  We identify a universal relation between the radial density slope \alpha (r)
and the velocity anisotropy \beta (r) for equilibrated structures. This
relation holds for a variety of systems, including disk galaxy mergers,
spherical collapses, cold dark matter (CDM) halos both with and without
cooling. We argue that the shape of the relation is reasonable from fundamental
principles when the dark matter or stars are assumed to obey Tsallis
statistics, and in that case we fit the \alpha - \beta relation with just one
free parameter. One can use this result to close the Jeans equations, for
example to construct mass models of elliptical galaxies from observational data
or to tune dark matter direct detection experiments. We also predict the
asymptotic central slope and anisotropy of CDM halos to be approximately -1 and
0.
