A Cusp Slope -- Central Anisotropy Theorem

For a wide class of self-gravitating systems, we show that if the density is cusped like 1/r^{gamma} near the center, then the limiting value of the anisotropy parameter beta = 1 - <v_T^2>/(2<v_r^2>) at the center may not be greater than (gamma/2). Here, <v_r^2> and <v_T^2> are the radial and tangential velocity second moments. This follows from the non-negativity of the phase space density. We compare this theorem to other proposed relations between the cusp slope and the central anisotropy to clarify their applicabilities and underlying assumptions. The extension of this theorem to tracer populations in an externally imposed potential is also derived. In particular, for stars moving in the vicinity of a central black hole, this reduces to gamma >= beta+(1/2), indicating that an isotropic system in Keplerian potential should be cusped at least as steep as 1/r^{0.5}. Similar limits have been noticed before for specific forms of the distribution function, but here we establish this as a general result.