---
abstract: 'It is well known that reverberation mapping of active galactic nuclei (AGN) reveals a relationship between AGN luminosity and the size of the broad-line region, and that use of this relationship, combined with the Doppler width of the broad emission line, enables an estimate of the mass of the black hole at the center of the active nucleus based on a single spectrum. This has been discussed in numerous papers over the last two decades. An unresolved key issue is the choice of parameter used to characterize the line width; generally, most researchers use FWHM in favor of line dispersion  (the square root of the second moment of the line profile) because the former is easier to measure, less sensitive to blending with other features, and usually can be measured with greater precision. However, use of FWHM introduces a bias, stretching the mass scale such that high masses are overestimated and low masses are underestimated. Here we investigate estimation of black hole masses in AGNs based on individual or “single epoch” observations, with a particular emphasis in comparing mass estimates based on line dispersion and FWHM. We confirm the recent findings that, in addition to luminosity and line width, a third parameter is required to obtain accurate masses and that parameter seems to be Eddington ratio. We present simplified empirical formulae for estimating black hole masses from the $\lambda4861$ and $\lambda1549$ emission lines.'
author:
- Elena Dalla Bontà
- 'Bradley M. Peterson'
- 'Misty C. Bentz'
- 'W. N. Brandt'
- 'S. Ciroi'
- Gisella De Rosa
- Gloria Fonseca Alvarez
- 'Catherine J. Grier'
- 'P. B. Hall'
- 'Juan V. Hernández Santisteban'
- 'Luis C. Ho'
- 'Y. Homayouni'
- Keith Horne
- 'C. S. Kochanek'
- 'Jennifer I-Hsiu Li'
- 'L. Morelli'
- 'A. Pizzella'
- 'R. W. Pogge'
- 'D. P. Schneider'
- Yue Shen
- 'J. R. Trump'
- Marianne Vestergaard
title: 'THE SLOAN DIGITAL SKY SURVEY REVERBERATION MAPPING PROJECT: ESTIMATING MASSES OF BLACK HOLES IN QUASARS WITH SINGLE-EPOCH SPECTROSCOPY'
---

Introduction {#section:intro}
============

Reverberation-Based Black Hole Masses
-------------------------------------

The presence of emission lines with Doppler widths of thousands of kilometers per second is one of the defining characteristics of active galactic nuclei [@Burbidge67; @Weedman76]. It was long suspected that the large line widths were due to motions in a deep gravitational potential and this implied very large central masses [e.g., @Woltjer59], as did the Eddington limit [@Tarter73]. Under a few assumptions, the central mass is $M
\propto V^2 R$, where $V$ is the Doppler width of the line and $R$ is the size of the broad-line region (BLR). It is the latter quantity that is difficult to determine. An early attempt to estimate $R$ by [@Dibai80] was based on the assumption of constant emissivity per unit volume, but led to an incorrect dependence on luminosity as in this case, luminosity is proportional to volume, so $R \propto L^{1/3}$. [@Wandel85] inferred the BLR size from the  luminosity. Other attempts were based on photoionization physics [see @Ferland85; @Osterbrock85]. [@Davidson72] found that the relative strength of emission lines in ionized gas could be characterized by an ionization parameter $$U= \frac{Q({\rm H})}{4\pi R^2 c n_{\rm H}},
\label{eq:Udef}$$ where $Q({\rm H})$ is the rate at which H-ionizing photons are emitted by the central source and $n_{\rm H}$ is the particle density of the gas. The ionization parameter $U$ is proportional to the ratio of ionization rate to recombination rate in the BLR clouds. The similarity of emission-line flux ratios in AGN spectra over orders of magnitude in luminosity suggested that $U$ is constant, and the presence of C[iii]{}\]$\lambda1909$ set an upper limit on the density $n_{\rm H} \la 10^{9.5}$ [@Davidson79]. Since $L \propto Q({\rm H})$, this naturally led to the prediction that the BLR radius would scale with luminosity as $R \propto L^{1/2}$. Unfortunately, best-estimate values for $Q({\rm
H})$ and $n_{\rm H}$ led to a significant overestimate of the BLR radius [@Peterson85] as a consequence of the simple but erroneous assumption that all the broad lines arise cospatially (i.e., models employed a single representative BLR cloud).

With the advent of reverberation mapping [hereafter RM; @Blandford82; @Peterson93], direct measurements of $R$ enabled improved black hole mass determinations. Attempts to estimate black hole masses based on early RM results and the $R \propto L^{1/2}$ prediction included those of [@Padovani88], [@Koratkar91], and [@Laor98]. The first multiwavelength RM campaigns demonstrated ionization stratification of the BLR [@Clavel91; @Krolik91; @Peterson91] and this eventually led to identification of the virial relationship, $R \propto V^{-2}$ [@PetersonWandel99; @PetersonWandel00; @Onken02; @Kollatschny03; @Bentz10], that gave reverberation-based mass measurements higher levels of credibility. Of course, the virial relationship demonstrates only that the central force has a $R^{-2}$ dependence, which is also characteristic of radiation pressure; whether or not radiation pressure from the continuum source is important has not been clearly established [@Marconi08; @Marconi09; @Netzer10]. If radiation pressure in the BLR turns out to be important, then the black hole masses, as we discuss them here, are underestimated.

Masses of AGN black holes are computed as $$\label{eq:masseqn}
M_{\rm BH} = f \left(\frac{V^2 R}{G}\right),$$ where $V$ is the line width, $R$ is the size of the BLR from the reverberation lag, and $G$ is the gravitational constant. The quantity in parentheses is often referred to as the virial product $\mu$; it incorporates the two observables in RM, line width and time delay $\tau = R/c$, and is in units of mass. The scaling factor $f$ is a dimensionless quantity of order unity that depends on the geometry, kinematics, and inclination of the AGN. Throughout most of this work, we ignore $f$ (i.e., set it to unity) and work strictly with the virial product.

While reverberation mapping has emerged as the most effective technique for measuring the black hole masses in AGNs [@Peterson14], it is resource intensive, requiring many observations over an extended period of time at fairly high cadence. Fortunately, observational confirmation of the $R$–$L$ relationship [@Kaspi00; @Kaspi05; @Bentz06a; @Bentz09a; @Bentz13] enables “single-epoch” (SE) mass estimates because, in principle, a single spectrum could yield $V$ and also $R$, through measurement of $L$ [e.g., @Wandel99; @McLure02; @Vestergaard02; @Corbett03; @Vestergaard04; @Kollmeier06; @Vestergaard06; @Fine08; @Shen08a; @Shen08b; @Vestergaard08]. Of the three strong emission lines generally used to estimate central black hole masses, the $R$–$L$ relationship is only well-established for $\lambda4861$ [@Bentz13 and references therein, but see the discussion in §\[section:hbpredictor\]]. Empirically establishing the $R$–$L$ relationship for $\lambda2798$ [@Homayouni20] and $\lambda1549$ [@Peterson05; @Kaspi07; @Trevese14; @Lira18; @Grier19; @Hoormann19] has been difficult.

Masses based on the $\lambda1549$ emission line, in particular, have been somewhat controversial. Some studies claim that there is good agreement between masses based on  and those measured from other lines [@Vestergaard06; @Greene10b; @Assef11]. On the other hand, there are several claims that there is inadequate agreement with masses based on other emission lines [@Baskin05; @Netzer07; @Sulentic07; @Shen08b; @Shen12; @Trak12]. [@Denney09a] and [@Denney13], however, note that there are a number of biases that can adversely affect single-epoch mass estimates, with low $S/N$ “survey quality” data being an important problem with some of the studies for which poor agreement between  and other lines is found. It has also been argued, however, that some fitting methodologies are more affected by this than others [@Shen19]. There have been more recent papers that attempt to correct  mass determinations to better agree with those based on other lines [e.g., @Runnoe13; @Coatman17; @Mejia18; @Marziani19].

Characterizing Line Widths
--------------------------

It is our suspicion that the apparent difficulties with -based masses trace back not only to the $S/N$ issue, but also to how the line widths are characterized. It has been customary in AGN studies to characterize line widths by one of two parameters, either FWHM or the line dispersion , which is defined by $${\mbox{$\sigma_{\rm line}$}}= \left[ \frac{ \int (\lambda - \lambda_0)^2 P(\lambda)\,d\lambda}{\int P(\lambda)\,d\lambda} \right]^{1/2},
\label{eq:Defsigl}$$ where $P(\lambda)$ is the emission-line profile as a function of wavelength and $\lambda_0$ is the line centroid, $$\lambda_0 = \frac{\int \lambda P(\lambda)\,d\lambda}{\int P(\lambda)\,d\lambda}.
\label{eq:Deflambda0}$$ While both FWHM and  have been used in the virial equation to estimate AGN black hole masses, they are not interchangeable. It is well known that AGN line profiles depend on the line width [@Joly85]: broader lines have lower kurtosis, i.e., they are “boxier” rather than “peakier.” Indeed, for AGNs, the ratio ${\rm FWHM}/{\mbox{$\sigma_{\rm line}$}}$ has been found to be a simple but useful characterization of the line profile [@Collin06; @Kollatschny13].

Each line-width measure has practical strengths and weaknesses [@Peterson04; @Wang20]. The line dispersion  is more physically intuitive, but it is sensitive to the line wings, which are often badly blended with other features. All three of the strong lines usually used to estimate masses — $\lambda4861$, $\lambda2798$, and $\lambda1549$ — are blended with other features: the $\lambda4570$ and $\lambda\lambda$5190, 5320 complexes [@Phillips78] and $\lambda4686$ in the case of , the UV  complexes in the case of , and $\,\lambda1640$ in the case of . The FWHM can usually be measured more precisely than  (although @Peterson04 note that the opposite is true for the rms spectra, described below, which are sometimes quite noisy), but it is not clear that FWHM yields more [*accurate*]{} mass measurements. In practice, FWHM is used more often than  because it is relatively simple to measure and can be measured more precisely while  often requires deblending or modeling the emission features, which does not necessarily yield unambiguous results.

There are, however, a number of reasons to prefer  to FWHM as the line-width measure for estimating AGN black hole masses. [@Fromerth00] point out that  better characterizes an arbitrary or irregular line profile. [@Peterson04] note that  produces a tighter virial relationship than FWHM, and [@Denney13] find better agreement between -based and -based mass estimates by using  rather than FWHM (these latter two are essentially the same argument). In the case of NGC 5548, for which there are multiple reverberation-based mass measures, a possible correlation with luminosity is stronger for FWHM-based masses than for -based masses, suggesting that the former are biased as the same mass should be recovered regardless of the luminosity state of the AGN [@Collin06; @ShenKelly12]. A possibly more compelling argument for using  instead of FWHM is bias in the mass scale that is introduced by using FWHM as the line width. [@Steinhardt10] used single-epoch masses for more than 60,000 SDSS quasars [@Shen08b] with masses computed using FWHM. They found that, in any redshift bin, if one plots the distribution of mass versus luminosity, the higher mass objects lie increasingly below the Eddington limit; they refer to this as the “sub-Eddington boundary.” There is no physical basis for this. [@Rafiee11] point out, however, that if the quasar masses are computed using  instead of FWHM, the sub-Eddington boundary disappears: the distribution of quasar black hole masses approaches the Eddington limit at all masses. Referring to Figure 1 of [@Rafiee11], the distribution of quasars in the mass vs. luminosity diagram is an enlongated cloud of points whose axis is roughly parallel to the Eddington ratio when  is used to characterize the line width. However, when FWHM is used, the axis of the distribution rotates as the higher masses are underestimated and the lower masses are overestimated. However, the apparent rotation of the mass distribution is in the same sense that is expected from the Malmquist bias and a bottom heavy quasar mass function [@Shen13]. Unfortunately, these arguments are not statistically compelling. Examination of the –$\sigma{*}$ relation using FWHM-based and -based masses is equally unrevealing [@Wang19].

In reverberation mapping, a further distinction among line-width measures must be drawn since either FWHM or  can be measured in the mean spectrum, $$\overline{F}(\lambda) =\frac{1}{N} \sum_{1}^{N} F_i(\lambda),
\label{eq:meanspec}$$ where $F_i(\lambda)$ is the flux in the $i^{th}$ spectrum of the time series at wavelength $\lambda$ and $N$ is the number of spectra, or they can be measured in the rms residual spectrum (hereafter simply “rms spectrum”), which is defined as $$\sigma_{\rm rms}(\lambda) = \left\{ \frac{1}{N-1}
\sum_{1}^{N}\left[ F_i(\lambda) -
\overline{F}(\lambda)\right]^2 \right\}^{1/2}.
\label{eq:rmsspec}$$ In this paper, we will specifically refer to the measurements of  in the mean spectrum as  and in the rms spectrum as . Similarly,  refers to FWHM of a line in the mean spectrum or a single-epoch spectrum and  is the FWHM in the rms spectrum. It is common to use  as the line-width measure for determining black hole masses from reverberation data — it is intuitatively a good choice as it isolates the gas in the BLR that is actually responding to the continuum variations. As noted previously, the strong and strongly variable broad emission lines can be hard to isolate as they are blended with other features. In the rms spectra, however, the contaminating features are much less of a problem because they are generally constant or vary either slowly or weakly and thus nearly disappear in the rms spectra.

Since the goal is to measure a black hole mass from a single (or a few) spectra, we must use a proxy for . Here we will attempt to determine if either  or  in a single or mean spectrum can serve as a suitable proxy for ; we know [*a priori*]{} that there are good, but non-linear, correlations between  and both  and . It therefore seems likely that either  or  could be used as a proxy for .

Investigation of the relationship among the line-width measures motivated a broader effort to produce easy-to-use prescriptions for computing [*accurate*]{} black hole masses using  and  emission lines and nearby continuum fluxes measurements for each line. We note that we do not discuss  RM results in this contribution as the present situation has been addressed rather thoroughly by [@Bahk19] and new SDSS-RM results will be published soon [@Homayouni20]. In §[2]{}, the data used in this investigation are described. In §[3]{}, the relationship between the  reverberation lag and different measures of the AGN luminosity are considered, and we identify the physical parameters to lead to accurate black-hole mass determinations. In §[4]{}, we will similarly discuss masses based on . In §[5]{}, we present simple empirical formulae for estimating black hole masses from  or . The results of this investigation and our future plans to improve this method are outlined in §[6]{}. Our results are briefly summarized in §[7]{}. Throughout this work, we assume $H_0 = 72$Mpc$^{-1}$, $\Omega_{\rm matter} =0.3$ and $\Omega_\Lambda = 0.7$.

Observational Database and Methodology
======================================

Data {#section:Data}
----

We use two high-quality databases for this investigation:

1.  Spectra and measurements for previously reverberation-mapped AGNs, for  (Table A1) and for  (Table A2). These are mostly taken from the literature (see also @Bentz15 for a compilation[^1]). Sources without estimates of host-galaxy contamination to the optical luminosity $L(5100\,{\rm \AA})$ have been excluded. This database provides the fundamental $R$–$L$ calibration for the single-epoch mass scale. In this contribution, we will refer to these collectively as the “reverberation-mapping database (RMDB)”.

2.  Spectral measurements from the Sloan Digital Sky Survey Reverberation Mapping Program [@Shen15 hereafter “SDSS-RM” or more compactly simply as “SDSS”]. We use both  (Table A3) and  (Table A4) data from the 2014–2018 SDSS-RM campaign [@Grier17b; @Shen19; @Grier19]. Each spectrum is comprised of the average of the individual spectra obtained for each of the 849 quasars in the SDSS-RM field.

In addition, because  RM measurements remain rather scarce, we augmented the  sample with measurements from [@Vestergaard06] (hereafter VP06), who combined single-epoch luminosity and line-width measurements from archival UV spectra with -based mass measurements of the objects in Table A1. The UV parameters are given in Table A5; we note, however, that we have excluded 3C273 and 3C390.3 because they both have uncertainities in their virial product larger than 0.5dex; the former was a particular problem because there were far more measurements of UV parameters for this source than for any other and the combination of a large number of measurements and a poorly constrained virial product conspired to disguise real correlations.

All SDSS-RM spectra have been reduced and processed as described by [@Shen15] and [@Shen16b], including post-processing with [PrepSpec]{} (Horne, in preparation). We note that only lags ($\tau$), line dispersion in the rms spectrum (${\mbox{$\sigma_{\rm R}$}}$), and virial products (${\mbox{$\mu_{\rm RM}$}}= {\mbox{$\sigma_{\rm R}$}}^2 c \tau/G$) are taken from [@Grier17b] and [@Grier19]; all luminosities and other line-width measures are from [@Shen19] (Tables A3 and A4 are included here for the sake of clarity).

For each SDSS AGN, there are two determinations of both  and ; one is the best-fit (BF) to the mean spectrum, and the other is the mean of multiple Monte Carlo (MC) realizations. For each MC realization, $N$ independent random selections of the $N$ spectra are combined and the line width is measured for both  and . After a large number of realizations, the mean $\langle V
\rangle$ and rms $\Delta V$, for $V = {\mbox{${\rm FWHM}_{\rm M}$}}$ and $V = {\mbox{$\sigma_{\rm M}$}}$ are computed, and the rms values are adopted as the uncertainties in each line-width measure.

For the purpose of mass estimation, we need to establish relationships based on the most reliable data. Many of the SDSS average spectra are still quite noisy, so we imposed quality cuts. Even though we are for the most part restricting our attention to the SDSS-RM quasars for which there are measured lags for  (44 quasars) or  (48 quasars), we impose these cuts on the entire sample for the sake of later discussion. The first quality condition is that $$\label{eq:minwidth}
V \geq 1000\,{\mbox{\rm km~s$^{-1}$}}$$ for both $V = {\mbox{${\rm FWHM}_{\rm M}$}}$ and $V ={\mbox{$\sigma_{\rm M}$}}$, since AGNs with lines narrower than 1000  are probably Type 2 AGNs; there are some Type 1 AGNs with line widths narrower than this, including several in Table A1, but these are low-luminosity AGNs [e.g., @Greene07], not SDSS quasars. The second quality condition is that the best fit value $V({\rm BF})$ must lie in the range $$\label{eq:consistency}
\langle V \rangle - \Delta V \leq V({\rm BF}) \leq \langle V \rangle + \Delta V$$ for both FWHM and . A third quality condition is a “signal-to-noise” ($S/N$) requirement that the line width must be significantly larger than its uncertainty. Some experimentation showed that $$\label{eq:s2n}
\frac{V}{\Delta V} \geq 10$$ is a good criterion for both $V = {\mbox{${\rm FWHM}_{\rm M}$}}$ and $V = {\mbox{$\sigma_{\rm M}$}}$ to remove the worst outliers from the line-width comparisons discussed in §[\[section:hblinewidths\]]{} and §[\[section:civfundamental\]]{}.

Finally, we removed quasars that were flagged by [@Shen19] as having broad absorption lines (BALs), mini-BALs, or suspected BALs in .

The effect of each quality cut on the size of the database available for each emission line is shown in Table 1. Of the 44 SDSS-RM quasars with measured  lags, 12 failed to meet at least one of the quality criteria, usually the $S/N$ requirement, thus reducing the SDSS-RN  sample to 32 quasars. Three quasars with  reverberation measurements (RMID 362, 408, and 722) were rejected for significant BALs, thus reducing the SDSS-RM  reverberation sample to 45 quasars. As we will show in §[\[section:massformulae\]]{}, another effect of imposing quality cuts is, not surprisingly, that it removes some of the lower luminosity sources from the sample.

[lcc]{} Original sample & 221 & 540\
(a) Minimum Line Width (eq. \[eq:minwidth\]) & 199 & 520\
(b) Consistency (eq. \[eq:consistency\]) & 194 & 368\
(c) $S/N$ (eq. \[eq:s2n\]) & 121 & 462\
(a) + (b) & 174 & 352\
(a) + (c) & 108 & 450\
(b) + (c) & 107 & 309\
(a) + (b) + (c) & 96 & 299\
All + BAL removal & 96 & 248 \[table:qcuts\]

Fitting Procedure
-----------------

Throughout this work, we use the fitting algorithm described by [@Cappellari13] that combines the Least Trimmed Squares technique of [@Rousseeuw06] and a least-squares fitting algorithm which allows errors in all variables and includes intrinsic scatter, as implemented by [@DallaBonta18]. Briefly, the fits we perform here are of the general form $$\label{eq:powerlaw}
y = a + b\left(x - x_0 \right),$$ where $x_0$ is the median value of the observed parameter $x$. The fit is done iteratively with $5 \sigma$ rejection (unless stated otherwise) and the best fit minimizes the quantity $$\label{eq:chi2line}
    \chi^2=\sum_{i=1}^N \frac{[a+b (x_i-x_0) - y_i]^2}
    {(b \Delta x_i)^2 + (\Delta y_i)^2 + \varepsilon_y^2},$$ where $\Delta x_i$ and $\Delta y_i$ are the errors on the variables $x_i$ and $y_i$, and $\varepsilon_y$ is the sigma of the Gaussian describing the distribution of intrinsic scatter in the $y$ coordinate; $\varepsilon_y$ is iteratively adjusted so that the $\chi^2$ per degree of freedom $\nu=N-2$ has the value of unity expected for a good fit. The observed scatter is $$\label{eq:Deltadef}
\Delta = \left\{ \frac{1}{N-2} \sum_{i=1}^N
\left[y_i - a - b\left(x_i - x_0\right) \right]^2 \right\}^{1/2}.$$ The value of $\varepsilon_y$ is added in quadrature when $y$ is used as a proxy for $x$.

The bivariate fits are intended to establish the physical relationships among the various parameters and also to fit residuals. The actual mass estimation equations that we use will be based on multivariate fits of the general form $$\label{eq:twoparameters}
z = a + b\left( x - x_0 \right) +c\left(y - y_0\right),$$ where the parameters are as described above, plus an additional observed parameter $y$ that has median value $y_0$. Similarly to linear fits, the plane fitting minimizes the quantity $$\label{eq:chi2plane}
    \chi^2=\sum_{i=1}^N \frac{[a + b (x_i-x_0) + c (y_i-y_0) - z_i]^2}
    {(b \Delta x_i)^2 + (c \Delta y_i)^2 + (\Delta z_i)^2 + \varepsilon_z^2},$$ with $\Delta x_i$, $\Delta y_i$ and $\Delta z_i$ as the errors on the variables $(x_i,y_i,z_i)$, and $\varepsilon_z$ as the sigma of the Gaussian describing the distribution of intrinsic scatter in the $z$ coordinate; $\varepsilon_z$ is iteratively adjusted so that the $\chi^2$ per degrees of freedom $\nu=N-3$ has the value of unity expected for a good fit. The observed scatter is $$\label{eq:twoparmDeltadef}
\Delta = \left\{ \frac{1}{N-3} \sum_{i=1}^N
\left[y_i - a - b\left(x_i - x_0\right)
-c\left(y_i - y_0\right)\right]^2 \right\}^{1/2}.$$

Masses Based on 
================

The $R$–$L$ Relationships
-------------------------

In this section, we examine the calibration of the fundamental  $R$–$L$ relationship using various luminosity measures. The analysis in this section is based only on the RMDB sample in Table A1 because all these sources have been corrected for host-galaxy starlight. To obtain accurate masses from , contaminating starlight from the host galaxy must be accounted for in the luminosity measurement, or the mass will be overestimated. For reverberation-mapped sources, this has been done by modeling unsaturated images of the AGNs obtained with the [*Hubble Space Telescope*]{} [@Bentz06a; @Bentz09a; @Bentz13]. The AGN contribution was removed from each image by modeling the images as an extended host galaxy plus a central point source representing the AGN. The starlight contribution to the reverberation-mapping spectra is determined by using simulated aperture photometry of the AGN-free image. In the left panel of Figure \[Figure:HbRL\], we show the  lag as a function of the AGN continuum with the host contribution removed in each case. This essentially reproduces the result of [@Bentz13] as small differences are due solely to improvements in the quality and quantity of the RM database \[cf. Table A1\]; we give the best-fit values to equation (\[eq:powerlaw\]) in the first row of Table 2.

![Left: The rest-frame  lag in days is shown as a function of the AGN luminosity $L_{\rm AGN}(5100\,{\rm \AA})$ in . The host-galaxy starlight contribution has been removed by using unsaturated [[*HST*]{}]{} images [see @Bentz13]. Right: The  lag in days is shown as a function of the broad  luminosity $L({\mbox{\rm H$\beta$}}_{\rm broad})$ in . The narrow component of  has been removed in each case where it was sufficiently strong (i.e., easily identifiable) to isolate. In both panels, the solid line shows the best-fit to the data using equation (\[eq:powerlaw\]) with coefficients given in Table 2. The short dashed lines show the $\pm1\,\sigma$ uncertainty (equivalent to enclosing 68% of the values for a Gaussian distribution) and the long dashed lines show the $2.6\sigma$ uncertainties (equivalent to enclosing 99% of the values for a Gaussian distribution). []{data-label="Figure:HbRL"}](Figure1.eps)

[llccccc]{} $\log L_{\scriptsize\rm AGN}(5100\,{\rm \AA})$ & $\log \tau({\mbox{\rm H$\beta$}})$ & $1.228 \pm 0.025$ & $0.482 \pm 0.029$ & $43.444$ & $0.213 \pm 0.021$ & $0.241$\
$\log L({\mbox{\rm H$\beta$}}_{\scriptsize\rm broad})$ & $\log \tau({\mbox{\rm H$\beta$}})$ & $1.200\pm 0.025$ & $0.492 \pm 0.030$ & $41.746$ & $0.218 \pm 0.022$ & $0.244$\
$\log L(1350\,{\rm \AA})$ & $\log \tau({\mbox{\rm C\,{\sc iv}}})$ & $1.915 \pm 0.047$ & $0.517 \pm 0.036$ & $45.351$ & $0.336 \pm 0.041$ & $0.361$\
$\log L_{\scriptsize\rm AGN}(5100\,{\rm \AA})$ & $\log L({\mbox{\rm H$\beta$}}_{\rm broad})$ & $41.797 \pm 0.017$ & $0.960 \pm 0.020$ & $43.444$ & $0.158 \pm 0.014$ &$0.171$\
$\log L({\mbox{\rm H$\beta$}}_{\rm broad})$ &$\log L_{\scriptsize\rm AGN}(5100\,{\rm \AA})$ & $43.396 \pm 0.018$ & $1.003 \pm 0.022$ & $41.746$ & $0.161 \pm 0.015$ & $0.174$ \[table:RL\]

Accounting for the host-galaxy contribution in the same way for large number of AGNs, such as those in SDSS-RM (not to mention the entire SDSS catalog), is simply not feasible. It is well-known, however, that there is a tight correlation between the AGN continuum luminosity and the luminosity of [e.g., @Yee80; @Ilic17], and it has indeed been argued that the  emission-line luminosity can be used as a proxy for the AGN continuum luminosity for reverberation studies [@Vestergaard06; @Greene10a]. However, in some of the reverberation-mapped sources, narrow-line contributes significantly to the total  flux; NGC 4151 is an extreme example [e.g., @Antonucci83; @Bentz06a; @Fausnaugh17]. Whenever the narrow-line component can be isolated, it has been subtracted from the total  flux. In Figure \[Figure:LAGNLHb\], we show the tight relationship between $L_{\rm AGN}(5100\,{\rm \AA})$ and $L({\mbox{\rm H$\beta$}}_{\rm broad})$; the best-fit coefficients for this relationship are given in Table 2.

In the right panel of Figure \[Figure:HbRL\], we show the  lag as a function of the luminosity of the broad component of , with the narrow component removed whenever possible. We give the best-fit values to the equation (\[eq:powerlaw\]) in the second row of Table 2, which shows that the slope of this relationship is nearly identical to the slope of the $R$–$L$ relationship using the AGN continuum. The luminosity of the  broad component is thus an excellent proxy for the AGN luminosity and requires only removal of the narrow component (at least when it is significant) which is much easier than estimating the starlight contribution to the continuum luminosity at 5100Å. Moreover, by using the line flux instead of the continuum flux, we can include core-dominated radio sources where the continuum may be enhanced by the jet component [@Greene05]. This is therefore the $R$–$L$ relationship we prefer for the purpose of estimating single-epoch masses and we will focus on this relationship through the remainder of this contribution.

![The relationship between the broad  emission line luminosity and the starlight-corrected AGN luminosity for the sources in Table A1. The black solid line is the regression of $L({\mbox{\rm H$\beta$}}_{\rm broad})$ on $L_{\rm AGN}(5100\,{\rm \AA})$; the red dotted line is the regression of $L_{\rm AGN}(5100\,{\rm \AA})$ on $L({\mbox{\rm H$\beta$}}_{\rm broad})$, which we use in equation (\[eq:LHbLAGN\]). The coefficients for both fits are given in Table 2.[]{data-label="Figure:LAGNLHb"}](Figure2.eps)

Line-Width Relationships {#section:hblinewidths}
------------------------

We now consider the use of  and  as proxies for [cf. @Collin06; @Wang19]. The left panel of Figure \[Figure:windowhbwidths\] shows the relationship between ${\mbox{$\sigma_{\rm R}$}}({\mbox{\rm H$\beta$}})$, the  line dispersion in the rms spectrum, and ${\mbox{$\sigma_{\rm M}$}}({\mbox{\rm H$\beta$}}),$ the  line dispersion in the mean spectrum. The relationship is nearly linear (slope $ = 1.085\pm0.045$) and the intrinsic scatter is small ($0.079$dex). The fit coefficients are given in the first row of Table 3.

![The relationship between  line dispersion in the rms ${\mbox{$\sigma_{\rm R}$}}({\mbox{\rm H$\beta$}})$ and mean ${\mbox{$\sigma_{\rm M}$}}({\mbox{\rm H$\beta$}})$ spectra is shown on the left. The relationship between  line dispersion in the rms spectrum () and FWHM in the mean spectrum () is shown on the right. Blue filled circles are for the RMDB sample (Table A1) and open green triangles are for the SDSS sample (Table A3). The solid lines are best fits to equation (\[eq:powerlaw\]) with coefficients in Table 3. The short dashed and long dashed lines indicate the $\pm 1 \sigma$ and $\pm 2.6 \sigma$ envelopes, respectively, and the red dotted lines indicate where the two line-width measures are equal. Crosses are points that were rejected at the 2.6$\sigma$ (99%) level and are colored-coded like the circles. The relationship on the left is nearly linear (slope $= 1.085 \pm 0.045$) and the scatter $\varepsilon_y$ is low (0.079dex). It is clear in the right panel that () and () are well-correlated, but the relationship is significantly non-linear (slope $= 0.535 \pm 0.042$), the scatter $\varepsilon_y$ is slightly larger (0.106dex), and there are several significant outliers.[]{data-label="Figure:windowhbwidths"}](Figure3.eps)

[llccccc]{} $\log {\mbox{$\sigma_{\rm M}$}}({\mbox{\rm H$\beta$}})$ & $\log {\mbox{$\sigma_{\rm R}$}}({\mbox{\rm H$\beta$}})$ & $3.260 \pm 0.008$ & $1.085 \pm 0.045$ & 3.297 & $0.079 \pm 0.006$ & 0.087\
$\log {\mbox{${\rm FWHM}_{\rm M}$}}({\mbox{\rm H$\beta$}})$ & $\log {\mbox{$\sigma_{\rm R}$}}({\mbox{\rm H$\beta$}}) $ & $3.205 \pm 0.011$ & $0.535 \pm 0.042$ & 3.559 & $0.106 \pm 0.001$ & 0.114\
$\log {\mbox{$\sigma_{\rm M}$}}({\mbox{\rm C\,{\sc iv}}})$ & $\log {\mbox{$\sigma_{\rm R}$}}({\mbox{\rm C\,{\sc iv}}})$ & $3.436 \pm 0.009$ &$0.822 \pm 0.059$ & 3.394 & $0.064 \pm 0.008$ & 0.067\
$\log {\mbox{${\rm FWHM}_{\rm M}$}}({\mbox{\rm C\,{\sc iv}}})$ & $\log {\mbox{$\sigma_{\rm R}$}}({\mbox{\rm C\,{\sc iv}}})$ & $3.447 \pm 0.016$ & $0.445 \pm 0.101$ & 3.580 & $0.121 \pm 0.014$ & 0.121 \[table:LW\]

We also show in the right panel of Figure \[Figure:windowhbwidths\] the relationship between () and the FWHM of  in the mean spectrum, (). The fit coefficients are given in the second row of Table 3. The relationship is far from linear (slope $= 0.535 \pm 0.042$), and the scatter $\varepsilon_y$ is larger than it is for the ()–() relationship, even after removal of the notable outliers. While it is clear that () is an excellent proxy for (), the value of () is less clear. Nevertheless we will fit both versions in order to understand the relative merits of each.

Single-Epoch Predictors of the Virial Product {#section:hbpredictor}
---------------------------------------------

In the previous subsections, we have re-established the correlations between $\tau({\mbox{\rm H$\beta$}})$ and $L({\mbox{\rm H$\beta$}}_{\rm broad})$ and between () and both () and (). As a first approximation for a formula to estimate single-epoch masses, we fit the following equations: $$\log {\mbox{$\mu_{\rm RM}$}}({\mbox{\rm H$\beta$}}) = a + b\left[\log L({\mbox{\rm H$\beta$}}_{\rm broad}) - x_0\right]
+c\left[\log{\mbox{$\sigma_{\rm M}$}}({\mbox{\rm H$\beta$}}) - y_0\right],
\label{eq:Fit_139}$$ and $$\log {\mbox{$\mu_{\rm RM}$}}({\mbox{\rm H$\beta$}}) = a + b\left[ \log L({\mbox{\rm H$\beta$}}_{\rm broad}) - x_0\right]
+c\left[ \log {\mbox{${\rm FWHM}_{\rm M}$}}({\mbox{\rm H$\beta$}}) - y_0\right].
\label{eq:Fit_141}$$

The results of these fits based on the combined RMDB data (Table A1) and SDSS data (Table A3) are given in the first two rows of Table 4, and illustrated in the upper panels of Figure \[Figure:Fit\_139\_141\]. Using these coefficients, we have initial fits $$\begin{aligned}
\log {\mbox{$\mu_{\rm SE}$}}({\mbox{\rm H$\beta$}}) & = &
6.975  + 0.566\left[ \log L({\mbox{\rm H$\beta$}}_{\rm broad}) - 41.857\right] \nonumber \\
& & + 1.757\left[ \log {\mbox{$\sigma_{\rm M}$}}({\mbox{\rm H$\beta$}}) - 3.293\right],
\label{eq:SEsigm}\end{aligned}$$ and $$\begin{aligned}
\log {\mbox{$\mu_{\rm SE}$}}({\mbox{\rm H$\beta$}}) & = &
6.981 + 0.587\left[ \log L({\mbox{\rm H$\beta$}}_{\rm broad}) - 41.857\right] \nonumber \\
& & + 1.039\left[ \log {\mbox{${\rm FWHM}_{\rm M}$}}({\mbox{\rm H$\beta$}}) - 3.599\right],
\label{eq:SEfwm}\end{aligned}$$ for () and (), respectively. The luminosity coefficient $b$ and the line-width coefficient $c$ are roughly as expected from the virial relationship and the $R$–$L$ relationship, and we note that the line-width coefficient for  ($ c = 1.039$) is much smaller than that of  ($c = 1.757$), as expected from Figure \[Figure:windowhbwidths\]. It is clear that both equations (\[eq:SEsigm\]) and (\[eq:SEfwm\]) overestimate masses at the low end and underestimate them at the high end, thus biasing the prediction. This suggests that another parameter is required for the single-epoch virial product prediction.

![The two upper panels show single-epoch (SE) virial product predictions based on equations (\[eq:Fit\_139\]) and (\[eq:Fit\_141\]) on the left and right, respectively, with coefficients from Table 4 compared with the actual RM measurements for the same sources. Blue filled circles represent RMDB data (Table A1) and green open triangles represent SDSS data (Table A3). The solid line shows the best fit to the data, and the red dotted line shows where the two values are equal. The short and long dashed lines show the $\pm 1 \sigma$ and $\pm 2.6 \sigma$ envelopes, respectively. It is clear that this is an inadequate virial product predictor as it systematically underestimates higher masses and overestimates lower masses. The two lower panels show the same relationship after the empirical corrections as embodied in equations (\[eq:predictHbsigm\]) and (\[eq:predictHbFWHM\]) for  and , respectively. The best fit lines cover the equality lines. []{data-label="Figure:Fit_139_141"}](Figure4.eps)

[lllccccccc]{} $\log L({\mbox{\rm H$\beta$}}_{\rm broad})$ & $\log {\mbox{$\sigma_{\rm M}$}}({\mbox{\rm H$\beta$}})$ & $\log {\mbox{$\mu_{\rm RM}$}}({\mbox{\rm H$\beta$}})$& $6.975 \pm 0.029$ & $0.566 \pm 0.035$ & $1.757 \pm 0.160$ & $41.857$ & $3.293$ & $0.273 \pm 0.025$ & $0.314$\
$\log L({\mbox{\rm H$\beta$}}_{\rm broad})$ & $\log {\mbox{${\rm FWHM}_{\rm M}$}}({\mbox{\rm H$\beta$}})$ & $\log {\mbox{$\mu_{\rm RM}$}}({\mbox{\rm H$\beta$}})$ & $6.981 \pm 0.033$ & $0.587 \pm 0.040$ & $1.039\pm 0.128$ & $41.857$ & $3.559$ & $0.323 \pm 0.028$ & $0.352$\
$\log L(1350\,{\rm \AA})$ & $\log {\mbox{$\sigma_{\rm M}$}}({\mbox{\rm C\,{\sc iv}}})$ & $\log {\mbox{$\mu_{\rm RM}$}}({\mbox{\rm C\,{\sc iv}}})$ & $7.664 \pm 0.039$ & $0.599 \pm 0.033$ & $1.014 \pm 0.265$ & $44.706$ & $3.502$ & $0.364 \pm 0.033$ & $0.397$ \[table:mfits\]

We investigated the possibility of another parameter by plotting the residuals $\Delta \log \mu =
\log {\mbox{$\mu_{\rm RM}$}}- \log {\mbox{$\mu_{\rm SE}$}}$ against other parameters, specifically luminosity, mass (virial product), Eddington ratio, emission-line lag, line width and line-width ratio ${\rm FWHM}/{\mbox{$\sigma_{\rm line}$}}$ for both mean and rms spectra. The most significant correlation between the virial product residuals and other parameters was for Eddington ratio, which has been a result of other recent investigations [@Du16; @Grier17b; @Du18; @Du19; @Alvarez19; @Martinez19]. To determine the Eddington ratio, we start with the Eddington luminosity $$\label{eq:Eddingtonlimit}
L_{\rm Edd} = \frac{4 \pi G c m_e M}{\sigma_e}
= 1.257 \times 10^{38} \left(\frac{M}{{\mbox{$M_\odot$}}}\right),$$ where $m_e$ is the electron mass and $\sigma_e$ is the Thomson cross-section. The black hole mass is $\log M = \log f + \log \mu$ and, as explained in the Appendix, we assume $\log f = 0.683 \pm 0.150$ [@Batiste17] so the Eddington luminosity is $$\label{eq:Ledd}
\log L_{\rm Edd} = \log f + 38.099 + \log {\mbox{$\mu_{\rm RM}$}}= 38.782 + \log {\mbox{$\mu_{\rm RM}$}}.$$ The bolometric luminosity can be obtained from the observed 5100Å AGN luminosity plus a bolometric correction $$\label{eq:bolometriclum}
\log L_{\rm bol} = \log L_{\rm AGN}(5100\,{\rm \AA})+ \log k_{\rm bol} .$$ We ignore inclination effects and, following [@Netzer19], we use $$\label{eq:bolometriccorrection}
\log k_{\rm bol} = 10 - 0.2\log L_{\rm AGN}(5100\,{\rm \AA}).$$ Since we are using $L({\mbox{\rm H$\beta$}}_{\rm broad})$ as a proxy for $L_{\rm AGN}(5100\,{\rm \AA})$, we substitute $L({\mbox{\rm H$\beta$}}_{\rm broad})$ for $L_{\rm AGN}(5100\,{\rm \AA})$ by fitting the luminosities in Table A1, yielding (see Table 2) $$\log L_{\rm AGN}(5100\,{\rm \AA}) = 43.396 
+ 1.003\left[\log L({\mbox{\rm H$\beta$}}_{\rm broad}) - 41.746\right],
\label{eq:LHbLAGN}$$ so we can write the bolometric luminosity as $$\log L_{\rm bol} = 44.717 + 0.802\left[\log({\mbox{\rm H$\beta$}}_{\rm broad}) - 41.746 \right].
\label{eq:boldef}$$ The Eddington ratio  is given by[^2] $$\log {\mbox{$\dot{m}$}}= \log L_{\rm bol} - \log L_{\rm Edd}.
\label{eq:Eddratio}$$ Using equations (\[eq:boldef\]) and (\[eq:Ledd\]), the Eddington ratio can then be written as $$\log \dot{m} = 5.935 + 0.802 \left[ \log L({\mbox{\rm H$\beta$}}_{\rm broad}) - 41.746\right]
- \log {\mbox{$\mu_{\rm RM}$}}.
\label{eq:Eddalt}$$

[lllccccc]{} All  & $\log \dot{m}$ & $\Delta \log \mu ({\mbox{$\sigma_{\rm M}$}})$ & $-0.010 \pm 0.022$ & $-0.422 \pm 0.045$ & $-0.951$ & $0.187 \pm 0.021$ & $0.246$\
All  & $\log \dot{m}$ & $\Delta \log \mu ({\mbox{${\rm FWHM}_{\rm M}$}})$ & $-0.007 \pm 0.023$ & $-0.543 \pm 0.046$ & $-0.951$ & $0.191 \pm 0.021$ & $0.247$\
All  & $\log \dot{m}$ & $\Delta \log \mu$ & $-0.049 \pm 0.026$ & $-0.557 \pm 0.048$ & $-1.155$ & $0.213 \pm 0.027$ & $0.282$\
All  & $\log {\mbox{$\mu_{\rm RM}$}}$ & $\Delta \log \mu$ & $-0.012 \pm 0.026$ & $0.297 \pm 0.024$ & $7.481$ & $0.000 \pm 0.000$ & $0.139$\
All  & $\log {\mbox{$\mu_{\rm RM}$}}({\mbox{\rm H$\beta$}},{\mbox{$\sigma_{\rm M}$}})$ & $\log {\mbox{$\mu_{\rm SE}$}}({\mbox{\rm H$\beta$}},{\mbox{$\sigma_{\rm M}$}})$ & $7.025 \pm 0.025$ & $0.805 \pm 0.038$ & $7.041$ & $0.249 \pm 0.021$ & $0.279$\
All  & $\log {\mbox{$\mu_{\rm RM}$}}({\mbox{\rm H$\beta$}},{\mbox{${\rm FWHM}_{\rm M}$}})$ & $\log {\mbox{$\mu_{\rm SE}$}}({\mbox{\rm H$\beta$}},{\mbox{${\rm FWHM}_{\rm M}$}})$ & $7.012 \pm 0.028$ & $0.749 \pm 0.042$ & $7.007$ & $0.278 \pm 0.023$ & $0.290$\
All  & $\log {\mbox{$\mu_{\rm RM}$}}({\mbox{\rm C\,{\sc iv}}}) $ & $\log {\mbox{$\mu_{\rm SE}$}}({\mbox{\rm C\,{\sc iv}}}) $ & $7.483 \pm 0.033$ & $0.787 \pm 0.041$ & $7.481$ & $0.321 \pm 0.028$ & $0.347$ \[table:residfits\]

To correct the single-epoch masses for Eddington ratio, we fit the equation $$\Delta \log \mu = \log {\mbox{$\mu_{\rm RM}$}}- \log {\mbox{$\mu_{\rm SE}$}}=
a + b(\log \dot{m} - x_0),
\label{eq:residuals}$$ and use this as a correction to our initial fits, equations (\[eq:SEsigm\]) and (\[eq:SEfwm\]). The best-fit parameters for  and -based predictors of  are given in Table 5 and shown in Figure \[Figure:windowhbresiduals\]. Combining the correction equation (\[eq:residuals\]) with the best-fit coefficients in Table 5 and equations (\[eq:SEsigm\]) and (\[eq:SEfwm\]) yield the corrected single-epoch masses $$\begin{aligned}
\log {\mbox{$\mu_{\rm SE}$}}({\mbox{\rm H$\beta$}}) & = &
6.965  + 0.566\left[ \log L({\mbox{\rm H$\beta$}}_{\rm broad}) - 41.857\right] \nonumber \\
& & + 1.757\left[ \log {\mbox{$\sigma_{\rm M}$}}({\mbox{\rm H$\beta$}}) - 3.293\right]
- 0.422\left[ \log {\mbox{$\dot{m}$}}+ 0.951 \right],
\label{eq:SEsigmcorr}\end{aligned}$$ and $$\begin{aligned}
\log {\mbox{$\mu_{\rm SE}$}}({\mbox{\rm H$\beta$}}) & = &
6.974 + 0.587\left[ \log L({\mbox{\rm H$\beta$}}_{\rm broad}) - 41.857\right] \nonumber \\
& & + 1.039\left[ \log {\mbox{${\rm FWHM}_{\rm M}$}}({\mbox{\rm H$\beta$}}) - 3.599\right] 
- 0.543 \left[\log {\mbox{$\dot{m}$}}+0.951 \right],
\label{eq:SEfwmcorr}\end{aligned}$$ for  and , respectively.

Once the dependence on Eddington ratio is removed, the residuals do not appear to correlate with other properties. The intrinsic scatter about the final residuals is $0.197$dex for -based masses and $0.204$dex for -based masses.

![Upper left: fit to mass residuals (equation \[eq:residuals\]) vs. Eddington ratio $\dot{m}$ (equation \[eq:Eddalt\]) for single-epoch masses based on (). Upper right: fit to mass residuals vs. Eddington ratio $\dot{m}$ for single-epoch masses based on (). Lower panels: residuals after subtraction of the best fit in the panel above. The $\varepsilon_y$ scatter in the residuals is 0.197dex for the -based virial products and 0.204dex for the -based virial products. In all panels, the solid blue circles represent RMDB data (Table A1) and the open green triangles represent SDSS data (Table A3). The solid line shows the best fit to the data. The short dashed and long dashed lines are the $\pm 1 \sigma$ and $\pm 2.6 \sigma$ envelopes, respectively. The coefficients of the fits are given in Table 5. Error bars are measurement uncertainties only, without systematic errors.[]{data-label="Figure:windowhbresiduals"}](Figure5.eps)

Masses Based on 
================

Fundamental Relationships {#section:civfundamental}
-------------------------

As noted in §\[section:intro\], the veracity of -based mass estimates is unclear and remains controversial. The ideal situation would be to have a large number of AGNs with both and  reverberation measurements to effect a direct comparison. There are, unfortunately, very few AGNs that have both; indeed Table A2 of the Appendix lists all  results for which there are corresponding  measurements in Table A1. For the few sources with both  and  reverberation measurements, we plot the virial products $\mu_{\rm RM}({\mbox{\rm C\,{\sc iv}}})$ and $\mu_{\rm RM}({\mbox{\rm H$\beta$}})$ in Figure \[Figure:RMcompare\]; these are in each case a weighted mean value of $$\label{eq:mudef}
\mu_{\rm RM} = \left( \frac{c \tau {\mbox{$\sigma_{\rm R}$}}^2}{G}\right)$$ for each of the observations of  and  for the AGNs that appear in both Tables A1 and A2. The close agreement of these values reassures us that the -based RM masses can be trusted, at least over the range of luminosities sampled.

![Virial products based on and  for the few cases in the RMDB sample for which both are available. The solid line is the locus where the two virial products are equal. The values are weighted means of ${\mbox{$\mu_{\rm RM}$}}({\mbox{\rm H$\beta$}})$ and ${\mbox{$\mu_{\rm RM}$}}({\mbox{\rm C\,{\sc iv}}})$ for individual AGNs that appear in both Tables A1 and A2. The Spearman rank coefficient for these data is $\rho = 0.805$. []{data-label="Figure:RMcompare"}](Figure6.eps)

We now need to consider whether or not luminosities and mean line widths are suitable proxies for emission-line lag and rms line widths in the case of . In Figure \[Figure:civradlum\], we show the relationship between the UV continuum luminosity $L(1350\,{\rm \AA})$ and the  emission line lag $\tau({\mbox{\rm C\,{\sc iv}}})$ based on the  data in Table A2, plus the SDSS-RM  data in Table A4. The coefficients of the fit are given in Table 2. We note again that we have removed from the [@Grier19] sample in Table A4 three quasars with BALs, thus reducing the sample size from 48 to 45. The slope of the  $R$–$L$ relation ($0.517$) is consistent with that of  ($0.492$), though the $\varepsilon_y$ scatter is substantially greater ($0.336$dex for  compared to $0.213$dex for ). Definition of the relationship does not depend on the two separate measurements of very short  lag measurements for the dwarf Seyfert NGC 4395 [@Peterson05]. Thus it seems clear that we can use $L(1350\,{\rm \AA})$ as a reasonable proxy for $\tau({\mbox{\rm C\,{\sc iv}}})$.

![Relationship between the  rest-frame emission-line lag $\tau({\mbox{\rm C\,{\sc iv}}})$ and the continuum luminosity at 1350Å. Blue filled circles represent RMDB data (Table A2) and green open triangles represent SDSS data (Table A4). The solid line is the best fit to the data using equation (\[eq:powerlaw\]) with coefficients given in Table 2. The short dashed and long dashed lines are the $\pm 1 \sigma$ and $\pm 2.6 \sigma$ envelopes, respectively. The Spearman rank coefficient for these data is $\rho = 0.503$. If the two lowest luminosity points (both measurements of the dwarf Seyfert NGC4395) are omitted, the Spearman rank coefficient is decreased to $\rho = 0.481$. []{data-label="Figure:civradlum"}](Figure7.eps)

We show the relationship between the  line dispersion measured in the rms spectrum () and the line dispersion in the mean spectrum () in Figure \[Figure:windowcivwidths\]. The best-fit coefficients are given in Table 3. The correlation is good. However, the correlation between () and (), also shown in Figure \[Figure:windowcivwidths\], is rather poor [see also @Wang20] and demonstrates that () is a dubious proxy for (). Measurement of () is clearly not a reliable predictor of (), so we will not consider () further.

![Left: Relationship between line dispersion in the mean and rms spectra of reverberation-mapped AGNs. The Spearman rank coefficient is $\rho = 0.873$. Right: Relationship between () and () for reverberation-mapped AGNs. The Spearman rank coefficient for these data is $\rho = 0.524.$ In both panels, blue filled circles represent RMDB sources in Table A2 and green open triangles represent SDSS-RM sources in Table A4. The red dotted line shows the locus where the two line-width measures are equal. The solid line is the best fit to equation (\[eq:powerlaw\]) and the coefficients are given in Table 3. The short dashed and long dashed lines show the $\pm 1 \sigma$ and $\pm 2.6 \sigma$ envelopes, respectively.[]{data-label="Figure:windowcivwidths"}](Figure8.eps)

Single-Epoch Masses {#section:civ}
-------------------

Following the same procedures as with , we use the RMDB data (Table A2) and the SDSS-RM data (Table A4) to fit the equation $$\log {\mbox{$\mu_{\rm RM}$}}= a + b\left[ \log L(1350\,{\rm \AA}) - x_0 \right]
+ c \left[ \log  {\mbox{$\sigma_{\rm M}$}}({\mbox{\rm C\,{\sc iv}}}) - y_0 \right].
\label{eq:Fit_244}$$ The resulting fit is shown in Figure \[Figure:civvps\] and the best-fit coefficients are given in Table 4.

![Left: Comparison of single-epoch virial products () and reverberation measurements () for the data in Table A2 (blue filled circles), the SDSS-RM  reverberation data from Table A4 (green open triangles), and data from Table A5 (red open circles). The solid line is the best fit to the data and has slope $0.787 \pm 0.041$. As was the case with , masses are overestimated at the low end and underestimated at the high end, excepting the three very low mass measurements. Right: Comparison of single-epoch virial products after empirical correction as given in equation (\[eq:predictciv\]). In both panels, the solid line is the best fit to equation (\[eq:Fit\_244\]). The short dashed and long dashed lines define the $\pm 1 \sigma$ and $\pm 2.6 \sigma$ envelopes, respectively. The diagonal red dotted line is the locus where  and are equal. []{data-label="Figure:civvps"}](Figure9.eps)

With the coefficients from this fit and equation (\[eq:Fit\_244\]), we can generate predicted virial masses ${\mbox{$\mu_{\rm SE}$}}({\mbox{\rm C\,{\sc iv}}})$. We compare the measured reverberation mass  with the single-epoch prediction  based on this fit in the left panel of Figure \[Figure:civvps\]. As was the case for  (Figure \[Figure:Fit\_139\_141\]), the distribution of points is slightly skewed relative to the diagonal, and, guided by our result for , we plot the residuals in $\log {\mbox{$\mu_{\rm RM}$}}- \log {\mbox{$\mu_{\rm SE}$}}$ versus Eddington ratio  in the upper left panel of Figure \[Figure:window244\_247\]. The Eddington ratio for the UV data is $$\log {\mbox{$\dot{m}$}}= -33.737 + 0.9 \log L(1350\,{\rm \AA}) - \log \mu_{\rm RM},
\label{eq:Edddefciv}$$ where again we have used a bolometric correction from $L(1350\,{\rm \AA})$ from [@Netzer19], $$\log k_{\rm bol} = 5.045 - 0.1 \log L(1350\,{\rm \AA}).
\label{eq:bolometricciv}$$ We fitted equation (\[eq:residuals\]) for  and the coefficients of the fit are given in Table 5.

![Mass residuals $\Delta \log \mu =\log {\mbox{$\mu_{\rm RM}$}}- \log {\mbox{$\mu_{\rm SE}$}}$ versus Eddington rate  (left column) and virial product  (right column). The upper left panel shows the residuals between () and () versus Eddington ratio  (equation \[eq:Eddratio\]). The upper right panel shows the residuals versus virial product . The middle panels show the residuals versus  (left) and  (right) after subtracting the fit in the upper left panel. The middle right panel shows a best fit to the residuals versus mass; coefficients are given in Table 5. Note that the intrinsic scatter in this relationship is $\epsilon_y = 0.000 \pm 0.000$ because the error bars are so large. The bottom panels show the mass residuals versus and  after subtracting the fit in the middle right panel. The scatter in the bottom panels is 0.138dex. In all panels, the blue filled circles represent RMDB data (Table A2), the green open triangles are SDSS data (Table A4), and the red open circles are VP06 data (Table A5). Best fits are shown as solid lines and the short dashed and long dashed lines indicate the $\pm 1 \sigma$ and $\pm 2.6 \sigma$ envelopes. []{data-label="Figure:window244_247"}](Figure10.eps)

The offset between the residuals in the upper left panel of Figure \[Figure:window244\_247\] between the RMDB and VP06 data on one hand and the SDSS data on the other might seem to be problematic and we were initially concerned that this might be a data integrity issue. However, upon examining the distribution of mass and luminosity for these three samples as seen in Figure \[Figure:masshist\], we see clearly that the mass distribution of the SDSS sources is skewed toward much higher values than for the RMDB and VP06 sources, which are relatively local and less luminous than the SDSS quasars. We will thus proceed by examining mass residuals versus both Eddington ratio and .

![Distribution in virial product  for the RMDB (Table A2, blue solid line), SDSS (Table A3, green dotted line), and VP06 (Table A4, red solid line) samples. The VP06 sample is a subset of the RMDB sample, which is dominated by the relatively low-mass Seyfert galaxies that were the first sources studied by reverberation mapping. The SDSS quasars are comparatively more massive and more luminous. []{data-label="Figure:masshist"}](Figure11.eps)

Figure \[Figure:window244\_247\] illustrates the process by which we eliminate the mass residuals in successive iterations. We compute the mass residuals $\Delta \log \mu = \log {\mbox{$\mu_{\rm RM}$}}- \log {\mbox{$\mu_{\rm SE}$}}$ from equation (\[eq:Fit\_244\]); these are shown versus  (left column) and ${\mbox{$\mu_{\rm RM}$}}$ (right column). We fit these residuals versus  (top left) and subtract the best fit to equation (\[eq:residuals\]), whose coefficients are given in Table 5. We subtract this fit from the mass residuals to get the corrected residuals in the middle panels. Examination of these residuals as a function of other parameters revealed that they are still correlated with  (middle right), suggesting that the importance of the Eddington ratio depends on the black hole mass. We therefore fit the residuals a second time, this time as $$\Delta \log \mu = a + b(\log {\mbox{$\mu_{\rm RM}$}}\ - x_0).
\label{eq:massresiduals}$$ The best fit to this equation is shown in the middle right panel and the coefficients are given in Table 5. Subtraction of the best fit yields the residuals shown in the bottom two panels. We would under most circumstances consider this procedure with some trepidation from a statistical point of view, since ${\mbox{$\mu_{\rm RM}$}}$ appears explicitly in one correction and is implicitly in the Eddington ratio. A generalized solution would have multiple degeneracies as both mass and luminosity appear in multiple terms. However, the residual corrections are physically motivated; several previous investigations have also concluded that Eddington ratio is correlated with the deviation from the [@Bentz13] $R$–$L$ relationship, and the middle panels of Figure \[Figure:window244\_247\] suggests that the impact of Eddington ratio varies slightly with mass. Nevertheless, one would prefer to work with parameters that are correlated with or indicators of  and , as we will discuss in §[\[section:discussion\]]{}.

It is worth noting in passing that after correcting for Eddington ratio (Figure \[Figure:windowhbresiduals\]), the residuals in the -based mass estimates show no correlation with either mass or luminosity.

Computing Single-Epoch Masses {#section:massformulae}
=============================

To briefly reiterate our approach so far, we started with the assumption that ${\mbox{$\mu_{\rm SE}$}}= f(R,L)$ only. This proved to be inadequate, so we examined the residuals in the $\log {\mbox{$\mu_{\rm SE}$}}$–$\log {\mbox{$\mu_{\rm RM}$}}$ relationship and found that these correlated best with Eddington ratio : fundamentally, at increasing , the [@Bentz13] $R$–$L$ relationship overpredicts the size of the BLR $R$ [@Du19]. In the case of , we found additional residuals that correlated with , although we cannot definitively demonstrate that some part of this is not attributable to inhomogeneities in the data base (a point that will be pursued in the future). While we believe this analysis identifies the physical parameters that affect the mass estimates, there are multiple degeneracies, with both mass and luminosity appearing in more than one term.

Instead of trying to fit coefficients to all the physical parameters that have been identified, we can do a purely empirical correction to equations (\[eq:Fit\_139\]), (\[eq:Fit\_141\]), and (\[eq:Fit\_244\]) since the residuals in the $\log {\mbox{$\mu_{\rm RM}$}}$–$\log {\mbox{$\mu_{\rm SE}$}}$ relationships (upper panels in Figure \[Figure:Fit\_139\_141\] and left panel of Figure \[Figure:civvps\]) are rather small. We can combine the basic $R$–$L$ fits (equations \[eq:Fit\_139\], \[eq:Fit\_141\], and \[eq:Fit\_244\]) with the residual fits (equations \[eq:residuals\] and \[eq:massresiduals\]) to obtain prescriptions that work over the mass range sampled. Renormalizing for convenience, we can estimate single-epoch masses based on () from $$\begin{aligned}
\log M_{\rm SE} & = & \log f + 7.530  
+ 0.703 \left[ \log L({\mbox{\rm H$\beta$}})-42 \right]
 + 2.183  \left[\log {\mbox{$\sigma_{\rm M}$}}({\mbox{\rm H$\beta$}}) - 3.5 \right],
\label{eq:predictHbsigm}\end{aligned}$$ with associated uncertainty $$\label{eq:predictHbsigmerror}
\Delta \log M_{\rm SE} = \left\{ (\Delta \log f)^2 +
\left[0.703\ \Delta \log L({\mbox{\rm H$\beta$}}) \right]^2 +
\left[2.183\ \Delta \log {\mbox{$\sigma_{\rm M}$}}({\mbox{\rm H$\beta$}})\right]^2 \right\}^{1/2}
\; .$$ Here $f$ is the scaling factor which is discussed briefly in the Appendix, and $\Delta \log P$ is the uncertainty in the parameter $\log P$. The intrinsic scatter in this relationship is $0.309$dex, and this must be added in quadrature to the random error. For the case of (), a single-epoch mass estimate is obtained from $$\begin{aligned}
\log M_{\rm SE} & = & \log f + 7.015  + 
0.784 \left[ \log L({\mbox{\rm H$\beta$}})-42 \right]
 + 1.387  \left[\log {\mbox{${\rm FWHM}_{\rm M}$}}({\mbox{\rm H$\beta$}}) - 3.5 \right],
\label{eq:predictHbFWHM}\end{aligned}$$ with associated uncertainty $$\label{eq:predictHbFWHMmerror}
\Delta \log M_{\rm SE} = \left\{ (\Delta \log f)^2 +
\left[0.784\ \Delta \log L({\mbox{\rm H$\beta$}}) \right]^2 +
\left[1.387\ \Delta \log {\mbox{${\rm FWHM}_{\rm M}$}}({\mbox{\rm H$\beta$}})\right]^2 \right\}^{1/2} \;
.$$ In this case, the intrinsic scatter is $0.371$dex.

A comparison of the reverberation-based virial products () and the single-epoch masses () based on equations (\[eq:predictHbsigm\]) and (\[eq:predictHbFWHM\]) is shown in the lower two panels of Figure  \[Figure:Fit\_139\_141\].

Similarly, single-epoch masses based on  can be computed from $$\begin{aligned}
\log M_{\rm SE} &  = & \log f +7.934  
+ 0.761 \left[ \log L(1350\,{\rm \AA})-45 \right]
+ 1.289  \left[\log {\mbox{$\sigma_{\rm M}$}}({\mbox{\rm C\,{\sc iv}}}) - 3.5 \right],
\label{eq:predictciv}\end{aligned}$$ with associated uncertainty $$\label{eq:predictciverror}
\Delta \log M_{\rm SE} = \left\{ (\Delta \log f)^2+
\left[0.761\ \Delta \log L(1350\,{\rm \AA} \right]^2 +
\left[1.284\ \Delta \log {\mbox{$\sigma_{\rm M}$}}({\mbox{\rm C\,{\sc iv}}})\right]^2  \right\}^{1/2}\;
.$$ The intrinsic scatter in this relationship is 0.408dex. Single-epoch predictions and reverberation-based masses for the AGNs in Tables A2, A4, and A5 are compared in the right panel of Figure \[Figure:civvps\].

In Figure \[Figure:zdistsdssselect\], we show the distribution in bolometric luminosity and black hole mass for the entire sample of SDSS-RM quasars for which  or  single-epoch masses can be estimated.

![Distribution of masses (upper panel) and bolometric luminosities (lower panel) for the entire SDSS-RM sample for which  or  single-epoch masses can be computed using equations (\[eq:predictHbsigm\]) and (\[eq:predictciv\]). Here we assume $f=4.28$ [@Batiste17]. Bolometric corrections were made using equations (\[eq:bolometriccorrection\]) and (\[eq:bolometricciv\]). On the left side, the quality cuts of §\[section:Data\] have been imposed. On the right side, no quality cuts have been made. []{data-label="Figure:zdistsdssselect"}](Figure12.eps)

Discussion {#section:discussion}
==========

Single-Epoch Masses {#single-epoch-masses}
-------------------

Our primary goal has been to find simple, yet unbiased, prescriptions for estimating the masses of the black holes that power AGNs. Our underlying assumption has been that the most accurate measure of the virial product is given by using the emission-line lag $\tau$ and line width in the rms spectrum  (e.g., equation \[eq:explicitmu\] in the Appendix) as that quantity produces, upon adjusting by the scaling factor $f$, an $M_{\rm BH}$–$\sigma_*$ relationship for AGNs that is in good agreement with that for quiescent galaxies. Given that both $\tau$ and  average over structure in a complex system [cf., @Barth15], it is somewhat surprising that this method of estimation works as well as it does.

Here we have shown that the broad component of the  emission line is a good proxy for the starlight-corrected AGN luminosity (Figure \[Figure:HbRL\]). This is useful since it eliminates the difficult task of accurately modeling the host-galaxy starlight contribution to the continuum luminosity. Moreover, the line luminosity and  reflect the BLR state at the same time; a measurement of the continuum luminosity, by contrast, better represents the state of the BLR at a time $\tau$ in the future on account of the light travel-time delay within the system [@Pogge92; @Gilbert03; @Barth15]; this is, however, generally a very small effect. For the sake of completeness, we also note that there is a small, but detectable, lag between continuum variations at shorter wavelengths and those at longer wavelengths [@McHardy14; @Shappee14; @Edelson15; @Fausnaugh16; @Edelson17; @McHardy18; @Edelson19].

We have also confirmed that, for the case of , both  and  are reasonable proxies for , though  is somewhat better than .

On the other hand, the case of  remains problematic, as it differs in a number of ways from the other strong emission lines:

1.  The equivalent width of  decreases with luminosity, which is known as the Baldwin Effect [@Baldwin77];  is driven by higher-energy photons than, say, the Balmer lines and the Baldwin Effect reflects a softening of the high-ionization continuum. This could be due to higher Eddington ratio [@Baskin04] or because more massive black holes have cooler accretion disks [@Korista98].

2.  The  emission line is typically blueshifted with respect to the systemic redshift of the quasar, which is attributed to outflow of the BLR gas [@Gaskell82; @Wilkes84; @Wilkes86; @Espey89; @Wills93; @Sulentic07; @Richards11; @Coatman16; @Shen16a; @Bisogni17; @Vietri18].

3.  BALs in the short-wavelength wing of , another signature of outflow, are common [@Weymann91; @Hall02; @Hewett03; @Allen11]. We remind the reader that in §[\[section:Data\]]{} we removed $\sim17$% of our SDSS  sample because the presence of BALs precludes accurate line-width measurements.

4.  The pattern of “breathing” in  is the opposite of what is seen in  [@Wang20]. Breathing refers to the response of the emission lines, both lag and line width, to changes in the continuum luminosity. In the case of , an increase in luminosity produces an increase in lag and a decrease in line width [@Gilbert03; @Goad04; @Cackett06]. In the case of , however, the line width increases when the continuum luminosity increases, contrary to expectations from the virial theorem (equation \[eq:masseqn\]).

We must certainly be mindful that outflows can affect a mass measurement, though the effect is small if the gas is at escape velocity. Notably, in the cases studied to date there is good agreement between -based and -based virial products (Figure \[Figure:civradlum\]), though, again, these are local Seyfert galaxies that are not representative of the general quasar population.

The  breathing issue is addressed in detail by [@Wang20], building on evidence for a non-reverberating narrow core or blue excess in the  emission line presented by [@Denney12]. In this two-component model, the variable part of the line is much broader than the non-variable core. As the continuum brightens, the variable broad component increases in prominence, resulting in a larger value of . As the broad component reverberates in response to continuum variations,  will track  much better than , thus explaining the breathing characteristics and why  is a poor line-width measure for estimating black hole masses. Physical interpretation of the non-varying core remains an open question: [@Denney12] suggests that it might be an optically thin disk wind or an inner extension of the narrow-line region.

The Role of Eddington Ratio
---------------------------

It is well known that there are strong correlations and anticorrelations among the UV-optical spectral features of AGNs as revealed by Principal Component Analysis (PCA) [@Boroson92; @Sulentic00; @Boroson02; @Shen14; @Sun15; @Marziani18 and references therein]. The strongest of these correlations, , is most clearly characterized by the anticorrelation between (a) the strength of the $\lambda4570$ and $\lambda\lambda5190$, 5320 complexes on either side of the broad  complex and (b) the strength of the $\lambda\lambda 4959$, 5007 doublet. There is consensus in the literature that is driven by Eddington ratio; our own analysis supports this. The studies cited above have noted that an Eddington ratio correction is required for single-epoch masses based on . We find, as did [@Marziani19], that a similar correction is required for -based masses as well.

One extreme of  is populated by sources with strong  and very weak . The broad emission lines in the spectra of these objects also have relatively small line widths. By combining the $R$–$L$ relation with eq. (\[eq:masseqn\]), the line width dependence is seen to be $$\label{eq:linewidth}
V \propto\left( \frac{M}{L^{1/2}} \right)^{1/2} \propto 
\left( \frac{M}{{\mbox{$\dot{m}$}}} \right)^{1/4},$$ where ${\mbox{$\dot{m}$}}\propto L/M$ is the Eddington ratio (eq. \[eq:Eddratio\]). Thus AGNs with the highest Eddington ratios have the smallest broad-line widths; many such sources are classified as “narrow-line Seyfert 1 (NLS1) galaxies” [@OsterbrockPogge85]. The Super-Eddington Accreting Massive Black Holes (SEAMBH) collaboration has focused on high- candidates in their reverberation-mapping program [@Du14; @Du16; @Du18; @Du19]. An important result from these studies, as we have noted earlier, is that the  lags are smaller than predicted by the current state-of-the-art $R$–$L$ relationship [@Bentz13]. This implies that in these objects the ratio of hydrogen-ionizing photons to optical photons is lower than in the lower  sources; this is also consistent with the relative strength of the low-ionization lines such as  in SEAMBH sources, the weakness of high-ionization lines, such as , and their soft X-ray spectra [@Boller96]. [@Du19] choose to make their correction to the BLR radius through adding a term that correlates with the deficiency of ionizing photons. In our approach, we absorb the correction directly into the virial product computation.

As noted in §\[section:civ\], from a statistical point of view, it would be preferrable to replace the Eddington ratio with a parameter strongly correlated with it. The PCA studies referenced above find that the ratio of the equivalent widths (EW) or fluxes of  to , ${\cal R} = {\rm EW}({\mbox{\rm Fe\,{\sc ii}}})/{\rm EW}({\mbox{\rm H$\beta$}})$, correlates well with Eddington ratio. In the UV, it is also found that the  blueshift correlates with Eddington ratio [@Baskin05; @Coatman16; @Sulentic17]. However, we find that the scatter in these relationships is so large that any gain in the accuracy of black hole mass estimates is offset by a large loss of precision. We therefore elect at this time to focus on the empirical formulae given in §\[section:massformulae\].

Future Improvements
-------------------

While we believe our current single-epoch prescription for estimating quasar black hole masses is more accurate than previous prescriptions, we also recognize that there are additional improvements that can be made to improve both accuracy and precision, some of which we became aware of near the end of the current project. We intend to implement these in the future. Topics that we will investigate in the future include the following:

1.  Replace those reverberation lag measurements made with the interpolated cross-correlation function [@Gaskell87; @White94; @Peterson98b; @Peterson04] with lag measurements and uncertainties from [JAVELIN]{} [@Zu11]. Recent tests [@Li19; @Yu20] show that while the [JAVELIN]{} and interpolation cross-correlation lags are generally consistent, the uncertainties predicted by [JAVELIN]{} are more reliable.

2.  Utilize the expanded SDSS-RM database, which now extends over six years, not only to make use of additional lag detections, but to capitalize on the gains in $S/N$ that will increase the overall quality of the lag and line-width measurements and result in fewer rejections of poor data.

3.  Expand the database in Table A1 with recent results and other previous results that we excluded because they did not have starlight-corrected luminosities.

4.  Update the VP06 database used to produce Table A5. There are now additional reverberation-mapped AGNs with archived [*HST*]{} UV spectra. Some of the poorer data in Table A5 can be replaced with higher-quality spectra.

5.  Consider use of other line-width measures that may correlate well with , but are less sensitive to blending in the wings. Mean absolute deviation is one such candidate.

6.  Improve line-width measurements. There appear to be some systematic differences among the various data sets, probably due to different processes for measuring ; for example, the bottom panels of Figure \[Figure:window244\_247\] show that the SE mass estimates for the VP06 sample are slightly higher than those from SDSS (compare also the last two columns in Table A5). Work on deblending alogrithms would aid more precise measurement of , in particular.

Summary
=======

The main results of this paper are:

1.  We confirm that the luminosity of the broad component of the  emission line $L({\mbox{\rm H$\beta$}}_{\rm broad})$ is an excellent substitute for the AGN continuum luminosity $L_{\rm AGN}(5100\,{\rm \AA})$ for predicting the  emission-line reverberation lag $\tau({\mbox{\rm H$\beta$}})$. It has the advantage of being easier to isolate than $L_{\rm AGN}(5100\,{\rm \AA})$, which requires an accurate estimate of the host-galaxy starlight contribution to the observed luminosity.

2.  We confirm that the line dispersion of the  broad component () and the full-width at half maximum for the  broad component () in mean, or single-epoch, spectra are both reasonable proxies for the line dispersion of  in the rms spectrum () for computing single-epoch virial products (). We find that () gives better results than (), but both are usable.

3.  In the case of , we find that the line dispersion of the  emission line () in the mean, or single-epoch, spectrum is a good proxy for the line dispersion in the rms spectrum () for estimating single-epoch virial products (). We find that (), however, does not track () well enough to be used as a proxy.

4.  Although the $R$–$L$ relationship based on the continuum luminosity $L(1350\,{\rm \AA})$ and  emission-line reverberation lag $\tau({\mbox{\rm C\,{\sc iv}}})$ is not as well defined as that for , the relationship appears to have a similar slope and it appears to be suitable for estimating virial products ().

5.  We confirm for both  and  that combining the reverberation lag estimated from the luminosity with a suitable measurement of the emission-line width together introduces a bias where the high masses are underestimated and the low masses are overestimated. We confirm that the parameter that accounts for the systematic difference between reverberation virial product measurements  and those estimated using only luminosity and line width is Eddington ratio. Increasing Eddington ratio causes the reverberation radius to shrink, suggesting a softening of the hydrogen-ionizing spectrum.

6.  While the virial product estimate from combining luminosity and line width causes a systematic bias, the relationship between the reverberation virial product  and the single-epoch estimate is still a power-law, but with a slope somewhat less than unity (upper panels of Figure \[Figure:Fit\_139\_141\], left panel of Figure \[Figure:civvps\]). We are therefore able to empirically correct this relationship to an unbiased estimator of  by fitting the residuals and essentially rotating the power-law distribution to have a slope of unity (lower panels of Figure \[Figure:Fit\_139\_141\], right panel of Figure \[Figure:civvps\]). We present these empirical estimators for () and () in §[\[section:massformulae\]]{}.

EDB is supported by Padua University through grants DOR1715817/17, DOR1885254/18, and DOR1935272/19 and by MIUR grant PRIN 2017 20173ML3WW\_001. EDB and BMP are grateful for the hospitality of STScI early in this investigation. JVHS and KH acknowledge support from STFC grant ST/R000824/1. YS acknowledges support from an Alfred P. Sloan Research Fellowship and NSF grant AST-1715579. CJG, WNB, JRT, and DPS acknowledge support from NSF grants AST-1517113 and AST-1516784. KH acknowledges support from STFC grant ST/R000824/1. PBH acknowledges support from NSERC grant 2017-05983. YH acknowledges support from NASA through STScI grant HST-GO-15650. SW, LJ, and LCH acknowledge support from the National Science Foundation of China (11721303, 11890693, 11991052) and the National Key R&D Program of China (2016YFA0400702, 2016YFA0400703). MV gratefully acknowledges support from the Independent Research Fund Denmark via grant number DFF 8021-00130. Funding for the Sloan Digital Sky Survey IV has been provided by the Alfred P. Sloan Foundation, the U.S. Department of Energy Office of Science, and the Participating Institutions. SDSS-IV acknowledges support and resources from the Center for High-Performance Computing at the University of Utah. The SDSS web site is www.sdss.org. SDSS-IV is managed by the Astrophysical Research Consortium for the Participating Institutions of the SDSS Collaboration including the Brazilian Participation Group, the Carnegie Institution for Science, Carnegie Mellon University, the Chilean Participation Group, the French Participation Group, Harvard-Smithsonian Center for Astrophysics, Instituto de Astrofísica de Canarias, The Johns Hopkins University, Kavli Institute for the Physics and Mathematics of the Universe (IPMU) / University of Tokyo, the Korean Participation Group, Lawrence Berkeley National Laboratory, Leibniz Institut für Astrophysik Potsdam (AIP), Max-Planck-Institut für Astronomie (MPIA Heidelberg), Max-Planck-Institut für Astrophysik (MPA Garching), Max-Planck-Institut für Extraterrestrische Physik (MPE), National Astronomical Observatories of China, New Mexico State University, New York University, University of Notre Dame, Observatário Nacional / MCTI, The Ohio State University, Pennsylvania State University, Shanghai Astronomical Observatory, United Kingdom Participation Group, Universidad Nacional Autónoma de México, University of Arizona, University of Colorado Boulder, University of Oxford, University of Portsmouth, University of Utah, University of Virginia, University of Washington, University of Wisconsin, Vanderbilt University, and Yale University.

Database of Reverberation-Mapped AGNs {#database-of-reverberation-mapped-agns .unnumbered}
=====================================

Reverberation-mapped AGNs provide the fundamental data that anchor the AGN mass scale. We selected all AGNs from the literature (as of 2019 August) for which unsaturated host-galaxy images acquired with [[*HST*]{}]{}are available, since removal of the host-galaxy starlight contribution to the observed luminosity is critical to this calibration, and measurements of  time lags. It is worth noting, however, that since our analysis shows that the broad  flux is a useful proxy for the 5100Å continuum luminosity, this criterion is over-restrictive and we will avoid imposing it in future compilations. In many cases, there is more than one reverberation-mapping data set available in the literature. In a few cases, the more recent data were acquired to replace, say, a more poorly sampled data set or one for which the initial result was ambiguous for some reason. In other cases, there are multiple data sets of comparable quality for individual AGNs, and in these cases we include them all. The particularly well-studied AGN NGC 5548 has been observed many times and in some sense has served as a “control” source that provides our best information about the repeatability of mass measurements as the continuum and line widths show long-term (compared to reverberation time scales) variations.

The final reverberation-mapped sample for  is given in Table A1. It consists of 98 individual time series for 50 individual low-redshift ($z <
0.3$) AGNs. They span a range of AGN luminosity $41.46 \leq \log
L(5100\,{\rm \AA}) \leq 45.81$, in . Luminosities have been corrected for Galactic absorption using extinction values on the NASA Extragalactic Database, which are based on the [@Schlafly11] recalibration of the [@Schlegel98] dust map. Line-width and time-delay measurements are in the rest-frame of the AGNs. Luminosity distances are based on redshift, except the cases noted by [@Bentz13], for which the redshift-independent distances quoted in that paper are used. For two of these sources, NGC 4051 and NGC 4151, we use preliminary Cepheid-based distances (M.M. Fausnaugh, private communication), and for NGC 6814, we use the Cepheid-based distance from [@Bentz19]. Individual virial products for these sources are easily computed using the  time lags (Column 6) and line dispersion measurements (Column 12) and the formula $$\label{eq:explicitmu}
\mu = 0.1952\left( \frac{\tau({\mbox{\rm H$\beta$}})}{{\rm days}}\right)
\left( \frac{{\mbox{$\sigma_{\rm R}$}}({\mbox{\rm H$\beta$}})} {{\rm km\,s}^{-1}} \right)^2\,{\mbox{$M_\odot$}}.$$

Further conversion to mass requires multiplication by the virial factor $f$, i.e. $\log M = \log f + \log \mu$, a dimensionless factor that depends on the inclination, structure, and kinematics of the broad--emitting region — indeed, detailed modeling of 9 of these objects [@Pancoast14; @Grier17a] shows that $f$ depends most clearly on inclination [@Grier17a]. Since such models are available for only a very limited number of AGNs, it is more common to use a statistical estimate of a mean value of $f$ based on a secondary mass indicator, specifically the well-known $M_{\rm BH}$–$\sigma_*$ relationship [@Ferrarese00; @Gebhardt00; @Gultekin09], where $\sigma_*$ is the host-galaxy stellar bulge velocity dispersion. The required assumption is that the AGN $M_{\rm BH}$–$\sigma_*$ is identical to that of quiescent galaxies [@Woo13]. In fact, it is found that the $\mu$–$\sigma_*$ has a slope consistent with the $M_{\rm BH}$–$\sigma_*$ slope for quiescent galaxies [@Grier13], and the zero points disagree by only a multiplicative factor, which is taken to be $f$. Here we take $\langle \log f \rangle =
0.683 \pm 0.150$ [@Batiste17] where the error on the mean is $\Delta \log f = 0.030$ — this error must be propagated into the mass measurement error when comparing AGN reverberation-based masses to those based on other methods.

[lccccccccccc]{} \[table:RMDBhb\] Mrk335 &1 & 49156-49338& 0.02579& 109.5& $  16.8^{+4.8}_{-4.2}$&$    43.802 \pm  0.010$ &$  43.703 \pm 0.013$ &$   42.083\pm  0.010$ &$1792     \pm  3 $&$ 1380 \pm 6$ & $    917  \pm 52$\
Mrk335 &1 & 49889-50118& 0.02579& 109.5& $  12.5^{+6.6}_{-5.5}$&$   43.861 \pm  0.010$ &$  43.777 \pm 0.013$ &$  42.124\pm  0.010$ &$1679    \pm 2  $&$ 1371 \pm 8$ & $     948   \pm 113 $\
Mrk335 &1 & 55431-55569& 0.02579& 109.5& $  14.3^{+0.7}_{-0.7}$&$   43.791 \pm  0.007$ &$  43.683 \pm  0.061$ &$  41.940\pm    0.009$ &$1273 \pm 3  $&$1663 \pm 6$ & $     1293  \pm 64  $\
Mrk1501 &2 & 55430-55568& 0.08934& 402.5& $  12.6^{+3.9}_{-3.9}$&$   44.314 \pm  0.011$ &$  43.980 \pm  0.053$ &$  42.719\pm    0.015$ &$3106 \pm 15 $&$3494 \pm35$ & $     3321  \pm 107 $\
PG0026+129 &3 & 48545-51084& 0.14200& 653.1& $  111.0^{+24.1}_{-28.3}$&$ 44.977 \pm  0.010$ &$ 44.911 \pm  0.011$ &$  42.867\pm    0.016$ &$2544 \pm 56 $&$1738 \pm 100$ &$     1773 \pm 285 $\
PG0052+251 &3 & 48461-51084& 0.15445& 751.9& $  89.8^{+24.5}_{-24.1}$&$  44.964 \pm  0.013$ &$ 44.791 \pm  0.020$ &$  43.113\pm    0.016$ &$5008 \pm 73 $&$2167 \pm 30$ & $     1783  \pm 86  $\
Fairall9 &4 & 49475-49743& 0.04702& 202.8& $  17.4^{+3.2}_{-4.3}$&$   44.224 \pm  0.007$ &$  43.920 \pm  0.026$ &$  42.393\pm    0.007$ &$5999 \pm 60 $&$ 2347\pm 16$ &$     3787  \pm 197 $\
Mrk590 &1 & 48090-48323& 0.02639& 112.1& $  20.7^{+3.5}_{-2.7}$&$   43.842 \pm  0.010$ &$  43.544 \pm  0.029$ &$  41.855\pm    0.011$ &$2788 \pm 29 $&$ 1942 \pm 26$ & $     789   \pm 74  $\
Mrk590 &1 & 48848-49048& 0.02639& 112.1& $  14.0^{+8.5}_{-8.8}$&$   43.666 \pm  0.011$ &$  43.075 \pm  0.073$ &$  41.522\pm    0.011$ &$3729 \pm 426$&$ 2168 \pm 30$ & $      1935 \pm 52  $\
Mrk590 &1 & 49183-49338& 0.02639& 112.1& $  29.2^{+4.9}_{-5.0}$&$   43.743 \pm  0.010$ &$  43.320 \pm  0.043$ &$  41.690\pm    0.010$ &$2743 \pm 79 $&$1967 \pm 19$ & $     1251  \pm 72  $\
Mrk590 &1 & 49958-50122& 0.02639& 112.1& $  28.8^{+3.6}_{-4.2}$&$   43.865 \pm  0.010$ &$  43.589 \pm  0.026$ &$  41.857\pm    0.010$ &$2500 \pm 43 $&$ 1880 \pm 19$ & $     1201  \pm 130 $\
3C120 &1 & 47837-50388& 0.03301& 140.9& $  38.1^{+21.3}_{-15.3}$&$ 44.078 \pm  0.012$ &$  44.010 \pm  0.014$ &$  42.306\pm    0.012$ &$2327 \pm 48 $&$ 1249\pm21$ & $     1166  \pm 50  $\
3C120 &5 & 54726-54920& 0.03301& 140.9& $  27.9^{+7.1}_{-5.9}$&$   44.116 \pm  0.013$ &$  44.094 \pm  0.013$ &$  42.453\pm    0.012$ &$2386 \pm 52 $& $\ldots$ & $     1689  \pm 68  $\
3C120 &2 & 55430-55569& 0.03301& 140.9& $  25.9^{+2.3}_{-2.3}$&$   43.993 \pm  0.012$ &$  43.903 \pm  0.052$ &$  42.298\pm    0.015$ &$1430 \pm 16 $&$1687 \pm 4$ & $     1514  \pm 65  $\
Akn120 &1 & 48148-48344& 0.03271& 139.6& $  47.1^{+8.3}_{-12.4}$&$  44.254 \pm  0.010$ &$  43.921 \pm  0.032$ &$  42.553\pm    0.010$ &$6042 \pm 35 $&$1753 \pm 6$ & $     1959  \pm 109 $\
Akn120 &1 & 49980-50175& 0.03271& 139.6& $  37.1^{+4.8}_{-5.4}$&$   44.131 \pm  0.010$ &$  43.569 \pm  0.067$ &$  42.390\pm    0.010$ &$6246 \pm 78 $&$1862 \pm 13$ & $     1884  \pm 48  $\
MCG+08-11-011&6& 56639-56797& 0.02048& 86.6 & $  15.72^{+0.50}_{-0.52}$&$ 43.574 \pm  0.009$ &$  43.282 \pm 0.045$ &$  41.706\pm     0.006$&$1159\pm 8$&$1681 \pm 2$ &$     1466  \pm 143 $\
Mrk6 &7 & 49250-49872& 0.01881& 80.6 & $  21.2^{+4.}_{-3.2}$&$   43.576 \pm   0.009$ &$  43.351 \pm  0.033$ &$  41.591\pm     0.011$&$   \ldots $&$ 2813 \pm 13$ &$        2836    \pm 48  $\
Mrk6 &7 & 49980-50777& 0.01881& 80.6 & $  20.7^{+3.0}_{-2.4}$&$   43.578 \pm   0.009$ &$  43.354 \pm 0.033$ &$  41.632\pm     0.010$&$   \ldots $&$2804 \pm 6$ &$     2626 \pm 37  $\
Mrk6 &7 & 50869-51516& 0.01881& 80.6 & $  20.5^{+5.6}_{-7.0}$&$   43.523 \pm   0.011$ &$  43.258 \pm 0.042$ &$  41.584\pm     0.013$&$   \ldots $&$2808 \pm 14$ &$      2626   \pm 37  $\
Mrk6 &7 & 51557-53356& 0.01881& 80.6 & $  23.9^{+17.0}_{-7.3}$&$   43.431 \pm   0.007$ &$ 43.070 \pm 0.058$ &$  41.449\pm     0.018$&$  \dots  $&$2870 \pm 13$ &$       3222   \pm 39  $\
Mrk6 &7 & 53611-54804& 0.01881& 80.6 & $  20.4^{+4.6}_{-4.1 }$&$   43.613 \pm   0.005$ &$ 43.413 \pm 0.027$ &$  41.579\pm     0.012$&$   \ldots $&$2807\pm 8$&$         2864   \pm 35  $\
Mrk6 &2 & 55340-55569& 0.01881& 80.6 & $  10.1^{+1.1}_{-1.1}$&$   43.719 \pm   0.008$ &$43.507 \pm   0.029$ &$  41.849\pm     0.012$&$2619 \pm 24 $& $4006 \pm 6$ &$     3714  \pm 68  $\
Mrk79 &1 & 47838-48044& 0.02219& 94.0 & $  9.0^{+8.3}_{-7.8}$&$   43.668 \pm   0.011$ &$ 43.569 \pm   0.014$ &$  41.818\pm     0.011$&$5056 \pm 85 $&$2314 \pm 23$ & $     2137  \pm 375 $\
Mrk79 &1 & 48193-48393& 0.02219& 94.0 & $  16.1^{+6.6}_{-6.6}$&$   43.754 \pm  0.010$ &$  43.675 \pm   0.012$ &$  41.851\pm     0.010$&$4760 \pm 31$ & $ 2281 \pm   26$ & $     1683    \pm 72  $\
Mrk79 &1 & 48905-49135& 0.02219& 94.0 & $  16.0^{+6.4}_{-5.8}$&$   43.695 \pm  0.010$ &$  43.602 \pm   0.013$ &$  41.820\pm     0.010$&$4766 \pm 71$ & $2312 \pm 21$ &$     1854    \pm 72  $\
Mrk374 &6 & 56663-56795& 0.04263& 183.3& $  14.84^{+5.76}_{-3.30}$&$ 43.994 \pm 0.009$ &$  43.752 \pm  0.036$ &$  41.764\pm     0.013$&$3250 \pm 19 $ &$1490 \pm 4$ &$     1329  \pm 373 $\
PG0804+761 &3 & 48319-51085& 0.10000& 447.5& $  146.9^{+18.8}_{-18.9}$&$ 44.905 \pm  0.011$ &$  44.849 \pm 0.011$ &$  43.230\pm     0.012$&$3053 \pm 38 $&$1434 \pm  18$ &$     1971 \pm 105 $\
NGC2617 &6 & 56639-56797& 0.01421& 59.8 & $  4.32^{+1.1}_{-1.35}$&$   43.099 \pm 0.011$ &$ 42.610 \pm   0.096$ &$  41.173\pm     0.012$&$5303 \pm 48$&$2709 \pm 6$ &$     2424    \pm 89  $\
Mrk704 &8 & 55932-55980& 0.02923& 124.5& $  12.65^{+1.49}_{-2.14}$&$ 43.708 \pm 0.005$ &$ 43.517 \pm   0.025$ &$  41.800\pm     0.007$&$3502 \pm 31$ &$ 2650 \pm 4$ & $     1860 \pm 120 $\
Mrk110 &1 & 48953-49149& 0.03529& 150.9& $  24.3^{+5.5}_{-8.3}$&$   43.711 \pm   0.011$ &$43.618 \pm   0.014$ &$  42.055\pm     0.011$&$1543 \pm 5$&$962 \pm 15$ &$     1196 \pm 141 $\
Mrk110 &1 & 49751-49874& 0.03529& 150.9& $  20.4^{+10.5}_{-6.3}$&$   43.771 \pm   0.010$ &$43.691 \pm  0.012$ &$  41.960\pm     0.010$&$1658 \pm 3 $ & $953  \pm 10$ &$     1115    \pm 103 $\
Mrk110 &1 & 50010-50262& 0.03529& 150.9& $  33.3^{+14.9}_{-10.0}$&$ 43.594 \pm  0.012$ &$  43.468 \pm  0.017$ &$  41.905\pm     0.012$&$1600 \pm 39$&$987 \pm 18$ &$     755 \pm 29  $\
Mrk110 &9 & 51495-51678& 0.03529& 150.9& $  23.4^{+3.6}_{-3.2}$&$  43.340 \pm  0.007$ &$  43.225 \pm   0.011$ &$  41.769\pm     0.007$&$ \ldots $ & $\ldots$ &$     \ldots$\
PG0953+414 &3 & 48319-50997& 0.23410& 1137.2& $  150.1^{+21.6}_{-22.6}$&$ 45.193 \pm  .010$ &$  45.126 \pm  0.011$ &$  43.390\pm     0.012$&$3071 \pm 27$ &$ 1659 \pm 31$ &$     1306 \pm 144 $\
NGC3227 &10& 54184-54269& 0.00386& 23.7 & $  3.75^{+0.76}_{-0.82}$&$ 42.629 \pm 0.035$ &$  42.243 \pm   0.068$ &$  40.387\pm     0.035$&$3972 \pm 25$ &$ 1749 \pm 4$ &$     1376  \pm 44  $\
NGC3227 &8 & 55933-56048& 0.00386& 23.7 & $  1.29^{+1.56}_{-1.27 }$&$42.757 \pm  0.006$ &$  42.424 \pm  0.051$ &$  40.487\pm     0.010$&$1602 \pm 2$ &$1402 \pm 2$ &$     1368    \pm 38  $\
Mrk142 &11& 54506-54618& 0.04494& 193.5& $  2.74^{+0.73}_{-0.83}$&$   43.709 \pm 0.010$ &$  43.543 \pm 0.015$ &$  41.639\pm     0.010$&$1462 \pm 2 $ &$ 1116 \pm 22$ &$     859  \pm 102 $\
Mrk142 &12& 56237-56413& 0.04494& 193.5& $  6.4^{+0.8}_{-2.2}$&$   43.610 \pm  0.010$ &$  43.443 \pm   0.016$ &$  41.586\pm     0.010$&$1647 \pm 69 $& $\ldots$ & $     \ldots  $\
NGC3516 &14,15& 54181-54300& 0.00884& 37.1 & $  11.68^{+1.02}_{-1.53}$&$43.299 \pm  0.055$ &$  42.726 \pm  0.133$ &$  40.995\pm     0.057$&$5236 \pm 12$&$ 1584 \pm  1$ &$     1591 \pm 10  $\
NGC3516 &8 & 55932-56072& 0.00884& 37.1 & $  5.74^{+2.26}_{-2.04}$&$ 43.272 \pm  0.007$ &$  42.529 \pm  0.196$ &$  41.022\pm     0.008$&$3231 \pm 14 $ & $ 2633 \pm 3  $&$     2448   \pm 69  $\
SBS1116+583A&11& 54550-54618& 0.02787& 118.5& $  2.31^{+0.62}_{-0.49 }$&$ 42.995 \pm  0.021$ &$  42.076 \pm 0.224$ &$  40.788\pm     0.015$&$3668 \pm 186$ &$ 1552 \pm 36$ &$ 1528    \pm 184 $\
Arp151 &11,13& 54506-54618& 0.02109& 89.2 & $  3.99^{+0.49}_{-0.68}$&$ 42.979 \pm 0.010$ &$  42.497 \pm   0.047$ &$  40.931\pm     0.011$&$3098 \pm 69$&$2006 \pm 24$ &$     1252   \pm 46  $\
NGC3783 &14,15& 48607-48833& 0.00973& 25.1 & $  10.2^{+3.3}_{-2.3}$&$   42.791 \pm 0.025$ &$  42.559 \pm   0.051$ &$  41.009\pm     0.021$&$3770 \pm 68$&$1691 \pm 19  $&$     1753  \pm 141 $\
Mrk1310 &11& 54550-54618& 0.01956& 82.7 & $  3.66^{+0.59}_{-0.61 }$&$ 42.937 \pm 0.018$ &$  42.231 \pm  0.120$ &$  40.646\pm     0.012$&$2409 \pm 24$&$ 1209 \pm  42 $&$     755  \pm 138 $\
NGC4051 &16& 54180-54311& 0.00234& 15.0 & $  1.87^{+0.54}_{-0.50}$&$42.290 \pm   0.015$ &$  41.847 \pm  0.080$ &$  40.079\pm     0.018$&$799 \pm 2 $ & $ 1045 \pm 4$ &$     927   \pm 64  $\
NGC4051 &6 & 56645-56864& 0.00234& 15.0 & $  2.87^{+0.86}_{-1.33}$&$42.265 \pm   0.005$ &$  41.732 \pm  0.106$ &$  39.882\pm     0.012$&$765 \pm 3 $ &$470 \pm 2  $&$     493 \pm 35  $\
NGC4151 &17& 53430-53472& 0.00332& 15.0 & $  6.59^{+1.12}_{-0.76}$&$42.549 \pm   0.012$ &$  42.004 \pm  0.113$ &$  40.499\pm     0.013$&$5840 \pm 863 $&$ 6158 \pm    47 $&$     2680 \pm 64  $\
NGC4151 &6 & 55931-56072& 0.00332& 15.0 & $  6.82^{+0.48}_{-0.57}$&$42.685 \pm   0.007$ &$  42.315 \pm  0.060$ &$  40.956\pm     0.008$&$992 \pm 4 $& $1833 \pm  2  $&$     1894  \pm 9   $\
Mrk202 &11& 54550-54617& 0.02102& 88.9 & $  3.05^{+1.73}_{-1.12}$&$42.946 \pm   0.016$ &$  42.198 \pm  0.126$ &$  40.477\pm     0.010$&$1471 \pm 18 $&$ 867 \pm  40 $&$     659  \pm 65  $\
NGC4253 &11& 54509-54618& 0.01293& 54.4 & $  6.16^{+1.63}_{-1.22}$&$42.948 \pm   0.012$ &$  42.509 \pm  0.044$ &$  40.873\pm     0.010$&$1609 \pm 39 $&$ 1088 \pm 37 $&$     \ldots  $\
PG1226+023 &3 & 48361-50997& 0.15834& 737.7& $  306.80^{+68.5}_{-90.9}$&$45.935\pm  0.011$ &$  45.907 \pm  0.011$ &$  44.072\pm     0.014$&$3509 \pm 36$ &$ 1778 \pm 17 $&$     1777 \pm 150 $\
3C273 &18& 54795-58194& 0.15834& 737.7& $146.3^{+8.3}_{-12.1}$& $45.864 \pm 0.011$ & $45.848 \pm 0.011$ &$44.056 \pm 0.010$&$3256 \pm 36$ & $1701 \pm  15$ &$1090 \pm 121 $\
PG1229+204 &3 & 48319-50997& 0.06301& 274.9& $  37.8^{+27.6}_{-15.3}$&$44.053 \pm   0.010$ &$  43.636 \pm  0.040$ &$  42.275\pm     0.011$&$3828 \pm 54 $& $1608 \pm 24 $&$     1385 \pm 111 $\
NGC4593 &19& 53391-53580& 0.00900& 37.7 & $  3.73^{+0.75}_{-0.75}$&$43.242 \pm   0.013$ &$  43.005 \pm  0.035$ &$  41.237\pm     0.013$&$5143 \pm 16 $&$1790 \pm  3 $&$     1561  \pm 55  $\
NGC4748 &11& 54550-54618& 0.01463& 61.6 & $  5.55^{+1.62}_{-2.22}$&$43.072 \pm   0.012$ &$  42.557 \pm  0.060$ &$  41.047\pm     0.010$&$1947 \pm 66$ & $ 1009 \pm 27$ &$     657 \pm 91  $\
PG1307+085 &3 & 48319-51042& 0.15500& 718.7& $  105.6^{+36.0}_{-46.6}$&$44.849 \pm   0.012$ &$  44.790 \pm 0.013$ &$  43.096\pm     0.020$&$5059 \pm 133 $ & $ 1963 \pm 47 $&$      1820 \pm 122 $\
MCG-06-30-15&20& 55988-56079& 0.00775& 25.5 & $  5.33^{+1.86}_{-1.75}$&$42.393 \pm   0.009$ &$  41.651 \pm  0.197$ &$  39.793\pm     0.011$&$1958 \pm 75 $ &$ 976 \pm 8$ &$     665   \pm 87  $\
NGC5273 &21& 56774-56838& 0.00362& 15.3 & $  2.21^{+1.19}_{-1.60}$&$42.000 \pm   0.009$ &$  41.465 \pm  0.106$ &$  39.702\pm     0.010$&$5688 \pm 163 $&$ 1821 \pm    53 $&$     1544 \pm 98  $\
Mrk279 &22& 50095-50289& 0.03045& 129.7& $  16.7^{+3.9}_{-3.9}$&$43.882 \pm   0.021$ &$  43.643 \pm    0.036$ &$  42.242\pm     0.021$&$5354 \pm 32$&$ 1823 \pm 11 $&$     1420  \pm 96  $\
PG1411+442 &3& 48319-51038& 0.08960& 398.2& $  124.3^{+61.0}_{-61.7}$&$44.603 \pm   0.012$ &$  44.502 \pm 0.014$ &$  42.792\pm     0.014$&$2801 \pm 43 $&$1774 \pm  29$ &$     1607 \pm 169 $\
NGC5548&23,24,25&47509-47809& 0.01718& 72.5 & $  19.7^{+1.5}_{-1.5}$&$ 43.534 \pm   0.021$ &$  43.328 \pm   0.042$ &$  41.728\pm     0.018$&$4674 \pm 63$&$ 1934 \pm 5 $&$     1687   \pm 56  $\
NGC5548 &24,25& 47861-48179& 0.01718& 72.5 & $  18.6^{+2.1}_{-2.3}$&$ 43.390 \pm   0.029$ &$  43.066 \pm   0.068$ &$  41.546\pm     0.029$&$5418 \pm 107$ &$ 2227 \pm 20 $&$    1882 \pm 83  $\
NGC5548 &24,26& 48225-48534& 0.01718& 72.5 & $  15.9^{+2.9}_{-2.5}$&$ 43.496 \pm   0.017$ &$  43.264 \pm   0.042$ &$  41.645\pm     0.026$&$5236 \pm 87$ &$ 2205 \pm 16 $&$     2075 \pm 81  $\
NGC5548 &24,26& 48623-48898& 0.01718& 72.5 & $  11.0^{+1.9}_{-2.0}$&$ 43.360 \pm   0.020$ &$  42.999 \pm   0.070$ &$  41.457\pm     0.030$&$5986 \pm 95$&$ 3110 \pm 53 $&$     2264  \pm 88  $\
NGC5548 &24,27& 48954-49255& 0.01718& 72.5 & $  13.0^{+1.6}_{-1.4}$&$ 43.497 \pm   0.016$ &$  43.267 \pm   0.040$ &$  41.691\pm     0.016$&$5930 \pm 42$&$ 2486 \pm 13 $&$     1909  \pm 129 $\
NGC5548 &24,28& 49309-49636& 0.01718& 72.5 & $  13.4^{+3.8}_{-4.3}$&$ 43.509 \pm   0.022$ &$  43.287 \pm   0.043$ &$  41.649\pm     0.022$&$7378 \pm 39 $&$ 2877 \pm 17 $&$     2895 \pm 114 $\
NGC5548 &24,28& 49679-50008& 0.01718& 72.5 & $  21.7^{+2.6}_{-2.6}$&$ 43.604 \pm   0.012$ &$  43.436 \pm   0.026$ &$  41.746\pm     0.013$&$6946 \pm 79$&$ 2432 \pm 13 $&$     2247  \pm 134 $\
NGC5548 &24,28& 50044-50373& 0.01718& 72.5 & $  16.4^{+1.2}_{-1.1}$&$ 43.527 \pm   0.020$ &$  43.317 \pm   0.039$ &$  41.656\pm     0.018$&$6623 \pm 93 $&$ 2276 \pm 15 $&$     2026 \pm 68  $\
NGC5548 &24,29& 50434-50729& 0.01718& 72.5 & $  17.5^{+2.0}_{-1.6}$&$ 43.413 \pm   0.018$ &$  43.113 \pm   0.054$ &$  41.622\pm     0.015$&$6298 \pm 65 $&$ 2178 \pm 12$ &$     1923 \pm 62  $\
NGC5548 &24,29& 50775-51085& 0.01718& 72.5 & $  26.5^{+4.3}_{-2.2}$&$ 43.620 \pm   0.020$ &$  43.459 \pm   0.032$ &$  41.762\pm     0.018$&$6177 \pm 36 $&$ 2035 \pm 11 $&$     1732 \pm 76  $\
NGC5548 &24,29& 51142-51456& 0.01718& 72.5 & $  24.8^{+3.2}_{-3.0}$&$ 43.565 \pm   0.017$ &$  43.376 \pm   0.034$ &$  41.719\pm     0.016$&$6247 \pm 57 $&$ 2021 \pm 18 $&$     1980 \pm 30  $\
NGC5548 &24,29& 51517-51791& 0.01718& 72.5 & $  6.5^{+5.7}_{-3.7}$&$  43.327 \pm   0.019$ &$  42.918 \pm   0.081$ &$  41.521\pm     0.017$&$6240 \pm 77 $&$2010 \pm 30 $& $     1969  \pm 48  $\
NGC5548 &24,29& 51878-52174& 0.01718& 72.5 & $  14.3^{+5.9}_{-7.3}$&$ 43.321 \pm   0.027$ &$  42.903 \pm   0.089$ &$  41.428\pm     0.026$&$6478 \pm 108 $&$ 3111 \pm 131 $&$       2173 \pm 89  $\
NGC5548 &24,30& 53432-53472& 0.01718& 72.5 & $  6.3^{+2.6}_{-2.3}$&$  43.263 \pm   0.016$ &$  42.526 \pm   0.211$ &$  40.967\pm     0.017$&$6396 \pm 167$&$3210 \pm 642$ & $    2388 \pm 373 $\
NGC5548 &10,24& 54180-54332& 0.01718& 72.5 & $  12.4^{+2.7}_{-3.9}$&$ 43.287 \pm   0.008$ &$  42.665 \pm   0.140$ &$  40.660\pm     0.070$&$12575\pm 47$&$4736 \pm   23 $&$     1822 \pm 35  $\
NGC5548 &11,24& 54508-54618& 0.01718& 72.5 & $  4.18^{+0.86}_{-1.30}$&$43.214 \pm  0.010$ &$  42.621 \pm   0.129$ &$  41.157\pm     0.017$&$12771 \pm 71$& $ 4266 \pm 65 $&$     4270    \pm 292 $\
NGC5548 &8,24 & 55931-56072& 0.01718& 72.5 & $  2.83^{+0.88}_{-0.90}$&$43.433 \pm  0.005$ &$  43.070 \pm   0.058$ &$  41.543\pm     0.010$&$10587 \pm 82$&$ 3056 \pm 4 $&$     2772  \pm 34  $\
NGC5548 &31 & 56663-56875& 0.01718& 72.5 & $  4.17^{+0.36}_{-0.36}$&$43.612 \pm  0.003$ &$  43.404 \pm   0.027$ &$  41.666\pm     0.004$&$9496 \pm 418$&$ 3691 \pm 162 $&$    4278 \pm 671 $\
NGC5548 &32& 57030-57236& 0.01718& 72.5 & $  7.18^{+1.38}_{-0.70}$&$43.175 \pm  0.005$ &$  42.787 \pm   0.063$ &$  41.630\pm     0.003$&$9912 \pm 362 $&$ 3350 \pm    272 $&$    3124 \pm 302 $\
PG1426+015 &3 & 48334-51042& 0.08657& 383.9& $  95.0^{+29.9}_{-37.1}$&$44.690 \pm  0.012$ &$  44.568 \pm   0.019$ &$  42.764\pm     0.015$&$7113  \pm 160 $&$ 2906 \pm 80$&$    3442 \pm 308 $\
Mrk817 &1 & 49000-49212& 0.03146& 134.2& $  19.0^{+3.9}_{-3.7}$&$ 43.848 \pm   0.010$ &$  43.726 \pm   0.015$ &$  42.010\pm     0.010$&$4711 \pm 78$&$ 1984\pm 8 $&$     1392    \pm 78  $\
Mrk817 &1 & 49404-49528& 0.03146& 134.2& $  15.3^{+3.7}_{-3.5}$&$ 43.761 \pm   0.087$ &$  43.608 \pm   0.124$ &$  41.936\pm     0.089$&$5237 \pm 67 $&$ 2098 \pm 13 $&$     1971 \pm 96  $\
Mrk817 &1 & 49752-49924& 0.03146& 134.2& $  33.6^{+6.5}_{-7.6}$&$ 43.762 \pm   0.009$ &$  43.609 \pm   0.016$ &$  41.860\pm     0.010$&$4767 \pm 72 $&$ 2195 \pm 16 $&$     1729 \pm 158 $\
Mrk817 &10& 54185-54301& 0.03146& 134.2& $  14.04^{+3.41}_{-3.47}$&$43.901 \pm 0.006$ &$   43.776 \pm  0.010$ &$  41.710\pm     0.016$&$5906 \pm 34$&$2365 \pm 9 $&$     2025    \pm 5   $\
Mrk290 &10& 54180-54321& 0.02958& 126.0& $  8.72^{+1.21}_{-1.02}$&$ 43.451 \pm   0.028$ &$  43.157 \pm 0.036$ &$  41.747\pm     0.030$&$4521 \pm 24 $&$ 2071 \pm 24 $&$     1609 \pm 47  $\
PG1613+658 &3 & 48397-51073& 0.12900& 588.4& $  40.1^{+15.0}_{-15.2}$&$ 44.948 \pm   0.010$ &$  44.713 \pm 0.019$ &$  42.943\pm     0.014$&$9074 \pm 103 $&$ 3084 \pm    33$&$      2547 \pm 342 $\
PG1617+175 &3 & 48362-51085& 0.11244& 507.4& $  71.5^{+29.6}_{-33.7}$&$ 44.445 \pm   0.011$ &$  44.330 \pm 0.014$ &$  42.682\pm     0.023$&$6641 \pm 190 $&$ 2313 \pm 69$&$     2626 \pm 211 $\
PG1700+518 &3 & 48378-51084& 0.29200& 1463.3& $  251.8^{+45.9}_{-38.8}$&$45.600 \pm   0.010$ &$  45.528 \pm 0.011$ &$  43.717\pm     0.020$&$2252\pm 85$&$ 3160 \pm 93 $&$     1700   \pm 123 $\
3C382 &6 & 56679-56864& 0.05787& 251.5& $  40.49^{+8.02}_{-3.74}$&$44.193 \pm   0.008$ &$  43.792 \pm 0.069$ &$  42.264\pm     0.011$&$3619\pm 203 $&$ 3227 \pm 7$&$       4552 \pm 190 $\
3C390.3 &33& 49718-50012& 0.05610& 243.5& $  23.60^{+6.2}_{-6.7}$&$43.902 \pm   0.018$ &$  43.620 \pm   0.039$ &$  42.222\pm     0.015$&$12694 \pm 13 $&$ 3744 \pm    42$ &$     3105 \pm 81  $\
3C390.3 &34& 50100-54300& 0.05610& 243.5& $  97.0^{+17.0}_{-17.0}$&$44.028 \pm  0.016$ &$  43.913 \pm   0.020$ &$  42.287\pm     0.021$&$11918\pm 325$& $\ldots $&$      \ldots  $\
3C390.3 &35& 53631-53714& 0.05610& 243.5& $  46.4^{+3.8}_{-3.2}$&$   44.485 \pm 0.007$ &$  44.434 \pm   0.008$ &$  42.695\pm     0.012$&$13211\pm 278 $&$ 5377 \pm    37 $&$     5455 \pm 278 $\
NGC6814 &11& 54545-54618& 0.00521& 21.6 & $  6.64^{+0.87}_{-0.90}$&$ 42.500 \pm 0.017$ &$  42.058 \pm   0.057$ &$  40.443\pm     0.010$&$3323 \pm 7 $&$1918 \pm 36 $&$     1610   \pm 108 $\
Mrk509 &1 & 47653-50374& 0.03440& 147.0& $  79.6^{+6.1}_{-5.4}$&$ 44.240 \pm   0.027$ &$  44.130 \pm   0.028$ &$  42.545\pm     0.027$&$3015 \pm 2 $&$ 1555  \pm 7  $&$     1276 \pm 28  $\
PG2130+099 &36& 54352-54450& 0.06298& 274.7& $  22.9^{+4.7}_{-4.6}$&$ 44.406 \pm   0.012$ &$  44.368 \pm   0.012$ &$  42.667\pm     0.011$&$2853 \pm 39 $&$ 1485 \pm 15 $&$     1246 \pm 222 $\
PG2130+099 &2 & 55430-55557& 0.06298& 274.7& $  9.6^{+1.2}_{-1.2}$&$  44.237 \pm   0.032$ &$  44.150 \pm   0.033$ &$  42.584\pm     0.033$&$1781 \pm 5  $&$ 1769\pm 2    $ &$    1825    \pm 65  $\
NGC7469 &37& 55430-55568& 0.01632& 68.8 & $  10.8^{+3.4}_{-1.3}$&$ 43.768 \pm   0.009$ &$  43.444 \pm   0.051$ &$  41.557\pm     0.013$&$4369 \pm 6 $&$ 1095  \pm 5 $&$     1274  \pm 126 $ \[table:hbrm\]

[lccccccccc]{} DESJ003-42&1 &56919-57627 & 2.593 & 20723 & $123^{+43}_{-42}$ &$46.510 \pm 0.020$& $4944 \pm 93$ & $3917 \pm 29 $& $6250 \pm 64$\
Fairall9 &2,3 & 49473-49713 &0.04702&202.8&$29.6^{+12.9}_{-14.4}$ &$44.530 \pm 0.030$ &$2968 \pm 37 $& $3068 \pm 27$ & $3201 \pm 285 $\
DESJ228-04&1&56919-57627 & 1.905 & 1686.4 &$95^{+16}_{-23}$ & $46.430 \pm 0.098$ & $5232 \pm 57$ &$3932 \pm 22$ & $6365 \pm 66$\
CT286 &4 &54821-57759& 2.556 & 20,366 &$459^{+71}_{-92}$ & $46.798 \pm 0.009$ & $6256$ & $\ldots$ &$\ldots$\
CT406 &4 &54355-57605& 3.183 & 26,533 &$115^{+64}_{-86}$&$46.910 \pm0.040$ & $6236$ &$\ldots$ &$\ldots$\
NGC3783 &5,3 & 48611-48833 &0.00973&25.1 &$3.8^{+1.0}_{-0.9}$ &$43.081 \pm  0.017$ &$2784 \pm 24$& $2476 \pm 18$ &$2948 \pm 160 $\
NGC4151 &6,7 & 47494-47556 &0.00332&15.0 &$3.44^{+1.42}_{-1.24}$ &$42.412 \pm 0.016$ &$2929 \pm 154$&$4922 \pm 51$ &$5426 \pm 196 $\
NGC4395 &8 & 53106 &0.00106&4.0 &$0.033^{+0.017}_{-0.013}$ &$39.494 \pm 0.007$ &$1214 \pm 2$&$1727 \pm 78$ &$3025 \pm 201 $\
NGC4395 &8 & 53190 &0.00106&4.0 & $0.046^{+0.017}_{-0.013}$ &$40.030 \pm 0.012$ &$1532 \pm 6$&$1662\pm34$ &$2859 \pm 376 $\
NGC5548 &9,3 & 47510-47745 &0.01718&72.5 &$9.8^{+1.9}_{-1.5}$ &$43.635 \pm 0.016$ &$5248 \pm 428$&$4351 \pm 37$& $3842 \pm 210 $\
NGC5548 &10,3 & 49060-49135&0.01718&72.5 &$6.7^{+0.9}_{-1.0}$ &$43.552 \pm 0.007$ &$4201 \pm 101$&$3738\pm17$ & $3328 \pm 104 $\
NGC5548 &11 & 56690-56866&0.01718&72.5 &$5.8^{+0.5}_{-0.5}$ &$43.625 \pm 0.007$ &$5236 \pm 87 $&$2205 \pm16$ &$2075 \pm 81  $\
3C390.3 &12,3 & 49718-50147 &0.05610&243.5&$35.7^{+11.4}_{-14.6}$&$44.013 \pm 0.045$ &$6180 \pm 638$&$4578 \pm 65$ &$4400 \pm 186 $\
J214355 &4 & 54729-57605& 2.620&20,985& $128^{+91}_{-82}$& $46.962 \pm 0.048$ & $6895$ & $\ldots$ &$\ldots$\
J221516 &4 & 54232-57689& 2.706&21821& $165^{+98}_{-13}$ & $47.155 \pm 0.057$& $5888$ & $\ldots$ &$\ldots$\
NGC7469 &13,3 & 50245-50293&0.01632&68.8 &$2.5^{+0.3}_{-0.3}$ &$43.719 \pm 0.016$ &$3112 \pm 54 $&$3650 \pm 27$ &$2619 \pm 118 $ \[table:civrm\]

[lcccccccc]{} 16 & $0.848$& 5240.9 & $32.0^{+11.6 }_{-15.5}$ & $44.7779 \pm 0.0012$ & $43.0718 \pm 0.0600$ & $7042 \pm 43  $ & $4804 \pm 41 $ & $6477 \pm 54$\
17 & $0.456$& 2466.9 & $25.5^{+10.9 }_{-5.8 }$ & $44.3552 \pm 0.0005$ & $42.1756 \pm 0.0064$ & $7847 \pm 203 $ & $4295 \pm 47 $ & $6101 \pm 48$\
101 & $0.458$& 2479.8 & $21.4^{+4.2  }_{-6.4 }$ & $44.3758 \pm 0.0005$ & $42.7316 \pm 0.0449$ & $2207 \pm 7   $ & $1178 \pm 5  $ & $976  \pm 32$\
160 & $0.359$& 1859.7 & $21.9^{+4.2  }_{-2.4 }$ & $43.7613 \pm 0.0009$ & $42.0456 \pm 0.0047$ & $3988 \pm 23  $ & $2914 \pm 36 $ & $1909 \pm 12$\
177 & $0.482$& 2635.8 & $10.1^{+12.5 }_{-2.7 }$ & $44.1735 \pm 0.0009$ & $42.2813 \pm 0.0125$ & $4808 \pm 32  $ & $2224 \pm 32 $ & $2036 \pm 39$\
191 & $0.442$& 2377.0 & $8.5 ^{+2.5  }_{-1.4 }$ & $43.9111 \pm 0.0015$ & $41.7344 \pm 0.0131$ & $2023 \pm 32  $ & $1078 \pm 79 $ & $1030 \pm 18$\
229 & $0.47 $& 2557.5 & $16.2^{+2.9  }_{-4.5 }$ & $43.8259 \pm 0.0017$ & $41.9083 \pm 0.0166$ & $3089 \pm 261 $ & $2178 \pm 156$ & $1781 \pm 38$\
265 & $0.734$& 4388.8 & $8.5 ^{+3.2  }_{-3.9 }$ & $44.3809 \pm 0.0019$ & $42.4400 \pm 0.0273$ & $3655 \pm 323 $ & $2526 \pm 55 $ & $7165 \pm 36$\
267 & $0.587$& 3342.0 & $20.4^{+2.5  }_{-2.0 }$ & $44.3013 \pm 0.0008$ & $42.5166 \pm 0.0237$ & $2395 \pm 23  $ & $1229 \pm 32 $ & $1202 \pm 33$\
272 & $0.263$& 1298.0 & $15.1^{+3.2  }_{-4.6 }$ & $43.9119 \pm 0.0009$ & $42.3449 \pm 0.0017$ & $2595 \pm 10  $ & $1590 \pm 5  $ & $1697 \pm 10$\
300 & $0.646$& 3754.6 & $30.4^{+3.9  }_{-8.3 }$ & $44.6130 \pm 0.0008$ & $42.5889 \pm 0.0379$ & $2376 \pm 33  $ & $1303 \pm 29 $ & $1232 \pm 30$\
305 & $0.527$& 2933.9 & $53.5^{+4.2  }_{-4.0  }$ & $44.2995 \pm 0.0008$ & $42.5025 \pm 0.0365$ & $2208 \pm 28  $ & $1647 \pm 20 $ & $2126 \pm 35$\
316 & $0.676$& 3968.3 & $11.9^{+1.3  }_{-1.0  }$ & $44.9958 \pm 0.0004$ & $43.4279 \pm 0.0020$ & $2988 \pm 10  $ & $1884 \pm 5  $ & $7195 \pm 40$\
320 & $0.265$& 1309.4 & $25.2^{+4.7  }_{-5.7 }$ & $43.6876 \pm 0.0010$ & $41.8663 \pm 0.0096$ & $4061 \pm 26  $ & $3110 \pm 37 $ & $1462 \pm 26$\
371 & $0.472$& 2570.5 & $13  ^{+1.4  }_{-0.8 }$ & $44.0638 \pm 0.0009$ & $42.3726 \pm 0.0086$ & $3506 \pm 26  $ & $1682 \pm 18 $ & $1443 \pm 11$\
373 & $0.884$& 5516.4 & $20.4^{+5.6  }_{-7.0 }$ & $44.9025 \pm 0.0012$ & $42.7743 \pm 0.0191$ & $5987 \pm 268 $ & $1897 \pm 48 $ & $2491 \pm 26$\
377 & $0.337$& 1727.4 & $5.9 ^{+0.4  }_{-0.6 }$ & $43.7819 \pm 0.0011$ & $41.5130 \pm 0.0156$ & $2746 \pm 118 $ & $1576 \pm 23 $ & $1789 \pm 23$\
392 & $0.843$& 5202.8 & $14.2^{+3.7  }_{-3.0 }$ & $44.4249 \pm 0.0032$ & $42.4894 \pm 0.0427$ & $2419 \pm 82  $ & $2446 \pm 110$ & $3658 \pm 56$\
399 & $0.608$& 3487.6 & $35.8^{+1.1  }_{-10.3}$ & $44.3272 \pm 0.0020$ & $42.2823 \pm 0.0281$ & $2689 \pm 88  $ & $1989 \pm 89 $ & $1619 \pm 38$\
428 & $0.976$& 6233.7 & $15.8^{+6.0  }_{-1.9 }$ & $45.4013 \pm 0.0015$ & $43.2816 \pm 0.0048$ & $2795 \pm 29  $ & $1836 \pm 18 $ & $7568 \pm 70$\
551 & $0.68 $& 3997.0 & $6.4 ^{+1.5  }_{-1.4 }$ & $44.1196 \pm 0.0021$ & $42.4389 \pm 0.0842$ & $2101 \pm 45  $ & $1255 \pm 59 $ & $1298 \pm 36$\
589 & $0.751$& 4513.8 & $46  ^{+9.5  }_{-9.5 }$ & $44.4877 \pm 0.0015$ & $42.6421 \pm 0.0107$ & $3738 \pm 62  $ & $2835 \pm 62 $ & $5013 \pm 49$\
622 & $0.572$& 3238.9 & $49.1^{+11.1 }_{-2.0 }$ & $44.3737 \pm 0.0006$ & $42.5966 \pm 0.0062$ & $2389 \pm 36  $ & $1147 \pm 11 $ & $1423 \pm 32$\
645 & $0.474$& 2583.6 & $20.7^{+0.9  }_{-3.0 }$ & $44.1342 \pm 0.0008$ & $42.2965 \pm 0.0047$ & $6428 \pm 163 $ & $2799 \pm 13 $ & $1438 \pm 17$\
720 & $0.467$& 2538.0 & $41.6^{+14.8 }_{-8.3 }$ & $44.3176 \pm 0.0008$ & $42.4324 \pm 0.0029$ & $2829 \pm 15  $ & $1679 \pm 17 $ & $1232 \pm 16$\
772 & $0.249$& 1219.6 & $3.9 ^{+0.9  }_{-0.9 }$ & $43.7867 \pm 0.0005$ & $41.5251 \pm 0.0081$ & $2381 \pm 33  $ & $1983 \pm 40 $ & $1026 \pm 14$\
775 & $0.172$& 805.9 & $16.3^{+13.1 }_{-6.6 }$ & $43.7943 \pm 0.0003$ & $41.7848 \pm 0.0021$ & $2744 \pm 36  $ & $2028 \pm 10 $ & $1818 \pm 8 $\
776 & $0.116$& 524.6 & $10.5^{+1.0  }_{-2.2 }$ & $43.3829 \pm 0.0004$ & $41.4179 \pm 0.0220$ & $3060 \pm 20  $ & $3178 \pm 19 $ & $1409 \pm 11$\
781 & $0.263$& 1298.0 & $75.2^{+3.2  }_{-3.3 }$ & $43.7604 \pm 0.0034$ & $41.8863 \pm 0.0155$ & $2506 \pm 19  $ & $1290 \pm 17 $ & $1089 \pm 22$\
782 & $0.362$& 1877.9 & $20  ^{+1.1  }_{-3.0 }$ & $44.0941 \pm 0.0006$ & $41.9722 \pm 0.0044$ & $3027 \pm 35  $ & $1527 \pm 16 $ & $1353 \pm 23$\
790 & $0.237$& 1153.2 & $5.5 ^{+5.7  }_{-2.1 }$ & $43.8222 \pm 0.0014$ & $41.8443 \pm 0.0272$ & $8365 \pm 44  $ & $5069 \pm 47 $ & $6318 \pm 38$\
840 & $0.244$& 1191.8 & $5   ^{+1.5  }_{-1.4 }$ & $43.6987 \pm 0.0005$ & $41.5724 \pm 0.0074$ & $6116 \pm 267 $ & $3286 \pm 254$ & $4457 \pm 60$ \[table:sdssrmhb\]

[lccccccc]{} 0 & 1.463 &10283 &$131.1 _{-36.6 }^{+42.9}$& $44.847 \pm 0.004$& $3967   \pm 107$& $1968   \pm 160$& $2144   \pm 46 $\
32 & 1.72 &12554 &$22.8  _{-3.6  }^{+3.5 }$& $44.492  \pm 0.021$& $2999   \pm 34 $& $1770 \pm 24 $& $2017 \pm 10 $\
36 & 2.213 &17094 &$188.4 _{-29   }^{+15.6}$& $45.909 \pm 0.001$& $4830   \pm 24 $& $2890 \pm 24 $& $3900 \pm 34 $\
52 & 2.311 &18020 &$56.5  _{-5.9  }^{+3.1 }$& $45.499  \pm 0.002$& $2258   \pm 14 $& $1809 \pm 15 $& $1322 \pm 22 $\
57 & 1.93 &14461 &$208.3 _{-5.6  }^{+10.6}$& $45.393 \pm 0.003$& $2692   \pm 11 $& $1626 \pm 8  $& $1682 \pm 12 $\
58 & 2.299 &17906 &$186.1 _{-7.4  }^{+5.9 }$& $45.353  \pm 0.002$& $3627   \pm 45 $& $2611 \pm 31 $& $3412 \pm 30 $\
130 & 1.96 &14737 &$224.3 _{-37.9 }^{+12.4}$& $45.534 \pm 0.001$& $5619   \pm 30 $& $4078 \pm 55 $& $4324 \pm 36 $\
144 & 2.295 &17868 &$179.4 _{-42.3 }^{+31.2}$& $45.516 \pm 0.001$& $6153   \pm 53 $& $2762 \pm 19 $& $2792 \pm 19 $\
145 & 2.138 &16390 &$180.9 _{-4.7  }^{+4.7 }$& $45.113  \pm 0.004$& $4472   \pm 74 $& $3287 \pm 40 $& $3408 \pm 16 $\
158 & 1.477 &10405 &$36.7  _{-26.1 }^{+18.6}$& $44.999 \pm 0.004$& $3603   \pm 101$& $2099   \pm 60 $& $2136 \pm 31 $\
161 & 2.071 &15764 &$180.1 _{-6.4  }^{+5.6 }$& $45.491  \pm 0.001$& $3163   \pm 28 $& $2323 \pm 25 $& $2524 \pm 20 $\
181 & 1.678 &12177 &$102.6 _{-10.1 }^{+5   }$& $44.545  \pm 0.015$& $2998   \pm 35 $& $2127 \pm 44 $& $2721 \pm 34 $\
201 & 1.797 &13248 &$41.3  _{-19.5 }^{+32  }$& $46.240  \pm 0.001$& $5438   \pm 56 $& $1833 \pm 9  $& $2408 \pm 117$\
231 & 1.646 &11892 &$80.4  _{-7.5  }^{+6.3 }$& $45.736  \pm 0.001$& $5975   \pm 98 $& $3267 \pm 102$& $3803   \pm 18 $\
237 & 2.394 &18810 &$49.9  _{-4.4  }^{+6.6 }$& $45.866  \pm 0.001$& $5455   \pm 39 $& $2734 \pm 18 $& $2779 \pm 23 $\
245 & 1.677 &12168 &$107.1 _{-28.6 }^{+22.9}$& $45.351 \pm 0.004$& $9496   \pm 107$& $4174   \pm 54 $& $3953 \pm 86 $\
249 & 1.721 &12562 &$24.9  _{-3.1  }^{+9.7 }$& $44.984  \pm 0.010$& $1871   \pm 15 $& $1432 \pm 12 $& $1640 \pm 15 $\
256 & 2.247 &17414 &$43    _{-11.9 }^{+16.3}$& $45.089 \pm 0.003$& $2544   \pm 54 $& $1742 \pm 29 $& $1802 \pm 24 $\
269 & 2.4 &18868 &$197.2 _{-12.6 }^{+2.4 }$& $45.193  \pm 0.003$& $3930   \pm 312$& $3280   \pm 50 $& $3547 \pm 30 $\
275 & 1.58 &11307 &$81    _{-24.4 }^{+8.2 }$& $45.611  \pm 0.001$& $3213   \pm 20 $& $2108 \pm 9  $& $2406 \pm 5  $\
295 & 2.351 &18400 &$163.8 _{-5.3  }^{+8.2 }$& $45.605  \pm 0.001$& $4311   \pm 41 $& $2501 \pm 23 $& $2446 \pm 19 $\
298 & 1.633 &11777 &$106.1 _{-31.7 }^{+18.7}$& $45.596 \pm 0.001$& $3160   \pm 30 $& $2066 \pm 26 $& $2549 \pm 35 $\
312 & 1.929 &14452 &$56.9  _{-6.7  }^{+11.4}$& $45.077 \pm 0.004$& $7663   \pm 166$& $4273   \pm 74 $& $4291 \pm 30 $\
332 & 2.58 &20598 &$81.6  _{-11.4 }^{+5.6 }$& $45.551  \pm 0.002$& $3799   \pm 14 $& $3009 \pm 63 $& $4277 \pm 33 $\
346 & 1.592 &11413 &$71.9  _{-11.3 }^{+23.8}$& $44.905 \pm 0.003$& $3389   \pm 168$& $2220   \pm 131$& $3055   \pm 29 $\
386 & 1.862 &13838 &$38.2  _{-19.3 }^{+13.2}$& $45.279 \pm 0.002$& $2972   \pm 40 $& $1782 \pm 38 $& $2187 \pm 41 $\
387 & 2.427 &19126 &$30.3  _{-3.4  }^{+19.6}$& $45.687 \pm 0.001$& $3676   \pm 24 $& $2123 \pm 14 $& $2451 \pm 23 $\
389 & 1.851 &13738 &$224.3 _{-18   }^{+7.1 }$& $45.564  \pm 0.002$& $5222   \pm 111$& $3839   \pm 16 $& $4064 \pm 15 $\
401 & 1.823 &13484 &$47.4  _{-8.9  }^{+15.2}$& $45.564 \pm 0.002$& $3273   \pm 21 $& $2457 \pm 12 $& $3321 \pm 12 $\
411 & 1.734 &12679 &$248.3 _{-39   }^{+21.1}$& $44.887 \pm 0.007$& $4256   \pm 67 $& $2511 \pm 61 $& $2490 \pm 39 $\
418 & 1.419 &9903 &$82.5  _{-16.9 }^{+27.6}$& $45.040 \pm 0.003$& $3143   \pm 44 $& $2662 \pm 94 $& $3110 \pm 23 $\
470 & 1.883 &14030 &$19.9  _{-4    }^{+43.2}$& $44.821 \pm 0.006$& $4022   \pm 52 $& $2441 \pm 34 $& $2317 \pm 60 $\
485 & 2.557 &20376 &$133.4 _{-5.2  }^{+22.6}$& $46.119 \pm 0.001$& $5342   \pm 48 $& $2924 \pm 32 $& $3961 \pm 41 $\
496 & 2.079 &15839 &$197.9 _{-6.6  }^{+9.7 }$& $45.560  \pm 0.001$& $2364   \pm 27 $& $2137 \pm 34 $& $2409 \pm 45 $\
499 & 2.327 &18172 &$168.5 _{-35.9 }^{+20.4}$& $45.058 \pm 0.003$& $3261   \pm 41 $& $2968 \pm 41 $& $3085 \pm 26 $\
506 & 1.753 &12850 &$231.6 _{-11.1 }^{+13.3}$& $45.075 \pm 0.003$& $5046   \pm 52 $& $3507 \pm 27 $& $3510 \pm 24 $\
527 & 1.651 &11937 &$52.3  _{-12.2 }^{+15.1}$& $44.788 \pm 0.003$& $5154   \pm 110$& $3384   \pm 62 $& $3587 \pm 34 $\
549 & 2.277 &17698 &$69.8  _{-7.2  }^{+5.3 }$& $45.369  \pm 0.002$& $3907   \pm 59 $& $1818 \pm 47 $& $2176 \pm 21 $\
554 & 1.707 &12437 &$194   _{-12.2 }^{+20.4}$& $45.573 \pm 0.002$& $3690   \pm 65 $& $2253 \pm 47 $& $2229 \pm 35 $\
562 & 2.773 &22476 &$158.5 _{-34.2 }^{+18.2}$& $46.302 \pm 0.001$& $4379   \pm 113$& $2036   \pm 29 $& $2078 \pm 27 $\
686 & 2.13 &16315 &$64.7  _{-6.3  }^{+12.6}$& $45.444 \pm 0.002$& $3827   \pm 34 $& $2135 \pm 25 $& $2203 \pm 27 $\
689 & 2.007 &15170 &$157.6 _{-42.2 }^{+22.9}$& $45.223 \pm 0.003$& $2258   \pm 23 $& $1292 \pm 8  $& $1407 \pm 5  $\
734 & 2.324 &18144 &$87.2  _{-11   }^{+13.9}$& $45.530 \pm 0.001$& $5701   \pm 121$& $2982   \pm 65 $& $3405 \pm 40 $\
809 & 1.67 &12106 &$108.6 _{-50.7 }^{+27.7}$& $45.204 \pm 0.005$& $4811   \pm 38 $& $5210 \pm 60 $& $4749 \pm 96 $\
827 & 1.966 &14792 &$137.7 _{-19.4 }^{+18.3}$& $44.999 \pm 0.006$& $2542   \pm 35 $& $971  \pm 13 $& $1443 \pm 13 $ \[table:sdssrmciv\]

[lccccc]{} Mrk335 &$ 2291 \pm 27   $ & $ 2116 \pm  160  $&$ 44.173 \pm 0.020$ & $ 6.663 \pm   0.337 $ &$7.079 \pm   0.145$\
Mrk335 &$ 1741 \pm 99   $ & $ 1806 \pm  360  $&$ 44.291 \pm 0.078$ & $ 6.588 \pm   0.375 $ &$7.080 \pm   0.187$\
Mrk335 &$ 2023 \pm 17   $ & $ 2140 \pm  93   $&$ 44.262 \pm 0.013$ & $ 6.720 \pm   0.332 $ &$7.153 \pm   0.140$\
PG0026+129 &$ 1837 \pm 136  $ & $ 3364 \pm  70   $&$ 45.165 \pm 0.025$ & $ 7.591 \pm   0.331 $ &$8.092 \pm   0.140$\
PG0052+251 &$ 3983 \pm 370  $ & $ 5118 \pm  486  $&$ 45.265 \pm 0.037$ & $ 8.009 \pm   0.341 $ &$8.402 \pm   0.150$\
PG0052+251 &$ 5192 \pm 251  $ & $ 5083 \pm  437  $&$ 45.176 \pm 0.041$ & $ 7.956 \pm   0.339 $ &$8.331 \pm   0.149$\
Fairall9 &$ 2593 \pm 65   $ & $ 2981 \pm  197  $&$ 44.470 \pm 0.028$ & $ 7.118 \pm   0.335 $ &$7.496 \pm   0.144$\
Fairall9 &$ 2831 \pm 40   $ & $ 3532 \pm  92   $&$ 44.582 \pm 0.011$ & $ 7.325 \pm   0.331 $ &$7.676 \pm   0.139$\
Fairall9 &$ 2370 \pm 151  $ & $ 2978 \pm  508  $&$ 44.759 \pm 0.126$ & $ 7.270 \pm   0.368 $ &$7.715 \pm   0.193$\
Mrk590 &$ 4839 \pm 59   $ & $ 3574 \pm  141  $&$ 44.119 \pm 0.029$ & $ 7.089 \pm   0.332 $ &$7.330 \pm   0.141$\
3C120 &$ 3302 \pm 75   $ & $ 3199 \pm  169  $&$ 44.943 \pm 0.039$ & $ 7.430 \pm   0.334 $ &$7.895 \pm   0.144$\
3C120 &$ 3278 \pm 105  $ & $ 3409 \pm  286  $&$ 44.617 \pm 0.056$ & $ 7.312 \pm   0.339 $ &$7.682 \pm   0.152$\
Ark120 &$ 3989 \pm 451  $ & $ 3795 \pm  165  $&$ 44.634 \pm 0.021$ & $ 7.414 \pm   0.332 $ &$7.755 \pm   0.141$\
Ark120 &$ 3945 \pm 42   $ & $ 3240 \pm  149  $&$ 44.482 \pm 0.022$ & $ 7.197 \pm   0.333 $ &$7.551 \pm   0.141$\
Mrk79 &$ 3182 \pm 521  $ & $ 3344 \pm  222  $&$ 43.879 \pm 0.039$ & $ 6.904 \pm   0.336 $ &$7.110 \pm   0.146$\
Mrk79 &$ 3049 \pm 128  $ & $ 2971 \pm  248  $&$ 43.495 \pm 0.058$ & $ 6.598 \pm   0.339 $ &$6.752 \pm   0.152$\
Mrk79 &$ 3113 \pm 122  $ & $ 3803 \pm  388  $&$ 43.726 \pm 0.065$ & $ 6.935 \pm   0.343 $ &$7.065 \pm   0.157$\
Mrk110 &$ 2990 \pm 64   $ & $ 2601 \pm  272  $&$ 43.770 \pm 0.050$ & $ 6.628 \pm   0.343 $ &$6.887 \pm   0.155$\
Mrk110 &$ 1638 \pm 59   $ & $ 2576 \pm  231  $&$ 43.876 \pm 0.081$ & $ 6.676 \pm   0.342 $ &$6.962 \pm   0.159$\
PG0953+414 &$ 2873 \pm 57   $ & $ 3512 \pm  361  $&$ 45.588 \pm 0.031$ & $ 7.853 \pm   0.342 $ &$8.438 \pm   0.151$\
NGC3516 &$ 4675 \pm 538  $ & $ 3311 \pm  372  $&$ 42.830 \pm 0.093$ & $ 6.340 \pm   0.348 $ &$6.306 \pm   0.167$\
NGC3516 &$ 4875 \pm 17   $ & $ 3132 \pm  64   $&$ 42.823 \pm 0.017$ & $ 6.288 \pm   0.331 $ &$6.270 \pm   0.139$\
NGC3516 &$ 5147 \pm 103  $ & $ 3245 \pm  84   $&$ 43.192 \pm 0.013$ & $ 6.514 \pm   0.331 $ &$6.570 \pm   0.139$\
NGC3516 &$ 4729 \pm 28   $ & $ 3430 \pm  92   $&$ 43.143 \pm 0.013$ & $ 6.536 \pm   0.331 $ &$6.564 \pm   0.139$\
NGC3516 &$ 4525 \pm 97   $ & $ 3137 \pm  79   $&$ 43.030 \pm 0.012$ & $ 6.399 \pm   0.331 $ &$6.428 \pm   0.139$\
NGC3516 &$ 3940 \pm 18   $ & $ 2834 \pm  95   $&$ 42.485 \pm 0.034$ & $ 6.022 \pm   0.332 $ &$5.957 \pm   0.142$\
NGC3516 &$ 4912 \pm 23   $ & $ 3973 \pm  36   $&$ 42.793 \pm 0.012$ & $ 6.479 \pm   0.330 $ &$6.380 \pm   0.138$\
NGC3783 &$ 2831 \pm 22   $ & $ 3273 \pm  100  $&$ 43.601 \pm 0.014$ & $ 6.738 \pm   0.331 $ &$6.886 \pm   0.139$\
NGC3783 &$ 2308 \pm 17   $ & $ 3179 \pm  185  $&$ 43.744 \pm 0.022$ & $ 6.789 \pm   0.334 $ &$6.979 \pm   0.143$\
NGC4051 &$ 1319 \pm 13   $ & $ 1713 \pm  227  $&$ 41.373 \pm 0.058$ & $ 4.995 \pm   0.351 $ &$4.830 \pm   0.163$\
NGC4151 &$ 6929 \pm 76   $ & $ 5220 \pm  123  $&$ 43.224 \pm 0.010$ & $ 6.944 \pm   0.331 $ &$6.860 \pm   0.139$\
NGC4151 &$ 5418 \pm 150  $ & $ 4604 \pm  249  $&$ 43.340 \pm 0.019$ & $ 6.896 \pm   0.333 $ &$6.878 \pm   0.142$\
NGC4151 &$ 5062 \pm 51   $ & $ 4651 \pm  371  $&$ 43.396 \pm 0.029$ & $ 6.935 \pm   0.338 $ &$6.926 \pm   0.147$\
NGC4151 &$ 5246 \pm 44   $ & $ 4675 \pm  397  $&$ 43.396 \pm 0.031$ & $ 6.939 \pm   0.339 $ &$6.929 \pm   0.148$\
NGC4151 &$ 5752 \pm 144  $ & $ 4585 \pm  321  $&$ 43.418 \pm 0.023$ & $ 6.934 \pm   0.336 $ &$6.935 \pm   0.144$\
NGC4151 &$ 5173 \pm 593  $ & $ 4664 \pm  475  $&$ 43.354 \pm 0.044$ & $ 6.915 \pm   0.342 $ &$6.896 \pm   0.153$\
NGC4151 &$ 3509 \pm 10   $ & $ 4384 \pm  66   $&$ 43.038 \pm 0.006$ & $ 6.694 \pm   0.330 $ &$6.621 \pm   0.138$\
PG1229+204 &$ 3391 \pm 205  $ & $ 3241 \pm  457  $&$ 44.654 \pm 0.028$ & $ 7.288 \pm   0.352 $ &$7.682 \pm   0.160$\
PG1307+085 &$ 3465 \pm 168  $ & $ 3687 \pm  290  $&$ 45.012 \pm 0.039$ & $ 7.590 \pm   0.338 $ &$8.027 \pm   0.148$\
Mrk279 &$ 4126 \pm 487  $ & $ 3118 \pm  414  $&$ 43.795 \pm 0.118$ & $ 6.799 \pm   0.355 $ &$7.007 \pm   0.181$\
Mrk279 &$ 3876 \pm 99   $ & $ 3286 \pm  511  $&$ 43.754 \pm 0.127$ & $ 6.823 \pm   0.363 $ &$7.005 \pm   0.189$\
NGC5548 &$ 4790 \pm 67   $ & $ 4815 \pm  257  $&$ 43.654 \pm 0.022$ & $ 7.102 \pm   0.333 $ &$7.142 \pm   0.142$\
NGC5548 &$ 4096 \pm 14   $ & $ 3973 \pm  34   $&$ 43.568 \pm 0.006$ & $ 6.889 \pm   0.330 $ &$6.969 \pm   0.138$\
NGC5548 &$ 3280 \pm 27   $ & $ 5050 \pm  787  $&$ 43.773 \pm 0.069$ & $ 7.206 \pm   0.359 $ &$7.259 \pm   0.171$\
PG1426+015 &$ 3778 \pm 448  $ & $ 4101 \pm  391  $&$ 45.295 \pm 0.023$ & $ 7.832 \pm   0.340 $ &$8.301 \pm   0.149$\
Mrk817 &$ 4027 \pm 71   $ & $ 4062 \pm  289  $&$ 44.123 \pm 0.022$ & $ 7.203 \pm   0.336 $ &$7.404 \pm   0.145$\
PG1613+658 &$ 5902 \pm 136  $ & $ 3965 \pm  215  $&$ 45.221 \pm 0.023$ & $ 7.764 \pm   0.334 $ &$8.226 \pm   0.142$\
PG1617+175 &$ 4558 \pm 1763 $ & $ 3383 \pm  1036 $&$ 44.784 \pm 0.108$ & $ 7.394 \pm   0.428 $ &$7.805 \pm   0.234$\
Mrk509 &$ 5035 \pm 298  $ & $ 3558 \pm  205  $&$ 44.641 \pm 0.029$ & $ 7.362 \pm   0.334 $ &$7.725 \pm   0.143$\
Mrk509 &$ 4345 \pm 49   $ & $ 3426 \pm  115  $&$ 44.532 \pm 0.015$ & $ 7.272 \pm   0.331 $ &$7.621 \pm   0.140$\
Mrk509 &$ 4973 \pm 233  $ & $ 3647 \pm  172  $&$ 44.803 \pm 0.020$ & $ 7.469 \pm   0.333 $ &$7.862 \pm   0.141$\
Mrk509 &$ 4961 \pm 218  $ & $ 3127 \pm  226  $&$ 44.552 \pm 0.033$ & $ 7.203 \pm   0.336 $ &$7.585 \pm   0.146$\
Mrk509 &$ 3716 \pm 228  $ & $ 3174 \pm  448  $&$ 44.706 \pm 0.071$ & $ 7.297 \pm   0.354 $ &$7.710 \pm   0.168$\
PG2130+099 &$ 2113 \pm 119  $ & $ 2390 \pm  184  $&$ 44.692 \pm 0.025$ & $ 7.044 \pm   0.337 $ &$7.541 \pm   0.146$\
NGC7469 &$ 3094 \pm 53   $ & $ 3379 \pm  182  $&$ 43.774 \pm 0.016$ & $ 6.858 \pm   0.333 $ &$7.036 \pm   0.142$\
NGC7469 &$ 2860 \pm 12   $ & $ 3266 \pm  110  $&$ 43.679 \pm 0.015$ & $ 6.778 \pm   0.331 $ &$6.945 \pm   0.140$ \[table:civVP06\]

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[^1]: The database is regularly updated at http://www.astro.gsu.edu/AGNmass

[^2]: Strictly speaking, the Eddington ratio is defined as ${\mbox{$\dot{m}$}}= \dot{M}/\dot{M}_{\rm Edd}$. Since ${\mbox{$\dot{m}$}}= L_{\rm bol}/\eta c^2$, ${\mbox{$\dot{m}$}}= L_{\rm bol}/L_{\rm Edd}$ as long as the efficiency $\eta$ is constant and not a function of the accretion rate, which we will assume for simplicity.
