Tycho and Kepler

by Kitty Ferguson

  In the introduction to the book, Kepler spoke of another discovery he had made: the inverse square law of light. If a burning candle is set on a table, the lighted area surrounding it on the table is a circle, with the candle in the center of the circle. Kepler, thinking in three dimensions rather than two, reasoned that light, starting from one point in space (the candle flame), spreads out not just in a circle but in all directions, in the form of a sphere. Wherever you are, within reach of the light, you can think of yourself as being at the edge of a sphere centered on the light source. Someone a little farther away can also imagine himself or herself being at the edge of a sphere centered on the light source. At that second location, the sphere is bigger, and the light looks dimmer. How much dimmer? was the question. Kepler reasoned that the light’s brightness was related to the size (the area) of the sphere. If two observers were both looking at the light, and observer B was twice as far away as observer A, then observer B’s sphere was four times as large as observer A’s sphere. B saw the light only a fourth as bright as A did. If B was three times as far away as A, B’s sphere was nine times as large as A’s sphere, and B saw the light only a ninth as bright as A did. As Kepler’s inverse square law of light states, the intensity of light is inversely proportional to the square of the distance. The square of two (two times as far away from the light source) is four. The square of three (three times as far from the light source) is nine, and so forth.

  In January 1604, a little more than a year after the Christmas for which he had promised it, Kepler presented the completed manuscript of Astronomiae Pars Optica10 to the emperor, and the book went into publication. The ideas and discoveries about light and optics that Kepler wrote about in this book and later applied to the telescope in another work, Dioptrice, became the foundation for seventeenth-century optical theory.

  After their clash in the autumn of 1602, Kepler and Tengnagel were often at loggerheads, but they managed to make some joint progress on the completion and publication of Tycho’s posthumous works. There had been an extremely uncomfortable moment in the spring of 1603 when Tengnagel, turning his attention to the Rudolfine Tables, had discovered that the Mars observations were missing. Kepler reluctantly surrendered them. He was at that time writing Astronomiae Pars Optica, and the removal of the possibility of working on Mars may have caused him to go as deeply as he did into that other subject.

  Tengnagel’s most impressive talents lay in politics and diplomacy, which Tycho had recognized and put to use when courting favor with the royalty of Europe. Hence more and more of Tengnagel’s time was taken up with Hapsburg politics and the deteriorating political situation in Bohemia and Germany. In the summer of 1604 he conceded that he could not possibly complete the Rudolfine Tables by himself. In return for a promise to complete them in a manner satisfactory to Tengnagel and to seek his approval before publishing anything based on the manuscripts, Tengnagel allowed Kepler to use some of Tycho’s observational journals.

  The precious Mars observations were once more in Kepler’s hands. However, promises to Tengnagel notwithstanding, he did not set to work on the Rudolfine Tables. He was again engrossed in his study of Mars’s orbit. His book about that, promised for Easter 1603, was well behind schedule. It would, in fact, not be ready to go to print until late in 1605, and it would have a new and highly appropriate title, Astronomia Nova—New Astronomy.

  In the autumn of 1604 the Keplers, with Barbara six months pregnant, moved house again, this time to Wenzel College in the Old Town, nearer the palace but still across the river. One of Kepler’s dearest friends, Martin Bachazek, rector of the University of Prague, lived there. Kepler relished the opportunity to converse with him daily.

  That same autumn, not long after the move and while Kepler was wrestling with one of the most difficult problems in his book about Mars, a celestial event occurred that left him no choice but to abandon his writing desk and his columns of calculations. The event began inauspiciously when a court official in a state of great agitation roused the family at dawn on October 11. When Kepler could make sense of the man, he learned that on the previous evening he had seen a brilliant new star through a gap in the clouds. Kepler was skeptical. Six days of overcast skies followed. He had almost forgotten the incident when on the evening of October 17 the sky cleared. He saw the star himself and realized that the messenger’s excitement had been justified. As bright as Jupiter, sparkling like a diamond in all the colors of the rainbow, this nova appeared in the sky near Saturn and Jupiter, which were near conjunction. Mars was also close by. Tycho had had his “star,” and now Kepler had his.

  As Kepler’s fateful drawing for his class in Graz had demonstrated, Jupiter and Saturn come into conjunction every twenty years. The regular pattern in which the conjunctions occur means that ten conjunctions happen within each of four areas of the zodiac. Astrologers associate the areas with the four elements identified by Aristotle: fire, water, earth, air (see figure 12.1). It takes eight hundred years for the conjunctions to pass through all four areas, known as “trigons.” The conjunction of Jupiter and Saturn at the time of the appearance of “Kepler’s star” marked the beginning of the two-hundred-year period in which the conjunctions would occur in the trigon associated with the element fire, the “fiery trigon.” Any conjunction was considered to have important effects on human events, mostly bad; a conjunction in the fiery trigon presaged even greater calamity.

  In view of the astrological implications of such a conjunction and a new star at the same time, Rudolph would not rest until he and all the other nervous citizens of the empire could be informed by the imperial mathematician what they should make of this wonder. Bachazek built a small wooden tower so that Kepler and he could see the star better, and Kepler almost immediately produced a delightful short report to reassure the emperor and the populace of Prague. Among other possibilities, Kepler predicted (with tongue in cheek) good sales for booksellers, because every theologian, philosopher, physician, mathematician, and scholar would want to publish his own ideas on the matter.

  To help fulfill that prediction, two years later Kepler himself dedicated a book about the nova to the emperor, based on research and continual observation as it faded. De Stella Nova’s subtitle was A Book Full11 of Astronomical, Physical, Metaphysical, Meteorological and Astrological Discussions, Glorious and Unusual. There was a widely held opinion that the planets had ignited the nova. Kepler insisted it was much farther away than the planets, at the distance of the fixed stars, and he made a good case (based on erroneous data) that the fixed stars were not suns. He also rejected the suggestion that a group of atoms had come together by pure chance to form a new star. That, he wrote, was like thinking that “if a pewter dish,12 leaves of lettuce, grains of salt, drops of water, vinegar, oil and slices of egg had been flying around in the air for all eternity, it might at last happen by chance that a salad would result.” He even had mentioned the matter to his wife as she set a salad on the table before him. As he wrote: “‘Yes,’ responded my dear, ‘but not so nice as this one of mine.’”

  Kepler ruminated a bit about the astrological implications of the star, but he ended by telling his readers that the best advice he could give them was to examine their sins and repent. Star or no star, that could certainly do them no harm.

  Modern research shows that Kepler’s Star, like Tycho’s in 1572, was a Type 1 supernova. There have been three in our galaxy in the last thousand years. (The other was in 1006.) Kepler’s was the last supernova that would be visible to the naked eye until 1987, when one occurred in the Large Magellanic Cloud, a satellite galaxy of the Milky Way. Tycho and Kepler had no idea how extraordinarily fortunate they were each to see one. The serendipitous salad was perhaps not so much less likely after all.

  The Kepler family kept growing. In early December 1604, Barbara gave birth to a son, Friedrich, who was to be a great favorite of Kepler’s. The domestic disruption surrounding his birth caused Kepler to exclaim in a letter, “For what a business,
13 what an activity, does it not make to invite fifteen or sixteen women to visit my wife, who lies in childbed, to receive them hospitably, to see them out!” Perhaps he should have reconsidered his lament that Barbara had neither heart nor means to make herself further known in society. Sadly, as Barbara’s house became livelier with children, she herself withdrew further into melancholy.

  For Kepler the astronomer, 1604 was both a frustrating and an exhilarating year as he struggled daily to solve the riddle of Mars’s orbit. He came to think of it as a war with Mars, by ancient tradition the most warlike of the planets. By the time Friedrich was born, Kepler still wasn’t sure whether he was nearing success, whether victory might still be several years off, or whether the orbit of Mars was perhaps not mathematically describable at all. Astronomia Nova was not a report on the results of completed research. Kepler had written fifty-eight chapters of the book in almost final form before he discovered that Mars’s orbit is elliptical.

  Tycho’s records contained plenty of data on Mars, for the “problem of Mars” had caused him over a long period of time to make that planet the focus of many observations. That “problem” was the catalyst that led Kepler to his first two laws of planetary motion.

  Already in antiquity, observations of Mars had made it clear that the speed of the planet does not remain uniform throughout its orbit. Astronomers in the intervening centuries had used ingenious devices to describe such irregularities in a mathematical/geometrical way. One such device was an “eccentric” orbit—an orbit not precisely centered on the center of the system (not precisely on Earth for Ptolemy; not precisely on the Sun for Copernicus). A straight line drawn through the center of the system (Earth or Sun) and the center of the eccentric orbit was called the apsidal line. Extended farther, the apsidal line passed through the point where the planet was farthest from the center of the system (at aphelion) and the point where it was closest (at perihelion) (see figure 19.1).

  Common experience indicates that objects seem to move much more slowly the farther away they are. A bird flying close overhead can easily appear to win a race with a plane far higher above it in the sky. Closer looks faster. Ptolemaic astronomers, Copernicus, and Tycho had all considered the possibility that what appears to be a variation in the speed of a planet, as viewed from the center of the system (Earth or Sun, depending on whether one followed Ptolemy or Copernicus), is only an illusion created by the fact that the planet, traveling on an eccentric orbit, is sometimes closer than at other times. But they had realized that this explanation was not sufficient to account for the extent of speeding up and slowing down revealed by observations. Ptolemy experimented with an equalizing point or “equant” that was not the center of the system (Earth in his case), and not the center of the eccentric orbit. It was a third point along the apsidal line, a point from which an observer would find that the planet appeared to be moving with constant speed.

  Figure 19.1 (Apsidal line): An off-center orbit was said to be “eccentric.” The distance between the center of the orbit and the center of the system (Earth or Sun) was known as the “eccentricity” of the orbit. A straight line drawn through both of those two points is the apsidal line. Extended farther, the apsidal line passes through the point where the orbiting planet is farthest from the center of the system (at aphelion) and the point where it is closest (at perihelion).

  The “problem of Mars” was that it was the most difficult of the three outer planets to accommodate in this manner. Mars has by far the greatest eccentricity. Tycho must have been well aware, when he decided to train the instruments of Uraniborg and Stjerneborg on Mars, that that planet provided not only the stickiest problems but also the best opportunity to define what the problems were and, one might hope, to solve them. As Kepler put it, “I consider it a divine decree14 that I came at exactly the time when Longomontanus was busy with Mars. Because assuredly either through it we arrive at the knowledge of the secrets of astronomy or else they remain forever concealed from us.”

  Although scholars had been studying the heavens for centuries, no one had discovered that Mars’s orbit was not a circle, but ancient and medieval astronomers cannot be accused of ignoring data to adhere to an erroneous assumption. The amount by which a planet’s orbit, even the orbit of Mars, departs from a circle is extremely small. The errors in the observations available prior to Tycho were at least as great as ten minutes of arc, and this margin of error—of which astronomers were well aware—made it impossible to discern that the orbit was other than circular. Tycho’s observations were trustworthy to within two or three minutes of arc, a great improvement on the ten-minute error tolerated before, but the discovery that Mars’s orbit was elliptical would not come—as one might naively suspect—from Kepler’s simply plotting a number of points where Tycho had found Mars and then failing in the attempt to draw a circle whose rim would pass through all of them. To discover the true orbit of Mars from Tycho’s observations required a level of subtlety, insight, and inventiveness from Kepler that arguably has not been surpassed in the history of science.

  The attack on the problem of Mars as it was originally assigned by Tycho to Longomontanus and Kepler was not intended to address the question whether or not Mars’s orbit was circular. It involved two basic calculations. The first was the position of Mars’s apsidal line—the straight line passing through its aphelion, equant, eccentric, Sun (for in the Tychonic system Mars orbited the Sun), and perihelion. The second was the extent of Mars’s eccentricity (how far the center of Mars’s orbit was from the Sun). If Ptolemy was right about the position of the equant in relation to the center of the system and the eccentric, they could expect to find that Mars’s equant was twice as far from the Sun as the eccentric was.

  From the time Kepler first began working with Tycho’s data at Benatky, he chose to let the tight constraints of mathematical/geometrical logic and precise observations be his primary guides and to give them, for a while, precedence over the ideals of symmetry and harmony. However, he had by no means abandoned those ideals. He would continue to measure his theories against them and be uncomfortable when his results did not survive the test. Kepler was setting a precedent still followed in science, where symmetry, harmony, and logical beauty are not the most important criteria for judging whether a theory is correct, but where there is suspicion if those hallmarks are absent. Kepler had also not given up on his former theories: He hoped with Tycho’s observations to be able to find out whether his polyhedral and harmonic theories were correct.

  In other ways as well, Kepler did not begin his assault on the orbit of Mars with his mind a tabula rasa. Though he described his efforts as a voyage of exploration, he, like most explorers, knew the direction he thought he was heading, if not exactly what he would find there. He had already come to believe that understanding planetary motion required knowing the physical explanation for the motion, and he had already reached the conclusion that an essential part of the physical explanation was a force residing in the Sun that caused the planets to move in their orbits.

  Kepler did not think that mathematical rigor, ideals of symmetry and harmony, and the search for a physical explanation were incompatible. He dove into a body of data that he trusted implicitly, though it was not his own, placing his bets that if his math was good enough and his instincts correct, he would come out the other side with his convictions about a physical explanation confirmed, and also clutching the trophies of symmetry and harmony. No one had traveled this particular route before. Kepler was not merely using science to find answers; he was working out what “science” was and would be, for himself and future generations.

  Unlike Galileo when he wrote his famous Dialogo, Kepler did not intend the book he first referred to as “Commentaries on the Theory of Mars,” which later evolved into Astronomia Nova, for the popular market. His target audience were his fellow early-seventeenth-century astronomers, men well versed in mathematics and planetary astronomy, including Copernicus’s astronomy, although most of the
m did not think Copernicus had been suggesting anything “real” when he put the Sun in the center. Kepler realized that a battle to discredit the ancient models of astronomy had best take place on familiar ground, with familiar weapons, and not look like a battle. Hence his book would have to spend some time meandering benignly through intellectual landscapes where his contemporaries felt comfortable, not to mention where they were capable of recognizing and coming to trust his own knowledge and skill.

  Kepler’s mind was well suited to this kind of discourse. At age twenty-six he had written, “There was nothing I could state15 that I could not also contradict.” He had no problem setting more traditional models in a fair and serious manner against Tycho’s data as a way of discovering for himself and leading his readers to see that no astronomer, even with the utmost mathematical and geometrical maneuvering, could rest his case with these theories. He hoped to convince his readers that a theory had to be able to survive the commonsense questions of what might actually be going on in the heavens among these huge, real bodies . . . and why. Thus Kepler thought he would set the stage for his new astronomy, and it would prove to be accurate to the limits of Tycho’s observations. At the outset, a confident Kepler who had promised to finish this book by Easter of 1602 had no way of knowing how long it would take him, how doubtful its outcome, how much ingenuity it would require of him, how many times he would fail, and how new his new astronomy would have to be.

  To understand Kepler’s achievement in writing Astronomia Nova, one must bear in mind that for most of Kepler’s contemporaries there was no reason to wonder what the orbit of Earth was like. In the Ptolemaic system and even in the Tychonic system, Earth had no orbit. It sat still. Copernicus had implied that Earth orbited the Sun like the other planets, but he had not carried through with this in his mathematical analysis. It was a matter of extreme interest to Kepler whether or not Earth did indeed orbit like the other planets, speeding up when closer to the Sun (at perihelion) and slowing down when farther away (at aphelion). He believed it must and that if he used the true, visible Sun as his reference point he would find he was right. It had been a significant move when, at Benatky in the spring of 1600, he asked for and obtained Tycho’s permission to use the true Sun when calculating Mars’s orbit. He again used the true Sun when he determined that Earth does move more quickly when it is closer to the Sun and more slowly when it is farther away. Earth, at least in this regard, is nothing unique. It is just a planet. Kepler would spend many pages persuading his readers that it was better to use the true Sun. A physical explanation demanded it.