Had the planet mars not existed: Kepler's equant model and its physical consequences

TL;DR

This paper explores the classical equant model of planetary motion, demonstrating its implications for Kepler's laws and acceleration, highlighting its historical and educational significance in understanding planetary dynamics.

Contribution

It shows that the equant model, up to first order in eccentricity, naturally implies Kepler's second law, Hamilton's hodograph, and an inverse-square law of acceleration, linking ancient models to modern physics.

Findings

Equant model satisfies Kepler's second law.

Implicates inverse-square law of acceleration.

Connects ancient Greek astronomy with modern physics.

Abstract

We examine the equant model for the motion of planets, which has been the starting point of Kepler's investigations before he modified it because of Mars observations. We show that, up to first order in eccentricity, this model implies for each orbit a velocity which satisfies Kepler's second law and Hamilton's hodograph, and a centripetal acceleration with an inverse square dependence on the distance to the sun. If this dependence is assumed to be universal, Kepler's third law follows immediately. This elementary execice in kinematics for undergraduates emphasizes the proximity of the equant model coming from Ancient Greece with our present knowledge. It adds to its historical interest a didactical relevance concerning, in particular, the discussion of the Aristotelian or Newtonian conception of motion.