Linearized Analysis of Barotropic Perturbations around Spherically Symmetric Gaseous Stars Governed by the Euler-Poisson Equations

TL;DR

This paper analyzes the linear stability of spherically symmetric gaseous stars governed by the Euler-Poisson equations, revealing that the spectral properties depend on the nature of the perturbations, with Sturm-Liouville structure only for irrotational cases.

Contribution

It provides a detailed functional analytic study of the spectrum of linearized Euler-Poisson operators, clarifying conditions under which Sturm-Liouville theory applies.

Findings

Spectrum is not Sturm-Liouville for general perturbations.

Spectrum is Sturm-Liouville for irrotational perturbations.

Highlights the importance of perturbation type in stability analysis.

Abstract

The hydrodynamic evolution of self-gravitating gaseous stars is governed by the Euler-Poisson equations. We study the structure of the linear approximation of barotropic perturbations around spherically symmetric equilibria based on functional analytic tools. In contrast to folklore, we show that the spectrum of the linearized operator for general perturbations is not of the Sturm-Liouville type unless the perturbations are restricted. In particular, we prove that it is of the Sturm-Liouville type for irrotational perturbations.