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Astron. Astrophys. 363, 1055-1064 (2000)

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4. Numerical results

Model atmosphere computations of a B3 V star illuminated by hard X-rays, with temperature [FORMULA] almost 4 orders of magnitude higher the [FORMULA], represents a very difficult numerical problem. We have computed two series of Comptonized model atmospheres of the single effective temperature [FORMULA] K and gravity [FORMULA] (cgs units). We have begun our computations adopting models illuminated by isotropic X-rays of diluted thermal spectral distribution with [FORMULA] K, which were previously computed under assumption of coherent Thomson scattering (Paper II). Isotropy implies, that the true intensity of external irradiation is given by


which defines the dilution factor w.

Model computations with Compton scattering lasted for 16-20 iterations, after which temperature corrections [FORMULA] diminished below 0.005, except for the narrow zone of large temperature drop. In all cases our code reproduces the structure of illuminated models and their spectra for wavelengths ranging from infrared to hard X-rays.

Model atmospheres of the first series assume the presence of hydrogen and helium, with solar number abundance [FORMULA]. In contrast to Paper II, here the hydrogen-helium models can interact with incoming X-ray photons very efficiently via Compton scattering process.

Second series of our models assume hydrogen, helium and iron chemical composition, with number abundances [FORMULA] and [FORMULA]. In this case iron ions are able to absorb energy of X-ray photons also via thermal absorption.

Similarly as in Paper II, our actual model atmospheres take into account bound-free opacities of 9 lowest levels of hydrogen, 30 levels of neutral helium, and 10 levels of singly ionized helium, supplemented by free-free opacities of their positive ions. The code includes also opacity of hydrogen lines of Lyman, Balmer and Paschen series. The corresponding subroutines were taken from ATLAS 7 code by Kurucz (1985). Iron rich models include rather complex fits to b-f opacities of all existing iron ions (Verner & Yakovlev 1995).

The code includes also very extensive set of LTE bound-free opacities from almost 1000 excited levels of all iron ions, with their threshold energies taken from the Opacity Project database (Seaton 1987). Our computations include also free-free opacities of each iron ion present in the model.

All model atmospheres were computed on very large grid of 1657 wavelength points ranging from 0.03 Å up to 25 000 Å , spaced equidistantly in logarithmic scale. The mesh of standard optical depths consisted of up to 169 points. Standard optical depths were distributed from [FORMULA] on the top of an atmosphere to few times [FORMULA] at the bottom, depending on the model.

Let us remind here that the outgoing bolometric flux [FORMULA] cgs roughly equals the external irradiating flux [FORMULA] at the value of dilution factor [FORMULA] (Paper II).

In our computations values of the factor [FORMULA] were either set to 0.33333 as in the Eddington approximation (for the computations of model atmospheres themselves), or iterated precisely as variable Eddington factors (for the final spectrum synthesis).

Iteration of temperature in Comptonized model atmospheres were quite efficient. Particularly in case of H-He models, initial temperatures were always very low, of the order [FORMULA] K in a wide range of optical depths above the photosphere. After illuminating that model by hard X-rays, relative temperature corrections initially exceeded one order of magnitude, and we had to diminish them artificially to max. 3-4 in order to ensure stability of temperature iterations. After at least 16 iterations, temperatures in the upper atmosphere could raise to [FORMULA] K, which represents a rather dramatic change of physical conditions there. In case of models containing iron, initial temperatures in the uppermost layers were already very high, about [FORMULA] K, and Compton scattering could exert either heating or cooling effect there, depending on the dilution factor w.

As in Paper II, we have averaged temperature corrections over three neighbouring optical depths to prevent temperature fluctuations on the optical depth scale. Moreover, we always computed geometrical average of two temperature distributions corresponding to the last two iterations. This approach enabled us to ensure perfect temperature constancy ([FORMULA]) both above the temperature drop and at large depths. Only for the optical depths surrounding temperature drop, temperature corrections did not reduce satisfactorily even after 16-20 iterations. For all the models we did not exceed 20 iterations, since they were very time consuming. Flux constancy was almost perfect in case of iron reach models, with [FORMULA]%. Unfortunately, in some H-He models flux errors have stabilized on rather high levels exceeding 30%, and further iterations apparently did not tend to lead to better flux constancy.

Flux errors of the above value usually are not acceptable in model atmosphere computations of single stars. In the present research, however, such errors occur in models irradiated by the external X-ray flux of bolometric value 1 - 100 times larger then the own bolometric flux of the atmosphere (equal to 100%). That error value means, that the huge incoming external flux of 100% - 10000% is not perfectly compensated numerically by the same outgoing excess flux. Moreover, we attribute this error also to the fact, that H-He models were computed with too shallow standard optical depth grids, not reaching [FORMULA].

We have used Pentium II computers, with clock speed of 366 and 600 MHz. A single model atmosphere with Compton scattering has been obtained after 2.5 - 4 days of computations.

4.1. Temperature structure

In all considered models irradiated by the external X-rays, Compton scattering causes very significant effect on temperature run in a stellar atmosphere. In case of pure hydrogen-helium models flux of incoming X-rays heats up atmosphere even by more than three orders of magnitude, as presented in Fig. 1. This is because Compton scattering opacity is the only efficient absorber of X-ray energy, contrary to very small absorption opacity provided by hydrogen and helium ions. Paper II gives an extensive presentation of rather small heating effects, which external X-rays exert in H-He models in presence of coherent Thomson electron scattering (see dotted lines in Fig. 1 of this paper). This is a rather dramatic change of physical conditions in an illuminated atmosphere. In particular, when the irradiating X-ray flux is extremely strong, then the temperature rises by more then 10 times even in the deepest layers, [FORMULA].

[FIGURE] Fig. 1. Run of temperature T vs. standard optical depth, for models with hydrogen and helium. The solid line denotes [FORMULA] for nonilluminated atmosphere, whereas dashed lines correspond to atmospheres illuminated by X-rays (Compton scattering). Negative integers attached to dashed lines denote decimal logarithm of the dilution factor w. Dotted lines represent temperatures in illuminated models with Thomson scattering, for the same dilution factors, respectively.

In case of iron rich models, temperatures stabilize also on very high values of the order of millions K (Fig. 2). However, Compton scattering can sometimes diminish temperature of illuminated atmosphere, as compared with models assuming Thomson scattering (Fig. 3). This can occur for weaker external X-ray fluxes, in which case Compton scattering reradiated excess of thermal energy. In previous models with Thomson scattering, iron ions were rather poor emitters of reprocessed energy.

[FIGURE] Fig. 2. Run of temperature T vs. standard optical depth, for models with hydrogen, helium, and iron. The solid line denotes [FORMULA] for nonilluminated atmosphere, whereas dashed lines correspond to atmospheres illuminated by X-rays, with Compton scattering and various dilution factors.

[FIGURE] Fig. 3. Comparison between temperature distributions vs. standard optical depth for pairs of models with hydrogen, helium, and iron, for two sample values of the dilution factor w. Dotted lines represent [FORMULA] in illuminated atmospheres with coherent Thomson scattering, whereas dashed lines correspond to models with Compton scattering. Note, that Compton scattering can cause either heating or cooling effect, depending on the value of dilution factor (i.e. the value of irradiating X-ray flux).

4.2. Outgoing spectra

Fig. 4 presents X-ray and extreme UV spectra for the series of hydrogen-helium models with various dilution factors w. The solid line is the spectrum of LTE nonilluminated model, showing outgoing flux for wavelengths [FORMULA] Å . Dashed lines represent spectra of illuminated models with Compton scattering, whereas dotted lines represent two sample outgoing spectra computed with coherent Thomson scattering (Paper II). One can see dramatic difference between both series of models. Our code with Compton scattering is able to reproduce very significant redistribution effects: practically all the irradiating hard X-ray photons are backscattered as softer photons with energies reduced by few orders of magnitude. Thermal absorption of X-rays by H and He ions is of marginal significance. Flux can increase even by over 15 orders of magnitude due almost solely to Comptonization. Therefore, Compton scattering in our models just creates soft X-ray and extreme UV continuum with distinct hump, which obviously can correspond to soft X-ray excess in AGN. We can also note the existence of distinct Compton cut-off of the outgoing flux.

[FIGURE] Fig. 4. Spectra of H-He atmospheres in EUV and X-ray wavelengths, for models with Compton scattering (dashed lines), and two sample Thomson scattering models (dotted lines). Compton scattering just creates huge EUV continuum bump, where most of energy is emitted.

All spectral features, as hydrogen and helium ionization edges completely disappear due to Comptonization even for low external illumination. This effect can be seen in Fig. 5, which shows series of synthetic spectra around hydrogen Lyman jump at 912 Å . It is obvious, that even for moderate external illumination, [FORMULA], Lyman jump practically disappears. For stronger external X-ray flux, absorption Lyman lines also disappear and even can reverse to emission.

[FIGURE] Fig. 5. Set of far UV spectra of Comptonized models at wavelengths around the hydrogen Lyman jump (dashed lines). Solid line represents spectrum of the nonilluminated LTE atmosphere. Lyman b-f jump disappears in all illuminated models due to very strong heating effects. Lyman lines appear in emission for the strongest illumination.

Fig. 6 presents X-ray and EUV spectra for series of iron rich models. One can easily trace the evolution of spectra with increasing [FORMULA]. Continuum of those spectra raises very distinctly, however, we did not claim the existence of dominating soft X-ray hump. There exist numerous iron b-f opacity jumps in emission, which can contribute to excess X-ray emission. In all spectra we note the existence of Compton scattering cut-off, with its threshold energy increasing with increasing [FORMULA].

[FIGURE] Fig. 6. Spectra of Comptonized iron-rich atmospheres in EUV and X-ray domain. Note huge iron and He II b-f opacity jumps in emission, and Compton cut-off with its energy increasing with increasing dilution factor w.

Fig. 7 presents two pairs of synthetic X-ray spectra computed either with the account of Compton scattering (dashed lines) or with Thomson scattering (dotted lines). One can easily note that even in iron rich models, Compton scattering raises that part of continuum spectra even by factor 100.

[FIGURE] Fig. 7. Comparison between two pairs of iron-rich model spectra, one pair computed with Compton scattering (dashed lines), and remaining pair - with Thomson scattering (dotted lines). Redistribution of hard X-rays by Compton scattering additionally increases EUV and soft X-ray flux by a factor approaching 100.

In case of iron rich models, hydrogen Lyman edge behaves in a slightly different way. Comparison of Fig. 5 and Fig. 8 shows, that significant reduction of the Lyman jump requires much stronger external X-ray flux in iron rich models. At the same time emission cores in Lyman lines are stronger.

[FIGURE] Fig. 8. Evolution of spectra near the hydrogen Lyman jump with increasing external illumination for iron-rich atmospheres (compare to Fig. 5). Again, external X-ray flux efficiently reduces Lyman jump to zero and creates emission cores in numerous Lyman lines.

We believe, however, that redistribution effects of the incoming X-ray flux are apparently exaggerated, see the outgoing Comptonized spectra in Fig. 4 and Fig. 6. Our artificial numerical assumption (Eq. 25) probably causes, that the high energy cut-off exhibited in both figures is too steep. Rejecting of such an assumption probably will result with more soft decrease of the outgoing X-ray flux with increasing photon energy.

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© European Southern Observatory (ESO) 2000

Online publication: December 5, 2000