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An equatorial bulge is a difference between the
equator The equator is a circle of latitude, about in circumference, that divides Earth into the Northern and Southern hemispheres. It is an imaginary line located at 0 degrees latitude, halfway between the North and South poles. The term can also ...
ial and polar
diameter In geometry, a diameter of a circle is any straight line segment that passes through the center of the circle and whose endpoints lie on the circle. It can also be defined as the longest chord of the circle. Both definitions are also valid f ...
s of a
planet A planet is a large, rounded astronomical body that is neither a star nor its remnant. The best available theory of planet formation is the nebular hypothesis, which posits that an interstellar cloud collapses out of a nebula to create a you ...
, due to the
centrifugal force In Newtonian mechanics, the centrifugal force is an inertial force (also called a "fictitious" or "pseudo" force) that appears to act on all objects when viewed in a rotating frame of reference. It is directed away from an axis which is paralle ...
exerted by the rotation about the body's axis. A rotating body tends to form an oblate spheroid rather than a
sphere A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is the c ...
.


On Earth

The
Earth Earth is the third planet from the Sun and the only astronomical object known to harbor life. While large volumes of water can be found throughout the Solar System, only Earth sustains liquid surface water. About 71% of Earth's sur ...
has a rather slight equatorial bulge: it is about wider at the equator than pole-to-pole, a difference which is about 1/298 of the equatorial diameter. If the Earth were scaled down to a globe with diameter of 1 meter at the equator, that difference would be only 3 millimeters. While too small to notice visually, that difference is still more than twice the largest deviations of the actual surface from the ellipsoid, including the tallest mountains and deepest oceanic trenches. The rotation of the earth also affects the
sea level Mean sea level (MSL, often shortened to sea level) is an average surface level of one or more among Earth's coastal bodies of water from which heights such as elevation may be measured. The global MSL is a type of vertical datuma standardis ...
, the imaginary surface that is used to measure
altitude Altitude or height (also sometimes known as depth) is a distance measurement, usually in the vertical or "up" direction, between a reference datum and a point or object. The exact definition and reference datum varies according to the context ...
s from. This surface coincides with the mean water surface level in oceans, and is extrapolated over land by taking into account the local gravitational potential and the centrifugal force. The difference of the radii is thus about 21 km. An observer standing at sea level on either pole, therefore, is 21 km closer to Earth's center than if standing at sea level on the Equator. As a result, the highest point on Earth, measured from the center and outwards, is the peak of Mount Chimborazo in
Ecuador Ecuador ( ; ; Quechua: ''Ikwayur''; Shuar: ''Ecuador'' or ''Ekuatur''), officially the Republic of Ecuador ( es, República del Ecuador, which literally translates as "Republic of the Equator"; Quechua: ''Ikwadur Ripuwlika''; Shuar: ' ...
rather than
Mount Everest Mount Everest (; Tibetan: ''Chomolungma'' ; ) is Earth's highest mountain above sea level, located in the Mahalangur Himal sub-range of the Himalayas. The China–Nepal border runs across its summit point. Its elevation (snow hei ...
. But since the ocean also bulges, like Earth and its atmosphere, Chimborazo is not as high above sea level as Everest is. Similarly the lowest point on Earth, measured from the center and outwards, is the Litke Deep in
Arctic Ocean The Arctic Ocean is the smallest and shallowest of the world's five major oceans. It spans an area of approximately and is known as the coldest of all the oceans. The International Hydrographic Organization (IHO) recognizes it as an ocean, a ...
rather than Challenger Deep in
Pacific Ocean The Pacific Ocean is the largest and deepest of Earth's five oceanic divisions. It extends from the Arctic Ocean in the north to the Southern Ocean (or, depending on definition, to Antarctica) in the south, and is bounded by the conti ...
. But since the ocean also flattens, like Earth and its atmosphere, Litke Deep is not as low below sea level as Challenger Deep is. More precisely, the surface of the Earth is usually approximated by an ideal oblate ellipsoid, for the purposes of defining precisely the
latitude In geography, latitude is a coordinate that specifies the north– south position of a point on the surface of the Earth or another celestial body. Latitude is given as an angle that ranges from –90° at the south pole to 90° at the north ...
and
longitude Longitude (, ) is a geographic coordinate that specifies the east– west position of a point on the surface of the Earth, or another celestial body. It is an angular measurement, usually expressed in degrees and denoted by the Greek let ...
grid for
cartography Cartography (; from grc, χάρτης , "papyrus, sheet of paper, map"; and , "write") is the study and practice of making and using maps. Combining science, aesthetics and technique, cartography builds on the premise that reality (or an i ...
, as well as the "center of the Earth". In the WGS-84 standard Earth ellipsoid, widely used for map-making and the GPS system, the radius of the Earth is assumed to be at the equator and center-to-pole; meaning a difference of in the radii and in the diameters, and a relative
flattening Flattening is a measure of the compression of a circle or sphere along a diameter to form an ellipse or an ellipsoid of revolution ( spheroid) respectively. Other terms used are ellipticity, or oblateness. The usual notation for flattening ...
of 1/298.257223563. The sea level surface is much closer to this standard ellipsoid than the surface of the solid Earth is.


The equilibrium as a balance of energies

Gravity In physics, gravity () is a fundamental interaction which causes mutual attraction between all things with mass or energy. Gravity is, by far, the weakest of the four fundamental interactions, approximately 1038 times weaker than the stro ...
tends to contract a celestial body into a
sphere A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is the c ...
, the shape for which all the mass is as close to the center of gravity as possible. Rotation causes a distortion from this spherical shape; a common measure of the distortion is the
flattening Flattening is a measure of the compression of a circle or sphere along a diameter to form an ellipse or an ellipsoid of revolution ( spheroid) respectively. Other terms used are ellipticity, or oblateness. The usual notation for flattening ...
(sometimes called ellipticity or oblateness), which can depend on a variety of factors including the size, angular velocity,
density Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematicall ...
, and elasticity. A way for one to get a feel for the type of equilibrium involved is to imagine someone seated in a spinning swivel chair and holding a weight in each hand; if the individual pulls the weights inward towards them, work is being done and their rotational kinetic energy increases. The increase in rotation rate is so strong that at the faster rotation rate the required centripetal force is larger than with the starting rotation rate. Something analogous to this occurs in planet formation. Matter first coalesces into a slowly rotating disk-shaped distribution, and collisions and friction convert 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, though gradually, by about two thousandths of a second per rotation every 100 years. Estimates of how fast the Earth was rotating in the past vary, because it is not known exactly how the moon was 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.


Effect on gravitational acceleration

Because of a planet's rotation around its own axis, the
gravitational acceleration In physics, gravitational acceleration is the acceleration of an object in free fall within a vacuum (and thus without experiencing drag). This is the steady gain in speed caused exclusively by the force of gravitational attraction. All bodie ...
is less at the
equator The equator is a circle of latitude, about in circumference, that divides Earth into the Northern and Southern hemispheres. It is an imaginary line located at 0 degrees latitude, halfway between the North and South poles. The term can also ...
than at the
poles Poles,, ; singular masculine: ''Polak'', singular feminine: ''Polka'' or Polish people, are a West Slavic nation and ethnic group, who share a common history, culture, the Polish language and are identified with the country of Poland in ...
. In the 17th century, following the invention of the
pendulum clock A pendulum clock is a clock that uses a pendulum, a swinging weight, as its timekeeping element. The advantage of a pendulum for timekeeping is that it is a harmonic oscillator: It swings back and forth in a precise time interval dependent on i ...
, French scientists found that clocks sent to
French Guiana French Guiana ( or ; french: link=no, Guyane ; gcr, label= French Guianese Creole, Lagwiyann ) is an overseas department/region and single territorial collectivity of France on the northern Atlantic coast of South America in the Guianas ...
, on the northern coast of
South America South America is a continent entirely in the Western Hemisphere and mostly in the Southern Hemisphere, with a relatively small portion in the Northern Hemisphere at the northern tip of the continent. It can also be described as the sou ...
, ran slower than their exact counterparts in Paris. 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 actually 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 because objects located on the Equator are about further away from the
center of mass In physics, the center of mass of a distribution of mass in space (sometimes referred to as the balance point) is the unique point where the weighted relative position of the distributed mass sums to zero. This is the point to which a force may ...
of the Earth than at the poles, 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% of the difference is contributed by the fact that objects circumnavigate the Earth's axis, and about 30% is due to the non-spherical shape of the Earth. The diagram illustrates that on all latitudes the effective gravitational acceleration is decreased by the requirement of providing a centripetal force; the decreasing effect is strongest on the Equator.


Effect on satellite orbits

The fact that the Earth's gravitational field slightly deviates from being spherically symmetrical also affects the orbits of
satellite A satellite or artificial satellite is an object intentionally placed into orbit in outer space. Except for passive satellites, most satellites have an electricity generation system for equipment on board, such as solar panels or radioiso ...
s through secular orbital precessions. They depend on the orientation of the Earth's symmetry axis in the inertial space, and, in the general case, affect ''all'' the Keplerian orbital elements with the exception of the
semimajor axis In geometry, the major axis of an ellipse is its longest diameter: a line segment that runs through the center and both foci, with ends at the two most widely separated points of the perimeter. The semi-major axis (major semiaxis) is the lo ...
. If the reference ''z'' axis of the coordinate system adopted is aligned along the Earth's symmetry axis, then only the longitude of the ascending node Ω, the argument of pericenter ω and the mean anomaly ''M'' undergo secular precessions. Such perturbations, which were earlier used to map the Earth's gravitational field from space, may play a relevant disturbing role when satellites are used to make tests of
general relativity General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...
because the much smaller relativistic effects are qualitatively indistinguishable from the oblateness-driven disturbances.


Formulation

The flattening coefficient f for the equilibrium configuration of a self-gravitating spheroid, composed of uniform density incompressible fluid, rotating steadily about some fixed axis, for a small amount of flattening, is approximated by: f = \frac = \frac \frac = \frac \frac where *G is the universal
gravitational constant The gravitational constant (also known as the universal gravitational constant, the Newtonian constant of gravitation, or the Cavendish gravitational constant), denoted by the capital letter , is an empirical physical constant involved in ...
, *a is the mean radius, *a_e = a\, (1 + \tfrac) and a_p = a\,(1 - \tfrac) are respectively the equatorial and polar radius, *T is the rotation period and \omega = \tfrac is the angular velocity, *\rho is the body density and M \simeq \tfrac \pi \rho a^3 is the total body mass. Real flattening is smaller due to mass concentration in the center of celestial bodies.


See also

* * Clairaut's theorem (gravity) *
Earth's gravity The gravity of Earth, denoted by , is the net acceleration that is imparted to objects due to the combined effect of gravitation (from mass distribution within Earth) and the centrifugal force (from the Earth's rotation). It is a vector qua ...
* Planetary flattening


References

{{DEFAULTSORT:Equatorial Bulge Planets Geodesy Topography