Geosynchronous and geostationary orbits 




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: 

a_{g} = a_{c}.  equation 1  
The gravitational acceleration a_{g} 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,  
a_{g} = G M/R^{2}  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 a_{c}, is omega^{2} 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: 

a_{c} = omega^{2} R  equation 3  
a_{c} = 4 pi^{2} R/T^{2}.  equation 4  
We can then set these accelerations equal to each other and combine terms:  
a_{g} = a_{c}  equation 5  
G M/R^{2} = 4 pi^{2} R/T^{2}  equation 6  
G M/R^{3} = 4 pi^{2}/T^{2}  equation 7  
R^{3}/(G M) = T^{2}/(4 pi^{2})  equation 8  
R^{3} = [G M/(4 pi^{2})] T^{2}.  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, R_{o}: 

H == R  R_{o}.  equation 10  
Table.  
Using this equation, we can build the following table for Earth and Mars:

planet  mass, M  period, T  radius, R_{o}  radius of geostationary orbit, R  height of geostationary orbit, H 

Earth  5.97 x 10^{24} kg  8.62 x 10^{4} s  6.37 x 10^{6} m  4.22 x 10^{7} m = 42 200 km  3.57 x 10^{7} m = 35 700 km 
Mars  6.46 x 10^{23} kg  8.86 x 10^{4} s  3.38 x 10^{6} m  2.05 x 10^{7} m = 20 500 km  1.71 x 10^{7} m = 17 100 km 
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