# Steep Balmer decrement in weak AGNs may be not caused by dust extinction:
clues from low-luminosity AGNs and changing-look AGNs

Jiancheng Wu Department of Astronomy, School of Physics, Huazhong University
of Science and Technology, Luoyu Road 1037, Wuhan, China  Qingwen Wu∗
Department of Astronomy, School of Physics, Huazhong University of Science and
Technology, Luoyu Road 1037, Wuhan, China  Hanrui Xue Department of Astronomy,
School of Physics, Huazhong University of Science and Technology, Luoyu Road
1037, Wuhan, China  Weihua Lei Department of Astronomy, School of Physics,
Huazhong University of Science and Technology, Luoyu Road 1037, Wuhan, China
Bing Lyu Kavli Institute for Astronomy and Astrophysics, Peking University,
Beijing 100871, Peoples Republic of China

###### Abstract

The hydrogen Balmer decrement (e.g., $\rm H\alpha/H\beta$) is widely adopted
as an indicator of the internal reddening of active galactic nuclei (AGNs).
This is challenged by some low-luminosity AGNs (LLAGNs) and changing-look AGNs
(CLAGNs), which have steep Balmer decrement but without strong evidence for
absorption. We compile a sample of normal AGNs and CLAGNs with a wider
distribution of bolometric Eddington ratio ($\lambda_{\rm Edd}=L_{\rm
bol}/L_{\rm Edd}$) and find a strong negative correlation between $\rm
H\alpha/H\beta$ and $\lambda_{\rm Edd}$, which suggests that the Balmer
decrement is also accretion-rate dependent. We further explore the Balmer
decrement based on the photoionization model using the Cloudy code by
considering spectral energy distribution (SED) from the accretion disk with
different accretion rates (e.g., disk/corona and truncated disk at high and
low Eddington ratios, respectively). Both the standard disk and truncated disk
predict a negative correlation of $\rm H\alpha/H\beta-\lambda_{\rm Edd}$,
where the relation is steeper in the case of the truncated disk. The negative
correlations are also explored in two single CLAGNs. The measured negative
correlation of $\rm H\alpha/H\beta$ – $\lambda_{\rm Edd}$ is mainly caused by
the lower responsivity $({\rm dlog}L_{\rm line}/{\rm dlog}L_{\rm cont})$ in
$\rm H\alpha$ relative to that in $\rm H\beta$, due to the larger optical
depth in the former. We propose that the steep Balmer decrements in low-
Eddington-ratio AGNs (e.g., some Seyferts 1.5-1.9 and CLAGNs) are not simply
caused by absorption but mainly caused by the relatively low flux of ionizing
photons.

Active galactic nuclei (16), Seyfert galaxies (1447), Accretion (14),
Supermassive black holes (1663), Line intensities (2084)

††software:  Cloudy (Ferland et al., 2017).

##  1 Introduction

Active galactic nuclei (AGN) are powered by the accretion of matter onto a
supermassive black hole (SMBH), which are the most luminous objects in our
universe. The AGNs are empirically divided into type 1 and type 2 sources
according to the presence of broad emission lines (e.g., Full Width at Half
Maximum, $\rm FWHM\geq 1000km/s$) or not. The broad emission lines are
produced in the broad line region (BLR) that stay $\sim$pc scale from the
central SMBH, and the narrow emission lines are emitted from the gas located
at much larger distances (narrow line region, NLR). The type 1 and type 2 AGNs
are unified based on the inclination of the accretion disk to the line of
sight, where the BLR is obscured in type 2 AGNs by a putative dusty torus. The
highly polarized components of broad emission lines as detected in some type 2
AGNs do support the above unification scenario (e.g., Antonucci, 1993; Urry &
Padovani, 1995). Intermediate-type AGNs (e.g., type 1.2, 1.5, 1.8, or 1.9)
based on the relative flux of narrow and broad emission lines are also
classified, which complicate the simple dichotomy (e.g., Osterbrock et al.,
1976; Osterbrock, 1981). For example, the broad H$\beta$ component is weak or
disappears in type 1.9 AGNs, where broad H$\alpha$ is still evident. A further
challenge to the unification model comes from the so-called changing-look
AGNs(CLAGNs), where a significant change in optical broad emission lines
and/or strong variation of line-of-sight column densities are found in a
single AGN within several years or several decades (see a recent review, Ricci
& Trakhtenbrot, 2022). The physical mechanism of CLAGNs is not well
understood, and possible mechanisms include the variation of the obscuring
material in clumpy torus (e.g., Marin et al., 2013; Agís-González et al.,
2014; Rivers et al., 2015; Turner et al., 2018; Wang et al., 2019) or the
variation of accretion rate (e.g., Noda & Done, 2018; Yang et al., 2021; Lyu
et al., 2021, 2022; Liu et al., 2022).

The optically thick, geometrically thin standard accretion disk (SSD, Shakura
& Sunyaev, 1973) is believed to be presented in bright AGNs, where the big
blue bump provides good evidence for this disk component. However, the SSD
likely transitions to an optically thin, geometrically thick, advection-
dominated accretion flow (ADAF) in the low luminosity AGNs (LLAGNs, Yuan &
Narayan, 2014). Many LLAGNs show only a red “bump”, which possibly comes from
the outer SSD (e.g., the truncated disk model, Ho, 2008). Broad H$\alpha$
lines are observed while H$\beta$ and other high-ionization lines are weak or
absent in many LLAGNs (see more details in Ho, 2008). The possible physical
reason may be the deficit of ionization photons in ADAF case (e.g., Elitzur &
Ho, 2009) or the BLR cannot be sustained due to the decrease of mass outflow
rate of disk winds in such a low accretion state (e.g., Elitzur & Shlosman,
2006).

Balmer decrement has long been adopted as an indicator of the internal
reddening of AGNs, which is measured from the strength of the hydrogen
emission corresponding to different excitation states. It is normally assumed
that the conditions (e.g., density, temperature, etc.) are fixed and high-
ionization lines may be easily affected by the obscuring material since the
optical depth is wavelength-dependent. Therefore, the deviation from the
expected flux ratio of $\rm H\alpha/H\beta$ will indicate the amount of
obscuration toward the line-emitting regions (e.g., Ward et al., 1987;
Gaskell, 2017; Osterbrock, 1989). Gaskell (2017) proposed that, with typical
parameters of BLR, the intrinsic Balmer decrement in extremely blue AGNs is
$\rm H\alpha/H\beta\sim 2.7$ for broad emission lines, which is more or less
consistent with a Baker-Menzel Case B value. However, the classical nebular
theory of hydrogen line emission (Case B) is not expected to be applicable in
the dense and optically thick conditions thought to exist in the line-emitting
gas within the BLRs of AGN (e.g., Netzer, 1975; Ferland et al., 1979; Korista
& Goad, 2004; Ferland et al., 2020). We also note that a Balmer line flux
ratio measurement that happens to correspond to a prediction of Case B does
not indicate the line-emitting gas conditions are those required by Case B:
modestly optically thick in the Lyman lines, optically thin in the excited-
state transition series, and spectrum completely dominated by recombination
cascade. In general, there is no such one-to-one mapping of the hydrogen line
flux ratio to physical line-emitting gas conditions. From the observations,
the Balmer decrements were explored with several samples from Sloan Digital
Sky Survey (SDSS). For example, La Mura et al. (2007) derived $\left\langle\rm
H\alpha/H\beta\right\rangle=3.45\pm 0.65$ based on 90 Seyfert 1 galaxies. Dong
et al. (2008) selected 446 Seyfert 1 galaxies and found $\left\langle\rm
H\alpha/H\beta\right\rangle=3.06$ with standard deviation 0.03 dex. Lu et al.
(2019) found $\left\langle\rm H\alpha/H\beta\right\rangle=3.16$ with a
standard deviation of 0.07 dex for 554 low-redshift AGNs. It is still unknown
whether the higher observational ratio of $\rm H\alpha/H\beta$ compared to the
theoretical prediction is intrinsic or caused by the internal reddening of
AGNs.

Furthermore, in some local Seyferts, the ratio of $\rm H\alpha/H\beta$ ranges
from 4 to 7 (e.g., Schnorr-Müller et al., 2016), which is even larger in some
type 1.8/1.9 AGNs (Crenshaw et al., 1988; Osterbrock et al., 1976). The $\rm
H\alpha/H\beta$ in some optical CLAGNs showing the appearance and
disappearance of broad emission lines can reach up to 10 or even higher with a
decrease of luminosities (Trakhtenbrot et al., 2019, e.g., $\rm
H\alpha/H\beta=8-15$ at 150 days after the transient in 1ES 1927+654,; $\rm
H\alpha/H\beta=8-10$ in NGC 2992, Guolo et al., 2021; $\rm H\alpha/H\beta=4-6$
in NGC 3516, Shapovalova et al., 2019; $\rm H\alpha/H\beta=4-9$ in Mrk 1018,
Cohen et al., 1986; McElroy et al., 2016). The effects of collisional
excitation and/or dust extinction are suggested to be the reason for such a
large deviation of the Balmer decrement Gaskell & Ferland (1984); Osterbrock &
Ferland (2006); Shapovalova et al. (2019). It should be noted that the
absorption is always low in some CLAGNs (e.g., Mrk 1018, LaMassa et al., 2017)
and some Seyfert 1.8/1.9 galaxies, (e.g., Barcons et al., 2003). Therefore,
the absorption scenario cannot fully explain the steep Balmer decrement in
these AGNs.

The broad emission lines in AGNs are sensitive to the spectral energy
distribution (SED) from the ultraviolet to far-ultraviolet or even X-ray
wavebands. Therefore, the emission lines carry the information of the part SED
of AGNs (e.g., Boroson & Green, 1992). Korista & Goad (2004) explored the
effects of a global change in the ionizing continuum level on the behavior of
the stronger optical hydrogen and helium broad emission lines based on a
distributed cloud photoionization model of a BLR in the LOC scenario (Baldwin
et al., 1995). They found that the differing responsivities$({\rm dlog}L_{\rm
line}/{\rm dlog}L_{\rm cont})$ among the emission lines can explain several
observed phenomena, including the anti-correlation of Balmer decrement with
continuum state within an individual AGN. However, they neither considered an
evolution of the SED with a continuum state (which can further change Q(H)),
nor SED differences across the populations of AGN with different masses and
$L/L_{\rm Edd}$. These considerations are central to the present study.
Ferland et al. (2020) also investigated the relationship between the hydrogen
and helium broad emission lines and AGN SEDs, and proposed that systematic
changes in the SEDs with Eddington ratio alone cannot fully explain the
hydrogen Balmer to He II emission line flux ratios and that the integrated BLR
covering factor may be decreasing systematically with increasing Eddington
ratio. Guo et al. (2020) studied the variability of Mg II, H$\alpha$, and
H$\beta$ lines in changing-look quasars and proposed that the dramatic changes
of the broad lines are consistent with photoionization responses to extreme
continuum variability.

In recent years, accumulating observational evidence supports the scenario
that CLAGNs undergo strong changes in continuum state (e.g., Noda & Done,
2018), as well as potentially accretion mode transitions during the changing
look phase (e.g., MacLeod et al., 2019; Guolo et al., 2021; Lyu et al., 2021;
Liu et al., 2022). In this work, we explore the possible influence of the
evolution of ionizing SED on the Balmer decrement in CLAGNs and LLAGNs, which
can shed light on the physical reason for the higher Balmer decrement and the
possible physics of the disappearance/re-appearance of broad emission lines.
In Section 2, we present our sample. The model and results are presented in
Sections 3 and 4, respectively. Conclusion and Discussion are shown in Section
5. In this work we use the standard cosmological model ($H_{0}={\rm
70\,km\,s^{-1}\,Mpc^{-1}},\Omega_{\Lambda}=0.7,\Omega_{\rm M}=0.3$).

##  2 Sample

There are many reported AGN samples for exploring the Balmer decrements in the
literature, and most of them focus on the bright blue AGNs. According to the
purpose of our work, we need an AGN sample with a wider distribution of
Eddington ratios. In this work, our sample consists of normal AGNs and
changing-look AGNs with measurements of both $\rm H\alpha$ and $\rm H\beta$
intensities.

We select AGNs from La Mura et al. (2007) and Jaffarian & Gaskell (2020),
which include AGNs with a wider distribution of Eddington ratios (e.g., type
1-1.9 or even some true type 2 sources with weak absorption). There are 90
AGNs in La Mura et al. (2007), which have a black hole (BH) mass in the range
of $10^{5-9}M_{\odot}$ and bolometric Eddington ratio ($\lambda_{\rm
Edd}=L_{\rm bol}/L_{\rm Edd}$) in range of 0.01-1. We also select 33 AGNs with
bolometric Eddington ratio $0.001\leq\lambda_{\rm Edd}\leq 1$ and low X-ray
column density (i.e. $N_{\rm H}\leq 10^{21}{\rm cm^{-2}}$) from Jaffarian &
Gaskell (2020), which imply no prominent intrinsic reddening and extinction.
The bolometric Eddington ratio is estimated from the optical (e.g., $L_{\rm
bol}\sim 9L_{\rm 5100}$, Kaspi et al., 2000; Elvis et al., 1994) or X-ray
observations with the empirical bolometric corrections (see Table 1 and
references).

For CLAGNs, we include 6 sources with multiple optical spectral observations
(NGC 5548, NGC 4151, 3C 390.3, NGC 3516, Mrk 1018, and 1ES 1927+654) and 16
sources with only two observations from Jin et al. (2022). We note that only
the emission-line intensities at a high state (or turn-on state) with evident
broad lines are adopted in our work because the emission lines are weak or
absent in a low state (or turn-off state). The multiple observational data for
NGC 5548, NGC 4151, and 3C 390.3 are selected from Rakić et al. (2017). For
Mrk 1018, we adopt quasi-simultaneous observation of $\rm H\alpha$ and $\rm
H\beta$ lines from Cohen et al. (1986) and McElroy et al. (2016). The emission
line data of NGC 3516 are selected from Shapovalova et al. (2019), where we
exclude the data from 2014 to 2018 due to the $\rm H\beta$ line being too weak
and the measurements on the broad component are not accurate. The data of 1ES
1927+654 is selected from Li et al. (2022). For CLAGNs with only two spectra
in Jin et al. (2022), we exclude the objects observed by XinLong Telescope for
the bad spectral quality and several sources with very large differences of
velocity even in active state (i.e., $\rm FWHM(H\beta)/FWHM(H\alpha)\geq 1.5$
or $\rm FWHM(H\alpha)/FWHM(H\beta)\geq 1.5$). The source ZTF18aajupnt is
considered to be a candidate for tidal disruption event (Jin et al., 2022),
which is also removed from our sample. The BH mass ${\rm log}M_{\rm BH}$=8.61,
7.72, 7.81, 7.28, 7.70, and 7.90 are adopted for the CLAGN of 3C 390.3
(Afanasiev et al., 2019), NGC 4151 (Du et al., 2015), NGC 5548 (Afanasiev et
al., 2019), 1ES 1927+654 (Trakhtenbrot et al., 2019), NGC 3516 (Vasudevan et
al., 2010) and Mrk 1018 (Noda & Done, 2018), respectively. It should be noted
that Mrk 590 is a well-known CLAGN, which is not included in our AGN sample
because we only find one simultaneous flux of $\rm H\alpha$ and $\rm H\beta$
lines that are reported in the literature.

Table 1: Objects with low absorption of $N_{\rm H}<10^{21}\rm cm^{-2}$
selected from Jaffarian & Gaskell (2020).

Objects |  z |  H$\alpha$/H$\beta$ |  ${\rm log}\lambda_{\rm Edd}$ |  log$M_{\rm BH}(M_{\odot})$ |  log$N_{\rm H}$ |  Reference
---|---|---|---|---|---|---
Mrk 110 |  0.035 |  4.22 |  -1.89 |  8.32 |  20.17 |  1
I Zw 1 |  0.061 |  4.66 |  -1.13 |  7.46 |  20.81 |  1
Mrk 335 |  0.026 |  2.60 |  -0.92 |  7.49 |  20.55 |  1
Mrk 79 |  0.022 |  5.53 |  -1.00 |  7.45 |  20.97 |  1
Mrk 1044 |  0.016 |  2.40 |  -0.49 |  6.45 |  20.62 |  2
Mrk 1310 |  0.020 |  3.97 |  -1.48 |  6.62 |  20.48 |  2
Mrk 142 |  0.045 |  2.90 |  -0.17 |  6.59 |  20.40 |  2
Mrk 205 |  0.071 |  4.28 |  -1.52 |  8.32 |  20.35 |  14
Mrk 304 |  0.066 |  2.86 |  -0.80 |  8.04 |  20.16 |  1
Mrk 40 |  0.021 |  2.65 |  -1.19* |  7.09 |  20.36 |  7,16
Mrk 42 |  0.024 |  3.63 |  -0.04* |  6.00 |  19.90 |  7,8
Mrk 474 |  0.037 |  3.09 |  -1.04* |  7.41 |  20.49 |  7,16
Mrk 493 |  0.031 |  2.71 |  -0.17 |  6.14 |  20.47 |  2
Mrk 876 |  0.121 |  5.20 |  -0.80 |  8.36 |  19.63 |  1
NGC 985 |  0.043 |  4.67 |  -1.24 |  8.36 |  20.77 |  4
PG 1001+054 |  0.160 |  3.20 |  -0.30 |  7.74 |  19.37 |  6
PG 1244+026 |  0.048 |  2.95 |  -0.10 |  7.21 |  19.49 |  9
PG 1448+273 |  0.064 |  3.40 |  -0.12 |  7.29 |  19.64 |  10
Fairall 9 |  0.046 |  2.59 |  -1.26 |  8.09 |  20.30 |  2
Mrk 50 |  0.024 |  3.30 |  -0.10* |  6.18 |  19.90 |  7,16
UGC 6728 |  0.007 |  7.00 |  -1.04 |  5.32 |  19.74 |  11
Mrk 509 |  0.034 |  2.69 |  -1.46 |  8.56 |  20.81 |  4
Mrk 817 |  0.031 |  4.00 |  -1.11 |  7.65 |  20.06 |  1
NGC 1566 |  0.005 |  4.18 |  -1.57* |  6.65 |  19.73 |  7,16
Mrk 1218 |  0.029 |  9.04 |  -2.96 |  8.75 |  20.65 |  5,16
Mrk 841 |  0.036 |  4.80 |  -1.62 |  8.76 |  20.20 |  1
Mrk 609 |  0.034 |  4.95 |  -1.48 |  7.80 |  20.77 |  12
NGC 3660 |  0.012 |  7.50 |  -2.26 |  6.85 |  20.26 |  15
NGC 5273 |  0.004 |  4.24 |  -2.74 |  7.14 |  20.95 |  2
NGC 5929 |  0.009 |  5.13 |  -0.89 |  7.25 |  20.71 |  3
NGC 7590 |  0.005 |  3.65 |  -1.40 |  6.51 |  20.96 |  3
Mrk 266SW |  0.028 |  6.88 |  -2.04 |  8.30 |  20.61 |  13
Mrk 590 |  0.026 |  4.19 |  -1.19 |  7.55 |  20.54 |  2

Note. — References: (1) Afanasiev et al. (2019); (2) Du et al. (2015); (3)
Bian & Gu (2007); (4) Vasudevan et al. (2010); (5) Singh et al. (2011); (6)
Wang et al. (2022); (7) Rudy (1984); (8) Wang & Lu (2001); (9) Jin et al.
(2013); (10) Laurenti et al. (2021); (11) García-Bernete et al. (2019); (12)
Lyu et al. (2022); (13) Iwasawa et al. (2020); (14) Laha et al. (2019); (15)
Rivers et al. (2016); (16) The BH mass is calculated from the $M-\sigma_{*}$
relation Ferrarese & Merritt (2000) where the dispersion $\sigma_{*}$ is
adopted from the HyperLeda database http://leda.univ-lyon1.fr/. The stars in
column (4) indicate that $L_{5100}$, from which $L_{\rm bol}$ is estimated,
was calculated from $L_{\rm H\alpha}$ using $L_{\rm H\alpha}-L_{\rm 5100}$ as
proposed by Greene & Ho (2005).

##  3 Photoionization Model

To explore the properties of broad emission lines, we simulate a grid of
photoionization models based on Ferland’s spectral synthesis code Cloudy
(version 17.03, Ferland et al., 2017) and consider a wide variation of the
incident continuum SEDs associated with ranges in the mass of SMBH and the
Eddington ratio. More details will be presented below.

Figure 1: The upper panel shows the incident SED for a typical BH mass of
$M_{\rm BH}=10^{8}M_{\odot}$, which includes a blackbody component and a
power-law component with cutoff at ${\rm log}\nu=19.5$. The red solid line and
the blue dot-dashed line represent the SED with no disk truncation with
dimensionless accretion rate $\dot{m}=$ 1 and 0.1, respectively, while the
orange dashed line and the red dotted line represent the case for the
truncated disk with $R_{\rm tr}=12R_{\rm g}$ at $\dot{m}=0.01$ and $120R_{\rm
g}$ at $\dot{m}=0.001$, respectively. Typical quasar SED that is widely
adopted in the photoionization model is plotted with black double dots dashed
line (Mathews & Ferland, 1987). The vertical grey dashed line represents the
position of 1 Ryd. The number of hydrogen ionizing photons emitted by the
source per second, Q(H), is presented for corresponding SEDs. The lower panel
shows the contour map of $\rm H\alpha/H\beta$, as predicted by the
photoionization model clouds in the grid, where the SED corresponds to the
blue dot-dashed line in the upper panel. The red dotted lines show the outer
and inner boundary of BLR that correspond to the case of $\dot{m}=0.1$ and
${\rm logQ(H)=54.94}$, the blue dot-dashed line in the top panel.

###  3.1 Incident continuum

It is found that the blackbody component and power-law component are strongly
evolved with the change of the Eddington ratios in both AGNs and BH X-ray
binaries (XRBs). Apart from the energy dissipation in the SSD, a small
fraction of the gravitational energy will be released into the corona, which
is needed to explain the power-law component as found in both high-state XRBs
and luminous AGNs. The SSD contribution will become weak and the power-law
component will become dominant in low-state XRBs and LLAGNs, where the cold
SSD has possibly transitioned into the hot ADAF when the accretion rate is
lower than a critical value (e.g., $\dot{m}_{\rm c}=\dot{M}/\dot{M}_{\rm
Edd}\sim 2\%$, Yuan & Narayan, 2014). In this work, we simply adopt that the
transition radius follows $R_{\rm tr}=6(\dot{m}/\dot{m}_{c})^{-1}R_{\rm g}$
when the accretion rate is lower than the critical value ($R_{\rm g}=GM/c^{2}$
is the gravitational radius, Taam et al., 2012). The blackbody emission is
calculated based on either the full SSD or the outer SSD in the truncated
disk. The physical formation of the corona/ADAF is still unclear, and we
calculate the power-law component from two observed empirical correlations: 1)
the bolometric correction on the 2-10 keV luminosity $k_{\rm
bol}{\rm(2-10keV)}=7\times[L({\rm 2-10keV})/10^{42}{\rm erg\,cm^{-2}}]^{0.3}$
by Netzer (2019); 2) relation between X-ray photon index and bolometric
luminosity $\Gamma_{2-10\rm{keV}}-L_{\rm bol}$ (Qiao & Liu, 2018).

In the upper panel of Figure 1, we present examples of incident SED with a
blackbody component and a power-law component with $\dot{m}=0.001-1$ for
$M_{\rm BH}=10^{8}M_{\odot}$. For comparison, the typical SED widely adopted
in the photoionization model in the literature is also presented.

###  3.2 Photoionization models and line emission

Locally Optimally Emitting Cloud (LOC) scenario as proposed by Baldwin et al.
(1995) can successfully reproduce the emission-line strengths and line ratios
in both BLR and NLR of different types of AGNs (e.g., Ferland, 2003; Ferguson
et al., 1997; Korista & Goad, 2000). In this scenario, only a relatively
narrow range of parameters (e.g., gas density and incident hydrogen ionizing
photon flux and their ratio that is the ionizing parameter) will result in
high continuum reprocessing efficiencies for a particular emission line. Thus
a broad range in such parameters is then indicated to explain the strengths of
the wide range of commonly found broad emission lines in AGN. The integrated
emission line luminosity can be approximated as (see also Bottorff et al.,
2002)

$L_{\rm line}\propto\int{\int{r^{2}F(r,n)f(r)g(n)dndr}},$ (1)

where $F(r,n)$ is the emission line surface flux of a cloud at distance $r$
from the central ionizing source with constant density $n({\rm H})=n$, for a
choice of a total column density $N({\rm H})$. The functions $f(r)$ and $g(n)$
are weighting functions, which are normally chosen as power laws in each
variable. The function $f(r)\sim r^{\Gamma}$ represents the radial
differential covering fraction ${\rm d}C(r)/{\rm d}r$ of the line-emitting
clouds, while function $g(n)\sim n^{\beta}$ describes the weighting of clouds
of differing gas densities to the total solid area covered by clouds at
distance $r$. We calculate the line luminosity (i.e., $L_{\rm line}$) by
integrating over a logarithmic grid of photoionization models in parameter
space of $(r,n)$, with proper outer boundaries. In this work, we adopt the
typical values of $\Gamma=-1,\beta=-1$, as proposed in Baldwin et al. (1995),
Korista & Goad (2000) and Nagao et al. (2006).

Based on the reverberation mapping observations, the typical BLR size is
correlated with the luminosity (Blandford & McKee, 1982; Peterson, 1993;
Peterson et al., 2004; Bentz et al., 2009). However, the inner and outer
boundaries of BLR are still unclear, even though the torus size is sometimes
adopted as the outer BLR boundary ($R_{\rm torus}\sim 5R_{\rm BLR}$, Koshida
et al., 2014). Naddaf et al. (2020) found that the ratio $R_{\rm out}/R_{\rm
in}\sim 20-60$ for $\dot{m}=0.01$ to 1 by assuming that the BLR clouds are
formed from the Failed Radiatively Accelerated Dusty Outflow (FRADO). In this
work, we simply set $R_{\rm out}/R_{\rm in}=40$, and the other boundary
conditions will be discussed. The volume density of BLR clouds should be
larger than $\sim 10^{9}{\rm cm}^{-3}$, due to no observed broad forbidden
lines, e.g. [O III] $\lambda 4363$. Schnorr-Müller et al. (2016) gave the BLR
hydrogen density $\sim 10^{11}\rm{cm}^{-3}$ by studying some nearby Seyfert
galaxies(see also, Ferland et al., 1992; Shields & Ferland, 1993). Korista &
Goad (2000) proposed an upper limit of the BLR density $\sim 10^{12}{\rm
cm}^{-3}$, since clouds of higher gas densities transition to thermal
continuum emitters rather than sources of emission lines (see also Guo et al.,
2020). In this work, we set the BLR hydrogen density to vary from $10^{8}{\rm
cm^{-3}}$ to $10^{12}{\rm cm^{-3}}$ with grid steps of 0.2 dex in simulations.
Finally, we adopt a fixed value in the cloud total hydrogen column density
$N_{\rm H}=10^{23}{\rm cm^{-2}}$, as representative of such within the BLRs of
AGN (Dumont et al., 1998), and assume solar gas elemental abundances
$Z=Z_{\odot}$. To calculate the line luminosity, the typical cloud covering
factor 0.5 is adopted in this work (Korista & Goad, 2000; Guo et al., 2020),
which may decrease as the accretion rate increases (Ferland et al., 2020). We
note that while the integrated covering factor affects the predicted emission
line luminosities, it does not affect the predicted emission line luminosity
ratio.

In the lower panel of Figure 1, we present an example of a contour map of $\rm
H\alpha/H\beta$ in the density-distance plane with a dimensionless accretion
rate $\dot{m}=0.1$. The contours are those of the flux ratio $F({\rm
H\alpha})/F({\rm H\beta})$ predicted by the photoionization model clouds in
the grid. The larger radius will lead to a smaller hydrogen-ionizing photon
flux ${\rm\Phi(H)=Q(H)/4\pi}r^{2}$ and a steeper Balmer decrement (see also
Korista & Goad, 2004). With the above typical model parameters, we present
several typical line luminosities in Table 2, and compare the theoretical
prediction of H$\alpha$ and H$\beta$ lines with the observations (e.g., the
observed relation of $L_{5100}-L_{\rm H\alpha}$ and $L_{5100}-L_{\rm
H\beta}$), where the line luminosities are roughly consistent with those from
observations. For the line ratios, our model predicts $\rm Ly\alpha/H\beta\sim
20$, Mg II $\lambda 2800/\rm H\beta\sim 1.5$, C IV $\lambda 1549/\rm
H\beta\sim 4$ and He II $\,\lambda 1640/\rm H\beta\sim 0.5$ for above typical
parameters with $\dot{m}\sim 0.1$. This is roughly consistent with the
observations of three Seyferts with $L_{\rm bol}/L_{\rm Edd}\sim 0.1$, where
all above lines are observed and the line ratios are $\rm
Ly\alpha/H\beta\gtrsim 8-18$, Mg II $\lambda 2800/\rm H\beta\sim 1-2$, C IV
$\lambda 1549/\rm H\beta\sim 2-8$ and He II $\lambda 1640/\rm H\beta\sim
0.4-0.7$ respectively (Marziani et al., 2010). Therefore, our model can
roughly reproduce the observational line luminosities and line ratios with the
typical model parameters.

Figure 2: The correlation of $\rm H\alpha/H\beta-\lambda_{\rm{Edd}}$. The
black dots represent the AGN sources selected from La Mura et al. (2007) and
Jaffarian & Gaskell (2020). The open symbols represent the CLAGNs with
multiple observations, and the orange open circles represent the active state
of some CLAGNs selected from Jin et al. (2022). The red solid triangles with
error bars represent the average ratios of $\rm H\alpha/H\beta$ for the
sources in three bins ($10^{-3}<\lambda_{\rm Edd}<10^{-2},10^{-2}<\lambda_{\rm
Edd}<10^{-1},10^{-1}<\lambda_{\rm Edd}<1$). The solid lines from top to bottom
represent the theoretical results for
$10^{10}M_{\odot},10^{9}M_{\odot},10^{8}M_{\odot},10^{7}M_{\odot}$ and
$10^{6}M_{\odot}$, respectively for $R_{\rm out}/R_{\rm in}=40$. For
comparison, we also represent two cases with $R_{\rm out}/R_{\rm in}=60$
(upper dashed line) and 20 (lower dashed line) for $M_{\rm
BH}=10^{8}M_{\odot}$.

##  4 Result

In Figure 2, we present the relation between $\rm H\alpha/H\beta$ and
$\lambda_{\rm{Edd}}$ for our sample. It can be found that the sources with low
Eddington ratios have steeper Balmer decrement, where
$\rm{H\alpha/H\beta}=4-10$ when the Eddington ratio is less than a few
percent, while the average value of $\rm{H\alpha/H\beta}$ is 3.24 for bright
AGNs with $\lambda_{\rm{Edd}}\geq 0.1$. The Spearman correlation coefficient
is $r=-0.51$ with $p=1.32\times 10^{-9}$ for only AGN sample, and $r=-0.65$
with $p<1\times 10^{-20}$ for all sources including CLAGNs. This negative
trend is also found in a single CLAGN, but its slope may be different for
different sources. The average Balmer decrement for 16 CLAGNs in turn-on state
from Jin et al. (2022) is $\rm{H\alpha/H\beta}=4.73$, which is also
statistically higher than that of normal bright AGNs. Except for the evident
deviation of $\rm H\alpha/H\beta$ for sources with $\lambda_{\rm{Edd}}<$1-2%,
the slope of the $\rm H\alpha/H\beta-\lambda_{\rm{Edd}}$ correlation also
become steeper in CLAGNs of 3C 390.3 and 1ES 1927+654 at lower Eddington
ratios. Therefore, the change of the slope in $\rm
H\alpha/H\beta-\lambda_{\rm{Edd}}$ correlation is found in both single CLAGNs
and AGN samples.

In Figure 2, we also present the theoretical prediction for $\rm
H\alpha/H\beta-\lambda_{\rm{Edd}}$, where the solid lines represent the
$M_{\rm{BH}}=10^{6},10^{7},10^{8},10^{9},10^{10}M_{\odot}$ (from bottom to
top), respectively. For given BH mass, the correlation becomes steeper when
the standard disk to hot ADAF at low Eddington ratios, which is roughly
consistent with the observations. For a given Eddington ratio, we find that
more massive BHs predict higher values of $\rm{H\alpha/H\beta}$, where
$\rm{H\alpha/H\beta}=6-8$ for $M_{\rm BH}=10^{10}M_{\odot}$ and
$\rm{H\alpha/H\beta}=2.6-3.3$ for $M_{\rm BH}=10^{6}M_{\odot}$ in the case of
$\lambda_{\rm{Edd}}=0.01-1$. We find that the BLR boundary also slightly
affects the results (see the dashed line in Figure 2), where the larger value
$R_{\rm out}/R_{\rm in}$ predicts a slightly steeper Balmer decrement. The
average logarithmic mass of our whole sample is 7.2 with a standard deviation
of 0.7, and most sources are roughly in the range of
$M_{\rm{BH}}=10^{6}-10^{8}M_{\odot}$. It should be noted that the scatter in
observational relation is still large (e.g., $\rm H\alpha/H\beta\sim 5$ for
some bright AGNs with $\lambda_{\rm{Edd}}\sim 0.1$), where better
observational sample with wider distribution of BH masses is wished to further
explore this relation.

We further explore the correlation of ${\rm H\alpha/H\beta}-\lambda_{\rm Edd}$
in two CLAGNs (Mrk 1018 and NGC 3516). Mrk 1018 shows strong evolution of SEDs
in four quasi-simultaneous observations from optical to X-ray (e.g., Noda &
Done, 2018; Lyu et al., 2021, and references therein). To build the SED for
each observation, we fit the optical spectrum with SSD or truncated SSD model
and fit the X-ray spectrum with a power-law component, where we simply assume
the cutoff at ${\rm log}\nu=19.5$. We only fit four optical data points in
optical-UV bands because the contribution from a galaxy is not important in
these bands. For two bright states, the SSD can roughly reproduce the optical
spectra, while two low-state observations need the truncated SSD with a
transition radius of $70R_{\rm g}$ and $110R_{\rm g}$, respectively (see the
top-left panel of Figure 3). We calculate the ratio of $\rm H\alpha/H\beta$
based on the fitted SEDs from observations, where the different BLR boundaries
($R_{\rm out}/R_{\rm in}$) are also considered(see the left-bottom panel in
Figure 3). For NGC 3516, we don’t have simultaneous optical to X-ray data and
we build the SED based on the optical data together with the empirical
correlations $\Gamma_{2-10\rm{keV}}-L_{\rm bol}$ relation(Qiao & Liu, 2018)
and the theoretical relation $k_{\rm bol}{\rm(2-10keV)}-L{\rm(2-10keV)}$ by
Netzer (2019), where $k_{\rm bol}{\rm(2-10keV)}$ is the bolometric correction
factor for X-ray luminosity in 2-10keV. It should be noted that ${\rm
H\alpha/H\beta}-\lambda_{\rm Edd}$ relation in NGC 3516 is a little bit
steeper compared with the other CLAGNs (see Figure 2), where the parameters of
$(\dot{m},R_{\rm tr})=(0.07,6R_{\rm g}),(0.05,6R_{\rm g}),(0.03,6R_{\rm
g}),(0.03,50R_{\rm g})$ are adopted to reproduce the observed correlation.

Figure 3: The top-left panel presents the SEDs of Mrk 1018 based on modeling
the multi-wavelength quasi-simultaneous observations and the red solid, blue
dot-dashed, green dotted, orange dashed lines represent the $(\dot{m},R_{\rm
tr})=$ $(0.11,6R_{\rm g})$, $(0.07,6R_{\rm g})$, $(0.03,70R_{\rm g})$ and
$(0.03,110R_{\rm g})$, which correspond to ${\rm\log
Q(H)}=54.92,54.61,51.67,50.27$, respectively. The grey dashed line represents
the position of 1 Ryd. The Balmer decrement calculated from the
photoionization model based on the above SEDs and comparison with observations
is shown in the bottom-left panel. The grey dots are the observation data, and
the green stars are calculated by simulation. The right two panels represent
the result for NGC 3516. In the right upper panel, the red solid, blue dashed,
green dotted, and orange double dots dashed lines represent the
$(\dot{m},R_{\rm tr})=(0.07,6R_{\rm g}),(0.05,6R_{\rm g}),(0.03,6R_{\rm
g}),(0.03,50R_{\rm g})$, which correspond to ${\rm\log
Q(H)}=54.47,54.31,54.08,52.51$, respectively. In the lower panels, the blue
solid, green dashed, and orange dot-dashed lines are calculated by varying the
BLR boundaries $R_{\rm out}/R_{\rm in}=60,40,20$, respectively.

##  5 Conclusion and Discussion

After including more AGNs with low-Eddington ratios, we find that the
intensity ratio of $\rm H\alpha$ to $\rm H\beta$ is strongly anti-correlated
to the Eddington ratios in both AGN sample and in individual CLAGNs. We model
the Balmer decrement based on the photoionization model using the Cloudy code,
where the incident AGN SEDs are assumed to be evolved with the accretion
rates. Considering the possible disk evolution, we find that the observed
anti-correlation can be reproduced by the theoretical model. We propose that
the SEDs with lower accretion rates can also lead to a higher ratio of $\rm
H\alpha/H\beta$, where the steeper Balmer decrement may be not fully caused by
the internal reddening since some LLAGNs and CLAGNs have negligible
absorption.

The Balmer lines are the most well-known and strong emission lines in the
optical spectrum, which are sometimes used to shed light on the amount of dust
extinction due to its insensitivity to the gas conditions (e.g., temperature
and density, Osterbrock, 1989). However, the Balmer decrement of broad lines
cover a wide range from one sample to another (e.g., $\rm
H\alpha/H\beta=3.06\pm 0.22$ based on 446 blue Seyfert 1 from the SDSS, Dong
et al., 2008; $3.2\pm 0.6$ for 554 bright AGNs in SDSS DR7, Lu et al., 2019;
$3.45\pm 0.65$ for 90 nearby Seyfert 1, La Mura et al., 2007). La Mura et al.
(2007) and Lu et al. (2019) found that the Balmer decrement may be anti-
correlated with the accretion rate or Eddington ratio, where they attributed
to the possible stronger absorption with accretion rate decreases. After
including more sources with low-Eddington ratios and some CLAGNs, we find this
anti-correlation becomes stronger, where the Spearman correlation coefficient
$r=-0.65$ with $p<10^{-20}$. It should be noted that the steep Balmer
decrement in some sources with low Eddington ratios might be not completely
caused by the absorption since that many types I sources have $\rm
H\alpha/H\beta=5-10$ but with little absorption of $N_{\rm H}<10^{21}\rm
cm^{-2}$ (e.g., Jaffarian & Gaskell, 2020; Crenshaw et al., 1988; Osterbrock
et al., 1976). Furthermore, the anti-correlation of ${\rm
H\alpha/H\beta}-\lambda_{\rm Edd}$ is found in most CLAGNs with a strong
variation of optical emission lines, where the decrease of broad $\rm H\beta$
line is normally faster than $\rm H\alpha$ line (Kim et al., 2018, e.g., Mrk
1018, and Li et al., 2022, 1ES 1927+654,). More importantly, the absorption is
always weak in some CLAGNs (e.g. Mrk 1018, LaMassa et al., 2017) during the
changing look stage. Mehdipour et al. (2022) explored the possible physical
reason for the changing look phenomenon in NGC 3516 based on optical to X-ray
observations, where they found its low-flux spectrum variability does not
require any new or variable obscuration. Therefore, the variation of Balmer
decrement cannot be fully attributed to the variation of absorption even
though we cannot exclude it in all sources.

The properties of broad emission lines in AGNs are also affected by the
continuum state, where the changes in AGN SEDs will affect the flux of photons
for ionization. The evolution of the broad emission lines in a single CLAGN
also supports this scenario. In CLAGNs, the $\rm H\beta$ lines normally become
weak faster than the $\rm H\alpha$ lines during the decay of CLAGNs. This is
similar to the case that the broad $\rm H\alpha$ lines are observed in some
LLAGNs and the broad $\rm H\beta$ lines are weak or disappear. The UV
ionization photons will quickly decrease with the decrease in accretion rate,
where the UV emission will further decrease if the inner SSD is truncated to
ADAF when the accretion rate is lower than a critical value. Based on the
putative evolution of nuclear SEDs with the variation of the accretion rate,
our model naturally predicts the anti-correlation of $\rm
H\alpha/H\beta-\lambda_{\rm Edd}$, where the $\rm H\beta$ line decreases
faster due to the fast decay of UV photons in the case of low-accretion rate.
The anti-correlation of $\rm H\alpha/H\beta-\lambda_{\rm Edd}$ is most
possibly triggered by the optical depth effects that have been mentioned in
earlier works (e.g., Netzer, 1975; Rees et al., 1989; Korista & Goad, 2004).
The optical depths of the Balmer emission lines within the BLR of an AGN are
generally proportional to the flux of hydrogen ionizing photons, and will
therefore correlate with changes in the AGN’s ionizing photon luminosity Q(H).
At a given luminosity of the central source, the optical depth of $\rm
H\alpha$ is larger than that of $\rm H\beta$. Thus, an increase in the Q(H)
will lead to a relatively smaller increase in $\rm H\alpha$ photons escaping
the clouds than $\rm H\beta$ photons. That is, the responsivity $({\rm
dlog}L_{\rm line}/{\rm dlog}L_{\rm cont})$ of $\rm H\alpha$ is expected to be
smaller than that of $\rm H\beta$ (which is smaller than that in $\rm
H\gamma$), and this results in a flattening of the Balmer decrement in high
continuum states and steepening in low continuum states. The anti-correlation
shown in Figure 2 or Figure 3 can also be understood by referring to the lower
panel of Figure 1. How the gas emits the Balmer lines depend mostly on the
flux of hydrogen ionizing photons $\rm\Phi(H)$ striking on the cloud. So if
the source changes from a higher continuum state to a lower continuum state,
the contours in this panel will slide down along the distance-axis owing to
${\rm\Phi(H)\propto Q(H)}/r^{2}$. But the $R_{\rm BLR}$ relying on $L_{5100}$
decrease slower than the flux of hydrogen ionizing photons which have higher
energy. This leads to the contours sliding down relative to the BLR boundaries
(i.e., the red dashed lines) in the panel. Finally, the whole BLR will emit a
steeper Balmer decrement than the case in which the source is in the high
state.

Korista & Goad (2004) found that their BLR model predicts the broad $\rm
H\alpha/H\beta$ flux ratio to change from 4.9 in a low continuum state to 3.7
in a high continuum state, for the far-UV luminosity of NGC 5548 changes a
factor of 8 for over 13 years period. In this, they assumed a constant SED,
such that Q(H) scales 1:1 with the observed far-UV luminosity (although they
discussed possible impacts on the emission line responsivities for a variable
SED). Had they explored quantitatively changes in the SED with continuum
state, such that Q(H) is further enhanced in high continuum states and further
reduced in low continuum states, they would have found a greater range in the
flux ratio of $\rm H\alpha/H\beta$ – likely even closer to the range observed
in NGC 5548 (see Figure 2). It should be noted that the physical mechanism for
the SSD transition is still unclear, and we adopt a simple relation of $R_{\rm
tr}-\dot{m}$ in calculating the blackbody emission. For the X-ray spectrum, we
adopt the empirical correlation of $\Gamma-L_{\rm bol}$ and theoretically
calculated relation $k_{\rm bol}{\rm(2-10keV)}-L{\rm(2-10keV)}$, which will
not affect our main conclusion. For a given mass black hole, the number of
ionizing photons emitted per second Q(H) is strongly correlated with the
Eddington ratio, where UV photons become weaker or roughly disappear as a
decrease in accretion rate (in particular for the case of a truncated disk).
The parameter Q(H) decreases much faster if the inner SSD transition to ADAF
since UV photons dominantly originate from the cold SSD (e.g., see top panel
of Figure 1). The correlation of $\rm H\alpha/H\beta-\lambda_{\rm Edd}$
becomes steeper for $\lambda_{\rm Edd}<$1%, which is mainly caused by the
faster decrease of $\rm Q(H)$ in the ADAF case. NGC 3516 follows a steeper
correlation of $\rm H\alpha/H\beta-\lambda_{\rm Edd}$ with $\log\lambda_{\rm
Edd}<-1.5$, which may be triggered by a little bit higher critical accretion
rate for disk transition (see Figure 3). Furthermore, the relation of $R_{\rm
tr}-\dot{m}$ may be different from source to source, which may be one the
reason why some CLAGNs follow their own $\rm H\alpha/H\beta-\lambda_{\rm Edd}$
track.

We note that our model also predicts that the AGNs with more massive BHs will
have larger $\rm H\alpha/H\beta$ (see Figure 2), where the $\rm
H\alpha/H\beta$ is expected to be 5-6 for a black hole with $M_{\rm
BH}=10^{10}M_{\odot}$, even if they have high accretion rates (e.g., $\geq
0.1\dot{M}_{\rm Edd}$). This is caused by the peak energy of the SEDs from SSD
moving into sub-ionizing UV band, similar to the cases of low accretion rates
or truncated disks for lower masses. Further observational tests with more
massive BHs will be the subject of future works.

The LOC model is adopted to explore the properties of emission lines in this
work, where the line emission originates from the combination of all clouds
but is dominated by those with the highest efficiency of reprocessing the
incident ionizing continuum (Baldwin et al., 1995). The LOC models provide
good fits to the measured line fluxes. Similar to the traditional LOC model,
we simply set $\Gamma=-1$ and $\beta=-1$ for the distribution functions of
clouds. Korista & Goad (2000); Nagao et al. (2006) found that $\Gamma$ value
may be slightly smaller than the typical value of -1 based on the fitting of
emission lines (e.g.,$\Gamma\sim-1.4$ to $-1$). We find that the Balmer line
luminosities will increase about 0.6 times and the line ratio $\rm
H\alpha/H\beta$ will increase about 0.4 when $\Gamma$ varies from -1.4 to -1,
which is mainly caused by the weight biasing to the inner part of clouds
within the BLR. It should be noted that a slight change of $\Gamma$ and
$\beta$ will not change our conclusion on the anti-correlation of $\rm
H\alpha/H\beta-\lambda_{\rm Edd}$ correlation. Apart from the parameters of
$\beta$ and $\Gamma$, the line luminosities and line ratios are also
correlated with the inner/outer BLR radius in the integration. In Figures 2
and 3, we can find that the larger value of $R_{\rm out}/R_{\rm in}$ will also
lead to a little bit steeper Balmer decrement. However, the anti-correlation
trend in relation of $\rm H\alpha/H\beta-\lambda_{\rm Edd}$ is unchanged. The
BLR boundary may be also evolved with the accretion rate. If this is the case,
the slope of $\rm H\alpha/H\beta-\lambda_{\rm Edd}$ will also slightly change.
Furthermore, the covering factor, BLR dynamics and BLR geometry may also
change during the changing look in AGNs. Monitoring the shape of broad lines
may help to understand this issue, which can shed light on the variations in
different lines. In the LOC model we adopted, high density and very low
ionization clouds at a large distance from the SMBH were included in our
computations, where weakly-ionized gas may emit strong low-ionization features
(e.g., Na I and Ca II, see Ferland & Persson, 1989). This expectation should
be more evident in low continuum states, where the flux of ionizing photons is
much reduced and the ionization parameter becomes lower for gas of a given
density. To avoid this issue, we also test the LOC model with a distance-
dependent density distribution, where we adopt $n\propto r^{s}$ with $s=-1.5$
(e.g., Kaspi & Netzer, 1999; Rees et al., 1989). We find this model can
reproduce our main results (e.g, Figures 2 and 3) but with much weaker neutral
emission lines, where the ratio of Ca II/H$\beta$ decreases more than one
order of magnitude compared to that in the former LOC model. The uncertainties
of the LOC model do not affect the main conclusion of anti-correlation of $\rm
H\alpha/H\beta-\lambda_{\rm Edd}$ in this work.

We appreciate our referee for very constructive comments and suggestions,
which help us a lot to improve our paper. We also thank Luis C. Ho, Jianmin
Wang, Xuebing Wu, Xinwu Cao, Junjie Jin, Hengxiao Guo, and Zhicheng He for
very helpful discussions on the results, samples, and Cloudy code. This work
is supported by the NSFC (grants 12233007, U1931203, U2038107) and the science
research grants from the China Manned Space Project (No. CMS-CSST-2021-A06).
W.H.Lei. acknowledges support by the National Key R&D Program of China (Nos.
2020YFC2201400) and by the science research grants from the China Manned Space
Project with NO.CMS-CSST-2021-B11. The authors acknowledge Beijng PARATERA
Tech CO., Ltd. for providing HPC resources that have contributed to the
results reported within this paper.

##  Appendix A Prediction for some emission lines

Table 2:  Model predictions for emission-line luminosity.

${\rm log}L({\rm line})$ |  $\dot{m}=1$ |  $\dot{m}=0.1$ |  $\dot{m}=0.01$ |  $\dot{m}=0.001$
---|---|---|---|---
$\rm H\alpha$ |  43.66 |  42.80 |  41.96 |  39.81
$L_{5100}-L_{\rm H\alpha}$ observation |  43.55 |  42.74 |  41.88 |  39.55
$\rm H\beta$ |  43.18 |  42.24 |  41.22 |  38.97
$L_{5100}-L_{\rm H\beta}$ observation |  42.96 |  42.17 |  41.32 |  39.05
Ly$\alpha$ |  44.57 |  43.54 |  42.23 |  40.00
C IV |  44.04 |  42.82 |  41.95 |  39.91
Mg II |  43.36 |  42.46 |  41.76 |  39.84
He II |  43.20 |  41.82 |  40.91 |  38.79

Note. — The model prediction for luminosities of several typical broad lines
for $M_{\rm BH}=10^{8}M_{\odot}$ with different accretion rates. For
comparison, we also present the observed line luminosities at give optical
continuum luminosity based on $L_{\rm 5100}-L_{\rm H\alpha}$ and $L_{\rm
5100}-L_{\rm H\beta}$ relation for an AGN sample (Greene & Ho, 2005).

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