The Radius-Luminosity Relationship Depends on Optical Spectra in Active Galactic Nuclei

The radius-luminosity (R-L) relationship of active galactic nuclei (AGNs) established by the reverberation mapping (RM) observations has been widely used as a single-epoch black hole mass estimator in the research of large AGN samples. However, the recent RM campaigns discovered that the AGNs with high accretion rates show shorter time lags by factors of a few comparing with the predictions from the R-L relationship. The explanation of the shortened time lags has not been finalized yet. We collect 8 different single-epoch spectral properties to investigate how the shortening of the time lags correlate with those properties and to understand what is the origin of the shortened lags. We find that the flux ratio between Fe II and H$\beta$ emission lines shows the most prominent correlation, thus confirm that accretion rate is the main driver for the shortened lags. In addition, we establish a new scaling relation including the relative strength of Fe II emission. This new scaling relation can provide less biased estimates of the black hole mass and accretion rate from the single-epoch spectra of AGNs.


Introduction
In the past 40 yr, reverberation mapping (RM; e.g., Bahcall et al. 1972;Blandford & McKee 1982;Peterson et al. 1993) has become a powerful tool to investigate the geometry and kinematics of the broad-line regions (BLRs) in active galactic nuclei (AGNs) and to measure the masses of supermassive black holes (BHs). Through long-term spectroscopic monitoring of an AGN, the size scale (R BLR ) of its BLR can be directly obtained by measuring the delayed response (t BLR ) of the emission line with respect to the variation of the continuum, where R BLR = t c BLR and c is the speed of light. Fortunately, the RM observations of ∼100 objects (e.g., Peterson et al. 1993Peterson et al. , 1998Peterson et al. , 2002Peterson et al. , 2004Kaspi et al. 2000Kaspi et al. , 2007Bentz et al. 2008Bentz et al. , 2009bDenney et al. 2009;Barth et al. 2011Barth et al. , 2013Barth et al. , 2015Grier et al. 2012Grier et al. , 2017Ilić et al. 2017;Shapovalova et al. 2009;Rafter et al. 2011Rafter et al. , 2013Du et al. 2014Du et al. , 2015Du et al. , 2016aDu et al. , 2018aDu et al. , 2018bWang et al. 2014a;Shen et al. 2016;Fausnaugh et al. 2017;De Rosa et al. 2018;Woo et al. 2019) lead to a correlation between the time lag of Hβ emission line (or the radius b R H of the Hβ-emitting region) and the monochromatic luminosity (l l L ) at 5100 Å (hereafter L 5100 ) with the form of where ℓ 44 =L 5100 /10 44 erg s −1 (e.g., Kaspi et al. 2000;Bentz et al. 2009aBentz et al. , 2013 Reverberation of broad emission lines to the continuum confirms photoionization as the major radiation mechanism in the BLR. As a canonic model of BLR, photoionization defined by the ionization parameter X µ L R n T ion BLR 2 e directly indicates R BLR ∝ L ion 1/2 , where L ion is the ionizing luminosity, and n e and T are electron density and temperature of the ionized gas, respectively. If we take L 5100 as a proxy of L ion , we have µ R L BLR 5100 1 2 , agreeing with the observations (see also Bentz et al. 2013). We would like to emphasize here the necessary conditions for this canonic relation: (1) the ionizing source should be isotropic or at least quasi-isotropic so that the BLR clouds receive the same luminosity with observers; (2) L ion ∝ L 5100 should always work; (3) ionizing luminosity comes from a point source, which is much smaller than the distances of the BLR clouds to the central BH. Condition (1) is broken in the AGNs powered by slim accretion disks (Wang et al. 2014c), where the puffed-up inner region may lead to nonisotropic ionizing radiation (Wang et al. 2014c). Condition (2) relies on spectral energy distributions (only holds for prograde accretion AGNs powered by the Shakura-Sunyaev disks), and does not work in ones with retrograde accretion (Wang et al. 2014b;Czerny et al. 2019). Low spin (also low-accretion rate and large BH mass) may lead to the deficit of the UV photons and nonlinear relation between L ion and L 5100 (Wang et al. 2014b;Czerny et al. 2019). And the L 5100 variation shows a little lag with respect to the L ion variation in the RM of accretion disks (Edelson et al. 2015;Cackett et al. 2018;McHardy et al. 2018). Regarding Condition (3), the size of accretion disk, although small, has been successfully measured and is not infinitesimal (Edelson et al. 2015;Cackett et al. 2018;McHardy et al. 2018). Therefore, the b R L H 5100 -relationship is expected to depend on the accretion situation or some other properties.
Recently, the super-Eddington accreting massive black hole (SEAMBH) campaign discovered that many objects with strong Fe II and narrow Hβ emission lines, which are thought to be the AGNs with high-accretion rates, lie below the b R L H 5100 -relationship (Du et al. , 2016a(Du et al. , 2018b. They found that the time lags of the AGNs with high-accretion rates become shortened by factors of 3∼8 relative to the normalaccretion-rate AGNs with the same luminosities, and the shortening itself shows correlation with the accretion rate (Du et al. , 2016a(Du et al. , 2018b. Wang et al. (2014c) proposed that the anisotropic radiation of the slim accretion disk may result in the shortened time lags. The Sloan Digital Sky Survey Reverberation Mapping (SDSS-RM) Project also reported that many AGNs have time lags shorter than expected from the b R L H 5100 relationship ), but cautioned that selection effects may arise at least in some cases (see Grier et al. 2019).
Although the detailed physical explanation causing the shortened time lags is not yet finalized (Wang et al. 2014c;Grier et al. 2017Grier et al. , 2019, more and more objects deviating from the traditional b R L H 5100 relationship are being discovered Du et al. 2018b). It is urgent to investigate the origin of the shortened lags in more detail. In this paper, we investigate how the deviation of an AGN from the b R L H 5100 relationship correlates with the properties in the single-epoch spectrum, and try to establish a new scaling relationship including the influence of single-epoch spectral properties. We describe the sample, data, and measurements in Section 2. A new scaling relation is established and presented in Section 3. Some discussions are provided in Section 4, and a brief summary is given in Section 5. We adopt the standard ΛCDM cosmology and the parameters of H 0 =67 km s −1 Mpc −1 , Ω Λ =0.68, and Ω m =0.32 (Planck Collaboration et al. 2014 in this paper.

Sample
The analysis in the present paper is mainly based on the samples: (1) the RM measurements compiled in Bentz et al. (2013) from the previous literature, (2) the AGNs with highaccretion rates of the SEAMBH campaign published in Du et al. (2014Du et al. ( , 2015Du et al. ( , 2016aDu et al. ( , 2018b, Wang et al. (2014a), and Hu et al. (2015), and (3) some other AGNs published after 2013: Mrk1511 from Barth et al. (2013), NGC5273 from Bentz et al. (2014), KA1858+4850 from Pei et al. (2014), MCG+06-30-015 from Bentz et al. (2016b), Hu et al. (2016), UGC06728 from Bentz et al. (2016a), and MCG+08-11-011, NGC2617, 3C382 and Mrk374 from Fausnaugh et al. (2017). 4 The collection in Bentz et al. (2013) includes 41 AGNs monitored successively since the late 1980s, most of which have relatively weaker Fe II emission and broader Hβ lines compared to the SEAMBH objects (Du et al. 2018b Table 1. Some objects have been mapped more than once, in order to understand the population properties better, we have to equalize the weights of the individual objects in the following analysis. We average the multiple measurements by taking into account their measurement uncertainties (see more details in Du et al. 2015). The average measurements, lags, FWHM, luminosities, etc., are also listed in Table 1.

Data of Single-epoch Spectral Properties
To investigate how the single-epoch spectral properties control the deviation of AGNs from the b R L H 5100 relationship, we compile eight different parameters from the spectra around the Hβ region in the optical band: (1) the flux ratio between Fe II and Hβ, which is denoted as   Boroson & Green 1992;Brotherton 1996;Du et al. 2018a), (7) the EW of He II (EW He II ), and (8) the EW of Hβ ( b EW H ). These parameters are referred to as "single-epoch spectral properties", because they can be measured simply from the single-epoch spectra rather than from the time-domain observations like RM. 5 In order to establish some new scaling relationships which can be applied to the large AGN samples obtained in the spectroscopic surveys of SDSS or Dark Energy Spectroscopic Instrument (DESI) in the near future, we need to find the correlations between the single-epoch properties and the deviation of AGNs from the  Table 1. We use them (and the average values for the objects with multiple RM measurements) directly in the following analysis. We search the other parameters in the literature and list the values in Table 2. For the parameters that we cannot find in the literature, we fit the spectra of the objects found in the public 4 We also include the new RM observations of the previous mapped objects after 2013 in the following analysis: NGC4593 (Barth et al. 2013), NGC7469 (Peterson et al. 2014), NGC5548 (Lu et al. 2016;Pei et al. 2017), NGC4051 ), PG1226+023 (3C 273, Zhang et al. 2019, and PG2130+099 (C. Hu et al. 2019, in preparation). 5 Of course, if we have the RM data, we can definitely measure them from an individual spectrum in the RM campaign or the mean spectrum (can be treated as the average of the values from the individual spectra). But if we do not have the RM data, we can still measure them from the single-epoch spectra found in other literature or databases.    (This table is available in machine-readable form.)  Table 2).

Fitting the Spectra
We use the following components in the spectral fitting: (1) a power law to model the AGN continuum, (2) two Gaussians to model the broad Hβ emission line, (3) a template constructed from the Fe II spectrum of the narrow-line Seyfert 1 (NLS1) galaxy I Zw 1 by Boroson & Green (1992) for the Fe II emission, (4) one or two Gaussians for each of the narrow emission lines, e.g., [O III]λλ4959,5007, Hβ, He II (if necessary), (5) one or two Gaussians to model the broad He II and (6) a simple stellar population model 6 from Bruzual & Charlot (2003) as a template for the contribution of the host galaxy if necessary. The fitting is mainly performed in the windows of 4170-4260 Å and 4430-5550 Å in the rest frame. If the host contribution is significant, we supplement the window of 6050-6200 Å to give a better constraint to the fitting of the stellar template. All of the narrow-line components in each object are fixed to have the same velocity width and shift, except for those showing very different width/shift. [O III] λ4959 is fixed to have one-third of the [O III]λ5007 flux (Osterbrock & Ferland 2006). NLS1s always show very weak narrow emission lines (in particular, the narrow Hβ). In the fitting for the spectra of NLS1s, we also fix the flux of narrow Hβ to be one-tenth of the [O III]λ5007 flux (Kewley et al. 2006;Stern & Laor 2013). Two examples (SDSS J081456 and Mrk 1310) of the multicomponent spectral fitting are shown in Figure 1. The contribution of the host galaxy in the spectrum of SDSSJ081456 is weak, while Mrk1310 is host-dominated. The fitting of these two objects is fairly good. We measure the spectral properties (see Section 2.2) from the fitting results and list them in Table 2.

Pairwise Correlations between Different Properties
Before discussing the correlations between the single-epoch spectral properties and the deviation from the b R L H 5100 relationship, we first present the pairwise correlations between different properties in Figure 2. Although there are many similar discussions using different samples in historical literature (e.g., Boroson & Green 1992), it is still valuable to do this demonstration for the RM objects. Spearman's rank correlation coefficients (ρ) and the corresponding two-sided p-value for a null hypothesis test (two sets of data are uncorrelated) are marked in the panels of  H is the most significant one, which has Spearman's correlation coefficient ρ=0.70. This means that if the width of the Hβ line is smaller, its profile tends to be more Lorentzian-like. This correlation has been demonstrated by, e.g., . The sample in the present paper is larger than that used in     ) are the prominent correlations in the famous AGN eigenvector 1 sequence (e.g., Boroson & Green 1992;Sulentic et al. 2000;Marziani et al. 2001Marziani et al. , 2003Marziani et al. , 2018Shen & Ho 2014;Sun & Shen 2015). The detailed physical process of this sequence is still under some debate (e.g., Panda et al. 2019a). The correlation between  Fe versus Hβ Asymmetry has been demonstrated using the PG quasar sample in Boroson & Green (1992), and is also associated with the eigenvector 1 sequence. The RM sample reproduces this correlation. Besides, there are some other weak correlations, please see Figure 2. More discussions about the pairwise correlations are provided in Section 4. relationship as

Deviation from the
where R Hβ,R−L is the prediction from the b R L H 5100 relationship. Here, we adopt logR Hβ,R-L =1.53+0.51 log ℓ 44 obtained by Du et al. (2018b) for the AGN with dimensionless accretion rate < 3 M (Ṁ is defined by the following Equation ( FWHM H , see, e.g., Boroson & Green 1992;Sulentic et al. 2000;Marziani et al. 2001Marziani et al. , 2003Shen & Ho 2014;Sun & Shen 2015, or  The eigenvector 1 sequence (or main sequence) of AGNs has been extensively investigated in the past decades, and contains the information of the evolution or systematic variation of AGNs (see the recent review in Marziani et al. 2018). Through the analysis of the eigenvector 1 sequence,  Fe has been demonstrated as a probe of accretion rate/Eddington ratio (e.g., Boroson & Green 1992;Sulentic et al. 2000;Marziani et al. 2001Marziani et al. , 2003Shen & Ho 2014;Sun & Shen 2015), thus a primary physical driver of the shortened time lags is the accretion rate. As a simple test, we provide here the linear regression of the correlation between Δ b R H and  Fe . We adopt the BCES method (Akritas & Bershady 1996, the orthogonal least squares) to perform the linear regression, which takes into account both of the error bars in x-and y-axes. MCG+06-26-012 has a relatively low sampling cadence in the first 80 days in its light curve of Wang et al. (2014a) and Hu et al. (2015), which means its time lag may be biased toward a longer value. We do not use it in the regression. And the intrinsic reddening of MCG+06-30-015 is strong in light of its high Balmer decrement . We correct its intrinsic reddening and use the corrected luminosity and the corresponding R Hβ,R-L . The MCG+06-26-012 and MCG+06-30-015 are marked as gray points in Figure 4 (also in the following Figure 5). The linear regression is yielded as The regression and the corresponding confidence band (2σ) are shown in Figure 4. We have also tested that the residual ) does not show any correlations with all of the spectral properties (with Spearman's coefficients

A New Scaling Relation
Because the strongest correlation is the relation between Δ b R H and  Fe , we can add  Fe as a new parameter into the b R L H 5100 relationship to establish a new scaling relation with smaller scatter. We fit the RM sample with the following new scaling relation: In order to obtain the uncertainties of the parameters, we employ the bootstrap technique. A subset is generated by resampling N points from the RM sample with replacement (N is the number of the objects in the RM sample). Then, we calculate the best parameters for this subset using the Levenberg-Marquardt method (Press et al. 1992), and repeat this procedure 5000 times to generate the distributions of α, β, and γ. The final best parameters and the corresponding uncertainties are obtained from the α, β, and γ distributions. The fit is shown in Figure 5, and the best parameters are:

Some Discussions on Pairwise Correlations
In Section 3.1, we showed the pairwise correlations between the parameters we compiled. Some of the correlations have been presented in the literature using different samples of AGNs, and some have been discussed directly or indirectly. A σ Hβb  H correlation, which is a width-profile correlation of the Hβ line similar to the b FWHM Hb  H in this paper, was presented in Collin et al. (2006) using the RM sample at that time. It was also discussed by  and explained as the different contributions from the rotation/Keplerian motions and the turbulent velocities in the objects with different line widths . The correlation in Figure 2 is generally the same as that in , but has more objects at the narrow-width (small b FWHM H ) end because the current sample has more NLS1s or high-accretion-rate objects. However, it should be noted that the parameter   The comparison between FWHM Fe II and b FWHM H has been presented for the quasar sample in the Sloan Digital Sky Survey (SDSS) in, e.g., Hu et al. (2008aHu et al. ( , 2008b and Cracco et al. (2016). The FWHM Fe II is systematically smaller than b FWHM H , which was demonstrated and explained by the contribution from a intermediate-line region in Hu et al. (2008aHu et al. ( , 2008b (Hu et al. 2008a(Hu et al. , 2008b. In addition, Hu et al. (2015) shows a comparison between the time lags of Fe II and Hβ using the SEAMBH sample, which is also direct evidence for the relatively larger size of the Fe II-emitting region and the smaller Hβ-emitting region. And Hu et al. (2015) also shows that the lag ratio between Fe II and Hβ correlates with  Fe . The  Figure 2) because of the  Feb FWHM H correlation (Eigenvector 1 sequence, e.g., Boroson & Green 1992;Sulentic et al. 2000;Marziani et al. 2001Marziani et al. , 2003Marziani et al. , 2018Shen & Ho 2014).
The principal component analysis in Boroson & Green (1992) has shown a weak correlation between  Fe and b EW H using the PG quasar sample, however, with a relatively small correlation coefficient of −0.425 (see Boroson & Green 1992 for more details). The  Fe and b EW H of the RM sample presented in this paper show a slightly stronger correlation with Spearman's coefficient of ρ=−0.60. This correlation may be related to the Baldwin effect of the Hβ line (e.g., Baldwin 1977;Korista & Goad 2004), and especially with the intrinsic Baldwin effect (e.g., Gilbert & Peterson 2003;Rakić et al. 2017). The intrinsic Baldwin effect shows an anticorrelation between EW of the emission line and the luminosity, and is also equivalent to an anticorrelation between EW and the accretion rate because the BH mass keeps a constant during the observation campaign (e.g., Gilbert & Peterson 2003;Rakić et al. 2017). The  Fe parameter is correlated with Eddington ratio/accretion rate, thus is naturally correlated with the b EW H . The asymmetry- Fe correlation has been shown in Boroson & Green (1992), and discussed in the context of the eigenvector 1 sequence (Sulentic et al. 2002). The high- Fe objects tend to have stronger blue Hβ wings, and vice versa. The FWHM Fe II / b FWHM H -asymmetry and b FWHM H -asymmetry correlations can also attribute to the asymmetry- Fe correlation. The origin of the Hβasymmetry must be subject to the geometry and kinematics of the BLRs, but is still under some debate because of the degeneracy of Hβ profiles with different BLR geometry and kinematics. A recent dedicated RM campaign project for the BLR kinematics of the AGNs with Hβ asymmetry 7 has started (Du et al. 2018a), and may provide more observations for the velocity-resolved RM measurements in the future.
The  II, respectively), and thus are sensitive to the variation of the AGN circumstances (e.g., spectral energy distribution, SED) in a similar way.

The New Scaling Relation and BH Mass Measurement
Through the analysis in this paper, we found that the  Fe parameter is the dominant observational property in the scatter of the b R L H 5100 relationship. This confirms the statement that the AGNs with high-accretion rates tend to have shortened lags in Du et al. (2015Du et al. ( , 2016aDu et al. ( , 2018b. The shortened time lags in high- Fe /high-accretion-rate AGNs imply smaller BLR scale sizes and smaller BH mass estimates with respect to the relationship was heavily utilized as a singleepoch BH mass estimator in large AGN samples, and helped establish our paradigm for AGN evolution (McLure & Dunlop 2004;Kollmeier et al. 2006;Vestergaard & Peterson 2006;Greene & Ho 2007;Shen et al. 2011). The shortened time lags in high- Fe /high-accretion-rate AGNs make it vital to take this into account in the BH mass estimation. Therefore, we suggest calculating the BH masses from single-epoch spectra following the steps below: (1) obtain the b R H from the luminosities and the strength of Fe II ( Fe ) using Equation (5), (2) get the BH mass by the following Equation (7) from the line width and the estimated b R H . Then, the dimensionless accretion rate can be easily estimated by the following Equation (8) in Section 4.3.
In combination with the velocity width (ΔV ) of the emission line, RM measurement yields an estimate of BH mass as Wandel et al. 1999;Peterson et al. 2004), where G is the gravitational constant, and f BLR is the virial factor related to the geometry and kinematics of the BLR (e.g., Onken et al. 2004;Park et al. 2012;Grier et al. 2013;Ho & Kim 2014;Woo et al. 2015). Although measuring BH mass through RM technique is feasible for a small number of objects, it is not easy to apply RM to large AGN samples because RM is fairly time-consuming (always continuing for months to years). Some multiobject RM campaigns based on fiber-fed telescopes, e.g., the RM campaigns of SDSS  and OzDES (King et al. 2015), are committed to enlarging the sample of RM objects, but they are still ongoing. Fortunately, the b R L H 5100 -relationship can be used to obtain R BLR from the single-epoch spectra very simply.
The geometry and kinematics of the BLRs determine the virial factor f BLR in BH mass estimate (in Equation (7)). Comparing the RM AGNs with stellar velocity dispersion measurements with M • -σ * relation of inactive galaxies gives f BLR ∼1 if the velocity width of Hβ is measured from b FWHM H (e.g., Onken et al. 2004;Ho & Kim 2014;Woo et al. 2015). The virial factor for high- Fe /high-accretion-rate AGNs is still a matter of some debate.  Williams et al. (2018), the virial factor is roughly consistent with the value derived from the M • -σ * relation, and the virial factors of individual objects do no show significant correlation with the Eddington ratios (or show potential and weak anticorrelation, namely a smaller virial factor for higher Eddington ratio). NLS1s (thought to have smaller BH masses and higher accretion rates) tend to host pseudobulges (e.g., Mathur et al. 2012). Ho & Kim (2014) classified the AGN sample to classical bulges/ellipticals and pseudobulges, and derived a virial factor of the AGNs with pseudobulges smaller than 1. Woo et al. (2015) found that NLS1s have no significant differences from the other AGNs, and derived f BLR =1.12 if using b FWHM H as the line width measurement. Thus, adopting f BLR ∼1 and acknowledging that its large uncertainty is acceptable at present.
As a simple test, we compare the BH masses measured by RM (M •,RM ) with the masses estimated from the single-epoch spectra (M •,SE ) using both of the new scaling relation in Section 3.3 and the traditional b R L H 5100 relationship (see Section 3.2) in Figure 6. Here, we adopt f BLR =1 for simplicity and list the RM BH masses in Table 1 relationship, which means the BH masses are overestimated if using this simple relationship.
Recently, Martínez-Aldama et al. (2019) found a correction for the time delay based on the dimensionless accretion rate (Ṁ in the following Section 4.3) considering the anticorrelation between f BLR and line width, established a correlation between the corrected time lag (R BLR corr ) and L 5100 , and discussed the measurements of the cosmological distances using this correlation. Their correction (R BLR corr ) relies on the measured L 5100 , b FWHM H , and R BLR itself. The new scaling relation in the present paper is established from a different perspective (based on spectral properties), and can deduce R BLR directly from L 5100 and  Fe .

Accretion Rate and Eddington Ratio
Because of the shortened lags, the accretion rates or Eddington ratios would be underestimated by factors of a few if using the traditional b R L H 5100 relationship. From the standard disk model (Shakura & Sunyaev 1973), the accretion rate can be estimated by the formula of  Table 1) in Figure 7, where L Bol is the bolometric luminosity and L Edd =1.5×10 38 (M • /M e ) is the Eddington luminosity for the gas with solar composition. Here we simply adopt L Bol =10 L 5100 (Kaspi et al. 2000), but it should be noted that the bolometric correction factor depends on accretion rate or BH mass (Jin et al. 2012). We do not draw the error bars in order to show the differences at the high-Ṁ end more clearly. It is obvious that the points of the new scaling relation at the high-accretion-rate end are much closer to the diagonal, while those estimated by the traditional  H ), which can be used to estimate the accretion rate directly from the BLR properties. It is called the fundamental plane (FP) of BLR. The FP can deduceṀ or Eddington ratio estimates without any luminosity measurements (see also Negrete et al. 2018); however, it has fairly large uncertainties (the scatter of the FP is 0.7 forṀ and 0.48 for Eddington ratio, respectively, see more details in Du et al. 2016b). As a comparison, we plot theṀ estimates from the FP and the new scaling relation in Figure 8. The single-epochṀ and Eddington ratios corresponding to the FP method are estimated from the  Fe and b  H listed in Table 2 and the FP in Du et al. (2016b). It should be noted that we switch the x-and y-axes of Figure 8 (with respect to Figure 7) for easier comparison with the figures in Du et al. (2016b). Again, we do not draw the error bars of theṀ from the new scaling relation for clarity. The scatters of theṀ and L Bol /L Edd estimated from the FP are much larger. The FP connect the BLR physics with the accretion status of AGNs; however, the different temperature, number density, metallicity of the BLR in different AGN introduces large uncertainties. The FP (Du et al. 2016b) is a relationship are biased toward to higher values with respect to those from the new scaling relation. The standard deviations (simply denoted by σ) of the distributions are provided in the upper-left corner. We do not plot the error bars in order to show the differences more clearly. good beginning that searches for a direct indicator of accretion rate from the BLR observational properties, but its scatter and accuracy should be improved by including more observational properties in the future.
The strong correlation between accretion ratesṀ and  Fe has been explored by Panda et al. (2018Panda et al. ( , 2019aPanda et al. ( , 2019b   asymmetric dynamics (Begelman & Shlosman 2009) will have different dependence on metallicity; however,  Fe is not a unique function of metallicity (Baldwin et al. 2004;Verner et al. 2004). More details of photoionization are necessary to investigate  Fe dependence on BLR clouds (density, temperature, and metallicity) and the SEDs of accretion disks.

Accretion Rate or Orientation?
The eigenvector 1 sequence can help break the degeneracy of accretion rate and orientation. It was demonstrated that the accretion rate or Eddington ratio drives the variation of  Fe , while the orientation effect dominantly controls the dispersion in b FWHM H at fixed  Fe (e.g., Marziani et al. 2001;Shen & Ho 2014). We plot the eigenvector 1 sequence of the RM sample color-coded by Δ b R H in Figure 9. It is obvious that the objects with the most significant lag deviations are located in the lower right corner (with the strongest  Fe ). Furthermore, there is no significant trend in the b FWHM H -axis at fixed  Fe . This means that the accretion rate definitely plays the primary role in the shortening of the time lags but the orientation does not contribute much. In Figure 4, the dispersion of the Δ b R H at lower b FWHM H is larger, this is caused by the higher  Fe in those objects, which is clearly shown in Figure 9.

Summary
In this paper, we systematically investigate the dependence of the -relationship correlates with those properties and to determine the origin of the shortened lags.
1. The flux ratio between Fe II and Hβ lines ( Fe ) is confirmed to be the most prominent property that correlates with the deviation of an AGN from the b R L H 5100 relationship.  Fe is thought to be the indicator of accretion rate; therefore, accretion rate is the driver for the shortened lags.   (Marziani et al. 2001;Shen & Ho 2014). It is obvious that accretion rate definitely plays the primary role in the shortening of the time lags (see more details in Section 4.4).