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

TL;DR

This study investigates the variability of the virial factor $f$ in two active galactic nuclei, NGC 5548 and NGC 4151, revealing correlations with observational parameters and linking $f$ variations to BLR dynamics and inclination changes.

Contribution

It provides the first detailed analysis of $f$ variability over multiple epochs, demonstrating its dependence on line width, Eddington ratio, and inclination, and models these variations with a damped random walk.

Findings

$f$ correlates with line width and Eddington ratio.

Inclination changes can explain $f$ variability.

$f$ variation timescales are consistent with BLR dynamical timescales.

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.