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

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2. The shock wave model

Consider a steady, plane-parallel shock wave propagating through the homogeneous medium which is at rest and consists of a pure hydrogen gas. No radiation and gravitational forces from external sources are assumed to be present. The problem to be solved is that to describe the spatial structure of the shock wave in terms of the self-consistent solution of the equations of fluid dynamics, the rate equations for hydrogen atomic level populations and the radiation transfer equation. The problem is characterized by three input parameters: the temperature [FORMULA] and the density [FORMULA] of the unperturbed gas as well as the speed [FORMULA] at which the gas material flows into the shock.

The structure of radiative shock waves is schematically divided into four zones: (1) a precursor, where the gas is heated and is partially ionized by radiation emerging from the postshock region; (2) a very narrow zone, where a major part of the kinetic energy of the upstream flow is converted due to viscosity and conductivity into the thermal energy of translational motions of heavy particles, that is, neutral atoms and ions; (3) a thermalization zone, where the kinetic energy of translational motions of heavy particles is redistributed among various degrees of freedom; (4) a radiative relaxation zone, where hydrogen atoms recombine and the gas radiatively cools. Because of its extremely small width (a few mean free paths of gas particles), the second zone cannot be correctly described in terms of the fluid dynamics and, hence, should be considered as a discontinuous jump across which the Rankine-Hugoniot equations are applied. Thus, the present study is confined by consideration of the spatial structure of the radiative precursor as well as the thermalization and relaxation zones.

Let the origin of the comoving frame to coincide with infinitesimaly thin discontinuous jump dividing the medium into the preshock and the postshock regions. The spatial coordinate is [FORMULA] at the discontinuous jump, is negative in the preshock region and is positive in the postshock region. The shock wave model is represented by a flat finite slab comoving with discontinuous jump. Thus, the velocity of the gas material flowing through the slab is always positive: [FORMULA]. At the outer boundary of the preshock region with spatial coordinate [FORMULA] the gas is assumed to be unperturbed. The spatial coordinate of the postshock outer boundary is [FORMULA] and physical properties of the gas at this point are not known.

The radiation transfer equation is solved for the whole slab in the framework of the two-point boundary value problem, therefore the slab is represented by a set of spatial cells. The discontinuous jump locates at the J -th cell boundary with spatial coordinate [FORMULA]. The cells are inequally spaced, are smallest at the discontinuous jump and increase outwardly in both directions from the discontinuous jump according to the geometrical progression. Following the traditional conventions of computational radiation hydrodynamics (see, for example, Mihalas & Mihalas 1984), all thermodynamic variables are defined at the cell centers [FORMULA] and are denoted by half-integer subscripts. The spatial coordinates of the cell centers nearest to the discontinuous jump hereafter are referred as the inner boundaries of the preshock and postshock regions, respectively, and are denoted as [FORMULA] and [FORMULA].

The assumption of the steady shock wave allows us to reduce the equations of fluid dynamics to a system of ordinary differential equations. In order to take into account the coupling between hydrodynamic motions and the radiation field we employ the iteration procedure comprising the consecutive solution of the transfer equation and integration of the rate and fluid dynamics equations. The starting point for integration of ordinary differential equations is the outer boundary of the preshock region [FORMULA] where the gas is assumed to be unperturbed. The preshock integration is done within the interval [FORMULA]. Then we solve the Rankine-Hugoniot equations

[EQUATION]

where h is the enthalpy,

[EQUATION]

where [FORMULA], [FORMULA], [FORMULA], [FORMULA], [FORMULA] and [FORMULA] are the translational energy; rotational, vibrational, electronic excitation internal energies and dissociation and ionization potential energies respectively. In this paper only [FORMULA] and [FORMULA] are different of zero. [FORMULA] is the total radiative flux, [FORMULA], [FORMULA] and [FORMULA] are the mass, momentum and energy fluxes across the discontinuous jump. In the present study we assume that the radiation pressure [FORMULA] and the radiation energy density [FORMULA] can be neglected in comparison with gas pressure and internal energy of the gas material. Applicability of this assumption is shown below.

Let us designate, for the sake of convenience, the quantities defined at [FORMULA] by the superscript minus and the quantities defined at [FORMULA] by the superscript plus. Eqs. (1) - (3) are solved for the temperature of heavy particles [FORMULA] and the inverse compression ratio [FORMULA] [FORMULA]. Assuming that hydrogen atoms undergo across the discontinuous jump neither excitation of the bound levels nor ionization and that the hydrogen ions have the same temperature as that of neutral hydrogen atoms, we obtain

[EQUATION]

where k is the Boltzmann constant, [FORMULA] and [FORMULA] are the total numbers of hydrogen atoms and free electrons per unit volume, respectively. It should be noted that jump conditions (1) - (3) are not applied to the infinitesimally thin discontinuous jump coinciding with the cell interface [FORMULA] but rather relate the physical variables at cell centers [FORMULA] and [FORMULA]. That is why the term containing [FORMULA] into Eq. (5) does not cancel and has to be taken into account. The inverse compression ratio is obtained from

[EQUATION]

where

[EQUATION]

Eqs. (5) and (6) imply that the postshock electron temperature [FORMULA] at the cell center [FORMULA] is known. Across the discontinuous jump the elctron gas undergoes the adiabatic compression and the electron temperature increases by a factor of [FORMULA], where [FORMULA] is the ratio of specific heats (Zel'dovich & Raizer 1966). Because the adiabatic compression has a weak effect and the electron heat conduction is out the scope of the present study, we assumed that the electron temperature does not change across the discontinuous jump, that is, [FORMULA].

Eqs. (5) and (6) are solved to determine the initial conditions for the postshock integration which is done within the interval [FORMULA]. Integration of ordinary differential equations provides with spatial distibutions of electron temperature [FORMULA], gas density [FORMULA], the number densities of free electrons [FORMULA] and atomic level populations [FORMULA]. These quantities are used in solution of the radiation transfer equation which gives improved radiation intensities within the whole slab. The global iteration procedure consisting of the integration of ordinary differential equations and solution of the radiation transfer equation continues until the relative changes of all quantities become less than the convergence criterion. Below we discuss details of the global iteration procedure.

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© European Southern Observatory (ESO) 1998

Online publication: April 20, 1998
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