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Astron. Astrophys. 326, 99-107 (1997)

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4. Results and discussion

4.1. The set of parameters

The method described above have been used to obtain spectra emitted by the accretion disk and the hot source. In the Newtonian case, as shown in Paper I, the disk emission depends only on the total luminosity [FORMULA] and the height [FORMULA] of the hot source above the disk. Furthermore, one finds a universal spectrum as a function of a reduced frequency [FORMULA] and reduced luminosity [FORMULA] where


corresponding to the characteristic temperature


In the relativistic calculations, one must also specify the mass M and the angular momentum by unit mass a of the black hole. Actually, the disk emission in reduced units depends only on a and [FORMULA]. However, one needs a value of [FORMULA] comparable to the observations, i.e. about [FORMULA]. As an example, for [FORMULA] and [FORMULA], one gets [FORMULA] eV that is [FORMULA] about 70.

The high energy spectrum depends also on the relativistic particle distribution, which was taken as a exponentially cut-off power law (cf. Paper I):


Thus, one needs also to specify the spectral index s, and the cut-off Lorentz factor [FORMULA] or equivalently the high energy cut-off frequency [FORMULA]. Again, the total spectrum is universal for a given value of a, [FORMULA], [FORMULA] and s. The OSSE/SIGMA observations favor the values [FORMULA] keV and [FORMULA]. We have kept these values for all simulations.

4.2. Angular distribution of radiation

As already mentioned, the angular distribution of high energy radiation is entirely determined by the two parameters [FORMULA] and [FORMULA], solutions of the linear system (2.3). Thus, it depends only on the [FORMULA] 's values, which depend at turn on geometrical factors. Hence, the only relevant parameter is the ratio [FORMULA]. We

plot in Fig. 3 the curves [FORMULA] and [FORMULA] as a function of [FORMULA] for [FORMULA]. The differences with the Newtonian case become important for [FORMULA], reaching about [FORMULA] at [FORMULA]. The closer the source to the black hole is, the smaller [FORMULA] and [FORMULA] are. This corresponds to less anisotropic photon field. This is due to two effects: first the presence of a hole in the accretion disk inside the marginal stability radius; second the curvature of geodesics making the photons emitted near the black hole arrive at larger angle than in the Newtonian case. As shown in Fig. 3, the first effect has a weaker influence than the second one. The polar plot of [FORMULA] is sketched in Fig. 4 for [FORMULA] and [FORMULA].

[FIGURE] Fig. 3. Parameters [FORMULA] and [FORMULA] versus [FORMULA]. The Newtonian values without central hole are plotted in solid line and Newtonian values with central hole of radius [FORMULA] (the marginal stability radius corresponding to [FORMULA]) in dashed line. The Kerr values with [FORMULA] are plotted in dash-dotted line.

[FIGURE] Fig. 4. Polar plots of [FORMULA] for [FORMULA] (solid line) and [FORMULA] (dashed line). The photon field at the hot source location is less anisotropic as the hot source is closer to the black hole. The bold line corresponds to the Newtonian case.

4.3. The radial temperature distribution

We have plotted in Fig. 5 the radial temperature distribution of three models: the Newtonian model of Paper I, the

[FIGURE] Fig. 5. Effective temperature profile of the disk versus r in 3 cases:
a) Our model in Newtonian metrics
b) Our model in Kerr metrics
c) Standard accretion disk
We suppose the same total luminosity in each model.

present relativistic model with [FORMULA] and [FORMULA], and the standard accretion disk model including relativistic effects (Novikov & Thorne 1973 ) for the same total luminosity in each cases. The temperature profile is markedly different between the two illumination models and the standard accretion disk one. At large distances, all models give the same asymptotic behavior [FORMULA] (cf. Paper I). In the inner part of the disk ([FORMULA]), in the illumination models, the temperature saturates around the characteristic temperature [FORMULA]. On the other hand, it keeps increasing in the accretion model, where the bulk of the gravitational energy is released at small radii. Thus, for rapidly rotating black hole, the main difference comes from the smaller marginal stability radius ([FORMULA] for [FORMULA], whereas [FORMULA] for [FORMULA]). This increases a lot the accretion efficiency that goes from [FORMULA] for a Schwarzschild black hole ([FORMULA]), to [FORMULA] for a maximally rotating Kerr black hole. In the same time the central temperature reaches much higher values. As seen in Fig. 5, these effects have much less influence in the illumination model. Indeed, the power radiated by the disk surface is essentially controlled by [FORMULA], which is approximately constant for [FORMULA] (i.e. [FORMULA]) as shown in Fig. 4. So, the differences with the Newtonian model comes only from gravitational and Doppler shifts which are only appreciable for small radii ([FORMULA]). Thus, they concern only a small fraction of the emitting area at [FORMULA], unless [FORMULA] is itself small enough.

4.4. Overall spectrum

4.4.1. Influence of the hot source's height

Fig. 6 shows the overall spectrum, in reduced units, predicted by the model for different values of [FORMULA], for [FORMULA], [FORMULA] and [FORMULA]. The frequency shift at both ends of the spectrum is due to the variations of the characteristic frequency [FORMULA] with [FORMULA] (cf. Eq. (45)). The relativistic effects themselves become important for values of [FORMULA] smaller than about 50. They produce a variation of intensity lowering the blue-bump and increasing the hard X-ray emission. The change in the UV range is due to the transverse Doppler effect between the rotating disk and the observer, producing a net redshift, the influence of this redshift being more important for small [FORMULA] as already explained in the last paragraph. In the X-ray range, the variation is due to the change of the parameters [FORMULA] and [FORMULA] when [FORMULA] decreases (cf. Fig. 3). The observed UV/X ratio can then be strongly altered by these effects. Quantitatively, the ratio between the maximum of the "blue-bump" and the X-ray plateau of our spectra, goes from [FORMULA] in the Newtonian case (or, equivalently, for high values of [FORMULA] in the Kerr metrics), to [FORMULA] for [FORMULA] and [FORMULA] in the Kerr maximal case, as shown in Fig. 7. This ratio is highly dependent on the inclination angle [FORMULA]. By taking the maximum of the"blue-bump" ,which may be not observed, we evidently overestimate the UV/X ratio compared to the observations. It appears also that a small value of

[FIGURE] Fig. 6. Differential power spectrum for different values of [FORMULA] for the Kerr maximal case. We use reduced coordinates and logarithmic scales.

[FIGURE] Fig. 7. UV/X ratio versus [FORMULA] for different values of the inclination angle, in the Kerr maximal case.

[FORMULA] could explain the comparable UV and X-ray fluxes observed in few Seyfert galaxies (Perola et al. 1986 , Clavel et al. 1992 ). This behavior is clearly the opposite of what we would expect for a hot source whose emission is independent of the disk emission, and thus does not depend on [FORMULA]. In such a case, the smaller the height of the hot source is, the larger the bending effects on the ray of light emitted by the hot source are, increasing the illumination of the disk and thus increasing the UV/X ratio (Martocchia & Matt 1996). It does not take into account the changes in the hot source emission due to the same bending effects and our model shows that, in this case, the global result is an increase of the X-ray flux toward the observer.

4.4.2. Influence of the inclination angle

One can see on Fig. 8 Newtonian and Kerr maximal spectra for different inclination angles for [FORMULA]. For small inclination angles, the Kerr spectra are always weaker in UV and brighter in X-ray than the Newtonian ones. However, the difference tends to be less visible for the highest inclination angles. These results can be easily explained: in the X-ray range, as shown in Fig. 4, it is due to the decreasing of the relative difference of the angular distribution between Newtonian and Kerr metrics, as the inclination angle increases. But, for very small values of [FORMULA], the gravitationnal shift can be so high that the Kerr X-ray spectra appears weaker than the Newtonian one. In the UV band, the relativistic effects (the gravitational shift and the Doppler transverse effect) produce a net redshift in the face-on case ([FORMULA]) compared to the Newtonian case. For higher inclination angle, the redshifted radiation is compensated by the blueshifted one, coming from the part of the disk moving toward the observer.

[FIGURE] Fig. 8. Differential power spectrum for different inclination angle, in the Newtonian (solid lines) and the Kerr maximal (dashed lines) cases for [FORMULA]. We use reduced coordinates.

These effects are much less pronounced for high [FORMULA] values because the emission area is much larger, and thus is less affected by relativistic corrections.

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© European Southern Observatory (ESO) 1997

Online publication: April 20, 1998