Geosynchronous and geostationary orbits
 30Es2 Essays

Geosynchronous and geostationary orbits.
A geostationary orbit is one where the orbit has the same period as its primary's rotation period, and remains stationary over a single point on the Earth's surface. A geosynchronous one only has the first restriction; that is, geosynchronous orbits can be elliptical, but geostationary ones have to be circular and stationed over the equator.

Now all you have to do is find an orbit where these conditions is satisfied (all you're interested in his how high up it is). One way of phrasing an orbit (a circular one, anyway) is that it's a path where the centripetal acceleration always equals the gravitational acceleration:

ag = ac. equation 1
The gravitational acceleration ag is merely F/m, where F is the Newtonian force due to gravity, and m is the mass of the test particle. This is the standard Newtonian gravitational field strength,
ag = G M/R2 equation 2
where G is the universal constant of gravitation, M is the mass of the primary, and R is the radius of our geostationary orbit.

The centripetal acceleration ac, is omega2 R, where omega is the angular velocity. The angular velocity is 2 pi/T, where T is the period of the geostationary orbit, or, equivalently, the rotation period of the primary:

ac = omega2 R equation 3
ac = 4 pi2 R/T2. equation 4
We can then set these accelerations equal to each other and combine terms:
ag = ac equation 5
G M/R2 = 4 pi2 R/T2 equation 6
G M/R3 = 4 pi2/T2 equation 7
R3/(G M) = T2/(4 pi2) equation 8
R3 = [G M/(4 pi2)] T2. equation 9
This is merely Kepler's third law, although strictly speaking we've only demonstrated it for circular orbits. Here you have what you need to find the characteristics of the geostationary orbit: You have the mass of the primary M, the rotation period of the primary T, and the radius of the geostationary orbit R.

This R gives you the radius measured from the center of the primary. To find the height of geostationary from the surface of the planet, merely subtract the radius of the primary, Ro:

H == R - Ro. equation 10
Table.
Using this equation, we can build the following table for Earth and Mars:

orbit, R
height of geostationary
orbit, H
Earth 5.97 x 1024 kg 8.62 x 104 s 6.37 x 106 m 4.22 x 107 m
= 42 200 km
3.57 x 107 m
= 35 700 km
Mars 6.46 x 1023 kg 8.86 x 104 s 3.38 x 106 m 2.05 x 107 m
= 20 500 km
1.71 x 107 m
= 17 100 km
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