The Relationship Between Luminosity and Broad-Line Region Size in Active   Galactic Nuclei
  We reinvestigate the relationship between the characteristic broad-line region size (R_blr) and the Balmer emission-line, X-ray, UV, and optical continuum luminosities. Our study makes use of the best available determinations of R_blr for a large number of active galactic nuclei (AGNs) from Peterson et al. Using their determinations of R_blr for a large sample of AGNs and two different regression methods, we investigate the robustness of our correlation results as a function of data sub-sample and regression technique. Though small systematic differences were found depending on the method of analysis, our results are generally consistent. Assuming a power-law relation R_blr \propto L^\alpha, we find the mean best-fitting \alpha is about 0.67+/-0.05 for the optical continuum and the broad H\beta luminosity, about 0.56+/-0.05 for the UV continuum luminosity, and about 0.70+/-0.14 for the X-ray luminosity. We also find an intrinsic scatter of about 40% in these relations. The disagreement of our results with the theoretical expected slope of 0.5 indicates that the simple assumption of all AGNs having on average same ionization parameter, BLR density, column density, and ionizing spectral energy distribution, is not valid and there is likely some evolution of a few of these characteristics along the luminosity scale. 
Shai Kaspi, Dan Maoz, Hagai Netzer, Bradley M. Peterson, Marianne   Vestergaard, and Buell T. Jannuzi