= Solution
The <Roche lobe> of star 1 is the volume around it bounded by the critical closed equipotential passing through the inner <Lagrange point> $L_1$. Material on a lower potential surface remains confined to star 1; at the critical surface a path opens toward star 2.
Let $\mathbf g=-\nabla\Psi$ and take $d\mathbf S$ outward. For a nearly spherical interior surface, the <divergence theorem> gives
$$
\int_S\mathbf g\mathbin\cdot d\mathbf S
=-\int_V\nabla^2\Psi\,dV.
$$
The self-gravity term contributes $-4\pi GM_1$. The companion lies outside $S$, so its potential is harmonic inside and contributes zero net flux. For the centrifugal term, $\nabla^2(-\Omega^2s^2/2)=-2\Omega^2$, so its acceleration has divergence $2\Omega^2$ and contributes $2\Omega^2V$. Using $V=4\pi r^3/3$ and $S=4\pi r^2$,
$$
\boxed{
\langle g\rangle
=\frac1S\int_S\mathbf g\mathbin\cdot d\mathbf S
\simeq-\frac{GM_1}{r^2}+\frac23\Omega^2r}.
$$
The sign is the outward-normal component; the dominant self-gravity is inward and therefore negative.
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