  We investigate the effects of a cD galaxy's gravity and AGN heating of the
host galaxy cluster. We consider a standard prescription for the hydrodynamics,
with the structures determined by mass continuity, momentum and energy
conservation equations in spherical symmetry. The cluster comprises a dark
matter halo (DM) and ionised X-ray emitting intracluster gas (ICM), which
jointly determine the gravitational potential. The cD galaxy is an additive
gravitational potential component. The DM assumes a polytropic equation of
state (determined by its microphysics), which could be non-radiative
self-interacting particles or more exotically interacting particles. The AGN
provides distributed heating, counteracting radiative cooling. Stationary
density and velocity dispersion profiles are obtained by numerically
integrating the hydrodynamic equations with appropriate boundary conditions.
The minimum gas temperature in the cluster core is higher when a cD galaxy is
present than when it is absent. The solutions also yield a point-like mass
concentration exceeding a minimum mass: presumably the AGN's supermassive black
hole (SMBH). Consistency with observed SMBH masses constrains the possible DM
equations of state. The constraints are looser when a cD galaxy is present.
Distributed (AGN) heating alters cluster global properties, and also reduces
the lower limits for the central point-mass, for the preferred DM models in
which the dark particles have greater heat capacity than point particles.
Eluding these constraints would require dominant non-spherical or anisotropic
effects (e.g. bulk rotation, non-radial streaming, asymmetric lumps, or a
strong magnetic field).
