# The Relation between the Optical Emission and the Dimensionless Accretion
Rate for Active Galactic Nuclei

Yan-Sheng Liu  School of Physics and Technology, Nanjing Normal University,
Nanjing 210023, China  Wei-Hao Bian  School of Physics and Technology, Nanjing
Normal University, Nanjing 210023, China

(Received XXXX XX, 2020)

###### Abstract

It was suggested that the prominent feature of the optical emission has a
connection with the accretion process in active galactic nuclei (AGN). For a
large sample of 4037 quasars ($z<0.8$) with measured H$\beta$ line dispersion
($\sigma_{\rm H\beta}$ ) selected from the Sloan Digital Sky Survey (SDSS) and
120 compiled reverberation-mapped (RM) AGN, we use $\sigma_{\rm H\beta}$ and
the extended $R_{\rm BLR}({\rm H\beta})-L_{\rm 5100}$ relation to calculate
supermassive black holes masses ($M_{\rm BH}$ ) from the single-epoch spectra
for the SDSS subsample, and $\sigma_{\rm H\beta}$ from the mean spectra for
the RM subsample. We find a strong correlation between the relative optical
strength $R_{\rm Fe}$ and for the SDSS subsample with the Spearman correlation
coefficient $r_{s}$ of $0.727$, which is consistent with that derived from the
mean spectra for the RM subsample. The magnitude of velocity shift of the
optical emission has a strong anticorrelation with , whenever there is inflow
or outflow. These strong correlations show that the optical emission has an
intimate connection with the accretion process. Assuming that the difference
of $M_{\rm BH}$ is due to the variable virial factor $f$ for adopting $\rm
FWHM_{H\beta}$ as the velocity tracer, we find that there is a relation
between $f$ and $\rm FWHM_{H\beta}$, $\log f=-(0.41\pm 0.002)\rm\log
FWHM_{H\beta}+(1.719\pm 0.009)$ for the single-epoch spectrum. The relation
between $\log f$ and $\sigma_{\rm H\beta}$ is not too strong, suggesting that
$\sigma_{\rm H\beta}$ does not seem to depend much on the broad-line region
inclination and a constant $\sigma$-based $f$ is suitable for $\sigma_{\rm
H\beta}$ as the velocity tracer.

galaxies: active – galaxies: nuclei – galaxies: Seyfert – quasars: emission
lines – quasars: general

##  1 Introduction

emissions shown in the optical and ultraviolet (UV) spectra are prominent
features in most active galactic nuclei (AGN; e.g., Boroson & Green, 1992; Hu
et al., 2008; Shen & Ho, 2014; Zhao et al., 2020). With the principal
component analysis of a low-$z$ sample of 87 Palomar–Green quasars, Boroson &
Green (1992) found that principal component 1 (PC1) is related to the relative
strength of optical to the broad H$\beta$ ($R_{\rm Fe}$ , the ratio between
the strength of emission within $4434-4684$ Å and the broad H$\beta$ ), the
FWHM of the broad H$\beta$ ($\rm FWHM_{H\beta}$), [O iii]5007 strength, and
principal component 2 (PC2) links optical luminosity and $\alpha_{\rm ox}$
(optical-X-ray spectral index). PC1 and PC2 have relations with the accretion
process around the supermassive black holes (SMBHs). With $\rm
FWHM_{H\beta}$-based SMBH masses ($M_{\rm BH}$ ), Boroson (2002) suggested
that PC1 is mainly correlated with the Eddington ratio ($L_{\rm Bol}/L_{\rm
Edd}$, $L_{\rm Bol}$ is the bolometric luminosity and $L_{\rm Edd}$ is the
Eddington luminosity) and PC2 has a strong connection with $M_{\rm BH}$ and
$L_{\rm Bol}/L_{\rm Edd}$ . Using a large sample of the Sloan Digital Sky
Survey (SDSS) quasars, Hu et al. (2008) did a detailed line-fitting technique
and found the connections between the optical strength, velocity shift, and
the Eddington ratio. Using a two-dimensional plane of $\rm FWHM_{H\beta}$ and
$R_{\rm Fe}$ for the SDSS quasars, Shen & Ho (2014) suggested that the
Eddington ratio is also a key factor to be considered in the orientation-based
unification of quasar phenomenology (Urry & Padovani, 1995).

In order to investigate the relation between features and the accretion
process, $M_{\rm BH}$ is a key parameter needing to be determined. With effort
over nearly two decades, the reverberation-mapping (RM) method with the
H$\beta$ spectral monitoring has been successfully applied for about 120 AGN
(e.g., Blandford & McKee, 1982; Peterson et al., 2004; Du et al., 2016a; Grier
et al., 2017; Yu et al., 2020a, b; Hu et al., 2021). The RM method can measure
the time lag between the line variation ( e.g., H$\beta$ , Mg ii, C iv) and
the corresponding continuum variation, and give the broad-line region (BLR)
distance from the central SMBH ($R_{\rm BLR}$). For type 1 AGN, the BLR clouds
emitting the broad H$\beta$ line can be used as a probe to calculate the
virial mass (e.g., Peterson et al., 2004; Netzer, 2013):

$M_{\rm BH}=f\times\frac{R_{\rm BLR}~{}(\Delta V)^{2}}{G}.$ (1)

where $\Delta V$ is the velocity of the BLR clouds, $f$ is a corresponding
virial factor, and $G$ is the gravitational constant. Through the RM AGN, an
empirical $R_{\rm BLR}-L_{5100}$ relation ($L_{\rm 5100}$ is the 5100 Å
monochromatic luminosity, $R_{\rm BLR}\propto L_{5100}^{0.5}$) was given and
used to estimate $R_{\rm BLR}$ from a single-epoch spectrum (e.g., Kaspi et
al., 2000; Bentz et al., 2013; Du et al., 2016b; Dalla Bontà et al., 2020;
Khadka et al., 2022; Maithil et al., 2022). The changes of the spectral energy
distribution, connected to changes in the accretion rate, would likely lead to
the breadth (an increment in the scatter) in the empirical $R_{\rm
BLR}-L_{5100}$ relation in the first-order consideration (Wang et al., 2014;
Martínez-Aldama et al., 2019; Du & Wang, 2019; Yu et al., 2020a; Maithil et
al., 2022). AGN with strong emission show a smaller $R_{\rm BLR}$ than
expected (e.g., Du & Wang, 2019). Since $R_{\rm Fe}$ has a relation with the
accretion rate (e.g., Netzer & Trakhtenbrot, 2007; Du et al., 2016a; Yu et
al., 2020a; Maithil et al., 2022), an extended $R_{\rm BLR}({\rm
H\beta})-L_{\rm 5100}$ relation ($R_{\rm BLR}\propto L_{5100}^{0.5}+R_{\rm
Fe}$) was recently suggested (Du & Wang, 2019; Yu et al., 2020a; Khadka et
al., 2022). The reduction in the scatter in the extended $R_{\rm BLR}({\rm
H\beta})-L_{\rm 5100}$ relation (Yu et al., 2020a) is in agreement with the
one ($\sim 0.13$ dex) previously reported by Kilerci Eser et al. (2015), who
used the UV luminosity instead of $L_{5100}$ as a substitute for the ionizing
luminosity in the empirical $R_{\rm BLR}({\rm H\beta})-L_{\rm 5100}$ relation.
The extended $R_{\rm BLR}({\rm H\beta})-L_{\rm 5100}$ relation would result in
a better measurement of $M_{\rm BH}$ .

For the RM AGN, $\Delta V$ is usually traced by four kinds of velocities,
i.e., the broad H$\beta$ FWHM or the line dispersion ($\sigma_{\rm H\beta}$)
measured from the mean/rms spectrum (e.g., Peterson et al., 2004; Ho & Kim,
2014; Yu et al., 2020b). From Equation 1, $f=M_{\rm BH}/(R_{\rm BLR}(\Delta
V)^{2}/G)$. For four kinds of velocities $\Delta V$, the corresponding four
kinds of virial factors $f$ are defined, which are usually calibrated through
the $M_{\rm BH}-\sigma_{\ast}$ relation ($\sigma_{*}$ is the bulge stellar
velocity dispersion) or other independent methods to derive the SMBH masses
(e.g., Onken et al., 2004; Ho & Kim, 2014; Yu et al., 2020b). For several
multiple RM AGN, it was found $\sigma_{\rm H\beta}$ is better than $\rm
FWHM_{H\beta}$ used to calculate $M_{\rm BH}$ with a constant factor $f$
(Peterson et al., 2004). For the 120 RM AGN sample (Yu et al., 2020b), with
respect to $M_{\rm BH}$ from $\sigma_{*}$ or $\sigma_{\rm H\beta,rms}$, it was
found that we can obtain $M_{\rm BH}$ from $\sigma_{\rm H\beta,mean}$ with the
smallest scatter than from $\rm FWHM_{H\beta}$. Dalla Bontà et al. (2020) also
suggested that the use of $\sigma_{\rm H\beta}$ gives better results, while
the use of $\rm FWHM_{H\beta}$ introduces a bias, stretching the mass scale
such that high masses are overestimated and low masses are underestimated,
although both velocity tracers are usable. A variable $\rm FWHM$-based $f$ was
suggested (Collin et al., 2006; Mejía-Restrepo et al., 2018; Yu et al., 2019,
2020b). The cumulative fraction of $f$ was consistent with the a simple model
of thick-diskBLRs, which implied that, as the tracer of BLRs velocity, $\rm
FWHM_{H\beta}$ has some dependence on the BLRs inclination, while $\sigma_{\rm
H\beta}$ is insensitive to the inclination (Collin et al., 2006; Yu et al.,
2019). For AGN with the single-epoch spectrum, $\rm FWHM_{H\beta}$ or
$\sigma_{\rm H\beta}$ is used to trace $\Delta V$, which was used in large
surveys, such as SDSS (Shen et al., 2011). With respect to the H$\beta$ FWHM,
$\sigma_{\rm H\beta}$ was preferred to calculate $M_{\rm BH}$ from the single-
epoch spectrum (Yu et al., 2020b). With Equation 1, $M_{\rm
BH}=f_{\sigma}\times\sigma_{\rm H\beta}^{2}R_{\rm BLR}({\rm H\beta})/G=f_{\rm
FWHM}\times{\rm FWHM}_{H\beta}^{2}R_{\rm BLR}({\rm H\beta})/G$. Considering a
variable $f_{\rm FWHM}$ in $\rm FWHM_{H\beta}$-based $M_{\rm BH}$ , it means

$f_{\rm FWHM}=f_{\sigma}(\sigma_{\rm H\beta}/\rm FWHM_{H\beta})^{2}.$ (2)

About the accretion strength, there are mainly two parameters, the Eddington
ratio $L_{\rm Bol}$ /$L_{\rm Edd}$ and the dimensionless accretion rate ,
where $\mdot\equiv\dot{M}/\dot{M}_{\rm Edd}$, $\dot{M}_{\rm Edd}=L_{\rm
Edd}/c^{2}$. The accretion rate $\dot{M}$ can be derived from the disk model
of Shakura & Sunyaev (1973), which has been extensively applied to fit the
spectra of AGN (e.g., Collin et al., 2002; Bian & Zhao, 2003; Davis & Laor,
2011; Mejía-Restrepo et al., 2018). Considering that the radius $R$
distribution of the effective disk temperature is given by $T_{\rm eff}\propto
R^{-3/4}$, is (e.g., Du et al., 2016a)

$\mdot\equiv\dot{M}/\dot{M}_{\rm
Edd}=20.1\left(\frac{l_{44}}{\cos\,\it{i}}\right)^{3/2}m_{7}^{-2}.$ (3)

where $l_{44}=L_{5100}/10^{44}{\rm erg\ s}^{-1}$ , $m_{7}=M_{\rm
BH}/10^{7}M_{\odot}$. An average value of ${\rm cos}~{}i=0.75$ is adopted,
which corresponds to the opening angle of the dusty torus (e.g., Davis & Laor,
2011; Du et al., 2016a). is inversely proportional to $M_{\rm BH}$ to the
second power. This Equation applies to AGN that have $\log(\mdot/M_{\odot})$
between -2 and 3.5, namely excluding the regimes of advection-dominated
accretion flows ($\log(\mdot/M_{\odot})<-2$) and of flows with hyperaccretion
rates ($\log(\mdot/M_{\odot})>3.5$). Using FWHM-based $M_{\rm BH}$ and $L_{\rm
Bol}$ /$L_{\rm Edd}$ for 63 RM AGN, Du et al. (2016a) found $R_{\rm Fe}$
correlated with $L_{\rm Bol}$ /$L_{\rm Edd}$ and (also see Netzer &
Trakhtenbrot, 2007; Hu et al., 2008; Maithil et al., 2022). They also
suggested the BLRs ”fundamental plane”, i.e., a strong bivariate correlation
of with $R_{\rm Fe}$ and the H$\beta$ shape $\rm FWHM_{H\beta}/\sigma_{\rm
H\beta}$.

In this paper, we use a large sample of 4037 SDSS quasars ($z<0.8$; Hu et al.,
2008) with measured broad H$\beta$ line dispersion $\sigma_{\rm H\beta}$ and
$R_{\rm Fe}$ to investigate again the relation between emission and the
accretion process, as well as a subsample of 120 RM AGN. About the accretion
strength, we use instead of $L_{\rm Bol}/L_{\rm Edd}$ . We also derive the
relation between $f_{\rm FWHM}$ and $\rm FWHM_{H\beta}$ for the single-epoch
spectrum. This paper is organized as follows. Section 2 presents the adopted
samples and data analysis. Section 3 is our results and discussions. Section 4
is our conclusions. All of the cosmological calculations in this paper assume
$\Omega_{\Lambda}=0.7$, $\Omega_{\rm M}=0.3$, and $H_{0}=70~{}{\rm km\
s}^{-1}{\rm Mpc}^{-1}$.

##  2 Samples and data Analysis

Figure 1: Distributions of $R_{\rm Fe}$ and $L_{5100}$ for the subsample of
4037 SDSS quasars (top panels, blue) and the subsample of 120 RM AGN (bottom
panels, red). The dashed lines are their mean values. The number, the mean
value, and the standard deviation are shown in panels.  Figure 2:
Distributions of $\rm FWHM_{H\beta}$, $\sigma_{\rm H\beta}$ , and their ratio
$\rm FWHM_{H\beta}/\sigma_{\rm H\beta}$ for the broad H$\beta$ . The blue
lines are for the subsample of 4037 SDSS quasars. The red lines are for the
subsample of 120 RM AGN. The fractional numbers are used to compare the
distributions for two subsamples.

Because we use $\sigma_{\rm H\beta}$ instead of $\rm FWHM_{H\beta}$ to
calculate $M_{\rm BH}$ , the spectral decomposition is crucial. We use a
subsample of 4037 quasars ($z<0.8$) from SDSS presented by Hu et al. (2008)
with criteria of signal-to-noise ratio $(\rm S/N)>10$ and $\rm
EW(Fe~{}II)>25~{}\AA$. We briefly introduce their spectral fitting (Hu et al.,
2008). The continuum is decomposed into three components: a single power law,
Balmer continuum emission, and a pseudocontinuum due to blended emission. For
the pure emission-line spectrum subtracting the continuum, the narrow H$\beta$
component, [O iii]4959, 5007 are modeled using Gaussians, and the broad
H$\beta$ component is modeled using a Gauss-Hermite function, whose best fit
yields the $\rm FWHM_{H\beta}$, $\sigma_{\rm H\beta}$ . At the same time, we
also use a compiled subsample of 120 RM AGN with measured four kinds of
velocity tracers from mean and rms spectra (Yu et al., 2020b, and references
therein). About the 120 RM subsample (Yu et al., 2020a), we mainly adopted the
data from Du et al. (2016a) and Shen et al. (2019). For 63 AGN in this RM
subsample, Du et al. (2016a) used the same fitting scheme as Hu et al. (2008).
They included the host correction in the fitting described in Hu et al.
(2015). For other 44 SDSS AGN in the RM subsample, Shen et al. (2019) used
almost the same fitting scheme as Hu et al. (2015). For this subsample of 120
RM AGN, the flux of broad optical is all measured by integration from 4434 to
4684 Å, $z$ is less than 1.026, host-corrected $L_{5100}$ is between
$10^{41.54}$ and $10^{45.95}$ erg s-1 . In order to use all of available RM
AGN, no criterion on $\rm EW(Fe~{}II)$ or $\rm S/N$ is used.

For the SDSS subsample, $L_{5100}$ , the luminosity, the broad H$\beta$
luminosity, $\rm FWHM_{H\beta}$ and $\rm\sigma_{H\beta}$ are adopted,
respectively, from columns (15), (3), (9), (11), (41) in Table 2 in Hu et al.
(2008). The ratio of the luminosity of to the broad H$\beta$ is calculated as
$R_{\rm Fe}$ with its error calculated from the error transfer formula. With
respect to [O iii] $\lambda$5007, the velocity shift and its error are adopted
from columns (7)-(8) in Table 2 in Hu et al. (2008). For the RM subsample,
$L_{5100}$ and $R_{\rm Fe}$ are adopted frm columns (3) and (7) in Table 1 in
Yu et al. (2020a), as well as their errors. Using the relative variance
fraction $F_{\rm var}$, Hu et al. (2015) gave RM for 10 narrow-line Seyfert 1
galaxies (NLS1s). They found that all 10 objects show variations with an
amplitude of a few to 10%. On average, the variability of is about 10% smaller
than the variability of the H$\beta$ line. As the variability of H$\beta$ is
unusually much larger than that of , the uncertainties of $R_{\rm Fe}$ are
mainly governed by H$\beta$ variability, which on average is $\sim 20\%$ (Hu
et al., 2015; Du et al., 2016a). $\rm FWHM_{\rm mean}$ , $\sigma_{\rm
H\beta,mean}$ , $\rm FWHM_{\rm rms}$ , $\sigma_{\rm H\beta,rms}$ and their
errors are adopted from columns (4)-(7) in Table 1 in Yu et al. (2020b).

In Figure 1, we show the distributions of $L_{5100}$ and $R_{\rm Fe}$ for the
SDSS subsample and the RM subsample. It is clear that the RM subsample has
averagely smaller $L_{5100}$ than the SDSS subsample. We perform the
Kolmogorov– Smirnov (K–S) test 111We used python (scipy.stats.ks_2samp,
scipy.stats.spearmanr) to do our following analysis for the K-S test and the
Spearman correlation test.  on distributions in Figure 1. The statistic d and
the significant level probability for the null hypothesis ($p$-value) are
-0.18 and 0.0001 for $R_{\rm Fe}$ , 0.52 and $2.22\times 10^{-16}$ for
$L_{5100}$ , respectively. The distribution of $R_{\rm Fe}$ has a difference
between the two subsamples and the difference of the $L_{5100}$ distribution
is more significant. In Figure 2, we show the distributions of $\rm
FWHM_{H\beta}$, $\sigma_{\rm H\beta}$ , and the broad H$\beta$ shape $D_{\rm
H\beta,mean}\equiv\rm FWHM_{H\beta}/\sigma_{\rm H\beta}$ from the mean
spectrum for the RM subsample (red lines). It was found that $D_{\rm
H\beta,mean}$ has a relation with $L_{\rm Bol}/L_{\rm Edd}$ or (e.g., Collin
et al., 2006; Du et al., 2016a; Yu et al., 2020a; Panda, 2022). We also show
the distribution of $\rm FWHM_{H\beta}$ and $\sigma_{\rm H\beta}$ from the
single-epoch spectrum for the SDSS subsample (blue lines). We perform the K–S
test on distributions in Figure 2 and find no significant differences between
the two samples. The statistic d and p-value for the null hypothesis are 0.17
and 0.03 for $\rm FWHM_{H\beta}$, 0.20 and 0.005 for $\sigma_{\rm H\beta}$ ,
0.24 and 0.0004 for $\rm FWHM_{H\beta}/\sigma_{\rm H\beta}$, respectively. The
value of $\rm FWHM_{H\beta}/\sigma_{\rm H\beta}$ is 2.35, 3.46, 2.45, 2.83,
and 0 for a Gaussian, a rectangular, a triangular, an edge-on rotating ring,
and a Lorentzian profile, respectively (Collin et al., 2006; Du et al.,
2016a). For the SDSS subsample, the mean value and its standard deviation for
$\rm FWHM_{H\beta}/\sigma_{\rm H\beta}$ are $1.78$ and $0.28$ (see right panel
in Figure 2). For the RM subsamples, the mean value and its standard deviation
are $1.94$ and $0.61$. They show the deviation from the value of $2.35$ for
the single Gaussian profile. There are 37 NLS1s with $\rm
FWHM_{H\beta}<2000~{}{\rm km\ s}^{-1}$ from the mean spectrum for the RM
subsample and 564 NLS1s from the single-epoch spectrum for the SDSS subsample
(e.g., Bian & Zhao, 2004). With respect to the proportion of the number of
NLS1s in the SDSS subsample ($\sim 14\%$), the higher proportion in the RM
subsample ($\sim 31\%$) is due to including an RM project searching for super-
Eddington accreting massive black holes (SEAMBHs; Wang et al., 2013; Du et
al., 2016a). The mean and its standard deviation of $D_{\rm H\beta,mean}$ for
NLS1s are $1.53$ and $0.27$ for the RM subsample, $1.50$ and $0.15$ for the
SDSS subsample, which are much deviated from the value of $2.35$ for the
single Gaussian profile.

For the SDSS subsample, $L_{5100}$ is from $\sim 10^{44}$ to $\sim 10^{46}$
erg s-1 , the mean value and the standard deviation of $\log L_{5100}$ are
44.55 and 0.37. We use the empirical formula to correct the host contribution
(Shen et al., 2011; Ge et al., 2016),

$\frac{L_{\rm 5100,host}}{L_{\rm
5100,QSO}}=0.8052-1.5502x+0.9121x^{2}-0.1577x^{3}.$ (4)

for $x+44\equiv\log(L_{\rm 5100,total}/{\rm erg\ s}^{-1})<45.053$. There are
3590/4037 AGN with luminosities below this value, and no correction is needed
for luminosities above this value. For host-corrected $\log(L_{5100}/{\rm erg\
s}^{-1})$, the mean value is 44.40, which is smaller than uncorrected mean
values of 44.55 by 0.15 dex.

##  3 Results and Discussions

###  3.1 $M_{\rm BH}$ , and $L_{\rm Bol}$ /$L_{\rm Edd}$

Figure 3: Distributions for $M_{\rm BH}$ and derived from $\sigma_{\rm
H\beta}$ (top two panels) and $\rm FWHM_{H\beta}$ (bottom two panels) for the
subsample of 4037 SDSS quasars (blue) and the subsample of 120 RM AGN (red).
The dashed lines are mean values. The number, the mean value, and the standard
deviation are shown in panels. Figure 4: $L_{\rm Bol}/L_{\rm Edd}$ versus for
the subsample of 4037 SDSS quasars (blue points) and the subsample of 120 RM
AGN (red points). The black and orange solid lines are the best linear
fittings for the SDSS subsample and the RM subsample.

An extended empirical $R_{\rm BLR}({\rm H\beta})-L_{\rm 5100}$ relation
including $R_{\rm Fe}$ was found for the RM AGN subsample (Du & Wang, 2019; Yu
et al., 2020a):

$\begin{split}\log\frac{R_{\rm BLR}({\rm H\beta})}{ltd}=(0.48\pm 0.03)\log
l_{44}-(0.38\pm 0.04)R_{\rm Fe}\\\ +(1.67\pm 0.09)\end{split}$ (5)

For $\log R_{\rm BLR}({\rm H\beta})$, it has an intrinsic scatter of 0.17 dex
(Yu et al., 2020a). We adopt a systematic error of 0.2 dex in our following
calculation. With the extended $R_{\rm BLR}({\rm H\beta})-L_{\rm 5100}$
relation, when $\sigma_{\rm H\beta}$ is used as the tracer of the virial
velocity, and $f$ is adopted as 5.5 (e.g., Yu et al., 2020a, b), the single-
epoch virial mass formula can be expressed as

$\begin{split}\log\frac{M_{\rm BH,\sigma_{\rm
H\beta}}}{M_{\odot}}=7.7+2\log\frac{\rm\sigma_{H\rm\beta}}{\rm 1000~{}{\rm km\
s}^{-1}}&\\\ +(0.48\pm 0.03)\log l_{44}-(0.38\pm 0.04)R_{\rm Fe}\end{split}$
(6)

When we use $\rm FWHM_{H\beta}$ as the tracer of the virial velocity, and
adopt $f$ as 1 (e.g., Du & Wang, 2019), the single-epoch virial mass formula
can be expressed as

$\begin{split}\log\frac{M_{\rm
BH,FWHM_{H\beta}}}{M_{\odot}}=6.96+2\log\frac{\rm FWHM_{H\beta}}{\rm
1000~{}{\rm km\ s}^{-1}}&\\\ +(0.48\pm 0.03)\log l_{44}-(0.38\pm 0.04)R_{\rm
Fe}\end{split}$ (7)

Substitute Equation 6 for Equation 3, the formula is:

$\begin{split}\log(\mdot_{\sigma_{\rm H\beta}})=(0.54\pm 0.06)\log
l_{44}-4\log\frac{\sigma_{\rm H\beta}}{\rm 1000~{}{\rm km\ s}^{-1}}&\\\
+(0.76\pm 0.08)R_{\rm Fe}+0.09\\\ \end{split}$ (8)

When using $\rm FWHM_{H\beta}$, a similar formula is:

$\begin{split}\log(\mdot_{\rm FWHM_{H\beta}})=(0.54\pm 0.06)\log
l_{44}-4\log\frac{\rm FWHM_{H\rm\beta}}{\rm 1000~{}{\rm km\ s}^{-1}}&\\\
+(0.76\pm 0.08)R_{\rm Fe}+1.57\\\ \end{split}$ (9)

We use a correction factor to calculate $L_{\rm Bol}$ from $L_{5100}$ ,
$L_{\rm Bol}=BC_{\rm 5100}L_{5100}$, where $BC_{\rm
5100}=53-\log(L_{5100}/{\rm erg\ s}^{-1})$ (Marconi et al., 2004; Netzer,
2013). In the following sections, if not stated, $\rm FWHM_{H\beta}$ and
$\sigma_{\rm H\beta}$ are from the mean spectrum for the RM subsample, and we
will adopt the $\sigma_{\rm H\beta}$-based $M_{\rm BH}$ and from the single-
epoch spectrum for the SDSS subsample and from the mean spectrum for the RM
subsample.

For the SDSS subsample of 4037 AGN, using $\sigma_{\rm H\beta}$ or $\rm
FWHM_{H\beta}$ from the single-epoch spectrum and the extended $R_{\rm
BLR}({\rm H\beta})-L_{\rm 5100}$ relation, from Equations 6 or 7, we calculate
$M_{\rm BH}$ . Based on the error transfer formula , the error of $M_{\rm BH}$
is calculated from the errors of $\sigma_{\rm H\beta}$ or $\rm FWHM_{H\beta}$,
$L_{5100}$ , $R_{\rm Fe}$ , including a systematic error of 0.2 dex for
$R_{\rm BLR}({\rm H\beta})$ . For the RM subsample of 120 RM AGN, using
$\sigma_{\rm H\beta}$ or $\rm FWHM_{H\beta}$ from the mean spectrum and
$R_{\rm BLR}({\rm H\beta})$ measured through RM technique, from Equation 1,
$M_{\rm BH}$ was presented in Yu et al. (2020b), as well as its error. We use
Equation 3 to calculate , as well as its error. In Figure 3, we show the
distributions of $M_{\rm BH}$ (left panels) and (right panels) for the SDSS
subsample and the RM subsample. Left two top panels show the mean value and
the standard deviation of $\log M_{\rm BH}$ from $\sigma_{\rm H\beta}$ are
8.13 and 0.51 for the SDSS subsample; 7.73 and 0.73 for the RM subsample. The
former is averagely larger than the latter by 0.4 dex. It is mainly due to a
large $L_{5100}$ for the SDSS subsample. The left two bottom panels show that
for $M_{\rm BH}$ calculated from $\rm FWHM_{H\beta}$. For the SDSS subsample ,
the mean values of $\sigma_{\rm H\beta}$-based $M_{\rm BH}$ and $\rm
FWHM_{H\beta}$-based $M_{\rm BH}$ are 8.13 and 7.88. For the RM subsample ,
the mean values of $\sigma_{\rm H\beta}$-based $M_{\rm BH}$ and $\rm
FWHM_{H\beta}$-based $M_{\rm BH}$ are 7.73 and 7.5. The $\sigma_{\rm
H\beta}$-based $M_{\rm BH}$ is averagely larger than $\rm FWHM_{H\beta}$-based
$M_{\rm BH}$ for the SDSS subsample (by 0.25 dex) or the RM subsample (by 0.23
dex). It is smaller than $0.5$ dex found in Bian et al. (2008). The right two
top panels show calculated from $\sigma_{\rm H\beta}$ . Right two bottom
panels shows that for calculate from $\rm FWHM_{H\beta}$. The $\sigma_{\rm
H\beta}$-based is averagely smaller than $\rm FWHM_{H\beta}$-based for the
SDSS subsample (by 0.5 dex) or the RM subsample (by 0.43 dex). We perform the
K-S tests on the distributions in Figure 3 between the SDSS subsample and the
RM subsample. For $\sigma_{\rm H\beta}$-based $M_{\rm BH}$ , $\rm d=0.344$,
p-value=$8.60\times 10^{-13}$; for $\sigma_{\rm H\beta}$-based , $\rm
d=0.123$, p-value=$0.053$; for $\rm FWHM_{H\beta}$-based $M_{\rm BH}$ , $\rm
d=0.320$, p-value=$4.29\times 10^{-11}$; for $\rm FWHM_{H\beta}$-based , $\rm
d=0.165$, p-value=$0.0031$. The difference of for these two subsamples is not
significant while the $M_{\rm BH}$ difference for these two subsamples is
significant.

A criterion of $\mdot\geq 3$ is adopted to select super-Eddington accreting
massive black holes (e.g., Wang et al., 2013; Du et al., 2016a). Using
$\sigma_{\rm H\beta}$-based $\mdot$, there are 912/4037 super-Eddington
accreting AGN in the SDSS subsample, and 40/120 super-Eddington accreting AGN
in the RM subsample. A relative large proportion of super-Eddington accreting
AGN in the SDSS subsample is due to the criterion of $\rm EW(FeII)>25\AA$. For
the RM subsample, there are 25 AGN presented by the SEAMBH collaboration (Wang
et al., 2013; Du et al., 2016a).

In Figure 4, we show the $L_{\rm Bol}/L_{\rm Edd}$ and for the SDSS subsample
(blue dots) or the RM subsample (red dots). For the SDSS subsample,
considering the errors in both coordinates, the best linear fitting through
kmpfit 222https://www.astro.rug.nl/software/kapteyn/kmpfittutorial.html.
kmpfit is the Kapteyn Package Python binding for a piece of software that
provides a robust and relatively fast way to perform nonlinear least-squares
curve and surface fitting. is $\log L_{\rm Bol}/L_{\rm Edd}=-(0.808\pm
0.001)+(0.521\pm 0.002)\log\mdot$. For the RM subsample, the best linear
fitting is $\log L_{\rm Bol}/L_{\rm Edd}=-(0.876\pm 0.013)+(0.558\pm
0.013)\log\mdot$. The latter relation is steeper than the former. The slope
difference is due to the larger $L_{5100}$ for the SDSS subsample (see Figure
1). It is consistent with the result by Maithil et al. (2022) (see their
Figure 6). These slopes are consistent with the slopes of $0.53$ found by
Davis & Laor (2011) and $0.52$ by Huang et al. (2020). Using Equations 1 and
3, $\mdot\propto L_{5100}^{1.5}M_{\rm BH}^{-2}$, $L_{\rm Bol}/L_{\rm
Edd}\propto L_{\rm Bol}M_{\rm BH}^{-1}\propto L_{5100}M_{\rm BH}^{-1}$, so
$L_{\rm Bol}/L_{\rm Edd}\propto\mdot^{0.5}L_{5100}^{0.25}$. The dependence on
$L_{5100}$ is smaller than on . The slope of $0.5$ is well consistent with our
slope of $0.52$ or $0.56$. The nonlinear relation between $L_{\rm Bol}/L_{\rm
Edd}$ and (although it is linear when the values of the logarithm are
considered) would have an impact on the relations using instead of $L_{\rm
Bol}/L_{\rm Edd}$ . Considering $L_{\rm Bol}=\eta\dot{M}c^{2}$, where $\eta$
is the accretion efficiency, $L_{\rm Bol}/L_{\rm
Edd}=\frac{\eta\dot{M}c^{2}}{\dot{M}_{\rm Edd}c^{2}}=\eta\mdot$. Adopting
$L_{\rm Bol}/L_{\rm Edd}\propto\mdot^{0.5}L_{5100}^{0.25}$, it means that
$\eta\propto\mdot^{-0.5}L_{5100}^{0.25}\propto(L_{\rm Bol}/L_{\rm
Edd})^{-1}L_{5100}^{0.5}$. The nonlinear relation implies that the converting
efficiency $\eta$ is anticorrelated with or $L_{\rm Bol}/L_{\rm Edd}$ (
including a relatively weaker dependence on $L_{5100}$ ), i.e., smaller
efficiency with higher dimensionless accretion rate.

###  3.2 $R_{\rm Fe}$ versus

Figure 5: Left: $R_{\rm Fe}$ versus . The green line and black line are the
best linear fittings for the SDSS subsample (blue points) and the RM subsample
(red points). The blue triangles represent the mean $R_{\rm Fe}$ in bins of
$\Delta\log\mdot=0.6~{}\rm dex$, and the error bars represent the standard
deviations. The green solid line shows the best fit to the binned data with a
second-order polynomial function. The purple squares denote 44 RM SDSS AGN
from the RM subsample. The $r_{s}$ and $p_{\rm null}$ for the RM subsample are
the ones excluding the SDSS RM AGN. Right: $v_{\rm Fe}$ versus for the SDSS
subsample. The green line and black line are the best linear fittings for the
inflow AGN ($v_{\rm Fe}>0$, red points and red contours) and the outflow AGN
($v_{\rm Fe}<0$, blue points and blue contours).

In Figure 5, we show the relation between $R_{\rm Fe}$ and . For the RM
subsample, the Spearman correlation test gives a strong correlation with the
coefficient $r_{s}=0.423$ and the probability of the null hypothesis $p_{\rm
null}=1.48\times 10^{-6}$. Considering the errors in both coordinates, the
kmpfit fitting gives $R_{\rm Fe}=(0.713\pm 0.047)+(0.532\pm 0.066)\log\mdot$.
Excluding 44 SDSS RM AGN from the subsample of 120 RM AGN, the correlation is
stronger with $r_{s}=0.60,p_{\rm null}=9.3\times 10^{-9}$. Considering the
errors in both coordinates, the fitting gives $R_{\rm Fe}=(0.548\pm
0.053)+(0.564\pm 0.074)\log\mdot$. Considering errors, these slopes are
consistent. For the SDSS subsample, the correlation is strong with
$r_{s}=0.727,p_{\rm null}=0$. Considering the errors in both coordinates, the
best linear fitting through kmpfit gives $R_{\rm Fe}=-(0.936\pm
0.006)+(0.600\pm 0.008)\log\mdot$. There exists some difference in intercepts
for the SDSS subsample and the RM subsample. The slightly intercept difference
implies that $\sigma_{\rm H\beta}$-based $M_{\rm BH}$ for the SDSS subsample
is slightly overestimated leading to an underestimated in . For 44 SDSS RM AGN
from the RM subsample, the correlation between $R_{\rm Fe}$ and is very weak
with $r_{s}=0.06,p_{\rm null}=0.686$. This weak correlation is possibly due to
the large uncertainties in the RM measurement of $R_{\rm BLR}$ for SDSS-RM
AGN. Excluding these 44 SDSS RM AGN from the RM subsample, for the relation
between $R_{\rm Fe}$ and , $r_{s}$ increases from 0.423 to 0.60, and $p_{\rm
null}$ decreases from $1.48\times 10^{-6}$ to $9.3\times 10^{-9}$. Considering
the errors in both coordinates, for these 44 SDSS RM AGN from the RM
subsample, the kmpfit fitting gives $R_{\rm Fe}=(0.661\pm 0.072)+(0.466\pm
0.115)\log\mdot$. Although with a large scatter, its slope of $0.466$ is
flatter than for the fitting relation for the SDSS subsample ($0.6$) and the
total RM subsample ($0.532$). The underestimation of is smaller than for the
SDSS subsample. We recognize that the much larger correlation coefficient
($r_{s}=0.727$ for the SDSS subsample) is in part an effect of self-
correlation induced by $R_{\rm Fe}$ in the extended R-L relationship to
estimate $M_{\rm BH}$ and . Including $R_{\rm Fe}$ to calculate , the
distribution of the SDSS subsample is well consistent with that for the RM
subsample (see left panel in Figure 5).

We investigate this correlation for super-Eddington and sub-Eddington AGN (see
Table 1). For 912 super-Eddington AGN in the SDSS subsample ($\mdot>3$; e.g.,
Du et al., 2016a), $r_{s}=0.57$ and $p_{\rm null}=2.35\times 10^{-79}$. For
3125 sub-Eddington AGN in the SDSS subsample, $r_{s}=0.56$ and $p_{\rm
pull}=3.45\times 10^{-252}$. For super-Eddington accreting AGN, the slope is
steeper than for sub-Eddington accretion AGN. For the RM subsample, for 40
super-Eddington RM AGN, $r_{s}=0.43$ and $p_{\rm null}=0.0052$. For 80 sub-
Eddington RM AGN, $r_{s}=0.1$ and $p_{\rm null}=0.38$, which shows the
correlation is not significant.

Table 1: The Spearman correlation coefficient and the probability of the null
hypothesis for relations.

SDSS Subsample RM Subsample All $\mdot<3$ $\mdot\geq 3$ All $\mdot<3$
$\mdot\geq 3$ Number 4037 3125 912 120 80 40 $R_{\rm Fe}$ - (0.727, 0) (0.56,
$3.45\times 10^{-252}$) (0.57, $2.35\times 10^{-79}$ ) (0.423, $1.48\times
10^{-6}$) (0.1, 0.38) (0.43, 0.0052) $v_{\rm Fe}-\mdot^{*}$ (-0.612,0)∗ (0.52,
$2.37\times 10^{-190}$)∗ (-0.08, 0.02 ) – – – $L_{\rm Bol}/L_{\rm Edd}$ -
(0.98, 0) – – (0.962, $2.56\times 10^{-68}$) – –

Note: The first value in brackets is the Spearman correlation coefficient and
the second value is the probability of the null hypothesis.
*: For the 3556 AGN with inflow ($v_{\rm Fe}>0$) in the SDSS subsample with 2809 sub-Eddington ($\mdot<3$) and 747 super-Eddington ($\mdot\geq 3$).

The systematic uncertainty is about $0.3~{}\rm dex$ in $M_{\rm BH}$ ,
$0.6~{}\rm dex$ in (e.g., Yu et al., 2020b). For the relation between $R_{\rm
Fe}$ and , using $\Delta\log\mdot=0.6~{}\rm dex$ as the bin width, we
calculate the mean and the standard deviation as the error in each bin. It
shows a steeper slope with increasing . Using 2th-order polynomial function,
we find the best fitting is $R_{\rm Fe}=(0.863\pm 0.024)+(0.389\pm
0.014)\log\mdot+(0.077\pm 0.009)(\log\mdot)^{2}$, which can be used to
estimate from $R_{\rm Fe}$ (Du et al., 2016a).

For a slightly smaller RM sample of 63 AGN (Du et al., 2016a), using FWHM-
based , it was found that $R_{\rm Fe}=(0.66\pm 0.04)+(0.30\pm 0.03)\log\mdot$
($r_{s}=0.65$). For 9818 SDSS DR4 AGN, it was found that, using FWHM-based
$M_{\rm BH}$ , there was a correlation between $R_{\rm Fe}$ and $L_{\rm
Bol}/L_{\rm Edd}$ shown in Figure 5 of Netzer & Trakhtenbrot (2007), $R_{\rm
Fe}\propto(L_{\rm Bol}/L_{\rm Edd})^{0.7}$. Considering the nonlinear relation
between $L_{\rm Bol}/L_{\rm Edd}$ and , $L_{\rm Bol}/L_{\rm
Edd}\propto\mdot^{0.52}$, it implied $R_{\rm Fe}\propto\mdot^{0.36}$ for 9818
AGN by Netzer & Trakhtenbrot (2007), which is consistent with the result by Du
et al. (2016a). These slopes are flatter than ours, which is due to larger
$M_{\rm BH}$ and smaller using $\sigma_{\rm H\beta}$ instead of $\sigma_{\rm
H\beta}$ . The strong correlation between $R_{\rm Fe}$ and or $L_{\rm
Bol}/L_{\rm Edd}$ show that the relative strength of optical is driven by the
strength of the accretion process. We also can use $R_{\rm Fe}$ to trace or
$L_{\rm Bol}/L_{\rm Edd}$ though with a large scatter. Including the H$\beta$
shape $\rm FWHM_{H\beta}/\sigma_{\rm H\beta}$, for the SDSS subsample, the
$r_{s}$ increases from 0.727 to 0.75. For the RM subsample, the $r_{s}$
increases from 0.423 to 0.46; $p_{\rm null}$ decreases from $1.48\times
10^{-6}$ to $1.12\times 10^{-7}$. It is found that, including $\rm
FWHM_{H\beta}/\sigma_{\rm H\beta}$, we cannot improve significantly the
relation between and $R_{\rm Fe}$ (see also Du et al., 2016a).

###  3.3 $v_{\rm Fe}$ versus

For the SDSS subsample of 4037 AGN, a systematic redshift of optical relative
to [O iii]$\lambda 5007$ ($v_{\rm Fe}$ ) was found by Hu et al. (2008). The
errors of $v_{\rm Fe}$ were calculated from the fitting of the continuum and
the fitting of the [O iii] line, which was consistent with that given by the
simulations. The median of the $v_{\rm Fe}$ distribution is 116 km s-1 (Hu et
al., 2008). Sulentic et al. (2012) called into question the measurement of
$v_{\rm Fe}$ in Hu et al. (2008) . The contradictory results are possibly due
to the different ways to construct their composite spectra. Sulentic et al.
(2012) constructed them from bins by parameters of their 4D Eigenvector 1
(EV1) formalism, namely, H$\beta$ FWHM and $R_{\rm Fe}$ , while Hu et al.
(2008) constructed them from bins in $v_{\rm Fe}$ . Hu et al. (2012) confirmed
that the redshift measurements of are robust.

For the SDSS subsample of 4037 AGN, it is found there is a relation between
$v_{\rm Fe}$ and $\rm FWHM_{FeII}$ with $r_{s}=0.59,p_{\rm null}=0$. For the
relation between $v_{\rm Fe}$ and $\rm FWHM_{H\beta}$, $r_{s}=0.50,p_{\rm
null}=9.51\times 10^{-229}$ (also see Figure 9 in Hu et al., 2008). It was
found that the Eddington ratio is the main physical driver for $v_{\rm Fe}$
(Hu et al., 2008). In the right panel in Figure 5, for the SDSS subsample of
4037 AGN, we show the relation between $v_{\rm Fe}$ and . The correlations are
strong with $r_{s}=-0.612,p_{\rm null}=0$ for 3556 AGN with inflow ($v_{\rm
Fe}>0$), and $r_{s}=-0.419,p_{\rm null}=9.40\times 10^{-22}$ for 480 AGN with
outflow ($v_{\rm Fe}<0$), respectively. Considering the errors in both
coordinates, the kmpfit fitting shows that $\log v_{\rm Fe}=(2.610\pm
0.007)+(-0.578\pm 0.01)\log\mdot$ for $v_{\rm Fe}>0$ (green line in right
panel in Figure 5). It is $\log|v_{\rm Fe}|=(2.331\pm 0.018)+(-0.467\pm
0.022)\log\mdot$ for $v_{\rm Fe}<0$ (black line). For 3556 AGN with inflow
($v_{\rm Fe}>0$) in the SDSS subsample, there are 2809 sub-Eddington AGN
($\mdot<3$), $r_{s}=-0.52$ and $p_{\rm null}=2.37\times 10^{-190}$. For 747
super-Eddington AGN with inflow ($v_{\rm Fe}>0$), $r_{s}=-0.08$, and $p_{\rm
null}=0.02$, which shows the correlation is not significant (see Table 1).

Using $\sigma_{\rm H\beta}$ and $R_{\rm BLR}({\rm H\beta})\propto
L_{5100}^{0.69}$ (McGill et al., 2008), for inflow ($v_{\rm Fe}>0$), it was
found that $\log v_{\rm Fe}=(1.00\pm 0.05)+(-1.83\pm 0.05)\log L_{\rm
Bol}/L_{\rm Edd}$, $r_{s}=-0.53,p_{\rm null}<1.0\times 10^{-5}$ (Hu et al.,
2008). Considering the nonlinear relation between $L_{\rm Bol}/L_{\rm Edd}$
and , $L_{\rm Bol}/L_{\rm Edd}\propto\mdot^{0.52}$, it implied $v_{\rm
Fe}\propto\mdot^{-0.95}$. Here using $\sigma_{\rm H\beta}$ , the extended
$R_{\rm BLR}({\rm H\beta})-L_{\rm 5100}$ relation including $R_{\rm Fe}$ and
the host-corrected $L_{5100}$ to calculate $M_{\rm BH}$ , and the standard
accretion disk to calculate , we find that the slope is $-0.578\pm 0.01$. The
flatter relation is due to the smaller $R_{\rm BLR}({\rm H\beta})$ for larger
$R_{\rm Fe}$ , which leads to smaller $M_{\rm BH}$ and larger . For the
relation between $v_{\rm Fe}$ and shown in right panel in Fig. 5, excluding
extreme accretors with log $\mdot$ larger than 1.5, we found that the
correlation does not improve. It becomes slightly weaker with $r_{s}$ changing
from 0.612 to 0.602. For outflow ($v_{\rm Fe}<0$) , we also find a flatter
relation with a slope of $-0.467\pm 0.022$ (see the black line in the right
panel in Figure 5). It suggests that the inflow velocity is slightly larger
than the outflow velocity for AGN with the same . The velocity of is driven by
gravity toward the center and decelerated by the radiation pressure, as well
as its initial velocity. The redward shift of and the inverse correlation
between $v_{\rm Fe}$ and favor a scenario in which emission emerges from an
inflow (Hu et al., 2008). An increase in would enhance the radiation pressure
and lead to a decrease of the inward velocity of the inflow.

From the general form of the equation of motion for a cloud of mass $M_{c}$,
the acceleration $a(r)=a_{\rm
rad}(r)-g(r)-\frac{1}{\rho}\frac{dP}{dr}+f_{d}/M_{c}$ (Netzer, 2013), where
$g(r)$ is the gravitational acceleration, $a_{\rm rad}(r)$ is the acceleration
due to radiation pressure force, and $f_{d}$ is the drag force; $P$ is the
thermal pressure [$P=\rho v_{s}^{2}$, where $v_{s}=(2k_{B}T/\mu m_{\rm
H})^{1/2}\sim 16.6(T/10^{4}K)^{1/2}{\rm km\ s}^{-1}$, $\mu m_{\rm H}$ is the
average mass per particle, and $\mu=0.6$]. Neglecting the gas pressure
gradient and the drag force for pure wind flows, $a(r)=a_{\rm
rad}(r)-g(r)\approx\frac{\sigma_{T}L}{4\pi r^{2}\mu m_{H}c}[M(r)-\frac{L_{\rm
Edd}}{L}]=\frac{G}{r^{2}}M_{\rm BH}[M(r)\frac{L}{L_{\rm Edd}}-1]$. $L$ is the
total luminosity, and approximately equals $L_{\rm Bol}$ . $M(r)$ is force
multiplier, which is the ratio of the total radiation pressure to the
radiation pressure due to Compton scattering. $M(r)\geq 1$, and $M(r)=1$ for
fully ionized gas. Assuming the clouds falls down from torus (at $r_{1}$) with
an initial $v_{0}=0$ to region at $r_{2}$ ($\sim$ $R_{\rm BLR}({\rm H\beta})$
), for the cases of constant $[M(r)\frac{L}{L_{\rm Edd}}-1]$, we get
$v^{2}=2GM_{\rm BH}(1-M\frac{L}{L_{\rm
Edd}})(\frac{1}{r_{2}}-\frac{1}{r_{1}})$. Assuming the size of the innermost
part of the torus is $r_{1}=nr_{2}$, where $n\sim 3-4$ (Netzer, 2013), and
$M_{\rm BH}=f\rm FWHM_{H\beta}^{2}r_{2}/G$, we get $v_{\rm Fe}^{2}\approx
2\frac{n-1}{n}f{\rm FWHM^{2}_{H\beta}}(1-M(r)\frac{L}{L_{\rm Edd}})$. The
force multiplier $M(r)$ was found to be inversely correlated with the
ionization parameter (Chelouche & Netzr, 2005). Assuming $M(r)=M_{0}(L/L_{\rm
Edd})^{-\alpha}$, we get $v_{\rm Fe}^{2}\approx 2\frac{n-1}{n}f{\rm
FWHM^{2}_{H\beta}}[1-M_{0}(L/L_{\rm Edd})^{1-\alpha}]$. A smaller $v$ would be
found for AGN with larger $L_{\rm Bol}/L_{\rm Edd}$ , and smaller $\rm
FWHM_{H\beta}$ (also see Figure 9 in Hu et al., 2008).

However, we find that it is not the case for the blueward shift of , i.e., a
smaller velocity of outflow for larger . The reason for the relation between
the blueward velocity $v_{\rm Fe}$ and is not clear.

###  3.4 The relation between $f$ and $\rm FWHM_{H\beta}$

Figure 6: Left: $f_{\rm FWHM}$ versus $\rm FWHM_{H\beta}$. The black line and
orange line are the best linear fittings for the SDSS subsample (blue points)
and the RM subsample (red points and triangles). The triangles show 44 RM AGN
from the SDSS. The red points show other 76 compiled RM AGN excluding 44 RM
AGN from the SDSS. The pink line is the relation given by Yu et al. (2020b).
Right: $f_{\rm FWHM}$ versus $\sigma_{\rm H\beta}$ for the subsample of 120 RM
AGN from the mean spectrum. The symbols are the same as in the left panel. The
Spearman correlation coefficient and the probability of the null hypothesis
are shown in panels. The blue is for the SDSS Subsample, the purple is for 44
RM AGN from the SDSS; the red is for the RM subsample excluding 44 RM AGN from
the SDSS; the orange is for the RM subsample.

Assuming a variable $f$ in $\rm FWHM_{H\beta}$-based $M_{\rm BH}$ for the
single-epoch or the mean spectrum, from Equation 2, $\log
f=2\log(\rm\sigma_{\rm H\beta}/FWHM_{H\beta})+logf_{\sigma}$. In the left
panel in Figure 6, we show $\log f$ versus $\rm FWHM_{H\beta}$ where
$f_{\sigma}=5.5$ as suggested by Yu et al. (2020b). For the SDSS subsample
(blue points), the correlation is strong with $r_{s}=-0.583,p_{\rm null}=0$.
Considering the errors in both coordinates, the best fitting is $\log
f=(1.719\pm 0.028)-(0.414\pm 0.008)\rm\log FWHM_{H\beta}$ (black line in the
left panel in Figure 6) for the single-epoch spectrum. For $\rm
FWHM_{H\beta}=1000~{}{\rm km\ s}^{-1},2000~{}{\rm km\ s}^{-1},6000~{}{\rm km\
s}^{-1},14196~{}{\rm km\ s}^{-1}$, $f=3,2.25,1.42,1$, respectively. For the
SDSS subsample, assuming the same $M_{\rm BH}$ from $\rm FWHM_{H\beta}$,
$\sigma_{\rm H\beta}$ and $f_{\sigma}=5.5$, the flatter slope means $f$ has a
narrow range, i.e., from 3 to 1 for $\rm FWHM_{H\beta}$ from 1000 to 14196 km
s-1 . For the value of $2.35$ for the single Gaussian profile, $\log
f=2\log(\rm\sigma_{\rm H\beta}/FWHM_{H\beta})+logf_{\sigma}=0.0017$. The
relation between $\log f$ and $\rm FWHM_{H\beta}$ also shows the relation
between $\rm FWHM_{H\beta}/\sigma_{\rm H\beta}$ and $\rm FWHM_{H\beta}$,
$\log\rm FWHM_{H\beta}/\sigma_{\rm H\beta}=-0.49+0.207\log\rm FWHM_{H\beta}$.
In the left panel in Figure 6, almost all of the AGN with $\rm
FWHM_{H\beta}<4000~{}{\rm km\ s}^{-1}$ (Population A, Marziani et al., 2001)
in the SDSS subsample show $\rm FWHM_{H\beta}/\sigma_{\rm H\beta}<2.35$
deviated from that for the single Gaussian H$\beta$ profile.

For the RM subsample, the Spearman correlation test gives $r_{s}=-0.672,p_{\rm
null}=7.522\times 10^{-11}$. Considering the errors in both coordinates, the
best-fitting relation for RM subsample shows $\log f=-(1.348\pm 0.080)\log\rm
FWHM_{H\beta}+(4.851\pm 0.27)$ ( orange solid line in Figure 6). Excluding 44
RM AGN from SDSS, the correlation is stronger with $r_{s}=-0.81,p_{\rm
null}=1.5\times 10^{-15}$. The best-fitting relation is $\log f=-(1.357\pm
0.069)\log\rm FWHM_{H\beta}+(4.883\pm 0.232)$. For 17 RM AGN with measured
$\sigma_{\rm\ast}$ (Yu et al., 2020b), it was found that the relation is $\log
f=-(1.10\pm 0.4)\log\rm FWHM_{H\beta}+(4.13\pm 0.11)$ (pink solid line in the
left panel in Figure 6). Although our fitting slope for the RM subsample is
steeper than Yu et al. (2020b), the trend is similar, i.e., larger $f$ for AGN
with smaller $\rm FWHM_{H\beta}$. With our large number of AGN, the error of
our slope is smaller, and our slope is consistent with the slope from RM AGN
measured $\sigma_{\rm\ast}$ (Yu et al., 2020b) considering their large error.
For the SDSS subsample, the slope derived from the single-epoch spectrum is
flatter with respect to that for the RM subsample from the mean spectrum,
i.e., $-0.414$ vs. $-1.348$.

Using $\sigma_{\rm H\beta,rms}$ as the benchmark instead of $\sigma_{\rm
H\beta,mean}$ in Equation 2, for 120 RM AGN, $r_{s}=-0.482,p_{\rm
null}=9.18\times 10^{-4}$. Considering the errors in both coordinates, the
best-fitting relation is $\log f=-(1.418\pm 0.266)\log\rm
FWHM_{H\beta}+(5.148\pm 0.08)$. Excluding 44 RM AGN from SDSS, the correlation
is stronger with $r_{s}=-0.616,p_{\rm null}=1.01\times 10^{-7}$. Considering
the errors in both coordinates, the best-fitting relation is $\log f=-(1.05\pm
0.40)\log\rm FWHM_{H\beta}+(3.755\pm 0.40)$. Considering their errors, these
slopes are consistent with that using $\sigma_{\rm H\beta,mean}$ as the
benchmark. A smaller $\sigma_{\rm H\beta}$-based $f_{\sigma}$ was suggested to
be $4.47^{+1.41}_{-1.08}$ for the single-epoch spectrum (Woo et al., 2015), or
$4.31\pm 1.05$ (Grier et al., 2013) for the rms spectrum. Adopting these
smaller $f$, it would lead our best fitting of the relation between $f$ and
$\rm FWHM_{H\beta}$ to have a smaller intercept by $\sim 0.1~{}\rm dex$.

Because there is a strong relation between and $\rm FWHM_{H\beta}$ (e.g., Bian
& Zhao, 2003), it implies that, using $\rm FWHM_{H\beta}$ as a velocity tracer
in the $M_{\rm BH}$ calculation, the corresponding $f$ has a correlation with
, i.e., a larger $f$ is needed for AGN with larger . For a simple model of
thick-disk BLRs, neglecting the contribution of outflow in the H$\beta$
profile, we obtained $f=1/(a^{2}+sin^{2}\theta)$, where $a$ is the ratio of
the scale height of the thick disk to the radius $r$, and $\theta$ is the
inclination of the thick-disk of BLRs to the line of sight (Collin et al.,
2006; Yu et al., 2019). The cumulative fraction of this variable $f$ for 34 RM
AGN with measured stellar velocity dispersion $\sigma_{\rm\ast}$ was
consistent with this simple model of thick-disk BLRs (Yu et al., 2019). The
relation between $f$ and $\rm FWHM_{H\beta}$ implied that, as the tracer of
BLRs velocity, $\rm FWHM_{H\beta}$ has some dependence on the BLRs
inclination, while $\sigma_{\rm H\beta}$ is insensitive to the inclination
(Collin et al., 2006; Yu et al., 2019).

In right panel in Figure 6, we show $\log f$ versus $\sigma_{\rm H\beta}$ for
the RM subsample (red points and triangles) and for the SDSS subsample (blue
points). The Spearman correlation test gives $r_{s}=-0.190,p_{\rm
null}=1.08\times 10^{-1}$ for the RM subsample, which is not too significant.
Excluding 44 RM AGN from SDSS, the correlation is always not strong with
$r_{s}=-0.364,p_{\rm null}=3.59\times 10^{-3}$. Only for 44 RM AGN from SDSS,
the correlation is very weak with $r_{s}=-0.012,p_{\rm null}=9.392\times
10^{-1}$. For the SDSS subsample , $r_{s}=-0.33,p_{\rm null}=6.06\times
10^{-103}$, which is weaker than that with $\rm FWHM_{H\beta}$. Using
$\sigma_{\rm H\beta,rms}$ as the benchmark instead of $\sigma_{\rm
H\beta,mean}$ in Equation 2 (see the right panel in Figure 6), the Spearman
correlation test gives $r_{s}=-0.084,p_{\rm null}=5.88\times 10^{-1}$ for the
RM subsample, which is not too significant. Excluding 44 RM AGN from SDSS, the
correlation is always not strong with $r_{s}=-0.249,p_{\rm null}=5.09\times
10^{-2}$. If assuming the virial factor $f$ has a dependence on the BLR
inclination to the line of sight, thes not significant correlations in the
right panel show that $\sigma_{\rm H\beta}$ has no dependence on the
inclination. It is consistent with the constant $f$ using $\sigma_{\rm
H\beta}$ as the velocity tracer in the $M_{\rm BH}$ calculation.

##  4 Conclusions

For a large sample of 4037 SDSS quasars ($z<0.8$) with measured $\sigma_{\rm
H\beta}$ and $R_{\rm Fe}$ and 120 compiled RM AGN, we use $\sigma_{\rm
H\beta}$ instead of $\rm FWHM_{H\beta}$ and the extended $R_{\rm BLR}({\rm
H\beta})-L_{\rm 5100}$ relation (including $R_{\rm Fe}$ ) to calculate $M_{\rm
BH}$ from the single-epoch spectra for the SDSS subsample, and $\sigma_{\rm
H\beta}$ from the mean spectra for the RM subsample. We use instead of $L_{\rm
Bol}/L_{\rm Edd}$ to indicate the accretion strength to investigate the
relation with optical emission ($R_{\rm Fe}$ and $v_{\rm Fe}$ ). We also
derive the relation between the virial factor $f$ and $\rm FWHM_{H\beta}$ for
the single-epoch spectrum. The main conclusions can be summarized as follows:

  * •

For the subsample of 4037 SDSS quasars, with respect to $\rm FWHM_{H\beta}$,
$\sigma_{\rm H\beta}$-based $M_{\rm BH}$ are averagely larger by $0.25$ dex;
are averagely smaller by $0.50$ dex. There exists a nonlinear relationship
between $L_{\rm Bol}/L_{\rm Edd}$ and , $L_{\rm Bol}/L_{\rm
Edd}\propto\mdot^{0.52}$. For the subsample of 120 RM AGN, the results are
similar. With respect to $\rm FWHM_{H\beta}$, $\sigma_{\rm H\beta}$-based
$M_{\rm BH}$ (from the mean spectrum) for the RM subsample are averagely
larger by $0.23$ dex, are averagely smaller by $0.43$ dex, and $L_{\rm
Bol}/L_{\rm Edd}\propto\mdot^{0.56}$. This relation of $L_{\rm Bol}/L_{\rm
Edd}$ with can be derived assuming $L_{5100}$ is calculated from the standard
accretion disk model and $L_{\rm Bol}$ is proportional to $L_{5100}$ .

  * •

Adopting $L_{\rm Bol}/L_{\rm Edd}\propto\mdot^{0.5}L_{5100}^{0.25}$, it means
that the converting efficiency
$\eta\propto\mdot^{-0.5}L_{5100}^{0.25}\propto(L_{\rm Bol}/L_{\rm
Edd})^{-1}L_{5100}^{0.5}$. The nonlinear relation implies that $\eta$ is
anticorrelated with or $L_{\rm Bol}/L_{\rm Edd}$ (including a relatively
weaker dependence on $L_{5100}$ ), i.e., smaller efficiency with higher
dimensionless accretion rate. The efficiency is related to the SMBH spin.

  * •

For the subsample of 4037 SDSS quasars, a strong relation between $R_{\rm Fe}$
and is found: $R_{\rm Fe}=-(0.936\pm 0.006)+(0.600\pm 0.008)\log\mdot$ with
the Spearman correlation coefficient $r_{s}$ of $0.727$. For the subsample of
120 RM AGN, the results are similar with a larger scatter; $R_{\rm
Fe}=(0.713\pm 0.047)+(0.532\pm 0.066)\log\mdot$ with $r_{s}=0.423$. With
respect to [O iii] $\lambda$5007, the velocity shift $v_{\rm Fe}$ has an
strong anticorrelation with , whenever inflow ($r_{s}=0.612$) or outflow
($r_{s}=-0.419$). These strong correlations show that the optical emission has
an intimate connection with the SMBH accretion process.

  * •

Assuming the clouds fall down from torus with an initial $v_{0}=0$, we get the
inflow velocity of the optical $v_{\rm Fe}^{2}\propto{\rm
FWHM^{2}_{H\beta}}(1-M(r)\frac{L}{L_{\rm Edd}})$. A smaller $v$ would be found
for AGN with larger $L_{\rm Bol}/L_{\rm Edd}$ , smaller $\rm FWHM_{H\beta}$.
It is qualitatively consistent with the relation of the inflow velocity
$v_{\rm Fe}$ with $\rm FWHM_{H\beta}$ and $L_{\rm Bol}/L_{\rm Edd}$ or .

  * •

Assuming a variable $f$ in $\rm FWHM_{H\beta}$-based $M_{\rm BH}$ from the
single-epoch spectrum, we find a strong correlation between $\log f$ and $\rm
FWHM_{H\beta}$, $\log f=(1.719\pm 0.028)-(0.414\pm 0.008)\rm FWHM_{H\beta}$.
It is flatter than that derived from the mean spectrum for the RM subsample or
17 RM AGN with measured $\sigma_{\rm\ast}$ . However, the relation between
$\log f$ and $\sigma_{\rm H\beta}$ is not too strong, suggesting that
$\sigma_{\rm H\beta}$ doesn’t seem dependent much on the BLR inclination and a
constant $f_{\sigma}$ is suitable for $\sigma_{\rm H\beta}$ as the velocity
tracer.

We are very grateful to the anonymous referee for her/his instructive comments
which significantly improved the content of the paper. This work has been
supported by the National Science Foundations of China (Nos. 11973029 and
11873032). This work is supported by the National Key Research and Development
Program of China (No. 2017YFA0402703).

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