2002ApJ...573..533S

As more cooling flow clusters of galaxies with central radio sources are observed with the Chandra and XMM-Newton X-ray observatories, more examples of ``bubbles'' (low-emission regions in the X-ray coincident with radio emission) are being found. These bubbles are surrounded by bright shells of X-ray emission, and no evidence of current strong shocks has yet been found. Using an analytic approach and some simplifying assumptions, we derive expressions relating the size and location of a bubble, as well as the density contrast between the bubble and the ambient medium, with the shock history of the bubble. These can be applied straightforwardly to new observations. We find that existing observations are consistent with a mild shock occurring in the past and with the bulk of the cool material in the X-ray shells being cooled at the cluster center and then pushed outward by the radio source. Strong shocks are generally ruled out. We also discuss Rayleigh-Taylor instabilities, as well as the case of a bubble expanding into an older bubble produced from a previous cycle of radio activity.