Shedding New Light on Weak Emission-line Quasars in the C iv–Hβ Parameter Space

Weak emission-line quasars (WLQs) are a subset of type 1 quasars that exhibit extremely weak Lyα + N v λ1240 and/or C iv λ1549 emission lines. We investigate the relationship between emission-line properties and accretion rate for a sample of 230 “ordinary” type 1 quasars and 18 WLQs at z < 0.5 and 1.5 < z < 3.5 that have rest-frame ultraviolet and optical spectral measurements. We apply a correction to the Hβ-based black hole mass (M BH) estimates of these quasars using the strength of the optical Fe ii emission. We confirm previous findings that WLQs’ M BH values are overestimated by up to an order of magnitude using the traditional broad-emission-line region size–luminosity relation. With this M BH correction, we find a significant correlation between Hβ-based Eddington luminosity ratios and a combination of the rest-frame C iv equivalent width and C iv blueshift with respect to the systemic redshift. This correlation holds for both ordinary quasars and WLQs, which suggests that the two-dimensional C iv parameter space can serve as an indicator of accretion rate in all type 1 quasars across a wide range of spectral properties.

Multi-wavelength observations of sources of this class have shown that they are unlikely to be high-redshift galaxies with apparent quasar-like luminosity due to gravitational-lensing amplification, dust-obscured quasars, or broad-absorption-line (BAL) quasars (e.g., Shemmer et al. 2006Shemmer et al. , 2010)), but that their UV emissionlines are intrinsically weak.Furthermore, WLQs are typically radio-quiet, and have X-ray and mid-infrared properties inconsistent with those of BL Lac objects (Shemmer et al. 2009;Lane et al. 2011;Wu et al. 2012;Massaro et al. 2017).
About half of WLQs have notably lower X-ray luminosities than expected from their monochromatic luminosities at 2500 Å (e.g., Luo et al. 2015;Ni et al. 2018Ni et al. , 2022;;Timlin et al. 2020).One explanation for this phenomenon is that, at small radii, the geometrically thick accretion disks of these WLQs are 'puffed up' and prevent highly ionizing photons from reaching the broad emission-line region (BELR; e.g, Wu et al. 2011Wu et al. , 2012;;Luo et al. 2015;Ni et al. 2018Ni et al. , 2022)).The X-ray radiation is partially absorbed by the thick disk, resulting in low apparent X-ray luminosities at high inclinations (i.e., when these objects are viewed edge-on).When these objects are viewed at much lower inclinations, their notably steep X-ray spectra indicate accretion at high Eddington luminosity ratio (L bol /L Edd , hereafter L/L Edd ; e.g., Shemmer et al. 2008;Luo et al. 2015;Marlar et al. 2018).
The indications of high Eddington ratios in WLQs may provide a natural explanation for the weakness of their emission lines in the context of the Baldwin Effect.In its classical form, this effect is an anticorrelation between the EW(C iv) and the quasar luminosity (Baldwin 1977).Subsequent studies, however, have found that this relation stems from a more fundamental anti-correlation between EW(C iv) and Hβ-based L/L Edd (Baskin & Laor 2004, hereafter BL04;Dong et al. 2009).This anti-correlation, coined the Modified Baldwin Effect (MBE), was extensively studied and built upon by Shemmer & Lieber (2015, hereafter SL15) (however, see also Wang et al. 2022).SL15 utilized a sample of nine WLQs and 99 nonradio-loud, non-BAL ('ordinary') quasars spanning wide ranges of luminosity and redshift to analyze the relative strength of the C iv emission-line and the Hβbased Eddington ratio.They confirmed the findings of BL04 for the sample of ordinary quasars.However, all nine WLQs were found to possess relatively low L/L Edd values, while the MBE predicts considerably higher Eddington ratios for these sources.This finding led SL15 to conclude that the Hβ-based L/L Edd parameter cannot depend solely on EW(C iv) for all quasars.Such a conclusion may also be consistent with subsequent findings that WLQs possess strong Fe ii emission and large velocity offsets of the C iv emissionline peak with respect to the systemic redshift (hereafter, Blueshift(C iv)) (Martínez-Aldama et al. 2018), and that L/L Edd correlates with Blueshift(C iv) at high Blueshift(C iv) values (see Figure 14 of Rankine et al. 2020).
In this work, we explore two possible explanations for the findings of SL15.The first of these is that the traditional estimation of Hβ-based black-hole mass (M BH ) values, and therefore L/L Edd values, fails to accurately predict M BH , particularly in quasars with strong optical Fe ii emission (e.g., Shen 2013;Maithil et al. 2022).Such a case is typical for WLQs, and thus a correction via measurement of the strength of the Fe ii emissioncomplex in the optical band is required (Du & Wang 2019;Yu et al. 2020b).The second explanation is that EW(C iv), by itself, is not an ideal indicator of the quasar accretion rate.In addition to EW(C iv), we utilize a recently defined parameter, the 'C iv Distance' (Rivera et al. 2022, hereafter R22), which represents a combination of the EW(C iv) and Blueshift(C iv) (Richards et al. 2011;Rivera et al. 2020;McCaffrey & Richards 2021), and search for a correlation between that parameter and L/L Edd .
To investigate these explanations, we extend the WLQ sample of SL15 to nine additional sources available from the Gemini Near-IR Spectrograph -Distant Quasar Survey (GNIRS-DQS; Matthews et al. 2023, under review, hereafter Paper I).Furthermore, we study the distribution of WLQs in L/L Edd space versus a sample of ordinary quasars from SL15 and Paper I. We aim to investigate the underlying driver for the weak emission lines in WLQs and test the assertion that all WLQs have extremely high accretion rates.
The structure of this paper is as follows.In Section 2, we discuss our sample selection and the relevant equations used to estimate Hβ-based L/L Edd values.In Section 3, we analyze the samples' spectroscopic properties as well as the sources' black-hole masses and accretion rates.Subsequently, we discuss the correlation between the C iv parameter space and L/L Edd .In Section 4, we summarize our findings.Throughout this paper, we compute luminosity distances using a standard ΛCDM cosmology with H 0 = 70 km s −1 Mpc −1 , Ω M = 0.3, and Ω Λ = 0.7 (e.g., Spergel et al. 2007).
Table 1 provides basic properties for the 18 WLQs in our sample.Column (1) provides the source name; Column (2) gives the systemic redshift determined from the peak of, in order of preference, the  5); Column (9) gives Fe ii-corrected Hβbased L/L Edd, corr values (from Equation 5); Column (10) gives EW(C iv); Column (11) gives Blueshift(C iv); Columns (12) and ( 13) provide the references for the rest-frame optical and UV spectral measurements, respectively.All derived properties are discussed in detail in Section 2.4.
The two WLQs from Shemmer et al. (2010) and the two introduced in Appendix A do not have a reliable C iv line measurement in the literature, hence we perform our own measurements from their SDSS spectra, following the procedure in Dix et al. (2023, hereafter Paper II).Briefly, we fit the C iv emission line with a local, linear continuum and two independent Gaussians.These Gaussians are constrained such that the flux densities lie between 0 and twice the value of the peak of the emission line; the FWHM is restricted to not exceed 15000 km s −1 .Furthermore, we visually inspect the initial fit to correct for any additional residuals.The EW of the line emission can then be measured, as well as the blueshift, which is calculated from the difference between λ1549 and the rest-frame wavelength of the peak of the emission-line profile (see Equation 1).
Our WLQs appear to possess stronger relative optical Fe ii emission (indicated by the larger R Fe II values) compared to ordinary quasars from their respective samples.Since such sources are selected based only on their C iv emission-line strength (EW(C iv) < 10 Å), we are unable to assess any potential biases introduced by the rest-frame optical emission to their selection process.

Ordinary Quasar Sample Selection
In order to create a comprehensive comparison sample of quasars for our analysis, which requires measurements of both the Hβ and C iv emission lines, we select two catalogs of ordinary quasars from the literature.For the high-redshift quasars (1.5 z 3.5), C iv emission properties can be obtained from SDSS, but the Hβ emission line lies outside of the SDSS range, and therefore it has to be measured with NIR spectroscopy.In this redshift range, we utilize the GNIRS-DQS catalog in Paper I. GNIRS-DQS is the largest and most comprehensive inventory of rest-frame optical properties for luminous quasars, notably the Hβ, [O iii], and Fe ii emission lines.To complement this sample of highredshift, high-luminosity quasars, we include an archival sample of quasars in the low-redshift regime from the BL04 subsample also utilized in SL15.In this redshift range (z < 0.5), the Hβ emission properties can be obtained from optical spectra, but the C iv emission-line properties are more difficult to obtain, and are available primarily from the Hubble Space Telescope (HST) and the International Ultraviolet Explorer (IUE) archives.Below, we briefly discuss the selection process for our ordinary quasar sample.
The GNIRS-DQS sources were selected to lie in three narrow redshift intervals, 1.55 z 1.65, 2.10 z 2.40, and 3.20 z 3.50 to center the Hβ+[O iii] spectral complex in the NIR bands covered by GNIRS (i.e., the J, H, and K bands, respectively).In total, the survey comprises 260 sources with highquality NIR spectra and comprehensive Hβ, [O iii], and Fe ii spectral measurements (see Matthews et al. 2021, and Paper I for more details).We exclude 64 BAL quasars, 16 radio-loud quasars (RLQs), and one quasar, SDSS J114705.24+083900.6 that is both BAL and radio loud.We define RLQs as sources having radioloudness values of R > 100 (where R is the ratio between the flux densities at 5 GHz and 4400 Å; Kellermann et al. 1989).RLQs and BAL quasars are excluded to minimize the potential effects of continuum boosting from a jet (e.g., Meusinger & Balafkan 2014) and absorption biases (e.g., see BL04), respectively.Two quasars, SDSS J073132.18+461347.0 and SDSS J141617.38+264906.1, are excluded due to a lack of C iv measurements from Paper II.In total, 177 GNIRS-DQS quasars are included in our analysis; of these, seven sources with EW(C iv) < 10 Å can be formally classified as WLQs (see Section 2.1).We adopt values of FWHM(Hβ), νL ν (5100 Å), EW(Hβ), and EW(Fe ii) values from Paper I. The latter two parameters are used to derive R Fe II .Paper II reports the EW(C iv) values and the wavelengths of the C iv emission-line peaks for the quasars in Paper I, which are then used to derive the Blueshift(C iv) values (see Section 2.3).
Sixty quasars at z < 0.5 from BL04 are added to our analysis from the 63 BL04 quasars in SL15.PG 0049+171, PG 1427+480, and PG 1415+451 are excluded due to a lack of published Fe ii spectral measurements.The UV data in the BL04 sample comes, roughly equally, from both the HST and the IUE archives (see, Baskin & Laor 2005).Throughout this work, we check whether including only HST or IUE data changes the conclusion of the paper, but we find no statistical difference in the results of Section 3. Therefore, we include both subsets in this work.We obtain the FWHM(Hβ), νL ν (5100 Å), and R Fe II values for the BL04 sources from Boroson & Green (1992), and their EW(C iv) and Blueshift(C iv) values from Baskin & Laor (2005).The line measurements are expected to be roughly consistent across the different samples utilized in this work, since they all employed similar standard fitting procedures.
Table 2 lists the basic properties of the ordinary quasars in our sample with the same formatting as Table 1.

Systemic redshifts and the Blueshift(C iv)
We derive the Blueshift(C iv) values of GNIRS-DQS sources from the observed wavelengths of the C iv emission-line peaks reported in Paper II and the systemic redshifts reported in Paper I. The Blueshift(C iv) values are derived following Equation (2) in Dix et al. (2020) where z meas is the redshift measured from the wavelength of the C iv emission-line peak, and z sys is the systemic redshift with respect to the [O iii], the Mg ii, or the Hβ emission lines reported in Paper I. In this work, we report the Blueshift(C iv) ≡ −∆v values.
A non-negligible fraction (∼ 1/3) of luminous quasars have extremely weak or undetectable [O iii] emission (e.g., Netzer et al. 2004), so we must use alternative emission lines as the reference for z sys (as was done for many ordinary GNIRS-DQS sources; see, Paper I).In spite of the larger intrinsic uncertainties associated with using the Mg ii and Hβ emission lines as z sys indicators (∼ 200 km s −1 and ∼ 400 km s −1 , respectively; Shen et al. 2016), these uncertainties are typically much smaller than the Blueshift(C iv) values observed in the majority of luminous high-redshift quasars (see, Paper I).Therefore, the lack of [O iii]-based z sys values for such sources should not affect the conclusions of this work significantly.
(2)  Note-Only the first ten lines are shown; the entire table is available in the electronic version.
However, the Hβ RM sample was subsequently determined to be biased toward objects with strong, narrow [O iii] emission-lines, and, in effect, is biased in favor of low-accretion-rate broad-line AGNs (see, e.g., Robinson 1994;Shen & Ho 2014).Recent RM campaigns aimed at minimizing such bias, such as the Super-Eddington Accreting Massive Black Hole (SEAMBH; Du et al. 2014;2016;2018) and the SDSS-RM project (Shen et al. 2015), found deviations from the traditional size-luminosity relationship.In particular, SEAMBH found a population of rapidly accreting AGNs with a BELR size up to 3-8 times smaller than predicted by Equation 2, which implies an overestimation of super-Eddington-accreting M BH values from single-epoch spectra by the same factor.We apply a R Fe II correction to the traditional Hβbased M BH estimation, a method developed by Du & Wang (2019).The R Fe II paramater has been shown to correlate with L/L Edd (e.g., Netzer & Trakhtenbrot 2007).
For the Fe ii-corrected values of M BH (hereafter, M BH, corr ), we apply the size-luminosity scaling relation for R Hβ following Equation ( 5 (3) Subsequently, M BH (M BH, corr ) can be estimated using the following relationship: where we adopt f = 1.5 for the virial coefficient, consistent with results from Ho & Kim (2014); Yu et al. (2019Yu et al. ( , 2020a)); Maithil et al. (2022), R BELR ≈ R Hβ (R Hβ, corr ) is the size-luminosity relation from Equation 2 (3), ∆V is the velocity width of the emission line, which is taken here as FWHM(Hβ), assuming Doppler broadening (Wandel et al. 1999), and G is the gravitational constant.
The L/L Edd parameter can be computed from the corresponding M BH value following Equation (2) of Shem-mer et al. (2010) assuming that L Edd is computed for the case of solar metallicity: (5) where f (L) is the luminosity-dependent bolometric correction to νL ν (5100 Å), derived from Equation ( 21) of Marconi et al. (2004).
We note that a wide range of bolometric corrections for quasars is available in the literature (e.g., Richards et al. 2006;Nemmen & Brotherton 2010;Runnoe et al. 2012;Netzer 2019).However, in general, the range of these corrections is not large enough to affect the conclusion of our work.For example, Maithil et al. (2022) recently used a constant bolometric correction of L Bol /νL ν (5100 Å) ∼ 9; the bolometric corrections we derive are in the range of ∼ 5-6, which results in a relatively small systematic offset in the derivation of the L/L Edd parameter.
The uncertainties associated with the corrected M BH and L/L Edd values in this work are estimated to be at least ∼ 0.3 dex (see Table 2 of Maithil et al. 2022), but could be much larger (∼ 0.4 − 0.6 dex) for high L/L Edd objects such as WLQs (see also, SL15).

Black Hole Masses and Accretion Rates
For the 248 quasars included in this work, we determine the virial Hβ-based M BH, corr and corresponding L/L Edd, corr values from their derived optical properties, following the Fe ii-corrected BELR size-luminosity relation of Equation 3, applied to Equations 4 and 5.We also calculate these quasars' M BH and L/L Edd values following the traditional BELR size-luminosity relation of Equation 2to compare the two methods for estimating black-hole masses and accretion rates.

The Anti-correlation between EW(C iv) and L/L Edd
We use our sample to explore the anti-correlation between EW(C iv) and Hβ-based L/L Edd previously studied in SL15 (i.e., the MBE), as well as with L/L Edd, corr .Figure 2 shows EW(C iv) plotted against the traditional L/L Edd values (left) and against the Fe ii-corrected L/L Edd, corr values (right).The first four rows of Table 3 present the respective Spearman-rank correlation coefficients (r S ) and chance probabilities (p) of the ordinary quasar sample and the complete sample, including WLQs, for the correlation involving EW(C iv).We detect significant anti-correlations between EW(C iv) and L/L Edd both with and without WLQs (i.e., p 1%).However, the anti-correlation for the sample including WLQs is slightly weaker than without WLQs (both p values are roughly similar, but r S increases slightly).Our result reaffirms findings by SL15, who found WLQs to be outliers in this relation.
With a Fe ii correction, the L/L Edd, corr values provide a significantly stronger anti-correlation with EW(C iv) as the r S value decreases from −0.36 (for the L/L Edd case) to −0.48.Furthermore, the inclusion of WLQs no longer spoils the Spearman-rank correlation; in fact, the p value remains extremely low (p = 4.02×10 −20 for the entire sample), and the r S value decreases from −0.48 to −0.54, indicative of a stronger anti-correlation.Nevertheless, the L/L Edd, corr values of most of the WLQs in our sample still appear considerably smaller than a linear model would suggest (see Figure 2).To quantify the deviation of WLQs from the MBE, we fit a simple linear model, without considering the errors, to the log EW(C iv) and log L/L Edd, corr values of the ordinary quasar sample.Our WLQs deviate from the best-fit model by a mean of ∼ 3.4σ, with a range in deviation from 1.08σ to 8.02σ.Such a discrepancy paints WLQs as significant outliers in this correlation.
We also explore whether a bolometric luminosity correction based on the peculiarity of WLQs could account for this discrepancy.Although several methods for correcting bolometric luminosity are available in the literature (e.g., Richards et al. 2006;Nemmen & Brotherton 2010;Runnoe et al. 2012;Netzer 2019), if the Eddington ratios of WLQs were to be reliably predicted by the MBE, these corrections must be up to ∼ 10 5 times larger than those of Marconi et al. (2004) (as in the case of SDSS J1141+0219 with EW(C iv) = 0.4 Å).Such a discrepancy is larger than the difference expected by any of the current bolometric correction methods in the literature.These results reveal that EW(C iv) is likely not the sole indicator of accretion rate in all quasars, in agreement with SL15.3); however, WLQs' L/L Edd, corr values are still considerably (more than an order of magnitude) over-predicted by the MBE, suggesting that EW(C iv) is not the sole indicator of quasars' accretion rates.2020) used an independent component analysis (ICA) technique to analyze the spectral properties of the C iv emission line in 133 quasars from the SDSS-RM project (Shen et al. 2015).In particular, they fitted a piece-wise polynomial to trace the positions of these sources on the EW(C iv) and the Blueshift(C iv) plane.The projected position of a quasar along this curve is defined as its 'C iv Distance'.To calculate the value of this C iv Distance parameter, we follow the procedure summarized in R22 and detailed in McCaffrey & Richards (2021).In short, we first transform the values of the two axes (EW(C iv) and Blueshift(C iv)) to lie between 0 and 1, using the MinMaxScaler function within scikit-learn (Pedregosa et al. 2011).Then, the C iv Distance values are measured relative to the first point of the best-fit curve, located at EW(C iv) ≈ 316 Å and Blueshift(C iv) ≈ 50 km s −1 .This parameter essentially indicates the location along a non-linear first principal component of the C iv parameter space, and encodes information about the physics of the C iv-emitting gas (e.g., Richards et al. 2011Richards et al. , 2021;;Giustini & Proga 2019).

Rivera et al. (
The left panel of Figure 3 shows the distribution of EW(C iv) versus Blueshift(C iv) of the 248 quasars in our sample.The right panel of Figure 3 shows the same distribution in scaled space, following the procedure in McCaffrey & Richards (2021), and the piece-wise polynomial best-fit curve from Figure 2 of R22.Even though our sources are drawn from samples that are different from those of R22, the best-fit curve traces the C IV parameter space of sources across wide ranges of redshifts and luminosities.Since all quasars in our sample are selected photometrically, either in optical (for GNIRS-DQS quasars) or UV (for BL04 quasars) surveys, and were not selected based on their spectroscopic characteristics, there are no known biases associated with their selection, and thus they are expected to trace the C iv parameter space in a similar manner to larger samples of quasars in other studies (e.g., see also Rankine et al. 2020).
While the EW(C iv) parameter, on its own, is not an ideal accretion-rate indicator, the C iv Distance parameter appears to provide a robust indication of the accretion rates for all quasars including WLQs.We plot the C iv Distance versus Hβ-based L/L Edd (left) and L/L Edd, corr (right) for all quasars in our sample in Figure 4.The last four rows of Table 3 provide the Spearman-rank correlation coefficients and chance probabilities for the correlations involving the C iv Distance.Both the L/L Edd and L/L Edd, corr are significantly correlated with the C iv Distance parameter (i.e., p 1%).In the case of C iv Distance versus L/L Edd, corr , the correlation coefficient is considerably larger than the correlation involving L/L Edd (0.57 versus 0.36), indicating the importance of the Fe ii correction to M BH .Furthermore, the inclusion of WLQs in the sample both strengthens the correlation (r S increases from 0.52 to 0.57 while the p value remains extremely small, < 10 −16 ), and allows the high-L/L Edd, corr end of the correlation to be more populated.There is also no significant deviation of the WLQs from this correlation,  as opposed to their behavior in the MBE (see, Figure 2) as well as in the C iv Distance versus traditional L/L Edd (see left panel of Figure 4).To quantify this effect, we fit a linear model to the C iv Distance and L/L Edd (L/L Edd, corr ) space, taking into account only the ordinary quasars.Then, we calculate the mean scatter of the WLQs from this line.In the case of L/L Edd , we find the deviation from the best-fit line to range from 0.62σ to 2.96σ, and the mean deviation to be ∼ 1.8σ.Meanwhile, the deviation in the case of L/L Edd, corr ranges from 0.01σ to 2.18σ, with a mean deviation of ∼ 1.1σ.Thus, using L/L Edd, corr not only results in a stronger correlation with C iv Distance, but C iv Distance also serves as a better predictor for L/L Edd, corr than for L/L Edd .
The right panel of Figure 4 shows that WLQs are not a disjoint subset of quasars in the UV−optical space (see also, Martínez-Aldama et al. 2018).Our results indicate that WLQs possess relatively high accretion rates, due not only to their extremely weak C iv lines, but rather to their relatively large values of the C iv Distance parameter.Similarly, we observe quasars with high accretion rates (and large values of C iv Distance) that do not necessarily possess extremely weak C iv lines, some of which have Eddington ratios that are larger than those of several WLQs.Finally, while we are unaware of a large population of quasars that deviate significantly from the correlations of Figure 4, a future examination of, e.g., Hβ-based L/L Edd values of quasars with very large EW(C iv) (e.g., Fu et al. 2022) is warranted to further test our results.
In this work, we show that the C iv and Hβ parameter space provides important diagnostics for quasar physics.In particular, we found that the C iv Distance can serve as a robust predictor of quasar's accretion rate, especially after a correction based on R Fe II is applied.Within the limits of our sample, we also find that WLQs are not a disjoint subset of the Type 1 quasar population, but instead lie preferentially toward the extreme end of the C iv-Hβ parameter space.

CONCLUSIONS
We compile a statistically meaningful sample of ordinary quasars and WLQs to study the dependence of quasar accretion rates, corrected for the relative strength of Fe ii emission with respect to Hβ, upon source location in the C iv parameter space.Utilizing 18 WLQs, 16 of which are obtained from the literature and two of which are presented in this work, we confirm the findings of Maithil et al. (2022) that the tra-ditional approach to estimating the Eddington ratio for rapidly-accreting quasars systematically underestimates this property by up to an order of magnitude compared to Fe ii-corrected values of this parameter.
Using the Fe ii-corrected values of Hβ-based L/L Edd , we investigate the correlation between this parameter and the C iv parameter space.We confirm and strengthen the SL15 results by finding that WLQs spoil the anti-correlation between EW(C iv) and Hβ-based L/L Edd for quasars, whether the latter parameter is estimated using the traditional method, or whether a correction based on Fe ii emission is employed in the M BH estimate.In keeping with SL15, we conclude that the EW(C iv) cannot be the sole indicator of accretion rate in quasars.
We also investigate the relationships between a recently-introduced parameter, the C iv Distance, which is a combination of EW(C iv) and Blueshift(C iv), and the traditional Hβ-based L/L Edd and the Fe iicorrected L/L Edd, corr .Such relationships yield strong correlations, especially in the case of Fe ii-corrected L/L Edd, corr , and can accommodate all the quasars in our sample.Our finding suggests that WLQs are not a disjoint subset of sources from the general population of quasars.We find that many WLQs have extremely high accretion rates which is indicated by their preferentially higher values of the C iv Distance parameter.Similarly, we find several quasars in our sample that possess high Eddington ratios, and correspondingly large values of the C iv Distance, that do not have extremely weak C iv lines; some of these sources display Eddington ratios that are larger than those of a subset of our WLQs.
In the context of the C iv parameter space, it will be interesting to investigate whether the extreme X-ray properties of WLQs are the result of extremely large C iv Distance values rather than resulting only from extremely weak C iv lines.Such a test would require X-ray coverage of a large sample of sources with Hβ+Fe ii data across the widest possible C iv parameter space such as the GNIRS-DQS sample of Paper I. It would also be useful to determine whether the weakness of the broad Lyα+N v emission line complex (from which the first high-redshift WLQs were identified) also correlates with C iv Distance, which will require restframe ultraviolet spectroscopy (Paul et al. 2022).The results of these investigations will shed new light on the connection between the quasar accretion rate and the physics of the inner accretion disk and BELR.A.1.For both targets, we used the Short Blue Camera, with spatial resolution 0. 15 pix −1 , and a 1.0 slit to achieve a spectral resolution of R ∼ 600.An H-filter was applied, producing a spectral range of 1.5 -1.8 µm, corresponding to rest-frame ∼ 4500 − 5300 Å.Exposure times for each subintegration were 238 s and 220 s, and the total integration times were 7140 s and 7040 s for SDSS J1137+3919 and SDSS J2137−0039, respectively.These observations were performed using the standard "ABBA" nodding pattern of the targets along the slit in order to obtain primary background subtraction.
The spectra were processed using the standard procedure of the IRAF Gemini package based on the PyRAF Python-based interface.Exposures from the same nodding position were added to boost the signal-to-noise ratio, then the sum of exposures from two different nodding positions were subtracted to remove background noise.Wavelength calibration was done against an Argon lamp in order to assign wavelength values to the observed pixels.
Spectra of telluric standard stars with T eff ∼ 9700 K were taken immediately before or after the science exposures to remove telluric absorption features in the quasars' observed spectra.These spectra were processed in a similar fashion, followed by a removal of the stars' intrinsic hydrogen absorption lines by fitting a Lorentzian profile to each hydrogen absorption line, and interpolating across this feature to connect the continuum on each side of the line.The quasars' spectra were divided by the corrected stellar spectra.The corrected quasar spectra were then multiplied by an artificial blackbody curve with a temperature corresponding to the telluric standard star, which yielded a cleaned, observed-frame quasar spectrum.
We modeled the spectra following the methods of Shemmer et al. (2004) and Shemmer et al. (2010).Our model consists of a linear continuum through the average flux densities of two narrow (∼20 Å) rest-frame bands centered on 4750 Å and 4975 Å, a broadened Fe ii emission template (Boroson & Green 1992) In both sources we detected weak and relatively narrow Hβ lines as well as strong Fe ii features compared to quasars at similar luminosities and redshifts (e.g., see Netzer et al. 2007;Shen 2016).We also determined the systemic redshift (z sys ) values from the observed-frame wavelength of the peak (λ peak ) of the Hβ emission-line, a similar treatment as in Matthews et al. (2021) for sources that lack [O iii] emission.The z sys values are larger than the redshifts reported by Lyke et al. (2020) by ∆z = 0.008 in SDSS J1137+3919 and by ∆z = 0.013 in SDSS J2137−0039, corresponding to velocity offsets (blueshifts) of 700 km s −1 and 1184 km s −1 , respectively, which is consistent with typical velocity offsets between SDSS Pipeline redshifts and z sys values observed in luminous, high-redshift quasars (Dix et al. 2020, Paper I).The rest-frame spectra in Figure A.1 have henceforth been corrected by z sys .Rest-frame EWs of Hβ λ4861, Fe ii λλ4434 − 4684, and the upper limit on the EWs of [O iii] λ5007 were calculated for SDSS J1137+3919 to be 16 Å, 53 Å, and ≤ 4 Å, and for SDSS J2137−0039 to be 20 Å, 49 Å, and ≤ 5 Å, respectively.The flux densities at a rest-frame wavelength of 5100 Å are 7.77 × 10 −18 ergs cm −2 s −1 Å−1 and 8.18 × 10 −18 ergs cm −2 s −1 Å−1 , respectively.Green (1992), which was broadened by 1500 km s −1 for SDSS J1137+3919, and 1400 km s −1 for J2137−0039.The bold solid line is the entire fitted spectrum.

Figure 1
presents the traditional versus corrected M BH and L/L Edd values for the quasars in our sample, following the procedure ofMaithil et al. (2022).The Hβ-based M BH, corr values of ordinary quasars show small systematic deviations from the traditional BELR size-luminosity relation estimates (less than a factor of two for 222 out of 230 quasars).On the other hand, for a majority of the WLQs, due to the relative weakness in Hβ emission compared to the Fe ii emission, M BH, corr values deviate significantly from the traditional relation, by up to one order of magnitude.Since L/L Edd is inversely proportional to M BH , the L/L Edd, corr values are enhanced by a similar factor.

Figure 1 .
Figure 1.Black-hole mass (left panel) and accretion rate (right panel) calculated using the traditional (x-axis) and RFe II- corrected (y-axis) BELR size-luminosity relation for all quasars in our analysis.Diamonds mark ordinary quasars and squares mark WLQs.The dashed lines represent a one-to-one relation between the two methods.Typical uncertainties of 0.5 dex on the MBH and L/L Edd values are displayed in the bottom right corner of each panel.The traditional relation overestimates MBH in rapidly-accreting quasars by up to an order of magnitude.In turn, the traditional relation underestimates L/L Edd by a similar factor.In particular, the RFe II-corrected accretion rates are much larger for a considerably larger fraction of sources in the WLQ subset than in the ordinary quasars, due to their larger RFe II values.This result is in line with the Maithil et al. (2022) finding of a larger deviation from the one-to-one relation in high-accretion-rate quasars.
3.3.The C iv Distance as an Indicator of L/L Edd

Figure 2 .
Figure 2. Correlation between EW(C iv) and L/L Edd of ordinary quasars (diamonds) and WLQs from Table 1 (squares).The left panel presents the traditional L/L Edd values, and the right panel displays the Fe ii-corrected L/L Edd, corr values.The dotteddashed lines represent the EW threshold below which objects are defined as WLQs.The correlation for the ordinary quasar sample, obtained by fitting a linear model, is shown as a dashed line.The shaded regions represent the 1-and 2-σ deviations from the fitted correlation.Correcting the traditional L/L Edd values results in a stronger anti-correlation expected by the MBE (see Table3); however, WLQs' L/L Edd, corr values are still considerably (more than an order of magnitude) over-predicted by the MBE, suggesting that EW(C iv) is not the sole indicator of quasars' accretion rates.

Figure 3 .
Figure 3. Left panel: distribution of EW(C iv) versus Blueshift(C iv) for our sample.One quasar from BL04, PG 1202+281, is not shown in the left panel, due to its extremely high EW(C iv) = 290 Å.Right panel: illustration of the C iv Distance parameter.The data are first scaled so that the two axes share the same limit, then each data point is projected onto the best-fit curve obtained from R22.The C iv Distance value of each quasar is defined as its projected position (green point) along the solid black curve.Three of the WLQs, SDSS J114153.33+021924.4,SDSS J123743.07+630144.7,SDSS J094602.31+274407.0, and one ordinary quasar, PG 1202+281, are not shown in the right panel, for clarity, but only their projected positions onto the curve are relevant to our results.

Figure 4 .
Figure 4. C iv Distance versus L/L Edd of 248 quasars in our sample.In the left panel, the C iv Distance values are plotted against the traditional Hβ-based L/L Edd parameter, and in the right panel, against the Fe ii-corrected Hβ-based L/L Edd, corr parameter.The ordinary quasar PG 1202+281 with L/L Edd, corr = 0.06 and C iv Distance = 0.02 is not plotted, for clarity.The correlation for the ordinary quasar sample, obtained by fitting a linear model, is shown as a dashed line.The shaded regions represent the 1-and 2-σ deviations from the fitted correlation.While using the traditional size-luminosity relation to estimate accretion rates already yields a strong correlation, the Fe ii-corrected accretion rates show a much stronger correlation with the C iv Distance parameter for all quasars.Furthermore, this parameter serves as a better predictor for L/L Edd, corr than for L/L Edd .
, and two Gaussian profiles for the Hβ λ4861 emission-line.No [O iii] emission-lines are detectable in either spectrum, and we placed upper limits on their EWs by fitting a Gaussian feature where the [O iii] emission-lines should be such that they are indistinguishable from the continuum.The final, calibrated near infrared spectra of the two WLQs appear in Figure A.1.

Figure A. 1 .
Figure A.1.The NIR spectra of SDSS J1137+3919 (top) and SDSS J2137−0039 (bottom).In each panel, the continuous line is the observed spectrum of each quasar.The continuous straight line below the spectrum is the linear continuum fit.The dashed line is the Hβ λ4861 profile modelled with two Gaussians.The dotted-dashed line is the Fe ii template fromBoroson & Green (1992), which was broadened by 1500 km s −1 for SDSS J1137+3919, and 1400 km s −1 for J2137−0039.The bold solid line is the entire fitted spectrum.

Table 1 .
Basic Properties of the WLQ Sample

Table 2 .
Basic Properties of the Ordinary Quasar Sample