Cooling-to-free-fall equality curve ($n_{H,\mathrm{crit}}=\frac{32G{m}_p}{3\pi X{f}_g}{\left(\frac{3\chi k_{B}T}{2\Lambda }\right)}^2$)

Equating the optically thin cooling time to the uniform-sphere free-fall time gives a critical gas density proportional to $T^2/{\Lambda }^2$. With hydrogen mass fraction $X$, particle ratio $\chi$ and gravitating gas fraction $f_g$, it is $[32G{m}_p/(3\pi X{f}_g)][3\chi k_{B}T/(2\Lambda ){]}^2$. At fixed temperature, gas above this density cools faster than free fall. A smaller gravitating gas fraction moves the boundary upward because gravity is faster at a fixed gas density.