Roche lobe

Remarks

The two Roche lobes touch at $L_1$, and the level set $\Phi = \Phi(L_1)$ that bounds them is a figure-eight in the orbital plane. The last term of $\Phi$ is the potential of the centrifugal force; the Coriolis force does no work and does not enter $\Phi$. The Roche lobe radius $r_L$ is the radius of the sphere with the same volume as the lobe. Eggleton's approximation, accurate to about one per cent for all mass ratios, is $$\frac{r_L}{a} \approx \frac{0.49 \, q^{2/3}}{0.6 \, q^{2/3} + \ln\left(1 + q^{1/3}\right)}, \qquad q = M_1/M_2.$$

Example: for $M_1 = M_2$, the symmetry $r \mapsto -r$ exchanges the two masses and fixes $\Phi$, so $L_1$ is the centre of mass and the two lobes are mirror images. Eggleton's formula gives $r_L \approx 0.49/(0.6 + \ln 2) \, a \approx 0.38 \, a$.

Non-example: take a value $c > \Phi(L_1)$ slightly above the critical one. The component of $\set{\Phi < c}$ containing $r_1$ also contains $r_2$, so it is a common envelope around both stars and not the lobe of $M_1$.