Equatorial bulge

The Earth has an equatorial bulge of 42.72 km (26.5 miles) due to its rotation. That is, its diameter measured across the equatorial plane (12756.28 km, 7,927 miles) is 42.72 km more than that measured between the poles (12713.56 km, 7,900 miles).

The equilibrium as a balance of energies

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. However, given a rotation there is a corresponding equatorial bulge. The common measure of the distortion from spherical shape is called the flattening (or sometimes ellipticity or oblateness), and can depend on a variety of factors including the angular velocity, density, and elasticity.

Perfect spherical shape is the shape of least gravitational potential energy, The oblate shape of rotating bodies corresponds to a state of potential energy that is higher than the lowest possible state. However, for a rotating celestial body, relaxing to the state of a perfect sphere is not available.

To get a feel for the type of equilibrium that is involved, imagine you are seated in a swivel chair, with weights in your hands, and you are rotating. If you pull the weights towards yourself, you are doing work and your rotation rate increases as a consequence of that. Once your rotation rate has increased, pulling the weights yet closer is harder than initially, because of your increased rotation rate.

Something analogous to this occurs in planet formation. Matter first coalesces into a slowly rotating disk-shaped distribution, with a lot of the objects in non-circular orbits around the common center of gravity. There is an ongoing process of collisions and friction converting kinetic energy to heat. Any energy that has converted to heat will dissipate further: hot objects radiate and thus the energy dissipates, into all of space. This dissipation of energy is what allows the protoplanetary disk to self-gravitate to denser and denser distributions. By contrast, the motion of the particles of Saturn's rings has become so well aligned that loss of kinetic energy in collisions is negligable. Hence for the matter of Saturn's rings there is no opportunity to coalesce to a celestial body. Summerizing: in planet formation the protoplanetary disk can contract only if it can lose energy.

Let me elaborate on how orbital energy can dissipate. Some of the celestial bodies of the solar system have a highly eccentric orbit, for example comets. The motion of the orbiting object from apogee to perigee can be seen as a contraction of the rotating system, in this contraction gravitational potential energy is converted to kinetic energy. In our current solar system a collision is very unlikely, so nearly always a comet will move away from the Sun again, with kinetic energy converting to gravitational potential energy again. But: if a collision occurs at the apogee, some kinetic energy is lost, and then system cannot expand again to the height of the previous perigee.

The process that drains energy from the system is conversion of kinetic energy to heat. In the system as a whole there are two repositories of energy: kinetic energy and gravitational potential energy. Some of the loss of kinetic energy is replenished by conversion of gravitational potential energy to kinetic energy.

As long as the proto-planet is still too oblate to be in equilibrium the release of gravitational potential energy on contraction keeps driving increase in angular velocity. As the contraction proceeds the rotation rate keeps going up, hence the required force for further contraction keeps going up. There is a point where the increase of rotational kinetic energy on further contraction would be larger than the release of gravitational potential energy. The contraction process can only proceed up to that point, so it halts there.

Discussion

At the beginning of this section I announced that the equilibrium is to be understood in terms of a balance of energies. Intuitively, most people do sense that the Earth's equatorial bulge must be due to some kind of equilibrium, but recognizing what kind of equilibrium it is is not straightforward. Very often people try to understand the bulge in terms of a force equilibrium: often there are statements about a balance between a centripetal force and a centrifugal force. This article is not the place to explain why invoking some centrifugal force is erroneous when it is presented as a theory of physics. In this article I have presented the explanation of the equatorial bulge in terms of physics proceeding from cause to effect.

Dissipation

As long as there is no equilibrium there can be violent convection, and as long as there is violent convection friction can convert kinetic energy to heat, draining rotational kinetic energy from the system. When the equilibrium state has been reached large scale conversion of kinetic energy to heat ceases.

Generally speaking, as long as there is friction at work there is dissipation of energy. Dissipation is the process through which dynamical systems reach the lowest state of energy that is available to them. In the case of the Earth there is no large scale interconversion of potential energy and kinetic energy anymore, hence there is hardly any opportunity anymore for energy to dissipate.

The Earth's rotation rate is still slowing down, but very gradually, about a thousandth of a second every 100 years. Estimates of how fast the Earth was rotating in the past vary, because it is unknown how exactly the moon has formed. Estimates of the Earth's rotation 500 million years ago are around 20 modern hours per revolution.

The Earth's rate of rotation is slowing down mainly because of tidal interactions with the Moon and the Sun. Since the solid parts of the Earth are ductile, the Earth's equatorial bulge has been decreasing in step with the decrease in the rate of rotation.

Differences in gravitational acceleration

Because of a planet's rotation around its own axis, the gravitational acceleration is less at the equator than at the poles. In the 17th century, following the invention of the pendulum clock, French scientists found that clocks sent to French Guiana, on the northern coast of South America, ran slower than their exact counterparts in Paris.

Any object that is stationary with respect to the surface of the Earth is in actual fact following a circular trajectory, circumnavigating the Earth's axis. Pulling an object into such a circular trajectory requires a force. The acceleration that is required to circumnavigate the Earth's axis along the equator at one revolution per sidereal day is 0.0339 m/s². Providing this acceleration decreases the effective gravitational acceleration. At the equator, the effective gravitational acceleration is 9.7805 m/s². This means that the true gravitational acceleration at the equator must be 9.8144 m/s² (9.7805 + 0.0339 = 9.8144). At the poles, the gravitational acceleration is 9.8322 m/s². The difference of 0.0178 m/s² between the gravitational acceleration at the poles and the true gravitational acceleration at the equator is predominantly due to the fact that objects located on the equator are about 21 kilometers further away from the center of mass of the Earth than at the poles, which corresponds to a smaller gravitational acceleration. (I have glossed over differences in density now. The Earth's core is much denser, and there are other, smaller density variations.)

In summary, there are two contributions to the fact that the effective gravitational acceleration is less strong at the equator than at the poles. About 70 percent of the difference is contributed by the fact that objects circumnavigate the Earth's axis, and about 30 percent is due to the non-spherical shape of the Earth.

Centre of gravitation

Let a satellite be in an equatorial orbit. If all of the Earth's mass would be concentrated in a single point, where would that point mass have to be to exert exactly the same gravitational force on the satellite as the Earth does?

In the case of a celestial body that is a perfect sphere the answer is that all of the mass can be treated as concentrated at the geometrical center. In the case of an oblate spheroid, flattened due to rotation, this does not apply. In the case of a satellite orbiting the Earth in an equatorial orbit, the center of gravitational attraction is about 10 kilometers away from the Earth's geometrical center. For an orbiting satellite this means that at every point in time a different point inside the Earth is the effective center of gravitational attraction. For a satellite orbiting the Earth, the center of gravitation is a point that circumnavigates the geometrical center of the Earth at a distance of about ten kilometers away from the geometrical center of the Earth.

It is not a coincidence that Saturn's rings orbit Saturn in Saturn's equatorial plane. The fact that the effective center of gravitational attraction moves around in a circle tends to align satellites with the planet's equatorial plane.

Satellite orbits

The fact that the Earth's gravitational field slightly deviates from being spherically symmetrical also affects the orbits of satellites and changes their orbits away from pure ellipses. This is especially important in the case of the trajectories of GPS-satellites.

Other celestial bodies

The planet with the largest known equatorial bulge (11808 km, 7337 miles) is Saturn.
Many rotating astronomical bodies other than planets also exhibit equatorial bulges.

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