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An equatorial bulge is a difference between the equatorial and polar diameters of a planet, due to the centrifugal force exerted by the rotation about the body's axis. A rotating body tends to form an oblate spheroid rather than a sphere.
The planet Earth has a rather slight equatorial bulge; its equatorial diameter is about 43 km (27 mi) greater than its polar diameter, with a difference of about 1⁄298 of the equatorial diameter. If Earth were scaled down to a globe with an equatorial diameter of 1 metre (3.3 ft), that difference would be only 3 mm (0.12 in). 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.
Earth's rotation also affects the sea level, the imaginary surface used as a reference frame from which to measure altitudes. 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 (13 mi). An observer standing at sea level on either pole, therefore, is 21 km (13 mi) 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 rather than Mount Everest. 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 the Arctic Ocean rather than Challenger Deep in the Pacific Ocean. 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, Earth's surface is usually approximated by an ideal oblate ellipsoid, for the purposes of defining precisely the latitude and longitude grid for cartography, as well as the "center of the Earth". In the WGS-84 standard Earth ellipsoid, widely used for map-making and the GPS system, Earth's radius is assumed to be 6378.137 km (3963.191 mi) to the Equator and 6356.7523142 km (3949.9027642 mi) to either pole, meaning a difference of 21.3846858 km (13.2878277 mi) between the radii or 42.7693716 km (26.5756554 mi) between the diameters, and a relative flattening of 1/298.257223563. The ocean surface is much closer to this standard ellipsoid than the solid surface of Earth is.
Gravity tends to contract a celestial body into a sphere, 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 (sometimes called ellipticity or oblateness), which can depend on a variety of factors including the size, angular velocity, density, 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. [1] 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.
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. 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 21 km (13 mi) further away from the center of mass 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.
The fact that the Earth's gravitational field slightly deviates from being spherically symmetrical also affects the orbits of satellites through secular orbital precessions. [2] [3] [4] 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. 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. [5]
Such perturbations, which were earlier used to map the Earth's gravitational field from space, [6] may play a relevant disturbing role when satellites are used to make tests of general relativity [7] because the much smaller relativistic effects are qualitatively indistinguishable from the oblateness-driven disturbances.
The flattening 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: [8]
where
A related quantity is the body's second dynamic form factor, J2: [9]
with J2=−√5C20=1.08262668×10−3 for Earth, [10] where
Real flattening is smaller due to mass concentration in the center of celestial bodies.
| ae [km] | ap [km] | freal | T [h] | M [1026 kg] | fformula | |
|---|---|---|---|---|---|---|
| Jupiter | 71,492 | 66,854 | 0.066 | 9.9 | 18.98 | 0.104 |
| Saturn | 60,268 | 54,364 | 0.108 | 10.6 | 5.68 | 0.178 |
| Uranus | 25,559 | 24,973 | 0.023 | 17.2 | 0.87 | 0.036 |
| Neptune | 24,764 | 24,341 | 0.017 | 16.1 | 1.02 | 0.032 |
Nutation is a rocking, swaying, or nodding motion in the axis of rotation of a largely axially symmetric object, such as a gyroscope, planet, or bullet in flight, or as an intended behaviour of a mechanism. In an appropriate reference frame it can be defined as a change in the second Euler angle. If it is not caused by forces external to the body, it is called free nutation or Euler nutation. A pure nutation is a movement of a rotational axis such that the first Euler angle is constant. Therefore it can be seen that the circular red arrow in the diagram indicates the combined effects of precession and nutation, while nutation in the absence of precession would only change the tilt from vertical. However, in spacecraft dynamics, precession is sometimes referred to as nutation.
In celestial mechanics, an orbit is the curved trajectory of an object such as the trajectory of a planet around a star, or of a natural satellite around a planet, or of an artificial satellite around an object or position in space such as a planet, moon, asteroid, or Lagrange point. Normally, orbit refers to a regularly repeating trajectory, although it may also refer to a non-repeating trajectory. To a close approximation, planets and satellites follow elliptic orbits, with the center of mass being orbited at a focal point of the ellipse, as described by Kepler's laws of planetary motion.
Precession is a change in the orientation of the rotational axis of a rotating body. In an appropriate reference frame it can be defined as a change in the first Euler angle, whereas the third Euler angle defines the rotation itself. In other words, if the axis of rotation of a body is itself rotating about a second axis, that body is said to be precessing about the second axis. A motion in which the second Euler angle changes is called nutation. In physics, there are two types of precession: torque-free and torque-induced.
In celestial mechanics, escape velocity or escape speed is the minimum speed needed for an object to escape from contact with or orbit of a primary body, assuming:
In fluid mechanics, hydrostatic equilibrium is the condition of a fluid or plastic solid at rest, which occurs when external forces, such as gravity, are balanced by a pressure-gradient force. In the planetary physics of Earth, the pressure-gradient force prevents gravity from collapsing the planetary atmosphere into a thin, dense shell, whereas gravity prevents the pressure-gradient force from diffusing the atmosphere into outer space. In general, it is what causes objects in space to be spherical.
In astronomy, axial precession is a gravity-induced, slow, and continuous change in the orientation of an astronomical body's rotational axis. In the absence of precession, the astronomical body's orbit would show axial parallelism. In particular, axial precession can refer to the gradual shift in the orientation of Earth's axis of rotation in a cycle of approximately 26,000 years. This is similar to the precession of a spinning top, with the axis tracing out a pair of cones joined at their apices. The term "precession" typically refers only to this largest part of the motion; other changes in the alignment of Earth's axis—nutation and polar motion—are much smaller in magnitude.
A spheroid, also known as an ellipsoid of revolution or rotational ellipsoid, is a quadric surface obtained by rotating an ellipse about one of its principal axes; in other words, an ellipsoid with two equal semi-diameters. A spheroid has circular symmetry.
A Sun-synchronous orbit (SSO), also called a heliosynchronous orbit, is a nearly polar orbit around a planet, in which the satellite passes over any given point of the planet's surface at the same local mean solar time. More technically, it is an orbit arranged so that it precesses through one complete revolution each year, so it always maintains the same relationship with the Sun.
Clairaut's theorem characterizes the surface gravity on a viscous rotating ellipsoid in hydrostatic equilibrium under the action of its gravitational field and centrifugal force. It was published in 1743 by Alexis Claude Clairaut in a treatise which synthesized physical and geodetic evidence that the Earth is an oblate rotational ellipsoid. It was initially used to relate the gravity at any point on the Earth's surface to the position of that point, allowing the ellipticity of the Earth to be calculated from measurements of gravity at different latitudes. Today it has been largely supplanted by the Somigliana equation.
The surface gravity, g, of an astronomical object is the gravitational acceleration experienced at its surface at the equator, including the effects of rotation. The surface gravity may be thought of as the acceleration due to gravity experienced by a hypothetical test particle which is very close to the object's surface and which, in order not to disturb the system, has negligible mass. For objects where the surface is deep in the atmosphere and the radius not known, the surface gravity is given at the 1 bar pressure level in the atmosphere.
Gravity darkening, also referred to as gravity brightening, is an astronomical phenomenon where the poles of a star are brighter than the equator, due to rapid rotation and oblate shape. When a star is oblate, it has a larger radius at its equator than it does at its poles. As a result, the poles have a higher surface gravity, and thus higher temperature and pressure is needed to maintain hydrostatic equilibrium. Thus, the poles are "gravity brightened", and the equator "gravity darkened".
The Eötvös effect is the change in measured Earth's gravity caused by the change in centrifugal acceleration resulting from eastbound or westbound velocity. When moving eastbound, the object's angular velocity is increased, and thus the centrifugal force also increases, causing a perceived reduction in gravitational force.
A Maclaurin spheroid is an oblate spheroid which arises when a self-gravitating fluid body of uniform density rotates with a constant angular velocity. This spheroid is named after the Scottish mathematician Colin Maclaurin, who formulated it for the shape of Earth in 1742. In fact the figure of the Earth is far less oblate than Maclaurin's formula suggests, since the Earth is not homogeneous, but has a dense iron core. The Maclaurin spheroid is considered to be the simplest model of rotating ellipsoidal figures in hydrostatic equilibrium since it assumes uniform density.
The gravity of Earth, denoted by g, is the net acceleration that is imparted to objects due to the combined effect of gravitation and the centrifugal force . It is a vector quantity, whose direction coincides with a plumb bob and strength or magnitude is given by the norm .
The geodetic effect represents the effect of the curvature of spacetime, predicted by general relativity, on a vector carried along with an orbiting body. For example, the vector could be the angular momentum of a gyroscope orbiting the Earth, as carried out by the Gravity Probe B experiment. The geodetic effect was first predicted by Willem de Sitter in 1916, who provided relativistic corrections to the Earth–Moon system's motion. De Sitter's work was extended in 1918 by Jan Schouten and in 1920 by Adriaan Fokker. It can also be applied to a particular secular precession of astronomical orbits, equivalent to the rotation of the Laplace–Runge–Lenz vector.
Stellar rotation is the angular motion of a star about its axis. The rate of rotation can be measured from the spectrum of the star, or by timing the movements of active features on the surface.
An Earth ellipsoid or Earth spheroid is a mathematical figure approximating the Earth's form, used as a reference frame for computations in geodesy, astronomy, and the geosciences. Various different ellipsoids have been used as approximations.
Nodal precession is the precession of the orbital plane of a satellite around the rotational axis of an astronomical body such as Earth. This precession is due to the non-spherical nature of a rotating body, which creates a non-uniform gravitational field. The following discussion relates to low Earth orbit of artificial satellites, which have no measurable effect on the motion of Earth. The nodal precession of more massive, natural satellites like the Moon is more complex.
In geodesy and geophysics, theoretical gravity or normal gravity is an approximation of Earth's gravity, on or near its surface, by means of a mathematical model. The most common theoretical model is a rotating Earth ellipsoid of revolution.
A planetary coordinate system is a generalization of the geographic, geodetic, and the geocentric coordinate systems for planets other than Earth. Similar coordinate systems are defined for other solid celestial bodies, such as in the selenographic coordinates for the Moon. The coordinate systems for almost all of the solid bodies in the Solar System were established by Merton E. Davies of the Rand Corporation, including Mercury, Venus, Mars, the four Galilean moons of Jupiter, and Triton, the largest moon of Neptune. A planetary datum is a generalization of geodetic datums for other planetary bodies, such as the Mars datum; it requires the specification of physical reference points or surfaces with fixed coordinates, such as a specific crater for the reference meridian or the best-fitting equigeopotential as zero-level surface.
