Astronomy and Astrophysics Library

I The Basic Equations.- 1. Coordinates, Mass Distribution, and Gravitational Field in Spherical Stars.- 1.1 Eulerian Description.- 1.2 Lagrangian Description.- 1.3 The Gravitational Field.- 2. Conservation of Momentum.- 2.1 Hydrostatic Equilibrium.- 2.2 The Role of Density and Simple Solutions.- 2.3 Simple Estimates of Central Values Pc, Tc.- 2.4 The Equation of Motion for Spherical Symmetry.- 2.5 The Non-spherical Case.- 2.6 Hydrostatic Equilibrium in General Relativity.- 2.7 The Piston Model.- 3. The Virial Theorem.- 3.1 Stars in Hydrostatic Equilibrium.- 3.2 The Virial Theorem of the Piston Model.- 3.3 The Kelvin-Helmholtz Time-scale.- 3.4. The Virial Theorem for Non-vanishing Surface Pressure.- 4. Conservation of Energy.- 4.1 Thermodynamic Relations.- 4.2 Energy Conservation in Stars.- 4.3 Global and Local Energy Conservation.- 4.4 Time-scales.- 5. Transport of Energy by Radiation and Conduction.- 5.1 Radiative Transport of Energy.- 5.1.1 Basic Estimates.- 5.1.2 Diffusion of Radiative Energy.- 5.1.3 The Rosseland Mean for ??.- 5.2 Conductive Transport of Energy.- 5.3 The Thermal Adjustment Time of a Star.- 5.4 Thermal Properties of the Piston Model.- 6. Stability Against Local, Non-spherical Perturbations.- 6.1 Dynamical Instability.- 6.2 Oscillation of a Displaced Element.- 6.3 Vibrational Stability.- 6.4 The Thermal Adjustment Time.- 6.5 Secular Instability.- 6.6 The Stability of the Piston Model.- 7. Transport of Energy by Convection.- 7.1 The Basic Picture.- 7.2 Dimensionless Equations.- 7.3 Limiting Cases, Solutions, Discussion.- 8. The Chemical Composition.- 8.1 Relative Mass Abundances.- 8.2 Variation of Composition with Time.- 8.2.1 Radiative Regions.- 8.2.2 Diffusion.- 8.2.3 Convective Regions.- II The Overall Problem.- 9. The Differential Equations of Stellar Evolution.- 9.1 The Full Set of Equations.- 9.2 Time-scales and Simplifications.- 10. Boundary Conditions.- 10.1 Central Conditions.- 10.2 Surface Conditions.- 10.3 Influence of the Surface Conditions and Properties of Envelope Solutions.- 10.3.1 Radiative Envelopes.- 10.3.2 Convective Envelopes.- 10.3.3 Summary.- 10.3.4 The T-r Stratification.- 11. Numerical Procedure.- 11.1 The Shooting Method.- 11.2 The Henyey Method.- 11.3 Treatment of the First- and Second-Order Time Derivatives.- 12. Existence and Uniqueness of Solutions.- 12.1 Notation and Outline of the Procedure.- 12.2 Models in Complete Equilibrium.- 12.2.1 Fitting Conditions in the Pc - Tc Plane.- 12.2.2 Local Uniqueness.- 12.2.3 Variation of Parameters.- 12.3 Hydrostatic Models without Thermal Equilibrium.- 12.3.1 Degrees of Freedom and Fitting Conditions.- 12.3.2 Local Uniqueness.- 12.3.3 Variation of Parameters.- 12.4 Connection with Stability Problems.- 12.5 Non-local Properties of Equilibrium Models.- III Properties of Stellar Matter.- 13. The Ideal Gas with Radiation.- 13.1 Mean Molecular Weight and Radiation Pressure.- 13.2 Thermodynamic Quantities.- 14. Ionization.- 14.1 The Boltzmann and Saha Formulae.- 14.2 Ionization of Hydrogen.- 14.3 Thermodynamical Quantities for a Pure Hydrogen Gas.- 14.4 Hydrogen-Helium Mixtures.- 14.5 The General Case.- 14.6 Limitation of the Saha Formula.- 15. The Degenerate Electron Gas.- 15.1 Consequences of the Pauli Principle.- 15.2 The Completely Degenerate Electron Gas.- 15.3 Limiting Cases.- 15.4 Partial Degeneracy of the Electron Gas.- 16. The Equation of State of Stellar Matter.- 16.1 The Ion Gas.- 16.2 The Equation of State.- 16.3 Thermodynamic Quantities.- 16.4 Crystallization.- 16.5 Neutronization.- 17. Opacity.- 17.1 Electron Scattering.- 17.2 Absorption Due to Free-Free Transitions.- 17.3 Bound-Free Transitions.- 17.4 Bound-Bound Transitions.- 17.5 The Negative Hydrogen Ion.- 17.6 Conduction.- 17.7 Opacity Tables.- 18. Nuclear Energy Production.- 18.1 Basic Considerations.- 18.2 Nuclear Cross-sections.- 18.3 Thermonuclear Reaction Rates.- 18.4 Electron Shielding.- 18.5 The Major Nuclear Burnings.- 18.5.1 Hydrogen Burning.- 18.5.2 Helium Burning.- 18.5.3 Carbon Burning etc..- 18.6 Neutrinos.- IV Simple Stellar Models.- 19. Polytropic Gaseous Spheres.- 19.1 Polytropic Relations.- 19.2 Polytropic Stellar Models.- 19.3 Properties of the Solutions.- 19.4 Application to Stars.- 19.5 Radiation Pressure and the Polytrope n = 3.- 19.6 Polytropic Stellar Models with Fixed K.- 19.7 Chandrasekhar's Limiting Mass.- 19.8 Isothermal Spheres of an Ideal Gas.- 19.9 Gravitational and Total Energy for Polytropes.- 19.10 Supermassive Stars.- 19.11 A Collapsing Polytrope.- 20. Homology Relations.- 20.1 Definitions and Basic Relations.- 20.2 Applications to Simple Material Functions.- 20.2.1 The Case d ? = 0.- 20.2.2 The Case ? = ? = ? = 1, a = b = 0.- 20.2.3 The Role of the Equation of State.- 20.3 Homologous Contraction.- 21. Simple Models in the U-V Plane.- 21.1 The U-V Plane.- 21.2 Radiative Envelope Solutions.- 21.3 Fitting of a Convective Core.- 21.4 Fitting of an Isothermal Core.- 22. The Main Sequence.- 22.1 Surface Values.- 22.2 Interior Solutions.- 22.3 Convective Regions.- 22.4 Extreme Values of M.- 23. Other Main Sequences.- 23.1 The Helium Main Sequence.- 23.2 The Carbon Main Sequence.- 23.3 Main Sequences as Linear Series of Stellar Models.- 23.4 Generalized Main Sequences.- 24. The Hayashi Line.- 24.1 Luminosity of Fully Convective Models.- 24.2 A Simple Description of the Hayashi Line.- 24.3 The Neighbourhood of the Hayashi Line and the Forbidden Region.- 24.4 Numerical Results.- 24.5 Limitations for Fully Convective Models.- 25. Stability Considerations.- 25.1 General Remarks.- 25.2 Stability of the Piston Model.- 25.2.1 Dynamical Stability.- 25.2.2 Inclusion of Non-adiabatic Effects.- 25.3 Stellar Stability.- 25.3.1 Perturbation Equations.- 25.3.2 Dynamical Stability.- 25.3.3 Non-adiabatic Effects.- 25.3.4 The Gravothermal Specific Heat.- 25.3.5 Secular Stability Behaviour of Nuclear Burning.- V Early Stellar Evolution.- 26. The Onset of Star Formation.- 26.1 The Jeans Criterion.- 26.1.1 An Infinite Homogeneous Medium.- 26.1.2 A Plane Parallel Layer in Hydrostatic Equilibrium.- 26.2 Instability in the Spherical Case.- 26.3 Fragmentation.- 27. The Formation of Protostars.- 27.1 Free-Fall Collapse of a Homogeneous Sphere.- 27.2 Collapse onto a Condensed Object.- 27.3 A Collapse Calculation.- 27.4 The Optically Thin Phase and the Formation of a Hydrostatic Core.- 27.5 Core Collapse.- 27.6 Evolution in the Hertzsprung-Russell Diagram.- 28. Pre-Main-Sequence Contraction.- 28.1 Homologous Contraction of a Gaseous Sphere.- 28.2 Approach to the Zero-Age Main Sequence.- 29. From the Initial to the Present Sun.- 29.1 Choosing the Initial Model.- 29.2 Solar Neutrinos.- 30. Chemical Evolution on the Main Sequence.- 30.1 Change in the Hydrogen Content.- 30.2 Evolution in the Hertzsprung-Russell Diagram.- 30.3 Time-scales for Central Hydrogen Burning.- 30.4 Complications Connected with Convection.- 30.4.1 Convective Overshooting.- 30.4.2 Semiconvection.- 30.5 The Schonberg-Chandrasekhar Limit.- 30.5.1 A Simple Approach - The Virial Theorem and Homology.- 30.5.2 Integrations for Core and Envelope.- 30.5.3 Complete Solutions for Stars with Isothermal Cores.- VI Post-Main-Sequence Evolution.- 31. Evolution Through Helium Burning - Massive Stars.- 31.1 Crossing the Hertzsprung Gap.- 31.2 Central Helium Burning.- 31.3 The Cepheid Phase.- 31.4 To Loop or Not to Loop.- 31.5 After Central Helium Burning.- 32. Evolution Through Helium Burning - Low-Mass Stars.- 32.1 Post-Main-Sequence Evolution.- 32.2 Shell-Source Homology.- 32.3 Evolution to the Helium Flash.- 32.4 The Helium Flash.- 32.5 Numerical Results for the Helium Flash.- 32.6 Evolution after the Helium Flash.- 32.7 Evolution from the Zero-Age Horizontal Branch.- 32.8 Equilibrium Models with Helium Cores - Continued.- 33. Later Phases.- 33.1 Nuclear Cycles.- 33.2 Shell Sources and Their Stability.- 33.3 Thermal Pulses of a Shell Source.- 33.4 Evolution of the Central Region.- 33.5 The Core-Mass-Luminosity Relation for Large Core Masses.- 34. Final Explosions and Collapse.- 34.1 The Evolution of the C-O Core.- 34.2 Carbon Burning in Degenerate Cores.- 34.2.1 The Carbon Flash.- 34.2.2 Nuclear Statistical Equilibrium.- 34.2.3 Hydrostatic and Convective Adjustment.- 34.2.4 Combustion Fronts.- 34.2.5 Numerical Solutions.- 34.2.6 Carbon Burning in Accreting White Dwarfs.- 34.3 Collapse of Cores of Massive Stars.- 34.3.1 Simple Collapse Solutions.- 34.3.2 The Reflection of the Infall.- 34.3.3 Effects of Neutrinos.- 34.3.4 Numerical Results.- 34.3.5 Pair-Creation Instability.- VII Compact Objects.- 35. White Dwarfs.- 35.1 Chandrasekhar's Theory.- 35.2 The Corrected Mechanical Structure.- 35.3 Thermal Properties and Evolution of White Dwarfs.- 36. Neutron Stars.- 36.1 Cold Matter Beyond Neutron Drip.- 36.2 Models of Neutron Stars.- 37. Black Holes.- VIII Pulsating Stars.- 38. Adiabatic Spherical Pulsations.- 38.1 The Eigenvalue Problem.- 38.2 The Homogeneous Sphere.- 38.3 Pulsating Polytropes.- 39. Non-adiabatic Spherical Pulsations.- 39.1 Vibrational Instability of the Piston Model.- 39.2 The Quasi-adiabatic Approximation.- 39.3 The Energy Integral.- 39.3.1 The ? Mechanism.- 39.3.2 The ? Mechanism.- 39.4 Stars Driven by the ? Mechanism - The Instability Strip.- 39.5 Stars Driven by the ? Mechanism.- 40. Non-radial Stellar Oscillations.- 40.1 Perturbations of the Equilibrium Model.- 40.2 Normal Modes and Dimensionless Variables.- 40.3 The Eigenspectra.- 40.4 Stars Showing Non-radial Oscillations.- IX Stellar Rotation.- 41. The Mechanics of Rotating Stellar Models.- 41.1 Uniformly Rotating Liquid Bodies.- 41.2 The Roche Model.- 41.3 Slowly Rotating Polytropes.- 42. The Thermodynamics of Rotating Stellar Models.- 42.1 Conservative Rotation 4.- 42.2 Von Zeipel's Theorem.- 42.3 Meridional Circulation.- 42.4 The Non-conservative Case.- 42.5 The Eddington-Sweet Time-scale.- 42.6 Meridional Circulation in Inhomogeneous Stars.- 43. The Angular-Velocity Distribution in Stars.- 43.1 Viscosity.- 43.2 Dynamical Stability.- 43.3 Secular Stability.- References.