Spikes and accretion of unbound, collisionless matter around black holes

We consider the steady-state density and velocity dispersion profiles of collisionless matter around a Schwarzschild black hole (BH) and its associated rate of accretion onto the BH. We treat matter, which could be stars or dark matter particles, whose orbits are unbound to the BH, but still governed by its gravitational field. We consider two opposite spatial geometries for the matter distributions: an infinite, three-dimensional cluster and a two-dimensional razor-thin disk, both with zero net angular momentum. We demonstrate that the results depend critically on the adopted geometry, even in the absence of angular momentum. We adopt a simple monoenergetic, isotropic, phase-space distribution function for the matter as a convenient example to illustrate this dependence. The effect of breaking strict isotropy by incorporating an unreplenished loss cone due to the BH capture of low-angular momentum matter is also considered. Calculations are all analytic and performed in full general relativity, though key results are also evaluated in the Newtonian limit. We consider one application to show that the rate of BH accretion from an ambient cluster can be significantly less than that from a thin disk to which it may collapse, although both rates are considerably smaller than Bondi accretion for a (collisional) fluid with a similar asymptotic particle density and velocity dispersion (i.e., sound speed).