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Equatorial bulge - Wikipedia, the free encyclopedia

Equatorial bulge

From Wikipedia, the free encyclopedia

An equatorial bulge is a planetological term which describes a bulge which a planet may have around its equator, distorting it into an oblate spheroid. 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).

An often-cited result of Earth's equatorial bulge is that the highest point on Earth, measured from the center outwards, is the peak of Mount Chimborazo in Ecuador, rather than Mount Everest. But since the ocean, like the earth and the atmosphere, bulges, Chimborazo is not as high above sea level as Everest is.

Contents

[edit] The equilibrium as a balance of energies

When spun, the spring-metal band bulges at its equator and flattens at its poles in analogy with the Earth.
When spun, the spring-metal band bulges at its equator and flattens at its poles in analogy with the Earth.

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 the rotation there is a corresponding equatorial bulge. The common measure of this 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, so the oblate shape of rotating bodies corresponds to a higher gravitational potential energy. For any 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, imagine someone seated in a swivel chair, with weights in their hands, to whom a rotating motion of the chair is imparted. If the person in the chair pulls the weights towards them, they are doing work and their rotation rate increases. As they pull the weights closer and closer, the force required for yet more contraction keeps increasing.

Something analogous to this occurs in planet formation. Matter first coalesces into a slowly rotating disk-shaped distribution, and collisions and friction convert much kinetic energy to heat, which allows the disk to self-gravitate into a very oblate spheroid.

As long as the proto-planet is still too oblate to be in equilibrium, the release of gravitational potential energy on contraction keeps driving the increase in rotational kinetic energy. 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.

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 then large scale conversion of kinetic energy to heat ceases. In that sense the equilibrium state is the lowest state of energy that can be reached.

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 "day".

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.

[edit] 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. However, measurements of the acceleration due to gravity at the equator must also take into account the planet's rotation. 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 poles, the gravitational acceleration is 9.8322 m/s2. 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.

The difference of 0.0178 m/s2 between the gravitational acceleration at the poles and the true gravitational acceleration at the equator occurs because the Earth is not spherical in shape. Objects at the equator are about 21 kilometers further away from the center of mass of the Earth, which corresponds to a smaller gravitational acceleration.

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.

[edit] 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.

[edit] Other celestial bodies

Generally any celestial body that is rotating (and that is sufficiently massive to draw itself into spherical or near spherical shape) will have an equatorial bulge matching its rotation rate. However, bodies with slow rotation rates like Mercury, Venus, the Moon, Pluto and moons of the giant planets (eg. Titan and Ganymede) have such slow rotation rates that they have no measurable equatorial bulge. Saturn is the planet with the largest equatorial bulge in our solar system (11808 km, 7337 miles).

The following is a table of the equatorial bulge of major celestial bodies or our solar system apart from those mentioned above with no measurable bulge:

Body Equatorial diameter Polar diameter Equatorial bulge Flattening ratio
Earth 12,756.28 km 12,713.56 km 42.72 km 1:298.2575
Mars 6,805 km 6,754.8 km 50.2 km 1:135.56
Ceres 975 km 909 km 66 km 1:14.77
Jupiter 143,884 km 133,709 km 10,175 km 1:14.14
Saturn 120,536 km 108,728 km 11,808 km 1:10.21
Uranus 51,118 km 49,946 km 1,172 km 1:43.62
Neptune 49,528 km 48,682 km 846 km 1:58.54
Eris 2,400 km unknown unknown unknown

[edit] Mathematical expression

Mathematically, for flattening we have

f = \frac{a-b}{a} = 1-b\!:\!a\ \approx {3 \pi \over 2 G T^{2} \rho}

where a is the equatorial radius, b is the polar radius and b:a is the aspect ratio. The approximation is valid in the case of a fluid planet of uniform density; it is a function of the Newtonian constant of gravitation G, the rotation period T and the density ρ.

[edit] See also

[edit] References

Solar System Formation, Accretion, and the Early Thermal State of the Earth
(PDF-file, 312 KB) Available as part of the MIT OpenCourseWare initiative

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