On a quotient space of Keplerian orbits

Abstract

Several metrics were proposed during last 15 years which transform divers spaces of Keplerian orbits in metric ones. They are used to estimate a proximity of orbits of celestial bodies (usually comets, asteroids, and meteoroid complexes). An important role play quotient spaces. They allow us not to take into account those orbital elements which change in the secular mode under different perturbations. Three quotient spaces were just examined. Nodes are ignored in one of them; arguments of pericenters are ignored in the second one; both nodes and arguments of pericenters are ignored in the third one. Here, we introduce a 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) is fixed. The function ̺6 serving as a distance between pointed classes of orbits, and satisfying first two axioms of metric spaces is determined. An algorithm of its calculation is proposed. In general the most complicated part of the algorithm represents the solution of a trigonometric equation of third degree. The question on the validity of the triangle axiom for ̺6, at least in a relaxed variant, will be examined later.