1. Equation of the Orbit
The radial eqn gives r as a function of t, but to visualize the orbit, we want to plot r vs. phi
The r equation of motion: Two tricks:
Substitute
Rewrite the d/dt in terms of phi using the chain rule

2. Equation of the Orbit After some manipulations, we get:
Now we will show that for an inverse-square force law, the solutions are conic sections: ellipses, parabolas, and hyperbolas.

3. Equation of the Orbit
For an inverse-square law, the force is given by:
The radial eqn becomes
And the general solution is:

4. Equation of the Orbit
Invert to find r as a function of phi: where

5. Matlab Task Create file KeplerOrbits.m
Let c=1 and plot orbits corresponding to ε = 0:0.1:0.8
Summarize your results