If a mass of incompressible fluid is floating in space, subject only to its own gravity and centrifugal force, what shape will it adopt as its angular momentum is increased?

Initially, it will form an oblate spheroid, with two equal semi-axes longer than its semi-axis of rotation.

However, past a certain value for the angular momentum, L, the fluid can have a lower total energy by breaking that symmetry and forming a “tri-axial” ellipsoid, with the semi-axes perpendicular to the axis, a and b, taking on different lengths!

The first configuration is known as a “Maclaurin spheroid”, after Colin Maclaurin, who studied it in 1742.

The second is called a “Jacobi ellipsoid”, after Carl Gustav Jacob Jacobi, who in 1834 realised that a fluid could be in equilibrium in this less symmetrical state.

There are two separate issues here.

One is the question of when the “level surfaces” of constant gravitational-plus-centrifugal potential for an ellipsoid can match the shape of the ellipsoid itself. What shapes can “sea level” take for a spinning mass of fluid?