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 tR is dependent on the number of bodies N in the system, as tR / tc = N / logN, where tc is the crossing time= rc / V. For a galactic center, with a volumic density of stars of 107 M / pc3, this relaxation time is 3 108 yr.
The distribution of stars around a black hole can be described, according to the distance to the center:
dM / dttide = 1 M / yr M84/3
and by stellar collisions:
dM / dtcoll = 0.1 M / yr M83
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 Mcore / tR, 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, tcoll < tR, and collisions may destroy the cusp (Norman & Silk 1983).