Periapsis shifts in dark matter distribution around a black hole
  We consider the periapsis shifts of bound orbits of stars on static clouds
around a black hole. The background spacetime is constructed from a
Schwarzschild black hole surrounded by a static and spherically symmetric
self-gravitating system of massive particles, which satisfies all energy
conditions and physically models the gravitational effect of dark matter
distribution around a nonrotating black hole. Using nearly circular bound
orbits of stars, we obtain a simple formula for the precession rate. This
formula explicitly shows that the precession rate is determined by a positive
contribution (i.e., a prograde shift) from the conventional
general-relativistic effect and a negative contribution (i.e., a retrograde
shift) from the local matter density. The four quantities for such an orbit
(i.e., the orbital shift angle, the radial oscillation period, the redshift,
and the star position mapped onto the celestial sphere) determine the local
values of the background model functions. Furthermore, we not only evaluate the
precession rate of nearly circular bound orbits in several specific models but
also simulate several bound orbits with large eccentricity and their periapsis
shifts. The present exact model demonstrates that the retrograde precession
does not mean any exotic central objects such as naked singularities or
wormholes but simply the existence of significant energy density of matters on
the star orbit around the black hole.
