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

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.