Bar mode instability in relativistic rotating stars: a post Newtonian treatment

TL;DR

This paper develops analytic models of relativistic rotating stars in post-Newtonian gravity to study their stability against bar mode instabilities, revealing that relativistic effects tend to weaken the viscosity-driven instability.

Contribution

It provides an analytic solution for PN homogeneous ellipsoids and analyzes the stability of rotating stars, extending classical models to include relativistic effects at PN order.

Findings

Relativistic effects increase the eccentricity at instability onset.

Higher rotation is needed to trigger instability as stars become more compact.

GR effects tend to weaken viscosity-driven bar mode instability.

Abstract

We construct analytic models of incompressible, rigidly rotating stars in PN gravity and study their stability against nonaxisymmetric Jacobi-like bar modes. PN configurations are modeled by homogeneous triaxial ellipsoids and the metric is obtained as a solution of Einstein's equations in 3+1 ADM form. We use an approximate subset of the equations well-suited to numerical integration for strong field, 3D configurations in quasi--equilibrium. These equations are exact at PN order, and admit an analytic solution for homogeneous ellipsoids. In this paper we present this solution, as well as analytic functionals for the conserved global quantities, M, M_0 and J. By using a variational principle we construct sequences of axisymmetric equilibria of constant density and rest mass, i.e. the PN generalization of Maclaurin spheroids, which are compared to other PN and full relativistic sequences presented by previous authors. We then consider nonaxisymmetric ellipsoidal deformations of the configurations, holding J constant and the rotation uniform, and we locate the point at which the bar modes will be driven secularly unstable by a dissipative agent like viscosity. We find that the value of the eccentricity, as well as the ratios \Omega^2/(\pi\rho_0) and T/|W|, defined invariantly, all increase at the onset of instability as the stars become more relativistic. Since higher degrees of rotation are required to trigger a viscosity-driven bar mode as the star's compactness increases, the effect of GR is to weaken the instability, at least to PN order. This behavior is opposite to that found for secular instability via Dedekind-like modes driven unstable by gravitational radiation, supporting the suggestion that in GR, nonaxisymmetric modes driven unstable by viscosity and gravitational radiation may no longer coincide.