R-mode Instability of Slowly Rotating Non-isentropic Relativistic Stars

TL;DR

This paper studies the r-mode instability in slowly rotating relativistic stars using a low-frequency formalism and a near-zone boundary condition, revealing discrete unstable modes and the persistence of a continuous spectrum.

Contribution

It introduces a novel boundary condition approach to analyze r-mode instabilities in relativistic stars, highlighting the existence of discrete unstable modes under certain conditions.

Findings

Discrete r-mode solutions exist only for specific polytropic models.

The properties of these solutions are similar to those with traditional boundary conditions.

A continuous spectrum component persists even with gravitational radiation effects.

Abstract

We investigate properties of $r$-mode instability in slowly rotating relativistic polytropes. Inside the star slow rotation and low frequency formalism that was mainly developed by Kojima is employed to study axial oscillations restored by Coriolis force. At the stellar surface, in order to take account of gravitational radiation reaction effect, we use a near-zone boundary condition instead of the usually imposed boundary condition for asymptotically flat spacetime. Due to the boundary condition, complex frequencies whose imaginary part represents secular instability are obtained for discrete $r$-mode oscillations in some polytropic models. It is found that such discrete $r$-mode solutions can be obtained only for some restricted polytropic models. Basic properties of the solutions are similar to those obtained by imposing the boundary condition for asymptotically flat spacetime. Our results suggest that existence of a continuous part of spectrum cannot be avoided even when its frequency becomes complex due to the emission of gravitational radiation.