Formation of Supermassive Black Holes in Galactic Bulges: A Rotating Collapse Model Consistent with theMBH‐σ Relation

Motivated by the observed correlation between black hole masses $\mbh$ andthe velocity dispersion $\sigma$ of host galaxies, we develop a theoreticalmodel of black hole formation in galactic bulges (this paper generalizes anearlier ApJ Letter). The model assumes an initial state specified by a auniform 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). Theblack hole mass is determined when the centrifugal radius of the collapse flowexceeds the capture radius of the central black hole (for Schwarzschildgeometry). This model reproduces the observed correlation between the estimatedblack 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 weidentify 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 observedvalues. 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 showshow these results depend on the assumed initial conditions; the most importantquantity is the initial distribution of specific angular momentum in thepre-collapse state.Comment: 31 pages, 4 figures, accepted to Ap