Period-product invariant for conservative mass transfer ($P(M_1{M}_2{)}^3=\mathrm{constant}$)

Negligible spin and a circular orbit give $J=M_1{M}_2\sqrt{Ga/M}$. Fixed $M$ and $J$ imply $a(M_1{M}_2{)}^2$ is constant. Kepler's third law then gives the period-product invariant $P(M_1{M}_2{)}^3$.

At fixed total mass $M$ and period-product invariant, $P/M_d^2$ is proportional to $[M_d^5(M-M_d{)}^3{]}^{-1}$. Its unique global minimum occurs at donor mass $5M/8$ and accretor mass $3M/8$. It is useful for testing all possible conservative progenitors against a pre-overflow period bound.