# Understanding the Broad-line Region of Active Galactic Nuclei with
Photoionization. I. the Moderate-Accretion Regime

Qiaoya Wu<EMAIL_ADDRESS>Department of Astronomy, University of
Illinois at Urbana-Champaign, Urbana, IL 61801, USA  Yue Shen
<EMAIL_ADDRESS>Department of Astronomy, University of Illinois at
Urbana-Champaign, Urbana, IL 61801, USA  National Center for Supercomputing
Applications, University of Illinois at Urbana-Champaign, Urbana, IL 61801,
USA  Hengxiao Guo Shanghai Astronomical Observatory, Chinese Academy of
Sciences, 80 Nandan Road, Shanghai 200030, People’s Republic of China  Scott
F. Anderson Astronomy Department, University of Washington, Box 351580,
Seattle, WA 98195, USA  W. N. Brandt Department of Astronomy and Astrophysics,
The Pennsylvania State University, University Park, PA 16802, USA  Institute
for Gravitation and the Cosmos, The Pennsylvania State University, University
Park, PA 16802, USA  Department of Physics, 104 Davey Laboratory, The
Pennsylvania State University, University Park, PA 16802, USA  Catherine J.
Grier Department of Astronomy, University of Wisconsin-Madison, Madison, WI
53706, USA  Patrick B. Hall Department of Physics and Astronomy, York
University, 4700 Keele St., Toronto, ON M3J 1P3, Canada  Luis C. Ho Kavli
Institute for Astronomy and Astrophysics, Peking University, Beijing 100871,
China  Department of Astronomy, School of Physics, Peking University, Beijing
100871, China  Yasaman Homayouni Department of Astronomy and Astrophysics, The
Pennsylvania State University, University Park, PA 16802, USA  Space Telescope
Science Institute, 3700 San Martin Drive, Baltimore, MD 21218, USA  Keith
Horne SUPA Physics and Astronomy, University of St Andrews, Fife, KY16 9SS, UK
Jennifer I-Hsiu Li Michigan Institute for Data Science, University of
Michigan, Ann Arbor, MI, 48109, USA  Department of Astronomy, University of
Michigan, Ann Arbor, MI, 48109, USA  Donald P. Schneider Department of
Astronomy and Astrophysics, The Pennsylvania State University, University
Park, PA 16802, USA  Institute for Gravitation and the Cosmos, The
Pennsylvania State University, University Park, PA 16802, USA

###### Abstract

Over three decades of reverberation mapping (RM) studies on local broad-line
active galactic nuclei (AGNs) have measured reliable black-hole (BH) masses
for $>100$ AGNs. These RM measurements reveal a significant correlation
between the Balmer broad-line region size and the AGN optical luminosity (the
$R-L$ relation). Recent RM studies for AGN samples with more diverse BH
accretion parameters (e.g., mass and Eddington ratio) reveal a substantial
intrinsic dispersion around the average $R-L$ relation, suggesting variations
in the overall spectral energy distribution shape as functions of accretion
parameters. Here we perform a detailed photoionization investigation of
expected broad-line properties as functions of accretion parameters, using the
latest models for the AGN continuum implemented in qsosed. We compare
theoretical predictions with observations of a sample of 67 $z\lesssim 0.5$
reverberation-mapped AGNs with both rest-frame optical and UV spectra in the
moderate-accretion regime (Eddington ratio $\textit{$\lambda_{\rm Edd}$}\equiv
L/L_{\rm Edd}<0.5$). The UV/optical line strengths and their dependences on
accretion parameters can be reasonably well reproduced by the locally-
optimally-emitting cloud (LOC) photoionization models. We provide quantitative
recipes that use optical/UV line flux ratios to infer the ionizing continuum,
which is not directly observable. In addition, photoionization models with
universal values of ionization parameter ($\log U_{\rm H}=-2$) and hydrogen
density ($\log n({\rm H})=12$) can qualitatively reproduce the observed global
$R-L$ relation for the current AGN sample. However, such models fail to
reproduce the observed trend of decreasing BLR size with $L/L_{\rm Edd}$ at
fixed optical luminosity, which may imply that the gas density increases with
the accretion rate.

black hole physics — galaxies: active — quasars:

##  1 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
($z\lesssim 0.3$) that have successful mass measurements with H$\beta$ RM
(e.g. Blandford & McKee, 1982; Peterson, 1993; Peterson et al., 1998, 2002,
2004; Kaspi et al., 2007; Barth et al., 2015; U et al., 2022; Du et al., 2015,
2018; Hu et al., 2021; Malik et al., 2023; Woo et al., 2023). 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 (Lira et al., 2018; Grier
et al., 2019; Homayouni et al., 2020; Kaspi et al., 2021; Hoormann et al.,
2019; Malik et al., 2023; Yu et al., 2021). Earlier studies on nearby RM AGNs
have revealed a tight (i.e., scatter $\lesssim 0.15$ dex in BLR size)
correlation between the average distance from the BH to H$\beta$ -emitting BLR
clouds and the optical luminosity $\lambda L_{5100}$ of the AGN, known as the
radius-luminosity ($R-L$) relation (Kaspi et al., 2000, 2007; Bentz et al.,
2013). This $R-L$ relation provides the foundation for the so-called single-
epoch (SE) virial BH mass recipes (Vestergaard & Peterson, 2006; Shen, 2013)
that use a single monochromatic continuum luminosity to estimate the time lag.
Due to their simplicity, these single-epoch 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., Shen et al., 2011; Wu & Shen, 2022).
Furthermore, this tight local $R-L$ relation motivates its potential utility
as a standard candle to measure luminosity distances for cosmology (Watson et
al., 2011; Martínez-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., Du et al., 2014; Grier et al., 2017; Fonseca
Alvarez et al., 2020; Shen et al., 2024). In particular, the super-Eddington
accreting massive black hole (SEAMBH) collaboration (Du et al., 2014, 2015,
2018; Hu et al., 2021) as well as the Seoul National University (SNU) AGN
monitoring project (Woo et al., 2023) targeting high-accretion-rate AGNs have
found a surprising downward offset of the average H$\beta$ lag from the
canonical relation in these objects. The average time lags for these high-
accretion-rate AGNs are consistently shorter than predictions from the
canonical relation, and can reach as much as a factor of five times shorter in
individual objects, which translates to a similar factor of offset in BH mass.
The recent dynamical mass measurement of SDSS J092034.17+065718.0 ($z\approx
2.3$) from GRAVITY+ also reported a smaller BLR size than predicted by the
canonical $R-L$ relation for this super-Eddington accretion quasar (Abuter et
al., 2024). This spectro-interferometric method independently confirms that
super-Eddington AGNs may have smaller BLR sizes as measured from reverberation
mapping. Meanwhile, the Sloan Digital Sky Survey Reverberation Mapping (SDSS-
RM) project (Shen et al., 2015a, 2024; Blanton et al., 2017) targeting non-
local AGNs at $0.1<z<4.5$ also found a large dispersion of $\sim 0.3$ dex
around the mean H$\beta$ $R-L$ relation.

This increased scatter in the $R-L$ relation has gained significant interest
in recent work (Czerny et al., 2019; Du & Wang, 2019; Dalla Bontà et al.,
2020; 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 (Li et al.,
2019; Fonseca Alvarez et al., 2020). Several recent works have attempted to
use the optical Fe II strength (denoted by $R_{\rm FeII}\equiv{\rm EW}_{\rm
FeII,4434-4684}/{\rm EW}_{\rm H\beta}$) as a proxy to correct this offset (Du
& Wang, 2019; Martínez-Aldama et al., 2020), where $R_{\rm FeII}$ is
considered a proxy of the Eddington ratio through the Eigenvector 1 relations
(Boroson & Green, 1992; Shen & Ho, 2014).

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
black holes can be classified into three main regimes based on their accretion
rates (here we use the dimensionless Eddington ratio $\textit{$\lambda_{\rm
Edd}$}\equiv L_{\rm bol}/L_{\rm Edd}=L_{\rm bol}/(1.26\times
10^{38}\textit{$M_{\rm BH}$}/M_{\odot})$). Firstly, at low-to-moderate
accretion rates ($\textit{$\lambda_{\rm Edd}$}\lesssim 0.5$), the accretion
flow can be reasonably modeled by the geometrically thin Shakura-Sunyaev Disk
(SSD, Shakura & Sunyaev, 1973), where the optically-thick gas radiates multi-
temperature blackbody radiation. As the accretion rate substantially increases
to $\textit{$\lambda_{\rm Edd}$}\gtrsim 1$, the structure of the accretion
flow undergoes significant transformations due to increased radiation
pressure, leading to a slim disk (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 (Du et al., 2014). Conversely, at very low accretion rates
($\textit{$\lambda_{\rm 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 (ADAF, Yuan & Narayan, 2014).
Recognizing that the black-hole 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., Ho, 2008; Castelló-Mor et al.,
2016), and given that the exact transition $\lambda_{\rm Edd}$ between
different disk geometry remains uncertain, we focus on AGNs with low-to-
moderate accretion rates ($10^{-3}<\textit{$\lambda_{\rm 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 §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 H$\beta$ emission distance from RM and the observed
quasar spectral properties. In §2, we describe our theoretical models for the
AGN SEDs and our photoionization calculations using the LOC model (Baldwin et
al., 1995). In §3, we present our analysis of the observational data and
comparison with the theoretical framework. We discuss our results in §4 and
conclude in §5. Throughout this paper we adopt a flat $\Lambda$CDM cosmology
with $\Omega_{M}=0.3$ and $H_{0}=70\,\rm{km\,s^{-1}Mpc^{-1}}$.

##  2 Theoretical framework

A physically motivated photoionization model for the AGN BLR is the locally
optimally emitting cloud (LOC) model (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.

###  2.1 AGN SEDs with XSPEC

Figure 1: Example of output SEDs generated using Xspec qsosed. The vertical
blue and red dotted lines represent photon energies of $h\nu_{\rm
CIV}=64.5\,{\rm eV}$ and $h\nu_{H}=13.6\,{\rm eV}$, respectively. Top Panel:
SEDs with a constant accretion rate ($\textit{$\dot{m}$}=0.1$) and varying BH
mass ($\textit{$M_{\rm BH}$}=10^{6-10}M_{\odot}$) in logarithmic intervals of
0.5 dex. Bottom Panel: SEDs with fixed BH mass ($M_{\rm BH}=10^{8}M_{\odot}$)
while varying the accretion rate $\log\dot{m}$ from $-1.5$ to $0$ in
logarithmic intervals of 0.25 dex. Figure 2: SEDs computed for various
accretion rates $\log\dot{m}\in[-1.5,0]$ with 0.25 dex intervals at constant
BH mass $M_{\rm BH}$ (from left to right: $10^{6}$, $10^{7}$, $10^{8}$,
$10^{9}$, and $10^{10}\,M_{\odot}$). Each SED within the panel is normalized
relative to the SED with $\dot{m}$=0.1. Vertical lines are placed at photon
energies of $h\nu_{\rm CIV}=64.5$ eV (blue) and $h\nu_{H}=13.6$ eV (red),
respectively.  Figure 3: SEDs of two SMBHs with mass $M_{\rm BH}$ of
$10^{7.5}$ and $10^{8}M_{\odot}$ and accretion rate $\log\dot{m}$ of $-0.5$
and $-1.25$, respectively. These two SEDs with diverse SED shapes produce
similar optical luminosities $\log\textit{$\lambda L_{5100}$}\sim 44$ but
different ionizing fluxes. Within the SSD regime and at fixed optical
luminosity, AGNs with higher Eddington ratios (lower BH masses) have
relatively more ionizing flux compared with the 5100 Å flux, which would
presumably lead to longer BLR lags.  Figure 4: Continuum luminosities at
$5100{\rm\AA}$ (top), $3000{\rm\AA}$ (middle) and $1350{\rm\AA}$ (bottom) in
the $M_{\rm BH}$-$\lambda_{\rm Edd}$ plane, computed from the qsosed models.
Observed continuum luminosities of the compiled RM AGN sample are overlaid in
each corresponding panel for comparison. Note that $\textit{$\lambda_{\rm
Edd}$}<-1.5$ for certain theoretical SEDs is due to the constant bolometric
correction applied to the continuum luminosity at 5100 Å.

To generate a grid of ionizing SEDs with various BH parameters, we use the
qsosed (Kubota & Done, 2018) model in Xspec (Arnaud, 1996) that has three
featured components: one truncated standard thin-disk component providing
multi-temperature blackbody radiation; a hot Comptonized corona for non-
thermal power-law emission dominating the X-ray continuum above 1 keV; and a
soft X-ray excess from an intermediate warm Comptonizing component. Compared
to its predecessor OPTXAGNF (Done et al., 2012), which assumes a single
Comptonization spectrum for the seed photon energy due to the lack of
interstellar absorption observations, the updated, physically driven model
qsosed treats the warm Comptonization component more carefully and assumes
that the seed photon energy originates from the underlying cool material in
the mid-plane (Kubota & Done, 2018). Both Panda et al. (2019) and Ferland et
al. (2020) utilize the qsosed model in their BLR photoionization computations
to explore the impact of the Eddington ratio on the BLR photoionization and
demonstrate consistent results with observations. Moreover, comparison with
the stacked AGN spectra shows that the qsosed-predicted SEDs align well with
intermediate black hole masses in the range of $7.5<\log\textit{$M_{\rm
BH}$}<9.0$ while showing discrepancy with observations at optical/EUV for both
ends (Mitchell et al., 2023). The uncertainties and limitations of qsosed will
be further discussed in §4.4. The qsosed model fixes several disk parameters
and includes reprocessing; the free input parameters are: the BH mass $M_{\rm
BH}$, the co-moving distance from the object to the observer ($D_{c}$), the
Eddington-scaled accretion rate ($\dot{m}$$\equiv\dot{M}/\dot{M}_{\rm
Edd}=\eta\dot{M}c^{2}/L_{\rm Edd}$, where $\eta$ is the spin-dependent
radiative efficiency, fixed at 0.057 for a non-spinning black hole in qsosed),
the dimensionless spin parameter ($a_{*}$), the cosine of the inclination
angle for the warm Comptonising component and the outer disk ($\cos{i}$), and
the redshift ($z$). As an exploratory study, we consider non-spinning
($a_{*}=0$) BHs with a mass range of $10^{6}-10^{10}\rm{M_{\odot}}$ and
accretion rates $-1.5\leq\log\dot{m}\leq 0$, with $9\times 7$ logarithmically-
space grid points in each direction and intervals of $0.5$ and $0.25$ dex. We
fix the co-moving distance $D_{c}=1000\,{\rm Mpc}$, redshift $z=0$, and the
inclination at $\cos{i}=0.5$ to compute the source SEDs on the $M_{\rm
BH}$-$\dot{m}$ plane.

We focus exclusively on the moderate accretion rate regime of
$\log\textit{$\dot{m}$}\in[-1.5,0]$, because of substantial theoretical
uncertainties inherent in the slim disk SED model, as well as the imposed
lowest accretion rate limit of $-1.65<\log\textit{$\dot{m}$}$ in qsosed. To
compare with observables, we do not use the mass-accretion rate ($\dot{m}$)
for these theoretical SEDs but instead use the Eddington ratio ($\lambda_{\rm
Edd}$) based on the bolometric luminosity calculated using the $5100{\rm\AA}$
monochromatic luminosity. This detail would lead to $\lambda_{\rm Edd}$ values
at $\log\lambda_{\rm Edd}<-1.5$ for our theoretical SEDs.

Several example output SEDs from qsosed are displayed in Figure 1. The optical
and UV continuum wavelengths used for fiducial luminosity calculations are
marked by the grey dashed vertical lines, and the ionizing energy for C IV and
hydrogen are shown in blue and red vertical dotted lines, respectively
($h\nu_{\rm CIV}=64.5{\rm eV}$ and $h\nu_{H}=13.6{\rm eV}$). The top panel
illustrates how the SED varies as the BH mass changes at a fixed accretion
rate $\textit{$\dot{m}$}=0.1$. The lower panel displays the SEDs variation as
a function of accretion rate at a fixed BH mass $\textit{$M_{\rm
BH}$}=10^{8}M_{\odot}$. Figure 1 illustrates that the increases in both BH
mass and accretion rate result in increased optical and UV continuum
luminosities ($\lambda L_{5100}$ and $\lambda L_{1350}$), accompanied by an
increase in the ionizing luminosity. The ratio of line-ionizing fluxes C IV
/hydrogen changes with the BH mass and accretion rate as well.

Figure 2 shows SED comparisons for accretion rates
$\textit{$\dot{m}$}\in[-1.5,0]$ at fixed mass ($10^{6}$, $10^{7}$, $10^{8}$,
$10^{9}$ and $10^{10}\,M_{\odot}$). At each BH mass, the SEDs are normalized
by the SED with $\log\textit{$\dot{m}$}=-1$. 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 $\lambda
L_{1350}$ is more sensitive to accretion rate than $\lambda L_{5100}$,
especially for more massive black holes. To illustrate the significant
variations in the SED due to accretion rate changes, Figure 3 presents two
theoretical SEDs with similar optical luminosities $\log\textit{$\lambda
L_{5100}$}\sim 44$ but different accretion parameters (mass and accretion
rate). Although the optical luminosity is similar, the more massive BH with a
lower $\lambda_{\rm Edd}$ (in blue) produces significantly less hydrogen
ionizing flux and hence is expected to have a shorter H$\beta$ 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 ($\log\dot{m}$) ranges
from $-1.5$ to 0 in qsosed, the $\lambda_{\rm Edd}$ values may fall below
$-1.5$ for the theoretical SEDs due to the universal bolometric correction
(${\rm BC}=9.26$) applied to the continuum luminosity at 5100 Å. The continuum
luminosities show a diagonal increasing trend on the $M_{\rm
BH}$-$\lambda_{\rm Edd}$ 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.

###  2.2 Photoionization with CLOUDY

We construct detailed photoionization models to calculate line luminosities
for individual clouds using CLOUDY (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 §2.1 are used as the
incident radiation field input. This approach assumes 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 (Korista & Goad,
2000, 2004; Guo et al., 2020), we assume that the BLR clouds cover a broad
range of hydrogen density ($6.0\leqslant\log{n(\rm{H})}\leqslant 14.0$) and
surface ionization flux ($16.0\leqslant\log{\Phi(\rm{H})}\leqslant 25.0$),
both with logarithmic intervals of 0.125 dex. The ionization parameter $U_{\rm
H}$ is defined as a dimensionless ratio between the hydrogen-ionizing photon
flux $Q(\rm{H})$ and the total hydrogen density $n(\rm{H})$:

$U_{\rm H}\equiv\frac{Q({\rm H})}{4\pi R_{\rm BLR}^{2}n({\rm
H})c}\equiv\frac{\Phi(\rm{H})}{n({\rm H})c},$ (1)

where $R_{\rm BLR}$ 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 $N_{\rm H}=10^{23}\,{\rm cm^{-2}}$,
abundance $Z=Z_{\odot}$, and an overall covering factor of $CF=50\%$. Given
the full set of hydrogen densities and 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 $8.0\leqslant\log{n(\rm{H})}\leqslant 12.0$ and ionizing flux
$6.0\leqslant\log{U_{\rm H}c}\leqslant 11.25$ are considered to obtain the
radial surface emissivity ($F(r)$) (see details in Korista & Goad, 2000, 2004,
2019; Guo et al., 2020). For simplicity, we adopt the empirical assumptions
outlined in Baldwin et al. (1995): $f(r)\propto r^{\Gamma}$ and $g(n)\propto
n^{\beta}$ ($\beta=-1$) 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, Korista & Goad (2000) and
Korista & Goad (2019) proposed $-1.4<\Gamma<-1$ and fix $\Gamma=-1.2$ for NGC
5548; Guo et al. (2020) employed $\Gamma=-1.1$ to align with the observed Mg
II luminosity; therefore, after comparing with observed line luminosities in
§3.2, we fix $\Gamma=-1.1$ to match the observed lines. The observed line
luminosity is:

$\begin{split}L_{\rm line}&\propto\int\int^{R_{\rm out}}_{R_{\rm
in}}r^{2}F(r)f(r)g(n)dndr,\\\ &\propto\int\,d(\log{n})\int^{R_{\rm
out}}_{R_{\rm in}}r^{1.9}F(r)\,d(\log{r}),\end{split}$ (2)

where $F(r)$ is the radial surface emissivity of a single cloud. The BLR sizes
predicted in §2.3 are adopted as the average distance for the integral.

###  2.3 Predictions of BLR size

Figure 5: Dependence of the H$\beta$ BLR size on monochromatic luminosity at
5100 ${\rm\AA}$ across a range of mass bins (from $\log\textit{$M_{\rm
BH}$}=6$ to $\log\textit{$M_{\rm BH}$}=10$ with $\Delta\log\textit{$M_{\rm
BH}$}=0.5$). Predicted time lags are represented by shaded bands with varying
filling colors denoting different Eddington ratios. Observed H$\beta$ time
lags are presented as circles and triangles for low accretion
($\textit{$\lambda_{\rm Edd}$}<0.5$) local AGNs and SDSS-RM sources,
respectively, with the same color scheme as the theoretical predictions. The
grey dashed line indicates the canonical $R-L$ relation from Bentz et al.
(2013), with a slope $k=0.533$ and an intercept $b=1.527$ at $\log\lambda
L_{5100}=44$.

The relation between BLR radius $R_{\rm BLR}=c\tau$ 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:

$R_{\rm BLR}=\sqrt{\frac{Q({\rm H})}{4\pi\Phi({\rm
H})}}=\sqrt{\frac{\int^{\infty}_{\nu}\frac{\pi
L_{\nu}}{h\nu}d\nu}{4\pi\Phi({\rm H})}}$ (3)

where $\Phi({\rm H})=U_{\rm H}n({\rm H})c$ is the flux of ionizing photons
received by the clouds. The LOC model assumes that the clouds with the highest
reprocessing efficiency dominate the observed emissions. Hydrogen and Helium
emission lines, in particular, exhibit peak reprocessing efficiency under
conditions of low ionizing photon flux and high hydrogen density ($\log n({\rm
H})\gtrsim 12$) (e.g., Baldwin et al., 1995; Korista et al., 1997; Korista &
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 $\log
U_{\rm H}n({\rm H})\sim 10$ ($\log n({\rm H})\sim 11.6$ to $12.6$ and $\log
U_{\rm H}\sim-2.5$ to $-2.0$) (e.g., Sigut & Pradhan, 2003; Matsuoka et al.,
2008; Negrete et al., 2012). Therefore, constant ionization parameter $\log
U_{\rm H}=-2.0$ and hydrogen density $n(\rm{H})=10^{12}\,{\rm cm^{-3}}$ are
adopted here to calculate the BLR gas distance (Eqn. 3).

Figure 5 compares our predicted H$\beta$ BLR size as a function of the
monochromatic luminosity at $5100{\rm\AA}$ with our compiled RM AGN sample
(see details in §3). We divide the observed RM sample into narrow mass bins
with $\Delta\log\textit{$M_{\rm BH}$}=0.5$ and assign them to their respective
mass bins in Figure 5. Within each subplot, the predicted time lags are
depicted by colored bands, and all data points are color-coded based on their
Eddington ratio. As shown in Figure 5, our predicted time lag trends with
optical luminosity are in overall agreement with the observed lags given the
assumed universal values of the ionization parameter and cloud hydrogen
density. This consistency supports the general notion that the BLR 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 §4.

##  3 Comparison with observations

###  3.1 Sample and data

Figure 6: Black hole mass measurements from H$\beta$ RM and optical luminosity $\lambda L_{5100}$ for local AGNs at $z<0.5$ are represented by circles (Du & Wang, 2019), while the SDSS-RM sample is coded by triangles (Grier et al., 2017; Shen et al., 2019, 2024). Objects lacking ultraviolet spectra are indicated in black, whereas those with ultraviolet spectra are distinguished by colors, with local AGNs in red and the SDSS-RM sample in blue.  Figure 7: UV spectra of the 18 quasars observed by HST/STIS under the program HST GO-16171. The data have been corrected for the galactic reddening and shifted to the rest frame. The orange lines demonstrate our fitting results. Table 1: Target Properties RMID | R. A. (J2000) | Dec. (J2000) | $z$ | $\tau_{\rm H\beta}$ | log(${{M}_{\rm BH,RM}}$) | NUV
---|---|---|---|---|---|---
| (deg) | (deg) |  | (days) | ($\rm M_{\odot}$) | (mag)
RM101 | 213.059 | 53.430 | 0.458 | $21.4^{+4.2}_{-6.4}$ | $7.26^{+0.09}_{-0.12}$ | 19.35
RM160 | 212.672 | 53.314 | 0.360 | $21.9^{+4.2}_{-2.4}$ | $7.85^{+0.09}_{-0.07}$ | 20.77
RM177 | 214.352 | 52.507 | 0.482 | $10.1^{+12.5}_{-2.7}$ | $7.57^{+0.35}_{-0.11}$ | 20.80
RM229 | 212.575 | 53.494 | 0.470 | $16.2^{+2.9}_{-4.5}$ | $7.65^{+0.09}_{-0.12}$ | 20.77
RM267 | 212.803 | 53.752 | 0.588 | $20.4^{+2.5}_{-2.0}$ | $7.41^{+0.08}_{-0.08}$ | 20.16
RM272 | 214.107 | 53.911 | 0.263 | $15.1^{+3.2}_{-4.6}$ | $7.58^{+0.10}_{-0.13}$ | 19.40
RM301 | 215.043 | 52.675 | 0.548 | $12.8^{+5.7}_{-4.5}$ | $8.64^{+0.17}_{-0.14}$ | 21.10
RM305 | 212.518 | 52.528 | 0.527 | $53.5^{+4.2}_{-4.0}$ | $8.32^{+0.07}_{-0.07}$ | 20.28
RM320 | 215.161 | 53.405 | 0.265 | $25.2^{+4.7}_{-5.7}$ | $7.67^{+0.09}_{-0.11}$ | 20.60
RM371 | 212.848 | 52.225 | 0.473 | $13.0^{+1.4}_{-0.8}$ | $7.38^{+0.08}_{-0.07}$ | 20.21
RM622 | 212.813 | 51.869 | 0.572 | $49.1^{+11.1}_{-2.0}$ | $7.94^{+0.11}_{-0.06}$ | 20.55
RM694 | 214.278 | 51.728 | 0.532 | $10.4^{+6.3}_{-3.0}$ | $6.70^{+0.20}_{-0.15}$ | 20.83
RM720 | 211.325 | 53.258 | 0.467 | $41.6^{+14.8}_{-8.3}$ | $7.74^{+0.14}_{-0.10}$ | 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 | $75.2^{+3.2}_{-3.3}$ | $7.89^{+0.07}_{-0.07}$ | 20.47
RM782 | 213.329 | 54.534 | 0.363 | $20.0^{+1.1}_{-3.0}$ | $7.51^{+0.06}_{-0.09}$ | 19.50
RM790 | 214.372 | 53.307 | 0.238 | $5.5^{+5.7}_{-2.1}$ | $8.28^{+0.31}_{-0.15}$ | 20.10
RM840 | 214.188 | 54.428 | 0.244 | $5.0^{+1.5}_{-1.4}$ | $7.93^{+0.13}_{-0.12}$ | 19.94

We start from the sample of $\sim 140$ $z\lesssim 0.5$ broad-line AGNs with
H$\beta$ RM measurements, and select those with $10^{-3}<\textit{$\lambda_{\rm
Edd}$}<0.5$ in the SSD regime as our primary sample. Figure 6 shows the
distribution of BH mass $M_{\rm BH}$ measured from H$\beta$ RM versus their
optical luminosity $\lambda L_{5100}$ for this primary sample, including
objects with and without UV spectroscopy. For $M_{\rm BH}$ values of these
local RM AGNs, we combine the catalog in Bentz & Katz (2015) and Table 1 in Du
& Wang (2019), and correct $M_{\rm BH}$ using a nominal virial factor $f_{\rm
FWHM,H\beta}=1.12$ from Woo et al. (2015). The host contamination in $\lambda
L_{5100}$ 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 $\lambda L_{5100}$ from light curves as
an upper limit. We supplement the local RM AGNs with 72 more distant quasars
from the SDSS-RM sample (Shen et al., 2024). The host contamination for the
SDSS-RM quasars used in this work is removed based on a spectral decomposition
approach described in Shen et al. (2015b). We use a bolometric correction of
${\rm BC}=9.26$ (Richards et al., 2006) for the optical continuum at $5100$ Å
to calculate the bolometric luminosity $L_{\rm bol}$, and the RM mass $M_{\rm
BH}$ for the dimensionless Eddington ratio.

After assembling the primary sample of 135 low-to-moderate accretion AGNs with
H$\beta$ 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 $0.2\lesssim z\lesssim 0.6$ from a dedicated HST Space Telescope
Imaging Spectrograph (STIS) program in Cycle 28 (GO-16171; PI: Shen). These
quasars are selected from the SDSS-RM sample (Shen et al., 2015a) with direct
RM lags and BH masses based on the broad H$\beta$ line (Grier et al., 2017).
We restrict to quasars with $0.2\lesssim z\lesssim 0.6$ and GALEX NUV
$\lesssim 21$ mag while spanning a broad range in luminosity and H$\beta$ lags
to sample the parameter space. Table 1 summarizes the properties of the 18
SDSS-RM targets.

The eighteen 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 1570–3180
${\rm\AA}$, 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$\sim
19-21$: for the brightest twelve targets, one full orbit exposure time is
utilized; five quasars (RM160, RM177, RM229, RM320, and RM694) were observed
for two orbits; the faintest target with GALEX NUV magnitude of 21.1 (RM301)
was observed with three orbits. The initial visits of all targets lasted from
Dec 2020 to Feb 2022; however, four targets (RM177, RM229, RM320, and RM694)
suffered from the failure of guide-star acquisition at the start of the second
orbit, resulting in lost data. Repeat visits for those targets were approved
and recurred from September to November in 2022. The pipeline-processed STIS
spectra of those targets with multiple orbits of exposure were coadded to
improve the 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 (IUE), the STIS, and the Cosmic Origins Spectrograph
(COS) 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 high-accretion SMBHs will have a
dramatic impact on the ionizing continuum illuminating the BLR, we only focus
on the low-to-moderate accretion sample in this work.

Our final sample includes 15 SDSS-RM quasars and 52 local RM AGNs with
accretion rates $10^{-3}<\textit{$\lambda_{\rm Edd}$}<0.5$, We then apply the
spectral decomposition code PyQSOFit (Guo et al., 2018) with minor custom
adjustments to measure their UV spectral properties and summarize them in
Table 2. The optical spectral properties ($\log\textit{$\lambda L_{5100}$}$,
H$\beta$ and He II $\lambda$4687 ) for the SDSS-RM sample are from Shen et al.
(2019), while those of local AGNs are from Du & Wang (2019).

###  3.2 UV/optical line strengths

Table 2: RM AGN UV+optical spectroscopic properties

Objects | $\log{\textit{$\lambda L_{5100}$}}$ | $\log{\lambda L_{1350}}$ | $\log{L_{\rm CIV}}$ | REW(CIV) | CIV blueshift | $\log{L_{\rm CIII]}}$ | REW(CIII]) | $\log{L_{\rm HeII1640}}$ | REW(HeII1640) | $\log{L_{\rm MgII}}$ | REW(MgII) | $\log{L_{\rm H\beta}}$ | REW(H$\beta$) |  |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
| $\rm erg\ s^{-1}$ | $\rm erg\ s^{-1}$ | $\rm erg\ s^{-1}$ | $\rm\AA$ | $\rm km\,s^{-1}$ | $\rm erg\ s^{-1}$ | $\rm\AA$ | $\rm erg\ s^{-1}$ | $\rm\AA$ | $\rm erg\ s^{-1}$ | $\rm\AA$ | $\rm erg\ s^{-1}$ | $\rm\AA$ |  |
RM160 | $43.79\pm 0.01$ | $44.34\pm 0.02$ | $43.62\pm 0.01$ | $321.2\pm 5.7$ | $-595\pm 45$ | $42.47\pm 0.03$ | $33.0\pm 2.5$ | $42.19\pm 0.09$ | $13.0\pm 2.7$ | … | … | $42.07\pm 0.01$ | $96.5\pm 1.4$ | $40.9\pm 0.06$ | $6.4\pm 0.6$
RM177 | $43.97\pm 0.01$ | $44.08\pm 0.03$ | $42.64\pm 0.03$ | $47.7\pm 3.9$ | $145\pm 366$ | $42.32\pm 0.09$ | $29.5\pm 5.7$ | $41.37\pm 0.23$ | $2.7\pm 2.5$ | … | … | $42.31\pm 0.01$ | $62.0\pm 1.7$ | $41.4\pm 0.02$ | $7.5\pm 0.4$
RM229 | $43.56\pm 0.01$ | $44.25\pm 0.03$ | $43.0\pm 0.05$ | $100.1\pm 9.1$ | $198\pm 504$ | $42.35\pm 0.27$ | $36.4\pm 16.5$ | $41.83\pm 0.24$ | $7.8\pm 6.5$ | … | … | $41.93\pm 0.02$ | $62.0\pm 2.3$ | $40.86\pm 0.15$ | $5.2\pm 2.5$
RM267 | $44.11\pm 0.01$ | $44.76\pm 0.02$ | $43.26\pm 0.02$ | $46.7\pm 2.1$ | $332\pm 166$ | $42.94\pm 0.12$ | $47.6\pm 14.0$ | … | … | … | … | $42.54\pm 0.02$ | $81.1\pm 4.2$ | $41.33\pm 0.17$ | $4.8\pm 2.5$
RM272 | $43.94\pm 0.01$ | $44.26\pm 0.05$ | $43.16\pm 0.04$ | $145.4\pm 15.3$ | $-287\pm 127$ | $42.24\pm 0.05$ | $21.0\pm 2.6$ | $42.2\pm 0.32$ | $17.1\pm 9.5$ | … | … | $42.37\pm 0.01$ | $132.2\pm 0.6$ | $41.46\pm 0.01$ | $15.8\pm 0.5$
RM301 | $44.04\pm 0.01$ | $43.75\pm 0.09$ | $42.82\pm 0.03$ | $183.9\pm 12.4$ | $304\pm 193$ | $41.83\pm 0.23$ | $20.4\pm 11.7$ | … | … | … | … | $42.27\pm 0.01$ | $58.7\pm 1.2$ | $40.75\pm 0.08$ | $1.7\pm 0.3$
RM305 | $44.19\pm 0.01$ | $44.37\pm 0.04$ | $42.91\pm 0.04$ | $54.5\pm 5.2$ | $-1210\pm 881$ | $42.61\pm 0.18$ | $40.5\pm 17.8$ | $41.47\pm 0.33$ | $2.1\pm 2.4$ | … | … | $42.53\pm 0.04$ | $77.7\pm 6.2$ | $41.4\pm 0.02$ | $5.6\pm 0.3$
RM320 | $43.44\pm 0.01$ | $43.59\pm 0.06$ | $42.68\pm 0.04$ | $138.7\pm 13.4$ | $407\pm 266$ | $41.92\pm 0.05$ | $25.5\pm 2.9$ | $42.02\pm 0.26$ | $29.9\pm 13.2$ | … | … | $41.89\pm 0.01$ | $74.9\pm 1.9$ | $41.18\pm 0.12$ | $14.2\pm 4.7$
RM371 | $44.09\pm 0.01$ | $44.03\pm 0.1$ | $43.17\pm 0.04$ | $205.4\pm 24.3$ | $1165\pm 435$ | $42.42\pm 0.28$ | $37.0\pm 13.8$ | $42.27\pm 0.11$ | $25.4\pm 6.3$ | … | … | $42.4\pm 0.01$ | $101.6\pm 2.0$ | $40.66\pm 0.14$ | $1.8\pm 0.6$
RM622 | $44.3\pm 0.01$ | $44.49\pm 0.03$ | $43.03\pm 0.07$ | $49.8\pm 7.5$ | $-1222\pm 609$ | $42.54\pm 0.24$ | $20.8\pm 9.9$ | $41.85\pm 0.26$ | $3.5\pm 3.4$ | … | … | $42.62\pm 0.01$ | $81.1\pm 1.1$ | $41.14\pm 0.04$ | $2.6\pm 0.3$
RM720 | $44.3\pm 0.01$ | $44.22\pm 0.04$ | $43.1\pm 0.07$ | $114.3\pm 16.7$ | $-328\pm 240$ | $42.29\pm 0.14$ | $20.4\pm 5.2$ | $42.29\pm 0.09$ | $18.0\pm 3.8$ | … | … | $42.46\pm 0.01$ | $61.8\pm 0.4$ | $41.5\pm 0.03$ | $6.4\pm 0.5$
RM781 | $43.6\pm 0.01$ | $43.88\pm 0.09$ | $42.49\pm 0.05$ | $55.1\pm 5.9$ | $-189\pm 343$ | $41.63\pm 0.15$ | $10.0\pm 4.0$ | $41.36\pm 0.24$ | $4.2\pm 4.1$ | … | … | $41.91\pm 0.02$ | $65.8\pm 2.8$ | $41.27\pm 0.05$ | $14.6\pm 1.6$
RM782 | $43.93\pm 0.01$ | $43.04\pm 0.31$ | $42.82\pm 0.04$ | $66.3\pm 5.9$ | $-586\pm 431$ | $41.8\pm 0.11$ | $6.0\pm 1.5$ | … | … | … | … | $42.0\pm 0.01$ | $37.0\pm 0.5$ | $40.5\pm 0.06$ | $1.1\pm 0.2$
RM790 | $43.33\pm 0.01$ | … | … | … | … | $41.29\pm 0.18$ | $13.4\pm 4.7$ | $41.16\pm 0.3$ | $20.2\pm 15.1$ | … | … | $41.87\pm 0.03$ | $53.9\pm 3.2$ | $24.03\pm 8.55$ | $0.0\pm 5.0$
RM840 | $43.24\pm 0.01$ | $43.66\pm 0.13$ | $42.65\pm 0.05$ | $153.7\pm 17.4$ | $-218\pm 269$ | $42.07\pm 0.07$ | $53.3\pm 7.3$ | $41.47\pm 0.31$ | $10.8\pm 5.2$ | … | … | $41.6\pm 0.01$ | $38.9\pm 0.7$ | $39.47\pm 19.82$ | $0.3\pm 0.2$
Mrk335 | $43.76\pm 0.07$ | $43.72\pm 0.01$ | $42.63\pm 0.01$ | $108.2\pm 1.8$ | $-127\pm 23$ | $42.8\pm 0.02$ | $135.6\pm 7.5$ | $41.5\pm 0.02$ | $7.9\pm 0.3$ | $41.91\pm 0.03$ | $15.5\pm 1.0$ | $42.09\pm 0.09$ | $108.2\pm 0.1$ | … | …
PG0026 | $44.97\pm 0.02$ | $45.26\pm 0.01$ | $43.55\pm 0.01$ | $32.8\pm 1.1$ | $-993\pm 129$ | … | … | $42.8\pm 0.15$ | $6.5\pm 3.7$ | … | … | $42.93\pm 0.04$ | $46.2\pm 0.0$ | … | …
PG0052 | $44.81\pm 0.03$ | $45.36\pm 0.01$ | $44.18\pm 0.03$ | $113.7\pm 7.6$ | $670\pm 51$ | $42.53\pm 0.18$ | $3.8\pm 2.4$ | $42.47\pm 0.61$ | $2.5\pm 9.0$ | … | … | $43.13\pm 0.05$ | $107.4\pm 0.0$ | … | …
Fairall9 | $43.98\pm 0.04$ | $44.43\pm 0.01$ | $43.33\pm 0.01$ | $114.9\pm 0.9$ | $61\pm 24$ | $42.68\pm 0.12$ | $29.0\pm 10.2$ | $42.53\pm 0.18$ | $19.1\pm 5.4$ | $42.84\pm 0.01$ | $53.5\pm 0.5$ | $42.67\pm 0.04$ | $249.8\pm 0.0$ | … | …
Mrk590 | $43.5\pm 0.21$ | $42.92\pm 0.01$ | $42.27\pm 0.01$ | $710.5\pm 7.3$ | $356\pm 89$ | $40.18\pm 0.3$ | $2.4\pm 9.4$ | $40.92\pm 0.09$ | $32.4\pm 7.0$ | $41.92\pm 0.01$ | $48.1\pm 0.8$ | $41.85\pm 0.12$ | $108.6\pm 0.1$ | … | …
Mrk1044 | $43.1\pm 0.1$ | $43.42\pm 0.01$ | $41.85\pm 0.01$ | $42.7\pm 0.3$ | $865\pm 26$ | $41.08\pm 0.21$ | $10.0\pm 7.1$ | $41.58\pm 0.01$ | $24.8\pm 0.7$ | $40.93\pm 0.06$ | $10.9\pm 1.5$ | $41.39\pm 0.09$ | $101.4\pm 0.1$ | … | …
3C120 | $44.0\pm 0.1$ | $44.41\pm 0.01$ | $43.31\pm 0.01$ | $134.9\pm 1.5$ | $13\pm 38$ | $42.04\pm 0.1$ | $10.5\pm 2.4$ | $42.39\pm 0.02$ | $17.7\pm 0.8$ | $42.67\pm 0.01$ | $72.3\pm 1.3$ | $42.36\pm 0.04$ | $118.8\pm 0.0$ | … | …
Ark120 | $43.87\pm 0.25$ | $44.61\pm 0.01$ | $43.32\pm 0.01$ | $77.9\pm 1.0$ | $20\pm 25$ | $42.91\pm 0.04$ | $36.2\pm 3.3$ | $42.69\pm 0.12$ | $18.8\pm 4.1$ | $42.76\pm 0.01$ | $37.5\pm 0.4$ | $42.54\pm 0.13$ | $244.8\pm 0.1$ | … | …
MCG811 | $43.33\pm 0.11$ | $42.81\pm 0.04$ | $41.91\pm 0.01$ | $172.7\pm 5.5$ | $643\pm 170$ | $41.54\pm 0.16$ | $45.5\pm 13.7$ | $41.15\pm 0.04$ | $27.2\pm 2.8$ | $41.96\pm 0.01$ | $72.3\pm 1.1$ | $41.66\pm 0.09$ | $108.7\pm 0.1$ | … | …
Mrk374 | $43.77\pm 0.04$ | $43.72\pm 0.02$ | $42.8\pm 0.06$ | $206.6\pm 26.5$ | $-20\pm 227$ | $41.76\pm 0.11$ | $24.9\pm 7.5$ | $40.56\pm 0.63$ | $1.3\pm 9.5$ | $41.91\pm 0.04$ | $43.1\pm 3.5$ | $41.83\pm 0.04$ | $58.3\pm 0.0$ | … | …
Mrk79 | $43.68\pm 0.07$ | $43.59\pm 0.01$ | $42.46\pm 0.01$ | $115.7\pm 3.7$ | $2446\pm 75$ | $42.0\pm 0.05$ | $46.9\pm 4.7$ | $42.06\pm 0.05$ | $49.0\pm 4.5$ | $41.75\pm 0.02$ | $26.2\pm 1.5$ | $41.9\pm 0.05$ | $85.4\pm 0.0$ | … | …
Mrk382 | $43.12\pm 0.08$ | $43.15\pm 0.01$ | … | … | … | … | … | … | … | $40.15\pm 0.18$ | $16.4\pm 7.1$ | $41.01\pm 0.05$ | $39.6\pm 0.0$ | … | …
PG0804 | $44.91\pm 0.02$ | $45.36\pm 0.01$ | $44.04\pm 0.01$ | $87.5\pm 1.4$ | $-923\pm 596$ | $42.66\pm 0.19$ | $6.3\pm 4.1$ | $42.78\pm 0.06$ | $5.3\pm 0.7$ | $43.0\pm 0.04$ | $19.2\pm 1.7$ | $43.29\pm 0.03$ | $122.5\pm 0.0$ | … | …
PG0844 | $44.22\pm 0.07$ | $44.59\pm 0.01$ | $43.01\pm 0.02$ | $38.6\pm 1.5$ | $-141\pm 59$ | $42.8\pm 0.18$ | $28.3\pm 8.9$ | $42.67\pm 0.02$ | $18.6\pm 1.0$ | $42.27\pm 0.08$ | $11.3\pm 2.4$ | $42.56\pm 0.05$ | $111.2\pm 0.0$ | … | …
Mrk110 | $43.66\pm 0.12$ | $43.92\pm 0.01$ | $43.05\pm 0.01$ | $186.4\pm 2.1$ | $-762\pm 37$ | $41.92\pm 0.08$ | $14.5\pm 2.7$ | $42.02\pm 0.01$ | $17.3\pm 0.6$ | $42.4\pm 0.01$ | $65.2\pm 0.5$ | $42.03\pm 0.08$ | $123.8\pm 0.1$ | … | …
PG0953 | $45.19\pm 0.01$ | $45.79\pm 0.01$ | $44.71\pm 0.02$ | $334.5\pm 17.5$ | $52\pm 52$ | … | … | … | … | … | … | $43.29\pm 0.04$ | $64.7\pm 0.0$ | … | …
NGC3227 | $42.24\pm 0.11$ | $39.99\pm 0.12$ | … | … | … | $39.74\pm 0.1$ | $50.8\pm 9.4$ | $38.66\pm 0.15$ | $18.5\pm 9.3$ | $40.17\pm 0.01$ | $38.7\pm 1.0$ | $40.38\pm 0.1$ | $71.0\pm 0.1$ | … | …
NGC3516 | $42.79\pm 0.2$ | $43.11\pm 0.01$ | $41.8\pm 0.01$ | $69.3\pm 0.4$ | $-1711\pm 73$ | $41.74\pm 0.12$ | $71.4\pm 14.8$ | $41.04\pm 0.01$ | $12.3\pm 0.3$ | $41.5\pm 0.02$ | $77.4\pm 2.8$ | $41.06\pm 0.18$ | $94.7\pm 0.2$ | … | …
SBS1116 | $42.14\pm 0.23$ | $42.85\pm 0.01$ | $41.95\pm 0.02$ | $231.6\pm 13.5$ | $-220\pm 62$ | $41.12\pm 0.04$ | $39.8\pm 3.8$ | $40.72\pm 0.07$ | $14.7\pm 2.6$ | $41.02\pm 0.04$ | $38.2\pm 3.4$ | $40.7\pm 0.07$ | $186.8\pm 0.1$ | … | …
Arp151 | $42.55\pm 0.1$ | $42.6\pm 0.01$ | $41.76\pm 0.02$ | $233.0\pm 8.6$ | $-2167\pm 132$ | $41.02\pm 0.05$ | $52.1\pm 5.5$ | $40.39\pm 0.06$ | $10.6\pm 1.5$ | $40.89\pm 0.01$ | $47.3\pm 0.8$ | $40.95\pm 0.11$ | $130.0\pm 0.1$ | … | …
NGC3783 | $42.56\pm 0.18$ | $43.34\pm 0.01$ | $42.4\pm 0.01$ | $171.1\pm 2.5$ | $-137\pm 29$ | $41.7\pm 0.1$ | $32.7\pm 7.1$ | $41.29\pm 0.02$ | $13.4\pm 0.6$ | $41.83\pm 0.01$ | $54.9\pm 1.2$ | $41.01\pm 0.18$ | $144.0\pm 0.2$ | … | …
Mrk1310 | $42.29\pm 0.14$ | $41.76\pm 0.01$ | $41.23\pm 0.03$ | $485.7\pm 37.6$ | $-183\pm 250$ | $40.1\pm 0.27$ | $28.7\pm 31.6$ | … | … | $40.65\pm 0.06$ | $64.7\pm 9.4$ | $40.56\pm 0.1$ | $94.3\pm 0.1$ | … | …
NGC4051 | $41.9\pm 0.15$ | $41.36\pm 0.01$ | $39.99\pm 0.01$ | $62.1\pm 1.4$ | $482\pm 516$ | $39.79\pm 0.1$ | $35.7\pm 7.5$ | $39.35\pm 0.02$ | $13.9\pm 0.7$ | $40.03\pm 0.01$ | $61.1\pm 1.8$ | $40.09\pm 0.19$ | $78.6\pm 0.2$ | … | …
NGC4151 | $42.09\pm 0.21$ | $42.9\pm 0.02$ | $42.12\pm 0.01$ | $210.0\pm 2.6$ | $139\pm 44$ | $41.36\pm 0.03$ | $37.3\pm 2.4$ | $40.76\pm 0.02$ | $8.9\pm 0.4$ | $41.37\pm 0.01$ | $46.3\pm 0.7$ | $40.56\pm 0.2$ | $150.8\pm 0.2$ | … | …
PG1211 | $44.73\pm 0.08$ | $44.9\pm 0.01$ | $43.74\pm 0.02$ | $111.8\pm 5.5$ | $499\pm 28$ | $43.04\pm 0.07$ | $28.8\pm 4.5$ | $42.55\pm 0.04$ | $7.6\pm 0.7$ | $42.38\pm 0.06$ | $7.3\pm 1.0$ | $43.02\pm 0.06$ | $100.2\pm 0.1$ | … | …
NGC4253 | $42.57\pm 0.12$ | $41.9\pm 0.03$ | $40.48\pm 0.06$ | $61.6\pm 7.9$ | $35\pm 267$ | $39.6\pm 0.25$ | $8.2\pm 11.3$ | … | … | $40.41\pm 0.08$ | $30.7\pm 5.8$ | $40.77\pm 0.12$ | $81.1\pm 0.1$ | … | …
PG1229 | $43.7\pm 0.05$ | $44.56\pm 0.01$ | $43.19\pm 0.01$ | $62.3\pm 2.0$ | $1038\pm 183$ | $42.78\pm 0.04$ | $29.8\pm 2.3$ | $42.74\pm 0.12$ | $22.9\pm 4.9$ | $42.52\pm 0.02$ | $27.7\pm 1.5$ | $42.31\pm 0.06$ | $209.7\pm 0.1$ | … | …
NGC4593 | $42.62\pm 0.37$ | $42.36\pm 0.01$ | $41.5\pm 0.01$ | $198.8\pm 1.2$ | $-113\pm 32$ | $40.66\pm 0.21$ | $21.6\pm 11.8$ | $40.42\pm 0.24$ | $15.4\pm 14.0$ | $41.14\pm 0.01$ | $61.7\pm 0.6$ | $40.95\pm 0.33$ | $108.3\pm 0.3$ | … | …
NGC4748 | $42.56\pm 0.12$ | $42.87\pm 0.01$ | $41.52\pm 0.02$ | $70.4\pm 3.5$ | $-1069\pm 559$ | $41.66\pm 0.07$ | $110.9\pm 15.9$ | $41.22\pm 0.08$ | $36.5\pm 6.6$ | $40.99\pm 0.01$ | $31.3\pm 0.9$ | $40.98\pm 0.1$ | $136.8\pm 0.1$ | … | …
PG1307 | $44.85\pm 0.02$ | $45.34\pm 0.01$ | $44.06\pm 0.02$ | $90.0\pm 3.9$ | $975\pm 56$ | … | … | … | … | $42.6\pm 0.12$ | $6.8\pm 1.8$ | $43.13\pm 0.06$ | $98.4\pm 0.1$ | … | …
MCG630 | $41.64\pm 0.11$ | $40.25\pm 0.37$ | … | … | … | $39.92\pm 0.21$ | $80.6\pm 45.2$ | … | … | … | … | $39.85\pm 0.19$ | $91.0\pm 0.2$ | … | …
NGC5273 | $41.54\pm 0.16$ | $41.0\pm 0.05$ | $39.89\pm 0.1$ | $175.2\pm 37.2$ | $-352\pm 736$ | $39.62\pm 0.29$ | $277.1\pm 131.1$ | $39.29\pm 0.27$ | $54.9\pm 22.4$ | $38.54\pm 0.16$ | $9.6\pm 3.6$ | $39.74\pm 0.11$ | $82.2\pm 0.1$ | … | …
Mrk279 | $43.71\pm 0.07$ | $43.99\pm 0.01$ | $41.76\pm 0.02$ | $9.7\pm 0.5$ | $1160\pm 184$ | $41.79\pm 0.23$ | $13.7\pm 12.7$ | $41.53\pm 0.02$ | $6.1\pm 0.3$ | … | … | $42.12\pm 0.06$ | $132.2\pm 0.1$ | … | …
PG1411 | $44.56\pm 0.02$ | $44.65\pm 0.01$ | $43.33\pm 0.02$ | $72.1\pm 2.4$ | $-1619\pm 77$ | $43.15\pm 0.35$ | $57.8\pm 33.8$ | $42.31\pm 0.03$ | $7.2\pm 0.5$ | … | … | $42.85\pm 0.03$ | $99.7\pm 0.0$ | … | …
NGC5548 | $43.3\pm 0.19$ | $43.58\pm 0.01$ | $42.55\pm 0.01$ | $134.3\pm 0.5$ | $549\pm 28$ | $42.28\pm 0.03$ | $71.2\pm 4.9$ | $41.76\pm 0.01$ | $21.7\pm 0.4$ | $42.17\pm 0.01$ | $74.5\pm 0.4$ | $41.65\pm 0.21$ | $117.8\pm 0.2$ | … | …
PG1426 | $44.63\pm 0.02$ | $45.2\pm 0.01$ | $43.72\pm 0.05$ | $55.6\pm 6.0$ | $208\pm 215$ | $43.13\pm 0.05$ | $20.3\pm 2.4$ | $43.2\pm 0.05$ | $18.2\pm 2.2$ | $42.97\pm 0.04$ | $21.9\pm 2.1$ | $42.83\pm 0.04$ | $80.1\pm 0.0$ | … | …
Mrk817 | $43.74\pm 0.09$ | $44.29\pm 0.01$ | $42.82\pm 0.01$ | $58.1\pm 0.7$ | $357\pm 38$ | $41.72\pm 0.25$ | $7.0\pm 4.0$ | $42.25\pm 0.01$ | $17.4\pm 0.3$ | $42.37\pm 0.05$ | $58.9\pm 6.0$ | $41.93\pm 0.14$ | $78.5\pm 0.1$ | … | …
Mrk1511 | $43.16\pm 0.06$ | $43.48\pm 0.01$ | $42.32\pm 0.03$ | $112.0\pm 8.3$ | $-245\pm 238$ | $41.37\pm 0.15$ | $15.4\pm 4.9$ | … | … | $41.56\pm 0.09$ | $25.9\pm 6.5$ | $41.52\pm 0.06$ | $115.5\pm 0.1$ | … | …
Mrk290 | $43.17\pm 0.06$ | $43.77\pm 0.01$ | $42.71\pm 0.01$ | $129.0\pm 1.8$ | $321\pm 45$ | $42.15\pm 0.13$ | $39.0\pm 9.9$ | $41.86\pm 0.04$ | $18.3\pm 1.5$ | $42.1\pm 0.01$ | $54.9\pm 0.9$ | $41.64\pm 0.06$ | $153.0\pm 0.1$ | … | …
Mrk486 | $43.69\pm 0.05$ | $43.17\pm 0.02$ | $41.88\pm 0.16$ | $55.2\pm 15.3$ | $-1110\pm 70$ | $41.0\pm 0.21$ | $5.5\pm 3.7$ | $41.16\pm 0.09$ | $9.7\pm 2.1$ | $42.01\pm 0.07$ | $41.6\pm 7.2$ | $42.12\pm 0.04$ | $135.9\pm 0.0$ | … | …
PG1613 | $44.77\pm 0.02$ | $45.18\pm 0.01$ | $43.93\pm 0.01$ | $88.7\pm 1.3$ | $1117\pm 84$ | $42.8\pm 0.18$ | $9.6\pm 4.5$ | … | … | $43.35\pm 0.02$ | $49.6\pm 2.4$ | $43.0\pm 0.03$ | $86.7\pm 0.0$ | … | …
PG1617 | $44.39\pm 0.02$ | $44.87\pm 0.01$ | $42.98\pm 0.02$ | $21.5\pm 0.9$ | $-72\pm 123$ | $43.26\pm 0.04$ | $53.5\pm 4.7$ | $41.91\pm 0.07$ | $2.0\pm 0.3$ | $43.2\pm 0.03$ | $69.8\pm 5.1$ | $42.74\pm 0.05$ | $114.8\pm 0.0$ | … | …
3C382 | $43.84\pm 0.1$ | $44.48\pm 0.01$ | $43.52\pm 0.01$ | $170.3\pm 5.0$ | $39\pm 51$ | $42.21\pm 0.13$ | $11.1\pm 2.9$ | $41.92\pm 0.29$ | $4.6\pm 5.2$ | $42.8\pm 0.01$ | $64.3\pm 1.2$ | $42.54\pm 0.03$ | $259.3\pm 0.0$ | … | …
3C390.3 | $44.43\pm 0.58$ | $43.73\pm 0.01$ | $42.79\pm 0.01$ | $226.6\pm 3.4$ | $1474\pm 84$ | $42.4\pm 0.2$ | $112.6\pm 33.4$ | $42.1\pm 0.23$ | $52.5\pm 17.9$ | $42.64\pm 0.01$ | $113.5\pm 3.3$ | $42.6\pm 0.35$ | $108.8\pm 0.4$ | … | …
NGC6814 | $42.12\pm 0.28$ | $41.66\pm 0.02$ | $40.61\pm 0.03$ | $182.6\pm 11.7$ | $-167\pm 416$ | $40.31\pm 0.14$ | $132.6\pm 39.1$ | $38.95\pm 0.25$ | $4.6\pm 2.0$ | $40.29\pm 0.01$ | $87.3\pm 2.3$ | $40.5\pm 0.28$ | $121.6\pm 0.3$ | … | …
Mrk509 | $44.19\pm 0.05$ | $44.62\pm 0.01$ | $43.49\pm 0.01$ | $116.7\pm 0.9$ | $344\pm 31$ | $43.12\pm 0.03$ | $65.2\pm 4.7$ | $42.55\pm 0.05$ | $14.5\pm 1.6$ | $42.86\pm 0.01$ | $49.7\pm 0.7$ | $42.61\pm 0.04$ | $132.7\pm 0.0$ | … | …
NGC7469 | $43.51\pm 0.11$ | $43.85\pm 0.01$ | $42.49\pm 0.01$ | $66.7\pm 0.6$ | $-253\pm 40$ | $41.79\pm 0.06$ | $16.4\pm 2.1$ | $41.93\pm 0.02$ | $19.3\pm 1.1$ | $42.1\pm 0.01$ | $50.2\pm 1.7$ | $41.6\pm 0.1$ | $63.0\pm 0.1$ | … | …
Zw229 | $42.71\pm 0.05$ | $43.16\pm 0.01$ | $42.25\pm 0.04$ | $187.0\pm 20.8$ | $870\pm 485$ | $41.37\pm 0.04$ | $28.1\pm 2.7$ | $41.16\pm 0.37$ | $16.1\pm 13.8$ | $41.52\pm 0.01$ | $49.6\pm 1.6$ | … | … | … | …
Mrk1501 | $44.32\pm 0.05$ | $44.7\pm 0.01$ | $43.61\pm 0.01$ | $120.3\pm 3.0$ | $-108\pm 88$ | $43.18\pm 0.13$ | $48.6\pm 10.9$ | $43.08\pm 0.07$ | $36.0\pm 4.8$ | $42.99\pm 0.05$ | $35.0\pm 3.6$ | … | … | … | …
Mrk50 | $42.88\pm 0.07$ | $43.22\pm 0.01$ | $42.16\pm 0.02$ | $152.6\pm 6.4$ | $-57\pm 86$ | $41.01\pm 0.16$ | $14.4\pm 7.6$ | … | … | $41.4\pm 0.01$ | $45.4\pm 0.7$ | … | … | … | …
NGC4395 | $39.77\pm 0.02$ | $39.82\pm 0.01$ | $38.62\pm 0.05$ | $82.9\pm 11.5$ | $1187\pm 544$ | … | … | $37.83\pm 0.02$ | $14.0\pm 0.6$ | … | … | … | … | … | …
Mrk6 | $43.75\pm 0.06$ | $42.09\pm 0.1$ | $40.9\pm 0.16$ | $91.0\pm 29.6$ | $4497\pm 2534$ | $40.33\pm 0.18$ | $22.5\pm 10.5$ | $40.44\pm 0.08$ | $31.3\pm 5.5$ | $40.94\pm 0.04$ | $64.7\pm 6.3$ | … | … | … | …

Note. — Properties of the 18 quasars proposed in HST GO-16171. RM black hole
masses are based on H$\beta$ lags from Grier et al. (2017). Time lags are
rest-frame values.

Figure 8: Distributions of line luminosities in $M_{\rm BH}$-$\lambda_{\rm
Edd}$ plane. The left panel shows the integrated ionizing luminosity ($L_{\rm
ion}=\int^{\nu_{\rm destruct}}_{\nu_{\rm product}}L_{\nu}\,d\nu$ within the
production and destruction photon energies); the middle panel displays the
line luminosities predicted by the LOC model; and the right panel presents the
observed line luminosities as a visual assessment of the theoretical
photoionization. The positive correlations in all panels demonstrate the
consistency between our theoretical computations and observations.  Figure 9:
Distributions of emission line equivalent width (EW) in the $M_{\rm
BH}$-$\lambda_{\rm Edd}$ plane. The left panel displays the EW distributions
predicted by the LOC model. The middle panel presents the observed EW from the
AGN sample, and the right panel displays the binned EW.

Figure 8 presents the comparison of ionizing luminosities, LOC-predicted line
luminosities, and observed line luminosities for various emission lines in the
$M_{\rm BH}$-$\lambda_{\rm Edd}$ plane. In the left panel, we compute the
ionizing luminosities using the integral expression $L_{\rm
ion}=\int_{\nu_{\rm product}}^{\nu_{\rm destruct}}L_{\nu}\,d\nu$ 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 $\lambda$1640 , and 13.6 eV for
H$\beta$ . The middle panel displays the line luminosities predicted by the
LOC model using Eqn. 2 (see §2.2). The right panel presents the observed line
luminosities for both local RM AGNs and the SDSS-RM sample. For local RM AGNs,
H$\beta$ line luminosities are from Du & Wang (2019). The line luminosities
for H$\alpha$ , H$\beta$ and He II $\lambda$4687 in the SDSS-RM sample are
from Shen et al. (2019). There are offsets of $\sim 1$ dex in the observed C
III] , He II $\lambda$4687 , and He II $\lambda$1640 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 $\textit{$M_{\rm BH}$}-\textit{$\lambda_{\rm Edd}$}$ plane.
Most importantly, this coherent trend between the LOC predictions and
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
($\log{\textit{$\dot{m}$}}<-1.5$), 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 panel, we use the incident SED and LOC predicted line flux to
predict the EW. For massive BHs ($\textit{$M_{\rm BH}$}>10^{8}\,M_{\odot}$),
more luminous (higher accretion-rate) BHs exhibit weaker emission line
strength, particularly for UV lines, and vice versa for BHs with mass
$\textit{$M_{\rm BH}$}\leqslant 10^{8}\,M_{\odot}$, as predicted by the LOC
model. This pattern predicted by the LOC model indicates that the continuum
and line luminosities change non-linearly with $M_{\rm BH}$ and $\lambda_{\rm
Edd}$. This non-linear relationship suggests that the underlying SED shape may
be more complex and varies significantly with changes in these parameters. The
middle panel of Figure 9 shows the observed EW values, and the right panel
displays binned observed EW values. Although the predicted EWs in C III] , He
II $\lambda$4687 , and He II $\lambda$1640 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 Figure 4 and
Figure 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 $M_{\rm BH}$-$\lambda_{\rm
Edd}$ space, thus unable to demonstrate the same diagonal trend in the LOC
model prediction.

###  3.3 Comparisons of composite UV spectra

Figure 10: Median composite spectra for different accretion rate samples (top)
and different H$\beta$ lag samples (bottom) with roughly matched optical
luminosity $43\leqslant\log(\textit{$\lambda L_{5100}$})\leqslant 45$,
normalized at 1350 ${\rm\AA}$ (grey shaded area).

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 rest-frame UV spectra for RM AGNs within a matched optical
luminosity range ($43\leqslant\log(\textit{$\lambda L_{5100}$})\leqslant 45$)
but with different $\lambda_{\rm Edd}$(upper panel) and H$\beta$ lags (lower
panel). The fluxes are normalized at rest-frame wavelength 1350 ${\rm\AA}$ to
better illustrate the UV line strength. In the upper panel, subsets with
higher $\lambda_{\rm Edd}$ values (green) exhibit stronger Ly $\alpha$ 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 lower panel of Figure 10, we divide our sample into two
subsets based on their H$\beta$ 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 sublet (blue); while
these two spectra appear to have similar Ly $\alpha$ 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
FUV compared to the longer-lag subset, with the former corresponding to the
higher-accreting systems in Figure 3. However, it appear 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 §4.2.

##  4 Discussion

###  4.1 Line ratios as a proxy for SED shape

Figure 11: Contour plots showing the dependence on the mass and accretion rate
for monochromatic luminosity ratio at certain energies (upper panel) and
various predicted line ratios (lower panel).

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 $M_{\rm BH}$ and $\lambda_{\rm Edd}$, predicted from
photoionization calculations. In the top panel of Figure 11, the monochromatic
luminosities at the ionizing energies of C IV (64.5 eV), Mg II (15.0 eV),
H$\beta$ (13.6 eV) and He II $\lambda$1640 (54.4 eV) are computed from the
grid of SEDs and then their line ratios are placed in the $M_{\rm
BH}$-$\lambda_{\rm Edd}$ plane. The bottom panel shows the corresponding line
luminosity ratios of the same lines as in the top panel. The line luminosity
ratio is computed using CLOUDY and the LOC model, as explained in §2.2. Both
monochromatic luminosity ratios and line luminosity ratios demonstrate
complicated dependence on the mass and accretion rate. Given that C IV and Mg
II are collisionally excited lines with excitation energies of 8 eV and 4.4
eV, while H$\beta$ and He II $\lambda$1640 are recombination-dominated lines,
the line ratios involving C IV may not accurately trace the SED shape; in
contrast, He II $\lambda$1640 is more effective in tracing the SED shape in
Figure 11.

Figure 12: Scaling relations of monochromatic luminosity ratios (i.e. SED
shape) and observables ($M_{\rm BH}$, $\lambda_{\rm Edd}$ and lines flux
ratio) for the theoretical models, with the coefficients of determination
$R^{2}$ on the upper left corners.

Given the strong correlations between predicted continuum-luminosity ratios
and line-flux ratios in Figure 11, we use the BH mass $M_{\rm BH}$, accretion
rate $\lambda_{\rm Edd}$, and line flux ratios in our theoretical computation
models to provide empirical relations that map the underlying ionizing SED
shape from $\sim 10$ to $\sim 60$ eV. The concept of using UV emission line
strengths to probe the ionizing SED has been previously applied to the EW of
He II $\lambda$1640 as tracers of the EUV continuum (e.g., Mathews & Ferland,
1987; Korista & Goad, 2000; Leighly, 2004; 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 He II
$\lambda$1640 EW is not tracing the real ionizing continuum shape but the
ratio between ionizing energy for He II $\lambda$1640 and the continuum that
the observer receives (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:

$\begin{split}\log(L_{E_{1}}/L_{E_{2}})=&\theta_{0}+\theta_{1}\log{M_{\rm
BH}}+\theta_{2}\log{\textit{$\lambda_{\rm Edd}$}}\\\ &+\theta_{3}\log(L_{\rm
line1}/L_{\rm line2}),\end{split}$ (4)

where $L_{E}$ is the monochromatic luminosity at energy $E$, and $E_{1}$ and
$E_{2}$ are the ionizing energy of line1 and line2, 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
($R^{2}=1-{\sum_{i=1}^{n}(y_{i}-\hat{y}_{i})^{2}}/{\sum_{i=1}^{n}(y_{i}-\bar{y})^{2}}$)
of the four pairs of lines suggest that the line ratios can trace the ionizing
continuum shape at $\sim 10$ eV fairly well, especially for C IV /Mg II .

Table 3: Best-fit parameters of line excitation energy ratios
$\log(L_{E_{1}}/L_{E_{2}})$ defined in Eqn. 4

Line ratios | $\theta_{0}$ | $\theta_{1}$ | $\theta_{2}$ | $\theta_{3}$
---|---|---|---|---
CIV/H$\beta$ | -0.894 | -0.169 | 0.567 | 3.163
C IV /Mg II | -1.602 | -0.109 | 0.335 | 5.362
He II $\lambda$1640 /H$\beta$ | -0.327 | -0.045 | 0.106 | 2.535
He II $\lambda$1640 /Mg II | -0.451 | 0.040 | -0.257 | 3.163

Multi-wavelength 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 ($M_{\rm BH}$ and $\dot{m}$), chemical
abundance, and ionization parameters (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 Cann et al. (2018) demonstrate that several
pairs of CL ratios might trace intermediate-mass BHs in dwarf galaxies, as the
harder SEDs in IMBHs can excite stronger high-ionization lines. With the
advent of the James Webb Space Telescope (JWST), infrared spectroscopic
observations covering a wealth of IR coronal lines 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.

###  4.2 Dispersion in the H$\beta$ $R-L$ relation

Figure 13: The $R-L$ relation: observed H$\beta$ lag against the optical
luminosity for RM quasars with $\textit{$\lambda_{\rm Edd}$}<0.5$, color-coded
based on the relative $\log\textit{$\lambda_{\rm Edd}$}$ values calculated as
the ratio of $\lambda_{\rm Edd}$ relative to the median $\lambda_{\rm Edd}$ in
each $\lambda L_{5100}$ bin (suggested by grey dotted lines).

In §2.3 we predicted the H$\beta$ time lags with constant ionization parameter
and compared them with the observed lags. Since the BLR is sensitive to the
entire ionizing flux emitted by the central source, varying $M_{\rm BH}$ and
$\lambda_{\rm Edd}$ lead to different ionizing SEDs that are responsible for
different BLR sizes as well as different line-ionizing fluxes for different
line-emitting ions in the BLR clouds. The parameters ($M_{\rm BH}$ and
$\lambda_{\rm Edd}$) are degenerate at fixed optical luminosity in the $R-L$
relation, as shown in Figure 3 and Figure 4. It is therefore natural to expect
an intrinsic dispersion in time lags at fixed optical luminosity caused by
different accretion parameters and subsequently the underlying SED. So far
this model has been the most favored interpretation in recent work on this
topic (Du & Wang, 2019; Fonseca Alvarez et al., 2020; Shen et al., 2024;
Czerny et al., 2019; Martínez-Aldama et al., 2019; Dalla Bontà et al., 2020).
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.

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 (Shakura & Sunyaev,
1973), and become a slim disk (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 H$\beta$ time lags in high-accretion
quasars have been reported by the SEAMBH collaboration (Du et al., 2014, 2015,
2018; 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 on 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 and at fixed optical luminosity,
AGNs with larger BH masses or lower accretion rates tend to have shorter lags.
This prediction is in tension with the observed trend. Figure 13 presents the
observed $R-L$ relation for our RM AGN sample, color-coded by the relative
Eddington ratio at each bin. A decreasing trend of lags with $L/L_{\rm Edd}$
at fixed optical luminosity is observed even for the SSD regime studied in
this work. This discrepancy suggests that certain simple assumptions in our
photoionization modeling, e.g., a universal gas density, might be incorrect.
For example, if $n_{\rm H}$ increases with $L/L_{\rm Edd}$, and given that
$R_{\rm BLR}\propto n_{\rm H}^{-1/2}$ (Egn. 3), 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 $n_{\rm H}$ increases, broad-line
emissivity would also increase, resulting in higher predicted broad-line
fluxes for higher Eddington ratios, again consistent with observations.

###  4.3 Predicted hydrogen-ionizing photon flux

Figure 14: Linear regression scaling between the ionizing photon number
$Q({\rm H})$ and the observables for theoretical measurements. Figure 15:
Predicted H$\beta$ time lag following Eqn. (3) versus the observed optical
luminosity, color-coded by $Q({\rm H})/\textit{$\lambda L_{5100}$}$, where the
ionizing photon flux $Q({\rm H})$ is predicted from observed parameters (Eqn.
5).  Table 4: Best-fit parameters of the hydrogen-ionizing photon flux $\log
Q({\rm H})$ in Eqn. 5

Line ratios | $\theta_{0}$ | $\theta_{1}$ | $\theta_{2}$ | $\theta_{3}$
---|---|---|---|---
CIV/H$\beta$ | 53.035 | 0.475 | 1.636 | 0.358
C IV /Mg II | 53.140 | 0.479 | 1.581 | 0.200
He II $\lambda$1640 /H$\beta$ | 53.057 | 0.492 | 1.583 | 0.342
He II $\lambda$1640 /Mg II | 53.119 | 0.493 | 1.546 | 0.282

In our theoretical analysis, the time lag of H$\beta$ is modeled as a linear
function of the hydrogen-ionizing photon flux $Q({\rm H})$, as expressed by
Eqn. 3. 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 (Negrete et al., 2012;
Ferland et al., 2020; Buendia-Rios et al., 2023). As discussed in §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 $Q({\rm H})$ for observed quasars through the
following equation:

$\begin{split}\log Q({\rm H})=&\theta_{0}+\theta_{1}\log{M_{\rm
BH}}+\theta_{2}\log{\textit{$\lambda_{\rm Edd}$}}\\\ &+\theta_{3}\log(L_{\rm
line1}/L_{\rm line2}),\end{split}$ (5)

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 $Q({\rm H})$ could be well estimated using such
observables ($M_{\rm BH}$, $\lambda_{\rm Edd}$, and line flux ratios).

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

###  4.4 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. We emphasize that we have not
attempted to construct a model finely tuned to match the properties of any
specific AGN. This general approach quantitatively reproduces the behavior of
the BLR distance and line luminosity relative to BH mass and accretion rate
given universal ionization parameters. However, this photoionization modeling
is carried out for low-accretion quasars in the SSD regime with fixed
inclination $\cos{i}=0.5$ and for non-spinning BHs only. The impact of BH spin
on the BLR size is pronounced, which would affect the shape of the ionizing
continuum, especially in the UV band (Kubota & Done, 2018; Wang et al., 2014a;
Reynolds, 2021). The inclusion of BH spin in our analysis would introduce
additional complexity, potentially leading to a greater dispersion in the
$R-L$ relation (Czerny et al., 2019) and producing degenerating results with
BH mass and accretion rate.

In our SED modeling, because of the imposed accretion rate limits of
$-1.65<\log\textit{$\dot{m}$}<0.39$ in qsosed, this work is limited to mostly
the SSD-dominated accretion regime and only models $\log\textit{$\dot{m}$}\leq
0$. Compared to the original agnsed, the simplified model qsosed fixes the
electron temperature and spectral index at typical values for the warm and hot
Comptonization components as follows: $kT_{\rm e,hot}=100\,\textit{$\rm
keV$}$, $kT_{\rm e,warm}=0.2\,\textit{$\rm keV$}$, $\Gamma_{\rm warm}=2.5$,
$r_{\rm warm}=2r_{\rm hot}$, $r_{\rm out}=r_{\rm sg}$, $h_{\rm max}=100$, with
the power dissipated by the hot Comptonization region fixed at $0.02L_{\rm
Edd}$ (Kubota & Done, 2018). By comparing the stacked AGN SEDs from SOUX
(Kynoch et al., 2023) with qsosed predictions, Mitchell et al. (2023) found
that while the qsosed model matches well with the observed SEDs for
intermediate-mass AGNs ($\log\textit{$M_{\rm BH}$}=7.5-9.0$), it underpredicts
the X-ray flux at low-$M_{\rm BH}$ and shows discrepancies in the optical/UV
continuum for high-$M_{\rm BH}$ AGNs. Assuming that the X-ray emission is
generated from the hot Comptonization region and the EUV emission originates
from the warm Comptonization region (Petrucci et al., 2018; Kubota & Done,
2018), the absence of a predictive model with solid theoretical basis to
untangle the poorly understood warm Comptonization component and its relation
to the hot Comptonization coronal properties still limits the comparison
between observations and theoretical analysis, and thereby our understanding
of AGN SEDs. However, the demonstrated correlations between the X-ray, EUV,
and UV regions (e.g., Green, 1996; Steffen et al., 2006; Just et al., 2007;
Lusso & Risaliti, 2016; Timlin et al., 2020, 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 §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 disc that shadows outer regions, which will potentially soften
the SED received by the BLR clouds (Abramowicz et al., 1988; 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 (Panda et
al., 2019; Korista & Goad, 2019). While 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 $M_{\rm
BH}$-$\lambda_{\rm Edd}$ 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 §4.2.

Finally, 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 some of the highest energy photons (e.g., Wu et al., 2011, 2012;
Luo et al., 2015; Plotkin et al., 2015; Ni et al., 2018, 2022). In this work,
we do not see much evidence for this shielding gas, as the predicted line
strengths are consistent with the BLR seeing all the continuum flux. However,
the covering fraction (Ferland et al., 2020) and the impact of this shielding
gas may become more prominent for even higher Eddington ratios, i.e., in the
slim disk regime. We plan to investigate this possibility in future work.

##  5 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 (Kubota & Done, 2018) as the input continuum. We compare
our theoretical results with observed AGN properties for a sample of 70 low-
redshift AGNs with reverberation mapping measurements that are in the sub-
Eddington accretion regime. Our main conclusions are as follows:

  1. 1.

Assuming universal values of hydrogen density ($\log n({\rm H})=12$) and
ionization parameter ($\log U_{\rm H}=-2$), the photoionization models
reproduce the observed global relation between the Balmer broad-line lag and
optical luminosity at rest-frame 5100 Å (§2.3). However, at fixed $L_{\rm
5100}$, the model predicts longer lags for higher Eddington ratios, opposite
to the observed trend. These results may imply that the gas density increases
with the accretion rate to reduce the average size of the BLR. (§4.2)

  2. 2.

The photonization 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
broad-line fluxes in the $M_{\rm BH}$-$\lambda_{\rm Edd}$ plane are consistent
between observations and theoretical predictions (§3.2)

  3. 3.

Based on these comparisons, we provide empirical scaling relations (Eqn. 4 and
5) that utilize the observed BH mass, $L/L_{\rm Edd}$, and line flux ratios to
constrain the ionizing continuum flux, which is not directly observable.
(§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 coronal lines (with ionizing energy at $>150\,{\rm eV}$) 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.

Q.W. and Y.S. acknowledge partial support from NSF grant AST-2009947. Based on
observations with the NASA/ESA Hubble Space Telescope obtained from the Data
Archive at the Space Telescope Science Institute, which is operated by the
Association of Universities for Research in Astronomy, Incorporated, under
NASA contract NAS5-26555. Support for programs HST-GO-16171 and HST-GO-17433
was provided through a grant from the STScI under NASA contract NAS5-26555.
HXG is supported by the National Key R&D Program of China No. 2022YFF0503402,
2023YFA1607903. LCH was supported by the National Science Foundation of China
(11991052, 12233001), the National Key R&D Program of China (2022YFF0503401),
and the China Manned Space Project (CMS-CSST-2021-A04, CMS-CSST-2021-A06).
J.I.L. is supported by the Eric and Wendy Schmidt AI in Science Postdoctoral
Fellowship, a Schmidt Futures program.

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