  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 {\it unbound} to the BH, but still governed
by its gravitational field. We consider two opposite spatial geometries for the
matter distributions: an infinite, 3D cluster and a 2D 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 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).

Astrophysics of Galaxies;General Relativity and Quantum Cosmology