Smeyers Linear Isentropic Oscillations of Stars

Theoretical Foundations
1. Auflage 2011
ISBN: 978-3-642-13030-4
Verlag: Springer
Format: PDF
Kopierschutz: Wasserzeichen (»Systemvoraussetzungen)

E-Book, Englisch, 473 Seiten

This book surveys the theory of free, linear, isentropic oscillations in spherically symmetric, gaseous equilibrium stars, from basic concepts to asymptotic representations of normal modes and with slow period changes in rapidly evolving pulsating stars.

Autoren/Hrsg.

Smeyers, Paul

Inhaltsverzeichnis

1;Linear Isentropic Oscillations of Stars;1
1.1;Preface
;5
1.2;Contents
;7
1.3;Introduction;15
1.4;Chapter 1: Basic Concepts;23
1.4.1;1.1 The Lagrangian Displacement of a Mass Element;23
1.4.2;1.2 Lagrangian and Eulerian Perturbations of PhysicalQuantities;26
1.4.2.1;1.2.1 Definitions;26
1.4.2.2;1.2.2 Additional Relations;29
1.4.3;1.3 The Eulerian Perturbation of a Velocity Component;31
1.4.4;1.4 Perturbations of Mass Density, Gravitational Potential, Pressure, and Temperature;32
1.4.4.1;1.4.1 Perturbations of Mass Density;32
1.4.4.2;1.4.2 Perturbations of Gravitational Potential;33
1.4.4.3;1.4.3 Perturbations of Pressure;36
1.4.4.4;1.4.4 Perturbations of Temperature;37
1.5;Chapter 2: The Equations Governing Linear Perturbations in a Quasi-Static Star;38
1.5.1;2.1 System of Coordinates;38
1.5.2;2.2 Equation of Motion;39
1.5.3;2.3 Equilibrium State of a Quasi-Static Star;40
1.5.4;2.4 Eulerian Form of the Equations Governing Linear Perturbations;43
1.5.4.1;2.4.1 First Additional Equation;44
1.5.4.2;2.4.2 Second Additional Equation;45
1.5.4.3;2.4.3 Third Additional Equation;46
1.5.5;2.5 Lagrangian Form of the Equations GoverningLinear Perturbations;47
1.6;Chapter 3: Deviations from the Hydrostatic and Thermal Equilibrium in a Quasi-Static Star;49
1.6.1;3.1 Introduction;49
1.6.2;3.2 Resolution of the Force Acting upon a Moving Mass Element;49
1.6.3;3.3 The Dynamic Time-Scale of a Star;51
1.6.4;3.4 Energy Exchange Between Moving Mass Elements;53
1.6.5;3.5 Criterion for Local Stability with Respect to Convection;56
1.6.6;3.6 Deviations from the Thermal Equilibrium;62
1.7;Chapter 4: Eigenvalue Problem of the Linear, Isentropic Normal Modes in a Quasi-Static Star;63
1.7.1;4.1 Time-Dependent Equations and Boundary Conditions Governing Linear, Isentropic Oscillations;63
1.7.2;4.2 Vectorial Wave Equation with Tensorial Operator U;64
1.7.3;4.3 Separation of Time;65
1.7.4;4.4 Inner Product of Linear, Isentropic Oscillations;67
1.7.5;4.5 Symmetry of the Tensorial Operator U;68
1.7.5.1;4.5.1 Proof of Kaniel and Kovetz;68
1.7.5.2;4.5.2 Proof of Lynden-Bell and Ostriker;70
1.7.6;4.6 Orthogonality of the Linear, Isentropic Normal Modes;73
1.7.7;4.7 Global Translations of a Quasi-Static Star as Normal Linear, Isentropic Modes;74
1.7.8;4.8 Immovability of the Star's Mass Centre;76
1.8;Chapter 5: Spheroidal and Toroidal Normal Modes;78
1.8.1;5.1 Introduction;78
1.8.2;5.2 Radial Component of the Vorticity Equation;78
1.8.3;5.3 Convenient Form of the Governing Equations;80
1.8.4;5.4 Helmholtz's Resolution Theorem for Vector Fields;81
1.8.5;5.5 Resolution of the Vector Field ;84
1.8.6;5.6 Resolution of the Displacement Field into a Radial and a Horizontal Field;88
1.8.7;5.7 Expansion of the Displacement Field in Terms of Spherical Harmonics;91
1.8.8;5.8 Spheroidal Normal Modes;92
1.8.8.1;5.8.1 Definition;92
1.8.8.2;5.8.2 Eigenvalue Problem of the Spheroidal Normal Modes;93
1.8.8.3;5.8.3 Divergence-Free Spheroidal Normal Modes;97
1.8.9;5.9 Toroidal Normal Modes;100
1.8.10;5.10 Inner Products of Normal Modes;105
1.8.10.1;5.10.1 Inner Product of Two Spheroidal Modes;105
1.8.10.2;5.10.2 Inner Product of Two Toroidal Modes;106
1.8.10.3;5.10.3 Inner Product of a Spheroidal and a Toroidal Mode;106
1.9;Chapter 6: Determination of Spheroidal Normal Modes: Mathematical Aspects;108
1.9.1;6.1 Introduction;108
1.9.2;6.2 Convenient Fourth-Order Systems of Differential Equations in the Radial Coordinate;108
1.9.2.1;6.2.1 Pekeris' System of Equations;108
1.9.2.2;6.2.2 Ledoux' System of Equations;110
1.9.2.3;6.2.3 Dziembowski's System of Equations;113
1.9.3;6.3 Determination of Radial Normal Modes;114
1.9.3.1;6.3.1 Admissible Solutions from the Boundary Point r=0;115
1.9.3.2;6.3.2 Admissible Solutions from the Boundary Point r=R;117
1.9.3.3;6.3.3 Eigenvalue Equation;120
1.9.4;6.4 Determination of Non-Radial Spheroidal Normal Modes;120
1.9.4.1;6.4.1 Admissible Solutions from the Boundary Point r=0;121
1.9.4.2;6.4.2 Admissible Solutions from the Boundary Point r=R;125
1.9.4.3;6.4.3 Eigenvalue Equation;129
1.10;Chapter 7: The Eulerian Perturbation of the Gravitational Potential;131
1.10.1;7.1 As Solution of Poisson's Perturbed Differential Equation;131
1.10.2;7.2 Derivation from the General Integral Solution of Poisson's Equation;132
1.10.3;7.3 The Cowling Approximation;140
1.11;Chapter 8: The Variational Principle of Hamilton;142
1.11.1;8.1 Introduction;142
1.11.2;8.2 First- and Second-Order Energy Variations;143
1.11.3;8.3 Equality of the Mean Kinetic and the Mean Potential Energy of Oscillation over a Period;147
1.11.4;8.4 First- and Second-Order Variational Principles;149
1.11.4.1;8.4.1 First-Order Variational Principle;149
1.11.4.2;8.4.2 Second-Order Variational Principle;150
1.11.4.3;8.4.3 Takata's Reformulation of the Second-Order Variational Principle;154
1.11.4.4;8.4.4 The Lagrangian Density of Tolstoy;155
1.11.5;8.5 Approximation Method of Rayleigh–Ritz;157
1.11.5.1;8.5.1 Convenient Form of the Lagrangian;157
1.11.5.2;8.5.2 The Approximation Method;159
1.11.6;8.6 Weight Functions for Spheroidal Normal Modes;161
1.11.7;8.7 Energy Density and Energy Flux;162
1.11.8;8.8 The Equations that Govern Linear, Isentropic Oscillations, as Canonical Equations;165
1.12;Chapter 9: Radial Propagation of Waves;168
1.12.1;9.1 Introduction;168
1.12.2;9.2 Local Dispersion Equations;168
1.12.2.1;9.2.1 General Local Dispersion Equation;168
1.12.2.2;9.2.2 Local Dispersion Equation Applyingto Surface Layers;170
1.12.3;9.3 Local Radial Propagation of Waves;172
1.12.3.1;9.3.1 Radial Propagation of Wavesin an Incompressible Layer Subject to Gravity;172
1.12.3.1.1;9.3.1.1 In Absence of Any Density Stratification;172
1.12.3.1.2;9.3.1.2 In Presence of a Density Stratification;173
1.12.3.2;9.3.2 Radial Propagation of Wavesin a Compressible Layer not Subject to Gravity;175
1.12.3.2.1;9.3.2.1 In Absence of Any Density Stratification;175
1.12.3.2.2;9.3.2.2 In Presence of a Density Stratification;176
1.12.3.3;9.3.3 Radial Propagation of Wavesin a Compressible Layer with a Density Stratification that is Subject to Gravity;177
1.12.4;9.4 Global Representation of the Radial Propagation of Waves;180
1.13;Chapter 10: Classification of the Spheroidal Normal Modes;185
1.13.1;10.1 Origin from Propagating Waves;185
1.13.2;10.2 The Radial Modes;186
1.13.3;10.3 Cowling's Classification of the Non-Radial Spheroidal Modes;189
1.13.3.1;10.3.1 The Non-Radial p- and g-Modes;189
1.13.3.2;10.3.2 The Non-Radial f-Modes;193
1.13.4;10.4 Validity of Cowling's Classification;194
1.13.4.1;10.4.1 The Equilibrium Sphere of Uniform Mass Density;194
1.13.4.1.1;10.4.1.1 The Equilibrium Model;194
1.13.4.1.2;10.4.1.2 The Oscillations of the Incompressible Equilibrium Sphere of Uniform Mass Density;195
1.13.4.1.3;10.4.1.3 The Oscillations of the Compressible Equilibrium Sphere of Uniform Mass Density;196
1.13.4.2;10.4.2 Polytropic and Physical Models;205
1.13.5;10.5 Beyer's Study on the Nature of the Oscillation Spectra;209
1.13.5.1;10.5.1 System of Equations;209
1.13.5.2;10.5.2 The Radial Modes;210
1.13.5.3;10.5.3 The Non-Radial Modes;211
1.14;Chapter 11: Classification of the Spheroidal Normal Modes (continued);213
1.14.1;11.1 Additional Nodes for Models with a Larger Central Mass Condensation;213
1.14.2;11.2 Mode Bumping in Models with a Larger Central Mass Condensation;217
1.14.3;11.3 Theory of the Avoided Crossings of Modes;218
1.14.4;11.4 Implications of Avoided Crossings of Non-Radial Modes for Mode Identifications;226
1.14.5;11.5 Strange Radial Modes;229
1.14.6;11.6 The First-Degree f-Modes;234
1.15;Chapter 12: Completeness of the Linear, Isentropic Normal Modes;236
1.15.1;12.1 Status Questionis;236
1.15.2;12.2 Approach of Eisenfeld;238
1.15.2.1;12.2.1 Eisenfeld's Operator T;238
1.15.2.2;12.2.2 Symmetry of the Operator T;240
1.15.2.3;12.2.3 Existence of a Non-Empty Resolvent Set for the Operator T?;242
1.15.2.3.1;12.2.3.1 For r 0;246
1.15.2.3.2;12.2.3.2 For r R;249
1.15.2.4;12.2.4 Eisenfeld's Conclusion;252
1.15.3;12.3 Lower Bound of the Tensorial Operator U;252
1.15.3.1;12.3.1 Expression for the Lower Bound;252
1.15.3.2;12.3.2 Hunter's Derivation;254
1.15.3.3;12.3.3 Alternative Derivation;256
1.15.4;12.4 Spectral and Expansion Theorems;261
1.16;Chapter 13: N2(r) Nowhere Negative as Condition for Non-Radial Modes with Real Eigenfrequencies;264
1.16.1;13.1 Introduction;264
1.16.2;13.2 N2(r) Nowhere Negative as Sufficient Condition;265
1.16.2.1;13.2.1 Chandrasekhar's Equation for the Eigenvalue of a Spheroidal Normal Mode;265
1.16.2.2;13.2.2 The Sufficient Condition;267
1.16.3;13.3 N2(r) Nowhere Negative as Necessary Condition;271
1.17;Chapter 14 Asymptotic Representation of Low-Degree, Higher-Order p-Modes;274
1.17.1;14.1 State of the Art;274
1.17.2;14.2 Appropriate Equations;276
1.17.3;14.3 Two-Variable Expansions at Larger Distances from the Boundary Points;277
1.17.4;14.4 Boundary-Layer Expansions Near r = 0;284
1.17.5;14.5 Matching of the Boundary-Layer Expansions Valid Near r = 0;288
1.17.6;14.6 Boundary-Layer Expansions Near r = R;291
1.17.7;14.7 Matching of the Boundary-Layer Expansions Valid Near r = R;294
1.17.8;14.8 Eigenfrequency Equation;297
1.17.9;14.9 Condition on the Eulerian Perturbation of the Gravitational Potential at r = R;298
1.17.10;14.10 Uniformly Valid Asymptotic Expansions;299
1.17.11;14.11 Identification of the Radial Order of a p-Mode with a Given Eigenfrequency;301
1.17.11.1;14.11.1 Radial Modes;301
1.17.11.2;14.11.2 Non-Radial p-Modes;302
1.17.12;14.12 Concluding Remarks;304
1.18;Chapter 15: Asymptotic Representation of Low-Degree and Intermediate-Degree p-Modes;307
1.18.1;15.1 Frequency Separations Dn, for Solar 5 Min-Oscillations;307
1.18.2;15.2 Appropriate Equation;311
1.18.3;15.3 Two-Variable Expansion at Larger Distancesfrom the Turning Point and the Boundary Point r=R;313
1.18.4;15.4 Boundary-Layer Expansion on the Outer Side of the Turning Point;313
1.18.5;15.5 Two-Variable Expansion at Larger Distances from the Boundary Point r=0 and the Turning Point;316
1.18.6;15.6 Boundary-Layer Expansion on the Inner Side of the Turning Point;317
1.18.7;15.7 Boundary-Layer Expansion Near the Boundary Point r=R;319
1.18.8;15.8 Eigenfrequency Equation;321
1.18.9;15.9 Uniformly Valid Asymptotic Representation of the Divergence of the Lagrangian Displacement;322
1.18.10;15.10 Application to the Compressible Equilibrium Sphere of Uniform Mass Density;323
1.18.11;15.11 Eigenfrequency Equation with the Methodof the Phase Functions;328
1.19;Chapter 16: Asymptotic Representation of Low-Degree, Higher-Order g+-Modes in Stars Containing a Convective Core;333
1.19.1;16.1 Introduction;333
1.19.2;16.2 Appropriate Equations;333
1.19.3;16.3 Stars Consisting of a Convective Core and a Radiative Envelope;334
1.19.3.1;16.3.1 Two-Variable Solutions in the Radiative Envelope at Larger Distances from its Boundaries;334
1.19.3.2;16.3.2 Boundary-Layer Solutions on the Outer Side of the Boundary Between the Convective Core and the Radiative Envelope;339
1.19.3.3;16.3.3 Junction with the Solutions Validin the Convective Core;343
1.19.3.4;16.3.4 Boundary-Layer Solutions Nearthe Boundary Point r=R;346
1.19.3.5;16.3.5 Eigenfrequency Equation;350
1.19.3.6;16.3.6 The Condition on the Eulerian Perturbation of the Gravitational Potential at r=R;352
1.19.3.7;16.3.7 Uniformly Valid Asymptotic Solutions;353
1.19.3.8;16.3.8 Identification of the Radial Orderof a g+-Mode with a given Eigenfrequency;355
1.19.4;16.4 Stars Consisting of a Convective Core, an Intermediate Radiative Zone, and a Convective Envelope;358
1.19.4.1;16.4.1 Boundary-Layer Solutions on the Inner Side of the Boundary Betweenthe Intermediate Radiative Zoneand the Convective Envelope;358
1.19.4.2;16.4.2 Asymptotic Solutions in the Convective Envelope;360
1.19.4.2.1;16.4.2.1 Two-Variable Solutions in the Convective Envelope at Larger Distances from its Boundaries;361
1.19.4.2.2;16.4.2.2 Boundary-Layer Solutions on the Outer Side of the Boundary Between the Intermediate Radiative Zone and the Convective Envelope;362
1.19.4.2.3;16.4.2.3 Boundary-Layer Solutions Near the Boundary Point r=R;364
1.19.4.2.4;16.4.2.4 Main Result of the Asymptotic Solutionsin the Convective Envelope;367
1.19.4.3;16.4.3 Eigenfrequency Equation;368
1.19.4.4;16.4.4 The Condition on the Eulerian Perturbation of the Gravitational Potential at r=R;369
1.19.4.5;16.4.5 Uniformly Valid Asymptotic Solutions;370
1.19.4.6;16.4.6 Identification of the Radial Orderof a g+-Mode with a Given Eigenfrequency;372
1.20;Chapter 17: Asymptotic Representation of Low-Degree, Higher-Order g+-Modes in Stars Consisting of a Radiative Core and a Convective Envelope;375
1.20.1;17.1 Introduction;375
1.20.2;17.2 Asymptotic Solutions in the Radiative Core;375
1.20.2.1;17.2.1 Two-Variable Solutions at Larger Distances from the Boundaries of the Radiative Core;375
1.20.2.2;17.2.2 Boundary-Layer Solutions Nearthe Boundary Point r=0;376
1.20.2.3;17.2.3 Boundary-Layer Solutions on the Inner Side of the Boundary Between the Radiative Core and the Convective Envelope;380
1.20.3;17.3 Eigenfrequency Equation;382
1.20.4;17.4 The Condition on the Eulerian Perturbation of the Gravitational Potential at r = R;383
1.20.5;17.5 Uniformly Valid Asymptotic Solutions;384
1.20.6;17.6 Identification of the Radial Order of a g+-Mode;384
1.20.7;17.7 Global Conclusion from the Asymptotic Theory of Low-Degree, Higher-Order p- and g+-Modes;385
1.21;Chapter 18: High-Degree, Low-Order Modes;388
1.21.1;18.1 Introduction;388
1.21.2;18.2 High-Degree, Low-Order p- and g+-Modes Trapped Near the Surface;388
1.21.3;18.3 High-Degree, Low-Order g+-Modes Trapped Near a Maximum of the Boundary of the G-Region;394
1.22;Chapter 19: Period Changes in a Rapidly Evolving Pulsating Star;398
1.22.1;19.1 Introduction;398
1.22.2;19.2 Appropriate Equations;398
1.22.3;19.3 Two-Time-Variable Expansion Procedure;401
1.22.4;19.4 Rate of Change of an Isentropic Pulsation Period;405
1.22.5;19.5 Use of the Equality of the Mean Kinetic and the MeanPotential Energy of Pulsation;407
1.22.6;19.6 Explicit Expression for the Rate of Changeof a Period in a Radially Pulsating Star;408
1.22.6.1;19.6.1 The First Part;409
1.22.6.2;19.6.2 The Second Part;409
1.22.6.3;19.6.3 The Third Part;412
1.22.6.4;19.6.4 Global Result;412
1.22.7;19.7 Rate of Change of a Period in a Radially Pulsating Star Subject to a Homologous Contraction or Expansion;413
1.23;Appendix A:Green's Fundamental Formula of Potential Theory;416
1.24;Appendix B: The Thermodynamic Isentropic Coefficients;419
1.24.1;B.1 Reversible Thermodynamic Processes;419
1.24.2;B.2 Thermodynamic Relations;419
1.24.3;B.3 Definitions of the Isentropic Coefficients;420
1.24.4;B.4 Equations for the Isentropic Coefficients;422
1.24.4.1;B.4.1 Equation for 1;422
1.24.4.2;B.4.2 Equation for 2;423
1.24.4.3;B.4.3 Equation for 3;424
1.25;Appendix C: Lagrange's Equations of Motion;425
1.26;Appendix D: Spherical Harmonics;430
1.27;Appendix E Singular Perturbation Problems of the Boundary-Layer Type;432
1.28;Appendix F: Boundary Condition Relative to the Pressure on a Star's Surface;436
1.29;Appendix G: The Curl of a Vector Field;440
1.30;Appendix H: Eigenvalue Problem of the Vibrating String;444
1.31;Appendix I: The Euler–Lagrange Equations of Hamilton's Variational Principle for a Perturbed Star;447
1.32;Appendix J: Acoustic Waves;449
1.32.1;J.1 Acoustic Waves in a Uniform Gas;449
1.32.2;J.2 Vertical Propagation of Acoustic Waves in a Plane Isothermal Layer;450
1.33;List of Symbols;453
1.34;References;457
1.35;Index;468