Lecture 5 -- Galileo, Brahe, and Kepler
Although no one actually read Copernicus' book (which as hopelessly confused and incomprehensible), his idea of a heliocentric model, in which the Sun was the center of the "universe", spread. One of those who listened was an Italian experimentalist named Galileo Galilei. Galileo set out to test various precepts of motion. as handed down from ancient times. For instance, according to Aristotle, heavy things should fall faster than light things. Galileo tested the assertion by dropping objects off the tower of Pisa (or so he says in his book). According to Galileo, his wooden ball and lead ball fell at almost exactly the same speed, in direct conflict with Aristotle. Similarly, Aristotle had stated that all things have a property called inertia, i.e., once set in motion, all items want to come to rest. Galileo studied the motion of a ball, and hypothesized that, if not for friction, the ball would continue rolling forever (in a circle around the earth). Thus, according to Galileo, Aristotle was again wrong, and the preferred motion of an object was a great circle (rather than being at rest).

Galileo's most important contributions, however, came from his use of a telescope to observe the heavens. Even with a small aperture and poor optics, Galileo showed that the heavens were not perfect: the Sun had sunspots, the Moon had craters, and Saturn had rings (he called them ears). He also noticed that four moons revolved about Jupiter . This was proof that not everything revolved about the earth. Finally, Galileo observed that Venus showed phases, just like the Moon. This fit nicely into the heliocentric model of the Solar System. The church put Galileo under house arrest, tried him as a heretic, threatened him with death (a routine formality) and forced him to recant.

About the same time that Galileo was making his observations, two scientists in Denmark were working to uncover the mystery of planetary motions. The first was Tycho Brahe. Brahe was a rich nobleman, whose foster father once saved the life of the King of Denmark. In 1572, a very bright new star, a nova, appeared in the sky. By measuring its position precisely Brahe proved, through its lack of parallax, that it was really in the heavens, and not just in the upper atmosphere. This caused quite a stir, and to keep Brahe in the country, the King of Denmark granted Brahe his own island, complete with paper mills, printing press, castle, prison, and, of course, an extremely generous endowment (which made him one of the richest men in Denmark). For 20 years, Tycho Brahe resided as a feudal lord over his island, and with his observatory, Uraniburg, made extremely careful (eyeball) measurements of planetary positions. (His measurements were good to 2 arcminutes!)

After 20 years, Brahe hired a bright, extremely hard working young mathematician, Johannes Kepler to try and make sense of his data. Almost instantaneously, the two learned to hate each other. Brahe wanted Kepler to prove his own peculiar theory of the cosmos: that the Sun went around the earth, but that the planets went around the Sun. Kepler had his own ideas (which Brahe did his best to discourage), and to keep Kepler in line, Brahe only gave Kepler access to some of his observations. After a stormy three years (during which Kepler was fired and re-hired), Brahe died. Kepler grabbed Brahe's measurements, and, before Brahe's heirs could stop him, he was gone. For the next seven years, Kepler tried to fit the planetary motions with every law imaginable (circles, circles inscribed in polygons, egg-shaped orbits, etc). Finally, Kepler blundered in his calculations, made a wrong approximation, blundered again, and stumbled upon the solution. The path of the planets around the Sun seemed to obey three laws.

Law 1: Planets move around the Sun in ellipses, with the Sun at one focus. Ellipses are like elongated circles: instead of every point being equidistant from a central point (as in a circle), there are two focii associated with an ellipse, and the distance from one focus to the ellipse plus the distance from the other focus to the ellipse is a constant. The long axis of the ellipse is called the major axis; half of this is the semi-major axis. The point of closest approach to the Sun is called the perihelion, and the point of furthest distance is the aphelion. The planets' orbits are almost circles, but not quite.

Law 2: The planets sweep out equal areas in equal times. When a planet is close to the Sun, it travels fast; when it is far from the Sun, it moves slowly. The rate at which areas are swept out within the ellipse is constant.

Law 3: The time it takes a planet to go around the Sun, squared, is equal to the length of the semi-major axis, cubed. . Or, in mathematical terms, P^2 = K A^3 , where P is the period, A is the semi-major axis, and K is some number to make the units come out right. If we apply this to the earth, and measure time in years and distance in Astronomical Units, where 1 A.U. is the semi-major axis of the earth's orbit, then K = 1 . So, with these units, you don't have to worry about the constant. The law tells us that the farther planets are from the Sun, the longer they take to go around the Sun. For example, a planet 4 A.U. away will take 8 years for one revolution.

Note that Kepler did not give any reason why the planets obey these laws. They just do.