A Cusp Slope–Central Anisotropy Theorem

For a wide class of self-gravitating systems, we show that if the density iscusped like 1/r^{gamma} near the center, then the limiting value of theanisotropy parameter beta = 1 - /(2) at the center may not begreater than (gamma/2). Here, and are the radial and tangentialvelocity second moments. This follows from the non-negativity of the phasespace density. We compare this theorem to other proposed relations between thecusp slope and the central anisotropy to clarify their applicabilities andunderlying assumptions. The extension of this theorem to tracer populations inan externally imposed potential is also derived. In particular, for starsmoving in the vicinity of a central black hole, this reduces to gamma >=beta+(1/2), indicating that an isotropic system in Keplerian potential shouldbe cusped at least as steep as 1/r^{0.5}. Similar limits have been noticedbefore for specific forms of the distribution function, but here we establishthis as a general result.Comment: to appear in ApJ (May 2006). (v4 - updated for styles and typos; v3 - a new simpler proof for beta>1/2; v2 - results slightly changed for divergent potentials