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What nonlinear deformations will a fast rotating planet exhibit?

This is really a basic physics question that I wanted to check myself on because I'm not entirely sure of the correct answer. I asked a puzzler question on the last episode of my podcast about hollow Earth claims. I asked people to imagine a sphere with a solid shell that was filled with a compressible material, and this was spun very rapidly. (1) Would a hollow form, and (2) what shape would that hollow take on if it did form?

My thinking is that, yes, a hollow would form, but it would not be spherical as the hollow Earthers claim. It would be a biaxial ellipsoid with the short axis oriented along the poles and the long axes to the equator. This is because the centrifugal forces at the equator due to the spin would be much greater than at the poles, so there would be no "outward force" at the poles.

Is this correct? If so, is that a good explanation for a lay audience? If not correct, then what is the solution (it's been a long time since I took classical mechanics)? Thanks!


marked as duplicate by dmckee Oct 28 '11 at 18:51

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

  • $\begingroup$ Soort answer: without at high tensile strength skin there can be no hollow. If the force at radius $r_1$ is outward, the force at radius $r_2 > r_1$ will be more outward and the body flies apart. (Because the centrifugal pseudo-force grows as $r$, and the gravitation attraction gets smaller (more distance, but not more mass because there is a hollow). $\endgroup$ – dmckee Oct 28 '11 at 18:55
  • $\begingroup$ I see you closed this as a duplicate of something else. I tried searching a few times but didn't come up with anything that similar. Could you point me to where this duplicate is? $\endgroup$ – Stuart Robbins Oct 28 '11 at 19:07
  • $\begingroup$ The prior question you link to considers incompressible, not compressible, but that detail matters very little. Your podcast addressed conspiracy theories that planets are hollow - obviously we have consensus on that. If you spin an ordinary blob of matter fast enough, a torus is a valid solution, although the stability for any period of time is disputable. It's harder to ask if one could you make a planet with a closed cavity, because my own creativity may be insufficient. However, even if you did come up with something, it's a purely academic exercise. $\endgroup$ – Alan Rominger Oct 28 '11 at 19:12
  • $\begingroup$ Ah, okay, I see the duplicate. I swear that wasn't linked before? Or I just didn't see it ... anyway, yeah, while the podcast episode was about hollow Earth, the puzzler question specifically set up the scenario that you have a solid (read: sturdy) shell that won't deform, and then you have material inside that could deform. So, it could be that it's because I asked the question here, but I don't think it's an exact duplicate. $\endgroup$ – Stuart Robbins Oct 28 '11 at 19:24
  • $\begingroup$ I will definitely mention when I record the solution this weekend, though, that in reality the whole planet would deform and, if spun fast enough, would form a torus. $\endgroup$ – Stuart Robbins Oct 28 '11 at 19:25