The Sloan Digital Sky Survey Reverberation Mapping Project: Estimating Masses of Black Holes in Quasars with Single-Epoch Spectroscopy

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 AGN 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 Hbeta (4861 A) and C IV (1549 A) emission lines.

The presence of emission lines with Doppler widths of thousands of kilometers per second is one of the defining characteristics of active galactic nuclei (Burbidge & Burbidge 1967;Weedman 1976). 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., Woltjer 1959), as did the Eddington limit (Tarter & McKee 1973). Under a few assumptions, the central mass is M ∝ 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 Dibai (1980) 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 ∝ L 1/3 . Wandel & Yahil (1985) inferred the BLR size from the Hβ luminosity. Other attempts were based on photoionization physics (see Ferland & Shields 1985;Osterbrock 1985). Davidson (1972) found that the relative strength of emission lines in ionized gas could be characterized by an ionization parameter where Q(H) is the rate at which H-ionizing photons are emitted by the central source and n 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] λ1909 set an upper limit on the density n H 10 9.5 cm −3 (Davidson & Netzer 1979). Since L ∝ Q(H), this naturally led to the prediction that the BLR radius would scale with luminosity as R ∝ L 1/2 . Unfortunately, best-estimate values for Q(H) and n H led to a significant overestimate of the BLR radius  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; Blandford & McKee 1982;Peterson 1993), direct measurements of R enabled improved black hole mass determinations. Attempts to estimate black hole masses based on early RM results and the R ∝ L 1/2 prediction included those of Padovani & Rafanelli (1988), Koratkar & Gaskell (1991), and Laor (1998). The first multiwavelength RM campaigns demonstrated ionization stratification of the BLR Krolik et al. 1991;Peterson et al. 1991) and this eventually led to identification of the virial relationship, R ∝ V −2 , 2000Onken & Peterson 2002;Kollatschny 2003;Bentz et al. 2010), 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 (Marconi et al. 2008(Marconi et al. , 2009Netzer & Marziani 2010). 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 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 µ; it incorporates the two observables in RM, line width and time delay τ = 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 , 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 (Kaspi et al. , 2005Bentz et al. 2006aBentz et al. , 2009aBentz et al. , 2013 enables "single-epoch" (SE) mass estimates because, in principle, a single spectrum could yield V and also R, through measurement of L (e.g., Wandel, Peterson, & Malkan 1999;McLure & Jarvis 2002;Vestergaard 2002;Corbett et al. 2003;Vestergaard 2004;Kollmeier et al. 2006;Fine et al. 2008;Shen et al. 2008a,b;Vestergaard et al. 2008). Of the three strong emission lines generally used to estimate central black hole masses, the R-L relationship is only well-established for Hβ λ4861 , and references therein, but see the discussion in §3.3). Empirically establishing the R-L relationship for Mg II λ2798 (Homayouni et al. 2020) and C IV λ1549 Kaspi et al. 2007;Trevese et al. 2014;Lira et al. 2018;Grier et al. 2019;Hoormann et al. 2019) has been difficult.
Masses based on the C IV λ1549 emission line, in particular, have been somewhat controversial. Some studies claim that there is good agreement between masses based on C IV and those measured from other lines Greene, Peng, & Ludwig 2010;Assef et al. 2011). On the other hand, there are several claims that there is inadequate agreement with masses based on other emission lines (Baskin & Laor 2005;Netzer et al. 2007;Sulentic et al. 2007;Shen et al. 2008b;Shen & Liu 2012;Trakhtenbrot & Netzer 2012). Denney et al. (2009a) and Denney et al. (2013), 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 C IV and other lines is found. It has also been argued, however, that some fitting methodologies are more affected by this than others . There have been more recent papers that attempt to correct C IV mass determinations to better agree with those based on other lines (e.g., Runnoe et al. 2013;Coatman et al. 2017;Mejía-Restrepo et al. 2018;Marziani et al. 2019).

Characterizing Line Widths
It is our suspicion that the apparent difficulties with C IV-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 σ line , which is defined by where P (λ) is the emission-line profile as a function of wavelength and λ 0 is the line centroid, While both FWHM and σ line 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 (Joly et al. 1985): broader lines have lower kurtosis, i.e., they are "boxier" rather than "peakier." Indeed, for AGNs, the ratio FWHM/σ line has been found to be a simple but useful characterization of the line profile (Collin et al. 2006;Kollatschny & Zetzl 2013). Each line-width measure has practical strengths and weaknesses Wang et al. 2020). The line dispersion σ line 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 -Hβ λ4861, Mg II λ2798, and C IV λ1549 -are blended with other features: the Fe II λ4570 and Fe II λλ5190, 5320 complexes (Phillips 1978) and He II λ4686 in the case of Hβ, the UV Fe II complexes in the case of Mg II, and He II λ1640 in the case of C IV. The FWHM can usually be measured more precisely than σ line (although Peterson et al. 2004 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 σ line because it is relatively simple to measure and can be measured more precisely while σ line often requires deblending or modeling the emission features, which does not necessarily yield unambiguous results.
There are, however, a number of reasons to prefer σ line to FWHM as the line-width measure for estimating AGN black hole masses. Fromerth & Melia (2000) point out that σ line better characterizes an arbitrary or irregular line profile. Peterson et al. (2004) note that σ line produces a tighter virial relationship than FWHM, and Denney et al. (2013) find better agreement between C IV-based and Hβ-based mass estimates by using σ line 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 σ line -based masses, suggesting that the former are biased as the same mass should be recovered regardless of the luminosity state of the AGN (Collin et al. 2006;Shen & Kelly 2012). A possibly more compelling argument for using σ line instead of FWHM is bias in the mass scale that is introduced by using FWHM as the line width. Steinhardt & Elvis (2010) used single-epoch masses for more than 60,000 SDSS quasars (Shen et al. 2008b) 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. Rafiee & Hall (2011) point out, however, that if the quasar masses are computed using σ line 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 Rafiee & Hall (2011), 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 σ line 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 (Shen 2013). Unfortunately, these arguments are not statistically compelling. Examination of the M BH -σ * relation using FWHM-based and σ line -based masses is equally unrevealing .
In reverberation mapping, a further distinction among line-width measures must be drawn since either FWHM or σ line can be measured in the mean spectrum, where F i (λ) is the flux in the i th spectrum of the time series at wavelength λ 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 In this paper, we will specifically refer to the measurements of σ line in the mean spectrum as σ M and in the rms spectrum as σ R . Similarly, FWHM M refers to FWHM of a line in the mean spectrum or a single-epoch spectrum and FWHM R is the FWHM in the rms spectrum. It is common to use σ R 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 σ R . Here we will attempt to determine if either σ M or FWHM M in a single or mean spectrum can serve as a suitable proxy for σ R ; we know a priori that there are good, but non-linear, correlations between σ R and both σ M and FWHM M . It therefore seems likely that either σ M or FWHM M could be used as a proxy for σ R .
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 Hβ and C IV emission lines and nearby continuum fluxes measurements for each line. We note that we do not discuss Mg II RM results in this contribution as the present situation has been addressed rather thoroughly by Bahk, Woo, & Park (2019) and new SDSS-RM results will be published soon (Homayouni et al. 2020). In §2, the data used in this investigation are described. In §3, the relationship between the Hβ 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 C IV. In §5, we present simple empirical formulae for estimating black hole masses from Hβ or C IV. 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 km s −1 Mpc −1 , Ω matter = 0.3 and Ω Λ = 0.7.

Data
We use two high-quality databases for this investigation: 1. Spectra and measurements for previously reverberation-mapped AGNs, for Hβ (Table A1) and for C IV (Table A2). These are mostly taken from the literature (see also Bentz & Katz 2015 for a compilation 1 ). Sources without estimates of hostgalaxy contamination to the optical luminosity L(5100Å) 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 "reverberationmapping database (RMDB)".
2. Spectral measurements from the Sloan Digital Sky Survey Reverberation Mapping Program , hereafter "SDSS-RM" or more compactly simply as "SDSS"). We use both Hβ (Table A3) and C IV (Table A4) data from the 2014-2018 SDSS-RM campaign (Grier et al. 2017b;Shen et al. 2019;Grier et al. 2019). 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 C IV RM measurements remain rather scarce, we augmented the C IV sample with measurements from  (hereafter VP06), who combined single-epoch luminosity and line-width measurements from archival UV spectra with Hβ-based mass measurements of the objects in Table A1. The UV parameters are given in Table A5; we note, however, that we have excluded 3C 273 and 3C 390.3 because they both have uncertainities in their virial product larger than 0.5 dex; 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 Shen et al. (2015) and Shen et al. (2016), including post-processing with PrepSpec (Horne, in preparation). We note that only lags (τ ), line dispersion in the rms spectrum (σ R ),  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 Hβ (44 quasars) or C IV (48 quasars), we impose these cuts on the entire sample for the sake of later discussion. The first quality condition is that for both V = FWHM M and V = σ M , since AGNs with lines narrower than 1000 km s −1 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., Greene & Ho 2007), not SDSS quasars. The second quality condition is that the best fit value V (BF) must lie in the range for both FWHM and σ line . 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 is a good criterion for both V = FWHM M and V = σ M to remove the worst outliers from the line-width comparisons discussed in §3.2 and §4.1. Finally, we removed quasars that were flagged by Shen et al. (2019) as having broad absorption lines (BALs), mini-BALs, or suspected BALs in C IV.
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 Hβ lags, 12 failed to meet at least one of the quality criteria, usually the S/N requirement, thus reducing the SDSS-RN Hβ sample to 32 quasars. Three quasars with C IV reverberation measurements (RMID 362, 408, and 722) were rejected for significant BALs, thus reducing the SDSS-RM C IV reverberation sample to 45 quasars. As we will show in §5, another effect of imposing quality cuts is, not surprisingly, that it removes some of the lower luminosity sources from the sample.

Fitting Procedure
Throughout this work, we use the fitting algorithm described by Cappellari et al. (2013) that combines the Least Trimmed Squares technique of Rousseeuw & van Driessen (2006) and a least-squares fitting algorithm which allows errors in all variables and includes intrinsic scatter, as implemented by Dalla Bontà et al. (2018). Briefly, the fits we perform here are of the general form where x 0 is the median value of the observed parameter x. The fit is done iteratively with 5σ rejection (unless stated otherwise) and the best fit minimizes the quantity where ∆x i and ∆y i are the errors on the variables x i and y i , and ε y is the sigma of the Gaussian describing the distribution of intrinsic scatter in the y coordinate; ε y is iteratively adjusted so that the χ 2 per degree of freedom ν = N − 2 has the value of unity expected for a good fit. The observed scatter is The value of ε 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 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 with ∆x i , ∆y i and ∆z i as the errors on the variables (x i , y i , z i ), and ε z as the sigma of the Gaussian describing the distribution of intrinsic scatter in the z coordinate; ε z is iteratively adjusted so that the χ 2 per degrees of freedom ν = N − 3 has the value of unity expected for a good fit. The observed scatter is 3. MASSES BASED ON Hβ 3.1. The R-L Relationships In this section, we examine the calibration of the fundamental Hβ 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 Hβ, 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 . 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 1, we show the Hβ lag as a function of the AGN continuum with the host contribution removed in each case. This essentially reproduces the result of Bentz et al. (2013) 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 (10) in the first row of Table 2.
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 Hβ (e.g., Yee 1980;Ilić et al. 2017), and it has indeed been argued that the Hβ emission-line luminosity can be used as a proxy for the AGN continuum luminosity for reverberation studies Figure 1. Left: The rest-frame Hβ lag in days is shown as a function of the AGN luminosity LAGN(5100Å) in erg s −1 . The host-galaxy starlight contribution has been removed by using unsaturated HST images (see Bentz et al. 2013). Right: The Hβ lag in days is shown as a function of the broad Hβ luminosity L(Hβ broad ) in erg s −1 . The narrow component of Hβ 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 (10) with coefficients given in Table 2. The short dashed lines show the ±1 σ uncertainty (equivalent to enclosing 68% of the values for a Gaussian distribution) and the long dashed lines show the 2.6σ uncertainties (equivalent to enclosing 99% of the values for a Gaussian distribution).   . However, in some of the reverberation-mapped sources, narrow-line Hβ contributes significantly to the total Hβ flux; NGC 4151 is an extreme example (e.g., Antonucci & Cohen 1983;Bentz et al. 2006a;Fausnaugh et al. 2017). Whenever the narrow-line component can be isolated, it has been subtracted from the total Hβ flux. In Figure 2, we show the tight relationship between L AGN (5100Å) and L(Hβ broad ); the best-fit coefficients for this relationship are given in Table 2. In the right panel of Figure 1, we show the Hβ lag as a function of the luminosity of the broad component of Hβ, with the narrow component removed whenever possible. We give the best-fit values to the equation (10) 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 Hβ broad component is thus an excellent proxy for the AGN luminosity and requires only removal of the Hβ 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 (Greene & Ho 2005). This is therefore the R-L  Table A1. The black solid line is the regression of L(Hβ broad ) on LAGN(5100Å); the red dotted line is the regression of LAGN(5100Å) on L(Hβ broad ), which we use in equation (24). The coefficients for both fits are given in Table 2.

Line-Width Relationships
We now consider the use of σ M and FWHM M as proxies for σ R (cf. Collin et al. 2006;Wang et al. 2019). The left panel of Figure 3 shows the relationship between σ R (Hβ), the Hβ line dispersion in the rms spectrum, and σ M (Hβ), the Hβ line dispersion in the mean spectrum. The relationship is nearly linear (slope = 1.085 ± 0.045) and the intrinsic scatter is small (0.079 dex). The fit coefficients are given in the first row of Table 3.
We also show in the right panel of Figure 3 the relationship between σ R (Hβ) and the FWHM of Hβ in the mean spectrum, FWHM M (Hβ). The fit coefficients are given in the second row of Table 3. The relationship is far from linear (slope = 0.535 ± 0.042), and the scatter ε y is larger than it is for the σ R (Hβ)-σ M (Hβ) relationship, even after removal of the notable outliers. While it is clear that σ M (Hβ) is an excellent proxy for σ R (Hβ), the value of FWHM M (Hβ) 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
In the previous subsections, we have re-established the correlations between τ (Hβ) and L(Hβ broad ) and between σ R (Hβ) and both σ M (Hβ) and FWHM M (Hβ). As a first approximation for a formula to estimate single-epoch masses, we fit the following  (Table A1) and open green triangles are for the SDSS sample (Table A3). The solid lines are best fits to equation (10) with coefficients in Table 3. The short dashed and long dashed lines indicate the ±1σ and ±2.6σ envelopes, respectively, and the red dotted lines indicate where the two line-width measures are equal. Crosses are points that were rejected at the 2.6σ (99%) level and are colored-coded like the circles. The relationship on the left is nearly linear (slope = 1.085±0.045) and the scatter εy is low (0.079 dex). It is clear in the right panel that FWHMM(Hβ) and σR(Hβ) are well-correlated, but the relationship is significantly non-linear (slope = 0.535 ± 0.042), the scatter εy is slightly larger (0.106 dex), and there are several significant outliers. and 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 4. Using these coefficients, we have initial fits and for σ M (Hβ) and FWHM M (Hβ), 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 FWHM M (c = 1.039) is much smaller than that of σ M (c = 1.757), as expected from Figure 3. It is clear that both equations (18) and (19) 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.  (16) and (17) 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 ±1σ and ±2.6σ 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 (36) and (38) for σM and FWHMM, respectively. The best fit lines cover the equality lines.  We investigated the possibility of another parameter by plotting the residuals ∆ log µ = log µ RM − log µ SE against other parameters, specifically luminosity, mass (virial product), Eddington ratio, emission-line lag, line width and line-width ratio FWHM/σ 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 Grier et al. 2017b;Du et al. 2018;Du & Wang 2019;Fonseca Alvarez et al. 2019;Martínez-Aldama et al. 2019). To determine the Eddington ratio, we start with the Eddington luminosity where m e is the electron mass and σ e is the Thomson cross-section. The black hole mass is log M = log f + log µ and, as explained in the Appendix, we assume log f = 0.683 ± 0.150 (Batiste et al. 2017) so the Eddington luminosity is log L Edd = log f + 38.099 + log µ RM = 38.782 + log µ RM .
The bolometric luminosity can be obtained from the observed 5100Å AGN luminosity plus a bolometric correction We ignore inclination effects and, following Netzer (2019), we use Since we are using L(Hβ broad ) as a proxy for L AGN (5100Å), we substitute L(Hβ broad ) for L AGN (5100Å) by fitting the luminosities in Table A1, yielding (see Table 2) so we can write the bolometric luminosity as The Eddington ratioṁ is given by 2 logṁ = log L bol − log L Edd .
Using equations (25) and (21), the Eddington ratio can then be written as To correct the single-epoch masses for Eddington ratio, we fit the equation and use this as a correction to our initial fits, equations (18) and (19). The best-fit parameters for σ M and FWHM M -based predictors of µ SE are given in Table 5 and shown in Figure 5. Combining the correction equation (28) with the best-fit coefficients in Table 5 and equations (18) and (19) yield the corrected single-epoch masses and log µ SE (Hβ) = 6.974 for σ M and FWHM M , 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 σ M -based masses and 0.204 dex for FWHM M -based masses.

Fundamental Relationships
As noted in §1, the veracity of C IV-based mass estimates is unclear and remains controversial. The ideal situation would be to have a large number of AGNs with both C IV and Hβ reverberation measurements to effect a direct comparison. There are, unfortunately, very few AGNs that have both; indeed Table A2 of the Appendix lists all C IV results for which there are corresponding Hβ measurements in Table A1. For the few sources with both C IV and Hβ reverberation measurements, we plot the virial products µ RM (C IV) and µ RM (Hβ) in Figure 6; these are in each case a weighted mean value of for each of the observations of Hβ and C IV for the AGNs that appear in both Tables A1 and A2. The close agreement of these values reassures us that the C IV-based RM masses can be trusted, at least over the range of luminosities sampled. 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 C IV. In Figure 7, we show the relationship between the UV continuum luminosity L(1350Å) and the C IV emission line lag τ (C IV) based on the C IV data in Table A2, plus the SDSS-RM C IV data in Table A4. The coefficients of the fit are given in Table 2. We note again that we have removed from the Grier et al. (2019) sample in Table A4 three quasars with BALs, thus reducing the sample size from 48 to 45. The slope of the C IV R-L relation (0.517) is consistent with that of Hβ (0.492), though the ε y scatter is substantially greater (0.336 dex for C IV compared to 0.213 dex for Hβ). Definition of the relationship does not depend on the two separate measurements of very short C IV lag measurements for the dwarf Seyfert NGC 4395 . Thus it seems clear that we can use L(1350Å) as a reasonable proxy for τ (C IV).
We show the relationship between the C IV line dispersion measured in the rms spectrum σ R (C IV) and the line dispersion in the mean spectrum σ M (C IV) in Figure 8. The best-fit coefficients are given in Table 3. The correlation is good. However, the correlation between FWHM M (C IV) and σ R (C IV), also shown in Figure 8, is rather poor (see also Wang et al. 2020) and demonstrates that FWHM M (C IV) is a dubious proxy for σ R (C IV). Measurement of FWHM M (C IV) is clearly not a reliable predictor of σ R (C IV), so we will not consider FWHM M (C IV) further.

Single-Epoch Masses
Following the same procedures as with Hβ, we use the RMDB data (Table A2) and the SDSS-RM data (Table A4) to fit the equation The resulting fit is shown in Figure 9 and the best-fit coefficients are given in Table 4. With the coefficients from this fit and equation (32), we can generate predicted virial masses µ SE (C IV). We compare the measured reverberation mass µ RM with the single-epoch prediction µ SE based on this fit in the left panel of Figure 9. As was the Lower panels: residuals after subtraction of the best fit in the panel above. The εy scatter in the residuals is 0.197 dex for the σM-based virial products and 0.204 dex for the FWHMMbased 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 ±1σ and ±2.6σ envelopes, respectively. The coefficients of the fits are given in Table 5. Error bars are measurement uncertainties only, without systematic errors. case for Hβ (Figure 4), the distribution of points is slightly skewed relative to the diagonal, and, guided by our result for Hβ, we plot the residuals in log µ RM − log µ SE versus Eddington ratioṁ in the upper left panel of Figure 10. The Eddington ratio for the UV data is logṁ = −33.737 + 0.9 log L(1350Å) − log µ RM , where again we have used a bolometric correction from L(1350Å) from Netzer (2019), We fitted equation (28) for C IV and the coefficients of the fit are given in Table 5.
The offset between the residuals in the upper left panel of Figure 10 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 11, 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  Relationship between the C IV rest-frame emission-line lag τ (C 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 (10) with coefficients given in Table 2. The short dashed and long dashed lines are the ±1σ and ±2.6σ envelopes, respectively. The Spearman rank coefficient for these data is ρ = 0.503. If the two lowest luminosity points (both measurements of the dwarf Seyfert NGC 4395) are omitted, the Spearman rank coefficient is decreased to ρ = 0.481. are relatively local and less luminous than the SDSS quasars. We will thus proceed by examining mass residuals versus both Eddington ratio and µ RM . Figure 10 illustrates the process by which we eliminate the mass residuals in successive iterations. We compute the mass residuals ∆ log µ = log µ RM − log µ SE from equation (32); these are shown versusṁ (left column) and µ RM (right column). We fit these residuals versusṁ (top left) and subtract the best fit to equation (28), 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 µ RM (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 Figure 8. Left: Relationship between C IV line dispersion in the mean and rms spectra of reverberation-mapped AGNs. The Spearman rank coefficient is ρ = 0.873. Right: Relationship between FWHMM(C IV) and σR(C IV) for reverberation-mapped AGNs. The Spearman rank coefficient for these data is ρ = 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 (10) and the coefficients are given in Table 3. The short dashed and long dashed lines show the ±1σ and ±2.6σ envelopes, respectively.  Table A2 (blue filled circles), the SDSS-RM C IV 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 ± 0.041. As was the case with Hβ, 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 (40). In both panels, the solid line is the best fit to equation (32). The short dashed and long dashed lines define the ±1σ and ±2.6σ envelopes, respectively. The diagonal red dotted line is the locus where µRM and µSE are equal.
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 µ 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  Table 5. Note that the intrinsic scatter in this relationship is ǫy = 0.000 ± 0.000 because the error bars are so large. The bottom panels show the mass residuals versusṁ and µRM after subtracting the fit in the middle right panel. The scatter in the bottom panels is 0.138 dex. 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 ±1σ and ±2.6σ envelopes.
correlated with the deviation from the Bentz et al. (2013) R-L relationship, and the middle panels of Figure 10 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 ofṁ and µ RM , as we will discuss in §6.
It is worth noting in passing that after correcting for Eddington ratio (Figure 5), the residuals in the Hβ-based mass estimates show no correlation with either mass or luminosity. Figure 11. Distribution in virial product µRM 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.

COMPUTING SINGLE-EPOCH MASSES
To briefly reiterate our approach so far, we started with the assumption that µ SE = f (R, L) only. This proved to be inadequate, so we examined the residuals in the log µ SE -log µ RM relationship and found that these correlated best with Eddington ratioṁ: fundamentally, at increasingṁ, the Bentz  In the case of C IV, we found additional residuals that correlated with µ RM , 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 (16), (17), and (32) since the residuals in the log µ RM -log µ SE relationships (upper panels in Figure 4 and left panel of Figure 9) are rather small. We can combine the basic R-L fits (equations 16, 17, and 32) with the residual fits (equations 28 and 35) to obtain prescriptions that work over the mass range sampled. Renormalizing for convenience, we can Here f is the scaling factor which is discussed briefly in the Appendix, and ∆ 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.
In this case, the intrinsic scatter is 0.371 dex.
Here we assume f = 4.28 (Batiste et al. 2017). Bolometric corrections were made using equations (23) and (34). On the left side, the quality cuts of §2.1 have been imposed. On the right side, no quality cuts have been made.
with associated uncertainty The intrinsic scatter in this relationship is 0.408 dex. Single-epoch predictions and reverberation-based masses for the AGNs in Tables A2, A4, and A5 are compared in the right panel of Figure 9.
In Figure 12, we show the distribution in bolometric luminosity and black hole mass for the entire sample of SDSS-RM quasars for which Hβ or C IV single-epoch masses can be estimated. 6. DISCUSSION

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 emissionline lag τ and line width in the rms spectrum σ R (e.g., equation A1 in the Appendix) as that quantity produces, upon adjusting by the scaling factor f , an M BH -σ * relationship for AGNs that is in good agreement with that for quiescent galaxies. Given that both τ and σ R average over structure in a complex system (cf., Barth et al. 2015), 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 Hβ emission line is a good proxy for the starlight-corrected AGN luminosity (Figure 1). 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 σ R 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 τ in the future on account of the light travel-time delay within the system (Pogge & Peterson 1992;Gilbert & Peterson 2003;Barth et al. 2015); 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 ( We have also confirmed that, for the case of Hβ, both σ M and FWHM M are reasonable proxies for σ R , though σ M is somewhat better than FWHM M .
On the other hand, the case of C IV remains problematic, as it differs in a number of ways from the other strong emission lines: 1. The equivalent width of C IV decreases with luminosity, which is known as the Baldwin Effect (Baldwin 1977); C IV is driven by higher-energy photons than, say, the Balmer lines and the Baldwin Effect reflects a softening of the highionization continuum. This could be due to higher Eddington ratio (Baskin & Laor 2004) or because more massive black holes have cooler accretion disks (Korista, Baldwin, & Ferland 1998).
3. BALs in the short-wavelength wing of C IV, another signature of outflow, are common (Weymann et al. 1991;Hall et al. 2002;Hewett & Foltz 2003;Allen et al. 2011). We remind the reader that in §2.1 we removed ∼ 17% of our SDSS C IV sample because the presence of BALs precludes accurate line-width measurements.
4. The pattern of "breathing" in C IV is the opposite of what is seen in Hβ (Wang et al. 2020). Breathing refers to the response of the emission lines, both lag and line width, to changes in the continuum luminosity. In the case of Hβ, an increase in luminosity produces an increase in lag and a decrease in line width (Gilbert & Peterson 2003; Goad, Korista, & Knigge 2004;Cackett & Horne 2006). In the case of C IV, however, the line width increases when the continuum luminosity increases, contrary to expectations from the virial theorem (equation 2).
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 Hβ-based and C IV-based virial products ( Figure  7), though, again, these are local Seyfert galaxies that are not representative of the general quasar population. The C IV breathing issue is addressed in detail by Wang et al. (2020), building on evidence for a non-reverberating narrow core or blue excess in the C IV emission line presented by . 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 σ M . As the broad component reverberates in response to continuum variations, σ M will track σ R much better than FWHM M , thus explaining the breathing characteristics and why FWHM M is a poor line-width measure for estimating black hole masses. Physical interpretation of the non-varying core remains an open question:  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) (Boroson & Green 1992;Sulentic et al. 2000;Boroson 2002;Shen & Ho 2014;Sun & Shen 2015;Marziani et al. 2018, and references therein). The strongest of these correlations, Eigenvector 1, is most clearly characterized by the anticorrelation between (a) the strength of the Fe II λ4570 and Fe II λλ5190, 5320 complexes on either side of the broad Hβ complex and (b) the strength of the [O III] λλ4959, 5007 doublet. There is consensus in the literature that Eigenvector 1 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 Hβ. We find, as did Marziani et al. (2019), that a similar correction is required for C IV-based masses as well.
One extreme of Eigenvector 1 is populated by sources with strong Fe II and very weak [O III]. The broad emission lines in the spectra of these objects also have relatively small line widths. By combining the R-L relation with eq. (2), the line width dependence is seen to be whereṁ ∝ L/M is the Eddington ratio (eq. 26). 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" (Osterbrock & Pogge 1985). The Super-Eddington Accreting Massive Black Holes (SEAMBH) collaboration has focused on high-ṁ candidates in their reverberationmapping program (Du et al. , 2018Du & Wang 2019). An important result from these studies, as we have noted earlier, is that the Hβ lags are smaller than predicted by the current state-of-the-art R-L relationship ). This implies that in these objects the ratio of hydrogen-ionizing photons to optical photons is lower than in the lowerṁ sources; this is also consistent with the relative strength of the low-ionization lines such as Fe II in SEAMBH sources, the weakness of high-ionization lines, such as [O III], and their soft X-ray spectra (Boller, Brandt, & Fink 1996). Du & Wang (2019) 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 §4.2, 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 Fe II to Hβ, R = EW(Fe II)/EW(Hβ), correlates well with Eddington ratio. In the UV, it is also found that the C IV blueshift correlates with Eddington ratio (Baskin & Laor 2005;Coatman et al. 2016;Sulentic et al. 2017). 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 §5.

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 (Gaskell & Peterson 1987;White & Peterson 1994;Peterson et al. 1998bPeterson et al. , 2004 with lag measurements and uncertainties from JAVELIN (Zu, Kochanek, & Peterson 2011). Recent tests (Li et al. 2019;Yu et al. 2020) 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. Table A1 with recent results and other previous results that we excluded because they did not have starlight-corrected luminosities.

Expand the database in
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 σ line , 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 σ M ; for example, the bottom panels of Figure 10 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 σ M , in particular.

SUMMARY
The main results of this paper are: 1. We confirm that the luminosity of the broad component of the Hβ emission line L(Hβ broad ) is an excellent substitute for the AGN continuum luminosity L AGN (5100Å) for predicting the Hβ emission-line reverberation lag τ (Hβ). It has the advantage of being easier to isolate than L AGN (5100Å), which requires an accurate estimate of the host-galaxy starlight contribution to the observed luminosity.
2. We confirm that the line dispersion of the Hβ broad component σ M (Hβ) and the full-width at half maximum for the Hβ broad component FWHM M (Hβ) in mean, or single-epoch, spectra are both reasonable proxies for the line dispersion of Hβ in the rms spectrum σ M (Hβ) for computing single-epoch virial products µ SE (Hβ). We find that σ M (Hβ) gives better results than FWHM M (Hβ), but both are usable.
3. In the case of C IV, we find that the line dispersion of the C IV emission line σ M (C IV) in the mean, or single-epoch, spectrum is a good proxy for the line dispersion in the rms spectrum σ R (C IV) for estimating single-epoch virial products µ SE (C IV). We find that FWHM M (C IV), however, does not track σ R (C IV) well enough to be used as a proxy.
4. Although the R-L relationship based on the continuum luminosity L(1350Å) and C IV emission-line reverberation lag τ (C IV) is not as well defined as that for Hβ, the relationship appears to have a similar slope and it appears to be suitable for estimating virial products µ SE (C IV). 5. We confirm for both Hβ and C IV 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 µ RM 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 µ RM and the single-epoch estimate µ SE is still a power-law, but with a slope somewhat less than unity (upper panels of Figure 4, left panel of Figure 9). We are therefore able to empirically correct this relationship to an unbiased estimator of µ SE by fitting the residuals and essentially rotating the power-law distribution to have a slope of unity (lower panels of Figure 4, right panel of Figure 9). We present these empirical estimators for µ SE (Hβ) and µ SE (C IV) in §5.

DATABASE OF REVERBERATION-MAPPED AGNS
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 Hβ time lags. It is worth noting, however, that since our analysis shows that the broad Hβ 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 wellstudied 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 Hβ is given in Table A1.  Bentz et al. (2013), 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 Bentz et al. (2019). Individual virial products for these sources are easily computed using the Hβ time lags (Column 6) and line dispersion measurements (Column 12) and the formula Further conversion to mass requires multiplication by the virial factor f , i.e. log M = log f + log µ, a dimensionless factor that depends on the inclination, structure, and kinematics of the broad-Hβ-emitting region -indeed, detailed modeling of 9 of these objects (Pancoast et al. 2014;Grier et al. 2017a) shows that f depends most clearly on inclination (Grier et al. 2017a). 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 BH -σ * relationship (Ferrarese & Merritt 2000;Gebhardt et al. 2000;Gültekin et al. 2009), where σ * is the host-galaxy stellar bulge velocity dispersion. The required assumption is that the AGN M BH -σ * is identical to that of quiescent galaxies (Woo et al. 2013). In fact, it is found that the µ-σ * has a slope consistent with the M BH -σ * slope for quiescent galaxies , and the zero points disagree by only a multiplicative factor, which is taken to be f . Here we take log f = 0.683 ± 0.150 (Batiste et al. 2017) where the error on the mean is ∆ 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.        NOTE-Columns are 1: AGN name; 2: literature reference for data; 3: Julian Dates of observations; 4: redshift; 5: luminosity distance; 6: C IV time lag τ (C IV); 7: log continuum luminosity at 1350Å; 8: FWHM of C IV in the mean spectrum; 9: line dispersion of C IV in the mean spectrum; 10: line dispersion of C IV in the rms spectrum.