Modeling Orbital Motion in a Circular Conic Reference Frame

Abstract

Two-body orbital trajectories conform to conic sections. However, typically in the literature their motion is analyzed in a plane. Kepler modeled elliptic orbital motion in a plane, stated his second law, and derived the geometric position-time relationship as the uniform change in area with respect to time. Kepler’s equation has been applied extensively and proven to give time as a function of position for exact solutions to orbital problems. An identical equation has been derived without reliance on geometry alone by applying basic principle of classical mechanics and the calculus. When the elliptic orbit is analyzed as a section of a circular cone and represented in three dimensions, additional variables relate position and time. In a conical reference frame, the planar and conic representations merge. This paper combines the conic section knowledge and characteristics of the cone to introduce a third dimension to modeling orbital motion. Force and potential energy have indeterminate limits because potential energy approaches zero as the radius approaches infinity and the gravitational force approaches infinity as the radius vector approaches zero, a singularity. The conic frame includes the apex where the singularity of a radius of zero is an established point in the reference system.