Kepler's Laws

As is well-known, Johannes Kepler was the first astronomer to correctly describe planetary motion in the Solar System (in works published between 1609 and 1619). The motion of the Planets is summed up in three simple laws:

1. The planetary orbits are all ellipses which are confocal with the Sun (i.e., the Sun lies at one of the focii of each ellipse).
2. The radius vectors connecting each planet to the Sun sweep out equal areas in equal time intervals.
3. The squares of the orbital periods of the planets are proportional to the cubes of their orbital major radii.

Let us now see if we can derive Kepler's laws from Newton's laws of motion.