The M_BH - T relation for a black hole in thermodynamic equilibrium with   the surrounding intergalactic medium
  I consider a toy model of self-regulated black hole accretion. The black hole grows through Bondi accretion and a fraction of the accretion power is distributed as thermal feedback into the surrounding gas. The gas expands or contracts until AGN heating and radiative cooling balance each other. The balance of heating and cooling is used to determine a quasi-equilibrium temperature at which the black hole accretes in self-regulated equilibrium with the surrounding intergalactic medium. This temperature grows with the black hole mass. The temperature increase is very steep around a critical black hole mass due to the shape of the cooling function. The quasi-equilibrium temperature cannot exceed the virial temperature or the AGN will drive a thermal wind. This limits the black hole mass to a maximum value determined by the depth of the potential well. In the regime in which cooling is dominated by bremsstrahlung, this model determines a relation between black hole mass and halo characteristic velocity of the form M_BH ~ v^4. The predictions of the model are consistent with the observed black hole mass -- bulge velocity dispersion relation. 
Andrea Cattaneo