Accurate black hole masses are necessary to understand the growth of black holes and their role in galaxy evolution. In nearby (<100 Mpc) galaxies, it is possible to measure black hole mass directly from high spatial resolution observations of the dynamics of stars and gas (e.g., Kormendy & Ho 2013). But for distant active galactic nuclei 10 (AGN), the primary method to obtain reliable black hole masses is reverberation mapping (RM) from time-domain spectroscopy (Blandford & McKee 1982;Peterson et al. 2004).
RM measures the time delay between variability in the continuum emission and the corresponding variability in the broad-line region (BLR). In the environment around a supermassive black hole, light from the accretion disk is absorbed and re-emitted by the BLR with a delay due to the light travel time between the two emitting regions. The time delay, multiplied by the speed of light, gives a characteristic distance to the BLR, which is assumed to be in a virial orbit around the black hole. The mass of the black hole is thus given by a virial mass calculation as in Equation (1), using the emission-line broadening (ΔV ), characterized by the line-width FWHM or σ line , combined with the radius of the BLR
( )
The mass calculation includes a dimensionless factor "f," to account for the geometry of the orbit and kinematics of the BLR; this factor can be calibrated from comparing RM and dynamical masses (Onken et al. 2007;Grier et al. 2013), the M BH -σ relation (Woo et al. 2015;Yu et al. 2019), or from dynamical modeling of the BLR (Pancoast et al. 2014). The ffactor is of order unity and the exact value depends on assumptions like how the broad-line velocity is measured (e.g., Peterson et al. 2004;Collin et al. 2006;Yu et al. 2019).
From RM measurements taken over the last two decades, a correlation has been observed between the measured BLR time delay and the continuum luminosity of the AGN (e.g., Kaspi et al. 2000;Bentz et al. 2009Bentz et al. , 2013)). From this "radiusluminosity" (R-L) relation, we can estimate the radius of the BLR with just a luminosity measurement (e.g., Equation (2)) and estimate the black hole mass from single-epoch observations. This allows for the measurement of black hole masses for a large number of AGN without high spatial resolution or longterm monitoring. However, single-epoch estimates are only correct if the R-L relation accurately describes the diverse AGN population; therefore, it is necessary to measure this relation over a broad AGN sample and with the least bias possible. Bentz et al. (2013) used Hβ time-lag measurements and reliable subtraction of host-galaxy light for 41 AGN from different RM campaigns to determine the following R-L relation between the mean radius of the Hβ-emitting BLR and the AGN continuum luminosity at 5100 Å (lL 5100 ) :
The slope of this relation (α=0.533) is consistent with the R BLR ∝ L 0.5 expectation from basic photoionization models (Davidson 1972). Bentz et al. (2013) The Sloan Digital Sky Survey Reverberation Mapping (SDSS-RM) project is a dedicated multiobject RM campaign that has been monitoring 849 quasars with spectroscopy and photometry since 2014 (Shen et al. 2015a). Grier et al. (2017) published an Hβ R-L relation for 44 AGN from the first year of SDSS-RM monitoring. The time lags measured by SDSS-RM are often significantly shorter than those predicted by Equation (2) for their given AGN luminosity, and thus these sources fall below the Bentz et al. (2013) R-L relation. In addition, the Super-Eddington Accreting Massive Black Holes (SEAMBH) survey presented a R-L relation for a sample of rapidly accreting AGN that also differs from Bentz et al. (2013) in the same manner (Du et al. 2016(Du et al. , 2018;;Du & Wang 2019).
In this work we examine if this discrepancy is due to observational biases that restrict the allowable lag detections, or if the SDSS-RM and SEAMBH samples have properties that represent a broader population of AGN compared to previous RM studies; thus indicating a physical origin for the discrepancy, as suggested by recent work (Czerny et al. 2019;Du & Wang 2019). We explore this by simulating a R-L relation based on Bentz et al. (2013), while imposing the observational constraints of the SDSS-RM data set. We present the data included in our study in Section 2, and provide a detailed description of our simulated R-L relation and results in Section 3. In Section 4, we discuss possible causes for the discrepancy. Throughout this work we assume a standard ΛCDM cosmology with Ω Λ =0.7, Ω M =0. .
For our analysis, we compare Hβ lags, lL 5100 , and the best-
relation for the Bentz et al. (2013), Grier et al. (2017), and Du et al. (2016, 2018) data sets. The lags for the three RM campaigns were measured using different methods: Bentz et al. (2013) and Du et al. (2016Du et al. ( , 2018) ) used the interpolated cross-correlation function (ICCF; Gaskell & Peterson 1987;White & Peterson 1994;Peterson et al. 2004), while Grier et al. (2017) primarily used JAVELIN (Zu et al. 2011) and CREAM (Starkey et al. 2016). JAVELIN and CREAM use different assumptions than ICCF but are designed to produce similar results, so any deviations from the Bentz et al. (2013) R-L relation should not be due to the different lag-detection methods, as discussed in Section 2.3. We briefly describe the details of the lag measurement methods in Section 2.1.
Figure 1 presents the R-L relation for the Bentz et al. (2013), Grier et al. (2017), and Du et al. (2016, 2018) samples of AGN with Hβ RM lags. We describe these three samples in detail in the subsections below. The distribution of AGN properties in each sample is presented in Figure 2. For the Eddington ratio
) , we assume L bol =5.15lL 3000 and L bol = 9.26λL 5100 (Richards et al. 2006). Published 3000 Å luminosities are available only for 41 of the Grier et al. (2017) AGN; we use the 5100 Å luminosities for all other AGN in the three samples. We use black hole masses for the Bentz et al. (2013) sample from the compilation of Bentz & Katz (2015). In all three samples, the AGN luminosities are host-subtracted, and as such the luminosity uncertainties include a contribution from the uncertainty associated with the host-galaxy decomposition.
In general this means that the AGN luminosity uncertainties are largest for low-luminosity and host-dominated AGN, and are generally small for luminous AGN. We determine the best-fit R-L relation for each sample employing multiple linear regression with the python Markov Chain Monte Carlo (MCMC) software PyMC3, including uncertainties in both radius (y-axis) and luminosity (x-axis) and allowing for excess intrinsic scatter.
The ICCF determines the cross-correlation between two light curves, measured as the Pearson correlation coefficient r as a function of time delay τ. Because the data are unevenly spaced due to observational constraints, the ICCF linearly interpolates the first light curve to produce overlapping points to calculate r for any delay τ. The same process is repeated starting with the second light curve shifted by t -. The cross-correlation coefficient for a given τ is obtained by averaging the two values of r. The ICCF repeats this procedure for a range of τ, to obtain the final cross-correlation function (CCF). The likely (2017( ), and Du et al. (2016( , 2018)). The black line shows the R-L relation from Bentz et al. (2013), with a slope α=0.533 and a normalization K=1.527. The lag measurements from SDSS-RM (Grier et al. 2017) and SEAMBH (Du et al. 2018) frequently lie below the R-L relation established by Bentz et al. (2013).
time lag between the two light curves is given by the centroid of the CCF. The uncertainties are calculated using Monte Carlo methods with flux resampling and random subset sampling (Peterson et al. 2004).
Instead of using linear interpolation, JAVELIN assumes that the variability of the continuum light curve is best described by a damped random walk (DRW) model. JAVELIN then models the BLR light-curve response with the same DRW model combined with a top-hat transfer function centered at a lag τ, producing a BLR light-curve model that is a shifted, smoothed, scaled version of the continuum light curve. MCMC is used to identify the most likely lag and uncertainty. CREAM adopts a similar approach to JAVELIN to measure lags, with the same DRW assumption about variability, but with a slightly different treatment of the uncertainties. Detailed simulations by Li et al. (2019) and Yu et al. (2020) find that, for light curves of similar cadence and noise to SDSS-RM, JAVELIN produces more accurate lags and lag uncertainties than ICCF, and fewer false positives. Grier et al. (2017) measured Hβ lags using JAVELIN, ICCF, and CREAM; in this work we primarily utilize the lags from JAVELIN and CREAM, while noting that the ICCF lags of SDSS-RM quasars produce the same offset in the R-L relation (Figure 5).
2.2. Bentz et al. Bentz et al. (2013) collected a sample of 41 AGN from previous RM surveys, focusing on adding accurate host-galaxy subtraction from Hubble Space Telescope (HST) imaging. The sample primarily includes nearby AGN that were generally selected to be apparently bright and variable, with luminosities in the range 10 42 <lL 5100,AGN <10 46 ergs -1 . The AGN have lags measured from observing campaigns with monitoring durations that ranged from 64 to 120 days, with cadences as rapid as 1 day between observations. Lags were measured using the ICCF method, resulting in 70 Hβ time lags for 41 unique AGN in the range 2-100 rest-frame days.
The luminosity measurements are corrected for host-galaxy contributions; this is especially important for lower-luminosity AGN since galaxy contamination leads to an overestimation of lL 5100 , steepening the R-L relation. Previous RM surveys that did not correct for host-galaxy luminosity found a steeper R-L relation with a slope α∼0.70 (Kaspi et al. 2000). Bentz et al. (2013) measured the host-galaxy contribution for each AGN through morphological decomposition of HST Advanced Camera for Surveys (ACS) images, using the GALFIT software (Peng et al. 2002) to determine the best-fit point-source AGN and extended galaxy surface brightness profiles implementing a nonlinear leastsquares fit algorithm.
Figure 11 in Bentz et al. (2013) presents the R-L relation observed for their measured Hβ time lags, with a slope a = -+ 0.533 0.033 0.035 and a normalization = -+ K 1.527 0.031 0.031 for the best-fit line. Our fitting method yields a nearly identical slope α=0.52±0.03 and a normalization K=1.52±0.03 for the Bentz et al. (2013) Hβ lags. Spectra of the quasars were obtained using the Baryon Oscillation Spectroscopic Survey (BOSS) spectrograph (Smee et al. 2013) on the SDSS 2.5 m telescope (Gunn et al. 2006) at Apache Point Observatory. The initial observations include 32 epochs taken over a period of 6 months in 2014. The exposure time for each observation was ∼2 hr and the average time between observations was 4 days (maximum 16.6 days).
Photometric observations were acquired in the g and i filters with the Bok 2.3 m telescope and the Canada-France-Hawaii Telescope (CFHT). Additionally, synthetic photometric light curves were produced from the BOSS spectra in the g and i bands. All of the g-and i-band light curves were merged using the CREAM software (Starkey et al. 2016) to create a continuum light curve for each AGN (see Grier et al. 2017 for additional details of the light-curve merging procedure). Grier et al. (2017) measured Hβ reverberation lags using ICCF, JAVELIN, and CREAM. Each method used a lag search range between -100 and 100 days, given the length of the SDSS-RM observation baseline (∼200 days). This resulted in 32 lags from JAVELIN and 12 from CREAM, only including "reliable" positive time lags that have SNR>2, a single welldefined peak in the lag probability distribution function, and a correlation coefficient of r max >0.45. Shen et al. (2015b) used principal component analysis to decompose the quasar and host-galaxy spectra, assuming that the total spectrum is a combination of linearly independent sets of quasar-only and galaxy-only eigenspectra. The SDSS eigenspectra are taken from Yip et al. (2004). To obtain the quasar-only spectrum, Shen et al. (2015b) subtracted the bestfit host-galaxy spectrum from the total spectrum. Yue et al. (2018) independently estimated the host-galaxy contribution using imaging decomposition and found consistent results with the spectral decomposition.
Figure 3 presents the relation between the 44 SDSS-RM Hβ time lags and lL 5100 . Host-subtracted continuum luminosity (lL 5100 ) measurements were taken from Shen et al. (2015b). The points in red represent AGN luminosities that are hostsubtracted as described above. The observed rest-frame time lags are generally shorter than predicted from the Bentz et al.
(2013) R-L relation. The SDSS-RM data exhibit a positive correlation between radius and luminosity, with a Spearman's ρ=0.54 and a null probability of no correlation of p∼0.0. The R-L properties of the SDSS-RM quasars are best fit by a line with shallower slope and lower normalization, as shown as the red best-fit line of slope α=0.24±0.08 and a normalization K=1.24±0.04. However, the limited dynamic range of the SDSS-RM quasars means that the data could also be consistent with the same α ; 0.5 slope of the Bentz et al. (2013) data, with an average offset of shorter lags in SDSS-RM quasars over a range of continuum luminosities. Fitting the same SDSS-RM data, while fixing the slope to be 0.533, results in the same lower normalization K=1.24±0.05. For this and all subsequent least-squares fitting, we exclude the SDSS-RM data point with the longest lag and smallest fractional uncertainty as an outlier (RMID 781). We also exclude the hypervariable quasar RMID 017, as it increases in luminosity by a factor of ∼10 over the span of the SDSS-RM monitoring (Dexter et al. 2019).
Figure 3 Finally, to be certain that the different lag-detection methods are not the cause of the offset, we present the R-L relation using ICCF measured lags from SDSS-RM in Figure 5. The ICCF lags fall below the Bentz et al. (2013) relation just as seen in the JAVELIN and CREAM lags.
The SEAMBH project is an RM campaign spanning 5 years of monitoring (Du et al. 2016(Du et al. , 2018)). The AGN in the sample were selected from SDSS using a dimensionless accretion rate , derived from the standard thin-disk equations (Wang et al. 2014a):
The inclination of the disk is given by i and we assume cos i=0.75 (Du et al. 2018) ( )
´x L 10 5100 tot 44 ergs -1 . For spectra with > L 5100 tot 1.053 10 44 ergs -1 , the host-galaxy contribution was assumed to be zero.
The R-L relation for the 29 SEAMBH Hβ lags measured by Du et al. (2016Du et al. ( , 2018) ) is presented in Figure 6. Similar to the SDSS-RM data in Figure 3, the measured lags are shorter than expected from Equation (2), resulting in an R-L relation with a shallower slope α=0.29±0.07 and a lower normalization K=1.24±0.04. The SEAMBH data, like the SDSS-RM data, cover a limited dynamic range on both axes, and also appear consistent with a slope of α ; 0.5 with an average offset for shorter lags over a broad range of continuum luminosity.
The effects of the SDSS-RM observational limits on the observed R-L relation are not easily predictable. For instance, the sample is magnitude-limited in the i band, rather than limited by the luminosity used for the R-L relation. There are also constraints on the length of the measurable lags due to the duration and cadence of the observations. In order to examine how observational biases affect the R-L relation, we simulated an R-L relation starting from Equation (2), with Κ and α from Bentz et al. (2013) and including observational errors and limits appropriate for the SDSS-RM monitoring campaign.
To create a representative sample of AGN, we generated 10 7 random AGN luminosities in the range 10 42 -10 46 ergs -1 following the i-band luminosity function from Ross et al. (2013):
The L 3.37 and L 1.16 terms represent the bright and faint end of the distribution, respectively, with a break luminosity of
The SDSS-RM observational selection effects were applied to the simulations by adding observational uncertainties as well as lag and magnitude limits to the S1 sample. First, observational uncertainties were assigned to each of the simulated AGN by randomly drawing luminosity and lag uncertainties (σ L and σ τ ) from the actual 44 SDSS-RM lL 5100 and τ measurements (Shen et al. 2015b; Grier et al. 2017). 11 We then replicated the sample limits of SDSS-RM by imposing the same lag and magnitude constraints as the observations. Simulated AGN were restricted to observed-frame lags 4<τ obs <75 days, and i-band magnitude <21.7.
The average cadence for SDSS-RM observations was 4 days, which places a lower limit on the possible observed-frame time lags. Conversely the upper limit of 75 days comes from the longest measured time lag from SDSS-RM, related to the monitoring duration of 180 days and the need for overlap between the continuum and emission-line light curves.
While the observed-frame lag limit can be implemented by a simple redshift conversion, several additional steps were required to fully emulate the magnitude limits of the observed SDSS-RM sample. The SDSS-RM parent sample of quasars is restricted to total (AGN+host) magnitudes of i<21.7, but the S1 sample has AGN-only luminosities at rest-frame 5100 Å. We add a host-galaxy contribution to the simulated AGN luminosities following Equation (4) (measured for similar SDSS AGN spectra by Shen et al. 2011). We assume a 0.35dex scatter in this relation, since 0.35dex is the standard deviation of the actual host-galaxy luminosities of the SDSS-RM quasars. We convert this total lL 5100 to i-band magnitude before implementing a magnitude cutoff. However, there is an additional magnitude dependence of the lag detection that must be considered, as lags are easier to recover for brighter AGN: the fraction of AGN from SDSS-RM with detected lags by Grier et al. (2017) is roughly 1/3 as high for i>20 AGN as for i<20 AGN. We account for this by removing all AGN with i>21.7 and keeping all AGN with i<20, and only keeping 1/3 of AGN with 20<i<21.7.
We designate this "observation-limited" sample S2, shown as blue points in Figure 7. The boundaries in rest-frame lag and luminosity are smooth rather than sharp due to the range of redshifts applied to the simulated sample, and are slightly tilted because both the observed-frame lag and magnitude limits depend on redshift to convert to the rest-frame lag and luminosity.
Finally, to account for the limit in the number of actual lag detections in SDSS-RM (44 measured lags), we randomly selected 44 points from S2; we designate this "number-limited" sample S3. The S3 sample for one of the simulations is shown as the red points in Figure 7.
We repeated the random selection of 44 points and best-fit line 2000 times to see how observing specific AGN affected the slope of the simulated relation. We 8). Only <1% of the simulations have best-fit slopes and normalizations that are as extreme as the best-fit R-L relation for the observed SDSS-RM data. This result suggests that observational biases are unlikely to be the main cause of the different R-L relation represented by SDSS-RM AGN compared to previous RM samples. To examine if the number of detected lags by SDSS-RM affects the R-L relation, we can increase the number of selected points to reflect the future number of detected Hβ lags. The Black Hole Mapper (BHM) in the upcoming SDSS-V will allow RM of over 1000 quasars (Kollmeier et al. 2017). We estimate that this will increase the number of Hβ lags to ∼100.
Here we assume the SDSS-RM observational effects applied to the simulations are also a reasonable approximation for the SDSS-V observations. The distribution of best-fit lines for the 100 random points has a median slope of In general the simulations of observational bias produce a R-L relation that is statistically consistent with the Bentz et al. (2013) best-fit relation, with only marginally flatter slopes and lower normalizations. Less than 1% of the simulations produce best-fit R-L relations that are as extreme as the observed SDSS-RM and SEAMBH R-L data. Li et al. (2019) arrived at a similar conclusion using independent light-curve simulations, additionally noting that JAVELIN lags measured from SDSS-RM data are unlikely to include enough false positive detections to strongly influence the measured R-L relation.
Our simulations suggest that observational bias is unlikely to be the main cause of the SDSS-RM and SEAMBH AGN lags falling below the Bentz et al. (2013) R-L relation. In the next section we investigate the possibility that R-L offsets are instead driven by physical AGN properties.
The R-L differences between SDSS-RM and Bentz et al. (2013) may exist because the SDSS-RM sample spans a broader range of quasar properties (Shen et al. 2015a(Shen et al. , 2019)). The SEAMBH sample also occupies a very different parameter space compared to the Bentz et al. (2013) sample, as SEAMBH AGN were specifically selected to have higher Eddington ratios.
In this section, we investigate how the offset from the Bentz et al. (2013) R-L relation depends on various AGN properties. We define this offset as the ratio between the measured restframe Hβ lag τ obs and the expected time lag t - R L from Equation (2) for the given AGN lL 5100 . We calculate the offset (t t -
) for each of the AGN in Grier et al. (2017), Bentz et al. (2013), and Du et al. (2016, 2018). In the subsequent analyses, we report the significance of each correlation in terms of the factor of sigma by which its slope is inconsistent from zero, using 3σ as our threshold for a significant correlation. et al. (2016, 2018) propose that the R-L offsets are driven by accretion rate, with more rapidly accreting AGN having Figure 8. Top: the distribution of slopes and normalizations from fitting 44 random points from our simulated sample, shown as red contours that include 38% (0.5σ), 68% (1σ), 86% (1.5σ), 95% (2σ), 98% (2.5σ) and 99% (3σ) of the distribution. The red point represents the fitting results for SDSS-RM (Figure 3). The black point represents the result from Bentz et al. (2013). The dark red point represents the fitting result for SDSS-RM keeping the slope fixed to be the same as Bentz et al. (2013). The SDSS-RM measurement falls outside the 3σ contour and is only <1% likely to be produced by the simulation of observational bias. The Bentz et al. (2013) measurement falls just outside the 2σ contour and is consistent with 5% of the simulated R-L parameters. Bottom: the distribution of slopes and normalization for 100 random points from the simulated sample, using the same enclosed probabilities for the contour levels. The SDSS-RM point is outside the 3σ contour and so is again only <1% likely to be consistent with the simulation. In both cases, observational bias is insufficient to explain the R-L offsets of the SDSS-RM quasars.
shorter lags at fixed lL 5100 . They suggest that radiation pressure in rapidly accreting AGN causes the inner disk to be thicker (a "slim" disk), causing self-shadowing of the disk emission that reduces the ionizing radiation received by the BLR and thus decreases its radius (Wang et al. 2014b). The self-shadowing does not affect the optical continuum emission used in the R-L relation, so the broad-line lags are shorter than expected for a given observed lL 5100 . However, a correlation between offset and accretion rate is expected not just from quasar properties but simply because the axes are correlated: the y-axis (t t -
) is a log ratio of t lL 5100 0.5 , while the xaxes (λ Edd , ) include log ratios of lL 5100 /τ and l t L 5100 1.5 2 , respectively.
Despite these self-correlations, for direct comparisons to the previous SEAMBH results (see Du et al. 2018, Figure 5) we estimate accretion rates for all three samples using two dimensionless quantities: the Eddington ratio (calculated as described in Section 2) and (Equation (3), as defined in Du et al. 2016). The R-L offsets of all three samples as a function of λ Edd and are presented in Figure 9. Best-fit lines (with slope m and y-intercept b given in the figure legends) indicate significant (>5σ) anticorrelations between the R-L offset and both estimators of accretion rate, with Spearman's ρ∼-0.50 and p∼10 -11 .
The anticorrelations in both panels of Figure 9 are qualitatively consistent with the simple self-correlations. To avoid these self-correlations, we instead study the dependence of R-L offsets on accretion rate by using only the components of the Eddington ratio that are not computed directly from the the RM lag τ. Since l º µ , we examine the R-L offset against two measurements of linewidth v fwhm and v σ to determine if there are residual correlations beyond the self-correlations induced from lL 5100 and τ appearing in both axes; this is presented in Figure 10. For all samples and for both line-width indicators, there are only Bentz & Katz (2015). These observed quantities are related to Eddington ratio, and so are an attempt to connect R-L offsets with accretion rate while avoiding direct self-correlation with τ on both axes. The red lines show the best-fit relations to the Grier et al. (2017) SDSS-RM data, while the blue lines show the best-fit relations to all three samples. The R-L offset is only marginally anticorrelated with the b H line widths in each case.
of the three RM samples with Eddington ratio λ Edd (top) and the accretion rate (see Equation (3)). In both cases there is a significant anticorrelation between the two quantities, with the best-fit lines shown in red. The best-fit lines have slopes m that are >5σ different from zero, and a Spearman's ρ∼-0.50 with a null-probability value of p∼10 -11 . However, these trends are difficult to interpret since the two axes are selfcorrelated. We find much weaker correlations when comparing R-L offsets to uncorrelated quantities associated with accretion rate, as seen in Figures 10 and11. marginal (<2σ) anticorrelations between R-L offset and b H broad-line width.
We make a final attempt at studying the relation between R-L offset and accretion rate by using the relative Fe II . The relative Fe II strength is one of the "Eigenvector 1" quantities that separate quasars into different spectral categories (Boroson & Green 1992), and in particular R Fe II correlates positively with the Eddington ratio (Shen & Ho 2014). Thus we can use R Fe II as an independent estimate of accretion rate that avoids any self-correlation with t t -
. Figure 11 presents the relation between R-L offset and R Fe II for the SDSS-RM AGN of Grier et al. (2017). We find no anticorrelation between the offset and R Fe II , with a slope that is 1σ consistent with zero and Spearman's ρ=-0.11 and p=0.49. This is in contrast to the recent work of Du & Wang (2019), who found a significant correlation between R-L offset and R Fe II using the SEAMBH and Bentz et al. (2013) AGN samples. We do find a consistent slope in the relation (m=-0.24±0.23 compared to m=-0.42±0.06 in Du & Wang 2019), and our anticorrelation may be marginal rather than significant due to the limited sample size of SDSS-RM, the different lag uncertainties of JAVELIN, and/or the greater diversity of AGN properties in the SDSS-RM sample.
The R-L relation is parameterized with the optical luminosity at rest-frame 5100 Å, but the response and size of the BLR is governed by the incident ionizing photons (e.g., Davidson 1972). In particular, the b H recombination line is driven by the incident luminosity of E>13.6 eV photons. The basic photoionization expectation of R ∝ L 0.5 is valid for the optical luminosity only if changes in optical luminosity also correspond to identical changes in the ionizing luminosity; however, the shape of the SED and therefore the ratio of UV and optical luminosities depends on factors like mass, accretion rate, and spin (e.g., Richards et al. 2006). Modeling of the BLR by Czerny et al. (2019), with the assumption that the BLR radius is determined by the ionizing luminosity or number of incident photos, shows a diversity in the R-L relation due to changing UV/optical luminosity ratios, reproducing the range of observed lags from RM surveys. Additionally, emission-line lags relative to the optical continuum may underestimate the broad-line radius if there is a nonzero time lag between the UVcontinuum and optical-continuum variability, as observed in NGC 5548 (Pei et al. 2017). However, lags between the UV and optical continuum are short, so this would only significantly affect emission-line lags on the order of a few days. Similarly, geometric dilution of the BLR lag within a spatially extended BLR would cause the measured lags to be shorter than the mean BLR radius (Goad & Korista 2014), but this effect is generally small for all but the most extended BLR geometries.
None of our samples have published measurements of the > E 13.6 eV ionizing luminosity; however, the SDSS-RM sample has luminosity measurements at rest-frame 3000Å and Hβ, better probing the (near-)UV compared to the optical lL 5100 . Both of these quantities are shown with the R-L offset of SDSS-RM AGN in Figure 12. We fit lines to each, finding that there is no anticorrelation between the R-L offset and lL 3000 , with Spearman's ρ=-0.28, p=0. The ratio of luminosities of the [O III]λ5007 and Hβ emission lines is also frequently used as a proxy for the number of ionizing photons (e.g., Baldwin et al. 1981;Veilleux & Osterbrock 1987). Both are recombination lines, and [O III] has an ionization energy of 55eV compared to the H ionization energy of 13.6eV. We find a significant (3.7σ) correlation between offset and L O III [ ] /L Hβ , shown in Figure 13, with Spearman's ρ=0.36, p=0.02, and an excess scatter of ∼0.24. 12 We conclude that the shape of the UV/optical SED is likely to play a role in the R-L offset of AGN, as evident from the Since R Fe II correlates with the Eddington ratio (Shen & Ho 2014), a significant anticorrelation between R-L offset and R Fe II would suggest that, at fixed luminosity, more rapidly accreting AGN have shorter lags. We do not observe a significant anticorrelation between R-L offset and relative iron strength for SDSS-RM quasars, with a best-fit slope 1σ consistent with zero, and Spearman's ρ=-0.11 and p=0.49.
12 We tested that the fit in Figure 12 ratio is likely tied to the broader shape of the AGN SED, which in turn is related to the accretion rate and/or black hole spin (e.g., Du et al. 2018;Czerny et al. 2019). It is a bit surprising that we find a significant correlation of R-L offset with but only a marginal anticorrelation with R Fe II , given the observed anticorrelation between [O III] equivalent width and R Fe II (Figure 1 of Shen & Ho 2014). This may be due to the limited sample size of SDSS-RM AGN, and/or to the large uncertainties in its measured lags. Regardless of the root cause of the UV/optical SED changes, it would be valuable to add far-UV observations to the samples of SDSS-RM and SEAMBH AGN in order to directly compare their R-L offsets with the luminosity of photons responsible for ionizing the BLR.
While previous RM studies revealed a tight R-L relation between the broad-line radius t = R c and the optical luminosity lL 5100 , more recent studies (SDSS-RM and SEAMBH) frequently find shorter lags than expected for a given optical luminosity. We use Monte Carlo simulations that mimic the SDSS-RM survey design to show that the R-L offsets are not solely due to observational bias. Instead, we find that AGN R-L properties correlate most closely with AGN spectral properties: at fixed lL with a slope 3.7σ different from zero and a Spearman's ρ=0.36 and p=0.02.
The Astrophysical Journal, 899:73 (12pp), 2020 August 10 Fonseca Alvarez et al.
We repeated the simulation with the larger ICCF lag uncertainties rather than the JAVELIN uncertainties, and found nearly identical best-fit slope and normalization.