**2.5. Formation of a cusp of stars around the black hole**

From a numerical resolution of the time-dependent Boltzmann equation,
with the relevant diffusion coefficients, it can be shown that around
a black hole at the center of a globular cluster, the stellar density should
be of a power-law shape, with a slope of 7/4
(Bahcall & Wolf
1976).
The relevant two-body relaxation time
t_{R} is dependent on the number of bodies N in the system,
as t_{R} / t_{c} = N / logN, where t_{c} is
the crossing time= r_{c} / V.
For a galactic center, with a volumic density of stars of
10^{7} M_{} /
pc^{3}, this relaxation time is 3 10^{8} yr.

The distribution of stars around a black hole can be described, according to the distance to the center:

- first for the stars not bound to the black hole,
at r > R
_{a}, their velocity distribution is Maxwellian, and their density profile has the isothermal power law in r^{-2}. There are also unbound stars inside R_{a}, but with a density in r^{-1/2}. This allows to compute the penetration rate of these unbound stars in the tidal or collision radius, to estimate the accretion rate. With a core stellar mass of M_{core}= 10^{7}- 3 10^{8}M_{}, a density 10^{7}pc^{-3}, the accretion rate is, by tidal disruption:dM / dt

_{tide}= 1 M_{}/ yr M_{8}^{4/3}and by stellar collisions:

dM / dt

_{coll}= 0.1 M_{}/ yr M_{8}^{3} - the orbits bound to the black hole r< R
_{a}: due to the cusp, their density is in r^{-7/4}, there is an excess of stars inside R_{coll}, that favors stellar collisions.

More refined Monte-Carlo simulations with
a distribution function f(E, L, t), taking into account the velocity
anisotropy, the disparition of disrupted stars, etc..
show that the stellar cluster cannot fuel the black hole
indefinitely
(Duncan &
Shapiro 1983).
The growth rate of the black hole and its luminosity
decreases as 1/time.
The loss-cone theory and the simulations are in agreement:
the accretion rate due to tidal disruptions is
M_{core} / t_{R}, typically of 10^{-2}
M_{} / yr, with a
maximum lower than 1
M_{}/yr; this cannot explain
the luminosity of QSOs. QSOs might be explained only when stellar
collisions are included, the corresponding accretion
rate is typically a hundred times higher.

Triaxial deviations from spherical symmetry of
only 5% (due to a bar or binary black hole)
can repopulate the loss-cone, increasing tidal
disruption to QSOs levels.
However, t_{coll} < t_{R}, and collisions may destroy
the cusp
(Norman & Silk
1983).