Online Astronomy eText: Orbital Motions
Kepler's Laws of Planetary Motions
(also see Ellipses and Other Conic Sections)
 Kepler originally discovered two Laws of Planetary Motion, which allowed him to define the path followed by each planet and the way that it moved along that path. Together, they exactly solved the problem he had set out to solve -- how to calculate the positions of the planets relative to each other and the Sun, and their apparent position in our sky.
 Over a period of years, Kepler worked on a number of things, including a set of tables (the Rudolphine Tables) which eased the calculations needed to predict the motions of the planets; and theories of how and why the planetary orbits were spaced as they are. As a result of his investigations into the latter problem he discovered a third Law of Planetary Motion, the Harmonic Law. This Law was not necessary for the calculation of individual orbital motions; the first two Laws were adequate for that. Instead, it implied some deep relationship between the various orbits, and in particular enabled him to deduce the nature of the force which keeps the planets moving as they do.
 Kepler's Laws of Planetary Motion may be summarized as:

Kepler's First Law: The orbits of the planets are ellipses, with the Sun at one focus of each planet's orbit.

Kepler's Second Law: The line between the Sun and any planet sweeps out equal areas of space in equal periods of time.

Kepler's Third Law: The cube of the semi-major axis of a planet's orbit is directly proportional to the square of its orbital period.

 The First Law tells us what kind of orbit planets follow -- not the combination of circular paths the ancients and early Renaissance scholars proposed, but a single closed, smooth, symmetrical curve, well-understood by ancient geometers such as Euclid; and where the Sun is in relation to the orbit -- in a specific place, the same for all planets, whose position relative to each orbit is directly related to the size and shape of the orbit. The Second Law, called the Law of Areas because of the way it is stated, tells us how the planets move in their orbits -- not in uniform unchanging motions, requiring no force to motivate them, but in motions which are in a constant state of flux or change, going faster as they approach the Sun and slower as they move away from the Sun, with the implication that some force must be changing the motion of the planets. This is some ways the most important discovery that Kepler made. The fact that the orbits are ellipses instead of some complicated combination of circles is interesting, elegant, and to a mathematician, beautiful; but the fact that the motions change puts the lie to all previous theories of planetary motion. Unlike the stars, the planets do not move in unchanging, uniform motion, but in motion which goes faster and slower as they orbit the Sun; in motion which almost certainly requires some kind of force, most likely due to the Sun, to keep them moving that way.
 As already noted, the Third Law is not specifically required to solve the initial problem of planetary motion. But in some ways it is the most beautiful of all, for it shows that there is some deep force at work, which relates what happens to one planet to what happens to all of them; for if you know the relative sizes of two planets' orbits, their relative orbital periods and relative velocities are directly determined by the Third Law. The mathematical form of the relationship seemed to Kepler so beautiful -- mathematically beautiful in the same way that a melody is musically beautiful -- that he called it the Harmonic Law (music and mathematics are very closely related, both in the theory underlying musical forms and in the way the human brain treats them, so that mathematical and musical ability often go hand in hand).