# What nonlinear deformations will a fast rotating planet exhibit?

It is common knowledge among the educated that the Earth is not exactly spherical, and some of this comes from tidal forces and inhomogeneities but some of it comes from the rotation of the planet itself. The deformation from the rotational effect makes it more oblate spheroid-like, or as I would prefer, "like a pancake". Here is one site illustrating the behavior, and image:

Literature exists detailing the mathematical expectations for a rotating planet using just hydrostatic forces, for example, see Hydrostatic theory of the earth and its mechanical implications. I like to imagine a ball of water in space held together by its own gravity. I also don't want to deviate from consideration of only hydrostatic (and gravitational) forces because I think it is sufficient for this discussion.

It would seem that the solution of the described problem is in terms of a small change in radius as a function of the azimuth angle, or z-coordinate if you take the axis of rotation to be the z-axis. This is using rotational symmetry. In other words, Earth's deformation due to rotation does not depend on longitude.

I want to ask about the extreme case. Imagine a planet rotating so fast that it is a very thin pancake. What will occur in this case? I am curious:

• Will the center hollow out, creating a donut shape?
• Will it break up into a multi-body system?

It seems to me that it would be logical for the high-rotation case to break up into 2 or more separate bodies. The reason is that a 2 body system is stable an can host a very large angular momentum. But would it be an instability that leads to this case? When would such an instability occur and could a rotating planetary body deform in a different kind of shape from the beginning, such as a dumbbell-like shape, which would transition into a 2-body system more logically than the pancake shape?

To sum up, how would a pancake shape transition into a dumbbell shape? Or would it? What are the possibilities for the described system?

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This is a good question. I don't know the answer for sure, but I'd guess that, at high enough rotation rates, the most stable configuration is a dumbbell shape, which breaks the rotational symmetry. That is, the shape will be a nontrivial function of the longitude angle $\phi$, as well as $\theta$. One group of people who probably know a lot about this are nuclear physicists: I think that rapidly-rotating nuclei suffer all kinds of strange deformations. Moreover, at least in some circumstances it's considered appropriate to model nuclei with a "liquid drop model." – Ted Bunn Jun 1 '11 at 17:28
Discussed in a fictional context in Hal Clement's old war horse Mission of Gravity. – dmckee Jun 1 '11 at 17:38
@Ted, such deformations are dicussed for some heavy nuclei, and vibrations of such ellipsoids as a model for fission. But this is always for surface tension models (liquid). For soap bubbles such transformations are easily shown experimentally. For a gravity "drop" things will be different, I think. – Georg Jun 1 '11 at 17:54
You're probably right that the gravitational case is quite different, so the nuclear case may not be very relevant to teh question at hand (although it's interesting in its own right, of course). – Ted Bunn Jun 1 '11 at 18:07
It looks to me like the key buzzwords to look for here are the Maclaurin and Jacobi sequences. These seem to be equilibrium shapes for rotating self-gravitating bodies. At slow rotation rates, the stable equilibrium seems to be an oblate spheroid (as you'd expect), but at higher rates it switches to a prolate spheroid, breaking the azimuthal symmetry. At least, that's the impression I get from a quick scan of various web pages, but I haven't tried to understand the details. There seem to be other sequences too (e.g., ptp.ipap.jp/link?PTP/67/844). – Ted Bunn Jun 1 '11 at 22:40