On Rotating Axisymmetric Solutions of the Euler-Poisson Equations

TL;DR

This paper investigates stationary axisymmetric solutions of the Euler-Poisson equations for gaseous stars, establishing conditions for their existence, analyzing their properties, and exploring the relationship between central density and total mass.

Contribution

It provides a new framework for understanding rotating star solutions with differential rotation using nonlinear integral equations and the implicit function theorem.

Findings

Existence of slowly rotating configurations near spherical equilibria.

Oblateness of star surface confirmed.

Relationship between central density and total mass established.

Abstract

We consider stationary axisymmetric solutions of the Euler-Poisson equations, which govern the internal structure of barotropic gaseous stars. We take the general form of the equation of states which cover polytropic gaseous stars indexed by $6/5<\gamma <2$ and also white dwarfs. A generic condition of the existence of stationary solutions with differential rotation is given, and the existence of slowly rotating configurations near spherically symmetric equilibria is shown. The problem is formulated as a nonlinear integral equation, and is solved by an application of the infinite dimensional implicit function theorem. Oblateness of star surface is shown and also relationship between the central density and the total mass is given.