Subsequent to This question about the Earth's actual cross-sectional shape at the equator, I'm wondering about the stability of non-spherical planet-sized masses.
Suppose there is a planet of uniform construction (perhaps a core, mantle, and crust each of which is uniformly distributed) whose cross-section perpendicular to the axis of rotation is severely elliptical. The material near the extremes of the major axis is moving at greater linear speed than the material near the extremes of the minor axis. Is there a rotation rate at which the planet will essentially tear itself in two, leading at least in the short term to a pair of tidal-locked objects orbiting a common center?
Also, assuming the materials are at least a little ductile, implying that a very slow rotation rate would not stop the planet from deforming into a sphere, is there a way to estimate the rotation rate vs. ductility to maintain the elliptical shape (neither collapse into sphere nor break apart)?

  • $\begingroup$ You should look into the Roche limit. $\endgroup$ – my2cts Feb 14 at 14:58
  • $\begingroup$ @my2cts thanks- but that's kind of the reverse problem from my question. I'm starting with a single rotating body. If it turns out that my planet does break up, I can understand that the Roche limit will further destroy the parts. $\endgroup$ – Carl Witthoft Feb 14 at 16:24

What you are describing is the McLaurin and Jacobi spheroids. If a deformable self-gravitating initially spherical body rotates it will become an ellipsoid. For low rates of rotation this is an oblate spheroid with a circular cross-section, the McLaurin case. As the rate of rotation becomes higher this state becomes unstable, and it turns into an elongated Jacobi ellipsoid. For even higher angular momentum these become unstable, and the object does break in two.

The topic is called "figures of equilibrium" and has a long history (see also this). The classic book on it is Chandrasekhar, S. (1969). Ellipsoidal figures of equilibrium. New Haven: Yale University Press.

  • $\begingroup$ FWIW, neutron stars with extreme magnetic fields can be prolate. arxiv.org/abs/1504.03006 $\endgroup$ – PM 2Ring Feb 14 at 16:48
  • $\begingroup$ Fascinating reading in those references. $\endgroup$ – Carl Witthoft Feb 14 at 18:08

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