  In the past years, several observations of AGN and X-ray binaries have
suggested 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. 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. We investigate the properties of such warm, optically thick coronae
and put constraints on their existence. 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 it is cooled
by Compton scattering, with the underlying dissipative disk providing photons
to the corona. Our analytic calculations show that a dissipative thick,
(\tau_{cor} ~ 10-12) corona on the 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. But, in 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} < 5 . We show 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 ~ 20 when the
magnetic pressure is 100 times higher that the gas pressure. The observation of
warm coronae with Thomson depth larger than ~ 5 puts tights constraints on the
physics of the accretion disk/corona systems and requires either strong
magnetic fields or vertical outflows to stabilize the system.
