The Tychonic Method for Calculating the Ratio between the Eccentricities of Mars

Abstract

In Chapter 16 of Astronomia nova, Kepler describes and applies a method for finding the parameters of what he will call the vicarious hypothesis: a model that still assumes circular orbits and an equant point, but does not assume the bisection of the eccentricity, that is, that the center of the orbit is halfway between the equant point and the Sun. The method allows Kepler to find independently both centers in a very elegant way, but its application is tedious. He confesses that he had to apply it seventy times over a period of 5 years to obtain trustable results. Years earlier, when Kepler arrived to work with Tycho, he found that Tycho and Longomontanus had rejected bisection and somehow had obtained a ratio between eccentricities that, as Kepler himself highlights, happened to be very close to the one Kepler would later find after so much effort. Kepler does not say how Tycho and Longomontanus obtained their parameters and, to the best of my knowledge, there is no single published work that attempts to answer this question. Still, it is a very interesting question to ask how they arrived at values so close to those that took so much pain for Kepler to obtain. Recently, I published a paper describing a method Tycho used for finding Saturn’s parameters. In this paper, I show that by applying this method to the data of Tychonic observations of oppositions, it is possible to arrive at parameters very close to those that we know Tycho found. In this way, I argue that this is the method Tycho applied for obtaining Mars’s parameters. The simplicity of the Tychonic method contrasts with the complexity of Kepler’s.