= Cooling-to-free-fall equality curve
{title2=$n_{H,\rm crit}=\frac{32Gm_p}{3\pi Xf_g}\left(\frac{3\chi k_BT}{2\Lambda}\right)^2$}
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 $[32Gm_p/(3\pi Xf_g)][3\chi k_BT/(2\Lambda)]^2$. 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.
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