What nonlinear deformations will a fast rotating planet exhibit?

Question

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?

image link

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

Answer 1

For an experimental test see: Liquid marbles http://adsabs.harvard.edu/abs/2001Natur.411..924A

(paywall http://www.nature.com/nature/journal/v411/n6840/full/411924a0.html)

For a paper in General Relativity see: Accurate simulations of the dynamical bar-mode instability in full general relativity http://adsabs.harvard.edu/abs/2007PhRvD..75d4023B

Basically once you start increasing rotation the pancake will become unstable and will make a transition to a rotating bar (dumbbell).

Anyway I think that nobody ever saw the bar actually breaking into 2 pieces. Typically you loose matter from the external regions, redistribute angular momentum and go back to axisymmetry.

Cheers

Answer 2

I have stumbled upon a paper that presents a solution to this problem. First, I arrived at it via the following (recent) online summary of these shapes:

http://www.aleph.se/andart/archives/2014/02/torusearth.html

The paper is at Arvix:

Uniformly rotating axisymmetric fluid configurations bifurcating from highly flattened Maclaurin spheroids. Feb 2008.

Critically, the bifurcation referred to seems to be exactly what I referenced in this question. They always start with a Maclaurin spheroids, which is what I referred to as "pancake". Then they go through a process, and in the end wind up with a torus or multiple objects. Here is the image that illustrates the real meat of their discoveries:

You can see in the leftmost image, they go from a pancake to a simple torus. The rightmost process demonstrates one of the other types of processes that are physical. However, many configurations also hit upon a mass shedding limit. In that situation, adding more rotation causes the apparent gravity on the edge to go negative. Obviously, this doesn't work, so the material "flies off into space". But that's not entirely true, it's just subtracted out of mathematical necessity because it doesn't actually have escape velocity.

Moving on... I am surprised. I did not expect to see the "pinch" in the middle shown above. This still doesn't make sense to me, and I can not come up with a good argument for why it happens. In order for it to slope in, I need to be able to postulate a non-equilibrium configuration where the center is either flat or slopes out, and in that configuration, the forces/gravity pushes material away from the center. This is a very difficult proposition to accept. I see obvious in either the gravitational or hydrostatic physics that would do this. Nonetheless, the authors seem to have done an excellent and thorough job with full computer simulations backing them up with the full problem complexity included. So it appears that I'm wrong about that.

If Earth were spun fast enough from its present state, the south pole and north pole would dip inward. This is very strange, but it seems to be the correct answer, as per this paper.

Answer 3

@AlanSE: My reading of the paper is that, beyond a critical speed of rotation, the Maclaurin shape (C) becomes unstable to small perturbations. For the $\epsilon_1$ perturbation given in Table 2 and Figure 6, if the perturbation goes one way (thicker in the middle), you hit the Mass Shedding Limit (A), develop a sharp edge at the equator, and fling excess material off into space. If it goes the other way (thinner in the middle), then the pancake turns into a torus (I,J,K,L).

The other perturbations are essentially higher (axisymmetric) harmonics. A real perturbation could be a mix of several harmonics, or even violate axisymmetry completely. But if you stay axisymmetric, then the $\epsilon_k$ probably form an orthogonal basis set that spans the whole space of axisymmetric perturbations.

Answer 4

This is not strictly theoretically deduced, just experimentally shown, but for most configurations that don't rely on some sort of differentiated body or highly non-uniform density within the object, the shape you get is in reality the Jacobi ellipsoid, not the sharply creased pancakes and definitely not toroids. These are equilibria but they are probably not stable over a period of days, let alone billions of years. I've yet to get one simulation where a pre-flattened planet-sized olivine and iron toroid spun at the correct speed doesn't undergo one of a dozen different collapse and shedding modes within just 24 hours, which tells me that whatever earlier axisymmetric models might have predicted, the reality is that these things are probably as stable as a pencil balanced on its point.

It is not impossible to get a toroidal planet to form and remain in equilibrium for a few hours given a Maclaurin spheroid is hit by some extremely lucky hit by hypervelocity debris at a just so angle, but it usually won't survive even a few hours before it beads, pinches, or fills the hole, which is The same general timescale for ones poofed into existence by magic.