Ellipticity of Rotating Self-Gravitating Fluid Bodies in Hydrostatic Equilibrium

[Kibble]

Sorry for the overly loquacious title, but thats the most concise way to sum up this problem - enough mass concentrated in one place (like a planet) will eventually reach hydrostatic equilibrium, and be (approximately) a sphere. But if the body is rotating, it will become flattened along its rotational axis. For some reason I haven't been able to find anywhere online describing an formula to calculate the magnitude of this flattening effect! Not even on wikipedia </3

Do any of you guys know how?

[Armchair Rocket Scientist]

Sorry for the overly loquacious title, but thats the most concise way to sum up this problem - enough mass concentrated in one place (like a planet) will eventually reach hydrostatic equilibrium, and be (approximately) a sphere. But if the body is rotating, it will become flattened along its rotational axis. For some reason I haven't been able to find anywhere online describing an formula to calculate the magnitude of this flattening effect! Not even on wikipedia </3

Do any of you guys know how?

Here's a formula that approximates it for bodies of constant density. http://en.wikipedia.org/wiki/Equatorial_bulge#Mathematical_expression. Note, however, that only a very small, non-differentiated body will be anywhere near constant density; real-world planet-sized objects are much denser at the center due to a combination gravitational compression and denser compounds sinking to the core.