= Conservative pre-giant period-ratio minimum
{title2=$\min(P/m_d^2)\text{ occurs at }m_d=5m/8$}
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.
Back to article page
