The ``mean-photon'' approximation

The main difficulty arising in calculation of the shock wave structure is due to the radiation transfer. Indeed, as is shown in Fig., the absorption coefficient $\kappa_{\nu}$ strongly depends on the frequency especially for lines. That is why many investigations were limited to one atom with only one or two continua (for instance [5], [10], [11]). The ``mean-photon'' approximation consists of replacement each exponential $\nu$-decreasing continuum by an average photon of frequency $\overline{\nu}\simeq\,\nu_{th}+\widetilde{T}_{e}/h$, where $\nu_{th}$ is the threshold frequency of the continuum and $\widetilde{T}_{e}$ is the effective temperature of recombination region which characterizes the shape of the continum spectrum. This approximation is discussed in details in [5]. Without scattering and using the Eddington approximation in which the transfer equation is splitted into two first order differential equations [12], it is possible to obtain a self-consistent solution of the whole shock structure (precursor and wake). Figure shows a typical result [11]. Recently, a new calculation reported in [10] takes into account the hydrogen molecular state which was not considered in [11]. In addition, to its bad frequency description, the solution of the transfer equation by the Eddington approximation is strongly affected by exponentially growing errors ([5], [12]). Thus this method cannot provide an accurate determination of the structure of the diffrent shock layers.

  

Figure: Structure of a radiative shock wake in a pure hydrogen gas with an unperturbed density 10^-10g/cm^-3 and temperature 1000K and for a shock front velocity of 40 km/s (Mach number = 8). The ordinate scale of Balmer F⁺_B, F^-_B fluxes and the ionization coefficient $\alpha\equiv\,n_{e}/n$ is between 0 and 1. The F⁺_L and F^-_L Lyman fluxes are multipied by 10 [10].