Oblate Ellipsoid: Can Earth Be Modified?

[Ben2]

TL;DR

How far can a planet's equation be changed from that of a standard sphere, or x^2 + y^2 + z^2 = r^2? What are the possible consequences?

I'm told Earth is an oblate spheroid. Is it possible for a planet to be an oblate ellipsoid (equation modified from (x/a)^2 + (y/b)^2 + (z/c)^2 = 1)? What would be the possible consequences, to include "tumbling"?

 

[Baluncore]

The Earth is an ellipsoid, which is flattened at the poles. It is therefore described as being "oblate".

The Earth is stable and does not tumble because the polar radius is less than the equatorial radius, it is oblate.

If the polar radius was greater than the equatorial radius, then the Earth would be shaped more like a spindle. It would then be unstable, and could regularly tumble or flip.

 

[Ibix]

And I don't think it's possible to have a planet with three different values for $a$, $b$ and $c$. An asteroid or small moon, sure, but anything larger will collapse under its own weight into pretty near a sphere. That may be made oblate due to spin, but the spin implies rotational symmetry about the spin axis.

 

[Bandersnatch]

Oblate spheroid and oblate ellipsoid are the same thing, no? One axis shorter than the remaining (and equal) two.

Anyhow. Other than spin doing the oblate thing, tidal forces try to deform bodies into prolate ellipsoids. The possible consequences of this process are similar to those of excessive spin, and may include disintegration when self-cohesion forces are exceeded (cf. 'Roche limit'). Use tidal forces responsibly. Keep away from children dwarf planets.

 

[Baluncore]

The tennis racket theorem relates to the tumbling of rigid bodies that have three axes of rotation with differing moments of inertia.
https://en.wikipedia.org/wiki/Tennis_racket_theorem

 

[Hornbein]

There is a limit to how oblate a heavenly body can be. Neutron stars can rotate at 60,000 rpm. If they try to go faster than that they become asymmetrical and radiate gravitational waves. The spheroid is "trying" to become ring shaped, but this is unstable as the mass "wants" to aggregate in part of the ring.

It is believed that tumbling may occur in the cute little Trappist-1 solar system.

 

[Ben2]

Oblate spheroid and oblate ellipsoid are the same thing, no? One axis shorter than the remaining (and equal) two.

Anyhow. Other than spin doing the oblate thing, tidal forces try to deform bodies into prolate ellipsoids. The possible consequences of this process are similar to those of excessive spin, and may include disintegration when self-cohesion forces are exceeded (cf. 'Roche limit'). Use tidal forces responsibly. Keep away from children dwarf planets.

Brilliant responses. My post should have been r^2/a^2 + z^2/b^2 = 1. Will study the references, and thanks!

 

[mfb]

And I don't think it's possible to have a planet with three different values for $a$, $b$ and $c$. An asteroid or small moon, sure, but anything larger will collapse under its own weight into pretty near a sphere. That may be made oblate due to spin, but the spin implies rotational symmetry about the spin axis.

It's not a planet, but Haumea is in hydrostatic equilibrium or very close to it with three different axes. The fast rotation makes this a stable configuration.