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 _ { ff } ) in thermally unstable cooling .
We find that the local cooling time ( t _ { cool } ) is as effective an indicator of cold gas , traced through its nebular emission , as the ratio of t _ { cool } /t _ { ff } .
Therefore , t _ { cool } alone apparently governs the onset of thermally unstable cooling in hot atmospheres .
The location of the minimum t _ { cool } /t _ { 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 _ { cool } /t _ { 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 _ { cool } /t _ { ff } profiles within central galaxies .
No significant departures of t _ { cool } /t _ { 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 .