A Theoretical Model for the $M_{\rm bh}-\sigma$ Relation for Supermassive Black Holes in Galaxies

Abstract

We construct a model for the formation of black holes within galactic bulges. The initial state is a slowly rotating isothermal sphere, characterized by effective transport speed $\aeff$ and rotation rate $\Omega$. The black hole mass is determined when the centrifugal radius of the collapse flow exceeds the capture radius of the central black hole. This model reproduces the observed correlation between black hole masses and galactic velocity dispersions, $\mbh \approx 10^8 M_\odot (\sigma/200 \kms)^4$, where $\sigma = \sqrt{2} \aeff$. This model also predicts the ratio $\mrat$ of black hole mass to host mass: $\mrat$ $\approx$ 0.004 $(\sigma/200 \kms)$.