropic concentration xn^ « 11%, following the afore-described scenario. Figure 2 shows the composition profiles for 0 and 22Ne obtained with the diagrams shown on figure 1. We have included the previous results in the calculation of the cooling time of white dwarfs. In order to avoid the complicated behavior of hot white dwarfs we will study the evolution of white dwarfs starting from luminosities of order ~ 10" 1 LQ, corresponding to core temperatures of the order of ~ 6 X 107 K. Under these conditions neutrino losses can be

Garcia-Berro & Hernanz: White dwarf crystallization

167

neglected. Thermonuclear reactions in the outer layers are also unimportant (Iben and Tutukov, 1984). Also, central densities of most single white dwarfs are not high enough to allow pycnonuclear reactions to occur, so we can neglect this source of energy as well. Hence, the star luminosity arises only from the thermal and gravitational energy release:

where B = U + 0. is the so-called binding energy, being U and ft the thermal and gravitational energy content of the white dwarf which are given, respectively, by the expressions G—dm and U = /

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(4)

Jo

The crystallization of each layer, and the related latent heat and gravitational energy release due to chemical differentiation, are taken into account automatically as the temperature of the star decreases along evolution. On the other hand, a relation between the luminosity of the star and the temperature of its isothermal core is necessary to avoid the difficulties inherent to the isolating, semidegenerate envelope of the white dwarf. Several of such relations have been obtained for a C-rich envelope (Lamb and Van Horn, 1975), a He-rich envelope (Wood and Winget, 1989) and a H-rich envelope (D'Antona and Mazzitelli, 1989). We have chosen the following functional dependence

f

TF

(»)

which takes into account the leading term of the mass dependence of the luminosity and it is enough for our purposes. In the case of a He envelope, the results of Wood and Winget (1989) for a 0.6 M© white dwarf (their CO60400 sequence) can be fitted by: log F(T) =1.98695(logT6)5 - 8.15521(logT6)4+ 11.863(logT6)3 - 6.88515(logT6)2+ 3.16893(logr6) - 4.56649 .

168

Garcia-Berro & Hernanz: White dwarf crystallization

The time required to cool down to a temperature T, corresponding to a luminosity L(T), is then given by:

=r

/To L> JT

and the characteristic cooling time rcooj = dtcooi/dMf,oi reads n.dBdT

Tcool - - ° - 4

from this expression it is quite clear that the core feeds the cooling through the release of binding energy whereas the envelope controls it according to the dependence of the luminosity on the temperature of the isothermal core. In order to characterize the effect on white dwarf cooling of the different crystallization processes, we compare the different calculations with a reference cooling time, obtained when a C/O white dwarf in which there are not trace elements present is forced to crystallize with no chemical fractionation. The different calculations include: case (i) a pure C/O white dwarf with fractionation, and case (ii) a CO/22Ne white dwarf with Xi3Ne = 1.0%. We consider first a 0.6 M© white dwarf with a uniform equimassive (Xc = XQ = 0.50) C/O distribution in the fluid. The importance of an initial composition gradient in the fluid phase will be considered later. The different results are displayed in figures 3 and 4. The sudden crystallization of 22Ne at the azeotropic concentration is reflected by the abrupt release of gravitational energy at the crystallization temperature (luminosity). This energy source will sustain the star at the same luminosity for a considerable amount of time. Crystallization of C/O occurs through a more continuous process. Figure 3 shows the binding energy obtained when the gravitational energy released respectively by C/O crystallization, and CO/22Ne crystallization, is taken into account. The amounts of gravitational energy released between tc and £-4.5, where tc is the time where crystallization occurs, and t_4.5 is the time where the star reaches the cut-off luminosity L = 10~4#5I/©, are respectively AJ3 C / O = 2.35 1046 erg, and A5 N /a2 Ne = 7.13 1046 erg. Onset of crystallization at the center of the star is found to occur respectively at Tc/o = 3.59 106K, and TN/22Ne = 3.84 106K. Thus the energy release due to the differentiation of the minor element 22Ne is of the same order as the one due to the crystallization of the major constituents, C and O. The effect on the cooling time of a 0.6M© white dwarf is shown in figure 4. The crystallization of 1% of neon occurs at a higher luminosity, and produces a larger time delay than the crystallization of C/O. Time delays

Garcia-Berro & Hernanz: White dwarf crystallization

169

4.65 no separation Carbon-Oxygen separation 1% of Ne 0.1% of Fe 4.64

4.63

4.62

4.61

-3.5

-4.5

g(/g Fig. 8.3 Binding energy 56of a C/O star (dotted line), a CO/22Ne star (dashed line) and a CO/ Fe star (dotted-dashed line). Energy released by crystallization is shown by the kinks. The full line corresponds to the case with no fractionation at crystallization. produced by crystallization of C/O and 22Ne to reach L = 10~ 4 5 LQ are respectively 2.2 Gyr, and 3.4 Gyr. For C/O, our result is about a factor 2 larger than previous estimates (Barrat et al 1988; Ichimaru et al 1988), and stems from the larger density discontinuity at crystallization obtained with our phase diagram. Enrichment of the core by trace elements is sufficient to extend the cooling time by an amount comparable to that produced by C/O chemical fractionation, about 20% of the time obtained when ignoring chemical separation at crystallization (Wood, 1992). We note from figure 4 that the time delay produced by the crystallization of 22Ne and the remaining C/O mixture (dotted-dashed line) gives 5.35 Gyr at L = 10~ 4 5 I©, almost exactly the sum of the time delays obtained when treating separately 22 Ne crystallization (short-dashed line) and crystallization of the pure C/O mixture with no trace elements (dotted line), which gives 2.2+3.4=5.6 Gyr. This result has been confirmed for different concentrations of 22Ne. In figure 5, we show the characteristic cooling times for our reference model as a function of luminosity for white dwarfs within a mass range 0.5 M© to 1.2 M©. High mass white dwarfs crystallize at higher luminosities, thus having smaller time delays, and then reach the Debye cooling regime

170

Garcia-Berro & Hernanz: White dwarf crystallization No C-0 separation C-0 separation 1% Ne and No C-0 separation 1% Ne and C—0 separation 0.1X Fe

It -4

-4.5,

9.5

Fig. 8.4 Cooling time of a C/O star (dotted line), a CO/22Ne star with XaaNe = 0-01 (short-dashed line), and a CO/22Ne star where the remaining C/O mixture crystallizes after 22Ne has crystallized (dot-dashed line). The full line corresponds to the reference model, when the C/O star is forced to crystallize with no chemical separation.

before, thus cooling down faster. The characteristic cooling times for case (i) are shown in figure 6. The large bumps reflect the important release of gravitational energy at the crystallization luminosity. In figure 7, we compare the characteristic cooling times of a 0.6 M© white dwarf with 22Ne abundances by mass -XwNe — 0-01 an< i 0.03, case (ii), with the ones obtained for a white dwarf without neon, i.e. Xaa^e = 0. The peaks in the characteristic cooling times reflect the sudden release of gravitational energy associated with neon crystallization, which occurs over a very narrow temperature (luminosity) range. This is due to the fact that the azeotropic point of the phase diagram is only slightly depressed (see Segretain et al, 1993). The different positions of the peaks correspond to the different temperatures (luminosities) at which crystallization starts, according to the phase diagram. We now consider an initial composition gradient in the C/O distribution inside the star, resulting from previous evolutionary stages. We have performed the same calculations with the C/O distribution obtained by Mazzitelli and D'Antona (1986). The reference model is the same as men-

Garcia-Berro & Hernanz: White dwarf crystallization

171

10

1.2

CO

o

-1

""<&

-3

-4

Fig. 8.5 Characteristic cooling times vs luminosity, for C/O white dwarfs with masses ranging from 0.5 to 1.2 M 0> computed with the assumption of no fractionation at crystallization.

tioned above, i.e. does not include stratification and chemical separation. The three other models include chemical separation and no stratification, case (ii) already discussed, (iii) stratification and no chemical separation, and (iv) stratification and chemical separation. The results are shown on figure 8. Stratification leads to an oxygen-enriched core, and then to a smaller gravitational energy release at crystallization, and a smaller time delay. Moreover, the latent heat is released at higher temperature (see the C/O phase diagram), i.e. at larger luminosity. Note that stratification takes place for M < 1.0 M© only. The characteristic cooling times are similar to the ones displayed on figure 5, except that, because of the higher oxygen abundance at the center of the star, crystallization begins at a higher temperature, and then at a greater luminosity, for masses smaller than 1.0 M©. These cooling sequences agree with the results of Wood (1992), except that this author does not consider the dependence of the initial C/O profile on the mass of the white dwarf.

Garcia-Berro & Hernanz: White dwarf crystallization

172

o

Fig. 8.6 Same as figure 5, but taking into account the effect of carbonoxygen fractionation (case ii), which yields a larger bump at the crystallization luminosity.

8.3 The luminosity function The luminosity function including the effect of minor chemical species has been computed from a generalization of the method developed by Noh and Scalo (1990). The distribution function / ( / , m, Z) gives the number of white dwarfs per unit volume in a generalized phase space where the coordinates are the mass, the metallicity and / = — log(i/i©), being L the luminosity of the white dwarf. This distribution reads:

/(/,m,Z)=7 6(1 - lo)Bwd(t, m, Z)l(t)dt l(l,m,Z)J-oo

(7)

jr. —Bwd(h,m,Z) 1(1,m,Z) where / is proportional to the bolometric magnitude, S is the delta function, /o = - log(Z0/£©)> being LQ the white dwarf luminosity at birth (assuming that all white dwarfs were born with the same luminosity), 1/1(1, m,Z) is ls proportional to the characteristic cooling time. Bwd(totm^) * n e so ~ called source function for white dwarfs at time to, denned as the total num-

Garcia-Berro & Hernanz: White dwarf crystallization 11

o

-3 log(VU)

Fig. 8.7 Characteristic cooling times vs luminosity, for a 0.6 M© C/O white dwarf with XnNe - 0% (solid line), 1% and 3% respectively. ber of white dwarfs with mass m and metallicity Z born at time to, per mass, metallicity and time units. Thus

Bwd(tQ,m,Z)

=

tPN

- Z{tb))

Here M is the mass of the progenitor of a white dwarf of mass m, (M) the initial mass function, generally assumed to be constant in time, TJ){t) the star formation rate, expressed as mass of stars formed per unit volume (pc 3 ), to is the birth time of the white dwarf (with a luminosity LQ which we will consider to be large compared with the actual luminosity, LQ > L), and tf, is the birth time of the progenitor star, obtained from: h + tm,(M) + tcool(L, M, Z(tb)) = tiiak

(8)

where tdiak is the age of the disk and tms is the time spent on the main sequence. Note that by adopting this expression for the source function we are implicitly assuming that the metallicity of white dwarfs is that of the interstellar medium when their progenitors were born. The integration of (7) gives the white dwarf luminosity function n(L), i.e.

Garcia-Berro & Hernanz: White dwarf crystallization

174 -3

-3.5

-4

no Carbon—Oxygen separation Carbon-Oxygen separation Carbon-Oxygen separation + MD no Carbon-Oxygen separation + MD

-4.5

-5,

9.5

log(t)

10

Fig. 8.8 Effect of initial stratification on the cooling time of a C/O 0.6 MQ white dwarf. Full line: no separation, no stratification; dotted line: separation, no stratification; dotted-dashed line: no separation, stratification; dashed line: separation, stratification. MD stands for Mazzitelli and D'Antona (1986).

the number of white dwarfs per unit of bolometric magnitude, per cubic parsec, with luminosity L: rM,

n(L) = / JM

tip

TC00,(L,

M,

, M, Z(tb))-

tma(M))(M)dM where M9up and M, n / denote respectively the maximum and the minimum mass of the white dwarf progenitors which contribute at luminosity L. M, n / is obtained by setting tt, = 0 in expression (8):

teooi(L, Minf, ZQ) =

(10)

where Z o is the initial (t = 0) metallicity of the interstellar medium, usually taken as 0. If we focus on pure C/O white dwarfs, then eqn. (9) reads

Garcia-Berro & Hernanz: White dwarf crystallization

rM,Up

n(L) = / r c o o / ( i , M)il>(tdisk - tcool(L, M) JMini(L)

175

tma(M))

(11)

4>{M)dM where Af,n/ satisfies now: tma{Minj) + tcooi{L, Af,n/) = Conversely, if we focus on the influence of minor chemical species on the white dwarf luminosity function, the cooling rates depend on the mass of the white dwarf, its luminosity and its metal content. For a sake of simplicity, the dependence of cooling times and characteristic cooling times on the mass of the white dwarf will be ignored and calculations will be performed for a typical M* = 0.6 M© white dwarf only. This is justified by the weak mass-dependence of tcoo/ and rcoo/ when compared with their strong metallicity-dependence. In this case, the luminosity function reads:

n(L)= / Tcoo^M^ZWtdisktcooiiLtM^Z) JMinf(f(L) tms(M))<j>(M)dM

(12)

where Minf is now given by: tdiak = tcooi(L, Af*,Z0) + tms(Minf). A close look at equation (12) reveals that, in fact, the effect of different progenitor masses is taken into account correctly. As can be seen from the previous expressions, the computation of the white dwarf luminosity function requires not only accurate cooling sequences, but also a model of galactic evolution. A good standard model for our purposes is that of Clayton (1984), which takes into account the infall of matter over the galaxy for a certain amount of time. The reason of this choice is basically its simplicity. Of course, there are more elaborate models of galactic evolution (see for instance Abia et al, 1991, and Bravo et al, 1993) but for our purposes - that is, showing the importance of a proper treatment of crystallization - this SFR is enough. Concerning the initial mass function, IMF, the one proposed by Salpeter (1961) has been adopted. We have computed luminosity functions for the cooling sequences described in the previous section. Figures 9 and 10 show the luminosity function (11) for our reference model and case (iii), i.e. no separation at crystallization without and with initial stratification respectively, computed for two different ages of the galactic disk in each case, namely 8.8 and 105 Gyr and 8.5 and 10.3 Gyr, respectively. These values best reproduce the position of the observed cut-off, obtained with two different approximations for

176

Garcia-Berro & Hernanz: White dwarf crystallization

O

Fig. 8.9 Theoretical luminosity functions for C/O white dwarfs with no fractionation (reference model), for ages of the disk t
the bolometric correction of the coolest white dwarfs (the bright portion of the observational luminosity function has been taken from Fleming, Liebert and Green, 1986). For low luminosity objects, two approximations to the bolometric correction are considered, following Liebert, Dahn and Monet (1988): either model bolometric corrections for cool DAs and blackbody bolometric corrections for cool non-DAs (open circles), or zero bolometric corrections for all cool white dwarfs (open squares). This dubiousness in the faint end of the observed luminosity function leads to an uncertainty of about 2 Gyr in the age of the disk, as can be seen from the figures. It is important to stress that these two possible observational luminosity functions, are not equivalent to Wood's error boxes approach: curves inside the error boxes do not necessarily cross the error bars, even though the resulting uncertainty in the age of the disk is similar. The small difference between our reference model and case (iii) stems from the effect of initial stratification in

177

Garcia-Berro & Hernanz: White dwarf crystallization , . 1 1 1 1 1 1 1 T • -2

1

i

.

.

.

.

f

CO

o

-5

r

R\ V- <\ :

-3 -

-4

r

-

i

.

.

.

.

i

.

.

.

.

i

-log(VL.)

Fig. 8.10 Same as figure 9 but with an initial stratification for oxygen (Mazzitelli and D'Antona, 1986) - case (iii). The ages of the disk are respectively 8.5 (left) and 10.3 (right) Gyr. the oxygen profile. The difference in the cooling time at log(i/Z©) ~ —4.5, and then in the estimate of the disk age, is about 0.5 Gyr. When C/O separation is taken into account, the ages of the disk are 10.5 and 12.3 Gyr for case (i) - figure 11 - and 9.5 and 12 Gyr for case (iv) - figure 12. The effect of initial stratification is slightly larger than for our reference model and case (iii), as could be expected from the corresponding cooling times. These calculations show that C/O differentiation at crystallization increases the cooling time at the observed cut-off luminosity by 1.5 to 2 Gyr. Therefore, any estimate of the age of the galactic disk obtained without taking into account the C/O differentiation process in the calculation of the luminosity function, must be increased by at least 1.5 Gyr. The bump in the theoretical luminosity function at log(Z/£©) ~ —4 will be discussed later. The inclusion of selection effects due to the different scale heights of old and young white dwarfs leads to a reduction of the amplitude of this bump (see also Garda-Berro et al 1988b for a detailed discussion). We will now focus on the effect the crystallization of minor chemical elements and, in particular, on the effect of 22Ne. The corresponding luminosity function, computed according to equation (12) with the model of galactic chemical evolution already mentioned, is shown in figure 13. In

178

Garcia-Berro & Hernanz: White dwarf crystallization

O

Fig. 8.11 Same asfigure9 but with C/O fractionation - case (i). The ages of the disk are 10.5 and 12.3 Gyr. order to characterize the effect of 22Ne crystallization, we first ignore C/O fractionation, and compare the results with our reference model (dotted line). The ages for the disk are again 8.8 and 10.5 Gyr. Therefore, although the cooling is strongly delayed by the presence of even a small amount of 22 Ne, this effect is not reflected in the location of the cut-off of the white dwarf luminosity function. This apparently surprising result stems from the strong dependence of the white dwarf luminosity function upon the shape of the age-metallicity relation. For standard models of chemical galactic evolution, the initial abundance of CNO cycle elements is zero and there is a gradual enrichment along the evolution of the galaxy, leading to a maximum for 22 Ne abundance of about 2 - 3 % per mass, depending on the adopted yields. The stars old enough to reach the cut-off come from progenitors formed with the galaxy, with no 22Ne enrichment. Therefore the age of the disk derived from the location of the cut-off is not affected by the presence of minor chemical species. For this reason, the effect of 56Fe on the luminosity function, qualitatively quite similar to the effect of 22Ne, has not been considered explicitly. Its effect on the cooling time has been calculated in our previous paper (Segretain et al, 1993). This result demonstrates the impor-

Garcia-Berro & Hernanz: White dwarf crystallization

179

•30

o -4 -

-5 Fig. 8.12 Same as figure 9 but with an initial stratification tor oxygen (Mazzitelli and D'Antona, 1986) - case (iv). The ages of the disk are 9.5 and 12 Gyr.

tance of a careful computation of the white dwarf luminosity function, with a complete model of galactic chemical evolution, and shows that the delays in the cooling times do not imply automatically increases of the disk age, as was the case for pure carbon-oxygen white dwarfs. For a sake of completeness, we have also computed white dwarf luminosity functions for different initial metallicities of the galaxy. In figure 14 we compare the results for ZQ = 0 and ZQ = 0.01, for tdisk =10 Gyr. A nonzero initial metallicity may reflect an initial metal enrichment of the galaxy due to a previous (Population III?) generation of massive stars. The cutoff position is shifted to higher luminosities for higher initial metallicity, as expected. For ZQ = 0.01, the age of the disk must be increased by about 3 Gyr in order to reproduce the observed cut-off at the right luminosity. Our last calculation corresponds to the most general case. This model takes into account both 22Ne crystallization, and then crystallization of the remaining C/O mixture, as given by the phase diagrams described in Segretain et al. (1993). Even though such calculation remains slightly uncertain from the quantitative viewpoint, because of the lack of an exact treatment of the crystallization diagram of three-component mixtures, it certainly retains most of the physical process, and then gives reasonable estimates. The

180

Garcia-Berro & Hernanz: White dwarf crystallization -r

o

Fig. 8.13 Theoretical luminosity functions for C/O/Ne white dwarfs, taking into account 22Ne deposition at crystallization (solid lines). Luminosity functions ignoring this process (reference model) are also shown for comparison (dotted lines). The ages of the disk are the same as in figure 9 (8.8 and 10.3 Gyr).

corresponding luminosity functions are shown in figures 15 and 16 (dotted lines), for disk ages 10.5 and 13 Gyr respectively, and are compared with the observed luminosity functions, obtained with the two approximations for the bolometric correction of cool white dwarfs (Liebert et al, 1988): blackbody bolometric correction for non DAs and a correction based on model atmospheres for DAs (figure 15) or zero bolometric correction for all cool white dwarfs (figure 16). The ages of the disk are very similar to the results of case (ii), i.e. zero metallicity, as expected. It is worth noting that the bump around log(i/ZrQ) ~ —3.8 is larger than for carbon-oxygen separation only. We will now discuss the presence of a spike in the luminosity function (see figures 13-16) at log(X/X 0 ) ~ -3.8. This pattern is due to 22Ne sedimentation, and its associated release of gravitational energy at a given luminosity, and could be used as the observable signature of this process. However, this has to be taken very cautiously for several reasons. First, these luminosity functions have been computed from equation (12), which considers that cooling times are those of a 0.6 M0 white dwarf with different 22 Ne abundances, thus neglecting the effect of the mass of the white dwarf

Garcia-Berro & Hernanz: White dwarf crystallization

181

o

Fig. 8.14 Same as figure 13 for different initial metallicities of the galaxy: Zo = 0 (solid line) and Zo = 0.01 (dotted line), and *,j,-,i=10 Gyr.

on cooling. If the luminosity function were calculated for the whole white dwarf mass distribution, the spike would be spread over a larger range of luminosities, yielding a smoother luminosity function. Second, as already mentioned, old white dwarfs have larger scale heights over the galactic plane than young ones. This leads to a further reduction of the amplitude of the bump and the spike. This last effect has been included in our last calculation (see Garci'a-Berro et al, 1988b for details on the adopted scale height-age law). The corresponding luminosity functions are shown in figures 15 and 16 (solid lines). Furthermore, from an observational point of view, white dwarfs are actually binned in magnitude intervals. This must be taken into account for a correct comparison of theoretical and observational luminosity functions. Since the observed luminosity function is obtained by binning the data in bins of a certain amplitude in visual magnitude (usually, AAfv=0.5 or 1 magnitude), we have followed the same procedure with the theoretical luminosity functions. We have averaged our results in bins of a certain amplitude, following the prescriptions of Liebert et al: AM,,=0.5 for hot white dwarfs and AM(,O/=1 for the cool ones, with the same relationship between M\,oi and Mv for hot white dwarfs and the two already mentioned possibilities for the bolometric corrections of cool white dwarfs. The results

182

Garcia-Berro & Hernanz: White dwarf crystallization

OO O

Fig. 8.15 Theoretical luminosity functions when both 22Ne crystallization and then C/O crystallization are taken into account successively. The effect of scale height inflation is either included (solid line) or ignored (dotted line), for *d,,t=10.5 Gyr. The observational points for cool white dwarfs are obtained with blackbody BCs for non DAs and model BCs for cool DAs. The open asterisks are the values of the binned theoretical luminosity function (solid line) (see text for details). are shown by the asterisks in figures 15 and 16, for our most general case. The larger discrepancy at l o g ( I / I © ) ~ - 4 for *<<„*= 13 Gyr (figure 16) is

due to the much lower position of the observational point at this luminosity, when a zero bolometric correction is used. It is clear that a better knowledge of the faint end of the observed luminosity function, as well as a more complete treatment of the scale height problem, are crucial to withdraw reliable information on the age of the galactic disk. 8.4 Conclusions We have examined the modification of the cooling time of old white dwarfs, due to the energy release associated with the chemical separation of major and minor elements at crystallization. Our calculations include up-to-date equations of state as well as accurate crystallization diagrams. The cooling time is computed directly from the temperature dependence of the binding energy, taking into account automatically the gravitational energy and the

Garcia-Berro & Hernanz: White dwarf crystallization -2

183

i—.—i—.—.—i—i—.—i—•—i—•—•—>—TT—r

60 O

Fig. 8.16 Same as figure 15 but for <
latent heat released by crystallization along the evolution of the star. Our calculations consider also the possibility of an initial composition gradient in the C and 0 distributions. We believe these calculations to provide the best estimate for the time delay induced by the crystallization of the C/O mixture, a long time standing problem, and trace elements. We show that crystallization of minor elements has a drastic effect on the cooling time of the star, in spite of their low cosmic abundance. The energy released by crystallization of these elements produces a substantial extension of the cooling time, of the order or even larger than the one induced by the crystallization of C/O itself, which is between 1 and 2 Gyr, depending on whether initial stratification is considered or not. Even though the exact determination of these processes would require the knowledge of a /our-component phase diagram, our results demonstrate convincingly the importance of the crystallization of major and minor species in modern white dwarf cooling theory. The present calculations suggest that 22Ne crystallizes first, in a homogeneous C/O liquid, all 22Ne being collected ultimately into the core, which yields a time delay of about 3 Gyr. Our treatment of the crystallization of the remaining C/O/ 56 Fe mixture is certainly oversimplified, specially since we

184

Garcia-Berro & Hernanz: White dwarf crystallization

consider that the C, 0 and 56Fe abundances remain unchanged after 22Ne crystallization occurs, and we ignore a possible 56Fe phase separation in the liquid phase. For this reason, our calculations provide probably an upper limit for the time delay induced by crystallization of the afore-mentioned three-component-mixture. However, we can reasonably estimate this delay to be at least 1 Gyr. Therefore, if minor elements are present in white dwarf interiors, the minimum extension of the cooling age of a 0.6 M© white dwarf could be as large as 4 Gyr, a fairly important amount.

On the other hand, theoretical luminosity functions have been also computed for pure C/O white dwarfs and for white dwarfs with a chemical composition which includes minor chemical species such as 22Ne and 56Fe. The calculations take into account the variation of metallicity along the chemical evolution of the Galaxy. We believe that these calculations give the most complete treatment of the effect of the different crystallization processes occuring in white dwarf interiors on the cooling history and the luminosity function of these stars. The theoretical luminosity functions are compared with the observed ones to determine the age of the galactic disk. We have shown that the time delay due to C/O fractionation at crystallization directly translates into the luminosity function, and that any age of the disk obtained from calculations which do not include this effect must be increased by at least 1.5 Gyr. This gives tdiak ~ 9-5 to 12 Gyr, if we take Wood's results as reference. Although the presence of minor chemical species is found to modify appreciably the cooling time, this does not affect the age of the disk obtained from the luminosity function, a direct consequence of the low-metallicity of old white dwarfs. On the other hand, the important release of gravitational energy associated with the sedimentation of minor species produces a sharp peak in the luminosity function around log(i/i©) ~ —3.6. Though not detectable with present day observations, the presence of this peak provides a future observational test of the effect of minor chemical species on the cooling history of white dwarfs. If the initial metallicity of the Galaxy is non-zero, then the presence of minor chemical species modifies significantly the position of the cut-off and increases the age of the disk by about 3 Gyr for ZQ = 1%. These calculations demonstrate the necessity to include a proper treatment of crystallization in modern white dwarf cooling theory, and show convincingly the importance of galactic evolution for a proper determination of the age of the disk, stressing the need for further study in this direction.

Garcia-Berro & Hernanz: White dwarf crystallization

185

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9 Gravitational collapse versus thermonuclear explosion of degenerate stellar cores J.ISERN Ctntre d'Estudis Avancats Blanes (CSIC), Camt de Santa Barbara sn, 17300 Blanes, Spain

R. CANAL Departament d'Astroftsica i Meteorologia (Universitat de Barcelona), Diagonal 675, 08028 Barcelona, Spain

Abstract In this paper we review the behavior of growing stellar degenerate cores. It is shown that ONeMg white dwarfs and cold CO white dwarfs can collapse to form a neutron star. This collapse is completely silent since the total amount of radioactive elements that are expelled is very small and a burst of 7-rays is never produced. In the case of an explosion (always carbonoxygen cores), the outcome fits quite well the observed properties of Type la supernovae. Nevertheless, the light curves and the velocities measured at maximum are very homogeneous and the diversity introduced by igniting at different densities is not enough to account for the most extreme cases observed. It is also shown that a promising way out of this problem could be the He-induced detonation of white dwarfs with different masses. Finally, we outline that the location of the border line which separetes explosion from collapse strongly depends on the input physics adopted. Dans cet article on revise le comportement d'un noyau stellaire degenere qui grandit. On montre que les naines blanches d'ONeMg et celles de CO, froides et massives, peuvent s'effondrer pour former une etoile a neutrons. Cet effondrement est completement silencieux puisque la quantite tot ale d'elements radioactifs expulsee est tres petite et on ne produit pas d'eruption de rayons gamma. Dans le cas d'une explosion (toujours pour des noyaux de carbone-oxygene), le resultat des calculs reproduit assez bien les proprietes observees des supernovae de Type la. La courbe de lumiere et les vitesses 186

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187

correspondant au maximum sont tres homogenes et les variations introduites par des differences dans les densites d'ignition, ne suffisent d'expliquer les cas observes les plus extremes. On montre aussi qu'une solution prometeuse a ce probleme pourrait etre la detonation induite par l'ignition de l'helium a la surface de naines blanches de masses assez differentes. Finalement, on signale que la frontiere entre l'explosion et l'effondrement depends fortement de la physique que l'on introduit. 9.1 Introduction Exploding stars play a very important role in the evolution of galaxies since they inject about 1051 ergs of mechanical energy per event to the interstellar medium. They are at the origin of the neutron stars and, consequently, they are responsible of the existence of pulsars and galactic X-ray sources. Finally, during the explosion they inject several solar masses of newly synthesized elements, which completely shapes the chemical evolution of the galaxies. In all cases, a stellar core, supported by the pressure of degenerate electrons is involved. The reason is twofold: a) At high densities, the electron pressure is dominant. This pressure is composed by two terms, a leading one, only depends on the density alone, and another one that vanishes whith temperature, i.e.: Pe = P0(p) + P1(p,T)

(9.1)

Pi -+ 0 T -> 0 Under degenerate conditions Po is dominant. If thermonuclear reactions start at some point, matter cannot expand in order to control them and a thermonuclear runaway that incinerates all the matter to iron peak elements occurs. As a consequence of the high temperatures resulting from the process, a burning front appears and propagates through all the star. b) If electrons are not relativistic, the leading term is Po oc p 5 ^ 3 , and from the dimensional point of view it scales like P ~ M5/3R~S. If electrons are relativistic, Po oc p 4 / 3 and it scales as P ~ M4/3R~A. The condition of hydrostatic equilibrium requires a functional dependence P ~ M 2 ^ " 4 . Thus, in the extremely relativistic case there is not a definite scale length and, depending on how fast the energy is injected or removed, it is possible to obtain an explosion (observed as a supernova) or a collapse (to form a neutron star). In the case of stellar cores, the outcome depends on the rate at which energy is injected by the burning front as compared with the rate at which it is removed by the electron captures on the burned

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material. The velocity of the burning front is a function of the density, temperature and chemical composition. The rate of electron captures on the incinerated material, which is essentially made of ^Ni, is a function of the Fermi energy. Therefore, the outcome -explosion or collapse- depends on the density at which the fuel is ignited, for a given velocity flame. 9.2 Physics of the explosion 9.2.1 The thermonuclear runaway The basic condition to obtain an explosion is that enough energy must be released before matter can dynamically react. In stars, this condition can only be fulfilled in electron degenerate structures. The reason is that if pressure were dominated by the classical ideal gas law, any energy release would produce an adiabatic expansion that would quench the reaction. In the degenerate case, the energy generation rate will increase because of the temperature increase and the process will be reinforced. Eventually, electrons will become nondegenerate and matter will start to expand. At this point, however, temperature will be so high that energy will be released in a time short as compared with the hydrodynamic time and a violent expansion will ensue. The time necessary for the system to react to an overpressure is equal to the time spent by a sound wave to cross the system. Thus, the hydrodynamic time can be defined as: rhd * -

(9.2)

where h is a characteristic dimension and cs is the sound velocity. Since degenerate cores are in hydrostatic equilibrium, the balance ensures that the hydrodynamical time is of the order of the free fall time: 1 Tff

444

=

p> i.e. the natural time scale for the free fall of a uniform pressureless selfgravitating sphere. Thus, the condition for instability is r n < T^d, where the characteristic nuclear heating time is defined as:

Tn=C~f

(9.4)

cv is the specific heat at constant volume and k is the rate of energy generation. The temperature at which rn = Thd is called the deflagration temperature and it should not to be mistaken for the ignition temperature, defined

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as the temperature at which neutrino losses balance the energy produced by thermonuclear reactions. Typical stars burn their fuel quiescently because they stabilize their temperature below the deflagration temperature by means of adiabatic expansion. The efficiency of adiabatic cooling can be defined as the relative density change necessary to restore the pressure equilibrium upon the release of some quantity of energy during an adiabatic expansion (Mazurek and Wheeler, 1980). If the gas is ideal, the adiabatic expansion efficiency is: Po

kl

where Q is the energy released per reaction. Since Q is of the order of MeV, the cooling is very efficient for temperatures below 109 K (~ O.lMeV). If the electron component is strongly degenerate dPe

.. ) <

where Pe and Pi are the electronic and ionic pressures respectively and

&oc-3-*

(97)

and adiabatic cooling is only efficient in the region where Pi > Pe.

9.2.2 Propagation of the burning front When the thermonuclear fuel is ignited, burning propagates through all the star driven by the following mechanisms: a) Spontaneous burning. In a contracting core, the condition for burning can be reached quasi simultaneously in several points. Therefore, due to the existence of an initially small temperature gradient, the burning can spread over a wide region without any transport mechanism (Blinnikov and Khokhlov 1986; Woosley 1986). The time scale for acceleration of the nuclear burning is: T

nuc = YJl

7J7\

(9-8)

and the location of the burning front changes with a phase velocity that is given by vph = (d,Tnuc/dr)"1. This velocity increases when the absolute values of the temperature and density gradients decrease. Therefore, regions with Vph > cs will ignite spontaneously and the burning front will propagate supersonically there. b) Detonation. In order to describe the properties of a burning front

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it is assumed that the fuel and the burned material are separated by a region, where the reactions take place, of a width 8 much smaller than any characteristic length, /, of the system. If 6 << / it is possible to connect both sides of the front by means of conservation laws of mass, momentum and energy. In the reference frame associated to the front, these equations can be written as (Landau and Lifchitz 1959, Mazurek and Wheeler 1980): (9.9) Pi + Piu\ = Po + Po4

(9-10)

that are similar to those describing shock waves except for the presence of the term q that represents the energy released by reactions. The subindexes 0,1 apply to the unburned and burned material respectively and the remaining symbols have their usual meaning. The mass flux crossing the front is denned as j = PQUQ = p\Ui and can be written as: 7-2 37 =

P

° ~ P'l

((

3

V0-Vl which implies that the mass flux across the front is determined by the ratio between the difference of pressures and specific volumes at both sides of the burning front. Real solutions must satisfy (Pi > Po)(Vi < VQ) or (Pi < Po)(Vi > VQ). The first solution corresponds to a detonation and the second one to a deflagration. From the equation of conservation of energy it is possible to write eo + q - «i + \{PQ + Pi)(Vb - Vi) = 0

(9.13)

which is called the detonation adiabat (the case q=0 is called the shock adiabat). This equation, together with that defining the mass flux determines the final state once the characteristics of the burning front have been specified. The physical meaning of the intersection between the adiabat and the mass flux is clear. A shock heats and compresses the material to a state (P 3 , Vs) given by the intersection of the shock adiabat with the line defined by j 2 . Because of the increase in temperature, material burns and reaches the state (Pi,Vi) defined by the detonation adiabat and the jf2-line intersection. Since q > 0, Pi < Ps and V\ > Vs which implies that a rarefaction is associated to the postchock burning. The family of solutions obtained in this way, with j as a free parameter,

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has an extremum for which j and the front velocity are minima. This solution, called Chapman-Jouguet detonation, corresponds to the case where the j-line is tangent to the detonation adiabat. It has the important properties of being univoquely determined by the thermodynamic properties of the material (including q) and having a propagation velocity that is equal to the sound velocity of the burned material. All the remaining solutions, called strong detonations, move subsonically with respect to the burned material. In stars, due to the spherical simmetry, material must be at rest at the centre. Therefore, the velocity has to decrease from some positive value behind the front to zero at the centre. This means that a rarefaction wave, moving at the sound velocity, must follow the detonation. Since strong detonations are subsonic respect the burned material, they are overtaken by the rarefaction wave and only the Chapman-Jouguet one can survive. Although it has not been completely elucidated, it is generally accepted that due to the high densities existing in the central regions of CO white dwarfs, the overpressures induced by the burning front cannot give rise to a detonation (Mazurek, Meier and Wheeler 1977). However, as the subsonic flame propagates into regions of progressively decreasing density, it accelerates and it can eventually become a detonation (Blinnikov and Khokhlov 1986; Woosley 1986). It is interesting to notice here that a detonation does not necessarily produces the complete incineration of the material into iron. If the density is smaller than ~ 107 g/cm3, nuclear statistical equilibrium cannot be reached and, as a consequence, elements of masses intermediate between C-0 and Fe are produced. If the density is smaller than 106 g/cm3 even the fuel, the C-0 mixture, has no time to be exhausted (Khokhlov 1989). In the case that the fuel is He this is not true and Fe is synthesized. c) Deflagration. Deflagrations are the solution to the conservation laws that fulfill the condition (Pi < P2) {Vx > V2). The main dificulty with these solutions arises from the fact that matter is subsonic at both sides of the front and boundary conditions behind it can affect the front as well as matter ahead. In the case of a star with a subsonic burning front propagating outwards with a velocity D, the condition of matter being at rest at the center demands the existence of a shock precursor that boosts matter outwards (Mazurek and Wheeler 1980) and causes the expansion of the star. There are two modes whereby a nuclear deflagration can propagate inside a degenerate core: The laminar mode and the turbulent mode. In the laminar case (often known as conductive front), electrons transport the energy released in the burning regions to the surroundings, inducing their ignition. The velocity can be estimated in the following way (Landau and Lifchitz 1959): The width of the front is given by 6 ~ y/xr, where x is

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the thermometric conductivity and r is the time that the burning lasts. The velocity of the burning front is thus given by D ~ 6T = y/xjr. This velocity is always of the order or smaller than 10~2Cg, where ca is the sound velocity. Useful approximations are provided by (Timmes and Woosley 1992):

valid in the range 0.01 < />g < 10 in the case of a C-0 mixture and:

valid in the range 1 < pg < 14 in the case of an ONeMg mixture. In the turbulent case, a hot and less dense layer is formed below a cold dense layer. Because of gravity, the interface is Rayleigh-Taylor unstable and both layers are mixed, that noticeably increasing the propagation velocity due to the increase in the efficiency of the conductive transfer. Although at present there is not a satisfactory theory of turbulent flames in stellar interiors, it is possible to make an estimate of the velocity of a turbulent flame in the central regions of a star. Nomoto et al (1984) proposed from mixing length arguments, a turbulent velocity vt c± \fgeffl/2 where geff = GMr2/Sp/p is the effective acceleration, 6p is the difference of densities between both sides of the front and / is the mixing length taken as / = min(r,a.flp) where a is an adjustable parameter of the order of unity. The use of the mixing length theory has been questioned not only because of the jump in density across the front but also because all other characteristicsof the burning strongly violate the basic hypothesis of the theory itself. Woosley(1990) proposed a fractal description of the burning front to take into account its wrinkling and Livne and Arnett (1993) proposed to treat the turbulent deflagration in terms of an ablative front in order to correctly handle the growth of the different unstable modes. At present, the question is completely open and in fact the mixing- length model, despite being physically incorrect, provides for a = 0.7 the best agreement with observations. In practice, numerical treatments like those of Sutherland and Wheeler (1984) and Unno (1967) are used. It must be also taken into account that electron captures behind the burning front can completely inhibit the development of the Rayleigh-Taylor instability (Timmes and Woosley 1992).

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9.3 The collapse of degenerate cores

The evolution of the primary star in a close binary system can give rise to either a helium white dwarf, a carbon-oxygen white dwarf, or an oxygenneon-magnesium white dwarf, depending on the initial parameters of the system. However, not all compositions can be involved in the accretion induced collapse. Helium white dwarfs can immediately be discarded. Their mass growth would lead to explosive helium ignition at the center of the star for a central density pc < 4 108 g/cm3, no matter how low is the temperature (Sugimoto and Nomoto 1980). After incineration, electron captures would be too slow, the overpressures would be large enough to start a detonation and the star would thus be completely disrupted (Woosley and Weaver 1986). This leaves only carbon-oxygen and oxygen-neon-magnesium white dwarfs as possible candidates for a core collapse. These groups have to be considered separately as accretion induced collapse poses different problems for each of them. Both cooling in the stage between the formation of the white dwarf and onset of mass accretion and reheating by mass accretion are specially relevant in the CO case, whereas semiconvection associated with electron captures plays an important role in the ONeMg case. As it was stated in the Introduction, the collapse/ explosive behavior alternative for a degenerate core depends on the density at which the burning front starts, for a fixed velocity of the flame. Since there is not yet a theory giving the velocity of the burning front, it is not possible to decide which is the value of the density beyond which the collapse ensues. It is possible, however, to establish two firm bounds and a guess as to this critical value. The upper limit is determined by the maximum velocity of the flame: the sound velocity. In this case, the minimum density necessary to guarantee the collapse is p « 2 1010 g/cm3. The lower limit is determined by the minimum velocity of the flame: the conductive velocity. In this case, the minimum density necessary to get a collapse is 8.5 109 g/cm3. In a previous calculation by Canal et al (1992), this limit was set to 9.5 109 g/cm3 because the Coulomb corrections to the equation of state of the ions were not correctly included. The guess can be obtained by using everal recepies appeared in the literature and making some kind of average. For densities higher than 9 109 g/cm3 some of them predict a collapse, and for 9.5 109 g/cm3 all of them do.

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9.3.1 The CO case Carbon-oxygen white dwarfs can form in binary systems either by Roche lobe overflow just before or just after ignition of He in initially close binaries or by Roche lobe overflow during the early or the thermally pulsing asymptotic giant branch phases in initially wide binaries. In the latter case, common envelope evolution should allow enough orbital angular momentum to be lost so that the wide binary evolves into a close binary. An important question is the upper mass limit for CO white dwarfs formed in this way. Observations of classical novae give average masses of 1.23 M©, although this figure is not truly representative of the average since several selection effects favor the detection of massive white dwarfs, and models for the recurrent nova U Sco give a mass M~ 1.38 M© (Starrfield et al 1989). However, the most massive white dwarfs found in these systems may well be ONeMg and not CO white dwarfs. Theoretically, CO cores of single stars should ignite C non explosively when M core > 1.1 — 1.2M© (Iben and Tutukov 1986). The behavior of the central layers of a growing CO core is determined not only by the local balance among the nuclear energy released, the neutrino losses and the compressional work, but also by the properties of the outer layers. If such CO core is part of an isolated white dwarf star, it will cool down because of the photospheric losses. If this core is growing, as is the case of an accreting white dwarf in a binary system, it is heated not only by the H and He burning shells but also by the compression of the outer, partially degenerate layers. An important ingredient ofthe problem of determining the density at which central ignition happens comes from the fact that nonperturbed white dwarfs cool down and eventually solidify. For instance, a CO white dwarf with M > 1M© starts to solidify after 1 Gyr since its formation. That ~ 0.5, that implies that the adiabatic coefficient is small (d\nT/d\np)s the heating by compression is gentle (Hernanz et al 1988) and that, in the absence of the influence of the outer layers, the ignition of the carbon-oxygen mixture is entirely controlled by pycnonuclear reactions (Canal and Isern 1979). In that case, the density at which the runaway starts is pc ~ 10io g/cm3 instead of 4 109 g/cm3 typical of the fluid phase. The energy released by the compression of the outer layers is, by far, the most important heating mechanism. It can roughly be approximated by (Nomoto 1982): Lg/LQ = 1.4 10- 3 (T/10 7 /O(M/10- 10 M©/yr)

(9.16)

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9.00 B

C

A

8.50

8.00

— 7.50

7.00

6.5

I

.50

I

I

I

I

8.50

1 I

I

1

I

I

I

I

I

9.50

10.50

log(RHO) Fig. 9.1 Path followed by the center of a mass-accreting white dwarf with an initial mass of 1.15 M, T o = 4 106 K and z c = x 0 = 0.5. Cases A, B and C correspond tho M = 10~ 6 ,5 10~ 8 and 1O~10 MO/yr respectively. The dashed line is the ignition line

If the accreted rate is smaller than 3 10" 10 M 0 /yr, energy losses through the photosphere are dominant and the star cools down. If the accretion rate is higher, compressional heating is dominant and a thermal wave propagating inwards is generated. The consequence is that the inner layers are heated up, they cross the ignition line, denned by the condition that neutrino losses exactly balance the energy released by the carbon burning, in the (/), T) plane and a thermonuclear runaway starts. Nevertheless, if the initial mass of the white dwarf is M > 1.2MQ and the accretion rate is M > 5 10~8M©/3/r, the thermal wave coming from the surface has no time to reach the center and the thermonuclear runaway is entirely determined by the local properties of matter (Hernanz et al 1988). Figure 1 displays the evolution of the center of an accreting carbon-oxygen white dwarf. The three different behaviors already mentioned are clearly displayed: In case A, the thermal wave has no time to reach the central

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-10

-9

-6

log(M)

Fig. 9.2 Ignition density as a function of the mass accretion rate in the case of a white dwarf with the same characteristics as in Figure 1. The dashed line represents the minimum density necessary to have a gravitational collapse instead of a thermonuclear explosion

regions and the center follows a trajectory of slope 0.5 in the temperaturedensity plane. In case B, a strong thermal wave coming from the surface heats the material and finally induces its thermonuclear runaway. In case C, heating and surface cooling exactly balance each other after a transient phase and the white dwarf evolves isothermally. Figure 2 displays the ignition density as a function of the mass accretion rate for a 1.15 M© white dwarf. Only those accreting either at a very small rate or at a high rate have a chance to collapse. Therefore, if we take into account that some of them can be initially hotter, it can be concluded that the majority of those white white dwarfs will explode and only a small fration will collapse. Notice also that if more massive white dwarfs were considered, the ignition density would increase and the collapse would be favored.

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9.3.2 The ONeMg case White dwarfs made of ONeMg would result, in close binary systems from loss of the helium layer when it expands to read giant size during C shell burning (Iben and Tutukov 1984; Nomoto 1984) and they are not expected to form from single-star evolution (Habets 1985). In contrast, their presence in close binary systems might be indicated by the observation of Ne-novae (Starrfield 1990; Truran and Livio 1986). They should be, on average, more massive and less frequent, by a factor ~ 104, than CO white dwarfs (Iben and Tutukov 1984). When the ONeMg white dwarfs are compressed, the Fermi energy increases and nuclei undergo electron captures. The behavior of the temperature depends on the relationship: ^<X(EF

+ ET-EV-

Etht0)

(9.17)

where EF is the Fermi energy of electrons, Ey is the energy of the excited states produced during the capture process, Eu is the energy of the neutrino emiited and Eth,o is the threshold energy for the electron capture to the ground state (Miyaji et al 1980). If the Fermi energy is high enough, the derivative is positive and the temperature increases in such a way that electron captures can trigger by themselves the thermonuclear runaway. The first element starting to capture electrons is 24Mg (p ~ 4 109 g/cm 3 ), followed by 20Ne (p ~ 9.1109 g/cm 3 ) and 16O (p ~ 1.9 109 g/cm 3 ). Since electron captures produce an excess of entropy that tends to induce convection and, at the same time, a chemical gradient that tends to suppress it, the density at which the runaway actually starts is extremely sensitive to the way in which convection is handled (Mochkovitch 1984). If the Schwarzschild criterion is applied, which does not take into account the presence of chemical gradients (| V |>| Vad | where V is the actual gradient and Vad is the adiabatic gradient), the entropy generated by electron captures induces the formation of a convective zone that transports very efficiently the excess of entropy. As the captures proceed, the star gradually contracts until a density of 2 1010 g/cm3 is reached. When this happens, electron captures on 0 trigger the ignition of this element and matter is completely incinerated to 56Ni. Electron captures on 56Ni are so fast that, independently of the speed of the burning front, the white dwarf collapses to a neutron star. The Ledoux criterion takes into account the existence of chemical inhomogeneities. According to this criterion, the condition for the onset of convection becomes | V |>| VL, |, where V^, is equal to the adiabatic gradient plus a stabilizing term that depends on the gradient of the chemical

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composition (see, for instance, Cox and Giuli 1968). Consequently, convection is inhibited and strong local heating is produced. 24Mg is exhausted and the temperature drops due to thermal neutrino emission before the temperature for explosive ignition is reached, but the laster happens at the onset of electron captures on 20Ne. In this case, the ignition takes place at 9.2 109 g/cm 3 when the Takahara et al (1989) electron capture rates are adopted and the influence of the chemical potential on the electron threshold is properly taken into account. 9.3.3 The nonexplosive collapse of white dwarfs Once nuclear reactions start at the center, the burning propagates through all the star. The one-dimensional calculations made up to now assume that the flame propagates with a velocity determined by the fastest mode of burning: either spontaneous, conductive or turbulent. The detonation mode is not considered since the central density, pc > 8 109 g/cm3, of the models considered here is always very high. Except for minor differences, the behavior of the CO and ONeMg cores is always the same: burning propagates outwards, the electron captures reduce the mean electron mole number and induce the contraction of the star, the Chandrasekhar limit falls below the actual mass of the star, and finally the star collapses homologously. It should be stressed here that in the case of CO white dwarfs the ignition happens in the interior of a solid. The strength of this solid is enough (Hernanz et al 1988) to prevent or at least strongly delay the developpement of convection. However, the neutrinos emitted by the electron captures on the burned material deposit enough energy to melt the crystal (the latent heat per nucleon is / ~ kTm, where Tm is the melting temperature). The energy deposited by neutrinos in their interaction with relativistic electrons is given by (Gehrstein et al 1976; Chechetkin et al 1980):

where the units are erg/g/s, L51 is the neutrino luminosity in units of 1051 erg/s and 17 is the radius in units of 107 cm, x = E^/Ep, E,, being the energy of neutrinos and Ep the Fermi energy in MeV. To quantify this situation, Isern et al (1990) considered a burning front placed at 107 cm from the center and assumed that L51 = 0.1, Ep = 10 MeV and E,, = 9 MeV. They found that the crystal melted in less than 1 second, a quantity that is of the order of the time necessary to develop the Rayleigh-Taylor instability.

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The ejection of matter due to neutrino deposition following the collapse can be treated as a neutrino-driven wind (Hernanz et al 1993). A general description of these winds can be found in Duncan, Shapiro and Wasserman (1986) and in its relativistic form in Paczynski (1990). The equations that define the wind in its stationary form are: =M

HYM + LQO + LV = E

v2/c2)Y2

Y = yJ{\-TalT)lyl(\-v*l#)

(9.19)

(9.20)

(9.22)

(9.23)

with, T9 = 2GM/c2, H = c2 + (P + U)/p0, where Eo and Mo are the respective rates at which energy and mass are injected at the base of the wind, H is the enthalpy, po is the rest mass density, Loo is the photon luminosity measured by an observer at infinity, L is the photon luminosity in the comoving frame, Lj, is the neutrino luminosity and k is the opacity. The equation of state is that of a gas composed by radiation, nuclei and e~e+ pairs, and the energy deposited by neutrinos takes into account the captures by protons and neutrons, the scattering by electrons and the creation of e~e+. Figure 3 displays an example that is in remarkably agreement with the numerical models of Woosley and Baron (1992). The recent observations of 7-ray bursts by BATSE at the Gamma Ray Observatory have shown that these events are distributed isotropically but not uniformly in radius (Fishman et al 1991, Meegan et al 1991). These observations have opened the possibility of a cosmological origin. In this case, they must be placed at zw 1 on average and emit ~ 1051 ergs in ~ 15 s (Paczynski 1991). One of the scenarios that have been proposed is the accretion induced collapse of a white dwarf (Dar et al 1992). We have solved equations (19-23) for a set of reasonable values of the temperature and radius of the neutrinosphere. In all cases we have obtained a heavy wind characterized by: M ~ E{GM/RU)~X > E/c2, where M and R,, are the mass of the compact object and the radius of the neutrinosphere respectively. Since 7-rays can only emerge if the condition M < 10~2 (E/c2) is fulfilled (Paczynski 1990), we must conclude that the collapse of white dwarfs cannot explain the existence of 7-ray bursts (Hernanz et al 1993).

Isern & Canal: Degenerate stellar cores

-

2

-

T

-

1

o

sS, GO

o

log(r)

Fig. 9.3 Characteristics of a neutrino driven wind when Tv = 4.961010 K and Rv = 30 km. At the sonic point p, = 9.95105g/cm3, T, = 5.9109 K and R, = 210.4 km. The intensity of the wind is M = 1.161031 g/s.

9.4 The explosion of degenerate cores For a long time it has been assumed that the observational constraints that Type la supernovae should satisfy were the following ones: 1) The surfaces of their progenitors should be devoided of H at the time of explosion in order to explain the absence of Balmer lines in the spectra. 2) The progenitors should be long-lived stars in order to account for their occurrence in all types of galaxies, even the elliptical ones. 3) Their explosion should produce at least ~ 0.5 M© of 56Ni in order to account for the light curve and to explain the late time spectra. 4) Intermediate mass elements should be present in the outer layers in order to explain their spectra at maximum light. 5) The explosion should produce events with very homogeneous peak magnitudes (Miller and Branch 1990; Branch and Tamman 1992) whereas the light curve shapes and the photospheric expansion velocities might show some degree of variability.

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201

6) The abundance ratios of Fe and Ni isotopes should agree with the Solar System values after combining SNIa yields with those from gravitational collapse supernovae. 7) The death rates of the progenitors should agree with the observational estimates of the frequency of SNIa events. Mainly due to points 1), 2) and 3) it is thought that SNIa are due to the explosion of CO white dwarfs in a binary system. Point 4) strongly suggests that the burning front propagates subsonically in the inner parts of the star and that only in the very outer layers, where p < 107 g/cm3, it could propagate supersonically. The observational situation mentioned in point 5) is very puzzling. It was claimed for some time that the rate of fading and the magnitude at the peak of the light curves as well as the expansion velocities of the photosphere of Type la supernova outbursts displayed a continuous behavior, i.e. the most luminous supernovae were declining more slowly and expanding more quickly than the less luminous ones (deVaucouleurs and Pence 1976; Pskovskii 1977; Branch 1981, 1982). This was challenged by Cadonau et al (1985), who examined the shape of twelve SNI light curves in a sample of elliptical galaxies and reached the conclusion that the dispersion of the light curves was smaller than 0.3m when only photoelectric photometry was taken into account. Concerning the maximum of the light curve, Miller and Branch (1990) examined the Pskovskii's sample and found that the dispersion in maximum brightness is smaller than 0.4m if the inclinations of the galaxies are taken into account. Branch and Tammann (1992) proposed an absolute blue magnitude at maximum of MB = -19.6±0.4. Concerning the postpeak decline, it has been shown that contamination by the light of the background galaxy and the K-corrections (Boisseau and Wheeler 1991; Leibundgut et al 1991) might account for the dispersion. Nevertheless, there are clear evidences of SNIa displaying peculiar behaviors. SN1885A, in M31, was very fast (deVaucouleurs and Corwin 1985). SN1986G in NGC128 (Phillips et al 1987) seems to have been intrinsically dim, to have a low expansion velocity and a fast light curve decline. SN 1991T (Filippenko et al 1992; Ruiz-Lapuente at al 1992) seems to have been overluminous and extremely peculiar in several aspects. Finally, the existence of different expansion velocities near the maximum of the light curve has been confirmed in a number of cases (Branch 1987; Branch et al 1988; Schneider et al 1988; Philips et al 1987; Barbon et al 1990). These velocities range from 10,000 km/s in SN1986G, SN1986A and SN1989B to at least 15,000 km/s in SN1983G and SN 19841 (Branch and Tammann 1992). The most extreme case of peculiar behavior

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has been provided by SN1991bg in NGC 4374 (an elliptical galaxy in the Virgo cluster) which was clearly underluminous: at maximum light its B magnitude was ~ 2.5 m fainter and its V magnitude ~ 1,6m fainter than a normal SNIa in the same galaxy, it declined very fast after maximum and entered the nebular phase sooner than other SNIa (Filippenko et al 1992; Leibundgut et al 1993). Therefore, the question to elucidate is the following one: is there a bulk of very homogeneous events, with some "dissidents" which can be explained just by allowing some minor changes in the main parameters of the drflagration/detonation model or is it necessary to build a new scenario to account for the existence of the "anomalous cases"?. The discovery of SN1991bg seems to point towards the second alternative. Point 6) has been recently analyzed by Bravo et al (1993) and the combined yields of SNI and SNII seem to account fairly well for the observed abundances. Point 7) has recently turned out to be critical. A popular scenario for SNIa explosions involves the merging of two CO white dwarfs in a binary system due to the emission of gravitational radiation (Iben and Tutukov 1984). However, the negative results of searches for progenitor systems have raised serious objections to this scenario (Munari and Renzini 1992). This has renewed the interest in the single degenerate scenario (Wheelan and Iben 1973) where a white dwarf grows to the point of explosive ignition by accreting matter from a nondegenerate companion, typically a red giant or a supergiant, probably forming a symbiotic star. New estimates of the space density of symbiotic stars have increased their inferred numbers by a factor ~ 10 — 100. That means that the fraction of such stars that should reach the Chandrasekhar mass is just ~ 4% (Munari and Renzini 1992) or ~ 40% (Kenyon et al 1993) according to the higher and lower estimates respectively. The observation of transient hydrogen lines in at least two SNIa, provides aditional support to the symbiotic scenario. In this context, it is especially relevant the discovery of H-lines in the nebular spectrum of SNl991bg (RuizLapuente et al 1993), which can be interpreted as due to H-rich material, stripped from the companion by the kinematic interaction with the supernova ejecta, that appears as low velocity material in the late type spectra (Chugai 1986). Nevertheless, it is necessary to keep in mind the peculiar behavior of this supernova before generalizing this observational evidence. The symbiotic scenario also opens up a new and interesting possibility. After burning, the accreted hydrogen is converted into heliun and accumulates on the surface of the star. Depending on the accretion rate and on

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the initial mass of the star, helium eventualy detonates and produces an inwards shock wave whose strenght increases due to geometric effects. Two dimensional hydrodynamic simulations (Livne and Glasner 1991) shows that this shock wave ignites a detonation in the center of the CO core that completely disrupts the star. This has lead the suggestion (Woosley and Weaver 1993, Canal 1993) that the helium detonation of CO white dwarfs with initial masses in the range 0.5 - 1.3 MQ after accreting ~ 0.1 - 0.2 M© of hydrogen-rich material could be at the origin of SNIa. Therefore, the question is whether both models, central ignition or He-induced detonation of CO white dwarfs, can coexist or one of them must be eliminated. Notice that the symbiotic scenario is compatible with the central ignition model and that one of the criteria to discriminate among them is their ability to reproduce the emergent variety of SNIa events. 9.4.1 Models igniting carbon at the center This family of models assumes a mass accreting CO white dwarf in a binary system that approaches to the Chandrasekhar's mass and is partially incinerated by a subsonic burning front. A close examination of the preexplosion evolution reveals that the thermal runaway can happen at a density p in the range 2 109 < p < 1.3 1010 g/cm3, that depending on the history of the binary system (Hernanz et al 1988). The shape of the light curve and the luminosity at maximum depend on the kinetic energy as well as on the total amount of 56Ni newly synthesized and on its distribution accross the star, since 7-rays emitted by radioactive nuclei must be thermalized before escaping to contribute to the optical light curve. Both properties are affected by the total amount of electron captures undergone by the incinerated material, which depends on the ignition density, and by the total amount of matter that is completely incinerated (Graham 1987; Canal et al 1988). For a given chemical composition, there are two properties that can modify the characteristics of the light curve. One is the velocity of the burning front and the other is the density at which the thermal runaway starts. If the burning front propagates as a Chapman-Jouguet detonation, the velocity of the burning front is completely determined by the thermodynamic properties of the burned material or, equivalently, by the density of the white dwarf. If the front propagates as a deflagration, the flame velocity is not uniquely determined by the density and it can be very different from one event to another, even if all the objects have a similar structure. In this section we examine the dependence on the density of the explosion characteristics.

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The initial phases of the explosion, those encompassing from the ignition at the center to homologous expansion, have been modelled by several groups. The most critical, point as mentioned above, is the treatment of the burning front. A typical model is, for instance, that of Bravo et al (1993), who used an explicit difference scheme similar to that of Colgate and White (1966), an equation of state for the ion component taken from Ichimaru et al (1988) and for the electron component an ideal Fermi gas plus electronpositron pairs. Radiation was also included. The electron capture rates were taken from the compilation of Fuller et al (1982) and when not available there they were computed from the gross theory of /?—decay (Kodama and Takahashi 1975). The equation of state for the nuclear statistical equilibrium (NSE) material has been computed using a set of 722 nuclei with 0.39 56Ni and the rates were taken from Caughlan and Fowler (1988). NSE was assumed when T> 2 109 K or T> 5.5 109 K, for p > 710 7 g/cm3 and p > 106 g/cm3, respectively. It is also possible to obtain a rough bolometric light curve using the diffusion approximation and a finite difference scheme similar to that proposed by Falk and Arnett (1977). A flux limiter, of the form proposed by Alme and Wilson (1974), has to be used to avoid overestimate of the radiative flux in the outer layers. The equation of state is that of an ionized ideal gas plus radiation and the degree of ionization can be computed from the Saha equation (see Hoflich et al 1992 for details). The justification of the procedure only simplicity. Below 3500 K, the degree of ionization is very low. However, since the space is pervaded by energetic 7-photons, Ey ~ 1 MeV, coming from the radioactive nuclei, the degree of ionization of matter is higher than that corresponding to such temperature and the opacity is increased by several orders of magnitude. The resulting opacity is conveniently modelled by (Swartz 1991): KC = max[/csc, 1.4 lO" 1 0 ^) 1 ' 3 ] (9.24) P where KSC is the opacity obtained assuming local thermodynamic equilibrium, e is the radioactive energy locally deposed and p is the density. The sources of opacity that have to be be considered are scattering Thomson by free electrons, bound-free and free-free transitions, and the contribution from the lines modified by the expansion effects (Karp et al 1977). The total average Rosseland opacity obtained in this way lies in the range of 0.05 to

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0.1 cm 2 /g. Finally, the energy deposited by 7-photons can be handled in several ways, the simplest one being to treat them as a simple absorption process (Sutherland and Wheeler 1984) assuming a 7 opacity K~, = 0.03 cm 2 /g, which is accurate enough for the majority of purposes. The development of the Rayleigh-Taylor instability associated with the deflagration can be handled in several ways, no one being completely satisfactory. Table 1 displays the main characteristics of several models. The columns have the following meanings: />g is the density at which the central runaway starts, in units of 109 g/cm3; Mj, is the total burned mass, incinerated plus partially burned; M M is the ejected mass of radioactive Ni; K51 is the kinetic energy in units of 1051 ergs, and Mpe is the mass of 56Fe synthesized. In the models labelled R, the development of the Rayleigh-Taylor instability associated with the deflagration was computed in the way proposed by Sutherland and Wheeler (1984), taking a = 0.7. Models labelled J were designed to allow the burning front to propagate as a deflagration in the central regions and as a detonation in the outer layers. The development of the Rayleigh-Taylor instability was handled, in this case, with Unno's theory of time dependent convection (Unno, 1967). Concerning the two free parameters of the theory, the excess of temperature and the initial velocity of the "bubbles", ATo and VQ respectively, it was assumed that VQ = 0, and l r ,dT x dP

dT.

where / is the mixing length. The last choices implies that the transition regime is very short and that the steady state is attained almost instantaneously. The characteristic mixing length scale was taken to be equal to the density length scale. In all cases, when the density was p = 3 — 4 107 g/cm3 respectively, the deflagration spontaneously turned into a detonation that partially incinerated the material. The models show that despite the increase in the total burned mass, which monotonically increases with the ignition density, the total mass of Ni produced during the explosion decreases and is strongly reduced in model R8. This is due to the electron captures on the incinerated material near the center, which are more important at high densities. Models J, however, propagate the burning faster, thus reducing the time available for electron captures, and so the final amount of 56Ni is similar in all of them. The total kinetic energy dos not appreciably change with density because the increase of the burned mass is compensated by the energy losses due to electron captures and the initially greater binding energy. In the models J the total

206

Isern & Canal: Degenerate stellar cores Table 9.1. General characteristics

of the computed

models

Model

P9

M6(M0)

Mm{MQ)

K 51

M F e (M 0 )

R2 R4 R8

2.5 4.0 8.0

0.86 0.89 0.96

0.56 0.52 0.34

0.85 0.91 0.86

0.58 0.59 0.46

J2 J4 J8

2.5 4.0 8.0

1.00 1.06 1.19

0.63 0.63 0.51

1.42 1.42 1.40

0.68 0.73 0.66

Table 9.2. General characteristics of the computed light curves Model

tbol

Mio/

\ph

0o

Pi

R2 R4 R8

13 13 11

-19.23 -19.20 -18.81

10400 11000 11200

6.5 6.5 7.0

3.9 4.0 4.0

J2 J4 J8

12 12 10

-19.56 -19.54 -19.54

12424 12700 13800

7.1 7.7 8.5

3.3 3.4 3.4

amount of burned matter is very large and, consequently, the kinetic energy is also very large. Table 2 displays the parameters that characterize the models. M^,/ is the magnitude at maximum; tj, 0 / is the time in days, from the explosion to the maximum; vphot is the velocity of the photospheric layer, defined as the layer of optical depth unit, in km/s; /3o and /?i are the slopes in magnitudes per 100 days of the light curve 10 days after maximum and during the exponential tail. The most noticeable fact in Table 2 is that the magnitude at maximum is the same in all the families of models and only the case R8 displays a significant deviation from the average. The reason is twofold: in the models where the runaway starts at high densities, the destruction of 56 Ni by electron captures in the central layers is counterbalanced in part by the production of this element in the outer layers and the proximity of the radioactive material to the surface, which facilitates the scape of photons. The large amount of 56 Ni newly synthesized and the small opacities of the models translate into very bright light curves, with the exception of the R8 one. Leigbungut et al (1991) have determined that the average apparent

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207

magnitude, free from extinction, of six SNIa in the Virgo cluster is 11.92 ± 0.11. If the distance to that cluster is assumed to be 20.03 ± 3 Mpc, their absolute magnitude would be MB = -19.6 ± 0.4 and only models J, i.e. the delayed detonation models, would be fully consistent with this value. However, if a distance of 16.5 Mpc is adopted (Jacoby et al 1992), the magnitude at maximum would be MB — —19.2 and models R would be completely acceptable. Concerning the shape of the light curves and leaving aside the problem of their absolute calibration, models R perfectly fit the shape of the bolometric light curve since, on average, the observed rate of decline of the bolometric light curves is /3Q = 6 ± 0.5 and /? ~ 3.3, although their exponential tail is too step. On the contrary, models J display a first decline after maximum and an exponential tail with a slower slope than in the case R. In any case, however, the light curves of each family are very similar and differ by less than 0.5 m , which implies that the ignition density does not seem to be able, just by itself, to introduce significant changes in the shape of the light curves. The expansion velocity changes from model to model. In the family labelled R, the velocity differences are modest, less than 800 km/s. In the J models, this difference reaches 1500 km/s. In any case, these values define a range of variation that is smaller that the observed one. Due to the low opacities of the models, K ~ 0.05 — 0.1 cm 2 /g on average, the photospheres of models J are rather deep and the velocity rather low, 11,000 to 12,000 km/s during maximum. These quantities are in agreement with the observations (Branch et 1 1985), but there is a noticeable amount of matter, 0.01 M©, moving at high velocities. Because of the lack of ultraviolet observations of supernovae, it is hard to obtain any conclusion from these figures. On one hand, Harkness and Wheeler (1990) obtained from UV observations that the maximum expansion velocity was 25,000 km/s for SN1981B. On the other hand, there are several supernovae like SN1984A (Wegner and McMahan 1987), SN1983G (McCall et al 1984), and SN1990N (Leibungudt et al 1991) that display material moving at very high velocities. The nucleosynthesis can be computed with a post/processing code that uses the previously computed time evolution of the temperature and density for each shell. The most noticeable feature is the extraordinary increase of neutronized elements when the ignition density increases. This implies that the number of exploding high density white dwarfs which have contributed to the building of Solar System abundances, must had been very low in order to account for the observed values. It is also important to notice that even the models that ignite carbon at low densities produce an excess of

208

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50

Ti, 54 Cr, 54-58Fe, and 56Ni. The first two nuclei are essentially produced in the central region and their overabundances cannot be avoided if, as it actually happens, the velocity of the burning front is very small at the center. Nevertheless, the degree of acceptability of these excesses can only be ascertained in the context of a model of the chemical evolution of the galaxy and taking into account the uncertainties that characterize our knowledge of isotopic and elementary abundances in stars and in the solar system. A simplified model (Bravo et al 1993) shows that the constraints introduced by the cosmic abundances of the elements are much less restrictive than previously thought. 9.4.2 Models igniting He off-center The symbiotic scenario demands a non-violent burning of the freshly accreted hydrogen in order to effectively incorporate the newly synthesized helium onto the white dwarf. The usual limits for the behavior of hydrogen are as follow: Low rates, M < 10~9 MQyr"1, lead to a nova outburst that not only removes all the accreted matter but even erodes the original white dwarf. High accretion rates, M > 10~6 MQyr"1, lead to the formation of a red giant envelope, while intermediate rates in between these two limits lead to the formation of a common envelope. The nova limit is based on calculations that assume spherically symmetric and soft accretion. Soft means here that material is deposited at the surface at rest and with the same entropy as the underlying material. It is thus cold and becomes strongly degenerate before igniting. Most likely, however, this material will form a disk around the compact star, and the accretion process will thus include angular momentum and kinetic energy dissipation. When those effects are taken into account, the actual range of MH producing nova explosions becomes ill defined (Shaviv and Starrfield 1987, Sparks and Kutter 1987) and further research on this point is thus needed. Anyway, even in the most favorable cases, nova outbursts would limit the actual mass growth to at most 10% of MHThe development of a common envelope (either due to accretion above the Eddington limit or to formation of a red-giant envelope should induce mass-loss by the system as a whole and thus inhibit further growth of the white dwarf. Nonetheless, accretion rates in the range 10~9 < Mfj < 10~6 M©yr~x allows to convert H into He trough steady combustion or weak flashes. A further constraint is that the He layer resulting from the burning of the accreted H will explosively ignite only if it is accumulated at a rate 10~9 <

209

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e.6

1.2

Fig. 9.4 Explosion energies, in units of 1051 ergs (upper line) and B6Ni, in MQ (lower line) of detonated white dwarfs je < 5 10~8 Moyr" 1 . Above that limit it burns steadily (Nomoto 1982) and below this limit can accumulate safely depending on the parameters of the binary system. Of course, the same criterion applies if He is directly accreted from a companion (either degenerate or nondegenerate) that has previously lost its hydrogen envelope. The ignition of He at the bottom of the envelope induces the formation of two strong shock waves. The first one becomes immediately a detonation that converts He into iron-peak elements, propagates outwards and sweeps out the outer envelope. The second one propagates inwards, increases its strength because of the spherical geometry and eventually turns into a detonation (Nomoto 1982). Figure 4 displays the kinetic energies and masses of 56 Ni synthesized by the detonation of low mass white dwarfs. The models, taken from Ruiz-Lapuente et al (1993), were computed with the implicit, one dimensional hydrocode described by Canal et al (1992). The light curves displayed in Figure 5 have been obtained assuming the diffusion approximation and a constant opacity k = 0.2 cm 2 /g. It is clear that the range of

210

Isern & Canal: Degenerate stellar cores

20-

18-

O

H 14,

12H

0

20

40

60

t( days) Fig. 9.5 Light curves obtained from the detonation of CO-white dwarfs of different masses. They correspond to the detonation of MjyD = 1.2, 0.8, 0.6 M0 variability that they show is higher than that obtained for central ignitions. Nevertheless, this problem still requires further study.

9.5 Conclusions Mass accreting white dwarfs can either collapse or explode. The critical point is the density at which the thermonuclear runaway starts. In the case of white dwarfs made of ONeMg, the ignition is triggered by electron captures on neon. The density at which it happens is 9.2 1CP g/cm 3 and, despite the uncertainties on the properties of burning fronts, a collapse is almost guaranteed. The ignition density of carbon-oxygen white dwarfs depends on the parameters of the binary system (initial masses and separation) which determine the instant at which mass transfer will start, its rate and chemical composition, as well as the initial mass of the white dwarf. If the white

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211

dwarf is massive enough (M> 1.15 M©) and cool enough, the thermonuclear runaway is delayed to densities higher than 9 109 g/cm3 for both very high and very low accretion rates and a collapse ensues. In both cases, the collapse is non explosive. The energy deposited by neutrinos in the outer layers only produces a heavy wind and there is not any 7-ray signal. In the explosive case, the light curves and the photospheric velocities display some degree of variability due to the different ignition densities but that is not enough to account for the observations. One possible way out for this problem could be the induced detonation of CO white dwarfs with different initial masses. This work has been partiall financed by the DGICYT grant PB91-0060, the Spanish-French Action "Physics of white dwarfs and brown dwarfs", and the CESCA grant "Structure and evolution of galaxies".

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10 Neutron star crusts with magnetic fields D.G. YAKOVLEV Ioffe Institute of Physics and Technology, 194021 St. Petersburg, Russia

A.D. KAMINKER Ioffe Institute of Physics and Technology, 194021 St. Petersburg, Russia

Abstract The properties of plasma in neutron star crusts with strong magnetic fields B = 1010 — 1013 G are reviewed: thermodynamic properties (equation of state, entropy, specific heat), transport properties (electron thermal and electrical conductivity of degenerate electron gas, radiative thermal conductivity of very surface nondegenerate layers) and neutrino energy losses. Classical effects of electron Larmor rotation in a magnetic field are considered as well as quantum effects of the electron motion (Landau levels). The influence of the magnetic fields on density and temperature profiles in the surface layers of neutron stars and on neutron star cooling is briefly discussed. Nous presentons la revue des proprietes du plasma dans l'ecorce des etoiles neutroniques avec des champs magnetiques forts B = 1010 — 1013 G: proprietes thermodynamiques (equation d'etat, entropie, chaleur specifique), proprietes de transfer (conductivite electronique thermique et electrique du gaz electronique degenere, conductivite radiative thermique des couches non-degenees superficielles), et les pertes dues a l'energie des neutrinos. Nous examinons des effets classiques de la rotation Larmor d'un electron dans le champ magnetique, et aussi des effets quantiques (niveaux de Landau ). Nous discutons en bref l'influence des champs magnetiques sur la densite et la temperature des couches des etoiles neutroniques et sur les taux de refroidissement des etoiles neutriniques. 214

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215

10.1 Introduction Neutron stars are the densest stars known in the Universe. Their masses are M ~ 1.4M©, and radii R ~ 10 km. The mass density of matter p in neutron star cores is several times larger than the standard nuclear density, po = 2.8 X 1014 g cm" 3 . The properties of this superdense matter (equation of state, nuclear composition, etc.; see Shapiro and Teukolsky 1983) are known poorly and attract attention of many scientists. These properties cannot be reproduced in laboratory but can be studied by astrophysical methods comparing theoretical models of processes in neutron stars (e.g., cooling of an isolated star) with observational data. Neutron star cores which contain superdense matter are surrounded by envelopes (crusts) of lower density, p < po. The crusts play a key role in many processes which are observed and studied theoretically. Many neutron stars possess strong magnetic fields, B = 1010 — 1013 G. The aim of this work is to review briefly the effects of the magnetic fields in neutron star crusts. The subject is studied since the first works of Canuto and co-authors in 70-s (Canuto and Ventura 1977). 10.2 Neutron Star Crusts without Magnetic Fields Let us start with a brief description of neutron star crusts without magnetic fields. The crust extends from the the bottom of the atmosphere to the dense core; its thickness is several km (Shapiro and Teukolsky 1983). It can be divided into the outer crust, p < p^, and the inner crust p > pd, where Pd « 4 x 1011 g cm" 3 is the neutron drip density. Matter of the outer crust consists of electrons and atoms. The electrons can be nondegenerate in a very thin surface layer, and they are strongly degenerate deeper in the star. The atoms are fully ionized by the electron pressure (they are actually bare nuclei) everywhere except near the very surface. In the inner crust, free neutrons appear (in addition to the electrons and nuclei) owing to the drip from the nuclei. The neutrons are degenerate and superfluid. The superfluidity is caused by the Cooper pairing of the neutrons due to nuclear forces; corresponding critical temperature is about 108 - 1010 K (e.g., Wambach et al., 1991). The rotation of the superfluid component of matter in rotating neutron stars is realized in the form of quantized vortices which are parallel to the rotational axis. The interaction of the vortices and the nuclei (pinning and depinning) in the inner crust is thought to be responsible for pulsar glitches (e.g., Pines 1991). The properties of nuclei and free neutrons are model dependent (Baym et al 1971, Negele and Vautherin 1973) and rather uncertain. At the bottom

Yakovlev & Kaminker: Neutron star crusts with magnetic fields

216

9 8

:

T -"Fl 6 -- N

N

y'

/

?e•

N

y \

i

>

LOGt ^> , G/CM" Fig. 10.1 p-T diagram of Fe matter for B = 1012 G. 7> is the electron degeneracy temperature, 7} corresponds to F = 1, Tm is the melting temperature, Tp is the ion plasma temperature; TB is given by (8); pu is explained in Sec. 3. Lines N restrict the low-T low-/) domain of incomplete ionization and electron gas non-ideality. Dashes show the curves for B = 0 of the inner crust, the nuclei may form clusters and droplets with strongly nonspherical shapes (Lorenz et al. 1993). Below we shall mainly consider the outer crusts. The state of electrons is determined by the electron Fermi momentum PFO and 'relativistic parameter' x:

(

\ —)

1/3 .

(!)

where ne is the electron number density, p% is density in units of 106 g cm" 3 , and fie is the number of nucleons per one electron. For x •< 1 (p$ < 1) the electron gas is non-relativistic, while for x >• 1 it is relativistic. The electron degeneracy temperature Tp is (Fig. 1) TF = To(\/1 + x2 — 1), where To = mec2/ks « 5.930 X 109 K, ks is the Boltzmann constant, and m e the electron mass.

Yakovlev & Kaminker: Neutron star crusts with magnetic fields

217

The state of the ions is characterized by the ion-coupling parameter

akBT

a 0.2275^ 1 ^ 1 ,

(2)

where a = [3/(47rn,)]1/3 is the radius of ion sphere (the charge of electrons within the sphere compensates the ion charge), n,- is the ion number density, and Tg = T/(10 8 K). For simplicity, we consider one component plasma of ions. Some properties of multi-component ion mixtures in neutron star crusts are reviewed by Yakovlev and Shalybkov (1989). At sufficiently high temperatures the ions form a classical Boltzmann gas. With decreasing T, the gas gradually (without any phase transition) becomes a Coulomb liquid, and then (with a phase transition) a Coulomb crystal. The gaseous regime occurs (Hansen 1973) for F < 1 (T > T/, Fig. 1). The classical Coulomb crystal melts (Nagara et al. 1987) at F « 172 (T = T m ). The thermodynamics of strongly coupled Coulomb systems has been studied extensively by Monte Carlo and other methods (see Hansen 1973, Pollock and Hansen 1973, Hansen et al. 1977, Slattery et al. 1980, 1982, and references therein). At low T the quantum effects in ion motion (zero-point ion vibrations) become important. These effects are especially pronounced if T < Tp, where

T p =^«7.832xl0 6 (^)

K, <^ = = ^ i ,

(3)

Up is the ion plasma frequency, and m; is the ion mass. The amplitude of zero-point vibrations is commonly much smaller than the typical interion distance, a. With increasing density, the amplitude to o ratio becomes larger, and at large p zero-point vibrations can prevent crystallization (Mochkovitch and Hansen 1979, Ceperley and Alder 1980). This effect is especially important for H and He. Note that the Debye temperature of the classical Coulomb crystal is « 0.45Tp (Carr 1961). For small p and T, the effects of incomplete ionization and electron gas non-ideality become important. This domain is restricted schematically by lines N in Fig. 1. The lines correspond to max(TF,T) = |fe|, where TF is calculated in the free electron gas approximation and ee is the mean energy per electron for isolated atoms in the Thomas-Fermi approximation.

218

Yakovlev & Kaminker: Neutron star crusts with magnetic fields

10.3 Magnetic Field Effects The properties of matter in neutron star crusts can be affected by the magnetic fields. The magnetic field effects are numerous. Below we consider two major effects: the classical electron Larmor rotation, and the Landau quantization of electron motion. The classical effects are described by the Hall magnetization parameter, (4) where TQ is the effective electron relaxation time (Sec. 5.2), uB is the electron gyrofrequency, Bu = 2?/(1012G), and m* is the effective electron mass (m* = me for nonrelativistic electron gas; m* = \/ m c + (PF/C)2 for strongly degenerate gas). The classical effects are strong when 770 > 1 (electrons suffer many Larmor rotations between successive collisions). The p — T domain where the electrons are magnetized (770 ]> 1) is commonly wide. For instance the field B = 1012 G magnetizes the electrons almost for all p and T shown in Fig. 1. The quantum effects are associated with quantization of electron motion transverse to the magnetic field. If the Landau gauge of the magnetic field vector potential (A = (—By, 0,0)) is used, an electron state can be characterized (Klepikov 1954, Kaminker and Yakovlev 1981) by four quantum numbers, pz, n, s and px. In this case pz is an electron momentum along B, n = 0,1,... enumerates the Landau levels, s is the sign of the projection of the electron spin onto the momentum (5 = ±1 for n > 0; 5 = —sign(pz) for n = 0) and px determines an y-coordinate of the electron Larmor guiding center. Then the electron energy is D

£ = Jm\c* + c2pl + 2nmec2hijJB, h^B = mec2b, b = — , v &

(5)

where ug = \e\B/(mec) is the electron cyclotron frequency, and Bc = 4.414 x 1013 G is the 'relativistic' magnetic field {huge = fnec2). The number density of the free electron gas is related to the electron chemical potential \i by the equation

where / is the Fermi-Dirac distribution. The electron degeneracy temperature is TF = To(\/l + xB — 1), where the 'relativistic parameter' XQ — PFJ(mec) depends generally on B. Using (6) one can easily show that the strongly degenerate electrons (T
Yakovlev & Kaminker: Neutron star crusts with magnetic fields

219

level when density is not too high, p < PB, where ps = 2.066 x g cm" 3 . In this case 2x3

2*hne PF =

, mUB

XB

= -T7-, oo

(7)

where x is defined by (1). In the limit of p PB, strongly degenerate electrons populate many Landau levels and the Fermi energy is almost independent of B (XB « x). For further analysis, it is convenient to introduce the temperature KB

« 1.34 X I O W ^ K . m

(8)

When T ~^> TB the electrons populate many Landau levels for any p due to high thermal energy. In this case the thermal widths of the Landau levels (~ KBT) are higher than the inter-level spacing and the magnetic field acts as a non-quantizing one regardless of the electron degeneracy. In the domain of T < TB and p < ps (separated into subdomains of degenerate and nondegenerate gas) the electrons populate mostly the ground Landau level. In this domain the magnetic field acts as a strongly quantizing one and modifies essentially all the properties of matter. Finally, in the domain of T < TB and p~> PB the electrons are degenerate and populate many Landau levels but the inter-level spacing exceeds ksT. Then the magnetic field is slightly quantizing. It does not affect the bulk properties (pressure, chemical potential) which are determined by all the Fermi sea of electrons but affects quantities determined by thermal electrons near the Fermi level (heat capacity, entropy, transport properties). We will assume below that the magnetic fields do not affect motions of ions, for instance, the phonon spectrum of the crystal. This is so (e.g., Usov et al. 1980) if uBi = ZeB/(mic) < u>p, i.e. if B < l O 1 4 ^ G.

10.4 Thermodynamics of Matter in High Magnetic Fields Thermodynamic properties of plasma are affected by the magnetic fields if the fields are quantizing. We will discuss the case of strong degeneracy (T < TF). The basic approach consists in obtaining the grand potential Q = —PV

220

Yakovlev & Kaminker: Neutron star crusts with magnetic fields

Loq- 9 Fig. 10.2 Equation of state of matter at T = 0 (Rognvaldsson et al. 1993) (V is normalization volume) which generates other thermodynamic quantities (Landau and Lifshitz 1980). In the free electron gas approximation +o

P = ns

°

(9)

—00

which yields Eq. (6) since ne = dP/dfi. With increasing p (and/or decreasing 5 ) , the thermodynamic quantities show features or oscillations of van Alphen - de Haas type. The features occur when degenerate electrons populate new excited Landau levels n (because the electron density of states dp*/de possesses square root singularities at pz = 0 for these n, see (5)). The bulk thermodynamic quantities (pressure, chemical potential) are determined by all electrons (e < /i). Accordingly these quantities are greatly affected by strongly quantizing fields but slightly affected by weakly quantizing fields. The quantities determined by the electrons with energies \e — /i| < ksT show large oscillations under the action of strongly and weakly quantizing fields. These quantities are usually expressed as derivatives of the bulk quantities with respect to thermodynamic variables. The higher the derivative, the larger the oscillations. The examples of strongly oscillating quantities are: specific entropy and heat, magnetization, magnetic susceptibility, elec-

Yakovlev & Kaminker: Neutron star crusts with magnetic fields

221

U 5.0 LOG 9 , G/CM3 Fig. 10.3 Ratios of electron specific entropy S(B)/S(0) and specific heat C(B)/C(0) versus p at 5 = 1012 G for T -+ 0, T = 3 x 106 K and 107 K in the free electron gas model. If T —• 0, one has S = C and formally (in the adopted model) the curves show infinite jumps at those p at which new Landau levels are populated. The jumps displayed corresond to population of the levels n = 1, 2 and 3 tron screening length of an electric charge in a plasma (see Yakovlev 1980a, 1984; Blandford and Hernquist 1982 and the references therein). With increasing T, the oscillations become weaker which can be treated as the thermal broadening of the Landau levels. When T > TQ (and the field becomes nonquantizing) the oscillations are entirely smeared out, and the field-free results are reproduced. The free-electron approximation (9) can be inaccurate for low T at p
222

Yakovlev & Kaminker: Neutron star crusts with magnetic fields

ries.) Another approach to describe strong oscillations at low T is based on introducing some effective widths of the Landau levels caused by electron interactions. This broadening of the Landau levels was studied in the solid state physics (Kubo et al. 1965; Schoenberg 1984) and in the astrophysics (e.g., Yakovlev 1980, Hernquist 1984). In particular, the Landau levels can be broadened by the electron collisions (with width 7 ~ h/ro). Note that a self consistent theory of strong quantum oscillations does not yet exist. Fig. 2 shows the TF equation of state for cold matter. If p < ps the electrons populate the ground Landau level and the pressure is essentially lower than for B = 0: the magnetic field creates additional binding of electrons. The same effect reduces strongly the electron chemical potential fi. If B is strong, the pressure (in the TF approximation) vanishes at some finite density p8. For instance, Rognvaldsson et al. (1993) obtain pa = 942 g cm" 3 for B = 1012 G. The density pa grows with B and can be treated as the surface density of a cold (T = 0) neutron star. Fig. 3 demonstrates quantum oscillations of the electron specific entropy and specific heat. The oscillations of the electron magnetization M (and its derivatives) are even stronger (Blandford and Hernquist 1982). However broadening of the Landau levels prevents the appearance of spontaneous magnetization (the so called Landau orbital ferromagnetism, B = 4irM) under realistic conditions in neutron stars (O'Connel and Roussel 1972, Schmid-Burgk 1973).

10.5 Transport Properties 10.5.1 Electron transport in degenerate electron gas Transport properties of the degenerate gas in neutron star crusts are mainly determined by the electrons. The main electron scattering mechanisms are: (i) Coulomb scattering on ions at T > Tm; (ii) scattering on phonons at T < T m ; (iii) Coulomb scattering on charged impurities at T Tp (hightemperature phonons). In these cases the scattering is almost elastic: electron energy transfer is negligible in a collision event. For the Coulomb (c) and high-T phonon (ph) scatterings, the Fourier transforms Vq of the scattering potentials are q

2_U*6Ze*\ 2 2

" \q

+q)

h2

'

'

2

(47TZe) 392 '

K

'

where h/qs is the appropriate screening length of electric charge in a plasma (Yakovlev and Urpin 1980, Yakovlev 1980a,b; 1984), SZ = Z for case (i),

Yakovlev & Kaminker: Neutron star crusts with magnetic fields

223

6Z = Zimp — Z for case (iii) (Z, mp e being the impurity charge), (r 2 ) = 3kBTu/(miU>p) is the mean squared thermal displacement of an ion in its lattice cite at Tp < T < Tm, and u « 13 is the appropriate numerical formfactor for a bcc lattice (e.g., Yakovlev and Urpin 1980). In a magnetic field, all transport coefficients (electric and thermal conductivities a and K as well as the thermopower) are tensors. For instance the thermal conductivity tensor k is determined by the longitudinal, transverse, and Hall conductivities K||, K±, and «//, respectively. Accordingly the thermal flux density is q = — K|| V||T - K±V±T — K//h x VT, where Vy and Vx are the local temperature gradients along and across B, respectively, and h = 3/B. Sometimes the electric resistivity tensor V, = &-1 is more convenient for applications than the electric conductivity tensor a. The components of k, & and 7£ can be expressed as x2k2BTneTf 3m. '

K±

~ 3m,(l + rj2)' *" ~ 3m,(l + r,2) '

r-j , (11) v c|e|n e ' ' where r\K = WBT£ and r\o — wjgT* are the Hall parameters for thermal and electrical conduction, respectively; r.f and T» are the effective electron relaxation times for the longitudinal transport while r£ and rj[ are the effective relaxation times for the transverse transport. Thus k, a and V, are determined by four effective relaxation times T», T», r j , r£ which are generally different, and depend on T, p, and B. The Hall resistivity does not depend on the electron relaxation (under the assumed conditions). This is a nondissipative quantity which describes Hall drift of the magnetic field. The thermopower tensor is defined by two additional parameters (Yakovlev 1980a,b; 1984). 10.5.2 Nonquantizing magnetic fields Let us describe briefly the electron transport in the degenerate electron gas with nonquantizing magnetic fields (Urpin and Yakovlev 1980). In this case all four relaxation times are equal: r.p'^ = r£1
224

Yakovlev & Kaminker: Neutron star crusts with magnetic fields

discussion. For Coulomb scattering and scattering on high-temperature phonons, one has (Yakovlev and Urpin 1980), c

_

p2FvF

5.65 x 1 0 " 1 7

rPh_

where x is given by (1), vp is the electron Fermi velocity; ns = n,, x; = Z for case (i); na = nj m p , x,- = (SZ)2na/ne for case (iii) (n i m p being the impurity number density), and L is the Coulomb logarithm which is ~ 1 for the conditions in neutron star crusts. Note that L can be affected slightly by nonquantizing magnetic fields (Yakovlev 1980a). With T»'a = ro, the longitudinal conductivities are equal to those for B = 0: K|| = K0,
,—--., \ TT;-I«- ; x3

c m s K

(io^)'-1-

'

™

Moreover TZ± =
It was Lee (1950) who first expressed KO and OQ in case (i) through the Coulomb logarithm L and made an estimate of L. Later the problem was reconsidered by other authors; see Yakovlev and Urpin (1980) for the references to earlier works. Yakovlev and Urpin (1980) obtained simple analytic expression for L neglecting the electron screening in (10). This screening generally gives a small contribution except at low p (near curve N in Fig. 1). Itoh et al. (1983) recalculated L including the electron screening. Yakovlev (1987) showed a noticeable contribution to L (about 40 % for 56Fe ions at p > 106 g/cm 3 ) comes from non-Born corrections. However the Born approximation has been used so far in all other calculations. The scattering on phonons at B = 0 was also considered in a a number of works (early works are discussed by Yakovlev and Urpin 1980). Raikh and Yakovlev (1982) studied the problem for all T < Tm and pointed out that the Umklapp processes (which are most important for high T) are frozen out at T < Tu ~ TpZ1^e2/(hvF). Itoh et al. (1984,1993) reconsidered the problem ignoring the freezing but including the Debye-Waller factor (which was neglected earlier). The authors did not calculate «o and CTQ directly but used some intermediate results of Raikh and Yakovlev (1982) in an

Yakovlev & Kaminker: Neutron star crusts with magnetic fields

225

approximate manner. Accordingly the results of Itoh et al. (1984,1993) are invalid for T < T U . It would also be desirable to reconsider the problem for higher T. As follows from (11) the transverse conductivities decrease monotonically with B. The Hall conductivities increase linearly with B at small B and decrease at large B. The magnetic field strongly affects the transverse conduction when T/O > 1. If, in particular, X]Q > 1 then the transverse conductivities are greatly suppressed (K± W «||/^) due to rapid Larmor rotation of the electrons. For instance, in the nonrelativistic electron gas at B ~ 1012 G and T > Tm the Hall parameter can be as high as 770 ~ 10 2 -10 3 . Then the suppression reaches 4 - 6 orders of magnitude.

10.5.3

Quantizing magnetic fields

The theory of longitudinal and transverse transport properties of a degenerate electron gas in quantizing magnetic fields is based on the linearized relativistic kinetic equation which describes relaxation of the electron Landau states (Sec. 3) due to electron collisions. The longitudinal conductivities Ky or # (strongly quantizing field, the ground level is populated only) but T\\ is several times larger than TQ for p « PB- The oscillations are also visible when much higher Landau levels are populated (p ^ PB)- The oscillations are affected by the thermal broadening of the Landau levels as well as by other broadening mechanisms. Generally, the relaxation times r,f and r,f are different and the Wiedemann-Franz rule does not hold. However one has r.j4 = T» in the limit of T —* 0. When the temperature grows and reaches TB, the oscillations are smeared out and

226

Yakovlev & Kaminker: Neutron star crusts with magnetic fields

Fig. 10.4 Quantum oscillations of T« and r,f versus p/hug II

for Coulomb

ll

scattering in a nonrelativistic electron gas with 56Fe ions at very strong (kBT/fi •— 0) and finite (kBT/fi = 1/40) degeneracy (Yakovlev 1980a). The curves are self-similar as long as B < 4x lCr3 G; n is the nonrelativistic chemical potential. Population of the Landau levels n = 1 - 7 is depicted. Population of the level n = 1 is shown separately (top right)

the nonquantizing regime is restored, r,f1
Yakovlev & Kaminker: Neutron star crusts with magnetic fields

227

Fig. 10.5 Same as in Fig. 4 for quantum oscillations of r j and r£ (Yakovlev 1980b). The width of the Landau levels assumed in the calculation is 7 = 0.1JbBT

appropriate electron scattering mechanisms. The general expressions for the transverse conductivities in quantizing magnetic fields were obtained by Yakovlev (1980b) and Kaminker and Yakovlev (1981). The expressions include integrals over electron energies which contain logarithmic-type divergencies. The divergencies are removed by introducing the effective widths of the Landau levels (Sec. 4). The widths may be caused by collisional broadening, by some inelasticity of electron scattering and/or by deviations from the Born approximation. The calculations of the transverse conductivities were carried out by Yakovlev (1980b), and, in more detail, by Hernquist (1984) and Schaaf (1988). The transverse effective relaxation times r£ and r j suffer powerful quantum oscillations (Fig. 5) in the quantizing fields (T < TB). When the temperature increases to TB, the oscillations are completely smeared out

228

Yakovlev & Kaminker: Neutron star crusts with magnetic fields

by the thermal broadenenig of the Landau levels, and the nonquantizing results are reproduced, r,?>
where a is the Stefan constant, l\\t± is the generalized Rosseland mean free path of photons along or across B (Fy = 2 cos2 6, F± = sin2 0); / = l(u>, 0) is the mean free path of photons of frequency u; = zkBT/h averaged over ordinary and extraordinary photon modes which can propagate in a magnetized plasma, and d is angle between propagation direction and B. Pavlov and Yakovlev (1977) and Silant'ev and Yakovlev (1980) calculated K \\,± f° r * n e c a s e s when the photon mean free path is determined by free-free transitions and/or Thomson scattering. The effect of the magnetic field on the radiative conductivity is described by the parameter/? = HUJBK^BT) — TB/T (see (8)). If T > TB (P < 1), the field is nonquantizing and its effect is weak, «|| « K± « KQ. If T < TB (/? > 1), the magnetic field is quantizing, and the thermal conductivity is anisotropic and strongly enhanced by the magnetic field, K|| J_ ~ KO/32. The enhancement occurs because the main heat carriers at (3 ^> 1 are extraordinary mode photons with frequencies u> •< u g . The mean free path of these photons is a factor of (U;B/U>)2 larger than at B = 0. The enhancement K a /?2 for (3 > 1 was first outlined by Tsuruta et al. (1972) and Lodenquai et al. (1974) using a simplified approach.

Yakovlev & Kaminker: Neutron star crusts with magnetic fields

229

An important fact is that K± becomes larger than «|| in high magnetic fields. For instance, in the case of the Thomson scattering at /? > 1 one obtains K± « 2«|| « (32/(2n2). These asymptotes are valid for large /? while for intermediate ft Silant'ev and Yakovlev (1980) got analytic fits

where «o = 7.5 x 1020T^p/fie erg cm" 1 s" 1 K" 1 , and p is expressed in g cm" 3 . For example, consider a helium plasma with T = 4.5 X 10s K, and B = 2 x 1012 G. The Thomson scattering dominates at p < 1 g cm" 3 . In the latter case K|j « 100«o and n± « 180KO- If P > 1 g cm" 3 , free-free transitions dominate, and the conductivity enhancement is weaker, KJJ « 31/co and KJ. w 21KO (now with respect to the free-free conductivity «o for the same T and p). At very low p the radiative thermal conductivity is affected by electron - positron vacuum polarization in strong magnetic fields (Pavlov and Yakovlev 1982). 10.6 Neutrino energy losses in magnetized crusts Neutron stars become transparent to neutrinos in half a minute after their birth. Neutrino generation produces powerful energy losses in the cores and crusts of young neutron stars. The main neutrino production mechanisms in the crusts are: electron bremsstrahlung on atomic nuclei (e + Z —• e + Z + v + F), electron-positron annihilation (e~ + e"*" —• u + V), plasmon decay (hu>pe —• v + 17), and photon decay (7 + e —• e + v + V). These

processes have been studied in detail for 5 = 0 since the classic works of Beaudet et al. (1967) and Festa and Ruderman (1969) (see Itoh et al. 1989 for the references to further works). All processes can be affected by magnetic fields. Moreover magneticfieldslead to a new process, synchrotron radiation of neutrino pairs by electrons (e —• e + v +F), which is forbidden by momentum and energy conservation for B = 0. Neutrino synchrotron radiation was first studied by Landstreet (1967) but rather qualitatively (see Kaminker et al. 1992a, for critical remarks). So far two processes have been investigated in strong magnetic fields: the neutrino synchrotron and pair-annihilation radiations. These processes are described by similar matrix elements. According to the Weinberg - Salam theory both processes proceed via charged and neutral currents, and hence all neutrino flavors can be generated. For temperatures T <; Mjyc2 ~ 100 GeV (M\y being the mass of the intermediate boson), the matrix element

230

Yakovlev & Kaminker: Neutron star crusts with magnetic fields

is described by a 'four-tail' diagram. General (but complicated) expressions for the neutrino energy loss rates Qs and Qp [erg cm" 3 s"1] in synchrotron and pair annihilation processes were obtained by Kaminker et al. (1992a) using a formalism for relativistic electrons in quantizing magnetic fields. Kaminker et al. (1992a) also calculated Q3 and Qp from the general expressions for a nonrelativistic (degenerate and nondegenerate) plasma with strong fields. Kaminker et al. (1991) investigated Qa in a relativistic degenerate electron gas with nonquantizing magnetic fields. Kaminker and Yakovlev (1993) performed a similar investigation for a nondegenerate relativistic electron gas. Kaminker et al. (1992b) examined Qp for a hot nondegenerate relativistic plasma with high magnetic fields. The main results are as follows. The pair annihilation energy loss Qp is usually unaffected by the magnetic fields in the cases when Qp contributes significantly to the total neutrino energy losses. However the magnetic field does influence Qp if B ^ 1014 G and/or if B is quantizing (Sec. 3). In the latter case the temperature is not too high (T •< TB) and annihilation is generally ineffective due to the small number density of positrons. Synchrotron energy losses Qa can be quite large in a relativistic electron gas (Fig. 6). One should especially mention a wide domain of the relativistic degenerate electron gas with nonquantizing magnetic fields in which TB C T < TF and T < TpB, where TpB = l.bhuBx2/kB « 2.0 X 10 8 5i 2 x 2 K (a; is given by (1)). According to Kaminker et al. (1991) in this domain Qa = 0.00126

*

g

(kBTf

« 8.97 X 10145?3r95 - ^ ,

(16)

where G is the Fermi weak interaction constant; Qa includes generation of ve, i/fj. and vT. As follows from (16) Qa depends neither on the electron mass nor on the electron Fermi momentum (i.e. on p). Accordingly all charged relativistic degenerate fermions emit synchrotron neutrino pairs at the same rate, provided the assumed conditions are satisfied. For electrons, this regime occurs in a very wide domain of p, T and B. A similar regime is possible in the hot relativistic plasma (Kaminker and Yakovlev 1993). A comparison of the annihilation and synchrotron neutrino energy losses shows that the annihilation process dominates in a hot and not too dense plasma, while the synchrotron process dominates at higher p and large B. According to the estimates, the synchrotron process may be the most effective neutrino production process in neutron star matter for T ~ 109 K, B > 1013 G, and p ~ 107 - 108 g cm" 3 . Further work is required for studying other neutrino generation mechanisms in strong magnetic fields.

231

Yakovlev & Kaminker: Neutron star crusts with magnetic fields

To

20

o Or

;

T=10 K — —'

oi \5

•

-

PL_

_

-£^—

<-_

__. •—•<_ j

•

•

s

"" S2

\

—

UJ

S1

-— 3 io

1—

~i—

1

9

\

—^

? 1

8

11

Fig. 10.6 Neutrino energy loss rates versus p/fie for T = 109 K. Solid lines show synchrotron (s) and pair annihilation (p) losses at B = 1012, 1013 and 1014 G (curves 1, 2, and 3, respectively). Dashes display bremsstrahlung (6) (Soyeur and Brown 1979, for 56Fe ions), plasmon decay (pi) and photon decay (7) losses (Itoh et al. 1989) for B — 0. The pair annihilation is much more powerful for higher T

10.7 Structure and thermal evolution of magnetized crusts The above results are important for modeling the thermal structure of the surface layers of neutron stars with strong magnetic fields. The modeling is required for investigating cooling and thermal surface radiation (spectra, beaming, polarization) of isolated neutron stars. The cooling and radiation theories are interrelated; they are used for interpretation of observations of thermal radiation from some neutron stars. Magnetic fields affect most strongly the atmosphere and outermost crust layers of neutron stars. Of particular importance is the layer of the crust across which the bulk of the temperature drop between the interior of the star and the atmosphere occurs. The latter layer is thick (up to several meters) in hot neutron stars whose effective surface temperature Te is higher than some 106 K. However the layer is much thinner for cooler stars. The magnetic field becomes important provided it affects this layer. In the latter case the temperature distribution is not spherically symmetric within the isolating layer due to anisotropic heat transport. The atmosphere acts as a transmitter: it creates spectral, angular and polarization features of the

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Yakovlev & Kaminker: Neutron star crusts with magnetic fields

outgoing radiation which passes through the atmosphere from the crust. In particular, the radiation spectrum may deviate noticeably from the blackbody one (Romani 1987). The strongest effects of the magnetic fields occur in the atmosphere. Although we do not consider atmospheres in this paper we note that huge magnetic fields distort the structure of atoms, ions and molecules in atmospheric layers (see, e.g., Lai et al. 1992, Pavlov and Meszaros 1993, and references therein). The distortion takes place particularly by electric fields induced in the center-of-mass frames of particles moving across B. The magnetic fields influence the ionization equilibrium, equation of state, other thermodynamic properties of matter, and the radiative spectral opacities. Radiation transfer in magnetized atmospheres (Shibanov et al. 1992) differs significantly from that in non-magnetic atmospheres. Deeper in the crust, magnetic field effects become weaker. The full self-consistent theory of magnetized atmospheres and deeper surface layers of neutron stars has not yet been developed. It should take into account the effects of magneticfieldson the equation of state and anisotropic thermal conductivities of crust layers. It is most likely that the thermal conductivity is unable to keep the heat balance, and additional slow flows of matter of the meridional circulation type (e.g., Schwarzschild 1958) are generated. The circulation may drag the magnetic field, distort its configuration and induce magnetic forces sufficiently large that they are important in the balance of forces. The solution of these problems should yield the temperature variation at the bottom of the atmosphere, which provides the necessary boundary conditions for the radiative transfer problem in the atmosphere. So far the above problems have been solved in a simplified manner. Magnetic fields have been considered as force-free, although their effects on the equation of state (Sec. 4) have been included. Possible circulation flows and magnetic forces have been neglected. The density profiles in cold (T = 0) neutron star crusts with magnetic fields have been calculated recently by Rognvaldsson et al. (1993) on the basis of the TF equation of state (Sec. 4, Fig. 2). The equation of hydrostatic equilibrium has been integrated for constant magnetic field in the plane-parallel approximation. The binding of matter introduced by strong magnetic fields (p < PB) leads to higher surface densities. In deep layers, P ^ PBi the density profiles tend to those for B — 0. The temperature profiles in magnetized crusts were analyzed by Hernquist (1985), using the plane-parallel approximation. Later the most extensive studies of temperature profiles in a crust with constant vertical magnetic

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Fig. 10.7 Density profiles versus depth z (measured from the surface) in a cold (T = 0) neutron star crust with magnetic field (Rognvaldsson et al. 1993); 514 is surface gravity in units of 10 14 cm s~ 2

field were performed by Van Riper (1988), again in the plane-parallel approach. In this case heat is transported by the longitudinal (electron and radiative) conductivity K||. The effects of the magnetic fields on the equation of state were included. The relation between the internal temperature T, and the effective surface temperature Te obtained by Van Riper (1988) is shown in Fig. 8. In hot crusts (Te > 106 K), the isolating surface layer is rather thick (p < 106 g cm" 3 ) and the relation is slightly affected by the magnetic fields. Accordingly the Te(T{) dependence appears to be close to the field-free one (Urpin and Yakovlev 1979, Gudmundsson et al. 1983, Hernquist and Applegate 1984). For cooler stars (Te < 106 K), the isolating layer shifts to lower densities. It becomes thinner and field dependent. For a given T,-, Te grows with B. The growth occurs because the thermal isolation is weaker when B is larger. First, the low-density matter is 'removed' by the magnetic binding (Fig. 7); second the radiative thermal conductivity is enhanced (Sec. 5.4) making the isolation poorer. A two-dimensional analysis of thermal conduction in the surface layers of a neutron star has been performed by Schaaf (1990). The magnetic field has been assumed constant both in magnitude and direction in the layers under study. The anisotropic thermal conductivity has been taken into account. Many simplifications introduced in the calculation make the results rather

234

Yakovlev & Kaminker: Neutron star crusts with magnetic fields

5 6

LOG

BIG] = 14 .

Q)

H o

/

y Ky y,? 8 , K

Fig. 10.8 Effective surface temperature Te (not redshifted) versus internal temperature 7} for a neutron star with surface gravity g = 10 14 cm s~ 2 at several values of B (from Fig. 29 of Van Riper 1988)

indecisive. However the main conclusions seem to be qualitatively correct. The magnetic field affects the surface temperature distribution if the star is sufficiently cool (as discussed above). The surface temperature at the magnetic equator appears to be lower than at the pole. This is mainly because the thermal isolation of the stellar interior is stronger at the equator the heat propagates across B in the equatorial layers, and the transverse thermal conductivity of degenerate electrons is strongly suppressed by high fields (Sec. 5). For instance, according to Schaaf (1990) the surface temperature at the equator of the star with B — 1011 G (and g = 1014 cm s~2) is about half of that at the pole provided the pole temperature is about 4 x 105 K. Currently the problem is being reconsidered by Van Riper and Miralles (private communication). The magnetic fields in the neutron star crust influence cooling (Fig. 9), especially when the star is sufficiently cold (Te < 106 K). Detailed cooling calculations of magnetized neutron stars have been carried out by Van Riper (1991) using the Te(Ti) relationship obtained by Van Riper (1988). The magnetic field leads to higher surface temperatures Te(t) during the neutrino cooling stage (t < 105 — 106 yrs) when the neutrino luminosity is larger than the photon surface luminosity. At this stage, the internal stellar

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6 -

8

t , 1R.S Fig. 10.9 Decrease of the effective surface temperature (redshifted) with time for a neutron star (Friedman - Pandharipande equation of state in dense core) with standard neutrino luminosity («) or with the luminosity enhanced (e) by the presence of quark matter in stellar interior (from Fig. 10 of Van Riper 1991)

temperature is controlled by the neutrino emission and does not depend on the surface magnetic field. Accordingly Te is larger if B is stronger. Later the star cools via photon surface emission. The magnetic field reduces thermal isolation of the stellar interior. This accelerates photon cooling and leads to a lower Te{t) (Fig. 9). If some enhanced neutrino production mechanism, such as neutrino emission by free quarks or the direct URCA process in a hyperonic core, operates in the stellar interior (see Pethick 1993 and references therein), the effect of the field on cooling is naturally more pronounced.

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10.8 Conclusions Recent results of the ROSAT X-ray observatory (Finley et al. 1992, Becker et al. 1993, Anderson et al. 1993) indicate that surface thermal radiation has most likely been detected from at least five neutron stars: PSR 0656+14, PSR 1929+10, PSR 1055-52, Geminga, and the Vela pulsar. All these objects are known to possess large magnetic fields. The detected radiation contains information on neutron star masses, radii, magnetic fields, as well on dense matter in stellar interiors: on the equation of state, and the absence or presence of enhanced cooling mechanisms, etc. So far theoretical interpretations of the observations are rather uncertain. The effective surface temperatures of the neutron stars appear to be model dependent, but they do not deviate significantly from those predicted by standard cooling theories (without enhanced neutrino luminosity). The exception is PSR 0656+14 (t « 105 yrs) whose surface temperature seems to be lower than the standard one but higher than that predicted by the fully allowed enhanced cooling (Te = (4 - 8) X 105 K, for most realistic theoretical fits of the radiation spectra detected, e.g. Anderson et al. 1993). In this case, one possibility is that cooling is enhanced by the direct URCA process in the stellar core but the enhancement is partly suppressed by nucleon superfluidity (e.g., Page and Applegate 1992). More precise observations and more elaborate theories are required to arrive at definite conclusions. Future theoretical studies of neutron star crusts with strong magnetic fields should be based on the theoretical results described above. However there are still many theoretical problems to be solved. The most important of them are concerned with the effects of magnetic fields in the atmospheres, and the surface layers of neutron crusts as discussed in Sec. 7. The above results are also important for analyzing generation and evolution of magnetic fields in neutron stars. These problems are not discussed in the present work (see, e.g., Blandford et al. 1983, Urpin et al. 1986, Urpin and Van Riper 1993, and the references therein). We are grateful to Yu.A. Shibanov and K. Van Riper for discussions and comments. We are also extremely thankful to C. Pethick for careful reading of the manuscript and many important critical remarks. This work was supported partly by Russian Foundation of Fundamental Researches, grant No. 93-02-2916.

References Anderson S.B., Cordova F.A., Pavlov G.G., Robinson C.R., and Thompson R.J., Ap. J. 414, 867 (1993)

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Beaudet G., Petrosian V. and Salpeter E.E., Ap. J. 150, 979 (1967) Baym G., Pethick C.J. and Sutherland P., Ap. J. 170, 299 (1971) Becker W., Triimper J. and Ogelman H., in: Isolated Pulsars, K. Van Riper, R. Epstein, C. Ho eds., Cambridge: Cambridge Univ. Press, 104 (1993) Blandford R.D. and Hernquist L., /. Phys. C 15, 6233 (1982) Blandford R.D., Applegate J.H. and Hernquist L., MNRAS 204, 1025 (1983) Canuto V. and Ventura J., Fundam. Cosmic Phys. 2, 203 (1977) Carr W.J., Phys. Rev. 122, 1437 (1961) Ceperley D.M. and Alder B.J., Phys. Rev. Lett. 45, 566 (1980) Festa G.G. and Ruderman M.A., Phys. Rev. 180, 1227 (1969) Finley, J.P., Ogelman, H., Kiziloglu, U., Ap. J. 394, L21 (1992) Gudmundsson E.H., Pethick C.J. and Epstein R.I., Ap. J. 272, 286 (1983) Hansen J.-P., Phys. Rev. A8, 3096 (1973) Hansen J.-P., Torrie G.M. and Vieillefosse P., Phys. Rev. A16, 2153 (1977) Hernquist L., Ap. J. Suppl. 56, 325 (1984) Hernquist L., MNRAS 213, 313 (1985) Hernquist L. and Applegate J.H., Ap. J. 287, 224 (1984) Itoh N., Mitake S., Iyetomi H., and Ichimaru S., Ap. J. 273, 774 (1983) Itoh N., Kohyama Y., Matsumoto N., and Seki M., Ap. J. 285, 758 (1984); erratum 404, 418 Itoh N., Adachi T., Nakagawa M., Kohyama Y., and Munakata H., Ap. J. 339, 354 (1989) Itoh N., Hayashi H. and Kohyama Y., Ap. J. submitted (1993) Kaminker A.D. and Yakovlev D.G., Teor. Mat. Fiz. 49, 248 (1981) Kaminker A.D. and Yakovlev D.G., Zh. Eksper. Teor. Fiz. 103, 283 (1993) Kaminker A.D., Levenfish K.P. and Yakovlev D.G., Pisma Astron. Zh. 17, 1090 (1991) Kaminker A.D., Levenfish K.P., Yakovlev D.G., Amsterdamski P., and Haensel P., Phys. Rev. D46, 3256 (1992a) Kaminker A.D., Gnedin O.Yu., Yakovlev D.G., Haensel P., and Amsterdamski P., Phys. Rev. D46, 4133 (1992b) Klepikov N.P., Zh. Eksper. Teor. Fiz. 26, 19 (1954) Kubo R., Miyake S. and Hashitsume N., Solid State Phys. 17, 269 (1965) Lai D., Salpeter E. and Shapiro S.L., Phys. Rev. A45, 4832 (1992) Landau L.D. and Lifshitz E.M., Statistical Physics, Part I, Pergamon: Oxford (1980) Landstreet J.D., Phys. Rev. 153, 1372 (1967) Lee T.D., Ap. J. I l l , 625 (1950) Lodenquai J., Canuto V., Ruderman M. and Tsuruta S., Ap. J. 190, 141 (1974) Lorenz C.P., Ravenhall D.G. and Pethick C.J., Phys. Rev. Lett. 70, 379 (1993) O'Connell R.F. and Roussel K.M., Astron. Astrophys. 18, 198 (1972) Mochkovitch R. and Hansen J.-P., Phys. Lett. A73, 35 (1979) Nagara H., Nagata Y. and NakamuraT., Phys. Rev. A36, 1859 (1987) Negele J.W. and Vautherin D., Nucl. Phys. A207, 298 (1973) Page D. and Applegate J.H., Ap. J. 394, L17 (1992) Pavlov D.G. and Meszaros P., Ap. J. submitted (1993) Pavlov D.G. and Yakovlev D.G., Astrofizika 13, 173 (1977) Pavlov D.G. and Yakovlev D.G., Astrofizika 18, 119 (1982) Pethick C.J., Rev. Mod. Phys. 64, 1133 (1992)

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Pines D., in: Neutron Stars: Theory and Observation, J. Ventura and D. Pines eds., p. 57, Kluwer Acad. Publ.: Dordrecht (1991) Pollock E.L. and Hansen J.-P., Phys. Rev. A8, 3110 (1973) Raikh M.E. and Yakovlev D.G., Astrophys. Space Sci. 87, 193 (1982) Rognvaldsson O.E., Fushiki I., Gudmundsson E.H., Pethick C.J., and Yngvason J., Preprint No. 9S/9A Nordita, Copenhagen (1993); Ap. J., in press Romani R., Ap. J. 331, 718 (1987) Schaaf M.E., Asiron. Astrophys. 205, 335 (1988) Schaaf M.E., Asiron. Astrophys. 235, 499 (1990) Schmid-Burgk J., Astron. Astrophys. 26, 335 (1973) Schoenberg D., Magnetic Oscillations in Metals Cambridge Univ. Press: Cambridge (1984) Schwarzschild M., Structure and Evolution of the Stars, Princeton University Press: Princeton (1958) Shapiro S.L. and Teukolsky S.A., Black Holes, White Dwarfs, and Neutron Stars, Wiley - Interscience: New York (1983) Shibanov Yu.A., Zavlin V.E., Pavlov G.G., and Ventura J., Astron. Astrophys. 266, 313 (1992) Silant'ev N.A. and Yakovlev D.G., Astrophys. Space Sci. 71, 45 (1980) Slattery W.L., Doolen G.D. and DeWitt H.E., Phys. Rev. A21, 2087 (1980) Slattery W.L., Doolen G.D. and DeWitt H.E., Phys. Rev. A26, 2255 (1982) Soyeur M. and Brown G.E., Nucl. Phys. A324, 464 (1979) Tsuruta S., Canuto V., Lodenquai J., and Ruderman M., Ap. J. 176, 739 (1972) Urpin V.A. and Van Riper K., Ap. J. 411, L87 (1993) Urpin V.A. and Yakovlev D.G., Astrofizika 15, 647 (1979) Urpin V.A. and Yakovlev D.G., Sov. Astron. 24, 425 (1980) Urpin V.A., Levshakov S.A. and Yakovlev D.G., MNRAS 219, 703 (1986) Usov N.A., Grebenschchikov Yu.B. and Ulinich F.R., Zh. Eksper. Teor. Fiz. 78, 296 (1980) Van Riper K., Ap. J. 329, 339 (1988) Van Riper K., Ap. J. Suppl. 75, 449 (1991) Wambach J., Ainsworth T.L. and Pines D., in: Neutron Stars: Theory and Observation, J. Ventura and D. Pines eds., p. 37, Kluwer Acad. Publ.: Dordrecht (1991) Yakovlev D.G., Preprint No. 678, Ioffe Phys. Techn. Inst., Leningrad (1980a) Yakovlev D.G., Preprint No. 679, Ioffe Phys. Techn. Inst., Leningrad (1980b) Yakovlev D.G., Astrophys. Space Sci. 98, 37 (1984) Yakovlev D.G., Astron. Zh. 64, 661 (1987) Yakovlev D.G. and Shalybkov D.A., Sov. Sci. Rev. / Sec. E, R.A. Syunyaev ed., 7, 311 (1989) Yakovlev D.G. and Urpin V.A., Sov. Astron. 24, 303 (1980) Zyrianov P.S. and Klinger M.I., Quantum Theory of Electron Transport Phenomena in Crystalline Semiconductors , Nauka: Moscow (1976)

11

High pressure experiments for Astrophysics. P.LOUBEYRE Laboratoire PMC, Universite Paris 6, boite 77, 4 place Jussieu 75252 Paris. France.

Abstract In the past decade, measurements of the properties of Hj and He systems at very high pressures have made great progress, now reaching density at the limit of the plasma phase transition of hydrogen. The potentialities and limits of static and dynamic methods will be reviewed. Then, a survey of the major experimental results is presented. It is the intention of this article to show how these measurements can bring information to model low-mass astrophysical objects. Three levels of usefulness are distinguished on selected examples: data for codes of planetary interiors, constraints for theoretical descriptions of dense matter, observations of unsuspected properties at very high density.

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Abstract De grands progres ont ete faits ces dix dernieres annees dans la mesure des proprietes des systemes d*H2 et d*He sous tres fortes pressions. Des densites a la limite de la transition de phase plasma de lTiydrogene peuvent maintenant etre obtenues en laboratoire. Les possibility's et limites des methodes dynamiques et statiques seront tout d'abord discutees. Ensuite, les principaux resultats experimentaux seront presentes. Le but de cet article est de montrer comment ces etudes peuvent etre utiles a la modelisation des interieurs planetaires. Trois niveaux d'application seront degages: donnees pour les codes de structures internes; contraintes pour valider les descriptions theoriques; mise en evidence a tres haute densite de comportements inhabituels .

11.1 Introduction The properties of matter at very high density are central to our understanding of planetary interiors. Reliable calculations are difficult, even for the two simplest elements hydrogen and helium. On the other hand, high pressure properties are measured in laboratory by static and dynamic methods. Are these experiments useful for the models of interiors of low-mass astrophysical objects? The importance of high pressure experiments is fully acknowledge in geophysics (Manghani 1987). Remarkably, measurements of geophysical compounds can now be made under conditions that simulate those found within the deep interior of earth ( believed to be close to 350 GPa and 6000 K). In doing so, important issues can be set, such as the melting curve of iron that constrains the temperature of the core (Jeanloz 1990). In contrast, high pressure measurements are generally considered weakly relevant for astrophysics. Certainly, the PT conditions at the centre of giant planets are far from experimental reach. However, in the past decade few laboratories have been very active in measuring the properties of H2 and He at very high pressures. In ten years a factor of 100 has been achieved in the maximum static pressure. Multiple shock-wave measurements have been also performed on H2 and He. Now, the properties of these systems can be measured up to densities at the limit of the plasma phase transition of hydrogen. It is the intention of this article to show how these measurements can give information to model giant planets and brown dwarfs.

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This article is organized into three major sections. In section 2, the potentialities and the limits of dynamic and of static methods will be presented and compared. Section 3 is a survey of the recent measurements of the properties of hydrogen and helium at very high pressures. The search for metal hydrogen, that has been the focus of a great activity, will be discussed in details. In section 4, three levels of usefulness for astrophysics are presented with selected examples, namely: Input data for codes of planetary interiors. Test for calculations of astrophysical matter. Observation of unsuspected properties at very high density. Finally, an outlook of this rapidly growing field will be tentatively given.

11.2 Experimental methods Experiments on systems of hydrogen and helium at very high pressures present the maximum difficulties. A conjunction of three factors makes their compression and characterization difficult: A very high compressibility. The volume of the initial state in static or dynamic methods ( almost equal to the one of the low temperature liquid at ambient pressure) has to be reduced by a factor 10 and 15, respectively in hydrogen and in helium so as to reach 100 GPa. In dynamic studies, this causes a large increase of temperature along the Hugoniot that prevents compression further. In static studies, this is associated with a large deformation of the sample chamber, lessening its rigidity. Low optical tensor. Various optical techniques are used to characterize systems at very high pressures: X ray diffraction, Raman diffusion, infrared absorption... Because the impinging light acts on the electronic density, optical tensors are weak for these low Z elements. Methods of investigation have to be specially adapted, such as the single crystal x ray diffraction with a synchrotron source (Mao 1988). Reactivity ofhydrogen. Due to hydrogen embrittlement, special containers have to be used. The reactivity of hydrogen increases with pressure and its confinement at very high pressures is a problem: An attack of the diamond anvils is suspected around 200 GPa. Due to these difficulties, hydrogen and helium were not the first systems studied under pressure despite their fundamental interest and their importance for astrophysics. Nevertheless, it should be noted that their demanding pressurization has had a feed-back improvement on the high

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pressure techniques. 11.2.1 Dynamic methods: shock-waves. Shock-wave research has been developed after world war II, independently in the United States and in Russia. Shock-wave experiments create a disturbance propagating at supersonic speed in a material, preceded by an extremely rapid rise in pressure, density and temperature. Although irreversible, the process is well-understood and can be controlled to produce a desired response, as reviewed elsewhere (Zeldovich and Raizer 1966, Ross 1985). The measurement of two dynamic variables, the shock front velocity, vs, and the perturbation front velocity, vp, is sufficient to determine the thermodynamic state of the compressed system. The pressure, volume and entropy (P, V, E) behind the shock front are related to initial properties and the dynamic variables through the Rankine-Hugoniot equations that expresses conservation of mass, momentum and energy. The shock process is highly irreversible and is accompanied by a large increase in temperature. In He and in Hj, this limits the compression. The resulting P(V) curve is known as the Hugoniot. The determination of the shock pressure is absolute. A considerable variation in the path of the Hugoniot can be achieved with double or multiple shock experiments for which the primary shock is passed through the sample and reflected from a high impedance anvil. The reflected wave then compresses the already compressed material to a higher pressure. At the same density, the temperatures and pressures along the reflected Hugoniot are much lower than those reached by the single shock and tend toward the isentrope.

barrel

Two-stage light-gas gun

Schematic of a ahock wave traveling

Figure 1: schematic of two-stage light-gas gun and shock-wave traveling.

Shock-wave measurements have been carried out on most elements and on several compounds. Experiments on H2 and He were all carried at Lawrence Livermore National Laboratory, mostly after 1983. Shockwaves were generated by the impact of a planar projectile into cryogenic specimen holders. Maximum pressures of 21 GPa (T=4800 K), 76 GPa (T=7000 K) and 120 GPa (T=3000 K) have been achieved on liquid D2

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through respectively simple, double and multiple shocks ( Nellis 1983, Weir 1993). Maximum pressures of 16 GPa (T=12000 K) and 56 GPa (T=21000 K) have been achieved on He through respectively single and double shocks (Nellis 1984). Isentropic compression experiments have also been reported on hydrogen at pressures above 100 GPa in a magnetic flux compression device (Hawke 1978). The volume was measured from flash radiograph of the radii of the sample chamber. Pressures and temperatures were then calculated from a magneto-hydrodynamic code. Such experiments have been designed for conductivity measurements at ultra-high pressures. Hydrogen and deuterium have been compressed by this method to density above lg/cm3, where a conducting behaviour was detected. Although higher densities can be achieved by magnetic compression than with impact shock-wave, the characterization of the thermodynamic state and of the physical properties of the sample is unfortunately quite primitive.

11.2.2 Static methods. A revolution, the DAC. Before 1979, high pressure studies on Hj and He were performed in massive apparatuses, technically very demanding, relatively expensive and dangerous to operate. The highest pressures, about 2 GPa, were obtained in the so-called piston-displacement technique. Measurements of the equation of state, melting curve and sound velocity could be done then. In 1979, for the first time, H, (Mao 1979) and He (Besson 1979) were loaded in a diamond anvil cell (DAC). Their melting point at room temperature was measured respectively at 5.2 GPa and 11.6 GPa. Today, pressures above 200 GPa can be generated on H2 and on He in a DAC that can fit into the palm of the hand. The diamond anvil cell was first designed in 1959. In the history of its evolution that spans over a period of 20 years before He and H2 could be loaded, important innovations are to be recognized: The introduction of the metal gasket technique for confining fluid sample; The ruby fluorescence technique for the pressure calibration; The bevelling of the diamond flats to reduce the pressure gradients in the stone and to reach the megabar range (Jayaraman 1983, 1986). A diamond anvil cell is a mechanical device to push two opposed diamond anvils together. Single crystal diamonds are used as anvils not only because diamond is the hardest material available but also because perfect diamond is transparent to most electromagnetic radiation over a

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wide range of wavelengths from far infra-red to y rays, apart from the window [3.5eV, 8KeV]. As shown Force in figure 2, a shim of metal (rhenium for use with hydrogen) is compressed between the two flats of the diamond anvils. The centre hole is made by micro-drilling or sparkerosion after pre-indentation of the gasket up to the limit of the plastic flow of the metal shim. A small ruby Diamond Anvli chip, for use as the pressure gauge, is put in the cylindrical sample chamber before loading. To load Gasket sufficient H2 or He sample in the DAC, cryogenic or high pressure loading are used. Initial densities are roughly equal in both methods but only the latter one can be used for loading mixtures of gases. Most progress in the generation of megabar pressures has resulted from optimizing the distribution of the force, changing the configuration Figure 2: Schematic of the DAC. of the anvils and achieving high precision of the anvil alignment. There are many designs of DAC, but the recent membrane diamond anvil cell (LeToullec 1988) has appeared to be particularly well-adapted for the studies of dense H2 and He ( axial thrust on diamond, smooth variation of pressure, large optical aperture, easy use in a cryostat,...).

Mil

There is a limit to the load that can be exerted on a diamond anvil. Greater pressure is thus generated by reducing the area of the central flat of the diamond tip. Very high pressures are obtained at the expense of smaller volume of the sample. A maximum pressure of 560 GPa, calculated from the volume measured by x ray diffraction, has been reported on Mo and it seems that higher pressures could be generated with the DAC (Ruoff 1992). However, the maximum pressure reported at present on H2 is around 250 GPa. This is roughly the limit of the ruby pressure gauge which is used for these systems. The pressure shift of the R, luminescence band of ruby has been calibrated to 110 GPa by measuring with x ray diffraction the molar volume of copper, gold and tungsten embedded with ruby in the Ne or Ar hydrostatic pressure medium of the DAC sample chamber (Mao 1986). The absolute accuracy of the ruby pressure gauge is estimated to be ±2%. Various effects render the measurement of pressure by the ruby gauge above 200 GPa almost

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impossible: the intensity of the ruby fluorescence decreases markedly with pressure; Stressed diamonds produce a broad luminescence that overwhelms the ruby fluorescence; the closing gap of diamond gives a strong absorption of the laser line; finally, such pressures are far from the range of calibration of the ruby gauge. Due to the optical transparency of the diamond, the DAC is the tool par excellence for optical spectroscopy at very high pressure: The sample can be fully characterized. Generally, the very weak signals of the minute samples of H2 and He have to be separated from the strong background of the diamond anvils. Raman, infra-red, absorbance, reflectance, dielectric or Nuclear Magnetic Resonance measurements and single crystal x-ray diffraction of H2 and He solids had to be adapted. In the next section, the main measurements on H2 and He systems at high pressure are surveyed. In table I, we have tried to compare the potentialities, limits and achievements of the dynamic and of the static methods. It seems that the DAC is more suitable for the study of astrophysical matter than the dynamic methods because it combines the generation of higher densities with a full characterization of the sample under static and equilibrium conditions. The coupling of very high temperatures with very high pressures in a DAC, that is being developed, would reinforce this assertion. Table 1: Comparison between DAC and Shock-wave for studying H2 and He systems. METHODS Shock-wave. Single -» Hugoniot Reflected -> Isentrope

Diamond Anvil Cell

POTENTIALITIES Ahaolute measurement of P andV No limit of pressure.

Static. X ray measurement of V. Complete optical measurements. Studies of mixtures. Reversible P T path. Low cost, easy use.

LIMITS Dynamic process. Hard to reach very high density. P(T) curves; Hugoniot, isentrope. Always in the fluid phase. Optical measurements difficult Study of mixtures impossible. Irreversible. Expensive. Mechanical limit of diamond. Measurement of P < gSOGPa. Very small volume. Parasitic properties of diamond No conductivity measurement.

ACHIEVEMENT

DrISO GPa 3000 K .QO

t*ma/nwslA

*d*o cm /mow

HeS6GPa 21000K •6 cm'/mole

248 GPa T7K -2.1 cm'/mole He: 60 GPa 300 K 2.S cm'/mole

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11.3 A decade of measurements of the properties of H2 and He at very high pressures. 11.3.1 A review of the literature. The static studies have been performed essentially by four groups ( Geophysical, Harvard, Paris 6 and Van der Waals laboratories). All the shock-wave studies have been performed at Livermore. The present status of their experimental investigations is presented below. W.3.1.1 Helium. He is the simplest atom with a close Is electronic shell. It has been calculated that the electrons should remain tightly bound to the nucleus up to at least 40 Mbars (Klepeis 1991). Also, due to its very low atomic mass, low density He has been extensively studied as the archetype quantum system. The two following problems have motivated the investigations of the properties of He at very high pressures, namely: How to model the interactions of a dense insulator and in particular what is the validity of a pair potential description? What are the importance of quantum effects at high density? The melting point at room temperature was the first measurement performed on He in a DAC (Besson 1979). Subsequently, the melting curve measurements were extended up to 24 GPa and 460 K by the method of quasi-isochoric scans (Loubeyre 1982, Vos 1991). An anomaly on the melting curve was detected around 290K, interpreted as a triple point associated to the possible stability of a bcc phase along melting (Levesque 1983, Loubeyre 1986). Direct structural determination by x ray diffraction could only become feasible with the use of high flux synchrotron beam (Mao 1988a). The structural properties, equation of state and phase diagram of He, from low temperature to 400 K and up to 60 GPa, have recently been pinned down (Loubeyre 1993a). The equation of state in the fluid phase has been derived from Brillouin scattering measurements of the sound velocity (LeToullec 1989). The structural properties are a sensitive test of the models of interaction. Shock-wave measurements have aimed to determine a spherically symmetric effective interatomic potential with which statistical mechanics calculations of dense He would be easy and accurate. There exists a significant discrepancy between the best ab-initio or experimental pair potentials and the effective potential derived from shock-wave data (Ross 1989). The latter is softer indicating the existence of many-body interactions. Two theories have been proposed to calculate these manybody interactions: either from the spherical contraction of the electronic

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cloud of the He atom in a dense environment (Lesar 1989) or from an effective three-body interaction (Loubeyre 1987a). Refractive index measurements indeed show the contraction of the electronic cloud of He atoms. However, up to now, none of these two forms of interaction explains satisfactorily the phase diagram of He, in particular why is it so different from the ones of heavier rare gas solids. In dense He,, quantum effects should originate from the uncertainty principle; they should be well estimated by the h2 Wigner-Kirkwood expansion or from path integral simulations (Barrat 1989). This gives macroscopic isotopic differences. Measurements of the isotopic shift on the melting curve (Loubeyre 1992) and on the equation of state (Loubeyre 1993) have been reported. Very surprisingly, experiment is inverse of theory, as it will be discussed in section 4.3.

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20

HCP m 0)

10

I

r

F

i n

1

0

100

1

(

200

300

400

Temperature (K) Figure 3: X ray determination of the phase diagram of 4He. The large dots and large triangles, respectively, correspond to single crystals of the hep and fee structures, for which the orientation matrix was found.

248

Loubeyre: High pressure experiments for astrophysics

11.3.1.2 Mixtures of hydrogen and helium. Hj/He binary systems were the first mixtures studied in a DAC. Owing to the conceptually simple electronic configuration of the two components, they were considered the most amenable to a theoretical description. Also, its components have been largely investigated under pressure and this should faciltate the analysis of the data. These DAC measurements have been recently used to test various dense mixture theories (Ree 1989, Vos 1991). The studies of mixtures are generally harder than the ones of pure systems: Great care has to be paid in the control of the concentration loaded in the sample chamber (a mixture of known concentration is loaded with the high pressure vessel technique); Many loadings have to be done to investigate completely the binary phase diagram; Signals get weaker upon dilution.

0.2

0.4 0.6 0.8 Mole f r a c t i o n of He

1.0

Figure 4: The Ha/He binary phase diagram at 296 K. A cusp on the F,+F2 boundary line has been measured around 90 mol% He. This could indicate a high pressure evolution towards a separation of phases as indicated in the inset.

The binary phase diagram of H/He has been measured from 175 K to 360 K up to 15 GPa (van den Bergh 1987, Loubeyre 1987b 1991). The measurements essentially follow the evolution trend extrapolated from low pressure data (Streett 1973): The binary phase diagram is of eutectic

Loubeyre: High pressure experiments for astrophysics

249

type with a large fluid-fluid separation phase, between the two triple point at 23 mol% He and 99 mol% He, the pressure domain of which is increasing with temperature. The vibronic properties of the H2 molecule (intra-molecular frequency) were shown to depend strongly on the concentration of the surrounding mixture (Loubeyre 1985). This was calibrated for use as an in-situ concentration gauge. With it, finer details of the binary phase diagram were observed. In particular, a cusp on the boundary line of the fluid-fluid separation domain, around 90 mol% He, could indicate a different high temperature evolution of the fluid separation of phases, namely the existence of a close fluid-fluid domain at high pressures such as it is represented in the inset of figure 4. The sound velocity in a mixture of nearly equal concentration was measured by Brillouin scattering (Loubeyre 1993c). Increasing deviation from ideality with pressure is observed. No theory can reproduce, with the same binary potential, the binary phase diagram and the sound velocity of Hj/He mixtures. Finally, the changes of the properties of the H2 molecule under strong compression in going from pure hydrogen to the rare gas matrices have been measured (Loubeyre 1992b). In particular, the bond length was found to decrease strongly when the H2 molecule is embedded in a rare gas matrix (Loubeyre 1991b). 11.3.1.3 fl*

The simplicity of the hydrogen molecule and its large quantum effect in the condensed phase has led to a vast literature, describing subtle molecular processes. The physics at low pressure has been reviewed (Silvera 1980, van Kranendonk 1985). With the advent of the DAC, experimental work on hydrogen under pressure had accelerated sharply. The Hj molecule remains a free rotor up to very high pressure and consequently, the ortho-para concentration has a determining role in understanding the phase diagram. At very low temperature, parahydrogen ( or ortho-deuterium) is a J=0 molecular solid. With pressure, the anisotropic interaction potential becomes sufficiently large, relative to the splitting of the rotational states, so that there is a strong admixture of higher-lying rotational states into the ground state. The lattice spontaneously orders at T=0K into a structure which minimizes the anisotropic interaction energy. This transition is called the BSP (Broken Symmetry Phase) transition. It has been observed in deuterium at 28 GPa (Silvera 1981), with a transition pressure increasing with temperature, and in hydrogen at 110 GPa and 8 K (Lorenzana 1989). However, the crystal structure of the BSP phase is unknown.

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Single-crystal x ray diffraction with a synchrotron source at room temperature shows that the solid structure of normal hydrogen is hep up to 45 GPa (Hemley 1990, Finger 1991). This has been also obtained by neutron diffraction on deuterium up to 30 GPa (Besedin 1991). The equations of state determined by either neutron or x ray diffraction are identical and show that deuterium is slightly more compressible than hydrogen at the highest pressures. However, none of these structural determinations could give information on the evolution of the molecular bond length with pressure. A spectroscopic determination was made from an analysis of the roton bands at T=6K in para-H2: The intramolecular distance presents a minimum around 30 GPa (Loubeyre 1991c). Brillouin scattering measurements give the sound velocities from which the elastic constants, the bulk modulus and the equation of state can be calculated. Measurements in fluid hydrogen at room temperature have produced the equation of state up to solidification (Shimizu 1981). Measurements on a single crystal up to 24 GPa have shown that solid hydrogen is elastically anisotrope (Zha 1993). The equation of state derived from these measurements is in good agreement with the x ray determination. The melting curves of H2 and D2 have been measured up to 373 K and 8 GPa (Diatschenko 1985). HYDROGEN Ordered H-A

Disordered H-A -. /

150 Broken-Symmetry Phase

£

100

-

r

-

-

hep

j

50 Liquid.

n 0.1

i

/

i

10

100

T(K) Figure 5: The phase diagram of molecular H2 at high pressures.

1000

Loubeyre: High pressure experiments for astrophysics

251

The very high pressure phase diagram was explored by optical methods, essentially Raman and infra-red measurements of the fundamental excitations of the system, vibrons, rotons, phonons, (Hemley 1991a, Silvera 1991a). The vibron frequency, corresponding to the intramolecular vibration of the H2 molecule, changes with pressure due to essentially three contributions: The static shift, which is a measure of the crystal compression on the molecule; The vibrational coupling, which results from the resonance transfer of vibrational energy between two neighbour H2 molecules; The variation of the electronic density going from the intramolecular region to the intermolecular one. The Hj vibron frequency shows a large discontinuity at 145 GPa and 77K that reveals a first order phase transition to a new molecular phase, named H-A. The same phonon, vibron and roton modes were observed in the hep solid and in the H-A phase. Also, the frequency of the lattice phonon was observed continuous through the transition. So, it is reasonable to assume that the structure of the molecular centers should be hep in the H-A phase. The existence of the critical point on the H-A phase line ( 150 K, 165 GPa) implies that there cannot be a change of symmetry at the transition: The molecules should have cylindrically symmetric distribution around the c axis. Orientational transition at the H-A phase transition has been also reported. It has been suspected on a number of grounds that the H-A phase is the metallic molecular phase of hydrogen. Refractive index, absorbance and reflectance measurements have tried to characterize the changes of the electronic properties at the H-A transition, aiming to show the closure of the band gap at 150 GPa. However, at this time, some uncertainty has developed for the support of the conduction state of the H-A phase, as it will be discussed more lengthy in the next section. The infra-red measurements of the vibron mode have been measured up to 180 GPa by synchrotron infra-red spectroscopy (Hanfland 1992). The difference between the infra-red and the Raman mode indicates a dramatic increase in intermolecular coupling with pressure. Similar results have been obtained by isotopic dilution (Loubeyre 1992). Clearly, the phase diagram of hydrogen at very high pressure is much more complex than previously imagined. This is certainly a sensitive reference system for testing many-body quantum calculations. This is stimulating high precision theoretical works and the close interaction between theory and experiment should lead to a good description of molecular hydrogen approaching its plasma phase transition. The above review of the literature is summarized in the table below.

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Table II. Essential measurements on systems ofH2 and He at high pressures. SYSTEM

RESULTS.

MAXIMUM PT

He

Phase diagram, EOS. Melting carve. Isotope effects in dense He. Sound velocity. Refractive index. Hugoniot.

60GPa, 40K-400K 24GPa, 480K lOGPa, 280K 12GPa, 300K lSGPa, 300K 56GPa, 21000K

Loubeyre 1993, Mao 1989 Vos 1990, Loubeyre 1982 Loubeyre 1992 Polian 1986 LeToullec 1989 Nellis 1984

H,

Structure, EOS. Melting. Sound velocity. Bond length. Raman: vibron, phonon, roton. Infra-red: vibron, phonon. Isotopic dilution. Abcorbance Reflectance. Refractive index. Hugoniot. Conductivity in shocked H,.

46GPa, 300K 8GPa, 380K 23GPa, 300K 40GPa, 6K 230GPa, 10K300K 180GPa, 300K 120GPa, 80K 230GPa, 77K300K 177GPa, 300K 220GPa, 300K 76GPa, 7000K 120GPa, 3000K

Hemley 1989, Besedin 1991 Diatschenko 1985 Shimizu 1981, Zha 1993 Loubeyre 1991 Hemley 1991, Silvers 1991 Hanfland 1992 Loubeyre 1991,1993 Eggert 1991, Hanfland 1991 Mao 1990, Eggert 1991 Hemley 1991, Garcia 1992 Nellis 1983 Nellis 1992, Weir 1993

Binary phase diagram. Vibron. Sound velocity. Isolation in matrices.

lOGPa, 373K lSGPa, 300K 5GPa, 300K 40GPa, 77K

Van Bergh 1987, Loubeyre 1991 Loubeyre 1991 Loubeyre 1993 Loubeyre 1992

H,/He

REFERENCES

11.3.2 An important problem: The search for metal hydrogen. 11.3.2.1 Compression of solid H2. The formation of metal hydrogen by application of pressure and the characterization of its physical properties is considered a major problem in condensed matter physics (Ginzburg 1983). This should result in the simplest possible metal, with each lattice point in the crystal occupied by a bare proton immersed in a sea of electrons. Theoretical predictions suggest that it will behave as a quantum metal, with properties considerably different from those of simple metals (Ashcroft 1991). The formation of metal hydrogen is of quantum nature (based on the Pauli exclusion principle) as first predicted by Wigner and Huntington (1935). In the original treatment, a first order phase transition takes place from the insulating molecular phase to the monatomic metal. More recently, an other mechanism of metallization has been proposed where an intermediate molecular metal phase will exist (Friedli 1977). In this phase, the solid retains its distinct molecular components with the metallic character achieved by overlap of the valence and the conduction

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253

bands. The knowledge of the sequence of the metallization of hydrogen, the order of the phase transition and the associated latent heat are essential information for describing the interior of giant planets and brown dwarfs. There are three likely experimental approaches to metallic hydrogen: Static isothermal compression; Isentropic compression; Shock compression. Shock compression leads to temperature that are too high for reaching the required density. Isentropic compression, in a magnetic flux compression device, has been performed and experimentalists have claimed to have produced metallic hydrogen. Unfortunately, only the most primitive diagnostic measurements have been made and the claims remain unconvincing. It recently became clear that the issue of the metallic transition will be settled by DAC technology. At the low temperature onset of the H-A transition, the discontinuity of the vibron frequency was approximately 100 cm'1, much larger than what would be expected from an orientational transition (Hemley 1988, Lorenzana 1989). This discontinuity was not related to a structural phase transition and so could originate from a change in electronic properties. It was suggested that the H-A phase was the molecular metallic phase of hydrogen (Mao 1989, Eggert 1990). So, the decrease of the vibron frequency would correspond to the transfer of electrons from molecular bond states to crystalline band states. Since the inter-proton electronic charge density in hydrogen is responsible for the binding of forces, if this charge density were reduced, it would result in a weaker bond and a lower vibron frequency. The most rigorous mean of establishing this insulator-metal transition is to measure the dc electrical conductivity and to show that it remains finite in the limit that T -» OK. This is difficult to do in a DAC at megabar pressures. Two experimental groups have opted to study the H-A transition by optical means: dielectric, reflectance and absorbance measurements. These various experiments have been recently reviewed with critical consideration, independently by the leaders of these two groups (Mao 1992, Silvera 1991b). They agree that the experimental situation is unclear and that there is no direct proof of metallization. This is discussed below. Dielectric measurements. The refractive index and its dispersion should reflect changes in the electronic density. These measurements have been performed on hydrogen at visible frequencies up to 170 GPa (Hemley 1991). No evidence of dielectric catastrophe (divergence of the static electric polarizability), which should be associated with free electrons, nor even a measurable discontinuity at the H-A phase transition was observed.

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This suggests that if the H-A phase transition is a metal-insulator phase transition, it should be due to the closure of an indirect band gap giving a very small free carrier density. In fact, extracting information on the electronic properties from such measurements is not straightforward. Comparison with full electronic density calculation of the dielectric response was proven useful to interpret similar measurements up to 220 GPa (Garcia 1992). Absorption and reflexion measurements. In the metallic state, the optical reflectance and absorption could be reasonably described by a Drude free electron model. The Drude model is a three parameter model (plasma frequency, electron relaxation time and the difference in the real part of the dielectric constant at the diamond sample interface). The plasma frequency is proportional to the square root of the charge carrier density. Entire confidence in the applicability of the Drude model must be provided by the demonstration of the consistency of the absorption and reflection data. Reflectance measurements have been performed (Mao 1990) on solid H2 to 177 GPa from 0.5 eV to 3 eV. At the highest pressures, they observed a rising reflectivity in their low frequency spectral limit. This was interpreted as a free carrier reflection. Analysis of volume dependence of the plasma frequency obtained from Drude-model fits to the spectra indicates that the pressure of the insulator-metal transition is 149 GPa. Subsequently, the interpretation of these experimental data was questioned (Eggert 1991), by performing absorption and reflection measurements a much higher pressure, 230 GPa, from 0.7 eV to 3 eV. Extrapolating the fit of the plasma frequency of Mao, a strong absorption and reflection should be present in contrast to what was observed. Finally, absorption data were reported (Hanfland 1991) to be consistent with the reflection measurements only under the assumption that at very high pressure the refractive index of diamond gets greater than the one of hydrogen, which seems quite unphysical. The discrepancy between the measurements of these two groups could arise from various reasons: First, the samples of the two groups are different and anisotropy and cristalline order can significantly shift the optical spectrum; Second, in Mao's experiments, the sample was loaded with ruby powder that at very high pressure could chemically react with hydrogen (Ruoff 1991). Free Al would then dilute in the sample and change the electronic properties. Still, direct experiments seem to infirm these hypothesis (Mao 1991). Third, the simple Drude model used to analyse the data is inappropriate. In summary, these absorbancereflectance data are confusing and this essentially corroborates the dielectric measurements. No dramatic changes of the electronic properties

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255

are observed above the H-A transition. If H-A were a metallic phase, the number of free electrons should be very small. Darkening of the sample. In an experiment to 250 GPa at 77K, the hydrogen being in the interstices of a compact ruby powder, darkening of the hydrogen was observed (Mao 1989). However, darkening itself is not a proof of metallization since it could be due to a narrowing of the band gap below the visible wavelengths. Furthermore, the topology of the sample was not ideal for such a report since darkening might be due to the presence of Al liberated by the reaction of hydrogen on ruby. Agreement with calculations. Following the discovery of the H-A transition, various ab-initio electronic calculations have tried to confirm the interpretation of the closure of an indirect band gap. The structure was assumed to be hep. However, the orientational distribution of the molecules has a profound effect on the electronic properties. In LDA calculations (Garcia 1990), the hep structure with spherical charge distribution for the hydrogen molecules which simulates orientational disorder, has a gap closure around 400 GPa whereas, for the hep structure with the molecules aligned along the c axis, the gap closure is around 180 GPa. Using a quasi-particle calculation, known to give accurate gap results (Chacham 1991), it was found, for the order and disorder hep structures, band gap closure at respectively 150 GPa and 300 GPa. However, it was latter shown that the order hep structure is not the stabler one; a structure with a molecular orientation with the c axis is stabler but has a wider band gap (Kaxiras 1991). Finally, it was recently shown that the zero-point motion yield important contributions (Surh 1993). So far, no ab-initio theory has been able to identify the observed H-A transition. Other interpretations of the H-A phase transition could be even more interesting than what we are looking for. There are two conjectured mechanisms by which electrons might be liberated from molecular bonds and yet still not available for molecular conduction: excitonic pairing and dynamic localization. However, no proof for such unusual state of matter has been reported (Ashcroft 1991). By increasing further the pressure, there is almost no doubt that hydrogen will form the long-sought atomic metal. All recent calculations converge for a transition pressure around 300 GPa (Ceperley 1987). Unfortunately, no progress in the maximum pressure achieved on solid H2 has been reported since 1989: It could be that the chemical attack of the diamond anvils by hydrogen limits compression further. A new

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Raman feature appearing at 240 cm1, with an onset pressure from 150 GPa to 200 GPa only when hydrogen or deuterium is in contact with the diamond tip, sustains this assumption (Hemley 1992). Approaches for reducing the pressure of the atomic metal transition are needed to circumvent this problem. This has been achieved recently by the discovery of a new molecular compound, Ar(H2)2.

11.3.2.2 A new path: the compression ofAr(HJ2 compound. It has been recently discovered that under pressure simple molecular systems can form ordered alloys. The first two examples found were Ne(He)2 (Loubeyre 1993b) and He(N2)n (Vos 1992). Subsequently, the Ar(H2)2 compound was solidified at 4.3 GPa. Synchrotron single crystal x ray diffraction showed that its structure is hexagonal with 4 Ar atoms and 8 Ha molecules in the unit cell. This structure is well-known as a Laves phase. Its stability can be understood from considerations of the maximization of the packing fraction. Interestingly, the H2 molecules are arranged at the corner of tetrahedra which are joined point to point and alternately base to base throughout space, forming long chains. This quasi-1 dimensional alignment offers the possibility to reduce the insulator-metal transition pressure of the hexagonal sublattice of H2 molecules. Raman measurements of the vibron and roton modes and absorbance measurements (from 1.4 eV to 3.0 eV) were performed up to 200 GPa twice on AKH2)2 and once on Ar(D2)2 compounds (Loubeyre 1993c). A first order phase transition was observed at 175 GPa. It is associated with the disappearance of the vibron mode (signature of the molecular entity), a visual increase in absorbance of the sample, a possible Drude edge in the absorption spectra (indicative of free carriers) and in good agreement with theoretical predictions of metallization in the m-hcp struture of pure solid H2 that is similar to the sublattice of the H2 molecules in this compound. Although none of the facts alone is sufficient at present to prove definitively that metallization has occured, their conjunction argues strongly for the observation of the metallization/dissociation by pressure of this system of hydrogen molecules. The Ar atoms should remain neutral. The physics involved in the metallization of this hexagonal sub-lattice of Hj is consequently equivalent to what could be found in metallizing solid H2. So it is now very interesting to try to fully characterize the properties of this new metal of hydrogen.

Loubeyre: High pressure experiments for astrophysics

257

Ar(H2)2 compound

Structure

Metallization

Figure 6: Three observations on the Ar(H2)2 compound. Growth: facetted hexagonal singlecrystal in equillibrium with a fluid of equal concentration. Structure: hexagonal structure with 8 H2 molecules and 4 Ar atoms in the unit cell. Metallization: First order phase transition at 175 GPa (sample chamber 10 um diameter with a small ruby) associated to a disappearance of the vibron and a visual increase in absorbance.

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11.4 Usefulness of high pressure measurements for Astrophysics. Three levels of usefulness can be distinguished in the use of high pressure measurements for understanding the interior of giant planets and brown dwarfs. -First, experimental data on the equation of state, on the melting curve, on the sound velocity or on the miscibilty of hydrogen/helium systems can directly serve as input for interior models. Unfortunately, the PT domain already explored covers only a small fraction of the radius of these astrophysical objects, mainly because static measurements were done up to now below 400 K. However, if the data are presented in form of fits that have a physical meaning, the PT range of these data can be significantly extended. -Second, the experimental data can be used to test the assumptions made for calculating the properties of planetary matter. In such calculations, the significant parameter is density rather than pressure or temperature. Properties of hydrogen and helium systems have been probed to densities at the limit of the plasma phase transition in hydrogen. The accuracy of the description of the whole molecular domain of the interior of giant planets and brown dwarfs could now be assessed. -Third, the study of matter at the frontier of investigation can reveal subtle behaviours of matter, generally unsuspected. Below, we illustrate this classification with a selection of some of the experimental data that were reviewed in the preceding section.

11.4.1. Data for codes. 11.4.2.1 Equation of state. X ray diffraction measurements give a direct and accurate determination of the equation of state in the solid phase. Shock-wave experiments provide an indirect determination by the fit of an effective symmetric effective intermolecular interaction which can be used with accurate theoretical models to calculate the equation of state. In figure 7 , these two determinations are compared for H2 and for He solids. An increasing difference with pressure between them is observed. This is a consequence of the growing importance of many-body interactions. Their pair average, which is incorporated in the effective pair interaction, depends on the structural state of the system: the average in the high temperature shock-fluid differs from the one in the solid state at the same density. The x ray data, combined with previous low pressure data, were shown

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to be well-represented by the two-parameter Vinet equation: 32

-£)~ '(1 -(-£) )e*p(! With the parameter sets (V0(cm3/mole)=13.72, K0(GPa)=0.225, K
40

60

P(GPa) Figure 7: Equations of state of He and Hj. The dots and triangles are the x ray data respectively for He and H2. The dashed-lines are the Vinet fits. The solid line is the EOS calculated from the pair interactions derived from shock-wave.

11.4.1.2 Melting curve. The melting data of H2 and He, respectively up to 360 K and 400 K, are

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well represented by the Simon law, P=ATC+B. For He, P(GPa) = 1.607 10 3 KK)lsa For Hv PiGPa) = -0.052 + 3.436 10"4 r1
100 300 500 700 900 1100 1300 1500

T(K) Figure 8: Evolution at very high pressures of the melting curves of H2 and He. The solid lines represent the Simon equations. The dashed-lines are the calculation with the pair interactions adjusted on shock-wave. The arrows indicate the maximum temperature of the measurements.

Loubeyre: High pressure experiments for astrophysics

261

11.4.1.3 Sound velocity. In figure 9, the adiabatic sound velocities in liquid He (Polian 1986) and in liquid Kj (Shimizu 1981) are plotted versus density at room temperature. A linear relation is observed above 0.13 g/cm3 in liquid H2 and above 0.45 g/cm3 in liquid He. This linearity is well-known in geophysics as the Birch law. From analysis in the dense solid phase, it seems that this relation should also be valid at much higher density (Loubeyre 1990). Still, no theoretical justification has been given. Certainly, this form should be useful to extrapolate experimental data.

0.2

0.4

0.6

3 0.8

density(g/cm

l.o

Figure 9. Adiabatic sound velocity versus density at 296 K. A linear relation is observed above 0.13 g/cm3 in fluid H2 and above 0.45 g/cm3 in fluid He.

11.4.1.4 Miscibility. The binary phase diagram of the Hj/He binary mixtures is of eutectic type with a large fluid-fluid separation of phase, as shown in figure 4. At a given temperature, the pressure interval of the fluid-fluid domain is lower-bounded by the pressure of the critical point and upper-bounded by the pressure of the fluid-fluid-solid triple point, above which solid separation of phase exists. The PT loci of these points, namely the critical line and the triple line, give the essential information on the pressure

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evolution of the miscibility of the system. From figure 10, it is likely that the fluid-fluid equilibrium of Ha/He mixture will persist up to very high pressure (van den Bergh 1987). However, a small accident on the fluidfluid boundary line around 90 mol% He was interpreted as a preliminary sign of the evolution towards a re-entrant fluid domain (Loubeyre 1987), as drawn in the inset of figure 4. It is too early to conclude. High temperature experiments have to be performed beforehand. 7

-

50

100

150

200

2S0

300

350

Figure 10: Critical line and three phase line in the He/H2 system (van den Bergh 1987).

11.4.2 Constraints for theoretical description. The calculation of the physical properties of the Hjj/He molecular envelop of low mass star interiors up to the plasma phase transition is difficult. The ab-initio approach, which starts with the coulombic hamiltonian of the system, presents convergence uncertainties when the electrons are localized in molecular entities. As seen above, the difference in the EOS of molecular H2 by two state-of the art ab initio methods, QMC and LDA, amounts to 20% at 200 GPa. Also, these calculations are mostly limited to T=0K That is why the molecular approach is generally used instead. It is based on the description of the interactions between molecular

Loubeyre: High pressure experiments for astrophysics

263

entities. With well-known statistical methods, calculations of the physical properties of the system at arbitrary PT conditions can then be made. The interactions are generally assumed of pair potential forms, which are adjusted on high pressure measurements. It will be shown below that many-body interaction and charge transfer between H2 molecules approaching the plasma phase transition complicate this description. Sometimes, it could thus be better to use extrapolation of high pressure experimental data. The properties of Ha/He mixtures are harder to calculate and to measure than the ones of the pure systems. So, they are generally estimated from the properties of the two pure components with the law of ideal mixing, i.e extensive variables are additive. This approximation will be questioned experimentally below.

1 2 3 4 PRESSURE (GPa)

5

Figure 11: Adiabatic sound velocities in fluid He (solid line), fluid H2 (solid line) and Hj/He fluid mixture of 52 mol% in H2 (the dots are the Brillouin data and the dashed line is the ideal mixture calculation).

11.4.2.2 Ideality. For an ideal mixture, the extensive variables add. There is no phase separation. Applied to the EOS, it means the additivity of the volumes of the two components, vM(x) = x vHe + (1-x) vH2. The volume of mixing quantifies the deviation from this behaviour but it is a difficult quantity

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to measure at high pressure. A related property is the sound velocity that can be measured by Brillouin scattering experiment. In figure 11, the sound velocity measured in a fluid Hj/He mixture of 52 mol% Hj (Loubeyre 1993d) is compared to the ideal mixing calculation from the measurements in pure Hj and pure He. A discrepancy appears increasingly with pressure. Even at the moderate density of the experiment, the error already amounts to 6%. The volume is related to the integral of an expression which contains the sound velocity to the square. So, the deviation from ideal mixing could even be greater for the equation of state than for the sound velocity. This casts some doubts about the validity of the use of the law of additive volume for planetary interior models. 11.4.2.2 Interactions in the molecular phase. The pair potential adjusted on high pressure data differs from the truly pair interaction between two isolated molecules. The repulsive part of the former is significantly softer. This experimental potential is an effective potential that includes many-body interactions. Physically, the deformation of the electronic cloud of a molecular entity in a dense environment modifies the interaction energy between two such electronic cloud: This is the many-body contribution. Electrons of Hj molecules are loosely bound to nuclei and easily change their charge density when other molecules are placed nearby whereas in He, the tightly bound electrons are less subject to change. Many-body contributions should be greater in Hj than in He. It has been shown above that the T=0K solid EOS calculated with effective pair potential fitted on Hugoniot, increasingly differs with pressure from the x ray determination. This shows the limit of this effective pair potential approach: the environment of an interacting molecular pair is different in a dense solid than in a shocked high temperature fluid, even at the same density. So, the effective many-body contribution to the pair potential interaction should be structure dependent. Two models have been proposed to consider this many-body contribution more explicitly. In the first one, a sphericalized contraction of the electronic cloud is calculated by Gordon-Kim-Hartree-Fock method (LeSar 1988). In the second one, a three body interaction is assumed to be the essential term and is added to the pair potential (Loubeyre 1987a). As seen in figure 12, the EOS calculated with these two approaches are in fair agreement with experiment, although none will be entirely satisfactory at higher density. The strong difference, between the calculated EOS with the pure pair potential calculated EOS and the experimental one, quantifies the contribution of the many-body

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Loubeyre: High pressure experiments for astrophysics

interaction in dense He. In dense H2, this term is more important and the situation is further complicated by the charge transfer interaction. 1

4.5

— — xray , hep A A. A A A. VT*9 V ^ • ^ • ^ • ^ • ^ ^ A i olJr

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Figure 12: Volume versus pressure data of hep solid 4He at 304 K. The dots and squares indicate x ray measurements. Four calculated EOS are compared: self-consistent phonon calculations either with the HFD-B pair potential (dash-dotted line), the Ross-Young potential (full line), the HFD-B pair potential plus three body interaction (dashed-stars line). The dashed-squares line is the LeSar many-body calculation.

When two Ha molecules are brought close to one another with density, the probability increases that electrons exchange through a mechanism involving electron tunnelling. This leads to an attractive interaction like the beginning of a covalent bounding. An experimental indication of it is presented in figure 13. In hep solid, there are two vibron modes: one, which corresponds to the coupled in phase motion of the two intramolecular vibrations in the unit cell, is raman active and the other one, which corresponds to the uncoupled motion , is infra-red active. The difference between these two modes is due to the vibrational coupling. The associated vibrational coupling constant is negative and increases with density, as seen in the inset of figure 13. This is unexpected for intermolecular distances corresponding to the strongly repulsive overlap

Loubeyre: High pressure experiments for astrophysics

266

intermolecular interaction and could be the indication of a charge transfer interaction. It contributes to the extra softening of the pair interaction but no model has been proposed so far to explicitly describe this interaction. The situation is even more complicated in mixtures because the concentration parameter affects the environment of an interacting molecular pair. The effective pair potential should thus be concentration dependent. No approach based on effective pair potentials can satisfactorily describe the Ha/He binary mixtures: the H2-He potential that is fitted on the binary phase diagram cannot reproduce sound velocity measurements and vice versa. Since re-entrant fluid phases in binary mixtures are generally attributed to the existence of an attractive interaction (like the hydrogen bound in water-alcohol mixtures), the observed cusp on the fluid-fluid boundary line of the tij/He binary phase diagram would probably be reproduced by explicitly considering the attractive charge transfer interaction between H2 molecules.

Raman

0

0

20 40 60 80 100120140160180200

P(GPa)

Figure 13: Pressure dependence of the Raman and infra-red vibrons of hep solid H2. Their difference is due to the vibrational coupling. The negative of the vibrational coupling constant is plotted versus the intermolecular distance in the inset.

Loubeyre: High pressure experiments for astrophysics

267

In summary, deviations from the pure pair potential interaction are important at high density. Their pair average in an effective pair interaction fitted on experimental data only gives a first description: interesting properties can be missed. More works are needed tofindout mathematical forms that can satisfactorily represent many-body and charge transfer interactions.

11.4.3 Unsuspected properties. Observing properties that were unsuspected or doing measurements that are in strong disagreement with the best theoretical predictions generally creates new physical approaches. In the experimental investigation of the properties of matter under very high pressure, this is all the more so to happen in the studies of H2 and He systems due to their electronic simplicity and their quantum nature. Indeed, the measurements of the quantum properties of He at high density have recently offered such a surprise. 60 40

MPa)

20

ol o?

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experiment »»»** PIMC -120 • • • • • large calculation volume experiment

-140

0

50

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200 250

Temperature(K)

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Figure 14: Difference between the melting pressures of 3He and 4He versus temperature. The stars and the squares are experimental data. The circles are the results of the PIMC simulation.

With increasing density in He, the effect of statistics should become

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negligible, even at very low temperature, because of the hindered density fluctuations by the core potential. On the other hand, quantum effects associated with the uncertainty principle should give measurable isotopic differences between macroscopic properties of 3He and 4He. The isotopic shift in the melting curve of He has been measured from 70 K to 260 K in a DAC (Loubeyre 1992a), extending previous large volume experiments done up to 30 K. In figure 14, these measurements are compared to a path integral Monte Carlo calculation (PIMC) which should be very close to a quasi-exact answer under the following three assumptions (Barrat 1989): the effects of statistics are negligible; the many-body interaction can be included in an effective pair potential and finally the BornOppenheimer approximation is valid. A discrepancy occurs with pressure, leading to an opposite isotopic shift above 200 K. When plotted versus the nearest neighbour interatomic distance, the difference between theory and experiment is of a regular exponential form. So, at least one of the three assumptions made prior the PIMC should be violated. The very recent measurement of the isotopic shift on the EOS of He at T=300 K has enabled to specify a bit more this problem. The EOS of 3He and 4He are observed to cross around 22 GPa, 3He being notably more compressible than 4He at high pressures. This is unusual and corroborates the observation of the inverse isotopic shift on the melting curve. This seems to favour the interpretation of important effects of statistics at very high density. This unsuspected behaviour could be even more dramatic at high density. Further works are needed to come up with a definite interpretation.

11.5 Conclusion. I have tried to review the status of the experimental investigation of the equation of state for Astrophysics. The subject was limited to the properties of the two major components of the low-mass astrophysical objects, H2 and He. It should be noted that the properties of denser elements, such as ices, have been also thoroughly investigated at high density by shock-wave or static DAC experiments. Rapid progress have been made in the past decade and densities characteristic of the molecular envelop of planets can now be achieved in static experiments. Even though these experiments have been performed at low temperatures ( <400 K), the above discussion aimed at showing how these measurements are relevant for astrophysics. Due to the electronic simplicity of the He and H2 elements, two other levels of usefulness have been distinguished that broadens the sole exploitation of the raw data, namely: experimental references to test theoretical descriptions of dense

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matter and implications of unsuspected behaviour. Now, the experimental efforts will try to extend the PT range of the static exploration. Generation of static pressures over 300 GPa on Hj should enable the observation and characterization of atomic metal hydrogen (as discussed above, the chemical reactivity of hydrogen on the diamond tip will have to be suppressed). Also, an important goal is to couple high temperatures with high static pressures. This is required to extend at higher density most of the measurements presented here: melting curves, Hjj/He binary phase diagram, sound velocity in the fluid phase... Recent technological developments give confidence that temperatures up to 1500 K will be coupled to static high pressures in the near future. Generation of higher temperatures, that will requires the laser heating technique ( already use for studying geophysical compounds), is the challenge for the next decade. By achieving it, the molecular layers of the Jovian planets will become transparent. Observations of the Jovian planets will thus give direct information on the ionised layer, or equivalently on the physical properties of a strongly coupled plasma. One can say that the Jovian planets are the high pressure laboratories of tomorrow. Its exploitation will certainly be of considerable interest for the condensed matter community.

Acknowledgements. I thank R. LeToullec and J.P. Pinceaux for helpful discussions and encouragements. This work was supported by the Commissariat a l*Energie Atomique under grant N°: DAM/CEL-2561/3272, NATO under grant N°: CRG-890084 and CNRS.

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12 Equation of state of dense Hydrogen and the plasma phase transition; A microscopic calculational model for complex fluids F. PERROT Centn d'Etudes de Limeil-Valenton 94195 Villeneuve St. Georges CEDEX, France

C. DHARMA-WARDANA Institute for Microstructural Sciences National Research Council of Canada Ottawa, Canada, K1A 0R6

Abstract We discuss problems related to the electronic and ionic structure of fluid Hydrogen, for equation of state calculations in the domain where a "plasma phase transition" (PPT) may occur. It is argued that the ionization of an electron bound to a particular nucleus proceeds through a progressive delocalization involving "hopping" electron states (i.e. cluster states). A description of the plasma containing pseudoatoms, pseudomolecules and free electrons is proposed. The PPT, if it exists, might be a mobility edge transition across a percolation threshold. It is shown how the effect of electron density, field-particle distributions and temperature on the binding energy of these pseudoatoms and pseudomolecules, can be included. Finally the abundances of these objects is determined by a minimization which allows the self-consistent optimization of ionic as well as electronic parameters contributing to the total free energy. On discute les problemes associes a la structure electronique et ionique de l'Hydrogene en phase fluide, en vue de calculs d'equation d'etat dans le domaine d'une eventuelle transition de phase vers l'etat de plasma (TPP). L'argument essentiel est que l'ionization d'un electron lie attache a un atome se produit par une delocalisation progressive mettant en jeu des "etats de grappe" (cluster states). La TPP pourrait etre une transition de la mobilite se produisant au seuil de percolation. On propose une description du plasma ou "pseudoatomes", "pseudomolecules" et electrons libres coexistent. On 272

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montre comment la densite electronique, le profil des particules de champ et la temperature peuvent etre pris en compte dans l'energie de liaison de ces pseudoparticules. Finalement, l'abondance de celles-ci doit etre determined par une minimisation qui assure l'optimisation simultanee des parametres ioniques et electroniques, de maniere autocoherente. 12.1 Introduction Density functional theory (DFT) is a very effective many-body technique (Hohenberg and Kohn 1964, Kohn and Sham 1965) for calculating the electronic and structural properties of atoms, solids, liquids and plasmas (Lindqvist and March 1983, Gross and Dreizler 1994). Finite temperature DFT (Mermin, 1965) proceeds via the Mermin-Kohn-Sham variational principle which asserts that the thermodynamic potential (TP) is a unique functional of the one-particle densities of the system and that the TP is a minimum for the true (physical) densities. Thus a DFT calculation should provide all the ingredients needed for a microscopic calculation of the equation of state (EOS) of a given system. In an elemental plasma, e.g., an Al-plasma at high temperatures, one of the complications is the existence of several ionization states, e.g., Al*< , with Z{ = 0,1,2, ...,Z. Thus there are Z + 1 ionic species and electrons, i.e., a Z + 2 component system, with concentrations Zj and a mean ionic-charge Z* =< X{Zi >. In the "average neutral-pseudoatom" approach the Z + 1 ionic species are replaced by one ionic species with "average" charge Z* and the EOS is determined in that simplified model. The name "neutral pseudo-atom" (NPA) refers to the neutral object consisting of an ion in a suitable profile (e.g., a Wigner-Seitz cavity) plus its cloud of bound and free electrons that form a neutral object (a more rigorous definition is given in terms of a sum rule on the phase shifts). The bound and free-electron distribution etc., at the NPA are determined by the self-consistent solution of Kohn-Sham equations for the electrons and ions in the plasma (Dharma-wardana and Perrot, 1982, and 1987). In the average-NPA approach the individual species concentrations i{ are not evaluated and hence Z* is fixed by other considerations, e.g., as in the studies of the EOS of dense Al (Perrot 1990) and Be (Perrot 1993). Recently we have implemented a DFT calculation of the EOS of the Z + 2 component mixture without appealing to the so called "chemical pictures", hard-sphere models etc. In this approach we construct Z + 1 different neutral pseudoatoms (each self-consistently determined for the given plasma conditions) and use their interactions with one another and with electrons, instead of a single average-NPA which interacts with the electrons.

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The above discussion of metallic plasmas takes on a new dimension of difficulty at lower densities and temperatures when molecule formation becomes important. Alkali metal vapors and Hydrogen plasmas have some similarities in this regard. Thus, a hydrogen plasma at 0.25 Mbar and leV could contain H2, H, H + , e~ and possibly H~, H*, and small amounts of other clusters. As the density is increased the identity of these" chemical" objects becomes "blurred". Thus an excited H2 molecule having a "bond length" R begins to resemble two unbound hydrogen atoms separated by a distance R. In effect, if mean distances become comparable to bond lengths the simple "chemical" picture breaks down. Even if chemical species like H2, H+, etc., could be identified, their electronic energy levels, vibrational spectra etc., have to be calculated self-consistently, including the interactions with the medium, i.e., the field-particles (FP) that surround a given "molecule-like" entity. Progress in this type of problem is easier for vapors of simple metals for which pseudopotential theory is applicable. The problem is more difficult for protons where the full non-linear consequences of a point-charge (with no moderation effect arising from a finite core size) have to be taken into account and no meaningful construction of pseudopotentials is possible. Thus in our study of ionized hydrogen plasmas (Dharma-wardana and Perrot 1982) we retained electron- and ion- coordinates at every stage of the calculation, and avoided pseudopotentials or linear response. The objective of this paper is to examine a tractable microscopic approach to EOS of hydrogen fluid in the proposed plasma phase transition (PPT) regime (Saumon and Chabrier 1992) where molecular species exist. It is useful to examine not just the thermodynamics, but also the dynamical process of ionization. Such a study sharpens our concepts about the PTT and leads the way for a future numerical study of this difficult regime of hydrogen fluids. 12.2 The mechanism of ionization Wigner and Huntington (1935) noted the similarities between hydrogen and the alkalis and sowed the seeds leading to the concept of a plasma phase transition (Stevenson and Salpeter 1977, Ebeling and Richert 1985). The most elaborate study of the PPT is due to Saumon and Chabrier (1992). They considered a hydrogen fluid containing the four species H2, H, H + , e~ and found that a fluid phase which was predominantly H2, with very low ionization (xfj+ < 0.5%) and low monatomic H concentration undergoes a first order phase transition to a significantly ionized (xH+ « 25%)

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fluid phase, still containing H2, H, H+, e~, at Tc = 1.3 eV and Pc = 0.614 Mbar corresponding to a critical density of 0.347 g/cc. Since a sharp increase in ionization is proposed, it is important to understand the process of ionization. Ionization of a single atom in the vacuum is the promotion of an electron from some bound state v = n,l,m of energy — |£KI to a continuum state k,l, m with energy k2/2 (here we use atomic units: h = 1, |e| = 1, me = 1). In Hummer and Mihalas (1988), and in Mihalas, Hummer and Dappen (MHD, 1988) the ionization process is modeled by assuming that the field particles (FP) create a microfield which Stark ionizes the atom. However, if an effectively spherically symmetric potential were applied to the atom (this could happen from a cubic packing of a coordination shell of FP around the atom) the microfield is zero but a very large destabilization of the atomic bound state could arise. Thus the MHD model is incorrect from the outset. The MHD-estimates of the "critical fields" for ionization of atoms do not contain effects of possibly large potential fluctuations and electron-exchange effects which nevertheless have small microfields (note that the fields are vectors, while the potentials are scalars). Ionization from metal clusters, and the workfunction of metal surfaces are known to be strongly determined by the exchange energy of the ionizing electron, i.e., a quantum effect unrelated to the Stark effect. The discussion given below suggests that the ionization process involves cluster states of the FP in a fundamental way. Consider an atom in a plasma. For simplicity, let the inner shells of the atom be full and let the last occupied electron (in the ground state) be ns where n is a principle quantum number. The "radius" of the atom is about n2 atomic units (a.u.). Consider a dilute plasma of atom density pa where the mean separation rws = (3/)a/47r)1/3 is large, say 100 n2 a.u. An ionizing electron acquires energy from the FP via thermal and random fluctuations of the particle distribution. Most collisions are long-range, weak, multiple collisions which slowly raise the electron to higher energy levels. When the electron reaches the excited state with nj = lOn the electron is still in a bound state but the "size" of the atom is comparable to the inter-atomic distance rwa. From then on the electron gains energy by hoping to orbitals which span several atomic centers (i.e, clusters) prior to ionization. That is, the atomic electron becomes a "hopping electron" before it passes into the continuum to become a fully delocalized ("free") electron. Thus the ionization process involves three types of electrons, viz., bound, hopping and free electrons. The discussions in terms of only bound electrons and free electrons apply only in situations where hopping electron concentrations are negligible. Unfortunately, hopping electrons are important except

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in fully ordered solids at zero temperature, or in very hot strongly ionized plasmas. The theory of hopping electrons, and the relationship of the ionization process to the concept of the "mobility edge" transition (see Davis and Mott 1975) present in disordered materials apply to plasma situations as well (Dharma-wardana and Perrot 1992). The same ideas are relevant for an understanding of the PPT. The cluster ionization picture can be restated as a local-band theory of disordered materials. Consider a Hydrogen fluid in the weakly ionized prePPT phase of Saumon and Chabrier (to be denoted the prephase, while the phase with a higher ionization will be called the postphase). The prephase is mostly H2 molecules. If we sit on an H2 molecule (the origin), then there is local order as defined by the range Rc beyond which particle correlations die-off and the pair function g(r) becomes essentially unity. This is effectively the "cluster" to consider. We could either think of cluster states or of a local bandstructure in the region r < Rc around the molecule at the center. Electron hopping will be determined by a matrix element T(ry) linking atomic centers i and j , and determines the local bandwidth. In H2 this band is fully occupied at T=0K and hopping occurs by transferring to the unoccupied conduction band, or via an upper "Hubbard band" if that is energetically more favorable (the upper Hubbard band of H2 is approximately the valence band of HJ). However, the situation becomes radically different if even a small amount of H*, H + , etc., and electrons are also present, as in the prephase. Then we have a small number of holes in the H2 local-valence band, a few electrons in the local-conduction band, as well as "impurity gap states" of H^", H*, H + , H and H~ structures (the H and H~ bandstructures are essentially like the lower and upper Hubbard bands of H). The gap states provide a degree of hopping conductivity via overlapping localized states, but the mean free path of the hopping electron remains smaller than the cluster size Rc- However, when the percolation threshold (e.g. Stauffer 1979) is reached the hopping paths "percolate" through the whole volume and provide at least one electronic state which carries the electron out of the cluster. This is the onset of the so-called mobility edge and amounts to a phase transition in the sense that localized carriers have now become delocalized. The idea of the mobility edge was first presented by Mott and further developed by many authors (see Davis and Mott 1975). It is very likely that if the proposed PPT exists, then it is essentially a transition across the mobility edge, and its existence depends on the sharpness of the mobility edge. Note that crossing the mobility edge to reach the more conductive postphase does not require that the carriers have reached the conduction band - they have merely crossed into the per-

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colating impurity states in the local band gap. The postphase need not to be a fully ionized phase, as has been assumed by some authors. The cluster picture of ionization involves a gradual delocalization of a bound electron on a given atomic site to occupy increasingly larger-sized bound states defined on a local transient cluster (hopping states) and finally in the whole plasma (full ionization). This picture is more convenient than the local-band picture with gap states. However, the two models complement and clarify the nature of the ionization process and the PPT. In an earlier study (1982) we considered a hydrogen plasma where each proton supported only one bound state. A significant feature of the bound ls-state reported in that study was that its average radius was larger than the mean proton-proton separation i.e., a hopping state. The bound state was calculated with the proton interacting with an "average" cluster of field ions (FI) and electrons around a proton. The Fl-cluster is modeled by the ion distribution pg(r), where p is the average (bulk) nuclear density. Although such an "average cluster" is incorrect at short time scales, it is valid for thermodynamics which depend only on space and time averaged quantities. To treat the molecular species present in the prephase and the postphase, a multi-center model is needed and will be taken up next.

12.3 Construction of pseudoatoms and pseudomolecules In the so called "chemical" picture for EOS calculations one assumes (by "chemical intuition") that certain well defined chemical species, e.g., H2, H, etc., exist in a given fluid and that their "internal" electronic coordinates do not appear in the discussion. Instead, energy spectra and weights of isolated molecules appear in "internal" partition functions. The internal partition functions of isolated atoms or molecules contain divergencies which are removed by various prescriptions. In a more fundamental analysis (sometimes called the "physical picture") electrons and nuclei interact via the Coulomb interaction, and the interplay between the electron coordinates and the nuclear coordinates is retained right up to the final stage of free energy minimization which determines the mixture composition, particle distributions and the thermodynamics. Even in the physical picture we can talk of molecular species noting that these are "pseudomolecules" (PM) which are "coupled to each other" since the internal structure of a PM depends on the environment around it, i.e., the self-consistent distribution of field particles around it. In the low density limit these PM reduce to the molecules of the chemical picture. Also, the pair-interactions of the PM will

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be a function of the particle distributions in the medium which have to be generated in situ while the free-energy is minimized. If we consider the prephase and the postphase of the PPT, then we need to construct pseudomolecules like H2, H, H*, H + , H~, H^" in the presence of the equilibrium concentration of electrons, at a given temperature and total density. Here an H + is really a proton carrying no bound electrons, plus a distribution of electrons, ions, and other molecular species (the field-particles), while H and H" carry a singly or doubly occupied bound state which spills into the respective self-consistent FP-distribution (FPD). These field-ion distributions are given by the classical form of the Kohn-Sham equations, while the boundstates and field-electron distributions are given by the (quantum) Kohn-Sham equations which are coupled to the classical equations. The electron coordinates and the ion coordinates are retained through out the energy minimization. Unfortunately, this procedure is numerically too arduous since we need to self-consistently resolve a multi-center, multi-species problem as well as quantum mechanical problems for bound and continuous spectra. Hence we consider a less ambitious procedure based on using single-center DFT calculations to construct a tight-binding-type model to account for the molecular states of the pseudomolecules.

12.3.1 Tight binding model A hydrogen atom placed in a given field-particle distribution (FPD) will have a finite set of bound states u , v = n,ltm and energy €u as well as a spectrum of scattering states v = k,l,m with eu = k2/2. From our previous work (single center calculations) we know how to determine this Kohn-Sham energy spectrum of bound and free states for a given FPD. To simplify the discussion we assume that there is only one bound state, «i 9 , but several bound states would be needed in many applications. We

fit the uu(r) function to the form u(Z*,r) = (Z**/n)l/2exp(-Z*r). This simple form where Z* is the only fit-parameter is probably adequate for the plasma-problem. Although we are concerned with H-plasmas, we also carry out single-center calculations for the He + + and He+ ions in the given FPD and determine their Kohn-Sham spectra. He + + and He+ correspond to the "zero bond length" limit of two protons, or a proton and an H atom. This "united-atom" limit of the molecular species is required to correctly recover the interaction energies at smaller particle separations. Now we consider the construction of pseudomolecules.

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12.3.2 H2 pseudomolecule The electron Kohn-Sham equation for the H* molecule in the fluid is :

(1)

1 + Vxc{r, n(f), p(r)} ^(r, R) = E^{r, R)

Here R is the internuclear separation between the protons labeled A and B; the electron coordinate f referred to the midpoint, and to the centers A and B are related by rA = r + R/2 and ?B = f— R/2. Also Vp and Vxc are the Poisson potential and the exchange-correlation potential due to the FPD. The xc-potential is a density-functional potential which brings in the many-body effects of exchange and correlation with electrons, and correlations with ions. Their formulation has been discussed in our earlier work (e.g., see Perrot, Furutani and Dharma-wardana, 1990). An approximate construction of Vp and Vxc for the two-center problem is given below. Consider the isolated molecule where Vp and Vxc = 0. In the limit where R is large we have two (gerade and ungerade) solutions : 4>(r,Z*,R)g,u = [u(Z*,rA)±u(Z*,rB)]/21/2

large R limit

(2)

with Z* = 1 and the energy

_x [(1 + R)e~2R ± (1 - 2r 2 /3)e- fi ] Eg,u = f Is + R

f, I /I i .

However, this is a poor solution for small R. In the R « 0 limit we use a value of Z* = ZQ consistent with the solution of the He+ problem, i.e., the "united atom" obtained from H*. That is, we find Z*(R) variationally for each R for the isolated H* problem so that

interpolates from R = 0 to R = 00, with Zoo = 1- The minimum of Eg(R) a,t R = Ro is the binding energy with the optimal Z*(RQ). This approach provides an excellent solution to the H^ problem (i.e., to the accuracy we needed for the plasma-fluid problem) since the experimental (Herzberg 1954) binding energy Eb = Eg(R0) - ei3 = -2.79 eV and Ro = 2.003 a.u.,

280

Perrot & Dharma-wardana: Equation of state of dense hydrogen

while this method (Bransden et al 1983) gives Eb = -2.25 eV and # 0 = 2.0 a.u. with the optimal Z*(Ro) = 1.23. The vibration- and rotation- spectra can be calculated as usual from Eb(R). Now consider the pseudomolecule H* in the presence of the FP (e.g., H2, H, H*, H + , H~, HJ etc). A given FPD can be rewritten as a density p(r) of protons and a density n(f) of electrons. The p(f) and n(f) interacting with the pseudomolecule whose mid-point is at the origin of coordinates produce the potentials Vp(f) and Vxc(r). These potentials are constructed from the one-center functions Vp(f,H),Vp(f,'H.e+) and V xc (^H),V rc (f,He + ) which are known from the single center calculation for H and He + . In effect we define Vp(?) = Vp(rA,Re+)F(R) + ~{VP(rA, H) + Vp(rB, E)}{Vp(rA, H+) + Vp(fB, H+)}

(5)

x {1 - F(R)} where the superposition takes place in the independent atom limit for very large R, while for small R we recover the united-atom limit of He + . The interpolation function F(R) could be modeled by f(R) of Eq. (4) or a better form can be constructed. The same interpolation scheme can be used for constructing Vxc(f) from the results of the one-center calculation and hence all the potentials need in the two-center Kohn-Sham equation, i.e., Eq.(l) are known. Now in variationally determining Z* appearing in ij)(r, Z*) we do not use just Eq.(3) but include also the FP-energy contribution, i.e. < V(r, Z*)\Vp(r) + Vxc(r)\j>(r, Z*) >. This will give us a new Z*(R,p,n) to replace Eq.(4), where the limiting forms ZQ and ZQ© correspond to the He"*" and H atom wavefunction-exponents from the one-center calculations with the field particles self-consistently included. A new binding energy curve Ef,(R,p, n) dependent on the FP-distributions p(r), and n(f) and a new intermolecular equilibrium distance RQ will result from this calculation. If there are no H* molecules in the system under the assumed conditions, then there will be no stabilizing minimum and E\,{R) > 0. Thus the method by itself determines the pseudomolecules found in the fluid, independently of any "chemical intuition". The binding energy curve E\,(R,r,n) of H^ calculated here is basically the interaction potential between an H atom and an H + ion separated by a distance R in the fluid, inclusive of all the fluid effects. Given Ef,(R,r,n), the vibrational and rotational spectra and the "internal partition function" can be calculated as usual. The concept of an internal partition function has a meaning only if an

Perrot & Dharma-wardana: Equation of state of dense hydrogen

281

"internal part" and an "external part" can be defined. Since the KohnSham calculation includes the whole "correlation sphere" or cluster defined by the FPD, this issue is non-trivial. In the usual neutral-pseudoatom calculations (e.g. Perrot 1993) for simple metallic fluids, it is sufficient to use a spherical Wigner-Seitz cavity to represent the FPD at the pseudo-ion. The effect of the cavity is corrected for using linear response theory applied to the electron gas. In more complicated pseudomolecular systems the simplifications available for "simple metallic" fluids are not present. However, the effect of the FPD can be allowed for and factored out consistently by using a response function constructed from the Kohn-Sham basis of the pseudomolecule, instead of the response function constructed with plane wave states. Then the numerical work is more demanding. Simplified approaches using Wigner-Seitz volumes for each pseudomolecule and projecting out the contributions from neighboring cells according to some physical scheme etc may also be used. Since the electronic spectra contain only finite numbers of bound states, there are no spurious degeneracies appearing in the partition function summation. The electron density n(f) will be adjusted to self-consistence but the 2center FPD is not reevaluated except in the singe-center step. That is, the electron density results, bond lengths etc., of the two-center step are inputs to recalculate the self-consistent FPD of the one-center step. This approach is valid to second order in the density corrections, and has given good results in other problems (e.g., see Harris 1987). 12.3.3 H2 pseudomolecule Here again we begin with a simple one-parameter model of the isolated H2 molecule and re-optimize this parameter as a function of the field-particle potentials Vp(f) and Vrc(f). For constructing the H2 pseudomolecule we use as our tight-binding basis a generalization of the solutions ^(fj Z*, R) of the H* system. The simplest symmetric form for the spatial part of the H2 wavefunction (singlet spin state) is :

This can be rewritten in terms of the atomic ls-functions as $(ri,r 2 ) = $Cov(ri,r2) + $ion(ri,r2) where

(6)

282

Perrot & Dharma-wardana: Equation of state of dense hydrogen

Instead of Eq.(6) we use the trial form

$(n,r 2,Z*,A) = (1 - A)*cot,(n,r2) + (1 + A)$ ion (n,r 2 )

(7)

where A is the new variational parameter. That is, Z*(R) is the value determined to be optimal for the H* problem and A is specific to the H2 problem. We may also optimize both Z* and A, with improved results. Using this approach for isolated H2 a binding energy Eb(Ro) = —4.00 eV and an equilibrium bond distance Ro = 1.5 a.u, are obtained (cf. experimental values (Herzberg 1954) Eb = -4.7 eV, Ro = 1.4 a.u.). In the pseudo-H2 system inclusive of FPD we have to construct the twocenter potentials Vp(f) and Vxc(r). We interpolate between the infiniteR limit where Vp(r*) is a superposition of two single-center Vp(r) H-atom potentials calculated using the single-center DFT equations, and the R = 0 united-atom limit which is the He-atom. The interpolation function F(R) is similar to that of Eq.(5), and constructed using results of trial calculations. Once the potentials Vp(f) and Vxc(f), and the limiting Z*(R,f, n) values for R = 0 and R = 00, are prepared from the one-center calculations for H and He pseudoatoms, A is now minimized (for each R) including the FPterm, i.e. < {rx,r2,Z*,\)\Vp{r) + Vxc(f)\(n,r2, Z*, A) >. The resulting binding energy curve E(,(R,n,p) of pseudo-H 2 in the fluid can be used to obtain the vibrational and rotational spectra and the "internal partition function" inclusive of the effects of the fluid environment. If the binding energy function Ef,(R, n, p) has no minimum at some ifo then the fluid does not support the existence of H 2 molecules.

12.3.4 Other pseudo-molecules and clusters These methods can be applied to almost any simple hydrogenic cluster. We begin with a simple tight-binding model for the cluster in isolation and then include the FPD via the optimization of a relevant parameter in the Kohn-Sham functions used in the tight-binding model. The method also yields interaction potentials between pseudomolecules without resorting to perturbation theory. Thus we may consider the interaction between

Perrot & Dharma-wardana: Equation of state of dense hydrogen

283

an H2-pseudomolecule with an H + ion in the fluid. The H + ion has a FPD around it. The construction of the potential around the pseudo-H2 molecule was already discussed. Hence if we have to consider the interaction potential between a pseudo-H2 molecule and a pseudo-H+ ion, we use the $(ri,f2,Z*,A) solutions of the previous problem and taking a linear combination involving $ABi $BC, a n d $CA where A, B, and C are the three nuclei of the H2 and H + interacting system. This includes the binding due to electrons hopping among the three nuclei and goes beyond the usual polarization potential models for the H2-H+ interaction. A three-body interaction-potential (which is what the H2-H"*" system is) can be reduced to a 2-body potential either by imposing an equilibrium bond distance RQ to one of the three inter-nuclear separations (valid if the other two distances are >> RQ), or by averaging with pair-distributions functions, as in Aers and Dharma-wardana (1984). The method proposed here is free of hard-sphere models, cutoffs in the partition function etc. It provides a microscopic approach which is capable of (i) determining the stability of a given pseudo-molecular cluster in the fluid and hence deciding whether we need to include it in the thermodynamics, (ii) providing environment-dependent binding energy functions and interaction energy functions, (iii) providing the "internal partition function" of each species taking account of interactions with the plasma, (iv) obtaining triplet and singlet spin states, £ , II momentum states etc., and their correct thermal averages. 12.4 Minimization of the free energy function It is convenient to rewrite the free-energy minimization step outside the multispecies density functional calculation which involves the self-consistent resolution of a set of coupled Kohn-Sham equations for each species, as well as electron spectra for each species. Then the proposed procedure is as follows: (i) We prepare a trial "mixture" of species like H2, H, H*, H + , H~, HJ, e~ at mean nuclear density pn and compositions z;. The electron density is constrained by charge neutrality. (ii) Single-center DFT calculations are done for the united-atom and single atom limits, viz., for He, He+, H+, H, H", and He". The FPD can be simplified if desired, by replacing the FP-molecules by simpler distributions. Thus an H2 molecule is replaced by a distribution of electrons centered at the mid-point of the H2-bond and calculated from |$(ri,r2,Z*, A)|2. The electrons contribute U(T)HI to the total electron distribution n(r). We also

284

Perrot & Dharma-wardana: Equation of state of dense hydrogen

include the two nuclei which contribute />(r)#2 to the proton distribution p(r). These contributions TI{T)H2, P(T)H2 have no effect on the Poisson and xc-potentials (in the local density approximation) unless n(r)H2 n a s some overlap with the pseudomolecule at the origin whose detailed structure is being calculated. The existence of any overlap leads to hopping electronic states which cause modifications in the spectrum of the central particle. (iii) The output from the single center calculations will be electron distributions and FP-distributions. (iv) These are now fed into the multi-center calculations for pseudomolecules. We do not directly solve multicenter Kohn-Sham equations, but use the tight-binding approach where one or two carefully chosen parameters determining the bound state wavefunctions are optimized. The FPD and the continuous electron spectrum are interpolated from the single- and united-atom limits as discussed before, in the spirit of the Harris functionals. (v) The new electron distributions are introduced into step number (ii) above. (vi) Step (iv) is repeated and interaction potentials, binding energy.curves and the internal partition functions are calculated. (vii) A more refined evaluation of the FPD, using the interaction potentials obtained at this stage may be carried out using MD or integrals equations (N.B. The DFT equation for classical particles are formulated as an effective HNC equation, coupled to the electronic Kohn-Sham equation). (viii) A total free energy minimization is carried out to obtain the mixture composition x,-. (ix) These new X{ are fed into step (ii) and the procedure is repeated until self-consistency is attained. (x) At this stage the mixture composition, particle distribution functions, interaction potentials etc.. are completely converged. The equation of state, electrical conductivity (Perrot and Dharma-wardana 1987), thermal conductivity and other properties which could serve as a "diagnostic" of a phase transition can be calculated at this stage. Thus the free-energy minimization step (i.e., step viii) is carried out within the large iterative loop (steps ii to viii) where the adjustment of the internal electronic coordinate is carried out in consort with the evaluation of FPD and the structure and spectra of the pseudomolecules. This is exactly the "physical picture" of a microscopic theory. In this discussion we have not addressed a number of issues. We assume that the protons can be treated as classical particles. Also, when an ion is immersed self-consistently in a fluid containing continuum ("free") electrons, the continuum density of states becomes modified. The effec-

Perrot & Dharma-wardana: Equation of state of dense hydrogen

285

tive masses m* of the electrons near each pseudomolecule change from the usual value of m*/me = 1, and the existence of finite sized bound cores on each ion also affect the compressibility of the electron gas. All these subtle effects contribute to the electron partition function and have to be consistently treated. These issues are basically simpler than the pseudomolecular problems discussed here, and have been adequately treated elsewhere by us, in a recent (unpublished) study of Al-plasmas. 12.5 Conclusion We have presented a tractable microscopic approach to the study of complex fluids containing neutral and charged molecular species and electrons. The intuitive "chemical picture" based on data relevant to isolated molecules, hard spheres etc., is no longer needed. The model is able to take into account the electronic interactions between molecular species and the fluid environment. This allows room for three types of electrons, viz., (i) electrons fully localized on the molecular species, (ii) electrons which hop between the molecular species and the neighboring molecules in the fluid , and (iii) fully delocalized (i.e. "free") electrons. It is suggested that the prephase of the PPT is a system with a high population of hopping electrons, while the postphase is rich in delocalized electrons. The PPT is really a transition across the "mobility edge" and its physical reality depends on the sharpness of the associated percolation threshold. Acknowledgment We thank Gilles Chabrier for requesting us to contribute to this volume and for stimulating our interest in the plasma phase transition. References G. C.Aers and M.W.C. Dharma-wardana, Phys. Rev. A 29, 2734 (1984) B. H. Bransden and C. J. Joachain, Physics of atoms and molecules, Longman, London (1983) J. Davis and N. F. Mott, Theory of Disordered Solids Oxford, U.K. (1975) M. W. C. Dharma-wardana and F. Perrot, Phys. Rev. A 26, 2096 (1982); Phys. Rev. A 45, 5883 (1992) W. Ebeling and W. Richert, Phys. Lett, 108A, 80 (1985) E. K. U. Gross and R. M. Dreizler, Density Functional methods in Physics NATO ASI, II Ciocco, Italy (Plenum 1994) J. Harris, Phys. Rev. 5 31,1770 (1987) G. Herzberg, Molecular Spectra and Molecular Structure, van Norstand, New Jersey (1950); also K. P.Huber and Herzberg, Constants of diatomic molecules (van Norstand, 1979) P. Hohenberg and W. Kohn, Phys. Rev. B 136, 864 (1964)

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Perrot & Dharma-wardana: Equation of state of dense hydrogen

D. J. Hummer and D. Mihalas, Astrophys. J. 331, 794 (1988) W. Kohn and L. J. Sham, Phys. Rev. A 140, 1133 (1965). S. Lindqvist and N. March, ed\, Theory of the inhomogeneous Electron Gas (Plenum, New-York 1983). N. D. Mermin, Phys. Rev. A 137, 1141 (1965). D. Mihalas, D. J. Hummer, and W. v. Dappen, Astrophys. J. 331, 815 (1988) F. Perrot, Phys. Rev. £ 4 7 , 570 (1993); Phys. Rev. A 42, 4871 (1990) F. Perrot and M. W. C. Dharma-wardana, Phys. Rev. A 30, 2619 (1984); Phys. Rev. A 36, 238 (1987) F. Perrot, Y. Furutani and M. W. C. Dharma-wardana, Phys. Rev. A 41,1096 (1990) D. Saumon and G. Chabrier, Phys. Rev. A 46, 2084 (1992) D. Stauffer, Physics Reports, Scaling theory of percolation clusters 54, 1 (1979) D. J. Stevensen and E. E. Salpeter, Astrophys. J. Suppl. Ser. 35, 221 (1977) E. B. Wigner and H. B. Huntington, J. Chem. Phys. 3, 764 (1935)

13 The equation of state of fluid hydrogen at high density G. CHABRIER Laboratoire ie Physique, Ecolt Normak Superieure it Lyon, 69864 Lyon CtitxOl, France

Abstract We present a free energy model for fluid hydrogen at high-density and hightemperature. This model aims at describing pressure dissociation and ionization, which occur in partially ionized plasmas encountered in the interiors of giant planets and low-mass stars. The model describes an interacting mixture of H2,H,H+ and e~ in chemical equilibrium. The concentrations of H 2+ and H~ ions are found to be negligible for equation of state purposes. Our model relies on the so-called chemical picture approach, based on the factorization of the partition function into translational, internal and cohfigurational degrees of freedom. The present model is found to be unstable in the pressure-ionization regime and predicts the existence of a first-order plasma phase transition (PPT) which ends up at a critical point given by Tc = 15300 K, Pc = 0.614 Mbar, and pc = 0.35 gem" 3 . The transition occurs between a weakly ionized phase and a partially ionized (~ 50%) phase. Nous presentons un modele d'energie libre pour l'hydrogene fluide a haute densite et haute temperature. Le but de ce modele est de decrire la dissociation et l'ionisation en pression, telles qu'elles se produisent dans les plasmas partiellement ionises rencontres a l'interieur des planetes geantes et des etoiles de faible masse. Le modele decrit un fluide en interaction compose de H2,H,H+ et e~ en equilibre chimique. Les concentrations de H 2+ et H~ sont negligeables pour les calculs d'equation d'etat. Notre modele repose sur 287

288

Chabrier: Fluid hydrogen at high density

l'approche chimique, basee sur la factorisation de la fonction de partition en degres de liberte translationnel, interne et configurationnel. Le modele presente une zone d'instabilite dans le domaine d'ionisation en pression et predit l'existence d'une transition de phase plasma (PPT) du premier ordre, se terminant en un point critique donne par Tc = 15300 K, Pc = 0.614 Mbar, et pc = 0.35 gcm~3. La transition a lieu entre une phase faiblement ionisee et une phase partiellement ionisee (~ 50%). 13.1 Introduction The similarities between hydrogen and alkali metals lead Wigner and Huntington (1935) to suggest that pressure-ionized hydrogen would behave like a monovalent 'metal' even at zero-temperature, and it has often been argued that first-order phase transition must occur between the two states, given the large difference between the molecular and the metallic states (Stevenson and Salpeter 1977; Ebeling and Richert 1985). Most of the recent investigations have focused on the zero-temperature or the room-temperature, where static compression experiments are now available above the megabar domain. These results indicate that a transition between a molecular and a semi-metal state occurs around 2 Mbar, but true metallization has not been observed unambiguously yet. This state is believed to occur around 2-3 Mbar. Few models exist however at high-temperature, in the fluid range, where shock-wave experiments have clearly establish the stability of the fluid molecular phase up to 0.8 Mbar (Nellis et al. 1984). Fluid hydrogen is the main component of stellar interiors and atmospheres. In most of astrophysical objects, hydrogen is ionized by temperature, but pressure ionization occurs in the interior of giant planets, brown dwarf stars and partially in the outermost layers of white dwarfs and low-masss stars. The recent discovery of global oscillations in Jupiter (see Mosser, these proceedings), as well as the recent achievements of helio- and astero-seismology (see Dappen and Fontaine, these proceedings) give us new information on the structure of these objects, and then on the properties of matter under extreme thermodynamic conditions. This stress the need for a correct calculation of fluid hydrogen at high-density, including a proper treatment of pressure dissociation and ionization. A simplified phase diagram of hydrogen at high temperature is shown on Figure 1, with the internal temperature profiles of a few dense astrophysical objects. In section II, we give a short presentation of our free energy model, which has been presented extensively elsewhere (Saumon and Chabrier 1991,

J

289

Chabrier: Fluid hydrogen at high density

6

5

iog,oT[K] Fig. 13.1 Simplified (/>,T) phase diagram for hydrogen. A few physical regimes are identified : above the line F = 1, where F is the usual coupling parameter, the classical ionic plasma is strongly coupled whereas correlations are dominant in the quantum electron plasma below the line r, — 1, where r, is the mean inter-electronic distance in unit of Bohr radius. Electrons are degenerate above the line 0 = kT/ej = 1, where ep is the electron Fermi energy. Protons are classical below the line A, = 1, where A,- is the De Broglie wavelength in unit of the mean inter-ionic distance a. The curve F = 178 denotes the crystallization line of the H + plasma. The various temperature profiles are characterise of the interior of Jupiter (J), a brown dwarf (BD), a ZZ-Ceti white dwarf (WD) and the Sun (S).

1992). In section III, we discuss in detail the PPT predicted by our model whereas a comparison with other models is discussed in Section IV. Section V is devoted to the astrophysical applications and Section VI to the conclusion. 13.2 Description of the free energy model The model relies on the so-called chemical picture, in the sense that we assume the existence of independent, bound configurations such as H atoms, H2 molecules, interacting with pair potentials. At densities corresponding to pressure ionization, such a scheme is erroneous and the concept

290

Chabrier: Fluid hydrogen at high density

of individual pair potential fails, requiring the use of quantum-statistical many-body theory, i.e. a physical picture where only fundamental particles (electrons and nuclei) exist (see the reviews by Rogers, Alastuey and Perrot and Dharma-wardana). In particular, our approach does not take into account the possibility of excitonic states, i.e. clusters or pseudo-atoms and pseudo-molecules, as described in Perrot and Dharma-wardana (these proceedings). Although formally exact, the physical picture, however, involves either diagramatic expansion which converge only at low-density or hightemperature (see the reviews by Rogers and Alastuey), or, when extended to higher-density, involve a coupled treatment of classical and quantummechanical N-body theories, which renders practical applications for the calculation of astrophysical EOS nearly impossible. In view of these difficulties, the chemical picture, and its inherent factorization of the partiction function in terms of different particle interactions, remains a very powerful method. It can be view as the best compromise between the rigorous treatment and the practical application. It is why it is important to compare the results obtained with "chemical" models with the ones derived from "physical" ones. Our EOS consists of a general free energy model which applies in the regime of partial temperature- or pressure-ionization, and which reduces to a so-called "neutral" model and "plasma" model respectively at lowdensity /low-temperature and at high-density and/or high-temperature. 13.2.1 Model for neutral hydrogen At low-density (p £ 1 gem" 3 ), low temperature (T^ 104 K), hydrogen is adequately described as a mixture of H atoms and H2 molecules. The concentrations of H~ and H 2+ ions are found to be negligible (< 10~3) for EOS purposes. Because all particles are very nearly classical in this regime, we can factorize the partition function and treat the small quantum effects with a semi-classical approximation. If we make the additional assumption that the internal levels of atoms and molecules are only weakly affected by the presence of nearby particles, as suggested by available experimental results up to electronic densities as large as Ne/V ~ 1021 cm" 3 (Weise et al. 1972, Grabowski et al. 1987, Hashimoto and Yamaguchi 1983), the Helmoltz free energy separates into ideal, configurational, internal, and quantum contributions : F(NH, NH2 ,V,T) = Fid + Fconf + Fint + Fqm

(1)

Chabrier: Fluid hydrogen at high density

291

13.2.1.1 the configuration term This term arises from the interaction between the different particles in their ground state. Computation of these interactions requires the knowledge of three interaction potentials 4>H2-H2, H-H and <j>H2-H- For H2-H2 use an effective potential derived from shock tube experiments (Nellis et al. 1984; Ross et al. 1983) which implicitly includes many-body effects. Since no similar experimental data exist for 4>H-H and 4>H2-HI we have used ab-initio potentials (Kolos and Wolniewicz 1965; Porter and Karplus 1964). We treat the spin dependence of the H-H interaction by averaging the interaction potentials of the singlet and triplet states; the resulting H-H potential has no bound states. The three potentials have been fitted by generalized Morse potentials. The configuration free energy FCOnf is derived from theua interaction potentials in the framework of the WCA fluid perturbation expansion (Weeks, Chandler and Andersen 1971). In this theory, the interaction potential is split into a repulsive reference potential 4>re*(r) and a weak, attractive perturbation potential 4?ert{r). We approximate the free energy of the reference system by that of a hard sphere fluid, which is known analytically (Mansoori et al. 1975), whereas the contribution of the perturbation potential is given by the first term of the free energy expansion (High Temperature Approximation) :

Fconf(N,V,T) = f 4%p{r)g5${r)d?

J

(2)

Here the ga,p(f) are the hard sphere pair correlation functions (Griindke and Henderson 1972) and a\ and o-i are the density and temperature-dependent hard sphere diameters determined thermodynamically by the WCA criterion (Weeks, Chandler and Andersen 1971). The standard WCA scheme, derived originally for liquid state theory, has been extended for this particular approach to high-density and high-temperature (Saumon, Chabrier and Weis 1989). The excess internal energy and pressure derived from this expansion scheme agree within less than 3% with MC simulations for the density and temperature range of interest (Saumon, Chabrier and Weis 1989). This assesses the validity of the configuration energy (2) for the H-H2 mixture.

we

292

Chabrier: Fluid hydrogen at high density

13.2.1.2 The Internal Free Energy The effect of near-neighbor interactions on the internal structure of bound species is essential to a correct description of pressure dissociation and ionization. We have used in our model the so-called occupation probability formalism (OPF) derived by Hummer and Mihalas (1988). In this formalism, the internal free energy reads :

Fint = -kTLn J2 Na^2uaigQie-e^kT a=l

(3)

i

where i runs over all internal states of species a and u)ai,gai and €ai are respectively the occupation probability, the multiplicity and the unperturbed energy of state i. The density dependent uai are computed from the configuration term Fconf in the free energy. This ensures consistency of both the interactions and their effects on the IPF. It also provides a smooth cut-off of the IPF, and therefore a plausible pressure dissociation/ionization effect. Moreover the present method uses unperturbed energy eigenvalues and does not invoke hypothetical energy level shifts of doubtful validity. As a matter of fact, such shifts have been shown to be too small to be significant, as mentioned above (Wiese et al. 1972, Grabowski et al. 1987, Hashimoto and Yamaguchi 1983). In practice, however, one must resort to a linearization of Fconf to compute the occupation probability (see Hummer and Mihalas (1988) for details). Calculations including terms beyond the density-linear term have been computed recently for helium at high-density (Aparicio and Chabrier 1994). In addition our occupation probabilities include neutral particle interaction only. The effect of charged particles, i.e. Stark ionization, requires a knowledge of the plasma microfield distribution at high-density, which complicates tremendously the calculations. The effect of the microfield will be discussed in detail later in the paper. In our treatment of the IPF of H2, we have included all vibrational and rotational levels of each bound state of the molecule (Hiiber and Herzberg 1979). The term Fqm in Eq.(l) is the quantum contribution to the free energy, which is always a weak perturbation of the classical free energy in the domain of interest for the hydrogen EOS, and then has been calculated to the first non-vanishing order in the Wigner-Kirk wood h2 expansion.

Chabrier: Fluid hydrogen at high density

293

13.2.2 Model for fully ionized hydrogen For kT £ lRyd or p £ 2gcm~3 (corresponding to ra ~ 1, where rs is the mean inter-electronic spacing in units of Bohr radius a 0 ), the fluid is a fully ionized electron-proton plasma. At these densities, the mean electron-ion potential energy E{e = e2 /rsao is smaller than the electron Fermi energy Ep so that the plasma can be described as a superposition of an electronscreened ionic fluid and a rigid electron background (Ashcroft and Stroud 1978). Under these conditions, the free energy of the plasma reads : F = Fid - NkTIn I e/3U<"(rUf+ Fxc + Fqm

(4)

where Fid denotes the ionic and electronic perfect gas contributions, Fxc is the exchange and correlation free energy of the electron gas at finite temperature (Ichimaru et al. 1987). The second term on the r.h.s of equation (4) is the free energy of the screened ionic fluid, calculated within the framework of the hyper-netted chain (H.N.C) theory for the temperature and density deAir(Ze)2fk2e(k,V,T) pendent screened Coulomb potential Ueff(k,V,T) (Chabrier 1990). The dielectric function c is the finite temperature Lindhard function corrected with a local field correction for the short-range interaction between electrons (Utsumi and Ichimaru 1982). The model free energy (4) shows excellent agreement with existing Monte-Carlo calculations at finite and zero-temperature and with non-adiabatic calculations (see Chabrier 1990 for details). At very low-density and high-temperature, the electrons behave almost classically and the thermodynamics of the electron-proton plasma can be evaluated by the Debye-Huckel two-component limit, corrected for small degeneracy effects (DeWitt 1962) :

2/3 (1 2

x

a

-

where < Z > = X)a a^a> denoting the ionic and electronic species. The first term in Eq.(5) is the leading term of the classical cluster expansion, i.e. the Debye term, whereas the second term in the bracket represents the correction for electron quantum effects, given by the electron quantum diffraction parameter fe = •^(2irh2nee2/me). At intermediate densities, the free energy is interpolated smoothly between the high-density model (4) and the low-density model (5). The quantum correction Fqm for the ions is also calculated to leading order in h2, using a Wigner-Kirkwood expansion for the screened potential Ueff-

294

Chabrier: Fluid hydrogen at high density

13.2.3 Model for partial ionization The two models described above are combined to give a description of the thermodynamics of the plasma in the partial ionization zone. The interaction between charged and neutral particles in their ground state is treated through a polarization potential approach (Kraeft et al. 1986). The twobody potential is approximated by an interpolation between a hard-core repulsion at short distance and a screened dipolar potential outside the core : (6) The hard core radii R{ are chosen to be the radii for H and H2 obtained for the configuration energy from the WCA criterion, OJJ denotes the polarizability of species i, and K is the inverse screening length of the electron-ion plasma, which enters the screened potential Ue/f in Eq.(4) (see Saumon and Chabrier 1992 for details). The hard core contribution effectively reduces the volume available for the ionic and electronic ideal terms by a factor (1 — 77) where 77 is the hard core packing fraction for H and H2. The second contribution introduces an additional polarization term Fpoi to the free energy given by :

Fpol = UT^f- £ NQBQi

(7)

The Bai denote the virial coefficients of particle of species a, i.e. : Bitc-

= BitH+

(1 - e-^"" ^)r2dr

= Bi = 2x

(8)

JRi

The general model free energy for the hydrogen fluid finally reads: F(V, T, NH2, NH, NH+, Ne-) = Fid(V, T, NH» NH) + Fid((l-V)V,T,NH+,Ne.) + Fj?(V,T,NH2,NH) + Ffx(V,T,NH+,Ne.) + Fpol(V, T, NH2, NH, NH+,Ne-)

(9)

where the subscript " id" denotes the ideal contribution whereas Fff and Ff stand for the non-ideal contributions of the neutral model and the x

295

Chabrier: Fluid hydrogen at high density

6

8

10

12

3

V(cm /mole of D 2 )

Fig. 13.2 Single and double-shock Hugoniot curves of D 2 . The solid curves shows the theoretical Hugoniot curves derived from the model free energy (9) after suitable modification for deuterium. The squares, circles and triangles denot the experimental data.

fully ionized model developed in the previous sections, respectively. Having imposed the electroneutrality condition, we minimize the free energy (9) at fixed total density and temperature to obtain the chemical equilibrium of the four component mixture (H2,H,H+,e~) :

dF

dF

0Xi

where F = F(x\,X2,p,T)

=0

(10)

is the specific free energy per proton,

X2 = Xfj2 and P is the mass density.

13.3 Results and discussion Figure 2 shows the theoretical Hugoniot curves derived from our model for the single- and double-shock experiments on H and D2 respectively along with the experimental data. In view of the experimental uncertainty, the agreement is excellent. We calculated the limit of stability of the model free energy (9) as a function of the density along several isotherms and found a first order phase

296

Chabrier: Fluid hydrogen at high density

transition (dP/dp < 0) in the regime of pressure ionization. The transition is associated with a sudden, discontinuous shift of the chemical equilibrium toward a high degree of ionization. We emphasize that this phase change occurs strickly as a result of the minimization of our free energy model. It does not arise from any additional assumption. We have accordingly studied the phase equilibrium in the regime of pressure ionization systematically (Saumon and Chabrier 1992) : Ti

= Tn

. pi

=

pn

. ^ ( p , r ) = pg(P,T)

(11)

The characteristic of the coexistence curve are given in Table I. The phase transition ends in a critical point whose coordonates are : Pc = 0.614 Mbar, pc = 0.347 gem" 3 and Te = 15300 K. The slope of the coexistence curve dP/dT is negative, which is consistent with the positive entropy discontinuity #5 = S11 — S1, a likely consequence of the increasing contribution of thermal effects at higher temperature, and the larger entropy in the plasma phase than in the molecular phase. Figure 3 shows the concentration of molecules and charged particles as a function of density for a few isotherms. We draw the following conclusions: i) The system undergoes a first-order phase transition from a neutral phase (xe- < 10~2 for T < Te) to a partially ionized phase (xe- « 0.5) as density increases. ii) At the transition pressure, the degree of ionization increases discontinuously whereas the concentration of molecules drops drastically, indicating that molecular dissociation and pressure ionization occur at almost the same density. Pressure ionization does not occur by first dissociating the molecules into atomic hydrogen as believed usually. iii) Above the critical density, the system reaches complete ionization very gradually. This points out the qualitative difference found when treating pressure ionization with realistic potentials and with pure hard sphere interactions (Ebeling and Richert 1985). Even though the model for the neutral species is highly questionable above the transition density, we believe these qualitative features to be physicaly realistic. iv) The first-order phase transition persists even when no coupling exists between the neutral and the fully ionized models. This indicates that the source of the PPT does not lie in the interaction between neutral and charged particles but rather in the very nature of the difference of interactions in the respective neutral and plasma phase. We actually believe the PPT to be due to the large difference between the strongly repulsive, hardsphere type potential in the insulating, molecular phase and the smoother

Chabrier: Fluid hydrogen at high density

297

Table 13.1. Characteristic of the plasma phase transition. For each temperature, we give the transition pressure, the density and the ionization fraction for each phase. The change in entropy is AS = Sn - S1. pi

(K)

P (Mbar)

gem" 3

gem" 3

xlO" 3

3.70 3.78 3.86 3.94 4.02 4.10 4.18 4.185

2.14 1.95 1.62 1.39 1.13 0.895 0.631 0.614

0.75 0.70 0.64 0.58 0.51 0.43 0.35 0.35

0.92 0.88 0.80 0.74 0.65 0.55 0.38 0.35

1.4 2.1 3.0 5.1 8.8 20. 170. 180.

io g r

2x'H+

2*j/ +

AS ka/proton

0.48 0.50 0.50 0.51 0.52 0.50 0.33 0.18

0.615 0.590 0.544 0.508 0.464 0.421 0.142

0

Yukawa-type potential in the conducting, plasma phase. This is supported by the fact that the PPT disappears above a temperature for which the dominant species in the neutral phase is no longer molecular hydrogen but atomic hydrogen, since the repusive part of the H-H potential is very similar to the screened inter-ionic potential (see Saumon and Chabrier 1992). In this sense, the PPT can be related to the metal-insulator transition in metals associated with the liquid-vapor transition. The crucial difference between the two effective interactions in the metallic and insulating phases, which reflects the change of the nature of the electronic states, leads eventually to a polarization catastrophe and the impending insulator-metal transition (Goldstein and Ashcroft 1985). 13.3.1 Uncertainties in the plasma phase transition In order to examine importance of some physical inputs of our model on the final results, we have carried out a limited set of calculations of the PPT using variations on our free energy model. These various PPT results are summarized on Figure 4, where the solid curve is the PPT derived from the basic model (9). 13.3.1.1 Influence of coupling

The most basic approximation in our model is to forbid the neutral and the charged particles to mix. This corresponds to a simple comparison of the two free energies (1) and (4). The resulting phase transition is what

298

Chabrier: Fluid hydrogen at high density

2.5

3.0

3

p(gcm" ) Fig. 13.3 Concentration of H2 and of charged particles (H++e~) near the PPT. Isotherms are labeled according to logT = A, 3.70; 5,3.86; C.4.02; D, 4.18; £,4.34. The left panel shows the low-density behavior on a logdensity scale.

we call the "forced" transition. It is shown by the dashed line in Fig. 4. The transition line lies at P « 3.2 Mbar and is nearly independent of the temperature. A more realistic variation on our final model is indicated by the triangles in Fig. 4. For these calculations, the PPT is computed without any coupling between the two models (1) and (4). In this approximation, plasma and neutral particles are allowed to mix but the two fluids do not interact. This corresponds to the free energy model (9) with n = 0 and Fpoi = 0. The highest temperature point shown for this calculation (logTc = 4.215) indicates the corresponding critical point. As already mentioned, it is important to stress that the first-order phase transition persists in this approximation, when no coupling exists between the neutral and the fully ionized models, indicating that the source of the PPT does not lie in the interaction between neutral and charged particles. The next variation is provided using a different expression for the screening length of the plasma so that the polarization free energy Fpoi is increased by about a factor of 5. This results in the coex-

299

Chabrier: Fluid hydrogen at high density

-0.6

4.2

Fig. 13.4 The effects of various assumptions in the free energy model upon the plasma phase transition. The solid and dashed curves are coexistence curves for the PPT and for the forced phase transition, respectively. The crosses ( x ) indicate the PPT obtained in a calculation where Fpoi (Eq.(7)) was overestimated by a factor of 5. The dotted line shows the coexistence curve obtained when using the polarization radius of Redmer et al. (1987) in Fpoi. When softened interaction potentials are used, the PPT occurs at higher pressures, as shown by the asterisks (*). When no coupling is allowed, the transition is shown by triangles (A). The open circle (o) indicates the effect of softening the lowest vibrational frequency of the Hj molecule by 4%. See text for details.

istence curve indicated by crosses, with the critical point at logT c = 4.175. The fourth variation is to use the polarisation radius for atomic H determined by Redmer, Ropke and Zimmermann (1987), RH = 1.4565 a.u. The H2 polarization radius was estimated by scaling the H value with polarizability : RHl « -R/^ajyj/a/y) 1 / 3 w 1.55 a.u. These radii are about 50% smaller than the ones used in the original calculations, increasing Fpoi by about one order of magnitude. The free energy model still undergoes unstabilities and the resulting phase line is shown on Fig.4 by the dotted line. The transition pressure is lowered by about 40%. The new critical point lies near logT c = 4.0 and Pc = 0.76 Mbar. This is from far the most important variation in our results. We have examined ex post facto the effect of Stark ionization as a form of coupling between the neutral and the ionized free energy models on our

300

Chabrier: Fluid hydrogen at high density

results. The effect of the plasma microfield is estimated by modifying the occupation probability in introducing a Holtzmark distribution. This is reasonnable in the low-density phase where the concentration of charged particles remains small, as discussed above. We find the effect to be negligible for logT < 4.1. The importance rises rapidly near the critical point, however, following the increase in degree of ionization. 13.3.1.2 #2 internal partition function In this section we examined the effect of the recently observed softening of the vibron frequency in the IPF of the H2 molecule (Hemley and Mao 1988). We have recomputed the transition pressure with the model given by Eq.(9) after reducing the vibration constant We in the IPF of H2 by 4%, as suggested by the experiment. This raises the transition pressure by 1%, as shown by the open circle in Fig.4. An other important issue is the rotational partition function of H2. The asymetry of the H2-H2 potential is expected to hinder rotation at highpressure, which can be interpreted as an increase of the rotational temperature 0rot- That could decrease substantially the entropy of the molecular phase and affect the location of the PPT. Our treatment uses the rotation temperature of the free molecule (« 85 K) at all densities. Recent experiments (Hemley, Mao and Shu 1990) show that the H2 molecules undergo significant rotational motion up to 1.6 Mbar at 77 K, a pressure typical of the PPT coexistence curve. Interestingly, the Raman spectrum of the roton mode does not show any line shifts of consequence for the EOS : 0rot appears to be nearly independent of pressure, well above the PPT. Moreover, thermal effects are more important in our calculations so that we can expect rotational modes to freeze at much higher temperatures than is observed experimentally below room temperature. 13.3.1.3 Softening of the potential Recently, Hemley et al. (1990) derived a new experimentally determined H2-H2 potential based on X-ray diffraction measurements of solid H2 at T=300 K and P £ 0.26 Mbar. This potential is slighly softer than the potential used in our calculations (Ross et al. 1983; Saumon and Chabrier 1991). Given the lack of information for the H-H and H2-H potentials at high density, we mimicked many-body effects by softening arbitrarily the repulsive part of the potentials respectively by 20% and 35%. Results are shown by the asterisks in Fig.4. In all cases, the pressure and the density of transition increase slightly, as expected for a softened EOS in phase I. The

Chabrier: Fluid hydrogen at high density

301

qualitative features of he PPT are not affected, and the magnitude of the effect is less than 5%.

13.3.1-4 Strong ion-electron coupling Our free energy model in the fully ionized phase is based on the so-called linear response theory, which is valid as long as the ion-electron interaction is negligible compared with the ion-ion interaction, or the electron-ion potential energy is negligible compared with the electron kinetic energy. This fails at low density, where the electrons become strongly correlated, and where electron localization begins. We expect the ion-electron non-linear effects in the plasma phase to lower slightly the free energy of the plasma (by a few percents around the critical point), favoring ionization. The same effect is expected from a proper inclusion of micro-field effects on the excited states. Table II shows preliminary comparison between the present calculations and calculations by Perrot and Dharma-wardana (these proceedings) based on the so-called density-functional theory, where ion-electron interactions in the plasma are treated beyond the linear response. As expected the degree of ionization is underestimated in our calculations. However the pressure and the entropy obtained within the two formalisms are in very good agreement (^ 3%). This adds credibility to the present EOS for dense hydrogen since the results are based on two completely different type of calculations.

13.3.1.5 Band effects A potentially important aspect of pressure ionization we have ignored is the onset of delocalization of the electronic wave functions at high density, leading to possible electronic "conduction bands" in the neutral mixture, similar to band narrowing in solids. Given the lack of exact calculations of such electron delocalization effect in a finite-temperature fluid, we have estimated the size of the effect in an ex post-facto calculation. We used zero-temperature band-gap calculations (Friedli and Ashcroft 1977). The fraction of electrons thermally excited into the conduction band varies from less than 10% at 8000 K to 2% at the critical point. Consequently we find the effect on the PPT to be small. Moreover, as discussed previously, the H2 molecule is likely to undergo substantial rotation at pressures characteristic of the PPT, which probably widens the band gap.

302

Chabrier: Fluid hydrogen at high density

Table 13.2. Comparison between the present model (lower row) and calculations based on the Density Functional formalism (DFT) of Perrot and Dharma-wardana (upper row) for hydrogen at high temperature and density. For each temperature-density point, we give the effective charge, the pressure and the entropy. The effective charge Z* denotes the average degree of ionization. (units are in CGS) logT

\ogp

Z*

logP

log S

5.161

-1.269

0.777 0.712 1.000 0.695 1.000 0.883 1.000 1.000 0.923 0.812 1.000 0.781 1.000 0.904 1.000 1.000 0.992 0.938 1.000 0.910 1.000 0.960 1.000 1.000

12.045 12.033 12.837 12.811 13.85 13.826 15.182 15.169 12.313 12.296 13.169 13.154 14.104 14.088 15.273 15.263 12.908 12.893 13.802 13.787 14.702 14.693 15.652 15.643

9.209 9.213 9.125 9.128 9.012 9.009 8.899 8.879 9.264 9.263 9.178 9.178 9.071 9.067 8.957 8.943 9.344 9.340 9.269 9.266 9.179 9.176 9.075 9.066

-0.4755 0.4275 1.330 5.462

-1.3455 -0.4755 0.4275 1.331

6.06

-1.375 -0.4755 0.4275 1.331

13.4 Other models Marley and Hubbard (1988) also computed a forced transition between a neutral and a fully ionized model. Compared to other efforts, their two models are closest to our own in level of accuracy and detail. Their transition line lies slightly above the one obtained in our calculations for the forced transition (see Fig.4). The difference arises from the cruder treatment of the internal levels of the molecules in Marley and Hubbard's calculations. Ebeling and Richert published two calculations for the P P T . Their two models differ in detail but are very similar in spirit and give essentially the

Chabrier: Fluid hydrogen at high density

303

same critical point, at Tc = 16500 K, Pc = 0.228 Mbar and pc = 0.13 gem" 3 . In the first model (Ebeling and Richert 1985a), atoms and molecules interact through hard-sphere potentials with fixed diameters, plus a Van-der-Waals correction. Molecules are approximated as two atoms in the hard-sphere free energy. Internal states are not included in the treatment and there is no coupling between charged and neutral particles. Their approximate coexistence curve crosses the experimental double-shock Hugoniot curve for deuterium, where no evidence is found for a PPT. This rules out the model, based on too crude approximations. In a second paper (Ebeling and Richert 1985b), molecules are excluded from the model. Bound states are introduced in the form of the PlanckLarkin partition function. This has been demonstrated to be incompatible with a chemical picture (Dappen et al. 1987). They find a second critical point surprisingly close to their previous dtermination. Again this is incompatible with existing shock-wave experiments at high-temperature which must be reproduced by any model aimed at describing the EOS of dense fluid hydrogen. In particular, the critical pressure is lower than predicted by the model (9). This fact stems from the excessively repulsive hard-sphere potentials between neutral particles. These potentials are completely inapropriate in the density range where pressure ionization occurs. 13.5 Effect of the PPT on the structure and the evolution of low-mass stars and giant planets The PPT predicted by our model occurs at densities and temperatures characteristic of giant planets and low-mass brown dwarfs. New interior models of Jupiter and Saturn including the PPT have been computed recently (Chabrier et al. 1992). Interior models assuming homogeneous H/He envelopes (i.e. no PPT) could not satisfy the observational constraints, adding titillating support for the presence of a PPT between a molecular I^/He envelope and a metallic H + /He + + envelope. The phase transition modifies the thermal structure of the planets, leading to a hotter internal adiabat than the one obtained with no PPT, because of the positive entropy jump at the PPT (see Table I). The effect of the PPT on the evolution of Jupiter, Saturn and low-mass brown dwarfs has also been examined in detail (Saumon et al. 1992). For fully convective, adiabatic objects, the evolutionary timescale is given by : Ldt = Lsdt - j (TSS)dm Jo

(12)

304

Chabrier: Fluid hydrogen at high density

for a time interval dt. L is the luminosity of the planet, La = 4TTR2
Chabrier: Fluid hydrogen at high density

305

Huber K.P. and Herzberg G., Molecular Spectra and Molecular Structures, (Van Nostrand, Princeton), (1979) Hemley R.J. and Mao H.K., Phys. Rev. Lett, 6 1 , 857, (1988) Hemley R.J., Mao H.K. and Shu J.F., Phys. Rev. Lett, 65, 2670, (1990) Hemley R.J., Mao H.K., Finger L.W., Jephcoat A.P., Hazen R.M. and Zha C.S., Phys. Rev. B, 42, 6458, (1990) Hummer D.G. and Mihalas D., Ap. J. 331, 794, (1988) Ichimaru S., Iyetomi H. and Tanaka S., Phys. Rep., 149, (1987) Kang H.S, Lee C.S., Ree T. and F. Ree J. Chem. Phys. 82 (1), 414, (1985) Kollos W. and Wolniewicz, J. Chem. Phys., 43, 2429, (1965) Kraeft W.D., Kremp D., Ebelimg W. and Ropke G., Quantum Statistics of Charged Particle Systems, Plenum (1986) Mansoori G.A., Carnahan N.F., Starling K.E. and Leland T.W. J. Chem. Phys. 54 (4), 1523, (1971) Marley M.S. and Hubbard W.B., /earns, 73, 53, (1988) Nellis W.J., Holmes N.C., Mitchell A.C., Trainor R.J., Governo G.K., Ross M. and Young D.A., Phys. Rev. Letters 53 (13), 1248, (1984) Porter R.N. and Karplus M., J. Chem. Phys., 40, 1105, (1964) Redmer R., Ropke G and Zimmermann R., J. Phys. B, 20, 4069, (1987) Ross M, Ree F.H. and Young D.A., J. Chem. Phys., 79, 1487, (1983) Saumon D. and Chabrier G., Phys. Rev. A 44, 5122, (1991) Saumon D. and Chabrier G., Phys. Rev. A 46, 2084, (1992) Saumon D., Chabrier G. and Weis J.J., J. Chem. Phys., 90, 7395, (1989) Saumon D., Hubbard W.B., Chabrier G. and Van Horn H.M., Ap. J. 391, 827, (1992) Stevenson D.J. and Salpeter E.E., Ap. J. Suppl, bf 35, 221, (1977) Utsumi K. and Ichimaru S., Phys. Rev. A, 26, 603, (1982) Weeks J.D., Chandler D. and Andersen H.C., J. Chem. Phys. 54, 5237, (1971) Weise W.L., Kelleher D.E. and Paquette D.R., Phys. Rev. A, 6, 1132, (1972) Wigner E. and Huntington H.B., /. Chem. Phys., 3, 764, (1935)

14 A comparative study of hydrogen equations of state D. SAUMON Lunar and Planetary Laboratory, University of Arizona, Tucson, AZ 85721, USA

Abstract The numerous complexities underlying large tables of thermodynamic quantities act as a deterrent to a careful evaluation of their reliability. As a consequence, equations of state are often used as black boxes. To clarify this situation, some of the more critical issues of equation of state physics are discussed from the point of view of the user. They are illustrated by a comparison of four equations of state for hydrogen. The flaws and disagreements thus brought into light are explained and evaluated with simple physical arguments. Les tables d'equations d'etat utilisees en astrophysique decoulent de modeles d'une complexite telle qu'il est souvent difficile d'en evaluer la fiabilite. H en resulte une situation ou les equations d'etat sont souvent utilisees sans une analyse critique de leur contenu physique ni de leur precision. Dans le but de remedier a cette situation, une discussion des principaux elements physiques des equations d'etat est presentee dans l'optique de l'utilisateur. Quatre equations d'etat de l'hydrogene developpees pour etre appliquees a des problemes d'astrophysique stellaire sont comparees de fac,on critique. Cette comparaison illustre l'importance de certains elements des des equations d'etat et la nature des problemes qui subsistent. Les defauts et les differences observes entre ces quatre equations d'etat sont elucides en termes de physique de base. 306

Saumon: Hydrogen equations of state

307

14.1 Introduction The richness of stellar phenomena exposed by modern observational techniques calls for a quantitative understanding of more subtle, "second order" effects in stellar structure. Examples of phenomena requiring accurate modeling of the underlying physics include the solar oscillation spectrum, the solar neutrino problem, stellar pulsations, and the origin of the abundance of elements in photospheres of white dwarfs. The equation of state (EOS) \ constitutes one of the properties of matter, along with transport coefficients, which enter the equations of stellar structure (Clayton 1983, p. 436ff). The pressure P(p,T) and the entropy S(p,T) appear explicitly in these equations and dictate the mechanical and thermal equilibria of the star, respectively. Beyond the fact that an equation of state is necessary to compute a stellar model, the quantitative understanding of "subtle" stellar phenomena does depend, sometimes sensitively, on the assumed EOS (see for example, the review by Dappen 1994). The importance of this point is not always realized. The astrophysicist interested in EOS as an input to solve a particular problem is faced with a number of difficult choices. Most applications require such complex EOS that it is not practical to compute one locally. This raises a number of questions which are often overlooked. Which of the available EOS is most appropriate for a particular problem? Does it include the proper physics? Which one is the most reliable? How does it compare with other EOS? Because EOS calculations usually involve a large number of approximations and assumptions as well as some level of internal inconsistency, it is very difficult to answer these questions from published literature alone. In this context, a direct comparison of several EOS tables and of the underlying assumptions becomes a powerful tool to reveal flaws and poor approximations and to develop a healthy appreciation of the uncertainties which persist in some physical regimes. The complexity of an EOS calculation increases considerably when nonideal effects are introduced. In fact, such calculations can only be performed numerically, and the results usually presented in tabular form. Historically, the Lawrence Livermore National Laboratory and the Los Alamos Scientific Laboratory have invested much effort in the development of tabular equations of state which are frequently used in astrophysical applications. Significant progress in the computation of realistic equations of state under t Strictly speaking, the equation of state is the relation between pressure, temperature and density, P(p,T). In the context of this work, we loosely apply the term to the ensemble of equilibrium thermodynamic properties of matter, such as the entropy, internal energy, specific heat, etc.

308

Saumon: Hydrogen equations of state

stellar conditions was made by Graboske, Harwood and Rogers (1969), Kerley (1972), Fontaine, Graboske and Van Horn (1977), Lamb (1974), Lamb and Van Horn (1975), Magni and Mazzitelli (1979) and more recently, by Rogers (1981), Hummer, Mihalas and Dappen (Hummer and Mihalas 1988; Mihalas, Dappen and Hummer 1988; Dappen et al. 1988) and under conditions typical of the interior of giant planets by Stevenson (1975), Stevenson and Salpeter (1977), Hubbard and DeWitt (1985, and references therein), and Marley and Hubbard (1988). The past decade has seen tremendous progress in our understanding of dense matter physics, on both the experimental and theoretical fronts. Equations of state for dense plasmas are now becoming well understood, thanks in part to progress in computer technology, which has permitted simulations of ever increasing complexity. These studies have demonstrated the great utility of a variety of approximations for the computation of plasma properties. In particular, when a sample of any substance is sufficiently compressed, atoms (or molecules) are so closely packed that the exclusion principle promotes bound electrons into conducting states (for hydrogen, this occurs near lg/cm 3 ). This "pressure ionization" represents a thorny problem in the calculation of an equation of state. It is often avoided by simply interpolating between atomic and fully ionized limits. Recent advances in statistical physics offer the opportunity for significant improvements in our understanding of this poorly understood phenomenon. Although great progress is being made in the laboratory, most astrophysically interesting regimes are still weakly constrained by experimental data. Thus the validity of an EOS can only be established in an indirect fashion. It should, of course, reproduce known asymptotic limits. Computer simulations also provide useful but limited tests for theoretical equations of state. In addition, the EOS is subject to the fundamental thermodynamic constraints of mechanical and thermal stability, < 0, and — dV v respectively, and of thermodynamic consistency,

dP df

_ dS = v dV

where P and S represent the total pressure and entropy of a system occupying a volume V at temperature T. Equations of state which satisfies these constraints is not necessarily accurate. In regimes dominated by non-ideal effects, and where neither experiments nor computer simulations are available, it is very difficult to establish

Saumon: Hydrogen equations of state

309

their validity. Nevertheless, the magnitude of uncertainties and their relative merit can be estimated by comparing independent EOS calculations and understanding the effects of the respective underlying assumptions. In this review, we have chosen four hydrogen EOS for a detailed comparison of their respective thermodynamic variables. Hydrogen has a rich phase diagram and illustrates most of the situations encountered under astrophysical conditions. It is the most abundant element in the universe and constitutes 75% of the mass of the majority of stars. It is therefore representative of the bulk of stellar matter. The choice of a pure composition, as opposed to a cosmic mixture of hydrogen and helium, for example, greatly simplifies the discussion of the underlying physics. One of these EOS (Saumon and Chabrier 1991, 1992) is not fully published and a few remarks relevant to this comparison are given below. But first, we briefly review the phase diagram of hydrogen in an astrophysical context as well as the most frequently used method for computing EOS, the free energy minimization technique, on which all four EOS discussed here are based. 14.2 The phase diagram of hydrogen Phase diagrams play an essential role in illuminating the EOS physics relevant to a particular problem. The simplified phase diagram for hydrogen of Fig. 1 helps to make a few basic points. In the low-density, lowtemperature region, hydrogen is essentially neutral and forms atoms and molecules. Molecules dominate at low temperatures (logT < 3.5) | and they dissociate into atoms as the temperature is raised. At still higher temperatures, atoms ionize to form a low-density plasma of protons and electrons. The dashed curve delimiting these three regions indicate a degree of dissociation (or ionization) of 50% and is based on detailed chemical equilibrium calculations for a non-ideal mixture of H2, H, H + and e. At densities above logp « —2, atoms and molecules interact strongly and form a non-ideal fluid. In addition, the Saha equations become inappropriate for log/o > —1, so that it is not possible to estimate the chemical equilibrium in this dense fluid with simple theories. At even higher densities, near log/) = 0 for hydrogen, the mean distance between H atoms becomes comparable to twice the value of the Bohr radius and the electronic wave functions of neighboring atoms overlap. The electrons are forced into unbound states and the fluid becomes a pressure ionized plasma. A calculation of pressure ionization by Saumon and Chabrier (1992) reveals that pressure ionization t Throughout this work, log T is the logarithm of the temperature in K, and log p is the logarithm of the mass density in g/cm 3 .

310

Saumon: Hydrogen equations of state

£ o 00

O

-10

-

Fig. 14.1 Phase diagram for hydrogen. Heavy solid lines separate various physical regimes. Below the Praa = Pga, line, the pressure of the photon gas exceeds that of the matter (H+ and e). Electrons are degenerate above the 0 = 1 line, and protons form a strongly coupled plasma above the T = 1 line. The thick curve labeled PPT shows the metastable region of the Plasma Phase Transition. The abundance of atomic hydrogen, H, is 50% along the dashed curve which indicates regimes of partial dissociation and ionization. The dotted curves are interior models for a) Jupiter, b) 0.3 MQ main sequence star, c) the Sun, d) the outer hydrogen layer of a T = 12 500 K DA white dwarf and e) a 15 MQ main sequence star.

of hydrogen may not be a gradual process at all temperatures but could occur discontinuously through a first order phase transition, the so-called plasma phase transition (PPT). The metastable region of this transition is shown by the curved labeled "PPT" and ends at a critical temperature of logTc = 4.185. Two important issues pertaining to the plasma are the degree of electron degeneracy and the strength of the Coulomb coupling between the charged particles. Above the solid line labeled 9 = 1, where 9 = eF/kT, the Fermi energy of the electrons €F is larger than kT, and they are therefore degenerate. Protons, on the other hand, remain classical over most of this diagram and in all astrophysical conditions (except in neutron stars). Above the line T = 1, non-ideal Coulomb effects play an important role as the electrostatic potential energy between two protons, e2/a, where e is the

Saumon: Hydrogen equations of state

311

charge quantum and a is the mean interparticle distance, becomes larger than their kinetic energy, kT. At intermediate temperatures (logT « 5) and densities of log p « 0, temperature and pressure ionization are of comparable importance. In this regime, thermal excitation of hydrogen atoms is significant and they are immersed in a moderately coupled plasma (F « 1) where electrons are partially degenerate (6 « 1). This regime is particularly difficult to treat as the internal levels of the atoms are strongly perturbed by the surrounding plasma. For most elements, this is, along with pressure ionization, the regime where equations of state are most unreliable. At low densities and high temperatures, radiation pressure Prad becomes larger than the gas pressure, PgM- Finally, the upper left part of Fig. 1 represents conditions which are not realized in astrophysical contexts, where hydrogen is a high-T molecular solid or possibly forms a Coulomb lattice. Interior models of various hydrogen-rich objects are shown by dotted curves in the density-temperature plane of Fig. 1. The gaseous envelope of Jupiter is shown by the curve labeled 'a.' The envelope is dominated by molecular hydrogen and it goes through the region of pressure ionization. If the PPT calculated by Saumon and Chabrier (1992) occurs in nature, it should also be found in the envelope of Jupiter. Just below the PPT, the dense molecular fluid becomes strongly non-ideal due to the strongly repulsive intermolecular forces. Curves 'b', 'c', and 'e' represent main sequence stars with masses of 0.3, 1 and 15 Af©, respectively, where M© is the mass of the Sun. The 15 M© star has the simplest EOS physics. It is fully ionized throughout its interior and the plasma is very weakly coupled (F << 1). Electron degeneracy is also weak (0 << 1). The contribution of radiation pressure is significant, however and the ratio Prad/-Pgas is roughly constant in the interior. The solar model (c) is both cooler and denser. Accurate modeling requires attention to relatively weak non-ideal effects (F « 0.1) and partial electron degeneracy near the center. Recombination of the plasma into H atoms affects the structure near the surface. Low-mass stars, such as the 0.3 M© model shown, probe more complex areas of the phase diagram. Electrons are partially degenerate throughout most of the star and the electrostatic interactions in the plasma become significant. The model crosses the difficult regime where F and 0 are of order unity. In the outer part of the model, recombination forms atoms in a non-ideal regime and finally molecules form at the very surface. The curve labeled 'd' is a 12 500 K DA white dwarf envelope, stratified into hydrogen-rich and helium-rich layers surrounding a carbon core (H, He and C, respectively). Only the outermost layer, consisting of pure hydrogen, is

312

Saumon: Hydrogen equations of state

shown here. In this layer, hydrogen forms a weakly coupled, non-degenerate plasma. Atomic hydrogen is found at the very surface of the star. The relatively low densities and high temperatures characteristic of this layer indicate that non-ideal effects in the EOS are of moderate importance.

14.3 The Free Energy Minimization method Nearly all EOS involving chemical equilibrium, i.e. molecular dissociation or ionization of atoms and ions, are based on the free energy minimization technique (FMIN). One major exception is the EOS of Rogers (1981) which is based on an activity expansion. It is briefly discussed in § 4. The FMIN method is well described in Graboske, Harwood and Rogers (1969 and references therein), Fontaine, Graboske and Van Horn (1977) and Hummer and Mihalas (1988). The approach is particularly simple. Given a mathematical model for the Helmholtz free energy of the system as a function of total volume, temperature and particle numbers, F(V,T,{Ni}), the chemical equilibrium of the mixture is obtained by minimizing of F at fixed V and T, subject to the stoichiometric constraints imposed by the chemical reactions taking place in the system. This fixes the {iVj}, and the pressure and entropy can then be calculated by differentiation of the free energy with respect to V and T, respectively, at fixed {iV,}. These so-called first derivatives of the free energy are given by:

av

and

The specific heats, compressibility, thermal expansion coefficients, adiabatic gradients are obtained by further differentiating P and S with respect to V and T and are second derivatives of the free energy. Differentiation amplifies the features and defects in F and since the second derivatives are usually obtained numerically, they are also prone to numerical noise. The FMIN method becomes truly useful when the grand partition function of the system Z is written as the product of kinetic, internal and configuration contributions:

Fontaine, Graboske and Van Horn (1977) give an excellent discussion of the four approximations leading to this factorizability. In practice, small

Saumon: Hydrogen equations of state

313

deviations from exact factorizability are usually accommodated by corrections based on expansions in terms of a small parameter. However, when the particles of the system interact strongly, the spectrum of bound states is affected and Z\at and Zconf are not factorizable anymore. Similarly, the bound state configuration determines the interaction potentials and modifies Zconi • The total partition function is nevertheless factorized, with some modification of the spectrum of bound states entering the internal partition function Zjnt (IPF) based on the interaction potentials. A great variety of treatments of this problem have been used, some very crude and with very low internal consistency, some quite sophisticated. However, there is no formally exact treatment and this is the source of many disagreements between EOS computed with this method. Despite this shortcoming, the FMIN technique has several powerful advantages. In principle, it ensures thermodynamic consistency of the resulting equation of state. All the physics and approximations appear at the outset in the free energy model and are therefore quite visible. No additional approximations are required. Contrary to expansion techniques, contributions with strongly non-linear dependence on density or temperature can be included with no additional effort. Under the assumption of factorizability, the free energy model becomes a sum of terms, each involving a different physical contribution. This is extremely convenient, as each term becomes a subroutine in the EOS code. Terms can be added, removed and modified with great ease. Hummer and Mihalas (1988) point out that this method will work for any free energy model. However, because the validity of the resulting EOS is hard to test, unless it violates fundamental constraints or known limits, great care must be taken in constructing the free energy model to ensure internal consistency. 14.4 Choice of equations of state In the spirit of clarity and conciseness, the comparison is limited to four hydrogen equations of state: the table of Fontaine, Graboske and Van Horn (1977), the pure H case of the H/He EOS of Magni and Mazzitelli (1979), f a pure hydrogen calculation based on the model developed by Mihalas, Hummer and Dappen (Dappen et al. 1988 and references therein), and the EOS of Saumon and Chabrier (1991, 1992). Hereafter, these four equations of state will be referred to as FGVH, MM, MHD and SC, respectively. These EOS span over 15 years of effort in developing reliable EOS for t A study of the MM EOS table which we obtained in 1987 shows that it is much improved over the version published in Magni and Mazzitelli (1979).

Saumon: Hydrogen equations of state

314

5

-

6 u ttO

00

o

-10

-

4

5

6

log T (K) Fig. 14.2 Density-temperature domain covered by the four EOS considered. MHD.dashed line (note the extension above logp = -2); FGVH, dotted line; SC, solid line; MM, dot-dashed line. The MM EOS extends beyond the limits of the figure. Pairs of triangles show the isotherms selected for the EOS comparison. stellar envelopes and interiors and they are representative of the better EOS currently in use by the community. Except for the SC EOS, they have been used extensively in a variety of astrophysical contexts. All four equations of state are based on the FMIN technique. While they have a number of features in common, they differ greatly in detail, in the level of internal consistency of the model and in the accuracy of the various contributions to the free energy. The (p,T) domain covered by each EOS is shown in Fig. 2. Two contiguous domains are shown for the MHD EOS. While the free energy model was developed for logp < - 2 , the calculation was pushed to higher densities, as shown by the figures in Mihalas, Dappen and Hummer (1988) and in Dappen et al. (1988). This "extension" of the MHD EOS above logp = - 2 is shown in Fig. 2 and is discussed further in § 6. Additional comparisons between the SC EOS and other equations of state used in astrophysical problems have also been performed. The H/He EOS of Marley and Hubbard (1988) was developed for modeling the interior of giant planets. It shares many similarities with the model of SC and under

Saumon: Hydrogen equations of state

315

the low-temperature conditions relevant to giant planets (logT < 4), the two EOS are nearly identical. Differences arise in a narrow density domain centered on pressure ionization, where Marley and Hubbard simply interpolated between the dense molecular fluid and the fully pressure-ionized plasma. For logT > 4, their approximate treatment of temperature ionization leads to substantial disagreement. A more detailed discussion is given in Chabrier et al. (1992). The hydrogen EOS developed at the Los Alamos National Laboratory, known as the SESAME 5251 EOS, was developed by Kerley (1972). A comparison of pressures P and internal energies U (second derivatives are not directly available) from an EOS table obtained in 1984 with the SC EOS shows a relatively good agreement for logT > 3.7. This is somewhat surprising if we consider that the SESAME 5251 hydrogen EOS is actually a deuterium EOS scaled in density. Differences in pressure reach a maximum of 25% in the regime of pressure ionization. In the regime of temperature ionization differences are as high as 40%. At the lower temperatures where molecules dominate the EOS, systematic differences of « 6% are found in U (Saumon and Van Horn 1987). This arises from the density scaling procedure which is not appropriate in the molecular phase. The energy levels of the molecule depends on the moment of inertia and the reduced mass of D2 which are twice as large as for H2. Rogers (1981) has developed an EOS with an approach entirely different from FMIN, using an activity expansion which considers only protons and electrons interacting with the Coulomb potential. Bound states (atoms) arise naturally in this approach and are not treated as a separate chemical species, as in the FMIN method. This approach is very rigorous and fundamental (Rogers 1994). Over the (/>, T) domain where this complex method can be presently solved, it leads to a most accurate EOS. While we have not yet compared it with the SC EOS, it has been compared with the MHD EOS under the conditions found in the solar envelope (see Dappen 1994 and Rogers 1994). The two equations of state are in extremely good agreement, with differences of less than 0.1% in the second derivatives of the free energy. While such differences are important when comparing the computed solar oscillation spectrum to the wealth of extremely precise data, they are completely negligible in all other astrophysical situations. It is very satisfying that two equations of state based on entirely different approaches should agree so well. This indicates that our understanding of the EOS of normal stellar material is now excellent, at least over some parts of the phase diagram.

316

Saumon: Hydrogen equations of state

14.5 About the Saumon-Chabrier EOS The free energy model underlying the SC EOS is described in details in Saumon and Chabrier (1991, 1992) but the EOS itself is not yet available (Saumon, Chabrier and Van Horn 1993). The model is summarized in the review by Chabrier (1994) and a few additional remarks relevant to the present comparison follow. Pressure ionization received particular attention in the SC EOS calculation. It was found that the adopted free energy model (and a number of variants) becomes thermodynamically unstable and predicts the existence of a first order phase transition between a mostly molecular phase and a dense, partially ionized phase. This plasma phase transition (PPT) is shown in Fig. 1. It terminates at high temperature at a critical point located at Tc = 15300K, Pc = 6.14 x 10 u dyn/cm 2 , and pc = 0.35g/cm3. Pressure ionization is a most difficult problem in EOS calculations and much remains to be said on this challenging topic. There is currently no experimental result which bears on the existence of the PPT. To allow for the possibility that the PPT is not realized in nature, there is an "interpolated" version of the SC EOS where the discontinuities associated with the PPT have been smoothed by interpolation. It is otherwise identical to the SC EOS with PPT. This interpolated version used for the comparisons in § 6. The interpolated region has an irregular shape but it extends roughly over 3.50 < logT < 4.78 and -0.5 < logp < 0.5. Ideally, the interpolation of P and the entropy S (or P and U) should be constrained by the requirement of thermodynamic consistency, which reflects that P and S are not independent quantities but derive from the same thermodynamic potential, in this case the Helmholtz free energy. Fontaine, Graboske and Van Horn (1977) applied this constraint when interpolating across the regime of pressure ionization. In the case of the SC EOS, however, it was found that the requirements of 1) continuity of P, S and their derivatives at the boundaries of the interpolation region and of 2) thermodynamic consistency overconstrain the interpolation. This difficulty can be avoided by widening the density range of the interpolation but only to an unacceptable degree where parts of the EOS table believed to be reliable (based on experimental data and an assessment of the model) would be replaced by less accurate, interpolated values. This suggests that while there may not be a PPT in hydrogen, pressure ionization probably occurs rather suddenly. Reliable EOS values were preserved at the cost of losing thermodynamic consistency and P and S were interpolated separately along isotherms over as narrow a density range as possible.

Saumon: Hydrogen equations of state

317

The free energy model underlying the SC EOS can be improved upon and ameliorations are being considered. The most important of which involves the effect of charged particles on the bound states of hydrogen atoms. Interactions with neighboring particles, charged and neutral, affect the number of bound states, or the internal partition function, of atoms and molecules. A proper treatment of this effect is essential for an accurate description of partial dissociation and ionization, particularly at larger densities (logp > —3). While the activity expansion of Rogers (1981) accounts for this naturally and rigorously, the FMIN method is only weakly constrained in this respect. In its current form, the free energy model developed by Saumon and Chabrier (1992) accounts only for the effect of neutral particles on bound states (by an excluded volume effect). In reality, neighboring charged particles also affect the bound states by inelastic collisions with bound electrons and also through the fluctuation micro-electric field induced by their thermal motion. This microfield has the effect of a time dependent perturbation on the Coulomb potential of the nucleus and can induce Stark ionization of the upper levels of an atom. Collisions and microfield effects on hydrogenic atoms are discussed in great details in Hummer and Mihalas (1988) who conclude that for log/> < —1.5, the microfield is the dominant mechanism. Being caused by random thermal motions, the fluctuating microfield is described by a statistical distribution. Hummer and Mihalas have adopted the F = 0 Holtzmark distribution. We have found that this distribution, which does not account for the correlations which arise between charged particles at T > 0, has much too strong an effect on the IPF and leads to spurious results for F « 1. Generating microfield distributions for finite F is computationally involved and a suitable, parametrized form was not available when the SC EOS was computed. As a consequence, the effect of the microfield is ignored altogether until an adequate distribution function becomes available. The net effect of this omission is that as the gas becomes mostly ionized by temperature, the IPF is less affected by the neighboring particles than when it was surrounded by neutral particles. This creates a long tail of residual atoms in the partial ionization zone. We will return to this point in § 6. Two thermodynamic surfaces from the SC EOS are show in Figs. 3 and 4. Only second derivatives of the free energy are shown because they display the various physical regimes more clearly than first derivatives such as P and 5". They also amplify defects in F they are very useful to reveal flaws in an EOS as well as its degree of smoothness. Note that the MHD free energy model is sufficiently simple to allow for analytic differentiation of F and their EOS is consequently very smooth. As emphasized in the reviews

318

Saumon: Hydrogen equations of state

Fig. 14.3 Inverse compressibility, xP = 01og.P/dlogp|x, for the (p,T) range covered by the Saumon-Chabrier EOS (interpolated version). The (p, T) grid shown is that of the original tabular data and no smoothing has been applied.

by Fontaine (1994) and Dappen (1994), smoothness of the EOS can be more desirable than accuracy in the context of non-adiabatic stellar pulsations. Figure 3 shows Xp — dlogP/dlogpfr, which measures the stiffness of the EOS. For an ideal gas, Xp = 1> as can be seen over most of the lowdensity part of the figure. The photon gas pressure depends only on T and Xp = 0 in that limit. The degenerate electron gas is less compressible than the Maxwell-Boltzmann ideal gas and Xp Tlses t o a plateau at 5/3. The two shallow "valleys" seen on either side of logT = 4 are due to molecular dissociation and ionization. They clearly separate the regions dominated by H2, H and H + , respectively. The very steep rise seen at low-T and highp is caused by the repulsive core of the H2-H2 interaction potential. Like

319

Saumon: Hydrogen equations of state

log Fig. 14.4 Adiabatic gradient, Vad = dlog7'/01og.P|.s for the (p,T) range covered by the Saumon-Chabrier EOS (interpolated version). The (p,T) grid shown is that of the original tabular data and no smoothing has been applied. all liquids, dense fluid H2 is relatively incompressible, a property reflected by the high value of Xp- There are a few spurious features caused by the interpolation procedure at intermediate T and high p. As indicated in Fig. 2, the EOS does not extend to the low-T and high-p limit where Xp — 0. Figure 3 is best interpreted in reference to the phase diagram shown in Fig. 1. Most of the physical regimes discussed above can be identified in Fig. 4 which shows the adiabatic temperature gradient, V»d = d\ogT/dlogP\s. When expressed in terms of p and T, V ^ is a function of the four second

derivatives of F: dP/dp\T, dP/dT\p, dS/dp\T and dS/dT\p. It therefore combines all the defects and noise found in the second derivatives. This is

320

Saumon: Hydrogen equations of state

the origin of the several spikes seen in the interpolation region. Other peaks and oscillations found along the high-/) border of the EOS are caused by edge effects in the table. 14.6 The EOS comparison Even by limiting the comparison to four EOS, it is not possible to do justice to the great efforts which went in their development or to review the merits of each one of them. The discussion will be limited to the areas were the largest differences arise. The introduction of spurious errors was kept to a minimum by avoiding numerical interpolation in the tables as much as possible. For that purpose, six isotherms common to all four EOS were selected: logT = 3.70, 4.10, 4.50, 5.30, 6.10, and 6.90 (Fig. 2). The last three isotherms are not tabulated by MM and the necessary T-interpolation was performed with a program provided with the table. The figures show the density points of the original tables connected by a straight line. The EOS were not "smoothed", however, it has been remarked before that the FGVH EOS has a number of "bad points" where the second derivatives of the free energy show anomalous behavior. Since these points are isolated, they must not arise from deficiencies in the underlying thermodynamic description but represent some localized numerical quirk. A few of these points are found in the six isotherms under consideration and the discordant values were corrected by a simple interpolation in density. These points are located at (logr,logp): (4.10, -5.667), (4.50, -3.667), (5.30, -2.333) for Vad. The quantities compared are logP, logf/ and Vad for all six isotherms. Exceptions are the MHD EOS which is not shown for logT = 6.10 and 6.90, the table available being limited to logT < 6, and the MM EOS which gives only logP, Vad and C p , the specific heat. This last quantity is not used in the present comparison. Note that all quantities shown here are taken directly from the EOS tables and are not constructed from other quantities by using thermodynamic identities, for example. This avoids introducing potential errors due to thermodynamic inconsistency in the EOS or numerical inaccuracies in the procedure. In all four cases, the zero of energy is chosen as the ground state of the H2 molecule and the contribution of the photon gas is included. The four EOS are compared in Figures 5-9 where it is readily apparent that the differences can be substantial. As a point of reference, the SC EOS indicates that P and U are within 1% of their ideal gas value for log/9 < —2. It is easy to verify that the non-ideal terms are very small at this density

Saumon: Hydrogen equations of state

1

321

1

14 -

——"

a

12 -

W /

0,

on O

10 -

8

-4

/

• sc - FGVH MHD - MM

^

1

1

1

-3

-2

-1

_

1 0

3

log p (g/cra ) Fig. 14.5 Comparison of pressure isotherms from the four equations of state. The isotherms are (from top to bottom): logT = 6.90, 6.10, 5.30, 4.50, 4.10 and 3.70. by comparing the volume occupied by atoms (or molecules) to the total volume. Surprisingly, significant differences are found even for densities below logp = — 2.

14-6.1 The pressure Figure 5 shows the pressure from the four EOS along the six isotherms. At lower and at higher densities than shown in this figure, the agreement is satisfactory. The two hottest isotherms correspond to a fully ionized gas of H + and e interacting weakly in the Debye-Hiickel limit. The photon pressure dominates gas pressure when P becomes independent of p along the log T = 6.90 curve. The agreement is excellent in this relatively simple

322

Saumon: Hydrogen equations of state

regime but severe divergences are found at lower temperatures. Curiously, the MM EOS systematically overestimates P at low densities, where the gas is ideal for all practical purposes. This is most likely due to an overestimate of the degree of dissociation and perhaps ionization which arises from their treatment of the IPF of H and of H2. Their IPF for H2 has been corrected since we obtained the MM table in 1987 (Mazzitelli 1993). For the three lowest isotherms, the MHD and FGVH EOS predict much higher pressures than either the SC or the MM EOS at moderate densities. For temperatures up to logT = 3.6, the SC EOS reproduces experimental results and can be considered as a reference for this comparison. The high pressures of FGVH and MHD are caused by the hard sphere potential used to model the interactions between neutral particles. This potential qualitatively models the strongly repulsive cores of the actual potentials, but being infinitely repulsive, it fails to describe the softness of the repulsion. This feature of neutral-neutral interactions becomes important at high densities. The hard sphere potential is too repulsive at high densities and leads to overestimated pressures even in a regime where the gas should be nearly ideal. The authors of the MHD EOS point out that their EOS should be used for log/> < —2, a safe limit at low-T in view of the above observation. In the regime of pressure ionization (—0.5 < log/9 < 0.5), the SC, MM and FGVH EOS can differ by up to a factor of 2. In all three cases, thermodynamic quantities were smoothly interpolated between a low-density and a high-density regime where the authors felt that their respective EOS were reliable.

1^.6.2 The internal energy Most of the features discussed above can also be seen in the internal energy U, shown in Fig. 6. Again, we see that for logT < 4.50 and \ogp > —2, the hard sphere model used by FGVH and MHD leads to an overestimate of U. The logp = — 2 limit recommended by MHD is a sensible choice for U as well as for P. The two intermediate isotherms illustrate the importance of a careful treatment of the influence of neighboring particles on the IPF. For the logT = 4.50 isotherm, SC lies above MHD and the reverse is true for logT = 5.30. At these low densities, characteristic of the ideal gas, this arises from differences in the degree of ionization, which is directly affected by the IPF of atomic hydrogen. At these temperatures, thermal excitation of H becomes significant and the chemical equilibrium depends on how

323

Saumon: Hydrogen equations of state

cm u

(90

o

13

-

12 —

-2

0 log p (g/cm 3 )

Fig. 14.6 Comparison of internal energy isotherms. The isotherms are (from top to bottom): logT = 6.90, 6.10, 5.30, 4.50, 4.10 and 3.70. many states are allowed in the IPF sum.f This effect was not visible in the pressure because it is relatively insensitive to excitation energies of bound species. At logT = 4.50, the degree of ionization is sufficiently low for the finite "size" of atoms to be the main non-ideal contribution. This, in effect, is an excluded volume interaction which removes the upper levels of the IPF to ensure that the atoms do not "overlap." MHD adopted a fixed and somewhat arbitrary diameter for the H atom in its ground state (1.06 A) while SC use a thermodynamic criterion (Saumon and Chabrier 1991) to compute a temperature and density dependent value ranging from 1.1 to t The SC and MHD EOS do not use a cut off in the IPF sum but a gradual removal of bound states based on the occu pation probability formalism presented in Hummer and Mihalas (1988). It is nevertheless useful to think in terms of a sharp cut off in the present context.

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Saumon: Hydrogen equations of state

about 1.6 A. Fewer states are retained in the IPF when the hard sphere diameter is larger, favoring a higher degree of ionization in the SC EOS and a larger U. Because the SC EOS uses more realistic interaction potentials between neutral particles and a thermodynamic criterion to obtain the hard sphere diameters of H and H2, it is more reliable in this regime than the MHD EOS. The situation is quite different along the log T = 5.30 isotherm where the degree of ionization is high and atoms are surrounded mostly by charged particles. As discussed in § 5, the motion of the ions and electrons induces a fluctuating micro-electric field which can cause Stark ionization of the upper levels of the atom, thereby removing them from the IPF. Since this effect is missing in the SC EOS, the IPF retains too many states and the degree of ionization as well as the internal energy are underestimated. According to Fig. 6, this effect is not very large, but the MHD EOS is nevertheless more accurate in this regime. Along the logT = 5.30 isotherm, the MHD and the SC EOS differ most notably for —1 < logp < 1. Under these conditions, pressure ionization occurs where thermal excitation of the atoms is large. We discussed this regime in § 2. While the MHD model is well beyond its limit of validity (log/* < — 2), none of the EOS presented here can be considered reliable in this difficult regime. 14-6.3 The adiabatic gradient As discussed above, the second derivatives of the free energy are very sensitive to the choice of thermodynamic model and display itsflawsprominently. The adiabatic temperature gradient is particularly interesting since it forms the basis of the Schwarzschild criterion for convective instability in stars. The six isotherms for the adiabatic gradient are shown on Figs. 7-9. Except in a few well known limits, figures of Vad are particularly difficult to interpret physically. We will limit the analysis to listing the failures and problems with each EOS. Figure 7 shows the two lower isotherms over a wide density range. The overall wavy structure is caused by partial dissociation and ionization. A number of features are immediately apparent: • Even at very low densities where the gas is ideal, the agreement is not perfect. Differences of 10% are commonplace. • The FGVH EOS can be very noisy. • The MHD EOS shows pathological behavior for log/) > —2, once again reinforcing their warning about not using their EOS above this limit.

325

Saumon: Hydrogen equations of state

0.50

0.40

0.30

>

a

0.20

-

0.10

log P (g/cm ) Fig. 14.7 Comparison of adiabatic gradient isotherms showing the logT : 3.70 and 4.10 isotherms. • The SC EOS is not very smooth in the regime of the fully ionized plasma (log/) > 0.5). The next two isotherms are displayed on Fig. 8. Again, the wavy structure seen for logT = 4.50 is due to partial ionization. Hydrogen is nearly fully ionized everywhere along the logT = 5.30 isotherm and the drop to = 0.25 at very low densities is due to the photon gas. We find that: • There are still differences in the ideal gas regime, but they are below the 10% level. • The FGVH EOS appears smoother in this regime • Above log/7 = —1.5, the MHD EOS shows pathological behavior along both isotherms.

Saumon: Hydrogen equations of state

326

0.50

0.40

-

0.30

-

0.20

-

0.10

-

-8

-6

-4

-2

log P (g/cm 3 ) Fig. 14.8 Comparison of adiabatic gradient isotherms showing the log T — 4.50 and 5.30 isotherms. • For the logT = 4.50 isotherm, the MM EOS shows a "phase lag." This indicates an ionization zone which is displaced to comparatively higher densities. This originates in their treatment of the IPF. • At the high-density end of these isotherms, T > 10 and 0 < 1, conditions under which the Coulomb interactions are strong. In FGVH and MM, these are described with a Thomas-Fermi-Dirac model and both show Vnd rising as the density is increased. On the other hand, SC use a screened onecomponent plasma model (SOCP, Chabrier 1994), a much more accurate description of the plasma, and find that V«i decreases along the isotherm. The SC EOS remains rather noisy in this regime. Finally, Figure 9 shows the two hottest isotherms. For a pure photon gas, Vad = 0.25 and it approaches 0.4, the value for a non-interacting (ideal), classical, monoatomic gas, as the pressure of the plasma comes into play.

327

Saumon: Hydrogen equations of state

0.50

0.40

1

T

1

-

6. 10

/ /

0.30 —

/

1 '

1 '

1

1

/*"" / f // / / / / 6 .90 / /

0.20 — -

0.10

0.0 -8

SC FGVH MM

-

, 1

1 ,

1 .

•

1

- 6 - 4 - 2 0 log p (g/cm 3 )

Fig. 14.9 Comparison of adiabatic gradient isotherms showing the logT = 6.10 and 6.90 isotherms. Both of these limits are readily apparent on this figure. The adiabatic gradient of a mixture of photons and non-interacting protons and electrons can be calculated analytically (Cox and Giuli 1968, § 9.17), a result accurately reproduced by FGVH and SC. The divergence of the MM curves from the analytic expression cannot be explained on physical grounds. At high densities, Vad drops below 0.4 due to relatively weak to moderate Coulomb interactions (r < 1). Both the FGVH and SC EOS show a downward trend in Vad and agree quite well while the MM EOS displays an increase similar to that observed in Fig. 8. Because of its strong connection with convective instability, Vad plays an important role in models of stellar interiors and envelopes. Figures 7-9 show differences of the order of 10% in the ideal gas regime of partial dissociation and ionization, underscoring the sensitivity of Vad to the treatment of the

328

Saumon: Hydrogen equations of state

states in the IPF. When strong non-ideal effects come into play, it appears that the adiabatic gradient remains a rather poorly determined quantity. 14.7 Concluding remarks This exercise of comparing several equations of state developed for applications to astrophysical problems (mainly stellar interiors) reveals that the situation is not as satisfactory as is commonly assumed. Much progress has been accomplished over the time span represented by these EOS: the SC and the MHD EOS represent considerable improvement over the older FGVH and MM EOS. This is due in part to new high-pressure experiments which probe the H2-H2 potential to smaller interparticle separations, the development of a solid knowledge of dense plasmas through numerical simulation and a more acute awareness of the importance of consistency between the treatment of the internal partition function and the interactions between particles. It also shows that each of these EOS has flaws or limitations, most of which can be addressed in the near future. The most challenging areas remain associated with partial dissociation and ionization for hydrogen. The treatment of temperature ionization with the FMIN method has improved considerably in the last few years but we have seen that none of the EOS presented here is truly satisfactory in this respect. The more rigorous activity expansion technique may provide a definitive treatment of temperature ionization. On the other hand, pressure ionization remains by far the most poorly understood phenomenon and maintains a shroud of uncertainty over a part of the phase diagram which is important for low-mass stars, brown dwarfs and most critically, the jovian planets. The calculation of such equations of state is a complex problem. Those presented here each required a few man-years of effort, and still they display flaws and problems of various importance. This raises strong doubts about the validity of the much simpler and often crude equations of state used in many astrophysical problems. Any problem calling for an equation of state should first be cast in a phase diagram (Fig. 1) to determine the relevant physical regimes, the magnitude of non-ideal effects and whether partial ionization and dissociation are expected. In numerous cases, the EOS is sufficiently simple for semi-analytic treatments to be adequate. However, if non-ideal effects are expected and the accuracy of the final result is important, there is no justification for not using the appropriate tabular equation of state. In conclusion, astrophysicists should be more critical of the equations of state they use.

Saumon: Hydrogen equations of state

329

I am very grateful to F. D'Antona, and to D. G. Hummer, who kindly provided the MM and the pure hydrogen MHD EOS tables, respectively. I thank G. Fontaine who generated Figs. 3 and 4 and provided the white dwarf model shown in Fig. 1, and F. J. Swenson for sending me the 0.3 MQ main sequence star model (Fig. 1.). This research was supported in part by NSF grant AST-8910780 and by NASA grant HF-1051.01-93A from the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS526555.

References Chabrier, G., these proceedings (1994) Chabrier, G., Saumon, D., Hubbard, W.B., and Lunine, J. Ap. J., 391, 817 (1992) Clayton, D. D., Principles of Stellar Evolution and Nucleosynthesis, 2 nd Ed., (Chicago: University of Chicago Press) (1983). Cox, J.P., and Giuli, R.T. Principles of Stellar Structure, Vol. 1, (Gordon and Breach: New York) (1968) Dappen, W., these proceedings (1994) Dappen, W., Mihalas, D., Hummer, D.G., Mihalas, B.W. Ap. J., 332, 261 (1988) Fontaine, G. these proceedings (1994) Fontaine, G., Graboske, H.C., Jr., and Van Horn, H.M. Ap. J. Supp., 35, 293 (1977) Graboske, H.C., Jr., Harwood, D. J., and Rogers, F. J. Phys. Rev., 186, 210 (1969) Hubbard, W.B., and DeWitt, H.E. Ap. J., 290, 388 (1985) Hummer, D.G., and Mihalas, D. Ap. J., 331, 794 (1988) Kerley, G.I. Phys. Earth Planet. Inter., 6, 78 (1972) Lamb, D.Q. PhD Thesis, University of Rochester (1974) Lamb, D.Q., and Van Horn, H.M. Ap. J., 200, 306 (1975) Magni, G., and Mazzitelli, I. Astron. Astrophys., 72, 134 (1979) Marley, M.S., and Hubbard, W.B. Icarus, 73, 536 (1988) Mazzitelli, I. private communication (1993) Mihalas, D., Dappen, W., and Hummer, D.G. Ap. J., 331, 815 (1988) Rogers, F.J., Phys. Rev., A24, 1531 (1981) Rogers, F.J., these proceedings (1994) Saumon, D., and Chabrier, G. Phys. Rev. A, 44, 5122 (1991) Saumon, D., and Chabrier, G. Phys. Rev. A, 46, 2084 (1992) Saumon, D., Chabrier, G., and Van Horn, H.M. in preparation for Ap. J. (1993) Saumon, D., and Van Horn, H.M. in Strongly Coupled Plasma Physics, F.J.. Rogers and H. E. DeWitt, Eds. (Plenum: New York), p. 173 (1987) Stevenson, D.J. Phys. Rev. B, 12, 3999 (1975) Stevenson, D.J., and Salpeter, E.E. Ap. J. Suppl, 35, 229 (1977)

15 Strongly Coupled Ionic Mixtures and the H/He EOS Hugh E. DcWitt Lawrence Livermore National Laboratory, Livermore, CA 94550 Abstract This paper summarizes recent work on the strongly coupled OCP and Binary Ionic Mixture equation of state and other thermodynamic quantities in white dwarf interior conditions for both fluid and solid phases with the assumption of a uniform background. Conditions for phase separation of different elements in fluid or solid phases is strongly dependent on deviations from the linear mixing rule which gives the equation of state as an additive function of the OCP equation of state. These deviations turn out to be small (a few parts in 105) and always positive including the case where the fraction of the higher Z component approaches 0. Also the equation of state of strongly coupled light elements (H and He particularly) obtained from simulations with a linear response description of the electrons is given for conditions appropriate to brown dwarf star interiors. Recent Livermore work on a band structure calculation of the enthalpy of H and He mixtures under jovian conditions is discussed. This work leads to a prediction of a high temperature (15000 °K) for miscibility of He in ionized H at 10 Mb.

Resume Ce papier resume l'ouvrage recent sur le OCP a fort couplage et sur l'equation d'etat et d'autres quantites thermodynamiques pour le melange binarire ionique aux conditions interieure des nains blancs. Les conditions pour seperation de phae dan les elements divers dans l'6tat solide ou fluide sont tres sensible mix deviations de regie lineaire qui donne l'equation d'e"tat comme function additive sur celui du OCP. Cettes deviations sont tres petities (quelques parts dans 105) et toujours positive meme guard la fraction du composant a Z superieur va vers O. L'equation d'&at pour les elements leger (surtout H et He) a fort couplage obtenu par simulations avec response lineaire des electrons est donne p;our les conditions des nains brun. Nous discutons aussi calculs recent, fait a Livermore, predisent que le He dans H cst immiscible a haut temperature (15000 K) a la pression de 10 Mb. 330

DeWitt: Strongly coupled ionic mixtures

33\

15.1 Introduction Hydrogen and helium mixtures form the main components of jovian planets, brown dwarf stars, and ordinary main sequence stars. In these various conditions the state of the mixtures ranges from molecular and atomic fluids in the outer layers of the jovian planets, to ionized H and neutral He deeper in jovian planets, to fully ionized low Z elements in brown dwarfs and main sequence stars. Hydrogen and helium are found on the surfaces of white dwarf stars but most of the interiors are composed of heavier elements beginning with C, O, Mg, and up to Fe that are fully ionized. The relativistically degenerate electron gas provides a nearly uniform density neutralizing background and the positive pressure that balances the gravitational contraction force and thus determines the size of the white dwarf. For applications to these various astrophysical objects we need to know the equation of state of mixtures of light elements from the molecular-atomic region at low temperature and high density on up to the extreme high densities where the ions are bare nuclei. One can distinguish four density regions: i) Ionic fluid in a (nearly) uniform background of degenerate electrons. Ts -> 0. An example is the C and O fluid (bare nuclei, usually assumed to be classical) above the crystallizing core of a white dwarf star • Monte Carlo simulations give very accurate results for the ionic interactions in both the fluid and the solid state. This system is well understood classically, though quantum effects are now known to play a strong role in real white dwarfs^. ii) Ions in a responding background of degenerate electrons fa < 1). This is the situation in brown dwarf star interiors with pressures above 80 Mb so that both H and He are fully ionized. The equation of state is given with fairly good accuracy by using Monte Carlo simulations with the Coulomb potential screened by using an appropriate dielectric function in linear response theory. Also fairly accurate results can be obtained by solving coupled Hypernetted Chain (HNC) equations for the ionic mixtures^. At the moderate temperature ( a few eV) but density high enough that 0.1 < rs < 1) the electrons are highly degenerate (kT « Ep) but still polarizable. iii) Partially Ionized Mixtures (rs >) with the ions still strongly coupled. For P > 3 Mb as in the interior of most of Jupiter one finds H + along with neutral He, and consequently both free and bound electrons. In this density and temperature region both MC and HNC calculations are of questionable

332

DeWitt: Strongly coupled ionic mixtures

accuracy, and there is still considerable uncertainty. A plasma phase transition 4 is possible for the hydrogen at roughly kT = 1 eV. Such a transition may be modified by the presence of neutral helium5'6*7. This region is unfortunately still outside the possible region of experimental measurements. iv) Neutral H? and He as in jovian atmospheres, 0 < P < 3 Mb. This region has been studied experimentally to the 1 Mb region for H2 and is accessible to 3 Mb in future experiments. Most of this paper will deal with region i), the extremely high density region where one has some hope of obtaining some nearly exact results from very long numerical simulations, either Monte Carlo or molecular dynamics. Yakovlev and Shalybkov1 have given an excellent review of most known results up until 1988 for the OCP and ionic mixtures in the strong coupling region, and discussed the applications to very dense stars, particularly white dwarf interiors. Problems that remain to be addressed with greater quantitative precision include the questions of the value of the coupling parameter at which the fluid solid transition occurs, and the possibility of phase separation, i.e. separation of heavy from light elements in the freezing process or even in the fluid region. Associated with the phase separation process is the need for an accurate representation of the equation of state of the ionic mixtures in both fluid and solid phases. The well known linear mixing rule gives an excellent first approximation to the mixture equation of state in terms of the OCP results for the energy and the Helmholtz free energy. Small deviations from the linear mixing rule determine the phase diagram of mixtures, for example the possible separation of Fe from the C and O in the fluid region of the white dwarf interior. These questions need to be addressed by the generation of very accurate Monte Carlo energy data for ionic mixtures using several hundred to a thousand particles and averaging over as many as a few hundred million configurations. Some new results for mixtures will be given here. 15.2 Strongly Coupled Ionic Mixtures in a Uniform Background We will use the now standard definitions of parameters. For the OCP with number density n = N/V , temperature p = 1/kT, and classical point charges Ze moving in a uniform background the coupling parameter is V — (Ze)2/akT with a = (—n)"1^3, the Wigner-Zeitz or ion sphere radius. At the extreme densities id white dwarf stars the deviation from a uniform

DeWitt: Strongly coupled ionic mixtures

333

background due to ionic polarization of the relativistic electrons is measured by r s = ae/ae with OQ = a/Z1/3 the electron sphere radius. rs from about 0.01 for p = 106 gm/cc down to 0.001 for p = 109 gm/cc, so that to a good approximation electron screening effects on the ion-ion contribution to the equation of state is quite small. The interior of a white dwarf if it were a single element is believed to be crystallized with an energy of the form: U/NkT = (Uo + Ulh)/NkT = a M r + (3/2 + A i/T + A2/T2 + ...)

(1)

where aM = - 0.89592926 is the Madelung constant for the bcc lattice, 3/2 is the classical thermal energy of the harmonic lattice, and the first anharmonic energy 8 term has the coefficient Ai = 10.84. The second anharmonic energy term9 is known only from MC simulations and is A2 « 600. The Helmholtz free energy for the OCP bcc lattice can be obtained by temperature integration: F/NkT = aMF + J l n r + C - S H - Ai/T - - ^ +

(2)

where C =1 + ln(2(3/47i)1/3) and S H I S the entropy constant ( , averaged over lattice phonon frequencies which for the OCP bcc lattice is

2.4939. The OCP fluid energy, U/NkT, obtained from long Monte Carlo simulations 10 in the strongly coupled region, 1 < F < 200 again has the remarkable property of splitting to a good approximation into a static piece and a thermal piece, but with the thermal energy governed by a power law: U/NkT = (U0(p) + Uui(p, P))/NkT = a F r + b P + c

(3)

where s is a number ranging from 1/4 to 2/5 depending on the number of terms used in the fitting form in Eq. 3; the best estimate is s » 1/3. Integration over the temperature from p to 0, gives the OCP fluid free energy as; F/NkT = a F T + (1 /s)br s +clnr + D

(4)

Where D is an integration constant. The coefficient of T in Eqs 3 and 4 is the 'fluid Madelung coefficient' which Rosenfeld11 has shown to have an exact limiting value of a F = - 9/10, which is the value in the F - * «»for the fluid. If a F is allowed to be a fitting parameter for the OCP fluid data the

334

DeWitt: Strongly coupled ionic mixtures

coefficients for the fit to the best available OCP MC data is aF = - 0.89921, b = 0.596, c = 0.268 and s = 0.3253. Other equally accurate fits to the OCP fluid data with ap fixed at -9/10 are given in Ref. 10. Generally the OCP energy data for large F is known to about ± 0.0005 (which is a few parts in 10 6 ), and simple fitting functions like Eq. 3 can reproduce the known OCP data to about ± 0.001. This kind of accuracy is ultimately needed also for the ionic mixtures in order to determine the freezing line for ionic mixtures and the conditions for phase separation in the freezing process or in the fluid. For the OCP the fluid and solid Helmholtz free energies cross at F = 172, which should be regarded as the best available estimate of the OCP freezing transition F. The inclusion of the first order anharmonic energy in Eq. 1 has changed the estimate from the earlier value 12 of 178 to 172. Since the fluid and bcc solid free energies have so nearly the same slope, a very small change in one of the free energies can send the crossing point up or down very much.. This fact will be even more true for ionic mixtures which means that current estimates1^ of the phase diagrams for ionic mixtures must be regarded with some skepticism. It should also be noted that the classical OCP and the classical ionic mixture is a serious approximation for white dwarf star interiors since in fact the ions in these stars have serious quantum diffraction effects2. The measure of QM diffraction effects is r\ = hC0p/kT which ranges from 3 to 8 in white dwarf interiors. However, the location of the freezing transition is only slightly affected by quantum effects. The transition temperature is lowered (F increased), but even for Tj = 8 the transition F is changed to only abut 200. Rosenfeld has demonstrated a number of other exact limiting results for the OCP fluid and the ionic mixtures for large F. The screening function, H(x) with x = r/a, in the pair correlation function: g(x) = exp{-F/x + H(x)} is needed for calculations of the screening enhancement of thermonuclear reactions in very dense stellar interiors. H(0) gives the lowering of the Coulomb barrier for two rapidly approaching ions. The Onsager molecule method used by Rosenfeld gives a result for the screening function for finite F: H(x)/F = ho(F) - hix 2 + h2(F)x4 - h3(F)x6 +

(5)

DeWitt: Strongly coupled ionic mixtures

335

In this expansion around the x = 0 only the coefficient hi is a constant independent of F, namely hi = 1/4.112 and 113 must be determined from the best available MC data for g(x). For large F ( 100 to 200) h2 - 0.038 which is large enough to influence the determination of ho. H(0) = hoF is given by the difference of the Helmholtz free energy for N charges Z and the free energy of N - 2 charges Z and on charge of 2Z. The result can be shown to be: H(0) = 2f 0C p(F) - fOCP(25/3F) - @/ax2)Afocp(F)

(6)

= (9/10)(25/3 - 2) = 1.0573F as F -»<*> where X2 = N2AN1 + N2). The first line of Eq. 6 is the linear mixing rule result for H(0) with AfocP = ^mixture - ^LM > and f = F/NkT. Rosenfeld has shown that by inclusion of the x 4 term in Eq. 5 , that H(0) can be obtained very accurately without using the linear mixing rule 14 , and that the results agree remarkably well with the linear mixing result. The sign of Af is quite important for this discussion. The general statement of the linear mixing rule for binary ionic mixtures is: U(Zi, xi, Z 2) x2)/NkT = - ^ " x 1 foCP(F 1) + x2foCP(F2)

(7)

with Fi = Zi 5 / 3 F C . Linear mixing applies exactly in the large F limit but is at best a good approximation for the fluid mixture thermal energy. Thus the usual statement of the linear mixing rule in Eq. 7 is indicated as an approximation. It is the deviations from linear mixing in the binary ionic thermal energy that result in the possibility of phase separation, i.e. Z2 ions separating from the Zi ions, and similarly the possible separation of large Z ions from smaller Z ions in the freezing process. Ogata, et a l ] 3 reported conditions for phase separation of high Z ions in the fluid phase and phase diagrams for binary ionic mixtures upon freezing which were very much influenced by their MC observations of some negative deviations from linear mixing in the limit of a small fraction of the large Z component, i.e. X2—»0. Our recent results (Slattery and DeWitt 15 ) indicate that the deviation from linear mixing is only positive.

336

DeWitt: Strongly coupled ionic mixtures

Brami, Joli, and Hansen 16 did a detailed study of the conditions for phase separation in binary ionic fluids using the pure HNC mixture equations (no bridge function correction) and found always positive deviations with a magnitude of never more than 0.027 for X2 = 0.05 and Z2/Z2 = 8. Ogata et al 13 reported negative deviations for MC binary fluid runs with Ni = 990, Zi = 1, and N2 = 10, Z 2 = 3 and 5 for I*i = 20, thus X2 = 0.01. We first tried the Z2 = 3 case with HNC mixture equations and found that Umix/NkT = -17.4614 and that ULM/NkT = -17.4621 (also from HNC). The difference is AUHNC/NICT = + 0.0007. All mixture runs done with the HNC equations give positive deviations from linear mixing. Since the HNC equation is an approximation, the above results may be questioned with the presumably more accurate MC fluid mixture simulations. We did a few MC mixture runs for X2= 0.01 (Ni = 990 and N2 = 10) but with 150 million configurations which is 20 times the number of configurations reported by Ogata et al. For I*i = 20 our MC mixture result was Umjx/NkT = -17.7231 and the linear mixing result using the best available fit to the OCP fluid energy data, Eq. 3, is ULM/NkT = -17.7241. Thus we obtain a positive deviation of AUMC/NkT = + 0.0010, whereas Ogata et al report a value of - 0.002. A possible source of error in the Ogata et al results is their fitting function for the OCP data which is inaccurate by about 0.003 for some values of T. The deviations from linear mixing are obviously very small and the reported results depend very much on the accuracy of the MC energy results and the accuracy of the fit to the OCP fluid data. Our conclusion is that the MC mixture results are always positive. A positive deviation from linear mixing has a number of consequences. Ogata, Iyetomi, and Ichimaru16 evaluated H(0) from their MC data for g(r) on the assumption that Ii2 in Eq. 5 was 0, whereas Rosenfeld14 finds h2 = 0.038 in agreement with Alastuey and Jancovici 17 from their work on enhancement of thermonuclear reactions in 1978. The Ogata et al 16 results for H(0)/T are about 2% larger than the results of Rosenfeld and of Alastuey and Jancovici which are also in close agreement with the linear mixing rule. Ogata et al 13 (in their report to the Rochester Conference) cite the correction term in Eq. 6 to explain this difference. However, their agreement with linear mixing plus the correction requires that their deviation from linear mixing be negative, but our results indicate that Af is positive. The 2% difference in estimates of h 0 = H(0)/T makes a factor of 10 difference in the final enhancement rate for thermonuclear reactions at large F. As Isern has pointed out at this Conference18 a very small change in the enhancement of thermonuclear reactions in a white dwarf star at the Chandrasekhar limit can make the difference in the star becoming a supernova or to collapse into a

DeWitt: Strongly coupled ionic mixtures

337

neutron star. For reasons given above we think the Alastuey-Jancovici estimates17 of the screening enhancement of thermonuclear reactions are more accurate and reliable than the results of Ogata, Iyetomi, and

Ichimaru16.

The crystallization of binary ionic mixtures provides another example for which the sign of the deviation from liner mixing can make a qualitative difference. Using a density functional theory of freezing Segretain and Chabrier19 estimated the domains of three types of phase diagrams depending upon the ratio of the charges, Z1/Z2. For 0.72 < Z1/Z2 < 1 the phase diagram for solidification is a spindle type; for 0.58 < Z1/Z2 < 0.72 an azeotropic phase diagram results; and for an even greater charge disparity, Z1/Z2 < 0.58, a eutectic phase diagram results. This is the same progression of phase diagrams as is found for mixtures of hard sphere diameters with different diameters. Ogata et al 13 , however, based on their MC simulations that lead to negative values of Af, find that the azeotropic form can persist for any value of Z1/Z2 near 1 when the fraction of the higher Z component is very small, i.e. X2=s0.01. The positive values of Af that we find support the spindle shaped phase diagram for all values of X2 for Z1/Z2 near 1. the type of phase diagram for binary ionic mixtures in white dwarf stars, C and O, and C and Fe, has major astrophysical consequences for the possible separation and crystallization of trace elements19, and the resulting effect on the cooling time for white dwarfs. It is clear that a lot more very accurate numerical simulation of binary ionic mixture energies is needed to finally settle these questions. 15.3 Strongly Coupled Ionic Mixtures in a Responding Background For applications to brown dwarf stars and the jovian planets the light elements, especially H and He, may be partially to fully ionized. Although the free electrons are largely degenerate the r s value is typically in the vicinity of 0.5 to 1. Thus the electrons can screen the protons and alpha particles appreciably. The electrons may also have finite temperature effects measured by the parameter 6 = kT/EF which leads to complete degeneracy when 9 « 1, but in some stars we may have the worst of all possibilities, namely 9 = 1 . Both the density dependence (rs) and the temperature dependence (0) can be dealt with by using an appropriate density and temperature electron dielectric function in linear response theory. This has been developed by Chabrier and Ashcroft20 for solution in the HNC ion mixture equations. The energy and pressure of the ionic mixture in linear response (from their paper) is:

338

DeWitt: Strongly coupled ionic mixtures

N

1 ^^(^w*^)] 2{2n) :

(8)

2 (2/r)3

(9)

In Eqs. 8 and 9 E is the total ion-ion and ion-electron interaction energy and P e x is the corresponding pressure contribution. v(k) is the Fourier transform of the Coulomb potential, 4rce2/k2, Sz(k) is the structure factor, and e(k) is the density and temperature dependent dielectric function. The first term in u cx and p c x is the usual BIM functions appropriate for a uniform background, rs = 0. The second term is the linear response density contribution, and third term is the result of the temperature dependence in the dielectric function. The screened potential, v(k)/e(k), is used in the

DeWitt: Strongly coupled ionic mixtures

339

solution of the coupled HNC equations for the BIM in order to calculate Sz(k) which is then used in Eqs. 8 and 9 to evaluate the energy and pressure of the mixture. Chabrier and Ashcroft presented results for a variety of mixtures, H and He, H and O, with temperatures and densities that went from weak coupling to strong coupling and with X2 = 0, .25, .75, and 1. They checked the linear mixing rule and found that deviations from linear mixing ranged from 0 % to at most 3% and were positive. The largest deviations from linear mixing occurred for 9 = 1, the most difficult temperature region. For the very high temperatures in the center of the sun Iyetomi and Ichimaru21 have solved the HNC equations for H and Fe mixtures in order to construct a phase diagram for H and Fe. They find that the central temperature of the sun is too hot by a factor three to allow for thepossibility of any separation of Fe from the remaining solar plasma. Their results suggest that heavy elements, such as Fe, might be able to separate from small cool stars with interior temperatures of only a few million °K. Eqs. 8 and 9 could in principle be used for evaluation by Monte Carlo simulation, though this has not been done yet with a temperature dependent dielectric function. Hubbard and DeWitt have done extensive MC calculations on H and He mixtures22, assumed to be fully ionized, with the density dependent RPA dielectric function (the Lindhard function) and 8 = 0. 45 MC simulations were done on pure H, pure He, and mixtures of HHe with values of rs ranging from 0.2 up 1.4 and values of F e ranging from 7 to 155. Three mixtures were computed: 80%H and 20% He, 50% and 50%, and 20% and 80%. Unfortunately because of limited computer time the number of particles used was only Ni + N2 = 50, and the energies were obtained after thermalization with 105 configurations. By today's standards these simulations are far too short and too few particles to give accurate results for the energy and pressure. Nevertheless, enough data was obtained that a Helmholtz free energy model could be constructed: f = F m i x /NkT = -ar c + blV/4 - clnr e + d

(10)

which is similar to Eq. 4 except the exponent s was chosen to be 1/4 and the coefficients in Eq. 10 are all functions of rs. Specifically, the coefficient of F e was expanded to the second power in rs: a(rs) = ai[(l-x)Zi5/3 + xZ 2 5 / 3 ] + a 2 (l-x)r s + a3xrs + [a4(l-x)

(11)

DeWitt: Strongly coupled ionic mixtures

340

The coefficients b, c, and d were modeled with only the linear dependence in rs# All together there were 10 numerical coefficients (ai , a2 ,...) that were obtained numerically by a least squares fit to the 45 MC data points. These coefficients used in Eq. 10 gave a useful numerical model for the free energy mixture of H + , He + + , and the screening electrons. In order to obtain the equilibrium curves for the coexistence of two liquid phases at a pressure of 8 Mb, the equilibrium equations were solved:

^(r.N, / V,N21V) = (dF I dNx\yjlt

(12) ^(T,NXa I V,N2a IV) = li2(T,Nlb I V,N2b I V\ P(T,Nla I V,N2,1V) = P(J,Nlh I V,N2b IV), P(T,NU/V,N2JV)

= PS,

using Eq. 10. The resulting equilibrium curve for the coexistence of to liquid phases is shown in Fig. 1 (from the Hubbard-DeWitt paper) Stevenson toooo

5000

0.5

i.o

Fig. 1 - Kqnilibrium curve for coexistence of two liquid phases of H iind lie at S Mb, compared with earlier work by Stevenson23-

DeWitt: Strongly coupled ionic mixtures

™l

At x = 0.07, the He fraction in Jupiter, the demixing temperature is 8000 °K. This estimate of the demixing temperature in Jupiter is open to criticism because the helium atoms in Jupiter from 3 Mb where hydrogen is presumed to pressure ionize on the 35 Mb pressure at the center it is probable that the helium atoms are neutral, i.e. with two bound electrons. Strictly speaking the Hubbard-DeWitt simulation of H and He applies only to astronomical objects with pressures above 50 to 80 Mb in order to assure the pressure ionization of the He. The free energy model for the H - He mixture can probably be considerably improved with more accurate MC simulations involving up to 1000 charges and up to several million configurations for each value of F e and rs. It should also be mentioned that the ten fitting coefficients used in Eq. 10 can be somewhat reduced in number by using the scaling obtained in recent work by Rosenfeld on charged Yukawa mixtures24. Thus the coefficients a3 and 04 can be replaced by a single coefficient weighted with ?/3 and

7/3

15.4 Partially Ionized H and He The obvious problem for theoretical calculations of the H-He equation of state and the possibility of phase separation of the He in jovian planets is the fact that the hydrogen whether in molecular or atomic form become pressure ionized at approximately 3 Mb while the tightly bound helium electrons remain attached to their nuclei in the H-He mixture until some much larger pressure, ~ 50 Mb. Chabrier and Saumon4 have given a detailed model for the equation of state of hydrogen from the molecular region to the fully ionized region that exhibits a plasma phase transition with a critical point at T c = 15300 K, P c =0.614 Mb, and p c = 0.35 gm/cc. The Chabrier-Saumon EOS is probably the best theoretically based and most quantitative result available for use in modeling the jovian planets. Whether there is a true phase transition or simply a sharply defined region where pressure ionization occurs does not affect the accuracy of the EOS very much. Unfortunately this possible phase transition is out of range of current experimental measurements. Pure He is believed to be pressure ionized at some far higher pressure, perhaps as much as 80 Mb. He atoms mixed in high pressure ionized hydrogen will probably ionize at some lower pressure which may be greater than the interior pressure of Jupiter. To address this problem of the H- He mixture in the 10 to 30 Mb range it is essential to have a detailed understanding of the electronic structure of the mixture. Klepeis, Schafer, Barbee, and Ross at Livermore have approached this problem of the

DeWitt: Strongly coupled ionic mixtures

342

electronic structure with total energy calculational methods of condensed matter physics at T = 0 K In this approach the atoms are placed in a lattice, fee or bec, and electronic enthalpy, H(x) = E(x) + P(x)V with x as the He fraction, is calculated with the local density approximation. A brief description of the method and band structure results for the H-He mixture is given in Ref. 5. A complete report is available from Livermore. The main thrust of this work is an accurate complete T = 0 K calculation with the ions on a lattice from which one obtains the enthalpy of mixing from: (13)

AH(x) = H(x) - xH(x =1) - (l-x)H(x =0) Results for 10.5 Mb are shown in Fig. 2

1

I ' '

I

" P = 10.5 Mbar

s o

-

a • — \ A

/

^\

1 -

u

/

P.

/

CtO

\

a

:

0.5

/ /

0

a bee lattice (sc) A bec lattice (dia) o bee lattice (rhom) • fee lattice (sc) . I , , • I

0

•

\

\ \ \ -

ill

0.2 0.4 0.6 0.8 x (Atomic Percent Helium)

1

Fig. 2 The enthalpy of mixing per atom in ev from Eq. 13 obtained from first principles total energy calculations for four different lattices at T = 0 K and P = 10.5 M b .

DeWitt: Strongly coupled ionic mixtures

343

The different lattices used have only a small effect on the mixing enthalpy. Calculations were also done at different pressures to find AH(x = 1/2). This peak enthalpy was nearly constant between 5 and 20 Mb and dropped only by 35 % at 1000 Mb. The large values of the mixing enthalpies are a direct consequence of the fact the helium-derived electrons are more tightly bound to the nucleus than the loosely bound hydrogen-derived electrons. To go to finite T one needs a Gibbs free energy of mixing. This was done by adding a simple result for the ideal gas mixing entropy to obtain: (14)

AG(x) = AH(x) + kT [ x In x + (l-x)ln(l-x)]

and from Eq. 14 the Gibbs mixing free energy was obtained for several temperatures as shown in Fig. 3.

. 1 ' . . 1 .

1

1

1

1

•

1

i

• .

I • •

• 1

-p == 10.5 Ml)ar>

I u A

I

^ -

/ 10 000 K

/

0.5 |--

15,000

yy

^

—

-V

20,000

30,000

-0.5 -

•

y^

O

t

Nv

*yk

-it

0]

i . . .

0.2

0.4

0.6

i . . .

\ -

0.8

x (Atomic Percent Helium) Fig. 3 Gibbs free energy of mixing from Eq. 14 for several temperatures at 10.5 Mb. The double tangent construction is Shown lor T = 15000 K

344

DeWitt: Strongly coupled ionic mixtures

Finally using the double tangent construction at various temperatures the demixing temperature is obtained and shown in Fig. 4

I

I

I

I

I

I

I

I

I

l

l

)

P = 10.5 Mbar -

r u fl)

I «

This work OCP-LM Iindhard Ion—sphere

9

o 3 CO

I

0

Fig. 4

I

I

I

I

I

I

I

I

I

I

I

I

0.2 0.4 0.6 0.8 x (Atomic Percent Helium)

1

Immisibility temperature limit for H-He mixtures.

The demixing temperatures obtained by this procedure are very much larger than those obtained by Stevenson and by Hubbard and DeWitt. At x = 0.07 corresponding to the helium fraction in Jupiter Fig. 4 gives the demixing temperature as close to 15000 K vs. the 8000 K from MC with Lindhard dielectric function. The error estimate on this calculation is ± 3000 K. However, it is clear that the approach needs serious improvement for finite temperature since i) the ions are assumed to be in a lattice rather than in a fluid state, and ii) the Gibbs thermal energy of the mixture is not included. Clearly more work needs to be done with this approach. The earlier estimate of demixing at 8000 K is consistent with the belief that H and He in Jupiter remain mixed through out much of the planet. If the 15000 K

DeWitt: Strongly coupled ionic mixtures

345

estimate given by the Livermore work is correct, it would have serious astrophysical consequences since it would mean that substantial separation of the He in Jupiter has already happened. Work performed under the auspices of the U.S. Department of Energy by the Lawrence Livermore National Laboratory under contract number W7405-ENG-48. References 1. 2. 3. 4. 5. 6. 7. 8. 9.

10. 11. 12. 13.

14. 15. 16. 17.

D.G. Yakovlev and D. A. Shalybkov, Sov. Sci. Rev. E. Astrophys. &Space Phys. Vol.7, 311-386 (1989) G. Chabrier, N.W. Ashcroft, H.E. DeWitt, Nature 360,48 (5 Nov. 1992) W.B. Hubbard and H.E. DeWitt, Astrophysical J. 290, 388 (1985) D. Saumon and G. Chabrier, Phys.Rev. A46, 2084 (1992): Phys.Rev.A44. 5122(1991) J.E.KIepeis, K.J. Schafer, T.W. Barbee III, M. Ross, Science 254, 986 (15 Nov., 1991) D. Saumon, W.B. Hubbard, G. Chabrier, H.M. VanHorn, Astrophysical J. 391, 327 (1992) G. Chabrier, D. Saumon. W.B. Hubbard, andJ.I. Lunine, Astrophysical J. 391, 317 (1992) D.H.E. Dubin, Phys. Rev. A42, 4972 (1990) H.E. DeWitt, W.L. Slattery, J. Yang, Strongly Coupled Plasma Physics, ed.H.M. Van Horn and S. Ichimaru, University of Rochester Press: Rochester, NY, p.425 (1993) G.S.Stringfellow, H.E. DeWitt, W.L. Slattery, Phys. Rev.A41, 1105(1990) Y. Rosenfeld, Proceedings of this Conference; Phys. Rev. A33, 2025 (1986); Phys. Rev. A37, 3403 (1988) W.L. Slattery, G.D. Doolen, H.E.DeWitt, Phys. Rev.A26, 2255 (1982) S. Ogata,H.Iyetomi,S. Ichimaru, H.M.Van Horn, Strongly Coupled Plasma Physics, ed. H.M. Van Horn and S. Ichimaru, University of Rochester Press: Rochester, NY, p.53 (1993) Y. Rosenfeld, Phys. Rev. A46, 1059 (1992) W.L. Slattery and H.E. DeWitt, recent MC results at Los Alamos S. Ogata, H. Iyetomi, and S. Ichimaru, Astrophys. J. 372, 259 (1991) A. Alastuey and B. Jancovici, Astrophys.J 226,1034 (1978)

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18. 19. 20. 21. 22. 23. 24. 25.

DeWitt: Strongly coupled ionic mixtures

J. Isern,proceedings of this Conference L. Segretain and G. Chabrier, Astron. Astrophys. 271, L13 (1993) G. Chabrier and N.W. Ashcroft, Phys. Rev. A42, 2284 (1990) H. Iyetomi and S. Ichimaru, Phys. Rev. A34, 3203 (1986) W. B. Hubbard and H.E. DeWitt, Astrophys. J. 290, 388 (1985) DJ. Stevenson, phys. Rev. B12, 3999 (1975) Y Rosenfeld, Phys. Rev. E47.2676 (1993) J. E. Klepeis, KJ. Schafer, T. W. Barbee III, M. Ross, Phase Separation in Mixtures of Hydrogen and Helium at Megabar Pressures, UCRL-JC-107995, July 1991. See also the brief report by the same authors in Strongly Coupled Plasma Physics, ed. H.M. Van Horn and S. Ichimaru, University of Rochester Press: Rochester,NY, p.73 (1993)

16 White dwarf seismology: Influence of the constitutive physics on the period spectra. G. FONTAINE Departement ie Physique, Universite it Montreal, C.P. 6128, succursale A, Montreal, Quebec, HSC 5/7, Canaia.

P. BRASSARD Departement ie Physique, Universite de Montreal, C.P. 6128, succursale A, Montreal, Quebec, HSC 5/7, Canaia.

Abstract We present the results of numerical experiments aimed at demonstrating how the <7-mode period spectra of pulsating DA white dwarfs depend on the various components of the input physics. We take advantage of recent developments on many fronts of physics (equation of state, opacity, convection) to compare the theoretical pulsation periods of models with different pieces of the constitutive physics, but with otherwise fixed values of their stellar parameters. This exercise is necessary to assess the reliability of the pulsation analyses of white dwarfs which have started to come out. Nous presentons les resultats de simulations numeriques pour determiner comment les periodes de pulsation (type g) des etoiles naines blanches DA dependent des differentes composantes de la physique constitutive. A cet effet, nous avons utilise des resultats recents au niveau de la physique de base (equation d'etat, opacite, convection) pour comparer les periodes de pulsation de modeles stellaires ayant des parametres fixes, mais qui different au niveau de leur physique constitutive. Notre demarche est essentielle afin de pouvoir quantifier les premiers resultats d'analyses d'etoiles pulsantes qui commencent a etre publies. 16.1 Introduction It is now well established that white dwarf stars become intrinsically variable during certain phases of their evolution. For the majority of them, the 347

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Fontaine & Brassard: White dwarf seismology

so-called DA white dwarfs (with atmospheres dominated by hydrogen), luminosity variations are observed when the stars have effective temperatures in the rather narrow interval 13,000 K £ Teff £ 11,000 K (Wesemael el al. 1991). As a DA white dwarf cools, its hydrogen eventually recombines in the outermost layers, leading to a very significant increase of the opacity there and to the formation of a superficial convection zone. By the time such a star has cooled down to Tefj ~ 13,000 K, the superficial hydrogen convection zone has grown deep enough to be able to drive nonradial pulsation modes of the gravity (g) type. It is not known why the excitation mechanism should become inoperative in more evolved, older DA white dwarfs with Teff ~ 11,000 K and cooler. The g-mode pulsations manifest themselves in terms of multiperiodic luminosity variations with typical periods in the range 100-1200 s and light curve amplitudes (in the optical domain) ranging from about ~ 5 millimagnitude for the smallest amplitude variable known to upward of 0.3 magnitude for the larger amplitude variables. The existence of pulsating white dwarfs offers us the fascinating possibility of using them to test directly the predictions of stellar evolution theory. In this context, we recall that evolution theory suggests that the bulk of the mass of a typical isolated white dwarf should consist of carbon and oxygen (the exact proportions are still unknown because of uncertainties in the rates of helium thermonuclear burning). Also, DA white dwarfs must have retained some small amounts of hydrogen and helium in their outer layers, and it is believed that diffusion processes rapidly sort these elements out, leading to an onion-like, stratified structure with hydrogen floating on top of a helium mantle, itself floating on a core of heavier elements (presumed to be C and 0 ) . Standard evolutionary calculations such as those of Iben & Tutukov (1984) or Koester & Schonbenner (1986) for example, make definite predictions as to the actual values of the hydrogen and helium remnant masses. What asteroseismology offers us is the potential for probing the internal structure of white dwarfs and, hence, testing the expectations of stellar evolution theory. Of particular interest here is the fact that the chemical stratification of a white dwarf model leaves a characteristic signature on the period spectrum of the star (see, e.g., Brassard et al. 1992a). We can then hope to infer the vertical run of the chemical composition and, in particular, the thickness of the outer hydrogen layer in a DA star by comparing the observed periods with those of theoretical models. Moreover, as a white dwarf cools, its structure changes slightly, which results in the slow evolution of a given pulsation mode in that star (Winget, Hansen, k Van Horn 1983). This gives us the possibility, through long-term observations, of determining the

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cooling timescale of a pulsating white dwarf by measuring the rate of period change of a pulsation mode. In turn, we can use this result to infer the core composition of a white dwarf since the cooling rate is essentially proportional to the mean atomic mass of the core material in such a star. In the twenty-odd years since their discovery, enormous progress has been made in our understanding of the properties of pulsating DA white dwarfs. Key review papers narrating this progress have successively been presented by McGraw (1977), Robinson (1979), Winget & Fontaine (1982), Van Horn (1984), Winget (1988), and Kawaler & Hansen (1989). However, despite the advances reported in these papers, it has not been possible (until very recently) to exploit fully the potential of white dwarf seismology. Indeed, due to shortcomings in the quality of the observational data and the lack of sufficiently powerful theoretical tools, it has not been possible to unambiguously identify the pulsation modes in white dwarfs. Without mode identification, inferrences about the internal structure are not possible. This situation is changing, however, and, in a very recent effort, Brassard et al. (1993) have presented the first successful seismological analysis of a pulsating DA white dwarf, including complete mode identification. From their analysis, they have inferred a hydrogen layer mass log(M(H)/M*) = logg(H) ~ —5.90lo'2O an<* a helium layer mass log(M(He)/M*) = logg(He) ~ -2.61 ± 0.02 in the pulsating white dwarf G117-B15A. In order to assess the reliability of such determinations (and others to follow), it is essential to study the influence of the constitutive physics on the period distributions and rates of period change computed from theoretical stellar models. This can be done by comparing the pulsation properties of stellar models with fixed parameters, but computed with different sets of constitutive physics. Thanks to the recent advances made on several fronts in this area (many of which discussed at this Colloquium), such a comparative study is now possible for white dwarfs. In the spirit of this meeting, we thus present the results of our analysis of the effects of the input physics of the y-mode period spectra of representative models of pulsating DA white dwarfs. 16.2 (7-mode periods and constitutive physics To construct a stratified DA white dwarf model it is necessary to specify five basic parameters: (1) the effective temperature Teff, (2) the surface gravity ga (or, equivalently, the total mass M since the equation of state specifies the mass-radius relationship in a white dwarf), (3) the core com-

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position, (4) the mass contained in the helium mantle g(He), and (5) the mass of the outer hydrogen layer ?(H). The stellar structure equations are then solved with the help of a given set of constitutive physics, of which we can distinguish four components. An equation of state must first be known; for white dwarfs, the equation of state is made of an analytic part to describe the ideal gas atmosphere (Saha equations), a tabular part to describe the partially-ionized, partially-degenerate, nonideal envelope, and a tabular part to describe the strongly interacting, completely ionized, dense Coulomb fluid of the core. The opacity of the material must also be known. In addition to the radiative opacity which determines the transport properties of the envelope material, an accurate description of the conductive opacity must also be available to describe the energy transfer in the degenerate core. A physical model for convection is also required to describe how much flux is transported by convection in the outermost hydrogen layers in a pulsating DA white dwarf. Finally, a physical model is equally required to describe the composition transition zones at the H/He and He/C interface regions. To demonstrate how the constitutive physics affects the ^r-mode period spectra of stars, we first recall a well-known result of linear pulsation theory. In the asymptotic limit of high radial overtones (k ^> 1), and for a chemically homogeneous and purely radiative model, it has been shown (Tassoul 1980) that g modes belonging to the same value of £ have a uniform period spectrum with a constant period spacing given by (1) .n , /t(t \/£\l + I) o where P^^ is the period of a <7-mode with radial order k and spherical harmonic azimuthal index £, and II0 is an integral quantity which depends only on the structural properties of a star. This quantity is given by

Pk.e

n

•••

2 K

m

where N is the so-called Brunt-Vaisala frequency (i.e., the local natural oscillation frequency of fluid elements when buoyancy is the restoring force) whose square can be written in the following convenient way

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351

where g is the local gravity, p is the density, P is the pressure, XT = (din P/d\nT)p and xp — (dlnP/dlnp)T are the usual logarithmic pressure derivatives with respect to temperature and density, respectively, Vad is the adiabatic temperature gradient, V is the actual temperature gradient, and Y is the mass fraction of helium which uniquely specifies the local chemical composition in stratified models of white dwarfs with H/He and He/C transition zones. Equations (l)-(3) illustrate clearly how the constitutive physics enters the computations of <7-mode periods. First, it is obvious that the equation of state plays a major role through the various thermodynamic quantities which appear in the Brunt-Vaisala frequency. Second, the opacity plays an equally important (if indirect) role by specifying the temperature gradient V and by determining the structure paths followed by a given model in the (p — T) plane. Third, even though equation (2) is valid only for purely radiative models, it is possible to generalize it to models with thin convection zones in which N2 < 0. In that case, the integral must be carried out only over the radiative regions, and the main influence of convection is then to increase n o as compared to a purely radiative model. Fourth, the way the composition transition zones are treated affects directly the third term in parentheses in equation (3). This term gives an important contribution to N2 only in composition transition zones where Y varies. White dwarf stars are, of course, neither purely radiative nor chemically homogeneous. These circumstances lead, in fact, to 0-mode period spectra which are highly non-uniform. Nevertheless, when allowance is made for the presence of a convection zone in the evaluation of IIO (as indicated above), this quantity still bears relevance to the pulsation periods of a white dwarf model as it now gives a fairly good estimate of the mean period spacing, (Pk+i,e — Pk,e), of an otherwise nonuniform period spectrum.

16.3 Reference model We have constructed a standard, reference model (STD) with parameters Teff = 12,500 K, log^ a = 8.0, pure carbon core, logg(He) = —2.7, and logg(H) = - 6 . 1 . These parameters are comparable to those inferred by Brassard et al. (1993) for G117-B15A. We have used a new model generator code based on an Runge-Kutta scheme to calculate full static stellar models. We developed this code in order to be able to fine tune theoretical pulsation periods to the observations, a task that cannot be accomplished efficiently with a stellar evolution code. This new tool is briefly described by Brassard & Fontaine (these Proceedings). The reference model makes use of improved

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versions for the envelope equation of state of Fontaine, Graboske, & Van Horn (1977; FGV) and the core equation of state of Lamb (1974; L). It uses the radiative opacity tables provided by the Los Alamos Opacity Library Program (Huebner 1980; LAO). Moreover, it uses conductive opacity fits (IH) which we developed by combining the Itoh et al. (1983, 1984) data at high densities with the Hubbard and Lampe data (1969) at low densities. Convection is treated with the so-called ML2 version of the mixing-length theory (Fontaine, Villeneuve, & Wilson 1981). An important improvement was made with regard to the treatment of the composition transition zones. We took advantage of the development phase of the model code to incorporate a full and self-consistent description of diffusive equilibrium in the transition zones. All previous calculations have been based on approximate (and sometimes completely arbitrary) treatments of this important piece of physics (for the pulsation periods). Figure 1 shows the run of the Brunt-Vaisala frequency of the standard model from the center (log q = log(l — M(r)/M+) = 0) to the surface of the star. In addition to the rapid decrease of N2 toward the interior due to increasing degeneracy, the Brunt-Vaisala frequency profile is characterized by two structures which are important from a pulsation point of view because they lead to significant nonuniformities in the period spectrum. The most important of these is the structure centered on logq ~ —6 which is caused by the changing chemical composition in the thin H/He transition zone of our chemically-layered model (the third term in equation [3]). Compared to our earlier results based on an approximate treatment of the composition transition zones (Brassard et al. 1992a), we now find wider transition zones and, as a consequence, the weaker structure associated to the He/C transition zone plays a much less significant role than before. In Fig. 1, the effects of the deeper (log q ~ —2.7) He/C transition zone are practically invisible. The second feature of interest in Fig. 1 is the well around log q ~ —16 caused by the presence of a thin hydrogen convection zone in the outermost layers in which N2 < 0. Very much like problems in quantum mechanics, the structures around logg ~ —6 and —16 lead to partial reflection and transmission at their sites and modify considerably the propagation characteristics of the pulsation waves and, ultimately, the eigenperiods. Modes can be trapped or confined above or below these regions, leading to highly nonuniform period spectra. It should be said that, for the periods of interest, chemical layering at the H/He interface is largely responsible for the nonuniformities in periods, and those bear primarily the signatures of the strength and location of the H/He transition zone. The convection zone, be-

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-4.0 0.0

-20.0

Fig. 16.1 Square of the Brunt-Vaisala frequency as a function of fractional mass depth in the reference model. Note that the choice of abscissa log q strongly emphasizes the outer layers; it is only in these outer regions that g modes have nonnegligible amplitudes in white dwarfs.

cause of its position high in the envelope, can only affect high-order modes which have nodes reaching that high in the star. Figure 2 shows the period spectrum of the standard model for g modes with £ = 2 in the interval of periods from 100 to 1000 s. The period distribution is shown here in a AP vs. P diagram which is a plot of the period difference (Pk+i,i - Pk,t) between two adjacent modes (Afc = 1) belonging to the same value of £ as a function of the period of the mode (Pk,e) starting to the left with k = 1. Very clearly, the period spectrum is not uniform, and this is due primarily to the presence of the H/He transition zone near logg ~ — 6. Nevertheless, we can still define an average period spacing which takes the value 28.99 s (as indicated by the horizontal dotted line). A direct integration over the model (eq. [2]) leads, on the other hand,

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40.0 -

30.0 -

100.

300.

500.

700.

900.

P(s) \i3lU/

Fig. 16.2 Period spacing tw. period for modes with £ = 2 in the standard reference model.

to a value of IIO = 75.74 s, i.e., to an estimate of 75.74/>/6 = 30.92 s for the average period spacing for I = 2 modes. This is close to the true number and indicates that, indeed, IIO is still of value for interpreting the pulsation periods of chemically stratified models. We will use this concept in what follows. More insight can be obtained by examining, this time, the running integral / \N\/rdr as a function of depth in a model. Figure 3 shows the results for the standard model (continuous curve). The vertical dashed line marks the^ position of the degeneracy boundary, defined here as the layer where V = 5 (where rjkT is the usual chemical potential for free electrons). The layer where the degeneracy parameter rf = 5 corresponds to the condition of approximate equality between the radiative and conductive opacities We may note from Fig. 3 that more than 99.99% of the total mass of the standard model is located in the degenerate core. More importantly here, the

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Fontaine & Brassard: White dwarf seismology

•0

I 0.12 h

0.06 h

0.00 -20.

Fig. 16.3 Running integral f(\N\/r)dr vs. mass fraction from the center to the surface for the standard model. The vertical dashed line indicates the layer where the electron degeneracy parameter TJ = 5.

figure shows that both the degenerate core and the nondegenerate envelope contribute roughly equally to the integral, which implies that the weight of period formation is distributed about equally between these two parts. This result is of fundamental importance in the present context, and suggests that reliable constitutive physics must be available in the two regimes to derive reliable periods for white dwarfs.

16.4 Comparative study We have calculated stellar models with the same parameters as those of the standard model but by varying one piece or another of the input physics. For each modified model, we next computed the (/-mode periods for modes with £ = 1,2, and 3 in the period window 100 — 1000 s with the help of our

356

Fontaine & Brassard: White dwarf seismology Table 16.1. Parameters equilibrium models Model

Teft

(K)

Envelope Core Radiative Conductive Convection Opacity EOS EOS Opacity

STD

FGV

SC

SC

12,500 12,500 IDEAL 12,500 NC 12,500 2K 12,500 HOT 14,820 OPAL 12,500 CM 12,500

and input physics for the various

Ideal FGV FGV FGV FGV FGV

L L L Ideal

L L L L

LAO LAO LAO LAO 2xLAO LAO OPAL LAO

IH IH IH IH

2xIH IH IH IH

ML2 ML2 ML2 ML2 ML2 ML2 ML2 CM

n0 (•) 75.74 75.78 73.68 77.55 68.22 68.33 76.73 75.73

adiabatic finite element pulsation code (Brassard et al. 19926). These are the values of interest for pulsating white dwarfs. We finally compared the periods with those of the standard reference model. Table 1 summarizes the properties of the equilibrium models we used in our numerical experiments. The table includes the value of IIO obtained by direct integration of equation (2) for each model. This is a useful quantity to have because we can estimate from it how large should the period deviations be between two models. Neglecting the difference in periods for the first mode (k = 1) between two models, we find that, for k >> 1, the expected period difference for the same mode (same k) between the two models should be

"Jrrh^-^

(4)

We have used this last equation to obtain the results listed in Table 2 which apply to I = 2 modes with a period P ~ 1000 s (k ~ 31 for the standard model). These expected period differences can be contrasted with the results of the detailed calculations. We discuss each of our numerical experiments in what follows. Figure 4 summarizes part of our results for a first experiment in which the envelope equation of state has been varied. This is a plot of the period difference for the same mode (same value offc)computed, on the one hand, for a model in which the recent Saumon-Chabrier (SC; Saumon & Chabrier 1989, 1991, 1992) equation of state for the H/He envelope has been used and, on the other hand, for our standard model with the FGV equation

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357

Table 16.2. Expected period differences for £ = 2 modes at P=1000 s Models compared

AP(s)

SC-STD IDEAL-STD NC-STD 2K-STD 2K-HOT OPAL-STD CM-STD

0.5 -25.2 22.2 -92.1 -1.3 12.1 -0.1

of state, all other things being the same. The period difference is plotted in terms of the period of the standard model for the sequence of modes belonging to £ = 2. These results are quite typical of the other values of £ (the periods and period differences must be scaled by a factor \/£(£+ 1)). What the diagram shows is that the periods of the two models are quite similar with maximum relative deviations of ~ 3%. Typically, the differences are much less than that. The largest deviations occur for the low-order modes which are formed relatively deep in the star, and which are most sensitive to the thermodynamics of the helium plasma. The largest differences between the SC and the FGV data indeed arise for helium in the regime of interest for pulsating DA white dwarfs. Note that, even for hydrogen, there are substantial local differences in thermodynamic quantities between the two generations of equation of state tables (see the paper by Saumon in these Proceedings). However, as discussed above, pulsation modes are mostly sensitive to integrated properties, so that these local differences tend to average out. The very similar values of the integrals no(SC)=75.78 s and II0(STD)=75.74 s (see Table 1) indeed reflect the fact that the pulsation properties of the two models must be very similar. The expected period difference of 0.5 s for the £ = 2, 1000 s mode (Table 2) is quite consistent with the actual difference shown in Fig. 4. A happy consequence here, at least from our point of view, is that the newer SC equation of state data do not change significantly the period spectra of pulsating DA white dwarfs. The model parameters we have used in our experiments are typical of these stars, and varying the parameters within acceptable ranges will not change this conclusion. This means, in particular, that past calculations based on the FGV tables remain qualitatively and quantitatively reliable (see, e.g., Brassard et al. 1992a). The reader should not be left with the impression that any envelope

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Fontaine & Brassard: White dwarf seismology

envelope_eos 30.0

i

m

10.0 -

o

s u V

£

•g -IO.O •c

s. -30.0

100.

300.

500.

700.

900.

Period (STD) Fig. 16.4 Comparison of the pulsation periods of the SC and STD models. The figure shows the period difference of the two models for the same mode (same value of Jb) in terms of the period of the STD model. The results refer to g modes belonging to the I = 2 sequence.

equation of state is adequate to describe pulsating white dwarfs, however. In this context, we show, in Figure 5, some of the results of an experiment in which we have removed all nonideal terms and computed an envelope equation of state for pure H, He, and C on the basis of Saha equations only (IDEAL). Of course, as is well-known, this approach leads to complete recombination at sufficiently high densities, which is unphysical. In a format identical to that of Fig. 4, the diagram shows period differences which are large compared to the previous results and which, moreover, are systematic. Indeed, the period difference grows systematically larger (in an absolute sense) with increasing radial order. The periods of the model based on the ideal equation of state become smaller than the periods of the standard model. For the £ = 2, 1000 s mode, the expected period difference

Fontaine & Brassard: White dwarf seismology

359

envelope.eos 30.0

i

i

i

i

i

10.0 Q>

O 0 U V

•o -10.0 o

•c »

-30.0

100.

300.

500.

700.

900.

Period (STD)

Fig. 16.5 Same as Fig. 4, but for the IDEAL and STD models. is -25.2 s according to Table 2, which is comparable to the exact result shown in Fig. 5. Note also that, to a large extent, the jagged appearance of the curve is caused by mode trapping/confinement effects in the region where we rather crudely switch from our neutral, recombined ideal gas in the envelope to our totally ionized dense Coulomb fluid in the core. We found it also interesting to investigate the effects of nonideal terms in the core equation of state. In the next experiment, we have thus removed all the nonideal terms in the pure C equation of state of Lamb (1974; see also Lamb & Van Horn 1975). This could be easily done because Lamb's code, among many nice features, is modular in its construction, so only the kinetic energy terms, for example, can be retained. Figure 6 summarizes our results for the £ = 2 modes. The figure is on the same vertical scale as the previous ones, but there is a shift of the zero point. We can observe a systematic increase of the period with increasing radial order for the modified model (NC, which stands for "no Coulomb" terms)

360

Fontaine & Brassard: White dwarf seismology

core.eos

50.0

30.0 -

g o

a 0>

•o

1 -10.0 100.

300.

500.

700.

900.

Period (STD) Fig. 16.6 Same as Fig. 4, but for the NC and STD models.

as compared to the reference model. The accumulated period difference at 1000 s is comparable to the expected value of 22.2 s (Table 2). We note that by keeping the surface gravity constant, the modified model can accommodate a larger mass than the standard model because the ideal NC core equation of state is harder. The upshot is that the altered model is slightly more degenerate in its interior, \N\ is smaller, II0 is larger, and the periods are larger than for our reference model. We have carried out similar experiments with the opacity. For example, we have computed the period distributions of models in which the radiative opacity, the conductive opacity, and then the total opacity have been arbitrarily multiplied by a factor of 2. The results for the experiment with the total opacity are illustrated in Figure 7 (modes with £ = 2). Again, the period difference is plotted in terms of the period of the standard model; the symbol "2K" here stands for the modified, more opaque model. Note that

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Fontaine & Brassard: White dwarf seismology

total-opacity

-90.0 100.

300.

500.

700.

900.

Period (STD)

Fig. 16.7 Same as Fig. 4, but for the 2K and STD models. the vertical scale has been expanded to make allowance for the relatively large period differences. We can observe that the g-mode periods of the opaque model are systematically smaller than those of the standard model. This is easily understood since an increase of the global opacity leads, for a model with a fixed surface temperature, to a larger core temperature. This, in turn, implies that the overall degeneracy of the core is less, the Brunt-Vaisala, frequency is larger, and, consequently, the characteristic period spacing Tl0 is smaller in the modified model. We find indeed that the expected period difference of -92.1 s (Table 2) based on the n o arguments match rather well the exact result for a mode with a period of 1000 s. It is interesting to point out that the more opaque model mimics very closely a hotter model. In other words, we can recover almost exactly the period spectrum of the more opaque model by considering a model with standard input physics, but with a higher effective temperature. Such a

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Fontaine & Brassard: White dwarf seismology

equivalent_hot_model

I

-10.0 -

I

~-30.0 o

g I -50.0

I

s. -70.0 -

-90.0 100.

300.

500.

700.

900.

Period (HOT) Fig. 16.8 Same as Fig. 4, but for the 2K and HOT models. model is referred to here as "HOT", and is the only equilibrium model for which we have changed a parameter (Tefj in the present case). By trial and error, we have determined that a model with Teff = 14,280 K has nearly the same central temperature as the opaque model at Teff = 12,500 K, and a very similar value of the characteristic period spacing n o (see Table 1). Accordingly, it is not surprising that the period spectra of the two models are nearly the same as illustrated in Figure 8 (plotted on the same scale as Fig. 7). A better agreement could have been found by fine tuning even more the effective temperature of the hot model, but we believe this is not necessary to make our point. In another experiment, we have taken advantage of the recent availability of the OPAL Rosseland radiative opacity tables (Rogers & Iglesias 1992) to compute a model (OPAL) incorporating the new data for pure H, pure He, and pure C. The periods of this model (for I = 2 modes) are contrasted to those of the standard model in Figure 9. We find that the OPAL opacities

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363

radiative.opacity 30.0

-30.0

100.

300.

500. 700. Period (STD)

900.

Fig. 16.9 Same as Fig. 4, but for the OPAL and STD models. lead to systematically larger periods than the case based on LAO opacities. Again, the expected period difference of 12.1 s at 1000 s based on the integrals II 0 is consistent with the result of Fig. 9. On the whole, we find that the OPAL opacities are slightly smaller than the LAO opacities in white dwarf envelopes, leading to models with slightly lower central temperatures and increased values of II 0 . We think that, in a large part, the differences are mainly caused by a difference in chemical composition: our older LAO tables refer to mixtures which contain small traces of heavy elements (Z= 10~3), whereas we used the OPAL data for pure elements. We believe that the contribution of the Z elements to the LAO results makes the opacity somewhat larger, on the average, than the OPAL opacity (since the differences in opacity for pure H or pure He, in particular, are not expected to be very significant between the two generations of tables). The test should probably be redone with radiative opacity tables referring to exactly the same chemical compositions.

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The effects of varying the convective efficiency using the ML1, ML2, and ML3 versions of the mixing-length theory have been thoroughly investigated by Brassard et al. (1992a), and will not be repeated here. We simply recall that an increase of the convective efficicency leads to a systematic increase of the y-mode periods. The effect is negligible for low-order modes but increases with increasing radial order. We have recently incorporated in our model building code the parameterfree convection theory of Canuto & Mazzitelli (1991, 1992; CM). At Teff = 12,500 K, the CM model shows a hydrogen convection zone which is slightly less deep than the ML2 zone of the standard model. Since, at that effective temperature, the convection zone is quite high in the envelope (see Fig. 1), it does not contribute very much to the II0 integral. Accordingly, there is very little difference in the values of II 0 for the CM and STD models (Table 1). Figure 10 illustrates that period differences only appear for the high-order modes, i.e., for those which have nodes reaching out into the high envelope and interacting with the convection zone. In the present case, the period differences are quite small. They would actually increase for cooler models (which develop deeper convection zones), but low-order modes would remain unaffected by convection. Finally, we again refer the reader to Brassard et al. (1992a) for a discussion of the effects of varying the physical treatment used to describe the composition transition zones. In a nutshell, the main effect of, for example, decreasing the thickness of a composition transition zone is to increase the nonuniformities in the period spectrum; the average period spacing IIO remains largely unchanged. 16.5 Discussion We have presented the results of a number of numerical experiments to illustrate how theflf-modeperiod spectra of pulsating DA white dwarfs depend on the various components of the constitutive physics. In this exercise, we used stellar models with typical values for their parameters. In this connection, it should be reminded that, for reasons of distance, it is more complicated to read the asteroseismological records of pulsating stars than that of the Sun. For instance, when observing pulsating white dwarfs, we do not dispose a priori of independent and accurate estimates of basic stellar parameters such as total mass, total radius, surface gravity, effective temperature, and age. (We do not dispose either of thousands of well-identified modes!). Because these parameters are well known for the Sun, helioseismology can potentially be used to test the equation of state of the solar

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convective_efficiency

100.

300.

500.

700.

900.

Period (STD) Fig. 16.10 Same as Fig. 4, but for the CM and STD models. plasma as discussed by Dappen (these Proceedings). This is not a realistic claim to make for pulsating white dwarfs. In fact, what white dwarf seismology can really bring us are the means to infer the stellar parameters through the signatures they leave on the period distributions. In this approach, the stellar parameters are derived under the implicit assumption that the set of constitutive physics used in the model building phase is a good representation of the true physical conditions in the actual stars. The method we have used in this paper remains the only avenue for estimating the "internal errors" due to the constitutive physics. We note that Brassard et al. (1993) used a similar approach in their analysis of G117-B15A, and were able to show that their error budget is dominated by observational uncertainties (and not by shortcomings in the constitutive physics). We conclude by pointing out that the constitutive physics bears on two more aspects of white dwarf seismology, which could not be discussed in this paper due to space shortage. While we have focussed here on the influence

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of the input physics on the period spectra, the question of rates of period change is also quite important. These rates depend, of course, directly on the cooling rate, a problem which has received considerable new attention in recent years, and which has been reviewed by Chabrier, Garcia-Berro, and Mazzitelli at this Colloquium. We note that in the range of effective temperature where the pulsating white dwarfs are found, no "exotic" mechanism (neutrino cooling, crystallization, phase separation in the core, etc.) are expected to dominate, so the computed cooling rates across the narrow instability strip should be quite reliable. Finally, there is the question of mode excitation in pulsating white dwarfs. Nonadiabatic pulsation calculations by several independent groups all indicate that the driving region is located near the base of the hydrogen convection zone in pulsating DA white dwarfs. The question of mode driving and, in particular, the question of the location of the theoretical blue edge of the instability strip boil down to how deep is the hydrogen convection zone in a given model. Clearly, the most important component of the constitutive physics in the present context is the physical model used to describe convection. (The envelope equation of state as well as the radiative opacity also bear on the problem). A nonadiabatic survey to assess the influence of different convection models on the question of the location of the theoretical blue edge is a timely project. We have undergone such a project (Brassard, Fontaine, k Wesemael 1993). Acknowledgement We wish to thank friends and collaborators for providing invaluable support, input, and insight during the course of our ongoing investigations of pulsating white dwarfs. Special thanks are due to those who have provided the basic data used in this paper: Gilles Chabrier, Hal Graboske, Bill Hubbard, Walter Huebner, Naoki Itoh, Don Lamb, Italo Mazzitelli, Forrest Rogers, Didier Saumon, and Hugh Van Horn. This work was supported in part by the NSERC Canada and by the Fund FCAR (Quebec). References Brassard P., Fontaine G., Wesemael F., k Tassoul M. Ap. J. Suppl. 81, 747, (1992a) Brassard P., Pelletier C , Fontaine G., k Wesemael F. Ap. J. Suppl. 80, 725, (19926)

Brassard P., Fontaine G., k Wesemael F., in preparation, (1993) Brassard P., Fontaine G., Bergeron P., Wesemael F. k Vauclair G., in preparation, (1993)

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Canuto V., k Mazzitelli I. Ap. J. 370, 295, (1991) Canuto V., k Mazzitelli I. Ap. J. 389, 724, (1992) Fontaine G., Graboske H.C. Jr. k Van Horn H.M. Ap. J. Suppl. 35, 293, (1977) Fontaine G., Villeneuve B., k Wilson J. Ap. J. 243, 550, (1981) Hubbard W.B. k Lampe M. Ap. J. 163, 297, (1969) Huebner W.F., private communication, (1980) Iben I. Jr., k Tutukov A.V. Ap. J. 282, 615, (1984) Itoh N., Mitake S., Iyetomi H. k Ichimaru S. Ap. J. 273, 774, (1983) Itoh N., Kohyama Y., Matsumoto N. k Seki M. Ap. J. 285, 758, (1984) Kawaler S.D., k Hansen C.J., IAU Colloq. 114, White Dwarfs, ed. G. Wegner (New York:Springer-Verlag), 97, (1989) Koester D., k Schonberner D. Astr. Ap. 154, 125, (1986) Lamb D.Q., Ph. D. thesis, University of Rochester, (1974) Lamb D.Q. k Van Horn H.M. Ap. J. 200, 306, (1975) McGraw J.T., Ph. D. thesis, University of Texas at Austin, (1977) Robinson E.L., IAU Colloq. 53, White Dwarfs and Variable Degenerate Stars, eds. H.M. Van Horn k V. Weidemann (Rochester: University of Rochester Press), 343, (1979) Rogers F.J. k Iglesias, C.A. Ap. J. Suppl. 79, 507, (1992) Saumon D. k Chabrier G. Phys. Rev. Letters 62, 2397, (1989) Saumon D. k Chabrier G. Phys. Rev. A 44, 5122, (1991) Saumon D. k Chabrier G. Phys. Rev. A 46, 2084, (1992) Tassoul M. Ap. J. Suppl. 43, 469, (1980) Van Horn H.M., Proc. 25th Liege Astrophysical Colloquium, Theoretical Problems in Stellar Stability and Oscillations, eds. A. Noels k M. Gabriel (Liege:Universite de Liege), 307, (1984) Wesemael F., Bergeron P., Fontaine G. k Lamontagne R.L., Proc. Seventh European Workshop on White Dwarfs, eds. G. Vauclair k E.M. Sion (NATO ASI Series), 159, (1991) Winget D.E., IAU Symp. 123, Advances in Helio- and Asteroseismology, eds. J. Christensen-Dalsgaard k S. Frandsen (Dordrecht:Reidel), 305, (1988) Winget D.E. k Fontaine G., Pulsations in Classical and Cataclysmic Variable Stars, eds. J.P. Cox k C.J. Hansen (Boulder: University of Colorado Press), 46, (1982) Winget D.E., Hansen C.J. k Van Horn H.M., Nature 303, 781, (1983)

17 Helioseismology: the Sun as a stronglyconstrained, weakly-coupled plasma W. DAPPEN Department of Physics and Astronomy, University of Southern California, Los Angeles, CA 90089-1342, USA

Abstract Accurate measurements of observed frequencies of solar oscillations are providing a wealth of data on the properties of the solar interior. The frequencies depend on the solar structure, and on the properties of the plasma in the Sun. Except in the very outer layers, the stratification of the convection zone is almost adiabatic. There, the sound-speed profile is governed principally by the specific entropy, the (homogenous) chemical composition and the equation of state. It is therefore essentially independent of the uncertainties in the radiative opacities. The sensitivity of the observed frequencies is such that it enables to distinguish rather subtle features of the equation of state. An example is the signature of the heavy elements in the equation of state. This opens the possibility to use the Sun as a laboratory for thermodynamic properties. Les frequences observees des oscillations solaires constituent une base de donnees extremement riche qui nous permet d'etudier les proprietes de l'interieur du soleil. Les frequences dependent de la structure solaire et des proprietes locales du plasma (surtout de la vitesse du son). Sauf dans les couches tres exterieures, la structure de la zone convective du soleil est essentiellement adiabatique. Le profil de la vitesse du son est done donne par l'entropie specifique, la composition chimique (homogene) et l'equation d'etat. L'opacite radiative ne joue pas de role. Grace a la grande precision des frequences observees on arrive a distinguer des phenomenes assez sub368

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tiles dans l'equation d'etat, comme la signature faible des elements lourds. Le soleil est devenu un laboratoire de physique des plasmas stellaires.

17.1 Introduction Solar acoustic oscillations have opened a new window into the Sun. By their nature they link the local sound speed in the interior with the observed oscillation frequencies. The spatial resolution of the solar disk allows the identification of a large number of individual oscillation modes, which are classified in terms of spherical harmonics. Modes in a large range of angular degrees, between / = 0 and a few thousand, are observed. The frequencies of these modes are centered around 3 mHz, which corresponds to periods around 5 minutes. They have been determined with high precision: typical relative errors are of the order of 10~4. The modes are confined to a cavity, which extends, broadly speaking, from the surface of the Sun, where the waves lose their material support, to the inner turning point which lies deeper the lower the angular degree / is. Radial modes, with / = 0, have no inner turning point and their cavity is the entire Sun. The observed solar oscillation modes are standing acoustic waves; hence the quantity most obviously probed is sound speed. Since the oscillations are largely adiabatic (except very near the surface), the frequencies are determined predominantly by the local adiabatic sound speed, which is a thermodynamic quantity. In addition, the frequencies depend on the density distribution in the Sun. Therefore, these helioseismic frequencies can be used as a diagnosis of the plasma of the solar interior. A high-quality thermodynamic potential is needed for the pressure-density relation (i.e. the equation of state, which is essential for determining the hydrostatic equilibrium between pressure gradient and gravity) and for thermodynamic quantities (mainly adiabatic sound speed). Introductions to helioseismology are, for example, the reviews by Deubner & Gough (1984), Christensen-Dalsgaard, Gough & Toomre (1985), Bahcall & Ulrich (1988), Christensen-Dalsgaard (1988), Libbrecht (1988), Vorontsov & Zharkov (1989), Gough & Toomre (1991), Libbrecht & Woodard (1991), Christensen-Dalsgaard & Berthomieu (1991), Gough (1992), and TurckChieze et al. (1993). The reviews by Christensen-Dalsgaard (1991) and Christensen-Dalsgaard & Dappen (1992) specifically address the helioseismic determination of the equation of state.

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Fig. 17.1 Observed p-mode frequencies obtained from a 20-day sequence obtained at the 60-Foot Solar Tower of Mount Wilson Observatory.

17.2 Helioseismology: observations After the discovery of the solar five-minute oscillations by Leighton et al. (1962), it took 15 years before they were recognized as global oscillations. Figure 1 shows a typical display of the helioseismic data. While early data looked extremely noisy, the observational progress made since has been tremendous, resulting in very clean data. In a typical representation of helioseismic data, the frequencies of all observed oscillation modes are plotted against their angular degree /. In general, for a given angular degree one observes more than one frequency. They belong to modes of different numbers n of radial nodes. If one plots the observed frequencies, those belonging to modes with the same number of radial nodes can be connected with smooth lines; this is true for any vibrating gas sphere. Figure 1 shows such a v — I diagram obtained from current observations. Since modes with the same radial order n lie on the same ridge, one can therefore identify the radial order n with the different ridges of the diagram. Such an identification is possible up to an unkown global constant no. Duvall (1982) found a technique to resolve this remaining ambiguity and to identify the radial order uniquely. In the mid-seventies, the ridges in the v — l diagram began to emerge from the noise (Deubner, 1975); once they were seen, they definitely established the solar nature of the five-minute oscillations as a superposition of global oscillation modes, a suggestion made earlier by Ulrich (1969, 1970).

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If the Sun were spherically symmetric, then each mode frequency vn\ would be 2/ + 1 times degenerate. The solar rotation (like any other nonspherical perturbation, such as, e.g., magnetic fields) breaks this symmetry, thus each frequency is split into a multiplet. The splitting is small, since it is of the order of the angular frequency of the solar rotation, which has a period of a little less than a month. Therefore the rotational splitting is too small to be visible in a plot of absolute frequencies such as Figure 1. However, thanks to observational series of weeks and months, the splittings can be well observed for a wide range of / (see, e.g., Harvey, 1988; Rhodes et a/., 1990). Why did it take some 15 years before the oscillations were properly identified? The reason is that the oscillation velocities are tiny, less than 1™. And yet, such velocities are observed using the Doppler effect of light. From each wing of a given spectral line, a narrow piece is cut out and sent through an interferometer into a comparator. The intensity difference of the two parts then becomes a measure of the Doppler line shift, and thus radial velocity. Since the solar disk can be well resolved, such measurements can nowadays be made typically for 1024 X 1024 pixels simultaneously, and this at a rate of a few times per minute. Using the Doppler effect of light, velocities of the order of 1-jp are only marginally detectable. One might therefore wonder why one can obtain so clean a picture as in Figure 1. This question is even more in order if one considers the seemingly chaotic motion on the Sun, granulation, supergranulation, flares, rotation, and so on. The answer lies in the extreme regularity and the surprisingly long life time of the modes, which allow the observers to follow an individual oscillation mode for days and weeks. Therefore, the strict periodicity of the signal is exploited, so that in the end the frequencies can be determined very accurately against all initial odds. The data like those of Figure 1 allow a high precision analysis of the structure of the solar interior. Tabulated frequencies are given in the article by Libbrecht et al., 1990. The relative precision, with which each of the observed mode frequencies vn\ is determined, now attains 10~4, which is at least one order of magnitude better than the uncertainties of any current theoretical predictions. The reason for this inadequacy of the theoretical models is that they are not (yet) sufficiently sophisticated, because the usual simplifying assumptions on convection, opacity, equation of state, nuclear physics, internal rotation, and other physical ingredients are not good enough to explain all the details encountered in the seismological data.

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17.3 Helioseismology: theory Broadly speaking, theoretical inferences from the observed helioseismic frequencies can be made in two ways. In the forward approach, we build a solar model and compute its normal modes. Then the "best" model is the one that satisfies all observational constraints. Should there be more than one "best" model, an aesthetic principle such as Occam's razor is invoked to select the simplest of them. In the inverse approach, we try to make as few theoretical assumptions as possible to infer the physical state of the solar interior directly from the oscillation frequencies.

17.3.1 The general equations for evolution and oscillations For tutorial purposes I will go somewhat off the beaten track and discuss the evolution and oscillations of the star at the hand of the same set of hydrodynamic equations. Of course, the time scales of evolution and oscillations are so much different that in practical calculations one always separates the two parts. Making here as many simplifications as I dare, I refer the interested reader to the superb book by Unno et al. (1989). The solar case is extensively dealt with, for instance, in the reviews by ChristensenDalsgaard and Berthomieu (1990) and Turck-Chieze et al. (1993). I neglect viscosity, and assume that any treatment of turbulent motion, or convective heat transfer, is done in terms of a mean-field approach. This means that state variables are averaged over time-scales of turbulent motion. Such an approach is justified except in a thin layer beneath the solar photosphere. Under the assumptions of inviscid motion and mean-field variables, the resulting 9 equations are

^ + div(pv)] = 0

(2) (3)

= AirGp

(4)

Dappen: Helioseismology

373

F = K(VT;7>,X)

(5)

enuc = €nuc(T,p,X)

(6)

K=K(T,P,X)

(7)

p = p(T,p,X)

(8)

s = s(T,p,X)

(9)

Here, v is the (Eulerian) velocity field, p and p are pressure and density, respectively, 4> is the (self-) gravitational potential, s is specific entropy (per unit mass), enue is the nuclear energy generation rate, F is the energy flux (in the mean-field sense) through the star, the vector X = (X\,X2,...Xn) symbolizes the chemical composition, with X{ being the mass fractions of element t, and K is the opacity. Note the square brackets in Eqs. (1-3). They are there for the discussion of the separate issues of evolution and oscillation (see below). Taking into account the vector nature of Eq. (1) and (5), Eqs. (1-9) are 13 equations for the 13 hydrodynamic fields v, p, p, , T, s, F, € nuc , «• Eqs. (1-4) are partial differential equations, Eq. (5-9) are "material" equations, and it is no surprise that they are the hard part of the overall problem. The toughest among them is the expression of the "conductivity" for energy (Eq. 5), because it is the result of wholly different physical processes according to the physical conditions, given by T, p, VT. Energy transport by radiation, convection and electron conduction are the most familiar ones. As long as stellar matter is optically thick (which it is except near the stellar surface), Eq. (5) can be simplified with the help of the diffusion approximation

but when matter is optically thin, strictly speaking even the form of Eq. (5) is inappropriate, because then radiative transport becomes intrinsically nonlocal, and radiation hydrodynamic will have to be brought into the game (see the book by Mihalas and Mihalas, 1984).

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Eq. (5) also needs some "switch" to change to convective form of energy transport when the local conditions do not warrant a stable radiative stratification. For the equilibrium model, one usually assumes a stability criterion d la Schwarzschild or Ledoux plus some mixing-length formalism (see e.g. Cox & Giuli, 1968; Gough k Weiss, 1976; Unno et a/., 1989). For the oscillation part, the interplay between convective and oscillatory motion can become very complicated. Compared with the question of energy transport (Eq. 5), the rest of the material equations (6-9) look relatively harmless. The most difficult among them is opacity (7), which appears in the diffusion approximation (10).

17.3.2 Evolution Formally speaking, the problem of stellar evolution is the one of Eqs. (1-9) without the parts in the big square brackets, that is without the inertia term of Eq. (1) and the thermal term of Eq. (3). Just to illustrate with a familiar equation, note that in the approximation of a spherically symmetric configuration Eqs. (1) and (4) allow elimination of and become the wellknown equation of stellar structure (see, e.g., Schwarzschild, 1958). dp _ GMrp ( } dr~ r* * Here, as usual, Mr denotes the mass of the sphere of radius r and G the constant of gravitation. Similarly, in the spherical approximation, Eq. (3) becomes (with Lr = 4irr2F) — = 4wr2pcnuc, (12) ar and Eq. (5) [in the form of the special case of Eq. (10)] becomes the equally familiar equation of the radiative temperature gradient

dr 647T
Dappen: Helioseismology

375

the problem of stellar evolution easy. In fact, the evolutionary part is the hard part, not only because of its nonlinear nature (in contrast to the oscillation problem, here no linearization is available). The complexity of stellar evolution is due both to the rich variety of physical phenomena contained in the material equations (5-9) and to the varying chemical composition in the star.

17.3.3 Oscillations Having solved the hard problem of stellar evolution, we arrive at an equilibrium model (often assumed to be spherically symmetric). Again, the reader is referred to the book by Unno et al. (1989). Let po, po, To, o, so, Fo be the variables and X(r) the profile of the chemical composition that define this equilibrium model at some point to in the star's evolution. Introduce the Eulerian displacement variables of the kind p1 — p — po, p' = p — po, and so forth, and insert these difference variables into the original equations (1-9). We then obtain equations that look essentially the same, though now they are written for the displacement variables. As long as no linearization or other simplification is made, the new equations are of course equivalent to the old ones. However, the purpose of going to displacements is to make approximations. The two most important are those of a static and spherically stratified equilibrium. The assumption of a static equilibrium precisely reflects the vast gulf between oscillation and evolution time-scale, an assumption certainly valid until the very violent final phases of the star's life. The other assumption, that of a spherical equilibrium state, is a working hypothesis, not bad if rotation and magnetic fields do not distort the equilibrium state too much. According to Noether's (1918) theorem, the two assumed underlying symmetries (time translations, rotations) lead to conserved quantities, but they show up nicely only in a linear theory. Let us thus linearize the whole system (1-9), that is, neglect all second and higher-order terms in the displacement quantities. In the material equations (5-9) the complicated functional behavior is greatly simplified by linear expressions that involve the equilibrium quantities Po,£nuc,o>Ko and their partial derivatives evaluated at the equilibrium values. Under these assumptions (static and spherical equilibrium and linearization), there are special solutions that are products of a radial amplitude with an exponential time dependence and a (possibly vector-) spherical harmonic function. If we consider the example of pressure we write

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(^0)-

(14)

There are analogous expressions for all other variables. The general solution is a superposition of such particular solutions. If we assume that thermal heat losses are negligible small during the fast oscillations, then we deal with adiabatic oscillations. Drastic simplifications become possible. The whole equation (3) disappears, because the left-hand side vanishes identically, showing that the equilibrium condition is exactly preserved (this statement is true even in the nonlinear case. In the absence of equation (3), temperature, energy flux and opacity do not participate in the oscillation equation, though they are of course important in the equilibrium part. Thus equations (5-7) are also gone. Thermodynamics becomes ultra simple, especially in the linear case, where the co-moving Lagrangian pressure and density fluctuations (Sp and Sp) are simply related through the equilibrium adiabatic gradient Fi = (dlnp/d\np)a

.

(15)

P

In the nonlinear adiabatic case, this simple relation would have to be replaced by the function p(p) that follows from the integral of motion s, i.e. from the implicit equation s = s(p,p,X.) = const. A further simplification of the adiabatic problem is that the equation of continuity permits expressing the tangential component of the displacement field in terms of the pressure fluctuation (for details see Unno et a/., 1989). We thus arrive at the famous adiabatic eigenvalue problem of stellar oscillations, which plays a central role in helioseismology. The result is, for each angular degree /, an eigenvalue problem, which consists of coupled equations for the radial amplitudes of the displacement vector £ r , the fluctuation of pressure pf and of the gravitational field , and of the usual boundary conditions at the center and the surface of the star. For radial oscillations or in the so-called Cowling approximation for nonradial oscillations (where one neglects changes of the gravitational field during the oscillatory motion), the equations become especially simple. Their formal type is (again, see Unno et al., 1989, for details)

Dappen: Helioseismology

377

f = Dp + Et + F

(16)

dr The Coefficients A,B,C,D,E,F are not constant but functions of the radius. They contain the properties of the equilibrium model and, most importantly, the eigenvalue u\. As mentioned, boundary conditions complement the equations (16). Thus the problem of adiabatic stellar oscillations is, for each /, completely analogous to an inhomogeneous vibrating string. For each / there is a set of solutions with different radial nodes n and frequencies uni. It should be clear by now that the solution of the eigenvalue problem is much easier than finding the equilibrium model through stellar evolution. Of course one can make things more complicated here as well. By considering nonadiabatic motion, the energy equation (3) is coining back, and with it temperature, which forces us to bring in £nuc» K» a n ^ convection. Again, I refer to the book by Unno et al. (1989). Near the solar surface there are nonadiabatic effects that have to be treated properly before the theoretical data will match the observations shown in Figure 2. 17.3.4 Inverse analysis If one writes Eq. (16) very formally, the oscillation frequencies un\ can be written as functionals r), p(r),...]

(17)

of the structure of the Sun. So far we have discussed how to obtain the frequencies, given the structure. With the ability to do so, one can compare observed frequencies with computations based on different models, and in this way obtain some information about the solar structure. However, it is evidently desirable to attempt to invert the process, to obtain more extensive information about the properties of the solar interior from the observed frequencies. Such inverse analyses are, in a certain sense, implicit in any type of scientific measurement, since a raw measurement rarely supplies the quantity that one is interested in. However, in the present case the relation between the desired properties of the Sun, e.g. p(r), and the observed quantities is more complex, since each frequency is sensitive to the structure of a substantial part of the Sun; thus the inverse problem is correspondingly more difficult. Similar problems are encountered in other

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branches of science, such as geophysics and radiation theory, and there is a substantial literature dealing with them {e.g. Parker 1977; Deepak 1977; Craig k Brown 1986; Tarantola 1987). An alternative method of inversion is based on asymptotic theory, where local propagation properties for acoustic waves are approximately examined in the spirit of a JWKB analysis. The need for such an approximate discussion comes from the fact that, although the numerical solution of the equations of adiabatic oscillations is relatively simple, it does not immediately provide an understanding of the properties of the oscillations. Such a direct understanding can come from the approximate asymptotic analysis It was shown by Gough (cf. Deubner & Gough 1984; Gough 1986) how to write down an approximate form of the oscillation equations, from which it is straightforward to obtain the asymptotic behavior of the solution. It turned out immediately that this asymptotic approach also opens the door for elegant asymptotic inversion methods. I refer the reader to the papers by Gough (1985), Thompson (1991), Gough & Thompson (1991), Brodsky & Vorontsov (1993), and Gough & Vorontsov (1993). The last two papers deal with a nonlinear asymptotic inversion. The power of such inversions for the equation of state is illustrated in the article by Vorontsov et al. (tfiese proceedings).

17.4 Comparison of theory with observations The most direct way to compare theory and observation is to compute the analogue of Fig. 1 with the forward techniques mentioned above, so that the difference between each observed and computed frequency can be taken. Figure 2 shows four such diagrams of frequency differences, each for a different theoretical model. Two equations of state and two different opacity tables were used in the models. The two equations of state were (i) the Eggleton, Faulkner & Flannery (1973) (EFF) equation of state and (ii) the CEFF equation of state, which is, as explained below, an EFF plus a Coulomb term (Christensen-Dalsgaard, 1991; Christensen-Dalsgaard & Dappen, 1992). The opacities used were the Cox and Tabor (CT) (1976) and Los Alamos Opacity Library (LAOL) tables. Since I merely want to illustrate the sensitivity of the helioseismic method, it doesn't matter that these opacities are not the most current ones. A recent calculation based on Livermore opacitites can be found in Berthomieu et al. (1993a). I remark in passing that in such comparisons of observed with computed data ("0-C diagrams"), it is useful if an appropriate scale factor is taken out (see, e.g., Christensen-Dalsgaard 1988; Christensen-Dalsgaard

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I £

-30

a)

-10

-30

2000

3000 4000 »„ (/iHz)

2000

3000

4000

2000

3000 4000 V- (MHz)

2000

3000

4000

-30

Fig. 17.2 Frequency differences, scaled by the factor Qn\ (see text), between observed frequencies in the compilation by Libbrecht et al. (1990) and four sets of computed frequencies, in the sense (observation) - (theory). The abscissa is cyclic frequency i/n/. The points have been connected with lines according to the value of the degree /: / — 20,30: ; / = 40,50,60,80,100: ; / = 120,150,200,300,400: ; and / = 500,600,700,800,900,1000: . The models are distinguished by their equation of state and opacity (a) EFF equation of state, CT opacity; (b) EFF equation of state, LAOL opacity; (c) CEFF equation of state, CT opacity; (d) CEFF equation of state, LAOL opacity (from Christensen-Dalsgaard k. Dappen, 1992).

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& Berthomieu 1991). This scale factor Qni, which is essentially the inertia or kinetic energy - of the mode with quantum numbers nl (normalized to the same surface amplitude), contains the principal / and frequency dependence of the individual mode frequencies vn\. The purpose of the illustration in Fig. 2 is to show the sensitivity of the helioseismic analysis with respect to changes in the physics of the model. A perfect model would yield a horizontal line corresponding to all f>vn\ = 0. Note that the discrepancies between theory and observation are huge compared to the observational errors which are nowadays significantly below 1 fiUz. Such a combination of quantity and quality of astrophysical data is truly exceptional. 17.5 The equation of state As we have seen, the three basic material properties required in stellar models are the equation of state, opacity, and the nuclear-energy generation rate. At this meeting, the focus is on the equation of state. I shall use the term equation of state in a slightly broader sense than usual, so that it encompasses not only pressure as a function of temperature and density, but also all thermodynamic quantities. These quantities must be consistent with each other, that is, their appropriate Maxwell relations have to be satisfied. Such formal consistency is always achieved if the equation of state and the thermodynamic quantities stem from a single thermodynamic potential. In trivial models (e.g. in a plasma assumed to be fully ionized everywhere) it is possible to write down a consistent equation of state and thermodynamic quantities independently. However, in more realistic cases, modeling a thermodynamic potential is the only practical way to obtain the equation of state and thermodynamic quantities. A quick glance at Fig. 2 reveals that solar observations are indeed very sensitive to details of the equation of state. One might go further and conclude that the Sun prefers the CEFF to the EFF equation of state. However, such conclusions are fraught with danger, although probably not in this clear-cut case. The reason why one has to be prudent is that there are too many uncertainties in the solar model, coming, e.g., from convection or opacity, so that one has to be alert to the possibility that by changing the equation of state one could trigger changes in the other physical parameters. An illustration for this is found at each railroad crossing in France, where a sign warns: "un train peut en cacher un autre" (which, applied to our situation, means: proceed with caution, watch out for a hidden train of thought). If, say, the opacity is bad, one can not rule out that a worse equation of

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state could cause an overall better agreement with observations. Only when simultaneous progress with the other physical quantities is made (that is, if someone is watching the other track, to use the train metaphor), we will learn how to disentangle the different effects. However, for a sensitivity analysis, Figure 2 is already sufficient. The transitions from panels a to c and 6 to d, respectively, are obtained by putting some additional nonideal effects (the Coulomb pressure) into the equation of state with everything else unchanged. The response of the Sun, as seen through the "eyes" of helioseismology, is huge. I will not elaborate how the equation of state is modeled. Several authors of these proceedings do it (Rogers, Alastuey, Saumon, and Chabrier). My message is different: I intend to show why there is still a long way to go before rigorous theories (for instance that presented by Alastuey, these proceedings) can be used in solar and stellar models. I will begin with requirements for any solar or stellar equation of state. I insist that formal aspects (such as consistency and smoothness) play a crucial role. As a consequence, I would like to raise sympathy for the many home-grown formalisms that stellar modelers have been constantly developing. Then I will discuss the nonideal plasma effects that have to be included in realistic solar equations of state. Finally I will present a few selected results from equation of state comparisons. In the absence of a perfect equation of state, the comparisons can give us at least important information about the amount of the current uncertainty in the equation of state. Also, it will tell us at which temperatures and densities the uncertainty is most noticeable and to what degree solar observations can discriminate between various models. 17.5.1 Requirements on an equation of state for stellar models A stellar equation of state has to satisfy four conditions: (i) a large domain of applicability (in p, T), (ii) a high precision of its numerical realization, (iii) consistency between the thermodynamic quantities, and (iv) the possibility to take into account relatively complex mixtures with at least several of the more abundant chemical elements. More specifically, the first condition demands that the formalism can be used from the stellar surface (the photosphere), where T is typically a few 103 K and p some 10~7 g/cm3, to the center of a star where T is, again typically, about 107 K and p some 102 g/cm3. The second condition demands that a given formalism can be cast in an algorithm that converges without ambiguity and with sufficient precision, so that all required thermodynamic derivatives (such as adiabatic

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gradients) can be computed. Note, that for this only formal precision is required: reality of the physical description is a different issue. The third condition, consistency, states that all thermodynamic quantities stem from a single thermodynamic potential. This condition is often violated in twoor more-zone formalisms, which contains a different physical theory in different parts of a star. An example is the ad hoc imposition of full ionization in the central region, in order to mimic a pressure-ionization device, in combination with a conventional Saha equation in the envelope of the star. Such a formalism leads to a discontinuous thermodynamic potential and a violation of thermodynamic identities. Such violations of thermodynamic identities are inadmissible in calculations of stellar structure and oscillations. As we have seen, calculations of stellar oscillation frequencies often exploit thermodynamic quantities to transform one variable into another. Equation (15) shows such a transformation. There the adiabatic gradient Fi is used to establish a connection between density and pressure changes, and it is an absolute necessity that the Fi is consistent with the equation of state and other thermodynamic variables of the model. This example illustrates the necessity of formal consistency. Finally, the third and last condition, i.e. the possibility to describe rather realistic chemical compositions, is a bit less important for the equation of state itself. However, for opacity, heavy elements are very important, and a good equation of state plays an important role in any opacity calculation. 17.5.2 The role of the solar convection zone Energy transport by radiation is treated adequately in the solar interior in the diffusion approximation; on the other hand, energy transport by convection is usually treated in a rather crude way, with an a priori unknown parameter, the so-called mixing length (see, e.g., Cox and Giuli, 1968). Near the surface, convection is probably sufficiently vigorous to cause dynamic effects on the average hydrostatic equilibrium, yet such effects are often ignored. At the lower boundary of the convection zone, motion is normally supposed to stop at the point where convective instability ceases; there is no doubt, however, that motion extends into the convectively stable region through convective overshoot, although the extent of the overshoot is uncertain (see, e.g. Berthomieu et al., 1993b). Despite the complications it introduces, in a certain sense convection simplifies the structure of the outer parts of the Sun. Regardless of the uncertain details of convective energy transport, there is no doubt that

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except in a thin boundary layer near its top the convection zone is very nearly adiabatically stratified {e.g. Gough k Weiss 1976). One can show (Christensen-Dalsgaard, 1986) that the structure of the almost adiabatically stratified convection zone only depends on the equation of state, the composition and the constant value of the specific entropy, which in turn is essentially fixed by the value of the mixing-length parameter; particular, the convection zone structure is insensitive to the opacity. Another simplification of convection is that it makes the chemical composition homogeneous in the convection zone, although there is of course the possibility of gravitational settling (for a recent calculation, see Christensen-Dalsgaard et a/., 1993). Beneath the convection zone, the stratification becomes highly dependent on radiative opacity. It is difficult to disentangle the helioseismic effects of equation of state and opacity, but if opacity can be nailed down relatively accurately, an equation of state diagnosis can also become possible. An examlpe of an equation of state issure is the possibility of partial recombination of He+ ions in the solar center (see Christensen-Dalsgaard k Dappen, 1992). 17.6 Equation of state comparisons The most direct way to test the equation of state would be laboratory experiments. However, so far they have not yet helped to check realistic stellar equations of state. For instance, attempts to use constraints from a highprecision optical emission spectrum (e.g. Wiese, Paquette k Kelleher, 1973) have failed, because line-broadening effects were overshadowing the subtle details of statistical mechanics. It is therefore no wonder that - despite their difference in statistical mechanics - several of the currently popular equations of state have been able to reproduce that optical experiment (Dappen, Anderson and Mihalas, 1987; Seaton, 1990; Iglesias k Rogers, 1992). An alternative "experimental" approach is to use solar oscillation data. As the comparisons between observed and theoretical solar oscillation frequencies (Fig. 2) demonstrate, one can use the Sun to test the equation of state (for more details, see Christensen-Dalsgaard, Dappen k Lebreton, 1988; Christensen-Dalsgaard, 1991; Christensen-Dalsgaard k Dappen, 1992). Inversions of solar oscillation frequencies, such as those presented by Vorontsov et al. (these proceedings), have also demonstrated a high diagnostic potential for subtle effects, such as the location of the pressure-ionization region of helium and the influence of heavy elements in the equation of state. The disadvantage of a solar diagnosis is of course that we cannot vary the

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parameters: we have to accept solar conditions as they are. Only asteroseismology carries the promise to overcome this handicap (for a recent review, see e.g., Christensen-Dalsgaard, 1993). In the absence of a rigorous computation of the equation of state (to the needed accuracy), one can make comparisons between different models of the equation of state. Such comparisons will give us information about the overall uncertainty in the equation of state. But they also allow solar physicists to determine how uncertainties in the equation of state propagate into theoretically predicted oscillation frequencies. In this way, a "map" of the T—p plane can be drawn, showing localized "interesting" regions, where nonideal effects of one or another kind are important. I will briefly present the equations of state used in the comparisons. More details about them (and further references) can be found in the article by Christensen-Dalsgaard & Dappen (1992). I just recall that all currently used stellar equations of state can be classified in terms of the so-called "chemical picture" and the "physical picture" (Krasnikov, 1977). While in the more conventional chemical picture bound configurations (atoms, ions and molecules) are introduced and treated as new and independent species, only fundamental particles (electrons and nuclei) appear in the physical picture. In the chemical picture, reactions between the various species occur, and thus the thermodynamic equilibrium must be sought among the stoichiometrically allowed set of concentration variables by means of a maximum entropy (or minimum free-energy) principle. In contrast, the physical picture has the aesthetic advantage that there is no need for a minimax principle; the question of bound states is dealt with implicitly through the Hamiltonian describing the interaction between the fundamental particles. For exhaustive treatments of these issues, consult the three books by Ebeling, Kraeft & Kremp (1976), Kraeft et al. (1986), Ebeling et al. (1991).

17.6.1 EFF Eggleton, Faulkner & Flannery (1973) developed a simple equation of state in the chemical picture (EFF) that is formally consistent and includes an ad hoc pressure ionization device that works at least qualitatively correctly. The device is not based on a physical model (e.g. a description of an atom and its surrounding particles), but is imposed by forcing the anticipated result, i.e., full ionization at high densities. In addition, the EFF equation of state incorporates a correct treatment of the partially degenerate electrons according to Fermi-Dirac statistics. Bound systems (atoms and ions) are

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always assumed to be in their ground state; the ground-state energy is constant and equal to the free-particle value.

17.6.2 CEFF To overcome the lack of a Coulomb term in the EFF equation of state, J0rgen Christensen-Dalsgaard and I have added a Coulomb configuration^ term in the Debye-Hiickel approximation (taken from the MHD equation of state). Such an upgrade of the EFF equation of state was motivated by the fact that adding a Coulomb term to the EFF equation of state makes a significant contribution towards a more realistic equation of state (see below and the papers by Christensen-Dalsgaard, 1991; Christensen-Dalsgaard & Dappen, 1992). Of course the remaining disadvantages of the EFF equation of state still point to the need of more complete formalisms. However, the successful application of the CEFF equation of state to solar physics makes it very well suited as a reference equation of state.

17.6.3 MHD The Mihalas-Hummer-Dappen (MHD) equation of state (Hummer & Mihalas, 1988; Mihalas et al., 1988; Dappen et a/., 1988) is realized in the chemical picture with the free-energy minimization method. Occupation probabilities are introduced on the one hand to avoid the famous (or rather notorious) discontinuities that come along with simple cut-off recipes for internal partition functions. On the other hand they represent a result that should come from quantum mechanics, namely the fraction of atoms or ions for which a given state can exist (given the constraints of the surrounding particles). Only then, these "available" states are populated according to statistical mechanics. It is dear that such an approach is largely intuitive. However, its advantage is that complicated plasmas can be modeled, with detailed internal partition functions for a large number of atomic, ionic, and molecular species. All particles are allowed to interact with each other. Also, full thermodynamic consistency is assured by analytical expressions of the free energy and its first- and second-order derivatives. This not only allows an efficient Newton-Raphson minimization, but, in addition, the ensuing thermodynamic quantities are of analytical precision and can therefore be differentiated once more, this time numerically. Reliable third-order thermodynamic quantities are thus calculated. The MHD equation of state was realized for the international "Opacity Project" (see Seaton, 1987).

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17.6.4 OPAL The OPAL equation of state is realized in the physical picture. A detailed presentation is given by Rogers (these proceedings). In the physical picture, the concept of a perturbed atom in a plasma is not needed at all. Therefore, no assumptions about energy-level shifts or the convergence of internal partition functions have to be made. On the contrary, properties of energy levels and the partition functions come out from the formalism. The OPAL equation of state was developed by a group at Livermore as part of their opacity project (Rogers, 1986; Iglesias, Rogers & Wilson, 1987; Rogers, these proceedings). This equation of state does satisfy the requirements from stellar modelling that I mentioned above; however, a systematic application application of the OPAL equation of state to helioseimology is still awaiting. 17.6.5 Results from the comparisons Early comparisons showed a striking agreement between the MHD and OPAL equation of state for conditions as found in the hydrogen-helium ionization zones of the Sun (Dappen, Lebreton & Rogers, 1990; Dappen, 1990). For convenience, a representative result from this early comparison is shown in Figure 3, which compares the MHD and OPAL results with that of the simple EFF formalism (which is essentially a consistent ground-stateonly Saha equation of state under these conditions). The absolute curves of part a of Figure 3 are merely able to show the difference between MHD (or OPAL) and the simple EFF results. To see the difference between the MHD and OPAL results, one needs the magnified part 6, which shows the relative differences between MHD and EFF, and between OPAL and EFF values, respectively. This relative plot now not only allows one to see the difference between MHD and OPAL results, but also their striking similarity. Later, it turned out that this agreement was nearly accidental. The reason for this was found by varying the parameters of the MHD equation of state. It followed that on the chosen isochore, all thermodynamic quantities are dominated by the Coulomb pressure correction (Dappen, 1990; ChristensenDalsgaard, 1991; Christensen-Dalsgaard k Dappen, 1992). The Coulomb correction overshadows the effect of the excited states (which are of course treated differently in the MHD and OPAL approach). Note that the Coulomb term acts directly and indirectly, at least in the language of the chemical picture, because it is not mainly the free-energy of the Debye-Hiickel term itself, but rather also the Coulomb-induced shift in the ionization equilibrium, which is responsible for the deviation from the unperturbed EFF result.

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2.5

0.02 i

-^

0.00

i !

—«^^

f

|

\ \

1 1 1 !

0.02

N

f

I k

t f

•

,

b)

0.04 4.0

4.5

5.0

5.5

log T

Fig. 17.3 Comparison of XT = (dlnp/dlnT)fi for /> = 10"6 5 g cm"3. Absolute quantities (a) and relative differences (with respect to EFF) (b) are shown. See text for more details. Of course, solar physicists were happy that two completely different formalisms delivered the same equation of state, but, by the same token, a first attempt to use the Sun as an equation-of-state test was also thwarted. This discovery suggested to upgrade the simple EFF equation of state with the help of the Coulomb interaction term. The resulting equation of state (called CEFF) has become a useful tool for solar physics (Christensen-Dalsgaard, 1991; Christensen-Dalsgaard & Dappen, 1992); at the same time, however, it became also clear that a helioseismic test of the important issue of chemical versus physical picture would be more difficult than first thought. For reasons not yet fully understood it seems that in the chemical picture, the signature of internal partition functions, such as those employed in the MHD equation of state, is much less visible in the thermodynamic quantities than a naive estimation of the shift in the ionization equilibrium would predict. It is likely that there are accidental cancellations in the derivatives of the free energy. The cancellations of partition-function effects in the chemical picture seem to be greatest for the ionization zone of hydrogen and somewhat less for those of helium. A more recent comparison of MHD

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1.6B0

1.670

1.660

5.0

5.5

6.0

6.5

7.0

Fig. 17.4 Fi for p = 5.00 x 10~3g cm" 3 and a representative solar mixture of H, He, and O. Parts (a) and (b) as in Fig.3, but here with CEFF instead of EFF. See text for more details.

and OPAL values (Dappen, 1992) has examined selected cases of higher densities (where sizeable discrepancies appear) and a first case of a mixture involving a representative solar heavy element (oxygen). It appears that for the heavier elements, the internal partition functions finally lead to the intuitively expected consequences for the thermodynamic quantities. Figure 4 shows the result of this comparison with oxygen for the quantity Ti. Density was chosen as p = 0.005 g cm" 3 , a value suggested by a helioseismic study of the solar helium abundance (Kosovichev et al., 1992). Here, not only do the large MHD partition functions cause shifts in the ionization balance but these shifts also significantly propagate into the thermodynamic quantities. The effect is large enough so that it appears, despite the small relative number of the heavy elements in the mixture, to be within reach of helioseismology (for more details see Christensen-Dalsgaard & Dappen, 1992; Dappen et al. (1993)). To examine the MHD ionization fractions, a single case was examined (T = 2.10 X 10s K,p = 5.00 X 10~3g cm" 3 ), once with the full MHD equation of state, once with a "stripped-down" version of MHD, which does not

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contain any excited states (but is otherwise identical). The resulting ionization fractions of 0 3 + , 0 4 + , 0 5 + were, respectively, 0.314, 0.248, 0.364 for the stripped-down MHD (without excited states), and 0.304, 0.476, 0.182 for the full MHD. (The result for the stripped-down very closely reflects the ground-state weights of the ions). Not unexpectedly in view of the Planck-Larkin partition function (see Rogers, these proceedings), the OPAL equation of state predicts ionization fractions close to those of the strippeddown MHD equation of state (Rogers, private communication). This comparison for the first time establishes a clear case of disagreement between the MHD and OPAL results. Clearly, the origin of the discrepancy in the ionization degrees is due to the treatment of the excited states. Of course, only some 2 percent of the matter in the Sun consist of elements heavier that H and He, and therefore the signature of the MHD-OPAL discrepancy in Fi (Figure 4) is small (of the order of 10~3). Nevertheless, as has been demonstrated by Christensen-Dalsgaard & Dappen (1992), even the resulting tiny sound-speed differences are within reach of a helioseismic diagnosis.

17.7 Conclusions Even weakly-coupled plasmas can pose tough problems if high accuracy is demanded. Solar oscillations are an example of a case where the present observational material is much better than the theoretical models. The solar convection zone is especially well suited for a study of the equation of state. It was suggested in a number of early papers {e.g. Berthomieu et a/., 1980; Ulrich, 1982; Shibahashi et a/., 1983, 1984) that improvements in the equation of state can reduce discrepancies between theory and observations. Later, Christensen-Dalsgaard, Dappen & Lebreton (1988) showed that the MHD equation of state significantly reduced these discrepancies for a large range of oscillation modes. Since the MHD equation of state simultaneously incorporates several different types of non-ideal corrections, it did not become immediately clear which one of these corrections was mainly contributing to this success. iFrom selected comparisons of the MHD with the OPAL equation of state, it turned out, rather surprisingly, that the net effect of the hydrogen and helium bound states on thermodynamic quantities was to a large degree eclipsed beneath the influence of the Coulomb term, which was thus recognized as the dominant non-ideal correction in the hydrogen and helium ionization zones. This discovery led to an upgrade of the simple EFF

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equation of state through the inclusion of the Coulomb interaction term (CEFF). However, for the heavier elements it appears that, in the chemical picture, the internal partition functions finally lead to the expected consequences for the thermodynamic quantities. The heavy elements can thus become the ideal testing ground for the effects of bound states in partially ionized plasmas. The small abundance of heavy elements in the Sun will make a diagnosis difficult and stretch the power of helioseismology to its limits, but as the study by Vorontsov et al. (these proceedings) shows, there are encouraging signs that the difficulties can be overcome. Acknowledgement: I would like to thank Angel Alastuey, Vladimir Baturin, Gilles Chabrier, J0rgen Christensen-Dalsgaard, Werner Ebeling, Andreas Forster, Douglas Gough, Asher Perez, Ed Rhodes, Forrest Rogers, and Sergei Vorontsov for stimulating discussions. I specifically want to thank J0rgen Christensen-Dalsgaard for the results displayed in Figure 2, Forrest Rogers for the OPAL results contained in Figures 3 and 4, and Ed Rhodes for the v — I diagram shown in Figure 1.

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Dappen: Helioseismology

393

Vorontsov, S.V. & Zharkov, V.N., Sov. Set. Rev. E. Astrophys. Space Phys. 7, 1-103 (1989) Wiese, W.L., Kelleher, D.E. & Paquette, D.R., Phys. Rev. A6 1132-1153 (1972)

18 Transport processes in dense stellar plasmas NAOKIITOH Deparimant of Physics, Sophia University, 7-1, Kioi-cho, Chiyoda-ku, Tokyo 102, Japan

Abstract Transport processes in dense stellar plasmas which are relevant to the interiors of white dwarfs and neutron stars are reviewed. The emphasis is placed on the accuracy of the numerical results. In this review we report on the electrical conductivity and the thermal conductivity of dense matter. The methods of the calculations are different for the liquid metal phase and the crystalline lattice phase. We will broadly review the current status of the calculations of the transport properties of dense matter, and try to give the best instructions available at the present time to the readers. Nous presentons une revue des propoietes de transport dans les plasmas denses stellaires caracteristiques des interieurs de naines blanches et d'etoiles a neutrons. L'accent est mis sur la precision des resultats numeriques. Nous presentons la conductivite electrique et la conductivite thermique dans la matiere dense. Les methodes de calcul sont differentes dans la phase liquide et dans la phase cristalline. Nous donnons une revue generale des calculs des proprietes de transport dans la matiere dense, et nous essayons de donner les meilleures instructions quant aux donnees disponibles actuellement.

18.1 Introduction In recent years white dwarf asteroseismology opened up a new fertile land of astrophysics (Bradley k Winget 1991; Bradley, Winget, k Wood 1992). Consequently, the basic physics data which go into white dwarf models need 394

Itoh: Transport processes in dense stellar plasmas

395

to be sufficiently accurate that they should live up to the standard required by the asteroseismological data. In order to match the accuracy required by white dwarf asteroseismology, tremendous effort has been devoted in the recent years to improve the numerical accuracy of the calculations of the transport properties of dense matter. For the liquid metal phase the papers of Itoh et al. (1983) and Mitake, Ichimaru, k Itoh (1984) have given significantly improved results of the conductivities upon some of the previous results reported in the papers of Flowers & Itoh (1976), Flowers k Itoh (1981), and Yakovlev k Urpin (1981). The paper of Itoh et al. (1984) has significantly improved the results of the conductivities in the cyrstalline lattice phase reported in the papers of Flowers k Itoh (1976), Flowers k Itoh (1981), Yakovlev k Urpin (1981), and Raikh k Yakovlev (1982). Recently, Itoh, Hayashi, k Kohyama (1993) have included the lower densities 10° — 104gcm~3 for the calculation of the electrical and thermal conductivities of dense matter in the crystalline lattice phase. The impurity scattering contributions to the electrical and thermal conductivities of dense matter in the crystalline lattice phase have been recently recalculated by Itoh k Kohyama (1993). The viscosity of dense matter in the liquid metal phase has been calculated by Itoh , Kohyama , k Takeuchi (1987).

18.2 Electrical and Thermal Conductivities in the Liquid Metal Phase In this section we review the calculations of the electrical and thermal conductivities of dense matter in the liquid metal phase following the paper of Itoh et al. (1983). We consider the case that the atoms are completely pressure-ionized. The corresponding condition is expressed by EF(p)>Z2

Ry,

(1)

where Ep{p) is the electron Fermi level at zero temperature and at a given mass density p. Numerically, the condition (1) can be rewritten as p > 0.378AZ2

gem" 3 ,

(2)

where Z and A are the atomic number and mass number of the atom considered, respectively. Thus we have p > 6.05gcm~3 for 4He,/> > 1.63 x 102gcm~3 for 12 C,p > 3.87xl0 2 gcm" 3 for 16 O, and p > 1.43xl04gcm~3for

396

Itoh: Transport processes in dense stellar plasmas

56

Fe. We further restrict ourselves to the density-temperature region in which electrons are strongly degenerate. This condition is expressed as

T < TF = 5.930 x 109 [[l + 1.018 (Z/A)2/3 pl/3] ^ - ll K, (3) where Tp is the Fermi temperature and ^6 is the mass density in units of 106gcm"3. For the ionic system , we consider in this section the case that it is in the liquid state. The criterion for that is a subject of the recent debate (see the review paper of H.E. DeWitt in the present Proceedings); but we follow in this review the criterion given by Ogata & Ichimaru (1987)

where a = [3/ (47rnj)] ' is the ion-sphere radius and T& is the temperature in units of 108K. The analytic fitting formulae presented in the paper of Itoh et al. (1983) are valid for 2 < T < F m , irrespective of the precise value ofF m . The low-temperature quantum corrections for the ions can be neglected unless the parameter

where kp is the Fermi wavenumber of the electrons and M is the mass of the ion, becomes appreciable compared with unity. The prescriptions to take into account the low-temperature quantum corrections for the ions have been given in detail by Mitake, Ichimaru, & Itoh (1984). However, these prescriptions are not complete, because they are powerless for the case y > 1. This situation is quite ironical, as the correct prescrptions for low-temperature quantum corrections are badly needed for y > 1. When we can calculate the low-temperature quantum corrections accurately, they are small anyway. Therefore, if one is not worried by the small corrections, it would be probably wise to disregard altogether the low-temperature quantum corrections and use simply the results of Itoh et al. (1983) for the whole range of 2 < T < Tm. Then one would not be too wrong in getting the right behavior of the conductivities. On the other hand, if one uses the results of Mitake, Ichimaru, & Itoh (1984) for the cases y > 1, one would be led to unphysical results, because the calculation of Mitake Ichimaru,

Itoh: Transport processes in dense stellar plasmas

397

4h »——

^-r=i7i

i

i

2

3

i

. 4 log/ 7

i 3

5

6

(g-cm- )

Fig. 18.1 The density-temperature diagram for the XH matter.

& Itoh (1984) is based on the lowest-order expansions with respect to the small values of y. Thus the method completely breaks down for y > 1. The expressions for the electrical conductivity a and the thermal conductivity K in the liquid metal phase are a = 8.693 x 1 0 2 1 ^ A

[1 + 1.018 (Z/A)2/3 p26/3\

K = 2.363 x 10 17

< S >

1.018(Z/A)2/3

ergscm~ ~ 11s~ s~11K~ 1 , <S>

= I'd (±.\ (± Jo

(6)

\2kFj\2kF

1.018 (Z/A)2/3p26/3 1 + 1.018 (Z/A)2/3pl/3

(7)

S(k/2kF)

398

Itoh: Transport processes in dense stellar plasmas

logP

6

,

8

-3v

3 (gem ) 4

Fig. 18.2 The density-temperature diagram for the He matter.

log) 0

8

-x (g-cm3)

Fig. 18.3 The density-temperature diagram for the 12C matter.

10

399

Itoh: Transport processes in dense stellar plasmas

6

8 log/?

10 (gem )

12

3

Fig. 18.4 The density-temperature diagram for the 56Fe matter.

S(k/2kF) 2kFJ \2kFJ =<S-i

[(k/2kF)2e(k/2kFyQ)\

1.018 (Z/A)2/3pl/3 >— 1 + 1.018 (Z/A)2/3pl/3 *

>,

(8)

where hk is the momentum transferred from the ionic system to the electron, S(k/2kF) is the ionic liquid structure factor, and e(k/2kF, 0) is the static dielectric screening function due to the degenerate electrons. The first term in equation (8) corresponds to the ordinary Coulomb logarithmic term, and the second term is a relativistic correction term. In Itoh et al. (1983) explicit calculations have been carried out for Z = 1,2,6,8,10,12,14,16,20,26, and asymptotic formulae have been presented for Z > 27. Some of the results are shown in Figures 5-8. The difference between the results of Yakovlev and Urpin (1981) and those of Itoh et al. (1983) is the following: Itoh et al. (1983) carried out the integrations in equation (8) by using the IHNC structure factor of the classical one-component plasma calculated by Iyetomi & Ichimaru (1982) and Jancovici's (1962) relativistic dielectric function for the degenerate electrons, whereas Yakovlev & Urpin (1981) used an approximate form of the srtuc-

400

Itoh: Transport processes in dense stellar plasmas

2

„ 3 4 log/0 (g-cnr3)

Fig. 18.5 Comparison of Yakovlev and Urpin's results (dashed curves) with those of Itoh et al. (solid curves) for the lYL matter. ture factor and neglected the screening due to the degenerate electrons by setting e(k/2kF,0) = 1. Yakovlev and Urpin's (1981) approximations are quite satisfactory at high densities as can be seen from Figures 5-8, but bring about significant deviations from the correct results at low densities where the screening due to the degenerate electrons is appreciable. Accurate analytical fitting formulae which summarize the numerical results have been presented by Itoh et al. (1983) 18.3 Electrical and Thermal Conductivities in the Crystalline Lattice Phase In this section we review the calculations of the electrical and thermal conductivities of dense matter in the crystalline latttice phase (F > r m = 180) following the papers of Itoh et al. (1984a) and Itoh,Hayashi,&Kohyama (1993). The relativistic extension of the expressions for the electrical and thermal conductivities due to degenerate electrons has been given by Flowers & Itoh (1976). The electrical conductivity a and thermal conductivity K are related to the effective electron collision frequencies ua and vK by

401

Itoh: Transport processes in dense stellar plasmas

1.0 -

Fig. 18.6 Comparison of Yakovlev and Urpin's results (dashed curves) with those of Itoh et al. (solid curves) for the 4He matter.

a = - ^ - = 1.525 x 10 20 -/» 6 m*V(,

x 1 + 1.018 (f

A

*)'

101I88s. - 1 ergs cm x s lK

(9)

l

,

(10)

where ne is the number density of eletrons and m* is the relativistic effective mass of an electron at the Fermi surface. In this section we are interested in the scattering of electrons by phonons. The collision frequencies va and uK due to one-phonon processes can be calculated by the variational method (Flowers k Itoh 1976; Yakovlev & Urpin 1981; Raikh & Yakovlev 1982) as

Itoh: Transport processes in dense stellar plasmas

402

4

6. log/ 0

, 8 % (gem3)

10

Fig. 18.7 Comparison of Yakovlev and Urpin's results (dashed curves) with those of Itoh et al. (solid curves) for the 12 C matter.

kBT
1/2 1s

= 9.554 x 10 r 8 |i

• ff dSdS dSdS' 2 4 S 'J Jfc |e(A;,l

272

4=1

1.018 [(Z/A)P6}2/3

[ [

(I3k\2] \2kF) J V* - ir

s~\

(11)

j 2W{k)

(12)

In the above the integral is over the areas of the Fermi surface, k is the momentum transfer, ea(p) is the polarization unit vector of a phonon with momentum p and polarization s, and

7

5

= 7.832 X

403

Itoh: Transport processes in dense stellar plasmas

56

Fe

15 -

y

r=io

*\ \

•

1.0 --

-

|r=40f

•

4

6

.

log/

7

8 10 (g-cm3)

12

Fig. 18.8 Comparison of Yakovlev and Urpin's results (dashed curves) with those of Itoh et al. (solid curves) for the 56 Fe matter.

(14) (15) (16) 2TT 2

(17)

up being the ionic plasma frequency. The momentum conservation requires k = ± p + K, where K is the reciprocal-lattice vector for the Brillouin zone to which k is confined. In equation (12) we have included the dielectric screening function due to relativistically degenerate electrons e(fc,0), the Debye - Waller factor e-2W(k)^ a n d t n e a t o m j c form factor/(Jb). Yakovlev & Urpin (1981) and Raikh & Yakovlev (1982) have used the Thomas - Fermi screening and set e-2W(k) =

u

404

Itoh: Transport processes in dense stellar plasmas

0.0

Log p (g cm') Fig. 18.9 la for the 4He matter. RY stands for the results of Raikh k.

Yakovlev (1982). The static longitudinal dielectric function due to relativisticaJly degenerate electrons (Jancovid 1962) is given by

-In 2q2x2 6<jrx

- : 2

(1+

1-9

,(18)

9 =

(19)

hkF _ 1 /9jr\ 1 / 3 _ t x = r mec • i37.o v 4 ; * '

(20)

r, = 1.388 x 10"2 ( - — )

(21)


•

Itoh: Transport processes in dense stellar plasmas 2.5

1

1

T-5000

405 i

1

i

c w =i

y^\V

4He

2.0-

1.0

0.5

—->-

\

1.5 -

•£\T V

0.0

r-i80

i

i

i

i

i

i

2

4

6

8

10

12

Log p (g cm"3) Fig. 18.10 li2) for the 4He matter.

2.0

12

Log p (g cm') Fig. 18.11 lo for the

12

C matter.

Itoh: Transport processes in dense stellar plasmas

406 3.5 3.0

•

1

\

-

2.5 -

r-sooo

N

I -

2.0 -

11// /

i

PI /^^

12C

RY

^—

—r-soo

1.5 1.0 -v 0.5 ' 0.0 0

2

4

6

i

i

i

8

10

12

10

12

Log p (g cm"3) Fig. 18.12 li2) for the

12

C matter.

2.0

2

4

6

8 3

Log p (g cm" ) Fig. 18.13 /„ for the 56Fe matter.

407

Itoh: Transport processes in dense stellar plasmas 5

2

4

6

8

10

12

3

Log p (g cm" ) Fig. 18.14 li2) for the 56Fe matter. The electron Fermi wavenumber is expressed as

(Z \ 1 / 3 kv = 2.613 x 1010 f jpeJ

(22)

-l

The Debye - Waller factor is written as (23)

where M is the mass of an ion and N is the number of ions in the lattice. Itoh et al. (1984b) have obtained the practical formula for the Debye Waller factor for the pure Coulomb bcc lattice which satisfies the frequency moment sum rules for both kBT > hu>p and kBT < hup (ColdweU-Horsfall & Maradudin 1960; Pollock k Hansen 1973). The practical formula for the Debye - Waller factor reads as 0.4306-y

2W(k) =

T T + 2)

408

Itoh: Transport processes in dense stellar plasmas

n2kl a =

2MkBT .

(25)

The practical approximations of the Debye function that appears in equation (20) are written as follows:

X

l1+6-3!

30-5!*

+

42-7!X

30-9!

+

66-11!*

for x < 2.5,

) (26)

e-nx

/o

for x > 2.5.

(27)

The errors of formulae (26) and (27) at x = 2.5 are both smaller than 10~5. When the de Broglie wavelength of the electron becomes short enough to be comparable to the nuclear radius, we have to take into account the finite nuclear size corrections for the Coulomb scattering. This is done by multiplying the matrix element by the atomic form factor

(2kFrcq)3

The charge radius of the nucleus is represented by rc = 1.15 X 10" 13 A 1/3 cm.

(29)

The phonon spectra are modified by the screening due to electrons. The longitudinal optical phonon turns into an acoustic phonon in the longwavelength limit, whereas the original transverse acoustic phonons are little affected by the electron screening (Pollock & Hansen 1973). Because the low-frequency transverse phonons play dominant roles in the resistivity of dense stellar matter, we neglect the effects of the electron screening on

Itoh: Transport processes in dense stellar plasmas

409

the phonon spectra and use the frequency moment sum rules for the pure Coulomb lattice. As we consider the case in which the Fermi sphere is much larger than the Debye sphere, (kp/ko)3 = Z/2 > 1 , Umklapp processes contribute to the scattering dominantly, and the vector k in equation (12) most probably falls in a Brillouin zone distant from the first zone. When we perform an integration within a single distant zone corresponding to the reciprocallattice vector K, we can make an approximation k = K in the integrand and carry out an integration over p within the first zone only. Here we follow the semianalytical approach adopted by Yakovlev & Urpin (1981) and also by Raikh & Yakovlev (1982). We write

^ 4=1

(30) '

/

(31)

where n=0 or 2, and integration is carried out over the first Brillouin zone, whose volume is VQ. By the use of this approximation Fa and FK in equation (12) are expressed as (32) (33)

06) - u <7min = ^

(38)

410

Itoh: Transport processes in dense stellar plasmas

Here we have introduced a small momentum transfer cutoff qm\n corresponding to the unavailability of Umklapp processes for q < qmia. The contributions of the normal processes are very much smaller than those of the Umklapp processes. For the choise of qm\n we have followed Raikh & Yakovlev (1982). Yakovlev & Urpin (1981) derived the asymptotic expressions of G^d) and G^2)(7) for 7 < 1 and 7 > 1 , and proposed the following analytic formulae for arbitrary 7 , which fit the main terms of the asymptotic expressions: -1/2

« 13.00(1 + 0.01747 2 )- 1 / 2 ,

(39) "3/2

^2

r3/2,

(40)

where «_ 2 « 13.00 (Pollock & Hansen 1973) and c2 = 29.98 (ColdwellHorsfall & Maradudin 1960) are the numerical constants that are characteristic of the phonon spectrum of the bcc Coulomb lattice. Raikh & Yakovlev (1982) calculated G^(j) and G^(j) numerically with the exact spectrum of phonons for 7 < 100. It has been confirmed that the fitting formulae (39) and (40) have an accuracy better than 10 % even at 7 ~ 1 . We have carried out the numerical integrations of equations (34) and (35) for 4He, 12 C, 16 O, 20Ne, 24Mg, 28Si, 3 2 S, 40 Ca, and 56Fe. Some of the results are presented in Figures 1-6. For comparison we have also included the case where we have neglected the effects of the Debye - Waller factor and set e~2W = 1. We also show the results of Raikh & Yakovlev (1982), which are [/,]RY = 2 - /32, [4 2 ) ]RY

= In Z - P2 + 1.583.

(41)

(42)

It is readily seen that the Debye - Waller factor reduces the resistivities (enhances the conductivities) by a factor of 2 - 4 near the melting temperature.

411

Itoh: Transport processes in dense stellar plasmas

Log p ( g cm") Fig. 18.15 Thermal conductivities of the 4He matter at the crystallization point T = 180. In Figures 15 - 17 we show the thermal conductivities at the crystallization point F = 180 as functions of densities. Thermal conductivities in the crystalline lattice phase are the present results. Thermal conductivities in the liquid metal phase have been calculated by using the method of Itoh et al. (1983). It is found that the thermal conductivity in the crystalline lattice phase is generally higher than that in the liquid metal phase by a factor 2 - 4 at the crystallization point T = 180 . In Figure 18 we show the thermal conductivity of the "pure" 12C matter as a function of the parameter T for the densities 104 g cm" 3 , 106 g cm" 3 , 108 g cm" 3 , and 1010 g cm" 3 . The thermal conductivity in the crystalline lattice phase is the present result, and the thermal conductivity in the liquid metal phase has been calculated by using the method of Itoh et al. (1983). The end point of the curve in the crystalline phase corresponds to the condition

1

I 2/3

1.018 [(Z/A)H J

1/2

(43)

412

Itoh: Transport processes in dense stellar plasmas

Log p( gem 0 ) Fig. 18.16 Thermal conductivities of the n C matter at the crystallization point T = 180.

which must hold in order that the Umklapp processes remain effective ( Yakovlev & Urpin 1981; Raikh k Yakovlev 1982). We emphasize that in practice impurity scattering dominates over phonon scattering at low enough temperatures (Itoh & Kohyama 1993). Therefore, the thermal conductivity in the crystalline phase presented here is an "ideal" thermal conductivity. Itoh, Hayashi, & Kohyama (1993) have presented accurate analytical fitting formulae which summarize the numerical results of the calculation.

18.4 Impurity Scattering Contributions in the Crystalline Lattice Phase In this section we review the calculation of the impurity scattering contributions in the crystalline lattice phase following the paper of Itoh & Kohyama (1993). In the crystalline lattice phase, the phonon scattering and the impurity scattering contribute to the electrical and thermal conductivities in the following way: a

tot

°phonon '

a

impurity

(44)

Itoh: Transport processes in dense stellar plasmas

413

Log p ( g c m ' ) Fig. 18.17 Thermal conductivities of the 56Fe matter at the crystallization point T = 180. ""tot

(45)

""phonon ' *im\ impurity

The contributions of impurity scattering to electrical and thermal resistivities are thoroughly discussed in Ziman's (1960) authoritative book "Electrons and Phonons". We closely follow his ideas. We suppose that there is a concentration x of (Zi,A\) ions and (1 — x) of (Z2,A2) ions which forms a compositionally disordered lattice. This means that each lattice site is randomly occupied either by a (Z\,Ai) ion or a (Zi,A-i) ion. Thus we have X =

•,1

» 1 ~r fl2

x = Til ~i~

**2

where n\ and 112 are the number densities of the (Zi,A\) and (^2,^2) ions, respectively. We define an average charge and an average mass number < Z >= xZx + (1 - x)Z2 ,

(47)

< A>=xA1+(l-x)A2

(48)

.

As the unperturbed state we take a regular lattice whose lattice sites are

414

Itoh: Transport processes in dense stellar plasmas

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

LogT Fig. 18.18 Thermal conductivity of the "pure" 12 C matter as a function of the parameter T for the densities 10 4 gcm~ 3 ,10 6 gcm~ 3 ,10 8 gcm~ 3 ,10 10 gcm" 3 .

occupied by (< Z >,< A >) ions. Then we suppose that impurity ions are added at lattice sites to make up the compositionally disordered lattice. Thus at a site occupied by a (Zi,Ai) ion, the added impurity charge is Zx- < Z >= (1 - x)(Zi - Z2) = (1 - x)AZ

.

(49)

At a site occupied by a {Z2,A2) ion, the added impurity charge is

Z2-

-

-xAZ

(50)

Next we consider these charge impurities as static, and calculate their contributions to electrical and thermal resistivities. This is exactly the same as that has been done by Itoh et al. (1983) when they calculated the electrical and thermal conductivities of dense matter in the liquid metal phase. We consider the case that the atoms are completely pressure-ionized. We further restrict ourselves to the density-temperature region in which electrons are strongly degenerate. This condition is expressed as = 5.930 x 109

[[1

Itoh: Transport processes in dense stellar plasmas

415

where TF is the electron Fermi temperature, X\ and X2 are the mass fractions (Xi + X2 = 1) of the ions (Z\,A\) and {Z2,A2), a n d ps is the mass density in units of 106gcm~3. For the ionic system we consider the case that it is in the crystalline lattice state. The latest criterion corresponding to this condition for the case of the one-component system is given by (Ogata & Ichimaru 1987)

where a = [3/(47rnj)]1/3 is the ion-sphere radius, and T% is the temperature in units of \Q%K. Extending the work of Itoh et al. (1983) to the case of charged impurities, and assuming that the correlations between the two kinds of impurity charges are the same, we obtain the expressions for the electrical conductivity a and the thermal conductivity « limited by impurity scattering: a = 8.693 x 102Ve (%• + %

K = 2.363 X 1017T8P6 ( £ + %

\M

_ \\( \( JJoJo 1.018 1.018

k

\( ( k V )\2kF\)\2kFJJ

Ai

S(k/2kF)

416

Itoh: Transport processes in dense stellar plasmas U

\ 5

[(k/2kFye(k/2kF,0)}2

2kF)) \2kF) 1.018

(z

Z V 3 p 2 /3 2

6

1 + 1.018 (fj-Xx + %X2) p6 where S(k/2kF) is the structure factor for the impurity charges, and c(k/2kF, 0) is the static dielectric function due to degenerate electrons, kF being the electron Fermi wavenumber. The results (53) and (54) nicely reproduce the experimentally verified rules. The property that the resistivity is proportional to x(l — x) is Nordheim's rule; the property that the resistivity is proportional to (AZ) 2 is Linde's rule (Ziman 1960). In order to calculate the structure factor S(k) for the impurity charges, we employ a simple model. We adopt the ion-sphere model, and assume the following radial distribution function for the impurity charges :

where a is the ion-sphere radius a = {3/[47r(n1 + n 2 )]} 1 / 3 .

(57)

Then the structure factor S(k) is obtained as

S(k) =l + (ni+n2)J

d3r[g(r) - 1] exp(-tk • r)

3

= 1-

(sinfca — kacoska)

.

(58)

{kay

We notice that ^(Jfca) 2 ,asfc-»0 .

(59)

Since the screening effect due to electrons is negligible compared with the effect of the impurity charge correlation expressed by the structure factor (58), we set e(k/2kF,0) = 1 in equation (55). We further replace the structure factor (58) by a simplified form

Itoh: Transport processes in dense stellar plasmas

417

which satisfies the condition (59). Insertion of this into equation (55) gives

\

\f

,

(61)

The electron Fermi wavenumber kp is written as kF = 2.613 X 1010 (^rXx + ^-X2) ) \Ai A2 The ion-sphere radius a is written as

pl/3cm-1

a = 0.7346xl0- lo f4 1 + 4 i> )" 1/ %6 1/3 cm \Ai

.

.

(63)

(64)

A2J

Thus we have

f) l / 3 (^ f)- I/3 .

(^

(65)

It is straightforward to generalize the results in the above to higher component systems. Let the mass fraction of the (£j,Aj) component be Xi(52iXi = 1). The result is a = 8.693 X 1021ps \Y

-r1}

^s"1 '

(66)

Y, ^f )

K = 2.363 x IO17TSP6 \ , ^ f ) 1

*+1.018



(%)2/%1<3]< (67) (68) (69)

418

Itoh: Transport processes in dense stellar plasmas

*.• = = ? - .

(70)

The factor < S > is given by equations (55),(61),(62) with kFa=

/

y.\ 1 / 3 /

1.920 ( j > * )

x\~1/3

(Ef:)

•

(71)

18.5 Concluding remarks We have reviewed the recent developments in the field of the transport properties of dense matter. The modern calculations of the transport properties of dense matter make full use of the recent developments in statistical physics, plasma physics, liquid state physics, and solid state physics. Interparticle correlations play essential roles in the quantitative evaluation of the transport properties of dense matter. Now the time is ripe for the close examinations of our understanding of the physics of dense matter by using the observational data of the asteroseismology of white dwarfs.

18.6 Appendix In this Appendix we directly insert equation (58) into eqation (55) and carry out the integration. The result is

(72)

The differences between the < S > values calculated from eqations (61) and (62) and those calculated from equations (72) and (73) are typically 10 % - 20 % (the latter is generally larger). The structure factor (58) based on the ion-sphere model is an approximation of the real ionic correction, and its oscillatory behavior does not have a rigorous meaning. Thus, in this paper, we will adopt the smooth structure factor (60) which leads to a simpler expression for < 5_i >. In any case, we should bear in mind that the differences in the < S > values caused by the use of the different structure factors indicate the accuracy of the present calculation.

Itoh: Transport processes in dense stellar plasmas

419

18.7 References Bradley, P.A., k Winget D.E. 1991, ApJS, 75, 463 Bradley, P.A., Winget, D.E., k Wood, M.A. 1992, ApJ, 391, L33 Coldwell-Horsfall, R.A., k Maradudin, A.A. 1960, J. Math. Phys., 1, 395 Flowers, E., k Itoh, N. 1976, ApJ, 206, 218 Flowers, E., k Itoh, N. 1981, ApJ, 250, 750 Itoh, N., Hayashi, H., k Kohyama, Y. 1993, ApJ, in press Itoh, N., k Kohyama, Y. 1993, ApJ, 404, 268 Itoh, N., Kohyama, Y., Matsumoto, N., k Seki, M. 1984a, ApJ, 285, 758; erratum 404, 418 Itoh, N., Kohyama, Y., k Takeuchi, H. 1987, ApJ, 317, 733 Itoh, N., Matsumoto, N., Seki, M., k Kohyama, Y. 1984b, ApJ, 279 , 413 Itoh, N., Mitake, S., Iyetomi, H., k Ichimaru, S. 1983, ApJ, 273, 774 Iyetomi, H., k Ichimaru, S. 1982, Phys. Rev. A, 25. 2434 Jancovici, B. 1962, Nuovo Cimento, 25, 428 Ogata, S., k Ichimaru, S. 1987, Phys. Rev. A, 36, 5451 Pollock, E.L., k Hansen, J.P. 1973, Phys. Rev. A, 8, 3110 Raikh, M.E., k Yakovlev, D.G. 1982, Ap. Space Sci., 87, 193 Yakovlev, D.G., k Urpin, V.A. 1981, Soviet Astr., 24, 303 Ziman, J. 1960, Electrons and Phonons (Oxford Univ. Press)

19 Cataclysmic variables: structure and evolution J.-M. HAMEURY Observatoire it Strasbourg, 11 rut it I'Universite, 67000 Strasbourg. France (present address) DAEC, Observatoire it Paris, F-92195 Mtudon ceiez, France

Abstract I discuss the structure and evolution of cataclysmic variables, with a particular emphasis on the influence of the physics used in calculating the internal structure of the secondary. The available observational data is very rich, and can, in principle, be used to constrain the stellar physics. It is found that, in order to explain the lack of systems with periods in the range 2 3 hr, it is required that main sequence star become convective for masses below 0.3 M0. This has little consequences on the equation of state, but constrains the opacities and the treatment of subphotospheric layers. On discute la structure et revolution des variables cataclysmiques, en s'attachant plus particulierement a l'infiuence de la physique de l'etoile secondaire. Les donnees observationelles, tres abondantes, peuvent en principe etre utilisees pour contraindre la physique stellaire. On trouve que, pour expliquer l'absence de systemes entre 2 et 3 heures, il faut que les etoiles de la sequence principale deviennent convectives lorsque leur masse atteint 0.3 M©. Ceci a peu de consequences sur 1'equation d'etat, mais contraint les opacites et le traitement des couches sub-photospheriques.

19.1 Introduction Cataclysmic variables (CV's) are binary systems containing a white dwarf and a normal star, which fills its Roche and transfers mass onto the compact 420

421

Hameury: Cataclysmic variables

Accretion Disk

| Low Mass jSecondary

Hot Spot

White Dwarf Primary

Fig. 19.1 Schematic view of a cataclysmic variable (from Ritter, 1985)

object. The orbital period is in the range 1 to 10 hr, although we know a few cases in which it is longer than one day or shorter than 1 hr. These systems are quite similar to low-mass X-ray binaries, with the difference that the primary star is a neutron star or a black hole. Figure 1 shows a schematic view of a cataclysmic variable. Because the orbital angular momentum of the accreting matter is high, matter cannot flow directly onto the white dwarf, and an accretion disc forms, except in the particular case of strongly magnetic white dwarfs. In the disc, mass flows inwards, whereas angular momentum is transported outwards to the outer edge of the disc, where it is transferred to the orbit via tidal effects. The disc therefore extends up to a significant fraction of the Roche radius of the primary (typically 0.8 or 0.9 times the Roche radius). The mass transfer rate in these systems is of the order of 10~10 - 10~9 M© yr" 1 , and is strongly variable on time scales ranging from days to years or more, giving rise to bolometric luminosities in the range 1032 - 1034 erg s" 1 . Most of the optical light is emitted by the accretion disc itself, while infrared is emitted by the secondary (usually a K or M dwarf); the white dwarf and the boundary layer between the disc and the surface of the white dwarf are responsible for XUV emission.

422

Hameury: Cataclysmic variables

We know several hundreds of these systems; their typical distance is of the order of 100 pc to a few hundred pc, so that the total number of cataclysmic variables in the galaxy is estimated to be about 200,000. We therefore have a wealth of available observational data that could be used to constraint models of low mass stars, which are normally quite difficult to observe. An interesting characteristic of these system is that, because the secondary fills its Roche lobe, there is a relation between the secondary radius and its mass, as we shall see later. This is particularly valuable for low mass stars, since the mass and radius determination is quite a difficult task in isolated low mass stars, and is model dependant. Another advantage of cataclysmic variables as compared to low mass X-ray binaries is, apart from the fact that they are much more numerous and therefore closer, that the effect of illumination of the secondary by radiation emitted by the primary is almost negligible, although some effect may still be present (Sarna, 1990). This is not the case for LMXB's, which may have an evolution rather different from that of CV's, due precisely to the illumination effect (see e.g. Ruderman et al., 1989; Podsiadlowski, 1991; Hameury et al., 1993) 19.2 Structure of cataclysmic variables 19.2.1 Orbital period distribution Figure 2 shows the orbital period distribution of cataclysmic variables. It is seen that most systems have orbital periods below a few hours, with a peak at 1.5-4 hr. There are only 4 systems having a period shorter than 80 min, which is the so-called "minimum period", whereas there is a lack of systems between 2 and 3 hr; this interval is called the "period gap", although there are indeed a few systems in it. As we shall see later, the orbital period distribution is an essential tool for the understanding the evolution of cataclysmic variables. 19.2.2 Primary and secondary masses The masses of both stars are difficult to determine, since, in order to get a reliable information on the orbital velocity of both components, one requires double spectroscopic systems. In order to determine the masses, one moreover has to know the inclination i of the orbital plane of the system. This is for example the case in eclipsing systems, for which i has to be larger than 70 - 80°. Finally, one must also know where the observed lines are emitted from: this may be either from the whole surface of the secondary (this is usually the case of absorption lines as in normal stars), but lines

423

Hameury: Cataclysmic variables

20

6

15

_

5

_

J

o 6

0

2

4 6 Orbital Period (hr)

8

10

Fig. 19.2 Orbital period distribution of cataclysmic variables. Data from Ritter (1985) may also originate from the vicinity of the L\ region (in the case of emission lines due to heating of the secondary by hard radiation from the white dwarf). There are also emission lines from the accretion disc; in order to obtain the velocity of the white dwarf, one must assume circular symmetry for the accretion disc, which is certainly not true for its outer part, because of distortion due to tidal forces. With all these caveats, it is found that the average mass of the white dwarf in CVs is about 1 M©, significantly larger than that of isolated white dwarfs, which is about 0.6 M©. This difference, as has been shown by Ritter and Burkert (1986) is not due to differences in the formation mechanism of isolated and non-isolated white dwarfs, nor to the steady mass increase of the primary as a result of accretion (in fact as much mass is removed during novae explosions as is accreted in between), nor to systematic errors. This difference can be simply explained by selection effects, that tend to favour the observation of systems with a massive white dwarf, that has smaller radius, so that the luminosity for a given mass transfer rate is higher. After correction of the selection effects, the intrinsic average white dwarf mass turns out to be of the order of about 0.6 M©, identical to that of isolated stars. The secondary mass is always found to be in the range 0.1 - 1.0 M©

424

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19.2.3 Mass-radius relation Kepler's third law relates the orbital separation a and period i^,r measured in hours via:

-f- = 0.50(Afi + M 2 ) 1 / 3 P h 2 / 3

(1)

ft©

where Mi and M2 are respectively the primary and the secondary masses measured in solar masses. Because the secondary fills its Roche lobe, the secondary radius J22 is determined by the Roche geometry, so that:

(Paczynski, 1971) for M2 < Mi (see also Eggleton, 1983 for a more accurate and more general fit of R^. Combining Eqs. (1) and (2) gives the massradius relation:

= 0.23J<3Phy3

(3)

which is independent of the primary mass. In addition, the average density of the secondary < p > defined as M2/(4/37riZf) is 110 P ^ 2 g cm" 3 , and depends only on the orbital period of the system. For orbital periods of a few hours, < p > is of the order of a few g cm" 3 , typical of main sequence stars. If one makes the further assumption that the secondary is indeed on the main sequence, then R2 = M2R0 and the secondary mass is uniquely determined by the orbital period of the system via: M2 = O.llPhr

(4)

As we will see later, this assumption is not quite justified, but, at least for orbital periods larger than 3 hr, is not too extreme. A first important consequence can be deduced from this approximate relation: because mass is lost from the secondary, the orbital period of the system must decrease (again only for systems which have main sequence secondaries). Ritter (1985) has shown that the secondary in many CVs is not too far from the main sequence. Fig. 2 shows the mass and radius of several cataclysmic variables, as compared with theoretical predictions. One can see first that, because of the mass-radius relation resulting from the geometry of the system, the error box in the plane M 2 , R2 is reduced to a line segment, and, more important, that the secondary in those systems is quite close to the main sequence, the 10 - 20 % difference in radii can be either intrinsic, or due to a systematic error in theoretical models.

425

Hameury: Cataclysmic variables 1

-1.0

-OS

,

,

0.0

tog(M/Mo) Fig. 19.3 Comparison of determined masses and radii of the secondary in 13 systems with the prediction of theoretical models of main sequence stars.

19.3 Evolution of cataclysmic variables 19.3.1 A preliminary

approach

As we have seen, the typical mass transfer rate in cataclysmic variables is about 10~ 9 M© y r ~ l , so that for a 1 M© secondary, the mass transfer time scale tjfr = M2/M is about 109 yr, much shorter than the Hubble time. On the other hand, the nuclear time W is longer or much longer than the Hubble time for stars less massive than 1 M©. This means that nuclear evolution is negligible, and that systems evolve under the effect of mass transfer. The Kelvin-Helmholtz time fo-H ~ 107M2~3 becomes larger than tjfr when the mass of the secondary reaches 0.3 - 0.4 M©. This means that for larger masses, corresponding to periods larger than 3 - 4 hr, the secondary has time to adjust thermally to its new structure, modified by mass transfer, and remains on the main sequence. At shorter periods, it must deviate from main sequence, and it turns out that the secondary becomes degenerate. In these stars, the mass-radius relation is quite different from that of main sequence stars, and is R% oc M% , so that applying the same

426

Hameury: Cataclysmic variables

argument as in section 1.2.3, one finds that M2 oc P ^ 1 , so that the orbital period of short period systems must increase. The transition main-sequence - degenerate therefore corresponds to a minimum in the orbital period, as has been shown by Paczyriski and Sienkiewicz (1981) and Rappaport et al. (1982). 19.3.2 Mass transfer; stability The evolution of a binary system is governed by the variation of three important quantities: the total mass M\ + M2, the secondary radius R2 equal to the Roche radius RL, and the total angular momentum of the system J given by: Ga

J = MXM2 '

Since the secondary must fill its Roche lobe at any time, R2 = RL, where the dot denotes time derivative. If one furthermore assumes for simplicity that the total mass of the system remains constant, then differentiation of Eq. (5) yields: a

R

~J

L

nM2

/

M2

J M2/5 2 Z

U

In order to get the mass transfer rate, one needs to know the relation between M2 and R2; for a main sequence star, M2/M2 = R2/R2- Departure from thermal equilibrium must however be taken into account, which requires a mode detailed description analysis of the secondary. The surface of a star is however not a well defined concept; matter extends above the photosphere, defined as the point where the optical depth is 2/3. Thus the prescription R2 = R\, can only be a first order approximation, that is accurate to a few photospheric pressure scale height Hp. It turns out that in the lower main sequence, Hp is a very small fraction (typically 10~4) of the radius, so that the prescription R2 = RL is a very good approximation. This is not the case for systems in which the secondary is an evolved star; one would then require a more refined description of mass transfer. This is also the case if one is interested in short episodes of the secular evolution, during which the secondary radius does not vary by much more than a few scaleheight, as for example the turn on of mass transfer. The hydrodynamics

Hameury: Cataclysmic variables

427

of mass transfer have been studied in quite some details by Lubow and Shu (1975) who showed that the flow is approximately isothermal and reaches the sound velocity at the lagrangian point L\. This enabled Ritter (1988) to determine the mass transfer rate as: -M2 = Moe-(R*-R^/H*>

(8)

where Mo is a quantity that depends of the photospheric temperature and density, as well as on the binary parameters. 19.3.2.1 Dynamical stability A star subject to mass loss reacts on a short time scale according to:

where £<,<* is the adiabatic exponent of the star, defined solely by its structure (fad = —1/3 f° r a fully convective star). The mass transfer rate is thus given by:

Ml

j

It can be easily shown that mass transfer is stable only if the denominator in Eq. (10) is positive; otherwise, since J is negative (a binary system cannot gain angular momentum), the secondary mass would have to increase. If this is not the case, then mass transfer raises on a dynamical time scale, equal to the orbital period, leading to enormous mass transfer rates, much higher than what is observed in CVs. It is believed that, when mass transfer happens to be dynamically unstable, mass cannot be accreted onto the primary, and a common envelope forms. The secondary mass must therefore be smaller than or appratimately equal to the primary mass. As Mi cannot exceed the Chandrasekhar mass, M2 must be smaller than 1-1.4 M©, and applying the mass-period relation derived in section 1.2.3 the case of main sequence stars, yields a maximum period of 10 - 15 hr. 19.3.2.2 Thermal stability In a similar way, it can be shown that the secondary is thermally stable against mass loss only if its mass in less than about the primary mass. If this were not the case, the mass transfer timescale would be of order of the Kelvin-Helmholtz time. This type of instability therefore leads to much weaker mass transfer rates; however, because the criteria for dynamical and

428

Hameury: Cataclysmic variables

thermal stability appear to be quite similar, it is not much of a surprise that most systems appear to be thermally stable (Ritter, 1985).

19.3.3 Angular momentum losses The discussion in the previous section shows that angular momentum losses are required for mass transfer to occur at all, since Eq. (10) predicts M2 = 0 if J = 0. Two mechanisms are invoked to account for the required J. 19.3.3.1 Gravitational radiation It was realized by Kraft et al. (1962) that gravitational radiation must play an important role in cataclysmic variables. Two massive bodies orbiting around each other emit gravitational radiation, just as charged particles would emit electromagnetic waves. Gravitational radiation carries aways angular momentum at a rate:

4 = -3.75 x 10-9MiM2(Mi + M2)l/3P^/:i

yr" 1

(11)

For orbital periods less than 2 hr, J is large enough to account for the observed mass transfer rates; however, for longer periods, this is not the case, and another mechanism must be efficient above the period gap. 19.3.3.2 Magnetic braking Magnetic braking is the mechanism responsible for the slowing down of the rotation of isolated stars (Schatzman, 1962,1965), and has been applied to cataclysmic variables by Verbunt and Zwaan (1981) who used the observed slowing down rate of isolated young G stars fitted by Skumanich (1972). Main sequence stars are magnetized, and loose mass via winds. These winds are coupled to the magnetic field, and co-rotate with the star, up to some radius R\ such as the energy density of the magnetic field does no longer exceed the kinetic energy of the wind. J is thus equal to My,R^il, where Mw is the wind mass loss rate, and Q, is the angular velocity of the star. In a binary system, there must also be some wind from the secondary, and the magnetic field must be stronger than in most isolated stars, since the rotation rate is much faster and the field is proportional to fl. This mechanism brakes the rotation of the secondary, which co-rotates with the system because of tidal forces. The coupling of the secondary wind and magnetic field therefore leads to angular momentum losses from the binary

Hameury: Cataclysmic variables

429

system at a rate (Verbunt, 1984): 4 3

'**,„-• ,fju, /

- = -4xlfl J

V inn/'' I \o\)\)yji/

/ff,,\ (R2_\

8 3

/

/

M_

U

^ 3 rr" 1

\

1 / 3

\ D \KQ,

'

(12)

J may account for the observed mass transfer rates at periods larger than 3 hr if both the wind and the magnetic field are strong enough. Surface magnetic fields of the order of 300 G are expected in rapidly rotating stars (scaling the solar field to a rotation period of 3 hr precisely gives that value); strong winds are also expected in late type stars, which show intense coronal activity. Other determinations of angular momentum losses via magnetic braking exist (see e.g. Mestel and Spruit, 1987 and Tout and Pringle, 1992); these are more realistic, but still lead to comparable results. The angular momentum losses given by Eq. (12) is however too high for systems with periods less than 2 hr; this means that magnetic braking must either disappear or become inefficient below 3 hr. As can be noted from Eq. (4), the corresponding mass is of the order of 0.3 M©, equal to the mass at which a main sequence star becomes fully convective. This coincidence lead to conjecture (Rappaport et al., 1983; Spruit and Ritter, 1983) that magnetic braking ceases when the secondary becomes fully convective, as in a fully convective star, magnetic field lines are not rooted in the radiative core, and the dynamo must have a different effect as compared to more massive stars. It must be emphasized at this point that this hypothesis is not based on very firm grounds, and that although most people agree that the angular momentum losses are strongly reduced when the orbital period reaches 3 hr, the disappearance of the secondary magnetic field for a fully convective star is questionable. It has for example been proposed that the secondary wind could be severely reduced at 3 hr (see e.g. Hameury et al., 1987), or that the magnetic field topology changes from a dipolar configuration to a higher multipole one when the secondary becomes fully convective (Taam and Spruit, 1989). The main reason for the success of the disrupted magnetic braking hypothesis is that it explains in a natural way the period gap. As the characteristic time scale of mass transfer above 3 hr is only slightly larger than the Kelvin-Helmholtz time of the secondary, the secondary is bigger than what a main sequence would be. When J is strongly reduced, the secondary has time to adjust to the mass transfer, and joins the main sequence, therefore

430

Hameury: Cataclysmic variables

contracting. It detaches from the Roche lobe, and mass transfer stops. The systems now shrinks under the influence of gravitational radiation, until the Roche lobe catches the secondary surface, at which point mass transfer resumes. Other explanations of the presence of a period gap have been put forward, such as modification of the nuclear burning processes due to 3He mixing when the secondary becomes fully convective (D'Antona and Mazzitelli, 1982; Joss and Rappaport, 1983), but these are now much less favoured. It had also been proposed that a gap arises naturally because of the bimodal distribution of white dwarfs (He WD or CO WD), and the upper edge of the gap would then correspond to the minimum period of systems containing a CO white dwarf (Webbink, 1979; Paczynski and Sienkiewicz, 1983). Whereas the white dwarf composition is of importance, as will be discussed below, the required mass transfer rates required to get a minimum period at 3 hr seem to be excluded by observations. 19.3.4 Evolution of the primary: novae Matter that accumulates at the surface of the white dwarf is unstable versus nuclear burning for a range of mass transfer rates corresponding to those observed in cataclysmic variables. After the thermonuclear explosion, a large amount of matter is ejected from the system. Novae explosion have two main effects on the evolution of the system. First, the total mass of the system does not remain constant; instead, novae observations as well as theoretical calculations show that as much mass is ejected as is accreted, so that the primary mass remains approximately constant (Kovetz and Prialnik, 1985; Truran and Livio, 1986). Another piece of evidence that M\ remains constant comes from the period distribution of a subclass of CVs, the AM Her systems which have a strongly magnetic white dwarf rotating synchronously with the system. Hameury et al. (1989) have shown that this requires also that M\ remains constant throughout the evolution of the system. Ejection of matter leads to a small increase of the orbital separation of the system, and hence to some temporary reduction in the mass transfer rate. This effect, called the "hibernation scenario" (Shara et al., 1986) has been put forward to explain the fact that historical novae appear to have very low mass transfer rates. It does not however significantly affect the secular evolution of CVs, since the duration of the low mass transfer phase is much shorter than the interval between two novae explosions. Another important effect of mass ejection during novae explosions is the

Hameury: Cataclysmic variables

431

interaction of the ejected matter and the secondary. This leads to angular momentum losses, which are quite difficult to calculate, especially since 2D computations are needed (Livio et al., 1990). A rough estimate leads to the conclusion that the effect could be comparable to angular momentum losses due to magnetic braking or gravitational radiation (McDonald, 1986; Livio et al., 1991). This effect has however not been taken into account in secular evolution calculations, simply because its magnitude is quite uncertain. This could be an important drawback in all "classical" models for CV evolution. 10.4 Numerical results of secular evolution As we have seen, a simple analysis of mass transfer and angular momentum losses enables one to explain the basic characteristics of the orbital period distribution. The minimum period is due to the transition main sequence degenerate sequence; the period gap results from the cessation of magnetic braking when the secondary becomes fully convective, i.e. when its mass is reduced to about 0.3 M©, corresponding to an orbital period of 3 hr; the maximum period corresponds to the maximum secondary mass for dynamically stable mass transfer onto a white dwarf whose mass cannot exceed the Chandrasekhar limit. In order to proceed further, one requires a more detailed description of the internal structure of the secondary that takes into account departure from thermal equilibrium, so that one can get the secondary radius E.2(M2,t) as a function of both mass and time. Two approaches have been used to estimate the thermal response of the secondary. Polytropic models (Rappaport et al., 1982, 1983; Hameury et al., 1987; Kolb and Ritter, 1992) in which the secondary is modelled as the superposition of two polytropes (one for the radiative core and the other for the convective envelope) have been largely used. The independent variables in these models are the mass and specific entropy of each polytropic component; one can therefore determine the radius variation as a function of the mass transfer rate and of the entropy variation, proportional to the luminosity excess. The mass transfer rate is then simply given by an equation similar to (7). These models must be calibrated, as they contain several free parameters, and are therefore not very accurate. The computing time is however severely reduces as compared to more detailed models, which enables the calculation of a large number of evolutionary sequences with different initial parameters. Detailed stellar models have also been used (Paczyriski and Sienkiewicz, 1983; McDermott and Taam, 1989; D'Antona et al., 1989; Hameury, 1990). These are much more accurate, but quite slower; they are required for the calibration of polytropic codes. Contrary

432

Hameury: Cataclysmic variables

to poly tropic models, these models require a refined prescription for the mass transfer rate, which is usually taken from Eq. (8). This prescription with a small value of Hp/B,2 makes the evolution codes quite sensible to numerical noise, and for this reason, larger values of Hp are sometimes taken (see e.g. Hameury 1990). This approximation merely affects the mass transfer turn-on and turn-off phases, as the main effect of this prescription for mass transfer is to keep R^ and Ri close to within a few scaleheight.

19.4-1 Standard model Figure 1.4 shows the evolution of a CV in the framework of what will be referred to as the standard model. The ingredients of this models are the following: the primary mass is 0.7 M©, and is assumed to remain constant throughout the evolution; mass ejected during novae explosions is assumed to have the same specific angular momentum as that of the white dwarf. The initial secondary mass is 0.6 MQ. The magnetic braking law is taken from Mestel and Spruit (1987). Concerning the secondary structure, the ingredients are basically the same as in Dorman et al. (1989). The distortion of the secondary by tidal forces is neglected; this introduces a small systematic error: the minimum period for example is shifted by about 10 % (Nelson et al., 1985). The opacities are taken from Alexander (1975) for temperatures less than 104 K, and from Cox and Tabor (1976) at higher temperatures. The equation of state has been interpolated from the tables of Fontaine et al. (1977), and the nuclear reaction rates are taken from Harris et al. (1983) and Fowler et al. (1975), with screening corrections from Graboske et al. (1973). It must be noted that in short period systems, the 3He(3He,2p)4He must be considered separately, since the central temperature is too low for nuclear equilibrium to be reached during the mass transfer timescale. Convection is treated by the mixing length theory, with 1/Hp — 1.5. Finally, the scaled solar T(r) relation of Krishna-Swamy (1966) is used in the photosphere, which accounts for the departure of the photosphere from a gray photosphere. It is seen in Fig. 1.4 that the secondary is out of thermal equilibrium during the mass transfer phases, with a luminosity excess of about o factor 10. Similarly, 3He is far from nuclear equilibrium; however, this has little consequences on the evolution, precisely because nuclear reactions are unimportant in determining the luminosity of the secondary when mass transfer is effective. Note that the difference between the actual and equilibrium helium abundances for the initial main sequence phase is due to the pres-

Hameury: Cataclysmic variables

433

10"e

3 5 o r b i t a l period (hr) Fig. 19.4 Evolution of the standard model. Panel (a) shows the mass transfer rate, (b) the ratio of the nuclear versus total luminosity of the secondary, (c) the actual (solid line) and equilibrium (dashed line) 3He abundance at the centre of the secondary, together with the average 3He abundance (dotted line, and (d) the central density versus orbital period.

434

Hameury: Cataclysmic variables

Table 19.1. Effects of the stellar parameters on binary evolution. The columns list the opacities, equation of state and boundary conditions used, the mixing length, the initial period JP, for which mass transfer starts, the upper and lower edge of the gap, Pu and /*,, and the minimum period K

EOS

BC

l/Hp

P.

A A

FGVH FGVH FGVH FGVH

NG NG NG G NG NG NG

1.5 1.0 2.0 1.5 1.5 1.5 1.5

5.02 5.14 4.91 5.23 4.90 4.99 4.77

A

A A A CS

P

sc FGVH

3.04 3.07 3.02 3.51 3.11 3.04 2.83

F\

*imin

2.03 2.01 2.04 1.92 1.90 2.03 2.07

1.34 1.35 1.34 1.44 — 1.28 <1

ence of a small convective nucleus, so that He mixing leads to a different abundance from the local equilibrium value. As can be seen, the predicted secular evolution accounts rather well for the observed period distribution. Hameury (1991) has investigated the influence of the assumed parameters on the position of both the minimum period and period gap, in order to determine whether one could constrain the modelisation of the secondary. 19.4.2 Effects of the stellar physics Table 1.1 summarizes the main results concerning the position and width of the period gap, together with the minimum period, for various assumptions on the input stellar physics. 19.4 2.1 Equation of state In order to evaluate the uncertainty on the period gap and minimum period that results from errors on the equation of state, evolution has been calculated using the EOS proposed by Paczynski (1969). The ionization fraction of each element is calculated using the Saha equation, and the dissociation of molecular hydrogen is calculated using van't Hoff's equation, as described in Vardya (1960), so that pressure ionization and Coulomb corrections are ignored. These assumptions are quite valid in the outer layers of the secondary, but totally fail in the interior. This is quite an extreme EOS, and still the difference with the standard model is not huge, except below the period gap. It turns out that the secondary never becomes degenerate, but

Hameury: Cataclysmic variables

435

becomes fully neutral when the central temperature becomes low. The use of a decent equation of state, such as that of Saumon and Chabrier (1991) inclluding the phase transition does not yield significant differences from the standard models. 19.4.2.2 Opacities Table 1.1 shows that the opacities are the main source of error on the evolution. Using Cox and Stewart (1969) opacities instead those of Alexander (1975) results in a much shorter period gap, as well as a quite different minimum period, incompatible with observations. Cox and Stewart neglect the contributions of molecules such as H2O, and their opacities are therefore largely underestimated for the temperatures and densities appropriate to the surface of red dwarfs. A major source of uncertainty is the formation and destruction of grains in convective atmospheres, as had been noted by Paczyriski and Sienkiewicz (1983); this severely affect the position of the minimum period. The strong dependence of the secondary structure on opacities is not really a surprise, since the specific entropy of the convective envelope is determined by surface boundary conditions. 19.4.2.3 Treatment of the photosphere As for the opacities, the treatment of the photosphere strongly affects the evolution of the system. If instead of using the scaled solar T(r) of KrishnaSwamy (1966) one uses the simple gray atmosphere condition, KP = 2/3<jr, in which K is the opacity at the photosphere, defined at the position at which the optical depth is 2/3, P the pressure and g the gravity, quite significant changes are obtained for the position of the period gap. This implies that model atmosphere have to be constructed for low luminosity, low mass stars in order to get reliable evolutionary tracks. It should however be noted that the T(T) relation used here is a fair approximation for low mass main sequence stars, as shown by VandenBerg et al. (1983), and that more accurate atmosphere models should not lead to important changes in the calculations presented here. 19.4.2.4 Convection Changing the ratio of the mixing length and the pressure scale height only affects the superadiabatic layers, which, for the stars considered here, are not very important. Convection parameters have some importance only for relatively massive stars (above 0.5 M©), in which the subphotospheric layers are not very dense and the luminosity is high, so that the departure from

436

Hameury: Cataclysmic variables

adiabaticity is more pronounced. The value of the minimum period is, as expected, independent of the mixing length value.

19.4-3 Influence of binary

parameters

The evolution of a particular system depends indeed on the initial masses of the secondary and white dwarf, as well as on the magnetic braking law used. Because the exact determination of the mass transfer rate is impossible, the width of the period gap is a free parameter that is used to adjust the strength of magnetic braking. The position of the period gap is however not dependant on the braking low used, as shown by Hameury (1991), but is more directly related to the physics of the secondary. The initial secondary mass has little influence on the evolution, as long as it is sufficient for the secondary thermal disequilibrium to be large enough before crossing the period gap. This occurs for Mi > 0.4 MQ. On the other hand, evolution strongly depends on the white dwarf mass, since J/J depends on Mi both in the case of gravitational radiation and of magnetic braking. This means that an individual evolutionary track is not necessarily representative of the observed systems, but a weighted average must be performed, especially if one is interested in reproducing the orbital period distribution, and not only three characteristic parameters. This point will be discussed in more details in section 1.4.5.

19.4'4 Particular cases As can be seen from Fig. 1.2, all cataclysmic variables do not follow the evolutionary scheme described here. Although not very numerous, there are systems below the minimum period and inside the period gap; there are also some long period systems. These must have followed some special evolution, that will be briefly discussed in the following. 19.4-4-1 Long period (Potb > 10 hr) systems: evolved secondaries

There are three systems with an orbital period longer than 15 hr. GK Per (48 hr), U Sco (30 hr), and V394 Cra (18.2 hr). These cannot obviously contain main sequence stars, since they would have a mass by far exceeding that of the white dwarf, and mass transfer would be unstable. The secondaries in long period systems are subgiant whose nuclear evolution drive the evolution of the system. Their evolution has been discussed by Webbink et al. (1983). The mass-period relation for these systems is given by (King,

Hameury: Cataclysmic variables

437

1988):

(

7.65 ^

<13>

where Mc is the mass of the helium core. It is seen that the period depends very sensitively upon Mc, and that the orbital period of these systems increase with time, as the helium core grows; long periods (up to a few days) can be obtained. This is confirmed by numerical models of the evolution of binaries containing an evolved secondary (Pylyser and Savonije, 1988a,b). 19.4-4-2 Ultrashort systems (Porb < 1 hr) systems: He secondaries Two systems have confirmed periods below 80 min: GP Com (46 min) and AM Cvn (18 min). These systems cannot have evolved through the sequence described above; they are too compact to contain H-rich main sequence or degenerate secondaries, but instead contain He stars that may or may not be degenerate. These systems have been studied in quite some details (see e.g. Rappaport and Joss, 1984; Tutukov et al., 1985; Nelson et al., 1986; Fedorova and Ergma, 1989). He rich secondaries have a much smaller radius than H-rich secondaries, and hence have shorter orbital periods. The formation of such systems is not well understood; they are clearly very rare. 19.4-4-3 Systems in the period gap ^From the period histogram shown in Fig. 1.3, it can be seen that there are some systems inside the period gap. This is not really unexpected, since depending on the initial secondary mass, systems may form at periods between 2 and 3 hr. Except in very special cases, they are fully convective, and evolve down to the minimum period. It seems however that the period gap il less pronounced for a subclass of cataclysmic variables, the AM Her systems, that contain a strongly magnetic white dwarf rotating synchronously with the orbit. This will be further discussed in the next section. 19.4-4-4 The case of magnetic systems Cataclysmic variables containing a magnetized white dwarf are divided into two subclasses: systems in which the interaction between the dipoles of the primary and secondary is strong enough to force co-rotation of the white dwarf with the orbit, the so-called AM Her systems or polars, and systems in which the white dwarf spin period is shorter than the orbital period (the DQ Her systems, or intermediate polars). Magnetic systems have received much attention, still many questions are left unanswered (see King, 1993 for a recent review),

438

Hameury: Cataclysmic variables

The fact that the average orbital period of AM Her systems is shorter than that of DQ Hers lead Chanmugam and Ray (1984) and King et al. (1985) to propose that intermediate polars must synchronize when the orbital separation has become small enough and evolve into polars. The issue is still controversial (see e.g. Lamb and Melia, 1987; Hameury et al., 1987), but it now appears that, although most AM Her systems must have been born as intermediate polars, most observed intermediate polars will never become synchronous. The period distribution of AM Her systems have two interesting features: (1) there is an accumulation of systems at a period of 114 min, and (2) the period gap is much less pronounced for these systems. An explanation of (1) has been suggested by Hameury et al. (1988): as systems emerge from the gap, their mass transfer is high and they spend a long time at about that period (see Fig. 1.4). This imposes tight constraints on the white dwarf mass distribution in these systems, as well as on angular momentum losses (Hameury et al., 1988; Ritter and Kolb, 1992). It is however not clear whether the spike is real, or will disappear as the number of detected systems increases. Point (2) may indicate that there is mass transfer while AM Her systems evolve through the gap, and that magnetic braking is either not interrupted or never effective. Wu and Wickramasinghe (1993) and Wickramasinghe (1993) suggested that the strong white dwarf field would prevent open field lines from the secondary, and consequently magnetic braking would be impossible. On the other hand, Schmidt et al. (1986) and Frank et al. (1993) argued that the interaction of the secondary wind with the white dwarf magnetic field would lead to enhanced magnetic braking that would persist even after the secondary has become fully convective. 19.4'5 Predicted orbital distribution As mentioned earlier, the evolutionary tracks obtained for different binary parameters must be convolved by the distribution of initial masses and periods to obtain the intrinsic orbital period distribution of cataclysmic variables. This has been done by Hameury et al. (1990) for AM Her systems only, and more recently by Kolb (1993) and Shafter (1992) for all systems. Both Kolb (1993) and Shafter (1992) used Politano's (1988,1990) and/or de Kool (1992) differential formation rates of cataclysmic variables, but Shafter assumption that the secondary lies on the main sequence is extremely crude, whereas Kolb used a bi-polytrope code. Kolb (1993) main results are: (1) only 1% of CVs are located below the period gap; (2) the intrinsic period distribution depends weakly on the details of magnetic braking; realistic

Hameury: Cataclysmic variables

439

10

CU

o oo

Fig. 19.5 Predicted intrinsic (dotted line) and observed (solid line) period distribution of cataclysmic variables (from Kolb, 1993).

braking laws from Mestel and Spruit (1987) or Verbunt and Zwaan (1981) lead to essentially the same results and (3) there is a much more significant dependence on the details of the stellar structure. Once the intrinsic population of CVs is determined, it must be corrected for selection effects. These are quite difficult to estimate (see e.g. a discussion by Ritter and Burkert, 1986); in particular, they are different for each sub-class of cataclysmic variables, depending on how they are detected. Figure 1.5 shows one of Kolb's (1993) result in the case of a bolometric luminosity limited sample (i.e. the detection probability is proportional to the total luminosity to the power 3/2), which is not a very good approximation (most of the luminosity emitted by a CV is in the UV range, and hence undetectable). It is seen that this is in disagreement with observations, as far too many systems are found above the period gap, and that the predicted excess at the minimum period does not show up in the observed distribution. Assuming a visual magnitude limited sample would solve the first problem, whereas it is quite difficult to explain why one does not observe an accumulation of systems at the minimum period. It is possible that this is due to some unforeseen selection effect; it is nevertheless worth noting that the secondary mass is of the order of 0.05 M© at the minimum period, and that the stellar structure for those low masses is not well understood.

440

Hameury: Cataclysmic variables

19.5 Conclusion Models for the evolution of cataclysmic variables are relatively sensitive to the assumed stellar physics, the strongest dependance being that due to the opacities and the treatment of the photospheric layers. If one were to know with reasonable accuracy the angular momentum losses from these systems, one would be able to test stellar models for masses in the range 0.05 0.5 M©. The presence of a more massive compact companion certainly helps in determining the characteristics of the secondary. In order to test the stellar models, one would also need to know the white dwarf mass in a larger number of systems; observational selection effects should also be better understood. The subclass of magnetic systems appear very promising from that point of view, even though the presence of strong magnetic field from the white dwarf introduces a further complication. The disagreement of the observed and predicted orbital period distribution for systems close to the minimum period is also quite interesting in that it is unlikely to be solely due to selection effects, and might therefore tell us something on the structure of very low mass stars.

References Alexander D.R., Astrophys. J. Suppl. Ser. 29, 363 (1975) Chanmugam G., Ray A., Astrophys. J. 285, 252 (1984) Cox A.N., Stewart J.N., Nauchn. Informatsii, 15, 1 (1969) Cox A.N., Tabor J.E., Astrophys. J. Suppl. Ser. 31, 271 (1976) D'Antona F., Mazzitelli I., Astrophys. J. 260, 722 (1982) D'Antona F., Mazzitelli I., Ritter H., Astron. Astrophys. 225, 391 (1989) de Kool M., Astron. Astrophys. 261, 188 (1992) Dorman B., Nelson L.A., Chau W.Y., Astrophys. J. 342, 1003 (1989) Eggleton P.P., Astrophys. J. 268, 368 (1983) Fedorova A.V., Ergma E.V., Astrophys. Sp. Sci., 151, 125 (1989) Fontaine G., Graboske H.C.Jr., Van Horn H.M., Astrophys. J. Suppl. Ser. 35, 293 (1977) Fowler W.A., Caughlan G.R., Zimmerman B.A., Ann. Rev. Astron. Astrophys. 13, 69 (1975) Frank J., et al., in preparation (1993) Graboske H.C.Jr, De Witt H.E., Grossman A.S., Cooper M.S., Asirophys. J. 181, 457 (1973) Hameury J.M., Astron. Astrophys. 243, 419 (1991) Hameury J.M., King A.R., Lasota J.P., Ritter H., Astrophys. Sp. Sci. 131, 583 (1987) Hameury J.M., King A.R., Lasota J.P., Livio M., Mon. Not. R. astr. Soc. 237, 835 (1989) Hameury J.M., King A.R., Lasota J.P., Mon. Not. R. astr. Soc. 242, 141 (1990) Hameury J.M., King A.R., Lasota J.P., Raison F., Astron. Astrophys. 227, 81 (1993)

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441

Hameury J.M., King A.R., Lasota J.P., Ritter, H., Astrophys. J. 316, 275 (1987) Hameury J.M., King A.R., Lasota J.P., Ritter, H., Mon. Not. R. astr. Soc. 231, 535 (1988) Harris M.J., Fowler W.A., Caughlan G.R., Zimmerman B.A., Ann. Rev. Astron. Astrophys. 21, 185 (1983) Joss P.C., Rappaport S., Astrophys. J. 270, L73 (1983) King A.R., Q. Jl R. astr. Soc, 29, 1 (1988) King A.R. in the proceedings of the Monte-Porzio conference Evolutionary links in the zoo of interacting binaries, in press (1993) King A.R., Frank J., Ritter H., Mon. Not. R. astr. Soc. 213, 181 (1985) Kolb U., Astron. Astrophys. 271, 149 (1993) Kolb U., Ritter H., Astron. Astrophys. 254, 213 (1992) Kovetz A., Prialnik D., Astrophys. J. 291, 812 (19*) Kraft R.P., Matthews J., Greenstein J.L., Astrophys. J. 136, 312 (1962) Lamb D.Q., Melia F., Astrophys. Sp. Sci. 131, 511 (1987) Livio M., Govarie A., Ritter H., Astron. Astrophys. 246, 84 (1991) Livio M., Shankar A., Burkert A., Truran J.W., Astrophys. J. 356, 250 (1990) Lubow S.H., Shu F.H., Astrophys. J. 198, 383 (1975) McDermott P.N. Taam R.E., Astrophys. J. 342, 1019 (1989) McDonald J., Astrophys. J. 305, 251 (1986) Mestel L. Spruit H.C., Mon. Not. R. astr. Soc. 226, 57 (1987) Nelson L.A., Chau W.Y., Rosenblum A., Astrophys. J. 299, 658 (1985) Nelson L.A., Rappaport S.A., Joss P.C., Astrophys. J. 304, 231 (1986) Paczyriski B., Ada Astron., 19, 1 (1969) Paczyriski B., Ann. Rev. Astron. Astrophys. 9, 183 (1971) Paczynski B., Sienkiewicz R., Astrophys. J. 248, L27 (1981) Paczynski B., Sienkiewicz R., Astrophys. J. 268, 825 (1983) Podsiadlowski P., Nature 350, 136 (1991) Politano M., PhD thesis, University of Illinois, Urbana-Champaign (1988) Politano M., in Accretion-powered compact binaries, ed. Mauche C.W., Cambridge University Press, p. 421 (1990) Pylyser E.H.P., Savonije G.J., Astron. Astrophys. 191, 57 (1988a) Pylyser E.H.P., Savonije G.J., Astron. Astrophys. 208, 52 (1988b) Rappaport S., Joss P.C., Astrophys. J. 283, 232 (1984) Rappaport S., Joss P.C., Webbink R.F., Astrophys. J. 254, 616 (1982) Rappaport S., Verbunt F., Joss P.C., Astrophys. J. 275, 713 (1983) Ritter H. 1985, in High energy astrophysics and cosmology, eds. Yang J. and Zhu, C , Gordon and Breach Science Publ. Inc., New York, p. 207 (1985) Ritter H., Astron. Astrophys. 202, 93 (1988) Ritter H., Astron. Astrophys. Suppl. Ser. 85, 1179 (1990) Ritter H., Burkert, Astron. Astrophys. 158, 161 (1986) Ritter H., Kolb, U., Astron. Astrophys. 259, 159 (1992) Ruderman M., Shaham J., Tavani M., Eichler D., Astrophys. J. 342, 292 (1989) Sarna M., Astron. Astrophys. 239, 163 (1990) Saumon D., Chabrier, G. Phys. Rev. A, 44, 5122 (1991) Schatzman E., Ann. Astr., 25, 18 (1962) Schatzman E., in IAU Symp. No. 22, ed. Lust R., Reidel, Dordrecht, p. 153 (1965) Schmidt G.D., Stockman H.S., Grandi S.A., Astrophys. J. 300, 804 (1986) Shafter A.W., Astrophys. J. 394, 268 (1992) Shara M.M., Livio M., Moffat A.F.J., Orio M., Astrophys. J. 311, 163 (1986)

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Skumanich A., Astrophys. J. 171, 565 (1972) Spruit H.C., Ritter H., Astron. Astrophys. 124, 267 (1983) Truran J.W., Livio, M., Astrophys. J. 308, 721 (1986) Tutukov A.V., Fedorova A.V., Ergma E.V., Yungelson L.R., Pis'ma Astr. Zh., 11, 123 (1985) VandenBerg D.A., Hartwick F.D.A., Dawson P., Alexander D.R., Astrophys. J. 266, 747 (1983) Vardya M.S., Astrophys. J. Suppl. Ser. 42, 281 (1960) Verbunt F., Mon. Not. R. astr. Soc. 209, 227 (1984) Verbunt F., Zwaan C , Astron. Astrophys. 100, L7 (1981) Webbink R.F., in White dwarfs and variable degenerate stars, IAU Colloq. 53, p. 426, eds. Van Horn H.M. and Weidemann V., University of Rochester (1979) Webbink R.F., Rappaport S., Savonije G.J., Astrophys. J. 270, 678 (1983) Wickramasinghe D.T., in Cataclysmic variables and related physics, Annals of the Israel Physical Society No. 10, p. 208, eds. Regev O. and Shaviv G. (1993) Wu K., Wickramasinghe D.T., in Cataclysmic variables and related physics, Annals of the Israel Physical Society No. 10, p. 336, eds. Regev O. and Shaviv G. (1993)

20 Giant planet, brown dwarf, and low-mass star interiors W.B. HUBBARD Department of Planetary Sciences, Lunar and Planetary Laboratory, University of Arizona, Tucson, AZ 85721, USA

Abstract Astrophysical objects of low mass, ranging from giant planets to extreme dwarf main-sequence stars, have a number of physical characteristics in common due to properties of their equations of state. Their luminosities are low (much less than the solar luminosity X©) and their evolutionary timescales are typically measured in Gyr. So far there are few observational examples of these objects, although they are undoubtedly numerous in the galaxy. The lower mass limit is set by the object's ability to retain hydrogen during accumulation (about the mass of Saturn), while the upper mass limit is set by the lifting of electron degeneracy by high internal temperature. Objects confined within this broad range, which extends up to about 0.1 M©, are governed by the thermodynamics of liquid metallic hydrogen. In this paper, we discuss the implications of this feature of their interior structure for their radii, interior temperatures, thermonuclear energy generation rates, and luminosities. We conclude with a brief assessment of the confrontation between observations and theory in galactic clusters and in the solar system. L'equation d'etat des corps celestes de faible masse, qui vont des planetes geantes aux etoiles naines qui sont a la limite de la sequence principale, est a l'origine d'un ensemble commun de proprietes physiques. Leur luminosite est de beaucoup inferieure a celle du Soleil et leur temps caracteristique devolution se mesure en milliards d'annees. A ce jour, nous ne connaissons que quelques exemples de ces objets malgre la conviction qu'ils sont nom443

444

Hubbard: Giant planet, brown dwarf, and low-mass star interiors

breux dans la Galaxie. La limite inferieure en masse (approximativement la masse de Saturne) est fixee par la capadte de retenir l'hydrogene au cours du processus d'accumulation de la matiere. La limite superieure est atteinte lorsque la temperature interne est suffisamment elevee pour que les electrons ne soient plus degeneres (environ 0.1 M©). Les proprietes des corps qui se retrouvent dans ce domaine etendu en masse sont principalement determinees par la thermodynamique de l'hydrogene metallique liquide. Cet article presente les effets de ce point commun de leur structure interne sur leurs rayons, leurs temperature internes, leurs taux de generation d'energie thermonucleaire et leurs luminosites. Nous concluons par une breve discussion de la confrontation entre les observations et la theorie dans le cas des amas galactiques ainsi que dans le systeme solaire.

20.1 Introduction In this chapter, we shall discuss the implications of the equation of state (mainly that of hydrogen) for giant planets, brown dwarfs, and very lowmass stars. Although the mass range covered by these seemingly disparate objects is moderately large (about two orders of magnitude), the physics of the equation of state is basically the same, and leads to certain common characteristics. By giant planet, we mean the four largest planets of the solar system, Jupiter, Saturn, Uranus, and Neptune, as well as their so far hypothetical counterparts in other solar systems. The equation of state of hydrogen is particularly relevant to the largest two giant planets, Jupiter and Saturn. Giant planets have masses M which lie in the range 5 X 10~5M© < M < 1 x 10~3M© (M© = mass of the sun). The term brown dwarf (BD) has become standard usage for designating a class of hydrogen-rich objects with the following characteristics: (a) composition similar to the sun, i.e. dominated by hydrogen; (b) masses about ten times larger than that of Jupiter; (c) masses smaller than the critical mass for sustained thermonuclear fusion of hydrogen. Masses of brown dwarfs lie roughly in the range 1 x 10~2Af© < M < 1 X 10~1M©. At present there is some uncertainty about the possible modes of origin of objects in the mass range 1 x 10~3M© < M < 1 X 10"~2M©; such objects could be considered either very large giant planets or extremely small BD's. According to Boss (1986), the minimum mass for direct formation of a BD from collapse of an interstellar cloud of H and He is ~ 0.02M©, and objects of lower mass form through a different sequence of events, which begins with coagulation of planetesimals from solid particles. As we shall discuss, these lower mass

Hubbard: Giant planet, brown dwarf, and low-mass star interiors

445

objects lie below the critical mass for fusion of deuterium and it is therefore convenient to classify them as giant planets (GP's) rather than BD's. Objects which are more massive than ~ 10~1M© but still substantially less massive than the sun are termed very low mass stars (VLM), or extreme M dwarfs. In contrast to the situation for GP's and BD's, many observational examples of VLM's exist. All of the objects under discussion here, GP's, BD's, and VLM's, have intrinsic luminosities L which are small compared with the luminosity of the sun £©. Their luminosity and associated interior thermal state change very slowly with time, typically over time scales measured in Gyr. At the same time, depending on the relative efficiency with which these objects are formed, they may comprise an appreciable fraction of the mass of the Galaxy. This mass could thus be largely hidden in objects which are difficult to detect. In modeling objects across the indicated mass range, we assume that the composition is similar to that of the sun, i.e., predominantly hydrogen. Although there is some uncertainty about the precise composition and indeed there may be some variation with mass and age, for our purposes it is sufficient to take a uniform composition with a helium mass fraction Y = 0.25, and a hydrogen mass fraction X = 1 - Y — Z, where Z is the mass fraction of all elements heavier than helium (the so-called metals). The value of Z plays little or no role in the equation of state as it does not exceed 0.02 for solar composition. However, the value of Z affects photon opacities in the outermost layers of these hydrogen-rich objects, and hence has a large impact on their interior thermal state. In the lowest mass range, M ~ < 3 X 10~4M©, significant amounts of hydrogen are lost during accumulation of the object, and Z becomes large enough to play a significant role in the equation of state. The similarity of the physics of the equation of state in all of these objects ultimately arises from the fact that their interiors lie for the most part within the following limits: 6 = eF/kT > 1

(1)

T = e2/akT > 1

(2)

and

Here 8 is the electron degeneracy parameter, the ratio of the electron fermi energy ep to a typical thermal excitation energy kT. Similarly, the ion

446

Hubbard: Giant planet, brown dwarf, and low-mass star interiors

coupling parameter (which for hydrogen is the same as the electron coupling parameter) T measures the ratio of a typical ion coulomb energy e2/a to kT. Here e is the ion charge and a is the average distance between ions. Because both of these parameters are large, the object's equation of state is mainly governed by the physics of metallic hydrogen. And, because F typically lies in the range 1 < T < 100, the metallic hydrogen is in a strongly coupled liquid phase. A third dimensionless parameter of relevance to the equation of state is the density parameter rs = a/ao, where ao is the Bohr radius. This parameter ranges from r , « l for a giant planet such as Jupiter to ra « 0.1 for the most massive BD's. Under these circumstances, the relation between pressure P and mass density p is largely independent of temperature T, and can be expressed in the form P ~ P°

(3)

with 1.6 < a < 2. This result is universally true for hydrogen-rich objects in the relevant mass range, and leads to the remarkable result that such objects have very similar radii R, regardless of their mass. However, this similarity does not extend to quantities related to the object's interior thermal state, such as its total intrinsic luminosity L, its effective temperature Te, and its central temperature T centra i, which in general depend sensitively on both the mass and age of the object. Some of these points are illustrated in Table 1.1, in which we compare Jupiter, a well-studied metallic-hydrogen object with an age of about 5 Gyr, with a hypothetical BD of the same age and gross chemical composition. Since Jupiter rotates rapidly and is therefore nonspherical, the radius R which is given is the equatorial radius at the 1-bar pressure level. For Jupiter, the quantity /9Centrai 1S actually the highest density of the metallic hydrogen zone in the planet's interior, and does not refer to the density of a Z-rich central core. 20.2 Radius vs. mass; relation to equation of state The overall behavior of the R(M) curve for giant planets, BD's, and VLM's is extremely diagnostic of the equation of state of metallic hydrogen-helium mixtures, and its general shape is shown in Figs. 1.1 and 1.2. Figure 1.1 shows a maximum radius for solar composition of 75740 km at M = 0.004M© (about four Jupiter masses). For pure hydrogen, the maximum in R(M) moves to 0.006MQ and 87200 km. The maximum radius exists as a direct consequence of a competition between electrostatic con-

Hubbard: Giant planet, brown dwarf, and low-mass star interiors

447

Table 20.1. Comparison between Jupiter and a typical brown dwarf

M(0) /2(km) age (Gyr) L/LQ •^'nuclear/-'-'

T (10 bar) Pcentral -* central

16

(

I

I

14 -

1

I

I

Jupiter

brown dwarf

0.001 71492 ± 4 5 0.9 x lO" 9 0 124 K 337 K 4 g cm" 3 22600 K

0.070 55870 5 1.5 x 10" 5 0.28 1262 K 550 K 979 g cm" 3 1.683 x 106 K

1

1

1

1 '

1

1

1

1

1

age = 5 Gyr

1

1

1

yS

—

12 — o o o o

10

/ —

8 -

/

L

j

6 '-

s 1

1

I

I

| , .05

. 1 . .1

1

1

1

1

.15

1

1

1

1

.2

M/M0 Fig. 20.1 Solid line (—): radius vs. mass for solar-composition objects at a cooling age of 5 Gyr. Objects to the left of the short break at 0.01M o are considered giant planets. BD's extend from the break to 0.08A/ o . •: GP's, shown in more detail in Fig. 1.2.

tributions to the equation of state which contribute a negative component to the pressure, and the electron fermi pressure, which contributes positively. For M < 0.004AfQ (r, ~ < 0.6) the electrostatic contributions to the pressure are sufficiently important (pressure increases sufficiently rapidly with increasing density) that there is a positive slope to the M(R) curve in this mass range. For masses greater than 0.004M©, the degenerate electron fermi pressure begins to dominate the equation of state, causing a decrease in R with increasing M. As Fig. 1.1 shows, there is a minimum in the R(M) curve at about 0.07M©. The minimum is not especially related to the equation of state, but is instead produced by the onset of thermonuclear energy

448

Hubbard: Giant planet, brown dwarf, and low-mass star interiors i i i I i i i I i i i I i i i I i i i I i i i I i i i I

O

o o o

OS

-u I I I .0002 .0004 .0006 .0008

.001

.0012 .0014

M/Mo Fig. 20.2 Solid line (—): radius vs. mass for spherically symmetric solarcomposition objects at a finite interior temperature. Dashed line ( ): same, but for pure H. •: giant planets Jupiter, Saturn, Uranus, Neptune.

generation in the BD's core, which tends to raise the interior temperature and reduces the electron degeneracy. According to Burrows et al. (1993), the minimum mass for sustained thermonuclear energy generation in BD's is 0.0767M©; objects which are more massive are not BD's but are instead VLM's, and settle eventually on the main sequence. The minimum in R(M) lies slightly below this minimum mass because Fig. 1.1 is computed for a finite age of 5 Gyr, and the most massive BD's are able to burn hydrogen for several Gyr before they fail to settle on the main sequence. Figure 1.2 shows an expanded view of the extreme left-hand corner of Fig. 1.1. The solid curve again shows R(M) for solar-composition material, calculated for temperatures along an adiabatic compression curve corresponding to the interiors of Jupiter and Saturn, while the dashed curve is the same, but for pure hydrogen. At masses slightly below the mass of Saturn (0.0003 MQ), the metallic-hydrogen core vanishes, and R(M) is determined by the equation of state of dense molecular hydrogen. At masses slightly below the mass of Uranus or Neptune (5 x 10~5M©), the molecular hydrogen adiabat falls entirely in the ideal-gas region, and as a result R(M) begins to increase with falling mass. However, the latter behavior cannot be realized in nature. As Fig. 1.2 makes clear, giant planets less massive than Saturn cannot capture significant amounts of hydrogen when they are formed, and their interior equations of state are dominated by heavier nuclei. More detailed analysis of the interior compositions of Jupiter and Saturn (Chabrier et al., 1992) shows that both are more enriched in the Z-component than

Hubbard: Giant planet, brown dwarf, and low-mass star interiors

449

solar composition, with Saturn more enriched than Jupiter. In Fig. 1.2, Jupiter plots slightly above the solar-composition curve because correction for rotation of Jupiter has not been included in the solid curve; when this correction is included, Jupiter plots slightly below the theoretical curve for solar composition. 20.3 Variation of luminosity with mass The heat flow which corresponds to the observed luminosity L is presumed to be derived from two sources: (a) heat release from the object due to work done on the object's interior and due to changes in internal energy E; (b) heat release due to nuclear reactions in the object's interior. Both of the sources can be combined into the equation rM P An L = / —±dm-Jo

pL at

A rM rM / Edm+ / €N(p,T)dm at Jo

(3)

Jo

where dm is an element of mass, t is the time, and ejv is the rate of release of energy by nuclear reactions. For the giant planets, ejq is effectively zero because temperatures and densities are too low for fusion reactions to proceed, and the abundances of radioactive high-Z elements such as 40 K, 232 Th, and 238 U are too low for radioactive decay to be significant. In the giant planets, only the first two terms in Eq. 3 are important, although they must be carefully evaluated generalized in the case of Saturn to account for immiscibility of helium in liquid metallic hydrogen and consequent formation of a helium-enriched core (Stevenson, 1975). As masses increase into the BD range, one must consider contributions to CN from the following reactions (Burrows et al., 1993): p + p->d + e+ + ve p + d -* 3He + 7

(1.442 MeV) (5.494 MeV)

(4) (5)

In the BD mass range, these reactions stop at 3He. In addition, the following reaction does not contribute significantly to cN but does serve as a useful tracer of BD evolution: p + 7U^2a

(6)

Basically, the division between VLM's and BD's is defined by reaction (4). For masses above the critical mass of O.O767M0, central densities exceed about 103 g cm" 3 , and central temperatures exceed about 2x 10^ K. Under these circumstances, enough heat is liberated by reactions (4) and (5) to

Hubbard: Giant planet, brown dwarf, and low-mass star interiors

450

0 p—i—I—i—i—i—r

-i

i

i

i

|

I

i

i

i

|

J -4 00

o -6

-8

log t (Gyr) Fig. 20.3 Curves of luminosity vs. time for VLM's and BD's of various mass. The upper curve is for a VLM of O.2OOM0, while the lowest curve is for O.OIOMQ. A smaller mass interval has been used for objects near the critical mass for hydrogen burning. • shows the 0.070M© model presented in Table 1.1. balance the heat radiated by the VLM's atmosphere. For objects below the critical mass, reaction (5) can still proceed but reaction (4) does not. As a result, the BD has a relatively brief phase during which it burns primordial deuterium, but once the deuterium is gone, only heat release from the first two terms of eq. (3) plays a role in the BD's luminosity. At the lower end of the BD mass range, even deuterium does not burn. This second critical mass lies at about 0.015MQ and corresponds to central densities about ~ 20 g cm" 3 , and central temperatures ~ 0.5 X 106 K. Objects below this second critical mass can be considered GP's. Figure 1.3 illustrates the above-described effects, showing the evolution of luminosity with time on a double-logarithmic scale. The bifurcation of objects into VLM's and BD's at the right side of the figure is apparent. The model with the prolonged curve is just subcritical at 0.0765M©, and shows a gradual decline of luminosity with time over time periods longer than the age of the universe. Other features in Fig. 1.3 are worthy of note. A "ripple" in L(t) is apparent for —3 < log* < —1.5, with the onset of the "ripple" progressively later for smaller masses. This "ripple" is caused by thermonuclear fusion of

Hubbard: Giant planet, brown dwarf, and low-mass star interiors

451

primordial deuterium via reaction (5), assuming that the primordial deuterium mass fraction is Yd = 2 x 10~5. The "ripple" vanishes for masses between O.OlOAf© (lowest curve in Fig. 1.3) and 0.020MQ (second lowest curve in Fig. 1.3). Thus the critical mass for deuterium burning lies at M « 0.015M©, and we conclude that GP's, such as Jupiter, will contain the primordial deuterium abundance, but BD's older than ~ 0.1 Gyr will be depleted in deuterium. Another, fainter, "ripple" in Fig. 1.3 can be discerned at L ~ 10~ 4 .£Q. This "ripple" is not related to the equation of state or to nuclear reaction rates, but is caused by temporarily increased atmospheric opacity at this luminosity level, as a consequence of the formation of dust grains in the BD's atmosphere at levels where most photons are just able to escape to space (Burrows et al., 1989). This increase in opacity causes the BD luminosity to temporarily drop and the cooling age of the BD to be correspondingly extended. As Fig. 1.3 makes clear, L is a function of both M and t for BD's and VLM's, and thus identification of an object as a possible BD requires knowledge of its age as well as its luminosity. But there is, theoretically, an independent test which can be used to determine whether an object lies within the BD mass range. Fragile nuclei such as deuterium and lithium are destroyed via reactions (5) and (6) respectively, at significantly lower temperatures than those required for reaction (4) to proceed. We have already discussed the destruction of an initial deuterium component in a BD. The corresponding destruction of initial lithium requires a mass equal or greater than about O.O65M0 (Magazzu, Martin, and Rebolo, 1993, D'Antona and Mazzitelli, 1993), and occurs at t ~ < 0.1 Gyr. Because the initial lithium abundance is very small, reaction (6) does not contribute appreciably to 6;v- An object older than 0.1 Gyr with detectable lithium in its atmosphere could be safely considered to be a BD. However, detection of such a lowabundance atom in such intrinsically faint objects presents a formidable observational challenge which has not yet been overcome. 20.4 Observational tests 20.4.1 Cluster luminosity functions One of the best methods available at present to investigate properties of putative BD's in the solar vicinity is to examine low-luminosity objects in nearby galactic clusters. Such clusters are relatively young and have a known t as established by the evolution of their more massive members. One may use the function L(M,t) as displayed in Fig. 1.3 to calculate the so-

Hubbard: Giant planet, brown dwarf, and low-mass star interiors 1 . , , , , , , . , , ,

, , , ,

•

452

x

i. x

o

-I

" - 114

^""*"x«

•4.

^y

-

/ 0.200 V-

1

• -.6 -

0.030 1 _

1 1

V.

1 It

-

1

0

/

.8 -

-

-

•

-1.6

-3

-2-5

-2

-15

-.6

log L/L, Fig. 20.4 Luminosity functions for p Oph for various values of a, compared with data from Comeron et al. (1993).

called luminosity function N(L, t), where N is the number of objects within a given interval in logX at a given time t. To calculate N, one must also know the initial number of objects created within the cluster as function of M; this is given by the so-called initial mass function (IMF):

Z(M)dM = CM~adM

(7)

where £(M) is the number of initial objects created within M and M + dM. The exponential dependence of £ on M was discovered by Salpeter (1955), who also showed that a ss 2.35 for stars in the galaxy. It is at present uncertain whether a power law of the form of eq. (7) applies to BD's. Clearly, the proportion of the mass of the Galaxy comprised of BD's depends on the relevant value of a for the BD mass range. For VLM's (and possibly BD's) in the Hyades galactic cluster (t = 0.6 Gyr), Hubbard et al. (1990) found that the theory matched low-luminosity star counts best for a w 0. As is clear from Fig. 1.3, BD's are best detected in young clusters, when their luminosities are relatively elevated. Fig. 1.4 shows a comparison of star counts in the young galactic association p Oph (t = 0.003 Gyr; Comeron et al., 1993) with the theory of Burrows et al. (1993). In such a young cluster, deuterium burning is still important in the BD's. While the number of detected low-luminosity objects is still quite small, a = 1.14 (Comeron et al., 1993) is a reasonable mean value for the BD-VLM mass range. This result, when added to f(M) for objects more massive than VLM's, implies that at most 1/3 of the mass of the Galaxy is composed of BD's. Figure 1.4 shows a very indirect test of the equation of state of BD's and VLM's. The primary influence of the equation of state on this plot is via its effect on the radii of the metallic-hydrogen objects, and on their

Hubbard: Giant planet, brown dwarf, and low-mass star interiors

453

thermal properties such as heat capacities and thermonuclear reaction rates. However, of equal importance are the atmospheric opacities which regulate the escape of interior heat, and the value of a.

20.4-2 Luminosity of giant planets With the exception of Uranus, the solar system's giant planets have measurable intrinsic luminosities. These were determined with considerable precision by experiments on the Voyager spacecraft after encounters with all four bodies during the previous decade (Pearl et al., 1991). Values of both the intrinsic luminosity L and the specific luminosity (L/M) are given in Table 1.2. The theory of giant-planet luminosity is developed by integrating eq. (3) over t (after setting c^ = 0), thus determining the time interval required for the planet's luminosity to decline to the present observed value. This can be compared with the known ages of these planets (t = 4.6 Gyr). In the case of Jupiter and Saturn, the luminosities are governed by the thermal properties of hydrogen, and the most recent calculation of L(t) for these objects (Saumon et al, 1992) has made use of the Plasma Phase Transition (PPT) theory of Saumon and Chabrier (1989, 1991, 1992). The calculation finds that interior adiabats in both Jupiter and Saturn cross the PPT during their evolution, at an interior point where P « 1 Mbar. The evolutionary age of the planet t£ is defined by the value of t for which the planet's luminosity drops to the value given in Table 1.2. For Jupiter, the theory gives *# = 5 Gyr, but for Saturn %E = 2.5 Gyr. Thus Jupiter's heat flow is in accordance with the latest equation of state of hydrogen and eq. (3), but Saturn's is not. The traditional explanation of the discrepancy for Saturn is that additional gravitational energy is liberated in Saturn's interior if helium becomes immiscible in hydrogen over part of the pressuretemperature range traversed in Saturn's interior (Stevenson and Salpeter, 1977). In this case helium droplets would form and sink to deeper layers in Saturn, liberating gravitational energy. Eq. (3) must be modified to take this mechanism into account (Hubbard and Stevenson, 1984). While the theory of helium-hydrogen phase separation in Saturn is as yet not fully quantitative, it does predict the substantial depletion of helium which is observed in Saturn's atmosphere (Conrath et al., 1984). In the case of Uranus and Neptune, the equation of state is dominated by heavier materials than hydrogen or helium, as we discuss below. The most relevant quantity for the evolution of a giant planet's luminosity is its interior specific heat at constant volume per unit mass, Cy. Neglecting eyy

454

Hubbard: Giant planet, brown dwarf, and low-mass star interiors

Table 20.2. Measured luminosity and specific luminosity of GP's

Jupiter Saturn Uranus Neptune

L/LQ

L/M ( 1 0 - U W/kg)

0.9 x 10~ 9 0.2 x 10- 9 < 0.002 x 10" 9 0.01 x 10" 9

18 15

<0.8 3

and any effects of immiscibility, eq. (3) can be schematically expressed as

~CvTx where dT/dt is the time rate of change of the mean temperature T, and (L/M) is the specific luminosity. One may estimate CV * 3k/Am, where A is the mean atomic weight of the planet's interior material in amu. For Jupiter and Saturn, A « 1, while for Uranus and Neptune, A « 5. Thus according to eq. (8), Uranus and Neptune should cool more rapidly than Jupiter and Saturn because their average interior CV is substantially lower. Yet, paradoxically, the estimated CV is large enough to permit IE for both Uranus and Neptune to greatly exceed their present age. The resolution of this paradox probably does not depend on some poorly understood property of the equation of state. Rather, it may be produced by large chemical gradients in the interiors of these hydrogen-depleted planets, which may have the effect of greatly impeding interior heat transport, especially in Uranus (Hubbard et al., 1994). 20.4-3 Giant planet interior structure Determination of the interior structure of the giant planets in the solar system proceeds somewhat differently from the approach taken for BD's. Much more detailed constraints are available, but the solar system giant planets are also somewhat altered from pure solar composition (substantially altered in the case of Uranus and Neptune), and thus do not cleanly constrain the hydrogen equation of state. In fact, modeling studies have been directed toward accurate determination of the Z-component of Jupiter, Saturn, Uranus, and Neptune by comparing their inferred interior pressuredensity profile with that of pure hydrogen. The gravitational potential V(r) within a planet satisfies Poisson's equa-

Hubbard: Giant planet, brown dwarf, and low-mass star interiors

455

tion (r is a vector from the planet's center of mass to the point of observation, and G is the gravitational constant),

V2V = -4irGp

(9)

while if the planet is in hydrostatic equilibrium and rotates on cylinders, the equation of hydrostatic equilibrium can be written VP = PVU

(10)

where U = V + Q is a generalized total potential, composed of V and a rotational potential Q. Since the rotation of all four giant planets is observed to be axially symmetric and north-south symmetric, generally consistent with rotation on cylinders, we may assume that Q likewise has these symmetries. It follows that the overall planetary structure p(r) is also axially symmetric and north-south symmetric, as is F(r). Under these conditions, V(r) exterior to the planet can be expanded in a form with the corresponding symmetry:

where r is the magnitude of r, and 8 is the angle that r makes to the rotation axis. In eq. (11), the dimensionless coefficients Jit represent the response of the planet to the rotational potential Q, and they are a direct constraint on the interior equation of state P(p), via the equation of hydrostatic equilibrium. Their value obviously depends on the choice of normalizing radius, R. For the giant planets, R is defined to be the equatorial radius of a surface at one bar pressure. If the planet rotates as a solid body with period TS, one has

[]

(12)

and in this case one can show that Ji on Q, J4 oc Q2, etc. Although for none of the four giant planets do the atmospheric features rotate with any unique period T$, their magnetic field configurations rotate with a welldefined periods. It is the latter period which is taken to define TS, a period which corresponds to the deep interior of the planet. All of the fundamental parameters constraining the interior equation of state of the four giant planets are presented in Tables 1.3 and 1.4. Although attempts have been made to solve an inverse problem by inferring the interior equation of state from a knowledge of a giant planet's J21 and its Q, such inverse solutions are not well constrained with the limited

456

Hubbard: Giant planet, brown dwarf, and low-mass star interiors

Table 20.3. Parameters constraining interior structure of Jupiter and Saturn /M® = 5.98 X 10 27 g = mass of the earth]

M (M e ) fl(km) rs (hr)

J2(R) x 106 J4(R) x 106 J6(R) x 106

Jupiter

Saturn

317.735 71492 ± 4 9.92492

95.147 60268 ± 4 10.65622 16331 ± 18

14697 ± 1 -584 ± 5 31 ±20

-914 ±38 108 ± 50

Table 20.4. Parameters constraining interior structure of Uranus and Neptune Uranus

M (M$) ii(km) TS (hr)

J2(R) x 106 J4(R) x 106

14.53

Neptune 17.14

25559 ± 4 17.24 ±0.01

24764 ±20 16.11 ±0.05

3516 ± 3

3538 ± 9

-31.9 ± 5

-38 ±10

number of J^t that are available, together with relatively large error bars on the higher-degree terms. In practice, the modeling has proceeded by assuming that the planet is a barotrope (i.e., has a unique P(p) relation throughout its interior). The equation of hydrostatic equilibrium is then integrated, for a specified function Q, to obtain a model planet of prescribed M. A satisfactory P(p) must yield a model with the observed R and JitSince the equation of state P(p, T) also depends on T, one needs a relation T(p) or T(P) to obtain the barotrope. Because of the interior heat flow in BD's and GP's, these bodies remain fully convective, even at the low luminosity values given in Table 1.2. If the convection is efficient, as is certainly the case for BD's as well as for chemically homogeneous GP's, the appropriate T(P) is an adiabatic relation between temperature and pressure, which is uniquely specified by the starting temperature of the adiabat at some observationally accessible portion of the object's convective atmosphere. Figure 1.5 shows adiabatic relations for T(p), for solar composition, and for four different starting temperatures. The BD starting temperature is taken to be 550 K at P = 10 bar, as given in Table 1.1, while the starting

457

Hubbard: Giant planet, brown dwarf, and low-mass star interiors

105

to* BO (550 K) J(!65K) S(t34K) U.N (72 K) 101

10-J

10-'

10°

10'

IOJ

103

p(g/cmJ)

Fig. 20.5 Adiabats for (top to bottom) a BD, Jupiter, Saturn, and UranusNeptune. 103 IO« 10J 101 |

10' too 80 (550 K) J(165K) S(134K) U.N (72 K)

10-' 10-J

10-4 10-J

10-'

too to* p(g/cmJ)

101

109

Fig. 20.6 The same adiabats as in Fig. 1.5, but on the P-p plane. temperatures for the giant planets are evaluated at P = 1 bar, respectively for Jupiter (165 K), Saturn (134 K), and a common Uranus-Neptune adiabat (72 K). Figure 1.6 shows the same adiabats on the P-p plane. As this figure shows, in this range of adiabatic temperatures, the P(p) relation is significantly changed by temperature effects by only for p ~ < 1 g cm" 3 , outside the domain of stability of metallic hydrogen. As is clear from Fig. 1.2, a chemically-homogeneous solar composition P{p) relation cannot produce a satisfactory model of Uranus or Neptune. In fact, such a model fails to reproduce the Jit of Jupiter and Saturn as well. Figures 1.7 and 1.8 show P{p) and T(p) relations for the same BD model as given in Table 1.1, together with GP models which agree with the data of Tables 1.3 and 1.4. As is made clear in Fig. 1.9, the four giant planets differ substantially from solar composition in their deep interiors. The equation of state for Jupiter follows the BD equation of state most closely, deviating toward a

458

Hubbard: Giant planet, brown dwarf, and low-mass star interiors

o 2

10-'

10-'

100

10'

p(g/crrvJ)

Fig. 20.7 The empirical equations of state for the four giant planets, compared with the equation of state for the BD model given in Table 1.1. The rightmost points for each object give values at the center.

10-J

10-'

100

p(g/cm3)

io>

to>

Fig. 20.8 Temperature profiles corresponding to the models shown in Fig. 1.7. For Uranus and Neptune, the diamonds show experimental points measured in dynamical shock compression experiments on synthetic Uranus material. 102

Fig. 20.9 An enlarged view of the GP models shown in Fig. 1.7.

Hubbard: Giant planet, brown dwarf, and low-mass star interiors

459

higher-density profile only in a small innermost ice or rock-ice core. By ice, we mean a material composed of the molecules H2O, CH4, and NH3 in solar proportions, not necessarily with intact molecules, and not necessarily in the solid phase. For solar composition proportions of C, N, and 0 , the corresponding fractions are 56.5% H2O,32.5% CH4, and 11% NH3 by mass. In Saturn, the deviation from the BD equation of state begins in the metallic-hydrogen region. The enhanced density of Saturn in the region is interpreted as being caused by enrichment of helium due to immiscibility in metallic hydrogen, and possible enrichment of the ice component as well (Chabrier et al., 1992). Figure 1.9 shows that Uranus and Neptune have very similar interior equations of state. Satisfactory models for both planets have outer hydrogenhelium envelopes which extend to a maximum pressure of ~ 0.1 Mbar, below which point the equation of state very closely follows the P(p) curve for ice (between P ~ 0.1 Mbar and 8 Mbar. A small rocky core may exist in either Uranus or Neptune, but is not required to fit the data. Uranus and Neptune are composed primarily of ice, a substance which is accessible to laboratory shock compression experiments in the relevant pressure range. These experiments have been carried out on "synthetic Uranus", a solution of water, ammonia, and isopropanol with mole fractions of 0.71, 0.14, and 0.15 respectively, up to a pressure of 2.2 Mbar (Hubbard et al., 1991). In the limit of high pressure, this material should behave identically with solar proportions of H2O,CH4,NH3. Figure 1.10 shows theoretical adiabats for mixtures of hydrogen-helium (in solar proportions) with ice, together with the shock data on "synthetic Uranus". All of the adiabats are computed for a starting temperature appropriate to Uranus or Neptune, and assume constant chemical composition. The left-most adiabat (marked 0 ) is for pure hydrogen and helium in solar proportions. The next adiabat to the right (marked 0.2) represents a mixture of the solar-composition adiabat with 0.2 mass fraction of ice, etc., ending with an adiabat of pure ice composition. The diamond symbols shown on Fig. 1.10 show the shock compression data for "synthetic Uranus" of Nellis et al. (1988). Open diamonds represent data from single shock compression, while solid diamonds are data from double shock compression. The two points marked with larger symbols are points for which the temperature was simultaneously measured; the latter data are shown in Fig. 1.8 as well. Finally, Fig. 1.11 shows the curves and data points of Fig. 1.10, superimposed on interior models of Uranus (dashed curve) and Neptune (solid curve). Open squares and circles show transition points in two alternative

460

Hubbard: Giant planet, brown dwarf, and low-mass star interiors

too t. o a ^^ o.

solar composition + "ice"

.•."•** .•"

10-'

10-1

10-1

10-«t-i 10-J

O

.-y'O-Z

"0.7

/i.O

10'

10-1

p(g/crrv>)

Fig. 20.10 Theoretical adiabats for mixtures of H and He in solar proportions, mixed with ice. Diamonds show shock compression data on ice.

10-*

10-'

p(g/cm3) Fig. 20.11 Interior models of Uranus and Neptune. interior models of Neptune calculated by Zharkov and Gudkova (1991); in the latter models, the hydrogen-helium envelope extends to about 0.2 Mbar.

20.5 Conclusion The properties of a broad range of astrophysical objects, ranging from VLM's (masses ~ 0.1M©) to giant planets (masses ~ O.OOIMQ) can be investigated within the framework of a general equation of state for a mixture of hydrogen and helium. For the more massive objects, confrontation between theory and data comes primarily from a comparison of the predicted spectral properties of photons emitted from BD atmospheres, as a function of their interior thermal properties and age, with the (sparse) observational data set.

Hubbard: Giant planet, brown dwarf, and low-mass star interiors

461

For the giant planets within our solar system, it is possible to study not only the relation between intrinsic luminosities and interior thermal state, but also the relation between the gravitational potential coefficients Jit and the interior P(p) relation. A further constraint on the interior equation of state comes from the M(R) relation. A primary result of this study is that, commencing at masses comparable to that of Jupiter, giant planets form in a process which tends to lose some of the hydrogen-helium component as the planet accretes. As the object's mass decreases, ever more hydrogenhelium is lost, such that objects in the mass range of Uranus and Neptune contain only a small fraction (~ few % by mass) of hydrogen-helium. In the mass range of Uranus and Neptune, typical central pressures reach only a few Mbar. Likely interior material (ice) is susceptible to experimental determination of its equation of state in this pressure range. Thus, theory can be at least partially replaced with experiment when equations of state are tested for the lowest mass giant planets. This work was supported in part by NASA Grant NAGW-1555 and by NSF Grant INT-8907133.

References Boss A.P., in Astrophysics of Brown Dwarfs (M.C. Kafatos, R.S. Harrington, S.P. Maran, eds.), Cambridge Univ. Press, pp. 206-211, (1986) Burrows A., Hubbard W.B., Lunine J.I., Astrophys. J. 345, 939, (1989) Burrows A., Hubbard W.B., Saumon D., Lunine J.I., Astrophys. J. 406, 158, (1993) Chabrier G., Saumon D., Hubbard W.B., Lunine J.I., Astrophys. J. 391, 817, (1992) Comeron F., Rieke G.H., Burrows A., Rieke M.J., Astrophys. J. 416, 185, (1993) Conrath B.J., Gautier D., Hanel R.A., Hornstein J.S., Astrophys. J. 282, 807, (1984) D'Antona F., Mazzitelli I., Astr. J. Suppl., in press, (1993) Dyson F., Ann. Phys. 63, 1, (1971) Hubbard W.B., Burrows A., Lunine J.I., Astrophys. J. 358, L53, (1990) Hubbard W.B., Nellis W.J., Mitchell A.C., Holmes N.C., Limaye S.S., McCandless P.C., Science 253, 648, (1991) Hubbard W.B., Pearl J.C., Podolak M., Stevenson D.J., in Neptune and Triton (D. Cruikshank, ed.) Univ. of Arizona Press, in press, (1994) Hubbard W.B., Stevenson D.J., in Saturn (T. Gehrels, M.S. Matthews, eds.), Univ. of Arizona Press, pp. 47-87, (1984) Magazzu A., Martin E.L., Rebolo R., Asirophys. J. 404, L17, (1993) Nellis W.J., Hamilton D.C., Holmes N.C., Radousky H.B., Ree F.H., Mitchell A.C., Nicol M., Science 240, 779, (1988) Pearl J.C., Conrath B.J., J. Geophys. Res. 96, 18921, (1991) Salpeter E.E., Astrophys. J. 121, 161, (1955) Saumon D., Chabrier G., Phys. Rev. Lett. 62, 2397, (1989)

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Hubbard: Giant planet, brown dwarf, and low-mass star interiors

Saumon D., Chabrier G., Phys. Rev. A 44, 5122, (1991) Saumon D., Chabrier G., Phys. Rev. A 46, 2084, (1992) Saumon D., Hubbard W.B., Chabrier G., Van Horn H.M., Astrophys. 3. 391, 827, (1992) Stevenson D.3.,Phys. Rev. 12B, 3999, (1975) Stevenson D.J., Salpeter E.E., Astrophys. J. Suppl. 35, 221, (1977) Zharkov V.N., GudkovaT.V., Ann Geophysicae 9, 357, (1991)

21 Searches for brown dwarfs JAMES LIEBERT Steward Observatory and Department of Astronomy, University of Arizona, Tucson, AZ 85721, USA

Abstract This review attempts a brief summary of the numerous and diverse searches for the so-called brown dwarfs, substellar objects having masses between giant planets and the lowest mass M dwarf stars. Cette revue donne un bref apergu de l'etat actuel des diverses recherches de naines brunes, objects substellaires ayant des masses comprises entre les planetes geantes et les naines M de faible masse.

21.1 Introduction Between the giant planets such as Jupiter ( 10~ 3 MQ) and stars at the bottom of the hydrogen-burning main sequence (< 0.1M©) - spanning more than two orders of magnitude in mass - the sequence of brown dwarfs has yet to be discovered and analyzed in detail. The previous sentence carries the positive bias of this author that - despite the current lack of a single, unambiguous example for me to discuss at this meeting - the flurry of searches now underway by a variety of techniques will identify at least some genuine brown dwarfs during the present decade. Our motivation for thinking and speaking positively is to encourage advances in the theory of both the interiors and atmospheres of such gaseous objects, in order to make possible positive identifications among the candidates found by observers. Indeed, numerous candidates exist of different kinds, some with measured masses, 463

464

Liebert: Searches for brown dwarfs

luminosities and temperatures which straddle the stellar mass limit (SML) near 0.08 M©. This paper is the observational complement to Bill Hubbard's, in which recent theoretical modelling of objects near and below the SML mass limit is discussed. One area of rapid and important theoretical progress not addressed by Dr. Hubbard is the application of new stellar atmosphere analyses to fit the infrared and optical spectra of very low mass stars and, potentially, the more luminous brown dwarfs. I will review this work briefly in Section 2. The remaining sections are devoted to overviews of the different techniques used in current searches, and the recent results. Most of this material is discussed much more extensively in Burrows and Liebert (1993), but there are some updated references and new results. 21.2 Low Mass Stars or Brown Dwarfs? In the solar neighborhood, the most accurately determinable stellar parameter is often the luminosity. This is because a large, accurate trigonometric parallax combined with measurements of multiple colors yields a good estimate of the bolometric flux even if the temperature is poorly known. To look at the situation in simpler, observational terms: if the magnitude at K(2.2/x) is known, the bolometric correction is small, and the luminosity may be estimated. It might therefore seem appropriate to establish the relationship between mass and luminosity near the bottom of the main sequence, in order to determine whether a given candidate is below the SML. We shall see, however, that this is not possible. 21.2.1 The Mass - Luminosity Relation On the hydrogen-burning main sequence, the luminosity is of course a monotonic and steeply-increasing function of mass. Note that a low mass star has a hydrogen-burning lifetime that is much longer than the age of the Galaxy. Hence, the so-called "zero age" main sequence (ZAMS) position at a given mass and chemical composition remains unchanged for time scales of interest to us. The mass below which hydrogen-burning cannot continue indefinitely is near 0.075-0.08 M©for the solar composition. The main sequence luminosity is often expressed as a power law of the mass with the exponent varying from approximately three for massive stars to nearly five below the solar mass. Such a relationship is also valid on the low mass main sequence (with smaller slope) until the approach to the stellar mass limit. The predictions of stellar interiors models have been tested

Liebert: Searches for brown dwarfs

465

empirically by comparing stars with measured masses (and luminosities) in astrometric binary systems. Henry and McCarthy (1993) have added an impressive number of new binary components using their technique of infrared speckle interferometry. Using the absolute K magnitude (M#) which, as we pointed out, is closely related to the log of the luminosity, they found a simple power law fit for the mass, o = -0.166M* + 0.560 which is an excellent fit to the 0.1-1 M©range. The problem comes in extending the fit below 0.1 M©towards the stellar mass limit, where several complications arise. First, the theory predicts that, as the stellar mass limit is approached, the M-L function will steepen radically - that is there will be a much larger decrease in luminosity over a given interval decrease in mass. Second, the pre-main sequence phase prior to the ZAMS lasts longer with decreasing mass, and below 0.1 of solar the objects require over 109 years to reach the ZAMS. Thus, the M-L relation becomes a substantial function of the age of a star. Moreover, below the SML the (substellar) objects never reach the ZAMS. Again, their phase of gravitational contraction brings them slowly through the same luminosities as the dimmest ZAMS stars. Finally, there are the so-called "transition" objects predicted to have masses of 0.07-0.075 M© for solar composition, which undergo limited hydrogen-burning for up to a few Gyr, though this energy release is unable to halt the contraction and growing degeneracy of the core. Nonetheless, before entering the brown dwarf cooling sequence, they may linger for these relatively long times in the luminosity range of the faintest ZAMS objects. Thus, we must know the age of a stellar object at a given low luminosity, before a mass can be assigned to it from its luminosity.

21.2.2 Temperature Estimates from Spectra and Model Atmosphere Fits The effective temperature (Te) estimates for very low mass stars have been very poor up to now and, until recently, based almost entirely on fits of observed colors to blackbodies. Now it is well known that the blackbody shape is a very poor approximation to the energy distributions of these objects. However, there had been relatively little attention to studying the spectra and atmospheres of stars on the M dwarf sequence, the hydrogenburning main sequence stars of lowest mass. This neglect has been due

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Liebert: Searches for brown dwarfs

primarily to the complexity of especially the molecular opacity sources in such cool stars. In recent years there has been impressive progress on some of the most relevant of the molecular band systems, such as CO, TiO and HjO. The two PhD dissertations of Allard (1990) at the University of Heidelberg and Ruan (1991) at the Australian National Observatory have changed this bleak situation. These resulted in the first model atmosphere grids reaching down to temperatures appropriate to the SML, and both demonstrated fair success in matching infrared and optical spectra of M dwarfs. The result of the analysis using the Allard model atmospheres was a set of temperatures for low mass stellar "standards" - well-studied, bright stars in the solar neighborhood. When spectra extending from 0.6 to 1.55 \i were fitted with synthetic spectra from models with solar composition and log g = 5, the first real attempt at defining a temperature scale for low mass stars based on model atmospheres (Kirkpatrick et al. 1993). Now these can be combined with the more accurate luminosities to place the objects in the astronomers' favorite diagram, and compare these with the predictions of theory as a function of mass and chemical composition. 21.2.3 The Hertzsprung-Russell Diagram Fig. 1 is a Hertzsprung-Russell (HR) Diagram, a plot of log L/Z©vs. log Te, in which the stars from Kirkpatrick et al. (1993) are shown asfilledcircles. Examples of the blackbody temperature determinations are shown as open circles in the diagram. Also shown for comparison are theoretical interiors calculations for masses approaching the stellar mass limit of 0.08 M©. It is perhaps not too much of a surprise that the model fits give Te values closer to the predicted locations than the blackbody fits, except for the coolest stars of lowest luminosities. However, the spectroscopic fits are far from perfect over the wavelength range covered. Moreover, Tinney, Mould and Reid (1993) show that Allard models give a poor fit to the spectral energy distributions observed at longer infrared wavelengths. There is clearly much work to be done in refining the temperature and HR diagram determinations for stars of the lowest masses. Also shown in Fig. 1, however, are a few tracks showing the very real evolution in this diagram of substellar objects. These brown dwarfs fall towards the main sequence in their phase of gravitational contraction, and limited nuclear burning for those with "transition" masses. Finally, they enter the cooling sequence at a fixed radius with the onset of degeneracy. During these phases the figure shows how closely their evolution parallels the

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main sequence, although the brown dwarf always reaches a given luminosity in a shorter time. Thus, the ambiguity of the stellar luminosity is not resolved - at least not easily - by using the effective temperature. It is nearly impossible, therefore, to distinguish a brown dwarf fairly near the SML from a stable star, based on position in the HR Diagram alone. The temperature of the brown dwarf is typically only a few hundred degrees lower than the star at a given luminosity. We have already seen that the Te assignable to a low mass star is at least that uncertain. Either of two additional stellar parameters may provide enough information to resolve the ambiguity - that is, determine whether a given low luminosity object is a star or a brown dwarf. These are the mass or the age. We have already mentioned the growing sample of nearby stars where the masses are estimated from the solutions to the binary orbit. Likewise, in young clusters or associations where the age may be known, single objects may be analyzed.

21.3 Searching for Field Stars 21.3.1 Proper Motion Surveys Selection by motion on the sky has been the traditional way of finding the Sun's nearest neighbors. The most important survey to date was that of Luyten (1963) using plates taken with the Palomar 1.2-meter Schmidt telescope at two different epochs generally some 12 years apart; the first epoch was the original Palomar Sky Survey of the early 1950s. The sample one assembles has a kinematic bias - for example, the stars which happen to have the smallest tangential velocities with respect to the Sun might be missed. However, this has been a very efficient way of finding solar neighbors over the entire sky. The nearest, low luminosity stars to the Sun are those amenable to the most accurate followup observations - trigonometric parallaxes, searches for companions, etc. The least luminous of the solar neighbors currently are two stars cataloged in Luyten (1979) as LHS 2924 and 2065, each with absolute visual magnitudes (My) fainter than +19, implying luminosities of a few x 10~*Z/©. Such luminosities place them very close to the bottom of the ZAMS (Fig. 1), but also in the realm of the young brown dwarfs. 21.3.2 Optical and Infrared Color Surveys The red colors of the lowest-mass stellar objects provide a method of selection free of kinematical bias. With the availability of sensitive emulsions,

468

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Liebert: Searches for brown dwarfs

469

(1993) have combined both proper motion and color selection into a more comprehensive attempt at finding low mass stars. Their project includes the measurement of trigonometric parallaxes as well. This sample promises to be the best collection of "field" stars. Already they have identified stellar objects somewhat fainter than the LHS stars mentioned previously. Tinney (1993) argues that the faintest known stars may not be stably supported by nuclear burning. An example of a more specialized color survey is that of Kirkpatrick (1992) using photometric CCD data rather than photographic magnitudes. He presents evidence in agreement with Tinney's (1993) conclusion about the faintest field stars. Finally, we note that one of the coolest known objects of this type - called PC 0025+0447 - was found as part of a survey for highredshift QSOs (Schneider et al. 1991). This object is characterized by a color somewhat redder than the benchmark LHS stars, as well as extremely strong Ha emission. Unfortunately, its luminosity is not measured, nor is the survey characterized to find a complete sample of very red stellar objects which might lack strong emission.

21.4 Searching in Young Clusters The advantages of confining the search to members of a young stellar aggregation are obvious. First, the objects will be young and relatively luminous. Secondly, they should have at least approximately the same ages, so that it will be possible to assign masses based on an estimate of the luminosity alone. Furthermore, all candidates lie at the same known distance - that is, if they really are members of the cluster or group. Finally, there are several interesting aggregations close enough for observation - ranging from molecular clouds forming stars of order 106"7 years to clusters nearly as old at 109 years. The development of large format CCD and infrared arrays has made observation of accurate colors over large regions of a cluster possible. The disadvantages are also formidable. Star-forming regions may be in the galactic plane, so that heavily reddened background stars may be confused with genuinely cool members of an association. Newly formed stars may possess a remnant accretion disk and/or strong chromospheric activity, which may distort the spectral energy distribution from that of a simple photosphere. Furthermore, the youngest groups may have a significant spread in age. All of these problems are generally magnified the younger the aggregation is. Finally, the candidates themselves are relatively far away and faint compared to field objects found in the solar neighborhood or in

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the color surveys, such that the opportunities for followup observations are limited. 21.4.1 Star-Forming, Molecular Clouds The youngest star-forming regions where it might be worth looking for luminous substellar objects are of the order of 106 years old, and range in density from the loose Taurus-Auriga clouds to Rho Ophiuchus, a giant molecular cloud and newly-forming star cluster. Both happen to be approximately 150 pc away. No clear success has been achieved with the former, which is characterized by a treacherous, heavily-reddened stellar background. Rho Ophiuchus differs in having a very high internal extinction of Av > 50 magnitudes. The work has been carried out exclusively in the infrared, especially the K band, using wide-format arrays. An exhaustive review of the work on Rho Oph is beyond the scope of this paper, but I will mention a recent highlight. Comeron, Rieke, Burrows and Rieke (1993) have now completed a survey of 200 square arc minutes to a completeness of K = 15.5. They have found 91 faint sources, all with multiple observations in H(1.6//) and K. They used the color information to estimate the (highly variable) extinction to each object and hence the luminosity. Then, comparison with a theoretical isochrone for an assumed age of (up to) 2 x 106 years yields a unique mass for each luminosity. Perhaps seven objects have indicated masses at or below 0.05M©- well below the SML! The method is simple and straightforward, since the amount of information is quite limited. In order to pursue these candidates further, it may be necessary to have a better understanding of the infrared spectra and energy distributions of low mass stellar and substellar objects. 21.4.2 The Pleiades and Hyades These clusters have attracted the most attention in the search for substellar objects of known age - that is, 6-7 x 107 years for the Pleiades and 6 x 108 for the Hyades. Since the latter is closer to the Sun (44 vs. 125 pc) it turns out that the predicted apparent brightnesses of their respective brown dwarf sequences would be very similar. We have space here to discuss only a few of the many searches in the fields of these clusters. Stauffer et al. (1989) surveyed some 900 sq. arcmin at V and / using a CCD detector. They found several good candidates which are likely to be members based on radial velocities and Ho emission line activity (Stauffer

Liebert: Searches for brown dwarfs

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et al. 1994). However, the first paper concluded that the mass function peaked near 0.2MQ and hence it was unlikely that the cluster had a high density of substellar objects. A deeper CCD survey has been published by Simons and Becklin (1992), which can penetrate well into the brown dwarf luminosity regime. At the moment, the implication of this work is not yet clear to this author. The most comprehensive and complete Pleiades survey is that of Hambly & Jameson (1991) and Hambly, Hawkins and Jameson (1991; HHB), using Schmidt photographic plates at R and / to study a three degree diameter field - the core of the cluster. The second citation above includes proper motion measurements to determine with fairly high probability the stars which are astrometric members of the cluster - of those which have appropriate magnitudes and colors to fit the predicted Pleiades low mass pre-main sequence. To illustrate how this double-selection method works, we show in Fig. 2 the plot of proper motions measured in the Pleiades field by HHB. The vast majority of background stars form a huge "core" near zero velocity. But an excess of stars clearly appears at the Pleiades velocity (lower southeast circle), which fortunately is well offset from zero. Still, some allowance for a background having this same velocity is necessary, and a "control field" (eastern circle) helped HHB estimate that number. Secondly, they plotted the candidates surviving astrometric selection into an I-R / color magnitude diagram (Fig. 3), for both the Pleiades and astrometric control field. The diagonal line is the expected locus of the main sequence at the Pleiades distance. Clearly there is an excess of points in the set with Pleiades motions corresponding to possible pre-main sequence objects above the diagonal line, with very few such points in the control field. Nonetheless, HHB could not expect that 100% of the candidates surviving both tests are actual Pleiades low mass stars and brown dwarfs. Further confirmation can be achieved with radial velocity measurements: Stauffer, Liebert, Giampapa and Hambly (1994, in preparation) indicate that this survey has selected brown dwarf and very low mass stellar members with high efficiency. The Hyades is sufficiently old that the luminosity function (LF) should have separated into distinct lower main-sequence and brown dwarf components. Due to the size or relative looseness of the cluster - and, ironically, its proximity to the Sun - it is difficult to establish membership reliably from proper-motion measurements alone. Moreover, a new astrometric study (Reid 1992) shows evidence for mass segregation; thus the lower-mass objects may be less centrally concentrated and hence even more difficult to identify as cluster members against a difficult background field.

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Leggett and Hawkins (1988, 1989) selected field stars with large R —I colors for followup infrared (JHK) photometry and derived infrared LFs. Again, there was evidence of a peak in the LF near MK ~ +6.7 (or ~ 0 . 2 M Q ) . Bryja et al. (1992) identified several faint, red objects on multiple epochs of red Palomar Sky Survey plates as brown dwarf candidate members. The visual-infrared color measurements were somewhat puzzling, though spectroscopic followup work has strengthened the case for at least a few of these stars. This work is still in progress, as is an / and K band imaging project by Macintosh et al. (1992).

21.5 Discovery and Analysis of Substellar Companions For stellar objects in close binary systems, the masses may be determined directly by analysis of the binary orbits. The observational techniques include direct photography of visual binaries near the Sun, astrometric perturbation analyses, and speckle interferometry especially at infrared wavelengths. But wider pairs are even easier to find and at least the luminosity of a faint companion can be established.

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Fig. 21.3 The color magnitude diagram for the Pleiades candidates surviving astrometric selection (top) and from a control field, as discussed in the text, (from HHB)

21.5.1

Wide Optical

Companions

Photography of thefieldsaround nearby stars has been employed for decades to discover resolved companions sharing the space motion of the primary. Several early "benchmark" stars of low luminosity were found by van Biesbroeck (1961); vB8 and vBlO are among the best studied of these. The latter, at Mv ~ +18.7, was the least luminous known star until the 1980s.

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Fig. 21.4 Infrared images of the 10,000 K white dwarf GD 165 (top) in 7(1.2/i), H(l.6fi), and AT(2.2/J) from left to right, taken from Becklin and Zuckerman (1988).

21.5.2 Infrared Imaging and

Photometry

Continuation of the search for fainter, resolved companions in recent years has employed new infrared array detectors to search at friendlier wavelengths. Skrutskie, Forrest and Shure (1989) found only one new very low luminosity companion in their survey of known, nearby stars, and this Gliese 569 (Forrest, Skrutskie, and Shure 1988) is luminous enough to be a main sequence star. An even deeper survey to K = 15.5 of all northern stars to distances out to 8 pc (G.H and M.J. Rieke, 1992 private communication) found no plausible brown dwarf candidates. The most exciting discoveries by this method are infrared detections around white dwarf stars, by E.E. Becklin and B. Zuckerman. The white dwarf GD 165 has a companion at least 120 a.u. away, with a color temperature of 2100 K and luminosity of 8 x 10~s£Q(Becklin and Zuckerman 1988), substantially cooler and fainter than any well-studied field star. The infrared images (Fig. 4) show a companion star dramatically cooler than its white dwarf primary at a separation of 4.3 arcsec. The companion is not detected at 1.2/x (left frame), but is brighter than the primary at 2.2/x (right). GD 165B also exhibits a very late type spectrum (Kirkpatrick, Henry and Liebert 1993). Nonetheless, various authors have shown that GD 165B may fit tracks of marginally stellar mass. A more puzzling but potentially more decisive case is the unresolved infrared excess of the white dwarf G 29-38 (Zuckerman and Becklin 1988). If due to a brown dwarf companion, as these authors originally suggested, the separation on the plane of the sky cannot exceed several a.u. However, detections of flux longward of 2/x out to 10/x led Telesco, Joy and Sisk (1990)

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Liebert: Searches for brown dwarfs

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Fig. 21.5 Unresolved excess far-infrared emission from G 29-38, with an 800 K blackbody fit. Shortward of 2p thefluxof the white dwarf primary star dominates, (from Tokunaga, Becklin and Zuckerman 1990). and Tokunaga, Becklin and Zuckerman (1990) to attribute the infrared flux to some kind of cooler dust shell around the white dwarf. The far-infrared energy distribution presented in the latter reference is shown in Fig. 5. The fit suggests that the dust has a temperature near 800 K. It was logical to ask how a white dwarf with a cooling age of 109 years could retain such a warm dust shell? However, the age of the white dwarf poses a problem for a brown dwarf interpretation as well, since, if the infrared luminosity were attributed primarily to a companion, it could not be very old. Finally, Barnbaum and Zuckerman (1992) report that G 29-38 is probably a small amplitude (5-10 km s"1) radial velocity variable with a possible period near 11 months; this could be consistent with a substellar

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Liebert: Searches for brown dwarfs

mass companion less than an a.u. away. The study of this fascinating object must continue. 21.5.3 Infrared Speckle Interferometry Henry and McCarthy (1990, 1992) have used two-dimensional speckle interferometric observations to survey a complete sample of nearby stars out to 8 pc for faint companions most easily detectable at 2.2 \i. Multiple observations of the separation and position angle of newly-discovered and previously-known binary systems lead to improved determinations of the masses (and luminosities) of low mass stars and brown dwarf candidates. These serve as the data points for their mass-luminosity function discussed earlier. The Henry and McCarthy (1992) LF declines sharply at MK ~ +10. In most cases, if companions two magnitudes fainter existed, they would have been found over a wide range of separations. The Zuckerman and Becklin (1992) search for companions to white dwarfs produced a very similar LF and sharp decline at the faint end - despite the discovery of GD 165B. 21.5.4 Radial Velocity Surveys A complementary technique to the various imaging approaches for finding unresolved companions to nearby stars is to search for radial velocity variations due to the orbital motion of a visible component. The stars must be bright enough for precise, high-resolution line profiles to be measured. The most comprehensive search to date (Marcy and Benitz 1989) covered 70 low mass M dwarfs, some 80% of allo known single stars later than dM2 and brighter than V = 10.5 accessible from the Northern Hemisphere. Since the brightness constraint requires the stars also to be within 10 pc, there is considerable overlap with the speckle sample, but greater sensitivity to smaller orbital separation. There is also the advantage that the companion need not emit any radiation - it need not be a young brown dwarf. As we shall see shortly, however, the inability to study directly the companion is also a disadvantage. Marcy and Benitz (1989) uncovered only one companion that is possibly substellar - Gliese 623B (Marcy and Moore 1988) - which was discovered independently in the infrared speckle work. Again, this star has a possible mass range (0.067-0.087 M©) that straddles the SML cutoff. Several ongoing studies are sensitive to even smaller velocity variations. Campbell, Walker and Yang (1988) and McMillan (1992, private commu-

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ideation) have relative accuracy near 10 meters per second. The latter continues a multi-year monitoring of 16 bright solar-type stars. The most exciting discovery to date from this method happened somewhat circumstantially. Repeated observations of the G dwarf HD 114762 by Latham et al. (1989) were intended to establish this star as a radial velocity standard under an International Astronomical Union program. Instead, they found that the star is variable with a period of 84 days and an amplitude of 0.55 km s" 1 . The unseen companion has a mass of eleven Jupiters - divided by the sine of the unknown orbital inclination. Thus, if it were viewed less than 8 degrees from pole-on, the companion could still be stellar. A new analysis by Cochran, Hatzes and Hancock (1991) finds an upper limit of only 1 km s" 1 for the projected rotation rate - suggesting indeed that a pole-on orientation is possible. Now the Harvard-Smithsonian group are engaged in a systematic monitoring program, with a target list including 24 nearby M dwarfs (Mazeh et al. 1990). 21.6 Halo stars, brown dwarfs and MACHOS The stellar mass limit for a halo star of 1/100 solar metallicity is close to 0.1M©(D'Antona 1987); from stellar interiors calculations, we would expect such a boundary star to be substantially more luminous and hotter than its counterpart at solar metallicity. Indeed, a Pop II main sequence is observed to "cut off' near My ~ +14, as expected, some five magnitudes brighter than the end of the disk sequence (Monet et al. 1992). The Population II main sequence is up to three magnitudes subluminous in comparison to disk stars of the same color (but much higher mass) - hence the low mass halo stars are called Msubdwarfs. Any existing halo brown dwarfs must be far too low in luminosity to be detectable. Information on the halo LF and mass function at the faint end is sketchy. Several globular clusters have been observed down to near the mass limit using CCD detectors (Richer et al. 1991), without evidence of a flattening or turnover. However, this ground-based work has necessarily focussed on stars in outer regions of the cluster. Yet there is evidence that the dynamical relaxation would cause the lower mass objects to be less centrally concentrated than more massive stars, so that the mass function would be biased to a steeper slope in such studies. A repaired Hubble Space Telescope could perform an unbiased determination of the average cluster mass function. Likewise, Richer and Fahlman (1992) have now attempted a corresponding mass function for the field halo population from one large-format CCD field. There are formidable problems with this kind of study: even if the

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stellar needles can be successfully separated from the extragalactic haystack, it is difficult to estimate their space density because the metallicities and distances are poorly determined. The conclusion of these authors that the halo has a steeply-rising halo mass function accounting for the "missing mass" must be regarded with caution, especially since it appears to conflict with other studies. For example, there was no evidence for such cool stars in the comparable study of Tyson (1988, and private communication). Moreover, proper motion studies should have found very efficiently most representatives of this population in the immediate solar neighborhood, yet the ratio of low mass M stars of Pop II to Pop I does not appear to differ from that for more massive stars (Hartwick et al. 1984; C.C. Dahn and Liebert, unpublished). Nonetheless, the halo mass and luminosity functions remain poorly determined. I suppose that there is also the possibility that virtually ali the unseen halo mass is below the SML. To account for all of our galaxy's alleged massive halo, however, requires a density of something like 0.5 brown dwarf per cubic parsed It is worth asking if there are any other ways of detecting invisible or very faint stellar objects? The proposal of Paczynski (1986) to observe gravitational lensing events of more distant stars by compact, unseen objects in the foreground of our Galaxy generated much excitement as a possible answer to this question. The search for what are commonly called MACHOS - massive, compact, halo objects, a term coined by Kim Griest - has been undertaken by several groups. Since the St. Malo meeting, two groups have reported the first, probable microlensing events of background Large Magellanic Cloud giants by halo objects - the EROS group (Aubourg et al. 1993) and the Livermore group (Alcock et al. 1993). A Polish project tied more directly to Paczynski has also reported a similar detection from the Galactic bulge (Udalski, A. et al. 1993). What are these microlenses likely to be? It is too early to draw conclusions. A perfectly satisfactory explanation at the moment for the joint EROS-Livermore event with an estimated mass in the 0.03-0.3 M@ range, of course, is that this is a low mass star. Time will tell how many such events are found in the well-defined search programs. 21.7 The Bottom Line What is the net result of the application of all of these attempts to find brown dwarfs in the galactic field populations, in young stellar associations, and as companions to nearby stars? First, it has to be said that not a single.

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unambiguous case of a brown dwarf can be pointed to with confidence. Yet, the pessimists must also acknowledge that literally dozens of interesting candidates have been found with this variety of search techniques, and more are being added by the month. The lack of proven examples may be blamed on the great difficulty of establishing a substellar mass. There is a need to sharpen our theoretical tools - through more accurate equations of state and opacities - so that the candidates may be unveiled for what they are. Better theoretical atmosphere, envelope and interior models are needed in order to tell the difference between a brown dwarf and a very low mass star. The author wishes to acknowledge support from the National Science Foundation through grant AST 92-17961. I thank Adam Burrows for many discussions that contributed greatly to this work, and George Rieke and Fernando Comeron for results in advance of publication.

References Alcock, C , et al., Nature, 365, 621, (1993) Allard, F. PhD dissertation, University of Heidelberg, (1990) Aubourg, E. et al., Nature, 365, 623, (1993) Barnbaum, C , and Zuckerman, B., Ap.J.Let, 396, L31, (1992) Becklin, E.E., and Zuckerman, B., Nature, 336, 656, (1988) Bryja, C., Jones, T.J., Humphreys, R.M., Lawrence, G., Pennington, R.L., and Zumach, W. Ap.J.Let,, 388, L23, (1992) Burrows, A. and Liebert, J. Rev. Mod. Phys. 65, 301, (1993) Campbell, B., Walker, G.A.H., and Yang, S., Ap.J., 331, 902, (1988) Cochran, W.D., Hatzes, A.P., and Hancock, T.J., Ap.J.Let, 380, L35, (1991) Comeron, F., Rieke, G.H., Burrows, A., and Rieke, M.J., Ap.J., in press, (1993) D'Antona, F., Ap.J., 320, 653, (1987) Forrest, W.J., Skrutskie, M.F., and Shure, M. Ap.J.Let., 330, L119, (1988) Gilmore, G., and Reid, I.N., Mon.Not.R.A.S., 202, 1025, (1983) Hambly, N.C., Hawkins, M.R.S., and Jameson, R.F., Mon.Not.R.A.S., 253, 1, (1991) Hambly, N.C., and Jameson, R.F. Mon.Not.R.A.S., 249, 137, (1991) Hartwick, F.D.A., Cowley, A.P., and Mould, J.R., Ap.J., 286, 269, (1984) Hawkins, M.R.S., Mon.Not.R.A.S., 223, 845, (1986) Henry, T.J., and McCarthy, Jr., D.W. Ap. J. 350, 334, (1990) Henry, T.J., and McCarthy, Jr., D.W., in Complementary Approaches to Double and Multiple Star Research, Proc. IAU Coll. 135, eds. H.A. McAlister and W.I. Hartkopf, ASP Conf. Series, p. 10 (1992) Henry, T.J., and McCarthy, Jr., D.W., Astron. J. 106, 773, (1993) Kirkpatrick, J.D., PhD dissertation, University of Arizona, (1992) Kirkpatrick, J.D., Henry, T.J., and Liebert, J., Ap.J., 406, 701, (1993) Kirkpatrick, J.D., Kelly, E.M., Rieke, G.H., Liebert, J., Allard, F., and Wehrse, R. Ap.J. 402, 643, (1993) Latham, D.W., Mazeh, T., Stefanik, R.P., Mayor, M., and Burki, G., Nature, 339, 38,(1989)

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Leggett, S.K., and Hawkins, M.R.S., Mon.Not.R.A.S., 234, 1065, (1988) Leggett, S.K., and Hawkins, M.R.S., Mon.NoLR.A.S., 238, 145, (1989) Luyten, W.J., Proper Motion Survey with the 48-inch Schmidt Telescope, No. 1 (University of Minnesota: Minneapolis), (1963) Luyten, W.J., it The LHS Catalogue, (University of Minnesota: Minneapolis), (1979) Macintosh, B., Zuckerman, B., Becklin, E.E., and McLean, I.S., Bull. Am. Astr. Soc, 24, 773, (1992) Marcy, G.W., and Benitz, K.J., Ap.J., 344, 441, (1989) Marcy, G.W., and Moore, D., Ap.J., 241, 961, (1988) Mazeh, T., Latham, D.W., Mathieu, R.D., and Carney, B.W., in Active Close Binaries, ed. C. Ibanoglu, (Kluwer Academic: Dordrecht), p. 145, (1990) Monet, D.G., Dahn, C.C., Vrba, F.J., Harris, H.C., Pier, J.R., Luginbuhl, C.B., and Abies, H.D., Astron.J., 103, 638, (1992) Paczynski, B., Ap.J., 304, 1, (1986) Reid, I.N. 1992, Mon.Not.R.A.S., 257, 257. Richer, H.B., and Fahlman, G.G., Nature, 358, 383, (1992) Richer, H.B., Fahlman, G.G., Buonanno, R., Fusi Pecci, F., Searle, L., and Thompson, I.B., Ap.J., 381, 147, (1991) Ruan, Phd dissertation, Australian National University, (1991) Schneider, D.P., Greenstein, J.L., Schmidt, M., and Gunn, J.E., Astron. J., 102, 1180,(1991) Simons, D.A., and Becklin, E.E., Ap.J., 390, 431, (1992) Skrutskie, M.F., Forrest, W.J., and Shure, M. Astron. J., 98, 1409, (1989) Stauffer, J.R., Hamilton, D., Probst, R., Rieke, G.H., and Mateo, M., Ap.J.Let, 344, L21, (1989) Stauffer,J.R., Giampapa, M., Liebert, J., Hamilton, D., Macintosh, B. and Reid, N., Ap.J., in press, (1994) Telesco, CM., Joy, M., and Sisk, C , Ap.J.Let., 358, L17, (1990) Tinney, C.G., Astron. J., 105, 1169. (1993) Tinney, C.G., Mould, J.R., and Reid, I.N., Astron. J. 105, 1045, (1993) Tinney, C.G., Reid, I.N., and Mould, J.R., Ap.J., 414, 254, (1993) Tokunaga, A.T., Becklin, E.E., and Zuckerman, B., Ap.J.Let., 358, L21, (1990) Tyson, A., Astron.J., 96, 1, (1988) Udalski, A., et al. Ada Astron., in press, (1993) van Biesbroeck, G., Astron. J., 51, 61, (1961) Zuckerman, B., and Becklin, E.E., Nature, Nature, 330, 138, (1988) Zuckerman, B., and Becklin, E.E., Ap.J., 386, 260, (1992)

22 Jovian seismology BENOIT MOSSER Institut d'Astrophysique de Paris, 98bis, bd Arago, 75014 Paris, France ; [email protected]

Abstract This paper reviews a new astrophysical subject: seismology of the giant planets. Seismology is dedicated to the sounding of the interior structure of any object; on the other hand, the interiors of the Jovian planets need to be constrained, in order to improve our knowledge of their structure and of their evolution, as well as the thermodynamical laws involved at high pressures and low temperatures. The relationship between Jovian seismology and, first, Jovian internal structure, and second, high pressure physics, is examined, in order to determine the task of "dioseismology" f in the next years. We present then the seismological theoretical approaches developped since the pionnering work of Vorontsov et ai. (1976), who calculated the frequencies of the Jovian eigenmodes. We report the first observational attempts for the detection of the oscillations of Jupiter. We discuss the observational results and examine what can be done in the future. La sismologie des planetes geantes apparait comme un centre d'interet astrophysique d'avenir. Elle doit permettre en effet - et il s'agit en fait du seul outil dont l'on dispose - de sonder les interieurs de ces planetes, actuellement mal connus, mais dont la determination represente un interet majeur. Cet article recapitule aussi bien les diverses approches theoriques developpees depuis Particle precurseur de Vorontsov et ai. (1976) que les dit This neologism, constructed in the same manner as the substantive helioseismology, should represent the seismology of all four giant, or Jovian, planets. 481

482

Mosser: Jovian seismology

verses experiences menees pour detecter les oscillations de la planete Jupiter. L'accent est mis sur les liens reliant l'etude sismologique des planetes geantes avec d'une part leur structure interne, d'autre part la physique hyperbare gerant les equations d'etat utilisees pour decrire le comportement de l'enveloppe fluide. II apparait que les deux problematiques, que l'on souhaiterait distinctes, s'enchevetrent a l'envi. L'enjeu de la "diosismologie" consiste en parvenir, essentiellement par un developpement dans un premier temps des observations, a demeler l'imbroglio actuel. Des pistes sont proposees, qui permettent d'etablir, a partir de resultats sismologiques, des resultats univoques en terme de structure interne.

22.1 22.1.1

Introduction The interior structure of giant planets

The image we have from any astrophysics! object is two-dimensional. What we see from the giant planets is in fact a very thin layer from where the photons escape or are reflected. In Jupiter for instance, the upper atmosphere is sounded by spectroscopy from a few microbars down to the 10-bar level. Some physical data, as pressure, temperature or density, cannot be measured in the deep interior. However, the values of the mass, the gravitational moments J2, J\ and J&, the rotation period of the planetary core and the luminosity constrain interior models. But the density profiles obtained from these integral quantities are strongly non unique. In addition, the following points must be noted: • The giant planet interiors correspond to pressure and density ranges where the equation of state (EOS) of hydrogen, helium and heavier elements are very far from the perfect gas law (Chabrier, these proceedings). The determination of the pressure-density profiles in the planetary interior would be a unique tool for determining the EOS of a hydrogen-helium gas mixing at very high pressure. • The precise determination of the actual state of the giant planet interior is a clue for their former evolution. • The measurement of the concentrations of helium or other elements in the whole planet and not only in the upper atmosphere, as well as the determination of the structure discontinuities in all four giant planets are key points for planetology.

Mosser: Jovian seismology

483

22.1.2 Seismology Seismology is a very powerful tool for the investigation of the interior structure of any object. A simple seismological experience consists of sounding a wall by knocking it: the sound it makes helps determining its consistency. In the same way, we need to "listen" to the Jovian resonances in order to determine of what the planetary interior is made. Because of its fluid interior, Jupiter looks like a star or the Sun, and the Jovian seismological study is a priori very similar to the one of any spherical fluid object. The acoustic modes (e.g. sound waves) which are favored in a sphere are expressed very crudely by: „

a

\n+-\

I

i/0

2J

(22.1) v

'

where n is the radial order of the mode. The degree I is related to the first index of the spherical harmonics Y™ associated to the mode, and VQ is the acoustic characteristic frequency. The pattern described by Eqt. 22.1 is approximately followed, for example, by the solar modes, the one of a-Cen (Pottasch et ai. 1992) as well as Procyon (Gelly et al. 1986) and the Jovian modes (Schmider et al. 1991, Mosser et al. 1993). The signature of each object appears in the smaller terms which are not expressed by Eqt. 22.1. The calculation, measurement and interpretation of these terms are the task of dioseismology. 22.1.3 Historical review The first paper about the seismology of giant planets is from Vorontsov, Zharkov & Lubimov (1976). The authors present the two basic ideas of dioseismology: the measurements of the oscillation periods is a uniqua tool for investigating the planetary interiors; intensive energetics can lead to the excitation of such oscillations. Low degree and low order pressure modes of Jupiter and Saturn are calculated, as well as discontinuity modes due to the core. This paper was followed by four other articles (Vorontsov & Zharkov 1981; Vorontsov 1981, 1984a and b), which represent a very complete approach of the specific problems of Jovian seismology. Structure discontinuities, oblateness and differential rotation are considered as perturbations of a spherical and continuous state. Bercovici & Schubert (1987) have introduced a more simple approach which is based on the ray tracing theory. They propose some possible excitation mechanisms for the modes, and are the first to give an estimate of the Jovian mode amplitude, expressed by the observable velocity in the troposphere: about 0.5 m.s" 1 . The first

484

Mosser: Jovian seismology

Jovian echelle diagrams (see Table 22.7) are due to Mosser et ai. (1988) and Vorontsov et ai. (1989); they show how strong is the influence of the core on the oscillation pattern. The first attempt for detecting the Jovian oscillations came 13 years after the first theoretical paper. It is due to Deming et ai. (1989). The long delay between the first theoretical development of Jovian seismology and the first observations is surprising, when considering the importance of the subject as well as the efforts which have been made at the same time in helioseismology. It can be understood only because of the real difficulty of the observation. In order to achieve the necessary resolution, the detection has to be made, with a large telescope, continuously over several nights, and therefore needs a stable detector. The first tentative detection was negative. Due to the geometry of the IR detector, only "high" degree modes (£ > 10) with azimuthal order \m\ = l\ were searched. According to the conditions required for the detection of Jovian pulsations (Schmider et ai. 1991, Mosser 1993), it seems that the observation conditions were not favorable.

22.1.4

Summary

A review on planetary seismology by Lognonne & Mosser (1993) has presented general theoretical and observational results on Jovian seismology. In this review, we propose a more detailed approach of the interconnection of the giant planets seismology and their interior structure. The main questions concerning the interior of the giant planets and their seismological consequences are presented in Section 22.2. Seismological calculations and observations of pressure modes are developped in Section 22.3. Finally, the discussion proposed in Section 22.4 gives clues to what can be in the future the efficiency of seismology for disentangling the planetary interiors. We will restrict our attention principally to Jupiter, the only giant planet whose oscillations have been likely detected. There is no conceptual difference between Saturn and Jupiter. The case of Uranus and Neptune is somewhat different, since they have a greater core. Their theoretical oscillations spectra are depicted in Section 22.3. But, due to their low luminosity, it will be a long time before they can be observed. t Such degrees cannot be considered as high in helioseismology, since the Sun appears 40 times greater than Jupiter in the s Icy!

Mosser: Jovian seismology

485

22.2 Giant planets structure The aim of this chapter is twofold. It wants first to establish the link between the construction of giant planets models and high pressure physics; secondly between the models and seismology. In order to examine these two points, we will focus on the method currently used for the construction of the giant planets interior models.

22.2.1

Current

constraints

22.2.1.1 The gravitational moments The current parameters constraining the structure are the gravitational moments J2n (Table 22.1). These parameters express the decomposition of the non spherical gravitational potential of the planet. Jo is simply the mass of the planet. J2, J4 and Js of Jupiter have been measured by the Voyager spacecrafts (Campbell & Synnot 1985). J2 indicates the presence of a more dense core, which is supposed to be made of heavy materials, ices and rocks (Hubbard & Marley 1989). The uncertainty on J& is too high to permit any constraint. 22.2.1.2 Convection and composition The internal flux radiated by Jupiter is supposed to be transported everywhere in the planet by convection (Hubbard 1968). The radiative opacity of hydrogen and helium is too high to allow energy to be carried out by radiation. The presence of convection implies that the planet is adiabatically stratified (in fact, a very little of superadiabaticity is needed to evacuate the internal energy, but at such a low level that it is usually neglected). Jovian models are then supposed to be fully adiabatic. Since an adiabatic temperature profile has been measured under the 1-bar pressure level by Voyager 2 (Lindal et ad. 1981), the adiabat starts at the 1-bar pressure level, with a tropospheric composition as given by the observations (Gautier & Owen 1989), and a temperature of 165 K. Models follow the same adiabat from the 1-bar level to the center of the planet. The gradient in the core is surely not adiabatic, since the planetary flux goes to zero at the center. Furthermore, the adiabat surely changes with the composition gradients. However, the EOS of the heavy materials used in the core, extraplolated from lower pressure theoretical estimations (Hubbard & Marley 1989) are also assumed to be adiabatic.

486

Mosser: Jovian seismology

1 12 -

-0.5

0.0 log p (g/cm )

Fig 22.1 Pressure-density relation along a pure hydrogen adiabats (from Chabner et al. 1992, Fig. 2). Differences between the EOS (solid curvebaumon et al. EOS with plasma phase transition (PPT), dashed curvewithout PPT (interploation), dot-dashed: Marley & Hubbard EOS) are very small, except in the vicinity of the PPT.

22.2.1.3 The plasma phase transition (PPT) The influence of the plasma phase transition of hydrogen (PPT) on the Jovian structure has been presented by Chabrier et al. (1992) and Saumon et al. (1992). The unique role of the PPT in the Jovian thermal balance, if really a first order transition, must be emphasized. Even if, according to Saumon et ai. (1992), the actual latent heat release contributes to only about 1% of the radiated power, the hypothetical transformation of all the metallic hydrogen into molecular hydrogen should provide the actual Jovian flux during not less than 1010 years! The role of the latent heat in the planetary evolution has to be taken very carefully into account.

22.2.1.4 Saturn, Uranus and Neptune Standard models of Saturn do not differ qualitatively from Jovian models Because of the lower mass, the PPT occurs deeper in the planet. The low helium abundance measured in the troposphere is explained by the unmiscibiUty of helium in the metallic region, hence its depletion in the envelope. The pressure in the fluid envelope of Uranus and Neptune is not high enough to permit PPT. These two giant planets present a large ice shell surrounding a rock core, and a thin fluid envelope (Fig. 22.2, Table 22.3).

Mosser: Jovian seismology

22.2.2

487

Construction of the interior models

The models take into account the oblate shape of the planet (see Table 22.1) due to the high rotation velocity. To be consistent, the calculations must be two-dimensionally developped. Variables are functions of the inner radius r and of the colatitude 9 (axisymmetry is assumed). The theory offigures,as exposed in Zharkov & Trubitsyn (1978), shows how it is possible to reduce, through some coefficients describing the oblate shape of the planet, the twodimensional dependence to a pure radial dependence. All models assume a solid body rotation, with a rotation period equal to that of the magnetic field (system III). Then, two equations govern the evolution of density and pressure: hydrostatic equilibrium : EOS = adiabat :

— = —p g dr p = p(p)

(22.2)

The gravitational field g expresses simply in function of the density: g = 4wGfQpu2du/r2. The hydrostatic equilibrium equation carries no essential information, but the interior is fluid. What governs the model is the EOS which is used. The Jovian adiabat results p(p) from the ideal additive volume law: 1

Y y Z 773- + p(p)z THT P(p)x + p(p)y

(22-3)

where X, Y and Z are respectively the mass fraction of hydrogen, helium and heavier elements; p(p)i is the EOS of species i following the planetary adiabat. Y and Z in the outer envelope are in agreement with the observational results, whereas their mean value is inferred from the solar composition (YjUp =YQ, enrichment in Z, cf. Table 22.2). This implies a composition discontinuity, whose exact location remains undetermined (Gudkova et al. 1989, Zharkov & Gudkova 1991, 1992). Fig. 22.1 represents three pure hydrogen adiabats, which have been used for the construction of Jovian standard models (Fig. 22.2). It can be remarked that the difference in the pressure-density relations (as high as 20 % at the location of the PPT) leads to difference in the density profiles (Fig. 22.3) less than 2%, except at the core level. The main characteristics of a Jovian standard model (Table 22.3) are finally: • the rotation and the resulting oblateness of the planet. • the core, with a discontinuity constrast of about 4 with respect to the fluid envelope.

488

Mosser: Jovian seismology

Table 22.1. Jupiter: primary constraints Mass Equatorial radius (1 bar) Oblateness Rotation period

h

u

1.899xl027kg 71492±4 km

6.48% 9h55min33s

} xlO

6

h

14697± 1 1± 5 &84± 5 31±20

Table 22.2. Jupiter: composition and structure envelope PPT core

Y ~ 0.20 and Z ~ 0.02 ; adiabatic gradient transition H2-Hmetai around the 1.2Mbar level rocks and ices, mixed or separated

Table 22.3. Giant planet models Model JUPl JUP2 JUP4 JUP5 JUP6 JUP7, JUP8

SATl URAl NEPl

Authors

Core

Hubbard & Marley 1989

mixed ice and rock

Chabrier & Saumon 1992 rock core + ice shell

Envelope

AZ^O no PPT PPT PPT, AY^O

Gudkova & Zharkov 1989 mixed ice and rock Hubbard & Marley 1989 Hubbard & Marley 1989 Podolak 1991

rock core + ice shell rock core + ice shell

• the transition from molecular to metallic hydrogen, occuring around the 1.2 Mbar level in the Jovian interior. • convection everywhere in the planet. Finally, in the standard model frame, the use of a given EOS, associated with a given set of secondary hypothesis, corresponds to a given interior model (Table 22.3). Uniqueness of the model is not insured, so that new constraints, stronger than the gravitational constraints, are needed.

Mosser: Jovian seismology

489

JuplUr

-2.10* K - 5 0 Mbor

Uraaui

- 0 0 8 RV

Ncptuao

Saturn

1 bor

1 %

1 Mbor

0.79 I

H, + H.

IIP S Mbor 12.1CTK 12 Mbor

0.25

0.13 R.

Fig. 22.2 Standard model of the four giant planets. Models of Jupiter and Saturn have three layers: the core, and the two fluid envelopes. The pressure in the fluid envelope of Uranus and Neptune is not high enough to permit the transition to metallic hydrogen. The greatest density contrast is at the rock core for Jupiter and Saturn, and the ice core for Uranus and Neptune.

Fig. 22.3 Comparison of the density profile of current Jovian models (see Table 22.3). Differences in the fluid hydrogen-helium envelope are not perceptible, but the core structures are very different.

490

Mosser: Jovian seismology

Fig. 22.4 Comparison of the sound speed profile of current Jovian models (see Table 22.3). Differences in the fluid hydrogen-helium envelope are as high as 10%, despite the similarity between the density profiles.

22.2.3 iFrom the interior structure to seismology An important parameter for studying pressure oscillations is the sound speed. Its expression is easily derived from the interior structure: 1 def / Op

dpJs

adiabaticity

(22.4)

where p and p represent the pressure and density perturbation of the wave and po and po the non perturbed terms. The definition of the sound speed - a second order derivative of the free energy - first shows the capability of seismology to distinguish between different EOS. In fact, even if different adiabatic EOS used for Jupiter give very similar density profiles in the fluid envelope (Fig. 22.3), they lead to sound speed profiles that differ by about 10% (Fig. 22.4). Secondly, the calculation of the sound speed puts in evidence the importance of the hypothesis of adiabaticity. A non adiabatic gradient would lead to a different sound speed profile. Finally, one must emphasize the influence of the core. The high density constrast at the core frontier induces a strong sound speed contrast, which drastically affects the oscillations pattern (Section 22.3.). Furthermore, according to the manner the models are constructed, the core plays a very important role: its mass, even if very small in current models (about 1% of the mass of Jupiter) is adjusted so that the mass of the model fits the planetary mass. Therefore, the core mass and size - and therefore its seismological signature - depend crucially on the EOS used in the fluid envelope.

Mosser: Jovian seismology

491

22.3 Giant planets seismology The oscillations of the giant planets correspond to perturbations of the pressure, density and gravitational potential. What perturbs the static equilibrium state will be examined in subsection 3.4. We first present the linear analysis whose object is the determination of the eigenfrequency pattern. The evolution of the velocity field of the oscillations and of the perturbed pressure, density and gravity terms associated to the oscillations is governed by the following equations (the analytical expressions and the complete resolution are given in Unno et al. 1979): • the equation of the movement, which exhibits three restoring forces: pressure perturbation, buoyancy, and perturbation of the gravitational potential. The Cowling approximation considers that the last term is negligible. The pressure term dominates for sound waves. • the equation of continuity, which expresses the mass conservation. • the Poisson equation (in fact useless when neglecting the perturbation of the gravitational potential). • the equation governing the evolution of the perturbation. Since the perturbation evolves much more rapidly than the characteristic evolution time of the unperturbed state, the waves propagate adiabatically. The most simple solution is the ray tracing theory, which only considers the propagation of the wave vector. It is sustained by the assumption that the propagative wave is a pure plane wave. Mosser et a/. (1988) use the the helioseismologic analysis proposed by Gough (1986) with this method, but include the structure discontinuity of Jupiter. The theory is valid for the determination of Eqt. 22.1, but inadequate for a further description of the eigenfrequencies pattern. In the following, we will focus on the asymptotic method, which proved to be powerful for the understanding of the planetary oscillations (Provost et al. 1993), as well as on the numerical calculations developped for precisely taking into account all the features of the Jovian interior structure, namely the core, the PPT and the rapid rotation. 22.3.1 Asymptotic approach An asymptotic method for calculating the Jovian oscillation spectrum has been proposed by Provost et al. (1993). It follows the asymptotic development of Tassoul (1980), but includes the discontinuity of the Jovian core. The exact eigensolutions describing the movements are asymptotically developped in Bessel's functions to the second order in frequency, near the surface (where the evolution is dominated by the adiabatic index n e of the upper atmosphere) and near the center (where the evolution is dominated

492

Mosser: Jovian seismology

by the degree t of the mode), and finally connected by the discontinuity. The main advantage of this analytical procedure is to obtain seismological parameters which can be related to the internal structure parameters. On the other hand, the method does not permit a precise description of more than one discontinuity. 22.3.1.1 The modulation due to the core Asymptotic eigenfrequencies accounting for the discontinuity in the sound speed profile are given by:

e2N-2 - — ^

e . - -smanit

. sin2an>/

with L2 = £(£ + 1) and

The first part of Eqt. 22.5 reproduces the solar asymptotic expression, while the second expresses the influence of the core. • The characteristic frequency UQ measures the travel time of the sound along a planetary diameter: *b =

2 / -

(22.7)

R is the planetary radius, commonly defined at the 1-bar pressure level. As shown by Mosser (1990), the upper level which should be considered when integrating VQ is the tropopause. Eqt. 223 recapitulates the different steps for the exact calculation of UQ, taking into account the tropospheric contribution and also the effect of the oblate planetary shape: Interior

•*-

Troposphere /•tropo P Buse ( J r \1-l

L

T)

Rotation

X

r

g-|

he

(228)

The cavity where the modes propagate ends at the tropopause (a level which is not included in current interior models). The contribution of the planetary oblateness e is due to the fact that the planetary seismological mean radius R(l — e/3) differs from the geometrical mean radius R(l — 2e/9). The modes favor in fact the equatorial regions (Mosser 1992).

Mosser: Jovian seismology

493

• The second order characteristic frequency is also a simple function of the sound speed, accounting for all the variations of c:

Vi = fRJO

(22.9) T

whereas the other V{ are complex integrals of interior parameters. • The core modulation has a period N and a relative amplitude e. The integer N measures the acoustic radius of the core compared to the whole planetary acoustic radius:

* - / T/I T •/planet

c

/

./core

c

< 2210 >

• e measures the discontinuity at the core frontier (Table 22.4). Finally, the pertinence of the seismological parameters shown by the previous equations must be noted (Provost et a/. 1993). Their determination, extrapolated from a non-asymptotic but numerical oscillation pattern, agrees with their theoretical values. 22.3.1.2 The rotation The comparison of Jupiter and the Sun shows the importance of rotation for Jovian seismology. Both objects present similar freefall frequencies, but their rotational frequencies are respectively 28.2 and 0.4/xHz. The removal of degeneracy due to the Jovian rotation cannot be neglected. Mosser (1990) describes the four different effects of rotation: the non-galilean planetary referential, the oblate shape of the planet, the Coriolis and centrifugal forces. The two last terms become negligible for high overtones (vn UQ). But the influence of rotation is severe (Fig. 22.6). The non-degenerated eigenfrequencies vn,e,m express as a function of the degenerated frequencies vn,i (Mosser 1990): »n,t,m = Vn,l [l + £(m 2 )] - 771 l/rot

(22.11)

where the azimuthal order m varies from —I to I (m is the second index of the spherical harmonics Yf1 associated with the mode). E(m2) is a function related to the oblateness and with the same order of magnitude. 22.3.1.3 Oscillation spectra of the giant planets The asymptotic data which describe the modulation due to the core are given in Table 22.5. What differs principally between Jupiter and Saturn on one side and Uranus and Neptune on the other side is the period N. Jupiter and Saturn have a small rock core, which implies a large JV, whereas Uranus

494

Mosser: Jovian seismology

N

0

0.5 n=13

'.

*

* *° o *

o

m

1

a

*

•

o

o

°

*• *

0

m

°n-2

o

m

• a c? o o a* o a* * o a*

o

'

*

*

i

0

0.5

1

1

0

•

*

*

"•

i•

• in

,

o

o

0

*

1

aa a o a a a a

*o

N

x

d

*

o * *

o * «

2 a)

(73

,

1

•

:

°

n= 10

o

a

CO

a a)

. -

X

•

E

u

°n- 2 °

i

2 m '

0.5

1

i

n-10 o * o o* o o*

• *

•

*

o

1

*

a a

* _

a a a

n •O

• O

i

-

CM

a

•

n-2 o

Lune

m

0

°

as

:

1

0.5

Fig. 22.5 Asymptotic echelle diagrams for giant planets p-modes with degree t = 0 —• 3: v/v0 is plotted as a function of the reduced frequency hv = u/vQ - [n + int(£/2) + n e /2 + 1/4]. The modulation is function of the frequency, namely of the radial order n: its period is about N. The amplitude of the modulation is related to the coefficient t. For each planet, the radial order n varies from 2 to the maximal order of the trapped modes (cf. Table 22.5). (£ = 0: o; 1= 1: n ; £ = 2: • ; £ = 3: *)

Mosser: Jovian seismology

495

Table 22.4. Jovian sound speed discontinuities Discontinuity e Core 0.13 — 0.44 PPT 0 — 0.05

-0.5

Ac/r (mrad.s"1) 3 - 0 . 1 — 0.1

0 0.5 1 f0 - |n+int(je/2)+n#/2+6/4l

o 1.5

Fig. 22.6 Jovian echelle diagram including rotation, according to the perturbation theory (Mosser 1990). Only modes with (t + m) even are shown. (* = 0: O ; / = 1 : D ; / = 2 :

« ; / = 3: *)

and Neptune have a large ice shell and N ~ 2. Table 22.6 summarizes the various frequencies governing the qualitative aspect of the planetary oscillation spectrum. • up = \JGMlR3/2ir measures the mean density of the planet and varies as i/o- The ratios vp/vo are similar for the four giant planets. • i/rot: the rotation frequency, compared to VQ, measures the complexity of the spectrum due to the rotation. For Saturn, 1 — 1 modes already overlap (^n,2,2 — ^n-1,2,-2); this overlap occurs for I = 3 Jovian modes and I = 5 for Uranus et Neptune, as indicated by the ratio i/o/2frof • vc; the cutoff frequency at the tropopause represents the highest possible eigenfrequency. It is related mainly to the temperature. The ratio i/e/i/o gives an estimate of the number of modes effectively trapped; modes with n greater than this ratio cannot be reflected at the tropopause level. Finally, the echelle diagrams of all four giant planets are presented on Fig. 22.5. The echelle diagram representation exhibits the meaningful difference between the eigenfrequencies developped to the second order in frequency and Eqt. 22.1. The method to build the echelle diagram is explained

496

Mosser: Jovian seismology

Table 22.5. Asymptotic seismological data Planet

V7

N (/iHz)

Jupiter Saturn Uranus Neptune

155 111 164 195

V> =

12.6 10.0 2.0 2.2

0.40 0.1 0.41 -0.3 0.21 -0.4 0.13 -0.4

2.3 1.5 0.1 1.9

-0.8 -0.7 -0.1 -0.1

-1.3 -0.3 0.6 -3.9

Table 22.6. Characteristic frequencies Planet

vp

j/ 0 i/rot

ve

vo/vp

. . . . ( / i H z ) . . . . (mHz)

Jupiter Saturn Uranus Neptune

99 70 94 105

155 111 165 190

28 26 16 18

3.0 1.6 1.8 2.2

1.59 1.59 1.74 1.80

2.8 2.1 5.1 5.3

18 13 10 10

the Sun

99

136 0.4

7.0

1.37

165

50

Table 22.7. Principle of the echelle diagram frequencies

eigenfrequency v equ

asymptotic relation

equidistance Ai> ~ 6u\

= P

I

i echelle diagram

vertical axis

P€1N

horizontal axis

in Table 22.7f. The echelle diagram of Fig. 22.6 includes the rotational removal of degeneracy. 22.3.2 Numerical approach Different numerical codes have been developped for the calculations of Jovian pressure modes (p-modes), fundamental modes (f-modes) or surface modes (these one have significant amplitudes in the vicinity of the discontinuities). Numerical calculations (Fig. 22.7, 22.8, 22.9 and 22.10) can explore the whole [n, t] domain, contrary to asymptotic calculations. The precision is limited by the absence of the tropospheric contribution, except for the very low frequency modes (<1 mHz) which are trapped in the deep troposphere. t The echelle diagram exhibits the small frequency term Su, which carries the essential information of the eigenfrequency u = pi>o + Sv (p is an integer).

497

Mosser: Jovian seismology

8

/"c

- |n+lnt(V2)l

Fig. 22.7 Jovian numerical echelle diagram: same as Fig. 22.5, but using numerically computed eigenfrequencies (Provost et a/. 1993). (I = 0: o; / = 1 : n;£ = 2: » ; * = 3: *)

mo MO 2020

-tO

-20

0

20

60

80

6v

Fig. 22.8 Jovian numerical echelle diagram according to Vorontsov (from Vorontsov et a/. 1989, Fig. 5). This diagram represents the same low degree modes as the one of Fig. 22.7. Differences between the two figures are mainly related to differences between the interior models.

22.3.2.1 Normal mode theory Most of the approaches (Vorontsov et ai. 1989, Marley 1991, Provost et aJ. 1993) are derived from helioseismology, following the normal mode theory, but including density and sound speed discontinuities. Rotation (and differential rotation) are considered as perturbations (Vorontsov 1981,1984a and b, Vorontsov & Zharkov 1981). On the other hand, Lee (1993) introduces the rotation at the zeroth order of the calculations. Because of rotation,

498

Mosser: Jovian seismology 1.0

a

tc ZJ

1

-[

a • &

a a ra > a

A

a

A A A A

u a IX)

A A A

o

0J -

D O O A

1—

a o a o a o a

« «

A A

A A A A A

<;

qa ID A

DO O

A

a o

a

O

OA

©

o o

O "

o -

o o CO"

D DO

o

o a

A A

o o "

© © « 0 © © © 0 © © ©

A

0 \> a >D

o

0.0 -

i

A

0

a o

0

—l

T"

o

© Q © © O © © © © « © © «

I\ A A A A

o

o

A -

a ©

OJ

1 0.05

i

1

|

i

1

0.10

015

020

0.25

0J0

6v

(mHz)

Fig. 22.9 Jovian numerical echelle diagram calculated by Lee (from Lee 1993, Fig. 4). Frequencies are given in the Jovian corotating frame, for m = 0, even low degree p-modes. The couplage between the different degrees precludes the intersection of the modes, contrary to the perturbation approach. (I = 0: o; I = 2: ©; I = 4: A ; / = 6: D) the modes are no longer described by an only spherical harmonics, but by a series of such terms. The mode described by the numbers n, I and m is developped in terms of the spherical harmonics Y™, *7±2> *7±4 ••• ( t n e index m remains the same because of axisymmetry). The main result of the couplage between the modes is that, contrary to perturbation results, crossings between modes of different degrees are avoided (Fig. 22.9). However, the principal property of the rotational splitting remains true = 2m

(22.12)

22.3.2.2 The Saturnian ring system as a seismometer Marley (1991) focusses his interest on low frequency Saturnian f-modes. These low frequency oscillations perturb the gravitational potential of the planet and can open gaps in the rings. Therefore, the Saturnian ring system acts as a seismometer. The great advantage of ring seismometry consists in the possibility of detecting very low frequencies £ modes, which are not detectable with a ground based detector. However, such a detection can only be applied to Saturn! An important observational result reported by Marley is the fact that low frequency Saturnian modes do not open large unexplained gaps in the ring system, which implies a surface amplitude lower than about 1 meter for very low frequency modes (a few tens of /iHz).

499

Mosser: Jovian seismology

Oscillation period (min) Fig. 22.10 Saturnian low frequency £ = m f-modes (from Marley 1991, Fig. 10). £modes are modes without radial nodes. For each degree £, the different dashes correspond to different Saturnian interior models. As for Jupiter, the differences between interior models express by huge differences in the eigenfrequency pattern.

22.3.3 Observations 22.3.3.1 Three observations Three positive observation runs of Jovian oscillations have been conducted in 1987, 1990 and 1991, using two different seismometric techniques. The principle of the detection is as follows. The oscillations induce in the upper troposphere a vertical velocity field (Fig. 22.11). The spectral lines, reflected or formed at these levels, are Doppler shifted. The Doppler shift of the solar sodium line reflected by the planet has been observed with a sodium resonance cell (Schmider et ai. 1991), and the Fourier transform spectrometry method has analysed the Doppler shift in the interferogram of the Jovian methane lines at 1.1 /xm (Mosser et ai. 1993). The Doppler signal is recorded over several consecutive nights. The resulting temporal series is cleaned and its Fourier transform is calculated in order to search for the planetary eigenfrequencies. 22.3.3.2 Analysis The most obvious signature which appears in the Fourier spectra is the non continuity of the observations. Observations conducted in a single

Mosser: Jovian seismology

500

2.1

MM

•m 3.2

4.2

Ifl

Fig. 22.11 Oscillation velocity field in the upper troposphere, corresponding to the projection of the spherical harmonics Y™ on the view axis. site are limited to about 8 hours. The resulting effect (Fig. 22.12) on the oscillation pattern is desastrous, as seen on a theoretical oscillation spectrum (Fig. 22.13). The very high mode density precludes any treatment of the window effect; the solution which consists of separately considering the nights to obtain a continuous series is also inoperative, since the resolution after one single night is not sufficient to resolve the modes. 22.3.3.3 Detection of the oscillations A commonly used criterion for the detection of oscillations (Gelly et ai. 1986, Pottasch et ai. 1992) is the building of an echelle diagram which should show regular patterns aligned vertically. This criterion cannot be extensively used for Jovian modes, since the modulation excludes vertical alignments in an entire echelle diagram (Provost et ai. 1993). In fact, the signature of the oscillations has been given by the signature of the rotation. The rotation affects the signal in two ways: first, it modulates the photometric signal, when some atmospheric features pass through the field of view; secondly, it removes the degeneracy of the modes, with the relation indicated by Eqt. 22.11. The first effect cannot be responsible for the signal at frequencies much higher than the rotational frequency (Schmider et ai. 1991). The second rotational signature is a consequence of Eqt. 22.12. The signature of the rotational removal of degeneracy is represented in the

501

Mosser: Jovian seismology

-40

Frequency

G*Hz)

o

Time

(night)

Fig. 22.12 Fourier spectrum of the window function of the observation made at the CFHT in 1990 with the Fourier transform spectrometer (below: the shaded zone corresponds to effective observation). The diurnal signature appears in the spectrum as aliases at ±11.6/iHz. The duration of the observation determines the sharpness of the peaks in the Fourier spectrum. The extension of the spectrum is given by the duty cycle: less than 8 hours daily observation imply the dilution of one peak of the spectrum over about 50/iHz, with only 1/4 of the power in the central peak.

m

•

6

•d

1= I cqd

,1,

1, ,1. JlL

li.

d d d

o 500

1... LhJ. Him 1000

1500

2000

2500

Frequency G*Hz)

Fig. 22.13 Theoretical spectra, as continuously observed, or with a window of 8 hours observation per night. Amplitudes have been arbitrarily chosen: Gaussian envelopes; cutoff above the cutoff frequency at the tropopause; decreasing amplitude with increasing £ value; same amplitude for multiplets

Fourier spectrum of the observed Fourier spectrum by half of the rotation period of Jupiter, and by overtones (Fig. 22.14). Finally, this second effect

502

Mosser: Jovian seismology

1

7A 7 I

CO

• 10

Times (day)

Fig. 22.14 Fourier spectrum of the Fourier spectrum observed in 1991 at the CFH telescope. The signature of the removal of degeneracy due to rotation appears at periods multiple of half the Jovian period of rotation. The empty zones correspond to the absence of observations during daytime. 1

.

.

• •• 77 "" 0HP87 • ». J'J ir" 1 41 II1 I 11 IIi m 1IBWUTH T

i,

'

_

s

I31 1

1

ill1 J:ir III II llll

4 1 1 • 3

4 1

i 1 1 lid l i • 1 a iWli.ll til III

LfUllJl 500

i:

i

1 1 Lit • -

i L • 1

1000

••

0 0

RIJ J i l v 'oiH4

is 0 Illi ^ '

lill 1 M l i Ah?|"0*-0 1500

2000

17

0

0

A.

2500

CFH90

500

1000

1500

2000

2500

Frequency G*Hz)

Fig. 22.15 Power spectra obtained in 1987 at the OHP, with the sodium cell resonance technique (Schmider et ad. 1991), and at the CFHT in 1990, with the Fourier Transform Spectrometer (Mosser et sd. 1993). A possible identification of the modes [n, (, m] is proposed (± is for m = ±1).

has proved the detection of a propagative signal. Because the other possible propagative signals - tropospheric or stratospheric waves - have very low frequencies ( < 1 mHz), the signal has been identified with Jovian global oscillations.

Mosser: Jovian seismology

503

22.3.4 Observational results The Jovian oscillation spectra are presented on Fig. 22.15. As reported in Mosser et ai. (1991, 1993) the unambiguous identification of all individual eigenfrequencies is not possible. However, a plausible identification may be proposed, that agrees for all three spectra, and is based on the identification of the signature of the rotational removal of degeneracy, namely the [£ = 1 or 2, m = ±1] doublets. 22.3.4-1 Discrepancies In fact, what is more interesting is the extrapolation from the three oscillations patterns of some characteristic asymptotic parameters. We will focus on the characteristic frequency VQ, revealed by a possible equidistance frequency Av. The theoretical value of VQ calculated from current interior models lies, according to Eqt. 22.8, between 155 and 160 //Hz. The observed value of the equidistance Av is about 136 //Hz. It can be related to a characteristic frequency VQ in the frequency range [Av(l — 2e/N), Af(l + 2e/N)]. Typically, Av and VQ may differ from about 10//Hz. At least, the discrepancy between the theoretical and observed values of VQ is about 10 //Hz. This discrepancy should have a manifold explanation. It can be related to problems encountered in physics (the determination of the EOS), or in seismology (the calculation of the sound speed and of vo), or in planetology (the validity of the hypothesis of the standard model: adiabaticity and homogeneous composition). • EOS: The relationship between the sound speed c and the EOS has already been emphasized. The precision on c largely depends on the one of the EOS (1% inaccuracy on the EOS leads to a much bigger inaccuracy on c). If accuracy is now insured for the EOS of molecular and metallic hydrogen, that is not the case for helium and other heavier elements. • Calculation of the sound speed: The calculation of c supposes the structure to be adiabatic in the whole planet, what is only an approximation. The influence of the superadiabaticity needed for evacuating the internal flux is limited, but still exists. Furthermore, the continuity of the adiabat from the 1-bar level to the deep interior supposes no compositional gradient and convection everywhere. Any change in the value of the adiabat would substantially modify the temperature profile at all the deeper levels, and then modify significantly the sound speed profile. • Composition gradient: A composition gradient must exist somewhere in the planet to reconcile the observed abundances in the outer envelope with the values assumed to occur in the metallic region on the basis of cosmogonical arguments. Such a gradient modifies inevitably the adiabat.

504

Mosser: Jovian seismology

• Radiative gradient: As calculated in Guillot et ai. (1993), a radiative window should exist in Jupiter near the 3000 K level. This window would change the entire planetary temperature profile, conducting to lower inner temperature, and maybe to lower sound speeds. • Inhomogeneities: Let us consider in Jupiter a layer dr, composed of hydrogen and heavier elements, with volumic fractions respectively (1—a) et a. The acoustic radius of the layer 6r measures the inverse of the mean sound speed c (dr = dr/c) and depends on the constitution: [ Homogeneous mixing: H

I 2 distinct levels:

dr2

^

1+

(22.13)

dr

Cz

/.

Since the sound speed in hydrogen, CH, is much higher than the one in the heavier elements, c z , the acoustic radius in case of heterogeneity is larger, which implies a lower characteristic frequency. • Perturbation of the adiabat by clouds: The planetary adiabat is for many elements over the critical point, implying no phase separation. However, the presence of water clouds introduces a discontinuity in the adiabat, and leads to lower interior temperatures, and subsequently to smaller sound speed. Finally, the hypotheses supporting the standard model should be revised. 22.3.4-2 Excitation mechanism The velocity of the observed low degree modes has been estimated about a few m.s" 1 at the 0.5-bar pressure level. The dependence of the velocity of the wave with altitude is very important (Mosser et a/. 1992). A velocity of about 1 m.s" 1 at the 0.5-bar level corresponds to only a few mm.s" 1 at the PPT level. Depending on the frequency u of the mode compared with the cutoff frequency, this velocity v corresponds to the amplitude a of an isobar level (Mosser, in preparation): a =

PQCV

, or a = v/2v u

(22.14)

That leads to an amplitude of the 1-bar level of about 100 m for v=l m.s" 1 . The excitation mechanism proposed by Goldreich et aJ. (1988) for the solar oscillations gives an order of magnitude of the expected Jovian velocity, which is too low by about a factor 100 or 1000 when considering the convection in the upper Jovian atmosphere (parallely to what is done for the Sun). Then, it seems inadequate for Jupiter, and no other mechanism is able to quantitatively explain the high amplitudes detected. On the other hand, Mosser (1991) has shown that:

Mosser: Jovian seismology

505

• The estimated energy in the oscillations is compatible with the flux radiated by the planetf. • The PPT is a possible region of excitation of the modes. Since the entropy shift per proton at the PPT is very high (about k^/2 per proton), the PPT, if it is a first order transition, acts as an impermeable surface. Surface waves can then develop. They have exponentially decreasing amplitudes and are therefore not detectable, but can be coupled with sound waves. • The couplage is efficient, because it occurs between phenomena with similar periods, and the excitation too, because it appears in the deep planet, contrary to convection, which is too slow in the deep region or inefficient in the upper atmosphere. 22.4 Discussion 22.4-1 The seismological test Helioseismology was an unknown solar field 30 years ago, but has now provided measurements which are among the most precise in astrophysics, and have permitted to reconstruct the solar interior profile (ChristensenDalsgaard et ai. 1985). Can we expect the same improvements for the giant planets? Bearing in mind that the precise measurement of many eigenfrequencies will not be obtained rapidly, we focus the discussion on global asymptotic criteria. Pure hydrogen adiabats, as well as density and sound speed profiles resulting from the different EOS have already been compared, showing how the seismological parameter c is sensitive to the EOS. Measuring the sound speed profile would require what we have exluded, the measurement of numerous eigenfrequencies. However, as shown by Eqt. 22.7 and Eqt. 22.9, most of the asymptotic parameters are related to the sound speed and may be translated into interior structure parameters. 22.4.2

The signature of the PPT

224-2.1

Low degree modes

In current models, the influence of the PPT is hidden by that of the core. This is not only due to the fact that the core discontinuity is much stronger; this is also conceptually related to the way the models are made. Any f This estimation is problematic, since it needs the determination of a characteristic damping time of the oscillations! We can guess that this time is not short according to the following informations: the viscosity of molecular hydrogen is very low; there is no radiative zone in Jupiter, and therefore no radiative viscosity; the quality factor of the planet is very high.

506

Mosser: Jovian seismology

change of the EOS (for example, with or without PPT) leads to a change of the size of the core, which implies a huge change in the seismological modulation. This appears very clearly on Fig. 3 of Provost et ai. (these proceedings), where eigenfrequendes of different models based on different EOS are compared. The EOS modification affects principally the low degree modes modulation. The second effect is the change of the temperature profile, corresponding then to a small change of the characteristic frequency VQ, which causes the small mean slope in the diagram, visible for all degree modes. But this can be considered only as an indirect consequence of the PPT. In fact the frontier at the PPT induces qualitatively the same kind of modulation as the one due to the core, but with a much lower amplitude (Table 22.4). This modulation already appears on the numerical echelle diagram (Fig. 22.7). Due to the expected location of the transition, the period iVPPT is about 2. The amplitude is very small, so that finally the direct signature of the PPT corresponds to a slight and rapid modulation. The detection of this modulation requires a very high frequency resolution. 22.4.2.2

High degree modes

High degree modes, which propagate in the upper envelope, are less sensitive to the core. We have compared different models based on different equations of state in order to test the capability of high degree modes to sound the PPT. Results are presented on Fig. 22.16, an histogram of the number of modes refracted at a given level. The direct signature of the PPT appears with evidence on modes with degrees higher than 8. The number of modes refracted at the PPT strongly depends on the sound speed discontinuity. If it is negative, modes penetrate deeper in the planet; if positive, the sound speed jump acts as a wall where the propagation towards the planet is stopped. 22.4.3 How to disentangle interior structure and EOS ? The previous chapter has put in evidence a major problem: how to discriminate between the influence of interior structure and that of the EOS? An answer is given by the seismological parameters. The work of Provost et aJ. (1993) has first shown the dependence between seismological parameters and interior structure parameters, and second proved that the seismological parameters are observable. The results are recalled by Fig. 22.17 and Fig. 22.18, and demonstrate that the measurement of asymptotic parameters corresponds to the measurement of interior structure parameters.

507

Mosser: Jovian seismology

0 -

20

40

I

1

o

PPT : Ac >

60 '

1

ESS"

o o — CM

ft _r

o o

-^£

-i—i

n

J

" -f~i=- ~l

^ 1 r-i_

Fig. 22.16 Histograms counting the modes refracted at a given level in the planet, depending on the degree range. Three models have been used, which differ by the description of the PPT. The number of high degree modes refracted at the PPT strongly depends on the sound speed discontinuity. If Ac < 0, modes are refracted deeper in the planet; if Ac > 0, total reflection may occur. The last model has no PPT.

22.4.4

future

It appears now very important to define the next steps which must be done in dioseismology. The improvement of the description of the Jovian standard model is first required. The connection between the twofields- seismology and interior structure modelling - is already operative and fruitful. New

Mosser: Jovian seismology

508 2.3

•

'

i

i

•

2.2

JUP2+

i

juP7+

i

:

+ jup6

-

j"P8+

ci :

t

-

ju P 5 + .

154

156

i

.

i

.

i

162

160

158

G*Hz) Fig. 22.17 [i/0, r o m ] diagram, indicating how the measure of the frequency i/0 can be related to the interior structure. T0<m represents the adiabatic coefficient which fits the inner (metallic) region of the fluid envelope. Models with a real PPT (JUP6 and JUP7) are separated from the others because of a higher I/Q value.

J"PB?-\

oo

1 £

jiJp^-^ +

• sups

to -

JUp1

t

-2

.

0

^~~^-H.

1

.

|

2 V, (mHz)

Fig. 22.18 [Vi, re] diagram, where rc is the core radius. The quasi-linear dependence proves that the observation of the characteristic frequency Vi provides the direct measurement of the core radius.

observations of Jovian oscillations are urgently needed. In order to improve the quality of the observed spectra, three solutions might be considered: • The window effect can be reduced only with ground-based observations made with a network of site, or with space observation. • In order to disentangle the spectrum, imaging must be developped (IR photometry at 10 /nn). An image of the Jovian oscillationfieldpermits the calculation of different spectra corresponding to different degree ranges and with a much lower mode density.

Mosser: Jovian seismology

509

• The cruise flight of an interplanetary spacecraft is a unique solution for observing continuously stellar or planetary oscillations. This solution will be used during the cruise flight of Mars94 (Baglin 1991), for asteroseismology. Observing Saturn in the last years was far from favorable, with the ring system vignetting the major part of the Southern hemisphere. However, Saturn has been observed in July 1993 with the Fourier transform spectrometer at CFHT. The data reduction is under progress. During the opposition in August 1995, the rings will be seen edge on, providing a unique opportunity to obtain a regular and symmetric visibility of the modes.

22.5 Conclusion In this paper, we have reviewed the first development of a new domain: the seismological study of giant planets, or dioseismology. Theoretical results, derived from helioseismology, are much in advance compared to the observations, except for the understanding of the modes excitation. The observations tend to prove the detection of the oscillations on Jupiter, with pretty large amplitudes, but they cannot yet be translated precisely in terms of interior structure. Even if theoretical results have shown the complexity of the oscillation spectrum due to the structure discontinuities, dioseismology is able to interpret their signature. One current problem is to distinguish between the different signatures. The one due to the core is obvious, and will provide information on the core size and structure. The direct observation of the PPT is more difficult, being hidden by the core. Finally, the new seismological constraints imply the development of a new generation of Jovian interior models.

Acknowledgements I thank very much the organizers of the colloquium, first for having given the opportunity of a first review about dioseismolgy, second for having run the risk of inviting a novice in research. Je tiens egalement a remercier Daniel Gautier, pour ses directives toujours judicieuses ainsi que son aide precieuse. This work has been supported by the Programme National de Planetologie from the Institut National des Sciences de l'Univers (INSU) and by the Groupement de Recherches 'Structure interne des etoiles et des planetes geantes' from the Centre National de la Recherche Scientifique (CNRS).

510

Mosser: Jovian seismology

References Baglin A. 1991. Adv. Space Res., 11, 4, (4)133-(4)140. Bercovici D. and G. Schubert 1987. Icarus 69, 557-565. Campbell J. K. and S. P. Synnot 1985. Astron. J. 90, 364-372. Chabrier G., D. Saumon, W.B. Hubbard and J.I. Lunine 1992. ApJ 391, 817-826. Christensen-Dalsgaard J., T.L. Duvall Jr., D.O. Gough, J.W. Harvey, and E.J. Rhodes 1985. Nature 315, 378-382. Deming D., M. J. Mumma, F. Espenak, D. E. Jennings, Th. Kostiuk, G. Wiedemann, R. Loewenstein, and J. Piscitelli 1989. ApJ 343, 456-467. Gautier D. and T. Owen 1989. In Origin and Evolution of Planetary and Satellites Atmospheres, (S.K. Atreya, J.B. Pollack and M.S. Matthews, Eds.) University of Arizona Press, Tucson. Guillot T., D. Gautier, Chabrier G., B.Mosser. Submitted to AkA Gelly B., G. Grec and E. Fossat 1986. AkA 164, 383-394. Goldreich P. and P. Kumar 1988. Astrophys. Journal 326, 462-478. Gough D.O. 1986. In Hydrodynamics and MHD Problems in the Sun and Stars (Y.Osaki, Ed.), 117-143. University of Tokyo Press. GudkovaT.V., V.N. Zharkov and V.V. Leont'ev 1989. [Astron. Vestnik, 22,3 252-261], Solar Sys. Res 22, 159-166 Hubbard W.B. 1968. ApJ 152, 745-753. Hubbard W.B. and M.S. Marley 1989. Icarus 78, 102-118. Lee U. 1993. ApJ. 405, 359-374. Lindal G.F., G.E. Wood, G.S. Levy, J.D. Anderson, D.N. Sweetnam, H.B. Hotz, B.J. Buckles, D.P. Holmes, P.E. Doms, V.R. Eshelman, G.L. Tyler and T.A. Croft 1981. J. Geophys. Res. 86, 8721-8727. Marley M.S. 1991. Icarus 94, 420-435. Mosser B., D. Gautier and Ph. Delache 1988. In Seismology of the Sun and Sun like-stars. Tenerife, Spain, Sept. 1988. Proc. ESA, SP-286, 593-594. Mosser B. 1990. Icarus 87, 198-209. Mosser B., F.-X. Schmider, Ph. Delache and D. Gautier 1991. ApJ 251, 356-364. Mosser B., D. Gautier and Th. Kostiuk 1992. Icarus 96, 15-26 Mosser B. 1992. Etude des oscillations globales de Jupiter et des planetes geantes. PhD thesis, Universite Paris-XI, Orsay. Mosser B., D. Mekarnia, J.-P. Maillard, J. Gay, D. Gautier and Ph. Delache 1993. AkA 267, 604-622. Pottasch 1992. AkA 264, 138. Provost J., B. Mosser and G. Berthomieu 1993. AkA 274, 595-611. Saumon D. and G. Chabrier 1989. Phys. Rev. Letters 62, 2397-2400. Saumon D., W.B. Hubbard, G. Chabrier and H.M. Van Horn 1992. ApJ 391, 827-831. Schmider F.-X., B. Mosser and E. Fossat 1991. AkA 248, 281-291. Stevenson D.J. 1985. Icarus 62, 4-15. Tassoul M. 1980. ApJ Suppl 43, 469-490. Unno W. ,Y. Osaki, H. Ando and H. Shibashi 1979. Nonradial oscillation of stars, (W. Unno, Ed.), 149-159. University of Tokyo press. Vorontsov S.V., V.N. Zharkov and V.M. Lubimov 1976. Icarus 27, 109-118. Vorontsov S.V. and V.N. Zharkov 1981. Astron. Zh. 58, 1101-1114, [Sov. Astron. 25, 627-634, 1982]. Vorontsov S.V. 1981. Astron. Zh. 58, 1275-1285, [Sov. Astron. 25, 724-729, 1982]. Vorontsov S.V. 1984a. Astron. Zh. 61, 700-707, [Sov. Astron. 28, 410-414, 1984].

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511

Vorontsov S.V. 1984b. Astron. Zh. 61, 854-859, [Sov. Astron. 28, 500-503, 1984]. Vorontsov S.V., T.V. Gudkova and V.N. Zharkov 1989. Pis. Astron. Zur. 15, 646-653. Zharkov V.N. and V.P. Trubitsyn 1978. In Physics of the planetary interiors (W.B. Hubbard, Ed.), 221-284. Pachart publishing house, Tucson. Zharkov V.N. and T.V. Gudkova 1991. Ann. Geophysicae 9, 357-366. Zharkov V.N. and T.V. Gudkova 1992. In High-Pressure research: Application to Earth and Planetary Sciences, Y. Syono, M.H. Manghnani Eds, 357-366.

23 EVRIS A. BAGLIN DASGAL, Observatoirc de Paris. URA CNRS 335. 92125 Afeudon Ctdtx. France

Abstract The EVRIS experiment is an exploratory mission devoted to stellar seismology. It will observe approximately ten bright stars, for 20 days each, during the cruise of the Russian MARS 94 mission. The photometer will be able to detect amplitudes of modes as small as a few 10~6 magnitude. Some objects of masses lower than the solar one will allow to test the thermodynamics. EVRIS est la premiere experience devouee a la sismologie des etoiles. Elle sera lancee par la mission Russe MARS 94. Elle observera une dizaine d'objets, chacun pendant une vingtaine de jours, avec un seuil de detection de quelques 10~6 magnitude. Plusieurs etoiles de masse plus faible que celle du Soleil devraient permettre des tests significatifs de leur thermodynamique. 23.1 An exploratory instrument for asteroseismology. 23.1.1 Scientific objectives and strategy. After the success of detection, measurements and interpretation of the eigemodes of the Sun, it is tempting to try to achieve similar progress on other stars and to allow for a comparative and differential study of the seismical stellar behavior. The major difficulties when going from the Sun to stars is the lack of photons and the lack of spatial resolution. The rationale for such an aim 512

Baglin: EVRIS

513

has already been developed several times (i.e. Hudson et al. 1986, Praderie et al. 1988). The need to go to space has also been extensively documented (see i.e. Mangeney and Praderie 1984, Hudson et al. 1986, Baglin 1990, Weiss 1992).As an important and costy mission is difficult to defend and to launch, it was decided to propose first a small mission with a quite restricted program, but exploring the field. After 10 years with various attemps and versions, we have finally succeeded in being selected on the Russian MARS 94 mission. The EVRIS experiment will work during the 300 days cruise to MARS, then stop when the spacecraft enters the MARS neighbourhood and starts orbiting the planet.

23.1.2

The instrument.

The instrument is a classical photometer, working in photon counting mode, in a large bandpass (3000 A) centered on the visible region, to collect as many photons as possible. Due to the limited weight and space (and also to the small budget!) available, we had to restrict ourselves to a very small telescope; the diameter of the entrance pupil is 70mm. Associated to a stellar sensor, it is placed on a pointing platform (PAIS) designed and manufactured by IKI (Russia), and situated in the shadow of the solar panels of the spacecraft. The pointing accuracy is 30 arcsec.

23.2 Detection threshold for a coherent signal. The condition to detect a coherent signal in the Fourier space, if the noise level is limited by the Poisson photon noise is:

Np inf(T,t)

> A~2

(1)

A, amplitude of the coherent signal; Np, number of photons collected in Is.; T, duration of the observing run (in s.); t, time of coherence of the signal. EVRIS is able to detect amplitudes of a few 10~6 in 5 days for stars brighter than mv « 3.5 in the frequency range 0.1 to 10 mHz. At lower frequencies, the level of detection increases due to the unavoidable rise of instrumental and environment sources of noise (Fig 1).

514

Baglin: EVRIS

Power spectrum Hz" 1

coherent signal ot amplitude A

s - 0.5 A 2 0

b~ 2.3/N

confidence level 90 %

1/N

(20J)

mn

1

s

1

frequency

Fig. 23.1 Detection threshold of a coherent signal as a function of frequency.The power spectrum of a coherent signal is compared to the spectrum of a representative noise (white noise plus instrumental noise at low frequency).

23.3 Choice of the observing program. The observing program is not yet completely fixed. It is submitted to many constraints due to instrumental conditions, like visibility from the space craft, stellar sensor operations, level of instrumental noise and to the scientific specifications.

23.3.1 Possible Targets. Approximately 180 stars fullfill the preliminary conditions. Excluding giants and OB stars for which the characterisitic timescales are too long, 46 objects remain up to now. Their position in a luminosity/surface temperature diagram (the so-called HR diagram) is shown on Fig.2. A final choice will be done to improve the scientific return. Some stars lie in the lower part of the main sequence, and their structure is essentially controlled by thermodynamics. Their envelope is convective, then independent of the radiative transfer but governed by the equation of state. They have been observed by Hipparcos (See Baglin (1993), this

515

Baglin: EVRIS -2

4.05

4

3.95 3.9 3.85 3.8 3.75

3.7 3.65

Fig. 23.2 HR diagram of possible tar gets. Absolute visual magnitudes and effective temperature are computed through the Genava calibration (Hauck, 1973, 1985).

conference), and their modelisation will then be strongly constrained. If pulsations are detected, they will permit to test precisely the stratification in the convective zone.

23.3.2

Duration of the runs.

It will be fixed during the mission depending on the first results, but never shorter than 5 days. The term inf(T,t) in eq.l will be optimised.

516

Baglin: EVRIS

23.4 Post EVRIS projects. - COROT, a small mission proposed to CNES, to follow a few objects, detected by EVRIS as multimode pulsators. The runs on each object will be longer to study the statistic of the amplitude and frequency of the modes. If accepted, the launch is scheduled around 98. - STARS, a M3 ESA proposal to beflownaround 2005 (see Lemaire(1993), this conference), is now selected for an assessment study.

Acknowledgments. We want to thank all the Co-investigators of Evris and all the technical team who is now manufacturing the experiment and its pointing platform. References Baglin, A. : 1993, this conference Baglin, A. : 1990, Solar Phys., 133, 155. Baglin, A.: 1991, Adv. Space Res., 11, 133. Harvey, C. : 1988, in Advances in Helio-and Asteroseismology (J. Christensen-Dalsgaard and S. Frandsen eds.), p. 497. Hudson, H.S., Brown, T.M., Christensen-Dalsgaard, J., Cox, A.N., Demarque, P., Harvey, J.W., Me Graw, J.T., Noyes, R.W.: 1986, A concept for an Asteroseimology Explorer. Proposal submitted to NASA. Lemaire Ph. : 1993, this conference. Praderie, F., Mangeney, A., Lemaire, P., Puget, P., Bisnovatyi-Kogan, G. : 1988, in Advances in Helio-and Asteroseismology (J. Christensen-Dalsgaard and S. Frandsen eds.), p. 549. Mangeney, A., Praderie, F. : 1984, Proceedings of the workshop on Space Prospects in Stellar Activity and Variability, Paris Observatory Press. Weiss W. : 1992, in Inside the stars (A. Baglin and W. Weiss eds.) ASP Conferences Series 40, 708.

24 HIPPARCOS A. BAGLIN AND J. FERNANDES DASGAL, Observatoire de Paris. URA CKRS 335. 92125 Meudon Cedex. France

Abstract The HIPPARCOS mission will permit a decisive step forward in the comparison between observed and predicted global properties of stars, in producing distances and apparent magitudes with accuracies more than one order of magnitude higher than before. Nearby stars of intermediate and low mass will allow for statistical tests on the validity of the equation of state, like for instance the steepness of the main sequence. La mission HIPPARCOS va permettre un pas en avant fondamental dans les tests des propriees thermodynamiques des etoiles de masse intermediaire en fournissant des distances et des magnitudes apparentes beaucoup plus precises que celles obtenues au sol. 24.1 Introduction. Tests of the physical description of stellar interiors rely on a theory vs observation comparison. The stellar evolution theory predicts the variation with time of the state of the interior of a star and, also, of its fundamental, observable parameters, i.e. luminosity, surface temperature, for a given mass. The HIPPARCOS (High Precision PARallax Collecting Satellite) mission will permit a decisive step forward in this confrontation by producing distances and apparent magnitudes with accuracies more than one order of magnitude higher than before (Baglin, 1988). 517

518

Baglin & Fernandes: The HIPPARCOS mission

For a description of the mission see for instance Perryman et al., 1992, and the 'Hipparcos Input Catalogue' (Turon et al., 1992). 24.2 Distances measurements. HIPPARCOS measures parallaxes i.e. distances, and proper motions. Aproximately 120 000 stars brighter than mv w 12.5 are observed; the survey is complete up to the apparent magnitude 7.5. The accuracy is 2 milliarcseconds or better, i.e. 10 to 50 times better than from the ground. This high accuracy is reached by accumulation of individual measurements, i.e. it increases with the duration of the mission. After slightly more than 3 years of observations, the nominal accuracy is obtained despite the poor quality of the orbit, far from the expected one (Perryman, 1993). The results obtained from only one year of data is given in Figure 1. 24.3 Apparent H magnitudes. Hipparcos has proven to be an excellent photometric instrument, working in only one large H band. It will provide H apparent magnitudes with an accuracy of few milli-magnitudes. In addition, BT and UT Tycho magnitudes will be obtained for star brighter than about 11. 24.4 Luminosities. Accurate distances provide accurate luminosities for the closest objects, if an apparent magnitude is known also accurately, and if the so-called 'bolometric correction' is also well known. L == lH d2 BC

(1)

L total flux radiated in one second; Ifj flux received on earth in the H band in one second; d distance, expressed in 10 parsecs; BC bolometric correction associated to the H magnitude, which corrects the energy radiated in the electromagnetic spectrum outside the H band. At 5 parsecs the relative precision on the distance is 1 %; it generates an uncertainty of 2% on luminosities, i.e. 10 to 50 times better than presently. The progress of spectrophotometry and of the modelisation of stellar atmosphere have also reduced drastically the uncertainties on the effective temperature and on the bolometric correction.

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Baglin & Fernandes: The HIPPARCOS mission

DISTRIBUTION OF PARALLLAX ERRORS 25000

C/>

20000

15000

-

1 0000

-

5000

-

I I I I I I' I I I I I I I I I I ~ 8

10

12

14

16

18

20

22

24

mas Fig. 24.1 The accuracy of the parallax measurements after only one year of observations (2000 stars observed more than 5 times) is already of the order of 2 milliarcseconds.

24.5

Low m a s s - s t a r s .

One of the most fundamental reasons to study low-mass stars (M<1.0MQ), is that they form, by number and mass the largest class in the Galaxy. More than 80 % of stars closest than 10 pc belong to this domain (Figure 2). In Figure 3 we present the internal structure of 0.6 and 0.8M© stars compared to the Sun. In the domain of temperature and density covered by their interior, the plasma is far from ideal gas. The main contribution comes from the coulombien interaction. In this domain, it is described accurately by Debye-Hiickel formalism (Cox k Giulli, 1968). The gas pressure Pg is express in terms of the pressure Pp of the ideal gas (Guenther et al., 1992), Pp, ideal gas law by,

520

Baglin & Femandes: The HIPPARCOS mission

Total number of stars: 276

0.25

1.00

1.75

230

3.00

Fig. 24.2 Histogram for stars closest than 10 pc (Gliese et al., 1991), with mass between 0.25 and 3.OOM0.

P9 =

{5X + 3)

(2)

where p is density, X is the mass abundance of hydrogen and T& the temperature in 106 K.

Baglin & Femandes: The HIPPARCOS mission

-10

521

-2

Fig. 24.3 Internal structure of 0.6, 0.8 and 1.0M© main sequence stars in log p/log T plane. Full lines represents the limit of convective and radiative zones, for each model. Figure 4, gives the ratio between P9 and Pp as function of temperature, for two models of 1.0 and 0.6MQ. It shows that the importance of corrective term (here after DHc) decreases with increasing mass. A quantitative estimate of the influence of the DHc term on the global structure is given by comparing stellar main sequence models of 1.0 and 0.6M© computed using two different equations of state EFF formalism (Eggleton et al., 1972) and CEFF (Christensen-Dalsgaard, 1991), which is

Baglin & Fernandes: The HIPPARCOS mission

522 Pg/Pp

0.95

0.9

0.85

0.8

LogT 0.75

Fig. 24.4 Ratio between gas pressure corrected by DHc, according (2), Pg, and gas ideal pressure, Pp, as function of Log T. DHc becames very important in convective region for low-mass stars.

the EFF equation where the Coulomb effects have been included according to the Debye-Hiickel theory (see also Lebreton et al., 1992). On Figure 5, these results are compared to the observations of 10 close low-mass stars in the HR diagram. Vertical bars represent the errors on luminosity presently and using Hipparcos results. A mean error bar in effective temperature is also presented. Here we have assumed that convective transport is represented correctly by the mixing length theory, using a constant value for the mixing length

Baglin & Femandes: The HIPPARCOS mission

0.4

523

LogL/Lo

0.2

-0.2

-0.4

-0.6 before Hipparcos -0.8

after Hipparcos

-1 Log Teff -1.2

3.8

3.75

3.7

3.65

3.6

3.55

Fig. 24.5 HR diagram of 10 low-mass stars closer than 10 parsecs, for which HIPPARCOS will improve luminosities determinations. Vertical error bars correspond to the present accuracy and the accuracy after reduction of the HIPPARCOS data on luminosity. Horisontal bar represents a mean error in effective temperature for these stars.

parameter, a, the solar one. This hypothesis is confirmed by recent results on calibration of the a Centauri binary system (Fernandes et al., 1993).

524

Baglin & Fernandes: The HIPPARCOS mission

24.6 Conclusion. This example ilustrates the power of low mass stars to test the equation of state. Figure 5 shows already a better agreement of CEFF formalism. The reduction of the error bars on luminosity, as expected from Hipparcos will allow to test precisely more refined treatement of the thermodynamics. As nearby cool stars are numerous, the steepness of the main sequence will be very accuratly determined. A further improvement on luminosity measurements is foreseen as a new generation of astrometric missions, designed to increase the accuracy up to 1 micro mas (100 times better than HIPPARCOS) and is already under study.

Acknowlegements JF acknowleges the grant BD-2094/92-RM from Junta Nacional de Investigagao Cientifica e Tecnologica - Portugal. References Baglin A.: 1988, in: Scientific Aspects of Input Catalogue Preparation II. Eds. J. Torra, C. Turon Christensen-Dalsgaard J.: 1991, Challengs to Theories of Structure of Moderate-Mass Stars, Springer Verlag. Eds. D.O. Gough, J. Toomre Cox J.P., Giulli T.: 1968, 'Principles of stellar structure, volume I'. Eds. Gordon and Breach Eggleton P.P, Faulkner J., Flannery B.P.: 1993, A&A, 23, 325 Fernandes J., Neuforge C , work in progress Gliese W., Jahreiss H.: 1991, 'Catalogue of nearby stars, 3rd edition', Astron. Rechen-Institut, Heidelberg. Guenther D.B., Demarque P., Kim Y.C., fc Pinsonneault M.H.: 1992, ApJ, 387, 372 Lebreton Y., Dappen W.: 1988, Symp. Seismology of Sun and Sun-like Stars, ESA SP-286 Perryman M.A.C. et al.: 1992, A&A, 258, 1 Perryman M.A.C: 1993, 27th Symposium, ESTEC, 10-14 May 1993 Turon C. et al: 1992, 'Hipparcos Input Catalogue', ESA SP 1136

25 IRIS and GONG F.-X. SCHMIDER Departement d'Astrophysique, Universite it Nice, France

25.1 Introduction After the first success of helioseismology, it has been shown than new results could only be obtained from long set of continuous observations. Therefore different groups intented to set-up worldwide networks in order to observe the Sun 24 hours a day. This is the case of GONG and IRIS. Both are sixstations networks. Figure 1 shows the different sites of the two networks. They have one common site in Izafia. IRIS has already 5 stations installed and running. The last site, in Australia, is under selection, and very probably will be in Culgoora, more accessible than the Western site selected by GONG team, in Learmonth. It will be set-up at the beginning of 1994. The operation, already started since 1990, will continue untill at least 2000. So far, the best piece of data obtained covers a period of three monthes, during Summer 1991, with a duty cycle of 60%, with only 3 instruments working. With 6 sites, a duty-cycle of at least 85% is expected. The six instruments of the GONG network are constructed and under tests. The deployement is supposed to take place during the second half of 1994. The network will be fully operational in 1995. A duty-cycle higher than 90% is expected. IRIS provides only full-disk velocity measurements, whereas GONG is designed to record images of the Sun, providing a spatial resolution. GONG will measure non-radial modes, from 1=1 to 1=200, when IRIS is only sensitive to radial and low degree non-radial modes (1=0 to 3). Fig. 2 shows 525

526

Schmider: IRIS and GONG

Fig. 25.1 The selected sites of the two networks IRIS and GONG. the diagnostic diagram 1-n, which can be obtained with GONG instrument. IRIS will detect the the modes at the extreme left of this figure. They are doing a complementary job, a* all the modes are necessary to recover the physical parameters from an inversion technique. It is also important to note that only the low degree modes penetrate the deep interior of the Sun. The data obtained IRIS are then the best way to investigate the core of the Sun, where the nuclear reactions occure. The recent results that we wiU present here come from a single set of data obtained with IRIS, during Summer 1991, with a resolution of 0 . 1 4 ^ * . The duty-cycle is 57%. It is the best set of helioseismic data available so far Other results are expected very soon from IRIS data and from GONG when it wiU be operated. In particular, the density of probability of the peaks can be measured, as well as the noise level between the peaks. This will permit us to invesitgate the excitation mechanism and the interaction between the osdlations and the convection. Interaction with the chromosphere can be studied through the pseudo-modes (the energy above the atmospheric cut-off frequency). But the most important observational result to come are the tables of frequencies, available for each year. These values will provide not only extremely precise sound-speed profile, which could be used as a test of the models, but also the variations with time of the frequencies. The long-term variations will be a excellent indication of the nature of the solar-cycle. And

527

Schmider: IRIS and GONG

4500-

250050

Fig. 25.2 A typical / - v diagram of the solar p-modes as seen by the GONG experiment. Each peak correspond to an individual mode with a specific / and n value, where / is the spherical harmonic degree and n is the radial order. IRIS is only sensitive to low degree modes, i.e. the first vertical column on the left of the picture.

3060

3080

3120

Frequency. Fig. 3. Typical power spectra of solar oscillation daia from Big Bear Solar Observatory: each horizontal trace is a section of a power spectrum for different m with / - 20. The peaks in the m - 0 spectrum at v - 3047. 3080 and 3114 u.Hz are from modes with (n.l) = (IS.19), (15.20) and (15.21), respectively. Peaks spaced ±11.6 uiiz around these features are temporal sidelobes arising from the day/night observing window. The shift in the frequency of the peaks as a function of m illustrates the rotational frequency splitting.

Fig. 25.3 Due to the solar rotation, each individual white point is composed of 2/ + 1 modes, with slightly different frequencies depending on the tesseral order m, as visible in the rightern picture.

short-term variations might give us predictions of solar activity at the scale of the week.

528

Schmider: IRIS and GONG 2.154

mHi 2.150 2.1M

2.016

mHi 2.02 2.022

S. •pUlUnj • O.4HS yHs

Si 1.882

1.884 1.688 mfb

20 40 nbr of pti

Fig. 25.4 The three modes (/ = l,n = 12,13,14) are individually shown, with their weighted average on which the sum of two gaussian curves has been fit.

25.2 A precise measurement of low degree rotational splitting The internal rotation of the Sun causes the splitting of the peaks of a pmode in 2/+1 peaks. The measurement of this separation between different m values, for different / values provides an estimation of the rotation at different depth and latitude. Fig. 3 shows a peak / = 20, separated in fourty one peaks. In first approximation, the splitting is proportional to m. Discrepancies to the linearity are the signature of the differantial rotation at different latitudes. Side-lobes, due to daily interruptions, interference between consecutives degrees, and natural width of the peaks limit the precision of the measurement. For low degree modes, only 2 or 3 peaks are visible and generally unresolved. Therefore, the precision decreases as the depth increases. The core rotation was unknown so far. In particular, it has been impossible to discard any of the two theories, the fast rotating core or the rigid rotator. The line width of the modes decreases with frequency, but only reach small enough values at low frequencies, about 2 mHz, where the amplitude are very low. The very low noise level reached with IRIS measurements permited the detection of this low order modes and shows 3 resolved / = 1 modes around 2 mHz, which are the n = 12 to 14 modes. These modes are displayed on Fig. 4. The mean splitting measured by different methods on the three doublets is 0.490 ± Q.OZQfiHz. Fig. 5 shows the values of the splitting of / = 1 modes

Schmider: IRIS and GONG

529

0.6

1

K /'/ a /

m

m d "jj

7 *?, /-

.45

0.5

XI

IPHIR IRIS i

nc / n ,

Fig. 25.5 Variation of the rotational splitting for / = 1 and n = 12,13,14 (full lines) relative to the ratio Qc/Q, for two different models The observed splitting given by IRIS and IPHIR are reported, taking into account the error bars. The derived domain of core rotation rate are indicated at the bottom of the figure. for two different models, for different n values, as a function of the core rotation. It can be seen that the value found on IRIS data almost excludes the value encountered with IPHIR measurements. The possible rotation lies between 1.5 and 4 surface rotation, depending on the model. This is for / = 1. Other measurements has been made on / = 2 modes in the same frequency range. But, the signal to noise ratio is much lower and did not provide any convincing measurement. Presently, we are working on the modes between 2.5 and 3.5 mHz, where, following Libbrecht the half width is higher than ifiHz, but almost constant. Therefore, it is possible to average up to 7 modes in order to improve the statistic. On the averaged modes, we measure the width by fitting three mean / = 0 modes on the mean / = 2 mode. The same has been done on / = 3 also. The results tend to validate the hypothesis of slow rotating core, with a frequency not higher than twice the surface rotation frequency.

25.3 On the acoustic cut-off frequency of the Sun The solar p-modes are a standing wave of the whole Sun between an internal refracting point, depending upon the degree and an reflecting point at limit of the photosphere. The reflection occurs if the frequency is lower than the

530

Schmider: IRIS and GONG

Frequency (Hz)

Fig. 25.6 Average spectrum of 309 individual days of observation taken with the IRIS stations of Kumbel, La Silla, Oukaibden and Izana. atmospheric cut-off frequency, otherwise the wave propagates through the atmosphere and no mode can exist. That is what can be seen on the right side of a typical solar spectrum, as displayed on Fig. 6. The amplitude of the modes decreases while the line width increases. However it can be seen that a substancial energy above the noise level remains whereas the contrast between modes has entirely vanished. What happens here is that as the frequency increases the reflection coefficient decreases. Then a part of the energy contained in the mode is transmitted to the chromosphere, so the mode is damped and its natural width increases. When the reflection coefficient reaches zero, no standing wave remains and we have only an interference pattern between an ascending waves coming directly from the excitation zone, and another one which has been first refracted inside the Sun. The periodicity of the interference pattern is just the separation between modes of consecutive n values. In the case of full-disk measurements, it happens that / = 1 modes and the sum of / = 0 and 2 modes have almost the same amplitude, and a frequency shift equal to the half of the periodicity. Therefore, when we have a null reflection, the two cosine interference patterns added provide a flat level. This explains why the contrast vanishes on the Fig. 6 and gives us the opportunity of measuring very precisely the atmospheric cut-off frequency. Other independant methods have been apllied to the data. Phase differences between measeruments with the sodium and potassium lines, which

Schmider: IRIS and GONG

531

are formed at different level in the photosphere, have been measured for different frequencies. An unique value has been found through the different methods, thus giving us a high level of confidence on the measured value, which is

fat™. = 5.55 ± O.lmHz

References Loudagh, S., Provost, J., Berthomieu, G., Ehgamberdiev, S., Fossat, E., Gelly, B., Grec, G., Khalikov, S., Lazrek, M., Palle, P., Regulo, C , Sanchez, L., Schmider, F.-X., 1993, Astron. Astrophys., 275, L25 Fossat, E., Regulo, C , Roca Cortes, T., Ehgamberdiev, S., Gelly, B., Grec, G., Khalikov, S., Lazrek, M., Palle, P., Sanchez Duarte, L., 1993, Astron. Astrophys., 266, 532 Harvey, J.W., Hill, F., Kennedy, J.R., Leibacher, J.W., 1992, in Brown, T.M. (ed.) GONG 1992: Seismic Investigation of the Sun and Stars, Astronomical Society of the Pacific Conference Proceedings, San Francisco

26 The spatial GOLF project S. TURCK-CHIEZE DAPNIA/Service d'Astrophysique, CE Saclay, 91190 Gif sur Yvette, FRANCE

and the GOLF scientific and technical team R. BOCCHIA5, P. BOUMIER1, M. CANTIN2, J. CHARRA1, B. COUGRAND1, J. CRETOLLE2, N. DENIS2, R. DUC2, H. DZITKO2, M. DECAUDIN1, A. GABRIEL1, J. HERREROS3, G. GREC4, N. PETROU2, T. ROCA CORTES3, J.M. ROBILLOT5 1

IAS, Universite Paris XI, Bat 121, 91405 ORSAY Cedex, FRANCE

2

DAPNIA/Service d'Astrophysique, CE Saclay, 91190 Gif sur Yvette, FRANCE

3

Institute Astrofistca de Canarias, Tenerife, SPAIN

4

Dtp. d'Astrophysique, Universite de Nice, 06108 Nice Cedex 02, FRANCE

8

Observatoire de Bordeaux, BP 89, 33270 FLOIRAC, FRANCE

Abstract This spatial experiment is under construction and has been defined as a 2 years mission on board SOHO, a satellite dedicated to the Sun which will be launched in mid 95. The main objectives are the detection of solar low degree acoustic modes and solar gravity modes for improving our knowledge of the solar nuclear region. 26.1 Introduction The spatial experiment, GOLF (Global oscillations at Low Frequencies), has been accepted by ESA in March 1988 and should be boarded on the SOHO (SOlar and Heliospheric Observatory) satellite (Dame et al 1988, Gabriel et al 1989). This satellite will be launched by NASA in mid 1995. The objectives of this experiment is to enhance our knowledge of the solar interior by the measurement of the low degree acoustic modes 1=0, 1, 2, 3, i.e the most penetrating ones, and by the possible measurement of the gravity modes. These different types of modes correspond to frequencies between some 10~6 and 8 10~3 Hz. On the same satellite there will be two other helioseismic experiments: VIRGO and MDI, the first one is a variability solar irradiance measurement in different wavelengths which allows to reach acoustic modes of degree 1= 0, 7. The second one, using a Michel532

Turck-Chieze: The spatial GOLF project

533

son Doppler imager, is a complementary experiment which must be able to detect degree acoustic modes up to 1= 4500. The success of such mission is largely dependent on the stability of the measurements, which requires a pointing stability of the satellite better than 1 arc sec per 15 minutes. 26.2 The low degree acoustic modes With GOLF instrument, we hope to improve our knowledge on the absolute values of the low degree acoustic mode frequencies and on the variability and the width of these modes which will be followed with a duty cycle of nearly 100% during the total mission. If it is the case, one will better determine the solar sound speed, density, composition and rotation in the nuclear region and one will progress on the excitation of these modes. Presently, the sound speed extracted from the acoustic mode frequencies is well defined down to 0.3 .ft©, the 2700 detected acoustic modes have already allowed to reject some physical idea as large mixing or hypothetical particles as WIMPs (Turck-Chieze et al 1993), moreover, they have pushed us to improve the microscopic physics used in standard model, particularly the solar equation of state and the opacity coefficients. Doing so, one has significantly improved the intermediate region between the nuclear region and the convective outerzone (Turck-Chieze and Lopes 1993). Concerning the central region, we have shown the low sensitivity to such ingredients due to the luminosity constraint, but the important role of some specific reaction rates and of the screening effect in different parts of the calculation (see Dzitko et al these proceedings for the screening effect on the reaction rates). The central solar region is not presently well understood due to the unsufficient number of detected modes. One aim of the GOLF experiment is to better determine tha acoustic modes corresponding to frequencies between 10~3 to 2 10~3 Hz, in order to put some constraints on the neutrino fluxes emitted by this solar region and to confirm or infirm the possible mixing and central convection suggested by the present analysis (Gough and Kosovishev 1993, Turck-Chieze, Dzitko and Lopes 1993). To progress in the knowledge of this solar region, the spatial GOLF experiment presents several advantages: the satellite SOHO will be placed at the LI Lagrange point, located at 1 million km from the earth. This situation suppresses effects of variability of the atmosphere due to large clouds and due to the connection between different sites of a ground network. This situation allows also a good continuity of the data, without day night or season variations, moreover, the duty cycle will be improved by the adjonction

534

Turck-Chieze: The spatial GOLF project

of an on board memory to avoid discontinuities due to bad communication with antenna. One hopes to reach a great stability of the instrument (at a level of 10~6), first by a uniform temperature inside the instrument always precisely pointed towards the Sun, secondly, by high counting rate photomultipliers which must garantee a white noise of very low level in order to extract a velocity of about lmm/s after three weeks of continuous experiment and finally because the experiment is mainly sensitive to unstability inside the cycle of the experiment which is lower than 40s. 26.3 The research for gravity modes An important objective of the GOLF mission is to put some constraints on the amplitude of g-modes. If gravity modes have been already measured on white dwarfs, the detection is controversial in the solar case (van der Raay 1989). In this last case, the convective external layers largely limit their amplitudes, moreover, the superficial solar noise due to the supergranulation and the active regions is important in the range of frequencies where they are looked for (some 10~4 Hz). Consequently, one needs a large integration time to disentangle the different processes. Moreover, the analysis of the solar noise would be facilitated by the determination of the shape and the deformations of the sodium line coupled with specific measurements on ground during the mission. 26.4 Principle of the instrument The instrument is an optical resonance spectrometer which follows the variation of the Doppler velocity on the sodium absorption line. Such instrument has been using for a long time in Nice (Grec et al 1991). An entry filter selects a passing band of approximately 20 A bracketing the sodium Di and D2 lines located at 5896 and 5890 A. Via an optical system, the entrance pupil is reimaged at the center of a gaseous sodium cell and the image of the Sun is rejecting at the infinity, which drastically limits the influence of the pointing. The photons absorbed by the sodium atoms are reemitted towards the two photomultipliers located at 90 degrees on both sides of the cell. The presence of a magnetic field around the cell will allow, via the Zeeman effect, to split the selected portion of the line (25 mA). It is thus possible to choose, by modifying the circular polarization of the incident beam, a left or right portion of the line (fig la). The Doppler velocity will be deduced from the two dissymetric countings Ni and N2 as V = Vo(Ni — N2)/(Ni + N2) (fig.lb). By adding a modulation to the magnetic field, we investigate two

Turck-Chieze: The spatial GOLF project

535

modulation Na vapour can permanent

filtw

component* marked * can be rotatad in raps o» 90 dags

optical

photomultipliers

solar Na Line profile

(c)

c vapour cell Zeeman components

Fig. 26.1 Principle of the experiment and schematic design of the GOLF experiment

other bands, that makes possible to follow the shape and the distortions of the line and to determine Vo (fig.lc). 26.5 The present status The project is built under the responsability of IAS (Institut d'Astrophysique spatial) of Orsay (PI: A. Gabriel, PM: J. Charra), with the collaboration of IAC (Instituto de Astrofisica de Canarias) in Tenerife, the University of Bordeaux and Nice (PS: G. Grec), and the SAp (Service d'Astrophysique) in Saclay. Five models have been designed: the structural model, mechanically

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Turck-Chieze: The spatial GOLF project

representative, the engineering model: electrically representative (these two models have been already delivered at ESA), the qualification model which is the first complete model and will stay in our laboratory and finally the flight and spare models. The flight model will be delivered at ESA at the end of the year 1993. The requirements on the performances of the experiment led to the realisation of original tests at the sub-system level. The thermal and optical properties of the sodium cell (the heart of the instrument) have been especially studied together with the properties of the spectral characteristics of the entrance sodium filter. Concerning the detection chain, we have chosen phototubes which use the photon counting technique with a gain of 3 106. The nominal counting rate would be 6 106/s for each of the two photomultipliers in order to ensure the level of stability required to observe very small amplitude signal. This stability quality has been verified in laboratory during a three week acquisition with the nominal integration time of 4s. Using a filtered lamp in the wavelength of sodium, we have deduced from the counting rate an artificial velocity and performed its Fourier transform to garantee the absence of artificial peaks of the same amplitude than the signal we would like to look for. Functional tests have been performed on the qualification model, as well as vibration, electrical and thermal tests. References Dame L. et al, ESA SP-286, ed. E.J. Rolfe (ESA Noordwijk 1988), p 367. Gabriel A. et al., The Soho mission, ESA SP-1104, (1989), p. 13. Turck-Chieze S., Dappen W., Provost J., Schatzman E. and Vignaud D., Phys. Rep 230, (1993) vol 2-4. Turck-Chieze S. and Lopes I., ApJ., 408, (1993), 347. Gough D. and Kosovishev, A.G., Mon. Not. R. Astron. Soc, 264 (1993), 522. Turck-Chieze, H. Dzitko and I. Lopes, in Neutral Currents twenty years later, Paris July 1993, to appear in World Scientific, ed. U. Nguyen-Khac Grec G., Fossat E., Gelly B. and Smider F. X.,Solar Phys. 133, (1991), 13. van der Raay H.B., in Progress of Seismology of the Sun and Stars, (1989), Lecture Notes in Physics, 367, p 227.

27 DENIS T. FORVEILLE Observatoire de Grenoble, 38041 Grenoble Cedex, France

Abstract The DENIS survey will survey the southern sky in the near-IR J (1.2 micron) and K (2.2 microns) bands at 3" resolution and to limiting magnitudes in J and K of respectively 16 and 14.5 (lmJy in both cases), and at 1" resolution in the red I band (0.9 microns). Astrophysical motivation is provided by basic problems concerning structure and evolution of galaxies, of types ranging from our own to active galaxies, and concerning specific stellar populations including stars with low temperature photospheres, those still embedded in their protostellar envelopes, and those currently losing mass on the AGB. 27.1 Scientific objectives The release of large 2D detector arrays sensitive in the near infrared provides the first opportunity to undertake a deep survey of the sky in the nonthermal infrared range (1 to 2.5 microns). This underexplored spectral range will provide crucial insights into fundamental problems in stellar and galactic astrophysics. Theere is no recent all-sky atlas of data between the visible and the IRAS 12 microns band. The 25 year-old IRC catalog remains the state of the art effort in the near IR despite its limitations. Our objective is to carr y a 3 colour (UK) survey of the complete southern sky, improving on the pioneering IRC sensitivity by 4 orders of magnitude and improving on its spatial resolution by a factor of 20. 537

538

Forveille: The DENIS survey

There are two main motivations for a deep near IR sky survey: the near IR brightness is the best tracer of mass in stellar form, and the interstellar extinction is reduced by a factor of 10 with respect to the visible V band. It is therefore the most apropriate range for descriptions of galactic structure and of the local structure of the universe. A basic astrophysical problem involves understanding the structure and the evolution of galaxies through the physical properties of their stellar populations. The bulk of the known mass of an evolved galaxy is in the form of stars that predominantly radiate in the micron ranghe because of their low effective temperature (red dwarfs and subdwarfs). Our knowledge of the universe derived from optical images is biased towards blue populations of stars and galaxies. It is in addition sewverley limited by extinction. The stellar luminosity function is observationnaly poorly determined at the low mass end, and is needed to determine the contribution of low mass stars to the missing mass in our galaxy, locally, and globally. The near IR is ideally suited to study the space density of such stars. The search for stars with even lower masses, brown dwarfs, is an especially important challenge. Various arguments imply that the number of brown dwarfs detectable by DENIS may be as high as 105, but confusion will render it difficult to isolate them from the few 107 stars expected in the survey. Modelling suggests that the number of brown dwarfs that may be identified as such on the basis of their colours (effective temperatures less than 2000 K) is at most 10*. The positive identification of even a few brown dwarfs would however be a major breakthrough. DENIS observations of low mass protostars in molecular clouds will result in improved knowledge of the initial mass function and of the efficiency of star formation in these clouds. At the high luminosity end of the stellar distribution, DENIS will detect all AGB stars in the magellanic clouds, allowing for the first time studies of a large sample of such objects at a well known distance. Extragalactic research will also benefit from DENIS. It will provide a consistent sensus of galaxies in the local universe (z<0.1, which is needed to normalize the deeper counts which are becoming available. Since it will be largely free of galactic extinction problems, the galaxy catalog resulting from the survey will be a basic tool for studies of the large scale structure of the local universe.

Forveille: The DENIS survey

539

27.2 The DENIS project The DENIS objective is to survey the entire ESO-accessible sky in two near IR bands (J at 1.25 micron, and K at 2.2 micron) to a sensitivity limit of 1 mJy (mtf=14.5 and mj=16), with a spatial resolution of 3 arcseconds, and in one red band (I at 0.9 micron) with a one arcsecond resolution. We use the existing ESO 1-metre telescope equipped with a dedicated camera. The camera is equpped with two NICMOS-3 arrays (256x256) for the J and K channels, and with one CCD (1024x1024) for the I channel. The first test data have been obtained at the telescope in december 1993, and show the sensitivity is nominal. Commissioning of the instrument and its software is proceeding smoothly and we expect the first survey observations to take place in July 1994. The processing of the huge amount of data (a few terabytes) is a major concern. The project is aware that the data reduction effort requires resources similar to those required by the instrument building and observing efforts. The expected number of sources in the new survey is a few 107, two orders of magnitude more than the number of sources in the IRAS Point Source Catalog. Our goal is to provide the astronomical community with the first comprehensive star and galaxy catalogs in this range of wavelengths, and with a complete digitized infrared map of the sky. These will be produced and distributed through two data analysis centers located in Paris and Leiden.

28 PRISM A: A mission to study interior and surface of stars P. LEMAIRE Institut d'Asttvphysique Spatiale, Universite it Paris XI, Bailment 121, 91405 Orsay Cedex, France

Abstract New technics such as asteroseismology are able to sound the deep interior of stars and to provide the data that will constrain the modelisation of the core. This information will be combined with data collected from the stellar surface which give direct access to measurements of the radiative losses, angular momentum losses and distribution of active structures. From the two sets of data, the key role of the convection zone will be clarified, as the convection zone excites the waves that propagate through the whole star and generates the magnetic field that structures the stellar surface. The PRBMA mission was developed to collect the data needed for detecting the oscillations by very accurate photometry (micromagnitude) and to derive the surface activity and rotation from accurate ultraviolet spectroscopy. A short description of the model payload is given with the observational constraints related to the needed accuracy of measurements. Following the non-selection by ESA in may 1993, some following perspectives are described.

28.1 Introduction The sounding of the stellar interior can be traced either by neutrino detection or by reconstruction of the path of travelling waves perturbing the surface. Asteroseismology is the study of such waves detected either in brillance or in velocity fluctuation. Up-to-now the use of such fluctuations (Grec et al, 1980; Frohlich and Toutain, 1992) has been proven to be a 540

Lemaire: PRISM A

541

powerful diagnostic tool to modelise the solar interior (Gough, 1985). The stellar surface vibrates with modes of increasing number as they propagate down to the deep interior or are reflected near the surface. The detection of those modes is related to the resolution on the stellar surface. Low modes, such as / = 0,1 and 2 penetrate in the deepest part of the star, near the core, and can be detected from the global measurement of the star. High modes, e.g. 500-1000, penetrated only few hundredth of stellar radius and can only be detected with high resolution on the surface. All the range of modes have been detected on the Sun, but on star where the surface cannot be resolved we have only access to low modes (0,1,2 and 3). The oscillations of late-type stars are driven by stochastic fluctuations of the convection envelopes and the oscillations detected are believed to be excited in the layers immediately below the photosphere. The magnetic field generation inside the star is detected by the manifestation of star activity either in the form of dark spots in the photosphere or of bright plages in the chromosphere and the corona. The magnetic field generation is probably generated through a (or a multiple) dynamo process at the base of the convection zone in the case of the Sun (Belvedere et al, 1991). In late-type stars such a process can be envisaged. Following the path opened by EVRIS (Baglin, 1994) the PRISMA mission (Probing Rotation and Interior of Stars: Microvariability and Activity, Appourchaux et al, 1993a and 1993b) has been proposed to study the interior of stars using simultaneous observations of the global intensity fluctuation and of the manifestation of activity at the surface and in the atmosphere.

28.2 Scientific objectives and requirements The broad scientific objectives are: • the study of the structure, evolution and dynamics of stellar interiors. • a determination of constraints on models of magnetic field generation by dynamo processes. Through the use of two differents tools, asteroseismolgy and measurement of activity, the data needed to fulfill the scientific objectives will be collected: i) asteroseismology: - the purpose is to test and improve models of stellar structure and evolution. This will yield a better parameterization of physical processes (mixing length, equation of state, opacities), as well as a reliable determination of the basic stellar parameters (mass, age, chemical composition). - the second purpose is a measurement of stellar internal rotation. This

542

Lemaire: PRISM A

will give an observational basis to models of angular momentum loss and transport. - another goal is the study of mode excitation processes. These goals will be reached by the observation of the frequencies, amplitudes and lifetimes of eigen-modes of oscillation, ii) stellar activity: - the major purpose is to study the emergence of the magnetic field at stellar surface (which will provide tests of dynamo and convection models, when combined with the results of the asteroseismology experiment), and the processes of energy deposition in atmospheric layers. - another goal is the precise determination of surface rotation rates (which yield estimates of differential rotation when combined with the results of the asteroseismology experiment). These goals will be reached by monitoring, during the stellar rotation, the temporal evolution of several UV and X-ray emission lines, probing various levels in the stellar atmosphere. The resulting time-resolved 3D picture of the atmosphere will provide the topology of the surface magnetic field and the vertical structure of the atmosphere. 28.3 Constraints and payload The quantities to be measured (table 1) are: - a - the intensity of visible light within a very large band pass with a very good photometric accuracy with a sensitivity better than 10~6 magnitude in amplitude per frequency band with 1.- 0.4 /zHz resolution (10-30 days of continuous observation on the same stars). Two photometric telescopes are used to fulfil this objective. The Large Photometer (with a collecting area equivalent to 40 cm diameter) is able to collect simultaneously the flux from a few stars (m,, smaller than 8) within a field of 1.5 x 1.5. The Small Photometer (15 cm diameter equivalent area) points in another direction and collects simultaneously the flux from a few stars (mv smaller than 6) within a 3 x 3 field. - b - the flux and profiles in UV lines which measure the stellar activity level: e.g. Mgll 12796, Hell 11640, CIV 11550 and HI La 11216 with few percent photometric accuracy. It needs to be complemented by the flux in one X- ray channel (e.g. 1170 ). The activity segment comprises on one hand a telescope selecting stars within the 1.5 x 1.5 field of view of the Large Photometer to feed a cross-dispersed spectrograph (with performances similar to the IUE high resolution mode), and on the other hand a 15 cm diameter, normal incidence, multilayer telescope. The two telescopes are aligned with

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543

Table 28.1. Quantities to be measured (from Lemaire et al.,1991). WD and CV stand respectively for white dwarfs and cataclysmic variables. The last column indicates the instrument: (a) = photometer; (b) = UV spectrometer; (c) = XUV telescope Observable

Frequencies Amplitudes Rotational splittings

Sensitivity scale

Time

Sampling time

Typical stars

10" 7 mag 0.02-20 mHz 10"6 mag

1 month

1 min

1 month

1 min

10"4 mag

1 month

sec.

Bright solar type variables mv = 8 WD.CV

0.05

Prot

1 min

0.5-50 day

UV-line profiles

0.05 0.10

X-ray flux

0.10

Prot Prot

Prot/5

10 min.

flare Prot

Prot/5

Bright solar-type solar type Bright solar-type solar type

a a a b b c c

the Large Photometer pointing axis and the UV spectrometer samples stars in the field to cover the phase of the star rotation. To accomplish the scientific objectives of the PRISMA mission, within the two nominal years, a special scheme of observation has been studied that allows the recording of more than one hundred stars distributed in the Hetzprung- Russel diagram nearby the main sequence and including two open clusters. Within the choice of several hundreds of fields, a preliminary catalog has demonstrated the possibility of filling this set when staying one month over each field to obtain the photometric spectral frequency resolution.

28.4 Conclusion The coupling of asteroseismology and surface activity measurements is a powerful tool to diagnose the stellar interiors. The PRISMA mission proposed this goal. Following the selection procedure at ESA (European Space Agency) in April-june 1993 the mission was not selected. Nevertheless on the line are appearing 2 missions to study the stellar interiors: - COROT (Catala et al, 1993), a CNES (Centre National d'Etudes Spatiales ) study to continuously look at few bright stars during several months

544

Lemaire: PRISMA

in order to have a very accurate measurement of frequencies to derive precise informations on stellar structure. - STARS (Jones et al, 1993), an ESA study to look at several hundreth of stars with emphasis on clusters that can be used to calibrated several parameters of the internal structure.

References Appourchaux.T., Gough,D.O., Hoyng.P. , Catala, C.,Frandsen,S., Frohlich,C,.Jones, A., Lemaire, P., Tondello.P., Weiss, W., (1993a), PRISMA: a new space mission for stellar physics,in Inside the Stars, IAU colloquium /57.W.W. Weiss and A. Baglin Eds, ASP Conf. series, vol 40, pp. 812-819 Appourchaux.T., Catala,C, Cornelisse.J., Frandsen.S., Fridlung.M., Frohlich,C, Gough.D.O., Hoing,P., Jones.A., Lemaire,P., Roxburg.I., Tondello.G., Volonte.S., Weiss.W., (1993b), PRISMA - Probing Rotation and Interiors of Stars: Microvariability and Activity, Report on the Phase-A Study ESA-SCI(93)3 Baglin, A.: 1994, these proceedings Belvedere, G; Proctor, M.R.E. and Lanzafame, G.: 1991, Nature 350, 481 Frohlich, C. and Toutain, T.: 1992, Astron. & Astrophys. 257, 287 Gough, D.O.: 1985, Solar Phys. 100, 65 Grec G., Fossat E., and Pomerantz M: 1980, Nature 288, 541 Jones.A.R., Gough,D.O., Andersen,B.N., Baglin,A., Bisnovatyi-Kogan, G., Browns,T.M., Catalano.S., Catala.C., Chiosi,C, Dziembowski,W.A., Frandsen,S., Frohlich,C., Gahm.G., Grenon.M., Hoying.P., Hudson,H.S., Lago.M.T., Lemaire.P., Mattews.J., Osaki.Y., Petrov.P.P., Radler.K.H., Roca Cortes.T., Rodono.M., Roxburg.I.W., Schrijver.C.J., Tondello.G., Thome, A.P., Ulrich.R.K., Waelkens.C., Weiss,W.W.,(1993), STARS- Seismic Telescope for Astrophysical Research from Space, An investigation of stellar structure and evolution Catala.C., Combes,M., Mangeney.A., Mosser.B., Epstein.G., Encrenaz.T., Gautier,D., Rouan.D., Baglin,A., Auvergne,M., Michel.E., Buey,T., Cruvellier.P., Vuillemain.A., Lemaire,P., Gabriel.G., Boumier.P., Fort.B.: (1993) COROT -COnvection et ROTation stellaires: experience en sismologie stellaire Lemaire,P., Appourchaux,T., Catala.C, Catalano.S., Frandsen.S., Jones.A., Weiss, W. :1991, Adv. Space Res. 11,(4),141

29 Towards a helioseismic calibration of the equation of state of the plasma in the solar convective envelope S. V. VORONTSOV 12 , V. A. BATURIN 1 3 , D. O. GOUGH 4 1 & W. DAPPEN 5 1

Astronomy Unit, Queen Mary and Westfield College, Mile End Road, London El 4NS, UK 2 Institute of Physics of the Earth, B.Gruzinskaya 10, Moscow 123810, Russia (permanent address) 3 Sternberg Astronomical Institute, Universitetsky Prospect 13, Moscow 119899, Russia (permanent address) 4 Institute of Astronomy, and Department of Applied Mathematics and Theoretical Physics, Madingley Road, Cambridge CB3 OHA, UK 5 Department of Physics and Astronomy, University of Southern California, Los Angeles, CA 90089-13J,2, USA

Abstract We report the results of a nonlinear inversion of solar oscillation data that enable us to detect nonideal Coulomb interactions between particles, including pressure ionization, in the solar convection zone.

29.1 Introduction Precise measurements of solar oscillation frequencies provide data for accurate inversions for the sound speed in the solar interior. Except in the very outer layers, the stratification of the convection zone is almost adiabatic. There, the sound-speed profile is governed principally by the specific entropy, the chemical composition and the equation of state, being essentially independent of the uncertainties in the radiative opacities. The inversions thus reveal, via tiny effects on the adiabatic compressibility of the sola.r plasma, physical processes that influence slightly the equation of state. We have carried out a nonlinear inversion based on a recent accurate asymptotic description of intermediate- and high-degree solar p modes (Brodsky & Vorontsov 1993; Gough & Vorontsov 1993), using the observational data of Libbrecht, Woodard k. Kaufman (1990). 545

546

Vorontsov et al.: The solar convective envelope

29.2 The equations of state (EOS) used in the analysis In the reference models, we use the following eqxiations of state. We are mostly brief, with the exception of the pressure-ionization model used in the helioseismic calibration. • Saha EOS: a free-energy-minimization type realization for a mixture of reacting ideal gases, with ground-state-only partition functions of the bound species. Note that by assuming only ground states we are using the term 'Saha' in a rather restricted sense. • Debye-Hiickel (DH) EOS: the Saha EOS together with a consistent free-energy term for the Coulomb interaction in the Debye-Hiickel approximation. • PI(S) EOS: a simple model to reproduce the essential thermodynamic features of pressure ionization of hydrogen, neutral helium and singly-ionized helium (we neglect the effect of the pressure ionization of the elements beyond hydrogen and helium because of their small abundance). This model, inspired by the confined-atom approach {e.g. Dappen, 1980), includes only ground states of particles, and describes the main effect of pressure ionization as a smooth reduction of the effective partition function as density increases:

( In this expression R. denotes the mean distance between the centres of nearest neighbours (of all species, charged and neutral), i?., is the radius of species i (we adopt i?.jj = i?jj e = 1, and Rrie+ = 0.5), and A is a parameter that we adjust to solar data (in the following PI(S) EOS means such an adjusted equation of state with A = 1.667). Expression (1) for Z\^ is then incorporated into DH EOS in a consistent way. • MHD EOS: the free-energy-minimization realization of the Hummer & Mihalas (1988) occupation-probability formalism (Mihalas, Dappen h Hummer, 1988; Dappen et al., 1988). A large number of species are interacting with each other; the destruction of extended species is taken into account by both volume and charge effects. The MHD equation of state includes the DH terms too.

29.3 Results We show the results of our inversions for various reference models based on the different equations of state described above, and we compare each model with observations. We plot the sound-speed gradient, [in the dimen-

547

Vorontsov et al.: The solar convective envelope

-0.64

DH EOS

GM o dr-0.65 -0.66 -0.67 -0.68

Fig. 29.1 Sound-speed gradient in the solar convection zone. Dashed line: reference model computed with the DH EOS. Solid line: inversion of artificial data computed with reference model. Solid line with error bars: inversion of observational frequencies. The horizontal bar indicates the region where the inversion is most accurate and reliable. Note the significant discrepancy between theory and observation.

,-0.64

r2 dc GN/Ldr -0.65

Saha EOS

-0.66 -0.67 -0.68

0.7

0.8

r/R 0

0.9

1.0

Fig. 29.2 Same as Fig. 1, but with an equation of state that has no electrostatic, interaction term (Saha EOS). The discrepancy between theory and observation becomes larger. sionless form used in studies of the helium abundance (Gough, 1984; see also the recent review by Kosovichev et a/., 1992). The sound-speed gradient obtained for the Sun is compared with that of our reference models: DH-EOS (Fig. 1), Saha EOS (Fig. 2), PI(S) EOS (Fig. 3), DH EOS for two

Vorontsov et al.: The solar convective envelope

548

-0.64 2

r

dc

2

PI(S) EOS

-0.66 -0.67 -0.68

0.7

0.8

0.9

1.0

r/Re

Fig. 29.3 The discrepancy between the model and the inverted solar data can be removed by taking appropriate account of pressure ionization [PI(S)].

-0.64 2

r

DH EOS

dc

C-only

-0.66 -0.67 -0.68

0.7

0.8

,

0.9

1.0

r/Ro Fig. 29.4 Same as Fig. 1, but with two models in which all the heavy elements have been replaced by either oxygen (O-only) or carbon (C-only).

selected choices of heavy-element abundance (C only and 0 only) (Fig. 4), and in Fig. 5 the MHD equation of state (also for two different choices of heavy-element abundance).

29.4 Conclusion The accuracy of the inversion that is achieved with the observational data currently available permits us to detect the influence of Coulomb interac-

Vorontsov et al.: The solar connective envelope

-0.64

GN/Ldr-0.65

549

MHDEOS MHD-2 MHD-5

-0.66 -0.67 -0.68

0.7

0.8

0.9

1.0

r/Ro Fig. 29.5 Same as Fig. 1, but with two models computed with the MihalasHummer-Dappen (MHD) equation of state. MHD-5 refers to a more complete set in which (besides H and He) C,N,O, and Fe are treated in all details, and all species interact with each other. MHD-2 stands for a simpler chemical mixture, with oxygen alone replacing all heavy elements. tions between particles. At the same time we have established a clear diagnostic potential to probe the mixture of heavy elements in the convection zone. We are thus putting constraints on the physical foundation and the chemical composition of the equation of state. In this way we can assess the quality of the different versions of the equation of state that are currently used for modelling the solar interior. • References Brodsky, M. k. Vorontsov, S.V., Astrophys. ,/., 409, 455 (1993) Dappen, Astron. Astrophys., 91, 212 (1980) Dappen, W., Mihalas, D., Hummer, D.G. & Mihalas, B.W., Astrophys. J., 332, 261 (1988) Gough, D.O., Mem. Soc. Astron. Itai, 55, 13 (1984) Gough, D.O. & Vorontsov, S.V., Mon. Not. R. astr. Soc, (1993) submitted Hummer, D.G. & Mihalas, D., Astrophys. J., 331, 794 (1988) Kosovichev, A.G., Christensen-Dalsgaard, J., Dappen, W., Dziembowski, W.A., Gough, D.0..& Thompson, M.J., Mon. Not. R. astr. Soc, 259, 536 (1992) Libbrecht, K.G., Woodard, M.F. b. Kaufman, J.M., Astrophys. J. Snppi, 74, 1129 (1990) Mihalas, D., Dappen W. k Hummer, D.G., Astrophys. ./., 331, 815 (1988)

30 Thermal cyclotron and annihilation radiation in strong magnetic fields V.G. BEZCHASTNOV Ioffe Institute of Physics and Technology, 194021, St.Petersburg, Russia

A.D. KAMINKER Ioffe Institute of Physics and Technology, 194021, St.Petersburg, Russia

Abstract The cyclotron and the one-photon annihilation emissions are investigated for a strongly magnetized thermal electron-positron plasmas. The annihilation spectral component is significant when the particle number density N exceeds some critical value, Ncr(T,B). For T ~ 108 - 109 K and B ~ 1012 - 1013 G, this condition can be fulfilled at N < 1022 cm" 3 , which is realistic for neutron star magnetospheres.

30.1 Introduction The e~e+-plasma in strong magnetic fields of neutron stars can be thought to be responsible for X-ray and 7-ray radiation of radio pulsars and 7-ray bursters. In the emitting regions of these objects, the cyclotron emission and one-photon pair annihilation can be important. Separately, they have been investigated by many authors (see, e.g., Bezchastnov and Pavlov 1991, Harding 1986, 1991, and the references therein). However the comparison of these mechanisms has not been performed even for the simplest case of thermal plasmas. We consider the total emission spectra and find the domain of temperatures T and magnetic fields B where the annihilation component is significant for realistic particle number densities N < 1022 cm"'1. 550

Bezchastnov & Kaminker: Thermal cyclotron and annihilation radiation

551

30.2 Spectra of radiation Quantum cyclotron emission and one-photon pair annihilation are characterized by the emissivities (summed over polarizations) j c and j a , respectively. For the photon propagation across B in a tenuous nondegenera.te thermal plasma, we obtain

Jg

Here a = e2/h~c, Ac = h/mc, u is photon energy in units of me2, A normalizes the thermal distribution function, b = B/Bcr (Bcr = m2c^/eh = 4.41 x 10 13 G), t = T/rnc2, N^ denote the e^ number density,

wCla(i/, n) = \Jl + 2(n + v)b ^ Vl + 2n6,

(3)

D n /, n = b(n + n') [Fx2 + F2 - F2 - F2] - F2 - F2,

(4)

Fi = F n /_i i T l , F 2 = F n / ? n _ l 5 F 3 = F n / _ i , n _ i , F 4 = F n '_ n are normalized Laguerre functions of the argument u2 /2b, and u

The emissivity jf = j c + ja is shown in Figs. 1-3. The cyclotron features are produced in radiative transitions \n + u) -^ \n) + \u>) of the particles at u < uc(v,n), while the annihilation ones are caused by the reactions \n + v) + |n) —»• \UJ) between e~ and e + at u> > u>a(is, n) > wa(0, 0) = 2. At u ^> 6 ^> t, one ma,y keejj the only terms n — 0 in Eq. (1) and replace the sum over v by the integral. Then

These spectra are in good agreement with the exact spectra (1). Averaging Eq. (2) over energy interval 2 < u < 1 + \/l + 26 between the first and the second annihilation peaks we obtain (Ja)

N-N+ ~

exp (-^)

.

(7)

552

Bezchastnov & Kaminker: Thermal cyclotron and annihilation radiation

a) j / N for N=102,2 cm"3 b) 10 xj/N for N=1Cr° cm"3 c) K T ' V ' N for N=10'8 cm"3

0.1 1 Photon energy. Mev Fig. 30.1 Spectra of differential emissivity j/N as a sum of the cyclotron and one-photon pair annihilation radiations in e~e+— plasma with T = 0.05mc- and B = 0.2Bcr for different number densities JV_ = N+ = N. Smooth solid lines show the averaged cyclotron spectra (6).

The ratio of Eq. (7) to Eq. (6) at u> = 2 gives the critical number density, H

;— exp I —; . (8) b / \ b t) For N > Ncr, the annihilation spectral component forms a jump at the main threshold (E = 1.022 Mev) followed by a sequence of the fine structure peaks (Fig. 3). The annihilation features are more pronounced on the cyclotron background, when the condition N > Ncr is stronger. Ncr(T,B) shows sharp growth with increasing T arid/or with decreasing B (Fig. 4).

Bezchastnov & Kaminker: Thermal cyclotron and annihilation radiation -.22

aj)

553

-3

j'/N for N = 1CT c m . )) 10~lw xj/N for N=1Cr cm"J 3 to

° 10"'°xj/N

b) 10" xj/N

c) 10 I5|

for N = 1

°

° c_... m

b=0.2 t=0.07

f (

10 i

/

%| Nj,b)

*

/

/

< iJ , >

IT—jf

r

!/

I'°- / / /

7

NJ,C)

V\\i

10-4 / io-"[/ ir,

-»L_.

Tttl

0.1 . I Photon energy, Mev

Fig. 30.2 Same as in Fig. 1 for T = 0.07mc2. The spectrum inside the rectangle is presented separately in Fig. 3.

30.3 Discussion Soft thermal-like spectra have been observed in the X-ray pulsar state of the 5 March 1979 event, and in the soft 7-ray repeaters (Mazets et al., 1982, Golenetskii et a/., 1984). The fits yield T ~ (3 - 5) x 10 s K. Thus one can expect to observe the one-photon annihilation features in the spectra of these soft sources under quite realistic conditions (B > 1012 G, TV < 102" cm~' J ). The work was partly supported by the ESO C&EE Grant A-01-068.

554

Bezchastnov & Kaminker: Thermal cyclotron and annihilation radiation 10 * E u 10

Photon energy, Mev

Fig. 30.3 Spectrum of differential emissivity j near the annihilation jump for T = 0.07mc2, B = 0.2£ cr and N = 1020 cm" 3 . The bar between the first and the second annihilation peaks corresponds to the averaged value obtained from Eq. (7).

0.05 0.07 0.09 0.11 0.130.15

Fig. 30.4 Critical particle number density Ncr as a function of t = T/mc* for three values of h — B/Bcr (numbers near the curves). References Bezchastnov V.G. and Pavlov G.G., Astrophys.Sp.Sci. 178, 1, (1991) Golenetskii S.V. tt a/., Nature 307, 41, (1984) Harding A., AsUvphys.J. 300, 167, (1986) Harding A., Phys.Reports 206, 327, (1991) Mazets E.P. tt a/., Astrophys.Sp.Sci. 84, 173, (1982)

31 Modified adiabatic approximation for a hydrogen atom moving in a magnetic field V.G. BEZCHASTNOV loffe Institute of Physics and Technology, 194021, St.Petersburg, Russia

A.Y. POTEKHIN loffe Institute of Physics and Technology, 194021, St.Petersburg, Russia

Abstract Motion of a hydrogen atom across the magnetic field shifts center of election density distribution. For strong magnetic fields, the radiative transitions can be considered in the modified adiabatic approximation in which the shifts are taken into account. The method is illustrated by calculating the photoionization cross sections.

31.1 Introduction The presence of hydrogen atoms affects radiative transfer in cool atmospheres of neutron stars with strong magnetic fields B — 10 11 — 1013 G (Shibanov et al., 1993). If 7 = 5/(2.35 x 109 G) > 1, the atomic structure and radiative transitions have been considered by many authors for non-moving atoms (see, e.g., Potekhin and Pavlov, 1993 and the references therein). The effects of atomic motion across B have been studied using the adiabatic approximation (Gorkov and Dzyaloshinsky, 1968), variational calculations (Vincke et ul., 1992), and the perturbation theory (Pavlov and Meszaros, 1993). For strong fields, it is natural to use the basis of the states of free charged particles (Landau functions). In the adiabatic approximation, one keeps the main term of the wave function expansion. In the traditional approach, the transverse part of the wa.ve function is localized in a magnetic, well. However, the real shift of the electron density from the Coulomb center cannot coincide with the shift of the magnetic well. We 555

556

Bezchastnov & Potekhin: A hydrogen atom moving in a magnetic field

consider matrix elements for radiative transitions assuming that the adiabatic wave functions for initial and final states are arbitrary shifted. This approach is applied to calculating the photoionization cross sections.

31.2 Basic equations The structure of the hydrogen atom with generalized momentum hK in the magnetic field B = (0,0,5) is determined by the Schrodinger equation

(/?

s

I)

f

\r + i)ro\

E

K

(f) = 0,

(1)

where E is the atomic energy, M — mc + m p , n — (mcmp)/M; mf: and m p are electron and proton masses, respectively, f = fe — rv — qra is the shifted relative co-ordinate, FQ = -(ch/eB2) K± X B and pz = -ihVz. The Hamiltonian

(ae = (nip— rnc)/2M and p± = — ihV±) corresponds to harmonic motion in the "x-y"-plane; ;/ is an arbitrary parameter. Each atomic state is described by the family of eigenfunctions with various ?;,

fa } ^{f- f.),

(3)

where F* = (77 — r]')ro. If 7 > 1, one may construct the adiabatic solution

^r)

= yg]^nMr±)-

(4)

Here $n,Af is an eigenfunction of H±, n = 0 , 1 , 2 , . . . and N = 0 , 1 , 2 , . . . enumerate the electron and proton Landau levels, respectively. Then the. full set of atomic quantum numbers is |A', i) = \K,n, iV,f), where t (positive or negative) determines the "longitudinal" energy. The motion across B shifts the electron density distribution in the direction of Fo - see Vincke <•:/. ul. (1992). The accuracy of Eq. (4) should be better, if q corresponds to the real shift. Estimates and calculations for the bound states (Ipatova ct ui, 1984, Vincke ct «/., 1992) show that 7/ ~ 0 at K± < Kc, and // ~ 1 at. A'j_ > Kc, the critical value Kc being larger for higher binding energy.

Bezchastnov & Potekhin: A hydrogen atom moving in a magnetic field

557

31.3 Radiative transitions Let us assume that the initial and final states of the moving atom are defined by (4) with different shifts, q and 77', respectively. Then the dipole matrix element Dfj = (K,f\ r\K,i) contains the factors I± = ($n>,N>(r± + r»)|exp {iae(r? - i/) whose direct calculation is complicated because F* ^ 0. However, using the shift transformation *n',N'(r± + ?*) = exp ( ( r)f *

n',N'(r±),

(6)

we may rewrite Eq. (5) in the form I± = (n',N'\ exp | i ( i ; ' - //)

\n,N),

(7)

where the initial and final states belong to the same Landau basis. For this basis, p± and f± can be expressed through the annihilation and creation operators, which have the following properties: a | n,N) A\TI,N)

= y/Ti\ n - 1,7V),

A+

n, N-l),

= T/N\

a+ | n,N)

= y/n + 1 | n+

\ n , N) = VWTT

[d,a+] = [A,A+] = 1 .

\n,N+

1,/V), 1),

(8)

Expanding the exponent in Eq. (7) over powers of the operators a, A, u+ and A + , we obtain '2u) , (9)

I± = where 4> = TT/2 + arctan(A', / /A', ; ),

u = (q - i/)2(aMK±)2/2,

(10)

— \/ch/eB and F^ ^ is a normalized Laguerre function. This yields the cyclic components of .D/,;: D+1 = (Dx + iDy)/V2 ^TlFn,
= -

_! = (A, - i-Dv)/V2 = .w -

y/NTiFnl,nFN.tN+l]

558

Bezchastnov & Potekhin: A hydrogen atom moving in a magnetic field

10 6

y=1000

r

K=100 au - - - K=50 au K=10 au

5

10 : £ 10*

b

S 10 3 w 102 CO (0

O O

10 1 10- 1 10 "2 10

1000

100

Photon energy (Ry) Fig. 31.1 Comparison of traditional (?j' = IJ = 1, long-dashed lines) ami modified (?/' = 1,»/ = 0, short-dashed lines) adiabatic photoionization cross sections of moving hydrogen atom with more accurate results (solid lines) for B = 2.35 x 1012 G and A'x = 50 a.u. y. - 0 and n - +1 correspond to longitudinal and right circular polarizations of photons, respectively.

^^^lzl^^))

(11)

(the arguments of the F-functions with low-case and capital indices are the same as in Eq. (9)). Photoionization of the moving hydrogen atom can be determined by the transitions between the states with different shifts. If A'j. < A'o, then the ground-state electron is localized near the nucleus (according to Vincke at

Bezchastnov & Potekhin: A hydrogen atom moving in a magnetic field al. 1992, Kc ~ 150 a.u. for 7 = 1000). Hence the adiabatic approximation with rj = 0 is appropriate for this initial state. For the final (continuum) states, the representation with // = 1 should be applied in order to make the Hamiltonian (1) axially symmetric at large z. Eqs. (11) allow one to reduce the dipole matrix elements to the "longitudinal" overlap integrals to be found numerically. This gives good agreement with more advanced results computed with the exact wave functions of the initial state (Fig. 1). For comparison, we have calculated the conventional adiabatic cross sections (77' = // = 1). As seen in Fig. 1, this approach underestima.tes strongly the transition rates at K± < Kc.

31.4 Discussion Motion of the hydrogen atom across the magnetic field makes the problem essentially three-dimensional, which complicates the numerical approach. The adiabatic approximation simplifies the solution reducing the problem to one-dimensional. We have shown that the accuracy of such consideration is determined by a proper choice of shifts (77 and 77'). The dependence il(B, K±,i) is not yet known. However, we know that // ~ 0 at K± < Kc, and 77 ~ 1 at K± > Kc, where Kc is a function of the magnetic field and atomic state. An analytic, estimate of the critical value Kc is thought to be useful. We are grateful to G.G. Pavlov and D.G. Yakovlev for useful discussions. The work was partly supported by the ESO C&EE Grant A-01-068.

References Gorkov L.P. and Dzyaloshinsky I.E., Sov. Phys.-JETP 26, 449, (1968) Ipat.ova I.P., Maslov A.Y. and Subaslnev A.V., Sov. Pkys.-JETP GO, 1037, (1984) Pavlov G.G. and Meszaros P., Astioyliys. ,/., accepted, (1993) Potekhin A.Y. and Pavlov G.G., Astrophys.,/. 407, 330, (1993) Shibanov Y.A. c.t al, Physics of Isolated PnLsajs. Proc. Los Alamos Workshop, eds. K. Van Riper c.t al. Cambridge University Press, pp. 174-81, (1993) Vincke M., Le Dourneuf M. and Baye D., J. Phys. B. 25, 2787, (1992)

559

32 Computations of static white dwarf models: A must for asteroseismological studies P. BRASSARD Departement de Physique, Universite de Montreal, C.P. 6128, succursale A, Montreal, Quebec, H3C 3J7, Canada.

G. FONTAINE Departement de Physique, Universite de Montreal, C.P. 6128, succursale A, Montreal, Quebec, HSC SJ7, Canada.

Abstract

We present briefly a new generation of white dwarf models incorporating the latest developments of the constitutive physics. These are static models especially designed for accurate seismological studies. 32.1 Introduction The main goal of asteroseismology is the determination of the internal structure of a pulsating star through the analysis of its observed pulsation properties. One way to fulfill this goal is by producing a stellar model that reproduces to high accuracy the observed periods of oscillation. This is generally not possible through full evolutionary calculations as the parameters of a model must be tuned rather finely to satisfy the requirement of accuracy. However, computations of static models can be used with profit here. We have therefore developed the capacity to rapidly build complete static models of stratified H-rich (DA) or He-rich (DB) white dwarfs, especially suited for asteroseismological studies, by specifying the stellar mass, the H-layer thickness, the He-layer thickness, the convective efficiency and the effective temperature. 32.2 Method To build our models, we integrate with the help of a Runge-Kutta technique the equations of stellar structure and stellar grey atmosphere (see, e.g., Cox 560

Brassard & Fontaine: Static white dwarf models

561

& Guili 1968 and Mihalas 1978) from the high atmosphere (p £ 10" 13 ) down to the center of the star. We iterate this procedure until we find a model with Mr = 0 at r = 0. To have a good spatial resolution both in the interior and the external regions, we use the integration variable x[= ln(r/P)]. This is the same integration variable that we use when we solve the nonradial oscillation equations to find the oscillation periods of a stellar model (Brassard et al. 1992). Because we do not know the history of the star, we assume that L(r) is proportional to Mr. This approximation is very good for white dwarf stars (see, e.g., Lamb & Van Horn 1975). We also assume that the composition transition zones are given by diffusive equilibrium profiles (Vennes et al. 1988)t. In order to integrate the equations throughout the entire model we use a, version of the mixing-length theory that can be also used in optically thin regions (Bergeron et «/., 1992). We also use the following expression for

where 1), we have
32.3 E q u a t i o n of s t a t e For the low-density regions we use an analytic treatment of the equation of state by solving the Saha equations for a mixture of radiation and an ideal, nondegenerate, partially ionized gas for a mixture of H and He. In the region of partial ionization where nonideal and partial degeneracy effects are important we use, at our choice, an improved version of the tables of Fontaine et al. (1977, hereafter named FGV) or the new equation of state of Saumon & Chabrier (1993). In the deep interior, we use an improved version of Lamb's (1974) equation of state for a completely ionized pure plasma. Also, we have made sure that the transition between the envelope and the deep interior is made smoothly. For pulsation calculations, it is very important to ha,ve a model without f Equation (14) of that paper contains a misprint; this equation should read

562

Brassard & Fontaine: Static white dwarf models

0 01

0

0 0

Log

Fig. 32.1 Temperature-density diagram for the equation of state of He. The first two regions indicate from which table the data are extracted: Lamb (high density region), FGV (medium density region). Data from the Saumon-Chabrier equation of state is enclosed in the FGV region and is delimited by dots. Formalism for perfect gases is used in the low density and high temperature region of the diagram. The two thick lines refer to a typical {p,T) excursion of a pulsating DA white dwarf (TeH = 12,500 K, right line) and of a cool DA white dwarf (Tetf = 4000 K, left line). any artificial discontinuity. Composition interpolations are made with the "additive-volume" method of FGV.

32.4 Opacities At our choice, three set of radiative opacities can be used. For comparison with previous available models, we have included the older Coc and Stewart opacities (1970) arid the Huebner (1980) opacities. The third set is a combination of the new OPAL opacities of Rogers & Iglesias (1992) for T>6000 K

563

Brassard & Fontaine: Static white dwarf models

0

6

0

6

0

6

Log

Fig. 32.2 Temperature-density diagram for the conductive opacities of He. The three regions indicate from which table the data are extracted: Hubbard & Lampe (low density), liquid metal phase of Itoh ct al. (medium density) and crystalline phase of Itoh tt al. (high density). Values outside these three regions are extrapolated conductive opacities. The two thick lines refer to a typical (p, T) excursion of a pulsating DA white dwarf (Teff = 12, 500 K, left line) and of a cool DA white dwarf (Teft = 4000 K, right, line). We can point out here that the center of the cool DA white dwarf is crystallized. This is not apparent in this diagram because the crystallisation arises in C composition at higher density than He. We can also note that the low temperature model relies heavily on extrapolated data.

and of the molecular opacities of Lenzuni, Chernoff & Salpeter (1991) for T<6000 K. At high densities, where no new data were available, opacities were obtained, in order, from the Huebner data, the Cox &: Stewart data., or from linear extrapolation. For the conductive opacities, we use the tables of Itoh et ul. (1983) in the liquid metal phase with the low-temperature quantum corrections of Mitake et al. (1984). In the crystalline lattice phase, we use the data from Itoh et al. (1984). At low density, conductive opacities are supplemented

564

Brassard & Fontaine: Static white dwarf models

with the older tables of Hubbard k. Lampe (1969). Finally, in the remaining uncovered regions, conductive opacities are extrapolated from a fourth order polynomial fitted with data available in the valid regions. 32.5 Conclusion Applications for our models are potentially numerous. As an example, we have already built a white dwarf model that reproduces the observed periods of pulsation of the DA star G117-B15A within 0.1 s (Brassard et al. 1993). Acknowledgement We wish to thank A. Talon for providing continued inspiration. This research has been supported in part by the NSERC Canada and by the Fund FCAR (Quebec). References Bergeron P., Wesemael F. k Fontaine G. Ap. ,/. 387, 288, (1992) Brassard P., Fontaine G., Bergeron P., Wesemael F. k Vauclair G. in preparation, (1993) Brassard P., Pelletier C , Fontaine G. k Wesemael F. Ap. J. Suppl. 80, 725 (1992) Cox A.N. k Stewart, J.N. Ap. J. Suppl. 19, 261, (1970) Cox J.P. k Guili R.T. it Principles of Stellar Structure, (New York: Gordon k Breach) (1968) Fontaine G., Graboske H.C. Jr. k Van Horn H.M. Ap. J. Snppl. 35, 293, (1977) Hubbard W.B. k Lampe M. Ap. J. 163, 297, (1969) Huebner W.F., privatt communication, (1980) Itoh N., Mitake S., Iyetomi H. k Ichimaru S. Ap. J. 273, 774, (1983) Itoh N., Kohyama Y., Matsumoto N. k Seki M. Ap. J. 285, 758, (1984) Lamb D.Q., Ph. D. thesis, University of Rochester, (1974) Lamb D.Q. k Van Horn H.M. Ap. J. 200, 306, (1975) Lenzuni P., Chernoff D.F. k Salpeter E.E. Ap. J. Suppl. 76, 759, (1991) Mihalas D. Stellar Atmospheres, (San Francisco:Freeman) (1978) Mitake S., Ichimaru S. k Itoh N. Ap. ./. 277, 375, (1984) Rogers F.J. k Iglesias, C.A. Ap. J. Suppl. 79, 507, (1992) Saumon D. k Chabrier G., private communication, (1993) Vennes S., Pelletier C., Fontaine G. k Wesemael F. Ap. J. 331, 876, (1988)

33 The Chandrasekhar mass of a gravitating electron crystal D. ENGELHARDT AND I. BUES Dr. Remeis Sternviarte, 96049 Bamberg, Germany

Abstract The internal structure of a. white dwarf may be changed by a strong magnetic field. A local model of the electrons is constructed within a thermal density matrix formalism, essentially a Heisenberg magnetism model. This results in a matrix Fermi function which is used to construct an isothermal model of the electron crystal. The central density of the crystal is 108fc(//m3 independent of the magnetic field within the plasma and therefore lower than the relativistic density, whereas this density is constant until the Fermi momentum xf = 0.3 * me * c. Chandrasekhar masses up to 1.44 * \AMQ are possible for polarizations of the plasma zone lower than 0.5, if the temperature is close to the Curie point, whereas the crystal itself destabilizes the white dwarf dependent on temperature. 33.1 Introduction From the theory of magnetic phase transitions of solid state physics (Grosse 1988) it is expected, that the structure of a single white dwarf is changed drastically by a magnetic field. The polarized electrons throughout the star may interact due to a magnetic field. The nonlinear influence of a crystallization transition and the crystal itself may change the mass and radius of a white dwarf. We construct a thermal Heisenberg model of the electrons which results in a Fermi matrix function, which predicts a plasma crystal phase transition. This Fermi function is used within the standard white dwarf theory. An additional magnetic field stabilizes the star if the white dwarf is not totally crystallized. 565

566

Engelhardt & Bues: A gravitating electron crystal

33.2 The local model The quantum mechanical state of an election is determined by the symmetry of the magnetic field. The electron state with polarization may be written in terms of density matrices />, e.g.:

where \t e represents a four component spinor solution of the Dirac equation, summed over all degenerate states i. The spinors are transformed with a. Lorentz transformation. The density matrix represents a kinetic probability distribution. An interaction p. exists, which we assume to be proportional to />, e.g. f i a p . Thus the complete Hamiltonian may be written as iP

+ Vip,

(2)

where E{ and V{ represent numerical values of the kinetic and the interaction energy of the ith electron, respectively. This form is similar to the Heisenberg operator. The weighted mean of the energy of many electrons is taken in terms of a. Boltzmann distribution w under periodic boundary conditions, w = exp(-[3H),

(3)

where (3 is the inverse temperature T. The expectation value of the energy is calculated as

which represents a Fermi matrix function. For (i —* oo (T = OK) and H < 0 as well as for f3 —> 0 T —> oo the energy is proportional to /•>, e.g. E <x p, which represents the Maxwellian limit of a polarized nondegenerate gas. For H > 0 and /3 —> oo (T —*• 0) the electrons crystallize, since < E >oc p * cxp(-/3H)

oc /rmx-_K5o/>x.

(5)

Our model is restricted to 4x4 matrices of degenerate elections, thus the basic assumptions are valid for metallic hydrogen. The more realistic situation of a carbon or Fe crystal may be also be described by (5), since the model is a three dimensional lattice model. The matrix function is calculated by the use of the Jones (1941) calculus, which uses the fact that the function of a. matrix is similar to the matrix itself. This concept wa.s used succesfully for the calculation of the matrix Planck function and lea.ds to the Bose - Einstein

Engelhardt & Bues: A gravitating electron crystal th«ta-l

567

log(T)-7.7 I

•

I

•

I

•

I

Ckln-0.06 •

I

•

i

•

i

•

I

The mi microscopic magnetization M = 0.54 The functional

SThe he local F M and tnl T

ff" ^ ^ ^ °° UnC 1OU Sh ^

°

Wn

"

^ Bues 1992). The properties pp Where t h e ' ' magnetization

Fi61 2

M a n d t h e c n t i c a l i n d e x 7 | defined by M =< E * p*a >Oi\T-TF with T c t h e Curie temperature and a one Dirac matrix, are plotted T h e

r T^

tO Slid Ut dl tO SOlid SUte m d e l s and

S

—urements. ThereS

he model r , should T ^ provide a description° of magnetized — u rFee the 33.3 The white dwarf model

The global white dwarf model is constructed by the use of the standard

: Z e T T i ( L i e b T 7) and the matHx Feimi ^

i

F h :r;

the equation of state, then a density model is constructed and the effect m a g n e t k fieW U P n t h G S t l U C t U r e

° f t h e W h i t e ^ a r f is ***** C ° n s i s t e n t l y ^ -spect to Lorentz rotatran * ™ * « » - The pressure P is related to the °

enegy e g

Ep 1

(2JTAV J

d k

ex

(6)

568

Engelhardt & Bues: A gravitating electron crystal K*bata(tau-0)-10

1.6

P-.64

Ui«t*-1

1.4

mu-0

/

-

>/

1.8

-

1.1 1

-

.0 -

.a

-

/

/

-

.f

-

-.4

1 -.3

.

1

-.a

.

1

-.i

.

1

.

1

.

1

.

1

.

0 tau

Fig. 33.2 The critical exponent 7 is plotted versus r = (T-Tc)/Tc, where r c is the Curie temperature, for a microscopic polarization M = 0 54 The relation of kinetic energy per thermal energy is 10 at the Curie point The chemical potential mu is equal to zero. The critical exponent at M = 0 equals approximately 1.1, this agrees with measurements of solid state. The macroscopic density is defined as

n=

1

•I

f ...

1

where h is Planck's constant, k is the momentum and V is the volume. The eigen-values of E * p reflect the linear or quadratic energy/momentum relation and the isochronic Lorentz group. The interaction energy or the chemical potential relates via the Boltzmann distribution equation (7) and (6). The pressure in terms of the Maxwellian limit of the Fermi function has the form of a Lorentz force , e.g.:

Poc [d3kE*p,

(8)

which leads to the usual polytropic index 7' = 4/3..5/3 dependent on the relativistic energy. The behaviour of electron crystallization is more complex, because the phase transition depends on temperature and the electrons within the magnetized crystal interact with each other non locally in space. The nonlocal model assumes, that all states of the crystal arefilledup according to Pauli's principle due to the gravitation of the crystal itself until a relativistic Fermi momentum. A constant temperature of 5 * 107K inside the crystal is assumed. The number of states are counted consistently with the chemical potential. The behaviour of the total equilibrium of the star is considered in terms

569

Engelhardt & Bues: A gravitating electron crystal

I0OM

1000

J

10

100

J

1000

-I

• ' — '

10000

10*

-J

J11

10

10

Illlld

10

Fig. 33.3 The phase transition from the Maxwellian plasma to the magnetized crystal is plotted in terms of density and Curie temperature for different microscopic, magnetizations P = 0.96, P = 0.76, P — 0.17.

\

10'

r

10*

r

10'

r

10*

r

10*

r

/

r

; ^ ^

10000 1000 100 10

Fig. 33.4 The density versus Fermi momentum in units if riie * c2 for different polarizations P = 0.17,0.76,0.96 of the Plasma. The non polarized Fermi density as well as the zero temperature density is shown. The model is valid for xf > 0.1. It is remarkable, that the maximum polarization curve approaches the usual scalar Fermi function inside the crystal. The central density of the crystal approaches 10 8 kg/m 3 .

of the virial theorem. The gravitational energy is balanced by the pressure in matrix form. The additional pressure dependent on the magnetic, field is written according to Cox et al. (1968) in terms of the gravitational energy with magnetic flux conservation inside and outside of the crystal.

570

Engelhardt & Bues: A gravitating electron crystal

Fig. 33.5 The mass in units of 1.44Mo versus macroscopic, magnetization M for different polarizations P = 0.17,0.36,0.54 of the nondegenerate plasma. The phase transition enhances the pressure at M = 0, whereas the binding energy of the crystal is lower than the scalar Fermi energy. This picture avoids the singularity which occurs, when the pressure density relation is written in a polytropic form.

33.4 Conclusion From the analysis within the combined Chandrasekhar - Heisenberg model we conclude, that a white, dwarf with a magnetic field, which is beginning to crystallize has a larger Chandrasekhar mass of 1.4 * 1.44M0 for the plasma regime above the crystallization zone which is polarized at half maximum. Along the cooling sequence a larger magnetic field is needed to stabilize the white dwarf. References Cox, J.P., Giuli, R.T.,stellar structun, Science, New York, (1968) Engelhardt, D., Bues, I.,H7tt*c Dwarfs, Kluwer, Dordrecht, 229, (1993) Giosse,R,,Models in Statistical Physics and Quantum Fitld, Springer Berlin (1988) ' ' Jones, R.C., J.Opt.Soc.Am., 31,488, (1941) Lieb.E.H., Cornmun. Math. Phy.s.,112, 147, (1987) Sakurai 1.1.,Advanced Quantum Mechanics, Addison- Wesley, Reading, (1967)

34

Coulomb corrections in the nuclear statistical equilibrium regime D. GARCi'A Dpi de Fistca i Enginytria Nuclear, UPC, Barcelona, Spuin, and Laboratori d'Astrofisica, Insiitut d'Estudis Catalans.

E. BRAVO Dpt de Fisica t Enginyeria Nuclear, UPC, Barcelona, Spain. Laboratori d'Astrofisica, Institut d'Estudis Catalans and Centre d'Estudis Avancats, C.S.I.C, Blanes, Spain

Abstract The ionic contribution to the Equation of State (EOS) of a multicomponent plasma of nuclei in the nuclear statistical equilibrium regime is studied, and a method to compute the coulombic corrections within the framework of the linear mixing hyphothesis is proposed. Some consequences of including these corrections in the EOS are briefly analysed in relation with two concrete astrophysical scenarios, the Supernovae la explosion and the Accretion Induced Collapse of a massive white dwarf.

34.1 Introduction Since the pioneering work of Salpeter (1961) dealing with the corrections to an ideal plasma at zero temperature non ideal effects and, specially, the Coulomb corrections have been incorporated to the EOS of stella.r evolutionary codes in order to get a better understanding of the stellar evolution. The late stages of stellar evolution are specially sensitive to the non-ideal effects: the negative Coulomb pressure contributes to bound the cores of red giant stars and in white dwarfs the Coulomb and quantum corrections can drastically alter its cooling time. In the case of the progenitors of supernovae and neutron stars, the Coulomb pressure diminishes the Chandrasekhar mass limit below its nominal value of MQU = 5.8ye2 MQ. The importance of Coulomb corrections in a. finite temperature plasma, is measured by the plasma coupling constant, T, = (Ze) /r\k&T, where r\ 571

572

Garcia & Bravo: Coulomb corrections

is the mean interionic distance. Because of the relative independence of the Coulomb terms with respect to the temperature, traditionally this correction has been applied to moderately cold material at high density, like that found in white dwarfs or in the cores of giant stars, being important to determine the presupernova structure. Nevertheless, the Coulomb correction has usually been neglected once the explosive thermonuclear runaway starts in the cores of the progenitors of supernovae. In view of the Z 5 / 3 dependence of Fj, the Coulomb correction to the equation of state could be important at high temperature when matter is in the nuclear statistical equilibrium (NSE).

34.2 Coulomb corrections in NSE For a MCP, in the linear mixing approximation and for T\ > 1, the correction due to the electric interaction of the ions with an uniform electron background is given by a. mean over the correction each element would ha.ve if it was the only species present (one-component plasma, OCP). For Fj < 1 this assumption could not be true but in this case the corrections are not so relevant and a. calculation by using an interpolation formulae (Yakovlev &: Shalybkov 1987) probably suffice. Adopting the expression obtained by Ogata & Ichimaru (1987) for the OCP, we have calculated the coulombic correction to the free energy per nuclei in NSE, / c , from:

in units of k%T, being F e = e2/aek^T, a e the mean interelectronic distance, a = -0.898004, b = 0.96786, c = 0.220703, d = -0.86097, e = 2.5269. The '0i are averages over the chemical composition,

where Xu is the number fraction of the species v. In NSE the photodisintegration reactions a.re so fast that maintain the chemical abundances in equilibrium for densities greater tha.n ~ 10s g/cuv J , once a. high enough temperature (> 5 109 K) has been achieved. Thus, the chemical composition can be obtained a.s a function ofp, T and Ye only, and the 4>i can be evaluated a.s well. We have computed the NSE abundances for a set of values of p, T, and Ye (10 8 < p < 1012 g/cin 3 , 2 109 < T < 4 1O10 K, and 0.40 < Ye < 0.50), taking into account the nuclear binding energy

Garcia & Bravo: Coulomb corrections

573

and the chemical potential of each nuclei (Mochkovitch & Nomoto 1986) corrected by the electrostatic interaction. In Fig. 1 there is plotted the pressure correction relative to the ideal gas pressure, as a function of the temperature for different densities, and for Ye = 0.5 mol/g.

34.3 Specific scenarios 34.3.1 Supernovae la

(SNIa)

The ultimate effect of the coulombic corrections in NSE must be evaluated following a complete hydrodynamical calculation of a explosion model. Nevertheless, we can by now outline the main consequences on the dynamics of the explosion and the nucleosynthesis. As can be seen in Fig. 1 the corrections in NSE are specially important at densities near 1O10 g/cm3. In the standard model of SNIa a carbon deflagration sets in at a central density p ~ 3 — 4 109 g/cm3. Therefore, although no negligible, coulombic corrections will probably not alter too much the dynamics of the explosion. The composition of the ejecta of SNIa is altered by the inclusion of the Coulomb chemical potentials in the Saha, equation that governs the abundances of the nuclides in NSE. The negative contribution of the electrostatic, interaction to the chemical potential is more important for high Z nuclei, so its abundance is increased. In Fig. 2 there are compared the NSE abundances calculated with and without coulombic corrections for typical conditions at freeze-out (/> = 108 g/cm 3 , T = 5 109 K). The abundance of alpha particles is the most affected, being reduced by about 15% for Ye = 0.5 mol/g. This should reduce the amount of alpha-rich freeze-out in the. expanding material of the SNIa, reducing the abundance of nuclei such as 58Ni and 62 Ni, and allowing more 54Fe to be ejected to the interstellar medium. The abundance of neutronized nuclei can be affected in other way by reducing the net rate of electron capture in NSE. At p = 2 10'* g/cnr* and T = 8 109 K (typical conditions of the material when most of the electron captures take place) the main contribution to the neutronization comes from the Co isotopes. In Fig. 3 there are shown the variation of the Co isotopes abundances, due to the inclusion of the Coulomb chemical potentials. The differences can reach ~ 40%, and a. similar increase in the neutronization rate can be expected (except for the important contribution of free protons, whose abundance is nearly unaffected).

574

Garcia & Bravo: Coulomb corrections

34.3.2 Accretion Induced Collapse

(AIC)

For certain cases, the central ignition of a massive white dwarf destabilized by matter accretion from a. companion can take place at densities as high as 10 10 g/cm 3 (Hernanz et al. 1988). At such a high density the electron captures onto the nuclei present in the NSE regime can be fast enough to induce the collapse, rather than the explosion, of the white dwarf, forming a neutron star. Then, for a. fixed initial chemical composition of the white dwarf (Carbon and Oxygen, or Oxygen-Neon-Magnesium) there does exist a. critical density which separates the explosive and collapsing outcome. The value of this density is a function of the velocity of the conductive burning front that determines the advance of the flame near to the center. As a general rule, slow fronts lead to collapse and fast fronts lead to explosion. Recently, Timmes & Woosley (1992) have calculated the velocity of a. conductive burning front a.s a. function of the chemical composition prior to ignition, and of the ignition density. When this expression for the conductive velocity is used to follow the evolution of a massive white dwarf without taking into account the coulombic corrections, a value of pc ~ 8.5 109 g/cm 1 is obtained for the critical density. However: - Coulombic effects alter the velocity of the conductive combustion front itself. In order to obtain an idea of the importance of non-ideal corrections on the conductive velocity, the hydrodynamic evolution of a. conductive flame in a medium composed of C and 0 ha.s been followed by means of an hydrocode, in the form described in Garcia et al. (1990), and using a, small nuclear network to follow the changes in the elemental abundances. Although a, larger network is needed for a. suitable representation of the flame velocity (Timmes & Woosley 1992), our approach is sufficient to coiupaxe the effect of including the coulombic correction. Our calculations show that an overestimation of the flame velocity in about 10% results when the specific heat cv is not corrected from non ideal terms. - Near p = 1010 g/cm'1 the overpressure caused by the almost instantaneous rise in temperature is about 10%, and it is greatly due to the ionic contribution. In these conditions a coulombic correction of about 35% (see curve c in Fig. 1) represents a. substantial reduction in the total overpressure. In Fig. 4 there is shown the evolution of a. white dwarf after the explosive carbon ignition at />c = 7 10'' g/c.nrJ. Curve (1) represents the evolution obtained using a.n EOS without coulombic corrections in the NSE regime, and with the conductive burning velocity given by Timmes & Woosley. Curve (2) is the same, but including coulombic corrections in NSE, and with the

575

Garcia & Bravo: Coulomb corrections

front velocity multiplied by a factor 0.9, according with the above mentioned results. As can be seen, the track of the central density is substantially altered, changing from explosion to collapse for the same ignition density.

34.4 Conclusions A simple method to compute the Coulomb corrections in a complex system composed of several hundred of nuclei which are in nuclear statistical equilibrium has been presented. The main implications for SNIa models are of nucleosynthetic nature: the rate of neutronization at a fixed temperature can be increased by ~ 40% in the heavy nuclei contribution at Ye = 0.50 mol/g, and the amount of alpha-rich freeze-out reduced by about 15%. The AIC is favoured by the pressure decrease and specific heat increase due to the Coulomb terms. This fact, besides a slight reduction on the conductive velocity of the thermonuclear front, allows the minimum density at which a C-0 white dwarf could collapse be as low as 7 109 g/cm 3 . This work has been supported by the D.G.I.C.Y.T. grant PB00-0912, and the C.E.S.C.A. project Hydrodynamical Evolution of Compact Stars.

References Garcia, D., Labay, J., Canal, R. and Isern, J., Proceedings of the Symposium "Nuclei In The Cosmos", Baden (Austria), Eds H. Oberhummer and W. Hillebrandt, P 97, 1990. Herriariz, M., Isern, J., Canal, R. and Labay, 3.,ApJ, 324, p331, 1988. Mochkovitch, R. and Nomoto, K., Astronomy and Astrophysics,l54, pi 15, 1986. Ogata, S. and Ichimaru, S., Physical R.eview, A36, p5451> 1987. Salpeter, E., ApJ, 134, p669, 1961. Timmes, F.K. and Woosley, S.E., ApJ, 396, p649, 1992. Yakovlev, D.G. and Shalybkov D.A...SW 5c?'. Rev. E. Astrophysics and Space Physics,!, p311, 1989.

0.47

0.4a

0.49

Time (sj

Y. (mol/g) ~* Figure 1. Pressure correction relative to the ideal ion pressure. Densities: a) 10* , b) 1 0 . c) 10" g/cm

~* Figure 2. Abundances in NSE at freew out. With Ceulomo chemical ootentiols (solid line) and »ithout (dashed line)

j Abundances of the main n . „ , i n ,) o n < j «ith out n J c l , i , w i t h r,M

e-capture

(dasned line) Coulomo chemical potential

Hours 4 Evolution of a for o C™ .hue udwarf stortina at 7 10'a/crr ' w u " ="J™n« •"• '"!/»

35 Molecular Opacities: Application to the Giant Planets T. GUILLOT Observatotre de la Cote d'Azur, BP229, 06304 Ar«ce Cedex 4. France

D. GAUTIER Observatotre de Paris, 5 pi J.Janssen, 92195 Meudon Cedex. France,

G. CHABRIER Laboratoire de Physique, E.N.S. Lyon, 69364 Lyon Cedex 07. France

Abstract Present available interior models of giant planets assume that the internal transport of energy is entirely convective and, accordingly, rule out any possibility of radiative transport. New opacity calculations at temperatures and densities occurring within the giant planets, taking into account H2H2 and H2-He collision-induced absorption as well as infrared and visible absorption due to hydrogen, water, methane and ammonia are presented. These opacities are not high enough to exclude the presence of a. radiative zone in the molecular Hj envelope of Jupiter, Saturn and Uranus.

Abstract Les modeles de structure interne des planetes geantes developpes actuellement supposent que le transport de l'energie s'effectiie entierement, par convection, ce qui elimine toute possibility de transport radiatif. Des nouveaux calculs d'opacite aux temperatures et densites caracteristiques des planetes etudiees, tenant compte de l'absorption induite par collisions H2H2 et H/2-He ainsi que de l'absorption dans l'infrarouge et. dans le visible de l'hydrogene, l'eau, le methane et l'ammoniaque, sont presentees. Ces opacites ne sont pas suffisainment elevees pour exclure la presence d'une zone radiative dans l'enveloppe (l'hydrogene moleculaire de Jupiter, Saturne et Uranus. 576

Guillot et al.: Molecular opacities

577

35.1 Introduction Since the estimations of the conductive and radiative opacities in Jupiter by Hubbard (1968) and Stevenson (1976) all the interior models of the four giant planets have been calculated under the assumption that the energy is transferred by convection through the entire hydrogen-helium envelope. Consequently, the thermal profile is assumed to be adiabatic at all depths. This hypothesis is based on the fact that these conductive and radiative opacities are high and that at least Jupiter, Saturn and Neptune have a substantial intrinsic luminosity. New facts prompt us to reexamine the question. Firstly, new calculations have permitted to improve substantially the hydrogen-helium opacity. Secondly, progress in molecular spectroscopy allows one to take into account the opacity due to the most abundant minor atmospheric, components. Thirdly, Voyager measurements have provided a new upper limit of the intrinsic luminosity of Uranus which is significantly weaker than that previously thought. In the next Section, we present the method used to determine the presence of a radiative zone. Then we calculate radiative opacities. In the last section we comment our results.

35.2 Method Neglecting rotation and compositional gradients, we use the Schwarzschild criterion: the medium is convective when V a j < V r a j, and radiative otherwise. V a j = (d In T/d In P)s is the adiabatic gradient and Vra,j the radiative gradient. This latter is proportional to the intrinsic luminosity of the planet (taken from Pearl and Conrath, 1991) and to the Rosseland mean opacity:

where KU is the monochromatic absorption and Bv is the Planck function. Therefore, we have to calculate Rosseland opacity tables for each planet, with chemical abundances compatible with the infrared observations of their atmospheres (Gautier & Owen, 1989). In particular, the abundances of the CNO compounds and hea,vier elements are set to 2, 4 and 50 times the solar value for Jupiter, Saturn and Uranus, respectively. We use for this comparison the interior models of Chabrier ct al. (1992) for Jupiter and Saturn, and those of Hubbard & Marley (1989) for Uranus.

578

Guillot et al.: Molecular opacities

CM

U

c o o (0 <

1000

2000

3000

4000

3x10*

Wave Number [ c m"" ] Fig. 35.1 Synthetic, absorption spectra for Jupiter, at T=300 K (left) and T=30UU K (right). The cut-off is equal to 1000 cm" 1 . The heavy line represents the total absorption while the other lines show different contributions.

35.3 Opacities Rosseland opacity tables, adapted from the work of Lenzuni et al. (1991) in order to account for heavier elements than hydrogen and helium, are calculated for 200 < T < 5000K, and 10~ 5 < p < l g . c m " 3 . The following absorption sources are taken into account: • H2-He and H2-H0 Collision-Induced Absorption (CIA) (Boiysow & Frommhold, 1989, 1990) • Rayleigh scattering by H2 (Dalgaino & Williams, 1965) • Rayleigh scattering by H and He (Kurucz, 1970) • rLJ free-free absorption (Bell, 1980) • H~ bound-free absorption (John, 1988) • Infrared and visible absorption of H 2 O, CH4, NH 3 (GEISA data bank - Husson et a/., 1991) A chemical equilibrium is calculated, taking into account the following species: H, H+, H", H 2 , H+, H+, e~, Na, Na+, Mg, Mg+, Al, A1+, Si, Si+", K, K+, Ca, Ca+, Fe, Fe+, Cl, NaCl, KC1, CaCl. All the C,N,0 atoms are assumed to form CH 4 , NH3 and H 2 O, respectively, according to the results of Barshay & Lewis (1979). The absorption of H 2 O, CH 4 , NH3 is calculated assuming a Lorentz profile with a cut-off set to 100, 500, and 1000 cm" 1 , respectively: at this distance from the line-center, the absorption is supposed to be exponentially decreasing (Birnbaum, 1979).

579

Guillot et al.: Molecular opacities 1

r

x

i

•

•

i

•

•

i

•

a

Uranus

N

""•*.

1

. •-. '"• x\

••V\ H—

f—^ '•

:

'

.•'

/

r :

/

v

/

:

' \

t \

1

q d 1 . . . .

1000 2000 3000 4000

:

NS

/

500

1000

1 •

1500

. . .

2000

Temperature [K] Fig. 35.2 Comparison of the radiative (dot-dashed lines) to the adiabatic gradients (plain lines) in the case of Jupiter (left) and Uranus (right). The medium is expected to be radiative when Vad > Vrad- The dashed lines correspond to the radiative gradient calculated with the opacity of hydrogen and helium alone. The various dot-dashed lines correspond to values of the cut-off equal to 100, 500 and 1000 cm"1. The vertical bars show the uncertainties on the measured intrinsic, luminosities.

Non-idea] effects for the CIA of H2-H2 and H2-He are taken into account, following the method described by Lenzuni & Saumon (1992). Figure35.1 shows two synthetic spectra at r=300K and T=3000K. One can see that the contribution of H2O, CH4, NH3 to the total opacity is significant at low temperatures, as these molecules are strong absorbers in the infrared, region which has then the most important weight in the Rosseland opacity. At higher temperatures and larger densities, the CIA, proportional to p2, and H<J free-free and H~ bound-free absorptions dominate the spectrum.

35.4 Results We compare in Fig 35.2 the radiative and adiabatic gradients for Jupiter and Uranus. Saturn and Jupiter have similar internal structures. Therefore all the results for Jupiter are also true for Saturn, at least qualitatively. The case of Neptune has not been treated here. The dashed lines in Fig35.2 correspond to radiative gradients calculated with hydrogen and helium only. A comparison of these curves to the adiabatic gradients (plain lines) shows that hydrogen and helium cannot ensure convection in the entire molecular hydrogen-rich envelopes of Jupiter and

580

Guillot et al.: Molecular opacities

Uranus. In both planets, the presence of H2O, CH4 and NH3 (dot-dashed lines) restore convection at temperatures below 1200 K (corresponding to pressures below 1 and lOkbar for Jupiter and Uranus, respectively). According to our calculations, a deep radiative window is therefore present in Jupiter, Saturn and Uranus. In the case of Jupiter, convection appears again at temperatures above 2900 K for which the absorption of H~ and H^~ becomes preponderant. At these levels the contribution of metals is significant. In the case of Uranus the comparison ends at 2000K, as our calculations cannot apply to the "ices"-f "rocks" core of this planet. We emphasis that our opacity calculations are uncertain, due to the lack of knowledge on both the chemical composition and absorption of the medium considered. However the predicted radiative zones will vanish only if these opacities are underestimated by more than one order of magnitude. Moreover, the observations do not preclude a very low value for Uranus' intrinsic luminosity. This would lead to a still larger radiative zone in this planet. Non-adiabatic models of the giant planets are needed for studying the consequences of the presence of a radiative zone on their internal structure. This is the subject of our next communication. References Barshay S.S., Lewis J.S., Icarus 33, 593-611, (1978). Bell K.L., J. Phys. B 13, 1859-1865, (1980). BirnbaumG., J. Quant. Spectrosc. Radial. Transfer 21, 597-607, (1979). Borysow A., Frommhold L., Astrvphys. J. 341, 549-555, (1989). Borysow A., Frommhold L., Astrophys. J. 348, L41-L43, (1990). Chabrier G., Saumon D., Hubbard W.B., Lunine J.I., Astrophys. J. 391, 817-826, (1992). Dalgarno A., Williams D.A., Proc. Phys. Soc. 85, 585-589, (1965). Gautier D., Owen T., In Origin and Evolution of Planetary and Satellite. Atmospheres (eds. S.K. At.reya, J.B. Pollack, and M.S. Matthews), University of Arizona Press, Tucson, pp. 487-512, (1989). Hubbard W.B., Astrophys. J. 152, 745-753, (1968). Hubbard W.B., Marley M.S., Icarus 78, 102-118, (1989). Husson N., Bonnet, B., Scott N.A., Chedin A., J. Quant. Spectrosc. Radiat. Transfer 48, 509-518, (1992). John T.L., Astron. Astrophys. 193, 189-192, (1988). Kurucz R.L., Smithsonian Obs. Spec. Rep. 309, 1-291, (1970). Lenzuni P., Chernoff D.F., Salpeter E.E., Astrophys. J. Suppl. 76, 759-801, (1991). Lenzuni P., Saumon D., Rev. Mex. Astron. Astrofis. 23, 223-230, (1992). Pearl J.C., Conrath B.J., J. Geophys. Res. Suppl. 96, 18921-18930, (1991) . Stevenson D.J., Ph.D. thesis, Cornell University, (1976).

36 On Radiative Transfer Near the Plasma Frequency at Strong Coupling YU. K. KURILENKOV Institute for High Temperatures, Russian Academy of Science,, Moscow 1211,12

H.M. VAN HORN Department of Physics and Astronomy and C. E. Kenneth Mees Observatory, University of Rochester, Rochester, NY 14627-0011, U.S.A.

Abstract The effects of strong coupling on the frequency-averaged optical characteristics of plasmas, such as the Rosseland mean-free-path, are considered. The general expression for the Rosseland mean opacity has been analyzed in terms of the transverse dielectric function of a dense plasma, and the frequency-dependent effective collision frequency. The corresponding values of the absorption coefficient and the refractive index for a dense plasma are presented at u < u)p in obvious forms.

36.1 Introduction We are concerned with rather "cold," classical, strongly coupled plasmas (SCPs), where the energy hu>p of a plasmon is comparable to or higher than the thermal energy of the elections. The electron plasma frequency is up = (47rn e e 2 /m c ) 1 / 2 . The plasma under consideration is specified by two parameters: (1) the ion coupling parameter T = (Ze)2/3/a > 1,

(1)

where /? = l/A.-gT' is the inverse temperature, Z is the ionic charge, and a = (3/47rn{)1/3 is the ion sphere radius, and (2) the electron degeneracy parameter

0 = 1//3EF = 2(4/9n)2^Z^:irs/T

> 1,

(2)

where Ep - (3n2nc)2/9Ti2/2mc is the Fermi energy of electrons at the temperature T = 0, and rs = (m c e 2 /ri 2 )(3/47rn e ) 1 / 3 is the electron density parameter. Under these plasma, conditions, nonideality effects in the optical properties are expected to be important. On the other hand, crystallization — which occurs for T > 180 — does not yet take place in this modest 581

582

Kurilenkov & Van Horn: Radiative transfer near the plasma frequency

F range. We are thus dealing with an intermediate state between ideal and solid-density plasmas. Such plasmas occur frequently both in postmain-sequence phases of stelar evolution (stellar interiors, core-envelope boundaries, and surface areas) and in laboratory experiments with high local concentrations of energy (see Van Horn and Ichimaru 1993). SCP effects manifest themselves differently in the total absorption coefficient for different frequency domains. We restrict ourselves here mainly to the important, but partial, case when the frequency is u> < up. This frequency range is critical for radiative transfer, especially under conditions where hup ~ kT. The collective modes are important in spite of the collision-dominated character of the SCP, so we attempt to consider the Rosseland mean absorption in terms of the collision frequency and the dielectric function.

36.2 Dielectric Function and Dynamic Collision Frequency In fact, all the information on the optical properties of a plasma is included in the transverse dielectric function £tr(u). In turn, the dielectric function depends upon the dynamic, electron-ion collision frequency v(u): u2 etr(u)=l-

p

(3)

u[u + ii/(u)\ for v(u) < u. Explicit theoretical results for v{u) are well-known in two limiting cases: in classical dilute plasmas (F < < 1 and 0 > > 1) and in quantum cases (0 < < 1), when the electron-ion interaction is weak enough. The Born approximation, which is acceptable for u> < u>pc (Ichimaru, Iyetomi, and Tanaka 1987; Berkovsky, Kurilenkov, and Milchberg 1992) gives the following expression for the dynamic collision frequency: 2

2

* M?K<(
Imnec(q,u) Mgw)|2

•

(4)

Here S'u(q) is the ion structure factor, fl ec (^,cj) is the electron polarization function, and ec(q,u)) — 1 — '(>,-;fc(
Kurilenkov & Van Horn: Radiative transfer near the plasma frequency

583

0.00 0.20

Fig. 36.1 The SCP refractive index n(u>). Curve 1 gives the results for an ideal plasma; curve 2 for T = 0.5,0 < 1; curve 3 for T = 0.8,0 « 4; and curve 4 for the case v = uv. The cross-hatched areas give estimates based on the MD results from Sjogren, Hansen, and Pollock (1981; T = 0.5, S « l : area A) and from Valuev (1981; T w 1, 0 » 1: area B). field correction and of non-Born efects are discussed elsewhere (Ichimaru, Iyetomi, and Tanaka 1987; Berkovsky and Kurilenkov 1993).

36.3 Refractive Index and Absorption Coefficient at U <

Up

From the roots k(u) of the dispersion relation k2(u) = (u/c)2Etr(u,k), the optical properties of any medium are easily denned: the refractive index n(u) = (c/u)B.e k(u), the absorption coefficient K(U) = Im k(u), the skindepth for electromagnetic wave penetration into a plasma L = 1/Im k(u) (see Berkovsky, Kurilenkov, and Milchberg 1992), etc. We suppose that u > > kvTe, and we neglect the ^-dependence of the transverse dielectric function Ztr in the discussion below (long-wavelength approximation). The refractive index and absorption coefficient are essential for calculations of the SCP Rosseland mean-free-path. Some results for n(u) and K ff(.u) a r e * m i s presented in Figs. 1 and 2, respectively, corresponding to different SCP parameters. Some estimates of n(u) and «//(w) from available molecular dynamics (MD) data for the high-frequency conductivity are also presented. The SCP collective modes at different T and 0 [in particular, the values of v(u/up)} affect the functions n{u) and Kff(u), including anomalous absorption and refraction (curves 4 in Figs. 1 and 2).

584

Kurilenkov & Van Horn: Radiative transfer near the plasma frequency

0.2 -

0.0

0.2

Fig. 36.2 The SCP free-free absorption coefficient «//(w). Curve 1 gives the results for T = 0.1,0 ss 2; curve 2 for T = 0.5,0 « 4; curve 3 for F = 0.8,0 « 4; and curve 4 for the case v = up. The triangle gives an estimate based on the MD results of Furukawa et al. (1990).

36.4 Rosseland Mean-Free-Path in SCP The data presented above can be used to calculate frequency-integrated optical characteristics such as the Rosseland mean-free-path f.R. The SCP effects manifest themselves in different ways in the total absorption coefficient in different frequency domains (Berkovsky et al. 1993). Let us discuss some qualitative features of (.R. The following expression for the Rosseland mean-free=path, expressed in terms of the transverse dielectric function (i.e. in terms of n(w) and *//(<*;) of dense plasmas, is a new generalization of earlier results (Bekefi 1966; Van Horn 1992): R

fo°°{dBJdT)du

(5)

where BU(T) is the usual Planck function. Correspondingly, dense plasma effects in the opacity depend upon the combination of plasma parameters F, 0, and TiWp (i.e., upon the charge densities and temperature, as usual). If the peak of the Rosseland factor in eq. (5) corresponds t o w < up, then the strong absorption clue to collective effects can produce a decrease in 1.R (or increasing opacity) in spite of the collision-dominated character of the plasma. Note that this effectively corresponds to a cut-off of the integration in the numerator of eq. (5) at frequencies less than ~ up (Van Horn 1992). However, when the Planck peak corresponds to w >> u;p, the role of

Kurilenkov & Van Horn: Radiative transfer near the plasma frequency the collective modes discussed above disappears, and the role of bound-free absorption is reduced significantly (and the free-free absorption turns out to be even less than the Kramers values: Berkovsky et al. 1993). Under these conditions, the Rosseland mean-free-path (.R may increase in comparison with the ideal value (reduced opacity). Note that corresponding features of radiative transfer, in particular, near the core-envelope boundaries of white dwarfs, probably have important consequences for their cooling (Van Horn 1991).

References Bekefi, G. 1966, Radiation Processes in Plasmas (John Wiley & Sons, Inc.: New York). Berkovsky, M. A., and Kurilenkov, Yu. K. 1993, Physica, to be published. Berkovsky, M. A., Kurilenkov, Yu. K., Kelleher, D., and Skowronek, M. 1993, ./. Phys. B: At. Mol. Opt. Pkys., 26, 2475. Berkovsky, M. A., Kurilenkov, Yu. K., Kobzev, G. A., Milchberg, H. M., and Skowronek, M. 1993, in Strongly Coupled Plasma Physics, eds. Van Horn, H. M., and Ichimaru, S. (University of Rochester Press: Rochester, NY), p. 263. Berkovsky, M. A., Kurilenkov, Yu. K., and Milchberg, H. M. 1992, Phys. Fluids, B4, 2423. Furukawa, H., Nichihara, K., Kawaguchi, M., Sakagami, H., Hiramat.su, T., and Yasui, H. 1990, in Strongly Coupled Plasma Physics, ed. S. Ichimaru (Elsevier Sci. Publ.: Tokyo), p. 613. Ichimaru, S., Iyetomi, H., and Tanaka, S. 1987, Phys. Rep., 149, 91. Sjogren, Hansen, J. P., and Pollock, E. L. 1981, Phys. Rev., A24, 1544. Valuev, A. A. 1981, High Temp., 19, 200. Van Horn, H. M. 1991, Science, 252, 384. Van Horn, H. M. 1992, Bull. A. A. S., 24, 824. Van Horn, H. M., and Ichimaru, S., eds. 1993, Strongly Coupled Plasma Physics (University of Rochester Press: Rochester, NY).

585

37 Effects of Superfluidity on Spheroidal Oscillations of Neutron Stars UMIN LEE, T.J.B. COLLINS AND H.M. VAN HORN Department of Physics and Astronomy and C. E. Kenneth Mees Observatory, University of Rochester, Rochester, NY 14627-0011, U.S.A.

R.I. EPSTEIN Los Alamos National Laboratory, MS D436, Los Alamos, NM 8754-5, U.S.A.

Abstract In the limit of short wavelengths, it has been shown that superfluidity significantly affects wave propagation in neutron stars. Here we abandon the short-wavelength restriction and extend these calculations to global oscillation modes. In the present analysis, the solid crust of the neutron star is divided into an outer crust and an inner crust, and a superfluid of neutrons coexists with the solid la,ttice in the inner crust. We have computed several low-order global spheroidal modes for / = 2 both with and without superfluidity. We find that superfluidity in the inner crust affects the frequency spectra of acoustic (p-) modes, shear ($-) modes, and interfacial (i-) modes, although the surface gravity ((/-) modes are not affected at all.

37.1 Introduction Most previous calculations of the non-radial oscillations of neutron stars have completely neglected the effects of superfluidity (cf McDermott, Van Horn, and Hansen 1988 and references therein). Epstein (1988) has previously considered superfluid effects, but only in the short-wavelength limit, where the length scales for variations in equilibrium quantities are all assumed to be much longer than the typical wavelength of an oscillation. In general, global oscillations may be either spheroidal or toroidal in character. Va,n Horn and Epsetein (1990) extended Epstein's short-wavelength results to include the global toroidal oscillation modes of neutron stars. More recently, Mendell (1991) and his colleagues have also considered the effects of superfluidity, but they employed simple models for neutron stars, and their analysis did not reflect the variety of oscillation modes of realistic, neutron stars. Although our analysis does not consider superfluidity in the core, we 586

Lee et al.\ Effects of superfluidity

587

use a more realistic neutron star model to investigate the effects of superfluidity in the inner crust on various oscillation modes of neutron stars. The goal of the present calculations is to extend these previous analyses to the global spheroidal oscillation spectra of neutron stars. In the two-fluid model of superfluidity, matter is represented as a mixture consisting of a superfluid component coexisting with a normal fluid. For all but the youngest neutron stars the temperature kgT < lkeV (where kg is the Boltzmann constant) is much less than the energy gap S ~ 1 — 2MeV predicted for the superfluid (Epstein 1988 and references therein). Thus the normal fraction is vanishingly small, and we neglect it completely in the following pulsational analysis. An object traveling through a superfluid more slowly than a critical velocity encounters no resistance. Since the critical velocity of the superfluid in a neutron star is on the order of the speed of light, one would initially expect no coupling between the solid lattice in the inner crust and the neutron superfluid. However, the neutron superfluid also penetrates the nuclei, where it exists at a, much higher density. Thus, movement of the nuclei requires propagation of regions of increased density through the fluid. This results in an effective drag force on nuclei moving through the superfluid, coupling the two (Epstein 1988). As in Epstein (1988) and Van Horn and Epstein (1990), the interaction between the lattice and the fluid is modeled by a viscous coupling between the two media. The Lagrangian perturbations of the lattice and superfluid stress tensors are, respectively, t><* j

KNuut>

2(

unSij) + CwuSij,

(1)

and dafij = I<swltbij + CuHbi:n

(2)

where the quantities u\j and W{j are the strain tensors of the lattice and fluid (it// = un + «22 + U33)i I1 is the shear modulus of the solid lattice, the A's are bulk moduli, and C is the coupling coefficient between the two media. The superscripts ,$' and N denote the quantities associated with the superfluid and normal matter, respectively. The stress tensors defined above lead to coupled momentum equations, which govern the oscillations of the two media. In this study, nonradial oscillations of neutron stars are investigated within the framework of the Newtonian theory of stellar oscillations. Oscillations are assumed to be a,diaba.tic, and no thermal effects associated with

Lee et al: Effects of superfluidity

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V \ 1

1.0

Fig. 37.1 The densities and pressures as functions of the fractional radius. The solid lines denote the total density p(= pN + ps) and pressure p = (pN + PS)- The dash-dotted lines denote ps and ps in the inner crust, while the dotted lines correspond to pN and pN in the inner crust. oscillations are included. We neglect the effects of magnetic fields and rotation on the oscillations. We employ the Cowling approximation and ignore the Eulerian perturbation of the gravitational potential.

37.2 Numerical Results The physical parameters of the unperturbed neutron star model used in the present calculations are M* = 0.503M©, /?.* = 9.839km, Tc = 1.03 x 107K, and pc = 9.44 x 1014g cm"'. We have computed the shear modulus for this model from the expression /i = 0.2945Z2e2n^3 (Baym and Pines 1971), using the the relation between the number density n/v of nuclei with charge Z and the mass density /> as given by Negele and Vautherin (1973). Above the neutron-drip density (4.3 x 10 n g cm" 3 ), neutrons emerge from the nuclei to form a free neutron superfliiid. This density marks the boundary between the inner and outer crust. The quantities that describe the neutron superfliiid in the inner crust are calculated following the prescription given by Epstein (1988). The densities and pressures in and around the inner crust are shown in Fig. 1. In Table 1, we present results for several low-order / =• 2 spheroidal modes including the effects of superfluidity, as well as results for those without superfluidity, where a = Os/KUGM*. The results are quite dramatic. The increased speed of the transverse waves, ct = (/z/^yv)1/2, greatly changes the frequency spectrum of the shear modes trapped in the crust. It changes the frequency spectrum of acoustic modes as well.

Lee et al.\ Effects of superfluidity

589

Table 37.1. The dimensionless frequency a and the period IT for surface g-modes, i-modes, s-modes and p-modes for 1 = 2.

nx

Mode 92

9\ «2 »i

*2 *3 *4

F

f

V\ Vi P3 P4

n2

2.037(-3) 3.230(-3)

368.43 232.38

2.037(-3) 3.230(-3)

368.43 232.38

8.282(-3) 1.028(-l)

90.626 7.2956

8.282(-3) 1.716(-1)

90.626 4.3738

3.093(-l) 5.556(-l) 7.483(-l) 8.571(-1)

2.4266 1.3509 1.0030 0.8740

7.015(-l) 1.183(+0) 1.437(+0) 1.728(+0)

1.0700 0.6344 0.5224 0.4344

— 1.886(+0) 4.038(+0) 4.736(+0) 6.062(+0) 7.129(+0)

— 0.3979 0.1859 0.1585 0.1238 0.1053

2.065(+0) 1.482(+0) 4.893(+0) 6.231(+0) 7.837(+0) 9.755(+0)

0.3635 0.5064 0.1534 0.1205 0.0958 0.0768

The quantities d\ and IIi are without superfluidity, while a-z and II2 are with superfluidity in the inner crust. The periods II are given in ms. Fig. 2 shows the radial component of the displacement vector, £,/r, for /-modes (modes with no nodes of £r/r) with and without superfluidity in the inner crust. We have two /-modes in the case with superfluidity. One of the two /-modes, denoted by F in Table 1, has a large amplitude £r/r for the supeifluid component, which is comparable with that of the lattice. Note that the oscillation amplitudes of the supeifluid component are usually quite small compared with those of the lattice. In particular, this is true for the conventional /-mode, denoted by / in Table 1. As shown by Table 1, the surface g modes are not affected at all by superfluidity in the inner crust.

37.3 Conclusions We have computed several low-order, global / = 2 spheroidal oscillation modes of neutron stars with M = 0.503M©, including the effects of superfluidity. The presence of supeifluid neutrons in the inner crust increases the local transverse wave speed substantially. This decreases the global oscillation periods of those modes which have significant amplitudes in the

590

Lee et al: Effects of superfluidity

0.0

0.2

0.4

0.6

0.8

1.0

0.4

0.6

rlR Fig. 37.2 The eigenfunctions £,/r of the F- and / - modes. In each panel, the solid and dotted lines denote the eigenfunctions associated with the normal matter and the supeifluid, respectively. In the right-hand panel, the dash-dotted line is the /-mode for the case without superfluidity in the inner crust. inner crust, often by very large factors, thus effectively restricting motions associated with the pulsations to the outer crust. Conversely, modes which do not have significant amplitudes in the inner crust, such as the surface (/-modes, are completely unaffected. Acknowledgements This work has been supported in part by the National Science Foundation under grant AST 91-15132, in part by NASA grant NAGW-2444, both through the University of Rochester, and in part by the U.S. Department of Energy.

References Baym, G., and Pines, D., Annals of Phy*., 66, 816, (1971) Epstein, R. I., Ap. ./., 333, 880, (1988) McDermott, P. N., Van Horn, H. M., and Hansen, C. J., Ap. J., 325, 725, (1988) Mendell, G., Ap. J., 380, 515, (1991) Negele, J. W., and Vautherin, D., Nucl. Phys., A207, 298, (1973) Van Horn, H. M., ami Epstein, R. I., Bull. A. A. S., 22, 748, (1990)

38 Magnetic Field Decay in the Non-superfluid Regions of Neutron Star Cores A.G. MUSLIMOV AND H.M. VAN HORN Department of Physics and Astronomy and C.E. Kenneth Mees Observatory, University of Rochester, Rochester NY 14627-0011 USA

Abstract We consider a simple model for the evolution of a poloidal magnetic field initally trapped in a. region containing normal npe matter within the outer liquid core of a neutron star. We have performed numerical computations for neutron stars with masses of 1.4, 1.6, and 1.7 MQ that undergo very rapid cooling due to the direct Urea process. Because the timescale for the magnetic field deca.y is directly proportional to T2, such a cooling history produces a rapid decline in the magnetic-field strength B, even for B as low as ~ 1012 G. In particular, we show that an initially quasi-homogeneous magnetic field of strength B = 1012 G declines during the first ~ 1 Myr. 38.1 Introduction The calculations of Ba.ym, Pethick, and Pines (1969a) have shown that the electrical conductivity of matter in the core of a neutron sta.r is too large to permit ohmic decay of the magnetic field within the age of the Universe. Recently, Haensel, Urpin, and Ya.kovlev (1990; hereafter HUY) have pointed out. that the magnetic-field strength |B| ~ 1012 G typical of pulsars is sufficiently strong that the anisotropy of the transport coefficients cannot be neglected and that the "resistivity" for current flow perpendicular to B is many orders of magnitude larger than that for current flow parallel to B. Using a simple "toy" model, they found that internal fields B > 101'1 G can decline to ~ 1012 G in times ~ 107 years, but tha.t fields < 1012 G remain practically unchanged on this timescale. In this paper, we derive a.n equation for the. evolution of the dipole magnetic field, taking into account the anisotropy and nonlinearity of the conductivity tensor produced by the strong field. We solve this equation numerically for the case in which the field is entirely confined to the outer, 591

592

Muslimov & Van Horn: Non-super fluid neutron star cores

non-superfluid, normal-ytpe-matter core of the neutron star. We regard these calculations as a step toward the construction of more complex models of field evolution in neutron stars. We have used the new neutron-star cooling histories obtained by Page and Applegate (1992; hereafter PA) which allow the possibility of the direct Urea process in the neutron star cores. For our present calculations, we have chosen their 1.4, 1.6, and 1.7 MQ models with maximal cooling.

38.2 Basic Equations and Numerical Results In this section, we sketch briefly the derivation of an equation describing the evolution of the magnetic field confined to that part of the core of a neutron star containing normal (z.e., non-superfluid) npe matter. Explicit expressions for the components of the resistivity tensor 7£ under conditions appropriate to neutron star interiors with normal npe matter are presented in Haensel, Urpin, and Yakovlev (1990). For strong fields, they find TZ± = 7£|| + A.B2, where TZ± and 1Z\\ are the components of V, associated with current flow perpendicular arid parallel to B, respectively, and A is a coefficient given below. For this case, an appropriately generalized form of Ohm's Law can be rewritten as E = ft||j + ft//j x B + AB x (j x B),

(1)

where the first term is responsible for ohmic dissipation, and the second represents the Hall drift. ^Frorn the strong-field relations given by HUY, the coefficients 1Zy and A are ^

5

A

(

2

)

where in* ss Tikf(c)/c. is the effective mass of the election, including relativistic effects, m* is the effective mass of the proton (~ 0.8 mp, c.f. Frima.11 and Maxwell 1979), nc is the number density of electrons, r c p is the electron relaxation time (due to Coulomb scattering from protons), and rpn is the proton relaxation time due to scattering from neutrons. According to Baym, Pethick, and Pines (1969a,b), the relaxation times rep and r pil are, respectively, 1/2

s and Tpn « 10" 1 9 Ty2 s.(3) Here To = T/10 5 ' K; po = 2.8 x 1014 g • cm" 3 is the nuclear-matter density; and other symbols have their usual meanings.

Muslimov & Van Horn: Non-superfluid neutron star cores

593

We now consider the evolution of the dipole component of a purely poloidal magnetic field confined to the neutron star core. Note that the Hall effect disappears in this case. For a purely poloidal field, we can express B in the form [in spherical coordinates (r, 6,)]: / S B = Bo\2^rcos0e rV r1

1 OS -—sineeo r Or

\ , )

(4)

where Bo is some normalization magnitude of the field, and e r and e# are unit vectors in the radial and transverse directions. To obtain the partial differential equation satisfied by the stream function S, we first note tha.t the form assumed for B yields a purely azimuthal (and solenoidal) current density j , from Ampere's Law. With the Hall term neglected, eq. (1) yields E from B and j . For this case, the r- and ^-components of Faraday's Law can be shown to contain the same physical information. Using Fa.rada.y's La.w we arrive at the following nonlinear equation for evolution of the function S(r,t): dt ~ The main problem with eq. (5) is that even if the field were initially dipolar, higher-order multipoles may be generated due to the third term in expression (1), which we have not taken into account. While the presence of higher-order multipoles can, in principle, modify the evolution of the dipole magnetic field, this problem is beyond the scope of the present paper. To solve equation (5) we need to specify appropriate boundary conditions. Here, we consider a simplified model in which the internal magnetic field is entirely confined to the core liquid region with normal npe matter (i.e., without superfluidity or superconductivity). For the 1.4 M© neutron star model we have employed in these calculations (Page and Applegate 1992), the stellar radius is B* = 11.3 km, the radius of the core is ~ 10 km, with central density pc = 4.76po! aiul the normal ripe-matter fluid occupies the outer layer of the core, with thickness ~ 4 km. We take the magnetic field to be confined entirely to the layer of normal npe matter. We assume that both the function S\r, t) and its derivative OS/Or vanish at the outer and inner boundaries of that region. We also adopt the normalization value Bo = 0.5 x 1012 G (see eq. [4]) corresponding to an initial, quasi-homogeneous magnetic field with field strength 1012 G at the magnetic poles. For the 1.4 MQ model, the internal temperature remains relatively constant, at T ~ 108 K, during the first ~ 102 yr; then it suddenly drops to ~ 3 x 10' K. Thereafter, the temperature decreases monotonically until it

594

Muslimov & Van Horn: Non-superfluid neutron star cores

1.0

o

s 2

3

4 5 LOG t (yr)

6

Fig. 30.1 Lett, panel: The temporal evolution of the magnetic energy, £jif (<). normalized to its initial value, EM(Q)- The magnetic energy decays by a factor of 10 in the first, ~ 4 x 105 years and then drops rapidly toward zero. Curves for the 1.4, 1.6, and 1.7 M 0 models of Page and Applegate (1992, Fig. 1) are shown. For each case, the initial field strength is normalized by the parameter-value Bu = 0.5 x 1012 G. Right panel: The spatial and temporal evolution of the dimensionless Stokes stream function Y(x,t) - S{x,t)/r'{, where x = r/rc is the radial distance in units of the core radius r c . The calculations are performed for the 1.4 M& neutron star model, and the initial field strength is normalized by the parameter-value 5,, = 0.5 x 10 12 G. The stream function is confined to the outer (non-superfluid) core of npc matter, and at the initial time shown, t = 10" years, it peaks near x » 0.80. The shape of Y as a function of x changes very little as t increases, but the amplitude decreases with t, at first only gradually, but then more abruptly after ( > 3 x 10r> years. This is seen more clearly in the left panel.

leaches ~ 3 x 106 K at * ~ 3 x 105 yr. After t ~ 3 X 105 yr the internal temperature drops catastrophkally. The more massive models have similar cooling histories, but the temperature plateaus are somewhat lower, and the temperature drops happen somewhat earlier than for the 1.4 M o model (see PA). In the left panel of Fig. 1 we depict the temporal behavior of the ratio E M (t)/E M (0), where

is the magnetic energy of the normal fluid layer, with volume VB. The rate of field decay increases as the neutron star cools, and the field effectively disappears at t w 1 Myr. The reason for this can be seen by dimensional analysis of eq. (5): with Bn = 1; p = (>0\ (i = 30; T9 = 3 x 1 0 - \ corresponding to the temperature T = 3 x 106 K at the time

Muslimov & Van Horn: Non-super fluid neutron star cores

595

t ~ 3 x 105 yr when the internal temperature begins to drop rapidly; and R$ — 0.4, corresponding to the thickness of the layer of normal npe matter, the characteristic decay timescale can be shown to be r<]ecay = 0.7 x 106 years. (Note that HUY use a definition of r<jecay that is smaller by a factor of 7r2). This is comparable to the timescale on which our explicit numerical calculations show that the magnetic energy in the layer with normal npt matter actually does decline. As the left panel of Fig. 1 shows, EM(1) begins to drop rapidly at just the time when the core temperature starts to fall. The temporal evolution of the dimensionless Stokes stream function Y = S/r%, where rc is the radius of the neutron star core, is shown in the right panel of Fig. 1. Even if there is no proton superconductivity, the presence of superfluid neutrons decreases the value of the parameter A (see Yakovlev 1993) and makes diffusion of the magnetic field inefficient. If a region occupied by a poloidal magnetic field is surrounded by a superfluid, then the field will be trapped in that region for a time of the order of the Hubble time, provided that the field lines penetrate the superfluid.

38.3 Conclusions Our principal results a,re that (1) we have proposed a simplified but realistic model equation for the evolution of a dipolar component of the magnetic field in the outer, normal rape-matter core of a neutron star; and (2) we find that the neutron-star cooling, allowing for the direct Urea process, can result in decay of the dipole magnetic field in a normal rape-matter core of a neutron star on a timescale ~ 106 years, even if the initial field is < 1012G

Acknowledgements This work has been supported by NSF grant AST 91-15132 through the University of Rochester, which we gratefully acknowledge. We a,re also grateful to D. Page for providing us with the numerical data from his neutron sta,r cooling calculations, on which our present calculations are based.

References Baym G., Pethick C. and Pines D., Nature, 224, 674, (1969) Baym G., Pethick C. and Pines D., Nature, 224, 675, (1969) Friman B.L., and Maxwell O.V., Ap. ./., 232, 541, (1979) Haensel P., Urpin V.A., and Yakovlev D.G., Astron. Ap., 229, 133, (1990): HUY Lifshit.z EM., PiUevskii L.P., Physical Kinetics (Oxford: Peigainon Press), (1981) Page D., Applegate .l.H. Ap. ./., 394, L17, (1992): PA Yakovlev D.G., in Strongly Coupled Plasma Physics, ed. H.M. Van Horn and S. Ichimaru (Rochester, NY: University of Rochester Press), p. 157, (1993)

39 On the equation of state in Jovian seismology J. PROVOST Depariement Cassini, OCA, BP 229, 06304 NICE Cedex 4

B. MOSSER DESPA, Observatoire de Paris-Meudon, 5, place Jules Janssen,92195 MEUDON Principal Cedex

G. CHABRIER Ecole Normale Superieure de Lyon, 46 Allee d'ltalie, 69364 LYON Cedex 07

Abstract We have investigated the effect on the seismological properties of the giant planet Jupiter of different descriptions of the equation of state (EOS) of the fluid hydrogen helium envelop. Recent Jovian models by Chabrier et al. 1992 use the equation of state of Saumon et al. (1992) which predicts the PPT (plasma phase transition). We show that different hypothesis at the level of the PPT induce large differences in the internal structure of the corresponding models, specially the sound speed. This gives rise to substantial differences of the oscillation frequencies up to 100 /iHz, for modes of degrees P. up to 20. These,results show the great capability of Jovian seismology to test the physics involved in the interior of the giant planets.

39.1 Description of the Jovian models Models of Chabrier et al. (1992) are based on the following assumptions: hydrostatic equilibrium of the rotating planet, adiabatic equation of state for each region (rocks and ices in the core; mainly hydrogen and helium in the envelope, with small addition of denser elements). All these models are constrained by the values of the gravitational moments J2 and J4 measured by the Voyager mission. The models use. the equation of state of Saumon et al. (1992) which predicts the plasma phase transition (PPT) of molecular to metallic hydrogen near the 1.2 Mbar level. The models considered mainly 596

Provost et al.: Jovian seismology

597

Fig. 39.1 Variation of the pressure, density and sound speed along the radius normalized to the Jovian radius, for the models of Chabrier et al. 1992.

differ by the description of the EOS. Model (1) has a simply interpolated EOS and does not include the treatment of the PPT. It corresponds therefore to a Jovian interior model without PPT. Models (2) and (3) include the PPT, with a slightly different treatment (there is a small jump in the Helium content at the level of the PPT for model (3)). These models have a double core, constituted of a small rocky core surrounded by a thin ice layer. This induces two strong discontinuities, clearly

598

Provost et al.: Jovian seismology

Fig. 39.2 Variation of the Lamb frequency Si = y/l(£+l)c/r as a function of the radius for different degrees i= 1, 2, 4, 6, 8, 10, 20, 40, 100, for the model (1). The acoustic cavity extends from the surface to the point, where Si I'll: = v.

visible on the pressure p, density p and sound velocity c profiles, represented in Figure 1 (a, b, c). There are large differences of pressure, density and sound velocity between the different models (1), (2) and (3), roughly localized in the core of the planets for p and p. The differences of sound speed are much larger and occur in the whole planet: this will differently affect the frequencies of the oscillations according to their degrees. Note that the slightly different treatments of the external layers of the models (2) and (3) result in a rather large difference of the size of their core. As a result, the treatment of the equation of state in the external layers has an incidence on the deeper internal structure of the planet and specially on the variation of the sound speed. 39.2 Jovian oscillations Jovian seismology will be a powerful tool to obtain detailed informations about the different, layers of the planet as soon as oscillations of different degrees will be observed . The acoustic, oscillations are trapped in a cavity with an upper boundary located in the surface layers. The lower boundary of an oscillation of frequency v and degree I is located at the level where the Lamb frequency, equal to St. — \/t-{t- + l)c/r divided by 2TT is equal to the frequency v . Thus it depends on the degree of the mode and goes up from the center to the surface when the degree increases, as it. can be seen in Figure 2. Since the frequency of an acoustic oscillation is mainly

599

Provost et al.\ Jovian seismology

i

i

i/(mHz)

utmHr)

Fig. 39.3 Difference between the computed theoretical frequencies of two models from Chabrier et al. (1992) as a function of the frequency for degrees I =0, 1, 2, 3, 4, 5, 6 (full line), I = 7, 8, 9, 10 (dashed line), t. = 12, 14, 16, IS and 20 (dotted line): (a) difference between models including or not the PPT ((1) - (2)); (b) difference between models with a small difference in the description of the PPT ((2) - (3)).

determined by the sound speed in the cavity of the mode, the observations of oscillations of various degrees will enable us to test different regions of the interior of the pla.net. The frequencies of linear, adia.batic, global acoustic modes of the Jovian

600

Provost et al.: Jovian seismology

models (1), (2) a.nd (3) ha.ve been numerically computed for modes of degree t. from 0 to 20. In order to accurately compare the models from the seismological point of view, the frequency differences between two models with different EOS are plotted as a function of the frequency in Figure 3. We have neglected the rotation, which strongly affects the frequencies (Lee 1993), but not too much their differences. The frequency differences between models (1) and (2) (Figure 3 a.) are very large, up to 100 /AHZ. For the low degree modes, which penetrate deep into the core of the planet, the curves have an oscillatory behavior. This is mainly due to the difference of structure in the core and to its signature, as asymptotically analysed by Provost et al. (1993). For the higher degree modes, less sensitive to the core, the frequency differences are larger, but do not oscillate any more: they are dominated by the difference of structure at the PPT. The frequency differences between models (2) and (3), which have a slightly different description of the external layers, are shown in Figure 3 b. As expected, they are essentially due to the differences of structure in the core: for the low degree modes, one finds again an oscillatory behavior with values up to 30 /xHz, while the higher degree modes are almost unaffected by the different assumptions in the planet modelling. Thus, when the PPT is taken into account, the frequencies are substantially modified and the observation of oscillation modes of different degrees would provide a strong constraint on the physics of the interior of giant planets.

References Chabrier G., Saumoii D., Hubbanl W.B., Lunine J.I., 1992, Ap J 391, 817 Lee U., 1993, Ap J 405, 359-374 Provost J., Mosser B., Bertlioniieu G., 1993, A&A, 274, 595-611 Saiunon D., Hubbard W.B., Chabrier G., Van Horn H.M., 1992, Ap .1 391, 827

40 Analysis of the screening formalisms in solar and stellar conditions H. DZITKO AND S. TURCK-CHIEZE Service d'Astrophysique, DAPNIA. CE. Saday, 91191 Gif sur Yvette. France

P. DELBOURGO-SALVADOR AND CH. LAGRANGE Service PTN. CE. Bruyeres le Chdtel, BP 12, 91680 Bruyeres le Chdtel. France

Abstract We discuss the quality of the electronic screening prescriptions usable in stellar evolution codes during hydrogen burning. The assumption l o g / = H(0)/kT is compared to a precise formalism where the radial dependence of the screened potential of the 2 ions is introduced.

40.1

The neutrino puzzle and the solar model accuracy.

The microscopic, description of the solar interior is crucial for both neutrino and acoustic mode frequency predictions, as that was pointed out by Turck-Chieze and Lopes (1993, TCL93). In this way, the best input physics have been looked for: precise composition, detailed opacity coefficients, and best nuclear reaction rates (see review of Turck-Chieze et al). As some neutrino sources are strongly dependent on the temperature and not really constraint by the luminosity ($ t/ ( 8 J B) oc T 1 8 ), a good microscopic, description is required (see table below). At present, the area between the nuclear and convective zones is rather under control by the determination of the sound speed behavior, but the nuclear region is not as well constraint by helioseismology, and the presence of mixing or of a small convective core is not. excluded (Dzitko and Turck-Chieze, 1993). We have also noticed that the uncertainties on the nucleax reaction rates were still important (TCL93), and it is the reason why we consider here the screening effect produced by plasma elections on the nuclear reaction rates. 601

602

Dzitko et al.: Screening formalisms in solar and stellar conditions Table 40.1. Comparison of experimental results (in SNU units 10~ 3 6 captures/atom/s) with Sacluy neutrino predictions(TCL93). experimental results Chlorine experiment: 2.33 ±0.25 Kamiokande experiment: KII 0.28 ±0.03 evts/d Kill Gallium experiments: SAGE bStil ± 14 GALLEX I 83 ± 19 ± 8 GALLEX II 97 ± 19 ± 8

Predictions

ratio^

6.4 ±1.4

0.36 ± 0.04 ± 0.08

0.48 ±0.12

0.58 ± 0.062 ± 0.15 0.72 ±0.15 ±0.17

123 ± 7

0.47J:°-£ ± 0.03 0.68 ±0.17 ± 0.04 0.79 ± 0.19 ± 0.05

40.2 Screening effect in nuclear reactions

In the central region of stars, most of the atoms are entirely striped of their atomic electrons. The nuclei are immersed in a sea of free electrons which cluster in their vicinity, lowering the repulsive Coulomb barrier, so that the probability of wave functions to tunnel through this barrier is increased. Thus, the nuclear reaction rates are enhanced over their vacuum values by the so called screening factor / =< av >in p / a s m a / < av >,, acuum (1). 40.2.1 Analytic form of the screening factor.

The screening factor / can be expressed as / = exp[H(0)/kT] if the internuclear spacing is much larger than the classical turning point. For stellar evolution applications, the. analytic forms of the screening factors which have generally been used are those developed by Salpeter (1954, hereafter S54) arid Graboske et al (1973, GWCD), namely: /,„ = expA = exp(ZlZ2e1)/{B.DkT),

B.D =

R,D is the Debye-Huckel radius, £ = \ZJ2i(^i(^i + &c))Xi/Ai the rms charge of the plasma and 0c the electronic degeneracy factor. A is the natural screening parameter, ie the ratio of the Coulomb energy interaction to the kinetic, energy of the reacting particles. /„, is only valid when A < 1, that is, in the weak screening (WS) regime. Later GWGC presented a formalism recovering the limits of WS and SS (strong screening) regimes: log fG = %//(,A£[(Z1 + Z 2 ) 1 + 6 -Z 1 1 + 6 -Z,| + ( > ], A O =

Dzitko et al.\ Screening formalisms in solar and stellar conditions Ao is a charge independent variable characterizing the plasma, b lies between 1 and 2/3 for WS or SS and 0.860 for IS (intermediate screening), value obtained with the cluster expansion theory, r/j, depends on the appropriate charge plasma average £. For WS 7/j, = £, kb — 1/2 leading to the classical Salpeter's formula. Thus, 3 screening regimes are defined: A < 0.1 -> WS, O . K A < 5 -» IS, and 5 < A -> SS. In the previous standard assumptions, it was written l o g / = H(0)/kT = AU/kT where All is the electrostatic energy difference of the 2 reacting nuclei between infinite separation and fusion, but Mitler showed that AF, the difference in Helmholtz free energy, is better adapted than AU (Mitler 1977, hereafter M77). In fact, the first two terms of the expansion of the effective interaction energy are: ZiZ2e

AF(0) \AF(Q)\ —^, hence log/MO = —TTJTr kg! kT This calculation is carried out in the 2 fluid approximation, within the statistical equilibrium assumption, and with a better charge density where 2 spherical domains around the reacting ions are denned and limited by some radius r\ (roughly, the radius of the Wigner-Seitz sphere): if r < T\ the electron density is taken uniform (as in S54), and if r > ri the density distribution goes like e~Kr/r, as in the Debye theory. This leads to a prescription which recovers the previous S54 and GWGC results in WS, and which is also valid for all regimes, but without any discontinuities over the A range, as it is the case for the GWGC formalism (see figure 1.1). U(r) =

40.2.2

Numerical computation with radial dependence of the screened potential function H(r)

In this case, one checks the validity of the expression (1). H(0) is then replaced by H(r) which accounts for the distorted ion-electron distribution around the 2 ions in interaction. At short distance, their common cloud is assumed to be uniformly distributed following an ellipsoid of revolution, and a.t large distance H(r) turns into an usual Yukawa potential. This radial dependence involves a. recalculation of the screened reaction rates < ov >screened- The enhancement factor / is thereby deduced as follows: JMr

=<

^ O"U ^screened

I ^ &v ^

40.2.3 Accuracy of the screening formalisms. The analytic screening prescriptions were studied for the p+' Be and p+ 1 4 N reactions in the solar and stellar ca.ses (0.6 < M < 20MQ) and compared

603

604

Dzitko et al.: Screening formalisms in solar and stellar conditions 2.2

0.1

0.2

0.3

0.4

0.3

A

0.6

°- 7

Fig. 40.1 Comparison of the analytic screening factors with precise values given by the /M»• radial dependent, formalism for various ZAMS stars, and for the center of the present Sun. The p + 1 4 N reaction is here presented since the effects are larger than for the p+p one for example. 1.24

1.22

IO

•fw

—

-«M0

1.2.

1.11 1.16-

2k.

1.14. 1.12.

l . l - -.

1.01.

Fig. 40.2 f versus v/R0 for the 7 5f:(p,7) 8 5 reaction. Note the as 12 % difference between /; and /„, and the intermediate values given by /yv/,

with those calculated with JM,- for plasma conditions corresponding to the WS and IS regimes. The Figure above shows that all the prescriptions

Dzitko et ah: Screening formalisms in solar and stellar conditions converge, as expected, toward the same value when A is small enough. In low mass stars, high densities and low temperature increase the screening effect, and therefore the spread of these prescriptions is highlighted, but, in fact, in those cases, hydrogene burning is dominated by the pp reaction. In the WS limit (A < 0.1), Mitler's prescription quite agrees with fW2 (a second order approximation in A of the pair distribution functions involved in the WS calculation). The agreement between JMO and JMV is very good in the solar case (ss 1.5 %) while fo and /,„ respectively under and over estimate /MY by about 10 and 8 %. The influence on the &B neiitrino predictions may be deduced from figure 1.2 where the screening factors are calculated for the 7 Be(p,j)^B reaction. At the maximum rate of this reaction (which occurs at about I/B,Q = 0.06) fw and fc differ by 12 %, leading to the same difference on the chlorine and water predictions. This difference is smaller for the Gallium detectors ( « 3 %), since about 60 % of the predicted fluxes are due to pp neutrinos.

40.3 Conclusion The S54 and GWGC prescriptions do not seem to be precise enough in the solar case, therefore, we recommend to use the simple # ( 0 ) Mitler's formalism which avoids the discontinuities at A = 0.1 and reproduces the radial dependent H(r) prescription within 2 % accuracy.

References Turck-Chieze S.and Lopes I., Astrophys. J. 408, 34 (1993). Tuick-Chieze S., Dappen W., Fossat, E., Provost J., Schatzman E., Vignaud D., Pliys. Rep. 230, 2-4, (1993) Dzitko H. and Turck-Chieze. S., in Advances in solar physics, ed. by G. Belvedere and M. Rodono, in press Salpeter E., Australian ./. Phys. 7, 353, (1954) Giaboske et, al, Astrophys. J. 181, 457, (1973) Mitler H. E., Astrophys. ./., 212, 513, (1977)

605

41 Theoretical Description of the Coulomb Interaction by Pade-Jacobi Approximants W. STOLZMANN Institut fur Theoretische Phystk and Sttrnwarte der Universitdt Kiel, Olshausenstr. 40, D-24118 Kid, Germany

T. BLOCKER Institut fiir Theoretische Physik und Sternwarte der Universitdt Kiel, Olshausenstr. 40, D-24U8 Kiel, Germany

Abstract Coulomb interactions for the Free Helmholtz energy and the pressure are studied in a partial new formulation which described more exactly the numerical evaluation of many body theories.

41.1 Introduction With regard to the EOS many activities have been developed to yield results which consider different phenomena, for instance pressure dissozia.tion and ionisation, degeneracy, relativity, Coulomb- and non-Coulonibic interactions, pair production and charge mixing in different chemical compositions. Various theoretical approaches are used in order to include exchange and correlation effects for fully ionized or partially ionized matter (see e.g. Salpeter and Zapolski 1967, Graboske et al. 1969, Hansen 1973, Pokrant 1977, Fontaine et al. 1977, March and Tosi 1984, Perrot and Dharmawardana 1984, Hubbard and Dewitt 1985, Dandrea et al. 1986, van Horn 1987, Kraeft et al. 1986, Ichimaru et al. 1987, Rogers and DeWitt 1987, Dappen et al. 1988, Ichimaru 1990, Eliezer and Ricci 1991, Saumon and Chabrier 1992). For many applications (e.g. stellar evolution calculations or astroseismology) it. is necessary either to have algebraic formulae for the EOS or extensive tables which supply the input, informations at a.ny density and temperature. As a, first step we present an analytical EOS for fully ionized 606

Stolzmann & Blocker: Pade-Jacobi approximants

607

multicomponent plasmas covering a large density-temperature range. The EOS includes non-ideal effects due to exchange-correlation interactions of charged particles at any degeneracy and is applicable to any chemical mixture. Relativistic effects as well as ionic quantum corrections are taken into account. The aim of this contribution is to derive explicite exchange and correlation corrections for the Coulombic part of the EOS in an algebraic form. Recently, we presented results for the pressure (Stolzmann and Blocker 1993a,b) and a first application to the mass-radius relationship of white dwarfs (Stolzmann et al. 1993). This paper will be devoted to a brief overview of the theoretical background of our EOS concept.

41.2 Theoretical Background We start with the Helrnholtz free energy F of a fully ionized plasma, consisting of a species which is given by (1) with F^ being the ideal free energy and Fcoul representing the Coulomb interaction contributions of the free energy. We splitted the Coulomb contributions in the following parts: ^coul

px. , pc , ^x + pc , pen , pc

(2)

where x and c denote the exchange and the correlation term, resp. .Fee, jFii and Flti correspond to the electron-electron, ion-ion and ion-electron interaction. In this notation F^'1 and F* describe ionic quantum corrections. The pressure is given by the thermodynamic relation (3) or equivalent by the grand potential il

with the definition of the Gibbs free energy ( dF \ The equivalence between (3) and (4) is achieved by the elimination of the chemical potential from the pressure (4) by means of an inversion procedure

608

Stolzmann & Blocker: Pade-Jacobi approximants

described in detail by Kohn and Luttinger (1960), Stolzmann and Kraeft (1979), Perrot and Dharma-wardana (1984), Rosier and Stolzmann (1987). In equivalence to (2) the Coulombic pressure contribution is given by pcoul

px , pc , px. , pc , pcq , pc

/g\

1

-1 ee '

\ )

±

ee '

J

n '

J

n ~

J

n

'

x

le

Consequently, we apply in (2) OkT'V

, A) + |AJ 3 / 2 (-0,

j

)}

(7)

with the De Broglie wavelength Ae and the relativity parameter A me The election degeneracy parameter 0 is denned implicitly by the density (9)

(10) According to Kovetz et al. 1972 we used for the lowest order exchange interaction

^

4U^..(^A).

(11)

with the relativistic Hartree-Fock integral J^('ip,X). The non-relativistic conditions can be described by the fit formula of Perrot and Dharmawardana (1984). We determine for the electron-electron correlation term the new Jacobi-Pa,de approximation

FL _ „ _

aorlri-a2Tlec(rH,0)/kT

The ground state energy sc can be taken from Salpeter and Zalpolsky (19G7), Vosko et. al. (1980) or Ebeling and Richert (1985). The Coulomb coupling parameter F describes the ratio of the Coulomb potential to the kinetic

Stolzmann & Blocker: Pade-Jacobi approximants

609

energy. The Jacobi-Pade approximants of the ion-electron and ion-ion interaction are proposed by Ebeling (1990):

c 2 i1 / 2 £ i

r;j/2

l

;

i2 The energy eie is adopted from Ebeling (1990), Hubbard and DeWitt (1985), DeWitt and Hubbard (1976) and Galam and Hansen (1974) whereas £a consists of Madelung-like and thermal energy and is taken from Stringfellow et al. (1990). Ionic quantum corrections (for the near-classical limit) can be considered according to Chabrier et al. (1992) (see also Nagaia et aJ. 1987):

21n 1 e

C - " " 0 1 ) + l n C1 -e

- 3 1 n 0 i + 2.71848-Q D e b (0i)

(15)

with the approximation for the Debye integral »Deb(©i) = 7

Leaving the classical region symmetry effects become important. We. try to determine the exchange free energy in the lowest order approximation (Hartree-Fock exchange) by

i + h(f + hit

I

analogous to the treatment of the electrons by Perrot and Dha.rma.-warda.nri.

z)

e

"HI

i = 17F

(18)

Straightforward we obtain with (3) - (5) for the pressure '2kT

5

\

-AJ 5/2 (V;,A)J

(19) (20)

610

Stolzmann & Blocker: Pade-Jacobi approximants

- IL /^c(rs,0) = £ c ( r s , 0 ) - ——^

(21) (22)

The pressure of the ion-electron and the ion-ion interaction is given by

( lac6 - 1 + i ) if2

Die and D\i are the denominators of (13) and (14), resp., whereas /i|,n and d\m are denned by

Finally, the ionic quantum corrections can be determined by (26)

- /S

(27)

A detailed description of our investigations is given by Stolzmann and Blocker (1993c).

References Chabrier, G., Ashcroft, N.W., DeWitt, H.E.: 1992, Nature 360, 48 Dappen, W., Hummer, D. G., Mihalas, D., Weibel-Mihalas, B.: 1988, ApJ. 332, 261 Dandrea, R.D., Ashcroft, N.W., Carlsson, A.E.: 1986, Phys. Rev. B34, 2097 Ebeling, W.: 1990, Contr. Plasma Phys. 30, 553 Ebeling, W., Richert, W.: 1985, phys. stat. sol. (b), 128, 467 Eliezer, S., Ricci, R.A. (eds.): 1991, High Pressure Equation of State: Theory and Applications, Enrico Fermi International School of Physics, Vol. 113 Fontaine, G., Graboske, H. C , van Horn, H. M.: 1977, ApJS 35, 293 Galam, S., Hansen, J.P.: 1976, Phys. Rev. A14, 816

Stolzmann & Blocker: Pade-Jacobi approximants

611

Graboske, H. C , Harwood Jr., D.J., Rogers, F. J.: 1969, Phys. Rev. 186, 210 Hanseri, J. P.: 1973, Phys. Rev. A8, 3096 Hubbard, W. B., DeWitt, H. E.: 1985, ApJ 290, 388 Ichimaru, S. (ed.): 1990, Strongly Coupled Plasma Physics, North-Holland Delta Series Ichimaru, S., Iyetorni, H., Tanaka, S.: 1987, Phys. Rep. 149, 91 Kohn, W., Luttinger, M.: 1960, Phys. Rev. 118, 41 Kovetz, A., Lamb, D.Q., van Horn, H.M.: 1972, ApJ. 174, 109 Kraeft, W. D., Kremp, D., Ebeling, W., Ropke, G.: 1986, Quantum statistics of charged particle systems, Plenum, New York March, N. H., Tosi, M. P.: 1984, Coulomb Liquids, Academic Press, London Nagara, H., Nagata, Y., Nakamura, T.: 1987, Phys. Rev. A36, 1859 Perrot, F., Dharma-wardana, M. W. C : 1984, Phys. Rev. A30, 2619 Pokrant, M.A.: 1977, Phys. Rev. A16, 413 Rosier, M., Stolzmann, W.: 1986, phys. stat. sol. (b) 137, 149 Rogers, F.I., DeWitt, H.E. (eds.): 1987, Strongly Coupled Plasma Physics, Plenum Press, New York Salpeter, E.E., Zapolsky, H.S.: 1967 Phys. Rev. 158, 876 Saurnon, D., Chabner, G.: 1992 Phys. Rev. A46, 2084 Stolzmann, W., Blocker, T.: 1993a, in Inside the stars, IAU Coll. 137, Astron. Soc. Pac. Conf. Ser. 40, p. 269 Stolzmann, W., Blocker, T.: 1993b, in White. Dwarfs: Advances in Observation and Theory, ed. M. Barstow, NATO ASI Series C, Kluwer, Dordrecht, p. 133 Stolzmann, W., Blocker, T.: 1993c, A&A, submitted Stolzmann, W., Blocker, T., Rieschick, A.: 1993, in White Dwarfs: Advances m Observation and Theory, ed. M. Barstow, NATO ASI Series C, Kluwer, Dordrecht, p. 127 Stolzmann, W., Kraeft, W.D.: 1979, Ann. Phys. 36, 388 Stnngfellow, G. S., DeWitt, H. E., Slattery, W. L.: 1990, Phys.Rev. A41, 1105 Van Horn, H.M.: 1987, Mitt. Astron. Gesellschaft 67, 63 Vosko, S.H., Wilk, L., Nusair, M.: 1980, Can. J. Phys. 58, 1200

42 New Model Sequences from the White Dwarf Evolution Code M. WOOD Department of Physics and Space Sciences, Florida Institute of Technology, Melbourne, FL 32901-6988 USA

Abstract Model sequences computed with the recently-published OPAL radiative opacities, Itoh et al. conductive opacities, and Itoh et al. neutrino rates are presented. Cooling times for DA model sequences are found to be independent of metallicity for Z < 0.001.

42.1 Introduction In the past decade, many improvements in the constitutive physics relevant to white dwarf evolutionary models have been published. These include improved radiative opacities (Rogers & Iglesias 1992; Iglesias & Rogers 1993), conductive opacities for pure (Itoh et al. 1993 and references therein) and mixed (Itoh & Kohyama 1993) compositions, and updated neutrino rates (Itoh et al. 1992 a,nd references therein). We have incorporated these results into our white dwarf evolution code (=WDEC; see Lamb Ik Van Horn 1975, and Wood 1990), and present here selected C-core DA model sequences computed with the updated code. Stellar masses for the sequences are 0.4, 0.6, and 0.8 MQ and surface layer masses are \ogq(H) = —6 and \ogq(He) — —A. To determine the effect of metallicity on the evohitionary timescale, we computed parallel sequences with Z = 0.000 and 0.001. 612

Wood: The white dwarf evolution code

613

42.2 Opacities

The radiative opacities used in WDEC in the past (Cox k Stewart 1970) had an unrealistically-high metallicity of Z = 0.001 (Zot)S£,10~5) The new OPAL opacities span a wide range of metallicities and compositions, and therefore allow the inclusion of more plausible composition profiles in the models. The OPAL opacities only extend to a minimum temperature of 6000 K, however, so for DA models WDEC references the pure-H opacities of Lenzuni et al. (1991) below this point. Figure 1 shows the p-T domain of interest, with the crosshatched regions showing the extents of the various opacity tables. Also included on this plot are three 0.6 M(;, models with effective temperatures of 173,000 K, 12,300 K, and 3780 K. Note that for models cooler than ~15,000 K, the opacities for shells in the outer envelope must be extrapolated off the OPAL (and Itoh et al.) tables. The extrapolations are along an isotherm and are linear in the logarithms of density and opacity. Of hydrogen, helium, or carbon, it is the helium opacity which shows the greatest sensitivity to metallicity between Z — 0.000 and 0.001 with AK max ~ 5 dex (but only for T < 20,000 K); for hydrogen and carbon (and helium above 20,000 K), we find AK max « 0.5 dex. As a result of these small differences, we expect to also find a small sensitivity to metallicity in the model ages, and this indeed is the case (see below). Unfortunately, because no radiative or conductive opacities currently exist for this dual-extrapolation region, the absolute, ages of the low-luminosity models are still uncertain (see also Mazzitelli's contribution in these proceedings). However, one of the positive outcomes of this meeting was a. formal letter to F. Rogers from a few of us in the white dwarf modelling community requesting an extension of his tables to logi?. = +5. This will essentially fill in the current opacity "hole," and should be a, feasible extension, according to Rogers (private communication). We can anticipate tha.t within a few months' time the ages of the cool white dwarf models will be on much firmer ground than is now the case. The conductive opacities of N. Itoh and collaborators were introduced a decade ago, and these and subsequent, refinements have been included into WDEC. The low-density, boundary to these tables is approximately given by the condition T < O-lTFermi- It would be most useful if these calculations could be extended to lower densities.

614

Wood: The white dwarf evolution code 1

1

.

,

.

1 ,

.

.

1 I

I

.

1 .

Itoh et al

"

10

Q.

0

O

T T-f = 173.000 i2 3 -

'I III \\s

\\\x

10 %

— T^.--«* 3.780 ' °°--

LyuPXL ^

_

1 1

I i

i

i

1

_

I i

i

6

i

1

8

YmdJr^ i

i

i

1

i

10

log T Fig. 42.1 Opacity tables' domains in p-T plane. Also shown are the f>-T loci for models with the indicated effective temperatures.

42.3 Neutrino Rates The neutrino rates of Itoh and collaborators are comparable to the rates of Beaudet et al. (1967) and De Zotti (1972). The largest difference between the old and new rates is for the bremsstrahlung process, which is roughly equivalent at high temperatures, but a factor of three or more weaker at low temperatures.

42.4 Results In order to explore the importance of surface metallicity, I computed computed parallel 0.4, 0.6, and 0.8 M 0 C-core DA sequences with Z = 0.000 and 0.001. The evolutionary summary listings for the Z - 0.000 sequences are given in Tables 1-3, sampled every 0.5 dex in l o g ( Z / I 0 ) before crystallization onset, and every 0.2 dex thereafter!. Comparing, we find that t Tin? listings of these anil other sequences are available in electronic form. F'lease senlei\p.ss.fit.edu.

Wood: The white dwarf evolution code

615

Table 42.1. Evolutionary summary of a Z = 0.000 0 . 4 M Q model sequence

\og{L/LQ) 2.5 2.0 1.5 1.0 0.5 0.0 -0.5 -1.0 -1.5 -2.0 -2.5 -3.0 -3.5 -4.0 -4.2 -4.3 -4.4 -4.5 -4.6 -4.7 -4.8 -5.0 -5.2 -5.3

Age (yO 1.718 5.790 2.679 1.133 4.801 1.944 6.405 1.771 5.009 1.473 3.757 8.646 1.821 4.180 5.403 6.351 7.501 8.637 9.707 1.078 1.187 1.412 1.657 1.790

Teff

logT c

(2) 113900 7.674 (3) 96064 7.677 (4) 79034 7.681 (5) (5) (6)

(7) (7) (8) (8) (8) (9) (9) (9) (9) (9) (9) (9) (10) (10) (10) (10) (10)

63668 50631 39962 31311 24307 18703 14294 10870 8249 6244 4736 4225 3990 3769 3559 3361 3173 2996 2670 2380 2247

7.690 7.718 7.791 7.796 7.709 7.581 7.410 7.218 7.008 6.783 6.465 6.353 6.294 6.231 6.173 6.117 6.063 6.009 5.901 5.792 5.734

log(pe) log(/2) 5.890 5.894 5.900 5.908 5.919 5.937 5.965 5.993 6.014 6.029 6.039 6.045 6.049 6.052 6.053 6.054 6.054 6.054 6.054 6.054 6.055 6.055 6.055 6.055

9.506 9.404 9.323 9.261 9.210 9.166 9.128 9.098 9.075 9.059 9.047 9.036 9.028 9.018 9:017 9.017 9.017 9.016 9.016 9.016 9.016 9.016 9.016 9.016

\og{L,,/ LQ)

Ifxtal

-0.415 -0.392 -0.363 -0.324 -0.279 -0.286 -0.533 -1.072 -1.928 -3.263 -4.979 -6.502 -8.357

0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.02 0.22 0.46 0.64 0.77 0.86 0.91 0.97 0.99 1.00

— — — — — — — — — — —

the fractional age differences a,re always below 1% for the 0.4 and 0.6 M Q models, and 2% or below for the 0.8 M(., models — i.e., the cooling times are relatively insensitive to met alii city for Z < 0.001. We note, however, that a 0.6 model sequence with a pure-C composition is 20% older than the DA sequence at log(i/Z ( .)) = —4.4, and nearly a factor of 2 older at log(Z/X 0 ) = - 4 . 9 .

42.5 Future Work Future work includes, first, publishing the current white dwarf model sequences so they are available in the literature; this is progressing. Italo Mazzitelli has kindly provided source code which implements the Canuto k Mazzitelli (1991, 1992) convection theory, and this will be included. We will also use the Lamb (1974) equation of state code to compute, a.n interior EOS tables of He, Mg, and Ne so that we can compute more realistic low-

616

Wood: The white dwarf evolution code

Table 42.2. Evolutionary

) Age (y) 3.0 2.5 2.0 1.5 1.0 0.5 0.0 -0.5 -1.0 -1.5 -2.0 -2.5 -3.0 -3.5 -3.8 -3.9 -4.0 -4.1 -4.2 -4.3 -4.4 -4.6 -4.8 -4.9

2.163 1.065 5.024 2.215 6.853 1.512 2.939 5.796 1.306 4.411 1.682 4.662 1.052 2.168 3.358 4.381 5.641 6.724 7.708 8.634 9.492 1.116 1.278 1.360

summaiy

r eff

of a Z = 0.000 0.6M© model

sequence

logT, loy(p,) log(fl) \og(L,,/LQ) tfxtal

(3) 162459 8.012 (4) 140649 8.016 (4) 117608 8.018 (5) 96844 7.999 (5) 78425 7.941 (6) 61875 7.880 (6) 47968 7.826 (6) 36831 7.760 (7) 28113 7.678 (7) 21378 7.540 (8) 16205 7.344 (8) 12248 7.125 9238 6.902 6961 6.678 (9) 5880 6.534 (9) 5560 6.463 (9) 5253 6.383, (9) 4962 6.311 (9) 6.244 4687 (9) 4427 6.179 (9) 4179 6.117 (9) 3727 5.995 (10) 3322 5.876 (10) 3136 5.816 (10)

6.301 6.311 6.332 6.378 6.437 6.475 6.498 6.513 6.526 6.536 6.544 6.550 6.553 6.556 6,556 6.557 6.557 6.557 6.558 6.558 6.558 6.558 6.558 6.558

9.448 9.323 9.228 9.147 9.080 9.036 9.007 8.987 8.971 8.959 8.950 8.943 8.938 8.934 8.930 8.929 8.928 8.928 8.927 8.927 8.927 8.927 8.926 8.926

1.503 1.542 1.580 1.525 1.230 0.809 0.359 -0.099 -0.748 -1.835 -3.484 -5.447 _ — — — — _ _ — — — — —

0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.19 0.47 0.67 0.80 0.88 0.94 0.98 1.00 1.00

mass and high-mass models. Finally, we need to include convective mixing in our envelopes, so that we may investigate the effect of changing surface composition on evolutionary timescale. These new sequences will be used in our ongoing studies of the white dwarf luminosity function (see Wood 1992). Acknowledgments: It is a pleasure to thank to Paul Bradley for numerous discussions and Steve Ka,wa.ler for starting models evolved from the ZAMS. This work wa.s supported in part by NSF grant AST92-17988 and the 19921993 Earnest F. Fullani Award of the Dudley Observatory.

References Beaudet, G., Petiosian, V. and Salpeter, E. E. 1967, ApJ, 150, 979 Canute, V. M., and Mazzit.elli, I. 1991, ApJ, 370, 295 Canuto, V. M., and Mazzitelli, I. 1991, ApJ, 389, 724

Wood: The white dwarf evolution code

617

Table 42.3. Evolutionary summary of a Z = 0.000 0.8M© model sequence )

3.0 2.5 2.0 1.5 1.0

0.5 0.0 -0.5 -1.0 -1.5 -2.0 -2.5 -3.0 -3.4 -3.5 -3.6 -3.7 -3.8 -3.9 -4.0 -4.1 -4.2 -4.4 -4.6 -4.7

A g e (yr)

r eff

logT c

4.927 (3) 4.327 (4) 1.496 (5) 3.039 (5) 5.433 (5) 9.350 (5) 1.656 (6) 3.401 (6) 1.037 (7) 6.440 (7) 2.328 (8) 5.637 (8) 1.213 (9) 2.178 (9) 2.627 (9) 3.148 (9) 3.698 (9) 4.301 (9) 4.970 (9) 5.969 (9) 7.178 (9) 8.165 (9) 9.766 (9) 1.108 (10) 1.166 (10)

195469 172013 147600 119074 93185 71822 54885 41731 31651 23975 18110 13649 10271 8177 7724 7295 6891 6510 6151 5813 5492 5187 4626 4124 3894

8.248 8.171 8.054 7.982 7.928 7.884 7.843 7.791 7.691 7.495 7.270 7.047 6.822 6.652 6.605 6.562 6.519 6.477 6.434 6.367 6.279 6.199 6.049 5.909 5.839

6.646 6.752 6.875 6.928 6.954 6.970 6.980 6.988 6.996 7.005 7.012 7.016 7.018 7.019 7.019 7.019 7.020 7.020 7.020 7.020 7.020 7.021 7.021 7.021 7.021

log(*)

log(WI0)


9.287 9.148 9.031 8.968 8.930 8.907 8.890 8.878 8.868 8.860 8.853 8.849 8.846 8.844 8.843 8.843 8.842 8.842 8.841 8.840 8.840 8.839 8.839 8.838 8.838

3.049 2.910 2.438 1.949 1.481 1.044 0.596 0.130 -0.756 -2.410 -4.475 -6.234

0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.07 0.21 0.35 0.48 0.60 0.70 0.81 0.91 0.96 0.99 1.00 1.00

— — — — — — — — — — — — —

Cox, A. N., and Stewart, J. N. 1970, ApJS, 19, 261 DeZotti, G. 1972, Estratto dalle Memorie della Societa Astionomica Italiana, 43, 89 Iglesias, C. A., and Rogers, F. J. 1993, ApJ, 412, 752 Itoh, N., Mutoh, H., Hikita, A., and Kohyama, Y. 1992, ApJ, 395, 622 Itoh, N., Hayashi, H., and Kohyaina, Y. 1993, ApJ, 418, 405 Itoh, N., and Kohyama, Y. 1993, ApJ, 404, 268 1983, ApJ, 273, 774 Lamb, D. Q. 1974, Ph.D. thesis, University of Rochester Lamb, D. Q., and Van Horn, H. M. 1975, ApJ, 200, 306 Lenznni, P., Chernoff, D. F., and Salpeter, E. E. 1991, ApJS, 76, 759 Mazzitelli, I., 1994, these proceedings Rogers, F. J., arid Iglesias, C. A. 1992, ApJS, 79, 507 Wood, M. A. 1990, PhD thesis, The University of Texas at Austin Wood, M. A. 1992, ApJ, 386, 539

43 Low temperature opacities C. NEUFORGE Institut d'Astrophysique de I'Universite de Liege,5, avenue de Cointe, B-4000 Liege-Belginm

Abstract The importance of low temperature opacities in stellar calibrations led us to compute new sets of Rosseland mean opacities for different Z-values. For the solar metallicity, these tables have been compared to those of Alexander (1975), Cox (1983), Sharp (1991) and Kurucz (1992).

43.1 Introduction Opacities in the atmospheric layers are generally not considered of great importance in the calculation of theoretical evolutionary tracks since the atmosphere of a star only comprises a tiny part of its mass (see however, section 1.2). Until recently, the most commonly used "atmospheric" or "low-T" opacity tables were those of Cox & Stewart (1970), Alexander (1975) and Cox (1983) but there are rather large discrepancies between these different tables for typical T and p ranges encountered in stellar atmospheres of solar type stars. Furthermore, for pop I stars, low-T opacities are calculated for very few values of the metallicity, Z, and the solar chemical composition is generally used in the calculation of tracks, whatever the actual value of Z. 618

Neuforge: Low temperature opacities

619

43.2 Low-T opacities and stellar calibrations Theoretical evolutionary tracks depend on mass, age, chemical composition on the zero age main sequence and convection parameter, a (ratio of mixing length to pressure scale height in the convective layers). Calibrating a. star consists in computing evolutionary models that reproduce, at given age, chemical composition on ZAMS and convection parameter, the observed values of the luminosity and effective temperature of the star. It has been shown that uncertainties that may affect low-T opacities (due to missing or wrongly estimated contributors) lead to uncertainties on the convection parameter. Consequently, the calibrated value of a in the Sun actually contains an "opacity component". Unfortunately, problems may be encountered in binary systems, whose components have different effective temperatures. This is the ca,se of the Alpha Centaury system, in which the masses of the two components (alpha Cen A and B) can be determined accurately. Their age t and chemical composition on the ZAMS are assumed to be the same, since both stars have very probably the same origin. Alpha Cen A and B being solar type stars, it seems reasonable to assume that their convection parameter is identical. Then, Y, Z, t and a can, in principle, be adjusted to the two observed luminosities and effective temperatures. But if the low-T opacities a,re not known with accuracy, the "opacity component" in a may be different for the two stars (of different Teff) making a. calibration in the frame of a unique convection parameter probably inconsistent!

43.3 New low-T opacity tables Problems encountered in stellar calibrations led us to compute new opacity tables for different Z-values around the solar metallicity: Z= 0.04, 0.03, 0.02 and X= 0.6 and 0.7. The tables were computed for T values ranging from 3000 to 15000 K and for Q values, where Q = In ^ 7 " (where T7 = ^j) , ranging from -3.838 to 4.605. We included the following contributors: HI bf and ff (Karzas and Latter, 1961), H" bf and ff (John, 1988), H+H bf (Doyle, 1968), H2~ ff (Somerville, 1964), H+ bf (Mihalas, 1965), CI, Mgl, All, Sil, Hel bf and ff (Peach, 1970), e~ scattering, HI and Hel rayleigh scattering (Dalgarnq, 1962), He~ ff (Carbon et al., 1969). Line absorption is included under the form of opacity distribution functions (Gustafsson et al., 1975). Missing IJV opacity is also included, according to the relation given by Ma,ga,in (1983).

620

Neuforge: Low temperature opacities

Neuforge \ Kurucz

Alexander \ Kurucz

Cox \ Kurucz

Sharp \ Kurucz l.o 1.5

utt

•?.»

LOOM

Fig. 43.1 Comparison of our new opacities and the opacities of Alexander (1975), Cox (1983) and Sharp (1991) with those of Kurucz (1992). The ratio of the different opacities are given as a function of LogT and Log(/v).

43.4 Comparison with other tables For the solar mixture, we compared our results, those of Alexander (1975), Cox (1983) and Sharp (1991) with the results of Kurucz (1992). It is encouraging for us to see that our opacities give the best agreement with those of Kurucz calculations, although we used far less lines than he did. Our re-

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suits remain within less than 13 % of Kurucz's while the other computations show far larger discrepancies.

43.5

Conclusion

We have extended our calculations for X= 0.0, 0.2, 0.5, 0.7 and 0.9 and Z= 0.0001, 0.001, 0.005, 0.02 and 0.04. For X < 0.5, UV line absorption is not included, while for X=0.0 , only continuous absorption is calculated. These new low temperature opacity tables are available on request (e-mail : U2126CN @ BLIULG11).

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