Revisiting Non-Rotating Star Models: Classical Existence and Uniqueness Theory and Scaling Relations
Hangsheng Chen
TL;DR
This work develops a rigorous variational framework for non-rotating stellar models under the Euler-Poisson system for general equations of state, including polytropes. It extends classical existence/structure results and adapts Lieb–Yau’s uniqueness arguments to the Newtonian setting, establishing existence of energy minimizers $\sigma_m$ with precise symmetry and EL conditions, and deriving explicit scaling laws that relate solutions across total mass $m$. The key contributions are the confirmation of minimizer properties (radial symmetry, compact support, negative chemical potential) and the provision of exact mass-scaling relations $\sigma_m(x)=A^{-1}\sigma(x/B)$ with $A=m^{-2/(3\gamma-4)}$, $B=m^{(\gamma-2)/(3\gamma-4)}$, yielding $e_0(m)=m^{(5\gamma-6)/(3\gamma-4)}e_0$; these results illuminate the small-mass limit and guide understanding of single-star versus multi-body configurations, laying groundwork for rotating systems and related astrophysical models.
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.