The virial factor $f$ of the H$\beta$ Broad-line for NGC 5548 and NGC 4151

Abstract

The variability of virial factor $f$ is investigated for two active galactic nucleus, NGC 5548 and NGC 4151, which had been previously reverberation mapped (RM) over 20 times in the past 30 years. Using four velocity tracers from the broad H$\beta$ width at half-maximum ($\rm FWHM_{\rm H\beta} $) or the line dispersion from the mean or rms spectra, $f$ for each RM epoch are calculated. Significant correlations are found between $f$ and observational parameters, such as the broad line widths, the Eddington ratios and the line profile shapes. For NGC 5548, $f \propto \rm {FWHM}_{mean}^{-0.70\pm0.13}$ and for NGC 4151, $f \propto \rm {FWHM}_{mean}^{-3.31\pm0.59}$. This suggests that a variable $f$ should be included to weight the virial SMBH mass. Using a simple model of thick-disc broad line regions (BLRs), we show that changes in mean inclination can explain $f$ variation. The inclination range is $14.1-40.6$ deg for NGC 5548 and $14.0-55.1$ deg for NGC 4151. Modeling the light curves of $f$ with a damped random walk process yields mean timescales of 638 and 668 days, consistent with BLR dynamical timescales within uncertainties. This indicates that $f$ variations are linked to BLR dynamics, likely due to changes in geometry or inclination.