The Onset of Thermally Unstable Cooling from the Hot Atmospheres of
  Giant Galaxies in Clusters - Constraints on Feedback Models
  We present accurate mass and thermodynamic profiles for a sample of 56 galaxy
clusters observed with the Chandra X-ray Observatory. We investigate the
effects of local gravitational acceleration in central cluster galaxies, and we
explore the role of the local free-fall time (t$_{\rm ff}$) in thermally
unstable cooling. We find that the local cooling time (t$_{\rm cool}$) is as
effective an indicator of cold gas, traced through its nebular emission, as the
ratio of t$_{\rm cool}$/t$_{\rm ff}$. Therefore, t$_{\rm cool}$ alone
apparently governs the onset of thermally unstable cooling in hot atmospheres.
The location of the minimum t$_{\rm cool}$/t$_{\rm ff}$, a thermodynamic
parameter that simulations suggest may be key in driving thermal instability,
is unresolved in most systems. As a consequence, selection effects bias the
value and reduce the observed range in measured t$_{\rm cool}$/t$_{\rm ff}$
minima. The entropy profiles of cool-core clusters are characterized by broken
power-laws down to our resolution limit, with no indication of isentropic
cores. We show, for the first time, that mass isothermality and the $K \propto
r^{2/3}$ entropy profile slope imply a floor in t$_{\rm cool}$/t$_{\rm ff}$
profiles within central galaxies. No significant departures of t$_{\rm
cool}$/t$_{\rm ff}$ below 10 are found, which is inconsistent with many recent
feedback models. The inner densities and cooling times of cluster atmospheres
are resilient to change in response to powerful AGN activity, suggesting that
the energy coupling between AGN heating and atmospheric gas is gentler than
most models predict.
