[Paper Review] Stability and rotational mixing of modes in Newtonian and relativistic stars

Keith H. Lockitch|ArXiv.org|Sep 8, 1999

Pulsars and Gravitational Waves Research11 references3 citations

TL;DR

This dissertation investigates hybrid rotational modes in Newtonian and relativistic stars, focusing on the stability and mixing of r-modes and g-modes in perfect fluid models with a one-parameter equation of state. Using analytical and numerical methods, it computes the first relativistic hybrid modes and confirms their degeneracy splitting under rotation, extending prior work on Maclaurin spheroids to realistic stellar models including uniform density and n=1 polytropes.

ABSTRACT

Despite the recent excitement over the r-mode instability of rotating stars, these modes are not yet well-understood for stellar models appropriate to neutron stars - perfect fluid models in which both the equilibrium and perturbed configurations obey the same one-parameter equation of state. In spherical stars of this kind, r-modes and g-modes form a degenerate zero-frequency subspace. Rotation splits the degeneracy to zeroth order in the star's angular velocity $Ω$, and the resulting modes are generically hybrids, whose limit as $Ω o 0$ is a stationary current with axial and polar parts. Lindblom and Ipser have recently found these hybrid rotational modes in an analytic study of the Maclaurin spheroids. This dissertation studies them in Newtonian stars (both uniform density models and $n=1$ polytropes) and reports a first computation of the hybrid rotational modes of relativistic stars.

Motivation & Objective

• To understand the stability and rotational mixing of r-modes and g-modes in Newtonian and relativistic stellar models.
• To resolve the degeneracy between r-modes and g-modes in non-rotating stars by analyzing their splitting under rotation.
• To extend the analytic results of Lindblom and Ipser on Maclaurin spheroids to more realistic stellar configurations.
• To perform the first numerical computation of hybrid rotational modes in relativistic stars using a one-parameter equation of state.
• To characterize the structure of these modes as axial-polar hybrids in the zero-rotation limit.

Proposed method

• Solving the linearized equations of motion and continuity for stellar perturbations in rotating, perfect fluid stars.
• Using a one-parameter equation of state to ensure consistency between equilibrium and perturbed configurations.
• Applying spectral methods and numerical integration to compute eigenfrequencies and eigenfunctions of hybrid modes in Newtonian stars (uniform density and n=1 polytropes).
• Extending the numerical framework to relativistic stellar models, solving the perturbation equations in the relativistic hydrodynamic formalism.
• Analyzing mode structure in the limit Ω → 0 to identify axial and polar components of the hybrid modes.
• Validating results against known analytic solutions for Maclaurin spheroids to ensure numerical accuracy.

Experimental results

Research questions

• RQ1 How do r-modes and g-modes mix under rotation in Newtonian stars with a one-parameter equation of state?
• RQ2 What is the structure and stability of hybrid rotational modes in relativistic stars, and how do they compare to Newtonian counterparts?
• RQ3 How does the degeneracy between r-modes and g-modes in non-rotating stars break under rotation, and what is the nature of the resulting modes?
• RQ4 To what extent do the hybrid modes in relativistic stars resemble the analytic solutions found for Maclaurin spheroids?
• RQ5 What are the axial and polar components of the hybrid modes in the zero-rotation limit, and how do they evolve with increasing angular velocity?

Key findings

• The first numerical computation of hybrid rotational modes in relativistic stars was successfully performed, confirming the existence of such modes in general relativistic frameworks.
• In Newtonian stars, hybrid modes were computed for both uniform density and n=1 polytropes, showing consistent behavior with analytic expectations.
• The modes exhibit a mixed axial and polar character in the Ω → 0 limit, confirming their hybrid nature as predicted by Lindblom and Ipser.
• Degeneracy between r-modes and g-modes is lifted at zeroth order in Ω, with the splitting pattern matching theoretical predictions.
• The eigenfrequencies and eigenfunctions of the modes were computed with high numerical precision, enabling comparison with future observational constraints.
• The results support the stability of these modes under certain conditions, though full nonlinear stability remains an open question.