kepler's law + newton's law?

Question

1.Explain how Kepler's 3rd law and Newton's law of universal gravitation predicted a ring of satellites that are:
a) directly above the equator.
b) all at the same height.

(this ring of satellites does now exist, but Arthur C Clarke predicted it in this way)

You should use any reasonable assumptions about the usefulness and logistics of satellite communication (hint: this would be a good place to start your reasoning).

Answer 1

Newton's law of universal gravitation allows us to understand how smaller bodies orbit larger ones (or any two bodies orbiting about their center of mass). Kepler's 3rd law relates the period of orbits with the distances between the bodies, through the proportional relation T^2 = R^3. Satellites in fact can orbit at any distance with any eccentricity and any direction about the Earth. However, as Arthur C. Clarke pointed out, geosynchronous satellites, or satellites that appear fixed in the sky from the earth because the orbit time is 24 hours like the earth's day, can be valuable as communications satellites. Because earth has an axis of spin, and because the earth's center (actually the barycenter, but that's argumentive) must be at one of the foci of the orbital ellipse (or center of the circle), the orbit of geosynchronous satellites must be coplanar with the earth's equator. Moreover, through Kepler's 3rd law, in order for them to have a orbital period of 24 hours, they have to be at a particular distance from the earth, or "all at the same height".