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Linear Isentropic Oscillations of Stars Theoretical Foundations

:	Tim Van Hoolst, Paul Smeyers
:	Linear Isentropic Oscillations of Stars Theoretical Foundations
:	Springer-Verlag
:	9783642130304
:	1
:	CHF 189.80
:

:	Astronomie
:	English

:	473
:	Wasserzeichen
:	PC/MAC/eReader/Tablet
:	PDF

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.

	Linear Isentropic Oscillations of Stars	1
	Preface	1
	5	1
	Contents	1
	7	1
	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.