Historical Astronomy: Concepts: Kepler's Laws Derivations

In the early 17th century, Kepler discovered the following three laws of planetary motion:

  1. The planets orbit around the sun in an ellipse with the sun at one focus.
  2. As the planets orbit around the sun, they sweep out equal areas from a line drawn to the sun in equal times.
  3. The ratio of the period squared to semi-major axis cubed is the same for all the planets orbiting the sun.

Kepler discovered these laws empirically. While he searched for a reason behind the laws, he was never able to discover why the laws were true. He only new that they worked. It wasn't until Newton formulated his three laws of motion and law of gravity that the "why" behind Kepler's laws was found. Newton was the first person to be able to derive Kepler's laws from fundamental principles.

Follow the links for each derivation.

Derivation of 1st Law see note below.
Derivation of 2nd Law
Derivation of 3rd Law
All Derivations (PDF)

Note: The derivation for the 1st Law involves doing Newton's Laws in polar coordinates, which we don't talk about in class. Here is a brief summary. It also involves doing conic sections in polar coordinates, so here is a brief summary of that. If the Word-generated html is too messy, here are PDFs of the originals: Conic Sections and Polar Coordinates.

last updated 2/3/03 by david mcclung, copyright 2003. All rights reserved.