On a Quotient Space of Keplerian Orbits

Abstract

Several metrics transforming diverse spaces of Keplerian orbits into metric ones have been proposed in the last 15 years. They are used to estimate the proximity of orbits of celestial bodies (comets, asteroids, and meteoroid complexes). Quotient spaces, which allow us to leave out of account those orbital elements that change secularly under various perturbations, are of great importance. Three quotient spaces were examined previously. Nodes are ignored in one of these; arguments of pericenters are ignored in the second one; both nodes and arguments of pericenters are ignored in the third one. We introduce the fourth quotient space, where orbits with arbitrary longitudes of nodes and arguments of pericenters are identified under the condition that their sum (longitude of pericenter) remains fixed. Function $${{\varrho }_{6}}$$, which represents the distance between the indicated classes of orbits and satisfies the first two axioms of metric spaces, is determined. An algorithm for its calculation is proposed. In the general case, the most challenging part of the algorithm is finding the solution of a trigonometric equation of the third degree. The issue of validity of the triangle axiom for $${{\varrho }_{6}}$$ (at least in its relaxed version) will be examined later.