Virial pressure-radius curve with uniform-sphere binding energy ($p_{\mathrm{ext}}=A/r^3-B/r^4$)

For fixed cloud mass $M$ and isothermal sound speed $c_s$, approximate the binding energy by $-3G{M}^2/(5r)$. The boundary-pressure virial theorem gives $A=3M{c}_s^2/(4\pi )$ and $B=3G{M}^2/(20\pi )$. The required external pressure peaks at $r_c=4B/(3A)$. A homologous radial displacement is restoring on the branch $d{p}_{\mathrm{ext}}/dr<0$, or $r>r_c$; the smaller-radius positive-pressure branch is unstable. This is a one-radius virial model, not the exact hydrostatic isothermal-sphere sequence.

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.