On slowly rotating axisymmetric solutions of the Euler-Poisson equations

TL;DR

This paper constructs and analyzes slowly rotating axisymmetric solutions to the Euler-Poisson equations, revealing properties like star surface shape and vacuum conditions for certain adiabatic exponents.

Contribution

It introduces a method to prove existence and properties of slowly rotating stellar solutions with small angular velocity for specific polytropic indices.

Findings

Existence of solutions for b3 in (6/5, 3/2]

Solutions exhibit physical vacuum conditions

Star surfaces are oblate under rotation

Abstract

We construct stationary axisymmetric solutions of the Euler-Poisson equations, which govern the internal structure of polytropic gaseous stars, with small constant angular velocity when the adiabatic exponent $\gamma$ belongs to $(\frac65,\frac32]$. The problem is formulated as a nonlinear integral equation, and is solved by iteration technique. By this method, not only we get the existence, but also we clarify properties of the solutions such as the physical vacuum condition and oblateness of the star surface.