Galaxies become red and dead when the central supermassive black hole (SMBH) becomes massive enough to drive an outflow beyond the virial radius of the halo. We show that this final SMBH mass is larger than the final SMBH mass in the bulge of a spiral galaxy by up to an order of magnitude. The relations in the two galaxy types are almost parallel (M ∝ σ4 + β, with β < 1) but offset in normalization, with the extra SMBH mass supplied by the major merger transforming the galaxy into an elliptical, or by mass gain in a galaxy cluster. This agrees with recent findings that the SMBH in two brightest cluster Galaxies are ∼10 × the expected mass. We show that these results do not strongly depend on the assumed profile of the dark matter halo, so analytic estimates found for an isothermal potential are approximately valid in all realistic cases.
Our results imply that there are in practice actually three relations, corresponding to spiral galaxies with evolved bulges, field elliptical galaxies and cluster centre elliptical galaxies. A fourth relation, corresponding to cluster spiral galaxies, is also possible, but such galaxies are expected to be rare. All these relations have the form MBH = Cnσ4, with only a slight difference in slope between field and cluster galaxies, but with slightly different coefficients Cn. Conflating data from galaxies of different types and fitting a single relation to them tends to produce a higher power of σ.