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We analyse the radial structure of self‐gravitating spheres consisting of multiple interpenetrating fluids, such as the X‐ray emitting gas and the dark halo of a galaxy cluster. In these  dipolytropic  models, the adiabatic dark matter sits in equilibrium, while the gas develops a gradual, smooth, quasi‐stationary cooling flow. Both affect and respond to the collective gravitational field. We find that all subsonic, radially continuous, steady solutions require a non‐zero minimum central point mass. For Mpc‐sized haloes with 7–10 effective degrees of freedom ( F  2 ), the minimum central mass is compatible with observations of supermassive black holes. Smaller gas mass influxes enable smaller central masses for wider ranges of  F  2 . The halo comprises a sharp spike around the central mass, embedded within a core of nearly constant density (at  10 1 –10 2.5  kpc  scales), with outskirts that attenuate and naturally truncate at finite radius (several Mpc). The gas density resembles a broken power law in radius, but the temperature dips and peaks within the dark core. A finite minimum temperature occurs due to gravitational self‐warming, without cold mass dropout nor needing regulatory heating. X‐ray emission from the intracluster medium mimics a β‐model plus bright compact nucleus. Near‐sonic points in the gas flow are bottlenecks to the allowed steady solutions; the outermost are at kpc scales. These sites may preferentially develop cold mass dropout during strong perturbations off equilibrium. Within the sonic point, the profile of gas specific entropy is flatter than   s ∝ r  1/2  , but this is a shallow ramp and not an isentropic core. When  F  2  is large, the inner halo spike is only marginally Jeans stable in the central parsec, suggesting that a large non‐linear disturbance could trigger local dark collapse on to the central object.

Radial structure, inflow and central mass of stationary radiative galaxy clusters