Critical rotation of a spherical cloud (${\Omega }_c^2=GM/R_c^3,R_c=J^2/({\alpha }^{2}G{M}^3)$)

In the spherical approximation, equatorial centrifugal acceleration balances surface gravity when ${\Omega }_c^2{R}_c=GM/R_c^2$. For $I=\alpha M{R}^2$ and conserved angular momentum $J$, this gives the displayed critical radius. Real rotating gas can deform, lose matter and exchange torques before a spherical model remains accurate at this limit.