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 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:

• 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 = 10⁷- 3 10⁸ M, a density 10⁷ pc^-3, the accretion rate is, by tidal disruption:

  dM / dt_tide = 1 M / yr M₈^4/3

  and by stellar collisions:

  dM / dt_coll = 0.1 M / yr M₈³

• 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).