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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 Contents Page xiv xv Preface to the Internet Edition Preface to the W H Freeman Edition Part I Stellar Interiors Chapter Introduction and Fundamental Principles 1.1 Stationary or “Steady” Properties of matter a Phase Space and Phase Density b Macrostates and Microstates c Probability and Statistical Equilibrium d Quantum Statistics e Statistical Equilibrium for a Gas f Thermodynamic Equilibrium – Strict and Local 1.2 Transport Phenomena a Boltzmann Transport Equation b Homogeneous Boltzmann Transport Equation and Liouville’s Theorem c Moments of the Boltzmann Transport Equation and Conservation Laws 1.3 Equation of State for the Ideal Gas and Degenerate Matter Problems References and Supplemental Reading ii 5 6 11 15 15 15 17 18 26 32 33 Chapter Basic Assumptions, Theorems, and Polytropes 2.1 Basic Assumptions 2.2 Integral Theorems from Hydrostatic Equilibrium a Limits of State Variables b β* Theorem and Effects of Radiation Pressure 2.3 Homology Transformations 2.4 Polytropes a Polytropic Change and the Lane-Emden Equation b Mass-Radius Relationship for Polytropes c Homology Invariants d Isothermal Sphere e Fitting Polytropes Together Problems References and Supplemental Reading Chapter Sources and Sinks of Energy 3.1 "Energies" of Stars a Gravitational Energy b Rotational Energy c Nuclear Energy 3.2 Time Scales a Dynamical Time Scale b Kelvin-Helmholtz (Thermal) Time Scale c Nuclear (Evolutionary) Time Scale 3.3 Generation of Nuclear Energy a General Properties of the Nucleus b The Bohr Picture of Nuclear Reactions c Nuclear Reaction Cross Sections d Nuclear Reaction Rates e Specific Nuclear Reactions Problems References and Supplemental Reading iii 34 34 36 36 38 40 42 43 46 47 49 51 53 54 56 57 57 59 60 61 61 62 63 64 65 66 68 70 72 75 75 Chapter Flow of Energy through the Star and Construction of Stellar Models 4.1 The Ionization, Abundances, and Opacity of Stellar Material a Ionization and the Mean Molecular Weight b Opacity 4.2 Radiative Transport and the Radiative Temperature Gradient a Radiative Equilibrium b Thermodynamic Equilibrium and Net Flux c Photon Transport and the Radiative Gradient d Conservation of Energy and the Luminosity 4.3 Convective Energy Transport a Adiabatic Temperature Gradient b Energy Carried by Convection 4.4 Energy Transport by Conduction a Mean Free Path b Heat Flow 4.5 Convective Stability a Efficiency of Transport Mechanisms b Schwarzschild Stability Criterion 4.6 Equations of Stellar Structure 4.7 Construction of a Model Stellar Interior a Boundary Conditions b Schwarzschild Variables and Method c Henyey Relaxation Method for Construction of Stellar Models Problems References and Supplemental Reading iv 77 78 78 80 86 86 86 87 89 90 90 91 94 94 95 96 96 97 100 101 102 102 105 109 110 Chapter Theory of Stellar Evolution 5.1 The Ranges of Stellar Masses, Radii, and Luminosity 5.2 Evolution onto the Main Sequence a Problems concerning the Formation of Stars b Contraction out of the Interstellar Medium c Contraction onto the Main Sequence 5.3 The Structure and Evolution of Main Sequence Stars a Lower Main Sequence Stars b Upper Main Sequence Stars 5.4 Post Main Sequence Evolution a Evolution off the Lower Main Sequence b Evolution away from the Upper Main Sequence c The Effect of Mass-loss on the Evolution of Stars 5.5 Summary and Recapitulation a Core Contraction - Envelope Expansion: Simple Reasons b Calculated Evolution of a M⊙ star Problems References and Supplemental Reading Chapter Relativistic Stellar Structure 6.1 Field Equations of the General Theory of Relativity 6.2 Oppenheimer-Volkoff Equation of Hydrostatic Equilibrium a Schwarzschild Metric b Gravitational Potential and Hydrostatic Equilibrium 6.3 Equations of Relativistic Stellar Structure and Their Solutions a A Comparison of Structure Equations b A Simple Model c Neutron Star Structure v 112 113 114 114 116 119 125 126 128 129 129 136 138 139 140 143 144 145 149 150 152 152 154 154 155 156 158 6.4 Relativistic Polytrope of Index a Virial Theorem for Relativistic Stars b Minimum Radius for White Dwarfs c Minimum Radius for Super-massive Stars 6.5 Fate of Super-massive Stars a Eddington Luminosity b Equilibrium Mass-Radius Relation c Limiting Masses for Super-massive Stars Problems References and Supplemental Reading Chapter Structure of Distorted Stars 7.1 Classical Distortion: The Structure Equations a A Comparison of Structure Equations b Structure Equations for Cylindrical Symmetry 7.2 Solution of Structure Equations for a Perturbing Force a Perturbed Equation of Hydrostatic Equilibrium b Number of Perturbative Equations versus Number of Unknowns 7.3 Von Zeipel's Theorem and Eddington-Sweet Circulation Currents a Von Zeipel's Theorem b Eddington-Sweet Circulation Currents 7.4 Rotational Stability and Mixing a Shear Instabilities b Chemical Composition Gradient and Suppression of Mixing c Additional Types of Instabilities Problems References and Supplemental Reading Chapter Stellar Pulsation and Oscillation 8.1 Linear Adiabatic Radial Oscillations a Stellar Oscillations and the Variational Virial theorem vi 161 161 164 165 167 167 168 168 172 173 175 176 176 177 184 185 186 187 187 190 195 195 196 198 199 199 201 202 203 b Effect of Magnetic Fields and Rotation on Radial Oscillations c Stability and the Variational Virial Theorem d Linear Adiabatic Wave Equation 8.2 Linear Nonadiabatic Radial Oscillations a Adiabatic Exponents b Nonadiabatic Effects and Pulsational Stability c Constructing Pulsational Models d Pulsational Behavior of Stars 8.3 Nonradial Oscillations a Nature and Form of Oscillations b Homogeneous Model and Classification of Modes c Toroidal Oscillations d Nonradial Oscillations and Stellar Structure Problems References and Supplemental Reading 205 206 207 208 209 209 211 212 214 214 216 219 220 221 221 224 Epilogue to Part I: Stellar Interiors Part II Stellar Atmospheres Chapter The Flow of Radiation Through the Atmosphere 9.1 Basic Assumptions for the Stellar Atmosphere a Breakdown of Strict Thermodynamic Equilibrium b Assumption of Local Thermodynamic Equilibrium c Continuum and Spectral Lines d Additional Assumptions of Normal Stellar Atmospheres 9.2 Equation of Radiative Transfer a Specific Intensity and Its Relation to the Density of Photons in Phase Space b General Equation of Radiative Transfer c "Creation" Rate and the Source Function vii 225 227 228 228 229 230 231 233 233 235 236 9.3 9.4 d Physical Meaning of the Source Function e Special Forms of the Redistribution Function Moments of the Radiation Field a Mean Intensity b Flux c Radiation Pressure Moments of the Equation of Radiative Transfer a Radiative Equilibrium and Zeroth Moment of the Equation of Radiative Transfer b First Moment of the Equation of Radiative Transfer and the Diffusion Approximation c Eddington Approximation Problems Supplemental Reading Chapter 10 Solution of the Equation of Radiative Transfer 10.1 Classical Solution to the Equation of Radiative Transfer and Integral Equations for the Source Function a Classical Solution of the Equation of Transfer for the Plane-Parallel Atmosphere b Schwarzschild-Milne Integral Equations c Limb-darkening in a Stellar Atmosphere 10.2 Gray Atmosphere a Solution of Schwarzschild-Milne Equations for the Gray Atmosphere b Solutions for the Gray Atmosphere Utilizing the Eddington Approximation c Solution by Discrete Ordinates: WickChandrasekhar Method 10.3 Nongray Radiative Transfer a Solutions of the Nongray Integral Equation for the Source Function b Differential Equation Approach: The Feautrier Method 10.4 Radiative Transport in a Spherical Atmosphere viii 240 241 243 244 244 245 247 248 248 249 251 252 253 254 254 257 260 263 265 266 268 274 275 276 279 Equation of Radiative Transport in Spherical 280 Coordinates b An Approach to Solution of the Spherical Radiative 283 Transfer Problem 287 Problems 289 References and Supplemental Reading a Chapter 11 Environment of the Radiation Field 11.1 Statistics of the Gas and the Equation of State a Boltzmann Excitation Formula b Saha Ionization Equilibrium Equation 11.2 Continuous Opacity a Hydrogenlike Opacity b Neutral Helium c Quasi-atomic and Molecular States d Important Sources of Continuous Opacity for Main Sequence Stars 11.3 Einstein Coefficients and Stimulated Emission a Relations among Einstein Coefficients b Correction of the Mass Absorption Coefficient for Stimulated Emission 11.4 Definitions and Origins of Mean Opacities a Flux-Weighted (Chandrasekhar) Mean Opacity b Rosseland Mean Opacity c Planck Mean Opacity 11.5 Hydrostatic Equilibrium and the Stellar Atmosphere Problems References Chapter 12 The Construction of a Model Stellar Atmosphere 12.1 Statement of the Basic Problem 12.2 Structure of the Atmosphere, Given the Radiation Field a Choice of the Independent Variable of Atmospheric Depth ix 291 292 292 293 296 296 297 297 299 300 301 302 303 304 304 306 307 308 309 310 310 312 314 b Assumption of Temperature Dependence with Depth c Solution of the Equation of Hydrostatic Equilibrium 12.3 Calculation of the Radiation Field of the Atmosphere 12.4 Correction of the Temperature Distribution and Radiative Equilibrium a Lambda Iteration Scheme b Avrett-Krook Temperature Correction Scheme 12.5 Recapitulation Problems References and Supplemental Reading 314 314 316 318 318 319 325 326 328 Chapter 13 330 Formation of Spectral Lines 331 13.1 Terms and Definitions Relating to Spectral Lines a Residual Intensity, Residual Flux, and 331 Equivalent Width b Selective (True) Absorption and Resonance 333 Scattering c Equation of Radiative Transfer for Spectral 335 Line Radiation 336 13.2 Transfer of Line Radiation through the Atmosphere a Schuster-Schwarzschild Model Atmosphere for 336 Scattering Lines b Milne-Eddington Model Atmosphere for the 339 Formation of Spectral Lines 346 Problems 347 Supplemental Reading Chapter 14 348 Shape of Spectral Lines 14.1 Relation between the Einstein, Mass Absorption, and 349 Atomic Absorption Coefficients 350 14.2 Natural or Radiation Broadening 351 a Classical Radiation Damping x Index 491 Index 492 Index 493 Index 494 Errata for W H Freeman Edition Errata for Fundamentals of Stellar Astrophysics Circa 3/23/91 Page numbers referred to here are those of the book • Page 28: The exponent on h in the last term should be () 8πp5 p 8πp h2 / n / n (p)dp = ∫0 dp = 15mh = 20 m π h 3 • Page 37: The exponent of in the last term was left off p P = ∫0 p2 m • Page 47: The exponent of the last term was omitted • Page 48: Sign error in the middle two terms (1.3.8) (2.2.5) (2.4.20) (2.4.22) • Page 61: Sign changes should be made in equations (3.2.1-2) for consistency (3.2.1) • Page 71: Sign error in the second term of the exponential • Page 76: Bahcall, J N., Huebner, W F., Lubia,S H., Parker,P D., and Ulrich, R K., Rev Mod Phy 54, 1982, p 767 • Page 82: Equation (4.1.16) should read (3.2.2) (3.3.10) (4.1.16) 495 Errata for W.H Freeman Edition • Page 101: The last two of equations 4.6.1 should read • Page 103: Equation 4.7.3 (d) should read • Page 107: Quantity left out of (b) and an improvement made to (d) • Page 108: Runge-Kutta is mispelled in line 10 of ¶2 496 (4.6.1) (4.7.3) (4.7.10) Errata for W H Freeman Edition • Page 131: Lead coefficient is wrong, should be (5.4.5) • Pages 140-143: Equations (5.4.9) - (5.4.14) should be renumbered to agree with the new section 5.5 that was introduced • Page 142: (Equation 5.5.1 needs an integral of ε over volume to get the entire contribution of energy to the star to match the losses through L.) • Page 142: Last paragraph line two, replace potential energy with internal energy (5.5.1) • Page 145: problem should read Choose a representative set of models from the evolutionary calculations in Problem 4, (a) Calculate the moment of inertia, gravitational and internal energies of the core and envelope, and the total energy of the star (b) Determine the extent to which the conditions in Section 5.5a are met during the evolution of the star • The discussion of Neutron Star Structure on pp 158-160 should be expanded to include the work by Keith Olive (1991 Sci 251, pp.1197-1198) on the Quark-Hadron phase transition Nothing here is wrong; it could just be made more complete • Page 163: The ε on the right hand side was left out 14 • Page 168: Capriotti has evaluated the luminosity integral and gets (6.4.8) • Page 168: last paragraph c Limiting Masses for Supermassive Stars Let us add equations (6.4.19) and (6.4.20) and, taking care to express the relativistic integrals as dimensionless integrals by making use of the homology relations for pressure and density, get for the total energy: 497 Errata for W.H Freeman Edition (6.5.5) • Page 169: The exponent on M⊙ in the last term should be (6.5.6) • Page 170: last paragraph However, the only energy transportable by convection is the kinetic energy of the gas, which is an insignificant fraction of the internal energy Therefore, unlike normal main sequence stars, although it is present, convection will be a very inefficient vehicle for the transport of energy This is • Page 173: Capriotti is spelled wrong • Page 179: Sign should be changed for consistency r D = -∇Λ • Page 180: We may remove the unit vector from the s-component D z = D φ = , D s = ω2s (7.1.9) (7.1.12) • Page 183: sign error in second term of first eq ∂ψ ~ ~ −1 ψ ( r ) D r = (4πρc) + ψ(r ) Sin θ = A(r ) + B(r )P2 (Cosθ) ∂r r (7.1.31) ψ (r ) ∂P2 (Cosθ) ~ ∂P2 (Cosθ) = C( r ) r ∂θ ∂θ • Page 186: The last term should have a (1/r) in it ∂Ω (r ) ∂P0 (r ) = −ρ + ρ (r )A(r ) ∂r ∂r ∂Ω (r ) ∂Ω (r ) ∂P2 (r ) + ρ (r )B(r ) + ρ (r ) = −ρ ( r ) ∂r ∂r ∂r P2 (r ) = −ρ (r )Ω (r ) + ρ (r )C(r ) / r D θ = ((4πρc) −1 (7.2.5) • Page 191: last two lines - In the equilibrium model, there are no mass motions, the velocity in equation (7.2.6) is already a first-order term and so to estimate its value we need only 498 Errata for W H Freeman Edition • Page 204: eq 8.1.7 should read r r r 0 d[M (r )] = = ∫ 8πr0 ρδrdr + ∫ 4πr02 δρ dr + ∫ 4πr02 ρ d (δr ) • Page 206: The I0 got left out σ2 = - [(Ω0+ M0) - ω0 0]/I0 (8.1.7) (8.1.16) • Page 211: Sign error on the third term (8.2.5) • Page 234: dV = cdAcosθdt (9.2.4) • Page 237: eq 9.2.18 in the book v's are occasionally υ's see [h4υ3/c2 ] and (υ/υ')3 Should be dn = dn gt + dn gs + dnl = σ' h ∞ c h 4ν3 ε + ∫0 ∫ (ν / ν' ) R(ν, ν' , Ω, Ω' )dν' dΩ'−αI ν (Ω) dVdVp h ν c 4π 4πc (9.2.18) • Page 238: equation (9.2.20) should be (9.2.20) • Page 246: equation (9.3.9) should be ˆˆ sin 2θ cosφ sinφ ˆk sinθ cosθ cosφ ˆˆ i i sin 2θ cos φ ij iˆ π 2π ˆ k sinθ cosθ sinφ ji jj jˆ Kν = ˆ ˆ sin 2θ sin φ cos φ ˆˆ sin 2θ sin φ 4π ∫0 ∫0 ˆi ˆj ˆˆ kˆ sinθ cosθ cosφ kˆ sinθ cosθ sinφ kk cos θ × Iν (τν )sinθ dθ dφ (9.3.9) • Page 251: The last three words of problem should be "space is constant." 499 Errata for W.H Freeman Edition • Page 257: sign of first term r.h.s of 2nd equation should be negative (10.1.9) • Page 266: summation should run from i=1, not i=0 as: • Page 277: ν-subscript missing on the τ, should be (10.2.10) • Page 278: The µ on the right hand side should be u • Page 280: line 10 should read "for which it is suited." • Page 304: Table 11.1 non-gray equation (1) should have Sν not Jν so that • Page 306: subscript ν on B in the denominator is a subscript, should be 500 (10.3.10) (10.3.13) Errata for W H Freeman Edition (11.4.10) • Page 309: problem should read Use a Model Atmosphere Code to find how the state of ionization of hydrogen varies with physical depth in a star with Te = 10000#K and Log g = 4.0 Repeat the calculation for a star with Te= 7000#K and Log g = 1.5 Compare the two cases • Page 315: term in braces should be to the -1 power so that (12.2.7) • Page 321: In the book kν(τ0) is given as κν(τ0) in the first two terms of eq 12.2.6 • Page 322: - superscript on K' L.H.S is wrong should be • Page 338: the in τ0>>1 got lost The equation should read • Page 350: - equation 14.1.4 should read (see equation 14.5.2) (12.4.6) (12.4.12) (13.2.12) (14.1.4) 501 Errata for W.H Freeman Edition • Page 351:- Equation(14.2.1) the average symbol should extend over the (14.2.1) • Page 353: - paragraph 1: the i in iω0t got lost It should read If we assume that the photon encounters the atom at t=0 so that E(t)=0 for t ηB* but η > ~ (or εB ε vice versa), the line is said to be mixed • Page 411: equation (15.2.27) should read: • Page 413: equation 15.3.5 should read: (15.2.27) (15.3.5) • Page 414: line should read: For isotropic scattering, g(n',n) = 1, while in the ^ ^ case of Rayleigh Scattering g(n',n) = 3[1+(n'•n)2]/4 ^ ^ ^ ^ • Page 415: - there should be no ' on ξ' in the second of equations 15.3.10 on the right hand side It should read (15.3.10) 503 Errata for W.H Freeman Edition • Page 416: - Denominator of first fraction should end with γl2not ηl2 the numerator , , of the fourth fraction should be γl2 and one of the ξ's in the denominator should not have a prime, so that equation (15.3.12) should read (15.3.12) • Page 418: - the last of equations (15.3.18) should use vth rather than vd so that it is consistent with the first of those equations Thus it should read • Page 421: - equation (15.3.33) should have a 1/4π in the last term • Page 429: problem should read Show how equation (15.3.25) is implied by equation (15.3.15) • Page 430: Peytremann is misspelled in Ref.16 504 (15.3.18) (15.3.33) Errata for W H Freeman Edition • Page 442: - The equations (16.2.1) should read (16.2.1) • Page 457: equation 16.2.39 last equation LHS should be averaged to read (16.2.39) • Page 464: second para., line delete the "when" so that the sentence reads: polarization approaching or exceeding the gray value in the vicinity of the Lyman Jump • Page 482: three lines from the bottom of the page: should be: the conservation laws of physics, the fundamental • Page 484: Capriotti, E.R 168, 178 505 ... Certainly the advent of pulsars, black holes and the other unusual objects that are often called stars has necessitated broadening the scope of the theory of stellar astrophysics Then there are... Chandrasekhar, and many others echo down through the history of this subject as the definers and elucidators of stellar structure The outline of the theory of the structure and evolution of th stars clearly... application of the fundamental physics of stellar astrophysics along with the explosive expansion of computing power will lead to the solutions of these problems in the present century While the copyright