A log-quadratic relation between the nuclear black hole masses and velocity dispersions of galaxies

Abstract

We demonstrate that a log-linear relation does not provide an adequate description of the correlation between the masses of super massive black holes (SMBHs, Mbh) and the velocity dispersions of their host spheroid (σ). An unknown relation between and log σ may be expanded to second order to obtain a log-quadratic relation of the form. We perform a Bayesian analysis using the local sample described in Tremaine et al., and solve for and α, in addition to the intrinsic scatter (δ). We find unbiased parameter estimates of and. At the 90 per cent level the relation does not follow a uniform power law. Indeed, over the velocity range the logarithmic slope of the best-fitting relation varies between 2.7 and 5.1, which should be compared with a power-law estimate of. The addition of the 14 galaxies with reverberation SMBH masses and measured velocity dispersions to the local SMBH sample leads to a log-quadratic relation with the same best fit as the local sample. However, the addition of the reverberation masses increases the significance of the log-quadratic contribution, yielding a value of β2 that is non-zero at the 5σ level. Furthermore, assuming no systematic offset, single epoch virial SMBH masses estimated for active galactic nuclei (AGNs) follow the same log-quadratic relation as the local sample, but extend it downward in mass by an order of magnitude. The log-quadratic term in the relation has a significant effect on estimates of the local SMBH mass function at, leading to densities of SMBHs with that are several orders of magnitude larger than inferred from a log-linear relation. We also estimate unbiased parameters for the SMBH-bulge mass relation using the sample assembled by Häring and Rix. With a parametrization, we find and. We determined an intrinsic scatter which is ∼50 per cent larger than the scatter in the relation.