Kepler's Laws: Demonstration and Derivation without Calculus

A demonstration apparatus for Kepler's three laws of planetary motion consists of an air-supported “satellite” whose orbit on a 1.2×1.5-m level-table surface is determined by an inverse-square force generated by a Peaucellier linkage and long spring. The equipment is a feasibility model which works—not an ultimate design. Cost of the equipment is about $25 for parts, plus considerable labor. Several comments are made regarding tricky aspects of the design. The apparatus can be analyzed by using high-school algebra and geometry. From the principles of conservation of energy, conservation of momentum, and static equilibrium, Kepler's laws can be deduced that: (1) the oribt is an ellipse with center of force at one focus, (2) the time rate of area swept out by the radius vector is constant, and (3) the square of the period is proportional to the cube of the semimajor axis. This analysis (believed to be new) in essence integrates the force, $1/r2$⁠, to obtain the energy, $−1/r$⁠, without using calculus.