  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.
