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We present accurate mass and thermodynamic profiles for a sample of 56 galaxyclusters observed with the Chandra X-ray Observatory. We investigate theeffects of local gravitational acceleration in central cluster galaxies, and weexplore the role of the local free-fall time (t$_{\rm ff}$) in thermallyunstable cooling. We find that the local cooling time (t$_{\rm cool}$) is aseffective an indicator of cold gas, traced through its nebular emission, as theratio of t$_{\rm cool}$/t$_{\rm ff}$. Therefore, t$_{\rm cool}$ aloneapparently governs the onset of thermally unstable cooling in hot atmospheres.The location of the minimum t$_{\rm cool}$/t$_{\rm ff}$, a thermodynamicparameter that simulations suggest may be key in driving thermal instability,is unresolved in most systems. As a consequence, selection effects bias thevalue and reduce the observed range in measured t$_{\rm cool}$/t$_{\rm ff}$minima. The entropy profiles of cool-core clusters are characterized by brokenpower-laws down to our resolution limit, with no indication of isentropiccores. We show, for the first time, that mass isothermality and the $K \proptor^{2/3}$ entropy profile slope imply a floor in t$_{\rm cool}$/t$_{\rm ff}$profiles within central galaxies. No significant departures of t$_{\rmcool}$/t$_{\rm ff}$ below 10 are found, which is inconsistent with many recentfeedback models. The inner densities and cooling times of cluster atmospheresare resilient to change in response to powerful AGN activity, suggesting thatthe energy coupling between AGN heating and atmospheric gas is gentler thanmost models predict.

The Onset of Thermally Unstable Cooling from the Hot Atmospheres of Giant Galaxies in Clusters - Constraints on Feedback Models