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 on one of the focii of the ellipses).
2. Each planet sweeps out an equal area in an equal time interval.
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.