We determine simple analytical conditions for combined radiative and thermal equilibrium between the X-ray emitting plasma and the cold reprocessor in active galactic nuclei (AGN) and galactic black holes (GBHs). These conditions determine the pressure at the transition zone, which is not arbitrary in the static situation. The conditions for the static solution are derived analytically for different forms of plasma heating and for cooling provided by Compton cooling, bremsstrahlung and (optionally) advective cooling.
We conclude that if Compton heating is the only heating mechanism, we always achieve static equilibrium between the phases. For constant volume heating and radiatively cooled plasma the static solution is never achieved and evaporation or condensation takes place. However, static solutions with no evaporation/condensation are found for radiative cooling supplemented with lateral advection. Similar results are obtained for a two-temperature plasma. In the case of a general prescription of mechanical heating (Q+h0PmT−s) of radiatively cooled one-temperature plasma, we find that a static solution is only possible when 0<s<3/2.
Such conditions apply to all models, like disc/corona solutions, accretion-dominated advection flows (ADAFs) at their outer ADAF/disc boundary or cold clumps embedded in a hot medium. These models have to be reconsidered and supplemented either with an additional condition for the pressure at the transition zone or with an equation describing the mass exchange between the phases.