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Astron. Astrophys. 333, 687-701 (1998)

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1. Introduction

Radiative shock waves belong to the conspicuous phenomena demonstrating the tight interplay between hydrodynamic motions and the radiation field. The role of this interplay is strongest in the low gas density flows, so the shocks are of tremendous importance in astrophysics. They are observed in a wide variety of astrophysical phenomena: nova and supernova explosions, bright filaments in old supernova remnants, accretion flows in protostellar clouds. Shock waves are detected also in atmospheres of radially pulsating variables such as Cepheids, RR Lyr, W Vir, RV Tau and Mira type stars. Periodic shocks propagating through pulsating atmospheres lead to the distention of outer atmospheric layers and to the mass loss.

Importance of radiative shock waves attracted attention of many authors but nevertheless the shock properties are explained quite well still qualitatively (see, for example, Zel'dovich & Raizer 1966; Skalafuris 1968; Mihalas & Mihalas 1984; Liberman & Velikovich 1986). The principal difficulty in obtaining the correct quantitative description of the shock wave structure is that the model has to allow for the strong coupling between the gas flow and the radiation field, both them being characterized by substantial departures from LTE. Solution of this problem encounters serious difficulties, so that in immensely numerous studies available at present in the literature the authors used various assumptions and simplifications (e.g. local thermodynamic equilibrium, treatment of the radiation transfer in diffusion approximation, neglecting the opacity in the Balmer continuum etc.). Many of these assumptions were found later inadequate or leading to uncertain conclusions. For instance, Kogure (1962), Sachdev (1968) and Hill (1972) used the LTE approximation which does not hold as emphasized later Narita (1973). Considering the hydrogen gas, Whitney & Skalafuris (1963) relaxed this assumption but incorrectly assumed that the postshock region is transparent for all hydrogen continua. Finally, Narita (1973) took into account the opacity in the both Lyman and Balmer continua. The most elaborate numerical modelling of radiative shock waves based on the self-consistent solution of the equations of fluid dynamics, radiative transfer and atomic level populations was done by Klein et al. (1976, 1978). However, the coarse zonning [FORMULA] did not allow to authors to consider the detailed structure of the shock front including the radiative precursor and the thermalization zone where the electron temperature gradually equalizes with temperature of heavy particles. Nevertheless, this approach was found to be enough for consideration of shock dynamics in atmospheres of A-type stars because the radiative precursor is not so important due to the high temperature of the unperturbed gas.

Because astrophysical shocks in stellar atmospheres propagate through the partially ionized hydrogen gas, a substantial fraction of photons produced within the wake are absorbed in the radiative precursor. As was shown by Gillet & Lafon (1984, 1990) the structure of the radiative precursor is complex and should be treated with same degree of approximation as the postshock region. In their studies Gillet & Lafon (1984, 1990) treated the radiative transfer as an initial value problem which was solved using the shooting method. The principal obstacle in such an approach is that the transfer equation possesses a singularity in the postshock region (Gillet et al. 1989). Indeed, application of the eigenvalue methods for solution of the nongrey transfer problem is affected by exponentially growing errors (Mihalas 1978).

One of the first attempts to obtain the self-consistent solution for the shock wave structure was undertaken by Nelson & Goulard (1969) and Nelson (1973). They considered the shock waves propagating through the argon-like gas with upstream Mach numbers of [FORMULA] and [FORMULA]. The continuity, momentum and energy equations were written in the integral representation whereas the radiation transfer was treated in the simplified formulation. The studies of radiative shock waves in helium and nitrogen done by Clarke & Ferrari (1965), Farnsworth & Clarke (1971) and Foley & Clarke (1973) seem to be the best among known in the literature. The authors emphasized the crucial role of the radiation transfer treatment and employed the formal solution of the transfer equation. The self-consistent shock wave models were obtained in these studies with iteration procedure. Unfortunately, there is a problem of exponential factors when the formal solution is applied for optically thick layers.

In this paper we present a new approach based on the iterative solution of the equations of fluid dynamics, the rate equations and the radiation transfer equation. The momentum equation, the energy and rate equations are written in the form of ordinary differential equations. These equations are stiff and such a representation is most appropriate from the point of view of stability and small truncation errors. The radiation transfer is treated as a two-point boundary value problem. This allows us to obtain the stable solution of the transfer equation for the whole spectral range including both the opaque Lyman continuum and the more transparent higher order continua. The method of global iterations takes into account the coupling between the gas flow and the radiation field, so that the structure of the radiative shock wave is considered in terms of the self-consistent model.

In the framework of this first approach, only devoted to provide a new technique for obtaining the self-consistent solution, we consider the structure of steady, plane-parallel shock waves propagating through an infinite, isotropic, pure hydrogen plasma. The steady assumption is correct, for example, to a good degree of approximation in most applications to stellar atmospheres. Indeed, the time required for the gas flow to cross the shock wake with typical thickness of 104 -107  cm is much less than the characteristic time during of which the shock wave energy appreciably decreases. In pulsating stars, the radiative lifetime of a shock is between a few hours and a few months, which, consequently, is much larger than the 10-2 -10 s of the gas flow time to cross the shock wake. Accuracy of the plane-parallel approximation follows from the very small width of the shock wave in comparison with stellar radius.

After describing the shock wave model (Sect. 2) we derive the system of ordinary differential equations (Sect. 3). The radiation transfer equation is solved for the whole shock wave model using the Feautrier technique (Sect. 4). In Sect. 5 we show that the global iteration procedure comprising the initial value problem for ordinary differential equations and the two-point boundary value problem for radiation transfer converges to the self-consistent solution. Results of calculations demonstrating the applicability of the method are given in Sect. 6. Finally, in Sect. 7, we give some concluding remarks and discuss the future aspects of the problem.

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Online publication: April 20, 1998