AstroTutor - Kepler's Laws

Prior to the Copernican Revolution, which displaced the Earth from the center of the universe, the Ptolemaic System held sway, based upon the belief that all objects move around Earth in perfect circles. That made for a very orderly system, but not a very predictable one. The purpose of a scientific model is not only to provide an accurate description of how the sky appears and how objects move in the sky, but also to predict the positions of objects in the future. The development of two devices spelled doom for the accuracy of the Ptolemaic system: the astrolabe and the clock. The astrolabe is a device perfected by the Arabs for the purpose of determining the accurate positions of objects in the sky.

The Arabs used this information in determining the direction to Mecca, the holy city toward which they faced while reciting their daily prayers. This was especially important as they spread across northern Africa and into Spain, getting further and further away from Mecca. The Europeans became aware of the astrolabe when they ejected the Arabs from Europe during the Crusades.

The clock needs no introduction, but it may surprise you to learn that the early interest in the mechanical clock was to allow monks to be awakened for prayers in the middle of the night! Further refinement allowed for more accurate determination of feast days whose dates depended on astronomical data (the celebration of Easter is still fixed to an astronomical event).

With the discovery and gradual improvement of these two devices for measuring the positions of objects in the sky, by the time of Copernicus (1473-1543) it was common knowledge that the Ptolemaic model could not predict the future position of a planet with any reasonable degree of accuracy. Copernicus therefore looked for an alternative way to model the heavens. Although the placement of Sun at the center of the universe was a stroke of genius by Copernicus, his system was no more successful at planetary prediction than was his predecessor. Both Copernicus and Galileo (1564-1642) suffered from that same flaw in logic that Ptolemy had: motions in the sky must adhere to the perfect shape, the circle.

Eventually, the work of two men led to the important conclusion that the circle may be a perfect shape, but that it is not to be found in the sky. Planets move in elliptical orbits, not circular ones. This discovery resulted from the combined talents of two notable astronomers. The technical skills of a Danish nobleman, Tycho Brahe (1546-1601)--who received financial backing from the King of Denmark to build instruments capable of very accurately measuring the positions of the planets as they move relative to the stars)--were incorporated into the mathematical skills of Johannes Kepler (1571-1630)--who was able to fit the positions of the planets as recorded by Brahe into the mathematical shape of an ellipse.

Kepler took the discovery of the elliptical shape of planetary orbits one step further--he discovered that the properties of elliptical orbits allowed him to explain other properties of the solar system not explained by the Ptolemaic model. He expressed these properties, now known to be valid for any system of orbiting objects, in the form of three laws of motion. These are called Kepler's Laws of Planetary Motion. It had long been noticed that Sun does not move smoothly along the ecliptic. At some times of the year it appears to move faster than at other times. To account for this observation, Ptolemy had placed Earth off-center--but still retained Sun in circular orbit around it. Kepler made the concept more elegant in his 1st Law--the elliptical-orbit law.

Earth is located off-center at a point called the focus of the ellipse, but the orbit is no longer a perfect circle. Kepler then discovered mathematically that the varying speed of Sun relative to the stars--which is due to Earth's revolution around Sun--could be described by his 2nd Law, or the equal areas law. In the Animation, a planet moving for a period of time (for this example, a month) will, during both intervals a-b and c-d sweep out equal areas within the ellipse (count the number of squares in each area to verify the law). Of course, since the planet is closer to Sun during the interval a-b than when it is in c-d, it travels faster along the orbit and covers a larger portion of the orbit. But it sweeps out the same area as the slower-moving portion of the orbit at c-d.

Although it was well known that planets travel at different rates relative to the backdrop of the stars, it was not clear exactly what the rates had to do with distance from Earth (Ptolemaic model) or Sun (Copernican system). Kepler discovered that a mathematical relationship exists between the two variables. This is Kepler's 3rd Law:

P² = A³ (P squared is proportional to A cubed)

where P is the time in years needed to circle Sun, and A is the distance from Sun in units of Earth's average distance from Sun (the astronomical unit, AU). One AU is approximately 93 million miles.

The three laws explain (mathematically) the movements of objects in the solar system with great precision. In fact, all objects in space move along paths that derive from one of the four shapes obtained by cutting a cylinder in various ways. These shapes (circle, ellipse, parabola, hyperbola) are therefore called conic sections. Kepler was one of the first to recognize that nature is far from haphazard in the manner in which objects behave. Even a tossed baseball conforms to one of the conic section shapes. was revolutionary in Kepler's discoveries is that they not only contradict the theory that objects in the sky move in perfect circles, but that they lead to the radical idea that mathematics can describe events in the sky.

Kepler published these discoveries between 1609 and 1618, just about the time Galileo was confronting the Church with his ideas. Galileo knew of Kepler's discoveries, but rejected them because he could not bring himself to accept the possibility that the orbits of the planets did not conform to the perfect circle ideal of the Greeks. Isn't that ironic! Galileo was brave and stubborn enough to confront the Church with a radical theory of his own, and was quite critical of the Church's unwillingness to be open-minded enough to accept it based upon his telescopic evidence. But at the same time he was unwilling to accept Kepler's evidence for the elliptical shape of planetary orbits because it led to a radical conclusion. This is rather common in the history of science as well as other areas of human activity.