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

TL;DR

This paper introduces a new unification model for the clumpy obscurer in active galactic nuclei, using Monte Carlo simulations to reproduce X-ray spectra and incorporating a Compton-thick reflector to fit NuSTAR data.

Contribution

The authors developed a generalized Monte Carlo X-ray radiative transfer code and a unification model that accounts for complex scattering and reflection in AGN obscurers, enabling better spectral fits.

Findings

Model spectra resemble the TORUS model due to strong Compton scattering.

Adding a Compton-thick reflector improves fits to NuSTAR spectra.

The model includes self-consistent infrared spectra for multi-wavelength analysis.

Abstract

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.

Tables

Table 1. 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 $\left[\mathrm{keV}\right]$
NH	$N_H\left(\text{LOS}\right)$	${10}^{20-26}$	Total LOS column density $\left[{\mathrm{cm}}^{-2}\right]$
TORsigma	$\sigma$	$6-90$	Cloud dispersion
CTKcover	$C$	$0-0.6$	Covering fraction of inner ring
Theta_inc	${\theta }_{\text{inc}}$	$0-90$	Viewing angle

Table 2. Table 2: Model parameters.

Description	Distribution	Parameter	Value	Determined from
Total number of clouds	fixed	${𝒩}_{\mathrm{tot}}$	${10}^5$	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	${\theta }^{\mathrm{cloud}}$	$1$	Eclipsing events
Column density of clouds	Log-Normal	$N_H^{\mathrm{cloud}}$	${10}^{23.5\pm 1.0}$	Column density distribution
Number of clouds (plane)	(from above)	${𝒩}_0$	$2-9$	Computed
Volume filling factor	(from above)	$f_V$	$2-5\%$	Computed
Cloud mean optical depth	(from above)	${\tau }_V$	$150$	Assuming Galactic relation

Equations

$$\mathcal{N}={\mathcal{N}}_0\cdot \exp \left\{-{\left(\frac{\beta }{\sigma }\right)}^m\right\}$$

Full text

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 johannes.buchner.acad@gmx.com1122   

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.

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

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:

[TABLE]

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.

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

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

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.

(2) Opening angles: 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.

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

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.

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”.

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

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 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.