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$\beta$ and H$\alpha$
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.
