Copernicus's epicycles from Newton's gravitational force law via linear perturbation theory in geometric algebra

TL;DR

This paper derives Copernicus's epicycles from Newton's gravitational law using linear perturbation theory within geometric algebra, connecting classical astronomy with modern physics and mathematics.

Contribution

It introduces a novel derivation of Copernican epicycles from Newtonian gravity using geometric algebra and linear perturbation theory, linking historical models with modern physics.

Findings

Derivation of Copernicus's formulas from Newton's law

Connection between epicycles and Hill's oscillator equation

Validation of Kepler's law for small eccentricities

Abstract

We derive Copernicus's epicycles from Newton's gravitational force law by assuming that a planet's orbit is a perturbed circular orbit, with the perturbation defined to be co-rotating with the said orbit. We substitute this orbit expression into Newton's gravitation law and showed that the perturbation satisfies the linear part of Hill's oscillator equation for lunar motion. We solve this oscillator equation using an exponential Fourier series and impose the boundary conditions at the aphelion and perihelion to derive Copernicus's formulas for the eccentric, deferent, and epicycle. We show that for small eccetricity, the Copernican orbit expression also leads to Kepler's law of areas for planetary motion. The formalism we use is the Clifford (geometric) algebra Cl_{2,0}.