Formation of Supermassive Black Holes in Galactic Bulges: A Rotating Collapse Model Consistent with the $\mbh-σ$ Relation

Motivated by the observed correlation between black hole masses $\mbh$ and the velocity dispersion $\sigma$ of host galaxies, we develop a theoretical model of black hole formation in galactic bulges (this paper generalizes an earlier ApJ Letter). The model assumes an initial state specified by a a uniform rotation rate $\Omega$ and a density distribution of the form $\rho = \aeff^2 / 2 \pi G r^2$ (so that $\aeff$ is an effective transport speed). The black hole mass is determined when the centrifugal radius of the collapse flow exceeds the capture radius of the central black hole (for Schwarzschild geometry). This model reproduces the observed correlation between the estimated black hole masses and the velocity dispersions of galactic bulges, i.e., $\mbh \approx 10^8 M_\odot (\sigma/200 {\rm km s^{-1}})^4$, where $\sigma = \sqrt{2} \aeff$. To obtain this normalization, the rotation rate $\Omega \approx 2 \times 10^{15}$ rad/s. The model also defines a bulge mass scale $M_B$. If we identify the scale $M_B$ with the bulge mass, the model determines the ratio $\mrat$ of black hole mass to the host mass: $\mrat$ $\approx$ 0.0024 $(\sigma/200 {\rm km s^{-1}})$, again in reasonable agreement with observed values. In this scenario, supermassive black holes form quickly (in $\sim10^5$ yr) and are born rapidly rotating (with $a/M \sim 0.9$). This paper also shows how these results depend on the assumed initial conditions; the most important quantity is the initial distribution of specific angular momentum in the pre-collapse state.