ASTRONOMY ANDASTROPHYSICS LIBRARY

Series Editors: G. Brner, Garching, GermanyA. Burkert, Mnchen, GermanyW. B. Burton, Charlottesville, VA, USA and

Leiden, The NetherlandsM. A. Dopita, Canberra, AustraliaA. Eckart, Kln, GermanyT. Encrenaz, Meudon, FranceE. K. Grebel, Heidelberg, GermanyB. Leibundgut, Garching, GermanyA. Maeder, Sauverny, SwitzerlandV. Trimble, College Park, MD, and Irvine, CA, USA

Andre Maeder

Physics, Formation and Evolutionof Rotating Stars

1 3

Andre MaederUniversite GeneveObservatoire de Geneve1290 SauvernySwitzerlandandre.maeder@unige.ch

Cover image: Cluster NGC 3603. Hubble Space Telescope WFPC2. PRC99-20, STScI OPO, June 1,1999. Wolfgang Brandner (JPL/IPAC), Eva K. Grebel (Univ. Washington), You-Hua Chu (Univ.Illinois, Urbana Champaign) and NASA.

ISBN: 978-3-540-76948-4 e-ISBN: 978-3-540-76949-1

DOI 10.1007/978-3-540-76949-1

Astronomy and Astrophysics Library ISSN: 0941-7834

Library of Congress Control Number: 2008936872

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Preface

Rotation is often considered as a side effect in stellar evolution, a point not de-serving more than a section at the end of a book. However, at each step from starformation to the final stages of evolution, rotation is present and in some cases evendominates the course of evolution, the timescales and the nucleosynthesis. This is inparticular the case in star formation, where the initial angular momentum has to bereduced by a factor of 105 at least. Also during nuclear evolution the rotational in-stabilities drive internal mixing of the elements and rotation may enhance the massloss rates. Recent works (see Chap. 29) even suggest that rotation is a dominanteffect in the evolution and element synthesis of the first stars at zero and very lowmetallicities. Also, stellar rotation is an essential ingredient for the occurrence ofgamma-ray bursts (GRBs). This is why here we thoroughly examine the basic me-chanical and thermal effects of rotation during evolution, their influence on stellarwinds, the effects of differential rotation and associated instabilities and the possibledynamos generated in rotating stars. Also, the observational signatures of rotationaleffects are numerous, first from spectroscopy and now also from interferometricobservations, from chemical abundance determinations, from helioseismology andasteroseismology, etc.

To be useful at an introductory level, this book presents in a didactical way thebasic concepts of stellar structure and evolution in chapters indicated by a star ().These chapters form a basic course, while the other more specialized chapters forman advanced course. In general, I have given the step-by-step derivations of theanalytical developments for the readers comfort.

Three centuries ago, there were books covering all scientific domains, with evena touch of theology in addition. Then, science became more specialized. Half acentury ago, there were still books, like the one by Pecker and Schatzman, able topresent the whole astronomy at a specialized level. Nowadays, due to the explosionof scientific knowledge, it is becoming a considerable task to cover fields like stellarformation and evolution. Thus, despite the many subjects studied in this book, thereare still many topics not treated here, in particular the properties of stellar remnants,which deserve full books (see for example [83] and [281]). The same applies to theevolution of binary stars, the fact they are not treated here does not mean that theyhave not a certain importance. Indeed, most effects studied here also find an appli-cation in binaries, however with a higher degree of complexity due to the interaction

v

vi Preface

with tidal mixing, tidal generation of gravity waves, transport of angular momentumand mass transfer.

As a consequence of the extraordinary vitality of astrophysics, the numericalmodels and observational results tend soon to become obsolete, being supersededby new results from more detailed computations and modern techniques. Therefore,I usually tried to emphasize the analytical results, which express the fundamentalphysics of the problem and fortunately are not aging in the same way. Numericalmodels, whenever presented, are given mainly for providing illustrations of the gen-eral properties. For specific applications, the last (and hopefully best) precise valuesare always recommended.

I want to quote and express my gratitude, when possible, to eminent scientists,colleagues and friends for their major help in the course of my career: Profs. P.Bouvier, G. Burki, P. Conti, A.N. Cox, M. Golay, B. Hauck, R. Kippenhahn, J.Lequeux, M. Mayor, G. Meynet, F. Rufener, E. Schatzman, M. Schwarzschild, L.Smith, G. Tammann, J.P. Zahn. I kindly ask the readers to consider that this book isnot aiming at giving a historical perspective, nor to give quotations in proportion toauthor achievements. For that, it is better to consult the ADS databases. I also thankvery much many colleagues for fruitful collaborations and for participating in themanuscript correction: C. Charbonnel, P. Eggenberger, S. Ekstrom, C. Georgy, R.Hirschi, S. Mathis, G. Meynet, N. Mowlavi. I apologize for the unavoidable remain-ing mistakes, which are evidently my responsibility. I also thank Prof. C. Chiosifor most helpful remarks on the manuscript and Dr. Ramon Khanna of Springer forfruitful and constructive interactions. Last but not least, I express my deep gratitudeto my wife Elisabeth for her inalterable kindness and support.

Finally, I wish to the students in astrophysics and readers as much joy and funin their attempts to discover and understand the processes which rule the stars as Ihave myself, whether it concerns astrophysics or all the other marvels of Nature.

Geneva Observatory, Switzerland, Andre MaederMay 2008

Contents

Chapters marked with * may form the matter of a basic introductory course

Part I Stellar Equilibrium With and Without Rotation

1 The Mechanical Equilibrium of Stars . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.1 Momentum and Continuity Equations . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.1.1 Hydrodynamical Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.1.2 Hydrostatic Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.1.3 Mass Conservation and Continuity Equation . . . . . . . . . . . . . 51.1.4 Lagrangian and Eulerian Variables . . . . . . . . . . . . . . . . . . . . . . 61.1.5 Estimates of Pressure, Temperature and Timescales . . . . . . . . 7

1.2 The Potential Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.2.1 Relation to the Potential and Poisson Equation . . . . . . . . . . . . 111.2.2 The Potential Energy as a Function of Pressure . . . . . . . . . . . 121.2.3 The Internal Stellar Temperature . . . . . . . . . . . . . . . . . . . . . . . 13

1.3 The Virial Theorem for Stars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131.3.1 Star with Perfect Gas Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141.3.2 Star with a General Equation of State . . . . . . . . . . . . . . . . . . . 161.3.3 Slow Contraction, the KelvinHelmholtz Timescale . . . . . . . 17

2 The Mechanical Equilibrium of Rotating Stars . . . . . . . . . . . . . . . . . . . . 192.1 Equilibrium Configurations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.1.1 From Maclaurin Spheroids to the Roche Models . . . . . . . . . . 192.1.2 Hydrostatic Equilibrium for Solid Body Rotation . . . . . . . . . 202.1.3 Stellar Surface and Gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.1.4 Critical Velocities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242.1.5 Polar Radius as a Function of Rotation . . . . . . . . . . . . . . . . . . 27

2.2 Equations of Stellar Structure for Shellular Rotation . . . . . . . . . . . . . 292.2.1 Properties of the Isobars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302.2.2 Hydrostatic Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312.2.3 Continuity Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322.2.4 Equation of the Surface for Shellular Rotation . . . . . . . . . . . . 33

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3 The Energetic Equilibrium of Stars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353.1 The Radiative Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.1.1 Equation of Radiative Transfer . . . . . . . . . . . . . . . . . . . . . . . . . 363.1.2 Radiation Properties in Stellar Interiors . . . . . . . . . . . . . . . . . . 373.1.3 Transfer Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383.1.4 The Rosseland Mean Opacity . . . . . . . . . . . . . . . . . . . . . . . . . . 403.1.5 The MassLuminosity Relation . . . . . . . . . . . . . . . . . . . . . . . . 413.1.6 Photon Travel Times and ML Relation . . . . . . . . . . . . . . . . . 42

3.2 Energetic Equilibrium of a Star . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433.2.1 Why Are Stars Stable Nuclear Reactors? . . . . . . . . . . . . . . . . . 433.2.2 Energy Conservation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443.2.3 Combined Equation of Conservation and Transfer . . . . . . . . . 453.2.4 Relation with the Heat Conduction . . . . . . . . . . . . . . . . . . . . . . 46

3.3 Energy Generation Rate from Gravitational Contraction.Thermodynamic Expressions of dq . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473.3.1 Contraction of a Star with Perfect Gas . . . . . . . . . . . . . . . . . . . 483.3.2 Case of a General Equation of State . . . . . . . . . . . . . . . . . . . . . 483.3.3 The Entropy of Mixing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 503.3.4 The Difference of Specific Heats . . . . . . . . . . . . . . . . . . . . . . . 513.3.5 Adiabatic Gradient for Constant . . . . . . . . . . . . . . . . . . . . . . 523.3.6 Adiabatic Gradient for Variable . . . . . . . . . . . . . . . . . . . . . . 53

3.4 Changes of T and for Non-adiabatic Contraction . . . . . . . . . . . . . . . 533.4.1 Major Consequences for Evolution . . . . . . . . . . . . . . . . . . . . . 54

3.5 Secular Stability of Nuclear Burning . . . . . . . . . . . . . . . . . . . . . . . . . . 563.5.1 Shell Source Instability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

3.6 The Role of Radiation Pressure in Stars . . . . . . . . . . . . . . . . . . . . . . . . 583.6.1 The Radiative Pressure as a Function of Mass . . . . . . . . . . . . 603.6.2 The Eddington Luminosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

4 The Energy Conservation and Radiative Equilibrium in RotatingStars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 674.1 Radiative Equilibrium for Rotating Stars . . . . . . . . . . . . . . . . . . . . . . . 67

4.1.1 The Equation of Radiative Transfer . . . . . . . . . . . . . . . . . . . . . 674.1.2 Conservation of Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 684.1.3 Structure Equations for Rotating Stars . . . . . . . . . . . . . . . . . . . 68

4.2 Radiative Transfer in Rotating Stars . . . . . . . . . . . . . . . . . . . . . . . . . . . 704.2.1 Breakdown of Radiative Equilibrium . . . . . . . . . . . . . . . . . . . . 704.2.2 The Von Zeipel Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 714.2.3 Interferometric Observations of Stellar Distortion

and Gravity Darkening . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 744.3 Interactions of Rotation and Radiation Effects . . . . . . . . . . . . . . . . . . . 75

4.3.1 The , and Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 754.3.2 The Limit: Combined Eddington and Rotation Limits . . 76

4.4 Critical Rotation Velocities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 784.4.1 No Break-up Velocity for Differential Rotation ? . . . . . . . . . . 78

Contents ix

4.4.2 Classical Expression of the Critical Velocity . . . . . . . . . . . . . . 794.4.3 The Different Rotation Parameters . . . . . . . . . . . . . . . . . . . . . . 804.4.4 Critical Velocity Near the Eddington Limit . . . . . . . . . . . . . . . 81

5 Stellar Convection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 835.1 Gravity Waves and the BruntVaisala Frequency . . . . . . . . . . . . . . . . 83

5.1.1 Relation with the Entropy Gradient . . . . . . . . . . . . . . . . . . . . . 855.1.2 The Schwarzschild and Ledoux Criteria . . . . . . . . . . . . . . . . . 865.1.3 The Four T Gradients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

5.2 Mixing-Length Theory for the Convective Flux . . . . . . . . . . . . . . . . . 905.2.1 Orders of Magnitude . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

5.3 Convection in Stellar Interiors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 945.4 Non-adiabatic Convection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

5.4.1 Radiative Losses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 975.4.2 Thermal Adjustment Timescale . . . . . . . . . . . . . . . . . . . . . . . . 985.4.3 Solutions for Non-adiabatic Convection . . . . . . . . . . . . . . . . . 995.4.4 Limiting Cases, Fraction Carried by Convection . . . . . . . . . . 100

5.5 Convection in the Most Luminous Stars . . . . . . . . . . . . . . . . . . . . . . . . 1035.5.1 Convection Near the Eddington Limit . . . . . . . . . . . . . . . . . . . 1035.5.2 Density Inversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1045.5.3 Pressure and Flux of Turbulence . . . . . . . . . . . . . . . . . . . . . . . . 104

6 Overshoot, Semiconvection, Thermohaline Convection, Rotationand SolbergHoiland Criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1096.1 Convective Overshooting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

6.1.1 Overshooting in an MLT Non-local Model . . . . . . . . . . . . . . . 1096.1.2 The Roxburgh Criterion for Convective Overshoot . . . . . . . . 1126.1.3 Turbulence Modeling and Overshooting . . . . . . . . . . . . . . . . . 1146.1.4 Observational Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

6.2 Semiconvection and Thermohaline Convection . . . . . . . . . . . . . . . . . . 1186.2.1 Various Approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1196.2.2 Kato Equation, Thermohaline Convection . . . . . . . . . . . . . . . . 1216.2.3 Diffusion Coefficient for Semiconvection . . . . . . . . . . . . . . . . 123

6.3 Time-Dependent Convection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1246.4 Effects of Rotation on Convection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

6.4.1 Oscillation Frequency in a Rotating Medium . . . . . . . . . . . . . 1256.4.2 The Rayleigh Criterion and RayleighTaylor Instability . . . . 1276.4.3 The SolbergHoiland Criterion . . . . . . . . . . . . . . . . . . . . . . . . . 1286.4.4 Numerical Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

6.5 Convective Envelope in Rotating O-stars . . . . . . . . . . . . . . . . . . . . . . . 130

x Contents

Part II Physical Properties of Stellar Matter

7 The Equation of State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1377.1 Excitation and Ionization of Gases . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

7.1.1 Excitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1377.1.2 Ionization of Gases: The Saha Equation . . . . . . . . . . . . . . . . . 1397.1.3 The SahaBoltzmann Equation . . . . . . . . . . . . . . . . . . . . . . . . . 1407.1.4 Ionization Potentials and Negative Ions . . . . . . . . . . . . . . . . . . 142

7.2 Perfect Gas and Mean Molecular Weights . . . . . . . . . . . . . . . . . . . . . . 1437.3 Partially Ionized Stellar Medium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

7.3.1 Coupled Equations for a Medium Partially Ionized . . . . . . . . 1467.3.2 Thermodynamic Coefficients for Partial Ionization . . . . . . . . 148

7.4 Adiabatic Exponents and Thermodynamic Functions . . . . . . . . . . . . . 1497.4.1 Definitions of the Adiabatic Exponents . . . . . . . . . . . . . . . . . . 1497.4.2 Relation Between the i and Specific Heats . . . . . . . . . . . . . . 151

7.5 Thermodynamics of Mixture of Gas and Radiation . . . . . . . . . . . . . . . 1537.6 Electrostatic Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

7.6.1 The DebyeHuckel Radius . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1567.6.2 Electrostatic Effects on the Gas Pressure . . . . . . . . . . . . . . . . . 1587.6.3 Ionization by Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1597.6.4 Crystallization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

7.7 Degenerate Gases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1617.7.1 Partially Degenerate Gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1647.7.2 Non-Relativistic Partial Degeneracy . . . . . . . . . . . . . . . . . . . . 1647.7.3 Completely Degenerate Gas . . . . . . . . . . . . . . . . . . . . . . . . . . . 1667.7.4 Electrostatic Effects in a Degenerate Medium . . . . . . . . . . . . 1697.7.5 A Note on the Consequences of Degeneracy and on White

Dwarfs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1707.8 Global View on the Equation of State . . . . . . . . . . . . . . . . . . . . . . . . . . 172

8 The Opacities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1778.1 Line Absorption, Electron Scattering, Rayleigh Diffusion . . . . . . . . . 177

8.1.1 Recalls on the Atomic Oscillators . . . . . . . . . . . . . . . . . . . . . . 1778.1.2 Spectral Lines or BoundBound Transitions . . . . . . . . . . . . . . 178

8.2 Electron Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1798.2.1 Electron Scattering at High Energies . . . . . . . . . . . . . . . . . . . . 1808.2.2 Rayleigh Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181

8.3 Photoionization or BoundFree Transitions . . . . . . . . . . . . . . . . . . . . . 1818.3.1 Negative H Absorption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183

8.4 Hyperbolic Transitions or FreeFree Opacity . . . . . . . . . . . . . . . . . . . 1858.5 Electronic Conduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186

8.5.1 Electron Conduction in Non-degenerate Gas . . . . . . . . . . . . . 1868.5.2 Electron Conduction in Degenerate Gas . . . . . . . . . . . . . . . . . 188

8.6 Global View on Stellar Opacities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1898.6.1 Dependence on T and , Changes with Masses . . . . . . . . . . . 1898.6.2 Opacity Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191

Contents xi

9 Nuclear Reactions and Neutrino Processes . . . . . . . . . . . . . . . . . . . . . . . 1939.1 Physics of the Nuclear Reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193

9.1.1 Reaction Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1939.2 Nuclear Reaction Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195

9.2.1 Particle Lifetimes and Energy Production Rates . . . . . . . . . . . 1969.3 Nuclear Cross-Sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196

9.3.1 The Rate of Non-resonant Reactions . . . . . . . . . . . . . . . . . . . . 2009.3.2 The Rate of Resonant Nuclear Reactions . . . . . . . . . . . . . . . . . 204

9.4 Electron Screening . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2059.5 Neutrino Emission Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207

9.5.1 Photo-neutrinos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2089.5.2 Pair Annihilation Neutrinos . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2089.5.3 Plasma, Bremsstrahlung, Recombination Neutrinos . . . . . . . . 211

Part III Hydrodynamical Instabilities and Transport Processes

10 Transport Processes: Diffusion and Advection . . . . . . . . . . . . . . . . . . . . . 21510.1 General Properties of Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216

10.1.1 Absence of Global Mass Flux . . . . . . . . . . . . . . . . . . . . . . . . . . 21610.1.2 Continuity Equation: Atomic Diffusion and Motion . . . . . . . 21710.1.3 Fluxes of Particles, Velocities and Diffusion Coefficient . . . . 218

10.2 Diffusion by an Abundance Gradient . . . . . . . . . . . . . . . . . . . . . . . . . . 22110.2.1 Equation of Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22110.2.2 Boundary Conditions and Interpolations . . . . . . . . . . . . . . . . . 22310.2.3 Caution About the Use of Concentrations . . . . . . . . . . . . . . . . 224

10.3 Microscopic or Atomic Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22510.3.1 Gravitational Settling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22510.3.2 Equations of Motion of Charged Particles . . . . . . . . . . . . . . . . 22710.3.3 The Electric Field and the Diffusion Velocities . . . . . . . . . . . . 22910.3.4 Diffusion Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23110.3.5 Effect of a Thermal Gradient . . . . . . . . . . . . . . . . . . . . . . . . . . . 232

10.4 The Radiative Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23310.4.1 Radiative Acceleration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23310.4.2 Acceleration by Spectral Lines . . . . . . . . . . . . . . . . . . . . . . . . . 23510.4.3 Continuum Absorption, Redistribution, Magnetic Field . . . . 23610.4.4 Orders of Magnitude, Diffusion in A Stars . . . . . . . . . . . . . . . 23710.4.5 Atomic Diffusion in the Sun . . . . . . . . . . . . . . . . . . . . . . . . . . . 238

10.5 Transport of Angular Momentum in Stars . . . . . . . . . . . . . . . . . . . . . . 23910.5.1 Equation of Transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23910.5.2 Transport of Angular Momentum by Shears . . . . . . . . . . . . . . 24110.5.3 Some Properties of Shellular Rotation . . . . . . . . . . . . . . . . . . . 24210.5.4 Transport in Shellular Rotation . . . . . . . . . . . . . . . . . . . . . . . . . 24410.5.5 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246

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11 Meridional Circulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24911.1 The Energy Conservation on an Isobar . . . . . . . . . . . . . . . . . . . . . . . . . 249

11.1.1 Thermal Imbalance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25011.1.2 The Horizontal Thermal Balance . . . . . . . . . . . . . . . . . . . . . . . 253

11.2 Some Properties of Baroclinic Stars . . . . . . . . . . . . . . . . . . . . . . . . . . . 25511.2.1 The Fluctuations of T, , and . . . . . . . . . . . . . . . . . . . . . . . 25511.2.2 The Baroclinic Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25811.2.3 The Horizontal Fluctuations of Effective Gravity . . . . . . . . . . 259

11.3 The Velocity of Meridional Circulation . . . . . . . . . . . . . . . . . . . . . . . . 26211.4 Properties of Meridional Circulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 267

11.4.1 Simplified Expressions and Timescale . . . . . . . . . . . . . . . . . . . 26711.4.2 T and Excesses and Circulation Patterns . . . . . . . . . . . . . . . 269

11.5 The Major Role of the GrattonOpik Term . . . . . . . . . . . . . . . . . . . . . 27211.5.1 Departure from Solid Body and Initial Convergence . . . . . 27211.5.2 Stationary Circulation in Equilibrium with Diffusion . . . . . . 27311.5.3 The GrattonOpik Circulation and Evolution . . . . . . . . . . . . . 274

11.6 Meridional Circulation with Horizontal Turbulence . . . . . . . . . . . . . . 27611.6.1 Transport of the Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27711.6.2 Transport of the Angular Momentum . . . . . . . . . . . . . . . . . . . . 281

12 Rotation-Driven Instabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28312.1 Horizontal Turbulence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283

12.1.1 The Horizontal Fluctuations of . . . . . . . . . . . . . . . . . . . . . . 28412.1.2 A First Estimate of the Horizontal Turbulence . . . . . . . . . . . . 28712.1.3 Turbulent Diffusion from Laboratory Experiment . . . . . . . . . 28812.1.4 What Sets the Timescale of Horizontal Turbulence ? . . . . . . . 29012.1.5 Consequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291

12.2 Shear Instabilities and Mixing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29312.2.1 The Richardson Criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29412.2.2 Dynamical Shears . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29612.2.3 Thermal Effects at Constant . . . . . . . . . . . . . . . . . . . . . . . . . 29612.2.4 The T Gradient in Shears . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29912.2.5 Thermal Effects and Gradient . . . . . . . . . . . . . . . . . . . . . . . . 300

12.3 Shear Mixing with Horizontal Turbulence . . . . . . . . . . . . . . . . . . . . . . 30212.3.1 Richardson Criterion with Horizontal Turbulence . . . . . . . . . 30212.3.2 The Coefficient of Shear Diffusion with Turbulence . . . . . . . 303

12.4 Baroclinic Instabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30512.4.1 The GoldreichSchubertFricke or GSF Instability . . . . . . . . 30612.4.2 The ABCD Instability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 308

13 Magnetic Field Instabilities and Transport Processes . . . . . . . . . . . . . . . 31113.1 The Equations of Magnetohydrodynamics (MHD) . . . . . . . . . . . . . . . 311

13.1.1 The MHD Equations in Astrophysics . . . . . . . . . . . . . . . . . . . . 31113.1.2 Equations of Stellar Structure with Magnetic Field . . . . . . . . 31313.1.3 Alfven Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315

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13.1.4 Dynamos and the Solar Dynamo . . . . . . . . . . . . . . . . . . . . . . . 31613.1.5 Observed Fields and Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317

13.2 Magnetic Braking of Rotating Stars . . . . . . . . . . . . . . . . . . . . . . . . . . . 31913.2.1 Saturation Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32113.2.2 Mass Dependence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32213.2.3 Consequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323

13.3 Magnetic Field Properties in Radiative Regions . . . . . . . . . . . . . . . . . 32513.3.1 The Ferraro Law of Isorotation . . . . . . . . . . . . . . . . . . . . . . . . . 32613.3.2 Field Amplification by Winding-Up . . . . . . . . . . . . . . . . . . . . . 32713.3.3 Magnetic Field Evolution and Rotational Smoothing . . . . . . . 328

13.4 The Tayler Instability and Possible Dynamo . . . . . . . . . . . . . . . . . . . . 33013.4.1 The Tayler Instability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33013.4.2 The TaylerSpruit Dynamo and Questions . . . . . . . . . . . . . . . 33113.4.3 Conditions for Instability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33213.4.4 Thermal Diffusivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33413.4.5 Solutions of the Dynamo Equations . . . . . . . . . . . . . . . . . . . . . 336

13.5 Transports of Angular Momentum by the Magnetic Field . . . . . . . . . 33913.5.1 Viscous Coupling by the Field . . . . . . . . . . . . . . . . . . . . . . . . . 33913.5.2 Horizontal Coupling of Rotation . . . . . . . . . . . . . . . . . . . . . . . 34013.5.3 Check for Consistency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341

13.6 Models with Magnetic Field and Circulation . . . . . . . . . . . . . . . . . . . . 34213.6.1 Evolution of , B and the Diffusion Coefficients . . . . . . . . . . 34313.6.2 Evolutionary Consequences . . . . . . . . . . . . . . . . . . . . . . . . . . . 347

13.7 Other Magnetic Instabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34813.7.1 Magnetic Shear Instability and Transport . . . . . . . . . . . . . . . . 34913.7.2 Magnetic Buoyancy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 351

14 Physics of Mass Loss by Stellar Winds . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35514.1 Stellar Wind Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35514.2 Radiatively Line-Driven Winds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357

14.2.1 Simplified Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35714.2.2 Metallicity, Velocities and Other Effects . . . . . . . . . . . . . . . . . 361

14.3 Kudritzkis Wind MomentumLuminosity Relation . . . . . . . . . . . . . . 36214.3.1 Rotation and the WLR Relation . . . . . . . . . . . . . . . . . . . . . . . . 363

14.4 Rotation Effects on Stellar Winds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36414.4.1 Latitudinal Variations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36414.4.2 Mass Loss and Rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367

Part IV Acoustic and Gravity Waves. Helio- and Asteroseismology

15 Radial Pulsations of Stars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37115.1 Thermodynamics of the Pulsations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37115.2 Linear Analysis of Radial Oscillations . . . . . . . . . . . . . . . . . . . . . . . . . 373

15.2.1 Convection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37815.2.2 Boundary Conditions and Eigenvalue Problem . . . . . . . . . . . . 379

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15.3 Bakers One-Zone Analytical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 38115.3.1 Adiabatic Pulsations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384

15.4 Non-adiabatic Effects in Pulsations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38515.4.1 The and Mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38515.4.2 The Damping Timescale of Pulsations . . . . . . . . . . . . . . . . . . . 38915.4.3 Secular Instability: Conditions on Opacities and Nuclear

Reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39015.5 Relations to Observations: Cepheids . . . . . . . . . . . . . . . . . . . . . . . . . . . 391

15.5.1 The Period-Luminosity-Color Relations . . . . . . . . . . . . . . . . . 39115.5.2 Physics of the Instability Strip . . . . . . . . . . . . . . . . . . . . . . . . . 39415.5.3 The PeriodLuminosity Relation . . . . . . . . . . . . . . . . . . . . . . . 39715.5.4 Light Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 398

16 Nonradial Stellar Oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40116.1 Basic Equations of Nonradial Oscillations . . . . . . . . . . . . . . . . . . . . . . 401

16.1.1 Starting Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40116.1.2 Perturbations of the Equations . . . . . . . . . . . . . . . . . . . . . . . . . 40216.1.3 Separation in Vertical and Horizontal Components . . . . . . . . 40416.1.4 Decomposition in Spherical Harmonics . . . . . . . . . . . . . . . . . . 405

16.2 Nonradial Adiabatic Oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40916.2.1 Basic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40916.2.2 Some Properties of the Equations . . . . . . . . . . . . . . . . . . . . . . . 41116.2.3 Simplification to a Second-Order Equation . . . . . . . . . . . . . . . 41216.2.4 Domains of the Acoustic and Gravity Modes . . . . . . . . . . . . . 41416.2.5 The Degree and Radial Order n . . . . . . . . . . . . . . . . . . . . . . . 416

16.3 Properties of Acoustic or p Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41916.3.1 Inner Turning Points of p Modes . . . . . . . . . . . . . . . . . . . . . . . 41916.3.2 Properties of the Solar Cavity: Parabolic Relations . . . . . . . . 42116.3.3 Behavior of p Modes at the Surface . . . . . . . . . . . . . . . . . . . . . 42316.3.4 Excitation and Damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 427

16.4 The Asymptotic Theory of p Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . 42816.4.1 The Frequencies of p Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . 42816.4.2 Second-Order Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 431

16.5 Helioseismology and Asteroseismology . . . . . . . . . . . . . . . . . . . . . . . . 43316.5.1 Helioseismic Observations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43316.5.2 Asteroseismic Observations . . . . . . . . . . . . . . . . . . . . . . . . . . . 43616.5.3 The Asteroseismic Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . 43816.5.4 Effects of X , Z and Mixing Length on the Large

and Small Separations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44116.6 Rotational Effects: Splitting and Internal Mixing . . . . . . . . . . . . . . . . 441

16.6.1 The Rotational Splitting: First Approach . . . . . . . . . . . . . . . . . 44216.6.2 Further Steps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44316.6.3 The Tachocline and Inner Solar Rotation . . . . . . . . . . . . . . . . . 44416.6.4 Structural Effects of Rotation . . . . . . . . . . . . . . . . . . . . . . . . . . 447

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17 Transport by Gravity Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44917.1 The Propagation of Gravity Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . 449

17.1.1 Properties of Gravity Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . 44917.1.2 Propagation Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45117.1.3 Non-adiabatic Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454

17.2 Energy and Momentum Transport by Gravity Waves . . . . . . . . . . . . . 45817.2.1 Wave Excitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 461

17.3 Consequences of Transport by Gravity Waves . . . . . . . . . . . . . . . . . . . 46517.3.1 Shear Layer Oscillations SLO . . . . . . . . . . . . . . . . . . . . . . . 46517.3.2 The Solar Rotation Curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46717.3.3 Waves and the Lithium Dip . . . . . . . . . . . . . . . . . . . . . . . . . . . . 469

17.4 Transport by Gravity Waves and Open Questions . . . . . . . . . . . . . . . . 47017.4.1 Particles Diffusion by Gravity Waves . . . . . . . . . . . . . . . . . . . . 47017.4.2 Open Questions and Further Developments . . . . . . . . . . . . . . 471

Part V Star Formation

18 Pre-stellar Phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47518.1 Overview and Signatures of Star Formation . . . . . . . . . . . . . . . . . . . . . 47518.2 The Beginning of Cloud Contraction . . . . . . . . . . . . . . . . . . . . . . . . . . 477

18.2.1 The Jeans Criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47718.2.2 Various Expressions of the Jeans Criterion . . . . . . . . . . . . . . . 48018.2.3 Initializing the Cloud Collapse . . . . . . . . . . . . . . . . . . . . . . . . . 48218.2.4 The Timescale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 482

18.3 The Role of Magnetic Field and Turbulence . . . . . . . . . . . . . . . . . . . . 48418.3.1 Magnetic Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48418.3.2 The Major Role of Turbulence in Star Formation . . . . . . . . . . 486

18.4 Isothermal Collapse and Cloud Fragmentation . . . . . . . . . . . . . . . . . . 48718.4.1 Dust Grains and Cooling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48718.4.2 The Initial Cloud Structure and its Evolution . . . . . . . . . . . . . 48818.4.3 The Hierarchical Fragmentation . . . . . . . . . . . . . . . . . . . . . . . . 49118.4.4 The Opacity-Limited Fragmentation . . . . . . . . . . . . . . . . . . . . 49218.4.5 The Initial Stellar Mass Spectrum . . . . . . . . . . . . . . . . . . . . . . 494

19 The Protostellar Phase and Accretion Disks . . . . . . . . . . . . . . . . . . . . . . 49719.1 Accretion Disks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 497

19.1.1 Observations of Disks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49719.1.2 Disk Formation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49819.1.3 Disk Properties and Evolution . . . . . . . . . . . . . . . . . . . . . . . . . 50019.1.4 Stationary Disks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 502

19.2 Accretion in Low and Intermediate Mass Stars . . . . . . . . . . . . . . . . . . 50419.2.1 Theoretical Estimates of the Accretion Rates . . . . . . . . . . . . . 50519.2.2 Structure of the Protostar in the Accretion Phase . . . . . . . . . . 506

19.3 The Phase of Adiabatic Contraction . . . . . . . . . . . . . . . . . . . . . . . . . . . 50819.3.1 Evolution of the Central Object . . . . . . . . . . . . . . . . . . . . . . . . 510

19.4 Properties at the End of the Protostellar Phase . . . . . . . . . . . . . . . . . . . 511

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20 The Pre-main Sequence Phase and the Birthlines . . . . . . . . . . . . . . . . . 51320.1 General Properties of Non-adiabatic Contraction . . . . . . . . . . . . . . . . 513

20.1.1 The KelvinHelmholtz Timescale . . . . . . . . . . . . . . . . . . . . . . 51320.2 Pre-MS Evolution at Constant Mass . . . . . . . . . . . . . . . . . . . . . . . . . . . 514

20.2.1 The Hayashi Line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51420.2.2 Gravitational Energy Production and D Burning . . . . . . . . . . 51520.2.3 From the Hayashi Line to the ZAMS . . . . . . . . . . . . . . . . . . . . 519

20.3 Pre-MS Evolution with Mass Accretion . . . . . . . . . . . . . . . . . . . . . . . . 52020.3.1 The Birthline and Its Timescales . . . . . . . . . . . . . . . . . . . . . . . 52020.3.2 The Luminosity from D Burning . . . . . . . . . . . . . . . . . . . . . . . 521

20.4 Evolution on the Birthline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52120.5 Evolution from the Birthline to the ZAMS . . . . . . . . . . . . . . . . . . . . . . 52620.6 Lifetimes, Ages and Isochrones . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52920.7 Lithium Depletion in Pre-MS Stars . . . . . . . . . . . . . . . . . . . . . . . . . . . . 531

20.7.1 Model Predictions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53120.7.2 Li and D in T Tauri Stars and Residual Accretion . . . . . . . . . 53320.7.3 Li Depletion in Low-Mass Stars and Brown Dwarfs . . . . . . . 53420.7.4 Li Dating from Brown Dwarfs and Low-M Stars . . . . . . . . . . 536

21 Rotation in Star Formation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53921.1 Steps in the Loss of Angular Momentum . . . . . . . . . . . . . . . . . . . . . . . 539

21.1.1 From Interstellar Clouds to T Tauri Stars . . . . . . . . . . . . . . . . . 54021.1.2 From T Tauri Stars to the ZAMS . . . . . . . . . . . . . . . . . . . . . . . 54121.1.3 End of Pre-MS Phase and Early Main Sequence . . . . . . . . . . 542

21.2 Disk Locking and Magnetospheric Accretion . . . . . . . . . . . . . . . . . . . 54321.2.1 Observational Evidences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 545

21.3 Magnetic Braking and Rotation in Clusters . . . . . . . . . . . . . . . . . . . . . 54521.3.1 Predicted Magnetic Braking . . . . . . . . . . . . . . . . . . . . . . . . . . . 54521.3.2 Comparisons with Rotation Velocities in Clusters . . . . . . . . . 547

22 The Formation of Massive Stars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55122.1 The Various Scenarios for Massive Star Formation . . . . . . . . . . . . . . . 551

22.1.1 The Classical or Constant Mass Scenario . . . . . . . . . . . . . . . . 55122.1.2 The Collision or Coalescence Scenario . . . . . . . . . . . . . . . . . . 55222.1.3 The Accretion Scenario . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 554

22.2 Timescales for Accreting Stars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55722.3 Limits on the Accretion Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 558

22.3.1 The Upper Limit on Accretion . . . . . . . . . . . . . . . . . . . . . . . . . 55822.3.2 Conditions on Dust Opacity . . . . . . . . . . . . . . . . . . . . . . . . . . . 56122.3.3 The Lower Limit on Accretion Rates . . . . . . . . . . . . . . . . . . . . 56222.3.4 The Role of Rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 563

22.4 Accretion Models for Massive Star Formation . . . . . . . . . . . . . . . . . . . 56522.4.1 Formation with Initially Peaked Accretion . . . . . . . . . . . . . . . 56522.4.2 The ChurchwellHenning Relation . . . . . . . . . . . . . . . . . . . . . 570

Contents xvii

23 The Formation of First Stars in the Universe: Pop. III and Pop. II.5Stars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57123.1 The Pre- and Protostellar Phases at Z = 0 . . . . . . . . . . . . . . . . . . . . . . 572

23.1.1 Molecular H2 and Gas Cooling . . . . . . . . . . . . . . . . . . . . . . . . . 57223.1.2 Fragmentation of Metal-Free Clouds . . . . . . . . . . . . . . . . . . . . 57323.1.3 Formation of an Adiabatic Core . . . . . . . . . . . . . . . . . . . . . . . . 57423.1.4 Accretion on the Core . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 575

23.2 The MassRadius Relation of Z = 0 Stars . . . . . . . . . . . . . . . . . . . . . . 57623.3 Evolution of the Largest Masses at Z = 0 . . . . . . . . . . . . . . . . . . . . . . . 579

23.3.1 Critical Accretion for Massive Stars at Z = 0 . . . . . . . . . . . . . 58023.4 The HR Diagram of Accreting Stars at Z = 0 . . . . . . . . . . . . . . . . . . . 581

23.4.1 The Case of Non-zero Metallicities . . . . . . . . . . . . . . . . . . . . . 58123.4.2 The Role of Rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 582

23.5 The Upper Mass Limit at Z = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58223.5.1 Main Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58323.5.2 HII Region in a Free-Falling Envelope . . . . . . . . . . . . . . . . . . 58323.5.3 Radiation Effect on an HII Region . . . . . . . . . . . . . . . . . . . . . . 585

23.6 The Pop. II.5 Stars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58723.6.1 HD Formation and Gas Cooling . . . . . . . . . . . . . . . . . . . . . . . . 58723.6.2 The Masses of the Pop. II.5 Stars . . . . . . . . . . . . . . . . . . . . . . . 588

Part VI Main-Sequence and Post-MS Evolution

24 Solutions of the Equations and Simple Models . . . . . . . . . . . . . . . . . . . . 59324.1 Hydrostatic and Hydrodynamic Models . . . . . . . . . . . . . . . . . . . . . . . . 593

24.1.1 Hydrostatic Models and VogtRussel Theorem . . . . . . . . . . . 59324.1.2 Hydrodynamic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59524.1.3 Boundary Conditions at the Center and Surface . . . . . . . . . . . 59624.1.4 Analytical Solutions in the Outer Layers . . . . . . . . . . . . . . . . . 597

24.2 The Henyey Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59924.3 Homology Transformations: Relations MLR . . . . . . . . . . . . . . . . . . 601

24.3.1 Other Effects: Electron Scattering, Prad, Convection . . . . . . . 60424.4 The Helium and Generalized Main Sequences . . . . . . . . . . . . . . . . . . . 605

24.4.1 The Helium Sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60524.4.2 Generalized Main Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . 605

24.5 Polytropic Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60724.5.1 Interesting Polytropes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60924.5.2 Isothermal Sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 610

25 Evolution in the H-Burning Phases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61325.1 Hydrogen Burning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 613

25.1.1 The pp Chains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61425.1.2 Equations for Composition Changes . . . . . . . . . . . . . . . . . . . . 61525.1.3 The CNO Cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61825.1.4 Energy Production in MS Stars . . . . . . . . . . . . . . . . . . . . . . . . . 62025.1.5 The NeNa and MgAl Cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . 621

xviii Contents

25.2 Basic Properties of MS Stars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62425.2.1 Differences in Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62425.2.2 Main Parameters as a Function of Mass . . . . . . . . . . . . . . . . . . 62525.2.3 Evolutionary Timescales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 627

25.3 Solar Properties and Evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62925.3.1 Internal Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62925.3.2 The Evolution of the Sun . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63225.3.3 Solar Neutrinos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 634

25.4 Evolution on the Main Sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63725.4.1 Internal Properties, Tracks in the HR Diagram . . . . . . . . . . . . 637

25.5 The End of the Main Sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64025.5.1 The SchonbergChandrasekhar Limit . . . . . . . . . . . . . . . . . . . 64025.5.2 Isochrones and Age Determinations . . . . . . . . . . . . . . . . . . . . . 642

26 Evolution in the He Burning and AGB Phases of Low andIntermediate Mass Stars with Rotation . . . . . . . . . . . . . . . . . . . . . . . . . . 64526.1 Helium Burning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64526.2 He Burning in Intermediate Mass Stars . . . . . . . . . . . . . . . . . . . . . . . . 647

26.2.1 From Main Sequence to Red Giants . . . . . . . . . . . . . . . . . . . . . 64726.2.2 Evolution in the He-Burning Phase and Dredge-up . . . . . . . . 65126.2.3 From AGB to the White Dwarfs . . . . . . . . . . . . . . . . . . . . . . . . 65426.2.4 The Blue Loops . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 656

26.3 Some Metallicity Effects in Evolution . . . . . . . . . . . . . . . . . . . . . . . . . 65726.4 Central Evolution and Domains of Stellar Masses . . . . . . . . . . . . . . . . 658

26.4.1 The Mass Limits for Evolution . . . . . . . . . . . . . . . . . . . . . . . . . 66126.4.2 Evolution of the Entropy per Baryon . . . . . . . . . . . . . . . . . . . . 664

26.5 The Horizontal Branch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66526.6 Evolution and Nucleosynthesis in AGB Stars . . . . . . . . . . . . . . . . . . . 667

26.6.1 Structure and Instability of TP-AGB Stars . . . . . . . . . . . . . . . 66726.6.2 Third Dredge-Up and TP-AGB Nucleosynthesis . . . . . . . . . . 67126.6.3 AGB Classification and Chemical Abundances . . . . . . . . . . . 67426.6.4 Post-AGB Stars to Planetary Nebulae and White Dwarfs,

Super-AGB Stars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67626.7 Rotation and Mixing Effects in AGB stars . . . . . . . . . . . . . . . . . . . . . . 67826.8 Nucleosynthesis in AGB Stars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 681

26.8.1 Nucleosynthesis in E-AGB stars . . . . . . . . . . . . . . . . . . . . . . . . 68126.8.2 Nucleosynthesis in TP-AGB Stars . . . . . . . . . . . . . . . . . . . . . . 683

27 Massive Star Evolution with Mass Loss and Rotation . . . . . . . . . . . . . . 68527.1 The Need for Both . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68527.2 Evolution at Constant Mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68627.3 Internal Evolution and the HR Diagram . . . . . . . . . . . . . . . . . . . . . . . . 688

27.3.1 Mass Loss Parametrizations . . . . . . . . . . . . . . . . . . . . . . . . . . . 68827.3.2 Mass Loss Effects in the HR Diagram . . . . . . . . . . . . . . . . . . . 69027.3.3 Internal Evolution with Mass Loss . . . . . . . . . . . . . . . . . . . . . . 692

Contents xix

27.3.4 Effects of Rotation in the MS Phase . . . . . . . . . . . . . . . . . . . . . 69327.3.5 Lifetimes and Age Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . 69427.3.6 He-Burning: Blue and Red Supergiants at Different Z . . . . . . 696

27.4 Evolution of the Chemical Abundances . . . . . . . . . . . . . . . . . . . . . . . . 69727.4.1 Steps in the Peeling-Off by Mass Loss . . . . . . . . . . . . . . . . . . . 69727.4.2 Observed N/H Excesses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69927.4.3 Chemistry in Models with Rotation . . . . . . . . . . . . . . . . . . . . . 70027.4.4 Abundances and Massive Star Filiations . . . . . . . . . . . . . . . . . 702

27.5 WolfRayet Stars: the Daughters of O stars . . . . . . . . . . . . . . . . . . . . . 70327.5.1 WR Properties: the Zebras in the Zoo . . . . . . . . . . . . . . . . . . . 70327.5.2 Optically Thick Winds. MLRTeff Relations . . . . . . . . . . . 703

27.6 WR Star Chemistry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70627.6.1 Observations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70627.6.2 Mass Loss, Rotation and WR Chemistry . . . . . . . . . . . . . . . . . 70727.6.3 22Ne in WC Stars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 709

27.7 Number Ratios of WR Stars in Galaxies . . . . . . . . . . . . . . . . . . . . . . . . 71027.7.1 Observed Number Ratios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71027.7.2 Models with Mass Loss and Rotation . . . . . . . . . . . . . . . . . . . 711

27.8 Evolution of the Rotational Velocities . . . . . . . . . . . . . . . . . . . . . . . . . . 71327.8.1 Rotation of LBV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71527.8.2 WR Star Rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 716

28 Advanced Evolutionary Stages and Pre-supernovae . . . . . . . . . . . . . . . 71928.1 Nuclear Reactions in the Advanced Phases . . . . . . . . . . . . . . . . . . . . . 719

28.1.1 C Burning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71928.1.2 Ne Photodisintegration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72328.1.3 O Burning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72328.1.4 Silicon Burning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 724

28.2 The Advanced Phases with and Without Rotation . . . . . . . . . . . . . . . . 72528.2.1 Toward the Onion Skin Model . . . . . . . . . . . . . . . . . . . . . . . 72628.2.2 Decoupling of Core and Envelope . . . . . . . . . . . . . . . . . . . . . . 72728.2.3 Evolution of Central Conditions . . . . . . . . . . . . . . . . . . . . . . . . 72728.2.4 Lifetimes and Core Masses, Rotation . . . . . . . . . . . . . . . . . . . . 729

28.3 Chemical Yields: Z, Mass Loss and Rotation Effects . . . . . . . . . . . . . 73128.3.1 Chemical Yields of -Rich Nuclei . . . . . . . . . . . . . . . . . . . . . . 732

28.4 Toward the Supernovae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73428.4.1 Supernova Types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73428.4.2 Core Collapse and Explosion . . . . . . . . . . . . . . . . . . . . . . . . . . 73628.4.3 Final Masses and Remnants . . . . . . . . . . . . . . . . . . . . . . . . . . . 738

28.5 Explosive Synthesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74128.5.1 Elements with A < 56 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74128.5.2 The Fe Peak . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74328.5.3 The Heavy Elements A 60 . . . . . . . . . . . . . . . . . . . . . . . . . . 74528.5.4 The s-Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74628.5.5 The r-Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 747

xx Contents

28.6 Evolution of Rotation: Pulsars and GRBs . . . . . . . . . . . . . . . . . . . . . . . 75028.6.1 Distribution of the Specific Angular Momentum . . . . . . . . . . 75028.6.2 The Rotation of Pulsars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75228.6.3 GRBs: A Challenging Problem . . . . . . . . . . . . . . . . . . . . . . . . . 75228.6.4 Models for the GRB Progenitors . . . . . . . . . . . . . . . . . . . . . . . 753

29 Evolution of Z == 0 and Very Low Z Stars . . . . . . . . . . . . . . . . . . . . . . . . . 75529.1 Basic Properties and Evolution of Z = 0 Stars . . . . . . . . . . . . . . . . . . . 755

29.1.1 Differences in the Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75529.1.2 The HR and logTc vs. logc Diagrams . . . . . . . . . . . . . . . . . . . 75629.1.3 Low-Mass Stars (M < 3 M) . . . . . . . . . . . . . . . . . . . . . . . . . . 75729.1.4 Intermediate Mass Stars (3 M < M < 10 M) . . . . . . . . . . . 75829.1.5 High-Mass Stars (M > 10 M) . . . . . . . . . . . . . . . . . . . . . . . . 76029.1.6 Other Properties: Mass Limits and CO Cores . . . . . . . . . . . . . 760

29.2 Rotation Effects at Z = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76129.2.1 HR Diagram and Lifetimes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76129.2.2 Evolution of the Rotation, Final Masses . . . . . . . . . . . . . . . . . 762

29.3 Rotation Effects in Very Low Z Models . . . . . . . . . . . . . . . . . . . . . . . . 76429.3.1 Rotational Mass Loss in the First Generations . . . . . . . . . . . . 76529.3.2 Enrichments by the Winds of the First Generations . . . . . . . . 766

A Physical and Astronomical Constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 771A.1 Physical Constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 771A.2 Some Astronomical Constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 772A.3 Initial Solar Abundances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 772

B Complements on Mechanics and Electromagnetism . . . . . . . . . . . . . . . . . 773B.1 Equations of Motion and Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . 773

B.1.1 Equations of Continuity and of Motion . . . . . . . . . . . . . . . . . . 773B.1.2 Remarks on Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 774B.1.3 Vectorial Operators in Spherical Coordinates . . . . . . . . . . . . . 774B.1.4 Viscous Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 775B.1.5 NavierStokes Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 776B.1.6 Equation of Motion with Rotation . . . . . . . . . . . . . . . . . . . . . . 777B.1.7 Geostrophic Motions, TaylorProudman Theorem . . . . . . . . . 778

B.2 Maxwell Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 779B.3 Statistical Mechanics: Pressure and Energy Density . . . . . . . . . . . . . . 780

B.3.1 Non-relativistic Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 781B.3.2 Relativistic Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 781

B.4 Expressions of Viscosity, Conductivity and Diffusion . . . . . . . . . . . . 782B.4.1 Viscosity from Turbulence, Radiation and Plasma . . . . . . . . . 782B.4.2 Conductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 784B.4.3 Diffusion Coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 785

B.5 Dimensionless Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 785B.5.1 Reynolds Number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 785B.5.2 Prandtl Number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 786

Contents xxi

B.5.3 Peclet and Nusselt Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . 786B.5.4 The Rossby Number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 787

B.6 More on the Physics of Rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 787B.6.1 The Angular Velocity in Spherical Functions . . . . . . . . . . . . . 787B.6.2 Rotational Splitting for Non-uniform Rotation . . . . . . . . . . . . 790

C Complements on Radiative Transfer and Thermodynamics . . . . . . . . . . 795C.1 Radiation: Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 795

C.1.1 The Quasi-Isotropic Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 798C.2 Expression of the Heat Changes dq = dq(P,) . . . . . . . . . . . . . . . . . . 798C.3 Adiabatic Acoustic Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 799C.4 The Entropy of Radiation and Perfect Gas . . . . . . . . . . . . . . . . . . . . . . 800

C.4.1 Entropy of Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 800C.4.2 Entropy of a Mixture of Perfect Gas and Radiation . . . . . . . . 801C.4.3 Degenerate Gases and Minimum Entropy . . . . . . . . . . . . . . . . 802C.4.4 The Entropy of Mixing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 803

C.5 Recalls on Fundamental Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 804C.6 Thermodynamic Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 805

C.6.1 Reactions with Changes of State . . . . . . . . . . . . . . . . . . . . . . . . 805C.6.2 MaxwellBoltzmann Distribution . . . . . . . . . . . . . . . . . . . . . . 806

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 807

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 823

Part IStellar Equilibrium With and Without

Rotation

Chapter 1The Mechanical Equilibrium of Stars

If the stars would not be in equilibrium during most of their life, stable conditionspermitting life would not have been present on the Earth. Mechanical equilibriumis a necessary condition for stable luminosity and temperature over long periodsof time. It is a fundamental property of stars, implying the exact balance betweenthe gravity force which attracts the matter toward the center and the force due tothe thermal pressure, which resists gravity. Any departure from this equilibriumwill immediately lead to a strong reaction for restoring the equilibrium state. Letus suppose, for example, that the Sun is arbitrarily compressed to a smaller radius.The gas becomes hotter rising the internal pressure. The higher pressure provokesan expansion re-settling the star to its equilibrium state. Conversely, an arbitraryextension of the radius would decrease the internal temperature and pressure, theSun would then contract again. Such re-adjustments are very fast, they would occurat the dynamical timescale of the order of half an hour for the Sun, which is almostinstantaneous with respect to solar evolution.

The mechanical equilibrium of a star governs all its properties. It is always satis-fied except in very short phases, such as the initial collapse of interstellar clouds orin supernova explosions.

1.1 Momentum and Continuity Equations

1.1.1 Hydrodynamical Equations

The basic equations of hydrodynamics determining the mechanical equilibriumof a star are the equation of continuity, which expresses mass conservation, andthe equation of NavierStokes, which is the equation of motion. They are derivedin Appendix B.1. For a medium of density and velocity v, the equation ofcontinuity is

t

+div(v) = 0 , (1.1)

This chapter may form the matter of a basic introductory course.

A. Maeder, Physics, Formation and Evolution of Rotating Stars, Astronomy and 3Astrophysics Library, DOI 10.1007/978-3-540-76949-1 1,c Springer-Verlag Berlin Heidelberg 2009

4 1 The Mechanical Equilibrium of Stars

and the equation of NavierStokes

dvvd t

=vv t

+(vv ) v = a 1P+ 2 v , (1.2)

where a is the acceleration due to external forces, P the pressure and thekinematic coefficient of viscosity (B.50). Let us consider a spherical star, where ris the distance to the center and assume that the viscous effects are negligible. First,we do not suppose hydrostatic equilibrium, letting open the possibility of internalmotions. The spherical components of the NavierStokes equation (1.2) become

d vd t

= (r, 0, 0) , P =(P r

, 0, 0

), a = g = (g, 0, 0) , (1.3)

where g is the vector of gravity directed toward the stellar interior. Its modulus isg = GMr/r2, with G the constant of gravitation and Mr the mass interior to radius r(Fig. 1.1). Thus the NavierStokes equation leads to

r = 1

P r

GMrr2

. (1.4)

This is the momentum equation of hydrodynamic models. It expresses that the ac-celeration is the sum of two contributions:

the acceleration due to the pressure gradient which is directed toward the exterior,since P/ r < 0,

the gravity g directed toward the stellar interior.

Depending on the kind of motion, r may be positive or negative. For examplein a pulsating star, r changes sign during a pulsation cycle. In some stages, such asthe collapse of an interstellar cloud (Sect. 18.2.4), the internal pressure forces arenegligible, so that one is just left with the equation of free fall (18.20).

Fig. 1.1 Some definitions. dPis the difference of pressurebetween the levels in r andr +dr. Mr is the mass interiorto level r, dMr is the massbetween r and r +dr

1.1 Momentum and Continuity Equations 5

1.1.2 Hydrostatic Equilibrium

If there is no fast radial motions, we have a situation of hydrostatic equilibrium. Thisis the general situation of most stars. The internal pressure gradient is balancing thegravity everywhere in the star. The equations in vectorial and scalar forms are

P = g and1

d Pd r

= GMrr2

, (1.5)

consistently with definitions (1.3). This equation says that at any level in a star inequilibrium the gradient of pressure sustains the matter against the gravity forceby volume unity. Equation (1.4) is to be taken rather than (1.5) if the ratio |r/g|is not negligible. In practice, this applies only to stellar pulsations, early stages ofstar formation and advanced phases of evolution. The above equation of hydrostaticequilibrium (1.5) may also be found very simply by considering a thin shell betweenradius r and r + dr, with pressures P and P + dP, respectively. Let Mr be the massinside radius r (Fig. 1.1). The difference of pressure dP is

dP = gdr = GMrr2

dr , (1.6)

which just gives (1.5).

1.1.3 Mass Conservation and Continuity Equation

In spherical symmetry, the change of the mass Mr(t) in a sphere of radius r can bewritten as

dMr(r, t) = 4 r2 dr4 r2 v dt. (1.7)

The first term on the right represents the change of mass due to a variation of radiusr at a given time t, the second term expresses the flux of mass out of the sphere ofconstant r due to an outward motion with velocity v > 0. The differential dMr(r, t)can also be written as

dMr(r, t) =(Mr r

)t

dr +(Mr t

)r

dt . (1.8)

Comparing with (1.7), we make the identifications,(Mr r

)t= 4 r2 and

(Mr t

)r

= 4 r2 v . (1.9)

The first expression is the definition of the local density (r). It also allows us tomove from variable r to Mr and reciprocally at a given time.

6 1 The Mechanical Equilibrium of Stars

The second equation expresses the change of mass in a sphere due to the motionof matter which goes through its surface. At the stellar surface, this equationgives the mass loss rate (or gain) M as a function of the wind velocity v > 0 foran outward stellar wind or v < 0 for an accretion of matter. As dMr is an exactdifferential, one has(

t

(Mr r

)t

)r

=(

r

(Mr t

)r

)t

, (1.10)

which gives

4 r2 t

= 4 r

(r2 v) or t

+1r2

r

(r2 v) = 0 . (1.11)

This is just the spherical form of the continuity equation (1.1). Thus, (1.7) alsoexpresses the continuity equation.

In a static situation, the velocity v is zero, the derivative (Mr/ t)r = 0 and weare left with the first of the two equations (1.9). Expression (1.7) and the continuityequation become,

dMrdr

= 4 r2 . (1.12)

There is no partial derivative: in a static case, there is only one variable r.

1.1.4 Lagrangian and Eulerian Variables

In stellar evolution, the coordinate r is not always convenient as an independent vari-able. Except for particular cases of heavy mass loss, the stellar mass remains almostconstant, while the stellar radius may rapidly change. It is thus more appropriate(and simpler) to choose the mass Mr or the mass fraction (Mr/M) as an independentvariable. This choice is usually done in model computations. The transition betweenvariables (r, t) and (Mr, t) is made with the help of the first of equations (1.9):

(

Mr

)t=(

rMr

)t

r

=1

4 r2 r

. (1.13)

We may express the equation of motion (1.4) in the hydrodynamical case as

r4 r2

= PMr

GMr4 r4

. (1.14)

Equations (1.6) and (1.12) for hydrostatic equilibrium are

dPdMr

= GMr4 r4

anddr

dMr=

14 r2

, (1.15)

the time t being absent in these equations.

1.1 Momentum and Continuity Equations 7

The equations written as a function of (Mr, t) in the hydrodynamic case, or ofMr in the hydrostatic case, are in Lagrangian variables. In this case, one followsthe mass elements during evolution. The equations written as a function of (r, t) orof (r) are in Eulerian variables. One has the following relation between the timederivatives of a function f in the Lagrangian and Eulerian cases,

d fdt

= f r

( r t

)Mr

+( f t

)r

(1.16)

or more generally

d fdt

= u f +( f t

)r

. (1.17)

On the left side, one follows a given mass element in time, the first term on theright expresses the change due to the motion of the matter with a velocity u. Thesecond term on the right expresses the time derivative at a given location in space.The following expressions for the derivative attached to a given mass element aregenerally equivalent

ddt

( t

)Mr

DDt

. (1.18)

These are sometimes called the hydrodynamical derivatives.

1.1.5 Estimates of Pressure, Temperature and Timescales

1.1.5.1 Internal Pressure

The equation of hydrostatic equilibrium (1.6) leads to an estimate of the order ofmagnitude of the internal pressure P in terms of the stellar mass M and radius R. Bytaking rough average values of the quantities in (1.5), |dP/dr| Pc/R with Pc thecentral pressure, Mr M/2, r R/2, we get the following orders of magnitude:

PcR

2 GMR2

. (1.19)

The average density = 3M/(4 R3) leads to

Pc 3

2GM2

R4. (1.20)

This provides a rough order of magnitude of the central pressure, it can also beused as an estimate of the average pressure P. For the Sun, we get a value ofP 5.4 1015 g s2 cm1. However, the most useful result from (1.20) is thebehavior of the pressure with stellar mass M and radius R.

8 1 The Mechanical Equilibrium of Stars

1.1.5.2 Limits on Central Pressure

Some limits on the central pressure can also be obtained [111, 423]. Integration ofthe first equation of (1.15) gives

Pc P =G

4

r0

Mr dMrr4

, (1.21)

where Pc is the central pressure. Using (r) = 3Mr/(4r3), one has r4 =[3Mr/(4 (r))]

43 and

Pc P =G

4

(43) 4

3 r

0

43 (r)M

13r dMr . (1.22)

As the average density (r) inside a given radius r does not increase outward, onehas a lower bound for central pressure

Pc G4

(43) 4

3

43 (r)

M(R)0

M 13r dMr =

12

(43) 1

3

G43 M

23 , (1.23)

where and M apply to the whole star. One assumes that the total pressure at thestellar surface is negligible. On the other side, (r) is always inferior to the centraldensity c, thus one also has

Pc 12

(43) 1

3

G43c M

23 . (1.24)

Thus, one obtains an upper and a lower bound for the central pressure in a star.These bounds result from hydrostatic equilibrium and from the assumption that decreases outward. We use these limits in Sect. 3.6.1.

1.1.5.3 Interior Temperature

One can also make a simple estimate of the internal temperature T of a chemicallyhomogeneous sphere obeying the law of perfect gases (7.31),

Pg =R

T =k

muT , (1.25)

where Pg is the perfect gas pressure, R the gas constant, k the Boltzmann constant, the mean molecular weight, mu is the atomic mass unit, i.e., (1/12) of the massof the neutral 12C atom, mu = 1.66051024 g (cf. Appendix A.1). If the pressureP Pc (1.20), taking the average density M/R3, we get an order of magnitudefor the average internal temperature,

1.1 Momentum and Continuity Equations 9

T muk

GMR

. (1.26)

The interest of this expression is that it shows the functional dependence of T versusmass M and radius R. A better estimate is given below (cf. 1.51).

1.1.5.4 Dynamical Timescales

The dynamical timescale dyn characterizes the departures from mechanical equilib-rium. Let us suppose that the internal pressure gradient in a gravitationally boundconfiguration becomes negligible (a situation which occurs in cloud collapse or inthe core collapse leading to supernova explosion). Then, (1.4) or (1.14) leads to thefollowing scaling for a spherical object of mass M and radius R:

R

2dyn GM

R2, (1.27)

giving dyn (

R3

GM

) 12

(G)12 . (1.28)

If the internal pressure is negligible, the star collapses under its own gravity andthe dynamical timescale dyn is essentially the free-fall timescale. The integration of(1.4) for the case of free fall is given in Sect. 18.2.4. For the Sun, dyn = 1.8103 s,i.e., half an hour. For a red giant with 106 g cm3, dyn 40 days. For a whitedwarf with 106 g cm3, dyn a few seconds. For any gravitational configu-ration, we may estimate the dynamical timescale. For the Universe as a whole, thistime is of the order of the Hubble time, i.e., 1010 yr. The example of the Sun, wheredyn is very short with respect to the evolutionary timescale, shows that any depar-ture from mechanical equilibrium leads to an immediate reaction. This implies thatthe mechanical equilibrium is always very quickly and closely adjusted, due to thefast mechanical response of the star.

We have supposed above that the pressure becomes negligible, but we can alsoestimate dyn, if the effect of the internal pressure becomes large with respect togravity. In this case, the scaling of (1.4) leads to

R

2dyn 1

PR

, (1.29)

which gives, with (1.20) and M/R3,

Rdyn

(

P

) 12

(

GMR

) 12

. (1.30)

As above in (1.28), the dynamical timescale dyn 1/

G, which is quite con-sistent. The sound speed in a gas is cs =

1 P/ (see 32.26), where 1 is defined

10 1 The Mechanical Equilibrium of Stars

in (7.57). For a perfect gas, 1 g = cP/cV, i.e., 5/3 in a ionized medium. Thus,from (1.30), one has dyn R/cs, which means that the dynamical timescale ofa star is of the order of the time necessary for the sound speed to cross the stel-lar radius. This is not surprising since the sound velocity characterizes the pressureadjustments. For the same reason, the dynamical timescale is of the order of the fun-damental period in a pulsating star. In a perfect gas cs

T , thus the outer stellar

layers, which have lower T values, mainly determine the dynamical timescale aswell as the stellar pulsations periods.

1.2 The Potential Energy

The mechanical equilibrium of a star implies that the gravitational energy, i.e., thepotential energy, is of the same order as the thermal energy, which supports the staragainst gravitation.

Let us consider a non-rotating spherical star in the process of formation byaddition of new mass elements. Let Mr be the mass already collected at the inte-rior of radius r (Fig. 1.1). The work dW provided by the gravitational force F whenit brings a new mass element Mr from radius r +dr to r is

dW = F dr = GMr Mrr2

dr . (1.31)

This is positive since F and dr have the same direction. The work W to bring Mrfrom the infinity to radius r is

W = GMr Mr r

drr2

= GMrr

Mr . (1.32)

One defines the potential energy for a mass element as = W . The formationof an entire star of mass M represents a potential energy

= G M

0

Mr dMrr

. (1.33)

This is the energy lost by the reservoir of gravitational energy during the formationof a star. The energy lost by the initial cloud is gained, for example, by the thermalenergy of the gas. During the formation of a star, the potential energy becomesmore and more negative. The potential energy of an interstellar cloud dispersedover a very extended region is zero, this is the maximum value. As a protostellarcloud contracts and forms a star, the potential energy of the configuration decreases,becoming negative.

Often, one writes the potential energy of a star in the simplified form

= q GM2

R, (1.34)

1.2 The Potential Energy 11

where q is a numerical factor which depends on the internal density distributionof the star. For a star with a constant density, using dMr = 4r2dr and Mr =(4/3) r3, one immediately obtains = (3/5)GM2/R, i.e., a factor q = 3/5.In Sect. 24.5, we shall see the values of q for some other density distributions. Inparticular, a factor q = 3/2 is more appropriate to the density distribution of MainSequence (MS) stars.

1.2.1 Relation to the Potential and Poisson Equation

The components of g are (g, 0, 0). The gravity g is derived from the gravitationalpotential by the relation

g = with g = r

, (1.35)

according to the definition (1.3) of g. For spherical symmetry, one has

g = r

=GMr

r2(1.36)

and the equation of hydrostatic equilibrium can be written as

P = g = . (1.37)

Sometimes the potential is defined with a different sign. There is a relation betweenthe potential energy and the gravitational potential . This relation is not im-mediately useful here, but it is needed in Sect. 24.5. In spherical symmetry, onehas

(r) = r

0

GMrr2

dr + const., (1.38)

where the constant is chosen so that () = 0. One can write the potential energy as follows

= 12

G M

0

1r

dM2r =12

GM2

R+

12

G R(M)

0

M2r drr2

. (1.39)

With d/dr = GMr/r2, one gets

= 12

GM2

R+

12

(M)0

Mrd

=12

GM2

R+

12

M(M) 12

M0

dMr . (1.40)

The sum of the first two terms is zero since (M) =GM/R for a sphere accordingto Newtons theorem. One has

12 1 The Mechanical Equilibrium of Stars

=12

M0

dMr . (1.41)

The potential energy of a spherical star is the half of the average potential weightedby the mass.

From the equation of hydrostatic equilibrium (1.6), one may write Mr as Mr =[r2/(G)] (dP/dr). By inserting this expression into (1.12), we get

1r2

ddr

(r2

dPdr

)= 4G . (1.42)

We may express (dP/dr) with (1.37) and get

1r2

ddr

(r2

ddr

)= 4G . (1.43)

We recognize the radial component of the Poisson equation

= 4G . (1.44)

It is indeed quite consistent that our equations for stellar equilibrium imply the Pois-son equation, which is the fundamental relation between the material content ofspace and the potential exerted by the matter.

1.2.2 The Potential Energy as a Function of Pressure

A useful expression of the potential energy as a function of the internal pressureP can be derived for a star in hydrostatic equilibrium. From (1.33), one may alsowrite

= G2

M0

dM2rr

. (1.45)

The equation of hydrostatic equilibrium (1.6) multiplied on the right side bydMr/(4r2dr) = 1 yields

dPdr

= G8 r4

dM2rdr

, (1.46)

from which we may express dM2r in (1.45) and get

= 4 P(R)

0r3 dP =

[4r3P

]R0 12

R0

Pr2dr . (1.47)

If P(R) is the pressure at the surface of the configuration, we get

1.3 The Virial Theorem for Stars 13

= 4R3P(R)3 R

0PdV . (1.48)

In some cases, for example for a stellar core, one cannot take P(R) equal to zero,since the pressure of the surrounding layers is not negligible. However, in mostcases, for example for a star as a whole, one may consider that the pressure P(R)at the surface is negligible and thus ignore the first term on the right-hand side of(1.48) and get

= 3 R

0PdV . (1.49)

This is an expression frequently used for the potential energy of a configuration inhydrostatic equilibrium.

1.2.3 The Internal Stellar Temperature

Let us estimate the potential energy of a star of perfect gas obeying the equation ofstate P = [k/(mu)]T (1.25). From (1.49), one has

= 3 k mu

R0

T dV = 3 k mu

M0

T dMr 3k

muT M , (1.50)

where T , as defined above, is the internal temperature averaged over the stellar mass.With (1.34), one gets

T =13mu

kq

GMR

. (1.51)

One finds the same functional dependence as above (1.26). For q = 3/2, we wouldget in the case of the Sun an average temperature T 7 106 K, which is a satis-factory order of magnitude (see Fig. 25.8).

1.3 The Virial Theorem for Stars

The Virial theorem expresses a basic relation between the potential energy and theinternal energy in a star at equilibrium. Evidently, these two energies must be of thesame order in a star at equilibrium, but one can be more precise. The Virial theoremhas wide applications and many important results of astrophysics for stars, clusters,galaxies, etc., can be derived to the first order from this theorem [137]. We start froma basic result of statistical mechanics (Appendix C.3), which states that the ratio ofpressure P to the density of the kinetic energy u in a gaseous medium is limited by

14 1 The Mechanical Equilibrium of Stars

13

Pu

23

. (1.52)

The upper limit applies to a non-relativistic medium. For a perfect gas we haveu = (3/2) [k/(mu)]T and the pressure is given by (7.31). The potential energyof any non-relativistic medium becomes with (1.49)

= 2 R

0udV = 2Ecin , (1.53)

where Ecin is the total kinetic energy of particle translation in the configuration. TheVirial theorem is thus

2Ecin + = 0 . (1.54)

For a star in equilibrium, twice the kinetic energy of the particles is equal to the abso-lute value of the potential energy. The Virial theorem expresses the balance betweenthe effects of gravitation and those of pressure which support the star against gravi-tation. The hypothesis of equilibrium enters with the use of (1.46) in the expressionof the potential energy. The other possible internal motions (in addition to trans-lation) are not accounted for, as well as the other energy sources, such as atomicexcitation and ionization.

The lower limit in (1.52) applies to relativistic particles. For example, for the pho-tons one has u = aT 4 and the radiation pressure is P = (1/3)aT 4. For relativisticparticles with P = (1/3)u, we get with (1.49)

Ecin + = 0 . (1.55)

This is the Virial theorem for a star made of relativistic particles. There is no factor2 as in (1.54). For a mono-atomic gas, the total energy of the star is E = Ecin + ,which is zero. This means that a star made of relativistic particles is unstable, sincea negligible energy can spread it out.

1.3.1 Star with Perfect Gas Law

Let us first consider the Virial equilibrium for a star of perfect gas. The kineticenergy of an average particle is (1/2)muv2 = (3/2)kT , where T is the averagetemperature. For the ensemble N of particles in the star, the total kinetic energy is

Ecin =32

N kT . (1.56)

Ecin can be related to the internal energy, which for one particle is U = cV muT .There, cV is the specific heat at constant volume by unit of mass. For the N particlesin a star, the internal energy is

1.3 The Virial Theorem for Stars 15

U = cV N mu T . (1.57)

The internal energy U contains all forms of internal energy: thermal energy,radiation, atomic excitation, ionization, electron degeneracy, etc. Eliminating Tbetween (1.56) and (1.57), one gets U = cV mu 2Ec/(3k). For a perfect gas, onehas cP cV = k/( mu), where cP is the specific heat at constant pressure per unitof mass. Thus, (1.57) becomes

U =23

cVcP cV

Ecin =23

1g 1

Ecin , (1.58)

where g is the ratio of the specific heats g = cP/cV for a perfect gas. Thus, one has2Ec = 3(g 1)U . One verifies that for a mono-atomic perfect gas with g = 5/3the internal energy U is equal to the kinetic energy Ec. The Virial theorem for a starof perfect gas becomes

3(g 1)U + = 0 . (1.59)

It expresses the equilibrium between the internal energy which supports the star andgravitation. The total energy E = U + can be written as

E = (3g 3)

+ =(3g 4)(3g 3)

. (1.60)

If E > 0, the star is unstable since it can do some work to spread its matter out inspace. A negative E expresses the physical cohesion of a star. A necessary conditionfor stellar stability is E < 0. Since < 0, one must have

g >43

for stability . (1.61)

This shows that the thermodynamic properties of a medium are critical for stability.In the case of a perfect mono-atomic gas with g = 5/3, this condition is satisfied.If it is not, the characteristic time of the departure from equilibrium is the dynamicaltimescale (1.28).

A value g < 4/3 would imply, for example during a contraction, that the result-ing increase of the gravity force is larger than the increase of the pressure gradient.Thus, contraction would go on unimpeded. This is clear from the meaning of g,which is given by g = ( lnP/ ln)ad (see Sect. 7.4.1). A simple scaling of grav-ity effects (Sect. 1.1.5) shows that P M2/R4 and M/R3, so that at constant M,the changes of P and are related by P (4/3) for an equilibrium configura-tion. If g < 4/3, the thermal pressure increase in a density change is not sufficientto maintain equilibrium, which leads to collapse. We also note that if g < 1, sta-bility is also formally present. However, starting from 3/5, the limit = 4/3 iscrossed first.

16 1 The Mechanical Equilibrium of Stars

We may wonder about the effects of rotation on the Virial theorem. Models ofrotating stars on the Main Sequence show that the energy of rotation is negligiblewith respect to the potential energy. At the critical velocity, the energy of rotation isa few percents of the potential energy.

1.3.2 Star with a General Equation of State

For a medium, which is not a perfect gas, what happens to the Virial theorem ex-pressed in the form (1.59)? As stated above, a departure from the mechanical equi-librium is characterized by the dynamical timescale (1.28) which is very short andusually much smaller than the thermal timescale (1.73). This means that the per-turbations of the mechanical equilibrium can be considered as adiabatic. The totalchange E of energy due the perturbation is the sum of the changes of the internaland potential energies,

E = U + (1.62)

with the total internal energy U =M

0 udMr, where u is the internal energy per massunit. The First Principle of thermodynamics implies that

q = u P2

= 0 , (1.63)

for an adiabatic change. q is the energy provided to the system by mass unit and(P/2) is the work provided by the system to the exterior. One can define ageneralized adiabatic exponent