  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.
