 Critical mass-to-flux ratio

For fixed cloud shape, gravitational energy scales as $-C_{g}G{M}^2/L$ and magnetic energy as $C_B{\Phi }^2/({\mu }_{0}L)$. The magnetic-criticality threshold for negligible pressure is $M/\Phi >\sqrt{C_B/C_g}/\sqrt{{\mu }_{0}G}$. Ideal flux freezing keeps this ratio fixed during homologous collapse; geometry determines the numerical coefficient.