A soft X-ray excess above the 2-10 keV power law extrapolation is generally observed in AGN X-ray spectra. Presently there are two competitive models to explain it: blurred ionized reflection and warm Comptonisation. In the latter case, observations suggest a corona temperature $sim$ 1 keV and a corona optical depth $sim$ 10. Moreover, radiative constraints from spectral fits with Comptonisation models suggest that most of the accretion power should be released in the warm corona. The disk below is basically non-dissipative, radiating only the reprocessed emission from the corona. The true radiative properties of such a warm and optically thick plasma are not well-known, however. For instance, the importance of the Comptonisation process, the potential presence of strong absorption/emission features or the spectral shape of the output spectrum have been studied only very recently. We present in this paper simulations of warm and optically thick coronae using the TITAN radiative transfer code coupled with the NOAR Monte-Carlo code, the latter fully accounting for Compton scattering of continuum and lines. Illumination from above by a hard X-ray emission and from below by an optically thick accretion disk is taken into account as well as (uniform) internal heating. Our simulations show that for a large part of the parameter space, the warm corona with sufficient internal mechanical heating is dominated by Compton cooling and neither strong absorption nor emission lines are present in the outgoing spectra. In a smaller part of the parameter space, the calculated emission agrees with the spectral shape of the observed soft X-ray excess. Remarkably, this also corresponds to the conditions of radiative equilibrium of an extended warm corona covering almost entirely a non-dissipative accretion disk. These results confirm the warm Comptonisation as a valuable model for the soft X-ray excess.
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