Tycho and Kepler (44 page)

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 wearied
¹
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 goodness
²
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 where 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
un
physical 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 melt
³
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 stone
⁴
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” while
⁵
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.