The Fundamentals of Stellar Astrophysics George W. Collins, II Copyright 2003: All sections of this book may be reproduced as long as proper attribution is given. Page Preface to the Internet Edition xiv Preface to the W. Freeman Edition xv Part I Stellar Interiors Chapter 1 Introduction and Fundamental Principles 3 1.1 Stationary or “Steady” Properties of matter 5 a Phase Space and Phase Density 5 b Macrostates and Microstates.
6 c Probability and Statistical Equilibrium 6 d Quantum Statistics 9 e Statistical Equilibrium for a Gas 11 f Thermodynamic Equilibrium – Strict and Local 15 1. Boltzmann Transport Equation 15 b. Homogeneous Boltzmann Transport Equation and Liouville’s Theorem 17 c. Moments of the Boltzmann Transport Equation and Conservation Laws 18 1.3 Equation of State for the Ideal Gas and Degenerate Matter 26 Problems 32 References and Supplemental Reading 33 ii www.com Chapter 2 Basic Assumptions, Theorems, and Polytropes 34 2.2 Integral Theorems from Hydrostatic Equilibrium 36 a Limits of State Variables 36 b β* Theorem and Effects of Radiation Pressure 38 2.4 Polytropes 42 a Polytropic Change and the Lane-Emden Equation 43 b Mass-Radius Relationship for Polytropes 46 c Homology Invariants 47 d Isothermal Sphere 49 e Fitting Polytropes Together 51 Problems 53 References and Supplemental Reading 54 Chapter 3 Sources and Sinks of Energy 56 3.1 "Energies" of Stars 57 a Gravitational Energy 57 b Rotational Energy 59 c Nuclear Energy 60 3.2 Time Scales 61 a Dynamical Time Scale 61 b Kelvin-Helmholtz (Thermal) Time Scale 62 c Nuclear (Evolutionary) Time Scale 63 3.3 Generation of Nuclear Energy 64 a General Properties of the Nucleus 65 b The Bohr Picture of Nuclear Reactions 66 c Nuclear Reaction Cross Sections 68 d Nuclear Reaction Rates 70 e Specific Nuclear Reactions 72 Problems 75 References and Supplemental Reading 75 iii www.com Chapter 4 Flow of Energy through the Star and Construction of Stellar Models 77 4.1 The Ionization, Abundances, and Opacity of Stellar Material 78 a Ionization and the Mean Molecular Weight 78 b Opacity 80 4.2 Radiative Transport and the Radiative Temperature Gradient 86 a Radiative Equilibrium 86 b Thermodynamic Equilibrium and Net Flux 86 c Photon Transport and the Radiative Gradient 87 d Conservation of Energy and the Luminosity 89 4.3 Convective Energy Transport 90 a Adiabatic Temperature Gradient 90 b Energy Carried by Convection 91 4.4 Energy Transport by Conduction 94 a Mean Free Path 94 b Heat Flow 95 4.5 Convective Stability 96 a Efficiency of Transport Mechanisms 96 b Schwarzschild Stability Criterion 97 4.6 Equations of Stellar Structure 100 4.7 Construction of a Model Stellar Interior 101 a Boundary Conditions 102 b Schwarzschild Variables and Method 102 c Henyey Relaxation Method for Construction of Stellar Models 105 Problems 109 References and Supplemental Reading 110 iv www.com Chapter 5 Theory of Stellar Evolution 112 5.1 The Ranges of Stellar Masses, Radii, and Luminosity 113 5.2 Evolution onto the Main Sequence 114 a Problems concerning the Formation of Stars 114 b Contraction out of the Interstellar Medium 116 c Contraction onto the Main Sequence 119 5.3 The Structure and Evolution of Main Sequence Stars 125 a Lower Main Sequence Stars 126 b Upper Main Sequence Stars 128 5.4 Post Main Sequence Evolution 129 a Evolution off the Lower Main Sequence 129 b Evolution away from the Upper Main Sequence 136 c The Effect of Mass-loss on the Evolution of Stars 138 5.5 Summary and Recapitulation 139 a Core Contraction - Envelope Expansion: Simple Reasons 140 b Calculated Evolution of a 5 M⊙ star 143 Problems 144 References and Supplemental Reading 145 Chapter 6 Relativistic Stellar Structure 149 6.1 Field Equations of the General Theory of Relativity 150 6.2 Oppenheimer-Volkoff Equation of Hydrostatic 152 Equilibrium a Schwarzschild Metric 152 b Gravitational Potential and Hydrostatic Equilibrium 154 6.3 Equations of Relativistic Stellar Structure and Their Solutions 154 a A Comparison of Structure Equations 155 b A Simple Model 156 c Neutron Star Structure 158 v www.4 Relativistic Polytrope of Index 3 161 a Virial Theorem for Relativistic Stars 161 b Minimum Radius for White Dwarfs 164 c Minimum Radius for Super-massive Stars 165 6.5 Fate of Super-massive Stars 167 a Eddington Luminosity 167 b Equilibrium Mass-Radius Relation 168 c Limiting Masses for Super-massive Stars 168 Problems 172 References and Supplemental Reading 173 Chapter 7 Structure of Distorted Stars 175 7.1 Classical Distortion: The Structure Equations 176 a A Comparison of Structure Equations 176 b Structure Equations for Cylindrical Symmetry 177 7.2 Solution of Structure Equations for a Perturbing Force 184 a Perturbed Equation of Hydrostatic Equilibrium 185 b Number of Perturbative Equations versus Number of Unknowns 186 7.3 Von Zeipel's Theorem and Eddington-Sweet Circulation Currents 187 a Von Zeipel's Theorem 187 b Eddington-Sweet Circulation Currents 190 7.4 Rotational Stability and Mixing 195 a Shear Instabilities 195 b Chemical Composition Gradient and Suppression of Mixing 196 c Additional Types of Instabilities 198 Problems 199 References and Supplemental Reading 199 Chapter 8 Stellar Pulsation and Oscillation 201 8.1 Linear Adiabatic Radial Oscillations 202 a Stellar Oscillations and the Variational Virial theorem 203 vi www.com b Effect of Magnetic Fields and Rotation on Radial Oscillations 205 c Stability and the Variational Virial Theorem 206 d Linear Adiabatic Wave Equation 207 8.2 Linear Nonadiabatic Radial Oscillations 208 a Adiabatic Exponents 209 b Nonadiabatic Effects and Pulsational Stability 209 c Constructing Pulsational Models 211 d Pulsational Behavior of Stars 212 8.3 Nonradial Oscillations 214 a Nature and Form of Oscillations 214 b Homogeneous Model and Classification of Modes 216 c Toroidal Oscillations 219 d Nonradial Oscillations and Stellar Structure 220 Problems 221 References and Supplemental Reading 221 Epilogue to Part I: Stellar Interiors 224 Part II Stellar Atmospheres 225 Chapter 9 The Flow of Radiation Through the Atmosphere 227 9.1 Basic Assumptions for the Stellar Atmosphere 228 a Breakdown of Strict Thermodynamic Equilibrium 228 b Assumption of Local Thermodynamic Equilibrium 229 c Continuum and Spectral Lines 230 d Additional Assumptions of Normal Stellar Atmospheres 231 9.2 Equation of Radiative Transfer 233 a Specific Intensity and Its Relation to the Density of Photons in Phase Space 233 b General Equation of Radiative Transfer 235 c "Creation" Rate and the Source Function 236 vii www.com d Physical Meaning of the Source Function 240 e Special Forms of the Redistribution Function 241 9.3 Moments of the Radiation Field 243 a Mean Intensity 244 b Flux 244 c Radiation Pressure 245 9.4 Moments of the Equation of Radiative Transfer 247 a Radiative Equilibrium and Zeroth Moment of the Equation of Radiative Transfer 248 b First Moment of the Equation of Radiative Transfer and the Diffusion Approximation 248 c Eddington Approximation 249 Problems 251 Supplemental Reading 252 Chapter 10 Solution of the Equation of Radiative Transfer 253 10.1 Classical Solution to the Equation of Radiative Transfer and Integral Equations for the Source Function 254 a Classical Solution of the Equation of Transfer for the Plane-Parallel Atmosphere 254 b Schwarzschild-Milne Integral Equations 257 c Limb-darkening in a Stellar Atmosphere 260 10.2 Gray Atmosphere 263 a Solution of Schwarzschild-Milne Equations for the Gray Atmosphere 265 b Solutions for the Gray Atmosphere Utilizing the Eddington Approximation 266 c Solution by Discrete Ordinates: Wick- Chandrasekhar Method 268 10.3 Nongray Radiative Transfer 274 a Solutions of the Nongray Integral Equation for the Source Function 275 b Differential Equation Approach: The Feautrier Method 276 10.4 Radiative Transport in a Spherical Atmosphere 279 viii www.com a Equation of Radiative Transport in Spherical 280 Coordinates b An Approach to Solution of the Spherical Radiative Transfer Problem 283 Problems 287 References and Supplemental Reading 289 Chapter 11 Environment of the Radiation Field 291 11.1 Statistics of the Gas and the Equation of State 292 a Boltzmann Excitation Formula 292 b Saha Ionization Equilibrium Equation 293 11.2 Continuous Opacity 296 a Hydrogenlike Opacity 296 b Neutral Helium 297 c Quasi-atomic and Molecular States 297 d Important Sources of Continuous Opacity for Main Sequence Stars 299 11.3 Einstein Coefficients and Stimulated Emission 300 a Relations among Einstein Coefficients 301 b Correction of the Mass Absorption Coefficient for Stimulated Emission 302 11.4 Definitions and Origins of Mean Opacities 303 a Flux-Weighted (Chandrasekhar) Mean Opacity 304 b Rosseland Mean Opacity 304 c Planck Mean Opacity 306 11.5 Hydrostatic Equilibrium and the Stellar Atmosphere 307 Problems 308 References 309 Chapter 12 The Construction of a Model Stellar Atmosphere 310 12.1 Statement of the Basic Problem 310 12.2 Structure of the Atmosphere, Given the Radiation Field 312 a Choice of the Independent Variable of Atmospheric Depth 314 ix www.com b Assumption of Temperature Dependence with Depth 314 c Solution of the Equation of Hydrostatic Equilibrium 314 12.3 Calculation of the Radiation Field of the Atmosphere 316 12.4 Correction of the Temperature Distribution and Radiative Equilibrium 318 a Lambda Iteration Scheme 318 b Avrett-Krook Temperature Correction Scheme 319 12.5 Recapitulation 325 Problems 326 References and Supplemental Reading 328 Chapter 13 Formation of Spectral Lines 330 13.1 Terms and Definitions Relating to Spectral Lines 331 a Residual Intensity, Residual Flux, and Equivalent Width 331 b Selective (True) Absorption and Resonance Scattering 333 c Equation of Radiative Transfer for Spectral Line Radiation 335 13.2 Transfer of Line Radiation through the Atmosphere 336 a Schuster-Schwarzschild Model Atmosphere for Scattering Lines 336 b Milne-Eddington Model Atmosphere for the Formation of Spectral Lines 339 Problems 346 Supplemental Reading 347 Chapter 14 Shape of Spectral Lines 348 14.1 Relation between the Einstein, Mass Absorption, and Atomic Absorption Coefficients 349 14.2 Natural or Radiation Broadening 350 a Classical Radiation Damping 351 x www.com b Quantum Mechanical Description of Radiation Damping 354 c Ladenburg f-value 355 14.3 Doppler Broadening of Spectral Lines 357 a Microscopic Doppler Broadening 358 b Macroscopic Doppler Broadening 364 14.4 Collisional Broadening 369 a Impact Phase-Shift Theory 370 b Static (Statistical) Broadening Theory 378 14.5 Curve of Growth of the Equivalent Width 385 a Schuster-Schwarzschild Curve of Growth 385 b More Advanced Models for the Curve of Growth 389 c Uses of the Curve of Growth 390 Problems 392 References and Supplemental Reading 395 Chapter 15 Breakdown of Local Thermodynamic Equilibrium 398 15.1 Phenomena Which Produce Departures from Local Thermodynamic Equilibrium 400 a Principle of Detailed Balancing 400 b Interlocking 401 c Collisional versus Photoionization 402 15.2 Rate Equations for Statistical Equilibrium 403 a Two-Level Atom 403 b Two-Level Atom plus Continuum 407 c Multilevel Atom 409 d Thermalization Length 410 15.3 Non-LTE Transfer of Radiation and the Redistribution Function 411 a Complete Redistribution 412 b Hummer Redistribution Functions 413 15.4 Line Blanketing and Its Inclusion in the construction of Model Stellar Atmospheres and Its Inclusion in the Construction of Model Stellar Atmospheres 425 a Opacity Sampling 426 xi www.com b Opacity Distribution Functions 427 Problems 429 References and Supplemental Reading 430 Chapter 16 Beyond the Normal Stellar Atmosphere 432 16.1 Illuminated Stellar Atmospheres 434 a Effects of Incident Radiation on the Atmospheric Structure 434 b Effects of Incident Radiation on the Stellar Spectra 439 16.2 Transfer of Polarized Radiation 440 a Representation of a Beam of Polarized Light and the Stokes Parameters 440 b Equations of Transfer for the Stokes 445 c Solution of the Equations of Radiative Transfer for Polarized Light.
454 d Approximate Formulas for the Degree of Emergent Polarization 457 e Implications of the Transfer of Polarization for Stellar Atmospheres 465 16.3 Extended Atmospheres and the Formation of Stellar Winds 469 a Interaction of the Radiation Field with the Stellar Wind 470 b Flow of Radiation and the Stellar Wind 474 Problems 477 References and Supplemental Reading 478 Epilog 480 Index 483 Errata to the W.com Preface To the (2003) WEB Edition One may justifiability wonder why anyone would take the time to put a decade- old book on astrophysics on the WEB. Several events of the past few months have led me to believe that may well be some who wish to learn about the basics of stellar structure.