Kepler's Laws of Planetary Motion

Tycho Brahe spent his career laboriously collecting data on the positions of planets and other astronomical phenomena. Tycho's data were of unprecedented accuracy at the time. Near the end of his life Tycho hired a new assistant, Johannes Kepler. When Tycho died, Kepler usurped the data before Tycho's heirs could claim them. Kepler used Tycho's data to study the orbits of the planets and deduce three laws of planetary motion that drastically altered our understanding of the cosmos.

Battling Mars

Tycho had the most data on the planet Mars, so Kepler's first battle was with the god of war. In Kepler's time, two millennia of scientific tradition, since before Ptolemy, dictated that all heavenly objects travel in uniform circular motion. Kepler therefore tried various combinations of circular orbits for both Earth and Mars. He worked first on circular orbits. Nothing worked. No combination of circles could match Tycho's data.

Circles had worked before, but Tycho's data were far more accurate than any previous data. One of the reasons that Kepler was a first rate scientist was that he would toss out ideas, no matter how cherished, that did not stand up to the scrutiny of accurate data.

Kepler therefore gave up on circular orbits and tried elliptical orbits. Ellipses worked! Using elliptical orbits, Kepler was able to find a combination of elliptical orbits for Earth and Mars that allowed him to predict positions of Mars in the sky that agreed with Tycho's observed positions of Mars. Elliptical orbits also worked for accurately predicting the positions of the other planets. Kepler's battle with the god of war lasted eight years. Kepler was so pleased to win that he drew, in the final diagram of his proof, the winged goddess of victory riding down in a chariot to place a laurel wreath on his head.

Kepler deduced three laws of Planetary motion.

Kepler's First Law of Planetary Motion

Kepler's first law states that the planets orbit the Sun in elliptical orbits and that the Sun is at one focal point of the ellipse.

Elliptical orbits no longer seem like a big deal to us, but remember that in Kepler's time tradition required circular orbits at a constant speed. Hence at the time it was a radical suggestion to say that the planets had elliptical orbits.

Kepler's Second Law of Planetary Motion

Kepler's second law states that as a planet orbits the Sun, it sweeps out equal areas in equal times. To do so, the planet must move faster when it is closer to the Sun and more slowly when it is farther away. After discarding circular orbits, Kepler's second law discarded the constant speed.

Kepler's Third Law of Planetary Motion

Kepler's third law is a mathematical relationship between the time it takes the planet to orbit the Sun and the distance between the planet and the Sun. The square of the orbital period is proportional to the cube of the orbital distance.

Modern Astronomers use Kepler's third law to find the mass of astronomical objects that have something orbiting them.

Newton and Kepler's Laws

Kepler did not know why his laws worked. He discovered them by working with Tycho's data. To try to explain his laws, Kepler incorrectly suggested that a magnetic force between the Sun and planets caused the planets to orbit the Sun.

However Newton's laws and a few pages of algebra can be used to prove Kepler's laws. Newton therefore explained physically why Kepler's laws worked in terms of the gravitational force and laws of motion.

Further Reading

Zeilik, M., Astronomy: The Evolving Universe 9th ed. Cambridge, 2002.

Hawking, S., On the Shoulders of Giants, Running Press, 2002

Johannes Kepler

Kepler's Victory Drawing

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