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Table of Contents
Intro; Supervisor's Foreword; References; Preface; References; Acknowledgements; Contents; List of Figures; List of Tables; 1 Introduction; 1.1 Prologue: Music of the Spheres; 1.2 Asteroseismology; 1.3 Neutron Stars Inside Out; 1.4 Odes from Modes; References; 2 The Oscillation Modes: Linear Perturbation Scheme; 2.1 The Fluid Equations; 2.2 Linear Perturbation Formalism; 2.3 Nonrotating Stars; 2.3.1 *Separation of Variables; 2.3.2 *Boundary Conditions; 2.3.3 *Dimensionless Formulation; 2.3.4 *The Cowling Approximation; 2.3.5 *Mode Trapping; 2.3.6 *A Trivial Solution.
2.3.7 Mode Orthogonality, Decomposition, and Energy2.4 Classes of Modes; 2.4.1 Polar Modes; 2.4.2 Axial Modes; 2.4.3 Hybrid Modes; 2.5 *The Homogeneous Model; 2.6 Rotating Stars; 2.6.1 Mode Orthogonality, Decomposition, and Energy; 2.6.2 The Slow-Rotation Approximation; References; 3 The f-mode Instability; 3.1 *Figures of Equilibrium; 3.2 The Kepler Limit; 3.3 *Maclaurin, Jacobi, or Dedekind?; 3.4 *A Mechanical Example; 3.5 The CFS Instability; 3.6 The Instability Window; References; 4 Mode Coupling: Quadratic Perturbation Scheme; 4.1 Quadratic Perturbation Formalism.
4.2 Resonant Mode Coupling4.2.1 Coupled Triplet Equations of Motion; 4.2.2 *The Multiscale Method; 4.2.3 Coupling Selection Rules; 4.2.4 Mode Normalisation; 4.3 Parametric Resonance Instability; 4.4 Saturation; 4.4.1 Stability Conditions; 4.4.2 Possible Evolutions; 4.4.3 *Frequency Synchronisation; References; 5 Results; 5.1 Setup; 5.1.1 *Eigenfrequencies and Eigenfunctions; 5.1.2 *Growth/Damping Rates; 5.1.3 *Couplings; 5.1.4 Saturation; 5.2 Approximate Relations for the Parametric Instability Threshold; 5.3 Typical Neutron Stars; 5.3.1 Instability Evolution; 5.3.2 Models; 5.3.3 Discussion.
5.4 Supramassive Neutron Stars5.4.1 Instability Evolution; 5.4.2 Models; 5.4.3 Discussion; 5.5 Stochastic Background; 5.6 *Comparison with r-modes; References; 6 Final Remarks; 6.1 Summary and Conclusions; 6.2 Epilogue: Notes on the Cosmic Staff; References; Appendix A Polytropic Stars; A.1 Nonrotating Polytropes; A.2 Rotating Polytropes; Appendix B Polar Mode Rotational Corrections; B.1 First-Order Corrections; B.2 Second-Order Corrections; B.2.1 Generic Formulation; B.2.2 Application to Polytropes; Appendix C Polar Mode Growth/Damping Rates; C.1 Gravitational Waves; C.2 Shear Viscosity.
C.3 Bulk ViscosityC. 4 Mode Energy; Appendix D Equations of Motion; D.1 Equation of Motion for Quadratic Perturbations; D.2 Amplitude Equation of Motion; Appendix E Polar Mode Coupling Coefficient; E.1 Parametrised Form; E.2 Angular Part; E.3 Radial Part; Appendix F Parametrically Unstable Mode Triplet; F.1 Parametric Instability Threshold; F.2 Equilibrium Solution; F.3 Linear Stability Analysis; Appendix G Coupling Spectrum.
2.3.7 Mode Orthogonality, Decomposition, and Energy2.4 Classes of Modes; 2.4.1 Polar Modes; 2.4.2 Axial Modes; 2.4.3 Hybrid Modes; 2.5 *The Homogeneous Model; 2.6 Rotating Stars; 2.6.1 Mode Orthogonality, Decomposition, and Energy; 2.6.2 The Slow-Rotation Approximation; References; 3 The f-mode Instability; 3.1 *Figures of Equilibrium; 3.2 The Kepler Limit; 3.3 *Maclaurin, Jacobi, or Dedekind?; 3.4 *A Mechanical Example; 3.5 The CFS Instability; 3.6 The Instability Window; References; 4 Mode Coupling: Quadratic Perturbation Scheme; 4.1 Quadratic Perturbation Formalism.
4.2 Resonant Mode Coupling4.2.1 Coupled Triplet Equations of Motion; 4.2.2 *The Multiscale Method; 4.2.3 Coupling Selection Rules; 4.2.4 Mode Normalisation; 4.3 Parametric Resonance Instability; 4.4 Saturation; 4.4.1 Stability Conditions; 4.4.2 Possible Evolutions; 4.4.3 *Frequency Synchronisation; References; 5 Results; 5.1 Setup; 5.1.1 *Eigenfrequencies and Eigenfunctions; 5.1.2 *Growth/Damping Rates; 5.1.3 *Couplings; 5.1.4 Saturation; 5.2 Approximate Relations for the Parametric Instability Threshold; 5.3 Typical Neutron Stars; 5.3.1 Instability Evolution; 5.3.2 Models; 5.3.3 Discussion.
5.4 Supramassive Neutron Stars5.4.1 Instability Evolution; 5.4.2 Models; 5.4.3 Discussion; 5.5 Stochastic Background; 5.6 *Comparison with r-modes; References; 6 Final Remarks; 6.1 Summary and Conclusions; 6.2 Epilogue: Notes on the Cosmic Staff; References; Appendix A Polytropic Stars; A.1 Nonrotating Polytropes; A.2 Rotating Polytropes; Appendix B Polar Mode Rotational Corrections; B.1 First-Order Corrections; B.2 Second-Order Corrections; B.2.1 Generic Formulation; B.2.2 Application to Polytropes; Appendix C Polar Mode Growth/Damping Rates; C.1 Gravitational Waves; C.2 Shear Viscosity.
C.3 Bulk ViscosityC. 4 Mode Energy; Appendix D Equations of Motion; D.1 Equation of Motion for Quadratic Perturbations; D.2 Amplitude Equation of Motion; Appendix E Polar Mode Coupling Coefficient; E.1 Parametrised Form; E.2 Angular Part; E.3 Radial Part; Appendix F Parametrically Unstable Mode Triplet; F.1 Parametric Instability Threshold; F.2 Equilibrium Solution; F.3 Linear Stability Analysis; Appendix G Coupling Spectrum.