system
Given the following post, respond as if you're commenting on Reddit:
human
Why is geostationary orbit for a satellite fixed at about 36,000 km above the Earth? What makes this the optimum distance?
gpt
The orbital altitude of 36000 km isn't just the optimum distance, from Newtonian mechanics it follows that this is the only distance for a geostationary orbit. The derivation is fairly simple and can be found below:

A geostationary orbit is a circular orbit in which the satellite stays, as the name implies, over the same point on Earth. That means that the satellite completes a full orbit every 24 hours.

For a circular orbit with radius *r*, a centripetal force is required to maintain the orbit, described by:

F = m r w^2

(*m* is the mass of the object, *w* is the angular velocity, typically measured in radians per second (radians are a unit of angle, a full circle corresponds to 2 pi radians)).

In the Earth-satellite system, this force is provided by gravity, which is given by:

F = G M m / r^2

(*G* is the Newtonian constant of gravity, *M* the mass of the Earth)
Equating the two yields:

m r w^2 = G M m / r^2

the *m* terms cancel, shuffle some of the others:

r^3 = G M / w^2

Fill in the values for *G*, *M* and *w* and you can compute the value of *r*, which should give you an outcome just over 42000 km. If you subtract the radius of the Earth, you'll find the 36000 km value.

In general, the speed of a circular orbit is related to the radius of the orbit. The smaller the radius of the orbit, the higher the speed of the object will be. The orbital altitude of about 36000 km is the sweetspot where a satellite moves with just the right speed to remain above the same spot on Earth.