Geostationary orbit

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A geostationary orbit (GEO) is a geosynchronous orbit directly above the Earth's equator (0º latitude), with orbital eccentricity of zero. From the ground, a geostationary object appears motionless in the sky and is therefore the orbit of most interest to operators of artificial satellites (including communication and television satellites). Due to the constant 0° latitude, satellite locations may differ by longitude only.

The idea of a geosynchronous satellite for communication purposes was first published in 1928 by Herman Potočnik. The geostationary orbit was first popularised by science fiction author Arthur C. Clarke in 1945 as a useful orbit for communications satellites. As a result this is sometimes referred to as the Clarke orbit. 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.

1 Introduction
2 Derivation of geostationary altitude
3 Practical limitations
4 See also
5 References
6 External links

[edit] Introduction

Geostationary orbits can only be achieved very close to the ring 35,786 km directly above the equator. In practice this means that all geostationary satellites have to exist on this ring, which poses problems for satellites that will be decommissioned at the end of their service life (e.g. when they run out of thruster fuel). Such satellites will either continue to be used in inclined orbits (where the orbital track appears to follow a figure-of-eight loop centered on the Equator) or be raised to a "graveyard" disposal orbit.

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 are used to provide visible, as well as infrared images of Earth's surface and atmosphere. These satellite systems include:

the US GOES

Meteosat, launched by the European Space Agency and operated by the European Weather Satellite Organization, EUMETSAT

the Japanese GMS

India's INSAT series

Most commercial communications satellites and television satellites operate in geostationary orbits. (Russian television satellites have used elliptical Molniya and Tundra orbits due to the high latitudes of the receiving audience.) The first satellite placed into a geostationary orbit was Syncom-3, launched by a Delta-D rocket in 1964.

A statite, a hypothetical satellite that uses a solar sail to modify its orbit, could theoretically hold itself in a "geostationary" orbit with different altitude and/or inclination from the "traditional" equatorial geostationary orbit. However, this would rely on using the solar wind at high altitude outside the Earth's magnetosphere.

[edit] Derivation of geostationary altitude

In any circular orbit, the centripetal acceleration required to maintain the orbit is provided by the gravitational force on the satellite. To calculate the geostationary orbit altitude, one begins with this equivalence, and uses the fact that the orbital period is one sidereal day.

F centripetal = F gravitational

By Newton's second law of motion, we can replace the forces $F$ with the mass of the object multiplied by the acceleration felt by the object due to that force:

$m_\mathrm{sat} \cdot a_\mathrm{centripetal} = m_\mathrm{sat} \cdot a_\mathrm{gravitational}$

We note that the mass of the satellite, $m s a t$ , appears on both sides -- geostationary orbit is independent of the mass of the satellite. So calculating the altitude simplifies into calculating the point where the magnitudes of the centripetal acceleration derived from orbital motion and the gravitational acceleration provided by Earth's gravity are equal.

The centripetal acceleration's magnitude is:

$|a_\mathrm{centripetal}| = \omega^2 \cdot \mathrm{r}$

where $ω$ is the angular velocity in radians per second, and $r$ is the orbital radius in meters as measured from the Earth's center of mass.

The magnitude of the gravitational acceleration is:

$|a_\mathrm{gravitational}| = \frac{G \cdot M_\mathrm{Earth}}{r^2}$

where $M Earth$ is the mass of Earth in kilograms, and $G$ is the gravitational constant.

Equating the two accelerations gives:

$r^3 = \frac{G \cdot M_\mathrm{Earth}}{\omega^2}$

$r = \sqrt[3]{\frac{G \cdot M_\mathrm{Earth}}{\omega^2}}$

We can express this in a slightly different form by replacing $G \cdot M_{Earth}$ by $μ$ , the geocentric gravitational constant:

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

The angular velocity $ω$ is found by dividing the angle travelled in one revolution ( $360^\circ = 2 \cdot \pi \ \mathrm{rad}$ ) by the orbital period (the time it takes to make one full revolution: one sidereal day, or 86164.09054^[1] seconds). This gives:

$\omega = \frac{2 \cdot \pi}{86164 \, \mathrm{s}} = 7.29 \cdot 10^{-5} \, \mathrm{rad} / \mathrm{s}$

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 (how fast the satellite is moving through space) is calculated by multiplying the angular velocity by the orbital radius:

$v = \omega \cdot r = 3.07466 \, \mathrm{km/s} = 11,068 \, \mathrm{km/h} = 6,877 \, \mathrm{mph}$

[edit] Practical limitations

While a geostationary orbit should hold a satellite in fixed position above the equator, orbital perturbations cause slow but steady drift away from the geostationary location. Satellites correct for these effects with station keeping manuvers. In the absence of servicing missions, consumption of thruster propellant for station keeping places a limitation on the lifetime of a satellite.

[edit] See also

[edit] References

[edit] External links

Graphical derivation of the geostationary orbit radius for the Earth
ORBITAL MECHANICS (Rocket and Space Technology)
List of satellites in geostationary orbit
Clarke Belt Snapshot Calculator
3D Real Time Satellite Tracking

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