# What happens in the event that the cooling radius is shorter than the virial radius of a Cold Dark Matter Halo?

The cooling radius of a cold dark matter halo is defined to be the time at which the cooling time $t_{cool} = t_{free fall}$

where

$$t_{cool}=\frac{\rho \varepsilon }{\Lambda \left ( T \right )n_{H}^{2}},\quad t_{free fall}=\sqrt{\frac{3 \pi}{32 G \left ( \rho + \rho_{dm} \right )}}$$

In the case where there is a spherical gas cloud gravitationally drawn towards a gaseous halo with the cooling radius $r_{cool} > r_{virial}$, the spherical gas cloud cools rapidly and contracts rapidly, prohibiting hydrostatic equilibrium from taking place. The result is a spherical gas cloud collapsing by the free fall time $t_{free fall}$.

What actually happens then in the event that the cooling radius $r_{cool}$ is smaller than the virial radius of the gaseous halo? What is the resulting dynamics?