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At the critical Roche surface, $g=0$ at $L_1$, making $⟨g^{-1}⟩$ singular. Once star 1 overfills its Roche lobe, the relevant equipotentials are no longer nested closed surfaces belonging to that star: they open through the $L_1$ saddle and connect to star 2. Matter then undergoes Roche-lobe overflow with finite velocity, so advective momentum and energy transport replace hydrostatic and purely radiative equilibrium near the nozzle. Density, temperature, and composition need not remain uniform on an equipotential. The one-dimensional volume coordinate $r^{\prime }(\Psi )$ and its correction factors therefore cease to describe the three-dimensional mass-transfer flow.

For a circular binary with fixed total mass $M=M_1+M_2$, the orbital angular momentum isConservative binary mass transfer keeps $J$ and $M$ fixed, so $a\propto (M_1{M}_2{)}^{-2}$. Since $M_1=Mq/(1+q)$ and $M_2=M/(1+q)$,

Conservative transfer gives $d\log M_2/d\log M_1=-q$ and hence $d\log q/d\log M_1=1+q$. It follows thatand, using $R_L\simeq 0.4a{q}^{2/9}$,

Let $f=\log (R_1/R_L)$. While the system is detached, $M_1$ is constant, soDuring Roche-lobe overflow, $\dot{\log M_1}=-f/{\tau }_{\mathrm{dyn}}$, whileThereforeIf $\beta >\alpha$, the stable fixed point isThe corresponding mass-transfer rate isso the donor overfills its Roche lobe by only a tiny amount while losing mass on the slow nuclear timescale.

If $\beta <\alpha$, overflow is unstable. Starting at contact time $t_0$ with $f(t_0)=0$,The overflow and mass-loss rate grow exponentially on a dynamical timescale, leading toward unstable mass transfer or a common-envelope phase.