Yesterday we did a fun project on parametric equations and touched on the motion of planets at the end:

Playing with parametric equations in Desmos

Even though the discussion of planetary motion in the last project wasn’t even close to a complete picture, the kids seemed pretty interested in it, so I continued with that idea today. The goal for today was to show them why the ideas we talked about yesterday weren’t quite right and how we could use polar coordinates to study the same ideas.

The main idea I drew on for today’s project came from one of Katherine Johnson’s technical notes on NASA website:

The link to that paper is here:

NASA Technical Note D-233 by T. H. Skopinski and Katherine G. Johnson

and the specific equation I’ll be using from that paper is equation 1a which gives an equation for an ellipse in polar coordinates:

I’ll also be drawing from some lecture notes on planetary motion I found via a google search:

Richard Fitzpatrick’s lecture notes on planetary motion on the University of Texas website

I started the project by introducing polar coordinates – sort of a high level conceptual introduction for my younger son and a few more details for my older son:

Next I showed the them the polar coordinate description of an ellipse from the Katherine Johnson paper and how if we applied the velocity and acceleration ideas from yesterday that we’d see the acceleration wasn’t always directed at the same point:

Next we looked more carefully at the movement around the ellipse and (after a while) saw that the line from the ellipse back to the origin was moving with a constant angular velocity.

The boys were able to explain that the movement of planets in an elliptical orbit probably wouldn’t have a constant angular speed:

Finally, we used the paper from the University of Texas to explore the ellipses corresponding to the orbits of various planets.

Definitely some neat ideas to share with kids and also a fun way to bring some important pieces of math history to life!