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The Earth is a Maclaurin spheroid (aka. oblate). This affects orbits, most notably $J_2$ rotating the node line for inclined orbits, enabling thins like Sun-synchronous orbits.

But the Maclourin spheroid is only one of the hydrostatic equilibriums a body can be in. At higher angular momentum, it collapses into a Jacobi ellipsoid, having two different equatorial diameters.

How does this shape affect orbits? Clearly, there can be stable orbits as Haumea has moons. But what effects do such orbits experience? In particular, there seems to be the possibility of "exotic" cases, such as synchronised polar orbits where the orbital radius is less than the greater axis of the body. Also, the shape of Jacobi ellipsoids kinds resembles tidal bulges, so can tidal forces be used to model such effects?

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One factor to consider is the rotation of the ellipsoid. A rotating Jacobi ellipsoid averages its mass distribution to approximate a Maclaurin spheroid over the course of one rotation, and since a Jacobi ellipsoid rotates fast such an averaging likely takes place over small portions of the orbit. For example, Haumea rotates 100 times or more while either of its moons orbit the dwarf planet just once (based on data from Wikipedia). So the motion of the moons is similar to what it would be with a Maclaurin spheroid.

This does not mean the orbital precession exactly follows the formula used for a more nearly spherical parent body such as Earth. For a body with a small deviation from a spherical shape, the precession is controlled by the $J_2$ moment as in the modeling of sun-synchronous orbits. With a fast-rotating object such that forms a Jacobi ellipsoid, the "ovality" is great enough for $J_{2n}, n\ge 2$ to potentially contribute to the orbital precession even if the "Maclaurin spheroid average" can be applied.

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    $\begingroup$ This is an interesting point! The smaller a body is, the faster it can rotate before "rapid unscheduled disassembly" while at the same time the longer the period at a given distance. Of course for a low orbit the period is independent of size and only depends on average density). But I feel your answer is incomplete as it only explores one end of the parameter space; the easy end. What if the periods were comparable or the orbital period was shorter? $\endgroup$
    – uhoh
    Commented Aug 20, 2021 at 2:34
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    $\begingroup$ So essentially, since the rotation necessarily has to be rapid, most orbits are "high" orbits, which can be approximately modelled by J2. Nice! $\endgroup$
    – SE - stop firing the good guys
    Commented Aug 20, 2021 at 9:14

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