Calibration of the virial factor $f$ in supermassive black hole masses
  of reverberation-mapped AGNs
  Using a compiled sample of 34 broad-line active galactic nuclei (AGNs) with
measured H$\beta$ time lags from the reverberation mapping (RM) method and
measured bulge stellar velocity dispersions $\sigma_*$, we calculate the virial
factor $f$ by assuming that the RM AGNs intrinsically obey the same $M_{\rm
BH}-\sigma_*$ relation as quiescent galaxies, where $M_{\rm BH}$ is the mass of
the supermassive black hole (SMBH). Considering four tracers of the velocity of
the broad-line regions (BLRs), i.e., the H$\beta$ line width or line dispersion
from the mean or rms spectrum, there are four kinds of the factor $f$. Using
the \hb Full-width at half-maximum (FWHM) to trace the BLRs velocity, we find
significant correlations between the factor $f$ and some observational
parameters, e.g., FWHM, the line dispersion. Using the line dispersion to trace
the BLRs velocity, these relations disappear or become weaker. It implies the
effect of inclination in BLRs geometry. It also suggests that the variable $f$
in $M_{\rm BH}$ estimated from luminosity and FWHM in a single-epoch spectrum
is not negligible. Using a simple model of thick-disk BLRs, we also find that,
as the tracer of the BLRs velocity, H$\beta$ FWHM has some dependence on the
inclination, while the line dispersion $\sigma_{\rm H\beta }$ is insensitive to
the inclination. Considering the calibrated FWHM-based factor $f$ from the mean
spectrum, the scatter of the SMBH mass is 0.39 dex for our sample of 34 low
redshift RM AGNs. For a high redshift sample of 30 SDSS RM AGNs with measured
stellar velocity dispersions, we find that the SMBH mass scatter is larger than
that for our sample of 34 low redshift RM AGNs. It implies the possibility of
evolution of the $M_{\rm BH}-\sigma_*$ relation from high-redshift to
low-redshift AGNs.
