Kepler's Physical Astronomy (Princeton Paperback) [Paperback]

First we look briefly at Kepler's Mysterium Cosmographicum (1596). Here "[Kepler's] exuberance was not yet balanced by the self-criticism which distinguished his mature writings. His technical command of mathematics and astronomy was still insecure." (p. 8, Springer ed.). As an illustration we may consider his formula for relating the distances to the sun and orbital periods of two planets, R_1/R_2=((T_1+T_2)/2)/T_2 (p. 13). "Considered as a physical deduction [this result] is most peculiar ... it totally lacks the character of a general law. One can compute the radius of Venus's orbit compared to Mercury's from their periodic times, and likewise the radius of the earth's orbit compared to that of Venus; but computing the radius of the earth's orbit directly from that of Mercury would not give the same answer." (p. 14)

Kepler's Astronomia Nova (1609), on the other hand, is a first rate work. Here "Kepler reintroduced physical argument to astronomy, and thereby shifted the overall emphasis of his book from the mathematical representation of observations to the determination of how and why the planets, huge, physical bodies, moved through the heavens." (p. 22). "In this task he was almost entirely on his own. Contemporary physics was not going to offer any help, and he was essentially left free to speculate about the kinds of things which were required to impose order on the motion of planets travelling 'in pure aether, just as birds in the air'." (p. 27). "Kepler ... used some admittedly vague speculations, concerning the difficulty of controlling a planet's motion with information available at the planet itself, to suggest that at least part of this task took place elsewhere: presumably, therefore, at the central body." (p. 28). "Copernicus ... had rejected the equant hypothesis because of its physical absurdity (p. 28), but Kepler reintroduced it "as a convenient and transparent way of representing what was for him the critical phenomenon: that the planet moved swiftly when near the sun and slowly when distant from it" (p. 29). "The Copernican model, besides concealing the variation in speed behind a combination of uniform motions, would have required an intolerable amount of 'mental' activity to control the motion. The Ptolemaic equant, on the other hand, by openly displaying this variation, encouraged Kepler's attempt to locate an impelling and guiding force in the sun. If only some way could be found to explain the planet's approach to and withdrawal from the sun, the variation of speed would be easily understandable as a consequence of the weakening of some solar force with distance from its source. Variations in the planet's speed, which in earlier astronomy had been a blemish to be ignored or concealed, singled out the sun now as the heavenly body which had to be somehow involved in moving the planet." (p. 29).

With this in mind, "Kepler ... finally set out to construct a planetary model ... an equant model, 'in imitation of the ancients' as he said, but without Ptolemy's restriction that the eccentricity be precisely bisected" (p. 42). He model was very successful. "Had he stopped there ... Kepler would already have contributed much to the refinement of Copernican astronomy. Instead he immediately demonstrated ... that his own theory remained inadequate. Tu be sure, it performed the function of a theory of longitude. ... What it did not give was the right location for the planet itself." (p. 44-45). Trying to solve this problem in the case of "the distances between Mars and the sun led Kepler back to the hypothesis of bisected eccentricity", which was "no accident" we can see retrospectively because "The area law ... can be well represented by equant motion around the empty focus of the ellipse. Thus the center of the ellipse bisects the eccentricity of its pseudo-equant point at the empty focus" (pp. 45-46). This not being available to Kepler yet, he attempted to show "how a physical hypothesis, simple and plausible, accounted for the success of the Ptolemaic equant hypothesis. His physical explanation was ... that the planet moved slower when it was more distant to the sun, in proportion to the distance. In [Mysterium Cosmographicum] he had sketched out an argument that the Ptolemaic hypothesis described a motion of just this kind. Here he expanded his reasoning into a geometrical demonstration." (p. 62). "The 'distance law' holds exactly---at the apsides---for equant motion with bisected eccentricity, and, incidentally for Kepler motion on an ellipse. Kepler himself stated only that it was true quam proxime, and probably did not know, when writing the Astronomia Nova, of its exact validity. Outside the apsides the theorem is not exact. Kepler remarked this fact ... claiming it to be of little consequence." (p. 66).

However, while the distance law in isolation shows that Ptolemy got lucky with his equant, Kepler's physical perspective taken further rules out eccentric circles altogether. "[Kepler] was analyzing motion on a eccentric circle, a model that had been in general use for nearly two millennia, apparently the simplest possible model with any empirical accuracy. He took apart this beautifully simple model and showed that as a physical process ... it was really quite complicated, so complicated as to raise doubt about whether it could be real. He had performed so radical a reassessment by interpreting astronomy, for the first time, as a physical science. ... [H]e found novel and effective criteria for evaluating theories. No longer did it suffice that a theory was mathematically plausible. ... [R]eal bodies were moved by physical forces ... The convenience of the astronomer yielded to the constraint of objectivity" (p. 78).

So why did eccentric circles work so well? There must be a simple physical principle that explains their success. Ta-da: Kepler's law of equal areas. Armed with this new law Kepler tackled Mars, "most obstinate of the ancient planets, which would test the powers of his physical astronomy" (p. 87), where his law forced him to conclude that "the orbit of Mars was not a circle; it was an oval" (p. 90), or, more precisely, an ellipse, as Kepler would discover "accidentally" (p. 107) only after much "exceedingly tedious work" (p. 100) and "garbled physics" (p. 101). "Unsure of the exact geometry of the Martian orbit", "he temporarily had to assume the oval to be an ellipse ... in order to apply the area law ... When locating the planet ... he found it to lie precisely on the auxiliary ellipse he had been using" (p. 129). To explain this type of motion physically, Kepler likened the sun's motive force to "a circular river carrying a boat around its course. A steering oar, ... as Kepler said, '... turns around once in twice in the periodic time of the planet'" (p. 110), generating the oval orbit.

One remarkable application of Kepler's physical theory, which he put forth in the Epitome Astronomiae Copernicanae (1618-1621), is that it predicts the densities of the planets. "[A] plamet resisted motion because of the inertia of its matter ... Moreover, a planet that was physically larger experienced the effect of the solar virtue through its whole volume", so "since the general factors, length of path and strength of force, would together increase the period as the square of distance from the sun, while the actual periods only grew as 3/2 power of distance, it was clear that the planetary densities must decrease as the square root of distance, to explain the observed relation" (p. 143). "This alerts us to a distinction which cannot be overemphasised. For Kepler, his 'third law' was no law at all, at least not so far as concerned natural science ... it was an empirical fact", which had an interesting application to planetary densities, and which "was clearly of archetypal importance, and could not have been unintended by the Creator" (p. 144).

this is a great book for anyone interested in astronom