A (very) simple proof of the gravitational energy formula of polytropic spheres

TL;DR

The paper provides a concise, elementary derivation of the gravitational energy for self-gravitating regular polytropes with $0\le n<5$ by combining the density-potential relation $\rho=B\Phi^{n}\theta(\Phi)$ with Gauss' divergence theorem and Chandrasekhar's virial tensor. It avoids the standard Lane-Emden function properties and thermodynamic arguments, and yields the compact result $U = -{3 M \mathcal{E}_{\rm t} \over 5-n}$ (equivalently $U = -{3 G M^2\over (5-n) r_t}$ when $\mathcal{E}_{\rm t}=GM/r_t$). The method generalizes to configurations with external potentials via a virial identity, offering a transparent toolkit for teaching and extending to more complex polytropic structures.

Abstract

It is shown how the well-known formula for the gravitational energy of self-gravitating regular polytropes of finite mass can be obtained in an elementary way by using Gauss's divergence theorem and the Chandrasekhar virial tensor, without resorting to lengthy algebra, to specific properties of Lane-Emden functions, and to thermodynamics arguments, as is instead commonly found in standard treatises and in astrophysical literature. The present approach, due to its simplicity, can be particularly useful to students and researchers, and it can be easily applied to the study of more complicated polytropic structures.