Motivated by hydrodynamic simulations, we discuss the X-ray appearance of radio galaxies embedded in the intracluster medium (ICM) of a galaxy cluster. We distinguish three regimes. In the early life of a powerful source, the entire radio cocoon is expanding supersonically and hence drives a strong shock into the ICM. Eventually, the sides of the cocoon become subsonic and the ICM is disturbed by the sonic booms of the jet's working surface. In both of these regimes, X-ray observations would

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... d an X-ray shell. In the strong shock regime, this shell will be hot and relatively thin. However, in the weak shock (sonic-boom) regime, the shell will be approximately the same temperature as the undisturbed ICM. If a cooling flow is present, the observed shell may even be cooler than the undisturbed ICM due to the lifting of cooler material into the shell from the inner (cooler) regions of the cluster. In the third and final regime, the cocoon has collapsed and no well-defined X-ray shell will be seen. We discuss ways of estimating the power and age of the source once its regime of behavior has been determined.