Unstable normal modes of low T/W dynamical instabilities in differentially rotating stars

TL;DR

This paper analyzes low T/W dynamical instabilities in differentially rotating stars using linear perturbation theory, identifying unstable modes linked to corotation radii and validating results with hydrodynamical simulations.

Contribution

First comprehensive eigenvalue analysis of low T/W instabilities in differentially rotating stars, connecting linear modes with 3D simulation results.

Findings

Unstable modes are associated with corotation radii inside the star.

Eigenfrequencies and eigenfunctions agree with 3D hydrodynamical simulations.

Normal mode analysis effectively identifies unstable equilibrium stars.

Abstract

We investigate the nature of low T/W dynamical instabilities in differentially rotating stars by means of linear perturbation. Here, T and W represent rotational kinetic energy and the gravitational binding energy of the star. This is the first attempt to investigate low T/W dynamical instabilities as a complete set of the eigenvalue problem. Our equilibrium configuration has "constant" specific angular momentum distribution, which potentially contains a singular solution in the perturbed enthalpy at corotation radius in linear perturbation. We find the unstable normal modes of differentially rotating stars by solving the eigenvalue problem along the equatorial plane of the star, imposing the regularity condition on the center and the vanished enthalpy at the oscillating equatorial surface. We find that the existing pulsation modes become unstable due to the existence of the corotation radius inside the star. The feature of the unstable mode eigenfrequency and its eigenfunction in the linear analysis roughly agrees with that in three-dimensional hydrodynamical simulations in Newtonian gravity. Therefore, our normal mode analysis in the equatorial motion proves valid to find the unstable equilibrium stars efficiently. Moreover, the nature of the eigenfunction that oscillates between corotation and the surface radius for unstable stars requires reinterpretation of the pulsation modes in differentially rotating stars.