Introduction

Reverberation mapping (RM) is a well-demonstrated technique to measure the black hole (BH) masses in active galactic nuclei (AGNs), using the distance and kinematics of the broad-line region (BLR) clouds. After several decades of RM work by different groups, there are now more than 100 local AGNs (z  0.3) that have successful mass measurements with Hβ RM (e.g., R. D. Blandford & C. F. McKee 1982;B. M. Peterson 1993;B. M. Peterson et al. 1998B. M. Peterson et al. , 2002B. M. Peterson et al. , 2004;;S. Kaspi et al. 2007; A. J. Barth et al. 2015;P. Du et al. 2015;P. Du et al. 2018;C. Hu et al. 2021;V. U et al. 2022; U. Malik et al. 2023; S. Wang & J.-H. Woo 2024; J.-H. Woo et al. 2024). Recent dedicated RM programs are also pushing these measurements to higher redshifts and to cover additional broad lines such as C IV and Mg II (P. Lira et al. 2018; C. J. Grier et al. 2019;J. K. Hoormann et al. 2019;Y. Homayouni et al. 2020;S. Kaspi et al. 2021;Z. Yu et al. 2021; U. Malik et al. 2023). Earlier studies on nearby RM AGNs have revealed a tight (i.e., scatter 0.15 dex in BLR size) correlation between the average distance from the BH to Hβ-emitting BLR clouds and the optical luminosity λL 5100 of the AGN, known as the radius-luminosity (R-L) relation (S. Kaspi et al. 2000Kaspi et al. , 2007;;M. C. Bentz et al. 2013): ( ) ( ) ( ) / / R L log lt days log 10 erg s , 1 BLR 5100 44 1 a l b -= + where M. C. Bentz et al. (2013) measured the slope 0.533 0.033 0.533 a = -+ and the intercept β = 1.527 ± 0.031 at ( ) / L log erg s 44 recipes (M. Vestergaard & B. M. Peterson 2006; Y. Shen 2013) that use a single monochromatic continuum luminosity to estimate the time lag. Due to their simplicity, these singleepoch BH mass recipes have been widely adopted to estimate BH masses in high-redshift quasars and for large AGN samples with single-epoch spectra (e.g., Y. Shen et al. 2011; Q. Wu & Y. Shen 2022). Furthermore, this tight local R-L relation motivates its potential utility as a standard candle to measure luminosity distances for cosmology (D. Watson et al. 2011; M. L. Martìnez-Aldama et al. 2019), once the BLR size is measured directly via RM.

However, recent RM studies targeting AGNs across broad ranges of luminosities and Eddington ratios have found evidence for increased dispersion around the canonical R-L relation (e.g., P. Du et al. 2014; C. J. Grier et al. 2017;G. Fonseca Alvarez et al. 2020;Y. Shen et al. 2024). In particular, the super-Eddington accreting massive BH (SEAMBH) collaboration (P. Du et al. 2014Du et al. , 2015;;P. Du et al. 2018;C. Hu et al. 2021) and the Seoul National University AGN monitoring project (S. Wang & J.-H. Woo 2024; J.-H. Woo et al. 2024) targeting high-accretion-rate AGNs have found a surprising downward offset of the average Hβ lag from the canonical relation in these objects. This suggests that variations in the accretion parameters (mass, accretion rate, and BH spin), along with factors like cloud distribution geometry and inclination (resulting in the transfer function of RM), significantly influence the observed distribution around the mean R-L relation. The recent dynamical mass measurement of SDSS J092034.17+065718.0 (z ≈ 2.3) from GRAVITY+ also reported a smaller BLR size than predicted by the canonical R-L relation for this super-Eddington accretion quasar (R. Abuter et al. 2024). This spectrointerferometric method independently confirms that super-Eddington AGNs may have smaller BLR sizes as measured from RM. Meanwhile, the Sloan Digital Sky Survey RM (SDSS-RM) project (Y. Shen et al. 2015aShen et al. , 2024; M. R. Blanton et al. 2017) targeting nonlocal AGNs at 0.1 < z < 4.5 also found a large dispersion of ∼0.3 dex around the mean Hβ R-L relation.

This increased scatter in the R-L relation has gained significant interest in recent work (B. Czerny et al. 2019;P. Du & J.-M. Wang 2019;E. Dalla Bontà et al. 2020;G. Fonseca Alvarez et al. 2020). Understanding the origin of this dispersion and deriving corrections to potentially "tighten" the R-L relation would have tremendous value both in designing better single-epoch mass recipes and in using the R-L relation as a luminosity indicator. Detailed cadence and duration simulations of RM monitoring have ruled out selection bias as a major contributor to the observed scatter (I.-H. Li et al. 2019;G. Fonseca Alvarez et al. 2020). Several recent works have attempted to use the optical Fe II strength (denoted by R Fe II ≡ EW Fe II, 4434-4684 /EW Hβ ) as a proxy to correct this offset (e.g., P. Du & J.-M. Wang 2019; M. L. Martìnez-Aldama et al. 2019;L.-M. Yu et al. 2020;J. Maithil et al. 2022;Z. Yu et al. 2023), where R Fe II is considered a proxy of the Eddington ratio through the eigenvector 1 relations (T. A. Boroson & R. F. Green 1992;Y. Shen & L. C. Ho 2014). Meanwhile, S. Wang & J.-H. Woo (2024) also suggest that [O III] 5007 luminosity could potentially eliminate the offset between low-Eddington and high-Eddington objects, as L [O III] 5007 systematically traces the Eddington ratio.

In this work, we consider the standard scenario that the BLR emission is sensitive to the ionizing continuum, which depends on the accretion rate and, consequently, the structure of the accretion flow. Accretion flows around BHs can be classified into three main regimes based on their accretion rates (here we use the dimensionless Eddington ratio λ Edd ≡ L bol /L Edd = L bol /(1.26 × 10 38 M BH /M e )). First, at low-to-moderate accretion rates (λ Edd  0.5), the accretion flow can be reasonably modeled by the geometrically thin Shakura-Sunyaev disk (SSD; N. I. Shakura & R. A. Sunyaev 1973), where the optically thick gas radiates multitemperature blackbody radiation. As the accretion rate substantially increases to λ Edd  1, the structure of the accretion flow undergoes significant transformations due to increased radiation pressure, leading to a slim disk (M. A. Abramowicz et al. 1988) where the disk height becomes comparable to the disk radius. Many of the high-Eddington-ratio AGNs included in the SEAMBH sample likely are better described by this slim disk model (P. Du et al. 2014). Conversely, at very low accretion rates (λ Edd < 10 -3 ), the viscously dissipated energy heats the flow rather than being radiated away, resulting in an optically thin, hot advection-dominated accretion flow (F. Yuan & R. Narayan 2014). Recognizing that the BH accretion rate in different regimes can lead to dramatic changes in the accretion-flow geometry and the resulting shape of the spectral energy distribution (SED; e.g., L. C. Ho 2008;N. Castelló-Mor et al. 2016), and given that the exact transition λ Edd between different disk geometry remains uncertain, we focus on AGNs with lowto-moderate accretion rates (10 -3 < λ Edd < 0.5), where the accretion flow is predominately in the SSD regime. However, contributions from the hot corona (and a potential warm corona) are also included in the model continuum SEDs (see Section 2.1 for details).

To fully understand the connection between the underlying ionizing continuum and observables, detailed photoionization calculations are required. In this work, we construct a series of SEDs and photoionization calculations to compare with the average Hβ emission distance from RM and the observed quasar spectral properties. In Section 2, we describe our theoretical models for the AGN SEDs and our photoionization calculations using the locally optimally emitting cloud (LOC) model (J. Baldwin et al. 1995). In Section 3, we present our analysis of the observational data and comparison with the theoretical framework. We discuss our results in Section 4 and conclude in Section 5. Throughout this paper, we adopt a flat ΛCDM cosmology with Ω M = 0.3 and H 0 = 70 km s -1 Mpc -1 .

Theoretical Framework

A physically motivated photoionization model for the AGN BLR is the LOC model (J. Baldwin et al. 1995). This model assumes an axisymmetric distribution of continuous clouds with varying gas densities and distances from the central continuum source. The observed line emission is the collection of all illuminated clouds but is dominated by those with the highest efficiency of reprocessing the incident ionizing continuum. In the following section, we apply the LOC model to a series of SEDs with different BH parameters (mass and accretion rate) to predict broad-emission-line properties with photoionization calculations. Increasing the mass does not significantly alter the SED shape, but it does increase the overall luminosity and slightly shifts the peak of the multitemperature blackbody radiation to a lower frequency. The lower panel displays the SED variation as a function of accretion rate at a fixed BH mass M BH = 10 8 M e . The increasing  m increases the overall SED of the AGN, pushing the UV peak emission from the thermal disk to higher frequencies and demonstrating a weaker X-ray emission due to changes in the hot flow geometry (A. Kubota & C. Done 2018). Note that the BHs in the lower panel exhibit similar spectral fluxes at around ∼10 keV, which is due to the hot corona assumption employed in qsosed (see details in Section 4.4). Increasing the BH mass and accretion rate leads to higher optical and UV continuum luminosities (L 5100 and λL 1350 ), as well as a corresponding increase in the ionizing luminosity, affecting the ratio of line-ionizing fluxes such as C IV/hydrogen.

Figure 2 shows SED comparisons for accretion rates  [ ] m 1.5, 0 Îat fixed mass (10 6 , 10 7 , 10 8 , 10 9 , and 10 10 M e ). At each BH mass, the SEDs are normalized by the median SED with  m log 1 = -. In all panels, higher accretion rates have relatively higher optical and UV luminosities and overall more hydrogen-ionizing flux. Meanwhile, the change in UV luminosity λL 1350 is more sensitive to accretion rate than λL 5100 , especially for more massive BHs. To illustrate the significant variations in the SED due to accretion rate changes, Figure 3 presents two theoretical SEDs with similar optical luminosities L log 44 5100 l ~but different accretion parameters (mass and accretion rate).

Given that an increase in  m and a decrease of M BH shift the UV peak to a higher frequency and soften the SED (Figure 1), the two demonstrated SEDs cross at both optical and soft X-ray wavelengths. Although the optical luminosity is similar, the more massive BH with a lower λ Edd (in blue) produces significantly less hydrogen-ionizing flux and hence is expected to have a shorter Hβ time lag, if assuming the localization of the BLR is related to the ionizing flux and other properties of the BLR are the same for the two cases.

Figure 4 displays the continuum luminosities computed from qsosed SEDs at three optical/UV wavelengths, with slightly higher luminosities at shorter wavelengths. While the simulated mass accretion rate (  m log ) ranges from -1.5 to 0 in qsosed, the λ Edd values may fall below -1.5 for the theoretical SEDs due to the universal bolometric correction (BC = 9.26) applied to the continuum luminosity at 5100 Å. The continuum luminosities show a diagonal increasing trend on the M BH -λ Edd plane. This trend suggests that despite variations in their underlying SED shapes, BHs with different accretion parameters can produce similar continuum luminosities at fixed observed wavelengths. This exploratory investigation demonstrates the possible diverse SEDs from AGNs with different accretion parameters. Since the BLR gas clouds are ionized by the ionizing continuum flux, the qualitative changes in the SED shape conceivably can lead to dispersion in the observed R-L relation based on the optical luminosity. We now proceed to use photoionization calculations to make quantitative predictions.

Photoionization with CLOUDY

We construct detailed photoionization models to calculate line luminosities for individual clouds using CLOUDY (G. J. Ferland et al. 2017;version 17.02) with a spherically symmetric geometry to account for various excitation mechanisms and radiative transfer effects. SEDs computed from Section 2.1 are used as the incident radiation field input. This approach assumes that the BLR gas sees all the radiation from the central source; i.e., there is no inner shielding of the radiation.

Following previous works in a spherically symmetric geometry (K. T. Korista & M. R. Goad 2000, 2004;H. Guo et al. 2020), we assume that the BLR clouds cover a broad range of hydrogen density (

), both with logarithmic intervals of 0.125 dex. In this section, unless otherwise specified, all physical quantities, such as n(H), Φ(H), and c, are given in cgs units. The ionization parameter U H is defined as a dimensionless ratio between the hydrogenionizing photon flux Q(H) and the total hydrogen density n(H):

where R BLR is the distance between the central ionizing source and the illuminated surface of the cloud and c is the speed of light. For each cloud, we assume a hydrogen column density N H = 10 23 cm -2 , abundance Z = Z e , and overall covering factor CF = 50%. Given the full set of hydrogen densities and 1.5, 0 Îwith 0.25 dex intervals at constant BH mass M BH (from left to right: 10 6 , 10 7 , 10 8 , 10 9 , and 10 10 M e ). Each SED within the panel is normalized relative to the SED with  m = 0.1. Vertical lines are placed at photon energies of hν C IV = 64.5 eV (blue) and hν H = 13.6 eV (red). ionizing flux, we compute the photoionization results with 4745 calculations for each SED.

The total line luminosity is computed using the LOC model by summing over all grid points with proper weights determined from assumed distribution functions. When summing the grid emissivity, only ionized clouds with a density range of

are considered to obtain the radial surface emissivity (F(r); see details in K. T. Korista & M. R. Goad 2000, 2004, 2019;H. Guo et al. 2020). For simplicity, we adopt the empirical assumptions outlined in J. Baldwin et al. (1995): f (r) ∝ r Γ and g(n) ∝ n β (β = -1) for the cloud distribution, which represent the cloud coverage fractions as functions of radius and gas density, respectively. This assumption ensures equal weighting for each grid point in the density-flux plane on a logarithmic scale. In previous studies, K. T. Korista & M. R. Goad (2000, 2019) proposed -1.4 < Γ < -1 and fixed Γ = -1.2 for NGC 5548, and H. Guo et al. (2020) employed Γ = -1.1 to align with the observed Mg II luminosity; therefore, after comparing with observed line luminosities in Section 3.2, we fix Γ = -1.1 to match the observed lines. The observed line luminosity is

where F(r) is the radial surface emissivity of a single cloud.

The BLR sizes predicted in Section 2.3 are adopted as the average distance for the integral.

Predictions of BLR Size

The relation between BLR radius R BLR = cτ and monochromatic luminosity can be calculated using this grid of SEDs. We use the following expression to compute the average distance between the ionizing source center and the Balmer-line-emitting clouds: - ; e.g., J. Baldwin et al. 1995;K. Korista et al. 1997; K. T. Korista & M. R. Goad 2000). Moreover, comparisons of several AGN emission line ratios with the photoionization of the dominant cloud have consistently indicated a product of the ionization parameter and hydrogen density approximating Matsuoka et al. 2008;C. A. Negrete et al. 2012). Therefore, a constant ionization parameter U log 2.0 H =and hydrogen density n(H) = 10 12 cm -3 are adopted here to calculate the BLR gas distance (Equation (4)). responds to the ionizing continuum. While the assumed photoionization parameters could reproduce the overall trend in the R-L plane across a range of BH mass, discrepancies in measured accretion parameters and/or assumptions in our theoretical calculations for individual objects may lead to deviations from the predicted time lags; these observational and theoretical uncertainties are discussed in Section 4.

Comparison with Observations

Sample and Data

We start from the sample of ∼140 z  0.5 broad-line AGNs with Hβ RM measurements and select those with 10 -3 < λ Edd < 0.5 in the SSD regime as our primary sample.

Figure 6 shows the distribution of BH mass M BH measured from Hβ RM versus their optical luminosity λL 5100 for this primary sample, including objects with and without UV spectroscopy. For M BH values of these local RM AGNs, we combine the catalog in M. C. Bentz & S. Katz (2015) and Table 1 Wang (2019) and correct M BH using a nominal virial factor f FWHM,Hβ = 1.12 from J.-H. Woo et al. (2015). The host contamination in λL 5100 has been subtracted for the local AGNs, except for eight sources (Zw 229+015, Mrk 1501, H0507+164, Mrk 704, PG 0934+013, Mrk 50, NGC 4395, and Mrk 6), for which we adopt their average λL 5100 from light curves as an upper limit. We supplement the local RM AGNs with 72 more distant quasars from the SDSS-RM sample (Y. Shen et al. 2024). The host contamination for a = -+ and an intercept β = 1.527 ± 0.031 at ( ) / L log erg s 44 5100 1 l = -.

the SDSS-RM quasars used in this work is removed based on a spectral decomposition approach described in Y. Shen et al. (2015b). We use a bolometric correction of BC = 9.26 (G. T. Richards et al. 2006) for the optical continuum at 5100 Å to calculate the bolometric luminosity L bol and the RM mass M BH for the dimensionless Eddington ratio.

After assembling the primary sample of 135 low-tomoderate-accretion AGNs with Hβ RM measurements, we select a subsample of quasars with both rest-frame UV and optical spectra. We acquired UV spectroscopy for 18 SDSS-RM quasars at 0.2  z  0.6 from a dedicated Hubble Space Telescope (HST) Space Telescope Imaging Spectrograph (STIS) program in Cycle 28 (GO-16171; PI: Shen). These quasars are selected from the SDSS-RM sample (Y. Shen et al. 2015a) with direct RM lags and BH masses based on the broad Hβ line (C. J. Grier et al. 2017). We restrict to quasars with 0.2  z  0.6 and Galaxy Evolution Explorer (GALEX) near-ultraviolet (NUV) 21 mag while spanning a broad range in luminosity and Hβ lags to sample the parameter space. Table 1 summarizes the properties of the 18 SDSS-RM targets.

The 18 SDSS-RM quasars were observed with HST/STIS in spectroscopic mode with the NUV-MAMA G230L grating to optimize the spectral coverage. This instrumental configuration provides broad spectral coverage from 1570 to 3180 Å, which covers the dominant UV emission lines from Si IV to C III] for our targets. The brightnesses of our targets vary with GALEX NUV ∼ 19-21: for the brightest 12 targets, one full orbit exposure time is utilized, five quasars (RM160, RM177, RM229, RM320, and RM694) were observed for two orbits, and the faintest target with a GALEX NUV magnitude of 21.1 (RM301) was observed with three orbits. The initial visits of all targets lasted from 2020 December to 2022 February; however, four targets (RM177, RM229, RM320, and RM694) suffered from the failure of guidestar acquisition at the start of the second orbit, resulting in lost data. Repeat visits for those targets were approved and recurred from 2022 September to November. The pipeline-processed STIS spectra of those targets with multiple orbits of exposure were coadded to improve the signal-to-noise ratio (S/N) of our sample. The observed spectra are presented in Figure 7.

For the remaining local RM AGNs in our sample, in addition to our SDSS-RM quasars, we collect archival UV spectroscopy from the International Ultraviolet Explorer, the STIS, and the Cosmic Origins Spectrograph on HST. If multiple spectra exist in the public archives for a given object, either multiple epochs taken with the same instrument or from multiple instruments, we combined the spectra for each object via the standard weighted average method. Since the disk structure of highaccretion SMBHs will have a dramatic impact on the ionizing continuum illuminating the BLR, we only focus on the low-tomoderate-accretion sample in this work. Table 1 Target Properties RMID R.A. (J2000) Decl. (J2000) z τ Hβ log(M BH,RM ) NUV (deg) ( deg) ( days) ( M e ) ( mag) RM101 213.059 53.430 0.458 21.4 6.4 4.2 -+ 7.26 0.12 0.09 -+ 19.35 RM160 212.672 53.314 0.360 21.9 2.4 4.2 -+ 7.85 0.07 0.09 -+ 20.77 RM177 214.352 52.507 0.482 10.1 2.7 12.5 -+ 7.57 0.11 0.35 -+ 20.80 RM229 212.575 53.494 0.470 16.2 4.5 2.9 -+ 7.65 0.12 0.09 -+ 20.77 RM267 212.803 53.752 0.588 20.4 2.0 2.5 -+ 7.41 0.08 0.08 -+ 20.16 RM272 214.107 53.911 0.263 15.1 4.6 3.2 -+ 7.58 0.13 0.10 -+ 19.40 RM301 215.043 52.675 0.548 12.8 4.5 5.7 -+ 8.64 0.14 0.17 -+ 21.10 RM305 212.518 52.528 0.527 53.5 4.0 4.2 -+ 8.32 0.07 0.07 -+ 20.28 RM320 215.161 53.405 0.265 25.2 5.7 4.7 -+ 7.67 0.11 0.09 -+ 20.60 RM371 212.848 52.225 0.473 13.0 0.8 1.4 -+ 7.38 0.07 0.08 -+ 20.21 RM622 212.813 51.869 0.572 49.1 2.0 11.1 -+ 7.94 0.06 0.11 -+ 20.55 RM694 214.278 51.728 0.532 10.4 3.0 6.3 -+ 6.70 0.15 0.20 -+ 20.83 RM720 211.325 53.258 0.467 41.6 8.3 14.8 -+ 7.74 0.10 0.14 -+ 19.80 RM772 215.400 52.527 0.249 3.9 0.9 0.9 -+ 6.60 0.10 0.10 -+ 19.92 RM781 215.265 51.972 0.264 75.2 3.3 3.2 -+ 7.89 0.07 0.07 -+ 20.47 RM782 213.329 54.534 0.363 20.0 3.0 1.1 -+ 7.51 0.09 0.06 -+ 19.50 RM790 214.372 53.307 0.238 5.5 2.1 5.7 -+ 8.28 0.15 0.31 -+ 20.10 RM840 214.188 54.428 0.244 5.0 1.4 1.5 -+ 7.93 0.12 0.13 -+ 19.94 Note. Properties of the 18 quasars proposed in HST GO-16171. RM BH masses are based on Hβ lags from C. J. Grier et al. (2017). Time lags are rest-frame values.

Our final sample includes 15 SDSS-RM quasars and 52 local RM AGNs with accretion rates 10 -3 < λ Edd < 0.5. We then apply the spectral decomposition code PyQSOFit (H. Guo et al. 2018) with minor custom adjustments to measure their UV spectral properties and summarize them in

Table 2. The optical spectral properties ( L log 5100 l , Hβ, and He II λ4687) for the SDSS-RM sample are from Y. Shen et al. (2019), while those of local AGNs are from P. Du & J.-M. Wang (2019). product destruct ò n = n n n with the production and destruction edges: 47.9 and 64.5 eV for C IV, 7.6 and 15.0 eV for Mg II, 24.6 and 54.4 eV for He II λ1640, and 13.6 eV for Hβ. The middle panels display the line luminosities predicted by the LOC model using Equation (3) (see Section 2.2). The right panels present the observed line luminosities for both local RM AGNs and the SDSS-RM sample. For local RM AGNs, Hβ line luminosities are from P. Du & J.-M. Wang (2019). The line luminosities for Hα, Hβ, and He II λ4687 in the SDSS-RM sample are from Y. Shen et al. (2019). There are offsets of ∼1 dex in the observed C III],

UV/Optical Line Strengths

He II λ4687, and He II λ1640 emission line luminosities to the LOC prediction, which may be a result of the underestimated EUV SED and the simplified assumptions of the BLR cloud distribution. Despite these offsets, all panels in Figure 8 exhibit a positive correlation in the M BH -λ Edd plane. Most importantly, this coherent trend between the LOC predictions and

4687) RM160 43.79 ± 0.01 44.34 ± 0.02 43.62 ± 0.01 321.2 ± 5.7 -595 ± 45 42.47 ± 0.03 33.0 ± 2.5 42.19 ± 0.09 13.0 ± 2.7 L L 42.07 ± 0.01 96.5 ± 1.4 40.9 ± 0.06 6.4 ± 0.6 RM177 43.97 ± 0.01 44.08 ± 0.03 42.64 ± 0.03 47.7 ± 3.9 145 ± 366 42.32 ± 0.09 29.5 ± 5.7 41.37 ± 0.23 2.7 ± 2.5 L L 42.31 ± 0.01 62.0 ± 1.7 41.4 ± 0.02 7.5 ± 0.4 RM229 43.56 ± 0.01 44.25 ± 0.03 43.0 ± 0.05 100.1 ± 9.1 198 ± 504 42.35 ± 0.27 36.4 ± 16.5 41.83 ± 0.24 7.8 ± 6.5 L L 41.93 ± 0.02 62.0 ± 2.3 40.86 ± 0.15 5.2 ± 2.5 RM267 44.11 ± 0.01 44.76 ± 0.02 43.26 ± 0.02 46.7 ± 2.1 332 ± 166 42.94 ± 0.12 47.6 ± 14.0 L L L L 42.54 ± 0.02 81.1 ± 4.2 41.33 ± 0.17 4.8 ± 2.5 RM272 43.94 ± 0.01 44.26 ± 0.05 43.16 ± 0.04 145.4 ± 15.3 -287 ± 127 42.24 ± 0.05 21.0 ± 2.6 42.2 ± 0.32 17.1 ± 9.5 L L 42.37 ± 0.01 132.2 ± 0.6 41.46 ± 0.01 15.8 ± 0.5 RM301 44.04 ± 0.01 43.75 ± 0.09 42.82 ± 0.03 183.9 ± 12.4 304 ± 193 41.83 ± 0.23 20.4 ± 11.7 L L L L 42.27 ± 0.01 58.7 ± 1.2 40.75 ± 0.08 1.7 ± 0.3 RM305 44.19 ± 0.01 44.37 ± 0.04 42.91 ± 0.04 54.5 ± 5.2 -1210 ± 881 42.61 ± 0.18 40.5 ± 17.8 41.47 ± 0.33 2.1 ± 2.4 L L 42.53 ± 0.04 77.7 ± 6.2 41.4 ± 0.02 5.6 ± 0.3 RM320 43.44 ± 0.01 43.59 ± 0.06 42.68 ± 0.04 138.7 ± 13.4 407 ± 266 41.92 ± 0.05 25.5 ± 2.9 42.02 ± 0.26 29.9 ± 13.2 L L 41.89 ± 0.01 74.9 ± 1.9 41.18 ± 0.12 14.2 ± 4.7 RM371 44.09 ± 0.01 44.03 ± 0.1 43.17 ± 0.04 205.4 ± 24.3 1165 ± 435 42.42 ± 0.28 37.0 ± 13.8 42.27 ± 0.11 25.4 ± 6.3 L L 42.4 ± 0.01 101.6 ± 2.0 40.66 ± 0.14 1.8 ± 0.6 RM622 44.3 ± 0.01 44.49 ± 0.03 43.03 ± 0.07 49.8 ± 7.5 -1222 ± 609 42.54 ± 0.24 20.8 ± 9.9 41.85 ± 0.26 3.5 ± 3.4 L L 42.62 ± 0.01 81.1 ± 1.1 41.14 ± 0.04 2.6 ± 0.3 RM720 44.3 ± 0.01 44.22 ± 0.04 43.1 ± 0.07 114.3 ± 16.7 -328 ± 240 42.29 ± 0.14 20.4 ± 5.2 42.29 ± 0.09 18.0 ± 3.8 L L 42.46 ± 0.01 61.8 ± 0.4 41.5 ± 0.03 6.4 ± 0.5 RM781 43.6 ± 0.01 43.88 ± 0.09 42.49 ± 0.05 55.1 ± 5.9 -189 ± 343 41.63 ± 0.15 10.0 ± 4.0 41.36 ± 0.24 4.2 ± 4.1 L L 41.91 ± 0.02 65.8 ± 2.8 41.27 ± 0.05 14.6 ± 1.6 RM782 43.93 ± 0.01 43.04 ± 0.31 42.82 ± 0.04 66.3 ± 5.9 -586 ± 431 41.8 ± 0.11 6.0 ± 1.5 L L L L 42.0 ± 0.01 37.0 ± 0.5 40.5 ± 0.06 1.1 ± 0.2 RM790 43.33 ± 0.01 L L L L 41.29 ± 0.18 13.4 ± 4.7 41.16 ± 0.3 20.2 ± 15.1 L L 41.87 ± 0.03 53.9 ± 3.2 24.03 ± 8.55 0.0 ± 5.0 RM840 43.24 ± 0.01 43.66 ± 0.13 42.65 ± 0.05 153.7 ± 17.4 -218 ± 269 42.07 ± 0.07 53.3 ± 7.3 41.47 ± 0.31 10.8 ± 5.2 L L 41.6 ± 0.01 38.9 ± 0.7 39.47 ± 19.82 0.3 ± 0.2 Mrk 335 43.76 ± 0.07 43.72 ± 0.01 42.63 ± 0.01 108.2 ± 1.8 -127 ± 23 42.8 ± 0.02 135.6 ± 7.5 41.5 ± 0.02 7.9 ± 0.3 41.91 ± 0.03 15.5 ± 1.0 42.09 ± 0.09 108.2 ± 0.1 L L PG 0026 44.97 ± 0.02 45.26 ± 0.01 43.55 ± 0.01 32.8 ± 1.1 -993 ± 129 L L 42.8 ± 0.15 6.5 ± 3.7 L L 42.93 ± 0.04 46.2 ± 0.0 L L PG 0052 44.81 ± 0.03 45.36 ± 0.01 44.18 ± 0.03 113.7 ± 7.6 670 ± 51 42.53 ± 0.18 3.8 ± 2.4 42.47 ± 0.61 2.5 ± 9.0 L L 43.13 ± 0.05 107.4 ± 0.0 L L Fairall 9 43.98 ± 0.04 44.43 ± 0.01 43.33 ± 0.01 114.9 ± 0.9 61 ± 24 42.68 ± 0.12 29.0 ± 10.2 42.53 ± 0.18 19.1 ± 5.4 42.84 ± 0.01 53.5 ± 0.5 42.67 ± 0.04 249.8 ± 0.0 L L Mrk 590 43.5 ± 0.21 42.92 ± 0.01 42.27 ± 0.01 710.5 ± 7.3 356 ± 89 40.18 ± 0.3 2.4 ± 9.4 40.92 ± 0.09 32.4 ± 7.0 41.92 ± 0.01 48.1 ± 0.8 41.85 ± 0.12 108.6 ± 0.1 L L Mrk 1044 43.1 ± 0.1 43.42 ± 0.01 41.85 ± 0.01 42.7 ± 0.3 865 ± 26 41.08 ± 0.21 10.0 ± 7.1 41.58 ± 0.01 24.8 ± 0.7 40.93 ± 0.06 10.9 ± 1.5 41.39 ± 0.09 101.4 ± 0.1 L L 3C 120 44.0 ± 0.1 44.41 ± 0.01 43.31 ± 0.01 134.9 ± 1.5 13 ± 38 42.04 ± 0.1 10.5 ± 2.4 42.39 ± 0.02 17.7 ± 0.8 42.67 ± 0.01 72.3 ± 1.3 42.36 ± 0.04 118.8 ± 0.0 L L Ark 120 43.87 ± 0.25 44.61 ± 0.01 43.32 ± 0.01 77.9 ± 1.0 20 ± 25 42.91 ± 0.04 36.2 ± 3.3 42.69 ± 0.12 18.8 ± 4.1 42.76 ± 0.01 37.5 ± 0.4 42.54 ± 0.13 244.8 ± 0.1 L L MCG 811 43.33 ± 0.11 42.81 ± 0.04 41.91 ± 0.01 172.7 ± 5.5 643 ± 170 41.54 ± 0.16 45.5 ± 13.7 41.15 ± 0.04 27.2 ± 2.8 41.96 ± 0.01 72.3 ± 1.1 41.66 ± 0.09 108.7 ± 0.1 L L Mrk 374 43.77 ± 0.04 43.72 ± 0.02 42.8 ± 0.06 206.6 ± 26.5 -20 ± 227 41.76 ± 0.11 24.9 ± 7.5 40.56 ± 0.63 1.3 ± 9.5 41.91 ± 0.04 43.1 ± 3.5 41.83 ± 0.04 58.3 ± 0.0 L L Mrk 79 43.68 ± 0.07 43.59 ± 0.01 42.46 ± 0.01 115.7 ± 3.7 2446 ± 75 42.0 ± 0.05 46.9 ± 4.7 42.06 ± 0.05 49.0 ± 4.5 41.75 ± 0.02 26.2 ± 1.5 41.9 ± 0.05 85.4 ± 0.0 L L Mrk 382 43.12 ± 0.08 43.15 ± 0.01 L L L L L L L 40.15 ± 0.18 16.4 ± 7.1 41.01 ± 0.05 39.6 ± 0.0 L L PG 0804 44.91 ± 0.02 45.36 ± 0.01 44.04 ± 0.01 87.5 ± 1.4 -923 ± 596 42.66 ± 0.19 6.3 ± 4.1 42.78 ± 0.06 5.3 ± 0.7 43.0 ± 0.04 19.2 ± 1.7 43.29 ± 0.03 122.5 ± 0.0 L L PG 0844 44.22 ± 0.07 44.59 ± 0.01 43.01 ± 0.02 38.6 ± 1.5 -141 ± 59 42.8 ± 0.18 28.3 ± 8.9 42.67 ± 0.02 18.6 ± 1.0 42.27 ± 0.08 11.3 ± 2.4 42.56 ± 0.05 111.2 ± 0.0 L L Mrk 110 43.66 ± 0.12 43.92 ± 0.01 43.05 ± 0.01 186.4 ± 2.1 -762 ± 37 41.92 ± 0.08 14.5 ± 2.7 42.02 ± 0.01 17.3 ± 0.6 42.4 ± 0.01 65.2 ± 0.5 42.03 ± 0.08 123.8 ± 0.1 L L PG 0953 45.19 ± 0.01 45.79 ± 0.01 44.71 ± 0.02 334.5 ± 17.5 52 ± 52 L L L L L L 43.29 ± 0.04 64.7 ± 0.0 L L NGC 3227 42.24 ± 0.11 39.99 ± 0.12 L L L 39.74 ± 0.1 50.8 ± 9.4 38.66 ± 0.15 18.5 ± 9.3 40.17 ± 0.01 38.7 ± 1.0 40.38 ± 0.1 71.0 ± 0.1 L L NGC 3516 42.79 ± 0.2 43.11 ± 0.01 41.8 ± 0.01 69.3 ± 0.4 -1711 ± 73 41.74 ± 0.12 71.4 ± 14.8 41.04 ± 0.01 12.3 ± 0.3 41.5 ± 0.02 77.4 ± 2.8 41.06 ± 0.18 94.7 ± 0.2 L L SBS 1116 42.14 ± 0.23 42.85 ± 0.01 41.95 ± 0.02 231.6 ± 13.5 -220 ± 62 41.12 ± 0.04 39.8 ± 3.8 40.72 ± 0.07 14.7 ± 2.6 41.02 ± 0.04 38.2 ± 3.4 40.7 ± 0.07 186.8 ± 0.1 L L Arp 151 42.55 ± 0.1 42.6 ± 0.01 41.76 ± 0.02 233.0 ± 8.6 -2167 ± 132 41.02 ± 0.05 52.1 ± 5.5 40.39 ± 0.06 10.6 ± 1.5 40.89 ± 0.01 47.3 ± 0.8 40.95 ± 0.11 130.0 ± 0.1 L L NGC 3783 42.56 ± 0.18 43.34 ± 0.01 42.4 ± 0.01 171.1 ± 2.5 -137 ± 29 41.7 ± 0.1 32.7 ± 7.1 41.29 ± 0.02 13.4 ± 0.6 41.83 ± 0.01 54.9 ± 1.2 41.01 ± 0.18 144.0 ± 0.2 L L Mrk 1310 42.29 ± 0.14 41.76 ± 0.01 41.23 ± 0.03 485.7 ± 37.6 -183 ± 250 40.1 ± 0.27 28.7 ± 31.6 L L 40.65 ± 0.06 64.7 ± 9.4 40.56 ± 0.1 94.3 ± 0.1 L L NGC 4051 41.9 ± 0.15 41.36 ± 0.01 39.99 ± 0.01 62.1 ± 1.4 482 ± 516 39.79 ± 0.1 35.7 ± 7.5 39.35 ± 0.02 13.9 ± 0.7 40.03 ± 0.01 61.1 ± 1.8 40.09 ± 0.19 78.6 ± 0.2 L L NGC 4151 42.09 ± 0.21 42.9 ± 0.02 42.12 ± 0.01 210.0 ± 2.6 139 ± 44 41.36 ± 0.03 37.3 ± 2.4 40.76 ± 0.02 8.9 ± 0.4 41.37 ± 0.01 46.3 ± 0.7 40.56 ± 0.2 150.8 ± 0.2 L L PG 1211 44.73 ± 0.08 44.9 ± 0.01 43.74 ± 0.02 111.8 ± 5.5 499 ± 28 43.04 ± 0.07 28.8 ± 4.5 42.55 ± 0.04 7.6 ± 0.7 42.38 ± 0.06 7.3 ± 1.0 43.02 ± 0.06 100.2 ± 0.1 L L NGC 4253 42.57 ± 0.12 41.9 ± 0.03 40.48 ± 0.06 61.6 ± 7.9 35 ± 267 39.6 ± 0.25 8.2 ± 11.3 L L 40.41 ± 0.08 30.7 ± 5.8 40.77 ± 0.12 81.1 ± 0.1 L L PG 1229 43.7 ± 0.05 44.56 ± 0.01 43.19 ± 0.01 62.3 ± 2.0 1038 ± 183 42.78 ± 0.04 29.8 ± 2.3 42.74 ± 0.12 22.9 ± 4.9 42.52 ± 0.02 27.7 ± 1.5 42.31 ± 0.06 209.7 ± 0.1 L L NGC 4593 42.62 ± 0.37 42.36 ± 0.01 41.5 ± 0.01 198.8 ± 1.2 -113 ± 32 40.66 ± 0.21 21.6 ± 11.8 40.42 ± 0.24 15.4 ± 14.0 41.14 ± 0.01 61.7 ± 0.6 40.95 ± 0.33 108.3 ± 0.3 L L NGC 4748 42.56 ± 0.12 42.87 ± 0.01 41.52 ± 0.02 70.4 ± 3.5 -1069 ± 559 41.66 ± 0.07 110.9 ± 15.9 41.22 ± 0.08 36.5 ± 6.6 40.99 ± 0.01 31.3 ± 0.9 40.98 ± 0.1 136.8 ± 0.1 L L PG 1307 44.85 ± 0.02 45.34 ± 0.01 44.06 ± 0.02 90.0 ± 3.9 975 ± 56 L L L L 42.6 ± 0.12 6.8 ± 1.8 43.13 ± 0.06 98.4 ± 0.1 L L MCG 630 41.64 ± 0.11 40.25 ± 0.37 L L L 39.92 ± 0.21 80.6 ± 45.2 L L L L 39.85 ± 0.19 91.0 ± 0.2 L L NGC 5273 41.54 ± 0.16 41.0 ± 0.05 39.89 ± 0.1 175.2 ± 37.2 -352 ± 736 39.62 ± 0.29 277.1 ± 131.1 39.29 ± 0.27 54.9 ± 22.4 38.54 ± 0.16 9.6 ± 3.6 39.74 ± 0.11 82.2 ± 0.1 L L Mrk 279 43.71 ± 0.07 43.99 ± 0.01 41.76 ± 0.02 9.7 ± 0.5 1160 ± 184 41.79 ± 0.23 13.7 ± 12.7 41.53 ± 0.02 6.1 ± 0.3 L L 42.12 ± 0.06 132.2 ± 0.1 L L PG 1411 44.56 ± 0.02 44.65 ± 0.01 43.33 ± 0.02 72.1 ± 2.4 -1619 ± 77 43.15 ± 0.35 57.8 ± 33.8 42.31 ± 0.03 7.2 ± 0.5 L L 42.85 ± 0.03 99.7 ± 0.0 L L NGC 5548 43.3 ± 0.19 43.58 ± 0.01 42.55 ± 0.01 134.3 ± 0.5 549 ± 28 42.28 ± 0.03 71.2 ± 4.9 41.76 ± 0.01 21.7 ± 0.4 42.17 ± 0.01 74.5 ± 0.4 41.65 ± 0.21 117.8 ± 0.2 L L PG 1426 44.63 ± 0.02 45.2 ± 0.01 43.72 ± 0.05 55.6 ± 6.0 208 ± 215 43.13 ± 0.05 20.3 ± 2.4 43.2 ± 0.05 18.2 ± 2.2 42.97 ± 0.04 21.9 ± 2.1 42.83 ± 0.04 80.1 ± 0.0 L L Mrk 817 43.74 ± 0.09 44.29 ± 0.01 42.82 ± 0.01 58.1 ± 0.7 357 ± 38 41.72 ± 0.25 7.0 ± 4.0 42.25 ± 0.01 17.4 ± 0.3 42.37 ± 0.05 58.9 ± 6.0 41.93 ± 0.14 78.5 ± 0.1 L L Mrk 1511 43.16 ± 0.06 43.48 ± 0.01 42.32 ± 0.03 112.0 ± 8.3 -245 ± 238 41.37 ± 0.15 15.4 ± 4.9 L L 41.56 ± 0.09 25.9 ± 6.5 41.52 ± 0.06 115.5 ± 0.1 L L Mrk 290 43.17 ± 0.06 43.77 ± 0.01 42.71 ± 0.01 129.0 ± 1.8 321 ± 45 42.15 ± 0.13 39.0 ± 9.9 41.86 ± 0.04 18.3 ± 1.5 42.1 ± 0.01 54.9 ± 0.9 41.64 ± 0.06 153.0 ± 0.1 L L Mrk 486 43.69 ± 0.05 43.17 ± 0.02 41.88 ± 0.16 55.2 ± 15.3 -1110 ± 70 41.0 ± 0.21 5.5 ± 3.7 41.16 ± 0.09 9.7 ± 2.1 42.01 ± 0.07 41.6 ± 7.2 42.12 ± 0.04 135.9 ± 0.0 L L PG 1613 44.77 ± 0.02 45.18 ± 0.01 43.93 ± 0.01 88.7 ± 1.3 1117 ± 84 42.8 ± 0.18 9.6 ± 4.5 L L 43.35 ± 0.02 49.6 ± 2.4 43.0 ± 0.03 86.7 ± 0.0 L L PG 1617 44.39 ± 0.02 44.87 ± 0.01 42.98 ± 0.02 21.5 ± 0.9 -72 ± 123 43.26 ± 0.04 53.5 ± 4.7 41.91 ± 0.07 2.0 ± 0.3 43.2 ± 0.03 69.8 ± 5.1 42.74 ± 0.05 114.8 ± 0.0 L L 3C 382 43.84 ± 0.1 44.48 ± 0.01 43.52 ± 0.01 170.3 ± 5.0 39 ± 51 42.21 ± 0.13 11.1 ± 2.9 41.92 ± 0.29 4.6 ± 5.2 42.8 ± 0.01 64.3 ± 1.2 42.54 ± 0.03 259.3 ± 0.0 L L 3C 390.3 44.43 ± 0.58 43.73 ± 0.01 42.79 ± 0.01 226.6 ± 3.4 1474 ± 84 42.4 ± 0.2 112.6 ± 33.4 42.1 ± 0.23 52.5 ± 17.9 42.64 ± 0.01 113.5 ± 3.3 42.6 ± 0.35 108.8 ± 0.4 L L Table 2 (Continued) Objects L log 5100 l L log 1350 l L log CIV REW (C IV) C IV Blueshift ] L log CIII REW (C III]) L log HeII1640 REW (He II 1640) L log MgII REW (Mg II) L log Hb REW (Hβ) L log He II 4687

4687) NGC 6814 42.12 ± 0.28 41.66 ± 0.02 40.61 ± 0.03 182.6 ± 11.7 -167 ± 416 40.31 ± 0.14 132.6 ± 39.1 38.95 ± 0.25 4.6 ± 2.0 40.29 ± 0.01 87.3 ± 2.3 40.5 ± 0.28 121.6 ± 0.3 L L Mrk 509 44.19 ± 0.05 44.62 ± 0.01 43.49 ± 0.01 116.7 ± 0.9 344 ± 31 43.12 ± 0.03 65.2 ± 4.7 42.55 ± 0.05 14.5 ± 1.6 42.86 ± 0.01 49.7 ± 0.7 42.61 ± 0.04 132.7 ± 0.0 L L NGC 7469 43.51 ± 0.11 43.85 ± 0.01 42.49 ± 0.01 66.7 ± 0.6 -253 ± 40 41.79 ± 0.06 16.4 ± 2.1 41.93 ± 0.02 19.3 ± 1.1 42.1 ± 0.01 50.2 ± 1.7 41.6 ± 0.1 63.0 ± 0.1 L L Zw 229 42.71 ± 0.05 43.16 ± 0.01 42.25 ± 0.04 187.0 ± 20.8 870 ± 485 41.37 ± 0.04 28.1 ± 2.7 41.16 ± 0.37 16.1 ± 13.8 41.52 ± 0.01 49.6 ± 1.6 L L L L Mrk 1501 44.32 ± 0.05 44.7 ± 0.01 43.61 ± 0.01 120.3 ± 3.0 -108 ± 88 43.18 ± 0.13 48.6 ± 10.9 43.08 ± 0.07 36.0 ± 4.8 42.99 ± 0.05 35.0 ± 3.6 L L L L Mrk 50 42.88 ± 0.07 43.22 ± 0.01 42.16 ± 0.02 152.6 ± 6.4 -57 ± 86 41.01 ± 0.16 14.4 ± 7.6 L L 41.4 ± 0.01 45.4 ± 0.7 L L L L NGC 4395 39.77 ± 0.02 39.82 ± 0.01 38.62 ± 0.05 82.9 ± 11.5 1187 ± 544 L L 37.83 ± 0.02 14.0 ± 0.6 L L L L L L Mrk 6 43.75 ± 0.06 42.09 ± 0.1 40.9 ± 0.16 91.0 ± 29.6 4497 ± 2534 40.33 ± 0.18 22.5 ± 10.5 40.44 ± 0.08 31.3 ± 5.5 40.94 ± 0.04 64.7 ± 6.3 L L L L 10 The Astrophysical Journal, 980:134 (19pp), 2025 February 10 Wu et al. L d ion product destruct ò n = n n n within the production and destruction photon energies), the middle panels display the line luminosities predicted by the LOC model, and the right panels present the observed line luminosities as a visual assessment of the theoretical photoionization. The increasing trend in all panels demonstrates the consistency between our theoretical computations and observations.

observations suggests that our observed emission line flux can be explained by the simple LOC model and depends on the BH parameters and subsequently the ionizing SEDs. For even lower accretion rates (  m log 1.5 < -), the computational limitations of qsosed restrict us from evaluating the incident continuum and subsequently performing photoionization computations.

We also compare theoretical and observed rest-frame equivalent widths (EWs) of several emission lines as functions of BH mass and Eddington ratio in Figure 9. In the left panels, we use the incident SED and LOC-predicted line flux to predict the EW. For massive BHs (M BH > 10 8 M e ), more luminous (higher accretion rate) BHs exhibit weaker emission line strength, particularly for UV lines, and vice versa for BHs with mass M BH 10 8 M e , as predicted by the LOC model. This pattern predicted by the LOC model indicates that the continuum and line luminosities change nonlinearly with M BH and λ Edd . This nonlinear relationship suggests that the underlying SED shape may be more complex and varies significantly with changes in these parameters. The middle panels of Figure 9 show the observed EW values, and the right panels display binned observed EW values. The Baldwin effect could be observed, as the higher-accretion and more massive objects show lower EWs in all three panels. Although the predicted EWs in C III], He II λ4687, and He II λ1640 are slightly higher than observed, the overall observed EW values remain generally consistent with the predictions from the LOC model. This consistency is expected as the observed continuum luminosities and line flux agree with the prediction in Figures 4 and 8, respectively. However, due to sample selection biases for the RM AGNs (i.e., missing low-mass, low accretion rate objects), it is difficult to explore the EW in the lower left corner of the M BH -λ Edd space; thus, we are unable to demonstrate the same diagonal trend in the LOC model prediction.

Comparisons of Composite UV Spectra

Since observed (and measured) properties in individual objects can suffer from significant measurement uncertainties, we also use the higher-S/N composite spectra to compare with theoretical predictions. Figure 10 presents the composite restframe UV spectra for RM AGNs within a matched optical luminosity range ( ( ) L 43 log 45 5100 l   ) but with different λ Edd (top panel) and Hβ lags (bottom panel). The fluxes are normalized at rest-frame wavelength 1350 Å to better illustrate the UV line strength. In the top panel, subsets with higher λ Edd values (green) exhibit stronger Lyα line strengths compared to the lower accretion group (blue). According to the SED example in Figure 3 and our photoionization results in Figure 9, at a matched optical luminosity, BHs with higher accretion rates possess more ionizing energy for both hydrogen and C IV, resulting in stronger emission lines. This trend is evident in our composite spectra, where higher accretion rate samples tend to show stronger emission lines. This agreement demonstrates that our photoionization models can accurately generate line strengths based on the given BH parameters.

Likewise, in the bottom panel of Figure 10, we divide our sample into two subsets based on their Hβ time lag relative to the canonical R-L relation. The subset with lags shorter than the R-L relation (green) has stronger C IV line strength compared to the longer-lag subset (blue), while these two spectra appear to have similar Lyα line strength. Given that C IV is a collisionally excited line with an excitation energy of 8 eV, the shorter-lag subset suggests that it may have an additional ionizing flux at far-UV compared to the longer-lag subset, with the former corresponding to the higher-accreting systems in Figure 3. However, it appears to contradict our BLR size prediction in Figure 5 that higher-accreting objects should have more ionizing energy and thus longer lags. This discrepancy may be due to the constant parameters of our BLR size prediction, which will be discussed further in Section 4.2.

Discussion

Line Ratios as a Proxy for SED Shape

Our LOC model photoionization calculations have shown qualitative agreement with the observed line properties (line strengths, BLR distances, etc.). Since lines with different ionization potentials depend on different parts of the SED, it is desirable to infer the underlying ionizing SED (not directly observable) from observables, such as the flux ratios between low-and high-ionization lines.

Figure 11 displays several groups of continuum luminosity ratios and line flux ratios as functions of M BH and λ Edd , predicted from photoionization calculations. In the top panels of Figure 11, the monochromatic luminosities at the ionizing energies of C IV (64.5 eV), Mg II (15.0 eV), Hβ (13.6 eV), and He II λ1640 (54.4 eV) are computed from the grid of SEDs, and then their line ratios are placed in the M BH -λ Edd plane. The bottom panels show the corresponding line luminosity ratios of the same lines as in the top panels. The line luminosity ratio is computed using CLOUDY and the LOC model, as explained in Section 2.2. Both monochromatic luminosity ratios and line luminosity ratios demonstrate a complicated dependence on the mass and accretion rate. Given that C IV and Mg II are collisionally excited lines with excitation energies of 8 and 4.4 eV, while Hβ and He II λ1640 are recombination-dominated lines, the line ratios involving C IV may not accurately trace the SED shape, especially at the high-mass, low-accretion end; in contrast, He II λ1640 is more effective in tracing the SED shape in Figure 11.

Given the strong correlations between predicted continuum luminosity ratios and line flux ratios in Figure 11, we use the BH mass M BH , accretion rate λ Edd , and line flux ratios in our theoretical computation models to provide empirical relations that map the underlying ionizing SED shape from ∼10 to ∼60 eV. The concept of using UV emission line strengths to probe the ionizing SED has been previously applied to the EW of He II λ1640 as tracers of the EUV continuum (e.g., W. G. Mathews & G. J. Ferland 1987; K. T. Korista & M. R. Goad 2000; K. M. Leighly 2004; M. J. Temple et al. 2023). However, this approach does not consider the possibility that the ionizing continuum that the BLR gas receives is not the same as what observers see; thus, the He II λ1640 EW is not tracing the real ionizing continuum shape but the ratio between ionizing energy for He II λ1640 and the continuum that the observer receives (J. D. I. Timlin et al. 2021). To address this issue, we use the flux ratio of different UV/optical line pairs to directly map the underlying SED shape, which involves only the ionizing continuum received by the BLR gas.

To demonstrate the ability of observables to trace the SED shape with empirical relations, we use linear regression to fit the theoretical results with the following relation:

, 5

where L E is the monochromatic luminosity at energy E and E 1 and E 2 are the ionizing energy of line 1 and line 2, respectively. The best-fit parameters are listed in Table 3. The scaling relations between the observables and the ionization continuum shape are shown in Figure 12. The coefficients

) of the four pairs of lines suggest that the line ratios can trace the ionizing continuum shape at ∼10 eV fairly well, especially for C IV/Mg II.

Multiwavelength observations probing additional emission lines with different ionizing energies will allow mapping the ionizing continuum to an even wider range. For example, the line ratios of infrared coronal lines (CLs) from different elements and different excitation states have proven to be related to the intrinsic BH properties (M BH and  m), chemical abundance, and ionization parameters (J. W. Ferguson et al. 1997). Most importantly, CLs have the capability to map the ionizing continuum to energy levels exceeding 100 eV, making these data more powerful than UV data alone. Photoionization computations in J. M. Cann et al. (2018) demonstrate that several pairs of CL ratios might trace intermediate-mass BHs in dwarf galaxies, as the harder SEDs in intermediate-mass BHs can excite stronger high-ionization lines. With the advent of the James Webb Space Telescope, infrared spectroscopic observations covering a wealth of IR CLs will provide a viable approach to studying the connections between the ionizing continuum and CL emission and to understanding the physical driver of the diversity in quasar line strength. Martìnez-Aldama et al. 2019;E. Dalla Bontà et al. 2020;G. Fonseca Alvarez et al. 2020;Y. Shen et al. 2024). Moreover, smearing our theoretical predictions with observed uncertainties demonstrates that uncertainties in the measured time lag along with luminosity variability can introduce additional scatter in the R-L relation.

Dispersion in the

The scale of the BLR can also be affected by other accretion parameters that were not included in this preliminary study. For example, in the high-accretion regime (approaching the Eddington limit), the accretion disk structure may significantly depart from the SSD model (N. I. Shakura & R. A. Sunyaev 1973) and become a slim disk (M. A. Abramowicz et al. 1988), where the disk scale height is comparable to the radius. The slim disk geometry will produce anisotropic radiation fields due to the self-shadowing effect, which could reduce the ionizing flux and soften the SED that the BLR cloud receives. In particular, systematically shorter Hβ time lags in high-accretion quasars have been reported by the SEAMBH collaboration (P. Du et al. 2014Du et al. , 2015;;P. Du et al. 2018;C. Hu et al. 2021). Due to considerable theoretical uncertainties in the slim disk SED predictions and the relative paucity of UV spectroscopy for RM AGNs accreting in this regime, we defer a more complete investigation of AGNs with slim disk accretion to a future paper.

It is important to note, however, that our investigations suggest that in the low-to-moderate-accretion thin-disk regime, the BLR time lag exhibits a strong dependence on both the BH mass and accretion rate. As demonstrated in Figure 5, an increase in either the BH mass or accretion rate leads to a corresponding increase in both the optical luminosity and the time lag, and the accretion rate has a more significant impact on the lag than the BH mass. This prediction, however, is in tension with the observed lag dispersion at fixed optical luminosity, suggesting that additional factors may influence the observed lags. Figure 13 illustrates the observed R-L relation for our RM AGN sample (left panel) and highlights how the Eddington accretion rate influences the Hβ lag across different optical luminosities (right). The data points are divided into eight groups based on their optical luminosity λL 5100 , with observed data points within each group color-coded according to their relative Eddington ratio in each bin. Even within the SSD regime analyzed in this study, a subtle decreasing trend in Hβ lags is observed as L/L Edd increases in each luminosity bin. While our model does not explore a broad parameter space in the right panels, it nonetheless predicts an increasing Hβ lag with higher L/L Edd . This discrepancy suggests that certain simple assumptions in our photoionization modeling, e.g., a universal gas density, might be incorrect. For example, if n H increases with L/L Edd , and given that

µ - (Equation ( 4)), we expect a more compact BLR size from the LOC model, which would be more consistent with the observed trend in the R-L plane. Additionally, as n H increases, broad-line emissivity would also increase, resulting in even higher predicted broad-line fluxes for higher Eddington ratios, again consistent with observations. Other choices of photoionization parameters or assumptions may also yield alternative R-L trends, which we leave for future studies to further explore these dependencies.

Table 3 Best-fit Parameters of Line Excitation Energy Ratios ( ) / L L log E E 1 2 Defined in Equation (5) Line Ratios θ 0 θ 1 θ 2 θ 3 C IV/Hβ -0.894 -0.169 0.567 3.163 C IV/Mg II -1.602 -0.109 0.335 5.362 He II λ1640/Hβ -0.327 -0.045 0.106 2.535 He II λ1640/Mg II -0.451 0.040 -0.257 3.163 Figure 12. Scaling relations of monochromatic luminosity ratios (i.e., SED shape) and observables (M BH , λ Edd , and line flux ratio) for the theoretical models, with the coefficients of determination R 2 in the upper left corners.

Predicted Hydrogen-ionizing Photon Flux

In our theoretical analysis, the time lag of Hβ is modeled as a linear function of the hydrogen-ionizing photon flux Q(H), as expressed by Equation (4). Because the observed ionizing continuum may not be the same as the BLR receives, the flux ratios of emission lines from species with different ionization potentials are the most promising way to constrain the ionizing photon flux that directly determines the BLR size (C. A. Negrete et al. 2012;G. J. Ferland et al. 2020;T. M. Buendia-Rios et al. 2023). As discussed in Section 4.1, the line flux ratios, coupled with the mass and Eddington ratio, have demonstrated their potential in deriving the SED shape. Therefore, we adopt a similar approach to predict the Q(H) for observed quasars through the following equation:

We first fit a linear regression relation on the parameter grid derived from theoretical models, and the best-fit parameters are listed in Table 4. The linear regression scaling is shown in Figure 14 for the four line pairs, where the ionizing photon flux Q(H) could be well estimated using such observables (M BH , λ Edd , and line flux ratios).

Subsequently, we apply the regression parameters to the observed RM quasars. Figure 15 presents the color-coded R-L relation for low-accretion AGNs with their predicted Q(H)/λL 5100 value. Since our assumption in Section 2.3 implies that R BLR ∼ Q 0.5 , suggesting that at any given λL 5100 , Hβ lags would follow a vertical increase as Q(H)/λL 5100 , such a vertical trend can be seen in Figure 15. However, in Figure 13, Hβ lags show a decreasing trend as relative accretion rates increase, suggesting the opposite trend to Figure 15. This discrepancy is possibly due to oversimplified assumptions (e.g., the constant hydrogen density) in our BLR size prediction (see Section 4.2).

Uncertainties and Limitations

The primary goal of this paper is to explore the first-order effects of varying the BH mass and accretion rate on the BLR distance and the diverse line strengths within the LOC model framework. While the success of the LOC model in previous studies is often attributed to its flexibility in determining photoionization parameters (e.g., n(H), U H , and the covering factor) on a case-by-case basis for a small number of objects (e.g., K. T. Korista & M. R. Goad 2000, 2004), our goal is to investigate broader trends across a representative AGN population. The use of predetermined LOC model parameters in our approach allows for a more generalized statistical analysis, which is necessary to capture the impact of BH parameters on the observables for the AGN population studies. The parameters adopted in this study are well tested by multiple previous studies (e.g., W. G. Mathews & G. J. Ferland 1987; theoretical analysis and thereby our understanding of AGN SEDs. However, the demonstrated correlations between the X-ray, EUV, and UV regions (e.g., P. J. Green 1996; A. T. Steffen et al. 2006;D. W. Just et al. 2007;E. Lusso & G. Risaliti 2016;J. D. Timlin et al. 2020;J. D. I. Timlin et al. 2021) may provide crucial insights that could uncover the nature of the hot/warm corona in quasars. Despite that this simplified qsosed model may provide limited flexibility in SED shapes, our LOC photoionization computations indicate that this straightforward model can successfully reproduce the line strengths as observed for various BH mass and accretion rates. As discussed in Section 4.2, as the accretion rate substantially increases beyond the SSD-dominated regime, the accretion disk solution may be better described by the slim disk model with a puffed-up inner disk that shadows outer regions, which will potentially soften the SED received by the BLR clouds (M. A. Abramowicz et al. 1988;J.-M. Wang et al. 2014b). Due to the complexities involved in modeling such a scenario and the uncertainties in slim disk SED prediction, we postpone a comprehensive investigation of these high-accretion AGNs to future work.

While we demonstrate the potential of using line ratios to probe the SED shape and the ionizing flux, the predicted line ratios may deviate from observed line ratios. This discrepancy can arise from factors such as differences in metallicity and assumptions regarding various ionizing conditions (K. T. Korista & M. R. Goad 2019; S. Panda et al. 2019). For instance, the radiative pressure confinement (RPC) model (A. Baskin et al. 2014) offers an alternative BLR structure by assuming that the radiation pressure confines photoionized gas, leading to a universal U ∼ 0.1 and n(H) ∼ 10 11 for the inner layer, independent of distance and luminosity. Under RPC conditions, the predicted R-L trend remains unchanged since only the product of U and n(H) influences the scaling in Equation (4), which cannot explain the shorter lags for highaccretion AGNs. Although the gas is not at its highest emissivity at such U and n(H), the constraints on the photoionization parameters make the line flux ratio more dependent on the incident SED shape compared to the LOC model. Although mechanisms other than BH mass and accretion rate will have some effects on the BLR size and observed line strength, we have demonstrated that variations in the M BH -λ Edd have successfully reproduced the lags and emission line properties over many orders of magnitude in luminosity and BH mass. Nevertheless, discrepancies remain in the detailed predictions of the lag dispersion at fixed optical luminosity, as discussed in Section 4.2.

There is the possibility that the BLR gas sees a different ionizing continuum than the observer, due to a potential inner shielding gas component that blocks the ionizing X-ray and EUV continuum (e.g., J. Wu et al. 2011Wu et al. , 2012;;B. Luo et al. 2015;R. M. Plotkin et al. 2015;Q. Ni et al. 2018Q. Ni et al. , 2022)). In this self-shielding gas model, the disk component (<1 keV) in the incident SED is reduced by the shielding gas (J.-M. Wang et al. 2014b), whereas the warm and hot Comptonized components are not affected much. Consequently, the hydrogen-ionizing photon flux Q(H) is reduced as well. Given that Q(H) ∝ ΦR 2 (Equation ( 4)), this reduction means that, compared to our LOC model, gas at a fixed distance (R) would have a lower surface ionizing flux Φ, and gas with the same Φ would reside closer to the center. As a result, the overall line emission from the BLR would be reduced, and the BLR clouds would be more concentrated toward the center in this shielding gas model. In this work, we do not observe much evidence for this shielding gas component, as the predicted line strengths are consistent with the BLR seeing all the continuum flux. However, the covering fraction (G. J. Ferland et al. 2020) and the impact of this shielding gas may become more prominent for even higher Eddington ratios, where the inner region of the slim disk may be geometrically thicker. Due to the complexity of modeling self-shielded SEDs, we plan to investigate this possibility in future work.

Conclusions

In this paper, we have performed CLOUDY photoionization calculations with the empirical LOC model and typical parameters to predict the broad-line emission as functions of accretion parameters, BH mass, and accretion rate using the latest AGN SED models (A. Kubota & C. Done 2018) as the input continuum. We compare our theoretical results with observed AGN properties for a sample of 67 low-redshift AGNs with RM measurements that are in the sub-Eddington accretion regime. Our main conclusions are as follows.

1. Assuming universal values of hydrogen density ( 12) and ionization parameter ( U log 2 H = -), the photoionization models reproduce the observed global relation between the Balmer broad-line lag and optical luminosity at rest-frame 5100 Å (Section 2.3). However, at fixed L 5100 , the model predicts longer lags for higher Eddington ratios, opposite to the observed trend (Figure 13). This discrepancy implies that the assumption of constant gas density may not be realistic, or that other factors are at play, such as a self-shielding wind in high-Eddington-ratio AGNs that significantly reduces the ionizing flux seen by the BLR (Sections 4.2 and 4.4). 2. The photoionization models produce increasing optical and UV broad-line strengths with accretion rate at fixed BH mass or optical luminosity, which are consistent with observations. Overall, the distributions of different broadline fluxes in the M BH -λ Edd plane are consistent between observations and theoretical predictions (Section 3.2). 3. Based on these comparisons, we provide empirical scaling relations (Equations ( 5) and (6)) that utilize the observed BH mass, L/L Edd , and line flux ratios to constrain the ionizing continuum flux, which is not directly observable (Section 4.3).

The overall consistency between observations of AGN broad-line properties and photoionization modeling suggests that such analyses are important to fully decipher the accretion parameters and the resulting broad-line emission in broad-line AGNs. The optical and UV broad emission lines only sample a limited portion of the ionizing continuum, and future observations with high-ionization CLs (with ionizing energy at >150 eV) can lead to better constraints on the overall continuum SED. Further refinements of the LOC model and AGN continuum models, as well as extending to higher accretion rates better described by the slim disk model, will be crucial to fully understanding the dispersion of lags at fixed optical luminosity, an important step to designing more accurate single-epoch BH mass recipes using the R-L relation.