What is the hydrostatic shape of an ideal rotating planet?

Question

It is a well-known fact that rotating planets have a flattened spheroidal shape. However, the NASA site says about Haumea:

The fast spin distorts Haumea's shape, making this dwarf planet look like a football.

Haumea rotates especially quickly, and its shape is an ellipsoid with 3 different axes. Does one in fact cause the other? If a planet rotates very quickly, can its shape be distorted to something not symmetrical around the axis of rotation?

It looks counter-intuitive to me. Imagine a thought experiment: we spin up a big planet gradually, while letting it assume hydrostatic equilibrium all the time. How will its shape change until it falls apart? My common sense tells me it will be flattened more and more, and its larger radius will approach infinity. But NASA says that it will assume a different shape at some point.

What gives?

Some useful background info: https://en.wikipedia.org/wiki/Jacobi_ellipsoid — PM 2Ring, Mar 09 '24 at 10:52
For reasons of symmetry, I can't see how rotation could turn an initially spherical object into anything but an oblate spheroid unless it is distorted by external forces. — Thomas, Mar 09 '24 at 12:08
@Thomas For comparison, there is the case of the 1958 Explorer I. Explorer 1 was 203cm (80 in) in length and 15.2cm (6.0 in) in diameter. Upon orbit insertion the propulsion system gave the satellite a spin around its long axis. However, the rotation state of the spacecraft shifted to head-over-head tumbling. The head-over-head rotation has - for the same angular momentum - a lower kinetic energy. The four flexible whip antennae flexed sufficiently to dissipate energy. Any system that can dissipate energy will. — Cleonis, Mar 09 '24 at 12:28
@Cleonis I don't think you can compare an artificial satellite to a self-gravitating object supposed to be in hydrostatic equilibrium. — Thomas, Mar 09 '24 at 12:54
@Thomas To your comment: I have posted a reply in the physics.stackexchange general chat room, h-bar — Cleonis, Mar 09 '24 at 13:21
Possible duplicates: https://physics.stackexchange.com/q/10670/2451 and links therein. — Qmechanic, Mar 09 '24 at 15:36

score 2 · Accepted Answer · answered Mar 09 '24 at 13:26

Check out the nice illustrations of Jos Leys The shape of Planet Earth for Ghys’ presentation.

It’s actually an illustration of symmetry breaking. Yes, the problem is symmetric, but this only means that the set of solutions is stable by the symmetry. You can only conclude that an individual solution is symmetric if it is unique. Symmetry breaking therefore arises when you have a multiplicity of solutions.

In your case, the error in your reasoning is to assume that there is only one possible shape for the rotating fluid. The idea is that when it spins sufficiently fast, you get a bifurcation and many different shapes are possible. The set of all the possible shapes is invariant by rotation. Furthermore, paradoxically, the Maclaurin solution which is axisymmetric and which is the one you have in mind that flattens out, ceases to be stable at a finite value of angular momentum.

Note that at even higher rotation rates you can have even less symmetric stable configurations (pear shaped, etc.).

Hope this helps.

Great! I have no idea how I missed such a beautiful theory when I studied physics! — user27542, Mar 09 '24 at 15:06

What is the hydrostatic shape of an ideal rotating planet?

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