= Binary mass-transfer contact equation
{title2=$\dot M_2/M_2=2(\dot J/J)/(\zeta+5/3-2q)$}
For a <circular orbit>, fixed total <mass>, transfer from $M_2$ to $M_1$, negligible spin and $R_L\propto a(M_2/M)^{1/3}$, the orbital <angular momentum> $J=M_1M_2\sqrt{Ga/M}$ gives
$$
\frac{\dot R_L}{R_L}=2\frac{\dot J}{J}+\left(2q-\frac53\right)\frac{\dot M_2}{M_2},\qquad q=\frac{M_2}{M_1}.
$$
If the donor follows a radius sequence with <stellar radius response exponent> $\zeta$, has no additional radius change at fixed mass, and maintains contact, equating its radius derivative to this expression proves the displayed contact equation. Negative external $\dot J$ drives donor mass loss when $\zeta+5/3-2q>0$. The appropriate equilibrium or adiabatic response and its timescale must be specified separately; the equation alone is not a universal dynamical-stability criterion.
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