Kepler's Third Law
Kepler's laws of motion describe the orbits of planets around the Sun. They were were a bridge between the Aristotlean view of the Solar System, which described in error and did not explain, and the Newtonian view, which described (almost) correctly and explained in scientific terms. Kepler's Laws of motion described (almost) but did not explain any basis. Kepler's Laws of motion were based purely on astronomical observations carried out over many years.
Consider an artificial satellite in orbit around the Earth. Ideally they do not require engines to keep them in orbit – the gravitational pull from the Earth supplies the centripetal force.
For a circular orbit, centripetal force = gravitational force.
There are equations for each of these forces.
The equation for the centripetal force is
The equation for the gravitational force is
=mass of satellite.
=Mass of Earth.
=radius of orbit
It is the universal gravitational constant, often called 'big g'.
We can equate these to obtain
Simplifying gives
(1)
The satellite orbits the Earth, travelling a distance
(the circumference of the orbit) in a time
The equation
gives
hence
(2)
Equation (1) and (2) give
and from this we obtain
is a constant so
This is known as Kepler's Law. Though it is derived here for a circular orbit, it is true for any orbit.