An algebra and trigonometry -based proof of Kepler's First Law

TL;DR

This paper presents an elementary, calculus-free proof of Kepler's First Law, demonstrating that planetary orbits are elliptical using algebra and trigonometry, based on conservation laws and properties of ellipses.

Contribution

It introduces a novel proof of Kepler's First Law that avoids calculus, relying solely on algebra, trigonometry, and known properties of ellipses.

Findings

Proof confirms elliptical orbits without calculus

Uses conservation laws and geometric properties

Provides an accessible derivation for educational purposes

Abstract

An elementary proof of Kepler's first law, i.e. that bounded planetary orbits are elliptical, is derived without the use of calculus. The proof is similar in spirit to previous derivations, in that conservation laws are used to obtain an expression for the planetary orbit, which is then compared against an equation for the ellipse. However, we derive the equation that we match against, using trigonometry, from two well-known properties of the ellipse. Calculus is avoided altogether.