A (Very) Simple Proof of the Gravitational Energy Formula of Polytropic Spheres

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.