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JOHANNES KEPLER

(Jupiter Changer)

Johannes Kepler (1571-1630)

Johannes Kepler was born on December 27, 1571, in Weil der Stadt in Swabia, a wine region in south west Germany not far from France. His paternal grandfather, Sebald Kepler, was a respected craftsman who served as mayor of the city; his maternal grandfather Melchior Guldenmann, was an innkeeper and mayor of the nearby village of Eltingen. His father, Heinrich Kepler, was "an immoral, rough and quarrelsome soldier," according to Kepler, and he described his mother in similar unflattering terms.





As a 7-month-old child (Kepler was sickly from birth) he contracted smallpox. His vision was severely defective, and he had various other illnesses throughout childhood, some of which may have been hypochondria.

From 1547 to 1576 Johannes lived with his grandparents; in 1576 his parents moved to nearby Leonberg, where Johannes entered the Latin school. In 1584 he entered the Protestant seminary at Adelberg, and in 1589 he began his university education and the Protestant university of Tubingen. At Tubingen he studied mainly theology and philosophy, but also mathematics and astronomy. At the university, Kepler's exceptional intellectual abilities became apparent. Kepler's teacher in mathematical subjects was Michael Maestlin, whom Kepler admired greatly. Maestlin was one of the earliest astronomers to believe in Copernicus's heliocentric theory, the theory that the sun, not the earth, was the center of the universe and that all planets, including the earth, orbited it, although he did not teach it to his students because Martin Luther denounced it and he would lose his job if he did. He was forced to teach the Ptolemaic system, or the geocentric theory which said that the earth was the center of the universe and that everything orbited the earth. This was named after Ptolemy, the man who first arrived at this conclusion. Maestlin, however, was able to influence some of his students to subscribe to the Copernican system, among them was Kepler.

After graduation, Kepler was offered a professorship of astronomy in faraway Graz (in the Austrian province in Styria), where he went in 1594. One of his duties of this professorship was to make astrological predictions. Despite earlier failures at predictions, he predicted a cold winter, and an invasion by the Turks. Both predictions turned out to be correct, he was treated with new respect, and his salary was raised.

While lecturing to his math class in Graz, contemplating some geometric figure involving concentric (having a common center) circles and triangles on the blackboard, Kepler suddenly realized that the figures of the type shown (Illus. 3-1) determined a definite fixed ratio between the sizes of the two circles, provided the triangle has all sides equal, and a different ratio of sizes will occur for a square between the two circles, another for a regular (having all sides equal) pentagon, and so on.

He thought this might be the key to the Solar System. He truly believed in the Copernican system, so he saw the planetary orbits as six concentric circles (only six planets had been discovered then), meaning the planets revolve around the sun and have the sun as their common center but have different radii, or distances to the sun. Disappointingly, he found it just didn't work out---the ratios were wrong. Then he had real inspiration. The universe was really three-dimensional, and instead of thinking of circles, he should be thinking about spheres, with the planetary orbits being along the equators. Thinking in three dimensions, the analogue of the above diagram (Illus. 3-1) would be two concentric spheres with a tetrahedron, or a pyramid shaped four planed figure, between them, so that the outer sphere passes through the vertices, or points, of the tetrahedron, and the inner sphere touches all its sides, but is completely contained in the tetrahedron. There were just six planets, so five spaces between spheres, and there are just five regular solids. Thus, if the distances came out right, the theory provided a complete explanation in terms of a geometric model of why there were just six planets, and why they are spaced as we find them. Actually, the distances still didn't come out right, especially for Jupiter, but Kepler was so sure of the rightness of his work, that he blamed the discrepancies on errors in Copernicus' tables. He titled his work Mysterium Cosmographicum--the Mystery of the Universe (explained). The crucial illustration from his book (also his model) is shown (Illus. 4-1), the outer sphere being the orbit of Saturn.

Except for Mercury, Kepler's construction produced remarkably accurate results. Because of his talent as a mathematician, displayed in his work and his book, Kepler was invited by the great Tycho Brahe to Prague to become his assistant and calculate new orbits from Tycho's observations. Kepler moved to Prague in 1600.

Kepler and Brahe did not get along well. Brahe apparently mistrusted Kepler, fearing that his bright young assistant might eclipse him as the prominent astronomer of his day. He therefore only let Kepler see part of his numerous data.

He set Kepler to the task of understanding the orbit of the planet Mars, which was particularly troublesome. It is believed that part of the reason for giving the Mars problem to Kepler was that it was difficult, and Brahe hoped it would occupy Kepler while Brahe worked on his theory of the Solar System. Ironically, it was precisely the Martian data that allowed Kepler to formulate the correct laws of planetary motion, thus eventually achieving a place in the development of astronomy far surpassing that of Brahe. When Brahe died in 1601, Kepler stole the data Brahe had been keeping from him, and began to work with it.

Once Kepler had secured Tycho's data, he set himself to the task of once and for all determining the exact orbit of Mars. A preliminary analysis showed the orbit to be very close to a circle, a radius about 142 million miles, but the sun was not at the center of the circle---it was at a point 13 million miles away from the center. Also, it was clear that Mars varied in speed as it went around this orbit, moving fastest when it was closest to the sun (at perihelion) and slowest when it was furthest from the sun (aphelion). Everybody (including Kepler) believed that the motion of planets must be a simple steady motion, or at least made up of simple steady motions, if only it were looked at in the right way. They wondered how the motion of Mars described above could be seen as some kind of steady motion.

Actually, a possible solution to this problem had been given long before by Ptolemy. The method was to introduce another point, called the equant, on the line through the sun and the center of the circular orbit, the equant being on the opposite side of the center from the sun.

The idea is to try to position this point so that the planet moves around the equant at a steady angular speed. This steady motion about the equant is somewhat believable, because the planet is observed to be moving slowest when it is furthest from the sun, which is when it is closest to the equant, and vice versa, so if you imagine a spoke going out from the equant point to the planet and sweeping around with the planet, maybe this spoke could be turning at a steady rate.

Ptolemy had shown that observations of the movement of Mars in its orbit were in fact well accounted for by a model of this sort, with the equant point the same distance from the center as the sun, but on the other side (as in Illus. 5-1). Kepler, however, had far more accurate records of the movement of Mars, and he was interested in seeing if the model still held up under this closer scrutinizing. He found it didn't. Even adjusting independently the radius of the orbit, the distance of the sun from the center, and the distance of the equant from the center, he found the best possible orbit of this type was still in error by eight minutes of arc (8/60 of a degree) in accounting for the observations. Such an error could not have been detected before Tycho's work. Kepler knew Tycho's work was accurate to about one minute, and so the model had to be thrown out.

Having thrown out the equant model, though, it was difficult to see what to do next. The natural thought for an astronomer at that time would have been to add an epicycle, that is, to imagine Mars to be going in a small circular orbit about a point which itself goes along the orbit shown above. But Kepler didn't like that approach. The whole business with cycles and epicycles was purely descriptive---trying to account for the observed planetary motions with a suitable combination of circular motions. Kepler, in contrast, was trying to think dynamically, that is, to understand the planetary motions somehow in terms of a force stemming from the sun sweeping them around in their orbits. Thinking in those terms, adding an epicycle looks unattractive--what force could be pushing the planet around the small circle, which has nothing at its center?

Kepler realized that to get the kind of precision he needed in analyzing the orbit of Mars, he first needed to have a very accurate picture of the earth's orbit, since all measurements of Mars' position were conducted from the earth. So to pin down Mars' position relative to the sun, it was necessary to know the earth's position relative to the sun to the required precision. He wondered how he could pin down the earth's position in space accurately. This is like being in a boat some distance from shore. If you can see only one landmark, such as a lighthouse, and you have both a compass and a map, that is not enough to really fix your position, because you cannot tell very accurately just how far away the lighthouse is. On the other hand, if you can see two landmarks in different directions, and measure with your compass the exact directions they lie from your boat, that is enough to fix your position exactly without guessing about distances. You just take out your map, draw lines through the two landmarks on the map in the direction your boat lies from each of them in turn, and the point where the two lines intersect on the map is your location. Essentially, this is just Thales' method used in geometry-- the two landmarks form the base of a triangle, and we know the direction of the boat from the two ends of the base, so we can construct a triangle on this base with the boat at the other vertex. Just knowing the angle between the two lines isn't enough, we have to know their individual directions relative to the base, which is what we can find with the map and the compass.

The idea was to use this same technique repeatedly to find the location of the earth, and thereby map out its orbit. The problem was, he needed two fixed lighthouses to form the base, and he only had one, the sun. The fixed stars wouldn't do, they were infinitely far away and just play the role of the compass, giving a fixed direction. Kepler solved the problem of the second lighthouse by a very clever trick. He used Mars. Of course, Mars is moving all the time, and the orbit of Mars is what he was trying to find, so this didn't seem to be a promising approach. But one thing Kepler did know is that if Mars was in a certain location at a certain time, it would be in exactly that same place 687.1 days later. Kepler was able to use Brahe's volumes of data to find the exact direction of Mars from the earth at a whole series of times at 687.1 day intervals. By finding the direction of Mars and that of the sun at those times, he had a steady Mars-sun base to use in constructing the earth's orbit.

In contrast to the orbit of Mars, Kepler found the earth's orbit to be essentially a perfect circle. (It is actually off by about one part in 10,000.) However, the center of the circle is about 1.5 million miles away from the sun, and the speed of the earth in its orbit varies, being greatest at the closest approach to the sun. At the furthest point, the earth is 94.5 million miles from the sun, and it is moving around its orbit at a speed of 18.2 miles per second. At the point of closest approach to the sun, the earth is 91.4 million miles from the sun, and moving around at a speed of 18.8 miles per second. Kepler noticed that there was an interesting relationship among these numbers. The ratio of speeds, 18.8/18.2= 1.03, is the inverse of the ratio of the corresponding distances, 91.4/94.5= 1/1.03. Kepler's interpretation of this was that the force he believed to be emanating from the sun, pushing the planets around, was weaker at the greater distance, and that was why the earth was being pushed more slowly.

This relationship between speeds and distances at the extreme points of the orbit enabled Kepler to develop his second law of planetary motion (described later). Along with his knowledge about the orbit of the earth, he was able to give a sufficiently precise account of the earth's position in space as a function of time to be able to go on to the main business, the plotting of the more interesting Martian orbit.

Kepler knew the orbit of Mars was not a circle. In fact, he had plotted it and found it to be an oval shape that could fit inside a circle, as shown here, and deviated from the circle by at most 0.00429 of the oval's half-breadth (half-width) MC, about one-half of one percent (see Illus. 10-1).

This means the ratio of AC/MC=1.00429. Kepler's figure was constructed directly from Tycho's data. He also measured the angle CMS. Mars subtended on the baseline consisting of the sun and the center of the orbit when the planet was in the position shown. The value of this angle was 5 18' (5 degrees 18 minutes). He stumbled entirely by chance on the fact that the ratio of lengths SM/Cm, was 1.00429.

Kepler felt that this could not just be a coincidence--there must be a similar relationship between angle SMC and the distance from the sun at all points on the orbit. He found from data that this was so, but was still unable to figure out what the curve must be. Still, he had stared at his plot long enough to believe that the curve was an ellipse with the sun at one focus, he then constructed an ellipse by a different approach. At this point it dawned on him that his original analysis also led to an ellipse.

In fact, with hindsight, it is not difficult to show how the numerical coincidence Kepler encountered follows for an ellipse. Kepler found AC/MC=1.00429=MS/MC. This meant AC=MS.

The analysis of the Martian orbit, with many wrong turns and dead ends, took Kepler six years and thousands of pages of calculations. It led to two simple laws.

 
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