Springer Tracts in Modern Volume 133

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Editors: ]. Kiihn, Karlsruhe Th. Mfiller, Karlsruhe R. D. Peccei, Los Angeles F. Steiner, Ulm 1. Trfimper, Garching P. W61fle, Karlsruhe

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Matter at High Densities in Astrophysics Compact Stars and the Equation of State

With 86 Figures

~ Springer Harald Riffert Herbert Miither Heinz Herold Hanns Ruder Universit/it Tiibingen Auf der Morgenstelle lO D-TzO76 Tfibingen Germany

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Mailer at high densities In astrophysics : compact stars and the equation of state / H. Herold ... - Berlin ; Heidelberg ; New York ; Barcelona ; Budapest ; Hong Kong ; London ; Milan ; Paris ; Santa Clara ; Singapore ; Tokyo : Springer, 1996 (Springer tracts in modern physics ; Vol. 133) ISBN 3-540-60605-X NE: Herold, Heinz; GT

Physics and Astronomy Classification Scheme (PACS): 21.65.+f, 64.1o.+h, 91.3o.Ks, 96.6o.Ly, 97.2o.Rp, 97.6o.Jd

ISBN 3-54o-6o6o5-x Springer-Verlag Berlin Heidelberg New York

This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9,1965, in its current version, and permission for use must always be obtained from Springer-Verlag.Violations are liable for prosecution under the German Copyright Law. © Springer-Verlag Berlin Heidelberg 1996 Printed in Germany The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. 15/pesetting: Camera-ready copy from the authors using a Springer TEX macro package. Cover design: Springer-Verlag, Design & Production SPIN: 10493069 56/3144-5 4 3 2 r o - Printed on acid-free paper In Honor of Friedrich Hund' s woth Birthday

Friedrich Hund on his 100th Birthday

Friedrich Hund was born on the 4th of February 1896 in Karlsruhe. Thus we now have the rare opportunity to congratulate a distinguished on his 100th birthday and to honor his achievements. Let us first of all recall the major stages in his life as a physicist. Hund studied in GSttingen and Marburg, and in 1921 he took the state examination in GSttingen. He spent the following year as a teacher at a high school. Dur- ing this time, Hund established contact with , who had just been appointed to the Chair of Theoretical Physics in GSttingen. This led to his working on a doctoral thesis for which he was awarded a PhD in 1922. At the same time Born offered him a position as scientific assistant, a post which Hund held until 1927. In 1925 he completed his Habilitation (the qualification needed to become a lecturer at a German university) and in 1926/27 he spent one semester in Copenhagen. In 1927 he was appointed to a professorship at the . From 1929 until 1946 Professor Hund was at the University of Leipzig, for the majority of this time together with Heisenberg. He then moved to the , where he remained until 1951. He spent the years from 1951 until 1956 at the University of Frankfurt. In 1956 (after the death of Richard Becker) Hund accepted a chair at the University of GSttingen and thus returned to the place where he had begun his career as a physicist. Until only a few years ago, throughout the decades since his official retirement, Hund continued to hold regular lectures in GSttingen. Anyone who hears the name Friedrich Hund immediately thinks of the dis- covery and development of quantum theory, events which he helped to shape during his time in GSttingen. In 1922 Hund attended the famous GSttingen Lectures of . When, in 1925, the old quantum theory was cast aside in favor of Heisenberg's discovery of quantum mechanics, Hund experienced this at first hand. His work during the 1920s addressed the problem of un- derstanding atomic and molecular spectra, based initially on the idea of the correspondence principle. He made essential contributions to explaining the observed multiplet structure of complicated spectra. "Hund's Rule" is to be found in all standard textbooks on quantum mechanics. In his first book, whose German title "Linienspektren und periodisches System der Elemente" translates as "Spectral Lines and the Periodic Table of the Elements" [1], he presented the results of this work. VIII Friedrich Hund on his lOOth Birthday

Following the publication in 1926 of SchrSdinger's equation, Hund recog- nized how it is possible to understand the connection between the electron system of a molecule and those of separate atoms. This important insight led to the development of the Hund-Mulliken method of molecular orbitals. Works were published on the interpretation of spectra and other properties of molecules. We will have to be content here with these few keywords. Very much later, in his book on the history of quantum theory [2], Hund describes how he witnessed the creation of quantum theory. His own contributions to these developments are mentioned with his typical modesty. During his years in Leipzig, he first of all pursued the theory of molecules and of chemical bonding. His contribution to the "Handbuch der Physik" (En- cyclopedia of Physics) about the quantum mechanics of atoms and molecules was also written during this period. He then addressed his attention in par- ticular to the quantum theory of solids. A series of works dealing with the states of electrons in crystal lattices bears witness to this interest. He was also active in the field of nuclear physics and, in 1937, he published a one- particle model of the atomic nucleus. This was a forerunner of the later shell model. Towards the end of the 1930s, Hund turned his attention to the field properties of matter. He was able to show that a qualitative understanding of chemical bonding and of the production of matter could already be achieved in the classical wave picture. He began work on a monograph about the field theory of matter. However, due to the unfavorable circumstances that prevailed during and immediately after the Second World War, he was only able to complete this manuscript very much later. The book finally appeared in 1954 with the title "Materie als Feld" (Matter as Field) [3]. It contained a systematic description of the wave aspects of matter and also a particularly clear introduction to quantum field theory. In the second half of his life, Hund was particularly concerned to com- municate his knowledge and experience to younger . During his last years in Leipzig he had already begun to write a textbook of theoreti- cal physics based on his lectures there. The book appeared shortly after the Second World War, its final form being the three-volume work "Theoretische Physik. Eine Einftihrung" (Theoretical Physics. An Introduction) [4], which appeared in 1956/57. Shortly after this, in 1961, he published a book "The- orie des Aufbaues der Materie" (Theory of the Structure of Matter) [5]. In the preface to this book Hund expressed very clearly his desire to provide an understanding of the foundations of physics. Anyone who has read his books can confirm that he did indeed achieve his objective. Hund has documented his thoughts about general questions of physics in very many articles and published lectures. In the book "Das Naturbild der Physik" (The Physical Concept of Nature) [6] one finds an impressive selection of these contributions. Friedrich Hund on his 100th Birthday IX

Following his official retirement, Hund directed his attention in particular to matters concerning the history of physics, a subject that had always fas- cinated him. He wrote two further books: "Grundbegriffe der Physik" (Basic Concepts of Physics) [7] and "Geschichte der physikalischen Begriffe" (The History of Physical Concepts) [8], in addition to his history of quantum theory that was mentioned above. A large portion of Hund's work was always dedicated to teaching. He was invariably exceptionally well prepared when presenting his lectures and talks. He got everything just right, even ensuring that he made best use of the available blackboards. His contributions to discussions in colloquia were of such high quality that all those who were present will remember them. His commitment was not confined to lectures. It is with pleasure that we recall how, in 1949, we took part (as guests from Berlin) in a vacation seminar in the Thuringian Forest. The seminar was organized by Hund and a mathematician for the benefit of Jena students. In the 1950s Hund regularly took part in the Oberwolfach meetings in the Black Forest, which at that time were organized by young theoreticians. Much energy was expanded not only in the scientific program but also in long hikes, one of Hund's favorite pastimes. The large number of his students confirms the fact that, throughout his career, he succeeded in gathering around himself a circle of highly motivated young physicists. When it comes to personal affairs, Hund is rather reserved. What is con- spicuous, however, is his distinctive personality, characterized by an upright and incorruptible nature, calmness, and deep sense of duty. He lived hon- orably throughout the evil times of Nazi rule (indeed, one of us recalls his kindness during this time with immense gratitude). In the few years for which he remained in the Soviet Zone, he retained his independence and made no secret of his critical opinions. Together with his wife and five children he left Jena in an unconventional manner, in other words, secretly. At that time there was no other way in which he could accept the professorship offered to him in Frankfurt. His interest in physical research remains undiminished today. When one of us visited him in G5ttingen shortly after his 99th birthday, he began the conversation with the question "What do we really know now about the mass of the top quark?" Since 1936, together with E. Trendelenburg, Hund was the editor of the "Ergebnisse der exakten Naturwissenschaften" (Achievements of the Exact Natural Sciences), the forerunner of the "Springer Tracts in Modern Physics". It was here, in 1936, that his article "Materie bei sehr hohen Drucken und Temperaturen" (Matter Under Very High Pressures and Temperatures)was published. The scientific content of the present volume is written in connection with this article. Both the German original and an English translation of Hund's 1936 article are to be found starting with p. 175. X Friedrich Hund on his 100th Birthday

Together with the editors of the "Springer Tracts" and the authors of this monograph, we convey our warmest congratulations to Friedrich Hund on the occasion of his 100th birthday. 44t, Gerhard HShler

Harry Lehmann

Ernst Niekisch

References 1 Hund F. (1927) Linienspektrum und periodisches System der Elemente. Julius Springer, Berlin 2 Hund F. (1984) Geschichte der Quantentheorie, 3rd edn. Bibliographisches Institut, Mannheim 3 Hund F. (1954) Materie als Feld. Eine Einffihrung. Springer, Berlin 4 Hund F. (1956/57) Theoretische Physik. Eine Einfiihrung, 3 Vols. Teubner, Stuttgart 5 Hund F. (1961) Theorie des Aufbaues der Materie. Teubner, Stuttgart 6 Hund F. (1975) Das Naturbild der Physik. Ausgew~hlte Schriften. (ed. by Hajdu J., Lfiders G. with support of the Deutsche Physikalische Gesell- schaft) Druckerei der Kernforschungsanlage Jfilich GmbH. 7 Hund F. (1969) Grundbegriffe der Physik. Bibliographisches Institut, MannN heim 8 Hund F. (1972) Geschichte der physikalischen Begriffe. Bibliographisches Institut, Mannheim Preface

This book would not exist were it not for some of Friedrich Hund's for- mer students, whose idea it was to dedicate a special volume of the Springer Tracts in Modern Physics in honor of Hund's 100th birthday. For a long time Friedrich Hund was an editor of the "Ergebnisse der exakten Naturwis- senschaften" (now the Springer Tracts) where in 1936 he published a review article on matter under extreme conditions, which contained a detailed dis- cussion on astrophysical aspects and applications. In spite of the tremendous progress in laboratory experiments at high pressure and temperature, astro- physics of compact stars remained the realm of matter in genuinely extreme environments. In the 1930s when Hund wrote his article, white dwarf stars had already been observed and their stability was understood in terms of the properties of the degenerate electron gas. The existence of neutron stars had been predicted at about the same time but it took more than 30 years until rapidly pulsating radio and X-ray emitting sources were discovered and iden- tified as rotating neutron stars, which possess in their interiors the largest densities in the universe. Due to the enormous progress in observational tech- niques and computer power, it is now possible to explore both experimentally and theoretically many details of the internal structure of planets, ordinary stars, white dwarf stars, and neutron stars. Thus, today the physics of com- pact astrophysical objects is more fascinating than ever, and it could serve as a common playground for astrophysicists, solid-state physicists, and nuclear physicists. Some parts of this monograph have the character of short reviews on in- dependent research areas and we are grateful to many friends and colleagues who are specialists in those fields for offering numerous suggestions and com- ments that have helped to improve the manuscript considerably. Amongst others we thank D. Koester, P. Kumar, G. RSpke, and M. Schneider, and we are particulary indebted to J. Ruoff for contributing to the chapter on neutron-star oscillations. In addition we are grateful to a legion of students for their assistance in our continuous battle with ~TEX macros, postscript files, scanners, and other hardware demons that had to be mastered in order to meet a deadline that had already been set one hundred years ago. We are also grateful to A. Wahl for re-typing Friedrich Hund's article, to A. Lahee for XII Preface the English translation, and to J. Lenz and V. Wicks from Springer-Verlag for proof reading the manuscript.

T/ibingen H. Riffert January 1996 H. Miither H. Herold H. Ruder Contents

o Introduction ...... 1

2. The Equation of State and the Structure of Cosmic Objects ...... 3 2.1 Basic Structure Equations and Ideal Equations of State ...... 4 2.2 Non-ideal Equations of State ...... 7 2.3 Stellar Energy Sources ...... 14 2.4 Energy Transport ...... 16

. Nuclear Equations of State ...... 21 3.1 General Remarks ...... 21 3.2 Green's Yhnction and Many-Body Theory ...... 29 3.2.1 Time-Dependent Perturbation Theory ...... 29 3.2.2 Perturbation Theory and Feynman Diagrams ...... 33 3.2.3 Single-Particle Green's Function ...... 37 3.3 Nucleon-Nucleon Interaction ...... 43 3.3.1 The One-Boson-Exchange Model ...... 45 3.3.2 NN Scattering ...... 48 3.3.3 Medium-Range Attraction ...... 52 3.4 Hole-Line Expansion and Other Approaches ...... 55 3.4.1 Brueckner-Hartree-Fock ...... 55 3.4.2 Beyond BHF ...... 61 3.5 Relativistic Effects ...... 66 3.5.1 Walecka Model for Nuclear Matter ...... 66 3.5.2 Dirac-Brueckner-Hartree-Fock ...... 69 3.6 Subnucleonic Degrees of Freedom ...... 74 3.6.1 Excitations of the Nucleons ...... 74 3.6.2 Pion Condensation ...... 78 3.6.3 Effective Quark Models ...... 84

4. Neutron Stars: Spherically Symmetric and Rotating Models ...... 93 4.1 Spherically Symmetric Neutron Stars ...... 93 4.1.1 Relativistic Structure Equations ...... 95 XIV Contents

4.1.2 Solution Method and Results ...... 98 4.1.3 Stability and Maximum Mass ...... 99 4.2 Rapidly Rotating Neutron Stars ...... 101 4.2.1 Basic Formulation ...... 103 4.2.2 Numerical Solution Method ...... 110 4.2.3 Results ...... 113

. Asteroseismology ...... 121 5.1 Oscillations of Spheres ...... 121 5.2 Free Oscillations of the Earth ...... 136 5.3 Helioseismology ...... 142 5.4 Asteroseismology of White Dwarfs ...... 154 5.5 Oscillations of Neutron Stars ...... 162

Reprint of Friedrich Hund's Review Article ...... 175

English Translation of Friedrich Hund's Review Article ...... 217

References ...... 259

Subject Index ...... 271