  We prove that the gravitational binding energy {\Omega} of a self gravitating
system described by a mass density distribution {\rho}(x) admits an upper bound
B[{\rho}(x)] given by a simple function of an appropriate, non-additive
Tsallis' power-law entropic functional Sq evaluated on the density {\rho}. 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[{\rho}(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 the one hand, and
the statistical formalism associated with non-additive, power-law entropic
measures, on the other hand.
