Roche-lobe-filling period-density relation ($P\propto (R_2^3/M_2{)}^{1/2}$)

If $R_2=R_L\simeq Ca(M_2/M{)}^{1/3}$, Kepler's third law givesThus a Roche lobe-filling star's period measures its inverse square-root mean mass density, largely independently of the companion mass in this approximation. A local mass-radius relation $R_2\propto M_2^{\zeta }$ gives $P\propto M_2^{(3\zeta -1)/2}$.

For contact with $R_d/R_{\odot }=M_d/M_{\odot }$ and $R_L/a=f(M_d/M{)}^{1/3}$, Kepler's third law gives $P_0=2\pi \sqrt{R_{\odot }^3/(f^{3}G{M}_{\odot })}$. The total binary mass cancels. The linear period relation depends on that particular equilibrium radius sequence; a donor with a different stellar radius response exponent follows a different mass-period relation.

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 $M_d^2$.

For fixed total solar-unit mass $m$ and conserved orbital angular momentum, conservative binary mass transfer preserves $K=P[m_d(m-m_d){]}^3$. Hence $P/m_d^2=K/[m_d^5(m-m_d{)}^3]$. Its logarithmic derivative is $-5/m_d+3/(m-m_d)$, vanishing at $5m/8$ with positive second derivative. The ratio diverges at both endpoints, so this is its global minimum. Comparing this minimum to a pre-giant bound $P_0$ tests whether any such conservative antecedent can overflow before the giant branch; satisfying it is necessary, not a proof of an entire evolutionary history.