An explanation for planetary movement that endows a planet with a “mind” to help steer it through the heavens does not sound to modern ears like a “physical explanation.” Kepler was not fond of the idea himself, but he could not dismiss it.
At this point in his book, Kepler decided that he had, for the moment, done all he could in the way of pursuing physical explanations and needed to return to the problem of describing the planets’ true motions mathematically and geometrically.
How much time it took a planet to travel a given distance along its orbit depended on how far it was from the Sun, so much was clear. The planet was slightly changing its distance from the Sun, and hence its speed, continually all the way around the orbit. To get a handle on these changes, Kepler needed integral calculus, but that would not be invented for at least three-quarters of a century. He nonetheless found another way.
A circle has 360 degrees, so Kepler divided the circumference of a circular orbit into 360 equal arcs. Then he laboriously proceeded to calculate the distance from the Sun (off-center in that circle) to each of these separate arcs, as if measuring the length of every spoke of a wheel that has an off-center hub. Of course he had to calculate only 180 of the 360 “spokes,” since planet-Sun distances on the other side of the orbit would be the same. Having done that, he could, for instance, imagine a planet starting at aphelion and passing through the first 30 of the 360 degrees of its orbit. The sum of those 30 orbit-to-Sun distances was to the sum of all 360 distances, as the time it took the planet to move those 30 degrees was to the time it took the planet to complete an entire orbit of 360 degrees. Even for Kepler this procedure became unendurably tedious and complicated. He decided to look for a shortcut.
Kepler recalled a method that the ancient mathematician Archimedes had used to calculate the area of a circle. Archimedes reasoned that a circle was made up of an infinite number of isosceles trianglesfn3 with their bases on the rim of the circle and their apexes at the center. Knowing he must use something more manageable than an infinite number, Archimedes, like someone who has baked a pie for a few that many show up to eat, divided the circle instead into very fine, equal isosceles triangles, again with their bases on the rim of the circle and their peaks at the center.
It went without saying that when Archimedes combined a few adjacent triangles of equal size, he was doing two things. One, he was combining their bases along the rim of the circle. The more triangles he combined, the longer the portion of the rim taken up by those bases. If he combined all of the triangles, he would have taken up the complete circle. Two, he was combining the areas of the triangles. If he doubled the number of triangles in the group, the area doubled; triple the number, and the area tripled; and so forth. Clearly there was a relationship between the amount of the rim that the combined bases covered and the area of the combined triangles. For example, for an arc twice as long, the area would be twice as big.
Kepler suggested that combining the triangles does more than combine their areas and combine the lengths their bases take up on the rim of the circle. He thought of the circle as a spoked wheel with an infinite number of spokes representing center-to-rim distances. No matter what size one made the triangles, each triangle would contain an infinite number of these spokes. Kepler, like Archimedes, could not compute with infinite numbers. However, though it was not possible to say how many spokes were contained in any one triangle or combination of triangles, it was reasonable to conclude that the more triangles you combined, the more of these spokes there were in the combined area. Even more precise than that: Combine two triangles of equal area, and the number of spokes doubles, and so forth. With this idea, Kepler had found a way to think about the relationship between orbit-to-Sun distances, the time that passed as the planet moved along its orbit, and areas within the circle. Whether this line of reasoning could be applied to an off-center orbit, where the triangles were no longer isosceles triangles, was problematic. And of course the whole point of the exercise was to illuminate the workings of an off-center orbit.
Kepler persevered, and he reached a tentative conclusion: A straight line drawn from a planet to the Sun, as the planet orbits, would sweep out equal areas of the circle in equal times. When he tried this rule with Earth’s orbit, it worked. Though he was not yet nearly confident enough about it to declare that it was correct, and never even clearly stated it in Astronomia Nova, Kepler had arrived at what has come down through history as his “area rule,” his “second law of planetary motion.” Confusingly, he did not discover his “first law” until somewhat later.
Figure 20.3: If the Sun were in the precise center of the orbit (a), the triangles would be isosceles triangles. But with the Sun off-center (b), the triangles Kepler was considering were no longer isosceles triangles.
Figure 20.4: Kepler’s area rule, shown here in its final form, as his second law of planetary motion, with an elliptical orbit. When he first arrived at the area rule, he was still trying to apply it to a circular orbit.
Imagine the planet moving around the orbit with a straight line drawn from it to the Sun. As the planet moves, so does the straight line. Watch the line move for, let us say, two minutes, then measure the area of the pie wedge it has “swept out.” For every two-minute interval, the wedge will have that same area, but it will not always be the same shape, nor will the edge of it that touches the orbit always be the same length. Near the Sun the wedge will be fat and cover a long portion of the orbital path. Far from the Sun it will be thin and cover a much shorter portion, meaning that far from the Sun the planet is moving a shorter distance in the two-minute interval.
Kepler realized immediately that his old distance rule and this new area rule were not necessarily the same. The limits of observational accuracy made it impossible to judge which was correct for Earth. He knew he must look at the orbit of another planet.
At this critical moment in October 1602, Kepler was rudely interrupted by Tengnagel’s return to Prague and his conclusion that Kepler had made no progress in the use of Tycho’s observations. It is not difficult to understand why Kepler secretly kept the Mars data, judging, correctly, that Tengnagel would not notice at least for a while.
The old “problem of Mars” now offered a splendid opportunity. Because Mars’s orbit was farther from being centered on the Sun than Earth’s orbit was, a flaw in the area rule was more likely to reveal itself. Kepler used the area rule to compute where Mars should be in its orbit at given times during the 687 days the planet takes to complete the orbit, and then he checked these predictions against the heliocentric longitudes of his Vicarious Hypothesis. He found agreement when it came to certain parts of the orbit but not others. In fact, he was back to an eight-minute discrepancy! Again, it had come to a showdown: Either the circular orbit was wrong, or the area rule was wrong. Kepler could not rule out even the possibility that both might be wrong.
Though still far from completely trusting his area rule, Kepler decided to take the plunge and try a noncircular orbit. A triangulation like the one he had used earlier for Earth’s orbit indicated that Mars’s path was indeed not a circle but bowed in at the sides. Mars was like a racer who cheats by coming within the circle of a circular racetrack. Doing that while still having to make it around two goalposts (aphelion and perihelion) would change a circular race into an oval race.
The precise amount by which Mars was “cheating” in this race was fiendishly difficult to establish. Circles, except for size, are identical. “Oval,” on the other hand, is a much less precise term. An ellipse is one kind of oval, the best defined geometrically and the one a man obsessed with harmony and symmetry in nature might be expected to assume was correct. Kepler did not. Not only did movement in an elliptical orbit appear to defy a physical explanation, but it also seemed too easy an answer. Kepler wrote to his friend David Fabricius6 that surely if the orbit were a perfect ellipse the problem he had been struggling with would have been solved long ago by Archimedes or Apollonius.
At the
time he wrote that letter, in July 1603, Kepler had been forced to abandon the struggle with Mars, because Tengnagel had finally noticed that the Mars observations were missing and confiscated them. Kepler was working on Astronomiae Pars Optica instead. It was not until a year later that he had the Mars data again.
As Kepler resumed juggling ovals, which he would continue to do for the rest of 1604 and in the beginning of 1605, his frustration grew intense. His math was inadequate. He was suspicious of his area rule. He even had some doubts whether Mars’s orbit made sense mathematically at all. An attempt he made to calculate Mars’s positions degree by degree gave him unsatisfactory results and was the sort of procedure he despised. This was not geometry, and Kepler took issue with God on the matter, in words he might have used to comment about a human colleague: “Heretofore we have not7 found such an ungeometrical conception in his other works.” Kepler had not changed in his intolerance of procedures or results that insulted his geometrical sensibilities.
Kepler resorted to working with an ellipse that he called the “approximating ellipse,” to see what he might learn from the exercise. That presented a new problem. He had earlier (as he described it) been obliged to “squeeze in” his circular orbit as though he were holding a “fat-bellied sausage” in his hand and squeezing it in the middle so that the meat was forced out into the ends. With his approximating ellipse, he had squeezed the sausage too much. The correct orbit had to be something in between.
Kepler’s desire for a physical explanation made his efforts more difficult. He had begun to think that the force resembling a magnetic force might account not only for the motion of the planets around the Sun but also for their motion toward and away from it. That was out of the question with an elliptical orbit. One of Kepler’s uses of an epicycle as a computational device had led him to have rather high hopes for the magnetic hypothesis, so he brought yet another epicycle out of storage. That resulted in a “puffy-cheeked” orbit (via buccosa). It is one of the ironies of scientific history that it was an error in his calculations that caused Kepler to reject this orbit. Kepler had reached chapter 58 when he wrote, “I was almost driven to madness8 considering and calculating this matter. I could not find out why the planet would rather go on an elliptical orbit!”
And then, “As if I were roused from a dream and saw a new light,” a torrent of answers fell into place. An elliptical orbit halfway between his approximating ellipse and a circle had a feature that was deeply satisfying to one who loved geometric harmony. The Sun was one of its foci. Kepler had arrived at his “first law of planetary motion.”
This, as Kepler put it, was “the sort of thing nature does.” With this ellipse, the orbit made physical sense, supporting his conviction that a force residing in the Sun moves the planets. What was more, if the area rule was correct, this model agreed “to the nail” with the long-trusted heliocentric longitudes of his Vicarious Hypothesis. This one shape of orbit, and only this shape, got the planet to the right place at the right time. The man who had said of himself, “There was nothing9 I could state that I could not also contradict,” had discovered a piece of incontrovertible truth.
At Easter 1605, the second Easter after the one for which he had promised his book, Kepler decided definitely on the ellipse. He finished the manuscript that year, adding a subtitle to emphasize that his “New Astronomy” was “Based on Causes, or Celestial Physics.” Kepler ended Astronomia Nova with the hope that God, having so richly endowed his creatures with analytical brains and insatiable curiosity, and his Creation with surpassing beauty and ingenuity, would allow humans sufficient time on this Earth to resolve questions he, Kepler, had not yet been able to answer.
Figure 20.5: Kepler’s first law of planetary motion: A planet moves in an elliptical orbit, and the Sun is at one focus of the ellipse.
The adventure begun on Hven when Tycho first made the decision to train his fabulous instruments on Mars had taken Kepler to a new astronomy. He had made sense of the positions of Mars spread over many pages of observational logs. In this miraculous cohesion of observations and mathematical theory, the numbers, in the words of Kepler scholar Max Caspar, “no longer stand together10 unrelated but rather each can be calculated from the other.” The limits of accuracy of Tycho’s observations had turned out to be exactly right for the task Kepler undertook: “[They were] narrow enough so that Kepler could not afford to neglect those very important eight minutes . . . but had they been considerably narrower, he would certainly have been caught in a fine meshed net, because in many of his calculations he would no longer have been permitted to overlook certain inaccuracies, as was necessary for the progress of his research.” Nevertheless, the precision of Tycho’s observations made it possible for Kepler to find the elliptical orbit of Mars even though it is so near to being a circle that any drawing of it on the page (such as the ones in this chapter) that makes it look even slightly elliptical is a gross exaggeration.
Figure 20.6: Comparing the eccentric circle, the true ellipse, and the approximating ellipse. F designates the two foci of the approximating ellipse. E and the Sun are the foci of the true ellipse. The drawing greatly exaggerates the eccentricity of the ellipses. Correctly drawn at this scale, they would be impossible to distinguish from one another or from a circle.
Kepler’s discoveries that Earth behaves like a planet, and of his first and second laws of planetary motion, were towering landmarks in human intellectual and scientific history. He had indeed plumbed the depths of a complicated universe and found harmony. He had also given Tycho Brahe his earthly immortality.
fn1 Eight minutes of arc is the equivalent of a little less than the thickness of a penny held at arm’s length and viewed edgewise.
fn2 In order to appreciate fully the manner in which Kepler used Tycho’s observations, it would be necessary to follow him in far greater detail than is possible here. Appendix 3 describes in a much simplified fashion just this one short phase of the work to give a flavor of the interplay between Tycho’s data and Kepler’s use of it. The process did not “draw Earth’s orbit” for Kepler though it might seem it could have. It did not lead him to conclude that the orbit is elliptical. That would come later.
fn3 An isosceles triangle is a triangle having two sides of equal length. (See figure 20.3[a], though it does not show an infinite number of triangles.)
21
THE WHEEL OF FORTUNE CREAKS AROUND
1606–1618
THE AGREEMENT STRUCK earlier between Kepler and Tengnagel required Kepler to submit the manuscript of Astronomia Nova to Tengnagel for approval. Tengnagel was not pleased: The book clearly argued for the Copernican rather than the Tychonic system. The difficulty was settled when Kepler agreed to allow Tengnagel to write a preface. Hence Astronomia Nova, like Copernicus’s De Revolutionibus, begins with an unpromising warning, in this case that readers should “not be swayed1 by anything of Kepler’s, especially his liberty in disagreeing with Brahe in physical arguments.” Both Tengnagel and Osiander (in the case of De Revolutionibus) emerge looking foolish in prefaces to two of the most significant astronomy books in history.
The publication of Kepler’s book moved at a snail’s pace. The printing didn’t even begin until 1608. It was Rudolph’s right to distribute all copies of a book by his imperial mathematician, but when it appeared in the summer of 1609, Kepler had to give the entire edition back to the printer in Heidelberg to sell to cover unpaid costs.
January 1610 marked ten years since Kepler had first arrived in Prague in Hoffmann’s carriage. There was abundant reason to celebrate the anniversary. Kepler’s reputation as scientific heir to Tycho Brahe, and the books Kepler had written, had elevated him from the status of an impoverished provincial mathematics teacher to that of a celebrated figure in educated circles all over Europe and in Britain. This success redounded to the emperor’s credit as well. Rudolph lavished praise on Kepler and granted him a bonus of two thousand talers, which would have been splendid had it been paid.
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br /> Sadly, Kepler’s decade of superb scholarly achievement, warm friendships, and almost universal respect was marred not only by his problems collecting his salary, but also by the decline of both his wife and his patron. Barbara had made some friends among the more pious Protestant women of Prague, but after a decade in the city she was still miserably homesick. In spite of her discontent, the Keplers had seldom been out of Prague at all during the years since Tycho’s death, except for a sojourn in Moravia when the plague came back in 1606. In the autumn of 1607 they had moved to new lodgings near the great bridge over the river. Ludwig, their second son, was born there that December. In 1608 Regina, Barbara’s daughter and Kepler’s stepdaughter, married Philip Ehem, who was descended from a prominent Augsburg family and was currently a representative at the imperial court for Elector Frederick IV of the Palatinate. There were many reasons for happiness and pride, yet Barbara suffered chronic bad health and deepening depression.
As for the emperor, by the time he granted Kepler’s celebratory bonus, Rudolph had been stripped of almost all his power. Oddly enough, his habitual state of political indecision and inaction had stood him in rather good stead for many years. He had kept up an endless, stalemated war with the Ottoman Turks and held together, often only by failure to act or react, an empire forever threatening to disintegrate. Rudolph had always been quirky and pathologically shy, but over time his indecision and stubbornness had degenerated to paralysis, and his mental state had continued to deteriorate until he was rumored to be insane. Though he was such a recluse that it was difficult to confirm or deny the rumor, he had clearly become a threat to the royal house of Hapsburg and the empire.
Tycho and Kepler Page 31