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

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7. Conclusion

The primary goal of the work reported in this paper was to obtain the self-consistent solution of the equations of fluid dynamics, rate equations and radiation transfer equation for the structure of the steady shock wave. The procedure of global iterations described in the present paper in general resembles the compute of stellar atmosphere models. Indeed, like in stellar atmosphere calculations the shock wave model takes into account the coupling between the gas material and radiation field. The self-consistent model is obtained with iteration procedure comprising the solution of the radiation transfer equation and integration of the mass, momentum and energy conservation equations written in the form of the ordinary differential equations. Each cycle of iterations gives, in general, improved characteristics of the gas and radiation field.

At the same time, the problem of the shock wave structure compared to that of stellar atmosphere models contains a number of serious complications. First, atomic level populations are not only in strong departures from LTE but are also in significant departures from statistical equilibrium. Second, unlike the stellar atmospheres, where the divergence of radiative flux is [FORMULA] (the condition of radiative equilibrium), in shock waves the part of the energy of hydrodynamic flow is transformed into radiation and the radiative equilibrium is established only far away from the discontinuous jump. Furthemore, in stellar atmosphere models the total radiative flux is given as one of the boundary conditions, whereas in the shock wave model the emerging flux is obtained from the solution of the problem. The small optical depth increments in hydrogen continua of order [FORMULA] lead to the losses of the machine accuracy when the Feautrier technique is applied. The problem is so serious, that even the improved method by Rybicki & Hummer (1991) sometimes fails. Third, the rate equations are stiff and need the special treatment in their solution. In particular, the convergence of global iterations depends on the tolerance parameter determining the maximum error permitted during the integration.

The present paper is confined by consideration of the two-level atomic model, so that the radiation transfer is treated for the Lyman and Balmer continua, only. This approximation seems to be insufficient for the shock wave problem because the occupation numbers of levels [FORMULA] obviously deviate from LTE and the perceptible fraction of radiation is transported at frequencies lower than the Balmer edge frequency. Nevertheless, notwithstanding such a restriction, there is a qualitative agreement of our results with those obtained earlier by other authors. For example, according to calculations of Gillet & Lafon (1990) the electron temperature just ahead the discontinuous jump is [FORMULA] for the upstream velocity of [FORMULA]. Although in the present study the highest upstream velocity was [FORMULA], a very approximate comparison can be done with fitting formula (55) which gives the same electron temperature for the upstream velocity [FORMULA]. Thus, more detailed calculations are needed and in the forthcoming paper we are going to present the grid of the shock wave models computed for the larger number of hydrogen atomic levels and wider range of upstream velocities, the models of the present study being used as initial approximation for more correct shock wave models.

More realistic models, however, should take into account not only bound-free terms but also bound-bound terms in the rate equations and the radiation transfer problem should be solved for the both continuum and spectral line radiation. This is the perspective for the near future. It is certainly one of the most basic. Indeed, preceding shock studies show that radiative processes, which determine the wake cooling, have a strong influence on the resulting shock structure. Because in the model of this paper we consider a pure H-plasma without [FORMULA], and only include the bound-free photo- and collisional processes of H atoms, we expect that the absence of some predominant coolants such as neutral and singly ionized metal atoms might appreciably underestimate the radiative cooling rate of the gas. The importance of radiative heating and cooling rates in shocked circumstellar envelopes have been recently investigated (Woitke et al. 1996) but only a few transitions of the numerous metal lines were considered. At present such a basic study seems to be beyond our immediate abilities and is out of the scope of our paper which was to investigate the possibility of obtaining a self-consistent solution of the structure of radiative shock waves in dense atmospheric gas.

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© European Southern Observatory (ESO) 1998

Online publication: April 20, 1998
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