The goal of this chapter is to derive Kepler's laws of planetary motion from Newton's laws of motion and gravitation. We will analyze the motion of two massive particles (such as the sun and Mars), assuming that all forces other than mutual gravitational attraction are negligible. We want to find explicit formulas allowing us to predict motions. Will Mars crash into the sun? Will comet Hale-Bopp return? If so, when? Our formulas will be of use to engineers, who may want to maneuver satellites or send a space probe that is “parked,” i.e., orbiting a planet, off to orbit a different planet. During this chapter we will pretend to be physicists. We will make savvy use of conserved quantities and particular functional forms to pick coordinates that simplify our calculations.
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