X-ray emission from expanding cocoons

X-ray observations of extragalactic radiosources show strong evidences of interaction between the radio emitting plasma and the X-ray emitting ambient gas. In this paper we perform a detailed study of this interaction by numerical simulations. We study the propagation of an axisymmetric supersonic jet in an isothermal King atmosphere and we analyze the evolution of the resulting X-ray properties and their dependence on the jet physical parameters. We show the existence of two distinct and observationally different regimes of interaction, with strong and weak shocks. In the first case shells of enhanced X-ray emission are to be expected, while in the second case we expect deficit of X-ray emission coincident with the cocoon. By a comparison between analytical models and the results of our numerical simulations, we discuss the dependence of the transition between these two regimes on the jet parameters and we find that the mean controlling quantity results to be the jet kinetic power. We then discuss how the observed jets can be used to constrain the jet properties.