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Stellar Structure and Evolution Lecture Notes on Stellar Structure and Evolution Jørgen Christensen-Dalsgaard Institut for Fysik og Astronomi, Aarhus Universitet Sixth Edition Fourth Printing March 2008 ii Preface The present notes grew out of an introductory course in stellar evolution which I have given for several years to third-year undergraduate students in physics at the University of Aarhus. The goal of the course and the notes is to show how many aspects of stellar evolution can be understood relatively simply in terms of basic physics. Apart from the intrinsic interest of the topic, the value of such a course is that it provides an illustration (within the syllabus in Aarhus, almost the first illustration) of the application of physics to “the real world” outside the laboratory. I am grateful to the students who have followed the course over the years, and to my colleague J. Madsen who has taken part in giving it, for their comments and advice; indeed, their insistent urging that I replace by a more coherent set of notes the textbook, supplemented by extensive commentary and additional notes, which was originally used in the course, is directly responsible for the existence of these notes. Additional input was provided by the students who suffered through the first edition of the notes in the Autumn of 1990. I hope that this will be a continuing process; further comments, corrections and suggestions for improvements are most welcome. I thank N. Grevesse for providing the data in Figure 14.1, and P. E. Nissen for helpful suggestions for other figures, as well as for reading and commenting on an early version of the manuscript. I also thank Bent Christensen-Dalsgaard and T. M. Brown for their assistance in locating the Bradbury reference in Chapter 2. The High Altitude Observatory, Boulder, Colorado is thanked for hospitality during the Summer of 1990, where a substantial part of the notes were written. Aarhus, August 1991 Jørgen Christensen-Dalsgaard Preface to 4th edition Relative to earlier editions a number of mistakes have been corrected and some of the figures have been revised. I am grateful to all of those who have provided comments and corrections, in particular N. H. Andersen and H. Jørgensen, who provided extensive lists of errors and suggestions for improvements. A revised description of the origin of the “hook” in the evolution tracks for moderate- and large-mass stars at the end of central hydrogen burning resulted from work by E. Michel, and I am very grateful to him for discussions and computations leading to this correction. The computations on which some of the figures iii iv are based were supported by the Danish Natural Science Research Council. Once again I thank the High Altitude Observatory for Summer hospitality during the revisions. Boulder, August 23, 1995 Jørgen Christensen-Dalsgaard Preface to 5th edition The notes have been significantly revised and updated in this edition, including new re- sults on the solar neutrino problem and from helioseismology, as well as an update of the discussion of supernova explosions and nucleosynthesis. In addition, Chapter 16 on compact objects is new; I am very grateful to Jes Madsen for his permission to include it, and to Karlheinz Langanke for translating it from the Danish original (‘Stjerneudviklin- gens slutstadier’). I also thank Langanke for very substantial contributions to the revised Chapters 14 and 15. Discussions with A. V. Sweigart and J. Lattanzio were very helpful in bringing my ideas about late stellar evolution more up to date, and I thank the former for providing the data used in Figure 12.7. Further corrections and suggestions received from the students who have followed the course are also gratefully acknowledged. Aarhus, September 6, 2000 Jørgen Christensen-Dalsgaard Preface to 6th edition The only substantial change in this edition is an update of Section 11.5.3, including the spectacular results on neutrino oscillations from the Sudbury Neutrino Observatory. In addition, a number of minor corrections have been incorporated. Second printing: In this reprinting a few minor changes have been made, including a replacement of Figure 2.6 and an update of a few references. Third printing: In this reprinting some generally minor changes have been made, partic- ularly in Chapter 14, and a few references have been added. Fourth printing: This reprinting is essentially identical in content to the third printing, but the text has been reset in LATEX, and the figures have been included in the output file. Thus this version is being made available on the internet. Aarhus, 9 March, 2008 Jørgen Christensen-Dalsgaard Contents 1 Introduction 1 1.1 Stellar timescales . 2 1.1.1 The dynamical timescale . 2 1.1.2 The timescale for release of gravitational energy (or the thermal timescale) . 3 1.1.3 The nuclear timescale . 3 1.2 The life of stars . 4 1.3 The physics of stellar interiors . 8 1.4 Tests of stellar evolution calculations . 10 1.5 Bibliographical notes . 12 2 Observable properties of stars 13 2.1 Introduction . 13 2.2 Stellar positions and distances . 13 2.3 Stellar brightness . 15 2.4 Interstellar absorption . 20 2.5 Spectral analysis . 20 2.5.1 Spectral lines . 21 2.5.2 Spectral classification . 23 2.5.3 Stellar abundances . 24 2.5.4 The Doppler shift . 25 2.5.5 The gravitational redshift . 26 2.5.6 The Zeeman effect . 26 2.6 Colour-magnitude diagrams . 27 2.7 Stellar masses . 31 2.8 Stellar pulsations . 32 2.9 Stellar activity . 34 2.10 Other types of emission . 35 2.10.1 Observations of neutrinos . 35 2.10.2 Solar wind particles . 36 2.10.3 Meteoritic abundances . 36 3 The equation of state 39 3.1 Introduction . 39 3.2 The ideal gas . 40 3.2.1 Simple gas . 40 3.2.2 Fully ionized gas consisting of a mixture of elements . 44 v vi CONTENTS 3.2.3 Partial ionization . 45 3.3 The distribution function for a classical gas . 47 3.4 The radiation pressure . 47 3.5 Degenerate matter . 49 4 Hydrostatic equilibrium 53 4.1 Introduction . 53 4.2 Estimates of stellar internal pressure and temperature . 55 4.2.1 The estimates . 55 4.2.2 The importance of radiation pressure . 56 4.3 Strict limits on the central pressure . 57 4.3.1 A simple limit . 57 4.3.2 A stronger limit . 59 4.4 The virial theorem . 59 4.4.1 Introduction . 59 4.4.2 The nonrelativistic case . 60 4.4.3 The relativistic case . 61 4.5 Simple solutions to the equation of hydrostatic equilibrium . 62 4.5.1 The linear model . 62 4.5.2 The isothermal atmosphere . 63 4.6 Polytropic models . 64 5 Radiative energy transport 71 5.1 Radiative transport in stellar interiors . 71 5.2 Radiation from the stellar surface . 73 5.3 The opacity . 74 5.4 Stellar atmospheres . 76 5.5 The energy equation . 78 6 Energy transport by convection 81 6.1 Introduction . 81 6.2 The instability condition . 82 6.3 Where does convection occur? . 85 6.4 Energy transport by convection . 86 6.4.1 Introduction . 86 6.4.2 Estimate of the superadiabatic temperature gradient . 88 6.4.3 The convective time scale . 91 7 Mass-luminosity relations 93 7.1 Stars dominated by radiative transport . 93 7.2 Predominantly convective stars . 95 7.2.1 Introduction . 95 7.2.2 Convective envelopes . 96 7.2.3 Completely convective stars . 98 7.3 The “forbidden” region . 99 CONTENTS vii 8 Nuclear energy generation 103 8.1 Introduction . 103 8.2 The cross sections . 104 8.3 The release of energy . 109 8.4 The average reaction rate . 111 8.5 Hydrogen burning . 115 8.5.1 The PP-chains . 116 8.5.2 The CNO cycle . 118 8.6 Later reactions . 120 9 Numerical calculation of stellar structure and evolution 123 9.1 Introduction . 123 9.2 Equations and boundary conditions . 123 9.3 Numerical solution of differential equations . 124 9.4 Computation of stellar models . 126 9.5 The evolution with time . 127 9.6 Concluding remarks . 128 10 Evolution before the main sequence 129 10.1 Introduction . 129 10.2 The Jeans instability and star formation . 130 10.3 Hydrostatic contraction . ..
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