Virial masses of black holes from single epoch spectra of active galactic nuclei

We describe the general problem of estimating black hole masses of active galactic nuclei (AGNs) by calculating the conditional probability distribution of M_BH given some set of observables. Special attention is given to the case where one uses the AGN continuum luminosity and emission line widths to estimate M_BH, and we outline how to set up the conditional probability distribution of M_BH given the observed luminosity, line width, and redshift. We show how to combine the broad-line estimates of M _BH with information from an intrinsic correlation between M _BH and L, and from the intrinsic distribution of M_BH, in a manner that improves the estimates of M_BH. Simulation was used to assess how the distribution of M_BH inferred from the broad-line mass estimates differs from the intrinsic distribution, and we find that this can lead to an inferred distribution that is too broad. We use these results and a sample of 25 sources that have recent reverberation mapping estimates of AGN black hole masses to investigate the effectiveness of using the C IV emission line to estimate M_BH and to indirectly probe the C IV region size-luminosity (R-L) relationship. A linear regression of log L _λ (1549 Å) on log M_BH found that L ₁₅₄₉ ∝ M_BH^1.17±0.22. A linear regression also found that M_BH ∝ L₁₅₄₉ ^0.41±0.07FWHM_{C IV}², implying a C IV R-L relationship of the form R_{C IV} ∝ L₁₅₄₉ ^0.41±0.07. Including the C IV line FWHM resulted in a reduction of a factor of ∼ 1/3 in the error in the estimates of M _BH over simply using the continuum luminosity, statistically justifying its use. We estimated M_BH from both C IV and Hβ for a sample of 100 sources, including new spectra of 29 quasars. We find that the two emission lines give consistent estimates if one assumes R ∝ L _UV^1/2 for both lines.