# Continuum reverberation mapping and a new lag-luminosity relationship for
AGN

Hagai Netzer,1
1School of Physics and Astronomy, Tel Aviv University, Tel Aviv 69978, Israel
E-mail<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

High cadence, high quality observations of active galactic nuclei (AGN)
clearly show continuum variations with lags, relative to the shortest observed
variable UV continuum, that increase with wavelength (“lag spectra”). These
have been attributed to the irradiation and heating of the central accretion
disk by the central X-ray emitting corona. An alternative explanation,
connecting the observed lag-spectra to line and continuum emission from gas in
the broad line region (BLR), has also been proposed. In this paper I show the
clear spectral signature of the time-dependent diffuse gas emission in the
lag-spectrum of 6 AGN. I also show a new lag-luminosity relationship for 9
objects which is a scaled down version of the well known $\tau({\rm
H}\beta)$-${L_{5100}}$ relationship in AGN. The shape of the lag-spectrum, and
its normalization, are entirely consisted with diffuse emission from radiation
pressure supported clouds in a BLR with a covering factor of about 0.2. While
some contributions to the continuum lag from the irradiated disk cannot be
excluded, there is no need for this explanation.

###### keywords:

(galaxies:) quasars: general; galaxies: nuclei; galaxies: active; accretion,
accretion discs

††pubyear: 2021††pagerange: Continuum reverberation mapping and a new lag-
luminosity relationship for AGN–LABEL:lastpage

##  1 Introduction

Super-massive black holes (BHs), central accretion disks, and strong broad
emission lines originating from high density gas in the broad line region
(BLR), are some of the main characteristics of type-I active galactic nuclei
(AGN). The spectral energy distribution (SED) of the accretion disk, and the
line and continua emitted by the BLR gas, all show variations on time scales
of days-to-years, depending on the bolometric luminosity of the source
(${L_{AGN}}$). A highly variable central X-ray source, thought to be produced
by a hot corona in the vicinity of the BH, is another characteristic signature
of such objects. The spectra of such sources have been studied in great detail
for years. This includes hundreds of thousands of sources at all redshifts,
from zero to beyond 7 (see review and numerous references in Netzer, 2013).

Progress in reverberation mapping (RM) of the BLR gas provides an opportunity
to map the location and motion of the gas, and to estimate the BH mass
($M_{\rm BH}$). Such measurements are available for a large number of sources
and for several broad emission lines (e.g. Kaspi et al., 2000; Bentz et al.,
2010, 2013; Hu et al., 2015; Du et al., 2015; Grier et al., 2017; Pei et al.,
2017; Lira et al., 2018; Kriss et al., 2019; Bentz et al., 2021). They have
been used to derive observables such as “the mean emissivity radius” or “the
mean responsivity radius”, and to show that these radii are proportional to to
$L_{5100}^{1/2}$, where ${L_{5100}}$ is the monochromatic continuum luminosity
at 5100Å in erg s-1. For sources in the range $10^{44}<$${L_{AGN}}$$<10^{46}$
erg s-1, both the mean emissivity and the mean responsivity radii of the
H$\beta$ line, $r($H$\beta$), are about 34${L_{5100,44}^{1/2}}$ light days
(ld), where ${L_{5100,44}}$ refers to the value of ${L_{5100}}$/$10^{44}$ erg
s-1 (e.g. Kaspi et al., 2000; Bentz et al., 2013).

K-band interferometry (GRAVITY Collaboration: et al., 2020), and combined
V-band and K-band photometry, (e.g. Koshida et al., 2014), provide information
about the outer boundary of the BLR. This boundary is roughly at the
sublimation radius of graphite grains which is $\sim 3\times r({\rm H}\beta)$
(e.g. Netzer, 2015, and references therein). Detailed theoretical studies,
like Baskin & Laor (2018), show that a single, well defined outer boundary is
a simplified description of a more complex configuration connecting the inner
accretion disk to an outer, torus-like structure.

More recent RM studies follow the optical-UV continuum variations relative to
the central X-ray source (so-called “continuum RM”, see Fausnaugh et al.,
2016; Edelson et al., 2019; Cackett et al., 2020, and references therein).
Such variations are predicted to take place in sources with large X-ray
luminosity (${L_{X}}$) due to the illumination and heating of the central
accretion disk by radiation from the hot corona. Simple irradiated disk models
predict time delays which are proportional to $\lambda^{4/3}$, where the
wavelength $\lambda$ is used as a proxy for the surface temperature of the
disk. More advanced models, taking into account the transfer of the X-ray
signal across the disk and relativistic ray bending, result in much longer
predicted lags (Kammoun et al., 2021b; Kammoun et al., 2021a). Whether or not
such models can explain the observations is still an open question. In some
sources, the optical-UV variation are, indeed, following the X-ray variations.
However, in several others this is not the case and the lag is either ill-
defined, not following the predicted $\lambda^{4/3}$ dependence, or the UV
leads the X-ray in contrast to the prediction of the model (e.g. Edelson et
al., 2019; Cackett et al., 2020).

Alternative explanations to the observed continuum RM involve time-variable
emission from the diffuse BLR gas. The variations are the results of the time-
variable ionizing luminosity emitted by the disk which induce continuum
variations that are not related to X-ray irradiated. The effects of the time
dependent bound-free and free-free continuum emission have first been studied
by Korista & Goad (2001) and in greater detail by Lawther et al. (2018),
Chelouche et al. (2019) and Korista & Goad (2019). Detailed comparison of the
model prediction to the observations of several sources are still missing and
there are several open issues related to the size of the lags and the
comparison with broad emission line variations observed in the same BLR.

In this paper I present detailed evidence that most, and perhaps all of the
continuum RM observed in AGN are due to the time-dependent emission of the
diffuse BLR gas. This is done by comparing lag-spectra of 9 AGN with the
results of radiation pressure confined (RPC) cloud models. I also present a
new lag-luminosity relationship for AGN and suggest ways to use it as a test
of the model. In §2 I describe the sample. In §3 I present predicted
irradiated disk lag-spectra and contrast them with the observations. In §4 I
discuss diffuse emission and dust lag spectra and present a new lag-luminosity
relationship for AGN. §5 gives a discussion of the new results and §6 presents
the conclusions of the paper. Throughout this paper I assume a standard
cosmological model with $\Omega_{\Lambda}=0.7$, $\Omega_{m}=0.3$, and
$H_{0}=70\,\hbox{km s${}^{-1}$}\,\,{\rm Mpc}^{-1}$.

##  2 Sample and data

The sample used here contains 9 AGN with high quality broad band observations
from the hard X-ray to the near infrared (NIR). The main facilities used are
XMM, Swift, HST, and a number of ground based telescopes, including several
1-2m telescopes of the LCO observatory. The observations are described and
discussed in a large number of papers including Fausnaugh et al. (2016),
Fausnaugh et al. (2018), Edelson et al. (2019), Cackett et al. (2020) and Kara
et al. (2021).

The 9 objects in the sample are separated into two groups. Group A includes 6
AGN with high quality, high cadence observations. These are: NGC 5548, NGC
4593, NGC 2617, F9, Mrk 142 and Mrk 817. Group B includes 3 objects with lower
quality data which are still good enough to obtain reliable lags for both
H$\beta$ and the 5000-5500Å continuum. The object are NGC 4151, Mrk 509, and
NGC 7469.

Table 1 provides basic information about all the sources, including $M_{\rm
BH}$, ${L_{5100}}$, ${L_{AGN}}$, the H$\beta$ lag ($\tau({\rm H}\beta)$), the
5000-5500Å lag (hereafter $\tau_{5100,cont}$), and the references used to
obtain this information. The bolometric luminosity, ${L_{AGN}}$, was obtained
by applying the disk bolometric correction factor proposed by Netzer (2019) to
the observed ${L_{5100}}$. This may differ, slightly, from estimates that are
based on other methods. As for the X-ray luminosity, I assume that the sole
energy source is accretion through an optically thin, geometrically thick
accretion disk (Shakura & Sunyaev, 1973). Therefore, ${L_{disk}}$=${L_{AGN}}$
and ${L_{X}}$ cannot exceed a certain fraction of ${L_{disk}}$ which is
assumed to be 0.5. I also assume $L_{\rm Edd}=1.5\times 10^{38}$($M_{\rm
BH}$/$M_{\odot}$).

Several additional comments are in order:

  1. 1.

All objects are highly variable on time scales of months-to-years with
variability amplitude as large as a factor two at some wavelengths, mostly in
the UV band. The typical variation at 5100Å is of order 30% and the typical
uncertainties on the numbers shown in Table 1 are of this order. Moreover,
most of the objects in this small sample, which includes all AGN with high
quality continuum RM observations at the time of writing, show $\tau({\rm
H}\beta)$ which falls below the Bentz et al. (2013) relationship by up to a
factor two. Thus, the mean H$\beta$ lag is not a fair representation of the
entire population.

In several cases, e.g. NGC 4151, the listed $\tau({\rm H}\beta)$ and
$\tau_{5100,cont}$ were observed in different campaigns and there is clear
evidence for a large change in ${L_{5100}}$ between the two epochs. I have
adjusted ${L_{5100}}$ in such cases to the most recent campaign. The
uncertainty on the $\tau({\rm H}\beta)/\tau_{5100,cont}$ ratio in such cases
is larger than the one found by a simple combination of the two uncertainties.

  2. 2.

I do not discuss the problematic issue of the UV-optical lag relative to the
X-ray flux (§1). I simply assume that the distance of the X-ray illuminating
source from the central BH is very small, a fraction of a light day.

  3. 3.

Given point (ii), there is no reference point to the measured lags, only
inter-band time-delay (IBTD) measurements. The shortest wavelength band
observed by HST, at around 1150Å, provides the best reference point since it
is close in wavelength to both the ionizing and illuminating continua. When
available, all lags are measured relative to this band. Otherwise, I use the
Swift UVW2 band centered at around 1928Å and note a clear positive lag between
1150Å and this band, by typically 0.5-1 days. The difference in lag seems to
follow both the predictions of the standard illuminated-disk model and the new
model presented below hence I used these calculations to derive a “zero lag”
point at 1200Å. This approach is complicated due to the large uncertainties on
the Rayleigh scattering feature (Korista & Ferland, 1998) which is peaked
close to the Ly$\alpha$ line center (see further details in § 5.1). Finally, I
did not change the observed $\tau({\rm H}\beta)$ which is usually measured
relative to the 5100Å continuum. Because of the continuum RM, the lag of this
line relative to the driving UV continuum is longer than listed in table 1.

Table 1: Sample properties

Source | DL | $M_{\rm BH}$ | ${L_{5100}}$ |  ${L_{AGN}}$a | $\tau(5000-5500)^{c}$ | $\tau({\rm H}\beta)^{c}$ | $f_{irr}^{b}$ | References
---|---|---|---|---|---|---|---|---
| Mpc | $M_{\odot}$ | erg s-1 | erg s-1 | days | days |  |
Group A |  |  |  |  |  |  |  |
NGC 5548 | 74.5 | 5e7 | 2.3e43 | 2.8e44 | 2.16 | 10 | 0.8 | 1,2
NGC 4593 | 38.8 | 7.6e6 | 7.4e42 | 7.4e43 | 0.57 | 3.7 | 0.8 | 1,5
NGC 2617 | 61.5 | 3.2e7 | 5.5e42 | 1.0e44 | 0.75 | 4.3 | 0.8 | 1,3,10
Mrk 142 | 199 | 1.7e6 | 3.5e43 | 6.8e44 | 0.8 | 6.0 | 0.08 | 1,6
F9 | 203 | 2.6e8 | 7.9e43 | 1.3e45 | 4.0 | 18 | 0.28 | 1,9
Mrk 817 | 134.2 | 3.9e7 | 6.9e43 | 1.1e45 | 4.2 | 22 | 0.15 | 1,4
Group B |  |  |  |  |  |  |  |
NGC 4151 | 16.6 | 3.6e7 | 3.2e42 | 5.4e43 | 1.4 | 8 | 0.8 | 1,7
NGC 7469 | 70.8 | 9.0e6 | 3.6e43 | 7.1e44 | 1.7 | 11 | 0.19 | 1,8
Mrk 509 | 151 | 1.1e8 | 1.4e44 | 8.4e44 | 3.3 | 77 | 0.8 | 1,7

  * •

a Calculated from ${L_{5100}}$. b Calculated in eqn. 2 assuming $\eta=0.1$ and
requiring $f_{x}\leq 0.5$. c Uncertainties can be found in the original
papers.

  * •

1 The AGN BH Mass Database http://www.astro.gsu.edu/AGNmass/ 2 Fausnaugh et
al. (2016) 3 Fausnaugh et al. (2018) 4 Kara et al. (2021) 5 Cackett et al.
(2018) 6 Cackett et al. (2020) 7 Edelson et al. (2019) 8 Pahari et al. (2020)
9 Hernández Santisteban et al. (2020) 10 Fausnaugh et al. (2017)

##  3 Accretion disks and observed lag-spectra

###  3.1 Theoretical disk lag spectra

The accretion disk model used to calculate the SED is the one presented in
Slone & Netzer (2012). It includes relativistic corrections and Comptonization
in the disk atmosphere. This is rather standard and other codes used in the
literature result in similar SEDs. The total accretion energy is
${L_{disk}}$$=\eta\dot{M}c^{2}$ where $\eta$ is the mass-to-energy conversion
efficiency which changes between 0.0398 ($a=-1$) to 0.328 ($a=0.998$), where
$a$ is the spin parameter. Part of the accreted energy is converted to X-ray
powerlaw radiation in a hot corona. The only X-ray photon index considered
here is $\Gamma=2$. This is close to the mean of the present sample that range
between 1.4 and 2.4.

The observed lags were compared with two predicted irradiated disk lags. The
first is the standard equation (e.g. Fausnaugh et al., 2016) presented here
as:

$\tau_{\lambda,irr}=1.62\times
10^{-5}\lambda_{A}^{4/3}\left[\frac{X}{4.97}\right]^{4/3}[M_{7}\dot{M_{\odot}}]^{1/3}[3+f_{irr}]^{1/3}\,\,{\rm
days},$ (1)

where $M_{7}$ is the BH mass in units of $10^{7}$ $M_{\odot}$,
$\dot{M_{\odot}}$ is the accretion rate in units of $M_{\odot}$/yr, and
$\lambda_{A}$ is the wavelength in Angstroms. The scaling parameter $X$ is
4.97 for a simple Wien’s law and $\approx 2.49$ when averaging contributions
from the entire surface of the disk. In all cases shown here I use $X=2.49$
and note that this, by itself, introduces a factor of about 2.5 uncertainty on
$\tau_{\lambda}$.

The term $f_{irr}$ represents the irradiated X-ray flux absorbed by the disk.
It is given by:

$f_{irr}=2(1-al)f_{x}\eta h\,\,,$ (2)

where $al$ is the albedo of the disk atmosphere,
$f_{x}=$${L_{X}}$/${L_{disk}}$ is the fraction of the total produced power
emitted as X-ray radiation, and $h$ is the height of the X-ray source above
the plane of the disk in units of $r_{g}=GM_{BH}/c^{2}$. $f_{irr}$ is
basically wavelength independent except for cases where $h$ is very large. In
what follows I assume $al=0.2$, $\eta=0.1$, and $h=10$.

The total corona-produced emission is a question of much debate. It depends on
low (unknown) and high (known in many cases) cut-off energies. Here I use
$E_{max}=100$ keV, as observed in many cases, and $E_{min}$ which is model-
dependent and is probably in the range of 50-500 eV. Kammoun et al. (2021b)
discussed this issue at great length and I used their eqn. 5 to estimate the
total X-ray energy, ${L_{X}}$. In some of the sources the resulting $E_{min}$
is very small, 1 Rydberg or even lower. This can result in
$L_{X}\sim$${L_{disk}}$ which is inconsistent with the basic energy production
assumption. I therefore require $f_{x}\leq 0.5$${L_{disk}}$. Values of
$f_{irr}$ are listed in table 1.

I have also compared the lags with the predictions of the Kammoun et al.
(2021b) model. Such fits are not discussed here since they were shown in the
original papers.

###  3.2 Observed lag spectra

Fig. 1 shows observed lag-spectra of 4 group-A AGN. I compare them with eqn. 1
using the parameters listed in table 1. As explained, the observed lags were
shifted upward by a constant obtained from the extrapolation of eqn. 1 to 1200
Å to achieve a zero lag at this wavelength. The uncertainly introduced by this
adjustment is small compared with the measured continuum lags and makes little
difference to the conclusion of the paper. I do not plot the lag uncertainties
since they obscure the spectral features that are at the center of this paper.

Figure 1: Lag spectra for four sources with high quality observations. Note
two clear bumps in all lag-spectra resembling real spectral features in many
AGN. The 2000-4000Å feature is similar in shape and in wavelength to the
“small blue bump” observed in most type-I AGN and the peak of the 5000-10000Å
feature is close to the wavelength of the Paschen jump at 8200Å.

All lag spectra in Fig. 1 clearly show two distinct bumps with maxima at
around 3500Å and $\sim$ 8000Å, and a local minimum at around 5000Å. The
shorter wavelength bump resembles the commonly observed “small blue bump” in
the spectrum of most type-I AGN. The existence of a time-lag excess in the $u$
or $U$ bands was noted in earlier publications (e.g. Edelson et al., 2019;
Cackett et al., 2018). It was interpreted as due mostly to the Balmer jump at
3646Å. Closer examination shows that the feature is much broader and common to
all type-I AGN with high quality lag observations. Contributing to the
feature, on top of the Balmer continuum emission, are strong broad FeII lines
on the short wavelength side and a large number of high order Balmer lines
that extend the continuum-looking emission to longer wavelengths all the way
to about 4000Å, depending on the source (see e.g. Wills et al., 1985; Mejía-
Restrepo et al., 2016).

The second bump covers the 5000-9000Å wavelength range. The rise at shorter
wavelengths and drop at longer wavelength are clearly visible and the peak at
around 7500-8500Å is very noticeable suggesting a combination of Paschen jump
in emission at around 8200Å, Paschen continuum drop towards $\sim 5000$Å, and
the conglomeration of many broad Paschen lines longward of the jump. The broad
band photometry includes, in some of the sources, at least part of the strong
and broad H$\alpha$ line which also contributes to this feature. This is not
the case in NGC 4593 where the data used purposely avoid this and other strong
lines.

The lags due to simple disk irradiation are small in all cases and contribute
little, if anything, to the lag-spectrum. The Kammoun et al. (2021a) lags,
which are not shown here, are similar in shape but longer by about a factor 2.
None of the irradiated models explains the two humps observed in the lag-
spectra.

##  4 Diffuse emission and dust Lag spectra

The combined luminosity of the accretion disk and the central X-ray source are
marked here by $L_{incident}$. I also use two different symbols to describe
the luminosity produced by the BLR gas. $L_{diff}$ stands for the combined
luminosity of the broad emission lines, bound-free and free-free continua, and
scattering by ionized and neutral gas. For the diffuse continuum (DC), I use
$L_{DC}$. Thus $L_{total}=L_{diff}+L_{incident}$. Emission by hot dust close
to the center is counted separately as $L_{dust}$. All these luminosities are
wavelength dependent.

###  4.1 Diffuse BLR emission

In Netzer (2020) I presented extensive Cloudy and ION photoionization
calculations of BLR clouds whose structure was determined by the radiation
pressure force of the central source (radiation pressure confined clouds, RPC,
see Baskin & Laor (2018)). The calculations were applied to a source referred
to as a “typical RM sample” AGN, and to NGC 5548. For the RM sample source,
there were two SEDs which assumed $M_{\rm BH}$=$10^{8}$ $M_{\odot}$ and two
different accretion rates. These were combined with typical X-ray powerlaw
continua. Distance from the BH, ionizing flux and diffuse emission, were all
normalized to ${L_{AGN}}$ which, in itself, was normalized to ${L_{5100}}$.
The computations can directly be applied to the present sample by scaling to
the observed ${L_{5100,44}}$. Unless otherwise stated, the SED used in the
present paper is the one presented in Netzer (2020) as AD1.

The Netzer (2020) model resulted in calculated lags of several emission lines
assumed to correspond to the emissivity weighted radii of the lines. This is a
good approximation when the time scale for the ionizing continuum variations
is longer than the crossing time of the BLR. The most important parameter of
the model, which determines the line luminosity and lag, is the distant
dependent covering factor, $c_{f}(r)$. For comparison with the observations of
the Balmer lines I used the population mean $\tau({\rm H}\beta)$ (Bentz et
al., 2013) expressed in a somewhat different form as,

$\tau({\rm H}\beta)=34L_{5100,44}^{1/2}\,\,{\rm days}\,\,,$ (3)

The model was also used to calculate the emissivity weighted radii, and
therefore the lags of the integrated Balmer and Paschen continua that dominate
the 2000-9000Å range. For the chosen SED, this was found to be,

$\tau_{DC}\approx 17L_{5100,44}^{1/2}\,\,{\rm days}\,\,.$ (4)

A potential deviation at short wavelength is a strong Rayleigh scattering bump
centered around Ly$\alpha$. The intensity of this radiation was investigated
by Korista & Ferland (1998). It is proportional to the column density of
neutral hydrogen which, in the RPC model, is not well defined. This part of
the spectrum is briefly discussed in § 5.1.

Variable ionizing radiation results in time and wavelength dependent diffuse
emission, $L_{diff}$, and a corresponding diffuse emission lag,
$\tau_{\lambda,diff}$, relative to the ionizing continuum light curve. Here I
investigate the situation where the diffused emission and the incident
continuum light curves cannot be separated and appear as a single combined
light curve. This light curve is then cross-correlated with the incident
continuum light curve. The computed lag depends, therefore, on the relative
scaling of the two components.

In this paper I follow earlier studies (e.g. Korista & Goad, 2001; Lawther et
al., 2018) and assume that the lag of the combined diffuse-incident light
curve relative to the incident continuum light curve is proportional to
$\tau_{\lambda,diff}$ and the proportionality factor is $L_{diff}/L_{total}$.
This is easy to justify if there is a 1:1 correspondence between $L_{diff}$
and $L_{incident}$ across the entire BLR. However, for every line or diffuse
continuum, there are parts of the BLR where this is a good approximation and
others where it is not.

A related complication is the way used to measure lags from the peak, or
centroid, of the cross correlation function (CCF). The location of the peak
depends on the shapes of the driving and the combined light curves which can
introduce deviations from a simple scaling with $L_{diff}/L_{total}$.

These issues were discussed in Korista & Goad (2019) who analyzed three
different campaigns of NGC 5548 using the LOC framework. They showed (see §2.6
and figure 11 in their paper) that deviations by factors of up to 2-3, occur
in their model in cases where $L_{diff}/L_{total}$ is far from 0 or 1.

The RPC model of Netzer (2020) is different, considerably, from the constant
gas density and the constant ionization parameter models of Lawther et al.
(2018), and from the LOC model of Korista & Goad (2019). Thus, the lags
predicted below can differ, substantially, from the results presented in those
papers. In an appendix to this paper I show that the wavelength dependencies
of the DC emission in the RPC model is rather similar across the BLR and
deviations from a 1:1 relation are not necessarily large. This, however, is
not a replacement for a full investigation of the issue of linearity in a
general and global BLR model. Such an investigation is beyond the scope of the
present paper.

For the remaining of the paper I use the simple approximation described above.
I also add the disk illumination term by applying a similar scaling factor.
This results in a total lag of:

$\tau_{\lambda,tot}=\tau_{\lambda,irr}\left[\frac{L_{incident}}{L_{total}}\right]+\tau_{\lambda,diff}\left[\frac{L_{diff}}{L_{total}}\right]\,\,{\rm
days}$ (5)

Unless otherwise specified, I replace $\tau_{\lambda,diff}$ by $\tau_{DC}$
from eqn. 4 which means that wavelength dependencies are entirely due to
$L_{diff}/L_{total}$. The above expression is similar, but not identical to
eqn. 22 in Lawther et al. (2018).

Fig. 2 shows calculations of $L_{diff}/L_{total}$ obtained from the Netzer
(2020) model assuming a single RPC cloud with $c_{f}=0.2$ situated at the mean
emissivity radius of the Balmer and Paschen continua (eqn. 4). This distance
was chosen since DC emission is the largest contributers to $L_{diff}$ at most
distances. In the appendix to this paper I show that the differences between
this approximation and the full BLR model are small. The broad band photometry
of most of the objects in the present sample include also contributions from
emission lines, mostly FeII lines on the short wavelength side of the
2000-4000Å bump and several Balmer lines. The emissivity weighted radii of the
FeII lines are 2-3 times larger than the distance of the Balmer and Paschen
continua. The relative contributions of these features to the measured lags
are discussed in §5.

Figure 2: Calculated lag spectrum (blue), and averages in several broad bands
(red), for a standard RPC cloud situated $17\times L_{5100,44}^{1/2}$ ld from
the central BH (see text for details). The green dashed line is the
contribution of the hot torus dust normalized to the mean observed
${L_{5100}}$/$L_{2\mu m}$.

In what follows I use $\tau_{5100,cont}$ to refer to measurements over the
range of 5000-5500Å. $L_{diff,5100}$ refers to the BLR emission over the same
wavelength range. This band includes broad FeII lines that are very weak in
the present calculations mostly because of the chosen distance. According to
Fig. 2, $L_{diff,5100}/L_{total}\approx 0.23$ for ${L_{5100,44}}$=1 and
$c_{f}=0.2$. Since $L_{diff}\propto c_{f}$, and since at this wavelength
$L_{diff}\approx L_{DC}$, we can combine the calculated spectrum with eqns. 4
to obtain:

$\tau_{5100,cont}\approx
17L_{5100,44}^{1/2}\frac{1.5c_{f}}{1+1.5c_{f}}\,\,{\rm days}\,\,.$ (6)

For ${L_{5100}}$=$10^{44}$ erg s-1, $\tau_{5100,cont}\approx 3.9$ days. Fig. 2
can be used, in exactly the same way, to calculate the second term in eqn. 5
for all other wavelengths.

The constants in all these equations are based on observations that do not
take into account the diffuse flux at 5100Å and the fact that the H$\beta$ lag
is measured relative to the 5100Å continuum which, in itself, lags the short
wavelength continuum. Taking this into account would change the constants in
eqns. 3 and 4 by about 12%. The calculations presented below do not include
this correction.

###  4.2 The continuum lag vs. ${L_{5100}}$ relationship

I have tested the model predictions in two related ways: a comparison with
observed lag-spectra of individual sources, and a test of the predicted
correlation between $\tau_{5100,cont}$ and ${L_{5100}}$ for the entire sample.

I used the transmission curves shown in Fausnaugh et al. (2016), together with
the Swift bands UVW2, UVM2 and UVW1, to calculate broad-band time lags using
eqn. 5 and the spectrum shown in Fig. 2, and assuming $\tau_{\lambda,irr}=0$.
Specific examples are shown in Fig. 3 where I compare model predictions with
the observations of NGC 5548 and NGC 2617. The only scaling of the model is by
the observed ${L_{5100}}$ of the two sources. The agreement between model and
observations is very good suggesting that the entire lag spectra of these
sources, in shape and in amplitude, are due to the variable diffuse emission.
It also suggests that the mean covering factor of the BLR in these sources is
close to 0.2. Additional tests, not shown here, suggest similar agreement for
Mrk 817 and F9. In the case of NGC 4593, the shape of the lag-spectrum is
similar but $c_{f}=0.1$ gives a better agreement to the observations. This is
also the case with Mrk 142 although this source is somewhat different as
discussed below.

Figure 3: Observed and calculated broad-band lag-spectra for NGC 5548 and NGC
2617. The model lag-spectrum in both cases was scaled by the observed
${L_{5100}}$ according to eqn. 6 assuming $c_{f}=0.2$.

I also used the observed $\tau_{5100,cont}$, and the measured uncertainties,
to test for a correlation with ${L_{5100}}$. The results are sown in Fig. 4.
The line shown in the diagram is eqn. 6 with $c_{f}=0.2$ and no other
adjustments. The predicted ${L_{5100}}$1/2 slope gives a very good agreement
with the observations with only one source, Mrk 142, that deviates from the
curve by more than a factor 2. Continuum RM by Cackett et al. (2020) noted an
excess lag in the $U$ band of this source relative to a $\lambda^{4/3}$ line,
and also relative to a $\lambda^{2}$ line expected in slim accretion disks.
However, they did not take into account the clear two-humps lag-spectrum which
is very similar to the other group-A sources listed in table 1. The source is
one of the most extreme super-Eddington AGN. Such sources are known to show
large deviations from the canonical $\tau({\rm H}\beta)$-${L_{5100}}$
relationship (Du et al., 2015, and references therein). It seems that a
similar deviation also exists for $\tau_{5100,cont}$ vs. ${L_{5100}}$.

I have also tested the correlation of $\tau_{7500,cont}$ with ${L_{5100}}$.
The model prediction is $\tau_{7500,cont}/\tau_{5100,cont}\approx 1.8$ and a
slope 0.5 line shifted up by this amount gives a good fit to the data. The
exception, again, is Mkn 142 which lies well below the predicted correlation.
Unfortunately, $\tau_{7500,cont}$ observations are available for only 7
sources which makes this result not very meaningful.

Figure 4: $\tau_{5100,cont}$ vs. ${L_{5100}}$ for the entire sample. Mrk 142
is marked in blue.

The new results can further be tested by computing $\tau({\rm
H}\beta)/\tau_{5100,cont}$ which is predicted to be approximately
$2(1+1.5c_{f})/1.5c_{f}$ with no dependence on ${L_{5100}}$. For $c_{f}=0.2$,
this ratio is about 8.7. Fig. 5 shows this ratio for all objects. Except for
Mrk 509, the ratio is basically independent of source luminosity and is within
a factor $~{}1.5$ of the this prediction. This time, Mrk 142 is not an
exception confirming the smaller BLR in this source. Mrk 509 is a factor 2-3
above the line of constant ratio. This object is well known for being above
the Bentz et al. (2013) relationship by a factor $\sim 2$. Interestingly, this
is not the case for its $\tau_{5100,cont}$. Since the H$\beta$ observations by
Kaspi et al. (2000) were taken more than 15 years before the continuum lag
measurements of Edelson et al. (2019), it is possible that the H$\beta$
emission region was much larger in the past, as is common for many AGN with
similar luminosities.

Finally, there are additional, mostly minor complications that must be taken
into account when using such a simplified approach, e.g. the dependency on BH
mass (see Fig. 6 below).

Figure 5:  $\tau({\rm H}\beta)/\tau_{5100,cont}$ vs. ${L_{5100}}$ for all
sources. Mrk 142 is marked in blue. The source close to the top right corner
is Mrk 509 (see text for details).

###  4.3 Hot dust

Time dependent continuum due to reverberation of the dust emission from the
torus, can also contribute to the observed continuum lags, especially at long
wavelengths. This has been discussed, extensively, mostly with regards to the
variations of the K-band flux (e.g. Koshida et al., 2014). Here I address only
the contribution of this flux to the observed lags in the 1000-10000Å
wavelength range.

I have used the observations of Mor & Netzer (2012) and Lani et al. (2017) to
scale the mean NIR luminosity relative to ${L_{5100}}$. The mean observed
ratio is $\lambda L_{NIR}$/${L_{5100}}$$\approx 1.25$, where $\lambda
L_{\lambda}$ is roughly constant over the range of 2–10$~{}\mu m$. The scatter
around the mean is of order 0.2 dex. I then used a normalized $T=1700$ K
greybody, about the highest dust temperature that contributes significantly to
the K-band emission, to estimate the contribution of such dust to the observed
lag. This contribution is given by:

$\tau_{\lambda,dust}\approx
100L_{5100,44}^{1/2}\frac{L_{dust}}{L_{incident}+L_{dust}}\,\,{\rm days},$ (7)

where the constant $100L_{5100,44}^{1/2}$ is obtained from the mean observed
ratio of $\tau_{dust}/\tau({\rm H}\beta)\sim 3$ (e.g. GRAVITY Collaboration:
et al., 2020). Scattering by dust grains in the torus is not included in the
present calculations.

The ratio $L_{dust}/(L_{incident}+L_{dust})$ is shown with a dashed line in
Fig. 2. The estimate is independent of the covering factor of the torus which
is already included in the normalization of the plot. However, $L_{dust}$ may
be overestimated since a considerable fraction of the K-band emission is
contributed by lower temperature dust.

The above simplistic approach does not include other effects like a special
torus geometry, or a combination of a torus and a dusty polar wind (Hönig et
al., 2013). It shows that hot dust contribution to the measured lag can be
important at long enough wavelengths, although its predicted contributions
seem to results in lags that are too long compared with the observations at
$\lambda\geq 7500$Å.

##  5 Discussion

The lag-spectra of 6 out of the 9 objects studied here clearly show broad
features resembling commonly observed spectral features in type-I AGN. While
indications for the shorter wavelength feature were noted in earlier works,
and were connected to Balmer continuum emission from the BLR, the second
feature was unnoticed so far. Moreover, the overall wavelength dependence was
masked by large observational uncertainties as well as attempts to fit lag-
spectra with various illuminated accretion disk models that predict smooth
dependence on wavelength. In this work I show that the two broad features are
naturally explained by combining the lag of the BLR gas to ionizing continuum
variations and are not necessarily the results of the disk illumination. I
also show a new correlation between $\tau_{5100,cont}$ and ${L_{5100}}$ which
looks like a scaled down version of the known $\tau({\rm
H}\beta)-$${L_{5100}}$ correlation and has the same dependence on
${L_{5100}}$1/2. Finally, there is some evidence that lags due to hot dust
emission may contribute too at long enough wavelengths.

The model presented here includes several simplifications. The most important
ones are the assumptions that a single emission region can represent the
entire BLR, the neglect to include, accurately, the contributions of broad
emission lines to several of the bands, and the possibility that the
irradiated disk contribution is larger than assumed in eqn. 1.

As for the observations, these are limited by the small sample size (see
comments in §2), and the fact that there are only 6 sources with lag
measurements that are accurate enough to perform such tests. In addition,
variability-related issues suggest that the uncertainties attached to the lags
are too optimistic mostly because ${L_{5100}}$, $\tau$(H$\beta$) and
$\tau_{5100,cont}$ are not always measured at the same epoch.

###  5.1 Comparison with detailed BLR models

Diffuse emission lags in the 1000-10000Å wavelength range combine
contributions from small distances, where lines like He ii are formed (small
contribution to the lags), intermediate distances where most of the bound-free
continua are produced (the largest contribution to $L_{diff}$), and large
distances, where the Balmer and FeII lines are produced (important
contribution to several of the bands, especially those covering the 2000-4000Å
bump). Some contributions depend more on the distribution of neutral and
ionized columns as a function of distance and covering factor. An important
example is Rayleigh scattering (Korista & Ferland, 1998) which is peaked close
to the Ly$\alpha$ line center and is sensitive to the column density of
neutral hydrogen. Detailed computations of the dependence of several of these
features on the distance from the central BH are provided in the RPC model of
Netzer (2020).

Alternative BLR models, e.g. the constant density and constant ionization
parameter models of Lawther et al. (2018), and the LOC model of Korista & Goad
(2019), have been proposed and discussed. Both types of models are very
different from the RPC model used here. The wavelength dependencies due to DC
emission in some of them are similar to the ones shown here. In others, they
are very different. As explained in Netzer (2020), some calculated time-lags
in these models, especially for the Balmer lines, are much longer than assumed
here. This is the results of the different physical assumptions about the
cloud properties and the much larger dust-free region assumed in several
cases. Predicted continuum RM of such models are still to be compared with
observations of other group-A sources.

###  5.2 Lag-spectra and accretion disk SED

The disk SED which depends on BH mass, spin and accretion rate, can make a
significant difference to the calculated lags. The SED used here assumes an
accretion disk around a BH with $a=0.7$, $M_{\rm BH}$=$10^{8}$ $M_{\odot}$ and
$L/L_{\rm Edd}$=0.1. Most objects in the sample have smaller BHs and probably
a range of spins. Their SEDs are, therefore, typical of higher temperature
disks. To illustrate the expected changes I show in Fig. 6 calculations based
on a model SED of a disk around $10^{7}$ $M_{\odot}$ BH with the same spin,
covering factor, $L/L_{\rm Edd}$ and ionizing luminosity normalized to
$L_{5100,44}^{1/2}$. The lower mass results in a higher temperature and a
steeper 1000-10000Å continuum. The shape of the two lag-spectra are similar
but the lags in the smaller BH case are longer by a wavelength dependent
factor. At the longest wavelength band, this factor is about 1.5.

Fig. 6 also shows the extension of $L_{diff}/L_{incident}$ to shorter
wavelengths which was not included in previous diagrams. In both cases,
$\tau_{2000,cont}/\tau_{1500,cont}\approx 2$. This is similar to the observed
ratio in sources where both Swift and HST measurements are available. The
comparison with observed lag spectra at wavelengths below $\sim 1400$Å is
dominated by a strong Rayleigh scattering feature. As explained in § 4.1, the
treatment of this feature is complicated due to the large uncertainty on the
neutral hydrogen column and the lack of a well define zero lag wavelength.

Figure 6: A comparison of normalized diffuse emission lags (eqn. 4) between
accretion disks around $10^{7}$ and $10^{8}$ $M_{\odot}$ BHs, as marked. The
curve for the larger BH is identical to the one shown in Fig. 2. The
normalized $r_{DC}$ and the BLR covering factors are the same in both cases.
The disk around the smaller BH induces longer lags at all wavelengths above
1000Å because of its steeper $\lambda>1000$Å continuum. The strong bump at
1100-1400Å is mostly due to Rayleigh scattering.

For the SED used here, and $c_{f}=0.2$, the fractional additional flux to the
disk radiation ranges from about 30% at around 5000Å, to about 80% at around
7500Å. Such contributions are not taken into account when measuring the shape
of the intrinsic (disk) continuum using HST observations such as the ones
reported by Bentz et al. (2013) and Kara et al. (2021). This can result in
erroneous estimates of the incident continuum slope at wavelengths longward of
the Lyman edge. For example, if the measured slope between $\lambda=1400$Å and
a certain longer wavelength indicates $L_{\nu}\propto\nu^{-0.5}$, allowing for
the DC emission would result in an intrinsic slope of $\nu^{-0.3}$ or
$\nu^{-0.12}$ for $\lambda=5000$Å and $\lambda=7500$Å, respectively. Such
slopes are in much better agreement with the typical slopes of AGN accretion
disks.

The fraction of the diffuse line and continuum flux depends on the inclination
angle of the disk (assumed here to be 45 degrees) since this emission is
roughly isotropic. This by itself introduces a scatter of about a factor 0.7
between objects.

There are clear indications that in some of the sources discussed in this
paper (NGC 5548, Mrk 817) there are significant contributions from powerful
disk winds to the observed diffuse emission (see Dehghanian et al., 2020).This
emission, and its time dependent properties, are not fully understood and are
not included in the present calculations.

###  5.3 FeII and high order Balmer and Paschen lines

Strong blends of FeII emission lines are a common signature of many type-I AGN
(see Netzer, 2013, and references therein). The strongest lines are those
emitted over the 2000-3500Å band and the somewhat weaker blends on both sides
of the H$\beta$ line (e.g. Wills et al., 1985). The single component model
presented here cannot reproduce the intensities of those lines whose mean
emissivity radius is somewhat larger than the one for the Balmer lines. A more
detailed model would show them as a strong extension of the Balmer continuum
towards shorter wavelengths with a lag which is similar to that of the Balmer
continuum because of several different effects: The combined luminosity is
similar to the Balmer continuum luminosity, the line formation zone is about
twice as far, and the line response to the changing continuum is weak and sub-
linear. Detailed calculations of these lags are beyond the scope of the
present paper.

All lag spectra shown here appear steeper than observed longward of the Balmer
and Paschen jumps. These are the bands where high order Balmer and Paschen
lines would normally merge, smoothly, into the respective continua. Such
lines, that are unresolved in most cases, are present in typical BLR spectra
(see many detailed examples in Wills et al., 1985; Mejía-Restrepo et al.,
2016). However, their calculated luminosities by Cloudy fall short of the
observed features by large factors. This is related to the unexplained
weakness of the calculated broad H$\alpha$ and H$\beta$ lines discussed in
Netzer (2020). It is likely the result of the simplified radiation transfer
method used in Cloudy and similar photoionization codes that fails to
reproduce the hydrogen line spectrum for conditions combining high density
with high Balmer line optical depths.

Examples showing the weakening of the Balmer lines as a function of density
and optical depth are presented in appendix B. I also show that the calculated
luminosity of the Balmer and Paschen continua are much less affected under
such conditions. Thus, the calculated EW(H$\beta$) and other Balmer lines for
$c_{f}=0.2$ are much smaller than observed but L(Balmer
continuum)/${L_{5100}}$ is in good agreement with the observations. The
expected time-lags for the high order lines are similar to the H$\beta$ lag
which is about twice as long as the Balmer and Paschen continua. This will be
reflected in the lag spectra over the wavelengths in question.

Turbulent velocity inside the clouds (e.g. Bottorff & Ferland, 2002) works in
exactly the opposite way. It decreases line optical depth, increases line
intensities, and completely change the structure of the clouds. As shown in
appendix C, such an assumption is inconsistent with the RPC model considered
in this paper.

###  5.4 Contributions from illuminated disks

The newly found lag-luminosity relationship is not necessarily in
contradiction with the predicted dependence of $\tau_{\lambda,irr}$ on
$[M\dot{M}]^{1/3}$ (eqn. 1). This was tested for the 9 objects by using the
mass, accretion rate and $f_{irr}$ listed in table 1. A weak correlation was
found for 7 of the sources while the other two are located more than 0.3 dex
below the line. However, the lags predicted by this expression are too short
compared with the observations.

The relative intensity of the irradiated to local disk flux is about
$f_{irr}/3$ and is, therefore, barely detectable in most objects and certainly
below the noise level in objects like Mrk 142 (see table 1). Increasing $\eta$
could increase the ratio but will, at the same time, shorten the lags. Most
importantly, $f_{irr}$ is predicted to be wavelength independent which is in
clear contrast to the observations that show much larger amplitude variations
at shorter wavelengths. The longest observed lags can perhaps be explained by
the Kammoun et al. (2021b) model but not the fractional flux variations.
Combining with the previous discussion, one cannot reject the possibility that
X-ray irradiation can contribute, slightly, to the observed lag-spectra that
are probably dominated by diffuse, time variable BLR emission.

###  5.5 Consequences to AGN accretion disks and BLR structure

The main finding of this paper is related to the variations of the diffuse
emission from the BLR which seems to explain most, if not all of the so-called
IBRM. This does not exclude some contributions from disk illumination which
implies disk size which is 2-4 times larger than the canonical Shakura &
Sunyaev (1973) accretion disk. Independent evidence from microlensing also
suggest disks that are larger than the standard optically thick geometrically
thin accretion disks (e.g. Blackburne et al., 2011). Thus, there seems to be a
tension between RM as explained here and microlensing observations.

Mapping the region producing the high ionization emission lines has been a
problem for sometime since it seems to indicate sizes that are of the same
order as the size of the central disk (e.g. the He ii$\lambda 4686$ emission
region, see Bentz et al., 2021). A related issue is the very different lags
measured for the two He ii lines at 1640 and 4686Å (Bottorff et al., 2002).
These puzzling issues are partly solved when taking into account the lag of
the 5100Å continuum. Adding the typical numbers deduced in this paper for the
lag of the 5100Å continuum (about 4 ld for a source with
${L_{5100}}$=$10^{44}$ erg s-1), one finds high ionization emission regions
which far exceeds the size of the disk and similar lags for the two He ii
lines. This is directly confirmed by the Pei et al. (2017) observations of NGC
5548.

##  6 conclusions

This paper discusses the signature of diffuse BLR gas in the lag spectra of 9
AGN. The main results can be summarized as followed.

  1. 1.

The lag spectra of 6 AGN with high cadence, high quality observations clearly
show two features that resemble well known spectral feature in type-I AGN.
There are maximum time lags at around 3500 and 7500Å and a local minimum at
around 5000-5500Å.

  2. 2.

Radiation pressure confined BLR clouds, with $c_{f}\sim 0.2$, result in lag-
spectra which are in good agreement, in shape and in magnitude, with those
observed in 5 sources. They also correctly predict the 5000-5500Å lag for the
three sources with lower quality observations. The lags are scaled with the
covering factor and with ${L_{5100,44}^{1/2}}$ with a possible minor
dependence on BH mass.

  3. 3.

I present a new $\tau_{5100,cont}$-${L_{5100}}$ relationship which is similar
in slope to the well know $\tau({\rm H}\beta)$-${L_{5100}}$ relationship and
is scaled down by about a factor 6. This gives further support to the
suggestion that most observed continuum lags are due to the variable diffuse
BLR emission.

  4. 4.

Comparison of the calculated intensities of the high order Balmer and Paschen
lines with the observations suggests that the model under-predicts their
intensity by a large factor. This is similar to the general problem of the
hydrogen Balmer lines in most BLR models.

##  7 Data availability

The data used in this paper were obtained from two sources: tables in
published articles and The AGN BH Mass Database
http://www.astro.gsu.edu/AGNmass/. This is listed, object by object, in table
1 where a full list of references is provided.

The photoionization calculations were done with Cloudy which is an open source
code detailed in Ferland et al. (2017).

##  8 acknowledgements

I thank the referee, Kirk Korista, for many useful comments and suggestions
that improved the quality of this paper. I also thank Doron Chelouche, Rick
Edelson, Gary Ferland and Elias Kammoun for their useful comments.

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##  Appendix A Diffuse BLR spectra at different distances

Diffuse spectra at different distances in the standard RPC model are shown in
two ways: lag-spectra as used in this work (Fig. 7), and standard spectra,
$L_{\lambda}$ vs. $\lambda$ (Fig. 8).

Figure 7: $L_{diff}/L_{total}$ at various distances in the standard
${L_{5100,44}}$=1 RPC model. Note the relatively small variations with
distance. The model at 17 ld from the center is the one used to calculate all
lag-spectra in this work.  Figure 8: $L_{\lambda}$ vs. $\lambda$ at various
distances in the standard ${L_{5100,44}}$=1 RPC model.

##  Appendix B High order Balmer lines and the Balmer continuum

Calculated luminosities of Balmer lines, and the Balmer continuum, are shown
in Fig. 9. They assume constant density clouds, with
$n_{H}=10^{11}~{}cm^{-3}$, and various column densities, as marked. The clouds
are situated at the standard distance of 17${L_{5100,44}}$1/2 ld from the
nominal AD1 SED. Constant density was chosen since RPC clouds are
gravitationally bounded and hence must have very large column density, of
order $10^{23.5}~{}cm^{-2}$. The mean density of RPC clouds at this distance
is somewhat higher than $10^{11}~{}cm^{-3}$ but the differences in the lag-
spectrum are small. The diagram shows the decrease of the Blamer line
intensities, relative to the Balmer continuum, with increasing column density.

Figure 9: Continuum subtracted 1500-5000Å $c_{f}=1$ spectra of constant
density clouds with various column densities, as marked, located at the
standard distance of 17${L_{5100,44}^{1/2}}$ ld. Note the weakening of the
Balmer lines relative to the Balmer continuum with increasing column density.

Fig. 10 shows the formation efficiency of Balmer line and the Balmer
continuum, versus the fraction of ionizing photons absorbed by the clouds.
$L({\rm H}\beta)$ increases with the absorbed fraction up to about 50% where
it saturates, indicating line destruction due to a combination of high gas
density and large optical depth. $L$(Balmer continuum) increases with the
number of absorbed photons all the way to 100% The maximum H$\beta$ equivalent
width for $c_{f}=1$ is about 79Å. The high order Balmer lines (not shown here)
behave in a way similar to H$\beta$.

Figure 10:  $L({\rm H}\beta)$ and $L$(Balmer continuum) vs. the fraction of
ionizing photons absorbed by a cloud situated 17${L_{5100,44}^{1/2}}$ ld from
the standard SED considered in this paper. Each point represents a complete
$c_{f}=1$ model. The column density increases from left to right. Note that
the saturation of $L({\rm H}\beta)$ starts where the fraction absorbed by the
gas is as small as 50% compared with the super-linear increase of $L$(Balmer
continuum) all the way to 100% absorption.  Table 2:  Comparison of the
standard single location model of this work with a full-BLR model.

Propertya | Present Model |  Netzer (2020)b | Observedc
---|---|---|---
L(H$\beta$)/${L_{5100}}$ | 0.003 | 0.0034 | 0.02
$L$(Balmer)/${L_{5100}}$d | 0.23 | 0.29 | 0.22

  * •

a All properties are normalized to $c_{f}=0.2$ b AD1-SED p=2.4 model in Table
3 normalized to $c_{f}=0.2$. c Assuming Balmer continuum emission contributes
50% of the luminosity of the 2000-4000Å bump.

  * •

d Main references: Wills et al. (1985), Netzer (1985), Mejía-Restrepo et al.
(2016).

##  Appendix C Turbulence

Turbulent motion inside the BLR clouds increase the internal pressure,
decrease all line optical depths, and reduce the destruction of line photons
(Bottorff & Ferland, 2002). This increases the line luminosity and changes the
internal energy budget.

Fig. 11 shows two examples of constant pressure clouds with gas density at the
illuminated face of $\sim 10^{10}$ cm-3. In one case $v_{turb}=30$ km s-1 and
in the second case $v_{turb}=100$ km s-1. The emitted spectrum, especially in
the second case, is, indeed very different, since the cloud remains completely
ionized even at the large column density assumed here ($10^{23.5}$ cm-2). This
results in stronger emission lines and weaker diffuse continuum. In the second
case the internal gas pressure is only a small fraction of the total pressure
(see figure caption). Raising the gas density to $10^{11}$ cm-3 results in a
large column of neutral gas at the back of the cloud but a small change in the
spectrum. Finally, turbulence requires an additional source of dissipative
heating which is not part of the energy budget considered here. It is thus
difficult to construct well motivated, physically justified photoionization
models for the entire BLR with large internal turbulent velocity in some or
all the clouds. All this is inconsistent with the RPC model which is the main
assumption of this paper.

Figure 11: $L_{diff}/L_{total}$ for clouds with various internal turbulent
velocities, as marked (the red curve is the standard model discussed in this
paper) and gas density of $\sim 10^{10}$ cm-3 at the illuminated face. The
luminosity of all Balmer lines increase with increasing turbulent velocity and
the pressure inside the cloud deviates more and more from the pressure in the
canonical RPC model due to turbulent pressure, $p_{turb}$. For $v_{turb}$=30
km s-1, $p_{turb}$ is comparable to $p_{gas}$ close to the ionization front.
For $v_{turb}$=100 km s-1, $p_{turb}/p_{gas}\approx 20$(!).