Using broadband optical imaging and Chandra X-ray data for a sample of 46 cluster central dominant galaxies (CDGs), we investigate the connection between star formation, the intracluster medium (ICM), and the central active galactic nucleus (AGN). We
report the discovery of a remarkably sharp threshold for the onset of star formation that occurs when the central cooling time of the hot atmosphere falls below ~ 5x10^8 yr, or equivalently when the central entropy falls below ~ 30 keV cm^2. In addition to this criterion, star formation in cooling flows also appears to require that the X-ray and galaxy centroids lie within ~ 20 kpc of each other, and that the jet (cavity) power is smaller than the X-ray cooling luminosity. These three criteria, together with the high ratio of cooling time to AGN outburst (cavity) age across our sample, directly link the presence of star formation and AGN activity in CDGs to cooling instabilities in the intracluster plasma. Our results provide compelling evidence that AGN feedback into the hot ICM is largely responsible for regulating cooling and star formation in the cores of clusters, leading to the significant growth of supermassive black holes in CDGs at late times.
