Revisiting Non-Rotating Star Models: Classical Existence and Uniqueness Theory and Scaling Relations

TL;DR

This paper systematically studies non-rotating stellar models governed by the Euler-Poisson system, extending existence and uniqueness results, and applying scaling methods to analyze solution behaviors across different masses.

Contribution

It extends classical existence and uniqueness results for stellar models and introduces scaling relations to understand solution behaviors as mass varies.

Findings

Extended existence results for non-rotating stellar models.

Adapted uniqueness results from quantum mechanics to classical models.

Derived scaling relations and convergence rates for solutions with varying mass.

Abstract

This paper presents a systematic study of the properties of non-rotating stellar models governed by the Euler-Poisson system under general equations of state, including the case of polytropic gaseous stars. We revisit and extend existence results by Auchmuty and Beals \cite{AB71}, adapt the uniqueness results from the quantum mechanical framework of Lieb and Yau \cite{LY87} to the classical Newtonian mechanical setting. The results are also synthesized in McCann \cite{McC06} but without proof. The second work we do is applying a scaling method to establish relations between solutions with different total masses. As the mass tends to zero, we analyze convergence properties of the density functions and identify precise rates for the contraction or extension of their supports.