# Geostationary orbit

A geostationary orbit (abbreviated GEO) is a circular orbit in the Earth's equatorial plane, any point on which revolves about the Earth in the same direction and with the same period as the Earth's rotation. It is a special case of the geosynchronous orbit (abbreviated GSO), and the one which is of most interest to operators of artificial satellite.

The idea of a geosynchronous satellite for communication purposes was first published 1928 by Herman Potocnik. Geosynchronous and geostationary orbits were first popularised by science fiction author Arthur C. Clarke in 1945 as useful orbits for communications satellites. As a result they are sometimes referred to as Clarke orbits. Similarly, the "Clarke Belt" is the part of space approximately 35,786 km above mean sea level in the plane of the equator where near-geostationary orbits may be achieved.

Geostationary orbits are useful because they cause a satellite to appear stationary with respect to a fixed point on the rotating Earth. As a result, an antenna can point in a fixed direction and maintain a link with the satellite. The satellite orbits in the direction of the Earth's rotation, at an altitude of approximately 35,786 km (22,240 statute miles) above ground. This altitude is significant because it produces an orbital period equal to the Earth's period of rotation, known as the sidereal day.

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## Use in artificial satellites

Geostationary orbits can only be achieved very close to the ring 35,786 km directly above the equator. All other circular non-active geosynchronous orbits will cross the geostationary orbit and possibly collide with satellites there. In practice this means that all geostationary satellites have to exist on this ring, which poses problems for satellites needing to be decommissioned at the end of their service life (for example when they run out of thruster fuel).

A geostationary transfer orbit is used to move a satellite from Low Earth orbit (LEO) into a geostationary orbit.

A worldwide network of operational geostationary meteorological satellites provides visible and infrared images of Earth's surface and atmosphere. These satellite systems include:

A statite, a hypothetical satellite that uses a solar sail to modify its orbit, can theoretically hold itself in a geostationary orbit with different altitude and/or inclination from the "traditional" equatorial geostationary orbit.

## Derivation of geostationary altitude

To calculate the geostationary orbit altitude, one finds the point where the magnitudes of the centrifugal acceleration derived from orbital motion and the centripetal acceleration provided by Earth's gravity are equal.

The centrifugal acceleration's magnitude is:

[itex]|a_c| = \omega^2 \cdot r[itex]

...where [itex]\omega[itex] is the angular velocity in radians per second, and [itex]r[itex] is the orbital radius in metres as measured from the Earth's centre of mass.

The magnitude of the gravitational attraction is:

[itex]|a_g| = \frac{M_e \cdot G}{r^2}[itex]

...where [itex]M_e[itex] is the mass of Earth in kilograms, and [itex]G[itex] is the gravitational constant.

Equating the two gives:

[itex]r^3 = \frac{M_e \cdot G}{\omega^2}[itex]

[itex]r = \sqrt[3]{\frac{M_e \cdot G}{\omega^2}}[itex]

This is alternatively expressed as:

[itex]r = \sqrt[3]{\frac{\mu}{\omega^2}}[itex]

...where [itex]\mu[itex] is the geocentric gravitational constant.

The value of [itex]\omega[itex] is found by dividing the angle of a full circle ([itex]2 \cdot \pi[itex] radians) by the orbital period (86400 seconds, or one day). This gives:

[itex]\omega = \frac{2 \cdot \pi}{86400} = 7.27 \cdot 10^{-5} \mathrm{rad} \cdot \mathrm{sec}^{-1}[itex]

The resulting orbital radius is 42,164 km. Subtracting the Earth's equatorial radius, 6,378 km, gives the altitude of 35,786 km.

Orbital velocity is calculated by multiplying the angular velocity by the orbital radius:

[itex]v = \omega \cdot r = 3.07 \mathrm{km} \cdot \mathrm{s}^{-1}[itex]

## See also

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