 Critical mass from a virial pressure turning point

Eliminate cloud radius using its mean density and the critical radius $r_c=4GM/(15c_s^2)$. The displayed threshold scales as $T^{3/2}{\bar{\rho }}^{-1/2}$ at fixed mean particle mass. Equivalently, the maximum confining pressure is $10125c_s^8/(1024\pi G^3{M}^2)$. The numerical coefficient depends on the assumed density profile and on whether the density refers to the mean or the boundary.