Escape probability methods versus "exact" transfer for modelling the X-ray spectrum of Active Galactic Nuclei and X-ray binaries

In the era of XMM-Newton and Chandra missions, it is crucial to use codes able to compute correctly the line spectrum of X-ray irradiated thick media (Thomson thickness of the order of unity), in order to build models for the structure and the emission of the central regions of Active Galactic Nuclei (AGN), or of X-ray binaries. In all photoionized codes except in our code Titan, the line intensities are computed with the so-called "escape probability approximation". In its last version, Titan solves the transfer of a thousand lines and of the continuum with the "Accelerated Lambda Iteration" method, which is one of the most efficient and at the same time the most secure for line transfer. We first review the escape probability formalism and mention various reasons why it should lead to wrong results concerning the line fluxes. Then we check several approximations commonly used instead of line transfer in photoionization codes, by comparing them to the full transfer computation. We find that for conditions typical of the AGN or X-ray binary emission medium, all approximations lead to an overestimation of the emitted X-ray line spectrum, which can reach more than one order of magnitude. We show that it is due mainly to the local treatment of line photons, implying a delicate balance between excitations of X-ray transitions by the very intense underlying diffuse X-ray continuum (which are not taken properly into account in escape probability approximations) and the net rate of excitations by the diffuse line flux. The most affected lines are those in the soft X-ray range. Such processes are much less important in cooler and thinner media ( like the Broad Line Region of AGN), as the most intense lines lie in the optical and near ultraviolet range where the diffuse continuum is small. We conclude that it is very important to treat correctly the transfer of the continuum to get the best results for the line spectrum. On the other hand the approximations used for the escape probabilities have a relatively small influence on the computed thermal and ionization structure of the surface layers, but in the deep layers, they lead to an overestimation of the ionization state. As a consequence the computed continuum emitted by the back (non-irradiated) side is not correct, and might be strongly overestimated in the EUV range.