Cosmos
Page 8
After Tycho’s death, Kepler, now the new Imperial Mathematician, managed to extract the observations from Tycho’s recalcitrant family. His conjecture that the orbits of the planets are circumscribed by the five platonic solids were no more supported by Tycho’s data than by Copernicus’. His “Cosmic Mystery” was disproved entirely by the much later discoveries of the planets Uranus, Neptune and Pluto—there are no additional platonic solids* that would determine their distances from the sun. The nested Pythagorean solids also made no allowance for the existence of the Earth’s moon, and Galileo’s discovery of the four large moons of Jupiter was also discomfiting. But far from becoming morose, Kepler wished to find additional satellites and wondered how many satellites each planet should have. He wrote to Galileo: “I immediately began to think how there could be any addition to the number of the planets without overturning my Mysterium Cosmographicum, according to which Euclid’s five regular solids do not allow more than six planets around the Sun … I am so far from disbelieving the existence of the four circumjovial planets that I long for a telescope, to anticipate you, if possible, in discovering two around Mars, as the proportion seems to require, six or eight round Saturn, and perhaps one each round Mercury and Venus.” Mars does have two small moons, and a major geological feature on the larger of them is today called the Kepler Ridge in honor of this guess. But he was entirely mistaken about Saturn, Mercury and Venus, and Jupiter has many more moons than Galileo discovered. We still do not really know why there are only nine planets, more or less, and why they have the relative distances from the Sun that they do. (See Chapter 8.)
Tycho’s observations of the apparent motion of Mars and other planets through the constellations were made over a period of many years. These data, from the last few decades before the telescope was invented, were the most accurate that had yet been obtained. Kepler worked with a passionate intensity to understand them: What real motion of the Earth and Mars about the Sun could explain, to the precision of measurement, the apparent motion of Mars in the sky, including its retrograde loops through the background constellations? Tycho had commended Mars to Kepler because its apparent motion seemed most anomalous, most difficult to reconcile with an orbit made of circles. (To the reader who might be bored by his many calculations, he later wrote: “If you are wearied by this tedious procedure, take pity on me who carried out at least seventy trials.”)
Pythagoras, in the sixth century B.C., Plato, Ptolemy and all the Christian astronomers before Kepler had assumed that the planets moved in circular paths. The circle was thought to be a “perfect” geometrical shape and the planets, placed high in the heavens, away from earthly “corruption,” were also thought to be in some mystical sense “perfect.” Galileo, Tycho and Copernicus were all commited to uniform circular planetary motion, the latter asserting that “the mind shudders” at the alternative, because “it would be unworthy to suppose such a thing in a Creation constituted in the best possible way.” So at first Kepler tried to explain the observations by imagining that the Earth and Mars moved in circular orbits about the Sun.
After three years of calculation, he believed he had found the correct values for a Martian circular orbit, which matched ten of Tycho’s observations within two minutes of arc. Now, there are 60 minutes of arc in an angular degree, and 90 degrees, a right angle, from the horizon to the zenith. So a few minutes of arc is a very small quantity to measure—especially without a telescope. It is one-fifteenth the angular diameter of the full Moon as seen from Earth. But Kepler’s replenishable ecstasy soon crumbled into gloom—because two of Tycho’s further observations were inconsistent with Kepler’s orbit, by as much as eight minutes of arc:
Divine Providence granted us such a diligent observer in Tycho Brahe that his observations convicted this … calculation of an error of eight minutes; it is only right that we should accept God’s gift with a grateful mind … If I had believed that we could ignore these eight minutes, I would have patched up my hypothesis accordingly. But, since it was not permissible to ignore, those eight minutes pointed the road to a complete reformation in astronomy.
The difference between a circular orbit and the true orbit could be distinguished only by precise measurement and a courageous acceptance of the facts: “The universe is stamped with the adornment of harmonic proportions, but harmonies must accommodate experience.” Kepler was shaken at being compelled to abandon a circular orbit and to question his faith in the Divine Geometer. Having cleared the stable of astronomy of circles and spirals, he was left, he said, with “only a single cartful of dung,” a stretched-out circle something like an oval.
Eventually, Kepler came to feel that his fascination with the circle had been a delusion. The Earth was a planet, as Copernicus had said, and it was entirely obvious to Kepler that the Earth, wracked by wars, pestilence, famine and unhappiness, fell short of perfection. Kepler was one of the first people since antiquity to propose that the planets were material objects made of imperfect stuff like the Earth. And if planets were “imperfect,” why not their orbits as well? He tried various oval-like curves, calculated away, made some arithmetical mistakes (which caused him at first to reject the correct answer) and months later in some desperation tried the formula for an ellipse, first codified in the Alexandrian Library by Apollonius of Perga. He found that it matched Tycho’s observations beautifully: “The truth of nature, which I had rejected and chased away, returned by stealth through the back door, disguising itself to be accepted … Ah, what a foolish bird I have been!”
Kepler had found that Mars moves about the Sun not in a circle, but in an ellipse. The other planets have orbits much less elliptical than that of Mars, and if Tycho had urged him to study the motion of, say, Venus, Kepler might never have discovered the true orbits of the planets. In such an orbit the Sun is not at the center but is offset, at the focus of the ellipse. When a given planet is at its nearest to the Sun, it speeds up. When it is at its farthest, it slows down. Such motion is why we describe the planets as forever falling toward, but never reaching, the Sun. Kepler’s first law of planetary motion is simply this: A planet moves in an ellipse with the Sun at one focus.
In uniform circular motion, an equal angle or fraction of the arc of a circle is covered in equal times. So, for example, it takes twice as long to go two-thirds of the way around a circle as it does to go one-third of the way around. Kepler found something different for elliptical orbits: As the planet moves along its orbit, it sweeps out a little wedge-shaped area within the ellipse. When it is close to the Sun, in a given period of time it traces out a large arc in its orbit, but the area represented by that arc is not very large because the planet is then near the Sun. When the planet is far from the Sun, it covers a much smaller arc in the same period of time, but that arc corresponds to a bigger area because the Sun is now more distant. Kepler found that these two areas were precisely the same no matter how elliptical the orbit: the long skinny area, corresponding to the planet far from the Sun, and the shorter, squatter area, when the planet is close to the Sun, are exactly equal. This was Kepler’s second law of planetary motion: Planets sweep out equal areas in equal times.
Kepler’s first law: A planet (P) moves in an ellipse with the Sun (S) at one of the two foci.
Kepler’s first two laws may seem a little remote and abstract: planets move in ellipses, and sweep out equal areas in equal times. Well, so what? Circular motion is easier to grasp. We might have a tendency to dismiss these laws as mere mathematical tinkering, something removed from everyday life. But these are the laws our planet obeys as we ourselves, glued by gravity to the surface of the Earth, hurtle through interplanetary space. We move in accord with laws of nature that Kepler first discovered. When we send spacecraft to the planets, when we observe double stars, when we examine the motion of distant galaxies, we find that throughout the universe Kepler’s laws are obeyed.
Many years later, Kepler came upon his third and last law of planetary motion, a law that relates the
motion of various planets to one another, that lays out correctly the clockwork of the solar system. He described it in a book called The Harmonies of the World. Kepler understood many things by the word harmony: the order and beauty of planetary motion, the existence of mathematical laws explaining that motion—an idea that goes back to Pythagoras—and even harmony in the musical sense, the “harmony of the spheres.” Unlike the orbits of Mercury and Mars, the orbits of the other planets depart so little from circularity that we cannot make out their true shapes even in an extremely accurate diagram. The Earth is our moving platform from which we observe the motion of the other planets against the backdrop of distant constellations. The inner planets move rapidly in their orbits—that is why Mercury has the name it does: Mercury was the messenger of the gods. Venus, Earth and Mars move progressively less rapidly about the Sun. The outer planets, such as Jupiter and Saturn, move stately and slow, as befits the kings of the gods.
Kepler’s second law: A planet sweeps out equal areas in equal times. It takes as long to travel from B to A as from F to E as from D to C; and the shaded areas BSA, FSE and DSC are all equal.
Kepler’s third or harmonic law states that the squares of the periods of the planets (the times for them to complete one orbit) are proportional to the cubes of their average distance from the Sun; the more distant the planet, the more slowly it moves, but according to a precise mathematical law: P2 = a3, where P represents the period of revolution of the planet about the Sun, measured in years, and a the distance of the planet from the Sun measured in “astronomical units.” An astronomical unit is the distance of the Earth from the Sun. Jupiter, for example, is five astronomical units from the Sun, and a3 = 5 × 5 × 5 = 125. What number times itself equals 125? Why, 11, close enough. And 11 years is the period for Jupiter to go once around the Sun. A similar argument applies for every planet and asteroid and comet.
Not content merely to have extracted from Nature the laws of planetary motion, Kepler endeavored to find some still more fundamental underlying cause, some influence of the Sun on the kinematics of worlds. The planets sped up on approaching the Sun and slowed down on retreating from it. Somehow the distant planets sensed the Sun’s presence. Magnetism also was an influence felt at a distance, and in a stunning anticipation of the idea of universal gravitation, Kepler suggested that the underlying cause was akin to magnetism:
My aim in this is to show that the celestial machine is to be likened not to a divine organism but rather to a clockwork …, insofar as nearly all the manifold movements are carried out by means of a single, quite simple magnetic force, as in the case of a clockwork [where] all motions [are caused] by a simple weight.
Kepler’s third or harmonic law, a precise connection between the size of a planet’s orbit and the period for it to go once around the Sun. It clearly applies to Uranus, Neptune and Pluto, planets discovered long after Kepler’s death.
Magnetism is, of course, not the same as gravity, but Kepler’s fundamental innovation here is nothing short of breathtaking: he proposed that quantitative physical laws that apply to the Earth are also the underpinnings of quantitative physical laws that govern the heavens. It was the first nonmystical explanation of motion in the heavens; it made the Earth a province of the Cosmos. “Astronomy,” he said, “is part of physics.” Kepler stood at a cusp in history; the last scientific astrologer was the first astrophysicist.
Not given to quiet understatement, Kepler assessed his discoveries in these words:
With this symphony of voices man can play through the eternity of time in less than an hour, and can taste in small measure the delight of God, the Supreme Artist … I yield freely to the sacred frenzy … the die is cast, and I am writing the book—to be read either now or by posterity, it matters not. It can wait a century for a reader, as God Himself has waited 6,000 years for a witness.
Within the “symphony of voices,” Kepler believed that the speed of each planet corresponds to certain notes in the Latinate musical scale popular in his day—do, re, mi, fa, sol, la, ti, do. He claimed that in the harmony of the spheres, the tones of Earth are fa and mi, that the Earth is forever humming fa and mi, and that they stand in a straightforward way for the Latin word for famine. He argued, not unsuccessfully, that the Earth was best described by that single doleful word.
Exactly eight days after Kepler’s discovery of his third law, the incident that unleashed the Thirty Years’ War transpired in Prague. The war’s convulsions shattered the lives of millions, Kepler among them. He lost his wife and son to an epidemic carried by the soldiery, his royal patron was deposed, and he was excommunicated by the Lutheran Church for his uncompromising individualism on matters of doctrine. Kepler was a refugee once again. The conflict, portrayed by both the Catholics and the Protestants as a holy war, was more an exploitation of religious fanaticism by those hungry for land and power. In the past, wars had tended to be resolved when the belligerent princes had exhausted their resources. But now organized pillage was introduced as a means of keeping armies in the field. The savaged population of Europe stood helpless as plowshares and pruning hooks were literally beaten into swords and spears.*
Waves of rumor and paranoia swept through the countryside, enveloping especially the powerless. Among the many scapegoats chosen were elderly women living alone, who were charged with witchcraft: Kepler’s mother was carried away in the middle of the night in a laundry chest. In Kepler’s little hometown of Weil der Stadt, roughly three women were tortured and killed as witches every year between 1615 and 1629. And Katharina Kepler was a cantankerous old woman. She engaged in disputes that annoyed the local nobility, and she sold soporific and perhaps hallucinogenic drugs as do contemporary Mexican curanderas. Poor Kepler believed that he himself had contributed to her arrest.
It came about because Kepler wrote one of the first works of science fiction, intended to explain and popularize science. It was called the Somnium, “The Dream.” He imagined a journey to the Moon, the space travelers standing on the lunar surface and observing the lovely planet Earth rotating slowly in the sky above them. By changing our perspective we can figure out how worlds work. In Kepler’s time one of the chief objections to the idea that the Earth turns was the fact that people do not feel the motion. In the Somnium he tried to make the rotation of the Earth plausible, dramatic, comprehensible: “As long as the multitude does not err,… I want to be on the side of the many. Therefore, I take great pains to explain to as many people as possible.” (On another occasion he wrote in a letter, “Do not sentence me completely to the treadmill of mathematical calculations—leave me time for philosophical speculations, my sole delight.”*)
With the invention of the telescope, what Kepler called “lunar geography” was becoming possible. In the Somnium, he described the Moon as filled with mountains and valleys and as “porous, as though dug through with hollows and continuous caves,” a reference to the lunar craters Galileo had recently discovered with the first astronomical telescope. He also imagined that the Moon had its inhabitants, well adapted to the inclemencies of the local environment. He describes the slowly rotating Earth viewed from the lunar surface and imagines the continents and oceans of our planet to produce some associative image like the Man in the Moon. He pictures the near contact of southern Spain with North Africa at the Straits of Gibraltar as a young woman in a flowing dress about to kiss her lover—although rubbing noses looks more like it to me.
Because of the length of the lunar day and night Kepler described “the great intemperateness of climate and the most violent alternation of extreme heat and cold on the Moon,” which is entirely correct. Of course, he did not get everything right. He believed, for example, that there was a substantial lunar atmosphere and oceans and inhabitants. Most curious is his view of the origin of the lunar craters, which make the Moon, he says, “not dissimilar to the face of a boy disfigured with smallpox.” He argued correctly that the craters are depressions rather than mounds. From his own observations he no
ted the ramparts surrounding many craters and the existence of central peaks. But he thought that their regular circular shape implied such a degree of order that only intelligent life could explain them. He did not realize that great rocks falling out of the sky would produce a local explosion, perfectly symmetric in all directions, that would carve out a circular cavity—the origin of the bulk of the craters on the Moon and the other terrestrial planets. He deduced instead “the existence of some race rationally capable of constructing those hollows on the surface of the Moon. This race must have many individuals, so that one group puts one hollow to use while another group constructs another hollow.” Against the view that such great construction projects were unlikely, Kepler offered as counterexamples the pyramids of Egypt and the Great Wall of China, which can, in fact, be seen today from Earth orbit. The idea that geometrical order reveals an underlying intelligence was central to Kepler’s life. His argument on the lunar craters is a clear foreshadowing of the Martian canal controversy (Chapter 5). It is striking that the observational search for extraterrestrial life began in the same generation as the invention of the telescope, and with the greatest theoretician of the age.