Warm and optically thick dissipative coronae above accretion disks

Context. In past years, several observations of AGN and X-ray binaries suggested the existence of a warm (T similar to 0.5-1 keV) and optically thick (tau(cor) similar to 10-20) corona covering the inner parts of the accretion disk. These properties are directly derived from spectral fitting in UV to soft-X-rays using Comptonization models. However, whether such a medium can be both in radiative and hydrostatic equilibrium with an accretion disk is still uncertain. Aims. We investigate the properties of such warm, optically thick coronae and put constraints on their existence. Methods. We solve the radiative transfer equation for grey atmosphere analytically in a pure scattering medium, including local dissipation as an additional heating term in the warm corona. The temperature profile of the warm corona is calculated assuming that it is cooled by Compton scattering, with the underlying dissipative disk providing photons to the corona. Results. Our analytic calculations show that a dissipative thick corona (tau(cor) in the range 10-12) on top of a standard accretion disk can reach temperatures of the order of 0.5-1 keV in its upper layers provided that the disk is passive. However, in the absence of strong magnetic fields, the requirement of a Compton cooled corona in hydrostatic equilibrium in the vertical direction sets an upper limit on the Thomson optical depth tau(cor) less than or similar to 5. We show that this value cannot be exceeded independently of the accretion disk parameters. However, magnetic pressure can extend this result to larger optical depths. Namely, a dissipative corona might have an optical depth up to similar to 20 when the magnetic pressure is 100 times higher than the gas pressure. Conclusions. The observation of warm coronae with Thomson depth larger than similar or equal to 5 puts tight constraints on the physics of the accretion disk/corona systems and requires either strong magnetic fields or vertical outflows to stabilize the system.