Calculating Positions

By Liona Fan-Chiang and Merv Fansler

The age old problem that has confronted astronomers has been that of determining what the nature of the motion of heavenly bodies is such that it generates the sense-perceptual data upon our celestial sphere. The problem might be posed more precisely: How many degrees of freedom are necessary to uniquely describe the motion of a planetary body?

Beginning with an ocean of observations passed down to him, Johannes Kepler was able to identify the singularities in sense-perception which provided insight into the determining characteristics (degrees of freedom) of a planet's motion. What did Kepler find those determining characteristics to be? Harmonies! Yet, it was only by an intense course of hypothesis and experimentation (“a narrow path”) that he could arrive at such a vivid insight. These harmonic principles were not immediately apparent to the naked eye or ear. In fact, it was only by conceptually unifying the set of observations as effects of a “Solar System”, that the motions could be brought into the reach of human knowledge.

Yet, the question still remained: How do those solar harmonics manifest themselves to an observer here on Earth? Is the act of projecting these harmonies itself harmonic? This latter question is one which directly confronted Carl F. Gauß and others after the initial sighting of the first discovered asteroid, Ceres. Unlike Kepler, who had years and even centuries of sense-perceptual data tracking the motions of the planets, Gauß had only 22 observations, covering a time span of 41 days and a motion on the celestial sphere of just barely 3 degrees (See Monatliche Correspondenz).


This animation, generated in Celestia, shows the series of observations which Giuseppe Piazzi would have seen in 1801 over an interval of 41 days.

For Kepler, the problem of uniquely determining a planet's motion reduced to accounting for the characteristics of two distinct modes of change: the first and second inequality. The notion of accounting for the proper motion of a planet by distinguishing between the effects of two inequalities, could only be accomplished by way of observing periodicities, which require decades, if not centuries of diligent observations to perceive. Given Gauß' circumstances, such a distinction seemed impossible. To approach this problem, one needs to peer deeper into the underlying relationship between physical action and sense-perception. The question then becomes: Since the harmonies uniquely determine the physical motion of every planet, is this also uniquely reflected in the apparent motion of a planet relative to the Earth (or any other planet, for that matter)? That is, reflected in the interplay between the two inequalities?

We begin our path toward answering this question by first examining how the determining characteristics, which Kepler discovered, interact to define a single apparent position.

Before proceeding, the reader should take a moment to hypothesize as to what those determining characteristics might be.