Such an approach also made it ridiculous to use devices consisting of invisible circles centered on invisible points on invisible circles on invisible points. However, it was not only to keep his readers with him that Kepler continued to use traditional Ptolemaic tools such as epicycles and equants, for Kepler needed all the mathematical and geometrical help they could provide to find his way to an astronomy that could do without them.
While Kepler had a superb mathematical mind, and improved his skills continually as he wrote Astronomia Nova, much of modern mathematics, including calculus, had not been invented yet. He also lacked the concept of inertia, though his contemporary Galileo understood it. Kepler did not visualize a universe in which an object keeps moving in a straight line at the same speed unless something comes along to affect that movement. Instead he thought an object sat still unless something moved it, and if that something ceased to move it, it reverted to stillness. In looking for the causes of motion he had to ask not only, “Why does it move as it does?” but also, “Why does it move at all, and why does it keep moving?”
Perhaps most significant of all, as Kepler set out on his quest, he was, as he put it, “armed with incredulity”—and16 hence ready to question all assumptions from the past as well as the theories and discoveries he made himself along the way.
Kepler’s work with the Mars data had begun at Benatky in the unhappy spring of 1600. If the hand of God was in this enterprise, as Kepler believed—and Tycho Brahe would not have disagreed—then God’s work on it had begun considerably earlier, and Astronomia Nova, which would be Kepler’s masterpiece, was the result toward which divine purpose had been moving Tycho and Kepler with the most intricate and unlikely maneuvering for at least fifty years.
No one could possibly have obeyed Tycho’s dying plea—“Let me not seem to have lived in vain”—more magnificently than Kepler did, but as Tycho had feared, he did it his own way, not the way that Tycho had intended, and history would celebrate the Copernican revolution, not the Tychonic revolution.
20
ASTRONOMIA NOVA
1600–1605
AFTER BEGINNING ASTRONOMIA Nova with a strong demonstration of how important the role of the Sun is, something he intended to hammer home even more emphatically later, Kepler turned his and his readers’ attention immediately to Ptolemy. His strategy was to improve and generalize Ptolemy’s theories as Ptolemy himself might have done had he had Tycho’s data—an exercise of which no Ptolemaic astronomer could disapprove. Kepler appropriately titled this section of his book “In Imitation of the Ancients.”
Kepler felt obliged to preface his “imitation” with six chapters describing and justifying the rigorous reworking he had given Tycho’s observations in order to use them effectively. One problem was the necessity of undoing some choices and corrections made by Tycho and his assistants. For example, Kepler made extensive use of observations of Mars at opposition. Opposition is normally loosely defined as being when a planet is on the opposite side of Earth from the Sun. However, at opposition, Mars is rarely directly opposite the Sun, because of its latitude north or south of the ecliptic (review figure 7.6). Tycho and his assistants knew they had to compensate for this, but they had failed to do so consistently, nor was it always clear in the logs whether they had or had not.
Figure 20.1: Ptolemaic astronomers placed the center of a planet’s eccentric orbit precisely halfway between its equant and the Earth (left). Kepler wanted to find out for himself where the center of Mars’s orbit lay in relation to its equant and the Sun (right).
In imitating the ancients, Kepler began by pointedly choosing not to imitate them in one important detail: Ptolemaic astronomers placed the center of a planet’s circular eccentric orbit precisely halfway between the equant point and Earth. Kepler wished to make no such assumption but to discover for himself where Mars’s orbital center lay. His work on this model began when Tycho was still alive, and it continued in the months after Tycho’s death. Kepler was still thinking in terms of a circular orbit.
To develop his model, Kepler chose observations of Mars that Tycho had made when Mars was at opposition. At this time Mars, Earth, and the Sun were in line, and an earthly astronomer saw Mars in approximately the position it would appear at that moment were he or she standing on the Sun. There were ten oppositions of Mars in Tycho’s log, the first in 1580. Hoping to measure Mars’s parallax, he had made observations with extreme care. Kepler would later make two more himself. Determining the position and time an opposition occurred, with the precision Kepler required, was no easy matter. It was a considerable achievement, requiring a great deal of skill and understanding, to deduce this information, for it could not be found directly. No astronomer was able to see Mars and the Sun at the same time during opposition, when the two bodies are on opposite sides of Earth, nor was it possible to see the background zodiac stars behind the Sun, for the Sun is too bright.
Though the calculations involved in developing Kepler’s new model were long and difficult, the family budget did not allow him to hire a permanent assistant to share the mathematical drudgery. “If you are wearied1 by this tedious procedure,” Kepler begged his readers, “take pity on me who carried out at least seventy trials.” Finally he was able to make a simple model that agreed with four of Tycho’s observations of Mars at opposition. From this model he could calculate, for any given time, where Mars would be seen from the Sun—its “heliocentric longitudes.” Kepler checked the theory against the remaining six observations from Tycho, and later against the two of his own, and he found his model agreed within the limits of those observations.
However, Kepler declared the model unsatisfactory. It was true that if one were standing on the Sun, one would see Mars in the positions predicted by his theory. But there was more to finding the correct location of a planet than knowing its position against the background stars, as seen from the Sun. Kepler wanted to know how far the planet was from the Sun at these positions.
In answering this question, Kepler discovered a serious discrepancy. His new theory indicated that the center of Mars’s orbit was six-tenths of the way from the Sun to the equant point rather than halfway between, as Ptolemaic astronomers assumed. However, to obtain the correct distances, he had no choice but to put the center of the orbit right back where Ptolemaic astronomy had traditionally put it. And when he did that, his theory no longer predicted correct heliocentric longitudes (positions of the planet as seen from the Sun). The errors were as large as eight minutes of arc.fn1 Kepler’s faith in Tycho’s observations did not allow him to let this pass.
The failure was not a defeat. In fact, Kepler had his readers where he wanted them, forced to admit that something new was required: “After divine goodness2 had given us, in Tycho Brahe, so careful an observer that from his observations the error of calculation amounting to eight minutes betrayed itself, it is appropriate that we recognize and utilize in a thankful manner this good deed of God’s—that is, we should take pains to search out at last the true form of the heavenly motions.” Kepler’s model, which he dubbed his “Vicarious Model” or “Vicarious Hypothesis,” had brought him to a crossroads.
Kepler then told his readers that a “renovation of the whole of astronomy” must begin at home. Suppose Tycho’s observations—indeed all observations—were (as Copernicus had suggested) made from a moving Earth. It behooved astronomers to be sure that their picture of this motion was correct. A flaw in that understanding would cause errors in any other astronomical work. In that interest, Kepler now changed directions and asked his readers to look toward Earth as though they were standing on Mars. With a brilliantly conceived triangulation from Earth’s orbit to Mars,fn2 Kepler demonstrated that Earth’s orbit and motion were like that of the other planets. Though Ptolemy, Copernicus, and Tycho had all thought that the center of Earth’s orbit was the same as its equant point, Kepler’s results showed that it lay instead somewhere in the middle between the equant point and the Sun, and that was wher
e astronomers had traditionally put the center of the orbit of a planet. Even more significantly, Kepler had found that Earth was speeding up when it came closer to the Sun and slowing down as it moved away. In other words, it was moving like a planet. The discovery that Earth behaves like a planet was a truly momentous advance and a strong argument on behalf of Copernican astronomy. In his book, Kepler had cleverly introduced his readers to that argument before they had a suspicion of where they were headed.
Kepler saw that the speeding up and slowing down had a predictable mathematical regularity to it. The speed of Earth at aphelion and perihelion was inversely proportional to its distance from the Sun. He decided that this rule surely had to apply not only at aphelion and perihelion but to the entire orbit. Kepler had arrived at his so-called distance rule.
Whether or not this tentative “rule” would turn out to be correct, Kepler had clearly, in getting there, become a virtuoso in the use of Tycho’s observations, devising ingenious ways to exploit their unique accuracy and comprehensiveness, bringing together sets of observations so that the whole amounted to much more than the sum of the parts, honing his mathematical skills and his creativity against the constraints of this precise data. Such mastery of the creative nexus between observation and theory has seldom been achieved and never surpassed in all the history of science. Tycho, had he been alive and able to see beyond his bias for the Tychonic system, might well have cheered for joy, for Kepler was asking questions that no one had thought to ask before, and still the observations required to answer them were right there in Tycho’s log.
Kepler did not turn directly to the question of whether his distance rule was correct, for he was determined to prise open a door into a new era of science where the search for physical explanations was of paramount importance. He was convinced that the true motion of the planets would elude him and all other astronomers until they knew the answer to the question of what was causing that motion. So he chose at this juncture to shift his focus to the search for physical explanations, thinking of this as by far the most urgent part of his work.
Kepler was wrong to believe that understanding the physical explanation for planetary motion had to come before knowledge of what that motion was. His discoveries of his three laws of planetary motion would precede by about three-quarters of a century Newton’s discovery of the physics that lay behind them. However, it seems that Kepler’s attempts to discover and understand the physical explanations, though often futile, were a necessary step in the process through which he discovered his laws.
One immediate result of Kepler’s thinking along physical lines was that it pointed up how ridiculous and unphysical previous descriptions of planetary motion were. He reasoned that since changes in a planet’s distance from the Sun appeared to dictate changes in its speed, the cause of the motion must be in one of the two bodies. Though he had already made up his own mind on that score, in his book Kepler paused to consider that idea in the contexts of the different planetary models. In the Ptolemaic system, for example, if the force that moved a planet in a circle resided in a body at the center of the circle, then it was difficult to conceive how a planet could possibly move in a circle with no body at its center—an epicycle, for instance. It was even worse if the planet must change its speed as it circled in the epicycle. It had clearly become impossible to take Ptolemy seriously, and Kepler sent that ancient genius bumping off on a trick unicycle with the wheel attached off-center to nothing at all. “The Sun will melt3 all this Ptolemaic machinery like butter,” wrote Kepler, “and the followers of Ptolemy will disperse, partly into Copernicus’s camp, partly into the camp of Brahe.” Kepler carted the epicycle off the stage, but he didn’t throw it away.
The Tychonic system fared hardly better. The idea that a planet-moving force residing in the Sun caused the planets to orbit worked fine for the five planets that Tycho’s model had orbiting the Sun, but when the Tychonic model required the Sun, in turn, to orbit around a stationary Earth, the arrangement floundered unless there was a separate force in the Earth to move the Sun, a force that did not affect the other planets. The Tychonic system was geometrically equivalent to the Copernican system, but Kepler saw that it was no match in terms of the possibility of a physical explanation. It was simply, distressingly, unlikely. Kepler might have imagined Tycho whispering urgently in his ear that the Moon circles Earth, not the Sun, and this posed a parallel problem. But Kepler put that off until another time and another book.
Kepler considered what the planet-moving force might be. It had to emanate through space in the way he had discovered light does. The strength of gravity, like the brightness of light, does fall off as the inverse square of distance, but Kepler did not discover that it does, and hence failed to arrive at the modern concept of gravity, though he came so close as to state, “If one would place a stone4 behind the Earth and would assume that both are free from any other motion, then not only would the stone hurry to the Earth but the Earth would hurry to the stone; they would divide the space lying between in inverse proportion to their weights.” In spite of his comparison with light, he decided that the strength of the force felt by the planets fell off as the simple inverse of distance.
Kepler felt obliged to justify that conclusion for the obvious reason that when he studied the relative speeds of the planets, and the relationship for a single planet between its speeds in various parts of its orbit, he found that the planets’ speeds did reflect a simple inverse relationship to distance from the Sun. There was no empirical evidence for an inverse square law. Gravity works in a more indirect way, so that the inverse square law, though it is correct, shows up only indirectly in planetary distances and speeds.
Kepler speculated that the Sun must rotate. He asked his readers to think of a lecturer surrounded on all sides by an audience. Those in the direction he is facing “see his eyes” while5 others “lack the aspect of his eyes.” If he turns, his head turns, and his gaze sweeps the crowd.
Figure 20.2: If the strength of the force fell off as the inverse square of distance, planet B, twice as far away from the Sun as planet A, would feel only a fourth as much of the force as planet A.
Likewise, wrote Kepler, the rotation of the Sun caused the force that moved the planets to sweep around. The planets could not be rigidly tied to the Sun by this force, as though fixed at the ends of spokes of a wheel, because they moved at different speeds. The Sun therefore had to be rotating at a speed that got ahead of them. Because the planets were “prone to rest” (recall Kepler’s lack of the concept of inertia), they lagged behind.
A rotating body that exerts a force at a distance through empty space, affecting closer things more strongly and giving out its force in the shape of a sphere, reminded Kepler of reading he had done recently about magnets, in books by Jean Taisner and the Englishman William Gilbert. Kepler never decided that the planet-moving force was magnetism, but it was encouraging that such a force was known to exist and that Gilbert had recently shown that Earth was a magnet. It was reasonable to think that the Sun could exert a similar force. Kepler’s suggestion was that as the Sun rotated, a field of magneticlike emanations coming from it also rotated, in turn moving the planets.
Having stated that the force propagates in all directions, not just along the ecliptic, Kepler was obliged to account for the fact that the planets are not spread out in all directions from the sphere of the Sun but instead all orbit near the plane of the solar equator. Kepler’s answer, using an analogy in which Earth replaced the Sun, was that if a planet were too far “north” or “south” it would be affected by the motion on the other side of the rotating globe. It would feel conflicting directions in the force that reached it, which over the “poles” would result in complete confusion. Hence Kepler felt that the planets could not help but end up orbiting only near the plane of the Sun’s equator, where they were not affected by the opposite stream of motion on the other side of the Sun.
Kepler was still left with a stubborn problem. He
had an explanation for why a planet more distant from the Sun would move more slowly than one closer, and why a planet’s speed would vary as its distance from the Sun changed. But he lacked an explanation for why a planet’s distance from the Sun should ever change at all. If the only thing at work were a planet-moving force residing in the Sun, then the planets would be carried around in circles centered on the Sun, never speeding up or slowing down. Something as yet unknown was alternately moving the planets toward and away from the Sun, to distances where the Sun’s planet-moving force was stronger or weaker. He speculated that a planet might have a mind, or some other non-mental mechanism, yet if this were so, there was still a question how that mind or mechanism would know how distant it was from the Sun. Yet a planet seemed mysteriously able to decide how far it should move away before coming back, and when it was close enough to move away again. Kepler could think of only one way in which distance from the Sun shows up clearly, and that is in the apparent size of the Sun’s disk. He conceded that there might be ways of “perceiving” that humans knew nothing about.
Tycho and Kepler Page 30
