Issue 
A&A
Volume 690, October 2024



Article Number  A175  
Number of page(s)  20  
Section  Extragalactic astronomy  
DOI  https://doi.org/10.1051/00046361/202449686  
Published online  07 October 2024 
Xray occultations in active galactic nuclei: Physical properties of eclipsing clouds as part of the broadline region cloud ensemble
^{1}
Dipartimento di Fisica e Astronomia, Università di Firenze, Via G. Sansone 1, I50019 Sesto Fiorentino, Firenze, Italy
^{2}
INAF – Osservatorio Astrofisico di Arcetri, L.go E. Fermi 5, Firenze, Italy
Received:
21
February
2024
Accepted:
23
July
2024
Context. Shortterm Xray variability in active galactic nuclei (AGNs) can be explained as being due to varying Xray absorption induced by the temporary occultation of the primary Xray source, when moving absorbing clouds cross the line of sight to the Xray source itself. Earlier work suggests that these absorbing clouds have physical properties similar to those of broadline region (BLR) emitting clouds and are located in the same spatial region.
Aims. We intend to extract physical information on each individual absorber associated with any given occultation event detected in our sample and to analyse general properties of the cloud ensemble whose components can produce Xray eclipses.
Methods. From the analysis of previously detected occultation events, two ‘observables’ characterising each single occultation event can be derived: the peak fractional hardness ratio variation (ΔHR/HR) and the duration of the event normalised by a characteristic eclipse timescale evaluated for each AGN source. To determine the eclipsing cloud properties, we devised a procedure a) based on simplifying assumptions on the geometry of both the Xray source and the cloudlike gas condensations, and on the cloud Xray absorbing properties in the energy range of interest (2–10 keV), and b) relying on a set of simulated instrumental responses from both XMM and Suzaku relevant instruments to different incoming Xray spectra, absorbed with varying absorber column density and maximum covering factor during the occultation. Thus, we derived information on the individual cloud producing any given occultation event, determining the cloud radius normalised to that of the Xray source, the spatial location of the cloud, and an estimate of the cloud gas number density for reasonable values of the equivalent absorber column density.
Results. The physical properties of eclipsing clouds that we obtained are consistent with those of BLR clouds. We can exclude the dominance, in the ensemble cloud size distribution, of clouds larger than about 10–12 times the Schwarzschild radius characterising each AGN, and we do not find any significant dependence of the cloud physical size on the distance from the central black hole, in agreement with the results of our previous work. As for the number density of these gas condensations, with our procedure we obtained values within a range of ∼10^{9} − 10^{11} cm^{−3}, which is consistent with the estimates derived from broad emission line analysis.
Key words: galaxies: active / galaxies: Seyfert / Xrays: galaxies
© The Authors 2024
Open Access article, published by EDP Sciences, under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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1. Introduction
Significant Xray variability is a common property of active galactic nuclei (AGNs). It is generally accepted that AGN emission from the innermost regions is processed and selectively absorbed by intervening material in the nuclear and circumnuclear region. The multiwavelength analysis of the observed radiation allows one to picture a general structure of AGNs in which the inner region is surrounded by somewhat toroidal absorbers of a clumpy nature, whose specific physical and geometrical properties are still a matter of study and debate. These absorbers particularly affect Xray radiation and the analysis of Xray variability can be turned into an analysis of the absorber properties and their possible variations.
In the past, the detailed investigation of a small number of individual sources led to the idea that shorttime Xray variability can be interpreted in terms of varying absorption due to the passage of an absorbing clumplike (or cloudlike) structure of gaseous material intersecting the line of sight to the central primary Xray source, resulting in a temporary ‘occultation’ of the primary Xray source itself. From these investigations (Risaliti et al. 2007, 2009, 2011; Maiolino et al. 2010; Nardini & Risaliti 2011; Bianchi et al. 2009; Sanfrutos et al. 2013), a general scenario in which the clumpy absorbers re identified with broadline region (BLR)like clouds crossing the line of sight to the Xray source, and thus temporarily ‘eclipsing’ the Xray source itself, has been proposed. In this framework, these occultations are the origin of the observed changes in the Xray spectrum, causing its shortterm variability. Following this general interpretation, in TorricelliCiamponi et al. (2014, hereafter referred to as Paper I) we started developing a method of analysing observed Xray hardnessratio light curves that allows one to detect reliable and statistically significant candidate occultations, and we applied it to a representative sample of about 40 AGNs, selected as the brightest sources with ‘long’ XMMNewton and/or Suzaku observations. With this method, we identified a total of 65 reliable occultation events in 14 of the AGN sources in the sample.
In a second piece of work (Pietrini et al. 2019, from now on Paper II), we analysed the consequences of coupling the constraints on properties of the broadline emitting gas, inferred from spectroscopic observations in the framework of a BLR consisting of an ensemble of cloudlike components, with the constraints defined by the global analysis of the Xray source occultations detected in Paper I. The purposes of the Paper II analysis were to define the viability of the proposed scenario and to assess global properties of an orbiting cloud ensemble satisfying both types of constraints mentioned above.
In the present paper, we intend to fully exploit the capabilities of the method and the amount of information gained from Paper I analysis and from the results of Paper II regarding the properties of the distribution in size for clouds belonging to the ensemble orbiting in the inner region of the AGN. In fact, we want to find a relation between the measured observable quantities for each detected eclipse event and the eclipsing cloud properties, in order to obtain a better and more detailed description of clouds in the central region.
A huge amount of work has been devoted to the study of the broad emission lines observed in AGN spectra. These characteristic spectral features arise from photoionised gas moving in the inner region of the AGN, subject to the gravitational potential of the central black hole (Wang et al. 2017). However, the physical and dynamical conditions as well as the spatial distribution of the gas contributing to the line emission are still not fully understood (Ruff et al. 2012; Raimundo et al. 2020). Different theoretical models of the BLR have been proposed, mainly divided into two different general families: models based on a continuous distribution of emitting gas (outflows, winds or outer regions of the accretion disc) on the one hand and models based on the distribution of individual cloudlike gas condensations orbiting the black hole potential on the other. Both types of models give positive answers to some of the open questions about the BLR, but at the same time fail to meet other requirements (see, for instance, Czerny 2019).
However, a BLR composed of an ensemble of individual gas clumps can give an explanation to the shorttime variability of AGN Xray emission in terms of temporary occultation of the Xray source produced by a rather small cloud crossing the line of sight to the source itself.
In the framework of a BLR composed of a large number of small gas condensations (‘clouds’) orbiting around the central black hole, a successful approach to the explanation of broadline observations is that of the ‘LOC’ (Locally Optimally emitting Clouds) model, introduced by Baldwin et al. (1995) (see Ruff et al. 2012; SchnorrMüller et al. 2016) and subsequently used by several authors (see, for instance, Bottorff et al. 2002; Korista & Goad 2000, 2004). Within the LOC model approach, the typical range of gas number density for the gas in clouds can be inferred as 10^{8} cm^{−3} < n_{H} < 10^{12} cm^{−3} (see also SchnorrMüller et al. 2016; Netzer 2008, 2013). More recent analyses and simulations of line emission (especially based on 2017 CLOUDY version, Ferland et al. 2017) confirm a range of gas number density, n_{H}, with values in any case between 10^{9} and 10^{12} cm^{−3} (Panda 2020; Marziani et al. 2010), and a maximum range for the total column density of BLR individual clouds of around 10^{22} − 10^{24} cm^{−2}.
As a matter of fact, it seems very difficult to further improve information on cloud physical properties using these types of line analysis, because any inference about physical quantities’ values is sort of averaged over a very large number of individual gas condensations. On the contrary, within the interpretative framework of shortterm Xray variability that we adopt, in our present approach it is indeed possible to analyse the physical properties of an individual gas condensation, even though this is of course feasible only for a limited number of cases, corresponding to the eclipse events detected and identified. Therefore, assuming that it is correct to interpret AGN shortterm Xray variability in terms of temporary occultations by clouds crossing the line of sight to the Xray source, while moving in the gravitational potential of the central black hole, our present method of analysis, coupled with our previous work (Paper I and Paper II), may give interesting and significant information on individual gas condensations. Also, we can show that indeed gas condensations responsible for Xray absorption are consistent with the typical conditions inferred for BLR clouds for the vast majority of our sample of detected shorttime Xray absorption events.
To this purpose, our work is presented as follows. In Sect. 2, we recall the main features of the sample of Xray light curves we examined in Paper I. We also identify the two main ‘observables’ that can be extracted from each eclipse event analysis; namely, the fractional peak hardness ratio variation, ΔHR/HR, and the event duration, τ_{0}. In Paper I no attention had been devoted to the eclipse duration, but in Sect. 3 we analyse in detail its relevance, by introducing a simple representative geometry for the occultation event, together with a few simplifying but descriptive assumptions about the clouds in the hypothesis that they do belong to a population including BLR clouds as well. Section 3 also includes a brief discussion of the fractional hardness ratio variation as a function of the eclipse covering factor and of the Xray absorbing column density, as it stems from our analysis of synthetic light curves (simulated observations) with XMM and Suzaku. The relations determined in Sect. 3 represent useful tools for the interpretation of each detected event. The procedure adopted for obtaining information on the cloud occulting the Xray source in each of our observed eclipse events is then described in Sect. 4. In Sect. 5, we discuss the ranges of cloud physical parameters derived from the analysis of all the events detected in our sample, taking also the information on cloud physical size inferred from Paper II analysis into account. In the final part of this section, we also compare the results obtained with our procedure for two of Mrk 766 lightcurves with the outcome of a spectral analysis of the same lightcurves performed by Risaliti et al. (2011), obtaining good agreement. Our conclusions are presented in Sect. 6.
2. The sample and its analysis
We refer to Paper I (Sect. 3) for the details of the source sample definition, of the choice of the energy bands for the hardness ratio (HR), and, especially, of our method of analysis. Here, we just give a summary of the main points that are specific to our method (see also Appendix A).
In Paper I, for a sample of type1 to Comptonthin type2 AGNs consisting of about 40 of the brightest Xray sources with long archival XMMNewton and Suzaku observations, we analysed the hardness ratio (HR) light curves, where HR ≡ F(5 − 10 keV)/F(2 − 4 keV) (see Sect. 2 of Paper I for details).
It is important to remark that, as it is discussed in Sect. 2 of Paper I, we chose to restrict our analysis in the 2 − 10 keV energy range, and therefore our search is sensitive only to occultations by clouds with a column density in the range of ∼10^{23} − 10^{24} cm^{−2}. Also, we have limitations on the temporal range of the detectable occultations and our search is sensitive to durations between ∼10 ks, in order to have enough statistics, and ∼1 day, to be able to isolate and properly identify an HR variation event within a single observation. Thus, these conditions are limiting our analysis to a subsample of possible absorbers producing a temporary occultation of the Xray source. Of course, this does not exclude the existence of temporarily obscuring absorbers characterised by larger column densities and/or longer timescales; our search is just not sensitive to them, and thus we cannot discuss them properly on the basis of our observed occultation events.
The events considered here are those for which a) the HR variations were found to be statistically significant (with a probability of random variability < 0.001) and b) the peak fractional hardness ratio variation (ΔHR/HR) of the event is ΔHR/HR > 0.1, since a statistically significant event can be unambiguously interpreted as an eclipse event only if this latter condition is satisfied, as we discussed in Paper I (Sect. 4).
Our analysis will be based on two observational quantities: the fractional HR variation, ΔHR/HR, of a given eclipse, and the duration of the eclipse, τ_{0}. The fractional HR variation, ΔHR/HR, is straightforwardly defined (see Paper I and Appendix A). The eclipse duration, τ_{0}, is the total duration of the observed event from its start to the end; since each event is modelled with a Gaussian function, τ_{0} is related to the Gaussian width parameter, w, and we chose to define τ_{0} as the full width of the Gaussian at one twentieth of the Gaussian peak, so that τ_{0} = 4.9w (see Appendix A).
We note that our choice of a Gaussian function is arbitrary but quite effective. The shape of the HR variations in light curves is not predictable and our aim was to reproduce the most significant variations with simple functions, defined just by some measure of height and width: a Gaussian function is therefore a suitable and valid representation. However, the choice of the exact shape of the function does not affect our analysis and our results (see Paper I).
In Paper I we have defined for each observed source a characteristic timescale for the eclipse, Δt_{ecl}, as Δt_{ecl} ≃ 3R_{X}/v_{cloud}, where R_{X} is the Xray source radius, which we assume to be ${R}_{\mathrm{X}}\simeq 2.5{R}_{\mathrm{S}}$, where R_{S} is the Schwarzschild radius and v_{cloud} = (GM_{BH}/R_{BLR})^{0.5} is the Keplerian velocity of clouds at the characteristic BLR distance, R_{BLR}, from the AGN centre. As it is described in Paper II, for the sources of our sample we derived the value of R_{BLR} from Bentz et al. (2013) radius luminosity correlation, R_{BLR} ≃ 33.65[L_{5100}/10^{44}]^{0.533} light days (where L_{5100} ≡ λl_{λ}(erg s^{−1}) at λ = 5100 Å). We adopted, when available, the values of L_{5100} reported in the literature and, instead, derived L_{5100} as an extrapolation from the monochromatic 2 keV flux when the value of L_{5100} was not available in the literature (see also Paper I for details). The adopted values of M_{BH} are those reported in Table 1 of the present paper, as gathered, together with their uncertainties from the references mentioned in Table 1. It is interesting to compare the actual ‘measured’ durations, τ_{0}, of significant events that can be unambiguously classified as occultation candidates with such a characteristic eclipse timescale for the corresponding source. For the vast majority of our detected candidate eclipse events, the obtained value of τ_{0} falls in a range within an order of magnitude difference with respect to our generically estimated eclipse timescale for their corresponding source. Thus, it appears that the nondimensional parameter, τ_{0}/Δt_{ecl}, obtained by accounting for the estimated eclipse timescale of the source of the given event, can be a meaningful representation of our observable ‘duration of an eclipse event’. Also, normalising the duration, τ_{0}, of each eclipse event with a timescale (Δt_{ecl}) representative of the phenomenon for the source of our sample in whose light curves we detected the event itself, we can gather our observational data, {Δ(HR)/HR, τ_{0}/Δt_{ecl}}, all together in a plot that gives their visual representation. Therefore, we have chosen to plot in Fig. 1 the two observables of our analysis, showing the derived values of Δ(HR)/HR versus the corresponding τ_{0}/Δt_{ecl} for the statistically significant and reliable candidate eclipse events that we have revealed with our procedure. The values of Δ(HR)/HR and τ_{0}/Δt_{ecl}, which we have obtained for all the observed reliable occultation events and which are shown in Fig. 1, are also reported in Table A.1 in Appendix A.
Parameters and quantities, derived within the newly calculated Paper II model described in the text, for sources with detected reliable occultations.
Fig. 1. Observables plot: Fractional hardness ratio variability, Δ(HR)/HR, versus the normalised eclipse duration, τ_{ecl} = τ_{0}/Δt_{ecl}, for each reliable eclipse event detected in Paper I. 
3. Interpreting the observables in the broadlineregionlike eclipsing clouds scenario
To address the interpretation of the two observables identified in the previous section and characterising each detected event, we present in the following subsections a geometrical and kinematical representation of the occultation event and the results of our analysis of the function HR = HR(C_{F}, n_{H}) derived from the simulation of ‘synthetic’ observations performed with XMMNewtonEPIC/PN and SuzakuXIS instruments, where C_{F} is the maximum covering factor of the Xray source produced by the absorber and n_{H} is the Xray absorption column density (see also Appendix B.1); in the following we shall define N_{HX} = n_{H}.
3.1. The geometry of eclipsing clouds and of the occultation event: Our assumptions and eclipse duration
In order to obtain a relation between eclipsing times and cloud geometrical properties, we make the following simplifying assumptions.
(a) The Xray source is spherically symmetric.
(b) The eclipsing clouds, which we suppose to be components of the BLR cloud ensemble, are spherical as well, with the radius R_{cloud}, and are characterised by a constant gas number density, n_{H}.
(c) Since the range of ionisation states in the BLR does not affect the Xray absorption in the 2–10 keV energy range of our interest, we consider the whole cloud to be an Xray absorber, so that we set
$$\begin{array}{c}\hfill {R}_{\mathrm{cloud}}={R}_{\mathrm{abs}},\end{array}$$
and in the following we shall use both notations with the same meaning.
(d) The absorber velocity on the plane of the sky is a constant vector; that is, it is both constant in value (here denoted by v_{c}) and in direction, for the whole duration of the eclipse. There may of course be a velocity component along the line of sight as well, but it does not influence the occultation duration.
With these assumptions, we can depict a general geometry of an occultation event, as is delineated in Fig. 2. In the figure, both the spherical Xray source and the Xray absorber are projected on the plane of the sky, which is that of the page. With the adopted choice of Cartesian axes, the line of sight to the Xray source coincides with the z axis coming out of the page (to the observer). The absorber is moving on the plane of the sky with a constant velocity, v_{c}, in (a direction that we have identified as) the y direction. At time t_{1}, the absorber projection starts to touch and overlap the Xray source disc (of radius R_{X}), and this instant marks the beginning of the eclipse; symmetrically, when the rear border of the projected Xray absorber disc (of radius R_{abs} = R_{cloud}) leaves the source projection, at time t_{2}, the eclipse comes to its end. In principle, we identify the time lapse t_{2} − t_{1} ≡ τ_{0} with the duration of the occultation.
Fig. 2. Schematic illustration of an eclipse event as seen on the sky plane from the observer line of sight (zdirection, normal to the page). The distance, HC_{X} ≡ d, between the source centre, C_{X}, and the straight line trajectory of the absorber projection centre, C_{0}, during the eclipse, represents a sort of ‘impact parameter’, d, for the occultation and is depicted in red. 
We define the impact parameter, d, as the distance between the centre (C_{X}) of the projection on the plane of the sky of the Xray source and the straight line trajectory of the centre (C_{0}) of the eclipsing cloud projection on the sky (see Fig. 2).
We also introduced the nondimensional parameters
$$\begin{array}{c}\hfill {f}_{\mathrm{abs}}\equiv {R}_{\mathrm{abs}}/{R}_{\mathrm{X}}={R}_{\mathrm{cloud}}/{R}_{\mathrm{X}}={f}_{\mathrm{cloud}}\phantom{\rule{1em}{0ex}}\mathrm{and}\phantom{\rule{1em}{0ex}}{f}_{\mathrm{d}}\equiv d/{R}_{\mathrm{X}}.\end{array}$$(1)
From our simplified geometry, it is easy to write
$$\begin{array}{c}\hfill {\tau}_{0}=\frac{2{R}_{\mathrm{X}}}{{v}_{\mathrm{c}}}{[{(1+{f}_{\mathrm{abs}})}^{2}{f}_{\mathrm{d}}^{2}]}^{1/2}=\frac{2{R}_{\mathrm{X}}}{{v}_{\mathrm{c}}}{Y}_{0},\end{array}$$(2)
where Y_{0} = [(1+f_{abs})^{2}−f_{d}^{2}]^{1/2} and 2Y_{0}R_{X} is the distance between the positions of the eclipsing cloud centre at the beginning and at the end of the eclipse (C_{0}(t_{1}) and C_{0}(t_{2}) in Fig. 2.
Equation (2) can be rewritten to obtain the duration of the event normalised by the eclipse timescale for the source to which the event refers, Δt_{ecl} (defined in Paper I and here reminded in Sect. 2). We assume that the velocity of the cloud is Keplerian – that is, v_{c} = (GM_{BH}/r_{c})^{1/2}, where r_{c} is the distance of the eclipsing cloud from the central black hole – and we obtain
$$\begin{array}{c}\hfill \frac{{\tau}_{0}}{\mathrm{\Delta}{t}_{\mathrm{ecl}}}=\frac{2}{3}{\left(\frac{{r}_{\mathrm{c}}}{{R}_{\mathrm{BLR}}}\right)}^{1/2}{Y}_{0}.\end{array}$$(3)
This last form of the geometrical and kinematical relationship is one of the tools that we use in the following to get some insight in the physics of the absorbers and, in particular, to derive the distance, r_{c}, in units of the characteristic BLR radius, R_{BLR}, as
$$\begin{array}{c}\hfill {x}_{\mathrm{BLR}}\equiv \left(\frac{{r}_{\mathrm{c}}}{{R}_{\mathrm{BLR}}}\right)=\frac{9}{4}{\left(\frac{{\tau}_{0}}{\mathrm{\Delta}{t}_{\mathrm{ecl}}}\right)}^{2}\frac{1}{{Y}_{0}^{2}}.\end{array}$$(4)
We note that from Eq. (3), given the choice of Δt_{ecl} as a representative occultation timescale for each given AGN source and Δt_{ecl} dependence on the characteristic BLR radius ($\mathrm{\Delta}{t}_{\mathrm{ecl}}\propto {M}_{\mathrm{BH}}^{1/2}{R}_{\mathrm{BLR}}^{1/2}$, from its definition), normalising the radial distance of the cloud, r_{c}, with R_{BLR} comes as a natural choice. It is also important to remark that the dimensional radial cloud distance, r_{c}, bears no dependence on the value of R_{BLR}, as it can be seen from Eq. (4), just expressing Δt_{ecl} and R_{X} explicitly, so that
$$\begin{array}{c}\hfill {r}_{\mathrm{c}}={R}_{\mathrm{BLR}}\left(\frac{{\tau}_{0}^{2}G{M}_{\mathrm{BH}}}{{R}_{\mathrm{BLR}}}\right)\frac{1}{4{R}_{\mathrm{X}}^{2}}\frac{1}{{Y}_{0}^{2}}=\left(\frac{{c}^{4}}{{10}^{2}G}\right)\left(\frac{{\tau}_{0}^{2}}{{Y}_{0}^{2}}\right){M}_{\mathrm{BH}}^{1}\u037e\end{array}$$(5)
therefore, the only role of R_{BLR} is that of a scaling characteristic length parameter. Thus, the normalised distance, ${r}_{\mathrm{c}}/{R}_{\mathrm{BLR}}\equiv {x}_{\mathrm{BLR}}$, can be derived as a function of the observable τ_{0}/Δt_{ecl} (or, equivalently, the dimensional distance, r_{c}, can be derived from τ_{0} using the same equation multiplied by R_{BLR}, that is Eq. (5)), once the impact parameter, f_{d}, is chosen and the normalised absorber radius, f_{abs} = f_{cloud}, is determined: x_{BLR} = x_{BLR}(τ_{0}/Δt_{ecl}, f_{d}, f_{abs}).
Equation (2), or equivalently Eq. (3) above, can also be formally solved for the expected radius of the absorber normalised to the source radius, f_{abs} ≡ R_{abs}/R_{X}, obtaining a general form
$$\begin{array}{c}\hfill \frac{{R}_{\mathrm{abs}}}{{R}_{\mathrm{X}}}={({f}_{\mathrm{d}}^{2}+\frac{9}{4}\frac{{\tau}_{0}^{2}}{\mathrm{\Delta}{t}_{\mathrm{ecl}}^{2}}\frac{{R}_{\mathrm{BLR}}}{{r}_{\mathrm{c}}})}^{1/2}1,\end{array}$$(6)
where, similarly to what illustrated for r_{c}, the apparent dependence on R_{BLR} cancels in the ratio R_{BLR}/Δt_{ecl}^{2}. From Eq. (6), it is clear that for a fixed normalised distance, r_{c}/R_{BLR}, the minimum value of the normalised radius of the absorber that produces an occultation with the given duration, τ_{0}/Δt_{ecl}, is attained for d = 0 (f_{d} = 0), which is when the trajectory of the cloud centre crosses the centre of the source circular projection on the plane of the sky, a configuration that we can define as a ‘central’ geometry for the occultation (C_{X}H = d = 0 in Fig. 2).
3.2. Fractional hardness ratio variation, Δ(HR)/HR, as an indicator of the eclipse covering factor
In Paper I we analysed the ‘synthetic’ hardness ratio, HR, expected from ‘simulated’ observations performed with XMMNewtonEPIC/PN and SuzakuXIS instruments changing the values of N_{HX} (equivalent column density of the Xray absorbing BLR cloud) and C_{F}, covering factor of the eclipsing absorber in the model spectrum (see also Appendix B.1). The predicted hardness ratio, HR, can thus be evaluated as a function of the values of N_{HX} and C_{F}: $HR=HR({N}_{\mathrm{HX}},{C}_{\mathrm{F}})$. Based on these results, we find a rather wellestablished relation between the peak fractional variation of hardness ratio, Δ(HR)/HR, and the corresponding maximum covering factor attained during an occultation due to the passage of a ‘typical’ BLRlike cloud, characterised by a given value of N_{HX} (see Appendix B for more detail). Provided the absorber equivalent column density falls in a relatively narrow range around ∼10^{23} cm^{−2}, the relation between Δ(HR)/HR and C_{F} turns out to be rather tight and we can therefore somewhat ‘translate’ the ‘observed’ value of Δ(HR)/HR for a given event into an estimate of the maximum covering factor ${C}_{\mathrm{F}}={({C}_{\mathrm{F}})}_{\mathrm{max}}$ characterising that occultation event (or vice versa).
We remind here that information on the Xray absorber column density, N_{HX}, which clearly is a central quantity in the interpretation of the observed fractional variation of hardness ratio in an occultation, can be obtained as a result of Xray spectroscopical analyses performed on a number of different AGNs and available in literature (see Bianchi et al. 2012 and Paper II, Sect. 9): from these, values of N_{HX} are inferred and turn out to fall in a relatively narrow range of values around 10^{−23} cm^{−2}, so that a representative range of values for the Xray absorber equivalent column density, N_{HX}, can be taken approximately between 5 × 10^{22} and 2 × 10^{23} cm^{−2}.
Within the geometrical framework described in Sect. 3.1, the maximum covering factor attained during an occultation event can be geometrically computed (see Appendix B.2), depending on the parameters {f_{abs}, f_{d}} (defined in Eq. (1)). As is illustrated in Appendix B.2, extensive calculations for wide ranges of the parameters {f_{abs}, f_{d}} allowed us to derive the relation among $\{{f}_{\mathrm{abs}},{f}_{\mathrm{d}},{C}_{\mathrm{F}}\}$; in particular, f_{abs} can be thus determined for given {f_{d}, C_{F}} values. Conversely, any given value of C_{F} can be obtained in occultations due to absorbers with different values of the normalised projected radius f_{abs} = R_{abs}/R_{X}, depending on the specific value of the impact parameter, f_{d}, characterising the occultation event.
The inferred relationship between Δ(HR)/HR and the corresponding (C_{F})_{max}, combined with the dependence of the covering factor on the geometrical parameters of the occultation, has a couple of immediate consequences, that we mention here and discuss in Appendix B.3.
First, the minimum value of ${R}_{\mathrm{abs}}/{R}_{\mathrm{X}}$ for any given event corresponds to null ‘impact parameter’ for the occultation – that is, f_{d} = 0, (see also Eq. (6)) – and it is given by ${({R}_{\mathrm{abs}}/{R}_{\mathrm{X}})}_{\mathrm{relmin}}={({C}_{\mathrm{F}})}_{\mathrm{max}}^{1/2}$. Second, from our constraints for ‘bona fide’ candidate eclipse events, we derived (see Appendix B.3) a limiting condition on the size of any absorber responsible for our observed eclipse events, defined as
$$\begin{array}{c}\hfill \frac{{R}_{\mathrm{abs}}}{{R}_{\mathrm{X}}}\ge 0.4\equiv {\left(\frac{{R}_{\mathrm{abs}}}{{R}_{\mathrm{X}}}\right)}_{\mathrm{MIN}}.\end{array}$$(7)
Clouds with smaller size, normalised by the Xray source radius, cannot produce detectable temporary Xray absorption events that can reliably be interpreted as occultations. Thus, all the clouds corresponding to eclipse events shown in the plot of Fig. 1 must have a radius that satisfies the condition
$$\begin{array}{c}\hfill {R}_{\mathrm{abs}}\ge 0.4{R}_{\mathrm{X}}={({R}_{\mathrm{abs}})}_{\mathrm{MIN}}.\end{array}$$
The limiting condition expressed by relation (7) shows that within our framework for the eclipse scenario there certainly is a selection effect related to the central black hole mass of the AGN, M_{BH}, and the minimum physical size of an absorber capable of producing a reliable and detectable eclipse increases linearly with increasing black hole mass of the AGN (see Appendix C).
4. Extracting physical information information on the occulting clouds
In addition to the assumptions discussed in Sect. 3.1, we also assume N_{HX} to be a parameter of our analysis, so that we can choose its value in the inferred representative range mentioned in Sect. 3.2. The other fundamental physical parameter is the mass of the central black hole, M_{BH}, which also sets the natural lengthscale through the assumption
$$\begin{array}{c}\hfill {R}_{\mathrm{X}}\simeq 2.5{R}_{\mathrm{S}}=5G{M}_{\mathrm{BH}}/{c}^{2}.\end{array}$$
In the following, we illustrate and discuss our procedure to extract physical information on the specific cloud producing the occultation for each detected eclipse event.
Given the value of M_{BH} referring to the AGN in whose observed light curves we have detected any specific event, we chose the values of the two other relevant parameters,
$$\begin{array}{c}\hfill \{{N}_{\mathrm{HX}},{f}_{\mathrm{d}}\}\u037e\end{array}$$
that is, we chose the Xray absorbing equivalent column density and the geometrical impact parameter for the occultation.
We could then derive, for the given event, the physical parameters of the occulting cloud,
$$\begin{array}{c}\hfill \{{f}_{\mathrm{cloud}},{x}_{\mathrm{BLR}},{n}_{\mathrm{H}}\}\u037e\end{array}$$
that is, the normalised radius of the cloud (remembering that, with our assumptions, f_{cloud} = f_{abs}), its normalised distance from the AGN central black hole, and its (constant) gas number density, starting from the specific values of the two observables, Δ(HR)/HR and τ_{0}/Δt_{ecl}, and making appropriate use of the relation between the Xray absorption equivalent column density, N_{HX}, and the cloud gas number density, n_{H} (involving a representative estimate of the ‘effective’ absorption depth of the spherical cloud, which we must define; see Sect. 4.3). It is apparent that changing the value of the physical parameter N_{HX} and/or that of the geometrical one, f_{d}, we obtain different solutions for the occulting cloud properties {f_{cloud}, x_{BLR}, n_{H}} for any given detected eclipse event, so that there is a degeneracy in the eclipse explanation. Since the equivalent absorption column density, N_{HX}, is rather well constrained, as was discussed above, the unknown impact parameter of any given occultation is mostly responsible for such a degeneracy.
4.1. Deriving the eclipsing cloud radius for each event
In our scenario for any given event, the central black hole mass parameter, M_{BH}, is identified by the corresponding AGN. Once the values of the parameters {N_{HX}, f_{d}} are also chosen, the observed value of Δ(HR)/HR defines the corresponding value of the normalised cloud radius ${f}_{\mathrm{cloud}}={R}_{\mathrm{cloud}}/{R}_{\mathrm{X}}$. In fact, using our results of the analysis of the expected hardness ratio and its variations with varying absorbing column density, N_{HX}, and absorber covering factor, C_{F}, from ‘simulated’ observations of XMMNewton and Suzaku instruments (see Sect. 3.2 and Appendix B), we have determined a relation between Δ(HR)/HR and C_{F} for any given N_{HX} of interest; namely, Δ(HR)/HR = Δ(HR)/HR(N_{HX}, C_{F}). Also, the geometrically calculated maximum covering factor, C_{F}, is a function of the impact parameter, f_{d}, and of the absorbing cloud radius, f_{abs} = f_{cloud}; that is, C_{F} = C_{F}(f_{d}, f_{cloud}). Therefore, for each given event, once the parameters, {N_{HX}, f_{d}}, have been chosen, the observed value of Δ(HR)/HR, with the given N_{HX}, determines that of C_{F} (see Fig. B.2) and the latter, depending on the chosen value of f_{d}, defines the corresponding normalised cloud radius, f_{cloud} (see Fig. B.3, black lines). In other words, the normalised cloud radius, f_{cloud}, required to explain the eclipse event, is indeed the one that allows the cloud to carry out an occultation with impact parameter, f_{d}, causing a maximum covering factor, C_{F}(f_{cloud}, f_{d}), which, in turn, with the chosen value of the parameter, N_{HX}, produces just the observed value of the peak relative hardness ratio variation, $\mathrm{\Delta}(HR)/HR=\mathrm{\Delta}(HR)/HR({N}_{\mathrm{HX}},{C}_{\mathrm{F}})$, for the given event.
4.2. Deriving the eclipsing cloud spatial location for each event
Again, for any given occultation event in our sample (defining the AGN parameter M_{BH}), once we have determined the appropriate f_{cloud} for a specific choice of the occultation parameters $\{{N}_{\mathrm{HX}},{f}_{\mathrm{d}}\}$ as discussed above, we can immediately derive the distance of the occulting cloud from the central black hole, r_{c}, from Eq. (4), using the other observable of the eclipse event; namely, the normalised occultation duration, τ_{0}/Δt_{ecl}. In fact, Eq. (4) gives the distance normalised with the characteristic BLR size, R_{BLR}, x_{BLR} ≡ r_{c}/R_{BLR}, as a function of $\{{\tau}_{0}/\mathrm{\Delta}{t}_{\mathrm{ecl}},{f}_{\mathrm{d}},{f}_{\mathrm{cloud}}\}$.
4.3. Deriving the occulting cloud gas number density for each event
More elaboration is needed to infer a representative and reliable estimate of the cloud gas number density, n_{H}, which we have assumed as constant throughout the spherical cloud. To do this, we need to properly express the equivalent Xray absorbing column density, N_{HX}, in terms of the gas number density and of an appropriately ‘averaged’ effective depth of the cloud as an absorber, which we indicate as l_{eff}. We thus need to devise a procedure to evaluate this effective depth, l_{eff}, depending of course on the cloud normalised radius, f_{cloud}, and also on the specific geometry of the occultation, through the parameter f_{d}; in fact, f_{cloud} and f_{d} together define the portion of the spherical occulting cloud that happens to intercept the Xray source projection on the sky plane. We show in Appendix C our approach to a simplified but representative derivation of the normalised effective depth, ${l}_{\mathrm{eff}}/{R}_{\mathrm{X}}=2{f}_{\mathrm{eff}}$, where we have introduced a normalised ‘effective radius’, f_{eff}, for an immediate comparison with f_{cloud}. In this derivation, the normalised effective depth turns out to be a function of both f_{cloud} and f_{d}, since, as is shown in Appendix C, it is
$$\begin{array}{c}\hfill {f}_{\mathrm{eff}}={f}_{\mathrm{eff}}({f}_{\mathrm{cloud}},{f}_{\mathrm{d}}).\end{array}$$
This is the length scale that most representatively connects the parameter N_{HX} with our estimate of the gas number density, n_{H}, of the cloud of normalised radius determined in Sect. 4.1 that is occulting the Xray source with the chosen geometric impact parameter, f_{d}, through the relation
$$\begin{array}{c}\hfill {N}_{\mathrm{HX}}=2{f}_{\mathrm{eff}}{R}_{\mathrm{X}}{n}_{\mathrm{H}}={l}_{\mathrm{eff}}{n}_{\mathrm{H}},\end{array}$$
so that we obtain the gas number density of the cloud as
$$\begin{array}{c}\hfill {n}_{\mathrm{H}}=\frac{{N}_{\mathrm{HX}}}{{l}_{\mathrm{eff}}({M}_{\mathrm{BH}},{f}_{\mathrm{d}},{f}_{\mathrm{cloud}})}.\end{array}$$(8)
5. Matching solutions to detected events: Results for physical parameters of occulting clouds
Each one of the actual detected events (shown in Fig. 1) corresponds to an AGN with a central black hole mass, M_{BH}; for each of the detected events, we thus identified the physical properties, {f_{cloud}, x_{BLR}, n_{H}}, of a cloud with equivalent Xray absorbing column density equal to the chosen value, N_{HX}, and obscuring the given Xray source with the chosen value of the impact parameter, f_{d}, that determines the specific geometry of the eclipse.
We define this as a ‘best match’ solution for a given event, meaning that it is the one obtained by identifying the f_{cloud} value that, once given the parameters M_{BH} and {N_{HX}, f_{d}}, corresponds to the best reproduction of the Δ(HR)/HR value derived for that event from our analysis of Paper I. Figure 3 represents the best matches obtained for the detected events assuming a value of 10^{23} cm^{−2} for the absorbing column density parameter, N_{HX}, and central geometry occultations (f_{d} = 0).
Fig. 3. Best match solutions obtained for our detected events in the case of central geometry. Yellow dots represent best matches for NGC 3783 detected eclipse events, cyan dots show best matches for those of MCG 063015, green dots show matches obtained for events referring to all other sources with MBH < 6.5, red dots are for all other sources with MBH > 7, and finally blue dots are best match events referring to intermediate mass sources with 6.5 < MBH < 7, where MBH ≡ log(M_{BH}/M_{⊙}). 
The results shown in Fig. 3 are representative, but merely exemplifying; in fact, the properties of a cloud that can explain any given observed occultation event depend on the specific choice of the two parameters {N_{HX}, f_{d}} and for different values of these parameters we obtain different ‘solutions’ describing the specific observed event, whose interpretation is thus degenerate (see Sect. 4). Increasing the impact parameter, f_{d} – that is, choosing more and more offcentre occultation configurations – leads to an increase in the normalised absorber size, f_{abs}(=f_{cloud}), in order to maintain the maximum covering factor at the value corresponding to the measured Δ(HR)/HR for the given detected event (see Appendix B).
We can somewhat reduce the degeneracy of possible solutions explaining each event and extract information on the physical parameters of the occulting clouds, by taking also into consideration our results of Paper II on the global properties of the ensemble of clouds orbiting in the central region of the AGN.
From Fig. 3 it is clear that, taking only central geometry occultations into account, a number of events are explained with an eclipsing cloud that is located farther than the external boundary (set at ${r}_{\mathrm{c}}/{R}_{\mathrm{BLR}}=3$) of the region that in Paper II we considered as the one in which our cloud ensemble was spatially distributed. Allowing for noncentral geometries, we can find solutions corresponding to clouds closer to the central black hole (see Eq. (4) and Appendix B.2). However, even choosing an impact parameter of the occultation f_{d} = 4, for the majority of those events the only solutions that we obtain still correspond to clouds ‘external’ to the region occupied by the cloud ensemble with the specific choice of spatial boundary parameters adopted in Paper II. Clearly this would imply an inconsistency, if we want to directly take the outcome of Paper II into account with the aim to analyse and discuss our best matching solutions for eclipse events.
Thus, still following the prescriptions and procedure of Paper II, we have worked out again the properties of the global model of the cloud ensemble, by appropriately modifying the values of two quantities upon which the model depends, namely the external boundary of the region over which the clouds belonging to the model ensemble are spatially distributed and the number of detected occultation events taken into account, so as to guarantee full consistency in our use of the resulting relevant properties of the cloud ensemble model in the present analysis.
Our first step was to enlarge the spatial region for our cloud ensemble, with respect to the case presented explicitly in Paper II, pushing the outer boundary approximately to the sublimation radius. For our sample of sources showing eclipse events, typically this turns out to be R_{subl} ∼ 6R_{BLR} (using the relation R_{subl} ∼ 0.5(L_{bol}/10^{46})^{0.5}pc with L_{bol} ∼ 9L_{5100}, see Netzer 2013, 2015); as a consequence, we can reasonably assume a new outer boundary, R_{OUT} = g_{ex}R_{BLR}, for the spatial region where the clouds of our global model of Paper II are distributed as R_{OUT} = 6R_{BLR}, thus choosing a new value for the nondimensional parameter, g_{ex}, introduced in Paper II; namely, g_{ex} = 6.
Furthermore, we decided to discard from our sample analysis those occultation events that, even considering extremely offcentre (f_{d} ≥ 4) geometries, are still explained in our scenario by eclipsing clouds located farther than the new outer boundary of the cloud ensemble model region, being x_{BLR} > g_{ex} = 6. This choice changes the number of detected eclipse events, N_{ecl}, for a few sources and therefore also changes the total number of events of our sample that we take into account from 65 to 59, excluding a total of 6 events, of which 2 for NGC 4395, 1 for NGC 4051, 2 for MCG 063015 and 1 for NGC 1365. These four sources are those whose value of N_{ecl} is modified with respect to its original value in the sample. Also, the abovementioned ‘rejected’ events still are the same and only ones that would correspond to occulting clouds farther than 6R_{BLR} for even more extreme geometries; that is, f_{d} > 4.
Hence, we recalculated a global model for the cloud ensemble producing both contributions to broad lines and Xray occultations (see Paper II), assuming its external boundary at around the sublimation radius (i.e. choosing g_{ex} = 6) and excluding from the total number of events taken into account those that can only be explained with solutions that correspond to clouds located outside the region occupied by our model ensemble. This is an elementary but reasonable choice, and the few ‘eliminated’ events could of course be explained by gas condensations located farther away than the region occupied by broadline emitting clouds, and as such not belonging to the cloud ensemble we have modelled, or by some more complex description that cannot be provided within the limits of our present simple geometrical and dynamical assumptions. Numerical values for physical quantities derived for the cloud ensemble may be changed with respect to those referring to the published model of Paper II, but the ‘newly’ calculated model for the cloud ensemble does maintain the same features and properties derived and discussed in Paper II.
The specific results of this ‘new’ global model for the ensemble of clouds can be now taken into account in order to explore the possibility of defining quantitative constraints on cloud size, thus investigating the actual viability of each one of our bestmatching solutions. Of course from now on we discuss results referring only to the 59 events that we considered for the building of the global model of the cloud ensemble mentioned above (see Appendix A and Table A.1 therein) and all the figures illustrating our outcome in the next subsections only show solutions matching those same events.
5.1. Physical size of clouds and degeneracy of solutions
It is worth reminding ourselves here of the results of Paper II on the shape of the number density distribution in size characterising the ensemble of clouds orbiting the black hole in the central region of an AGN, comprising clouds that contribute to broadline formation and potentially produce occultations of the Xray source; such results in fact still maintain their validity within the modified version of the global model of the cloud ensemble that we are now dealing with. This distribution turns out to be a decreasing function of the cloud physical size (see Paper II), with a power law of exponent equal to −2, so that in general it is much less likely that an eclipse event is due to a ‘large’ (f_{cloud} > 4) cloud with respect to the case for the same event being explained by a ‘small’ (f_{cloud} ≲ 1) cloud occultation, because ‘large’ clouds are by far less numerous than ‘small’ ones.
Defining a limitation in physical normalised size of the occulting clouds would lead to a more stringent identification of the range of gas number density for the ensemble of clouds.
Adopting the new parameters discussed above for the model introduced in Paper II, from our recalculation of the global model for the cloud ensemble we can derive an estimate of the maximum physical size of clouds for each of the AGN sources showing Xray eclipse events; this estimate thus takes into account both the constraints from observationally inferred properties of clouds contributing to the broadline formation and those obtained from our analysis of Xray occultations (see Paper II) into account. The notation for such an estimate of the maximum cloud physical radius used in Paper II is r_{2}, but here we identify this same estimate with the symbol R_{2}, so that (f_{cloud})_{max} = (R_{cloud})_{max}/R_{X} = R_{2}/R_{X}. In Table 1 we report the values of R_{2}/R_{X} derived for each of the sources showing reliable occultations. These values for R_{2}/R_{X} are clearly larger than the ones reported in Paper II because of the change of parameters (g_{ex} = 6 mainly, and for some AGN sources also the number of detected events considered) that we adopted for full consistency of our present discussion. Nevertheless, for 6 sources (mostly those with a central black hole mass about or larger than 10^{7} M_{⊙}) it turns out that (f_{cloud})_{max} is in the range 4 − 7. Thus, for these sources, we have a significant limitation to occulting cloud size and, as a consequence, a constrain on degeneracy of eclipse event interpretation, since for eclipse events detected for these same sources we can exclude explanations that would require a cloud with f_{cloud} > (f_{cloud})_{max}. This limitation only pertains to those AGN sources for which (f_{cloud})_{max} ∼ 4 − 7, and therefore we cannot obtain a general strong constraint on the cloud size directly from our definition of (f_{cloud})_{max}. In fact, for the other AGN sources, we find an estimate of the cloud radius upper limit that is typically larger ((f_{cloud})_{max} ∼ {12 − 27}) or significantly larger for the most extreme cases of NGC 4395 and NGC 4051, corresponding to the smallest black hole masses in our AGN sample. Thus, in the following we discuss the issue of cloud size along a more general line of reasoning.
We can now take into consideration
(a) the number of reliable eclipse events, N_{ecl}, that we have effectively considered for each AGN source according to the discussion presented above (for each source N_{ecl} is shown in Table 1) as well as
(b) the properties of the number density distribution in cloud size for our model cloud ensemble, dN/dR, which was derived with Paper II procedure under the assumption that such a distribution does not depend on the distance from the central black hole. Thus, using the value of N_{ecl} together with dN/dR, for each AGN source in our sample we can determine the expected number of eclipses, among the N_{ecl} detected, that can be due to clouds with radius in the interval between a given value of the cloud radius, R_{i}, and the maximum radius R_{2} obtained (see Table 1) for that source. To obtain this result, we start using the number density distribution in radius for the clouds, dN/dR (see Sects. 4 and 7 of Paper II), to derive, for each of our AGN sources, the fraction of potentially occulting clouds in a given range of radii $[{R}_{\mathrm{i}},{R}_{2}]$, as
$$\begin{array}{c}\hfill Fra{c}_{[{\mathrm{R}}_{\mathrm{i}},{\mathrm{R}}_{2}]}=\frac{{\int}_{{R}_{\mathrm{i}}}^{{R}_{2}}(\mathrm{d}N/\mathrm{d}R)\mathrm{d}R}{{\int}_{{R}_{1\mathrm{x}}}^{{R}_{2}}(\mathrm{d}N/\mathrm{d}R)\mathrm{d}R},\end{array}$$(9)
where R_{1x} ≡ 0.4R_{X} is the minimum value of cloud radius for a cloud whose possible eclipsing effects could be detected with our method of analysis (see Sects. 4 and 5). Because of its derivation, the quantity Frac_{[Ri, R2]} defined above also bears no dependence on the distance from the central black hole, so that for each source we can use Frac_{[Ri, R2]} to evaluate the expected number of eclipses, among the N_{ecl} detected, that can be due to clouds with radius in the interval [R_{i}, R_{2}]. We define this number as
$$\begin{array}{c}\hfill {n}_{\mathrm{ecl}}[{R}_{\mathrm{i}}/{R}_{\mathrm{X}},{R}_{2}/{R}_{\mathrm{X}}]=Fra{c}_{[{\mathrm{R}}_{\mathrm{i}},{\mathrm{R}}_{2}]}{N}_{\mathrm{ecl}}.\end{array}$$
For each of the AGN sources in our sample, Table 1 shows the values of n_{ecl} determined for different intervals of cloud radius.
The range of f_{cloud} values explaining the events in our sample depends on the eclipse impact parameter value f_{d}, as it is apparent from Fig. 4, with f_{cloud} typically increasing with increasing f_{d}. Taking again the properties of the number density distribution in cloud size for our model cloud ensemble into account, for any of the sources in our sample the expected fraction of potentially eclipsing clouds with f_{cloud} > 5 − 6 turns out to be < 6 − 7% at most: smaller clouds are always strongly dominant in number. For f_{cloud} in the range [6, R_{2}/R_{X}] we obtain n_{ecl}(6, R_{2}/R_{X})∼0.5 only for MGC 063015, whereas for most of the other sources in that range of normalised cloud radii the estimated n_{ecl} is significantly lower or inexistent.
Fig. 4. Best match solutions for cloud physical size obtained for our detected events in different geometries: circles again refer to central geometry (f_{d} = 0), but we also show matches for f_{d} = 1 (filled hexagons), f_{d} = 2 (filled triangles) and f_{d} = 4 (large filled triangles). Colour code is the same as in Fig. 3. In each panel hollow circles or triangles represent event descriptions that, for the specific f_{d} value, must be discarded, since the occulting clouds are located farther than the chosen outer border of the cloud ensemble; the corresponding events are only explainable with f_{d} = 4 or larger. 
Therefore, we can reasonably conclude that, except for MGC 063015 (and possibly, NGC 1365), even for those sources for which the distribution in size is characterised by (f_{cloud})_{max} > 6 (see the fourth column of Table 1), it is highly unlikely that one occultation event is explained with clouds with size larger than ∼6R_{X}. Hence, the analysis of our sample of detected occultation events, consistently taking into account also the outcome of our study of a global model for the cloud ensemble performed following the method presented in Paper II, indicates that in any case very large (f_{cloud} > 6) clouds do not give a significant contribution to explaining occultations in our sample and we can rather safely exclude that such clouds are responsible for our detected events. This somewhat poses some limitations on degeneracy in the interpretation of any given occultation event, due to the unknown impact parameter of the eclipse.
Figure 4 shows the values of the normalised cloud radius R_{cloud}/R_{X} plotted as a function of the resulting cloud distance from the central ionising source r_{c}/R_{BLR} (in units of the AGN BLR radius) for the best match solutions describing our detected events obtained assuming N_{HX} = 10^{23} cm^{−2} for four representative choices of the geometrical configuration of the occultation, i.e four different values of the impact parameter f_{d}. Each of the four panels of Fig. 4 presents the best matching solutions obtained for our detected eclipse events, assuming the specific value of f_{d} labeling the panel. It is apparent that the events for which we find a solution in central geometry (f_{d} = 0, upper left panel) are also explained with an offcentre geometry occultation produced by a larger cloud (see, for instance, upper right panel, for the case f_{d} = 2). Inspecting Fig. 4, the lower right panel, showing solutions with impact parameter of the occultation f_{d} = 4, of course presents only event explanations corresponding to clouds located within the chosen external boundary R_{OUT}/R_{BLR} = 6 for the cloud spatial distribution (see beginning of Sect. 5); on the contrary, in the other panels, pertaining to smaller values of the occultation impact parameter, solutions with eclipsing clouds located at larger distances still appear (as hollow circles or triangles, instead of coloured filled ones), since they refer to events that, when choosing f_{d} = 4 can be explained by clouds belonging to our ensemble, redefined at the beginning of Sect. 5, that is, clouds located at r_{c} < 6R_{BLR}. We have chosen to keep these latter solutions in the plots, as an exemplification of another way to remove degeneracy of solutions for a given event: in these cases, even though we can find an explanation of the event with f_{d} < 4, for consistency with our global model for the cloud ensemble only strongly offcentre solutions, that is those for f_{d} = 4, which involve clouds placed within the ensemble spatial region, are going to be regarded as representative solutions for the given event.
From Fig. 4 it is also apparent that all the solutions shown (independently of the value of the impact parameter f_{d} characterising the occultation geometry) correspond to R_{cloud}/R_{X} within a range ∼0.4 − 5; this result contributes to strengthen the idea of a robust preponderance of clouds of limited size in the ensemble orbiting in the AGN central region. Moreover, this same figure also shows that the normalised cloud size of occulting clouds bears no significant dependence on the distance of the cloud from the central black hole, normalised with the characteristic length scale of the BLR, r_{c}/R_{BLR}, consistently with our previous assumptions of Paper II.
5.2. Gas number density for Xray eclipsing clouds
In Fig. 5, we show the corresponding values of the gas number density n_{H}, as derived from Eq. (8), for clouds explaining the detected occultations according to our model solutions, again plotted with respect to the resulting cloud distance from the central ionising source; each panel refers to one of the different geometrical configurations chosen to illustrate our search for model solutions matching the observed events. The gas number density is a physical parameter that has been extensively studied for the gas contributing to broadline formation and whose range of values has been inferred from detailed analysis and modelling of the emitted broad lines, even though each observed broad line is the product of the integrated contribution of the whole ensemble of emitting cloudlike gas structures. Thus, it is worth examining more closely our results regarding n_{H} for the individual clouds that can explain each of our detected eclipse events.
Fig. 5. Gas number density from best match solutions obtained for our detected events in different geometries. Here we impose the constraints on maximum normalised physical size of the cloud described for Fig. 4 and colour codes are the same as in Fig. 4. 
1) Even taking possible different geometries of the occultation into account, Fig. 5 shows that the range of resulting possible values of gas number density is in agreement with the one inferred from analysis of broad emission lines (typically ∼10^{9} − a few10^{11} cm^{−3} or, considering the most extended interval, 10^{8} − 10^{12} cm^{−3} ; see, for instance, Elvis 2017; Korista & Goad 2000) thus confirming our interpretation of the temporary Xray absorption as due to Xray source occultation by a cloud belonging to the same ensemble of BLR clouds and crossing the line of sight to the Xray source itself.
2) Reminding that the colour code refers to ranges or specific values of the AGN central black hole mass, Fig. 5 illustrates that for a fixed value of the Xray absorbing column density, namely N_{HX} = 10^{23} cm^{−2}, and taking into account a given geometry of the occultation into account, the range of gas number density depends on M_{BH}, again as a consequence of the different cloud physical size range detectable depending on M_{BH}. We have no definite information on whether the most representative value of N_{HX} bears some dependence on the central black hole value M_{BH}. We have thus explored the solutions for the extreme values of our interval of interest for the Xray absorbing column density, that is N_{HX} = 5 × 10^{22} cm^{−2} and 2 × 10^{23} cm^{−2}, and in Fig. 6 we show the corresponding results for the gas number density of occulting clouds explaining the events detected in our sample for two different values of the impact parameter (f_{d} = 0, 2). An inspection of Fig. 6, as compared to the two upper panels of Fig. 5 confirms the agreement of our derived values of n_{H} with the typical range inferred from broadline analysis, mentioned above, independently of the specific value of N_{HX} considered in the relevant interval. We just mention that, in order to explore N_{HX} values close to our sensitivity interval upper limit (see Sect. 2), we have also analysed possible descriptions of our occultation events with occulting clouds characterised by N_{HX} = 5 × 10^{23} cm^{−2}, still obtaining results compatible with BLRlike cloud properties.
Fig. 6. Gas number density from best match solutions obtained for our detected events and for two different values of the absorbing column density N_{HX}, as shown by the labels in each panel. Here we impose the constraints on maximum normalised physical size of the cloud described for Fig. 4 and colour codes are the same as in Fig. 4. 
3) As noted above, the gas number density of the cloud depends on the AGN black hole mass M_{BH}; from Eq. (8) it is apparent that this dependence enters through ${l}_{\mathrm{eff}}^{1}\propto {R}_{\mathrm{X}}^{1}\propto {M}_{\mathrm{BH}}^{1}$. Considering the statistical errors as reported in Table 1, their effect on n_{H} would be substantially negligible in most cases. Even allowing for a more conservative average factor of 2 uncertainty on black hole mass estimates, the resulting gas number densities would in any case remain within the typical interval for BLRlike clouds.
We note in passing that a similar line of reasoning could be applied to the effects of such a factor of 2 uncertainty on M_{BH} on the cloud linear size that scales linearly with M_{BH} (R_{abs} = R_{cloud} = f_{abs}R_{X} ∝ M_{BH}; see also Appendix C).
4) For each of our AGNs, imposing the upper limit for R_{cloud}/R_{X}, reported in Table 1 as R_{2}/R_{X} (see Sect. 5.1) allows us to evaluate a lower limitation on the gas number density for clouds in the ensemble orbiting around the central black hole. This is a general result for clouds of the ensemble orbiting on a range of distances approximately between ∼0.1R_{BLR} and ∼6R_{BLR} (see Sect. 5.1), independently of the phenomenology through which clouds manifest themselves (contributing to broadline formation or occulting the Xray source). In fact, we estimate (n_{H})_{min} by using Eq. (8) with f_{d} = 0, so that the ratio f_{eff}/f_{cloud} is independent on f_{cloud} (see Sect. 4.3 and Appendix C) as well as properly representative to our present purposes, and with f_{cloud} = (f_{cloud})_{max} = R_{2}/R_{X} for each of the AGN sources in our sample; thus we obtain minimum values for n_{H} ranging from ∼2 × 10^{8} cm^{−3} to ∼5 × 10^{9} cm^{−3}, a range that is consistent with the lower limit constraints derived from broadline analysis.
5.3. Spatial location of absorbing clouds
Going back to Fig. 3, it is clear that the original distribution in distance (normalised to R_{BLR}) of the matching solutions for eclipsing clouds would span a rather large range, with 0.05 ≲ r_{c}/R_{BLR} ≲ 10^{2}. As for the upper limit of normalised cloud distance x_{BLR} = r_{c}/R_{BLR}, we have explained our choice (based on consistency reasons as illustrated at the beginning of Sect. 5) to keep the discussion of results centred on a range of distances whose upper limit is the external boundary, (x_{BLR})_{max} = g_{ex} = 6, of the model here adopted for the spatial region occupied by the cloud ensemble that was introduced in Paper II. Turning our attention to Fig. 4, for the offcentre occultation solution panels (f_{d} > 0) the range in r_{c}/R_{BLR} is more limited than for central occultation solutions (f_{d} = 0) on the small values side, because of the coupling of our restriction on the normalised cloud size (see Table 1 and Sect. 5.1) with the fact that for any event increasing the impact parameter f_{d} implies a solution with a larger normalised cloud size. A closer inspection anyway shows that the majority of the possible event matching solutions corresponds to clouds located between r_{c}/R_{BLR} ∼ 0.1 and r_{c}/R_{BLR} ∼ 6.
The results shown in Fig. 4 are fully consistent with the assumption of no significant dependence of the distribution in size of clouds on spatial distance from the central black hole. From Eq. (4), it is clear that the occultation duration has a crucial role in defining the spatial position of the occulting cloud: the shorter the duration of the eclipse event (requiring in our scenario a larger transverse velocity component), the closer to the central black hole is the occulting cloud located. This is of course true independently of the chosen distance normalisation: the same conclusion holds for dimensional distance r_{c} and distance normalised to the gravitational radius of each source, i.e x_{g} = r_{c}/R_{g} with R_{g} = GM_{BH}/c^{2}); to this respect, the exemplifying Fig. 7 shows the bestmatching solutions with cloud size plotted against x_{g} = r_{c}/R_{g} for two different choices of eclipse impact parameter.
Fig. 7. Cloud physical size for occultation matching solutions plotted versus cloud distance normalised to the gravitational radius of the AGN source specific to the eclipse event. Symbols and colour codes are the same as in Fig. 4. 
Events that in our simple framework are only matched by solutions corresponding to clouds located very close to the central black hole (x_{g} ∼ 10^{2} or even x_{g} < 10^{2}) are most probably not correctly explained by our elementary model: keplerian velocities would be around or larger than c/10 and this would make the description through our very simple model unreliable and incorrect. This may be the case for the inner red dot, corresponding to an event of PG 1501+106, only appearing in the left panel of Fig. 7 and the two events of NGC 3783 explained by the solutions appearing as the two innermost yellow dots in Figs. 3 to 7. Figure 7 better shows that the same can be true for the NGC 4151 event whose solution is represented by the second innermost red dot in the f_{d} = 0 panel.
Plotting solutions with respect to ${x}_{\mathrm{g}}={r}_{\mathrm{c}}/{R}_{\mathrm{g}}$ also facilitates a discussion and a comparison with the outcome of Markowitz et al. (2014) work, searching for and analysing ‘discrete Xray absorption events’ in a sample of Seyfert AGNs from the archive of RXTE.
Among the sources of the sample of Markowitz et al. (2014) there are 9 of the 14 AGN sources for which we do find significant and reliable Xray absorption events (see Table 1 and Papers I and II). However, Markowitz et al. (2014) detected secure events for only 4 of these nine sources we have in common. Also, typically Markowitz et al. (2014) absorption events are characterised by a duration that is significantly longer than those characterising the majority of our events; this is both due to the nature of RXTE observations analysed and to the authors’ method of analysis, thus probing essentially a different and admittedly somewhat ‘complementary’ portion of the parameter space of absorbers (pag. 6 Markowitz et al. 2014) with respect to the portion that can be studied analysing XMMNewton and Suzaku observations, like we did.
The four sources showing secure Xray absorption events in both works (Markowitz et al. 2014 and the present one) are NGC 5506, NGC 3227, NGC 3783 and Mrk 79. We find some sort of agreement with the results for NGC 5506 and NGC 3227: for these sources the range of spatial location of BLR emitting clouds does overlap the region of estimated locations of eclipsing clouds. For Mrk 79 the single detected event in our work can be explained with eclipsing cloud solutions such that x_{BLR} > 1 (actually ∼2.75 for central geometry), but decreasing towards unity with increasingly offcentre geometries: again a qualitative agreement with Markowitz et al. (2014) results can be claimed for this source as well. As for NGC 3783, we have detected several (8) reliable events in this source light curves and they can be all explained with occultations due to clouds located in a rather large interval straddling the nominal BLR radius, thus supporting our scenario with an ensemble of clouds spatially distributed around this position, that can both contribute to broadline emission and potentially produce Xray eclipses. In Markowitz et al. (2014) work, the two secure detected events for NGC 3783 are both characterised by significantly longer durations and thus the corresponding absorbing clouds derived location turns out to be much farther away from the central black hole with respect to our work deductions; hence, we deal with different conditions and cannot find an agreement. In any case, it is interesting that, even though with different methods of analysis, these studies show a spatial distribution of cloudlike gas structures on a rather extended range of distances around the nominal estimate of the BLR radius, R_{BLR}.
In fact, we remind that (as exemplified in Fig. 3) we do find solutions explaining our detected eclipse events on a large interval of distances (between ∼0.05R_{BLR} and ∼10^{2}R_{BLR}). Nevertheless, for consistency with our global model of cloud ensemble built following Paper II ‘recipe’, we restrict our attention to solutions with r_{c}/R_{BLR} ≲ 6 when we want to apply and combine the properties of such a model cloud ensemble to our present results. This is a physically reasonable choice, given that our selection of the external boundary of the spatial region where our model system of clouds is distributed roughly represents the sublimation radius for the AGNs in our sample, that is a distance beyond which physical conditions of gas condensations can be expected to change, thus requiring a different and more composite description with respect to our simple model.
5.4. A comparison with spectral analysis results in the case of Mrk 766
Risaliti et al. (2011) analysed in detail 4 observations of XMMNewtonEPIC/PN (4 orbits), performing a spectral fitting of the portions of light curves showing a significant temporal variation of hardness ratio. Among the various observations of Mrk 766 that we have studied, both of Suzaku and of XMMNewton, we have analysed three of those discussed in Risaliti et al. (2011) (R11 from now on). We can thus compare our results for the expected physical parameters of clouds producing the occultations (that we suppose are at the origin of the temporary variation of hardness ratio in the light curves examined), with those of R11, obtained through the analysis of the Xray spectrum extracted on the time intervals corresponding to the same occultation events.
For a better comparison, we chose to perform a new fit (with respect to the one of Paper I) of the two light curves obtained from XMMNewtonEPIC/PN observations of may 23 2005 and may 25 2005 respectively, namely the first and second orbit studied by R11. In fact, still using our method of Paper I, we fitted the temporal variations of hardness ratio analysing the two light curves altogether, thus obtaining a better and more reliable definition of the ‘constant’ hardness ratio level underlying the temporal variations of HR representing significant Xray variability events that can be safely interpreted as occultations by a cloud like structure crossing the line of sight to the Xray source. We thus obtained, for the three events detected in those two light curves, observable values that can be different from those reported in Paper I for those same events, especially for what regards the fractional hardness ratio ΔHR/HR. Using these new values for the event observables, we determined the best matching solutions for the occulting cloud responsible for each event in our simple model scenario. First, from the ΔHR/HR value for each event we derived (see also Fig. B.2) the C_{F} values for the ranges of N_{HX} (in the interval ∼10^{23} − 2 × 10^{23} cm^{−2}) inferred from the spectral analysis performed by R11 for the various subintervals (see Table 2 of R11). The obtained values of C_{F} (around 0.62–0.73, see third panel of Fig. 8) are in accordance with those derived from spectral analysis of R11. Then, among the possible solutions (degenerate because of possible different geometries of the eclipse event), we selected our ‘best’ solutions for the other physical parameters of the three events; these solutions are shown in Fig. 8.
Fig. 8. Best matching solutions for the three events that we have detected and were also spectrally analysed by Risaliti et al. (2011), chosen so as to maximise the agreement with R11 results for N_{HX} and C_{F} values as reported in their paper. Blue points indicate a solution with N_{HX} = 10^{23} cm^{−2}, whereas red ones refer to cases in which N_{HX} = 2 × 10^{23} cm^{−2}; circular dots show central geometry solutions (f_{d} = 0) and triangles indicate a case with impact parameter of the occultation f_{d} = 2. 
The good agreement obtained for the cloud gas number density, that R11 estimate in the range ∼10^{10} − 10^{11} cm^{−3}, supports our method of analysis and substantiates its possible use on large samples of sources for which a complete spectral analysis is not possible.
As for the range of cloud spatial positions, R11 mention an interval of ${r}_{\mathrm{c}}/{R}_{\mathrm{g}}$ values around 10^{3} − 10^{4}, which the authors actually recognise to be not very strongly constrained; again, the lower right panel of Fig. 8 shows a rather good agreement.
6. Summary and conclusions
Our analysis of a composite sample of bright Xray AGN sources, performed in Paper I, shows that Xray variability events that are explainable as temporary occultations of the Xray source itself are quite widespread and frequent. The amount of data analysed deserves a more thorough and deeper analysis, with the aim of extracting physical information on the occulting absorbers.
As was brought up in Sect. 2 and discussed in detail in Paper I, the sample is composed of type1 to Comptonthin type2 AGNs, thus excluding Comptonthick sources, and the energy interval analysed is the 2–10 keV range, so our Paper I search for eclipses is sensitive only to occultations by clouds with a column density in the range of ∼10^{23} − 10^{24} cm^{−2} at most. Also, our analysis of Paper I is sensitive to occultation durations no longer than about 1 day. These conditions are of course confining our study to a subsample of all possibile temporary absorbers of the AGN Xray source. In fact, cases of longterm variability and/or Comptonthick eclipses are already known (Ricci et al. 2016; Marchesi et al. 2022). Indeed, our method has limitations on the range of eclipse duration and cloud column density. This point has been discussed in more detail in Paper I; however, it is worth remembering that the detection of the possible eclipse in a light curve requires a measurable change in hardness ratio within a timescale shorter than the typical duration of an observation. This excludes longterm variability due to, for example, more distant clouds. It also excludes cases like NGC 1068 (Zaino et al. 2020), where even the lowabsorption state is almost Comptonthick, implying that the hardness ratio in the 2–10 keV range is nearly constant during an eclipse. Instead, it doesn’t exclude extreme cases where a Comptonthick cloud obscures an otherwise Compton thin source. Based on the observations of NGC 1365 (Risaliti et al. 2005), we know that these cases may occur, but they are very rare.
The important point here is that these limitations do not affect our main conclusion: eclipses by Comptonthin BLR clouds are common in local AGNs, and allow us to investigate the physical properties of the absorbing BLR clouds. These conclusions hold even if there are additional sources of absorption by far away circumnuclear gas and or by thicker clouds.
In Paper II we started our study with a global approach, restricting our interest to the assessment of overall properties of a system of orbiting ‘clouds’ (gas condensations) that can both contribute to the emission of broad lines and produce temporary Xray source absorption, through occultation; thus, we concluded that such a system can exist (in the inner region of AGNs), being consistent with both BLR observational constraints and the conditions derived from the investigation of the sample of detected eclipse events described in Paper I from a general point of view.
In the present work, we adopted a different approach, by also taking into account and analysing the specific characteristics of each single eclipse event detected in our sample and devising a procedure that enables us to derive physical information on individual gas condensations (‘clouds’) producing the occultation, even though based on strongly simplifying assumptions. Our aim is to extract more knowledge on the properties of clouds that are part of the system orbiting in the central region of AGNs and defining the BLR and, thus, to strengthen and confirm our proposed scenario. To pursue this goal, we have examined the duration, τ_{0}, of each detected eclipse event, that was derived in our light curve analysis of Paper I, but not yet investigated in our previous papers.
We have devised (and described in detail, see Sects. 3 and 4) a procedure through which we can analyse each single detected eclipse event with the aim of defining ranges of physical parameters for the specific cloudlike structure producing the occultation, thus attempting at gaining insight into the physical conditions of the single unit of gas condensation of the ensemble. In fact, we analysed our detected events, trying to match them with possible theoretical conditions (physical properties) for eclipsing clouds. To this aim, we considered the results of Xray spectroscopical analysis present in literature and performed for several single sources, indicating an absorbing column density, N_{HX}, in a rather narrow range around 10^{23} cm^{−2} (see, for instance, Bianchi et al. 2012). Thus, we have discussed results referring to a fixed value N_{HX} = 10^{23} cm^{−2} that we assumed as representative of the Xray eclipse phenomenon (even though we explored a range in column density =[0.5 × 10^{23}, 2 × 10^{23}] cm^{−2}) to determine possible occulting cloud properties. Because the selected value of N_{HX} is right in the narrow range inferred from Xray absorption analysis, our results on clouds producing such absorption through Xray source occultation can be taken as representative and reliable.
The main aspects and results of the present work are the following.

From this analysis of our sample of eclipse events, except for very few cases, the properties (distance from the central BH, physical size, gas number density) of each individual cloud producing temporary Xray absorption, by means of a transitory occultation of the central Xray source, are indeed consistent with those of gas condensations in the BLR (see Fig. 3). Thus, we can safely conclude that BLR clouds do produce temporary absorption as observed in AGN Xray spectrum and significantly contribute to this observed absorption phenomenon. Very few cases are in fact explainable with occulting clouds located at distances from the AGN centre that are (even significantly) larger than the representative BLR size, R_{BLR}, and we may consider those as not belonging to the BLR itself. In our work, we consider as representative of the external boundary of the BLR ensemble of clouds a distance ∼6R_{BLR}, roughly corresponding to (comparable with) the sublimation radius for our AGN sources (see Sect. 5 and references therein). According to this choice, we have discussed in detail results pertaining to occultation events explainable with gas condensations that we regard as being part of the BLR (Sects. 5.1 and following). Thus, this sample of temporary Xray absorption events, whose detection we have discussed in Paper I, is indeed valuable, since the present analysis in terms of occultation events gives the opportunity to ‘see’ the individual gas condensation producing each occultation event, through its effect as an Xray absorber.

From Figs. 3 and 4 it is clear that, for any given geometry of the occultation, no significant trend of increase for R_{cloud}/R_{X} (=f_{cloud}) appears with increasing r_{c}/R_{BLR}. For example, in the case of central geometry it is R_{cloud}/R_{X} ∈ [0.4, 1] and this range is basically unchanged independently of the distance. It is just the degeneracy of explanations of each single event (due to unknown geometry of the occultation) that allows the possibility of clouds with normalised physical size R_{cloud}/R_{X} > 1 to be involved. Our present conclusion on the substantial lack of dependence of cloud size range on the distance from the central black hole is in agreement with the assumption of Paper II that the number density distribution in size of clouds is independent of distance.

Availing ourselves of this outcome, we have consistently used the results of Paper II to verify that indeed the analysis of our sample of occultations indicates that in general the size of occulting clouds turns out to be ‘limited’ to at most a few times the size of the Xray source, excluding the dominance of ‘very large’ clouds. In fact, for any eclipse event, the size of the occulting cloud depends of course on the geometry of the occultation, but in our analysis we identified the following two main points supporting our present conclusion. a) For a few of our sources (6 out of 14) the cloud size is certainly strongly limited by the estimate of cloud maximum size R_{2}, deriving from our results for the global model of the cloud ensemble devised following Paper II procedure with the adapted parameters for the cloud location outer boundary and for the number of eclipse events taken into account for the AGN sources in our sample (see Sect. 5 and 5.1). Indeed, for these six AGNs we obtain R_{2}/R_{X} in the range ∼4 − 7, that is a maximum size ‘small’ enough to define a significant limitation to the radius of an occulting cloud: for those sources it certainly must be R_{cloud}/R_{X} ≤ R_{2}/R_{X} ∼ 4 − 7. b) More in general, using the properties of the number density distribution in size of clouds, derived in Paper II, we showed (see Sect. 5.1) that in any case (i.e. for any of our AGN sources), it is highly improbable that an eclipse in our sample is produced by a cloud with R_{cloud}/R_{X} ≥ 5 − 6. Very large clouds may of course be present in the cloud ensemble, but their numbers are negligible with respect to those of clouds with characteristic size comparable with (or at most a few times) that of the Xray source and their presence is very unlikely to be revealed directly by shorttime Xray absorption events, like those we are studying.

From the analysis of solutions matching our detected events observables, we conclude that the inferred values for gas number density n_{H} turn out to be within the range of values derived from broad emission lines analysis, typically 10^{9} − 10^{11} cm^{−3}, and this outcome is independent of the geometry of the eclipse. Clearly, the specific value of the Xray absorber column density N_{HX} influences the resulting values of the gas number density; however, within the rather narrow range N_{HX} ∼ 5 × 10^{22} − 2 × 10^{23} cm^{−2}, identified by Xray spectroscopical analysis (see Bianchi et al. 2012), such an effect does not change our conclusions about the derived cloud gas number density, which is consistent anyway with the range of values implied by broad emission lines investigation.
Our results are also confirmed by the following considerations.
We have discussed a comparison of our results with those of Markowitz et al. (2014) work, finding no significant elements of contrast, especially since M14 work investigates discrete Xray absorption events whose parameters are expected to be in a range somewhat complementary to our case because of the RXTE data set the authors deal with (typically significantly longer events detectable and detected).
Also, we have exploited the results of the spectral analysis of Mrk 766 discussed in R11 by comparing them with those obtained with our present procedure for the 3 eclipse events that R11 and the present work have in common. We find substantial agreement on physical parameters derived using the two different approaches in our present work and in R11. In fact, even if the initial (first step) method used by R11, based on the analysis of the hardness ratio light curve is analogous to the one we used, the relevant difference is that in the case of R11 this method is used only to find the time intervals of the possible eclipses. All the subsequent analysis is done through a complete spectroscopic analysis of the data. Therefore, the parameters derived by R11 can be considered a ‘gold standard’ to calibrate our analysis of the light curves, which does not include the analysis of the spectral data. This certainly gives support to our method of search for and analysis of the eclipse events in the sample examined, despite the simple modelling adopted, and it is particularly relevant since our present procedure of analysis has the quality of being widely usable for many more sources than those for which spectral analysis can be performed. Overall, the consistency of most of our results again indicates the reliability of our proposed scenario for explaining Xray shortterm variations as due to temporary absorption by a cloudlike gas condensation eclipsing the Xray source while moving under the gravitational influence of the AGN central black hole. Also, it supports the validity of the procedure we devised to interpret the combination of observables that characterise each single detected eclipse event in order to extract physical information on the individual gas condensation crossing the line of sight to the Xray source and temporarily occulting it. The resulting physical properties of these gas condensations indeed confirm the plausibility of their being part of the same ensemble of cloudlike gas structures that contribute to the formation of the observed broad emission lines.
Acknowledgments
GTC thanks INAFOsservatorio Astrofisico di Arcetri for support. We also thank the anonymous referee for interesting and useful comments for improvement of the paper.
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Appendix A: Investigation method for reliable eclipse event detection: Light curve analysis
Referring to Paper I for a full description of our method of analysis, here we summarise its main features. For the sources in our sample, we extracted the light curves for each observation with the required duration available in XMMNewton and Suzaku archives. The choice the energy bands [soft and hard: (24) keV and (510) keV respectively] stems from our analysis of Paper I aiming to maximise the hardness ratio variations expected as a result of absorption variability induced by the temporary occultation of the source by a gaseous structure with Xray absorption equivalent column density n_{H} in the range [5 × 10^{22} − 2 × 10^{23}] cm^{−2}; this range of column density excludes absorbers typical of ‘Comptonthick’ sources, consistently with our sample choice, as explained in Sect. 2 of Paper I, and, conversely, is representative of physical conditions expected for typical ‘clouds’ of the BLR.
In Paper I we have fully analysed the hardness ratio light curves in our sample by fitting them with a constant plus a number of variation components modelled with Gaussian curves superimposed to the constant level. In general, we started from the simplest possible model, the ‘zerostep’, defined by just a constant in time (HR = const.=K_{0} for the whole duration of the observation), and then adding further components, thus generating a progressively more complex model. Each one of the events identified in the fitting procedure of a light curve has been evaluated in terms of its significance by applying the Ftest to the comparison of the two models fitting with and without that particular event component. Thus, the nth variation component in a given light curve is characterised by a null probability value F_{n}, representing the significance of the specific component (as it is explained in Paper I and reported explicitly in Table 2 of that same paper): the lower the value of F_{n} is, the higher is the significance of the corresponding event. At the end of the fitting process, each one of the analysed light curves is characterised by the parameters of the final fit (Nth step) defined by a constant, K_{n}, that represents the constant level of hardness ratio underlying the variation events, plus N Gaussian shaped variation components. Thus, the ensemble of the parameters defining the final fit comprises the constant K_{n}, the parameters defining each of the various Gaussian components (peak value of the nth component, A_{n}, width parameter w_{n} and location in time along the light curve, t_{n}, with index n = 1...N) and the respective significance, represented by the null probability value F_{n} associated with the nth component according to an Ftest analysis (see Paper I). Table 2 of Paper I includes the parameters relevant to the present analysis for all the light curves in our sample, but it does not report the width parameter of the single events detected.
From the ratio of the peak value, A_{n}, of each event to the underlying constant level of the final fit of a light curve, K_{n}, we evaluated the peak fractional hardness ratio variation, which is then ΔHR/HR ≡ A_{n}/K_{n} for the nth event of a light curve in which we detected a total of N distinguishable HR Gaussian shaped variation components. The width parameter w_{n} was not discussed in Paper I, but in the present work it is used to determine the duration of the eclipse event, that we define as τ_{0} (see Sect. 2). Within our interpretative framework, we measure the total observed duration of the eclipse, from its start (when the occulting absorber begins to intercept the source projection on the sky) to its end (when the rear brim of the moving absorber leaves the source projection) as the total time interval, τ_{0}, over which the HR variation event extends in the light curve. To evaluate τ_{0} from our fits we have chosen to define it operatively as the width of the Gaussian component, describing the HR variation event, at a reference fractional height of its peak value close enough to its base to be representative of the total time extension of the variation. Specifically, we define τ_{0} as the full width (FW) of the Gaussian at 1/20 of its peak value (A_{n} for the nth component in a given light curve), that is, τ_{0} = FW(A_{n}/20). In terms of the full width half maximum (FWHM ≡ 2.3548w_{n}) of the Gaussian component, we have τ_{0} ≃ 4.9w_{n} ≃ 2.08FWHM. With the definition above, we consider this estimate of τ_{0} as our measure of the actual duration of each occultation candidate event we have revealed in our analysis.
In Table A.1, for all the 59 detected events that fulfil the conditions discussed in Sect. 5 and are subsequently analysed in that same section, we show the relevant quantities as derived from the final fit parameters, including the event duration τ_{0} and the null probability value F, that defines the event significance.
Specifics of each event among the 57 under examination: AGN source, observing instrument (S for Suzaku, X for XMMNewton), date of the observation, order number of the event for any given source, ‘significance’ log F (as determined in Paper I), the derived observable quantity ΔHR/HR, normalised event duration and duration τ_{0} (in ks) of the event.
Appendix B: Event ‘observables’ and cloud parameters
B.1. Relation between C_{F} and Δ(HR)/HR
One of our observed quantities is the fractional hardness ratio variation Δ(HR)/HR of the detected eclipse event in the examined light curve. As we have shown in Paper I and mentioned in Sect. 2, it must be Δ(HR)/HR > 0.1 in order to guarantee an unambiguous interpretation of the corresponding event as an eclipse candidate.
In Paper I, convolving theoretical spectral models (absorbed powerlaw models) with the response matrices of both XMMNewtonEPIC/PN and SuzakuXIS instruments and using the XSPEC package, we have chosen the energy bands defining the most appropriate hardness ratio for our analysis, namely HR = F(5 − 10 keV)/F(2 − 4 keV) (where F represents the resulting photon flux in the specified energy range). We have then analysed the dependence of this hardness ratio HR on parameters γ, n_{H}, C_{F} (respectively, photon spectral index, hydrogen column density of the intervening absorber and covering factor of the absorber itself) and defined a ‘synthetic’ HR = HR(γ, n_{H}, C_{F}, R), where R is the ratio between the normalisation of the reflected to the direct power law components of the spectrum, for the case of no reflected component (R = 0). The result is (see Sect. 4.3 of Paper I) that the behaviour of the fractional variation in hardness ratio Δ(HR)/HR with changing column density of the eclipsing absorber, N_{H} = N_{HX}, for any fixed value of the covering factor C_{F}, is essentially independent of the specific value of γ within a reasonable range representative of AGN Xray sources (see in particular Fig. 6 of Paper I, where we exemplify the results of our analysis by showing explicitly how the value of Δ(HR)/HR for any given N_{HX} and C_{F} of interest for our event sample remains unchanged when assuming γ = 2 or 2.3).
We thus discuss the results pertaining to ‘simulations’ effected with synthetic spectra characterised by γ = 2 as a representative value and we consider the predicted hardness ratio HR as a function of of the values of N_{HX} and C_{F}, $HR=HR({N}_{\mathrm{HX}},{C}_{\mathrm{F}})$.
In Fig. B.1, we show the expected fractional variation in hardness ratio from synthetic spectra with γ = 2, for occultations by absorbers with different column density N_{HX}, here indicated with n_{H}, and with different covering factor conditions in the case of type 1 sources and of Comptonthin type 2 sources as well (the latter are represented by the dotted lines below each of the type 1 curves, corresponding to a given C_{F} value). This figure suggests the possibility of relating the observed value of the fractional variation of hardness ratio Δ(HR)/HR with the corresponding maximum covering factor for reasonable values of the eclipsing cloud column density. The inferred relationship between Δ(HR)/HR and C_{F} for the range of interest of n_{H} values, N_{H} = 5 × 10^{22} − 2 × 10^{23}cm^{−2}, is shown in Fig. B.2. Based on these results, we find a rather well established relation between the peak fractional variation of hardness ratio, Δ(HR)/HR, and the corresponding maximum covering factor attained during an occultation due to the passage of a ‘typical’ BLRlike cloud characterised by a given Xray absorbing column density N_{HX}.
Fig. B.1. Fractional variation of hardness ratio as a function of N_{HX} for fixed values of the covering factor, C_{F}. Blue labels specify the value of C_{F} for which each group of curves is obtained. For each C_{F} value chosen, solid black curves represent the results of SuzakuXIS simulations for type1 sources, whereas the longdashed red ones show those derived for type1 with XMMNewtonEPIC/PN. When present, the third shortdashed red line of each C_{F}value group of curves represents the case for type2 XMM simulations. 
The results discussed above are important to our purposes in a twofold way.
i) Provided the absorber column density falls in a relatively narrow range around ∼10^{23}cm^{−2}, the relation between Δ(HR)/HR and C_{F} turns out to be rather tight and we can therefore somewhat translate the observed value of Δ(HR)/HR for a given event into an estimate of the maximum covering factor ${C}_{\mathrm{F}}={({C}_{\mathrm{F}})}_{\mathrm{max}}$ characterising that occultation event.
ii) Since from a model occultation, characterised by a given geometry (normalised impact parameter f_{d}) and a value of the normalised projected size of the absorber (f_{abs}), the corresponding maximum covering factor C_{F} can be analytically calculated (see following Appendix B2), we can conversely evaluate the value of relative variation of the hardness ratio Δ(HR)/HR expected for that specific model occultation, when observed by XMM or Suzaku instruments.
B.2. Covering factor and geometry of occultation
In our simple hypothesis of spherical symmetry for both the Xray source and the cloud (see Sect. 3.1), the value of the maximum covering factor attained during an occultation event can be geometrically computed from the evaluation of the maximum intersection area of the circles that represent the projections on the sky of the eclipsing cloud and of the source during the occultation itself. The value of C_{F} is obtained by normalising this intersection area by the area ($\pi {R}_{\mathrm{X}}^{2}$) of the circular projection of the Xray source on the sky plane and it is a function of the two quantities $({f}_{\mathrm{abs}},{f}_{\mathrm{d}})$, that is, of the absorber radius and of the impact parameter, both normalised by the source radius R_{X} (see Eq. (1)).
Extensive calculations of the value of C_{F} for wide ranges of the parameters $\{{f}_{\mathrm{abs}},{f}_{\mathrm{d}}\}$ allowed us to derive the relation among {f_{abs},f_{d},C_{F}}, defining a matrix that enables us to obtain the couples {f_{abs},f_{d}} that produce any given value of the maximum covering factor C_{F}. From this matrix we obtained the dependence of f_{abs} on f_{d} and C_{F} values, namely ${f}_{\mathrm{abs}}=\mathcal{F}({C}_{\mathrm{F}},{f}_{\mathrm{d}})$, defining, for each given value of the maximum covering factor C_{F}, the corresponding function ${f}_{\mathrm{abs}}={\left({f}_{\mathrm{abs}}({f}_{\mathrm{d}})\right)}_{{C}_{\mathrm{F}}}$, and, as a consequence, the value of ${Y}_{0}={[{(1+{f}_{\mathrm{abs}})}^{2}{f}_{\mathrm{d}}^{2}]}^{1/2}$ as ${Y}_{0}={({Y}_{0}({f}_{\mathrm{d}}))}_{{C}_{\mathrm{F}}}$.
Fig. B.2. Δ(HR)/HR as a function of C_{F} for different values of the column density. The dotted line curve corresponds to N_{H} = 5 × 10^{22}cm^{−2}, the solid curve to N_{H} = 1 × 10^{23}cm^{−2}, the shortdashed curve to N_{H} = 2 × 10^{23}cm^{−2}, the longdashed curve to N_{H} = 3 × 10^{23}cm^{−2}, and the dashdotted curve refers to N_{H} = 5 × 10^{23}cm^{−2} 
Fig. B.3. Black curves represent f_{abs} as a function of the normalised impact parameter, f_{d}, for a given value of C_{F}. Red curves show Y_{0} (see Eq. 2) again as a function of f_{d} and constant C_{F}. Red curves are labelled with C_{F} values; in the same order, increasing upwards, the same labels also apply to the black curves. 
This is illustrated in Fig. B.3, where we represent the results for the intervals f_{abs} = 0 − 6 and f_{d} = 0 − 5. Referring to that exemplifying figure, once we have derived an estimate for the value of C_{F} from the observed Δ(HR)/HR and chosen the value for f_{d} (normalised impact parameter), we can determine the value of the parameter f_{abs} (normalised size of the absorber projected on the plane of the sky) from the black curve ${f}_{\mathrm{abs}}={f}_{\mathrm{abs}}({f}_{\mathrm{d}})$ identified by the specific value of C_{F} estimated, and, in turn, the value of Y_{0} must be on the corresponding curve ${Y}_{0}={\left({Y}_{0}({f}_{\mathrm{d}})\right)}_{{C}_{\mathrm{F}}}$. Thus, in Fig. B.3 each of the plotted black curves represents the locus of points defined by the couples of values $\{{f}_{\mathrm{abs}},{f}_{\mathrm{d}}\}$ that produce a given value of the maximum covering factor C_{F}, labelling the curve, and those couples identify the corresponding values of Y_{0} associated with that same C_{F}. Again referring to Fig. B.3, it is interesting to point out that :
(1) for any given value of the impact parameter f_{d} an increase in the value of the maximum covering factor C_{F} requires an increase in the absorber normalised projected radius f_{abs} and of the corresponding Y_{0};
(2) the value of C_{F} estimated from Δ(HR)/HR determines the range of possible values of Y_{0}, plotted in Fig. B.3 as a function of the impact parameter f_{d}; reasonably assuming eclipsing clouds such that the absorber radius R_{abs} is at most ∼ a few$\times {R}_{\mathrm{X}}$, for ${C}_{\mathrm{F}}<1$, this range of Y_{0} turns out to be rather limited, decreasing its extent with decreasing C_{F} value.
B.3. Determining the condition on minimum value of normalised size of Xray absorber ${R}_{\mathrm{abs}}/{R}_{\mathrm{X}}$
Based on the results discussed in the previous subsections of the present Appendix B, we can derive the absolute minimum value for ${f}_{\mathrm{abs}}={R}_{\mathrm{abs}}/{R}_{\mathrm{X}}$ for absorbers that can be detected with our procedure of analysis [see Eq. (7) in Sect. 3].
Under the conditions discussed in Appendix B.1, for a given event the observed value of Δ(HR)/HR can be translated into an estimate of the maximum covering factor ${C}_{\mathrm{F}}={({C}_{\mathrm{F}})}_{\mathrm{max}}$ characterising that occultation event. Therefore, once for any given event we have translated the observed value of Δ(HR)/HR into a representative value of C_{F}, we can also exploit our results on the dependence of the maximum covering factor C_{F} on the values of the geometrical parameters of the occultation $\{{f}_{\mathrm{abs}},{f}_{\mathrm{d}}\}$, discussed in the previous Appendix B.2, and immediately derive the corresponding minimum value of ${R}_{\mathrm{abs}}/{R}_{\mathrm{X}}$ (i.e. f_{abs}) for the given event. In fact, Eq. (6) immediately shows that for a given eclipse event the minimum value of ${R}_{\mathrm{abs}}/{R}_{\mathrm{X}}$ is attained for ${f}_{\mathrm{d}}=0$ and this is also clear from the curves ${f}_{\mathrm{abs}}={f}_{\mathrm{abs}}({f}_{\mathrm{d}})$ for each given value of C_{F} in Fig. B.3.
For a ‘central’ occultation geometry, that is for f_{d} = 0, the maximum covering factor value is geometrically given by the simple relation ${C}_{\mathrm{F}}={({R}_{\mathrm{abs}}/{R}_{\mathrm{X}})}^{2}$, so that, once determined the C_{F} characterising the given event from Δ(HR)/HR, the minimum value of the nondimensional size of the absorber is derived as ${({R}_{\mathrm{abs}}/{R}_{\mathrm{X}})}_{\mathrm{relmin}}={C}_{\mathrm{F}}^{1/2}$; the subscript ‘relmin’ indicates that this is a minimum value referring to the specific eclipse event considered.
A second significant consequence derives from the fact that for ‘bona fide’ candidate eclipse events it must be Δ(HR)/HR > 0.1 (see Sect. 2). On the basis of the inferred relationship between Δ(HR)/HR and C_{F} and taking into account that the values of interest for N_{HX} are between 5 × 10^{22} and 2 × 10^{23} cm^{−2}, the condition Δ(HR)/HR > 0.1 implies ${C}_{\mathrm{F}}\gtrsim 0.15$ (see Figs. B.1 and B.2). From this we conclude that for all the events plotted in Fig. 1 the ‘absorbers’ responsible for such eclipse events must have a radius R_{abs} such that the corresponding nondimensional value ${R}_{\mathrm{abs}}/{R}_{\mathrm{X}}$ is ≥0.15^{1/2} ∼ 0.4, that is
$$\begin{array}{c}\hfill \frac{{R}_{\mathrm{abs}}}{{R}_{\mathrm{X}}}\ge 0.4\equiv {\left(\frac{{R}_{\mathrm{abs}}}{{R}_{\mathrm{X}}}\right)}_{\mathrm{MIN}},\end{array}$$
corresponding to Eq. (7) of Sect. 3. The above condition [Eq. (7)] thus defines the absolute minimum value of the normalised Xray absorber radius consistent with the lower limit value for C_{F} required to have reliable eclipse candidates defined above. Hence, all the clouds corresponding to eclipse events shown in the plot of Fig. 1 must have a radius R_{abs} that satisfies the condition
$$\begin{array}{c}\hfill {R}_{\mathrm{abs}}\ge 0.4{R}_{\mathrm{X}}={({R}_{\mathrm{abs}})}_{\mathrm{MIN}}.\end{array}$$
The limiting condition expressed by relation (7) shows that within our framework for the eclipse scenario there certainly is a selection effect related to the central black hole mass of the AGN, M_{BH}. This selection effect comes from our analysing procedure: absorbers that are ‘very small’ with respect to the Xray source size R_{X} (${R}_{\mathrm{abs}}/{R}_{\mathrm{X}}<0.4$) cannot induce any detectable and reliable eclipsing effect on the Xray source, because of the very small maximum covering factor they can produce, with a resulting nonsignificant fractional hardness ratio variation (i.e. below 0.1, see Sect. 2); this, in turn, implies that the minimum physical size of an absorber capable of producing a reliable and detectable eclipse increases linearly with increasing black hole mass of the AGN, since we have assumed ${R}_{\mathrm{X}}\simeq 2.5{R}_{\mathrm{S}}$, with R_{S} Schwarzschild radius (see Sect. 2) and thus it is ${R}_{\mathrm{X}}\propto {M}_{\mathrm{BH}}$.
Appendix C: Derivation of a normalised effective depth to Xray absorption for a spherical cloud
In Sect. 4.3 we discuss the estimate of the occulting cloud gas number density n_{H}, that for a chosen value of the parameter N_{HX}, equivalent Xray absorption column density, depends on the size of the occulting cloud normalised by that of the Xray source, f_{cloud}, and on the geometry parameter of the eclipse, f_{d}. Indeed, since our clouds are supposedly spherical, we have to derive an evaluation of the representative ‘effective’ depth of the cloud as an Xray absorber, ${l}_{\mathrm{eff}}=2{f}_{\mathrm{eff}}{R}_{\mathrm{X}}$, through which we can reliably estimate the gas number density as
$$\begin{array}{c}\hfill {n}_{\mathrm{H}}=\frac{{N}_{\mathrm{HX}}}{{l}_{\mathrm{eff}}}.\end{array}$$
In the definition of l_{eff}, we have introduced the quantity f_{eff} as a sort of normalised effective ‘radius’, for immediate comparison with f_{cloud}. Considering the condition of maximum coverage, that is, when (referring to Fig. 2) ${C}_{o}{C}_{\mathrm{X}}={f}_{\mathrm{d}}{R}_{\mathrm{X}}$, a spherical gas condensation offers to the incoming Xray photons a different physical depth depending on the position {x, y} on the sky plane of any given point of the portion of the absorber projection superimposing on that of the Xray source, that is the occultation region on the sky plane. We can thus derive the effective depth l_{eff} by appropriately ‘averaging’ on the actual depths parallel to the line of sight (along the zdirection, as shown in panel (b) of Fig. C.1) in the region of superposition of the cloud projection on the sky plane onto that of of the Xray source. Here we illustrate our approach to a simplified but representative derivation of f_{eff} = l_{eff}/2.
Referring to Fig. C.1 for the coordinate system and using all lengths normalised to the source size R_{X}, we can introduce an angle θ on the sky plane {x, y}; each θ value identifies in the spherical eclipsing cloud a circular section parallel to the plane {x, z} by defining its distance, y(θ), from the maximum cross section of the cloud, lying on the plane {x, z} itself, as y(θ) = f_{cloud}sin(θ) and its radius r(θ) as r(θ) = f_{cloud}cos(θ). The maximum cross section of the cloud, lying on the plane {x, z} and whose centre is the cloud centre C_{0}, is defined by θ = 0 rad. On each of these circular sections of the cloud we can identify the height x of any of its chords parallel to the line of sight direction to the observer (i.e. the zaxis direction) from the plane {y, z} (on which the circular section centre lies), by introducing the angle ϕ from the circular section diameter parallel to the zaxis. On the circular section identified by a given value of θ, the height on the plane {y, z} of a given chord along the line of sight direction is x(θ, ϕ) = r(θ)sin(ϕ) = f_{cloud}cos(θ)sin(ϕ) and, most importantly, its length (again normalised to R_{X}) is l_{ch}(θ, ϕ) = 2r(θ)cos(ϕ) = 2f_{cloud}cos(θ)cos(ϕ). In order to properly average on all the lengths effectively representing a depth the Xray photons encounter along the line of sight, it is necessary to appropriately define the interval of values of θ and, for each given θ value, the interval of values of ϕ; these intervals depend on the specific occultation, namely on the parameters $\{{f}_{\mathrm{cloud}},{f}_{\mathrm{d}}\}$, that define the shape of the intersection region of the two circles representing respectively the projection on the sky plane of the source and of that of the eclipsing cloud; this region is the occultation region.
For a central geometry of the occultation, that is f_{d} = 0, (and provided f_{cloud} ≤ 1, which is always the case for our detected eclipse events when f_{d} = 0 is chosen) this averaging is very simply obtained as
$$\begin{array}{c}\hfill {f}_{\mathit{eff}}=\frac{{f}_{\mathrm{cloud}}}{{(\pi /2)}^{2}}{\displaystyle {\int}_{0}^{\frac{\pi}{2}}}cos\theta d\theta {\displaystyle {\int}_{0}^{}}\frac{\pi}{2}cos\varphi d\varphi =\left(\frac{4}{{\pi}^{2}}\right){f}_{\mathrm{cloud}},\end{array}$$(C.1)
In the present Appendix, we devote our attention to the exemplifying cases of noncentral geometry that we have shown in the main text, that is occultations with an impact parameter ${f}_{\mathrm{d}}\ge 1$. In these cases, for a realistic and reliable estimate of n_{H} it is more appropriate to determine the maximum value of the angle θ (see exemplifying Fig. C.1), and then derive the minimum value of ϕ as well as its maximum value for each of the cloud cross sections parallel to the plane {x, z} and characterised by a value of θ ∈ [0, θ_{max}], that is ϕ_{min}(θ) and ϕ_{max}(θ). These limiting values depend on the geometry of the occultation, that is on the specific values of f_{cloud} and f_{d}. Two different regimes can be identified (see also Fig. C.2a and C.2b). The first one pertains to the cases in which
Fig. C.1. Schematic representation of the maximum coverage condition for an exemplifying case of noncentral occultation geometry; here ${f}_{\mathrm{d}}=2$. Panel (a) shows in red the projection on the {x, y} sky plane of the Xray source, whose centre is ${C}_{\mathrm{X}}$, whereas the projection of the occulting cloud is in black, with centre in C_{0}. In panel (b) the same configuration is seen from a different perspective, in which the line of sight to the observer is shown as the zaxis direction, along which Xray photons are coming from the Xray source from the left. See the main text for the meaning of other notation. 
$$\begin{array}{c}\hfill A)\phantom{\rule{1em}{0ex}}{f}_{\mathrm{cloud}}<{(1+{f}_{\mathrm{d}}^{2})}^{1/2},\end{array}$$(C.2)
that is those geometrical conditions in which the intersection points between the two circumferences centred in C_{0} and C_{X} respectively are located at a distance from the yaxis (along which the cloud centre is supposed to move) smaller than the one between C_{0} and C_{X}, that is, f_{d} (see Fig. C.2a). In this case, θ_{max} is defined as the angle subtended by the arc from the point of intersection of the cloud projection circumference with the xaxis (θ = 0) to the point (identified as P in Fig. C.2a) of intersection of the two circumferences delimiting the source and cloud projections on the sky plane {x, y}. Therefore, we have
$$\begin{array}{c}\hfill {\theta}_{\mathrm{max}}=arctan\left(\frac{{y}_{\mathrm{P}}}{{x}_{\mathrm{P}}}\right),\end{array}$$(C.3)
where $({x}_{\mathrm{P}},{y}_{\mathrm{P}})$ are the coordinates of point P on the sky plane and are defined as
$$\begin{array}{c}\hfill {x}_{\mathrm{P}}=\frac{{f}_{\mathrm{d}}^{2}+{f}_{\mathrm{cloud}}^{2}1}{2{f}_{\mathrm{d}}}\phantom{\rule{1em}{0ex}}{y}_{\mathrm{P}}={({f}_{\mathrm{cloud}}^{2}{x}_{\mathrm{P}}^{2})}^{1/2}.\end{array}$$(C.4)
When condition (C.2) is satisfied, the maximum value for the angle ϕ is ϕ_{max}(θ) = π/2 independently of the specific θ value (i.e. of the circular section parallel to the plane {x, z} considered and whose portion of interest, projected on the sky plane, {x, y}, is identified as the green line parallel to xaxis in Fig. C.2a); on the other hand the corresponding minimum value of ϕ does indeed depend on the value of θ, as well as on those of f_{cloud} and f_{d}, following the relation
$$\begin{array}{c}\hfill {\varphi}_{\mathrm{min}}=arcsin\left(\frac{{f}_{\mathrm{d}}{(1{f}_{\mathrm{cloud}}^{2}{sin}^{2}(\theta ))}^{1/2}}{{f}_{\mathrm{cloud}}cos(\theta )}\right)\u037e\end{array}$$(C.5)
the angle ϕ thus varies in the interval [ϕ_{min}(θ),π/2]. In the second case we deal with
$$\begin{array}{c}\hfill B)\phantom{\rule{1em}{0ex}}{f}_{\mathrm{cloud}}\ge {(1+{f}_{\mathrm{d}}^{2})}^{1/2},\end{array}$$(C.6)
implying that the distance from the yaxis of the intersection points between the two circumferences centred in C_{0} and C_{X} is now larger than, or equal to, f_{d} (see Fig. C.2b). In the present case, the maximum value of the angle θ is the one defined by the xaxis and the radius C_{0}K, where point K is on the cloud projection circumference and at a distance from the xaxis equal to the source unit radius, as shown in Fig. C.2b, so that $sin({\theta}_{\mathrm{max}})=1/\overline{{C}_{0}K}$ and thus
$$\begin{array}{c}\hfill {\theta}_{\mathrm{max}}=arcsin\left(\frac{1}{{f}_{\mathrm{cloud}}}\right).\end{array}$$(C.7)
From Fig. C.2b it is apparent that ϕ_{max} on a given circular section of the spherical occulting cloud and parallel to plane (x, z) depends on the value of θ characterising the specific circular section. Defining θ_{p} as the value of θ formed by the xaxis and the radius from C_{0} to the point P of intersection between the two circumferences, it is
$$\begin{array}{c}\hfill {\theta}_{\mathrm{p}}=arctan\left(\frac{{y}_{\mathrm{P}}}{{x}_{\mathrm{P}}}\right),\end{array}$$
where ${x}_{\mathrm{P}}$ and ${y}_{\mathrm{P}}$ are again given by Eq. (C4); for $\theta \in [0,{\theta}_{\mathrm{p}}]$ we have ϕ_{max} = π/2, whereas for $\theta \in ({\theta}_{\mathrm{p}},{\theta}_{\mathrm{max}}]$ simple geometrical calculations give
$$\begin{array}{c}\hfill {\varphi}_{\mathrm{max}}=arcsin\left(\frac{{f}_{\mathrm{d}}+{(1{f}_{\mathrm{cloud}}^{2}{sin}^{2}(\theta ))}^{1/2}}{{f}_{\mathrm{cloud}}cos(\theta )}\right).\end{array}$$(C.8)
As for ϕ_{min}, its dependence on θ and on the parameters $({f}_{\mathrm{cloud}},{f}_{\mathrm{d}})$ is correctly defined by Eq. (C.5) also in the present case of ${f}_{\mathrm{cloud}}\ge {(1+{f}_{\mathrm{d}}^{2})}^{1/2}$.
With these evaluations of the new limits of integration, we can apply the same averaging method shown in Eq. (C.1), that in the general case for ${f}_{\mathrm{d}}\ge 1$ gives
$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {f}_{\mathit{eff}}& =\frac{{f}_{\mathrm{cloud}}}{{\theta}_{\mathrm{max}}}{\displaystyle {\int}_{0}^{{\theta}_{\mathrm{max}}}}d\theta \frac{cos\theta}{{\varphi}_{\mathrm{max}}(\theta ){\varphi}_{\mathrm{min}}(\theta )}{\displaystyle {\int}_{{\varphi}_{\mathrm{min}}(\theta )}^{{\varphi}_{\mathrm{max}}(\theta )}}cos\varphi d\varphi \hfill \\ \hfill & =\frac{{f}_{\mathrm{cloud}}}{{\theta}_{\mathrm{max}}}{\displaystyle {\int}_{0}^{{\theta}_{\mathrm{max}}}}d\theta \frac{cos\theta}{{\varphi}_{\mathrm{max}}(\theta ){\varphi}_{\mathrm{min}}(\theta )}G(\theta ),\hfill \end{array}\end{array}$$(C.9)
where
$$\begin{array}{c}\hfill G(\theta )=sin({\varphi}_{\mathrm{max}}(\theta )sin({\varphi}_{\mathrm{min}}(\theta )\end{array}$$
and θ_{max}, ϕ_{max}, ϕ_{min} depend on the specific case, as illustrated above. In each of the cases discussed, at least ϕ_{min} always depends on f_{cloud} and f_{d}, so that in general it is
$$\begin{array}{c}\hfill {f}_{\mathrm{eff}}={f}_{\mathrm{eff}}({f}_{\mathrm{cloud}},{f}_{\mathrm{d}}).\end{array}$$
Fig. C.2. Panels (a) and (b) illustrate the maximum coverage condition with Xray source and occulting cloud seen projected on the sky plane {x, y} respectively for case A), in which ${f}_{\mathrm{cloud}}<{(1+{f}_{\mathrm{d}}^{2})}^{1/2}$, and for case B), in which ${f}_{\mathrm{cloud}}\ge {(1+{f}_{\mathrm{d}}^{2})}^{1/2}$; see the main text for details. 
All Tables
Parameters and quantities, derived within the newly calculated Paper II model described in the text, for sources with detected reliable occultations.
Specifics of each event among the 57 under examination: AGN source, observing instrument (S for Suzaku, X for XMMNewton), date of the observation, order number of the event for any given source, ‘significance’ log F (as determined in Paper I), the derived observable quantity ΔHR/HR, normalised event duration and duration τ_{0} (in ks) of the event.
All Figures
Fig. 1. Observables plot: Fractional hardness ratio variability, Δ(HR)/HR, versus the normalised eclipse duration, τ_{ecl} = τ_{0}/Δt_{ecl}, for each reliable eclipse event detected in Paper I. 

In the text 
Fig. 2. Schematic illustration of an eclipse event as seen on the sky plane from the observer line of sight (zdirection, normal to the page). The distance, HC_{X} ≡ d, between the source centre, C_{X}, and the straight line trajectory of the absorber projection centre, C_{0}, during the eclipse, represents a sort of ‘impact parameter’, d, for the occultation and is depicted in red. 

In the text 
Fig. 3. Best match solutions obtained for our detected events in the case of central geometry. Yellow dots represent best matches for NGC 3783 detected eclipse events, cyan dots show best matches for those of MCG 063015, green dots show matches obtained for events referring to all other sources with MBH < 6.5, red dots are for all other sources with MBH > 7, and finally blue dots are best match events referring to intermediate mass sources with 6.5 < MBH < 7, where MBH ≡ log(M_{BH}/M_{⊙}). 

In the text 
Fig. 4. Best match solutions for cloud physical size obtained for our detected events in different geometries: circles again refer to central geometry (f_{d} = 0), but we also show matches for f_{d} = 1 (filled hexagons), f_{d} = 2 (filled triangles) and f_{d} = 4 (large filled triangles). Colour code is the same as in Fig. 3. In each panel hollow circles or triangles represent event descriptions that, for the specific f_{d} value, must be discarded, since the occulting clouds are located farther than the chosen outer border of the cloud ensemble; the corresponding events are only explainable with f_{d} = 4 or larger. 

In the text 
Fig. 5. Gas number density from best match solutions obtained for our detected events in different geometries. Here we impose the constraints on maximum normalised physical size of the cloud described for Fig. 4 and colour codes are the same as in Fig. 4. 

In the text 
Fig. 6. Gas number density from best match solutions obtained for our detected events and for two different values of the absorbing column density N_{HX}, as shown by the labels in each panel. Here we impose the constraints on maximum normalised physical size of the cloud described for Fig. 4 and colour codes are the same as in Fig. 4. 

In the text 
Fig. 7. Cloud physical size for occultation matching solutions plotted versus cloud distance normalised to the gravitational radius of the AGN source specific to the eclipse event. Symbols and colour codes are the same as in Fig. 4. 

In the text 
Fig. 8. Best matching solutions for the three events that we have detected and were also spectrally analysed by Risaliti et al. (2011), chosen so as to maximise the agreement with R11 results for N_{HX} and C_{F} values as reported in their paper. Blue points indicate a solution with N_{HX} = 10^{23} cm^{−2}, whereas red ones refer to cases in which N_{HX} = 2 × 10^{23} cm^{−2}; circular dots show central geometry solutions (f_{d} = 0) and triangles indicate a case with impact parameter of the occultation f_{d} = 2. 

In the text 
Fig. B.1. Fractional variation of hardness ratio as a function of N_{HX} for fixed values of the covering factor, C_{F}. Blue labels specify the value of C_{F} for which each group of curves is obtained. For each C_{F} value chosen, solid black curves represent the results of SuzakuXIS simulations for type1 sources, whereas the longdashed red ones show those derived for type1 with XMMNewtonEPIC/PN. When present, the third shortdashed red line of each C_{F}value group of curves represents the case for type2 XMM simulations. 

In the text 
Fig. B.2. Δ(HR)/HR as a function of C_{F} for different values of the column density. The dotted line curve corresponds to N_{H} = 5 × 10^{22}cm^{−2}, the solid curve to N_{H} = 1 × 10^{23}cm^{−2}, the shortdashed curve to N_{H} = 2 × 10^{23}cm^{−2}, the longdashed curve to N_{H} = 3 × 10^{23}cm^{−2}, and the dashdotted curve refers to N_{H} = 5 × 10^{23}cm^{−2} 

In the text 
Fig. B.3. Black curves represent f_{abs} as a function of the normalised impact parameter, f_{d}, for a given value of C_{F}. Red curves show Y_{0} (see Eq. 2) again as a function of f_{d} and constant C_{F}. Red curves are labelled with C_{F} values; in the same order, increasing upwards, the same labels also apply to the black curves. 

In the text 
Fig. C.1. Schematic representation of the maximum coverage condition for an exemplifying case of noncentral occultation geometry; here ${f}_{\mathrm{d}}=2$. Panel (a) shows in red the projection on the {x, y} sky plane of the Xray source, whose centre is ${C}_{\mathrm{X}}$, whereas the projection of the occulting cloud is in black, with centre in C_{0}. In panel (b) the same configuration is seen from a different perspective, in which the line of sight to the observer is shown as the zaxis direction, along which Xray photons are coming from the Xray source from the left. See the main text for the meaning of other notation. 

In the text 
Fig. C.2. Panels (a) and (b) illustrate the maximum coverage condition with Xray source and occulting cloud seen projected on the sky plane {x, y} respectively for case A), in which ${f}_{\mathrm{cloud}}<{(1+{f}_{\mathrm{d}}^{2})}^{1/2}$, and for case B), in which ${f}_{\mathrm{cloud}}\ge {(1+{f}_{\mathrm{d}}^{2})}^{1/2}$; see the main text for details. 

In the text 
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