An extended size-luminosity relation for the reverberation-mapped AGNs:
  the role of the accretion rate
  For a compiled sample of 120 reverberation-mapped AGNs, the bivariate
correlations of the broad-line regions (BLRs) size ($R_{\rm BLR}$) with the
continuum luminosity at 5100 \AA\ ($L_{5100}$) and the dimensionless accretion
rates ($\dot{\mathscr{M}}$) are investigated. Using our recently calibrated
virial factor $f$, and the velocity tracer from the H$\beta$ Full-width at
half-maximum (FWHM(H$\beta$)) or the line dispersion ($\sigma_{\rm H\beta}$)
measured in the mean spectra, three kinds of SMBH masses and
$\dot{\mathscr{M}}$ are calculated. An extended \RL relation including
$\dot{\mathscr{M}}$ is found to be stronger than the canonical $R_{\rm
BLR}({\rm H\beta}) - L_{\rm 5100}$ relation, showing smaller scatters. The
observational parameters, $R_{\rm Fe}$ (the ratio of optical Fe II to H$\beta$
line flux) and the line profile parameter $D_{\rm H\beta}$ ($D_{\rm H\beta}=\rm
FWHM(H\beta)/\sigma_{\rm H\beta}$), have relations with three kinds of
$\dot{\mathscr{M}}$. Using $R_{\rm Fe}$ and $D_{\rm H\beta}$ to substitute
$\dot{\mathscr{M}}$, extended empirical $R_{\rm BLR}({\rm H\beta}) - L_{\rm
5100}$ relations are presented. $R_{\rm Fe}$ is a better "fix" for the $R_{\rm
BLR}({\rm H\beta}) - L_{\rm 5100}$ offset than the H$\beta$ shape $D_{\rm
H\beta}$. The extended empirical $R_{\rm BLR}({\rm H\beta}) - L_{\rm 5100}$
relation including $R_{\rm Fe}$ can be used to calculate $R_{\rm BLR}$, and
thus the single-epoch SMBH mass $M_{\rm BH}$. Our measured accretion rate
dependence is not consistent with the simple model of the accretion disk
instability leading the BLRs formation. The BLR may instead form from the inner
edge of the torus, or from some other means in which BLR size is positively
correlated with accretion rate and the SMBH mass.
