FROM ELLIPTICAL MOTION KINEMATICS TO THE GRAVITATIONAL FORCE, Version 2

Abstract

The force law governing the motion of a material point along an elliptical orbit is derived using a purely kinematic approach. Starting from the differential equations of motion in a Cartesian coordinate system, an angular equation of motion is obtained, directly yielding Kepler's second law. It is shown that the acceleration is directed toward the focus of the ellipse and is inversely proportional to the square of the distance to it. The crucial result is that the mathematical uniqueness of the focus as the center of attraction for inverse-square fields strictly follows from the derived kinematic analogue of the Binet formula. The derivation is generalized to precessing orbits (rosette orbits) with an arbitrary parameter k ≠ 1. The obtained expressions are applied to the Earth–Moon system. The forces calculated on the basis of Newton's second law are compared with the forces determined by the law of universal gravitation. The relative discrepancy does not exceed 1.2%, which confirms the equivalence of the obtained expressions.