  Recent analysis shows that it is important to explicitly include the
gravitational potential of the central brightest central galaxy (BCG) to infer
the acceleration due to gravity ($g$) and the free-fall time ($t_{\rm ff}
\equiv [2r/g]^{1/2}$) in cool cluster cores. Accurately measuring $t_{\rm ff}$
is crucial because according to numerical simulations cold gas condensation and
strong feedback occur in cluster cores with min($t_{\rm cool}/t_{\rm ff}$)
below a threshold value close to 10. Recent observations which include the BCG
gravity show that the observed threshold in min($t_{\rm cool}/t_{\rm ff}$) lies
at a somewhat higher value, close to 10-30; there are only a few clusters in
which this ratio falls much below 10. In this paper we compare numerical
simulations of feedback AGN (Active Galactic Nuclei) jets interacting with the
intracluster medium (ICM), with and without a BCG potential. We find that, for
a fixed feedback efficiency, the presence of a BCG does not significantly
affect the temperature but increases (decreases) the core density (entropy) on
average. Most importantly, min($t_{\rm cool}/t_{\rm ff}$) is only affected
slightly by the inclusion of the BCG gravity. Also notable is that the lowest
value of min($t_{\rm cool}/t_{\rm ff}$) in the NFW+BCG runs are about twice
larger than in the NFW runs. We also look at the role of depletion of cold gas
due to star formation and show that it only affects the rotationally dominant
component (torus), while the radially dominant component (which regulates the
feedback cycle) remains largely unaffected. The distribution of metals due to
AGN jets in our simulations is predominantly along the jet direction and the
radial spread of metals is less. We also show that the turbulence in cool core
clusters is weak, consistent with recent Hitomi results on Perseus cluster.
