Kepler: Epitome of Copernican Astronomy

Johannes Kepler was a German astronomer who lived about 400 years ago. He was sure Copernicus was right that the planets go around the sun. But when he checked the very careful measurements of the planet Mars made by his teacher Tycho Brahe, the circles didn't quite fit. The error was tiny, about the width of your fingernail held at arm's length, but Kepler trusted Tycho's numbers.

After years of work he found the answer: Mars goes around the sun not in a circle but in an oval, an ellipse, and the sun is not in the middle of the oval but off to one side. When Mars is near the sun it speeds up; far away, it slows down.

Who, when and why

Johannes Kepler (1571–1630) grew up in Württemberg, in southern Germany, studied at the Lutheran university of Tübingen, and became a mathematics teacher in Graz, then Imperial Mathematician in Prague after Tycho Brahe died in 1601, and later a teacher in Linz. His life was hard: poverty, the deaths of his first wife and several children, religious wars and moves forced on him for his Lutheran faith, and a long trial in which he defended his mother against a charge of witchcraft, and won her release.

In A New Astronomy (1609) he described how he found the shape of Mars's orbit. The Epitome of Copernican Astronomy (1617–1621) is his textbook, written as questions and answers for students. Books IV and V, in the Great Books set, explain the whole sun-centered system in Kepler's new way: every planet, and the Moon, moves in an ellipse, driven by a force from the sun.

The problem Kepler faced

Copernicus had moved the sun to the center but kept the circles and many small epicycles. Tycho Brahe had then measured planet positions for twenty years with huge instruments and no telescope, accurate to about 1/30 of a degree. Kepler's best circle-based model of Mars still missed Tycho's positions by 8 minutes of arc (about 1/8 of a degree). Earlier astronomers would have shrugged at that. Kepler wrote that those eight minutes pointed the way to a complete reform of astronomy.

Idea 1: orbits are ellipses (the first law)

An ellipse is a stretched circle. It has two special points inside, the foci. For every point on an ellipse, the two distances to the foci add up to the same total. Kepler found that each planet moves on an ellipse with the sun at one focus. The other focus is empty. How stretched an ellipse is, its eccentricity $e$, runs from 0 (a circle) toward 1 (very long and thin). Mars's is about 0.09; Earth's only 0.017.

Idea 2: equal areas in equal times (the second law)

Kepler actually found this rule first, while still using circles. Picture a line from the sun to the planet. As the planet moves, the line sweeps out area like the hand of a clock sweeping a pie. The line sweeps equal areas in equal times. So near the sun, where the line is short, the planet must move fast; far away it moves slowly.

Idea 3: a force from the sun

Kepler was the first astronomer to ask what physically pushes the planets. Inspired by William Gilbert's book on the magnet, he imagined the sun spinning and sending out a force, like spokes of a turning wheel, that drags the planets around; magnetic-like pulls and pushes then make the orbit oval. The details were wrong, but the question was new: astronomy should explain motions by causes, not just describe them with circles.

Getting the history right

• The laws came in a different order, and not as "three laws". Kepler found the area rule first and the ellipse second (both in A New Astronomy, 1609); the third law came ten years later in The Harmonies of the World (1619). He never numbered them; later writers did.
• He tried many shapes first. Kepler tested circles with off-center points, then egg shapes (ovals that were not ellipses), before the ellipse fit. His book tells this long struggle openly, wrong turns included.
• Proof came through prediction. Kepler's Rudolphine Tables (1627), built on his laws, were far more accurate than any tables before. They predicted that Mercury would cross the face of the sun in November 1631, and Pierre Gassendi in Paris saw it happen.

What later science kept, and what it changed

Kept: ellipses with the sun at a focus and the equal-area rule, still taught in every astronomy class. Changed: the cause. Newton (1687) showed that both laws follow from one force of gravity pulling toward the sun, growing weaker with the square of the distance. The equal-area law turned out to be even more general: it holds for any force aimed at one center (today we say angular momentum is conserved). And the orbits are not perfect ellipses: the planets tug on each other a little.

Faith in the data. Kepler's whole discovery hung on trusting Tycho's eight minutes of arc. Tycho had spent a fortune building instruments and a lifetime using them. Why might a scientist decide that a small mismatch matters, when others would call it close enough?

Kepler's reasons. Kepler was a devout Lutheran who believed God had made the world on a mathematical plan, and that finding it was a kind of worship. His first book, The Cosmic Mystery (1596), fitted the six planets' spheres around the five regular solids of Euclid. That idea was wrong, but the search it began led to his laws. His beliefs are his own; readers of any faith or none can still follow his mathematics and his evidence.

A textbook as a revolution. The Epitome spread the new astronomy more than any other book of its time, because it was written to be taught. Explaining something to students forced Kepler to make the whole system fit together.

Big questions. When is a small error a clue rather than a nuisance? Is it enough for a model to predict well, or should it also say what causes the motion?