Stability of rotating gaseous stars

TL;DR

This paper investigates the stability of rotating gaseous stars using the Euler-Poisson system, establishing criteria for stability and instability, and revealing differences from non-rotating stars regarding the turning point principle.

Contribution

It provides a sharp stability criterion for axi-symmetric perturbations and demonstrates the failure of the turning point principle in certain rotating star families.

Findings

Stability criterion for Rayleigh stable rotation

Unstable modes and exponential trichotomy estimates

Counterexample to the turning point principle for rotating stars

Abstract

We consider stability of rotating gaseous stars modeled by the Euler-Poisson system with general equation of states. When the angular velocity of the star is Rayleigh stable, we proved a sharp stability criterion for axi-symmetric perturbations. We also obtained estimates for the number of unstable modes and exponential trichotomy for the linearized Euler-Poisson system. By using this stability criterion, we proved that for a family of slowly rotating stars parameterized by the center density with fixed angular velocity profile, the turning point principle is not true. That is, unlike the case of non-rotating stars, the change of stability of the rotating stars does not occur at extrema points of the total mass. By contrast, we proved that the turning point principle is true for the family of slowly rotating stars with fixed angular momentum distribution. When the angular velocity is Rayleigh unstable, we proved linear instability of rotating stars. Moreover, we gave a complete description of the spectra and sharp growth estimates for the linearized Euler-Poisson equation.