  H1821+643 is the nearest quasar hosted by a galaxy cluster. The energy output
by the quasar, in the form of intense radiation and radio jets, is captured by
the surrounding hot atmosphere. Here we present a new deep Chandra observation
of H1821+643 and extract the hot gas properties into the region where Compton
cooling by the quasar radiation is expected to dominate. Using detailed
simulations to subtract the quasar light, we show that the soft-band surface
brightness of the hot atmosphere increases rapidly by a factor of ~ 30 within
the central ~ 10 kpc. The gas temperature drops precipitously to < 0.4 keV and
the density increases by over an order of magnitude. The remarkably low
metallicity here is likely due to photo-ionization by the quasar emission. The
variations in temperature and density are consistent with hydrostatic
compression of the hot atmosphere. The extended soft-band peak cannot be
explained by an undersubtraction of the quasar or scattered quasar light and is
instead due to thermal ISM. The radiative cooling time of the gas falls to only
12 +/- 1 Myr, below the free fall time, and we resolve the sonic radius.
H1821+643 is therefore embedded in a cooling flow with a mass deposition rate
of up to 3000 Msolar/yr. Multi-wavelength observations probing the star
formation rate and cold gas mass are consistent with a cooling flow. We show
that the cooling flow extends to much larger radii than can be explained by
Compton cooling. Instead, the AGN appears to be underheating the core of this
cluster.
