Inverse-Square Orbits Revisited

Kepler’s laws are an essential part of any mechanics course at the introductory level. They are commonly used to illustrate the power of Newton’s laws at explaining and predicting the motion of planets and comets around the sun. While the derivation of the second and third laws are outlined in every standard textbook, that of the first law (and the shape of orbits in general) is invariably omitted, due mainly to the insufficient proficiency of the targeted students in differential equations. The paper offers a workaround this difficulty. Assuming familiarity with the basic properties of conics and employing Newton’s law of universal gravitation, we manage to obtain, with very little calculus, the classic result: orbits are conic sections whose nature is determined by the body’s total energy.