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

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4. Discussion

4.1. Nature of primary emission

We used our OTCM model from Paper I in order to analyze the requirements which result from the comparison of the three variants of the model to Laor et al. (1997) mean spectra of radio quiet quasars. The advantage of concentrating on quasars was in a reliable description of their broad band spectra from optical band to hard X-rays.

Model (A) reproduces well the entire broad band spectrum of quasars, including the optical slope of the data. On the other hand, present computations carried for that model allow only a minor contribution from the primary emission which results in a too high equivalent width of the iron [FORMULA] line and a need of some special geometry for cloud distribution in order to explain this level of primary emission (see Sect. 4.3).

Model (B) is the least attractive since it displays the same problems as model (A) with respect to the level of primary emission and it does not explain the optical slope of quasars.

Model (C) is also too steep (i.e. too hard) in the optical band in comparison with the mean quasar spectrum although it models well the UV/X-ray band. This may mean that model (C) well represents well the innermost part of the accretion flow but it has to be supplemented by the presence of more distant clouds and/or an outer accretion disk. However, this model also display the problem of too high equivalent width of the iron [FORMULA] line since it again allows only a marginal contribution of the direct primary emission.

4.2. Ionization level

The modelling of the overall shape of the spectrum was very sensitive to the choice of the ionization parameter [FORMULA] since this parameter determines the cloud temperature and the emission from their dark sides which dominate the optical/UV part of the spectrum.

Our analysis based on the overall spectral shape clearly favored an [FORMULA] of order of 300-500 to fit the mean quasar spectrum of Laor et al. (1997). The value [FORMULA] gave a cloud temperature too high to fill the gap in the optical band between the model predictions and the observed spectra if the soft X-ray band was modelled correctly. It also left no room for any presence of the primary emission since the soft X-ray slope of the reflected component was just marginally steep enough for model (C) to match the data.

We can compare this result with more detailed spectroscopic constraints.

The most direct estimate of the ionization level of the cool gas in quasars comes from the analysis of the position of the [FORMULA] line. ASCA results show that for moderately bright quasars the line is at 6.4 - 6.5 keV, while for bright quasars it is at 6.57 keV (Nandra et al. 1997b). This suggests that the ionization parameter for bright quasars is of the order of a few hundreds (e.g. Zycki & Czerny 1994). In fact, the flux averaged position of the iron line in our computations is 6.82 keV for [FORMULA], model (A), 6.43 keV for [FORMULA], and 6.61 keV for [FORMULA], model (C). Therefore, also from this point of view, all intermediate value of [FORMULA] better corresponds to the observations.

4.3. Geometry of clouds distribution and the level of primary emission

In the case of purely random distribution of clouds with a covering factor [FORMULA] and averaged observing angle the relative contribution of the primary emission should also be determined by the covering factor, thus giving a formula appropriate for modelling a mean spectrum:


where [FORMULA] is the amplification factor taking into account the multiple scattering


where A is the effective albedo approximately equal 0.85 (see Paper I). This formula gives the maximum efficiency since it assumes that the source of primary emission does not intercept repeatedly scattered radiation. Other quantities have the same meaning as in Paper I, i.e. [FORMULA] is the primary (incident) radiation, [FORMULA] is the emission of the dark sides of the clouds and [FORMULA] is the radiation reflected/reemitted by the bright sides of the clouds. In the case of model(C) the amplification factor is decreased by the presence of hot plasma intercepting a fraction of the photons and the appropriate formula is given by


Such an approach reduces the number of the original free parameters of the model as all weights are now expressed by the covering factor [FORMULA] which allows us to test the consistency of the derived model parameters with the random distribution of clouds.

In the case of model (A) and [FORMULA] the Laor et al. (1997) data were reproduced by the model with the relative weights of the components in Eq. (2) equal to 0.15, 1.2 and 1.0. If we rely on the relative weight of the second and third term, this result can be translated into a covering factor [FORMULA]. The role of the amplification is taken into account ([FORMULA]). Such a cloud distribution, while reproducing precisely the normalization of the last two components, predicts a significantly larger value of the primary contribution (0.90 instead of 0.15). Lower amplification still widens this gap. This means that either the cloud distribution is not random or our theoretical spectra are too hard (i.e. flat) in the soft X-ray band. Since the equivalent width of the Fe [FORMULA] line produced by the model is also too large if the primary level is as low as 0.15 it strongly suggests that the geometry is correct but the reflection spectra should be improved.

The size of the primary source could in principle be estimated from the statistical properties of the observational sample of objects. We can only infer that the source is probably not too compact in comparison with the clouds distance from the center since a point like source would give two distinct classes of objects: a fraction of objects dominated by primary emission and the rest of the sources with the primary completely hidden from an observer, which seems not to be the case.

In the case of model (C) a slightly higher value of [FORMULA] seems to be favored, of order of 500. The relative weights of the components are equal to 0.15, 5.0 and 1.0. Therefore, dark sides of the clouds are contributing more to the total spectrum. However, now the amplification is weaker by a factor [FORMULA] due to the size of the hot cloud. Combining the requirements for the relative contribution of the reflected component, dark side component and Eq. (1) we obtain [FORMULA] and [FORMULA]. Such a model predicts the contribution from the primary (hot plasma in that case), of order of 0.65, a factor of 4 higher than allowed by the data.

It is interesting to note that the condition for the size of the hot plasma cloud (Eq. 1) leads to a reasonable value of the [FORMULA] ratio for the value of the Compton parameter y which resulted from the choice of the plasma parameters giving the hard X-ray slope of index [FORMULA].

The model specifies the [FORMULA] ratio, but not [FORMULA] itself. However, if we assume that the mean quasar spectrum actually corresponds to a mean value of the accretion rate [FORMULA]yr-1 (i.e. total luminosity [FORMULA] erg s-1) and a mean value of the black hole [FORMULA] (after Zheng et al. 1997) the adopted value of the cloud density and the obtained value of the ionization parameter [FORMULA] suggest immediately that the representative value of [FORMULA] is of order of [FORMULA]. We cannot fully rely on the value of the accretion rate and the mass of the black hole since they depend on the choice of a specific model. However, any reasonable assumption about the luminosity to the Eddington luminosity ratio (of order of 0.1 in this case) would give a similar order of magnitude. Therefore, the kinematic effects are important. In Fig. 15 we show how the soft X-ray emission features are affected by the motion of the clouds. We see that the result is quite sensitive to the actual distances of the clouds from the center. Larger distances, of about [FORMULA] leave most of the spectral features almost unaffected while considerably smaller values, [FORMULA], smear out all the features apart from a strong OVIII line which is very broad. Therefore, if soft X-ray emission is really produced as a reflection spectrum the presence and the strength of the spectral features should help to test the model. Unfortunately, the available data for quasars are not yet of the appropriate quality and there is an additional problem of confusion with the spectral features due to warm absorber.

The spectral features in the optical/UV band are quite large in that case. The kinematic broadening of these features due to the motion of the clouds may slightly reduce and broaden those features (see Fig. 15 for this effect in the soft X-ray band) but will not really remove them and we intend to devote a special paper to that problem since it is one of the major issues in all realistic models, including accretion disks.

[FIGURE] Fig. 15. The importance of the kinematics for the spectral features in the soft X-ray spectrum of OTCM clouds. The upper dotted line shows the spectrum from Fig. 10 (model A) but coming from clouds located at randomly oriented orbits of the radius [FORMULA], the lower dotted curve shows the same spectrum but from clouds circulating at [FORMULA] and the continuous line shows model (C) from Fig. 13 but coming from clouds at [FORMULA]. The spectra were systematically shifted for convenience.

4.4. Future prospects

The comparison between the theoretical models and the data shown in Sect. 4 did not include spectral fitting in terms of determination of the [FORMULA] statistics since the cloud model is not ready yet for that kind of quantitative analysis. Clearly, very careful, more advanced computations of both the reflected spectrum and the emission of the dark side of a cloud are necessary in order to decide whether the soft X-ray part of the quasar spectrum can be identified with the radiation reflected by the irradiated sides of the clouds, with some contribution from the primary radiation.

Actually, preliminary computations taking into account multiple scattering (see Paper I) indicate that the amplification is wavelength-dependent as the spectra are steeper in soft X-rays due to this effect. More careful computations based on radiative transfer of soft photons and Monte Carlo computations of X-ray photons show that the slope of the spectrum for [FORMULA] has the slope as steep as 2.2 for the reflection component in the 0.2 - 2 keV band due to the multiple reflection (Abrassart et al., in preparation). Such a steep reflection spectrum would allow a primary contribution as high as 0.5 which is only lower by a factor two than expected from a random distribution of the clouds.

The geometrical arrangement of the origin of the primary radiation does not seem equally important as the physical input of the radiative transfer code since the geometrical parameters will adjust themselves to the shapes of the basic spectral components.

Considerable help may be provided by the variability constraints. In the OTCM there are three kinds of variations expected.

The first one is connected with variations of the primary source, or the comptonizing hot medium. Clouds respond to those variations both in optical/UV and in soft X-rays with an average delay of order of the travel time through the region they occupy, [FORMULA], of order of some ten thousand of seconds for a [FORMULA] black hole.

The second one is connected with a single cloud motion, i.e. the visibility of the primary. These variations are mostly limited to the primary variations and some changes in optical/UV or soft X-rays resulting from the temporary change in geometry are not expected to be delayed in any systematic way. The time-scale of these variations should be of order of the cloud period diminished by the factor describing the relative size of the X-ray source, [FORMULA], thus not considerably longer than the previous time-scale. Variations in quasars on those time-scales are unmeasurably weak in the optical band (below 1.2 % in time-scales of hours in 3C 273, von Montigny et al. 1997).

Finally, any systematic changes in the accretion rate in the innermost part of the flow which would both include a change of the level of primary emission as well as of the covering factor should happen on considerably longer time-scales. However, at the present stage the OTCM does not give quantitative predictions of the relative changes of these two factors, although we generally expect that the covering factor should grow with accretion rate, i.e. with bolometric luminosity, thus resulting in harder UV spectra and larger [FORMULA] for larger luminosity.

The variability in hard X-ray band for radio quiet quasars also seems to be the best and most direct probe of the nature of the primary source. High quality data for galactic sources allow us to compute the time delays as functions of the Fourier phase as well as the coherence function (e.g. Cui et al. 1997) which strongly support the Comptonization mechanism for the production of 'primary emission' and provide a potentially powerful method to constrain the distribution of the hot gas (e.g. Hua, Kazanas & Titarchuk 1997). The time-scales involved are in the range of [FORMULA]s to 1s for Cyg X-1 so simple minded scaling may suggest time-scales from days to years.

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© European Southern Observatory (ESO) 1998

Online publication: September 14, 1998