Mathematical aspects relative to the fluid statics of a self-gravitating perfect-gas isothermal sphere

TL;DR

This paper explores the mathematical properties and solution multiplicity of the Lane-Emden equation in modeling self-gravitating isothermal gas spheres, emphasizing numerical methods and boundary condition effects.

Contribution

It introduces novel boundary conditions for the Lane-Emden equation and compares numerical schemes to accurately capture multiple solutions in self-gravitating gas models.

Findings

Multiple solutions exist for the model under certain boundary conditions.

Numerical schemes can be optimized to detect solution multiplicity.

Analytical properties of the model are characterized.

Abstract

In the present paper we analyze and discuss some mathematical aspects of the fluid-static configurations of a self-gravitating perfect gas enclosed in a spherical solid shell. The mathematical model we consider is based on the well-known Lane-Emden equation, albeit under boundary conditions that differ from those usually assumed in the astrophysical literature. The existence of multiple solutions requires particular attention in devising appropriate numerical schemes apt to deal with and catch the solution multiplicity as efficiently and accurately as possible. In sequence, we describe some analytical properties of the model, the two algorithms used to obtain numerical solutions, and the numerical results for two selected cases.