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

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4. The radiation transfer equation

For the shock wave models considered in the present study the extinction coefficient is highest at the Lyman edge frequency [FORMULA]. In the preshock region the corresponding extinction coefficient is [FORMULA]. Behind the discontinuous jump the extinction coefficient initially decreases because of ionization of hydrogen atoms and then increases within the recombination zone. For all the models considered [FORMULA]. Thus, the time needed for photons to travel a mean free path is [FORMULA]  s, where c is the velocity of light. Because a characteristic structural length for Lyman photons is only a few mean free paths, a photon flight time is much shorter than the characteristic time of producing the changes due to hydrodynamic motions and, therefore, the time-derivative term in the transfer equation can be omitted because the radiation field is quasi static to a good accuracy.

In plane-parallel geometry the quasi static radiative transfer equation is written as

[EQUATION]

where [FORMULA] is the specific intensity of radiation depending on the frequency [FORMULA], the directional cosine µ and the monochromatic optical depth [FORMULA].

The present study is confined by treatment of the continuum radiation transfer with total extinction coefficient given by

[EQUATION]

where [FORMULA] and [FORMULA] are bound-free and free-free absorption coefficients, respectively. They can be found, for instance, in Mihalas (1978). [FORMULA] is the Thomson scattering cross section and [FORMULA] is the Rayleigh scattering cross section evaluated due to approximate formulae by Kurucz (1970). Because the scattering is assumed to be isotropic, the source function [FORMULA] can be written as

[EQUATION]

The source function [FORMULA] is determined from integration of the system of ordinary differential equations (20) - (24) and is evaluated at each cell center [FORMULA] as a function of frequency [FORMULA].

Solution of the radiation transfer equation when applied to the radiative shock waves is accompanied by some difficulties. First, because the shock wave propagates in the nearly neutral hydrogen gas, the slab has the appreciably large optical depth in the Lyman continuum [FORMULA], whereas in the Balmer and higher order continua the total optical depth is very small. For example, the ratio of the total optical depth at the Lyman continuum edge to that in the Balmer continuum is as high as [FORMULA]. Thus, in order to obtain the stable solution for the entire frequency range we have to treat the transfer equation as a two-point boundary value problem. Second, the optical depth increments

[EQUATION]

are extremely small for hydrogen continua of order [FORMULA]. Here [FORMULA] and [FORMULA] is the column mass contained in the cell. The straightforward application of the Feautrier method fails because of the limited machine accuracy. To alleviate this obstacle we employ an improved Feautrier solution proposed by Rybicki & Hummer (1991) and providing with much better numerical conditioning of the recurrence elimination scheme.

Thus, the transfer equation (46) is transformed (see, for example, Mihalas 1978) into the second-order differential equation

[EQUATION]

where [FORMULA] is a mean-intensity-like variable defined by

[EQUATION]

and µ changes in the range [FORMULA].

The transfer equation (50) is subject to boundary conditions at both surfaces of the slab. Assuming that radiation is produced only by the shock wave and that there is no incoming radiation from external sources, we have

[EQUATION]

The boundary conditions (52) are used in the second-order accuracy Taylor's expansion of the mean-intensity-like variable [FORMULA] at both boundaries of the slab and are sufficient to complete the system of the finite-difference transfer equations.

The transfer equation (50) is solved each cycle of global iterations for mean-intensity-like variable [FORMULA] defined at the cell centers. The total number of cells is [FORMULA] depending on the shock wave model. The frequency range [FORMULA] is divided into [FORMULA] intervals, where the upper boundary of the range is [FORMULA]. Boundaries of the intervals correspond to the threshold ionization frequencies and the lower boundary of the frequency range [FORMULA] is the ionization threshold frequency of the [FORMULA] -th bound state. Within each interval the integral with respect to frequency [FORMULA] is replaced by the Gaussian quadrature sum, so that the integral over the whole frequency range [FORMULA] is obtained by summation of interval integrals. The angular range [FORMULA] is also replaced by a set of angular points [FORMULA] at the Gaussian quadrature nodes. In the present study the number of quadrature nodes within each frequency interval and the number of angular points were [FORMULA]. Integrating [FORMULA] with respect to µ and [FORMULA] we evaluated in each cell center the mean intensity [FORMULA], the radiation energy density [FORMULA], the radiation pressure [FORMULA] and the divergence of radiative flux [FORMULA]. The total radiative flux was calculated in cell boundaries from

[EQUATION]

where

[EQUATION]

is an antisymmetric flux-like variable. It should be noted that according to our conventions the radiative flux is negative if the radiation propagates in negative direction. Thus, the radiative flux emerging ahead the discontinuous jump is always negative.

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© European Southern Observatory (ESO) 1998

Online publication: April 20, 1998
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