Star Formation-Regulated Growth of Black Holes in Protogalactic Spheroids

The observed relation between central black hole mass and spheroid velocity dispersion is interpreted in terms of a self-regulation model that incorporates a viscous Keplerian accretion disk to feed the black hole, embedded in a massive, self-gravitating star-forming disk that eventually populates the spheroid. The model leads to a constant ratio between black hole mass and spheroid mass that is equal to the inverse of the critical Reynolds number for the onset of turbulence in the accretion disk surrounding the central black hole. Applying the fundamental plane correlation for spheroids, we find that the black hole mass has a power-law dependence on the spheroid velocity dispersion with a slope in the range of 4-5. We interpret the larger scatter in the Magorrian relation with respect to the black hole mass-spheroid velocity dispersion relationship as a result of secular evolution of the spheroid that primarily affects its luminosity and to a much lesser extent its velocity dispersion.