Newtonian gravitation is (usually) much simpler than it might be because it can be shown that any spherically symmetrical mass attracts other masses as if it’s mass were concentrated in a point at it’s center. In a handful of situations however, the special case of spherical symmetry cannot be taken for granted. Perhaps the most common example is tidal effect. Since the Earth and moon are not infinitely rigid, their mutal gravity causes a bulge towards each other, with the effect of transferring angular momentum from the rotation of the Earth to the revolution of the moon around the Earth. Another is calculating the orbit of satellites around the Earth: because the Earth rotates, it has an equatoral bulge (‘olbate spheroid’), and this effects the satellites’ paths, most notably in causing the plane of the orbits (and if elliptical, the axis of the orbits) to precess.

For a fluid rotating object, there are different configurations it can be in. At low speed you have the oblate spheroid. At higher speeds the radial symmetry breaks down and the mass begins to form two opposite lobes. At higher speed still this forms a dumbell shape connected by a narrow neck and at some critical value it breaks apart into two separate masses orbiting each other. At that point however, anything that was orbiting the original mass is now orbiting two: you now have the famous three-body problem, which probably has no general solution.

My question is, at what point in this procession from perfect sphere to two orbiting bodies do we stop having a general solution for calculating the gravitational influence on another object?