Planetary Motions
Lecture 3 -- Geocentric and Heliocentric Systems

So far, the motions of the Sun and the Moon have been presented from a heliocentric point of view. We have described the earth as being round, we have said that it spins, and we have said that it moves around the Sun (and even precesses). But how do we know that?

It's fairly easy to prove that the earth is round. Look at the shadow of earth on the Moon during a lunar eclipse. It appears round. Look at the position of Polaris, then take a walk. As you walk north, Polaris will get higher and higher overhead. (OK, you'll have to walk a few hundred miles before you really notice, but its height is changing.) In fact, around 200 B.C. Eratosthenes not only showed that the earth is round, but he also measured its size. Eratosthenes noticed that at noon on the first day of summer in Syene, Egypt, the Sun appeared directly overhead. Its light would shine all the way down to the bottom of a deep well. Meanwhile, at the exact same time in Alexandria, Egypt, the Sun was not directly overhead: towers there would cast a shadow. From the length of the shadow, Eratosthenes determined that the Sun was a full 7 degrees away from the zenith (about 1/50 of a circle). Since the distance from Syene to Alexandria was 5000 stadia, the earth's circumference must be 50 x 5000 = 250,000 stadia. (Did he get the right answer? Maybe, but historians don't really know the definition of a stadia. He was either correct to within 1%, or off by 20%.)

Proving that the earth goes around the Sun is more difficult. We've always been told that the earth goes around the Sun, but how do we know it? Is there a simple experiment we can think of that can actually prove that the earth moves? Yes, there is.

Hold your thumb up at arms length. Look at it with one eye. Then look at it with the other eye. Its position appears to change, with respect to the background. This is called parallax. If the earth moves around the Sun, then the apparent positions of the stars must change between January and June. (Even if all the stars are at the same distance, as if drawn on a wall, their positions with respect to one other must change as you get closer and farther away from one side.) The effects of parallax were known as far back as Babylonian times. But, no matter how hard people looked, no parallax could ever be measured. This was taken as proof that the earth did not move. The only alternative was that the Sun must revolve about the earth.

For the Sun and Moon, this hypothesis does not present much of a difficulty. We see the Sun (and Moon) move from one constellation in the sky to another in an orderly west-to-east fashion. It is therefore relatively easy to imagine the earth being in the center, and the Sun and Moon moving in perfect circles about the earth. But what about the planets?

Aside from the Sun and the Moon, ancient astronomers knew of five special objects in the sky. These points of light looked like bright stars, but unlike normal stars, they did not appear in the same location in the sky every night. They wandered amongst the fixed stars, hence their name, wanderers, or planets. Like the Moon, these planets moved within the zodiac constellations, never straying too far from the Sun's path (i.e., the ecliptic). Also, like the Sun and the Moon, their travels were mostly in a west-to-east direction. However, the details of their motions with respect to the fixed stars were very strange.

The ancient astronomers divided the planets into two groups. Mercury and Venus were inferior planets. These objects never stray too far from the Sun: Venus is never seen more than 46 degrees from the Sun, while Mercury's greatest elongation is 23 degrees. These planets move very quickly in the sky, sometimes moving from one constellation to another in a matter of days. Mars, Jupiter, and Saturn were called superior planets; they moved through the constellations at a slower rate (Mars the fastest, Saturn the slowest), and, more importantly, they could be anywhere (on the ecliptic). The superior planets could be near the Sun, or could be very far away. We will now define two terms. When a planet is in opposition, it is located on the opposite side of the sky from the Sun. (This is like the Full Moon.) When a planet is in conjuction, it lies in the same direction in the sky as the Sun. (This is like the New Moon.) Note that these are geocentric terms: they refer to the position of the planet on the sky (with respect to the Sun) as seen from earth.

As stated above, all the planets move mostly west-to-east in the sky. However, when a superior planet is near opposition, it slows down, stops, and actually reverse itself. The planet undergoes retrograde motion, and travels east to west for a short while. Afterward, the planet straightens itself out, and begins its west-to-east motion again. Inferior planets are also observed to (sometimes) undergo retrograde motion, but for them, it (sometimes) occurrs shortly after their point of greatest elongation.

What could cause a planet to slam on the brakes, go backward, and then continue forward again? Aristotle in around 350 B.C. had an ingenious solution. He hypothesized that planets don't simply circle the earth. Instead he hypothesized that the planets moved at a constant speed around small circles, called epicycles, and these small circles then circled the earth at a constant speed along a big circle called a deferent. The combination of the motion of the planet around the epicycle and the motion of an epicycle around the deferent could cause a planet to make a loop in the sky.

Claudius Ptolemy in 140 A.D. went one step further. In laying out the motions of the planets in his book Almagest (``All Things''), Ptolemy noted that even deferents and epicycles were not enough to reproduce the motion of the planets exactly. In order to better predict the motions of the planets (which had been recorded in Babylonian texts for several centuries), Ptolemy put the earth slightly off the center of the deferent (at a point called the eccentric), and stated that the epicyles move at a constant speed only when viewed from another point inside the deferent, called an equant. Things were getting complicated! Unfortunately, even this model wasn't enough to precisely predict where a planet would be at a given day/month/year of time.

Ptolemy's version of the solar system (indeed, the universe) stayed gospel until Copernicus in 1530. In the book De Revolutionibus Copernicus created a mathematical model which explained the motions of the planets by putting the Sun in the center of the solar system, and having the earth (and the rest of the planets) go around the Sun. In his heliocentric model, retrograde motion was a natural consequence of the earth passing a superior planet in space. In other words, much like a car passing another car on the highway, a planet would appear to move backwards, as a consequence of the two planet's relative motion. Copernicus was given a very hard time by the church for this revolutionary idea, and the book was suppressed for 13 years. (When the book was finally published, Copernicus was on his deathbead, and the manuscript carried a disclaimer stating that the model was not a representation of reality.) Nevertheless, the publication of De Revolutionibus was an important moment in the history of science. Unfortunately, the book had a serious flaw: when it came to predicting the positions of planets, Copernicus' heliocentric model was no better than the model of Ptolemy.

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.