Tycho and Kepler: solid myth versus subtle truth

Social Research, Spring, 2005 by Owen Gingerich, James R. Voelkel

Kepler's "motive force hypothesis" (as we will call it) was conceived as a means to relate cosmological features of the Copernican system, the planets' mean distances from the sun and their periods of revolution. In chapter 22 of the Mysterium, Kepler extended its use to a more focused question of planetary theory, that of the variation of a planet's speed around its own orbit. In both Copernican and Ptolemaic planetary theory, the circular orbits of the planets are eccentric with respect to the sun (or earth), so that even if the planets moved uniformly around their orbits, they would still appear to move nonuniformly from the sun (or earth). In addition to this optical effect, both systems have mechanisms that cause the planets to move nonuniformly around their orbits, more slowly when more distant from the sun and more quickly when nearer. Kepler realized that his motive force hypothesis could account for this nonuniformity of a planet's motion around its own eccentric orbit. He was able to show for the specific case of a planet's closest and furthest distances from the sun that the proposed motive force would indeed produce the motion described purely mathematically in Ptolemaic and Copernican planetary theory. This remarkable demonstration represents the first, small step toward a physically derived planetary theory that Kepler would bring to fruition in the Astronomia nova.

Unfortunately, there were a few inconsistencies between the theories of the planets' motion put forward by Ptolemy or Copernicus and Kepler's physical account of planetary motion. The most fundamental of these inconsistencies concerned the motion of the sun, or, equivalently, the earth (for heliocentrists like Copernicus and Kepler). Because the fundamental change that Kepler introduced into the theory of the earth was arguably the most significant change he introduced into astronomical theory, we should take a moment to consider it carefully.

From the time of Hipparchus through Ptolemy and Copernicus up to Tycho Brahe, the theory of the sun had always been a simple eccentric; that is, the sun revolved uniformly around a circular orbit whose center was eccentric with respect to the sun. This simple system had been sufficient for all earlier astronomers to describe the nonuniform motion of the sun through the ecliptic. In the geocentric theory, Ptolemy had added another mechanism, the so-called equant, which added a physical--as opposed to apparent--nonuniformity in the motions of planets, and Copernicus had devised an equivalent alternative for his planetary theory. The equant provided uniform circular motion about a point not at the center of the orbit, which had the effect of generating nonuniform motion on the planetary circle itself. But the equant was not necessary for the sun because of the simplicity of its motion. (The equant was required in the theories of the planets in order to model their retrograde motion realistically; without them, the planets' retrograde arcs would have been noticeably exaggerated, the more so near a planet's aphelion or perihelion.) However, the treatment of the earth in Copernican theory did bring an additional complication: in his inversion of the Ptolemaic geocentric system into a heliocentric one, Copernicus had made the center of the earth's orbit--not the sun--the point of reference for all of the planetary theories. The center of the earth's circular orbit was also called the "mean sun" and was separated from the true sun by the earth's eccentricity. The difference was mostly technical, for in a general cosmological sense, Copernicus accepted the sun as the center of the system. But the ambiguity was unacceptable to Kepler.