TITLE:
On the Gravitational Two-Body System and an Infinite Set of Laplace-Runge-Lenz Vectors
AUTHORS:
Caesar P. Viazminsky, Piere K. Vizminiska
KEYWORDS:
Two-Body System; Laplace-Runge-Lenz Vector; Nesting Orbits; Laws of Equivalent Orbits; Total Areal Velocity
JOURNAL NAME:
Applied Mathematics,
Vol.4 No.5,
May
20,
2013
ABSTRACT: The current approach of a system of two bodies that interact through a gravitational force goes beyond the familiar expositions [1-3] and derives some interesting features and laws that are overlooked. A new expression for the angular momentum of a system in terms of the angular momenta of its parts is deduced. It is shown that the characteristics of the relative motion depend on the system’s total mass, whereas the characteristics of the individual motions depend on the masses of the two bodies. The reduced energy and angular momentum densities are constants of motion that do not depend on the distribution of the total mass between the two bodies; whereas the energy may vary in absolute value from an infinitesimal to a maximum value which occurs when the two bodies are of equal masses. In correspondence with infinite possible ways to describe the absolute rotational positioning of a two body system, an infinite set of Laplace-Runge-Lenz vectors (LRL) are constructed, all fixing a unique orientation of the orbit relative to the fixed stars. The common expression of LRV vector is an approximation of the actual one. The conditions for nested and intersecting individual orbits of the two bodies are specified. As far as we know, and apart from the law of periods, the laws of equivalent orbits concerning their associated periods, areal velocities, angular velocities, velocities, energies, as well as, the law of total angular momentum, were never considered before.