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In the past years, several observations of AGN and X-ray binaries havesuggested the existence of a warm T around 0.5-1 keV and optically thick, \tau~ 10-20, corona covering the inner parts of the accretion disk. Theseproperties are directly derived from spectral fitting in UV to soft-X-raysusing Comptonization models. However, whether such a medium can be both inradiative and hydrostatic equilibrium with an accretion disk is stilluncertain. We investigate the properties of such warm, optically thick coronaeand put constraints on their existence. We solve the radiative transferequation 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 it is cooledby Compton scattering, with the underlying dissipative disk providing photonsto the corona. Our analytic calculations show that a dissipative thick,(\tau_{cor} ~ 10-12) corona on the top of a standard accretion disk can reachtemperatures of the order of 0.5-1 keV in its upper layers provided that thedisk is passive. But, in absence of strong magnetic fields, the requirement ofa Compton cooled corona in hydrostatic equilibrium in the vertical directionsets an upper limit on the Thomson optical depth \tau_{cor} < 5 . We show thisvalue 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 ~ 20 when themagnetic pressure is 100 times higher that the gas pressure. The observation ofwarm coronae with Thomson depth larger than ~ 5 puts tights constraints on thephysics of the accretion disk/corona systems and requires either strongmagnetic fields or vertical outflows to stabilize the system.

The existence of warm and optically thick dissipative coronae above accretion disks