A new method to measure the virial factors in the reverberation mapping of AGNs
Based on the gravitational redshift, one prediction of Einstein's general relativity theory, of broad optical emission lines in active galactic nuclei (AGNs), a new method is proposed to estimate the virial factors $f$ in measuring black hole masses $M_{\rm{RM}}$ by the reverberation mapping of AGNs. The factors $f$ can be measured on the basis of two physical quantities, i.e. the gravitational redshifts $z_{\rm{g}}$ and full widths at half maxima $v_{\rm{FWHM}}$ of broad lines. In the past it has been difficult to determine the factors $f$ for individual AGNs. We apply this new method to several reverberation mapped Seyfert 1 galaxies. There is a correlation between $f$ and broad-line region (BLR) radius $r_{\rm{BLR}}$, $f=5.4 r_{\rm{BLR}}^{0.3}$, for the gravitationally redshifted broad lines He II, He I, H$β$ and H$α$ in narrow-line Seyfert 1 galaxy (NLS1) Mrk 110. This correlation results from the radiation pressure influence of the accretion disc on the BLR clouds. The radiation pressure influence seems to be more important than usually thought in AGNs. Mrk 110 has $f \approx$ 8--16, distinctly larger than the mean $\langle f\rangle \approx 1$, usually used to estimate $M_{\rm{RM}}$ in the case of $v_{\rm{FWHM}}$. NGC 4593 and NLS1 Mrk 486 has $f\approx 3$ and $f\approx 9$, respectively. Higher $f$ values of several tens are derived for three other NLS1s. There is a correlation between $f$ and accretion rate $\mathscr{\dot M}_{f=1}$, $f=6.8\mathscr{\dot M}^{0.4}_{f=1}$ for five objects, where $\mathscr{\dot M}_{f=1}=\dot M_{\bullet}/L_{\rm{Edd}}c^{-2}$ as $f=1$ is assumed to estimate $M_{\rm{RM}}$ used in the Eddington luminosity $L_{\rm{Edd}}$, $\dot M_{\bullet}$ is the mass accretion rate, and $c$ is the speed of light. These larger $f$ values will produce higher $M_{\rm{RM}}$ values and lower Eddington ratios.