Kepler's Third Law

Kepler's third law states that the ratio of the cube of the orbital radius R to the square of the period T is the same ratio for all the planets. We can easily use Newton's laws of gravity and motion to check this result for the case of circular orbits. The result, is
(1)
where M s is the mass of the sun. In this calculation, the mass m p of the planet, the orbital radius R, the speed v all cancelled, leaving only the sun mass M s as a variable. Since all the planets orbit the same sun, this ratio should be the same for all the planets.
When the planet is in an elliptical orbit, the length of the radius vector R changes as the planet goes around the sun. What Kepler found was that the ratio of R³ /T² was constant if you used the "semi major axis" for R. The semi major axis is the half the maximum diameter of the ellipse, shown in Figure (1).

Figure 1: The semi major axis of an ellipse.