A Celebration of Kepler’s Third Law: Harmony, Power Laws, and Keplerian Orbits

Abstract

Kepler’s greatest achievements are contained in his three laws, which today reappear constantly at the cutting edge of modern astrophysics. The modified third law has been the major workhorse of astronomy and astrophysics in determining the mass of objects throughout the universe. Kepler never revealed why he chose the integers, 2 and 3, to test using Tycho’s data. Because of his attachment to Pythagorean ideas and his knowledge of musical theory, I suggest he was guided by the musical interval known as the perfect fifth, for which the ratio of frequencies is 3/2. The perfect fifth is the most consonant of all intervals except the octave, and, as such, is the basis of all the tuning of stringed instruments. If Kepler had recognized the significance of the perfect fifth in analyzing Tycho’s data, it suggests a very pleasing historical parallelism between music and astronomy. In discovering the third law, Kepler also chanced upon the world's first known power law, which is now found in many forms throughout the earth and heavens. In discovering that the Galilean moons of Jupiter also obeyed the third law, Kepler encountered the phenomenon of scale independence, which is responsible for the ubiquity of power laws across the universe. The third law also played a crucial role in Newton’s discovery of the inverse square law of gravity in 1666. Not only did it provide Newton with a crucial mathematical step, but the third law also had the authority of Tyco’s observations. Christopher Wren and Edmund Halley relied similarly upon the third law for their apparent independent discoveries of the inverse square law.