This post is an aside. Here I show how to prove Kepler's third law of planetary motion from what we have done so far. Nothing which follows depends on it, and if you choose to skip this post then you will lose nothing of substance from the series on Orbital Mechanics.
Kepler's third law states that
the square of the period of a planetary orbit is proportional to the cube of its distance from the Sun.
In one orbit, the orbiting body travels a distance equal to the circumference of the circle with radius r. This distance is 2πr. So the time taken to travel this distance, or the period of the orbit in seconds, P, is given by
P = 2πr / VC
Taking the square of both sides gives
P² = 4π²r² / VC²
Now from our last post we know that
VC² = GM/r
Putting this into the equation for the period squared gives
P² = 4π²r² / (GM/r)
which simplifies to
P² = 4π²r3 / GM
Now if this equation is applied to planets orbiting the Sun, then the square of period of the orbit, P², is directly proportional to the cube of the distance from the Sun, r3. So we have proved Kepler's third law.