Oblate

From Freepedia

This article is about the geometric solid figure. For a member of a Roman Catholic religious order, see Oblate (religion).

An oblate spheroid is ellipsoid having a shorter axis and two equal longer axes.

Oblateness is a measure for a planet that is a spheroid in shape, bulging outward in the center due to its rotation. Earth is slightly oblate.

Gravity tends to contract celestial bodies into a perfect sphere, the shape where all the mass is as close to the center of gravity as possible. Perfect spherical shape is the shape of least gravitational potential energy, the oblate shape corresponds to a higher gravitational potential energy than that. For a rotating planet relaxing to the state of a perfect sphere is not available.

To get a feel for the type of equilibrium that is involved, one could imagine sitting on a swivel chair, with weights in one's hands, whilst rotating. If one pull the weights towards him or herself, one's rotation rate goes up (by conservation of angular momentum), so then one needs to pull harder on the weights. If there is an upper limit to the amount of inward force that one can bring into play, then at some point one cannot pull the weights in any closer.

Something analogous to that happens in planet formation. Matter is first coalescing into a slow rotating disk-shaped distribution, and collisions and friction convert kinetic energy to heat, allowing the disk to self-gravitate into an oblate spheroid.

As long as the proto-planet is still too oblate to be in equilibrium, gravitational potential energy is released as it contracts. The contraction increases the rotation rate, making further contraction harder. There is a point where further contraction would require more input of energy than the amount of gravitational potential energy that is released by the contraction. That point is the final equilibrium shape.

As long as there is no equilibrium, friction can convert kinetic energy to heat, draining energy from the rotational dynamics. When the equilibrium state has been reached large scale conversion of kinetic energy to heat has ceased. In that sense the equilibrium state is a state of lowest possible energy.

Mathematically, for oblateness we have

<math>\epsilon = {R_{e}-R_{p} \over R_{e}} \approx {3 \pi \over 2 G T^{2} \rho}</math>

where <math>R_{e}</math> the equatorial radius and <math>R_{p}</math> is the polar radius. The approximation is valid in the case of a fluid planet of uniform density; it is a function of the Newtonian constant of gravitation <math>G</math>, the rotation period <math>T</math> and the density <math>\rho</math>.