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:

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
Using this equation, we can build the following table for Earth and Mars:

planetmass, Mperiod, Tradius, Roradius of geostationary
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
Erik Max Francis -- TOP
     Welcome to my homepage.
Writing -- UP
     Various things I've written.
Essays -- UP
     Essays I've written.
Stellar evolution -- PREVIOUS
     The births, lives, and deaths of the stars.
Black hole evaporation -- NEXT
     A brief analysis of the mathematical results of Hawking radiation.
Quick links.
Contents of Erik Max Francis' homepages -- CONTENTS
     Everything in my homepages.
Feedback -- FEEDBACK
     How to send feedback on these pages to the author.
About Erik Max Francis -- PERSONAL
     Information about me.
Copyright -- COPYRIGHT
     Copyright information regarding these pages.

Copyright © 1996 Erik Max Francis. All rights reserved.
Web presence provided by
Alcyone Systems
Last updated
2017 Jan 01 19:34
Web design by
7 sisters productions

The Alcyone Systems Web Ring (13 sites) >>>
Hard Science
Hard science, the easy way.
Alcyone Systems | Erik Max Francis | Blackgirl International | | CatCam | Crank dot Net | Hard Science | Losers dot Org | Polly Wanna Cracka? | Realpolitik | sade deluxe | 7 Sisters Productions ]