Optimizing the Ptolemaic Model of Planetary and Solar Motion

TL;DR

This paper analyzes and proposes improvements to the ancient Ptolemaic geocentric model, demonstrating how tuning shifts can enhance its accuracy in approximating planetary orbits beyond first-order approximations.

Contribution

It introduces a method to optimize Ptolemy's shifts, improving the model's precision to quadratic order in eccentricities, thus extending its potential accuracy.

Findings

Optimal shift ratio of 5/3 improves model precision

Model accuracy can reach quadratic order in eccentricities

Full precision requires shifts in both epicycles and deferents

Abstract

Ptolemy-s planetary model is an ancient geocentric astronomical model, describing the observed motion of the Sun and the planets. Ptolemy accounted for the deviations of planetary orbits from perfect circles by introducing two small and equal shifts into his model. We show that Ptolemy-s choice of shifts allowed him to approximate the true eccentricity of planetary orbits only in the first order, linear in eccentricity. We show that, if the shifts were tuned in the ratio 5/3, the model precision could be improved substantially, perhaps delaying its rejection. The best achievable precision is quadratic in eccentricities. Although, to achieve it fully, one would have to introduce Ptolemaic shifts in the epicycles as well as in deferents the planets.