Orbital motion equations with dynamic models.

Kepler's orbital motion equations are of primary importance in prediction of a satellite's position. His approach is the foundation for the development of time-position equations for all orbits. In this paper, fundamentals of non-relativistic classical mechanics are used to analyze two-body trajectories, and this method is compared with Kepler's. Equations for power, energy, and time-displacement are derived by integration, beginning with a summation of the acting forces. In each case - elliptic, parabolic, and hyperbolic - the derivation produces models based on circles. Elliptic and hyperbolic equations are expressed as functions of conformance angles that result naturally from the integration. Motion equations for the parabolic orbit are derived that duplicate well-documented results in the literature. The models presented provide both time-displacement and velocity vector relationships for each orbit type. Dynamic models validate the integration approach by demonstrating compliance with conservation of energy. Additionally, these models show that three dimensions are necessary to represent all energy terms for general elliptic and hyperbolic motion. All the models can be constructed in a coordinate system that is fixed relative to the radius vector. Equations and models are derived for rectilinear trajectories. Each is a special case of the orbit's general model. Rectilinear equations and dynamic models expand our understanding of two-body trajectories.