Pre-giant Roche-lobe-filling period limit ($P_i<2\pi \sqrt{R_{\mathrm{BGB}}^3/(C^{3}G{M}_d)}$)

For $R_L=Ca(M_d/M{)}^{1/3}$, the period when a donor reaches radius $R$ is $P=2\pi \sqrt{R^3/(C^{3}G{M}_d)}$. Requiring overflow before a specified evolutionary radius therefore bounds the initial period independently of companion mass in this lobe approximation. If that radius scales as $M_d^{5/3}$, the bound scales as $M_d^2$.