Which brings us to the hero of this story. The German astronomer and astrologer Johannes Kepler (1571-1630) was a thoroughgoing Copernican. He was convinced that the Copernican or heliocentric (meaning “Sun-centered”) model was celestial fact, rather than just a computational facilitator. He was something of a mystic, and he spent much of his life trying (and failing) to demonstrate that the arrangement of the solar system was based on the five regular Platonic solids, which needn’t bother us here.
Kepler was also a staunch Protestant, which more than once got him in trouble with the local authorities and forced him out of his home. It was during one of these forced evacuations that Tycho Brahe invited Kepler to join him at Uraniborg, Tycho’s island observatory. Tycho (who, like Galileo, is best known by his first name) was a nobleman, with a patron to boot, which meant that he could observe in peace. He was also known as the best visual astronomer around. After more than a millennium, he was the first astronomer to surpass the accuracy of Ptolemy’s star catalogue in the Almagest. Whereas Ptolemy’s star positions were accurate to about 10 arcminutes, Tycho’s were accurate to 1 arcminute, the width of a basketball as seen from a kilometer away. All this without the use of a telescope—that instrument would not be used for astronomical purposes for another decade. Tycho’s offer appealed to Kepler, so at the beginning of 1600, he accepted.
As it happened, though, Tycho was long past his best days as an observer. His observational prowess had made him famous, and he was more interested now in entertaining barons than in collaborating with the younger Kepler. It’s unclear whether Tycho viewed Kepler as a competitor, but getting observational data out of him proved too difficult for Kepler. When Tycho died the following year following a bout of excess drinking, Kepler mourned, but lost no time. Tycho’s heirs had no intellectual interest in his reams of data, and they were anxious to sell them as quickly as possible. Kepler had to snap them up quickly before they were lost forever.
Kepler immediately set to work on using the data to prove the validity of the Copernican model. It was not easy work. If the solar system were geocentric (that is, “Earth-centered”), the Earth would be stationary. The apparent paths of the planets would then be due to the actual motions of those planets, and nothing else. But if the Earth were merely one of the planets revolving around the Sun, Tycho’s observations were then made from a moving platform. The apparent paths of the planets would be a combination of the actual motion of those planets, plus the motion of the Earth. Worse yet, before the development of astronomical telescopes, there was no good way to tell how close a planet was. Essentially, Kepler was trying to formulate a three-dimensional model from contaminated two-dimensional data.
Fortunately, he was Kepler. He realized that although the planets moved from moment to moment, their orbits didn’t. If the Earth was in one place at a given time, then it would return to that spot one year later. The same went for the other planets. Mars revolves around the Sun in one Martian year, a little less than 687 days. That meant that if Mars were in one place on a given day, it would again be in that place a little less than 687 days later.
But—and this was the key to the whole enterprise—the Earth would also have moved around the Sun in those 687 days. A Martian year is about 43 days less than two Earth years. That meant the Earth would be about 43 days behind in its orbit at a time when Mars had returned to its original spot. Kepler could then use simple triangulation to determine the true position of Mars in three-dimensional space.
After some months of furious scribbling, Kepler came up with the circular parameters for Mars that seemed to match up with most of Tycho’s observations reasonably well—to 2 arcminutes. Tycho’s observations were generally accurate to 1 arcminute, but Kepler was willing to overlook that. Alas, two of the observations were off from predictions by as much as 8 arcminutes. That’s about one-fourth of the width of the full Moon, and 8 times the error in Tycho’s observations. As much as he wanted to verify the Copernican model, Kepler could not see his way to ignoring this discrepancy. Tycho’s observations were simply too good.
It was at this moment that Kepler took his greatest leap and abandoned the circle. This meant that the Copernican model was not correct except in its broadest terms, but Kepler meant to salvage what he saw as its distinguishing feature: heliocentrism. He tried a number of different shapes—ovals of various proportions, egg-like shapes, anything that would fit Tycho’s data. Computing three-dimensional positions took pages and pages of tedious computation. In his 1609 Astronomia Nova (“New Astronomy”), he describes not only his results but his efforts as well, taking the reader through even erroneous attempts. “If you are wearied by this tedious procedure,” he wrote, “take pity on me who carried out at least seventy trials.” One of the computational errors initially led Kepler to reject the correct solution, until at last he, like Hipparchus, took a page from Apollonius of Perga and tried the formula for an ellipse.
Because one of the innovations of Apollonius was to characterize geometrical shapes in novel ways, Kepler didn’t realize at first that he had seen the ellipse before. This time, however, he carried out his calculations without error and to his astonishment and delight, the predictions matched the observations precisely.
An ellipse is a sort of stretched-out circle. In fact, one of the less useful ways to characterize an ellipse is to say that it is a circle with its aspect ratio changed. Here’s another way to characterize it: if you have two points, F1 and F2, called foci (singular, focus), and a distance d, then the ellipse is defined as the set of points P such that F1P + F2P is equal to d.
This leads to one simple way to draw an ellipse. If you hammer two nails into a wooden board, and loop a loose string loop around them, you can draw an ellipse by inserting a pencil inside the loop and drawing an oval around the nails, taking care to keep the string taut at all times.
Kepler discovered that the orbit of Mars was perfectly described as an ellipse. What’s more, he also discovered that the Sun was not at the center of the ellipse. Instead, it was off-center, at one of the foci (one of the nails, in other words). Actually, this was no surprise to Kepler. It was well-known that the planets were closer to the Sun at certain times than they were at others. Copernicus modelled this by placing the Sun off-center of his orbits, too, and that’s what Kepler tried first. However, because Copernicus used circular orbits, he couldn’t match the observations nearly as well as Kepler did with his elliptical ones. Kepler’s first law of planetary motion is simply this:
Planets move in elliptical orbits, with the Sun at one focus.
When Kepler took into account the date and time of each observation, he found that the planets didn’t move at the same speed throughout their orbits, but sped up when they were close to the Sun, and slowed down when they were further away. Again, this was known by Copernicus, and didn’t surprise Kepler. However, Kepler, with his elliptical orbits, could quantify the variation. Nowadays, we would say that the angular velocity of a planet is inversely proportional to its distance from the Sun, but Kepler was raised in the classical tradition that used geometrical arguments for analysis, and he put it differently. He looked at the wedge of space traced out during a unit of time by each planet at various places in its orbit. When the planet was close to the Sun, it travelled faster, so the wedge was short but fat. On the other hand, when it was far from the Sun, it travelled slower, so the wedge was long but thin.
Kepler found that these wedges had different shapes, but they all had the same area. This is Kepler’s second law of planetary motion:
Planets sweep out equal areas in equal times.
And I think you’ll agree with me that that is a much more elegant way of putting it.
Kepler wasn’t done yet. After ten more years, he had analyzed enough data to arrive at his third and final law of planetary motion. The third law was first published in his 1619 book, Harmonices Mundi (“Harmonies of the World”), but was laid out in complete detail, along with the first two laws, in Epitoma Astronomia Copernicanae (“Summary of Copernican Astronomy”). Here it is:
The square of a planet’s period of revolution is proportional to the cube of its average distance from the Sun.