On local normability of spaces of Keplerian orbits

Abstract

Geometric properties of spaces of Keplerian orbits are of interest for problems of Celestial Mechanics related with the search of groups of celestial bodies whose orbits are close to each other. Those groups include asteroid families and meteoroid streams. Their study brings an important information about the evolution of the solar system, characteristics of a family members and their parent bodies. The local properties of a distance function between orbits are of main importance for the problems of search of families of genetically related celestial bodies, because orbits of a family members cluster together in a small region of the orbits space. In this article we consider several metrics on the set of Keplerian orbits H and its quotient spaces. For each of these metrics we solve the question: does there exist a normed space locally isometric to the orbits metric space? In two of considered cases, the answer turns out to be positive: the quotient space of H by the equivalence relation, neglecting the magnitude of pericentre argument of the orbit, can be isometrically embedded into R4. The embedding into R3 also exists for the quotient space by the pair of elements: the longitude of ascending node and the pericentre argument. It is shown in the article, that the answer to the stated question is negative for other metrics we considered. The possibility of an isometric embedding of orbits space or of its part into Euclidean space is useful in application to aforementioned problems of Celestial Mechanics. The isometric map helps to define the mean of orbits family in a natural way: the arithmetic mean of images corresponds to the orbit with the minimal square deviation of distances from orbits of the family.