The M•–σ∗ relation for black holes fed by capturing stars and dark matter from an isothermal cusp

The capture rates of stars onto supermassive black holes depend strongly on the spatial and kinematical distribution of the stars at the centre of bulges and elliptical galaxies. We here explore the possibility that all ellipticals/bulges have initially isothermal cusps (ρ ∝ r). If the orbits can be adequately randomized a significant fraction of the total mass of black holes in the bulges of galaxies will be due to the capture of stars and dark matter. The dark matter fraction of the total mass captured may be as high as 20–40 percent for typical cold dark matter halos. A tight relation M• ∼ 10 8(σ∗/200 km s ) between black hole mass and stellar velocity dispersion can arise at the high mass end (M• ≥ 10 8M⊙) if these giant black holes grow primarily by the capture of stars without tidal disruption. For smaller black holes a shallower M• − σ∗ relation with larger scatter is expected. Efficient randomization of the orbits can be due to the dense central regions of infalling satellites which can avoid tidal disruption and sink to the sphere of influence by dynamical friction. The presence of an isothermal cusp and the reduction of the relaxation time scale at the sphere of influence enhance the estimated tidal disruption rate of stars to ∼ 10 – 10 yr per galaxy. Disruption flares in bright galaxies may thus be as frequent as a few percent of the supernovae rate at moderate redshifts when the galaxies still had an isothermal cusp. The efficient replenishment of the loss cone also explains why the supermassive binary black holes expected in hierarchically merging galaxies do generally coalesce as suggested by the observed relation between black hole mass and the inferred mass of stars ejected from an isothermal cusp.