 Solution

If gas cannot radiate enough energy to fall below the virial temperature, infall converts gravitational energy into heat through shocks and compression. Thermal pressure can support an extended atmosphere in the dark-matter halo. It need not have the same density profile as the collisionless dark matter, because its entropy and pressure matter.

A useful quantitative comparison is the radiative gas cooling time $t_{\mathrm{cool}}\sim (3/2)n{k}_{B}T/(n_e{n}_i\Lambda )$ against the halo dynamical time $t_{\mathrm{dyn}}\sim r_{\mathrm{vir}}/V_{\mathrm{vir}}$. If cooling remains slow or a heating source balances it, the gas stays predominantly hot and diffuse, rather than forming a compact, cold, self-gravitating stellar system. Without sufficient cooling, most baryons remain pressure-supported halo gas. Very slow cooling can still feed gradual central condensation; the condition is about energy loss relative to the evolution time, not an absolute prohibition on any inward motion.