The intimate relation between the low T/W instability and the co-rotation point

TL;DR

This study investigates the low T/W instability in differentially rotating stars, revealing that the instability is triggered by the development of a co-rotation point and is influenced by the star's rotation profile and mode properties.

Contribution

It demonstrates that the low T/W instability occurs when a co-rotation point develops, providing an empirical relation linking mode frequency, star parameters, and instability growth time.

Findings

The l=m=2 f-mode becomes unstable at the co-rotation point.

The instability grows faster within the co-rotation region.

An empirical relation correlates mode frequency with star parameters.

Abstract

We study the low T/W instability associated with the f-mode of differentially rotating stars. Our stellar models are described by a polytropic equation of state and the rotation profile is given by the standard j-constant law. The properties of the relevant oscillation modes, including the instability growth time, are determined from time evolutions of the linearised dynamical equations in Newtonian gravity. In order to analyse the instability we monitor also the canonical energy and angular momentum. Our results demonstrate that the l=m=2 f-mode becomes unstable as soon as a co-rotation point develops inside the star (i.e. whenever there is a point where the mode's pattern speed matches the bulk angular velocity). Considering various degrees of differential rotation, we show that the instability grows faster deep inside the co-rotation region and deduce an empirical relation that correlates the mode frequency and the star's parameters, which captures the main features of the l=m=2 f-mode growth time. This function is proportional to the product of the kinetic to gravitational energy ratio and the gradient of the star's spin, strengthening further the relationship between the co-rotation point and the low T/W instability. We briefly consider also the l=m=2 r-mode and demonstrate that it never moves far inside the co-rotation region even for significant differential rotation.