NationMaster - Encyclopedia: Geosynchronous orbit derivation

This is the derivation of the geosynchronous orbital distance for a body in circular orbit around the Earth. It also applies to geostationary orbits. The geosynchronous orbits have the same period as the Earth's rotation. A geosynchronous orbit is a geocentric orbit that has the same orbital period as the sidereal rotation period of the Earth. ... In astrodynamics or celestial mechanics a circular orbit is an elliptic orbit with the eccentricity equal to 0. ... Adjectives: Terrestrial, Terran, Telluric, Tellurian, Earthly Atmosphere Surface pressure: 101. ... A geostationary orbit (abbreviated GEO) is a circular orbit in the Earths equatorial plane, any point on which revolves about the Earth in the same direction and with the same period as the Earths rotation. ... The orbital period is the time it takes a planet (or another object) to make one full orbit. ...

The rotational period of the Earth is slightly shorter than a day (24 hours), because in one day the Earth does a complete rotation and a little extra due to it also moving round the Sun. Without this extra rotation speed the Sun would not quite appear in the same place at noon from day to day. Adjectives: Terrestrial, Terran, Telluric, Tellurian, Earthly Atmosphere Surface pressure: 101. ... The Sun is the star at the center of the Solar System. ...

A sidereal day is 23 hours, 56 minutes, 4.09054 seconds, or 86164.09054^[1] seconds. On a prograde planet like the Earth, the sidereal day is shorter than the solar day. ...

Constants and variables

$G = 6.67259 times 10^{-11}; mathrm{m}^3, mathrm{kg}^{-1}, mathrm{s}^{-2} ;$ gravitational constant^[2]

$M_e= 5.9742 times 10^{24}; mathrm{kg} ;$ mass of Earth^[3]

$omega = frac{2 pi}{86164.09054}; mathrm{rad}, mathrm{s}^{-1} ;$ Earth's angular speed

$r ;$ radius of geosynchronous orbit

$v ;$ orbital speed

$m_s ;$ mass of satellite

According to the law of universal gravitation, the attractive force between two bodies is proportional to the product of their masses and inversely proportional to the square of the distance between them. ... Angular frequency is a measure of how fast an object is rotating In physics (specifically mechanics and electrical engineering), angular frequency ω (also called angular speed) is a scalar measure of rotation rate. ... The orbital speed of a body, generally a planet, a natural satellite, an artificial satellite, or a multiple star, is the speed at which it orbits around the barycenter of a system, usually around a more massive body. ...

Derivation

$F = frac{m_s v^2}{r}$ (centrifugal force required to maintain circular orbit)

$F = frac{G M_e m_s}{r^2}$ (force of gravity from body

m e

acting on a body of mass

m s

)

$frac{v^2}{r} = frac{G M_e}{r^2}$ (equate and cancel previous formulae)

$omega = frac{v}{r}$ (rotational rate in radians per second as a function of v,r)

$omega^2 = frac{v^2}{r^2} = frac{G M_e}{r^3}$ (from previous two formulae)

$r = sqrt[3]frac{G M_e}{omega ^2} approx 42,164 mbox{ km }$ (rearrangement of the above formula)

This gives the distance of the circular geosynchronous (and hence geostationary) orbit from the centre of the Earth. Centrifugal force (from Latin centrum center and fugere to flee) is a term which may refer to two different forces which are related to rotation. ... A geostationary orbit (abbreviated GEO) is a circular orbit in the Earths equatorial plane, any point on which revolves about the Earth in the same direction and with the same period as the Earths rotation. ...

Evaluation and approximation

Evaluation of the above formula yields r = 42 173 531 m (to the nearest meter), with an inaccuracy of about 0.05% arising mostly from the approximations to $G$ and $m e$ . Note that this figure is measured from the centre of the Earth and not from the surface - to find the altitude of this orbit about the surface of the Earth we would simply deduct the Earth's radius at that point. This leaves an altitude of around 35 000 000 m (2sf) above the surface.

Category: Earth orbits

Results from FactBites:

Geostationary orbit - Wikipedia, the free encyclopedia (776 words)
	It is a special case of the geosynchronous orbit (abbreviated GEO), and the one which is of most interest to operators of artificial satellites (including communication and television satellites).
	Geosynchronous and geostationary orbits were first popularised by science fiction author Arthur C. Clarke in 1945 as useful orbits for communications satellites.
	A geostationary transfer orbit is used to move a satellite from low Earth orbit (LEO) into a geostationary orbit.

More results at FactBites »

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Constants and variables var ACE_AR = {Site: '756743', Size: '300250'};

Derivation

Evaluation and approximation

Constants and variables