# Fluid statics of a self-gravitating perfect-gas isothermal sphere

## TL;DR

This paper investigates the fluid static equilibrium of an isothermal, self-gravitating perfect gas sphere using a novel boundary condition approach, revealing multiple solutions and thermodynamic stability considerations.

## Contribution

It introduces a new method for solving the isothermal sphere problem by fixing the gravitational field at the shell boundary, differing from traditional central density prescriptions.

## Key findings

- Identification of a gravitational number upper bound.
- Existence of multiple equilibrium solutions.
- Preliminary thermodynamic stability analysis.

## Abstract

We open the paper with introductory considerations describing the motivations of our long-term research plan targeting gravitomagnetism, illustrating the fluid-dynamics numerical test case selected for that purpose, that is, a perfect-gas sphere contained in a solid shell located in empty space sufficiently away from other masses, and defining the main objective of this study: the determination of the gravitofluid-static field required as initial field ($t=0$) in forthcoming fluid-dynamics calculations. The determination of the gravitofluid-static field requires the solution of the isothermal-sphere Lane-Emden equation. We do not follow the habitual approach of the literature based on the prescription of the central density as boundary condition; we impose the gravitational field at the solid-shell internal wall. As the discourse develops, we point out differences and similarities between the literature's and our approach. We show that the nondimensional formulation of the problem hinges on a unique physical characteristic number that we call gravitational number because it gauges the self-gravity effects on the gas' fluid statics. We illustrate and discuss numerical results; some peculiarities, such as gravitational-number upper bound and multiple solutions, lead us to investigate the thermodynamics of the physical system, particularly entropy and energy, and preliminarily explore whether or not thermodynamic-stability reasons could provide justification for either selection or exclusion of multiple solutions. We close the paper with a summary of the present study in which we draw conclusions and describe future work.