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Carl J. Hansen Steven D. KawalerVirginia Trimble

Stellar InteriorsPhysical Principles, Structure,and Evolution

Second Edition

With 119 Illustrations and a CD-ROM

, Springer

Carl J. Hansen, Department of Astrophysics and Planetary Sciences, University of Colorado, Boulder, CO 80309-0440, USA. Email: chansen@jila.colorado.edu

Steven D. Kawaler, Departments of Physics and Astronomy, Iowa State University, Ames, IA 50011, USA. Email: sdk@iastate.edu

Virginia Trimble, Department of Astronomy, University of Maryland, College Park, MD 20740, and Department of Physics, University of California, Irvine, Irvine, CA 92717, USA

Cover illustration: Calligraphy by Sara Compton.

Library of Congress Cataloging-in-Publication Data Hansen, Cari J.

Stellar interiors : physical principles, structure, and evolution.-2nd ed. / C.J. Hansen, S.D. Kawaler, and V. Trimble.

p. em. - (Astronomy and astrophysics library) Inc\udes bibliographieal referenees and index.

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Preface

The first edition of this text appeared in 1994. Shortly after the third printing,our editor suggested that we attempt a second edition because new develop-ments in stellar structure and evolution had made our original work outdated.We (the original authors, CJH and SDK) reluctantly agreed but with reser-vations due to the effort involved. Our initial reluctance disappeared whenwe were able to convince (cajole, twist the arm of, etc.) our new coauthor-colleague Virginia Trimble to join us. (Welcome Virginia!) We (i.e., all threeof us) hope that you agree that the present edition is a great improvementcompared to the 1994 effort.

Our objectives in this edition are the same ones we set forth in 1994:

What you will find is a text designed for our target audience: the typi-cal senior undergraduate or beginning graduate student in astronomyor astrophysics who wishes an overview of stellar structure and evo-lution with just enough detail to understand the general picture. Sheor he can go on from there to more specialized texts or directly tothe research literature depending on talent and interests. To this end,this text presents the basic physical principles without chasing all the(interesting!) details.

For those of you familiar with the first edition, you will find that somethings have not been changed substantially (F = ma is still F = ma), whileothers definitely have. For example, Chapter 2 has been completely rewritten.In many respects this chapter is the key to the text because it gives an ex-tensive overview of the subject. The next eight chapters rely on the student’shaving absorbed large parts of Chapter 2, though complete understanding isnot necessary. Many students may wish to start with Chapter 2, although werecommend at least a once-through of Chapter 1, which contains some fun-damental material. And, in response to many requests, there is substantiallymore observational material.

We have also attempted to improve on the graphics and have includedmore than we did in the first edition. In addition, the instructor will findmany more “Exercises” at the end of chapters. They are a mixed bag (easy,moderate, difficult) but we hope they illuminate much of what we have tosay. (Chapter 2 has more than its share; and, in fact, Chapters 1 and 2, plusexercises, could be the basis of a mini-course.)

VI Preface

Also new is the inclusion on the inside back cover of a CD-ROM contain-ing computer programs that make decent “zero-age main sequence” stellarmodels and analyse those models for “pulsations” (radial and nonradial), andstellar evolution codes everyone can play with. All are in FORTRAN and shouldwork on most computer platforms. Some of these codes are of our doing andwe thank Andy Odell and Dean Pesnell (Nomad Research) for their generouscontributions. As an additional bonus we have included portions of a colorfuland informative Stellar Evolution Tutorial put together by John Lattanzioand his colleagues (as part of a commercial enterprise called Cantanout Ltd.).See the README files on the CD-ROM for more information on the programsand tutorial.

Acknowledgments: We wish to thank our many past and present senior col-leagues and students for numerous reprints, corrections, suggestions, com-ments, problems (i.e., exercises), book loans, help with computer glitches,and PostScript figure files. They made our task much easier and enjoyable.Blame the typos, mistakes, and confusion on us. In particular, for the secondedition, we thank Dave Arnett, Mitch Begelman, David Branch, Nic Brum-mell, Joe Cassinelli, Maurice Clement, Peter Conti, Ethan Hansen, HennyLamers, Michael McCarthy, Cole Miller, Sean O’Brien, Dean Richardson,Dimitar Sasselov, Ted Snow, Peter Stetson, Pat Thaddeus, Juri Toomre, DonVandenberg, Craig Wheeler, Matt Wood, and Ellen Zweibel. VT gives per-sonal thanks to those people from whom she first learned that stellar struc-ture and evolution is an exciting topic—namely, (the late) Thornton LeighPage, J. Beverly Oke, and Bohdan Paczynski. She also recognizes the pastencouragement and support of UCLA, CalTech, and the Stony Brook Sum-mer School. CJH and SDK wish to thank their families and especially theirwives Camille and Leslie: may they not become computer widows yet again.Finally, we send many kudos to our editors at Springer-Verlag.

The text was set in LATEX2ε by the authors.

Carl J. Hansen University of Colorado at BoulderSteven D. Kawaler Iowa State University at AmesVirginia Trimble University of California at Irvine,

University of Maryland at College Park

Contents

1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Hydrostatic Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 An Energy Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3 The Virial Theorem and Its Applications . . . . . . . . . . . . . . . . . . 7

1.3.1 Application: Global Energetics . . . . . . . . . . . . . . . . . . . . . 101.3.2 Application: The Kelvin–Helmholtz Time Scale . . . . . . 111.3.3 Application: A Dynamic Time Scale . . . . . . . . . . . . . . . . 131.3.4 Application: Estimates of Stellar Temperatures . . . . . . . 131.3.5 Application: Another Dynamic Time Scale . . . . . . . . . . . 14

1.4 The Constant-Density Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161.4.1 Calculation of Molecular Weights . . . . . . . . . . . . . . . . . . . 171.4.2 The Temperature Distribution . . . . . . . . . . . . . . . . . . . . . 21

1.5 Energy Generation and Transport . . . . . . . . . . . . . . . . . . . . . . . . 211.6 Stellar Dimensional Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241.7 Evolutionary Lifetimes on the Main Sequence . . . . . . . . . . . . . . 301.8 The Hertzsprung–Russell Diagram . . . . . . . . . . . . . . . . . . . . . . . . 301.9 Summary Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321.10 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331.11 References and Suggested Readings . . . . . . . . . . . . . . . . . . . . . . . 38

2 An Overview of Stellar Evolution . . . . . . . . . . . . . . . . . . . . . . . . . 432.1 Young Stellar Objects (YSOs) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 442.2 The Zero-Age Main Sequence (ZAMS) . . . . . . . . . . . . . . . . . . . . 47

2.2.1 Life on the Main Sequence . . . . . . . . . . . . . . . . . . . . . . . . . 482.2.2 Brown Dwarfs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

2.3 Leaving the Main Sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 512.3.1 Cluster HR Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

Cluster and Galactics Ages . . . . . . . . . . . . . . . . . . . . . . . . 562.3.2 Mass Loss From Massive Stars . . . . . . . . . . . . . . . . . . . . . 57

2.4 Red Giants and Supergiants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 602.5 Helium Flash or Fizzle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

2.5.1 Helium Core Burning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65Helium Core Exhaustion . . . . . . . . . . . . . . . . . . . . . . . . . . 66

2.5.2 Asymptotic Giant Branch . . . . . . . . . . . . . . . . . . . . . . . . . 672.6 Later Phases, Initial MassesM≤6–10M� . . . . . . . . . . . . . . . . 69

VIII Contents

2.6.1 A Bit About White Dwarfs . . . . . . . . . . . . . . . . . . . . . . . . 702.7 Advanced Phases, Initial MassesM >6–10M� . . . . . . . . . . . . 732.8 Core Collapse and Nucleosynthesis . . . . . . . . . . . . . . . . . . . . . . . . 75

2.8.1 Abundances and Nucleosynthesis . . . . . . . . . . . . . . . . . . . 782.9 Variable Stars: A Brief Overview . . . . . . . . . . . . . . . . . . . . . . . . . 83

2.9.1 Eclipsing and Ellipsoidal Variables . . . . . . . . . . . . . . . . . . 832.9.2 Spotted, Rotating Stars . . . . . . . . . . . . . . . . . . . . . . . . . . . 842.9.3 T Tauri Stars, FU Orionis Stars (FUORs), and

Luminous Blue Variables . . . . . . . . . . . . . . . . . . . . . . . . . . 842.9.4 Last Helium Flash and Formation of Atmospheric Dust 84

2.10 Pulsational Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 852.11 Explosive Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

2.11.1 Novae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 902.11.2 Supernovae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

SN1987A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97SN Type I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100SN Remnants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

2.12 White Dwarfs, Neutron Stars, and Black Holes . . . . . . . . . . . . . 1032.13 Binary Stars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

2.13.1 Types of Binaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1052.13.2 The Roche Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1072.13.3 Formation and Early Evolution . . . . . . . . . . . . . . . . . . . . 1092.13.4 The First Mass Transfer Phase and its Consequences . 1102.13.5 Systems With One Compact Component . . . . . . . . . . . . 1132.13.6 The Second Phase of Mass Transfer . . . . . . . . . . . . . . . . . 1132.13.7 Binaries With Two Compact Components . . . . . . . . . . . 114

2.14 Star Formation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1162.15 Supplemental Material . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1202.16 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1222.17 References and Suggested Readings . . . . . . . . . . . . . . . . . . . . . . . 138

3 Equations of State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1453.1 Distribution Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1463.2 Blackbody Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1503.3 Ideal Monatomic Gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1523.4 The Saha Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1553.5 Fermi–Dirac Equations of State . . . . . . . . . . . . . . . . . . . . . . . . . . 159

3.5.1 The Completely Degenerate Gas . . . . . . . . . . . . . . . . . . . 1603.5.2 Application to White Dwarfs . . . . . . . . . . . . . . . . . . . . . . 1633.5.3 Effects of Temperature on Degeneracy . . . . . . . . . . . . . . 166

3.6 “Almost Perfect” Equations of State . . . . . . . . . . . . . . . . . . . . . . 1713.7 Adiabatic Exponents and Other Derivatives . . . . . . . . . . . . . . . . 173

3.7.1 Keeping the Composition Fixed . . . . . . . . . . . . . . . . . . . . 174Specific Heats . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174Adiabatic Exponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175

Contents IX

Mixtures of Ideal Gases and Radiation . . . . . . . . . . . . . . 176Mixtures of Degenerate and Ideal Gases . . . . . . . . . . . . . 178

3.7.2 Allowing for Chemical Reactions . . . . . . . . . . . . . . . . . . . 1803.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1843.9 References and Suggested Readings . . . . . . . . . . . . . . . . . . . . . . . 189

4 Radiative and Conductive Heat Transfer . . . . . . . . . . . . . . . . . . 1934.1 Radiative Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1934.2 The Diffusion Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199

4.2.1 A Brief Diversion into “∇s” . . . . . . . . . . . . . . . . . . . . . . . 2014.3 A Simple Atmosphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2024.4 Radiative Opacity Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207

4.4.1 Electron Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2084.4.2 Free–Free Absorption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2094.4.3 Bound–Free and Bound–Bound Absorption . . . . . . . . . . 2124.4.4 H− Opacity and Others . . . . . . . . . . . . . . . . . . . . . . . . . . . 213

4.5 Heat Transfer by Conduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2144.6 Tabulated Opacities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2164.7 Some Observed Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2234.8 Line Profiles and the Curve of Growth . . . . . . . . . . . . . . . . . . . . 226

4.8.1 The Lorentz Profile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2264.8.2 Doppler Broadening . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2284.8.3 Curve of Growth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230

4.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2324.10 References and Suggested Readings . . . . . . . . . . . . . . . . . . . . . . . 236

5 Heat Transfer by Convection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2415.1 The Mixing Length Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241

5.1.1 Criteria for Convection . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2425.1.2 Radiative Leakage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2475.1.3 The Equation of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . 2495.1.4 Convective Efficiencies and Time Scales . . . . . . . . . . . . . 2505.1.5 Convective Fluxes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2535.1.6 Calculations in the MLT . . . . . . . . . . . . . . . . . . . . . . . . . . 2545.1.7 Numeric Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255

5.2 Variations on the MLT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2585.2.1 Beyond the MLT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2615.2.2 Semiconvection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261

5.3 Hydrodynamic Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2635.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2665.5 References and Suggested Readings . . . . . . . . . . . . . . . . . . . . . . . 267

X Contents

6 Stellar Energy Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2716.1 Gravitational Energy Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2716.2 Thermonuclear Energy Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . 273

6.2.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2746.2.2 Nuclear Energetics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2766.2.3 Astrophysical Thermonuclear Cross Sections and

Reaction Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2796.2.4 Nonresonant Reaction Rates . . . . . . . . . . . . . . . . . . . . . . . 282

Example: The 12C (p, γ) 13N Reaction . . . . . . . . . . . . . . . 2866.2.5 Resonant Reaction Rates . . . . . . . . . . . . . . . . . . . . . . . . . . 2896.2.6 Other Forms of Reaction Rates . . . . . . . . . . . . . . . . . . . . . 291

Neutron Capture and the S-Process . . . . . . . . . . . . . . . . . 292Weak Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294The Proton–Proton Reaction . . . . . . . . . . . . . . . . . . . . . . 296

6.2.7 Special Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2976.3 The Proton–Proton Chains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299

6.3.1 Deuterium and Lithium Burning . . . . . . . . . . . . . . . . . . . 3036.4 The Carbon–Nitrogen–Oxygen Cycles . . . . . . . . . . . . . . . . . . . . . 3036.5 Helium-Burning Reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3076.6 Carbon, Neon, and Oxygen Burning . . . . . . . . . . . . . . . . . . . . . . 3116.7 Silicon “Burning” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3136.8 Neutrino Emission Mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . 314

Pair Annihilation Neutrinos . . . . . . . . . . . . . . . . . . . . . . . 315Photoneutrinos and Bremsstrahlung Neutrinos . . . . . . . 316Plasma Neutrinos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317

6.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3186.10 References and Suggested Readings . . . . . . . . . . . . . . . . . . . . . . . 322

7 Stellar Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3297.1 The Equations of Stellar Structure . . . . . . . . . . . . . . . . . . . . . . . . 3297.2 Polytropic Equations of State and Polytropes . . . . . . . . . . . . . . 331

7.2.1 General Properties of Polytropes . . . . . . . . . . . . . . . . . . . 3327.2.2 Numerical Calculation of the Lane–Emden Functions . 337

Shooting for a Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338The Fitting Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 340

7.2.3 The U–V Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3437.2.4 Newton-Raphson or “Henyey” Methods . . . . . . . . . . . . . 3457.2.5 Eigenvalue Problems and the Henyey Method . . . . . . . . 3497.2.6 Dynamic Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3517.2.7 The Eddington Standard Model . . . . . . . . . . . . . . . . . . . . 3577.2.8 Applications to Zero-Temperature White Dwarfs . . . . . 361

7.3 The Approach to Real Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3627.3.1 Central Expansions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3627.3.2 The Radiative Stellar Envelope . . . . . . . . . . . . . . . . . . . . . 363

The Structure of the Envelope . . . . . . . . . . . . . . . . . . . . . 363

Contents XI

The Radiative Temperature Structure . . . . . . . . . . . . . . . 3667.3.3 Completely Convective Stars . . . . . . . . . . . . . . . . . . . . . . . 367

A Question of Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 370Application to Pre-Main Sequence Evolution . . . . . . . . . 373

7.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3747.5 References and Suggested Readings . . . . . . . . . . . . . . . . . . . . . . . 376

8 Asteroseismology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3798.1 Adiabatic Radial Pulsations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 380

8.1.1 The Linear Adiabatic Wave Equation . . . . . . . . . . . . . . . 3848.1.2 Some Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3858.1.3 Asymptotic Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 388

8.2 Nonadiabatic Radial Motions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3918.2.1 The Quasi-Adiabatic Approximation . . . . . . . . . . . . . . . . 3948.2.2 The κ- and γ-Mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . 397

The Epstein Weight Function and Cepheids . . . . . . . . . . 3998.2.3 Nonadiabaticity and the Cepheid Strip . . . . . . . . . . . . . . 400

A Footnote on Nonlinear Modeling . . . . . . . . . . . . . . . . . 4038.3 An Introduction to Nonradial Oscillations . . . . . . . . . . . . . . . . . 404

8.3.1 Linearization of the Hydrodynamic Equations . . . . . . . . 4048.3.2 Separation of the Pulsation Equations . . . . . . . . . . . . . . 4088.3.3 Properties of the Solutions . . . . . . . . . . . . . . . . . . . . . . . . . 412

Mode Classification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415The Eigenfunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 418

8.3.4 The Inverse Problem and Rotation . . . . . . . . . . . . . . . . . 419Probing for Internal Rotation . . . . . . . . . . . . . . . . . . . . . . 421Solid Stars?! . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423

8.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4248.5 References and Suggested Readings . . . . . . . . . . . . . . . . . . . . . . . 427

9 Structure and Evolution of the Sun . . . . . . . . . . . . . . . . . . . . . . . 4319.1 Vital Statistics of the Sun . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4329.2 From the ZAMS to the Present . . . . . . . . . . . . . . . . . . . . . . . . . . . 434

9.2.1 The Sun on the ZAMS . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4349.2.2 Evolution From the ZAMS . . . . . . . . . . . . . . . . . . . . . . . . 4379.2.3 The Present-Day Sun . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439

9.3 The Solar Neutrino “Problem” . . . . . . . . . . . . . . . . . . . . . . . . . . . 4429.4 The Role of Rotation in Evolution . . . . . . . . . . . . . . . . . . . . . . . . 448

9.4.1 von Zeipel’s Paradox . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4509.4.2 Rotational Mixing of Stellar Interiors . . . . . . . . . . . . . . . 452

9.5 Helioseismology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4549.5.1 Observed and Predicted Pulsation Frequencies . . . . . . . 4559.5.2 Helioseismology and the Solar Interior . . . . . . . . . . . . . . 456

Structural Inversions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 457Rotational Inversions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 459

XII Contents

9.6 References and Suggested Readings . . . . . . . . . . . . . . . . . . . . . . . 460

10 Structure and Evolution of White Dwarfs . . . . . . . . . . . . . . . . . 46710.1 Observed Properties of White Dwarfs . . . . . . . . . . . . . . . . . . . . . 46710.2 White Dwarf Evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 469

10.2.1 Cooling of White Dwarfs . . . . . . . . . . . . . . . . . . . . . . . . . . 47010.2.2 Realistic Evolutionary Calculations . . . . . . . . . . . . . . . . . 472

10.3 The Magnetic White Dwarfs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47710.3.1 Magnetic Field Decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 479

10.4 The Variable White Dwarfs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48110.4.1 The Observed Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . 48210.4.2 White Dwarf Seismology . . . . . . . . . . . . . . . . . . . . . . . . . . 482

White Dwarfs and the Whole Earth Telescope . . . . . . . 48510.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49010.6 References and Suggested Readings . . . . . . . . . . . . . . . . . . . . . . . 490

A Mini Stellar Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 497

B Table of Symbols and Physical Constants . . . . . . . . . . . . . . . . . 503

C List of Journal Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 513

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 515