Linear Isentropic Oscillations of Stars (eBook)

Linear Isentropic Oscillations of Stars 1
Preface 
5 
Contents 
7 
Introduction 15
Chapter 1: Basic Concepts 23
1.1 The Lagrangian Displacement of a Mass Element 23
1.2 Lagrangian and Eulerian Perturbations of PhysicalQuantities 26
1.2.1 Definitions 26
1.2.2 Additional Relations 29
1.3 The Eulerian Perturbation of a Velocity Component 31
1.4 Perturbations of Mass Density, Gravitational Potential, Pressure, and Temperature 32
1.4.1 Perturbations of Mass Density 32
1.4.2 Perturbations of Gravitational Potential 33
1.4.3 Perturbations of Pressure 36
1.4.4 Perturbations of Temperature 37
Chapter 2: The Equations Governing Linear Perturbations in a Quasi-Static Star 38
2.1 System of Coordinates 38
2.2 Equation of Motion 39
2.3 Equilibrium State of a Quasi-Static Star 40
2.4 Eulerian Form of the Equations Governing Linear Perturbations 43
2.4.1 First Additional Equation 44
2.4.2 Second Additional Equation 45
2.4.3 Third Additional Equation 46
2.5 Lagrangian Form of the Equations GoverningLinear Perturbations 47
Chapter 3: Deviations from the Hydrostatic and Thermal Equilibrium in a Quasi-Static Star 49
3.1 Introduction 49
3.2 Resolution of the Force Acting upon a Moving Mass Element 49
3.3 The Dynamic Time-Scale of a Star 51
3.4 Energy Exchange Between Moving Mass Elements 53
3.5 Criterion for Local Stability with Respect to Convection 56
3.6 Deviations from the Thermal Equilibrium 62
Chapter 4: Eigenvalue Problem of the Linear, Isentropic Normal Modes in a Quasi-Static Star 63
4.1 Time-Dependent Equations and Boundary Conditions Governing Linear, Isentropic Oscillations 63
4.2 Vectorial Wave Equation with Tensorial Operator U 64
4.3 Separation of Time 65
4.4 Inner Product of Linear, Isentropic Oscillations 67
4.5 Symmetry of the Tensorial Operator U 68
4.5.1 Proof of Kaniel and Kovetz 68
4.5.2 Proof of Lynden-Bell and Ostriker 70
4.6 Orthogonality of the Linear, Isentropic Normal Modes 73
4.7 Global Translations of a Quasi-Static Star as Normal Linear, Isentropic Modes 74
4.8 Immovability of the Star's Mass Centre 76
Chapter 5: Spheroidal and Toroidal Normal Modes 78
5.1 Introduction 78
5.2 Radial Component of the Vorticity Equation 78
5.3 Convenient Form of the Governing Equations 80
5.4 Helmholtz's Resolution Theorem for Vector Fields 81
5.5 Resolution of the Vector Field 84
5.6 Resolution of the Displacement Field into a Radial and a Horizontal Field 88
5.7 Expansion of the Displacement Field in Terms of Spherical Harmonics 91
5.8 Spheroidal Normal Modes 92
5.8.1 Definition 92
5.8.2 Eigenvalue Problem of the Spheroidal Normal Modes 93
5.8.3 Divergence-Free Spheroidal Normal Modes 97
5.9 Toroidal Normal Modes 100
5.10 Inner Products of Normal Modes 105
5.10.1 Inner Product of Two Spheroidal Modes 105
5.10.2 Inner Product of Two Toroidal Modes 106
5.10.3 Inner Product of a Spheroidal and a Toroidal Mode 106
Chapter 6: Determination of Spheroidal Normal Modes: Mathematical Aspects 108
6.1 Introduction 108
6.2 Convenient Fourth-Order Systems of Differential Equations in the Radial Coordinate 108
6.2.1 Pekeris' System of Equations 108
6.2.2 Ledoux' System of Equations 110
6.2.3 Dziembowski's System of Equations 113
6.3 Determination of Radial Normal Modes 114
6.3.1 Admissible Solutions from the Boundary Point r=0 115
6.3.2 Admissible Solutions from the Boundary Point r=R 117
6.3.3 Eigenvalue Equation 120
6.4 Determination of Non-Radial Spheroidal Normal Modes 120
6.4.1 Admissible Solutions from the Boundary Point r=0 121
6.4.2 Admissible Solutions from the Boundary Point r=R 125
6.4.3 Eigenvalue Equation 129
Chapter 7: The Eulerian Perturbation of the Gravitational Potential 131
7.1 As Solution of Poisson's Perturbed Differential Equation 131
7.2 Derivation from the General Integral Solution of Poisson's Equation 132
7.3 The Cowling Approximation 140
Chapter 8: The Variational Principle of Hamilton 142
8.1 Introduction 142
8.2 First- and Second-Order Energy Variations 143
8.3 Equality of the Mean Kinetic and the Mean Potential Energy of Oscillation over a Period 147
8.4 First- and Second-Order Variational Principles 149
8.4.1 First-Order Variational Principle 149
8.4.2 Second-Order Variational Principle 150
8.4.3 Takata's Reformulation of the Second-Order Variational Principle 154
8.4.4 The Lagrangian Density of Tolstoy 155
8.5 Approximation Method of Rayleigh–Ritz 157
8.5.1 Convenient Form of the Lagrangian 157
8.5.2 The Approximation Method 159
8.6 Weight Functions for Spheroidal Normal Modes 161
8.7 Energy Density and Energy Flux 162
8.8 The Equations that Govern Linear, Isentropic Oscillations, as Canonical Equations 165
Chapter 9: Radial Propagation of Waves 168
9.1 Introduction 168
9.2 Local Dispersion Equations 168
9.2.1 General Local Dispersion Equation 168
9.2.2 Local Dispersion Equation Applyingto Surface Layers 170
9.3 Local Radial Propagation of Waves 172
9.3.1 Radial Propagation of Wavesin an Incompressible Layer Subject to Gravity 172
9.3.1.1 In Absence of Any Density Stratification 172
9.3.1.2 In Presence of a Density Stratification 173
9.3.2 Radial Propagation of Wavesin a Compressible Layer not Subject to Gravity 175
9.3.2.1 In Absence of Any Density Stratification 175
9.3.2.2 In Presence of a Density Stratification 176
9.3.3 Radial Propagation of Wavesin a Compressible Layer with a Density Stratification that is Subject to Gravity 177
9.4 Global Representation of the Radial Propagation of Waves 180
Chapter 10: Classification of the Spheroidal Normal Modes 185
10.1 Origin from Propagating Waves 185
10.2 The Radial Modes 186
10.3 Cowling's Classification of the Non-Radial Spheroidal Modes 189
10.3.1 The Non-Radial p- and g-Modes 189
10.3.2 The Non-Radial f-Modes 193
10.4 Validity of Cowling's Classification 194
10.4.1 The Equilibrium Sphere of Uniform Mass Density 194
10.4.1.1 The Equilibrium Model 194
10.4.1.2 The Oscillations of the Incompressible Equilibrium Sphere of Uniform Mass Density 195
10.4.1.3 The Oscillations of the Compressible Equilibrium Sphere of Uniform Mass Density 196
10.4.2 Polytropic and Physical Models 205
10.5 Beyer's Study on the Nature of the Oscillation Spectra 209
10.5.1 System of Equations 209
10.5.2 The Radial Modes 210
10.5.3 The Non-Radial Modes 211
Chapter 11: Classification of the Spheroidal Normal Modes (continued) 213
11.1 Additional Nodes for Models with a Larger Central Mass Condensation 213
11.2 Mode Bumping in Models with a Larger Central Mass Condensation 217
11.3 Theory of the Avoided Crossings of Modes 218
11.4 Implications of Avoided Crossings of Non-Radial Modes for Mode Identifications 226
11.5 Strange Radial Modes 229
11.6 The First-Degree f-Modes 234
Chapter 12: Completeness of the Linear, Isentropic Normal Modes 236
12.1 Status Questionis 236
12.2 Approach of Eisenfeld 238
12.2.1 Eisenfeld's Operator T 238
12.2.2 Symmetry of the Operator T 240
12.2.3 Existence of a Non-Empty Resolvent Set for the Operator T? 242
12.2.3.1 For r 0 246
12.2.3.2 For r R 249
12.2.4 Eisenfeld's Conclusion 252
12.3 Lower Bound of the Tensorial Operator U 252
12.3.1 Expression for the Lower Bound 252
12.3.2 Hunter's Derivation 254
12.3.3 Alternative Derivation 256
12.4 Spectral and Expansion Theorems 261
Chapter 13: N2(r) Nowhere Negative as Condition for Non-Radial Modes with Real Eigenfrequencies 264
13.1 Introduction 264
13.2 N2(r) Nowhere Negative as Sufficient Condition 265
13.2.1 Chandrasekhar's Equation for the Eigenvalue of a Spheroidal Normal Mode 265
13.2.2 The Sufficient Condition 267
13.3 N2(r) Nowhere Negative as Necessary Condition 271
Chapter 14 Asymptotic Representation of Low-Degree, Higher-Order p-Modes 274
14.1 State of the Art 274
14.2 Appropriate Equations 276
14.3 Two-Variable Expansions at Larger Distances from the Boundary Points 277
14.4 Boundary-Layer Expansions Near r = 0 284
14.5 Matching of the Boundary-Layer Expansions Valid Near r = 0 288
14.6 Boundary-Layer Expansions Near r = R 291
14.7 Matching of the Boundary-Layer Expansions Valid Near r = R 294
14.8 Eigenfrequency Equation 297
14.9 Condition on the Eulerian Perturbation of the Gravitational Potential at r = R 298
14.10 Uniformly Valid Asymptotic Expansions 299
14.11 Identification of the Radial Order of a p-Mode with a Given Eigenfrequency 301
14.11.1 Radial Modes 301
14.11.2 Non-Radial p-Modes 302
14.12 Concluding Remarks 304
Chapter 15: Asymptotic Representation of Low-Degree and Intermediate-Degree p-Modes 307
15.1 Frequency Separations Dn, for Solar 5 Min-Oscillations 307
15.2 Appropriate Equation 311
15.3 Two-Variable Expansion at Larger Distancesfrom the Turning Point and the Boundary Point r=R 313
15.4 Boundary-Layer Expansion on the Outer Side of the Turning Point 313
15.5 Two-Variable Expansion at Larger Distances from the Boundary Point r=0 and the Turning Point 316
15.6 Boundary-Layer Expansion on the Inner Side of the Turning Point 317
15.7 Boundary-Layer Expansion Near the Boundary Point r=R 319
15.8 Eigenfrequency Equation 321
15.9 Uniformly Valid Asymptotic Representation of the Divergence of the Lagrangian Displacement 322
15.10 Application to the Compressible Equilibrium Sphere of Uniform Mass Density 323
15.11 Eigenfrequency Equation with the Methodof the Phase Functions 328
Chapter 16: Asymptotic Representation of Low-Degree, Higher-Order g+-Modes in Stars Containing a Convective Core 333
16.1 Introduction 333
16.2 Appropriate Equations 333
16.3 Stars Consisting of a Convective Core and a Radiative Envelope 334
16.3.1 Two-Variable Solutions in the Radiative Envelope at Larger Distances from its Boundaries 334
16.3.2 Boundary-Layer Solutions on the Outer Side of the Boundary Between the Convective Core and the Radiative Envelope 339
16.3.3 Junction with the Solutions Validin the Convective Core 343
16.3.4 Boundary-Layer Solutions Nearthe Boundary Point r=R 346
16.3.5 Eigenfrequency Equation 350
16.3.6 The Condition on the Eulerian Perturbation of the Gravitational Potential at r=R 352
16.3.7 Uniformly Valid Asymptotic Solutions 353
16.3.8 Identification of the Radial Orderof a g+-Mode with a given Eigenfrequency 355
16.4 Stars Consisting of a Convective Core, an Intermediate Radiative Zone, and a Convective Envelope 358
16.4.1 Boundary-Layer Solutions on the Inner Side of the Boundary Betweenthe Intermediate Radiative Zoneand the Convective Envelope 358
16.4.2 Asymptotic Solutions in the Convective Envelope 360
16.4.2.1 Two-Variable Solutions in the Convective Envelope at Larger Distances from its Boundaries 361
16.4.2.2 Boundary-Layer Solutions on the Outer Side of the Boundary Between the Intermediate Radiative Zone and the Convective Envelope 362
16.4.2.3 Boundary-Layer Solutions Near the Boundary Point r=R 364
16.4.2.4 Main Result of the Asymptotic Solutionsin the Convective Envelope 367
16.4.3 Eigenfrequency Equation 368
16.4.4 The Condition on the Eulerian Perturbation of the Gravitational Potential at r=R 369
16.4.5 Uniformly Valid Asymptotic Solutions 370
16.4.6 Identification of the Radial Orderof a g+-Mode with a Given Eigenfrequency 372
Chapter 17: Asymptotic Representation of Low-Degree, Higher-Order g+-Modes in Stars Consisting of a Radiative Core and a Convective Envelope 375
17.1 Introduction 375
17.2 Asymptotic Solutions in the Radiative Core 375
17.2.1 Two-Variable Solutions at Larger Distances from the Boundaries of the Radiative Core 375
17.2.2 Boundary-Layer Solutions Nearthe Boundary Point r=0 376
17.2.3 Boundary-Layer Solutions on the Inner Side of the Boundary Between the Radiative Core and the Convective Envelope 380
17.3 Eigenfrequency Equation 382
17.4 The Condition on the Eulerian Perturbation of the Gravitational Potential at r = R 383
17.5 Uniformly Valid Asymptotic Solutions 384
17.6 Identification of the Radial Order of a g+-Mode 384
17.7 Global Conclusion from the Asymptotic Theory of Low-Degree, Higher-Order p- and g+-Modes 385
Chapter 18: High-Degree, Low-Order Modes 388
18.1 Introduction 388
18.2 High-Degree, Low-Order p- and g+-Modes Trapped Near the Surface 388
18.3 High-Degree, Low-Order g+-Modes Trapped Near a Maximum of the Boundary of the G-Region 394
Chapter 19: Period Changes in a Rapidly Evolving Pulsating Star 398
19.1 Introduction 398
19.2 Appropriate Equations 398
19.3 Two-Time-Variable Expansion Procedure 401
19.4 Rate of Change of an Isentropic Pulsation Period 405
19.5 Use of the Equality of the Mean Kinetic and the MeanPotential Energy of Pulsation 407
19.6 Explicit Expression for the Rate of Changeof a Period in a Radially Pulsating Star 408
19.6.1 The First Part 409
19.6.2 The Second Part 409
19.6.3 The Third Part 412
19.6.4 Global Result 412
19.7 Rate of Change of a Period in a Radially Pulsating Star Subject to a Homologous Contraction or Expansion 413
Appendix A:Green's Fundamental Formula of Potential Theory 416
Appendix B: The Thermodynamic Isentropic Coefficients 419
B.1 Reversible Thermodynamic Processes 419
B.2 Thermodynamic Relations 419
B.3 Definitions of the Isentropic Coefficients 420
B.4 Equations for the Isentropic Coefficients 422
B.4.1 Equation for 1 422
B.4.2 Equation for 2 423
B.4.3 Equation for 3 424
Appendix C: Lagrange's Equations of Motion 425
Appendix D: Spherical Harmonics 430
Appendix E Singular Perturbation Problems of the Boundary-Layer Type 432
Appendix F: Boundary Condition Relative to the Pressure on a Star's Surface 436
Appendix G: The Curl of a Vector Field 440
Appendix H: Eigenvalue Problem of the Vibrating String 444
Appendix I: The Euler–Lagrange Equations of Hamilton's Variational Principle for a Perturbed Star 447
Appendix J: Acoustic Waves 449
J.1 Acoustic Waves in a Uniform Gas 449
J.2 Vertical Propagation of Acoustic Waves in a Plane Isothermal Layer 450
List of Symbols 453
References 457
Index 468