Feeding black holes at galactic centres by capture from isothermal cusps

We estimate the rate at which stars are captured by supermassive black holes (BHs) in the centres of bulges and elliptical galaxies assuming that these initially had an isothermal cusp (rho ~ r^{-2} with velocity dispersion sigma_*). If radial orbits can be adequately replenished a BH can grow significantly by capturing stars and particles of the dark halo; the latter could be up to 20--40 percent of the total mass captured. A tight relation M_bh 10^8 (sigma_*/200\kms)^5 between BH hole mass and stellar velocity dispersion arises at the high mass end M_bh >= 10^8 M_sun if these giant BHs grow primarily by swallowing stars. For smaller masses a gradual flattening of the relation and an increase of the scatter are expected due to tidal disruption of stars. At moderate redshifts when bright galaxies may still possess isothermal cusps tidal disruption flares may be as frequent as 10^{-4} -- 10^{-2} yr^{-1} per galaxy, a few percent of the supernovae rate. Efficient randomization of the orbits can occur when the dense cores of infalling satellites sink to the sphere of influence of the BH by dynamical friction. Efficient replenishment of the loss cone of radial orbits can also explain why the supermassive binary BHs expected in hierarchically merging galaxies do generally coalesce, as suggested by the observed relation between BH mass and the inferred mass of stars ejected from an isothermal cusp.