Astron. Astrophys. 338, 386-398 (1998)

2. The model

The overall outline of the cloud model and of the radiation transfer within a cloud was presented in Paper I so we only summarize here the basic assumptions.

We consider a distribution of clouds surrounding a hot plasma which is a source of incident (primary) radiation for the clouds. Sides of the clouds exposed to the central source are bright while the unexposed sides are relatively dark and cold because of the considerable optical depth of the clouds. The schematic geometry is shown in Figs. 1 and 2.

Fig. 1. The overall view of the geometry of the model (A) and (B). Clouds surround a central source of unspecified origin, with one of the clouds just accidentally partially blocking our direct line of sight to the source of primary emission.

Fig. 2. The overall view of the geometry of the the model (C). Hot plasma in the center acts as a source of 'primary emission' by Compton scattering soft photons emitted or reflected by the clouds.

All clouds are assumed to have the same density and to be exposed to the same incident flux since any assumption about the radial distribution of the cloud properties would be completely arbitrary without an underlying dynamical model. However, we carefully calculate the radiative transfer in the clouds. The resulting broad band spectra from the IR to X-rays are combinations of the radiation transmitted, emitted and scattered by clouds as well as a fraction of the primary radiation. The relative weight of these components depends on the covering factor and the size of the source of primary emission.

The radiation transfer code used in this paper (Dumont & Collin-Souffrin, in preparation) has been improved in comparison with the one used in Paper I. The ionization of all hydrogenic ions (not only hydrogen itself) proceeds from all 5 levels and interlocking (subordinate) lines are included. These modifications influence to some extent the line emission and the Lyman edge, particularly due to its coincidence with the second level of .

Since the radiative transfer code of Dumont & Collin-Souffrin is constrained in the present version to photons with energy below 24 keV (although Compton heating by harder X-rays is included) we calculate the shape of the reflected spectrum above 24 keV using the method given by Lightman & White (1988). We use the opacities for the neutral gas since the ionization level is unimportant for these high energy photons. The continuity between the two reflected spectra was achieved by adjusting the opacity parameter C of Lightman & White (1988) in their Eq. (17). The value of this parameter was in all cases about , by a factor of two lower than the value suggested by George (private communication) for a neutral medium and elemental abundances used in George & Fabian (1991).

2.1. Cloud parameters

Since, at this stage, we do not introduce a full dynamical model describing the cloud formation and disruption and their motion the radial distribution of the clouds themselves and of their properties is arbitrary. Therefore, in order to keep the parameterization as simple as possible we assume that all clouds have the same (constant throughout a cloud) density n and column density . Since clouds optically thin for electron scattering were ruled out as a model for optical/UV/soft X-ray continuum we study clouds with , usually concentrating on the case. Higher strongly suppress the emission from the unilluminated side of the cloud so only reflection is seen. The clouds are opaque for the X-ray radiation so even if some clouds are in our line of sight towards the primary X-ray source this would not lead to the presence of the transmitted, strongly absorbed spectral component. The primary emission intercepted by the clouds is partially reflected and partially reemitted by the dark side of the cloud (i.e. the side opposite to the central source) in the form of thermal emission.

The distribution of the clouds is described by the covering factor with respect to the X-ray source. If is close to 1, the distribution necessarily is almost spherical. If is smaller (e.g. 0.5) then the distribution may either be still spherical, or rather constrained to the symmetry plane. These two cases are different if Doppler effects connected with the motion of the clouds are taken into account since plane confinement would result in broader features for the same velocity field.

2.2. The primary radiation

The nature of the primary emission is still not known. In the past, non-thermal models were favored. In such models a (usually power law) distribution of relativistic electrons produces the radiation either by direct synchrotron emission or by Comptonization of soft photons. These soft photons were either their own synchrotron photons (SSC) or came from external source like an accretion disk. In these models the pair creation process was frequently important, particularly for the shape of the spectrum above .

These models are still very popular in the case of radio loud AGN. However, in the studies of radio quiet AGN the attention recently shifted towards thermal models. In these models the hot plasma is thermal, at a temperature of order of a few hundreds of keV, and it produces the radiation through Comptonization of soft external photons. In these models the effect of pair creation is usually negligible.

These two basic families of models differ also in the extension of the primary component into the low energy range. In non-thermal models the primary emission may (but does not necessarily have to) extend down to the far-IR/mm band. It was an attractive possibility in the late 70s and 80s when IR emission was generally thought to be dominated by non-thermal processes. Evidence of a significant contribution of dust to the IR emission as well as some arguments based on variability (e.g. Done et al. 1990 for NGC 4051) diminished the popularity of that view. However, a single underlying IR/X-ray continuum is still sometimes advocated on the basis of observational arguments (e.g. Walter & Fink 1993, Loska & Czerny 1990, Fiore et al. 1995). Thermal models, on the other hand, predict a contribution of the hard component only above the energy of the seed photons for Comptonization, i.e. mostly starting from the UV band.

As a result, in their computations of the overall continua, various authors used different specific assumptions about the shapes of the incident spectra for reprocessing. Lightman & White (1988) consider a non-thermal model with an energy index 0.7 and a spectrum extending from 1 eV to 3 MeV, Guilbert & Rees (1988) have the primary non-thermal spectrum with energy index 0.5 (they also include the effect of pair creation on the shape of the primary). Sivron & Tsuruta (1993) assume that the initial synchrotron radiation with an energy index 1 extending from 0.1 eV to 1 MeV is filtered by CFR cloudlets (Celotti et al. 1992). This special, very compact, population of cloudlets, with densities cm^-3, hydrogen column cm² and covering factor 1, absorbs the IR emission (through the free-free process) but it is transparent at the optical band and above. The absorbed energy is reemitted in the form of free-free emission with the adopted value of the temperature K. Such a preprocessed spectrum serves as incident spectrum for the main clouds occupying the space between and .

In order to study the influence of the shape of the incident flux on the resulting spectrum predicted by the model we choose three representative shapes of the primary continuum justified by general theoretical arguments. They are named models (A), (B) and (C), and they are shown in Fig. 3.

Fig. 3. Three examples of the IR-Xray incident (primary) spectra for OTCM models, as described in Sect. 2.2: model (A) - dotted line, model (B) - dashed line and model (C) - continuous line. Model C was computed assuming the hot medium temperature equal to K and its optical depth equal to 0.1.

2.2.1. Model A: synchrotron emission extending to IR

This incident radiation resembles closely the incident radiation adopted in our initial study (Paper I) as well as by Kuncic et al. (1997). We assumed a low frequency cut-off at 0.1eV as in Paper I. In order to reproduce well the observed spectrum (according the present knowledge) we assume an energy index either 0.9, after the classical paper on Seyfert 1 galaxies (Pounds et al. 1990) or steeper since in the case of quasars the situation is not clear: Laor et al. (1997) give a slope of 1 for radio quiet objects while Williams et al. (1992) give 0.92 when flat spectrum objects are excluded. We assume the values of the cutoff energy 100 keV and 280 keV since this last value seems to be suggested by GRO data for Seyfert 1 galaxies (Grandi et al. 1998 for MCG8-11-11; Madejski et al. 1995 for IC 44329A, Gondek et al. 1997 for a composite spectrum). No direct data constraints for cut-off energy in radio quiet quasars are available.

Such a model corresponds to the presence of a relatively strong magnetic field within the hot plasma, such that the energy density of the magnetic field is larger than the energy density of the soft photons available due to the presence of cool clouds. Models of that type were studied e.g. by Maraschi et al. (1982), with lower frequency cut-off Hz determined by self-absorption.

The hypothesis of the existence of such an IR/X-ray power law was observationally tested in the case of the source NGC 4051 (Done et al. 1990) and its existence was not confirmed since strong X-ray variability in this source was not accompanied by any optical variability. On the other hand NGC 4051 is not a typical example of an AGN and its optical emission may be strongly dominated by starlight. A number of other sources like NGC 5548 (Korista et al. 1995) and NGC 4151 (Edelson et al. 1997) show coherent day to day variations in optical, UV and X-ray band, with unmeasurable delays smaller than several hours.

We do not analyze the conditions for the production of such a primary emission in the present paper. We simply assume the parametrization by a single power law and we fix the low energy cut-off at 0.1 eV, as in Paper I.

The input model parameters are: the ionization parameter, (where L is the bolometric luminosity), the cloud density, n, and its hydrogen density column, , (or cloud size). The observational appearance of the system is further parameterized by the relative contribution of the primary (or incident) radiation and the emission from dark sides of clouds with respect to the reflected component. For purely random cloud distribution both parameters are uniquely determined by the covering factor, (see Eq. 2).

2.2.2. Model B: synchrotron emission extending to UV

Our second model is a power law characterized by the same energy index and high energy cut-off as before but the adopted low frequency cut-off is set at 30 eV i.e. in the UV band ( Hz). This simple model is supposed to represent a situation when the gas is not simply in equipartition with the magnetic field but dominates the behaviour of the matter (Rees 1987). As the expected magnetic field is in this case two orders of magnitude or more higher than in the previous case ( gauss, see Celotti et al. 1992) the self-absorption frequency is proportionally higher. Additionally, in such a strong magnetic field a new family of cloudlets, or filaments, may form in the innermost part of the flow. Such cloudlets are opaque to photons below, again, Hz due to their high density and free-free absorption so they filter the primary emission before this emission may reach the main clouds outside (Sivron & Tsuruta 1993). Therefore the incident radiation in the case of magnetic field dominating innermost flow would not extend beyond optical/UV. Absorption by cloudlets leads additionally to reemission of the absorbed radiation in the form of a black body radiation. However, as we cannot predict the temperature of such radiation and the covering factor by the inner cloudlets we simply neglect this component in the present consideration.

The input model parameters are the same as in the previous section.

2.2.3. Model C: self-consistent thermal model

This model is based on the assumption that amplification of the magnetic field within the flow is not efficient. Therefore, clouds are the only source of the soft photons. We also assume this time that the hot medium is thermal, i.e. basically characterized by the optical depth and the temperature. Hard X-ray emission in this model forms by Compton upscatter of a fraction of the soft photons from the clouds while soft photons result from interception of a fraction of hard X-ray emission by clouds.

In order to obtain the appropriate shape of the hard X-ray continuum as described above we have to adopt a temperature of the hot medium equal to K and an optical depth . Adopting higher (lower) value for the temperature would require a lower (higher) value of the optical depth of the hot medium. The Compton parameter y for such a plasma is 1.1.

The fraction of the soft photons intercepted by the hot medium results from the model (see Paper I) and is approximately given by

where is the fraction of photons reflected or reemitted by the illuminated side of a cloud, determined by the physical conditions in the gas. Our model is therefore more flexible than the original version of the corona model (Haardt & Maraschi 1991) which requires equal to 0.5 and , thus practically fixing the value of the Compton parameter y. Recently developed clumpy corona models are based on an additional covering factor for the hot medium which is equivalent to our ratio from the point of view of the efficiency of the Comptonization but the two models nevertheless differ with respect to overall geometry and its (angular-dependent) appearance.

The input parameters of the model are, as usual, the ionization parameter, the cloud density, its hydrogen density column (or size), and the weights of the primary and dark side components.

However, the computations of the model require an iterative procedure to achieve a self-consistent solution for the soft and hard emission. In order to avoid an arbitrarily adopted spectral shape in the optical band we start with a mechanically heated cloud with temperature K. Next we compute the Comptonized spectrum which results from the interception of soft photons by hot plasma with the parameters given above. For that purpose we use the method of Czerny & Zbyszewska (1991), appropriate for a thin plasma. This radiation now serves as incident radiation. Next we calculate the reflected (not emitted) component as the cloud faces the hot medium clearly with its hot hemisphere. The comptonization of this reflected component gives a second approximation to the hard X-ray emission spectrum and the model is iterated until it converges. The normalization is fixed by the adopted value of the ionization parameter and the ratio of the Comptonized reflected radiation to renormalized radiation gives us accurately the fraction of soft photons intercepted by the hot gas, or the relative extension of the two regions, roughly estimated by Eq. (1).

© European Southern Observatory (ESO) 1998

Online publication: September 14, 1998
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