 Virial pressure-radius curve with uniform-sphere binding energy

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.