Students are asked to explore the parameters that affect orbit shape using the supplied Maple worksheet or Mathematica notebook.
In the Mathematica Worksheet Conics.nb or the online Geogebra visualization at GMM: Graphs in Polar Coordinates, you will examine a three parameter family of curves described by the polar equation \[r(\phi)=\frac{\alpha}{1+\epsilon\cos(\phi+\delta)}.\] Describe in detail how the shape of the plot depends on the parameters \(\alpha\), \(\delta\), and \(\epsilon\). Pay particular attention to different values of \(\epsilon\).
This activity was originally designed to provide students an opportunity to explore polar plots of conic sections in a pure math environment, so that when they see the derivation of the formula for orbital motion, they will immediately recognize it as the polar formula for a conic section. In this implementation, you can do the activity any time before that derivation. Make sure to include an opportunity (in-class or homework) for the students to work out the relationship between the mathematical parameters (ellipticity, etc.) and the physical ones (angular momentum, etc.)
Other faculty have chosen to use this activity after a lecture derivation of the equations of motion the two-body central force problem \[r(\theta)=\frac{\frac{l^2}{\mu k}}{1+C' \cos{(\phi+\delta)}}.\] A discussion of polar plots (how they are generated, how they are different from the usual cartesian plots) takes place just before students are released to play with the Mathematica notebook or Geogebra applet. This latter order offers less opportunity for students to discover things for themselves.