 Binary mass-transfer contact equation

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 $J=M_1{M}_2\sqrt{Ga/M}$ givesIf 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.