# Ray-tracing simulations of the Soft X-ray Scattered Emission in obscured
Active Galactic Nuclei

Jeffrey McKaig1, Claudio Ricci2,3,1, Stéphane Paltani4, K. K. Gupta2,5,
Nicholas P. Abel6, Y. Ueda7
1George Mason University, Department of Physics and Astronomy, MS3F3, 4400
University Drive, Fairfax, VA 22030, USA
2Instituto de Estudios Astrofísicos, Facultad de Ingeniería y Ciencias,
Universidad Diego Portales, Av. Ejército Libertador 441, Santiago, Chile
3Kavli Institute for Astronomy and Astrophysics, Peking University, Beijing
100871, China
4Department of Astronomy, University of Geneva, 1205 Versoix, Switzerland
5STAR Institute, Quartier Agora - Allée du six Août, 19c B-4000 Liège, Belgium
6College of Applied Science, University of Cincinnati, Cincinnati, OH, 45206,
USA
7Department of Astronomy, Kyoto University, Kyoto 606-8502, Japan E-mail:
<EMAIL_ADDRESS>
(Accepted 2023 September 16. Received 2023 September 12; in original form 2023
May 9)

###### Abstract

Most Active Galactic Nuclei (AGN) in the local Universe are obscured. In these
obscured AGN an excess is usually observed in the soft X-rays below $\sim 2$
keV above the absorbed X-ray continuum. This spectral component is associated
with the scattering of X-ray photons off free electrons in the Narrow Line
Region (NLR), and/or to photoionised lines. Recent studies have found that in
highly obscured AGN this component has lower flux relative to the primary
X-ray continuum than in less obscured AGN. This is measured by the scattering
fraction, or $f_{\text{scatt}}$, which is the ratio of the scattered flux to
the continuum. Here, we use the ray-tracing platform RefleX to perform
simulations of scattered X-ray radiation to test two possible explanations for
this phenomenon: (1) sources with lower $f_{\text{scatt}}$ are viewed at
higher inclinations or (2) low $f_{\text{scatt}}$ sources are characterised by
larger covering factors. We consider a conical NLR of free electrons, while
allowing the column density and opening angle (and hence covering factor) to
vary. We also consider electron densities inferred from observations, and from
simulations carried out with the spectral synthesis code Cloudy. Our
simulations show $f_{\text{scatt}}$ is expected to be related to both the
inclination angle and covering factor of the torus; however, the observed
negative correlation between $f_{\text{scatt}}$ and $N_{\rm H}$ can only be
explained by a positive relation between the column density and the covering
factor of the obscuring material. Additional contributions to
$f_{\text{scatt}}$ can come from unresolved photoionised lines and ionised
outflowing gas.

###### keywords:

atomic processes – galaxies: Seyfert - galaxies: active - X-rays: general -
X-rays: galaxies

††pubyear: 2015††pagerange: Ray-tracing simulations of the Soft X-ray
Scattered Emission in obscured Active Galactic Nuclei–B

##  1 Introduction

It is now widely accepted that the high luminosity and wide ranging continuum
seen in active galactic nuclei (AGN) is the result of mass accretion onto a
Supermassive Black Hole (SMBH) at the centre of the host galaxy. In-flowing
material is expected to form an accretion disk around the SMBH (e.g., Shakura
& Sunyaev, 1973). This accretion flow produces copious optical / UV photons,
which can then be up-scattered into the X-ray band by a corona of hot
electrons (Haardt & Maraschi, 1991, 1993). These X-ray photons are absorbed
and reprocessed by various distributions of gas and dust surrounding the SMBH,
such as the broad line region (BLR) and the dusty torus in the plane of the
accretion disk, as well as the narrow line region (NLR), which is expected to
be perpendicular to the accretion disk (e.g., Urry & Padovani, 1995). The fact
that the system is axially symmetric is thought to be one of the main reasons
we observe different classes of AGN (i.e., type 1s and type 2s; e.g.,
Antonucci, 1993; Netzer, 2015; Ramos Almeida & Ricci, 2017).

The primary X-ray emission component, observed in AGN as a powerlaw
[$F(E)\propto E^{-\Gamma}$; Mushotzky et al., 1993; Dadina, 2008; Beckmann et
al., 2009], is thought to be produced in the X-ray corona where photons
created in the accretion disk are Compton up-scattered into X-ray energies via
optically thin, hot electrons surrounding the accretion disk. The powerlaw is
characterised by a photon index of $\Gamma\sim 1.8-1.9$ (Nandra et al., 2000;
Brightman & Nandra, 2011; Ricci et al., 2017a), and shows an exponential
cutoff at a few hundred keV (Dadina, 2008; Ricci et al., 2018; Baloković et
al., 2020). X-ray emission in AGN is accompanied by a reflection component,
which is produced by X-ray photons reprocessed by the circumnuclear material.
This is typically observed as an Fe K$\alpha$ line at 6.4 keV (Fabian et al.,
2000; Yaqoob & Padmanabhan, 2004; Shu et al., 2010; Fukazawa et al., 2011;
Ricci et al., 2014) and a reflection hump (known as the “Compton Hump”) at
$\sim$30 keV (Lightman & White, 1988; Ghisellini et al., 1994; Krolik et al.,
1994). X-ray absorption also has a significant effect on the observed spectra
of AGN. The main components driving absorption in the X-rays are: i)
photoelectric absorption which, for soft X-rays ($<10$ keV) can play a
significant role at column densities as low as
$\log{(N_{\text{H}}\big{/}\text{\,cm}^{-2})}\approx 21-22$ and ii) Compton
scattering from bound electrons at column densities of
$\log{(N_{\text{H}}\big{/}\text{\,cm}^{-2})}\approx 23.5$.

X-ray surveys have been increasingly used to detect and identify AGN,
especially obscured AGN, for which traditional optical diagnostics can be
uncertain (e.g., Brandt & Alexander, 2015; Koss et al., 2017; Ricci et al.,
2017a; Hickox & Alexander, 2018). This is due to the fact that hard X-rays
($E>10\,$keV) can penetrate high column densities of material as well as avoid
contamination from the host galaxy emission, which allows the detection and
study of highly obscured, even Compton thick AGN
[$\log{(N_{\text{H}}\big{/}\text{\,cm}^{-2})}\geq 24$; Matt et al., 2000; Ueda
et al., 2007; Yaqoob & Murphy, 2011; Brightman & Ueda, 2012; Ricci et al.,
2015; Odaka et al., 2016; Ricci et al., 2017a; Hikitani et al., 2018]. The
obscured phase could represent an important stage in the life of AGN, with
this period being associated to a significant fraction of black hole growth.
In some cases this heavily obscured phase is thought to be onset from a merger
event (e.g., Blecha et al., 2018; Ricci et al., 2017b, 2021; Yamada et al.,
2021). Thus, X-ray surveys are an important tool for studying SMBH and galaxy
evolution.

In all obscured AGN, an excess at $E\lesssim 2-3$ keV over the absorbed
continuum is observed. This soft excess is very different from that usually
observed in unobscured AGN, which is likely produced very close to the SMBH
(e.g., Gierliński & Done, 2004; Crummy et al., 2006; Done et al., 2012;
Boissay et al., 2016; Petrucci et al., 2018, 2020; Gliozzi & Williams, 2020).
The soft X-ray excess in obscured AGN has been interpreted as a combination of
scattered radiation off electrons around the AGN (Turner et al., 1997; Cappi
et al., 2006; Ueda et al., 2007; Ricci et al., 2017a) and photoionised lines
(Sako et al., 2003; Bianchi et al., 2006; Guainazzi & Bianchi, 2007). The fact
that photoionised gas can be traced with the [O iii] $\lambda 5007$ emission
out to hundreds of parsecs, along with the domination of photoionised lines,
points to the soft X-ray excess being associated with gas in the NLR (Sako et
al., 2000; Bianchi et al., 2006; Guainazzi & Bianchi, 2007; Gupta et al.,
2021). The amount of scattered radiation seen over the absorbed continuum can
be quantified by the scattering fraction $f_{\text{scatt}}$ (Turner et al.,
1997; Cappi et al., 2006; Ueda et al., 2007; Winter et al., 2009; Kawamuro et
al., 2016; Yamada et al., 2020), which is the ratio of the flux of scattered
radiation to the flux of the primary continuum.

Numerous studies have looked at the amount of scattered X-ray radiation in
obscured AGN. For example, studying hard X-ray selected AGN from the Swift/BAT
all-sky-survey, Ueda et al. (2007) reported two AGN with unusually low values
of $f_{\text{scatt}}$ ($<0.5\%$). They explained the low value as being due to
the fact that these AGN are buried in a geometrically-thick torus, or to the
absence of gas available for scattering. Ichikawa et al. (2012) used a sample
of 135 non-Blazar Swift/BAT AGN that included 13 sources with
$f_{\text{scatt}}<0.5\%$ to study their infrared properties. They found
enhanced fluxes at 9 $\mu$m in the averaged SED of these AGN with low
$f_{\text{scatt}}$ relative to the type 2 AGN in the sample, which could
suggest that the host galaxy is undergoing a starburst phase. Furthermore, AGN
with $f_{\mathrm{scatt}}<0.5\%$ tend to also have low [O iv] $\lambda$25.89 /
$L(12\,\mu\mathrm{m})$ ratio, giving a new method of identifying “buried” AGN
(Yamada et al., 2019). However, Hönig et al. (2014) argued that this new class
of AGN with low $f_{\text{scatt}}$ could possibly be caused by host-galaxy
contamination along the line of sight, since many of these AGN are in edge-on
galaxies. Ueda et al. (2015) used a complete sample of AGN from the Swift/BAT
9-month hard X-ray survey in order to study the correlation between the X-ray
and [O iii] $\lambda 5007$ luminosities. They concluded that AGN with low
$f_{\text{scatt}}$ also showed low [O iii] $\lambda 5007$ to X-ray luminosity
ratios, suggesting that their SMBHs are buried in tori with small opening
angles. A similar conclusion was drawn in Gupta et al. (2021) who, from a
sample of hard X-ray selected obscured AGN, found a positive correlation
between $f_{\text{scatt}}$ and the [O iii] $\lambda 5007$ to X-ray luminosity
ratio. Ueda et al. (2015) also noted that in their sample of AGN with low
$f_{\text{scatt}}$, more than half are free from host galaxy absorption by
matter in the galactic disk. Therefore, absorption from the host galaxy alone
cannot be the reason for the low $f_{\text{scatt}}$ values.

More recently, Gupta et al. (2021; hereafter G21) found a significant negative
correlation between $f_{\text{scatt}}$ and the observed column density
($N_{\text{H}}$; i.e., the line-of-sight column density through an obscuring
torus) in a sample of 386 hard X-ray selected obscured AGN from the
BASS111www.bass-survey.com survey (Koss et al., 2017, 2022; Ricci et al.,
2017a; see Figure 1 and Table 1 in G21). G21 performed spectral simulations of
dummy populations of AGN that indicate this relation is not due to an inherent
degeneracy between $f_{\text{scatt}}$ and $N_{\text{H}}$. They were also able
to reproduce this correlation in different bins of Eddington ratio, black hole
mass, and X-ray luminosity, further confirming that this correlation does not
depend on these physical properties of AGN. Similar to Noguchi et al. (2010),
G21 found a slight negative correlation between $f_{\mathrm{scatt}}$ and the
Eddington ratio with no correlation with black hole mass or X-ray luminosity.
G21 put forward two different explanations for the
$f_{\mathrm{scatt}}-N_{\mathrm{H}}$ relation: (1) the dependence of the
differential Compton cross-section on the inclination angle or (2) sources
with low $f_{\text{scatt}}$ might tend to have larger covering factors
($f_{\text{cov}}$; fraction of the sky covered by the obscuring torus) as
proposed by Ueda et al. (2007), reducing the amount of scattered radiation.
This would imply a relationship between the covering factor of a source and
its column density (e.g., Ramos Almeida et al., 2011; Ricci et al., 2011;
Mateos et al., 2016; Tanimoto et al., 2018; Panagiotou & Walter, 2019; G21).
The first explanation relies on the fact that the observed photons are Compton
scattered off material in the NLR and therefore are subject to the Klein-
Nishina (K-N) cross-section (Rybicki & Lightman, 1986). However, the photons
responsible for the soft X-ray excess have energies $\leq$2 keV and are
therefore subject to the low energy approximation to the K-N form. This is the
energy independent elastic Thomson cross-section with the differential
relation $d\sigma/d\Omega\propto 1+\cos^{2}(\theta)$, where $\theta$ is the
scattering angle and $d\Omega=\cos{(\theta)d\theta d\phi}$. Thus, when these
photons interact with the free electrons in the NLR, the electrons will behave
like a mirror while re-directing the incoming photons to the line of sight.
G21 then assumed a similar torus model for all the AGN in their sample while
predicting the covering factor of different layers of the torus using
statistical arguments from Ricci et al., 2015. Since the scattering fraction
is expected to depend on the inclination angle, sources with higher column
densities (which are being viewed at higher inclinations), will have lower
values of $f_{\text{scatt}}$, consistent with the relationship between
$f_{\text{scatt}}$ and $N_{\text{H}}$.

Our goal in this paper is to use the ray-tracing simulation software
RefleX222https://www.astro.unige.ch/reflex/ (Paltani & Ricci, 2017, see also
Ricci & Paltani, 2023) in order to study the origin of the scattered component
with respect to Thomson scattered emission, and to test the two different
explanations for the trend between $f_{\text{scatt}}$ and $N_{\text{H}}$: (1)
a relation between the column density and the inclination angle to the AGN,
and (2) a scaling relationship between the column density of a source and its
covering factor, where sources with higher column densities have larger
covering factors. To do this, we consider a NLR of various opening angles
consisting of free electrons exposed to a powerlaw X-ray source to test the
amount of scattered radiation at various inclination angles. We then use the
inclination angle as a proxy for the torus column density to compare our
relation to G21, discussing both the normalisation as well as the slope. Our
model setup can be found in $\S$2, results in $\S$3, discussion in $\S$4, and
our conclusions in $\S$5.

Figure 1: Cross section of the NLR geometry considered in our simulations and
a diagram showing the Thomson scattering process with the incoming photon
($\gamma$) and electron $e^{-}$. Yellow stars with orange borders represent
Thomson scattering events. The NLR traces a cone geometry as seen in optical
kinematic and narrowband imaging. Each shade of grey shows a different half
opening angle ($\alpha$; not all opening angles are shown). The inclination
angle to the detector, $i$, is also shown. The scattering fraction
$f_{\text{scatt}}$ is calculated by isolating the photons which undergo
scattering off the free electrons in the NLR as well as the photons which
undergo no interactions (the X-ray continuum), calculating the flux ratio of
these two groups of photons in 5 eV bins, and averaging the results. The
column density is calculated by multiplying the electron density ($n_{e}$) by
the height of the NLR ($H=1000\,$pc).

##  2 Model Setup

###  2.1 RefleX

In order to study the fraction of scattered X-ray radiation from AGN, we use
the X-ray simulation platform RefleX (Paltani & Ricci, 2017). RefleX uses ray-
tracing to track photons with energies ranging from 0.1 keV to 1 MeV
interacting with various distributions of gas and dust around an X-ray source.
The X-ray source can output different spectral distributions of photons such
as a powerlaw, Gaussian, or a blackbody. RefleX also allows the user to
specify different combinations of geometries of gas such as a disk, torus, or
a cone with user-specified sizes and densities. Material in these geometries
can be neutral, ionised, molecular, or in the form of dust (Ricci & Paltani,
2023). Photons can undergo many different physical processes when interacting
with the material, such as photo-ionisation, Compton scattering from free and
bound electrons, and Rayleigh scattering. Therefore, with its ability to
consider many different geometries and scattering processes, RefleX represents
a very powerful tool to study how photons will interact with the free
electrons in the NLR, while isolating the continuum (or non-interacting)
photons.

###  2.2 NLR Model: Geometry and Composition

####  2.2.1 Geometry and incident radiation

We consider the traditional unification model where there torus collimates the
radiation from the central accretion disk and forms a bi-cone shape for the
NLR. This has been confirmed via kinematic (e.g., Fischer et al., 2013) and
optical narrowbnd imaging of [O iii] $\lambda$5007 and H$\alpha$+[N ii]
$\lambda$6584 (e.g., Storchi-Bergmann et al., 2018). The geometry for our
simulated NLR can be seen in Figure 1 where the various shades of grey shown
represent different half opening angles ($\alpha$; not all shown) to be
considered. The NLR extends from an X-ray point source located at the center
of the system representing the accreting SMBH to a height of $H=1\,$kpc. This
range is generally observed in AGN (e.g., Heckman et al., 1981; McCarthy et
al., 1987; Liu et al., 2013; Chen et al., 2019). The X-ray source emits a
powerlaw spectrum incident on the NLR in the 0.3$-$200 keV range with a photon
index of $\Gamma=1.8$ and exponential cutoff at 200 keV, typical values for
nearby AGN (e.g., Nandra et al., 2000; Brightman & Nandra, 2011; Ricci et al.,
2017a, 2018).

####  2.2.2 Expected NLR column density

Thomson scattering occurs when a soft X-ray photon interacts with a free
electron (see Figure 1). Thus, the free electron density distribution observed
in the NLR may be an important parameter in our simulations. The properties of
the NLR in AGN have been widely studied (e.g., Tadhunter et al., 1989; Bianchi
et al., 2006; Bennert et al., 2006; Dadina et al., 2010; Fischer et al., 2013;
Gómez-Guijarro et al., 2017; Fabbiano et al., 2018). In particular, Bennert et
al. (2006) used optical spectra from a sample of six Seyfert-1 galaxies to
study the physical properties of their NLRs such as the electron density as a
function of radius using a powerlaw:

$n_{\text{e}}(r)=n_{\text{e,0}}\left(\frac{r}{r_{0}}\right)^{-\delta}$ (1)

where $n_{\text{e,0}}$ is the density of electrons at $r_{0}$, $r$ is the
distance to the SMBH, and $\delta$ is a decay parameter. Table 9 in Bennert et
al. (2006) shows the values of $n_{\mathrm{e,0}}$ and $\delta$ for all sources
observed in their sample. These range from
$3.1\leq\log{(n_{\mathrm{e,0}}\big{/}\mathrm{cm}^{-3})}\leq 4.1$ and
$-2.32\leq\delta\leq-0.94$. The density drop off with distance generally
observed in AGN (e.g., Kraemer et al., 2000; Barth et al., 2001; Walsh et al.,
2008; Stoklasová et al., 2009) could be explained by the compression of the
gas by the incident radiation pressure as shown by Stern et al. (2014). A
discussion on the effect that this density distribution has on the fraction of
scattered radiation can be found in Appendix A. There we show that the exact
density distribution as a function of depth has little effect on the fraction
of scattered radiation, and only the total column density of free electrons
needs to be considered. Thus, our simulated NLR will have a constant density
of free electrons333More accurately, RefleX allows for a population of fully
ionised hydrogen and helium with the ‘metallicity 0 temperature 1’ command.
This is what was used here. with distance from the SMBH. The column density of
free electrons as a function of the density distribution can be found using
the integral equation:

$N_{\text{H}}^{\text{NLR}}=\int_{r_{0}}^{H}fn_{\mathrm{e}}(r)dr$ (2)

where $f$ is the filling factor of the gas, assumed to be unity, and
$n_{\mathrm{e}}(r)$ is given by Equation 1. Considering different values of
$f$ will change the NLR column density and thus may also be an important
factor in our simulations. However, we will also consider different NLR column
densities later in our calculations so we do not vary $f$. To find an estimate
for the column density to use in our simulations, we average the values of
$n_{\mathrm{e,0}}$ and $\delta$ from Table 9 in Bennert et al. (2006) and use
these in Equation 2. This gives a column density of
$N_{\text{H}}^{\text{NLR}}=10^{22.43}$ cm-2. To use this value in our
simulations, we assume the column density is measured along the $z$ axis (see
Figure 1) from the SMBH to the height of the NLR and calculate the
corresponding density with $n_{\mathrm{e}}=N_{\mathrm{H}}^{\mathrm{NLR}}/H$.

Figure 2: Fraction of scattered radiation as a function of inclination angle
(bottom axis) and torus column density (top axis) for different NLR opening
angles ($\alpha=10^{\circ},30^{\circ}$, and $70^{\circ}$). Inclination angles
are translated into column densities via Equation 4 (see also §4.1.2 for
details) based on the G21 model. Dashed-dotted lines show relations for the
scattering fraction for the NLR column density ($N_{\text{H}}^{\text{NLR}}$)
as calculated from averaging the sources from Bennert et al.
(2006$;N_{\text{H}}^{\text{NLR}}=10^{22.43}\text{\,cm}^{-2}$). The dashed
black line is the best fit trend to the data points (purple points with grey
error bars) which are the mean $f_{\mathrm{scatt}}$ values per
$N_{\mathrm{H}}$ bin found in G21.

In addition to the column density of free electrons as obtained from the
method outlined above, we also use the spectral synthesis code
Cloudy444https://www.nublado.org (version c17.02; Ferland et al., 2017) in
order to infer a free electron column density in a 1-dimensional sphere of
photoionized gas using an input AGN spectral energy distribution (SED). Cloudy
allows for the input of an SED, gas density, and composition, as well as
various stopping criteria such as temperature, pressure, or cloud thickness.
We use the XSPEC (Arnaud, 1996) model optxagnf from Done et al. (2012)
assuming a black hole mass of $M_{\text{BH}}=10^{7.5}M_{\odot}$ and an
Eddington ratio of $\log{(L/L_{\text{Edd}})}=-1.5$, values typical of nearby
AGN (e.g., Koss et al., 2017; Ricci et al., 2017c), as the incident AGN SED.
The cloud is described by an input radius of 1 pc, a stopping radius of 1000
pc, a density law as seen in Equation 1 with $n_{\mathrm{e,0}}$ and $\delta$
taken as the average from the Bennert et al. (2006) sources. Additionally, the
cloud has depleted ISM abundances with Cloudy’s defult ISM grain set included.
A description of the other parameters used in the input SED, as well as other
important Cloudy parameters, is reported in Appendix B. The results of these
Cloudy simulations give a column density of $10^{22.38}\,\text{\,cm}^{-2}$ for
free electrons from Hydrogen, the main contributor to the population of free
electrons, at a depth of 1000 pc. This is in good agreement with what was
obtained by averaging the column densities from the Bennert et al. (2006)
sources using Equation 2. We note that while Cloudy produces results for a
1-dimensional spherical geometry, different from the 3-dimensional ray-tracing
implementation in RefleX, we calculate $N_{\mathrm{H}}^{\mathrm{NLR}}$ in a
geometry-independent way and this can be compared to the Cloudy calculations.

Figure 3: Contour plot showing the different opening angles
($\alpha=$10$-$70∘) and NLR column densities
[$\log{(N_{\text{H}}^{\text{NLR}}\big{/}\text{\,cm}^{-2})}=22.0-23.5$]
explored by our RefleX simulations, and the corresponding $f_{\text{scatt}}$
value on the color bar. The data shown here are for a single inclination
angle, $i=85^{\circ}$.

###  2.3 Calculating the Scattering Fraction

We set out to study the relationship between $f_{\text{scatt}}$ and the
properties of a dusty torus surrounding the accreting system. These properties
include the line-of-sight column density of the torus ($N_{\mathrm{H}}$) and
its covering factor ($f_{\mathrm{cov}}$). G21 used the $f_{\text{scatt}}$
obtained by Ricci et al. (2017a), where $f_{\mathrm{scatt}}$ is measured by
fitting the spectrum of each obscured AGN with nine models in the spectral
fitting package XSPEC including a cutoff powerlaw (CUTOFFPL), photoelectric
absorption from neutral material (ZPHABS), a reflection component (PEXRAV) as
well as a Thomson scattering component of the primary continuum.
$f_{\mathrm{scatt}}$ was used as a free parameter to renormalise the flux.

We calculate $f_{\mathrm{scatt}}$ using the following method. In a single
simulation, RefleX can isolate the spectrum of photons which undergo only
scattering interactions with a user defined resolution. In this study we
choose 5 eV, but this choice has little effect on our results.
$f_{\mathrm{scatt}}$ is then calculated by dividing the Thomson scattered
spectrum by the continuum flux in each energy bin and averaging the results.

In addition to the NLR column densities expected from observations found in
$\S$2.2.2, we also create a grid of RefleX calculations for a range of NLR
column densities and half opening angles. These range from $\alpha$ = 10$-$70∘
in steps of $5^{\circ}$ and
$\log{(N_{\text{H}}^{\text{NLR}}\big{/}\text{\,cm}^{-2})}=22.0-23.5$ in steps
of $0.1$ dex. This grid is created considering only a single inclination angle
($i$; also known as viewing angle, i.e., angle which the “detector” makes with
the $z$ axis in Figure 1) of $85^{\circ}$. We calculate $f_{\mathrm{scatt}}$
for each choice of $N_{\mathrm{H}}^{\mathrm{NLR}}$ and $\alpha$.

When using the NLR column density obtained by averaging the Bennert et al.
(2006) sources, we calculate $f_{\mathrm{scatt}}$ as a function of inclination
angle ranging from $45^{\circ}-90^{\circ}$ in steps of $5^{\circ}$ where the
method of calculating $f_{\mathrm{scatt}}$ above is repeated for each chosen
inclination angle. We take care the detector only samples from inclinations
which are larger than the half opening angle of the NLR to be consistent with
the idea that we are simulating type 2 (edge-on) AGN. This was done for the
half opening angles $\alpha=10,30,\mathrm{and\,\,}70^{\circ}$, in order to
explore the effect of a varying NLR covering factor on the scattering faction.
Note, while we do not include a torus in our simulations, the half opening
angle of the NLR determines the covering factor of the torus in the equatorial
region. This is due to the fact that obscuring material is thought to
collimate the NLR (e.g., Antonucci & Miller, 1985; Malkan et al., 1998;
Fischer et al., 2013; Storchi-Bergmann et al., 2018). Thus, increasing values
of $\alpha$ is equivalent to decreasing the covering factor of the torus
[i.e., $f_{\text{cov}}=\cos{(\alpha)}$].

###  2.4 Theoretical Estimate of the Scattering Fraction

Here we derive a theoretical estimate of the amount of Thomson scattered
radiation from an optically thin gas cloud to compare to our numerical
results. The optical depth of photons to electron scattering can be calculated
by $\tau_{e^{-}}=N_{\mathrm{H}}^{\mathrm{NLR}}\sigma_{\mathrm{T}}$, where
$\sigma_{\mathrm{T}}\approx 0.665\times 10^{-24}\,$cm2 is the Thomson cross-
section. The amount of scattered radiation will be proportional to
$\tau_{e^{-}}$, multiplied by the fraction of material available for
scattering, i.e., $f_{\mathrm{cov}}$, as shown below:

$f_{\mathrm{scatt}}\simeq
N_{\mathrm{H}}^{\mathrm{NLR}}\sigma_{\mathrm{T}}(1-f_{\mathrm{cov}})=N_{\mathrm{H}}^{\mathrm{NLR}}\sigma_{\mathrm{T}}(1-\cos{(\alpha)})$
(3)

where $\alpha$ is the half opening angle of the NLR. We can see
$f_{\mathrm{scatt}}$ is linearly dependant on both
$N_{\mathrm{H}}^{\mathrm{NLR}}$ and $f_{\mathrm{cov}}$. Considering
$\alpha=45^{\circ}$ and
$N_{\mathrm{H}}^{\mathrm{NLR}}=10^{22.43}\,\mathrm{cm}^{-2}$, we would expect
a scattering fraction of $f_{\mathrm{scatt}}=0.52\%$ from pure Thomson
scattering. This is within the limits of what should be expected using the
average $f_{\mathrm{scatt}}$ from Ricci et al. (2017a).

##  3 Results

###  3.1 Numerical Results

Figure 4: Panel (a): scattering fraction for a NLR of varying column density
as a function of the half opening angle at an inclination angle of
$i=85^{\circ}$. Panel (b): scattering fraction calculated for various NLR
opening angles as a function of NLR column density. The blue shaded region
represents the median value of the scattering fraction with errors from the
Swift/BAT 70-month all-sky survey taken from Ricci et al. (2017a). The indigo
dashed line shows the theoretical treatment in Equation 3.

Using our RefleX simulations, we derive $f_{\text{scatt}}$ as a function of
inclination angle for $\alpha=10^{\circ},30^{\circ}$, and $70^{\circ}$. This
was done for the averaged value of $N_{\mathrm{H}}^{\mathrm{NLR}}$ from
Bennert et al. (2006) as discussed in $\S$2.3. These results are illustrated
in Figure 2 while a contour plot of our full grid of simulations are shown in
Figure 3. Figure 2 also includes the Monte Carlo (MC) linear regression line
from G21 (black dashed line), as well as the MC simulation results per column
density bin (purple points with grey error bars) from G21. It can be seen from
Figure 2 that $f_{\mathrm{scatt}}$ increases with $\alpha$ due to the increase
in gas available for scattering. Some dependence on $i$ is expected due to the
differential Thomson cross-section (e.g., G21), with photons being more likely
scattered $0^{\circ}$ or $180^{\circ}$ to the incident radiation. However, our
results show the correlation of $f_{\text{scatt}}$ with inclination angle is
only very weakly negative, leading to significant discrepancies in the slopes
of the simulated and observed relations. Therefore, based on these
calculations, it is unlikely inclination angle alone accounts for the trend
observed by G21.

Figure 3 shows a contour plot of $f_{\mathrm{scatt}}$ as a function of both
$\alpha$ and $\log{(N_{\mathrm{H}}^{\mathrm{NLR}}\big{/}\mathrm{cm}^{-2})}$
for a inclination angle of $i=85^{\circ}$. This value of $i$ was chosen to be
consistent with observations typical of the most inclined type 2 sources. For
similar reasons as discussed above, $f_{\mathrm{scatt}}$ increases with both
$\alpha$ and $\log{(N_{\mathrm{H}}^{\mathrm{NLR}}\big{/}\mathrm{cm}^{-2})}$
due to the increase in free electrons. This is better seen in Figure 4, which
shows cross-sections of the contour plot. The shaded region is the median
scattering fraction value ($f_{\text{scatt}}=1.0\pm 0.5\%$) for the sample of
AGN taken from the 70-month Swift/BAT all-sky survey (Ricci et al., 2017a).
The theoretical estimate from Equation 3 is also shown in Figure 4 as an
indigo dashed line which agrees well with the trend and normalisation seen in
both panels.

Figure 5: $f_{\text{scatt}}$ as a function of covering factor (bottom axis)
and line-of-sight column density ($N_{\mathrm{H}}$; top axis) for chosen
values of $N_{\mathrm{H}}^{\mathrm{NLR}}$ along with the mean values of
$f_{\mathrm{scatt}}$ per $N_{\mathrm{H}}$ bin found in G21 (purple data points
with error bars). The black trend line is the relation found by G21 reproduced
by Equation 6. The Blue lines represent other possible choices of NLR column
density which do not reproduce the G21 relation when using Equation 6.

##  4 Discussion

###  4.1 Normalisation and slope of the $f_{\mathrm{scatt}}-N_{\mathrm{H}}$
relation

####  4.1.1 Normalisation

Here we discuss the values of $f_{\text{scatt}}$ obtained by our simulations
as compared to the observed values reported in G21, and to the median value of
a large sample of hard X-ray selected AGN from Ricci et al. (2017a). As can be
seen in Figure 2, the simulated scattered fraction only agrees with the
observed values in G21 for $\alpha\geq 30^{\circ}$, or $f_{\mathrm{cov}}\leq
0.87$. From our estimate shown in Equation 3, we expect $f_{\mathrm{scatt}}$
to be dependant on both $f_{\mathrm{cov}}$ (and thus $\alpha$) and
$N_{\mathrm{H}}^{\mathrm{NLR}}$. This can be understood physically due to the
fact that as the opening angle increases, keeping the column density fixed
causes an increase in the number of free electrons available for scattering.
This is shown in our numerical calculations in panel (a) of Figure 4 where,
for a chosen inclination angle of $i=85^{\circ}$ and fixed NLR column density,
$f_{\text{scatt}}$ increases as a function of the opening angle $\alpha$.
Additionally, higher column densities would also increase the number of
available electrons. We investigate this in panel (b) of Figure 4, where the
column density of the NLR is allowed to vary. This shows that NLR column
densities of $\log{(N_{\text{H}}^{\text{NLR}}\big{/}\text{\,cm}^{-2})}\geq
22.2$ are needed to achieve the required normalisation from Ricci et al.
(2017a), depending on the opening angle. Varying the height of the NLR would
likely have a similar effect. Equation 3, shown as the dash indigo line,
broadly agrees with our numerical results.

What could lead to different sources having different values of
$N_{\mathrm{H}}^{\mathrm{NLR}}$? There could be differences in the free
electron column densities of different AGN potentially associated to different
SEDs. An SED with a higher Eddington ratio might lead to a higher UV flux
relative to the X-ray flux, and thus create more free electrons (e.g., Done et
al., 2012). However, Noguchi et al. (2010) and G21 found a slight negative
dependence of $f_{\mathrm{scatt}}$ on $\log{(L/L_{\text{Edd}})}$, which might
suggest that in rapidly accreting AGN the radiative force has expelled all gas
responsible for the soft X-ray excess.

It should be noted that we have also neglected additional physical processes
which could significantly contribute to the fraction of soft X-ray radiation
such as photoionized emission lines. Photoionized lines dominate the soft
X-ray emission in many observations of type 2 AGN with non-detectable
continuum in the $\sim$0.1$-$2 keV range (e.g., Sako et al., 2003; Bianchi et
al., 2006; Guainazzi & Bianchi, 2007). Radiative recombination continuum
features (e.g., Bianchi et al., 2006; Guainazzi & Bianchi, 2007) could also
contribute extra scattered flux and increase the normalisation of
$f_{\text{scatt}}$ relative to what is expected when considering Thomson
scattering alone. Additionally, free electrons from the powerful ionised
outflows produced in the accretion disk (e.g., Tombesi et al., 2010; Fukumura
et al., 2010; Harrison et al., 2018) could contribute to the column density of
the scattering material in the polar region of the system, therefore enhancing
the fraction of scattered photons. However, our results only attempt to
explain the relation found in G21 in the context of Thomson scattering as
proposed by G21 with no contribution from these processes. If, for example,
photoionized lines were included in our calculations, they are likely to
contribute a proportional factor due to the possibility that emission lines
and Thomson scattered radiation could be co-spatial.

Fluorescent line emission is also expected in the X-ray spectrum of AGN as
seen most notably in the Fe K$\alpha$ line at $\sim$6.4 keV (e.g., Fabian et
al., 2000; Yaqoob & Padmanabhan, 2004; Shu et al., 2010; Fukazawa et al.,
2011; Ricci et al., 2014). It has been shown that the new polar component
recently discovered in the Mid-Infrared emission of many local AGN (e.g.,
Hönig et al., 2012; Burtscher et al., 2013; Asmus et al., 2016; López Gonzaga
et al., 2016; Leftley et al., 2019) can lead to the production of fluorescent
lines in the $\sim$0.3$-$5 keV range (McKaig et al., 2022). Thus this
component could possibly increase the normalisation of the scattered emission.
To estimate the increase in $f_{\mathrm{scatt}}$ resulting from this
component, we integrate the total contribution from all the lines found in
McKaig et al. (2022) and find, at inclination angles of
$80^{\circ}-90^{\circ}$, this component could contribute up to 0.126$\%$ to
the scattered flux. However, this contribution is subject to change based on
the column density, and extent of the polar region (see also Liu et al.,
2019).

####  4.1.2 Reproducing the $f_{\mathrm{scatt}}-N_{\mathrm{H}}$ slope from G21

The dashed black line in Figure 2 shows the relationship between
$f_{\text{scatt}}$ and the line-of-sight column density observed in G21 (top
axis). In order to explain this correlation, G21 drew a relation between the
torus column density and the inclination angle of a source. This was done by
assuming that the circumnuclear material in all sources of their sample had a
similar geometry. This geometry is a SMBH surrounded by an equatorial torus
with different vertical layers having different column densities (increasing
with inclination angle; see Figure 6 in G21). This implies the only parameter
that changes between sources is the inclination angle. The covering factors of
different layers of the torus were obtained by following the statistical
arguments given in Ricci et al. (2015) who studied a large sample of AGN and
the relationship between line of sight $N_{\mathrm{H}}$ and covering factor.
Ricci et al. took advantage of the fact that the probability of observing an
obscured AGN with $\log{(N_{\mathrm{H}}\big{/}\mathrm{cm}^{-2})}\geq 22$ is
directly related to its covering factor. Thus, the measure of a distribution
of covering factors and line-of-sight column densities will provide a
relationship between $N_{\mathrm{H}}$ and $i$ when assuming
$i=\arccos{(f_{\mathrm{cov}})}$ (G21). It was also assumed that the denser
material resides closer to the equatorial plane of the torus. Ricci et al.
(2015) found 70$\%$ of their sample was obscured, meaning that the average
covering factor of gas with $\log{(N_{\mathrm{H}}\big{/}\mathrm{cm}^{-2})}\geq
22$ in the AGNs in their sample was 70$\%$. This will correspond to an
inclination angle of $i\gtrsim 46^{\circ}$ for
$\log{(N_{\mathrm{H}}\big{/}\mathrm{cm}^{-2})}\geq 22$. Additionally, covering
factors of $52\%$ and $27\%$ were found on average for column densities
$\log{(N_{\text{H}}}\big{/}\text{\,cm}^{-2})\geq 23$ and
$\log{(N_{\text{H}}}\big{/}\text{\,cm}^{-2})\geq 24$, respectively. These
column densities correspond to inclination angles of $i\gtrsim 59^{\circ}$ and
$i\gtrsim 74^{\circ}$, respectively. We use these points to construct a linear
relation between the inclination angle (in degrees) and the torus column
density. This relationship is shown below in Equation 4

$\log{(N_{\text{H}}\big{/}\text{\,cm}^{-2})}=19.24+0.07i.$ (4)

Using this equation, we change variables to plot $f_{\text{scatt}}$ as a
function of $\log{(N_{\text{H}}\big{/}\mathrm{cm}^{-2})}$ on the top x-axis of
Figure 2. Comparing the results of our simulations with the observational
results of G21 (dashed black line in Figure 2) it is obvious that the
dependence of $f_{\rm scatt}$ on the inclination angle based on Equation 4
cannot reproduce the slope of the observed negative correlation between
$f_{\rm scatt}$ and $N_{\rm H}$ and another explanation is needed.

A second possible explanation for the slope of the $f_{\rm scatt}-N_{\rm H}$
relation is that the average covering factor of the circumnuclear material in
sources with higher column densities tend to be larger than in less obscured
sources. This would be in agreement with what was previously proposed by Ueda
et al. (2007) to explain the existence of sources with extremely low values of
$f_{\rm scatt}$. Moreover, several observational studies both in the IR and
X-rays have suggested that obscured type 2 sources with high column densities
also have higher covering factors (e.g., Ramos Almeida et al., 2011; Ricci et
al., 2011; Mateos et al., 2016; Tanimoto et al., 2018; Panagiotou & Walter,
2019; G21). Here we use the results of our simulations to assess what relation
between the covering factor and the torus column density would be needed to
explain the $f_{\rm scatt}-N_{\rm H}$ relation.

Figure 3 shows that in addition to the column of free electrons, the
scattering fraction also depends on the opening angle of the NLR (i.e.,
$f_{\text{cov}}$; also see Equation 3 and panel (a) of Figure 4). In order to
find a relation between the torus column density and its covering factor, we
first consider a NLR at an inclination of $85^{\circ}$ to the observer and
column density of
$\log{(N_{\text{H}}^{\text{NLR}}\big{/}\text{\,cm}^{-2})}=23.0$ at various
opening angles $\alpha$. We then assume a linear relation between
$f_{\mathrm{scatt}}$ and $f_{\mathrm{cov}}$ by using the fact that
$f_{\mathrm{cov}}=\cos{({\alpha})}$. This gives

$\log{(f_{\mathrm{scatt}}\big{/}\%)}=1.67-2.57f_{\mathrm{cov}}.$ (5)

We assume the relationship between $f_{\mathrm{scatt}}$ and $f_{\mathrm{cov}}$
from our calculations should give the same results as found in G21 and thus
Equation 5 was set equal to the $f_{\mathrm{scatt}}-N_{\mathrm{H}}$ trend
found by G21. Solving for $f_{\mathrm{cov}}$ yields

$f_{\text{cov}}=-3.59+0.18\log{(N_{\text{H}}\big{/}\text{\,cm}^{-2})}.$ (6)

Using this relationship to translate between $f_{\text{cov}}$ and
$N_{\text{H}}$, we plot in Figure 5 the scattering fraction as a function
$f_{\text{cov}}$ and $N_{\text{H}}$ (bottom and top axes, respectively), along
with the median value data points of $f_{\text{scatt}}$ per column density bin
from G21. Using Equation 6, we are able to perfectly reproduce the best fit
line to the data found in G21. This is shown as the black line in Figure 5.
Other values of $N_{\mathrm{H}}^{\mathrm{NLR}}$ which fail to reproduce the
G21 relation using Equation 6 are shown in blue. Thus, a positive correlation
between $f_{\text{cov}}$ and $N_{\text{H}}$ could easily explain the
$f_{\mathrm{scatt}}-N_{\mathrm{H}}$ trend. Now, when using this relation to
translate covering factors into observed column densities, one must note that,
because covering factors cannot be negative or exceed unity, the valid range
of column densities that Equation 6 can be used is
$\log{(N_{\text{H}}\big{/}\text{\,cm}^{-2})}=[19.89,25.44]$.

###  4.2 Co-spatial partially and fully ionised regions

Since we set out to study the soft X-ray excess that originates from scattered
radiation off free electrons in the NLR (e.g., Turner et al., 1997; Cappi et
al., 2006; Ueda et al., 2007; Ricci et al., 2017a) our simulations consists
only of fully ionised Hydrogen and Helium providing the population of
electrons responsible for the scattered X-ray radiation. However,
photoelectric absorption and transitions from bound electrons of metals such
as Oxygen and Silicon can absorb some the the X-ray emission (e.g., Osterbrock
& Ferland, 2006). To test the impact of these bound electrons in our
simulations, we consider the X-ray ionisation parameter given by:

$\xi=\frac{4\pi
F_{\mathrm{ion}}}{n_{\mathrm{H}}}=\frac{L_{\mathrm{ion}}}{n_{\mathrm{H}}r_{0}^{2}}$
(7)

where $n_{\mathrm{H}}$ is the Hydrogen number density, $r_{0}$ is the distance
from the X-ray source, and $L_{\mathrm{ion}}$ is the ionising luminosity.
Estimating $L_{\mathrm{ion}}=10^{44}\,\mathrm{erg\,s}^{-1}$, $r_{0}=1\,$pc and
considering a range of densities of $n_{\mathrm{H}}=10-10^{4}\,$cm-3 gives a
possible X-ray ionisation parameter of
$\xi=10^{3}-10^{6}\,\mathrm{erg\,cm\,s}^{-1}$. Thus, we expect that, close to
the SMBH, most metals would be ionised and would not absorb significantly the
X-ray radiation, particularly below 2 keV.

However, there must also exist partially ionised metals in the NLR due to the
fact that extended X-ray emission can be traced by [O iii] $\lambda$5007
emission (e.g., Young et al., 2001; Bianchi et al., 2006; Dadina et al., 2010;
G21). Indeed, the correlation between $f_{\mathrm{scatt}}$ and the
$L_{X}/L$([O iii] $\lambda$5007) ratio found by Ueda et al. (2015) and G21
also confirms the idea that these regions are co-spatial. Partially ionised
material would also absorb some of the X-ray emission through photoelectric
processes. To study how the presence of partially ionised material changes the
amount of scattered radiation, we plot in Figure 6 the scattered fraction as a
function of column density from two NLRs, one consisting only of free
electrons and the other consisting of solar metallicity material (abundances
taken from Lodders, 2003) with only hydrogen and helium fully ionized. In
RefleX, this is accomplished with various combinations of the metallicity and
temperature commands. The metallicity command determines the composition of
the material, metallicity 0 for hydrogen and helium only and metallicity 1 for
solar abundances. The temperature command determines the ionisation state of
the gas, temperature 0 for cold neutral material and temperature 1 for hot
material where hydrogen and helium are fully ionised. Thus, material
consisting of free electrons is achieved with ‘metallicity 0 temperature 1’
and solar metallicity material with hydrogen and helium ionised is given by
‘metallicity 1 temperature 1’.

It can be seen in Figure 4 that $f_{\mathrm{scatt}}$ from the fully ionised
material is always a factor of $\sim$2 larger, and higher column densities of
partially ionised of material are needed to reproduce the average
$f_{\mathrm{scatt}}$ as seen from Ricci et al. (2017a) (shaded region). This,
along with the expected presence of a photoelectric cutoff when bound
electrons are present, implies that highly ionised material would produce
significantly more scattered radiation in the soft X-rays. If the X-ray
emission can traced with the [O iii] $\lambda$5007, why does partially ionised
material not account for the observed $f_{\mathrm{scatt}}$?

When clouds of gas in the NLR are directly exposed to the AGN SED, absorption
of ionising photons will lead to compression of the photoionised layer (e.g.,
Dopita et al., 2002; Groves et al., 2004a, b; Stern et al., 2014). This
Radiation Pressure Confinement (RPC; Stern et al., 2014) leads to a highly
ionised surface which emits X-rays, and a layer closer to the Hydrogen
ionisation front which emits optical lines such as [O iii] $\lambda$5007\. RPC
also results in a thin photoionised region, many orders of magnitude smaller
than its distance to the photoionising source (Stern et al., 2014). Thus, RPC
predicts a common origin for optical and soft X-ray emitting regions (Stern et
al., 2014). Bianchi et al. (2019) studied the effects of radiation pressure
confinement on soft X-ray emitting gas in a sample of obscured AGN using the
XMM-Newton Reflection Grating Spectrometer. They found that radiation pressure
compression in the photoionised region reproduces the observed differential
emission measure slope, implying the NLR in these AGN are indeed dominated by
radiation pressure as opposed to another confinement mechanism such as a
constant gas pressure multiphase medium. It is therefore possible that the
illuminated faces of radiation pressure confined slabs, where the estimates
made from Equation 6 are true, are mostly likely what is driving the scattered
soft X-ray emission observed in G21. These slabs inevitably reach the Hydrogen
ionisation front where $\xi\ll 1$, the material becomes neutral and is no
longer able to efficiently scatter the X-ray radiation below $\sim$2 keV. This
will explain the spatial extent and overlap between the X-ray and optically
emitting regions. This highly ionised surface could also be the origin of
spectra from species of iron such as Fe xxv and Fe xxvi which have been
observed in Compton-thick AGN (e.g., Bianchi & Matt, 2002; Bianchi et al.,
2005).

Figure 6: $f_{\mathrm{scatt}}$ versus
$\log{(N_{\mathrm{H}}^{\mathrm{NLR}}\big{/}\mathrm{cm}^{-2})}$ for solar
metallicity material with Hydrogen and Helium ionised, and fully ionised
material. The partially ionised material could produce [O iii] $\lambda$5007
emission while the fully ionised material gives rise to the scattered
emission. The shaded region represents the median scattering fraction with
errors from Ricci et al. (2017a).

##  5 Summary and conclusions

We have used X-ray simulations on the platform RefleX to study the origin of
the soft X-ray excess in obscured AGN. G21 observed a negative correlation
between the fraction of scattered radiation relative to the primary continuum
with a sources column density. We attempt to discern between two possible
explanations for this observation: (1) the angle dependence on the
differential Thompson cross-section and (2) objects with higher column
densities have covering factors closer to unity, as suggested by previous
studies (e.g., Ueda et al., 2007; G21). We consider a full grid of opening
angles and free electron column densities from the NLR, where most of the
X-ray scattering takes place as traced by the [O iii] $\lambda 5007$ emission
(e.g., Bianchi et al., 2006, G21). We also consider values derived using
observations from Bennert et al. (2006) as well as Cloudy calculations. While
we assumed here that Thomson scattering in the NLR is the dominating process
in the soft X-rays, there could be significant contributions to the scattered
component from photoionised lines or recombination continuum which we do not
consider here. However, such components would be expected to be cospatial with
the gas responsible for the Thomson scattered radiation. Figure 2 and 3 show
the main results of our simulations. Our main conclusions are the following:

  * •

We find a very weak negative correlation between $f_{\text{scatt}}$ and the
inclination angle (see Figure 2). When using statistical arguments from Ricci
et al. (2015) to translate between inclination angles and observed column
densities, we are unable to produce the observed slope from G21. This,
combined with the weak dependence of $f_{\text{scatt}}$ on the inclination
angle, implies that a change in inclination angles alone cannot reproduce the
relation found in G21.

  * •

The column densities expected for the NLR from observations only reproduce the
observed values of $f_{\rm scatt}$ for a small range of $\alpha$ and $i$. We
explored possible scenarios in which extra scattering can contribute to
$f_{\mathrm{scatt}}$. This extra scattered flux could come from a higher
column density of the NLR (see Figure 4), from additional radiative
recombination continuum features (e.g., Bianchi et al., 2006; Guainazzi &
Bianchi, 2007), extra free electrons from highly ionised material associated
to outflows from the accretion disk (e.g., Tombesi et al., 2010; Fukumura et
al., 2010; Harrison et al., 2018), photoionized lines seen in the soft X-rays
of obscured type 2 AGN using high spectral resolution instruments (e.g., Sako
et al., 2003; Bianchi et al., 2006; Guainazzi & Bianchi, 2007), as well as
from the polar component seen in the MIR of many local AGN, (e.g., Hönig et
al., 2012; Burtscher et al., 2013; Asmus et al., 2016; López Gonzaga et al.,
2016; Leftley et al., 2019) which has been shown to contribute many
fluorescent lines to the soft X-ray spectrum of AGN (Liu et al., 2019; McKaig
et al., 2022). This latter component can produce up to 0.126$\%$ of the
primary continuum.

  * •

Using our full grid of simulations shown in Figure 3, we showed that a
positive relationship between line of sight column densities and covering
factors would be able to reproduce the relationship between $f_{\text{scatt}}$
and $N_{\text{H}}$ observed in G21. The relationship can be seen in Equation 6
and is plotted as the black line in Figure 5. This implies that the main cause
of the observed slope found in G21 might be that the covering factor in local
AGN is correlated to its column density, with more obscured AGN also having
larger covering factors of the circumnuclear material.

## Acknowledgements

The authors are very grateful to the anonymous referee for their insightful
comments and suggestions on the paper. JM acknowlages support from NASA grant
80NSSC20K0038. CR acknowledges support from the Fondecyt Regular grant 1230345
and ANID BASAL project FB210003.

## Data Availability

The datasets generated and/or analysed in this study are available from the
corresponding author on reasonable request.

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##  Appendix A Importance of the Electron Density Distribution

In $\S$2.2.2 we state that the exact distribution of free electrons with
distance is unimportant when calculating the scattering fraction and the only
parameter which needs to be considered is the total column density of the NLR.
We show this in Figures 7 and 8. In these figures, we use the six sources from
Bennert et al. (2006) and implement the electron density distribution in
Equation 1 by splitting up the NLR into 100 constant density sections and
calculating the density at the centre of the section. The scattering fraction
as a function of inclination angle, when considering this exact density
distribution, can be seen as the black triangles in Figures 7 and 8 with the
best fit curve being the dashed black line for each source. The blue triangles
in the figure represent the scattering fraction at different inclinations from
a NLR consisting of only one constant density section but with the same
overall column density as its counterpart with 100 sections. Each data point
as well as the best fit curve shows that, for each source, the scattering
fraction is similar irrespective of using the exact electron distribution seen
in Equation 1, or the average column density for that particular source.

Figure 7: Scattering fraction as a function of inclination angle for three of
the AGN from Bennert et al. (2006), calculated by using either the full
prescription of Equation 1 (black upside-down triangles; best fit trend shown
as black dashed line) or a single, constant density NLR with the same overall
column density (black triangles; best fit trend shown as blue dashed line).
Each panel in the figure represents the scattering fraction as a function of
the inclination angle for Fairall 51 (upper panel), NGC 6860 (middle panel),
and NGC 526a (lower panel).

Figure 8: Same as Figure 7 but for Mrk 915 (upper panel), MCG -05-13-017
(middle panel), and MCG -06-30-15 (lower panel)

##  Appendix B Parameters for Cloudy Simulations

Here we describe our Cloudy simulations to infer another value for the free
electron column density of the NLR. Our Cloudy model consisted of a 1D
spherical cloud geometry representing the NLR with ISM abundances, including
dust grains exposed to an ionising AGN SED. The Hydrogen density at the face
of the cloud was set consistent with the values from Bennert et al. (2006),
and the thickness of the cloud (used as the stopping criterion) being the same
as the NLR considered in our simulations (1000 pc). A powerlaw density profile
with distance was also used with an index of -1.46 (again found when averaging
the Bennert et al., 2006 sources). The ionising AGN SED that we considered is
from Done et al. (2012), which models the intrinsic disk emission from the
optical to X-ray band from an AGN. This is done by splitting up the accretion
disk into an outer disk which emits as a colour temperature corrected
blackbody; a warm, optically thick inner disk of free electrons responsible
for the soft excess, and a hot, optically thin corona responsible for the
power-law extending up to hundreds of keV. This model has been well tested in
fitting disk emission from AGN (Done et al., 2012). Thus, this model is
optimal as an input SED into Cloudy to study the number of free electrons in
the NLR. The values chosen as input parameters for the SED are the from the
best fit values from a sample of 51 AGN from Jin et al. (2012a) and Jin et al.
(2012b) (see Table 3 in Done et al., 2012). The parameters chosen for our
model are a black hole mass $\log{(M_{\rm BH}/M_{\odot})}=7.5$, an Eddington
ratio $\log{(L/L_{\text{Edd}})}=-1.5$ (Koss et al., 2017), a temperature of
the warm electrons in the inner disk $kT_{e}=0.31\,$keV, the power-law of the
hot electron corona $\Gamma=1.87$, the optical depth of the warm electrons
$\tau=13$, and the percent of the radiation emitted in the power-law
$f_{pl}=0.34$. From these simulations we found that the column density of
ionised Hydrogen, i.e. the largest contributor to the free electron column
density, was $N_{\text{H}}^{\text{NLR}}=10^{22.38}\text{\,cm}^{-2}$.