Geostationary Orbit: Cool Stuff

The geostationary orbit of satellites is one of my new favorite topics in physics; from one reference point on earth, a satellite will not appear to move at all because it is orbiting the planet at the same speed that the planet is revolving, and its altitude will not change.

It turns out that there is a specific altitude and angular speed that the satellite has to be moving at to make this happen when the satellite is orbiting directly above the equator. Below is the proof.

Assume that the satellite has mass m. In order for this to work, we want our centripetal force, F_centripetal, to equal the force of gravity, F_gravity. The expression we get is as follows:

F_centripetal = F_gravity.

The next step is to use Newton’s 2^nd law, ΣF=ma, making our expression then equal to:

ma_centripetal = mg, where g is the acceleration due to gravity.

We can see that the m’s cancel out leaving us with simplified:

a_centripetal = g

From here we use two expressions that we already know, a_centripetal = ω²r and g = (GM/r²), where G is the gravitational constant of the universe and M is the mass of the earth. We then get the expression:

ω²r =  (GM/r²), which, with a bit of algebra, simplifies to: r = (GM/ ω²)^(1/3)

This still leaves us with a problem, as we have two variables to solve for, ω and r. fortunately, we can figure out ω; in order for the satellite to be geostationary, it must rotate with the earth, meaning that it must make a full revolution, 2π radians, in 1 day, 86400 seconds. Now that we know ω, we can solve for r, which is just simple algebra. Our final altitude ends up being approximately 42200 kilometers above the center of the Earth, or when subtracting Earth’s radius of 6400 kilometers, 35800 kilometers, approximately.

In theory, satellites in this orbit would have an exact footprint on the Earth, but due to other forces other than the Earth’s gravity, satellites still shift, regardless of whether or not they are at the exact altitude of a geostationary orbit.

So yeah, geostationary orbit, fun stuff.