Entropic upper bound on gravitational binding energy

We prove that the gravitational binding energy Ω of a self gravitating system described by a mass density distribution ρ(x) admits an upper bound B[ρ(x)] given by a simple function of an appropriate, non-additive Tsallis’ power-law entropic functional Sq evaluated on the density ρ. The density distributions that saturate the entropic bound have the form of isotropic q-Gaussian distributions. These maximizer distributions correspond to the Plummer density profile, well-known in astrophysics. A heuristic scaling argument is advanced suggesting that the entropic bound B[ρ(x)] is unique, in the sense that it is unlikely that exhaustive entropic upper bounds not based on the alluded Sq entropic measure exit. The present findings provide a new link between the physics of self gravitating systems, on one hand, and the statistical formalism associated with non-additive, power-law entropic measures, on the other hand.