Supermassive Black Holes with High Accretion Rates in Active Galactic Nuclei. V.A New Size-Luminosity Scaling Relation for the Broad-Line Region

# Supermassive Black Holes with High Accretion Rates in Active Galactic Nuclei. V. A New Size-Luminosity Scaling Relation for the Broad-Line Region

Xu-Liang Fan66affiliation: Yunnan Observatories, Chinese Academy of Sciences, Kunming 650011, China , Xiang-Er Fang99affiliation: Department of Astronomy, University of Science and Technology of China, Hefei 230026, China , Jin-Ming Bai66affiliation: Yunnan Observatories, Chinese Academy of Sciences, Kunming 650011, China , Wei-Hao Bian88affiliation: Physics Department, Nanjing Normal University, Nanjing 210097, China , Ye-Fei Yuan99affiliation: Department of Astronomy, University of Science and Technology of China, Hefei 230026, China ,
Luis C. Ho44affiliation: Kavli Institute for Astronomy and Astrophysics, Peking University, Beijing 100871, China 55affiliation: Department of Astronomy, School of Physics, Peking University, Beijing 100871, China and Jian-Min Wang11affiliation: Key Laboratory for Particle Astrophysics, Institute of High Energy Physics, Chinese Academy of Sciences, 19B Yuquan Road, Beijing 100049, China 33affiliation: National Astronomical Observatories of China, Chinese Academy of Sciences, 20A Datun Road, Beijing 100020, China **affiliation: Corresponding author; wangjm@ihep.ac.cn
(SEAMBH collaboration)
###### Abstract

This paper reports results of the third-year campaign of monitoring super-Eddington accreting massive black holes (SEAMBHs) in active galactic nuclei (AGNs) between . Ten new targets were selected from quasar sample of Sloan Digital Sky Survey (SDSS), which are generally more luminous than the SEAMBH candidates in last two years. H lags () in five of the 10 quasars have been successfully measured in this monitoring season. We find that the lags are generally shorter, by large factors, than those of objects with same optical luminosity, in light of the well-known relation. The five quasars have dimensionless accretion rates of . Combining measurements of the previous SEAMBHs, we find that the reduction of H lags tightly depends on accretion rates, , where is the H lag from the normal relation. Fitting 63 mapped AGNs, we present a new scaling relation for the broad-line region: , where is 5100 Å  continuum luminosity, and coefficients of  lt-d, , and . This relation is applicable to AGNs over a wide range of accretion rates, from to . Implications of this new relation are briefly discussed.

black holes: accretion – galaxies: active – galaxies: nuclei
journal: To appear in The Astrophysical Journal.slugcomment: Received 2015 December 13; accepted 2016 April 20

## 1. Introduction

This is the fifth paper of the series reporting the ongoing large campaign of monitoring Super-Eddington Accreting Massive Black Holes (SEAMBHs) in active galaxies and quasars starting from October 2012. One of the major goals of the campaign is to search for massive black holes with extreme accretion rates through reverberation mapping (RM) of broad emission lines and continuum. Results from the campaigns in 2012–2013 and 2013–2014 have been reported by Du et al. (2014, 2015, hereafter Papers I and IV), Wang et al. (2014, Paper II) and Hu et al. (2015, Paper III). This paper carries out the results of SEAMBH2014 sample, which was monitored from September 2014 to June 2015. With the three monitoring years of observations, we build up a new scaling relation of the broad-line region (BLR) in this paper.

Reverberation mapping (RM) technique, measuring the delayed echoes of broad lines to the varying ionizing continuum (Bahcall et al. 1972; Blandford & McKee 1982; Peterson 1993), is a powerful tool to probe the kinematics and geometry of the BLRs in the time domain. Countless clouds, which contribute to the smooth profiles of the broad emission lines (e.g., Arav et al. 1997), are distributed in the vicinity of supermassive black hole (SMBH), composing the BLR. As an observational consequence of photonionization powered by the accretion disk under the deep gravitational potential of the SMBH, the profiles of the lines are broadened, and line emission from the clouds reverberate in response to the varying ionizing continuum. The reverberation is delayed because of light travel difference between H and ionizing photons and is thus expected to deliver information on the kinematics and structure of the BLR. The unambiguous reverberation of the lines, detected by monitoring campaigns from ultraviolet to optical bands since the late 1980s, supports this picture of the central engine of AGNs (e.g., Clavel et al. 1991; Peterson et al. 1991, 1993; Maoz et al. 1991; Wanders et al. 1993; Dietrich et al. 1993, 1998, 2012; Kaspi et al. 2000; Denney et al. 2006, 2010; Bentz et al. 2009, 2014; Grier et al. 2012; Papers I-IV; Barth et al. 2013, 2015; Shen et al. 2015a,b). The relation was first discussed by Koratkar & Gaskell (1991) and Peterson (1993). Robust RM results for 41 AGNs in the last four decades lead to a simple, highly significant correlation of the form

 RHβ≈α0ℓβ044, (1)

where is the 5100 Å  luminosity in units of (corrected for host galaxy contamination) and is the emissivity-weighted radius of the BLR (Kaspi et al. 2000; Bentz et al. 2013). We refer to this type of correlation as the normal relationship. The constants and differ slightly from one study to the next, depending on the number of sources and their exact luminosity range (e.g., Kilerci Eser et al. 2015). For sub-Eddington accreting AGNs, ltd and , but for SEAMBHs they are different (see Paper IV).

As reported in Paper IV, some objects from the SEAMBH2012 and SEAMBH2013 samples have much shorter H lags compared with objects with similar luminosity, and the relation has a large scatter if they are included. In particular, the reduction of the lags increases with the dimensionless accretion rate, defined as , where is the accretion rate, is the Eddington luminosity and is the speed of light. Furthermore, it has been found, so far in the present campaigns, that SEAMBHs have a range of accretion rates from a few to . This kind of shortened H lags was discovered in the current SEAMBH project (a comparison with previous campaigns is given in Section 6.5). Such high accretion rates are characteristic of the regime of slim accretion disks (Abramowicz et al. 1988; Szuszkiewicz et al. 1996; Wang & Zhou 1999; Wang et al. 1999; Mineshige et al. 2000; Wang & Netzer 2003; Sadowski 2009). These interesting properties needed to be confirmed with observations. We aim to explore whether we can define a new scaling relation, , which links the size of the BLR to both the AGN luminosity and accretion rate.

We report new results from SEAMBH2014. We describe target selection, observation details and data reduction in §2. H lags, BH mass and accretion rates are provided in §3. Properties of H lags are discussed in §4, and a new scaling relation of H lags is established in §5. Section 6 introduces the fundamental plane, which is used to estimate accretion rates from single-epoch spectra, for application of the new size-luminosity scaling relation of the BLR. Brief discussions of the shortened lags are presented in §7. We draw conclusions in §8. Throughout this work we assume a standard CDM cosmology with , and (Ade et al. 2014).

## 2. Observations and data reduction

### 2.1. Target Selection

We followed the procedures for selecting SEAMBH candidates described in Paper IV. We used the fitting procedures to measure H profile and 5100 Å  luminosity of SDSS quasar spectra described by Hu et al. (2008a,b). Following the standard assumption that the BLR gas is virialized, we estimate the BH mass as

 M∙=fBLRRHβV2FWHMG=1.95×106 fBLRV23τ10 M⊙, (2)

where , is the H lag measured in the rest frame, days, is the gravitational constant, and is the full-width-half-maximum (FWHM) of the H line profile in units of . We take the virial factor in our series of papers (see some discussions in Paper IV).

In order to select AGNs with high accretion rates, we employed the formulation of accretion rates derived from the standard disk model of Shakura & Sunyaev (1973). In the standard model it is assumed that the disk gas is rotating with Keplerian angular momentum, and thermal equilibrium is localized between viscous dissipation and blackbody cooling. Observationally, this model is supported from fits of the so-called big blue bump in quasars (Czerny & Elvis 1987; Wandel & Petrosian 1988; Sun & Malkan 1989; Laor & Netzer 1989; Collin et al. 2002; Brocksopp et al. 2006; Kishimoto et al. 2008; Davis & Laor 2011; Capellupo et al. 2015). The dimensionless accretion rate is given by

 ˙M=20.1(ℓ44cosi)3/2m−27, (3)

where (see Papers II and IV) and is the inclination angle to the line of sight of the disk. We take , which represents a mean disk inclination for a type 1 AGNs with a torus covering factor of about 0.5 (it is assumed that the torus axis is co-aligned with the disk axis). Previous studies estimate [e.g., Fischer et al. (2014) find a inclination range of , whereas Pancoast et al. (2014) quote ; see also supplementary materials in Shen & Ho (2014)], which results in from Equation (3). This uncertainty is significantly smaller than the average uncertainty on ( dex) in the present paper, and is thus ignored. Equation (3) applies to AGNs that have accretion rates , namely excluding the regimes of advection-dominated accretion flows (ADAF; Narayan & Yi 1994) and of flows with hyperaccretion rates (; see Appendix A for the validity of Equation 3 for SEAMBHs).

Using the normal relation (Bentz et al. 2013), we fitted all the quasar spectra in SDSS Data Release 7 by the procedures in Hu et al. (2008a, b) and applied Equations (2) and (3) to select high targets. We ranked quasars in terms of and chose ones as candidates with the highest . We found that the high quasars are characterized by 1) strong optical Fe ii lines; 2) relatively narrow H lines (); 3) weak [O iii] lines; and 4) steep 2–10 keV spectra (Wang et al. 2004). These properties are similar to those of NLS1s (Osterbrock & Pogge 1987; Boroson & Green 1992), but most of the candidates have more extreme accretion rates (a detailed comparison of SEAMBH properties with normal quasars will be carried out in a separate paper). Considering that the lags of all targets should be measured within one observing season, and taking into consideration the limitations of the weather of the Lijiang Station of Yunnan Observatory (periods between June and September are raining seasons there), we only chose objects with maximum estimated lags of about 100 days or so (the monitoring periods should be at least a few times the presumed lags). Also, to ensure adequate signal-to-noise ratio (S/N) for measurements of light curves, we restricted the targets to a redshift range of and magnitudes . The fraction of radio-loud objects with is not high. We discarded radio-loud objects111It has been realised that high-accretion rate AGNs are usually radio-quiet (Greene & Ho 2006), although there are a few NLS1s reported to be radio-loud. The fraction of radio-loud AGNs decreases with increasing accretion rate (Ho 2002, 2008). based on available FIRST observations, in order to avoid H reverberations potentially affected by nonthermal emission from relativistic jets, or optical continuum emission strongly contaminated by jets. We chose about 20 targets for photometry monitoring, which served as a preselection to trigger follow-up spectroscopic monitoring. The photometric monitoring yielded 10 targets with significant variations ( magnitudes), and time lags were successfully measured for 5 objects (Table 1). For an overview of our entire ongoing campaign, Table 1 also lists samples from SEAMBH2012 and SEAMBH2013.

To summarize: we have selected about 30 targets for spectroscopic monitoring during the last three years (2012–2014). The successful rate of the monitoring project is about 2/3. Our failure to detect a lag for the remaining 1/3 of the sample are either due to low-amplitude variability or bad weather that leads to poor monitoring cadence. In particular, the SEAMBH2014 observations were seriously affected by the El Niño phenomenon.

### 2.2. Photometry and Spectroscopy

The SEAMBH project uses the Lijiang 2.4m telescope, which has an alt-azimuth Ritchey-Chrétien mount with a field de-rotator that enables two objects to be positioned along the same long slit. It is located in Lijiang and is operated by Yunnan Observatories. We adopted the same observational procedures described in detail in Paper I, which also introduces the telescope and spectrograph. We employed the Yunnan Faint Object Spectrograph and Camera (YFOSC), which has a back-illuminated 20484608 pixel CCD covering a field of . During the spectroscopic observation, we put the target and a nearby comparison star into the slit simultaneously, which can provide high-precision flux calibration. As in SEAMBH2013 (Paper IV), we adopted a -wide slit to minimize the influence of atmospheric differential refraction, and used Grism 3 with a spectral resolution of 2.9 Å/pixel and wavelength coverage of 3800–9000 Å. To check the accuracy of spectroscopic calibration, we performed differential photometry of the targets using some other stars in the same field. We used an SDSS -band filter for photometry to avoid the potential contamination by emission lines such as H and H. Photometric and spectroscopic exposure times are typically 10 and 60 min, respectively.

The reduction of the photometry data was done in a standard way using IRAF routines. Photometric light curves were produced by comparing the instrumental magnitudes to those of standard stars in the field (see, e.g., Netzer et al. 1996, for details). The radius for the aperture photometry is typically (seeing ), and background is determined from an annulus with radius to . The uncertainties on the photometric measurements include the fluctuations due to photon statistics and the scatter in the measurement of the stars used.

The spectroscopic data were also reduced with IRAF. The extraction width is fixed to , and the sky regions are set to on both sides of the extracted region. The average S/N of the 5100 Å continuum of individual spectra are from 16 to 22, except for J085946, which only has S/N 12. The flux of spectroscopic data was calibrated by simultaneously observing a nearby comparsion star along the slit (see Paper I). The fiducial spectra of the comparison stars are generated using observations from several nights with the best weather conditions. The absolute fluxes of the fiducial spectra are calibrated using additional spectrophotometric standard stars observed in those nights. Then, the in-slit comparison stars are used as standards to calibrate the spectra of targets observed in each night. The sensitivity as a function of wavelength is produced by comparing the observed spectrum of the comparison star to its fiducial spectrum. Finally, the sensitivity function is applied to calibrate the observed AGN spectrum222The uncertainty of our absolute flux calibration is 10%. We multiply the fiducial spectra of the in-slit stars with the bandpass of the SDSS r filter and compare their synthesized magnitudes with the magnitudes found in the SDSS database. The maximum difference is 10%. . The procedures adopted here resemble the method used by Maoz et al. (1990) and Kaspi et al. (2000). In order to illustrate the invariance of the comparison stars, we show the light curves from the differential photometry of the comparison stars in Appendix B. It is clear that their fluxes are very stable and the variations are less than 1%.

The calibration method of van Groningen & Wanders (1992), based on the [O iii] emission line and popularily used in many RM campaigns (e.g., Peterson et al. 1998; Bentz et al. 2009; Denney et al. 2010; Grier et al. 2012; Barth et al. 2015), is not suitable for SEAMBHs. [O iii] tends to be weak in SEAMBHs (especially for the objects in SEAMBH2013-2014), and, even worse, is blended with strong Fe ii around 5016 Å. Applying the calibration method to SEAMBHs results in large statistical (caused by the weakness of [O iii]) and systematic (caused by the variability of Fe ii; see Paper III) uncertainties. The method based on in-slit comparison stars, used in our campaign, does not rely on [O iii] and provides accurate flux calibration for the spectra of SEAMBHs. For comparison, in Paper I we measured the [O iii] fluxes in the calibrated spectra of three objects in SEAMBH2012 with relatively strong [O iii]; the variation of their [O iii] flux is on the order of 3%. This clearly demonstrates the robustness of our flux calibration method based on in-slit comparison stars.

The procedures to measure the 5100 Å and H flux are nearly the same as those given in Paper I. The continuum beneath H line is determined by interpolation of two nearby bands (4740–4790 Å and 5075 -5125 Å) in the rest frame. These two bands have minimal contamination from emission lines. The flux of H is measured by integrating the band between 4810 and 4910 Å after subtraction of the continuum; the H band is chosen to avoid the influence from Fe ii lines. The 5100 Å flux is taken to be the median over the region 5075-5125 Å. Detailed information of the observations is provided in Table 1. All the photometry and continuum and H light curves for the five objects with successfully detected lags are listed in Tables and shown in Figure 1. We also calculated the mean and RMS (root mean square) spectra and present them in Appendix C.

### 2.3. Host Galaxies

Like the SEAMBH2013 sample, we have no observations that can clearly separate the host galaxies of the AGNs in SEAMBH2014. Shen et al. (2011) propose the following empirical relation to estimate the fractional contribution of the host galaxy to the optical continuum emission: , for , where and is the total emission from the AGN and its host at 5100 Å. For , , and the host contamination can be neglected. The host fractions at 5100 Å for the objects (J075949, J080131, J084533, J085946 and J102339) are (27.5%, 37.1%, 14.2%, 19.0% and 31.8%). The values of listed in Table 5 are the host-subtracted luminosities. We note that this empirical relation is based on SDSS spectroscopic observations with a fiber, whereas we used a -wide slit. It should apply to our observations reasonably well (see Paper IV for additional discussions on this issue). We will revisit this issue in the future using high-resolution images that can more reliably separate the host.

## 3. Measurements of Hβ Lags, Black Hole Masses and Accretion Rates

### 3.1. Lags

As in Papers I–IV, we used cross-correlation analysis to determine H lags relative to photometric or 5100 Å  continuum light curves. We use the centroid lag for H. The uncertainties on the lags are determined through the “flux randomization/random subset sampling” method (RS/RSS; Peterson et al. 1998, 2004). The cross-correlation centroid distribution (CCCD, described in Appendix E) and cross-correlation peak distribution (CCPD) generated by the FR/RSS method (Maoz & Netzer 1989; Peterson et al. 1998, 2004; Denney et al. 2006, 2010, and references therein) are shown in Figure 1. We used the following criteria to define a successful detection of H lag: 1) non-zero lag from the CCF peak and 2) a maximum correlation coefficient larger than 0.5. Data for the light curves of the targets are given in Tables 2–4. All the measurements of the SEAMBH2014 sample are provided in Table 5.

The -band light curves are generally consistent with the 5100 Å  continuum light curve, but the former usually have small scatter, as shown in Figure 1. We calculated CCFs for the H light curves with both -band photometry and with 5100 Å  spectral continuum for all objects. The quality of the H CCFs is usually better than the H CCFs. We show the H CCFs for all objects in Figure 1, except for J080131. We use the H lags in the following analysis. For J080131, the -band light curve between 200 and 220 days does not match the 5100 Å  continuum light curve, even though H does follow 5100 Å  continuum tightly. Notes to individual sources are given in Appendix D.

### 3.2. Black Hole Masses and Accretion Rates

There are two ways of calculating BH mass, base either on the RMS spectrum (e.g., Peterson et al. 2004; Bentz et al. 2009; Denney et al. 2010; Grier et al. 2012) or on the mean spectrum (e.g., Kaspi et al. 2005; Papers I–IV). Different studies also adopt different measures of the line width, typically either the line dispersion (second moment of the line profile) or the FWHM. In this study, we choose to parameterize the line width using FWHM, as measured in the mean spectra. The narrow H component may influence the measurement of FWHM. We adopt the same procedure as in Paper I to remove the narrow H. We fix narrow H/[O iii] to 0.1, and measure FWHM from the mean spectra with narrow H subtracted. Then we set H/[O iii] to 0 and 0.2 and repeat the process to obtain lower and upper limits to FWHM. The relatively wide slit employed in our campaign () significantly broadens the emission lines by , where is the instrumental broadening that can be estimated from the broadening of selected comparison stars. As in Paper IV, we obtain the intrinsic width of the mean spectra from . The FWHM simply obtained here is accurate enough for BH mass estimation. Our procedure for BH mass estimation is based on FWHM measured from the mean spectrum (see explanation in Papers I and II). As shown recently in Woo et al. (2015), the scatter in the scaling parameter () derived in this method is very similar to the scatter in the method based on the RMS spectrum. We use Equations (2) and (3) to calculate accretion rates and BH masses for the five sources listed in Table 5. For convenience and completeness, Table 5 also lists H lags, BH masses and accretion rates for the sources from SEAMBH2012 and SEAMBH2013. Our campaign has successfully detected H lags for 18 SEAMBHs since October 2012.

As described in Paper II, there are some theoretical uncertainties in identifying a critical value of to define a SEAMBH (Laor & Netzer 1989; Beloborodov 1998; Sadowski et al. 2011). Following Paper II, we classified SEAMBHs as those objects with . This is based on the idea that beyond this value, the accretion disk becomes slim and the radiation efficiency is reduced mainly due to photon trapping (Sadowski et al. 2011). Since we currently cannot observe the entire spectral energy distribution, we have no direct way to measure , and this criterion is used as an approximate tool to identify SEAMBH candidates. To be on the conservative side, we chose the lowest possible efficiency, (retrograde disk with ; see Bardeen et al. 1972). Thus, SEAMBHs are objects with . For simplicity, in this paper we use as the required minimum (Papers II and IV). We refer to AGNs with as SEAMBHs and those with as sub-Eddington ones. Paper IV clearly shows that the properties of the relation for and are significantly different.

Figure 2 plots distributions of , EW(H), and of all the mapped AGNs (41 from Bentz et al. 2013 and the 18 SEAMBHs from our campaign; see Table 7 in Paper IV and Table 5 here). As shown clearly in the diagrams, SEAMBH targets are generally more luminous by a factor of 2–3 compared to previous RM AGNs (Figure 2a). The BH masses of SEAMBHs are generally less smaller by a factor of 10 compared to previous samples, whereas, as a consequence of our selection, the accretion rates of SEAMBHs are higher by 2–3 orders of magnitude (Figures 2c and 2d). However, EW(H) of SEAMBHs are not significantly smaller (Figure 2d). On average, the high sources have lower mean EW(H), consistent with the inverse correlation between EW(H) and (e.g., Netzer et al. 2004).

## 4. Properties of Hβ Lags in SEAMBHs

The correlation was originally presented by Peterson (1993; his Figure 10, only nine objects). It was confirmed by Kaspi et al. (2000) using a sample of 17 low-redshift quasars. Bentz et al. (2013) refined the relation through subtraction of host contamination and found that its intrinsic scatter is only 0.13 dex. Paper IV (Table 7) provides a complete list of previously mapped AGNs, based on Bentz et al. (2013); we directly use these values333NGC 7469 was mapped twice by Collier et al. (1998) and Peterson et al. (2014). While their H lags are consistent, the FWHM of H is very different. We only retain the later observation in the analysis.. As in Paper IV, for objects with multiple measurements of H lags, we obtain the BH mass from each campaign and then calculate the average BH mass. Using the averaged BH mass, we apply it to get accretion rates of the BHs during each monitoring epoch, which are further averaged to obtain the mean accretion rates of those objects (Kaspi et al. 2005; Bentz et al. 2013). We call this the “average scheme.” On the other hand, we may consider each individual measurement of a single object as different objects (e.g., Bentz et al. 2013). We called this the “direct scheme.” Although the two approaches are in principle different, we obtain very similar results (see a comparison in Paper IV).

All correlations of two parameters shown in this paper are calculated with the FITEXY method, using the version adopted by Tremaine et al. (2002), which allows for intrinsic scatter by increasing the uncertainties in small steps until reaches unity (this is typical for many of our correlations). We also emplot the BCES method (Akristas & Bershady 1996) but prefer not to use its results because it is known to give unreliable results in samples containing outliers (there are a few objects with quite large uncertainties of ).

### 4.1. The RHβ−L5100 relation

As shown in Paper IV, the H lags of the SEAMBH2013 sample were found to significantly deviate from the normal relation, by a factor of a few. We plot the relation of all samples in Figure 3. For sub-Eddington AGNs () in the direct scheme, with an intrinsic scatter of (see Paper IV). Using FITEXY, we have

 log(RHβ/ltd)={(1.30±0.05)+(0.53±0.06)logℓ44(˙M≥3),(1.44±0.03)+(0.49±0.03)logℓ44(for all ˙M), (4)

with intrinsic scatters of . Clearly, the intrinsic scatter of SEAMBHs is much larger than the sample of sub-Eddington AGNs. In the averaged scheme, we have   for sub-Eddington AGNs, with an intrinsic scatter of (see Paper IV), and

 log(RHβ/ltd)={(1.32±0.05)+(0.52±0.06)logℓ44(˙M≥3),(1.44±0.03)+(0.49±0.03)logℓ44(for all ˙M), (5)

with intrinsic scatters of . The slope of the correlation for the SEAMBH sample is comparable to that of sub-Eddington AGNs, but the normalization is significantly different. It is clear that the SEAMBH sources increase the scatter considerably, especially over the limited luminosity range occupied by the new sources.

As in Paper IV, we also tested the correlation between H lag and H luminosity, namely, the relation. The scatter of the correlation is not smaller than that of the correlation, and we do not consider it further.

### 4.2. ˙M−dependent BLR Size

To test the dependence of the BLR size on accretion rate, we define a new parameter, , that specifies the deviation of individual objects from the relation of the subsample of sources (i.e., as given by Equations 4b and 5b for AGNs in Paper IV). The scatter of is calculated by , where is the number of objects and is the averaged value. Figure 3 provides plots for comparison.

Figure 4 shows versus , as well as distributions for the and subsamples in the direct (panels a and b) and averaged (panels c and d) schemes. A Kolmogorov-Smirnov (KS) test shows that the probability that the two subsamples are drawn from the same parent distributions is for the direct scheme and for the averaged scheme. This provides a strong indication that the main cause of deviation from the normal relation is the extreme accretion rate. Thus, a single relation for all AGNs is a poor approximation for a more complex situation in which both the luminosity and the accretion rate determine . From the regression for AGNs, we obtain the dependence of the deviations of from the relation in Figure 4:

 ΔRHβ={(0.39±0.09)−(0.47±0.06)log˙M(direct scheme),(0.34±0.09)−(0.42±0.07)log˙M(averaged scheme), (6)

with , respectively. We have tested the above correlations also for . The FITEXY regressions give slopes near 0, with very large uncertainties: and for Figure 4a and 4c, respectively, implying that does not correlate with for the group. All this confirms that is an additional parameter that controls the relation in AGNs with high accretion rates.

## 5. A New Scaling Relation for the BLR

We provide evidence H lags depend on luminosity and accretion rate. There are a total of 28 SEAMBHs (including those discovered in other studies). We now have an opportunity to define a new scaling relation for the BLR, one that properly captures the behavior of sub-Eddington and super-Eddington AGNs. Considering the dependence of (Equation 6), a unified form of the new scaling law can take the form444We have tried , which is continuous for the transition from sub- to super-Eddington sources. The fitting also yields a very rapid transition at , with and (the present sample is still dominated by sub-Eddington AGNs, with a ratio of 35/63). We prefer the form given by Equation (7).

 RHβ=α1ℓβ144min⎡⎣1,(˙M˙Mc)−γ1⎤⎦, (7)

where is to be determined by data. Equation (7) reduces to the normal relation for sub-Eddington AGNs and to for AGNs with . There are four parameters to describe the new scaling relation, but only two ( and ) are new due to the inclusion of accretion rates; the other two are mainly determined by sub-Eddington AGNs. The critical value of , which is different from the criterion of SEAMBHs, depends on the sample of SEAMBHs.

In order to determine the four parameters simultaneously, we define

 χ2=1NN∑i=1(RHβ−RiHβ)2(ΔiRHβ)2, (8)

where is the error bar of . Minimizing among all the mapped AGNs and employing a bootstrap method, we have

 α1=29.6+2.7−2.8;   β1=0.56+0.03−0.03;   γ1=0.52+0.33−0.16;   ˙Mc=11.19+2.29−6.22. (9)

This new empirical relation has a scatter of , smaller than the scatter (0.26) of the normal relation for all the mapped AGNs. The new scaling relation is plotted in Figure 5.

Equation (7) shows the dependence of the BLR size on accretion rates, but it cannot be directly applied to single-epoch spectra for BH mass without knowning . Iteration of Equation (7) does not converge. The reason is due to the fact that larger leads to smaller and higher , implying that the iteration from Equation (1) does not converge. Du et al. (2016b) devised a new method to determine from single-epoch spectra. Beginning with the seminal work of Boroson & Green (1992), it has been well-known that , the flux ratio of broad optical Fe ii to H, correlates strongly with Eddington ratio (Sulentic et al. 2000; Shen & Ho 2014). At the same time, the shape of broad H, as parameterized by , where is the line dispersion, also correlates with Eddington ratio (Collin et al. 2006). Combining the two produces produces a strong bivariate correlation, which we call the fundamental plane of the BLR, of the form

 log˙M=α2+β2DHβ+γ2RFe, (10)

where

 α2=2.47±0.34; β2=−1.59±0.14; γ2=1.34±0.20. (11)

## 6. Discussion

### 6.1. Normalized BLR Sizes

In order to explore the relation between BLR size and accretion rate, we define a dimensionless radius for the BLR, , where  cm is the gravitational radius. As in Paper IV, we insert Equation (3) into to replace , to obtain . This relation implies that increases with accretion rates as for sub-Eddington AGNs, whereas in SEAMBHs (as shown in Figure 6a) and and tends toward a maximum saturated value of , where is the minimum velocity width of H (see Equation 15 in Paper IV). We note that the minimum observed FWHM values of H (which is comparable to H) is among low-mass AGNs (Greene & Ho 2007; Ho & Kim 2016). Indeed, this limit is consistent with the saturation trend of (Figure 6a).

We note that the relatively large scatter in Figure 6a is mostly due to the uncertainties in BH mass. In order to better understand the relation between the BLR and the central engine, we define, as in Paper IV, the parameter , which reduces to

 Y=1.9×104 ˙M0.35. (12)

We would like to point out that Equation (12) describes the coupled system of the BLR and the accretion disks. It is therefore expected that is a synthetic parameter describing the photoionization process including ionizing sources.

It is easy to observationally test Equation (12) using RM results. Figure 6b plots versus . It is very clear that the observed data for objects with agree well with Equation (12). Furthermore, there is a clear saturation of for objects with objects. All these results strengthen the conclusions drawn in Paper IV. As in that work, we define an empirical relation

 Y=Ysatmin⎡⎣1,(˙M˙Mb)b⎤⎦, (13)

where

 Ysat=(3.5+0.6−0.5)×104,   ˙Mb=15.6+22.0−9.1,   b=0.27+0.04−0.04. (14)

In fact, we can get by inserting Equation (3) into (7), and find that it is in agreement with Equation (13). From the saturated, we have the maximum value of

 rHβ,sat=(3.5+0.6−0.5)×104m−0.297  or  RHβ,sat=(19.9+3.4−2.8)m0.717ltd. (15)

This result provides a strong constraint on theoretical models of super-Eddington accretion onto BHs.

### 6.2. The Shortened Lags

The shortened H lag observed in SEAMBHs cannot be caused by retrograde accretion. However, the strong dependence on accretion rate of the deviation from the standard lag-luminosity relation implies that the properties of the ionizing sources are somehow different from those in sub-Eddington AGNs. According to the standard photoionization theory, the observed relation can be explained if and is constant, where , is the ionizing luminosity, is gas density of BLR clouds, and is the average energy of the ionizing photons (Bentz et al. 2013). The relation holds for sub-Eddington AGNs, and the constancy of is determined by the clouds themselves. is not expected to vary greatly as a function of Eddington ratio. Therefore,

 RHβ=L1/2ionQc=SR0Hβ, (16)

where the factor describes the anisotropy of the ionizing radiation field, is the shadowed ionizing luminosity received by the BLR clouds, and is the BLR radius corresponding to , the unshadowed luminosity. Based on the classical model of slim disks, Wang et al. (2014c) showed that, for a given accretion rate, strongly depends on the orientation of the clouds relative to the disk, and that it range from 1 to a few tens. Therefore, the reduction of the H lag can, in principle, reach up to a factor of a few, even 10, as observed.

Furthermore, the saturated- implies that the ionizing luminosity received by the BLR clouds gets saturated. The theory of super-Eddington accretion onto BHs is still controversial. Although extensive comparison with models is beyond the scope of this paper, we briefly discuss the implications of the current observations to the theory. Two analytical models, which reach diametrically extreme opposite conclusions, have been proposed. Abramowicz et al. (1988) suggested a model characterized by fast radial motion with sub-Keplerian rotation and strong photon-trapping. Both the shortened lags and saturated may be caused by self-shadowing effects and saturated radiation from a slim disk. Both features are expected from the Abramowicz et al. model (Wang et al. 2014b). On the other hand, photon-bubble instabilities may govern the disk structure and lead to very high radiative efficiency (Gammie 1998). If super-Eddington accretion can radiate as much as (Equation 14 in Begelman 2002), the disk remains geometrically thin. In such an extreme situation, self-shadowing effects are minimal, H lags should not be reduced, and saturation disappears.

Recent numerical simulations that incorporate outflows (e.g., Jiang et al. 2014) and relativistic jets (Sadowski et al. 2015) also suggest that super-Eddington accretion flows can maintain a high radiative efficiency. However, most AGNs with high accretion rates are radio-quiet (Ho 2002; Greene & Ho 2006), in apparent contradiction with the numerical simulation predictions. Furthermore, evidence for -saturation also does not support the models with high radiative efficiency. Recent modifications of the classical slim disk model that include photo-trapping appear promising (e.g., Cao & Gu 2015; Sadowski et al. 2014), but the situation is far from settled. Whatever the outcome, the results from our observations provide crucial empirical constraints on the models.

### 6.3. Inclination Effects on ˙M

If the BLR is flattened, its inclination angle to the observer will influence , and hence (see Equation 3). To zero-order approximation, the observed width of the broad emission lines follow

 ΔVobs≈[(HBLRR)2+sin2i]1/2VK, (17)

where is the Keplerian velocity and is the height of the flatten BLR (e.g., Collin et al. 2006). For a geometrically thin BLR, , , and hence , which is extremely sensitive to the inclination can be severely overestimated for low inclinations. On the other hand, many arguments (e.g., Goad & Korista 2014) support , and the inclination angle significantly influences only for . Currently, the values of are difficult to estimate, but detailed modelling of RM data suggests (Li et al. 2013; Pancoast et al. 2014). If true, this implies that the BLR is not very flattened, and hence the inclination angle only has a minimal influence on and .

### 6.4. Comparison with Previous Campaigns

The objects in our SEAMBH sample are very similar to NLS1s. As previous RM AGN samples include NLS1s, why have previous studies not noticed that NLS1s deviate from the relation (e.g., Figure 2 in Bentz 2011)? We believe that the reason is two-fold. First, the number of NLS1s included in previous RM campaigns was quite limited (Denney et al. 2009, 2010; Bentz et al. 2008, 2009; see summary in Bentz 2011). The level of optical variability in NLS1s is generally very low (Klimek et al. 2004), and many previous attempts at RM have proved to be unsuccessful (e.g., Giannuzzo & Stirpe 1996; Giannuzzo et al. 1999). Second, not all NLS1s are necessarily highly accreting. Our SEAMBH sample was selected to have high accretion rates (see listed in Table 7 of Paper IV), generally higher than that of typical NLS1s previously studied successfully through RM. As discussed in Wang et al. (2014b) and in Paper IV, high accretion rates lead to anisotropic ionizing radiation, which may explain the shortened BLR lags.

### 6.5. SEAMBHs as Standard Candles

Once its discovery, quasars as the brightest celestial objects in the Universe had been suggested for cosmology (Sandage 1965; Hoyle & Burbidge 1966; Longair & Scheuer 1967; Schmidt 1968; Bahcall & Hills 1973; Burbidge & O’Dell 1973; Baldwin et al. 1978). Unfortunately, the diversity of observed quasars made these early attempts elusive. After five decades since its discovery, quasars are much well understood: accretion onto supermassive black holes is powering the giant radiation, in particular, the BH mass can be reliably measured. Quasars as the most powerful emitters renewed interests for cosmology in several independent ways: 1) the normal relation (Horn et al. 2003; Watson et al. 2011; Czerny et al. 2013); 2) the linear relation between BH mass and luminosity in super-Eddington quasars (Wang et al. 2013; Paper-II); 3) Eddington AGNs selected by eigenvector 1 (Marziani & Sulentic 2014); 4) X-ray variabilities (La Franca et al. 2014) and 5) relation (Risaliti & Lusso 2015). These parallel methods will be justified for cosmology by their feasibility of experiment periods and measurement accuracy.

The strength of SEAMBHs makes its application more convenient for cosmology. Selection of SEAMBHs only depends on single epoch spectra through the fundamental plane (Equation 10). BH mass can be estimated by the new scaling relation (Equation 7). We will apply the scheme outlined by Wang et al. (2014a) to the sample of selected SEAMBHs for cosmology in a statistical way (in preparation). On the other hand, the shortened H lags greatly reduce monitoring periods if SEAMBHs are applied as standard candles, in particular, the reduction of lags govern by super-Eddington accretion can cancel the cosmological dilltion factor of . Otherwise, the monitoring periods of sub-Eddington AGNs should be extended by the same factor of <