Why only one geostationary orbit?

Why only at one height and over the Equator?

A satellite that stays above one place on the Earth's surface has to sit above the Equator. Then the centre of the circle around which it is orbiting is the centre of the Earth, and the plane of orbit coincides with the equatorial plane. So the gravitational force is pulling the satellite towards the centre of its orbit: there is a centripetal force. If the plane of orbit of the satellite is moved closer to either pole then the gravitational force will be pulling the satellite towards the centre of the Earth, out of its plane of orbit.

The satellite sits at one height due to a balance of two factors, one depending on the speed in orbit and one on the gravitational field. Both of these depend on the radius of orbit, but in different ways. You'll recognise these as the two factors that affect the path without gravity, and the distance fallen towards the Earth.

An explanation

The time for an orbit is fixed at 24 hours, so the greater the radius of orbit, the greater the speed must be, because there is a greater distance to cover in the same time.

The gravitational field gets weaker as you go farther out, so the acceleration and the rate of fall towards the Earth also lessens.

Think about 1 second of motion of the satellite:

• The farther out you go, the longer the (imagined) gravity-free path, taking the satellite tangentially away from the Earth. (That is the path taken if the string is cut.)
• The farther out you go, the smaller the drop, due to gravity, that brings the satellite back to the same distance from the Earth, at the start of this second.

At one distance, and one distance only, these two factors balance, allowing a satellite to stay at the same height without having to fire rockets.