## 4. The radiation transfer equationFor the shock wave models considered in the present study the
extinction coefficient is highest at the Lyman edge frequency
. In the preshock region the corresponding
extinction coefficient is . 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
. Thus, the time needed for photons to travel a
mean free path is s, where In plane-parallel geometry the quasi static radiative transfer equation is written as where is the specific intensity of
radiation depending on the frequency , the
directional cosine The present study is confined by treatment of the continuum radiation transfer with total extinction coefficient given by where and are bound-free and free-free absorption coefficients, respectively. They can be found, for instance, in Mihalas (1978). is the Thomson scattering cross section and 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 can be written as The source function is determined from integration of the system of ordinary differential equations (20) - (24) and is evaluated at each cell center as a function of frequency . 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 , 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 . 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 are extremely small for hydrogen continua of order . Here and 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 where is a mean-intensity-like variable defined by and 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 The boundary conditions (52) are used in the second-order accuracy Taylor's expansion of the mean-intensity-like variable 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
defined at the cell centers. The total number of cells is
depending on the shock wave model. The
frequency range is divided into
intervals, where the upper boundary of the
range is . Boundaries of the intervals
correspond to the threshold ionization frequencies and the lower
boundary of the frequency range is the
ionization threshold frequency of the -th
bound state. Within each interval the integral with respect to
frequency is replaced by the Gaussian
quadrature sum, so that the integral over the whole frequency range
is obtained by summation of interval
integrals. The angular range is also replaced
by a set of angular points 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
. Integrating with
respect to where 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. © European Southern Observatory (ESO) 1998 Online publication: April 20, 1998 |