11institutetext: Pontificia Universidad Cat lica de Chile, Instituto de Astrof
sica, Casilla 306, Santiago 22, Chile22institutetext: Excellence Cluster
Universe, Boltzmannstr. 2, D-85748, Garching, Germany33institutetext: Cahill
Center for Astrophysics, California Institute of Technology, 1216 East
California Boulevard, Pasadena, CA 91125, USA 44institutetext: Max Planck
Institute for Extraterrestrial Physics, Giessenbachstrasse, 85741 Garching,
Germany55institutetext: National Optical Astronomy Observatory, 950 N Cherry
Avenue, Tucson, AZ 85719, USA66institutetext: Millenium Institute of
Astrophysics, Vicu a MacKenna 4860, 7820436 Macul, Santiago,
Chile77institutetext: Space Science Institute, 4750 Walnut Street, Suite 205,
Boulder, Colorado 80301

# X-ray spectral and eclipsing model of the clumpy obscurer in active galactic
nuclei

Johannes Buchner<EMAIL_ADDRESS>Murray Brightman 33 Kirpal
Nandra 44 Robert Nikutta 5511 Franz E. Bauer 116677

(-Received date / Accepted date)

We present a unification model for a clumpy obscurer in active galactic nuclei
(AGN) and investigate the properties of the resulting X-ray spectrum. Our
model is constructed to reproduce the column density distribution of the AGN
population and cloud eclipse events in terms of their angular sizes and
frequency. We developed and release a generalised Monte Carlo X-ray radiative
transfer code, XARS, to compute X-ray spectra of obscurer models. The geometry
results in strong Compton scattering, causing soft photons to escape also
along Compton-thick sight lines. This makes our model spectra very similar to
the Brightman & Nandra TORUS model. However, only if we introduce an
additional Compton-thick reflector near the corona, we achieve good fits to
_NuSTAR_ spectra. This additional component in our model can be interpreted as
part of the dust-free broad-line region, an inner wall or rim, or a warped
disk. It cannot be attributed to a simple disk because the reflector must
simultaneously block the line of sight to the corona and reflect its
radiation. We release our model as an Xspec table model and present
corresponding CLUMPY infrared spectra, paving the way for self-consistent
multi-wavelength analyses.

##  1 Introduction

X-ray spectroscopy is a powerful method for characterising the accretion onto
supermassive black holes, even when the accretion disk is hidden behind thick
columns of gas and dust. For the most heavily obscured Compton-thick active
galactic nuclei (CT AGN, ${N_{\rm{H}}}\gtrsim 10^{24}{\rm cm}^{-2}$) inference
on the accretion and nature of the obscurer requires a correct spectral model
for the obscurer (the ‘torus’). To date, the explored range of X-ray spectral
models has been limited to simple geometries, while observations have shown
that the real geometry is more complex.

For instance, strong evidence exists today that the obscurer is clumpy and has
a large covering factor. The former is found through X-ray eclipse events
(e.g. Risaliti et al., 2002; Markowitz et al., 2014), while the latter is
evident from the large fractions of obscured AGN found in flux-limited X-ray
surveys (e.g. Treister et al., 2004; Brightman et al., 2014; Ueda et al.,
2014; Buchner et al., 2015; Aird et al., 2015). Following the orientation-only
version of the unification paradigm (straw-person model, Antonucci, 1993), a
single obscuring structure may explain the diversity of obscured and
unobscured AGN as well as eclipse events through sampling different viewing
angles. However, such a strict unification across all AGN is ruled out because
the accretion luminosity is anti-correlated with the obscured fraction (e.g.
Ueda et al., 2003; Hasinger et al., 2005; Buchner et al., 2015). In both the
infrared and X-ray regime, many studies have therefore attempted to constrain
the covering factor of individual AGN.

Figure 1: Simulated X-ray spectra, with geometries shown on top. The _left
panel_ presents spectra of a Compton-thick bi-cone cut-out geometry in edge-on
view. Liu & Li (2015, shown as ①) demonstrated that the model of Brightman &
Nandra (2011a, shown as ②) should not leak soft ($\lesssim 4{\rm keV}$)
photons. We obtained with our code XARS (shown as ③) the same spectrum as Liu
& Li (2015). The _right panel_ compares this uniform density toroid with a
gradient toroid (shown as ④) and our unified clumpy model (shown as ⑤). The
arrow highlights that, unlike the smooth geometry, leakage of soft photons is
possible in the clumpy torus. In the main plots, we removed spectral lines to
highlight the continuum shape. The _insets_ show a zoom around the Fe
K$\alpha$ and $\beta$ features, normalised at $7.07\mathrm{keV}$. They feature
the strongest scattering low-energy tails in the ⑤unified clumpy model.

Therefore, physically motivated geometric models are needed. Early modelling
efforts (see e.g. Matt et al., 2000, and references therein) of the global,
low-resolution X-ray spectra at the time employed simplistic models (e.g. slab
appropriate for cold reflection from accretion disks; Magdziarz & Zdziarski,
1995). As knowledge and capabilities in spectral and spatial resolution
progressed, more realistic modelling was developed, allowing refined parameter
constraints and interpretations (e.g. Murphy & Yaqoob, 2009; Brightman &
Nandra, 2011a; Ikeda et al., 2009). Studies of nearby (e.g. Matt, 2000, and
references therein) and distant (e.g. Buchner et al., 2014) obscured AGN show
that X-ray spectra are to first order consistent with a photo-electrically
absorbed power law, but additionally require varying degrees of Compton-
scattering to fit the generally flat spectra and the $\sim 30\mathrm{keV}$
Compton hump. This can be seen in the literature, for example by many fits
preferring the model of Brightman & Nandra (2011a), BNTORUS, which produces a
flat spectrum for Compton-thick obscured AGN, similar to that created by cold
disk reflection. Some studies even find additional Compton reflection beyond
this model, with examples both in individual local AGN (e.g. Arévalo et al.,
2014; Bauer et al., 2015) and large X-ray surveys (e.g. Rivers et al., 2013;
Buchner et al., 2014). The arrival of _NuSTAR_ (Harrison et al., 2013)
revealed, in detail, the complexity of spectral shapes above $10\mathrm{keV}$
for many local AGN. The analysis of these data is limited by the currently
freely available, easy-to-install X-ray spectral models of AGN obscurers. The
two physically motivated models most commonly used today are: the BNTORUS
model of a bi-conical sphere cutout, and the ‘donut’ geometry of Murphy &
Yaqoob (2009), MYTORUS. Both assume simple geometries and a smooth
distribution of obscuring material, which cannot explain AGN with time-
variable line-of-sight column densities, commonly interpreted as cloud eclipse
events (e.g. Risaliti et al., 2002). A few clumpy geometry models have been
recently proposed (e.g. Liu & Li, 2014; Furui et al., 2016), but have not been
used to explain eclipse events. To explore various possible geometries,
interest has grown in the availability of Monte Carlo codes (e.g. Paltani &
Ricci, 2017; Baloković et al., 2018). An extendable open source Monte Carlo
simulation package is not available yet.

To compute X-ray spectra we developed a new Monte Carlo simulation code, XARS.
We begin in 2 by describing this code and basic results for inhomogeneous
media. In 3 we construct a clumpy, unified AGN obscurer model which reproduces
(1) the column density distribution of AGN at various luminosities and (2) the
distribution of cloud eclipse events. Although our initial model spectra
produce strong Compton scattering signatures, in 5 we demonstrate that we must
add another reflection component to fit _NuSTAR_ data. We interpret this
result and discuss physical origins in 6.2. Finally, we show how our clumpy
obscurer model allows self-consistent infrared & high-energy studies in
combination with Clumpy models (Nenkova et al., 2008a, b).

##  2 Computation of X-ray spectra in inhomogenous media

###  2.1 Simulation code XARS

We compute the emerging X-ray spectrum for several torus geometries of
interest through Monte Carlo simulations. We assume that an isotropically
emitting point-source generates photons at the centre of our obscurer
geometry. Photo-electric absorption, Compton scattering and line fluorescence
(Fe K$\alpha$+K$\beta$; K$\alpha$ of C, O, Ne, Mg, Si, Ar, Ca, Cr and Ni) are
simulated self-consistently as photons pass through matter. The simulation
method has already been described in detail in Brightman & Nandra (2011a), but
we provide some computational details in A. Solar abundances (Anders &
Grevesse, 1989) are assumed with cross-sections from Verner et al. (1996).

We make our modular, Python-based simulation code XARS (X-ray Absorption Re-
emission Scattering) publicly available at
https://github.com/JohannesBuchner/xars. Users can specify their obscurer
geometries by implementing a class which prescribes how photons propagate
through the medium, and easily change the input abundances and emission lines
without modifying the code. Once the Green’s functions at each energy grid
point and viewing angle have been computed with adequate signal-to-noise, an
input photon spectrum, for example a powerlaw, can be applied, and an Xspec
grid model compiled.

We verified our code first by comparing with other implementation. We also
verified generated spectra from (1) the SPHERE geometry of Brightman & Nandra
(2011a) and Liu & Li (2015), discussed below, (2) disk reflection off a semi-
infinite plane-parallel slab (Magdziarz & Zdziarski, 1995), discussed in the
Appendix, (3) reflection off individual spheres (Nandra & George, 1994, see
Appendix B) and (4) the biconical cut-out geometry studied in the next
section. As a benchmark, for a Compton-thick sphere, XARS can compute very
high-quality spectra ($10^{9}$ photons input across 1000 energy bins) in $90$
minutes on a single $2{\rm GHz}$ laptop CPU, and is trivial to parallelise
through independent executions.

###  2.2 Inhomogenous geometries leak soft photons

An interesting test case studied in previous works is a sphere with a
biconical cut-out, illustrated in the top left of Figure 1. The left panel of
Figure 1 compares ③ our results with those of the Geant4-based code of Liu &
Li (2015) (①, grey thick line). For the spectra shown in Figure 1 we have
chosen an edge-on line of sight with column density ${N_{\rm{H}}}=10^{24}{\rm
cm}^{-2}$ and a photon index of $\Gamma=1.8$, consistent with Liu & Li (2015).
We find consistent results. In Brightman & Nandra (2011a), the geometry code
erroneously111J. Buchner & M. Brightman, priv. comm. let photons back-
scattered from the inner wall escape without considering re-entry into the
torus ②. This affects the geometric interpretability of fits with this model,
and potentially the parameters inferred from jointly fit components.
Nevertheless, it is very interesting that across the literature this model is
found to describe observational data well (e.g., Brightman & Nandra, 2011a, b;
Rivers et al., 2013; Buchner et al., 2014; Ricci et al., 2017).

Next we simulated the spectrum of a clumpy model (similar to Clumpy
geometries, see next section for details). We assume spheres of constant
density222See Appendix B for the impact of assuming different density
profiles.. Our cloud geometry code uses LightRayRider (see Buchner et al.,
2017, for details) with parallelised and optimised C routines to quickly solve
the individual photon paths through millions of clouds exactly. However, our
tests indicate that beyond $10^{4}$ clouds, the resulting spectrum is not
affected any more by the number of clouds. We generated $\sim 10^{9}$ photons
for each geometry. For the photons not absorbed inside the obscurer and
escaping to infinity we record energy and direction. The LOS column density
${N_{\rm{H}}}$ of that direction from the central source is then computed, and
the photon assigned to that particular ${N_{\rm{H}}}$ and altitude angle
bin333The models considered here are azimuthally symmetric.. In the right
panel of Figure 1 we plot the energy distribution of photons emerging from
directions with LOS column densities of ${N_{\rm{H}}}=10^{24}{\rm cm}^{-2}$.
Photons originally on paths of lower LOS column densities can be scattered
into the LOS. This leads to a substantial component of soft photons
proportional to the scattering surface (covering fraction) in ⑤ clumpy torus
models, as noted in previous works (Liu & Li, 2014; Furui et al., 2016). We
also consider a smooth torus with gradually varying density layers ④
reproducing the same column density distribution. In contrast to the clumpy
model, this geometry blocks soft photons from escaping. In the clumpy model,
fluorescent lines are also scattered farther and emerge with strong horns in
the spectrum (inset plot of Figure 1).

##  3 Clumpy Torus Geometry

###  3.1 Cloud population geometry from obscured fractions

Figure 2: Column density distribution and associated covering fractions
produced by our clumpy model. The curves represent variations of the angular
width parameter $\sigma$. Obscured fractions from AGN surveys are shown as
black and grey symbols, for bright and faint AGN, respectively. Varying
$\sigma$ allows a single instance of our model to reproduce the column density
distributions of the population at various luminosities. Thin grey curves show
the comparatively narrow column density distributions of a single sphere and a
torus, each with a maximum crossing column density of
${N_{\rm{H}}}=10^{22}\mathrm{cm}^{-2}$. Figure 3: Predicted rates of eclipses
by our model. The probability for an eclipse in a sightline after a given
delay is computed, with model clouds on random, circular Keplerian orbits. The
systematic study of eclipse event rates measured with the RXTE satellite by
Markowitz et al. (2014) are drawn with orange/cyan error bars (type-1/type-2,
respectively), with steps approximating their observation window function. We
analyse two cases: How frequently an obscured sightline becomes more obscured
(red curve) in a high-covering geometry, and how frequently an unobscured
sightline becomes obscured in a low-covering geometry (dashed blue curve). The
eclipse probability increases with the covering factor. The angular diameter
distribution of the model clouds is shown in the inset (blue: low-covering
model, red: high-covering model). Sizes of individual observed eclipsing
clouds are shown as a black histogram (Nikutta et al., in prep), covering the
low-end tail of the intrinsic distribution.

In principle, the construction of a clumpy obscurer model has infinite degrees
of freedom, as each location can have an arbitrary density. To simplify this
problem, we assume symmetry around one axis and that the obscurer column
density is, at least on average, a monotonic function with decreasing column
density towards the poles. We use the Nenkova et al. (2008b, N08 hereafter)
formalism to describe the distribution and properties of clouds. In that
formalism, the number of clouds ${\cal N}$ seen along a radial line-of-sight
is axi-symmetric, and decreases from ${\cal N}_{0}$ at the equatorial plane
with inclination angle $\beta$ towards the poles, according to a Gaussian
function:

${\cal N}={\cal
N}_{0}\cdot\exp\left\\{-\left(\frac{\beta}{\sigma}\right)^{m}\right\\}$ (1)

The angular width, $\sigma$, controlling the torus scale height and the
average number of clouds in the equatorial plane, ${\cal N}_{0}$, are free
parameters a priori. We follow previous studies (e.g. N08) which adopted a
Gaussian profile ($m=2$). Using an exponential distribution instead does not
affect our conclusions.

Figure 4: Cross-sections of clumpy torus model. The left and right geometries
differ in the vertical extent $\sigma$, which controls the Gaussian
distribution away from the equatorial plane. The red cross indicates the
central point source. The clouds vary in size and column density (darkness). A
fly-through video is available at https://doi.org/10.5281/zenodo.602282.

To narrow down the possible geometries in a realistic way, we first attempt to
fit the model to several known constraints. The first constraint applied is
the column density distribution. For luminous (moderately luminous) AGN, we
should reproduce the fractions of $25\%$ ($35\%$) for $N_{{\rm H}}\gtrsim
10^{24}{\rm cm}^{-2}$ and $30\%$ ($65\%$) at $N_{{\rm H}}>10^{22}{\rm
cm}^{-2}$ (from Buchner et al., 2015; Aird et al., 2015; Ricci et al., 2015).
These are intrinsic fractions (observed, but corrected for selection bias)
with the obscuration of the host galaxy subtracted (see Buchner & Bauer, 2017,
for details). Simultaneously, the obscurer should allow, albeit rarely, column
densities up to $N_{{\rm H}}=10^{26}{\rm cm}^{-2}$, since sources with
$N_{{\rm H}}>10^{25}{\rm cm}^{-2}$ are known (e.g. NGC 1068, Matt et al.,
2000). However, when assuming each cloud has the same column density, the
column density distribution produced by Equation 1 is too narrow.

Widening the model column density distribution can be achieved by allowing at
a given luminosity a diversity of opening angles or by giving the clouds a
diversity in column densities. We choose the latter option, as we prefer to
remain with a unification model. Additionally, absorber variability studies
show a wide range of preferentially Compton-thin column density changes (e.g.
Risaliti et al., 2002, 2005; Markowitz et al., 2014), and Compton-thick
variations have also been seen when telescopes with good sensitivity and
angular resolution at $>10\mathrm{keV}$ became available (e.g., Ricci et al.,
2016; Marinucci et al., 2016). We therefore allow a range of values for the
column density of individual clouds, $N_{{\rm H}}^{{\rm cloud}}$. Clouds are
placed randomly according to the distributions (and re-drawn when
overlapping). Fitting the angular width $\sigma$, the total number of clouds
and the parameters of a log-normal distribution simultaneously, we find a
suitable fit with $10^{5}$ clouds with a 1 dex wide distribution centred at
$10^{23}{\rm cm}^{-2}$. This is shown in Figure 2. Varying the angular width
$\sigma$ makes our model (coloured thick curves) reproduce the column density
distribution of the bright or faint AGN population (black and grey points,
respectively).

The radial cloud distribution was assumed to be uniform across two orders of
magnitude. While the model is scale-free, we can define the ratio of
outermost-to-innermost radial distances of clouds,
$Y=R_{\mathrm{out}}/R_{\mathrm{in}}$. Observations (e.g. Lira et al., 2013;
Fuller et al., 2016; García-Burillo et al., 2016; Ichikawa et al., 2016)
indicate that the clouds relevant for the infrared radiation extend out up to
approximately one order of magnitude farther than the dust sublimation radius,
that is, $Y=R_{\mathrm{out}}/R_{\mathrm{d}}\approx 10-20$. For X-rays however,
inner ionised clouds444e.g. those in the broad-line region (BLR) can also act
as absorbers. We choose $Y=100$ for the X-ray model computation, so that the
X-ray obscurer spans two radial orders of magnitude, while the UV/optical,
dusty obscurer spans only the last magnitude in radial scale (motivated by
Tristram & Schartmann, 2011 and Markowitz et al., 2014). For a consistent
infrared model, each cloud outside the sublimation radius can be assigned an
optical depth of $\tau_{V}$, related to the column density via
$\tau_{V}=N_{{\rm H}}^{{\rm cloud}}/\left(2\cdot 10^{21}{\rm cm}^{-2}\right)$,
the galactic relationship (Predehl & Schmitt, 1995; Nowak et al., 2012, see
also Burtscher et al., 2016).

###  3.2 Properties of clouds from eclipse events

Cloud eclipse events provide constraints on the angular size of individual
clouds. In general, the clouds are observed to be very small in angular extent
subtended at the black hole ($\sim 0.1^{\prime}-1\text{ }$, Nikutta et al., in
prep.). However, observational biases due to limited observing periods
preferentially select clouds with high angular velocities and small angular
extents. It is furthermore reasonable to assume that the sizes of clouds show
a powerlaw-like distribution which spans several orders of magnitude. In other
words, we expect the observed distribution to only cover the low-end tail of
the intrinsic powerlaw angular size distribution. For simplicity, we adopt an
exponential distribution of the angular sizes centred around $\theta^{{\rm
cloud}}=1\text{ }$, shown in the inset of Figure 3. This encompasses observed
sizes (black histogram). The clouds are assumed to be spherical blobs with
diameter $D^{{\rm cloud}}=d\cdot\sin\theta^{{\rm cloud}}$ for a distance $d$.

Our model is also constructed to reproduce eclipse event rates. The systematic
study of cloud events by Markowitz et al. (2014) found that the probability to
see a cloud event increases with the number of days between observations.
Figure 3 indicates detected events (error bars) and the sensitivity of their
survey based on their windowing function (steps), taken from their Figure 7.
The frequency of events in type 1 (cyan) and type 2 (orange) appear to
approximately follow a power law relation. To predict such observables from
our clumpy geometry, we assume for simplicity circular Keplerian orbits on
random planes of orientation. We set the orbital period of the innermost cloud
to one day, which for the outer-most clouds translates to a period of eight
years. The corresponding distances555The inner most cloud has a orbital radius
of $42\,\mathrm{au}\,\left(M_{\mathrm{BH}}/10^{7}M_{\odot}\right)^{1/3}$. are
consistent with observationally inferred distances (approximately light days,
see Markowitz et al., 2014, for a review of constraints). We then advance all
orbits by a time $\Delta t$ (x-axis) and determine what fraction of 20,000
random viewing sight-lines change column density significantly (for type-1:
unobscured become obscured, for type-2: obscured at least double the column;
obscured refers to the ${N_{\rm{H}}}=10^{22}\mathrm{cm}^{-2}$ threshold).
Because of the different sample selection, we do not assume the same $\sigma$
values as in Figure 2. Instead, we assume that type-2 detections are more
likely to be created in a more covered torus ($\sigma=40\text{ }$ gives the
best fit) than type-1 AGN ($\sigma=15\text{ }$). The slope and eclipse event
rates of type-1/type-2 AGN are reproduced, as Figure 3 demonstrates. When
reproducing the data of Markowitz et al. (2014) in Figure 3, we were free to
chose the clumpy model parameter $\sigma$ and the time normalisation. These
control the normalisation in the y and x-axis, respectively. The key result is
however that a powerlaw over sufficient orders of magnitudes is predicted,
with the correct slope. This result stems from the combination of our assumed
exponential angular size distribution and our wide range in distances for the
clouds ($Y=100$) leading to a wide range of orbital and eclipse durations.

A visualisation of our model is presented in Figure 4. This shows a cross-
section through the vertical mid-plane for the two models with angular widths
$\sigma=15\text{ }$ and $40\text{ }$. Table 2 in the appendix summarises the
assumptions and parameters of our clumpy model.

###  3.3 Final model geometry with optional inner ring

Figure 5: Compton-thick cloud ring around central point source (red). Here six
touching clouds are shown, with a second row of gap-filling clouds. This
configuration has a covering factor of $30\%$. Our model allows variations in
the covering factor between $0\%$ (no cloud ring) and $60\%$ (3 touching inner
clouds).

However, the clumpy model constructed above cannot fit some local heavily
obscured AGN. As discussed below and in more detail in a companion paper, a
highly covering, high-column density (${N_{\rm{H}}}\gtrsim
10^{25}\mathrm{cm}^{-2}$) reflector is necessary to reproduce the narrow high-
energy Compton hump of some local heavily obscured AGN, as traced by
$8-10-20\mathrm{keV}$ X-ray colours (see Buchner et al., in prep). Briefly
speaking, the clumpy structure is composed of a wide range of column
densities, together producing a broad Compton hump. See Appendix B for further
details regarding the spectrum of reflection off individual clouds, as well as
earlier work by Nandra & George (1994). Introducing an accretion disk that
provides reflection to the powerlaw spectrum proved ineffective, because both
the intrinsic and reflected emission is strongly attenuated by the torus.
Instead, a reflecting mirror is needed that has a virtually unobscured sight-
line both to the corona and the observer, while the sight-line between
observer and corona is obscured. To address this, we introduce into our clumpy
model an optional Compton-thick obscurer. While there are multiple equally
satisfactory geometries one could choose for this component, for computational
efficiency we compose it also of spheres. As illustrated in Figure 5, two
rings of Compton-thick clouds, just touching each other, form a thick donut.
The covering fraction, $C$, determines the radius and number of touching inner
clouds used (three to sixteen). _This component remains optional so that its
presence can be tested by parameter fitting._ The discussion section 6.2
covers physical interpretations of this component.

We compute our final, two-component clumpy model grid across two parameters:
(1) the dispersion of the cloud population, TORsigma $\sigma$, and (2) the
covering factor of the Compton-thick inner ring, CTKcover $C$, by changing the
number of clouds. For computational efficiency, and because the X-ray spectrum
remains largely uneffected by this choice, we reduce the number of clouds by a
factor of ten to ${\cal N}_{{\rm tot}}=10^{5}$ and proportionally increase
their angular area, setting $\theta^{{\rm cloud}}=\sqrt{10}\cdot 1\text{
}\approx 3\text{ }$. Column density distributions and cloud eclipse
predictions when this component is included are shown in Figure 13 in the
appendix. To achieve a smoother column density distribution we also altered
the cloud log-normal distribution to $N_{{\rm H}}^{{\rm cloud}}=10^{23.5\pm
1.0}{\rm cm}^{-2}$.

##  4 X-ray spectral model

Figure 6: Clumpy model spectra. Viewing angles (dashed & solid lines) and LOS
column densities (labelled at their respective turn-overs energies) gives
diversity in spectral shapes. The highest column density,
${N_{\rm{H}}}=10^{26}\mathrm{cm}^{-2}$ is shown in red. The lines present
typical face-on viewing angles (dotted lines, preferentially low column
densities, $0\text{ }$) and edge-on viewing angles (solid and dashed lines,
preferentially high column densities, $90\text{ }$). The dashed lines have the
inner ring described in 3.3 enabled.

###  4.1 Model spectrum

Figure 6 presents various spectra produced by one model geometry
$\sigma=30\text{ }$. We input a powerlaw source of photon index $\Gamma=2$
without a high-energy cut-off. Five LOS column densities are shown, as well as
two viewing angles. Towards the Compton-thick regime, a scattering component
appears and allows soft photons to escape. Higher column densities also create
a deeper trough at $\sim 10\mathrm{keV}$ relative to the Compton hump. The
trough also becomes deeper with $C$, as comparing solid and dashed black
curves reveals. In a companion paper, Buchner et al. (in prep.), we explore
this trough in detail with a $8-10-20\mathrm{keV}$ X-ray colour-colour
diagram. There we show that local Compton-thick AGN show remarkably flat
spectra at $8-10\mathrm{keV}$, similar to the dashed red curve in Figure 6,
which are difficult to reproduce by existing models. We demonstrate this below
in 5.

###  4.2 X-ray spectral model release

We release our spectral model, UxClumpy (Unified X-ray Clumpy model), as an
Xspec table model for download at https://doi.org/10.5281/zenodo.602282. The
parameters of the model are listed in Table 1.

The model can be used as follows in Xspec syntax:

⬇

model atable{uxclumpy.fits}

The model also allows for an exponential high-energy cut-off to more
realistically represent the X-ray continuum. This, and the addition of other
components, can make model fitting more degenerate and create multiple local
optima in parameter space. Therefore, we recommend to initially freeze Ecut to
the maximum value. Once a reasonable fit has been obtained, constraining Ecut
can be attempted. Unrealistic fits show low photon indices ($\Gamma<1.4$) and
low energy cut-offs to mimic a Compton hump.

Compared to previous spectral models (e.g. BNTORUS, MYTORUS), there are some
important differences:

(1) _The meaning of ${N_{\rm{H}}}$ and viewing angle_: In our model, the NH
fitting parameter is the LOS column density. A given geometry (TORsigma,
CTKcover) produces a diversity of LOS column densities. To understand the
implications of the NH and Theta_inc parameters better, a few technicalities
should be mentioned: For each simulated photon escaping to infinity, its
direction is noted. For the same direction from the X-ray point source, the
LOS column density is computed. The photon is then placed in a bin
corresponding to column density and viewing angle (0-30 : face-on, 30-60 :
intermediate, 60 -90 : edge-on). This binning scheme partitions the sky around
the corona by column density and viewing angle. In practice, this makes
spectral fitting easier, as the LOS column density is the primary influence on
spectral shape. Furthermore, physically meaningful fits of column density
variations over multiple observation epochs are possible: Unlike existing
models, it is not required that the geometry (or equatorial torus column
density) changes from source to source, or from observation to observation, or
that the viewing angle changes drastically. Instead, multiple LOS column
densities are possible at very similar viewing angles, allowing realistic
eclipse events by slight rotation self-consistently in our model. We recommend
that multi-epoch fits keep the geometry constant, but allow variations in NH.

Table 1: Clumpy model spectral parameters. Name |  Symbol | Range |  Description
---|---|---|---
PhoIndex |  $\Gamma$ | $1-3$ |  Photon index
Ecut |  $E_{\text{cut}}$ | $60-400$ |  Energy cut-off $[{\rm keV}]$
NH |  ${N_{\rm{H}}}(\text{LOS})$ | $10^{20-26}$ |  Total LOS column density $[{\rm cm}^{-2}]$
TORsigma |  $\sigma$ | $6-90\text{ }$ |  Cloud dispersion
CTKcover |  $C$ | $0-0.6$ |  Covering fraction of inner ring
Theta_inc |  $\theta_{\text{inc}}$ | $0-90\text{ }$ |  Viewing angle

(2) _Opening angle_ s: A key geometric parameter of torus models is the
opening angle, $\theta_{{\rm op}}$. In our model, the inner Compton-thick
covering, CTKcover, can be directly interpreted as a covering factor for that
component. The cloud population has however a fuzzy opening angle boundary,
which is defined by the angular width $\sigma$ from the plane where the LOS
number of clouds decrease to their $1\sigma$ value. The corresponding opening
angle depends on the considered column density and can be read off Figure 2.
Our angular width definition is consistent with the existing Clumpy model
definition (see 6.1). In the BNTORUS model, the opening angle
$\Theta_{\text{tor}}$ can change slightly the Compton hump of that model, and
this has been used in some studies to constrain the torus opening angle. In
our model, the CTKcover parameter provides a more powerful, geometrically
consistent way to probe the reflector geometry. Unfortunately, there exists no
simple geometry mapping between the models, as the best-matching photon index
is different at the corresponding LOS column densities.

###  4.3 Warm mirror component

Absorbed AGN spectra frequently show excess soft photons. This can be
interpreted as a warm mirror: hot electrons Compton scattering the intrinsic
powerlaw past the obscurer (e.g. Matt et al., 2000; Bianchi et al., 2006,
2010; Brightman et al., 2014). Frequently this component is simply modelled as
a powerlaw with the same photon index as the intrinsic powerlaw but a fraction
of the normalisation, following the Thomson scattering approximation. In
Xspec, this can be added with:

⬇

model atable{uxclumpy.fits}+const*zpower

The parameters of the additional powerlaw should be set to the same values as
that of the UxClumpy obscurer model.

However, if the warm mirror is volume-filling inter-clump hot gas or the
narrow-line region, the spectrum is probably more complex, combining warm (hot
gas) and cold reflection (dense clouds). As an improved approximation, we sum
the emission over all angles ($4\pi$) for each geometry. To first order, this
angle-averaged spectrum is a powerlaw, because it is dominated by photons
escaping without absorption. To second order, a mild Compton hump and Fe K
feature are present, as in the unobscured sight-lines in Figure 6. To
represent the warm mirror more accurately, we provide a second table model
that can be added instead of a simple powerlaw:

⬇

model atable{uxclumpy.fits} \+

const*atable{uxclumpy-scattered.fits}

All table parameters of the angle-averaged table model (uxclumpy-scattered,
warm reflected component) should be linked to their counterparts in the
obscurer model (uxclumpy, which includes transmitted and cold reflected
component). The relative normalisation constant should not exceed $\sim 10\%$
to avoid ambiguous powerlaw shapes and unrealistic scattering efficiencies.
Additional galactic and warm absorption layers can be added on top as usual.

Figure 7: _NuSTAR_ observations of _Circinus_ fitted with clumpy torus model
without inner ring component. For visualisation, the data are binned here to
1000 counts and deconvolved with the model. Additionally, a warm mirror
component (orange dotted line, see 4.3) and lines at rest-frame energy $6.5$
and $7.1{\rm keV}$ were added to the fit. The best fit has a LOS column
density of $N_{{\rm H}}=3\times 10^{24}{\rm cm}^{-2}$ and a photon index of
$\Gamma=1.69$. This model rises already at $\sim 8{\rm keV}$ and produces a
shallow Compton hump. In contrast, the data remain flat until $10{\rm keV}$
before rising steeply. This leads to substantial systematic residuals
(positive at $6\mathrm{keV}$, negative at $10\mathrm{keV}$, positive at
$20-30\mathrm{keV}$) in the _lower panel_.

Figure 8: _Left panels_ : Clumpy torus model fit of Circinus, NGC 424 and ESO
103-G035 _NuSTAR_ observations, respectively. In each case a reasonable fit
(black line) is obtained with our unified clumpy torus model including an
inner ring (see 3.3). We added a Thomson scattered component (see Section 4
for details), which contributes substantially in Circinus. _Right panels_ show
constraints on the geometry parameters of the inner ring (CTK covering factor,
y-axis) and the cloud population ($\sigma$, x-axis), within the allowed ranges
of our model. High Compton-thick covering factors are preferred for all three
sources.

##  5 Fitting local Compton-thick AGN

The nearby Seyfert 2 Circinus Galaxy (hereafter Circinus) is a prototypical
Compton-thick AGN, and due to its proximity has X-ray observations of the
nucleus with high signal-to-noise ratios and spatial resolution (see Arévalo
et al., 2014, for a recent analysis). The _NuSTAR_ spectrum was extracted
following standard procedures with the _NuSTAR_ pipeline (see Brightman et
al., 2015, for details).

First, we disable the inner ring component ($C=0$, see 3.3). Figure 7 presents
the best fit666We use the X-ray fitting tools Xspec (Arnaud, 1996), and Sherpa
(Freeman et al., 2001) with BXA (Buchner et al., 2014), bin spectra to 20
counts, and employ $\chi^{2}$ statistics. of this simple clumpy model. In the
_NuSTAR_ data the Compton hump begins to rise at relatively high energies
($\sim 10\,{\rm keV}$), with the model producing a shallower Compton hump.
This is caused by the diversity of clouds creating Compton reflection: In low
column density clouds, singly and doubly Compton scattered emission can
escape. High column density clouds lead to self-absorption and a trough at
$8-15\,{\rm keV}$. In Appendix B we discuss the reflection spectra arising
from single clouds in more detail. We find similar results with other Compton-
thick AGN (NGC 1068, NGC 424, NGC 3393), and discuss this problem in more
detail in a companion paper, using a X-ray colour-colour diagnostic to
characterise Compton hump shapes. The spectral shape mismatch results in a
poor fit ($\chi^{2}/\mathrm{dof}=2774/1703\approx 1.6$). A “pure” high-column
density reflector dominating over all contamination by lower column density
reflectors (the majority of our current cloud population) is needed. We
anticipated the need for such a component in 3.3. To reduce strong line
residuals distracting from the main point of Figure 7, we included additional
gaussian lines. While our model already includes line emission, extended line
emission from beyond the torus ($\gg$10pc) is known in this source (Arévalo et
al., 2014). Our results and conclusions are consistent with and without these
additional lines.

We now refit the data, with the inner ring component enabled. The fit is much
better ($\chi^{2}/\mathrm{dof}=1765/1714\approx 1.0$, top left panel in Figure
8), with the residuals (lower panel) not showing systematic deviations. The
best-fit has a photon index of $\Gamma=2.11$, assuming no energy cut-off,
behind a line-of-sight column density of ${N_{\rm{H}}}(\text{LOS})=4\times
10^{24}{\rm cm}^{-2}$. Because the NuSTAR extraction region exhibits
substantial contamination by other X-ray sources (Arévalo et al., 2014), we
now focus on fitting only the $\geq 8\,{\rm keV}$ energy range to derive
physical properties of the reflector. In this case, we find a photon index of
$\Gamma=\operatorname{1.81\pm 0.10}$, behind a line-of-sight column density of
${N_{\rm{H}}}(\text{LOS})=\operatorname{31_{-16}^{+25}\times{10}^{24}}{\rm
cm}^{-2}$. The high-energy cutoff uncertainties, $E_{cut}=\operatorname{209\pm
69}$, are degenerate with those of $\Gamma$. Importantly, we are able to
constrain the geometry of the obscurer (contours in top right panel of Figure
8). The clumpy component has a covering angle $\sigma$ below $45\text{ }$, and
the Compton-thick covering factor of the inner ring is $C>40\%$, that is, the
corona is extensively covered by Compton-thick material.

To study the soft energy AGN emission as well, we consider NGC 424. Other
nearby Compton-thick AGN (such as Circinus) are often contaminated by nearby
sources in the host galaxy, such as supernova remnants, ultraluminous X-ray
sources and X-ray binaries. NGC 424 is much more luminous than Circinus and
therefore its nuclear emission dominates over such host galaxy contaminants.
The soft energies show no contamination in _XMM-Newton_ observations down to
energies of $\sim 3{\rm keV}$ (Marinucci et al., 2011). The middle left panel
of Figure 8 presents our best fit to _NuSTAR_ spectra in the $3-65{\rm keV}$
range. The fit is of good quality ($\chi^{2}/\mathrm{dof}=84/79\approx 1.1$)
and gives a photon index of $\Gamma=\operatorname{2.29\pm 0.14}$ without the
need for a high-energy cutoff (parameter unconstrained). Such a high photon
index is expected from this high-Eddington accretion rate source
($L_{\text{bol}}/L_{\text{Edd}}\approx 0.1$, Bian & Gu, 2007; Brightman et
al., 2013, 2017). This X-ray emitter is behind a line-of-sight column density
of ${N_{\rm{H}}}(\text{LOS})=\operatorname{26_{-20}^{+25}\times{10}^{24}}{\rm
cm}^{-2}$. We are able to constrain the geometry of the obscurer (contours in
the centre right panel of Figure 8): The clumpy component has a covering angle
$\sigma<60\text{ }$, and the Compton-thick covering factor of the inner ring
is $C>30\%$.

We further consider ESO 103-G035, a Compton-thin source. The _NuSTAR_ spectrum
of this source is soft without strong obscuration or host contamination,
therefore we consider the entire _NuSTAR_ spectrum ($3-70\,{\rm keV}$). The
bottom left panel of Figure 8 shows our best fit for this source, which is
again of good quality ($\chi^{2}/\mathrm{dof}=978.58/1020\approx 0.96$) and
gives a photon index of $\Gamma=\operatorname{1.84\pm 0.04}$ with a high-
energy cutoff $E_{cut}=\operatorname{209\pm 69}$. This X-ray emitter is behind
a line-of-sight column density of
${N_{\rm{H}}}(\text{LOS})=\operatorname{(1.6\pm 0.1)\times{10}^{23}}{\rm
cm}^{-2}$. We are also able to constrain the geometry of the obscurer in this
case (contours in the bottom right panel of Figure 8): The clumpy component
has a covering angle $\sigma>60\text{ }$ under a face-on viewing angle and the
Compton-thick covering factor of the inner ring is $C=\operatorname{28\pm
7}\%$.

All three spectra prefer large covering fractions of the high-column density
inner ring reflector. In Compton-thick AGN the cloud population, by means of
low $\sigma$ values, tends to be hidden away in the shadow of this inner ring,
avoiding low-column density reflection. Indeed, the emerging spectrum and fits
change insignificantly when the torus clumps are removed ($\sigma=0$). In
Compton-thin AGN, the cloud population provides the Compton-thin line-of-sight
absorber, while the inner ring is again significantly needed as a Compton-
thick reflector.

##  6 Discussion

We have presented a clumpy torus model geometry, UxClumpy, from which we
simulate X-ray spectra. UxClumpy provides (1) a wide coverage of energies
($0.1-1000\mathrm{keV}$), (2) a wide range of LOS column densities
($10^{20-26}\mathrm{cm}^{-2}$), (3) a variable energy cut-off and (4)
fluorescent emission lines self-consistently. Additionally, the clumps in our
geometry reproduce the frequency, duration and sizes of eclipse events and
X-ray eclipse event observations can be modelled self-consistently without
requiring a drastic change in the viewing angle.

The spectral shapes around the Compton hump and trough above the Fe edge
differ for the various geometries and viewing angles, and are diverse (see
Figure 6). We additionally compute a better approximation of the warm mirror
component. Overall however UxClumpy produces spectral shapes similar to the
BNTORUS model, which has been found to fit observations well. The added
benefit of UxClumpy is the ability to infer novel physical characteristics
from the spectral fits and consider the energy cut-off. Below we discuss the
multi-wavelength spectrum of our model and interpret our inner ring component.

###  6.1 Multi-wavelength Spectrum

Figure 9: Multiwavelength spectrum of our unified clumpy geometry. The clumpy torus with $\sigma=30\text{ }$ (left panel of 4) is seen through a obscured, Compton-thick line-of-sight either edge-on (red) or face-on (blue). Dashed lines indicate the case of a unobscured view of the accretion disk.  Table 2: Model parameters. Description | Distribution | Parameter | Value | Determined from
---|---|---|---|---
Total number of clouds | fixed | ${\cal N}_{{\rm tot}}$ | $10^{5}$ | Column density distribution
Vertical distribution | Gaussian | $m$ | 2 | Column density distribution
Vertical dispersion | variable | $\sigma$ | 6 - 90 | Column density distribution
Radial distribution | uniform | $q$ | 0 | Assumed
Radial extent | fixed | $Y$ | 100 (X-ray), 10 (IR) | Assumed
Cloud sizes | Exponential | $\theta^{{\rm cloud}}$ | $1\text{ }$ | Eclipsing events
Column density of clouds | Log-Normal | $N_{{\rm H}}^{{\rm cloud}}$ | $10^{23.5\pm 1.0}$ | Column density distribution
Number of clouds (plane) | (from above) | ${\cal N}_{0}$ | $2-9$ | Computed
Volume filling factor | (from above) | $f_{V}$ | $2-5\%$ | Computed
Cloud mean optical depth | (from above) | $\tau_{V}$ | $150$ | Assuming Galactic relation

Infrared and X-ray data can be self-consistently analysed under our clumpy
geometry model, by construction. The emerging infrared model was computed
using CLUMPY (Nenkova et al., 2008a, b) and is depicted in Figure 9. CLUMPY
performs probabilistic ultraviolet-optical-infrared radiative transfer. This
approximation requires that the volume filling factor is small ($<10\%$) and
the cloud sizes are small, which is the case given our assumed small angular
sizes.

Figure 10: Multiwavelength fit of our unified clumpy geometry to data of NGC
1365. Infrared photometry on the left overlayed with two CLUMPY fit solutions,
and spectra from three _NuSTAR_ observations are shown on the right with
UxClumpy fits. Insets show parameter probability distributions for each fitted
dataset in corresponding colours. The infrared fit solution shown in brown is
overall compatible with the X-ray fits (blue, green, orange).

Table 2 presents the model parameters and assumptions that went into
constructing our clumpy cloud population. For a self-consistent physical model
in the infrared, the Clumpy torus parameters should be set as follows (see
Table 2): $Y\approx 10-20$ ($<100$), $\sigma={\rm TORsigma}$, $N_{0}=2-5$
(depending mildly on $\sigma$), $q=0$, and $\tau_{{\rm V}}\approx 120-160$.

Figure 9 presents the X-ray model spectrum with the reprocessed infrared
emission from the . We normalised at $12{\rm\mu m}$ and use the empirical
ratio of 1.7 between the $12{\rm\mu m}$ and intrinsic $2-10\,{\rm keV}$
luminosity (e.g. Asmus et al., 2015). In the optical and ultraviolet the
intrinsic AGN disk emission dominates along unobscured sight-lines (dashed
lines in Figure 9). At long wavelengths, the reprocessed emission is gray-
body-like and dominated by the coldest dust regions. In the mid-infrared,
self-absorption in edge-on systems hides the hot inner dust emission, while
face-on views expose a range of temperatures. The silicate feature is visible
around $10$ and $18{\rm\mu m}$. In the X-ray regime very energetic photons can
penetrate the obscurer and escape. Between $\sim 0.2-1\mathrm{nm}$ a tail of
soft photons circumvents the line-of-sight obscurer by being reflected off
distant material. In addition to the continuuum, K shell fluorescence features
are clearly visible, the most prominent being Fe K.

We now attempt a multi-wavelength fit of the obscurer from X-rays to infrared.
To illustrate the clumpy obscurer, we select NGC 1365, notorious for changing
its LOS column density several times over time-scales of days to years (e.g.,
Risaliti et al., 2005, 2007; Rivers et al., 2015). A successful model should
allow the obscurer geometry and inclination to stay the same with only small
changes in the sight-line to the corona. Simultaneously, we expect the same
geometry to also fit the infrared photometry. Archival photometric data
(McAlary et al., 1983; Skrutskie et al., 2006; Ramos Almeida et al., 2009;
Asmus et al., 2014, we imposed a 10% minimum flux uncertainty) and _NuSTAR_
spectra (Rivers et al., 2015, ObsIDs 60002046005,7,9; 5-77 keV) are presented
in Figure 10. In the three observations (blue, orange, green) in the left of
Figure 10, the spectra differ noticibly in shape. In the insets, the UxClumpy
parameter probability distributions of a fit to each spectrum are shown with
corresponding colours. Fitting the infrared photometry with CLUMPY leads to
two solutions, shown in brown dashed and purple dot-dashed curves. The
parameter distributions are again shown in the insets. We compare the solution
shown in brown, with a face-on inclination $i<20{{}^{\circ}}$, a substantial
torus thickness $\sigma>30{{}^{\circ}}$ (see central insets of Figure 10) to
the constraints from the X-ray fits (or assumed UxClumpy parameters shown in
the right insets). Overall, the derived parameters are compatible. This allows
the interpretation that a clumpy structure surrounds NGC 1365 whose inner-most
clouds vary the X-ray column density and whose outer, dusty clouds irradiate
in the infrared. A single geometry, with the same inclination angle was able
to simultaneously reproduce column density variations over time and the broad-
band X-ray to infrared spectral energy distribution. However, the presented
fit does not model all of the complexities present in the data. For example,
the least obscured spectrum (orange) shows complex, warm absorption (see
Rivers et al., 2015). Additionally, in other variable-obscuration sources we
find that a multi-wavelength fit gives incompatible geometry constraints,
perhaps indicating that the structures differ. This indicates that further
modelling is needed and will be addressed in future work.

###  6.2 Interpretation of the inner ring component

When the inner ring component is present, the X-ray spectrum around the
Compton hump is similar to pure cold reflection (e.g. off an infinitely thick
slab, or a smooth donut). In the literature such a bulged component has been
proposed and interpreted for a variety of reasons. It could be a Compton-thick
cloud wind launch site (Krolik & Begelman, 1988), an inner wall (Lightman &
White, 1988) or a puffed up inner rim of a hot disk as seen in proto-planetary
disks (Dullemond et al., 2001; Dullemond & Monnier, 2010). The spectral shapes
are also similar to the spectrum of a warped disk (see Buchner et al., in
prep), and can also be interpreted as one. This is probably the most suitable
interpretation for Circinus, where an edge-on maser disk is seen (Greenhill et
al., 2003). In the companion paper Buchner et al. (in prep), a warped disk
model provides the best fit.

For compatibility with CLUMPY IR models, we can assume that the inner Compton-
thick reflector is dust-free and does not contribute significantly to the
infrared spectrum (see simulations of Gallagher et al., 2015, for a component
of similar column density). In this case, if it is interpreted as part of a
dust-free broad emission line region, very smooth line profiles do not
necessitate that it has further discrete clumpy substructure (e.g., Laor et
al., 2006). Alternatively, a similar inner ring component was introduced by
Pier & Krolik (1993) to explain the observed, hot ($\sim 1300{\rm K}$) dust
component (see Netzer, 2015, for a review); see also Kishimoto et al. (2011)
and Hönig et al. (2012) who considered transferring the “puffed-up inner rim”
model from proto-planetary disks (Dullemond et al., 2001; Dullemond & Monnier,
2010) to AGN.

Recent interferometric observations in the mid-infrared indicate that the dust
emission is in several sources primarily polar with respect to the disk (Asmus
et al., 2016; Tristram et al., 2014). Incorporation of these results goes
beyond this work. However, clumpy geometries that address these can be
calculated rapidly with our XARS code.

##  7 Conclusion

We constructed a clumpy model for the obscurer of AGN which reproduces 1) the
column density distribution of the AGN population and 2) observed cloud
eclipse events. However, when applying the clumpy model to several nearby,
heavily obscured AGN, we find differences between the observed _NuSTAR_
spectra and the model, in particular in the energy region between the Fe K
edge and the Compton hump. To resolve this, it is necessary to insert close to
the corona a Compton-thick reflector with a substantial covering factor. This
new inner component could be interpreted as an inner rim, or a warped or
bulged disk. It cannot be attributed to reflection off a simple disk because
the reflector must simultaneously block the line of sight to the central
engine and reflect it.

The need for the inner Compton-thick reflector (from X-ray spectral fits) and
the need for the clumpy component (from eclipses and infrared observations)
together could imply three classes of AGN of increasing inclination angle: 1)
Unobscured; 2) Obscured by clumps; and 3) Heavily obscured by the inner
reflector. The former two can have strong Compton reflection from the inner
ring and show eclipses. The latter is reflection-dominated and may produce
Compton-thick column density variations (as seen for instance in NGC 1068,
Marinucci et al., 2016).

We have developed a powerful Monte Carlo X-ray radiative transfer code, XARS,
which can deal with arbitrary geometries. We release this open source software
at https://github.com/JohannesBuchner/xars.

We release our UxClumpy model as Xspec tables available at
https://doi.org/10.5281/zenodo.602282. The spectral model is similar in its
ease of use, parameters and spectral shape as the popular BNTORUS model (see
4). It also allows a physical interpretation of the underlying geometry
including covering factors and enables self-consistent analysis of variable
column densities (see also Simm et al., 2018). We also note that our geometry
has corresponding CLUMPY infrared spectra777Available at
https://www.clumpy.org, paving the way for self-consistent multi-wavelength
analyses.

##  8 Acknowledgements

We thank the anonymous referee for careful reading of the manuscript and their
insightful comments. JB thanks Marc Schartmann and Leonard Burtscher for
discussions on tori in the infrared. JB thanks Eva Lefa for sharing her Monte
Carlo code to verify our Compton scattering implementation. JB thanks Mislav
Baloković for insightful conversations.

We acknowledge support from the CONICYT-Chile grants Basal-CATA PFB-06/2007 &
AFB-170002 (JB, FEB), FONDECYT Regular 1141218 (FEB), FONDECYT Postdoctorados
3160439 (JB) and the Ministry of Economy, Development, and Tourism’s
Millennium Science Initiative through grant IC120009, awarded to The
Millennium Institute of Astrophysics, MAS (JB, FEB). This research was
supported by the DFG cluster of excellence “Origin and Structure of the
Universe”.

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##  Appendix A Green function computation

We briefly outline how XARS computes the response spectrum to an input energy
as seen from a distant observer. The user provides a function that describes
the geometry, and a function that specifies how emerging photons are gridded
(e.g. in viewing angle).

Given an input energy, a set of photons is created, typically originating at
the geometry origin and directed at all angles uniformly. Then these photons
are pumped through the geometry as follows.

The distance traveled by the photon is calculated following an exponential
distribution with expectation unity in units of optical depth. The optical
depth is defined as the cross-section to both photo-electric absorption and
Compton scattering. Because the energies are sufficiently high, electrons
appear unbound during scattering, allowing the Klein-Nishina approximation to
within a small error (Liu & Li 2014). With the distance in units of
${N_{\rm{H}}}$ to travel, the geometry user function is called with the
current position, direction and distance to travel. It computes the
coordinates of the end point. This is achieved by integrating from the start
point in the direction until the ${N_{\rm{H}}}$ value has been accumulated. In
a simple geometry such as a single unit sphere, the end point is the direction
times distance from the current position, where the distance is related to the
ratio of ${N_{\rm{H}}}$ to the radial column density through the sphere. In
more complex geometries (e.g. many spheres), gaps between objects have to be
considered. In any case, the user function returns both the end point and
whether the photon has escaped to infinity. After this propagation, all
photons still in the medium are interacting with it. Proportional to the
relative photo-electric absorption to Compton scattering cross-section at the
current energy, the photon is randomly either absorbed or Compton scattered.
Compton scattering follows standard formulae (see Rybicki & Lightman 1986;
Brightman & Nandra 2011a; Paltani & Ricci 2017) and assigns a new energy and
direction. In the case of absorption, there is a chance for the photon to be
re-emitted as a fluorescent line. The ratio of the line yield multiplied with
abundance to the total absorption cross-section of that element gives the
relative probability of line emission (we use the energies and line yields
specified in Brightman & Nandra 2011a). Based on these proportionalities, we
distribute the photon to one of the fluorescence lines, with a new randomly
drawn direction. In either case (Compton scattering, fluorescent emission),
the pumping repeats from the start. If the photon however left the medium, the
photon energy, location and direction are passed to the second user function
to accumulate the Green function response to the current input energy.

In each pumping step, millions of photons can be processed simultaneously
through the use of numpy arrays, making XARS efficient even though it is
Python-based.

##  Appendix B Irradiation of a single blob under various density profiles

Figure 11: Blob irradiation geometry. A isotropic source irradiates the blob
from a certain distance, and the spectrum of all reflected photons is assessed
in Figure 12.

Figure 12: Reflection off blobs with various profiles and column densities.
For comparison, reflection off a semi-infinite disk is shown in all panels
(orange curve: 50 inclination angle; red curve: angle-averaged, our code).

In this work we have approximated clumps as spheres filled with uniform
density profiles. In this section we determine if more realistic profiles
without sharp cut-offs (e.g. polytropes) would yield different results. In
particular we are interested in the shape of the Compton reflection, and if
varying the density profile could alleviate the need for our inner ring
component.

Our test geometry is simple and shown in Figure 11: X-rays from an isotropic
source irradiate a single, isolated blob. We adopt (1) a uniform density
profile, (2) a Gaussian profile and (3) an exponential profile. For the last
two we require that the density at the border is $10^{-6}$ of the centre
density. The profiles are normalised so that maximum column density is
${N_{\rm{H}}}=10^{23,24,25,26}/{\rm cm}^{2}$ (i.e. through the centre). Any
reflected/reprocessed X-rays escaping to infinity, from any angle, are
collected and their X-ray spectrum determined.

The reflection spectra are shown in Figure 12, with the panels corresponding
to the three profiles. In each panel we also include reflection off a plane-
parallel semi-infinite disk under a 50 inclination angle (orange line). This
is taken from the pexmon spectrum of (Magdziarz & Zdziarski 1995; George &
Fabian 1991; Nandra et al. 2007), and comparable to the angle-averaged
spectrum produced by our XARS code of the same geometry (red line). In the top
panel, reflection off high-column density uniform blobs (black lines)
resembles the disk reflection. At lower column densities (grey) the spectrum
peaks at lower energies. The same results were obtained in Nandra & George
(1994). Varying the density profile yields significantly different shapes, as
shown in the other panels of Figure 12 for Gaussian and exponential profiles.
The Compton hump is much weaker, and the spectrum flatter towards low
energies. This is not just because these blobs have a lower mass: high-column
density results differ throughout also from results of lower column density
uniform spheres. The location where reflection primarily occurs is at the
surface of uniform blobs, but deeper inside for the other profiles. In the
latter, a photon can pass into the blob further, and after scattering may
escape off the more common lower column density routes. A single scattering is
more likely to put the flight direction of a photon away from the centre than
towards it. In the Gaussian/exponential profiles, this decreases the effective
column density after each scatter, and subsequently allows high-energy photons
to pass through much more easily than in a uniform density profile.

The spectral shape of _NuSTAR_ data studied in this work is similar to the
black lines in the left panel, that is, reflection dominated with a strong
Compton hump. The addition of the torus ring component emphasises reflection
off high-column clouds and surpresses the reflection off low-density clouds
(grey in left panel). As Figure 12 shows, introducing Gaussian/exponential
profile clumps would worsen the situation (as their spectra peak at lower
energies) and would under-predict the Compton hump.

##  Appendix C Clumpy model distributions

Figure 13: Column density distribution and associated covering fractions
produced by our clumpy models including the inner ring component. Data points
(as in Figure 2) represent intrinsic AGN obscured fractions at different
luminosities. Compared to Figure 2, the Compton-thick column densities are
more frequent because of the inner Compton-thick ring. Figure 14: Eclipse
rates predicted when including the inner ring component. Curves as in Figure
3, with orange and cyan error bars and curves representing eclipse
measurements by Markowitz et al. (2014). Now we show UxClumpy with
$\sigma=15\text{ }$ and $\sigma=35\text{ }$ to reproduce Sy1 and Sy2,
respectively.

Figure 13 presents the column density distribution of the final model. Similar
to Figure 2, a wide range of column densities are produced by each geometry.
Now, extremely thick column densities are more common.

Figure 14 presents the frequency of eclipse of the final model. Similar to
Figure 3, observed cloud sizes and eclipse frequencies can be reproduced,
albeit with different model parameters. This is because the inner ring
component shades clouds below its covering factor, requiring higher $\sigma$
values.