 Solution

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.