= Donor-wind period invariant with fixed retention fraction
{title2=$P_{\rm orb}M_1^{3f}M_2^3(M_1+M_2)^2=\mathrm{constant}$}
In a circular <binary star> let the <donor star> lose <mass> $\dot M_1<0$, with constant retained fraction $f$ and $\dot M_2=-f\dot M_1$. A <donor star> <stellar wind> escaping with the <donor star>'s orbital specific <angular momentum> has $\dot J/J=(1-f)\dot M_1M_2/[M_1(M_1+M_2)]$. Differentiating $J=G^{2/3}(P_{\rm orb}/2\pi)^{1/3}M_1M_2(M_1+M_2)^{-1/3}$ and using this loss law gives
$$
\frac{\dot P_{\rm orb}}{P_{\rm orb}}=-3f\frac{\dot M_1}{M_1}-3\frac{\dot M_2}{M_2}-2\frac{\dot M_1+\dot M_2}{M_1+M_2}.
$$
For constant $f$ this integrates to $P_{\rm orb}M_1^{3f}M_2^3(M_1+M_2)^2=\mathrm{constant}$. If $f$ varies, the differential relation remains valid but the first factor must be replaced by $\exp(3\int f\,d\log M_1)$; treating a time-dependent exponent as a fixed power introduces an extra $\dot f\log M_1$ term.
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