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

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1. Introduction

Electron scattering is an important opacity source in atmospheres of hot stars. An usual approximation to that process is known as Thomson scattering. This is quite a standard approximation, which assumes that the scattering of radiation on free electrons is a fully coherent event, in which the frequency of a photon is kept fixed before and after scattering. Since Thomson scattering is perfectly grey, it does not leave any specific signature in the spectrum of radiation emerging from the star.

The assumption of Thomson scattering causes a very important implication, that there is no energy and momentum exchange during photon-electron collision. One can say, that the properties (temperatures) of radiation and matter are perfectly decoupled in a strongly scattering atmosphere. This is the case of hot stellar atmospheres, in which matter is strongly ionized and then electron scattering dominates the opacity. Thomson scattering does not influence the equation of radiative equilibrium, because scattering terms there precisely cancel out (Mihalas 1978).

Compton scattering is the same physical photon-electron process which is described with a very precise mathematical formulation, in which both photon and electron change their energies and momenta as is required by the conservation principles. A correct mathematical description of Compton scattering from thermal electrons includes the following physical effects: reddening of scattered photons due to electron recoil effect, broadening (in frequency scale) of the incident radiation spectrum due to the chaotic thermal motion of electrons, and the blue shift of the incident photons known as the inverse Compton effect. Moreover, there exists nonzero induced Compton scattering. A full description of Compton scattering is given by Pomraning (1973). In case of [FORMULA] keV there exists also the Fokker-Planck expansion of scattering integrals referred to as Kompaneets equation (Kompaneets 1957; Rybicki & Lightman 1979). However, derivation of the Kompaneets equation requires at the very beginning that the scattered radiation field is isotropic, whereas Pomraning's equation does not require such an assumption.

Compton scattering opacity remains practically grey in X-ray domain. However, the nonzero exchange of energies between colliding particles implies a coupling between radiation and matter. Consequently, Compton scattering can influence the temperature structure of a stellar atmosphere. In case that the white dwarf atmosphere contains hydrogen alone, this effect is most significant and appears as a distinct rise of temperature in the outermost layers (Madej 1993, 1994a, 1994b). The heating effect results from scattering of hard radiation from deep and hot layers of thermalization with cooler gas at the external boundary of an atmosphere.

The other problem is the Compton scattering of X-ray photons in atmospheres of hot stars, [FORMULA] K or more. In this case the energy of scattered photons (0.1-1 keV) strongly exceeds thermal energy of electrons ([FORMULA] eV). Therefore [FORMULA], and then the effect of electron recoil dominates. Compton scattering of X-rays proceeds with loss of photon energy to the electron pool (Rybicki & Lightman 1979). It causes a redistribution of X-ray photons to lower energies and, consequently, decreases the X-ray flux (cf. Madej 1993, 1994a, 1994b).

A discussion of this spectral feature in still cooler pure hydrogen (DA) white dwarfs is one of the subjects of this paper. Presentation of both Compton scattering effects in atmospheres of hot neutron stars (Type I X-ray bursters) is given by Madej (1991), cf. also London et al. (1986).

1.1. Impact on spectral lines

Scattering by free electrons has to be treated as noncoherent in cases, when the spectrum contains narrow spectral lines. Then the width of a line can get smaller than the electron Doppler width. There exists a series of papers, which investigated the importance of noncoherent continuum scattering for the formation and broadening of spectral lines in the visual and UV spectral region (cf. Rybicki & Hummer 1994 for the reference list).

Generally, such an interaction between electron Doppler frequency shifts and thermally broadened spectral lines can cause observable broadening of lines in spectra of the hottest stars. Recent papers by Hillier (1991) and Hamann et al. (1992) demonstrate the effect in theoretical spectra of Wolf-Rayet stars. Rybicki & Hummer (1994) present a new numerical method for the computations of NLTE populations in presence of noncoherent electron scattering. The authors predict the existence of a whole pattern of spectral effects due to noncoherent electron scattering and claim, that the effect influences the NLTE level populations and therefore both visual and UV continuum lines and spectra of stars with [FORMULA] around 50 000 K and low surface gravities.

However, all the above papers investigate noncoherent scattering due solely to thermal motion of free electrons, which is only a single component of the Compton scattering. Effects of photon redshift due to electron recoil and the Doppler blueshift of scattered photons, as well as induced scattering are not considered there. Therefore the above approach cannot be used to investigate Compton heating effects nor the study of continuum X-ray spectra.

1.2. Diffusion equation

There exist quite numerous research papers in which Compton scattering in the diffusion approximation has been included in the radiative transfer. Ross (1979) developed and applied his numerical method to study the radiative transfer in spherical shells around a compact object, being the source of X-rays. A discussion of the Kompaneets equation and the relevant radiative transfer models was extensively presented by Sunyaev & Titarchuk (1980). Also London et al. (1986) applied their computer code to compute a set of theoretical model atmospheres of X-ray burst sources (see Lewin & Joss 1981, 1983; and Joss & Rappaport 1984 for a description of this class of objects).

However, there exists an important limitation of the computer codes based on the diffusion approximation (or Kompaneets equation). As Rybicki & Hummer (1994) point out, the diffusion equation can be used only if the radiation field varies slowly with frequency, i.e. changes of the radiation field are small on the frequency shift after Compton scattering. Therefore the diffusion approximation to Compton scattering is appropriate to the radiative transfer in continua, whereas it is not useful to both LTE and NLTE radiative transfer involving narrow spectral lines.

Such a statement implies also, that unfortunately NLTE model atmosphere computations with Compton scattering have to be ruled out, when using otherwise a physically correct diffusion equation. This is because NLTE model computations can yield physically false conclusions on the NLTE occupation numbers, if line transitions were ignored (Mihalas 1978).

This research follows a series of papers on the treatment of Compton scattering in white dwarfs and X-ray bursters, in which two different numerical methods have been defined and tested already (Madej 1991, 1994b). This paper presents the third numerical technique for that purpose, which is certainly the most realistic and accurate among them. Moreover, the following method can be used as an extension of other algorithms which used coherent Thomson scattering only. This allows one for the investigation of both the continuum X-ray spectra of hot stars, and the effects of Compton heating on the uppermost atmospheric layers. However, narrow spectral lines still cannot be included there, which prevents realistic computations of NLTE effects.

There exists a number of various formulations of the radiative transfer equation including Compton scattering (cf. Pomraning 1973; Rybicki & Lightman 1979), suitable either to classical or to fully relativistic plasmas. The following chapters present a discussion of stellar atmospheres at [FORMULA] of the order [FORMULA] K, and make use of the nonrelativistic diffusion approximation to the Compton scattering following Pomraning (1973). The diffusion equation adopted from Pomraning (1973) is more general than the Kompaneets equation in Rybicki & Lightman (1979), but both are identical in case when the angular distribution of scattered photons is isotropic.

The diffusion equation takes into account all the physical effects present in the scattering process, and is not restricted to the electron Doppler broadening alone. However, further discussion will be restricted to LTE model atmospheres with no spectral line opacities. This yields only conclusions concerning effects of Compton heating in the uppermost layers and X-ray spectra of the models.

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Online publication: November 9, 1998