# How does a Maclaurin spheroid become a Jacobi ellipsoid? What happens?

A Maclaurin spheroid can become a Jacobi ellipsoid but I don't understand how this happens. From the plot I see that it must be spinning fast enough and from the text there seems to be some viscous damping necessary, in the Jacobi ellipsoid I'd assumed that the fluid rotates as if it were a rigid body.

I'm also wondering if the fluid motion described in the Veritasium video The Bizarre Behavior of Rotating Bodies, Explained (linked in this question) applies.

image Source

Caption in Wikipedia:

The equatorial (a, b) and polar (c) semi-principal axes of a Jacobi ellipsoid and Maclaurin spheroid, as a function of normalized angular momentum, subject to abc = 1 (i.e. for constant volume of 4π/3).

The broken lines are for the Maclaurin spheroid in the range where it has dynamic but not secular stability - it will relax into the Jacobi ellipsoid provided it can dissipate energy by virtue of a viscous constituent fluid.

Description on the source page:

A plot of the long (a), middle (b) and polar (c) semi-principal axes of a Jacobi ellipsoid and Maclaurin spheroid (spinning bodies of homogeneous self-gravitating fluid in equilibrium), against normalized angular momentum, subject to abc = 1 (i.e. for constant volume of 4π/3).

The broken lines are for the Maclaurin spheroid in the range where it has dynamic but not secular stability - it will relax into the Jacobi ellipsoid provided it can dissipate energy by virtue of a viscous constituent fluid.

The lines for the Jacobi ellipsoid _go_beyond_ the point where the piriform (pear-shaped) equilibrium body could exist. According to Cartan the Jacobi ellipsoid loses all stability at this point. However, it unclear how this is to be reconciled with the fact that the piriform body is itself unstable and has a higher energy than the ellipsoid (see Christodoulou, "Phase-Transition Theory of Instabilities. III. The Third-Harmonic Bifurcation on the Jacobi Sequence and the Fission Problem", https://arxiv.org/abs/astro-ph/9505008 )...