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Astron. Astrophys. 340, 617-625 (1998)

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3. Convergence: sample high temperature models

Compton scattering causes two distinct effects in hot stellar atmospheres and in their spectra. First, energy exchange between electron gas and energetic photons coming from deep layers causes an increase of gas temperature in the outermost (mostly scattering) layers of a stellar atmosphere. Second, hard energy photons which lose their energy in the first process are then redistributed to lower frequencies, and this causes a depression of the monochromatic flux of radiation emerging from the atmosphere. The existence of both effects in hot DA white dwarfs was demonstrated in earlier papers with two earlier (independent) computer programs (Madej 1993, 1994a, 1994b).

Both the above effects are caused by Compton scattering of the most energetic photons available in the radiation spectrum, which have energies significantly higher than the average thermal energy of a gas. At the [FORMULA]'s corresponding to hot white dwarfs both effects are rather marginal as long as the overall spectrum and radiative equilibrium is considered. However, they can be important for the X-ray spectra and the temperature structure in or above line-forming region in the hot atmosphere.

3.1. Pure H models at Teff = 1 [FORMULA] 105 K

The overall quality and convergence properties of the new computer code have been tested in a series of LTE model stellar atmospheres computed at [FORMULA] K and various surface gravities, corresponding either to gravities of white dwarfs or subdwarf O stars. The chemical composition of the models included hydrogen only.

All the model atmospheres discussed in this paper were computed over a large range of the standard optical depths, ranging from [FORMULA] to [FORMULA] or even [FORMULA], depending on the model. The number of depth points D approached 130-140, with 12 points per decade. The number of frequency points I always extended 850. Such a choice of model parameters allowed for the reproduction of radiative equilibrium with the accuracy of [FORMULA] better than [FORMULA] in 6-8 iterations, starting from a rough "grey" temperature stratification. However, the convergence of [FORMULA] was apparently fastest in the highest layers, where Compton scattering dominated absorption by many orders of magnitude.

Fig. 1 displays the run of temperature T vs. the Rosseland optical depth [FORMULA] in a family of pure hydrogen, LTE model atmospheres. Dashed and dotted lines in the Figure present models with [FORMULA], 6.0, and 5.5, respectively. In all the models Compton scattering produces a characteristic increase of temperature T in the uppermost layers. Note, that the model with [FORMULA], which is the gravity typical of white dwarfs, exhibits rather remnant Compton heating effect even at the uppermost [FORMULA]. In the models of lower [FORMULA] the effect extends to deeper layers of the atmosphere, still it does not extend deeper than to [FORMULA].

[FIGURE] Fig. 1. Run of gas temperature T vs. Rosseland optical depth [FORMULA] in pure H models, computed with account of Compton scattering at [FORMULA] K and various [FORMULA]. Effects of Compton heating and cooling by free-free absorption are well pronounced in all the models. Compton heating reaches the maximum extent in model of the lowest gravity, [FORMULA]. For comparison, the solid line presents the run of [FORMULA] in the model [FORMULA] computed with coherent Thomson scattering.

The solid line in Fig. 1 presents the run of temperature in the LTE model at [FORMULA], in which the standard Thomson scattering was assumed. The heating effect obviously is absent in the Thomson scattering model.

Fig. 2 presents synthetic continuum spectra of two models, both with a low surface gravity [FORMULA]. The solid line is the spectrum which includes Compton scattering, whereas the dashed line represents the spectrum of a model atmosphere computed with the standard Thomson scattering. Both spectra are practically identical except for the X-ray branches (at [FORMULA], i.e. at [FORMULA] keV), where one can note a distinct depression of the flux in Comptonized atmosphere, and an apparent cutoff of the flux at [FORMULA] (0.33 keV).

[FIGURE] Fig. 2. Spectra of sample pure hydrogen model atmospheres with [FORMULA] K and [FORMULA]. Solid line represents the spectrum of the atmosphere which includes effects of Compton scattering. Dashed line represents the spectrum of the model which assumes coherent Thomson scattering. Note the depression of flux at [FORMULA] and the cutoff at [FORMULA], which illustrate the fact that Compton scattering flushes out hard X-ray photons to longer wavelengths.

The deficiency of X-ray flux in the Comptonized model reflects in fact the difference between photon energies (higher than 400 eV) and the mean thermal energy of scattering electrons (of the order of 10 eV). In this case repeated Compton scatterings cause systematic loss of photon energy and shifts them to longer wavelengths. However, Compton scattering does influence neither visual/UV nor far EUV continuum spectra of the discussed model.

3.2. Influence of heavier elements

Chemical composition of the models included hydrogen alone, since pure hydrogen atmospheres exhibit the most evident effects of Compton scattering. This is due to the fact, that both hydrogen b-f absorption and free-free opacity quickly vanish with increasing frequency [FORMULA]. Therefore in pure H atmospheres outgoing radiation from the hard-energy branch of the spectrum has been created by thermal processes in particularly deep and hot layers of thermalization. Photons escaping into empty space undergo then numerous Compton scatterings, in which they efficiently lose their energy to cooler electron gas and simultaneously move in the frequency space to lower energies.

In case that helium or heavier elements are present, efficient b-f absorption continua cover the high-energy tails of stellar spectra which moves the thermalization depth upward, just slightly below the stellar photosphere. In such case the significance of Compton scattering very seriously decreases, and both effects (temperature rise and deficiency of X-ray photons) do not develop.

3.3. The Compton y parameter

The Compton y parameter denotes the relative value of energy change which a photon experiences after repeated Compton scatterings in a gas. Rybicki & Lightman (1979) compute y as the product of mean energy change after a single Compton scattering times the number of such events. If the value of y is of the order unity of larger, than the Compton scattering can be generally considered as the dominant agent in the radiative transfer in some medium.

Rybicki & Lightman (1979) derive the following expression describing the relative energy transfer per single scattering, for a photon of initial energy [FORMULA]


The expression is valid for nonrelativistic electrons in thermal equilibrium. For photons penetrating a purely scattering finite layer this yields the approximate expression


where [FORMULA] denotes the total optical depth of the medium with respect to Thomson scattering. Eq. (3.2) is the expected change of energy only for photons of initial energy [FORMULA], and not for the general pool of incident photons.

In a stellar atmosphere of infinite optical depth [FORMULA] has the meaning of the scattering optical depth of the layer, where emerging photons were created by thermal processes (at the depth of thermalization). Thermalization of radiation occurs in a layer of the effective optical depth [FORMULA] of the order unity (cf. also Rybicki & Lightman, 1979, for the definition of the effective optical depth). The value of [FORMULA] has been assumed as the depth of thermalization in the following considerations. Since the free-free absorption (dominating in X-rays in pure H atmosphere) changes as [FORMULA], the depth of thermalization, [FORMULA], and the Compton parameter [FORMULA] are strongly frequency-dependent quantities.

The useful prescription for computing of [FORMULA] in a real inhomogeneous stellar atmosphere follows from the incremental definitions


which yields


We can then integrate Eq. (3.5) to obtain


which can be easily computed given the run of monochromatic opacities in a model atmosphere interpolated to the effective optical depth [FORMULA] at the frequency [FORMULA]. Substituting [FORMULA] to Eq. (3.2) yields the final value of the Compton parameter [FORMULA].

In case of the pure H model of [FORMULA] K and [FORMULA], the above procedure yields [FORMULA] at the cutoff [FORMULA]. This is a crude estimate which shows, that such photons get substantially redistributed to longer wavelengths when travelling from the layer of thermalization outwards and vanish from that energy range.

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© European Southern Observatory (ESO) 1998

Online publication: November 9, 1998