On local normability of spaces of Keplerian orbits

Abstract

Geometric properties of spaces of Keplerian orbits are of interest for celestial mechanics problems related to the search for groups of celestial bodies with close orbits. Those groups include asteroid families and meteor streams. Studying these groups provides important information on the evolution of the Solar System, as well as on the characteristics of objects within a family and their parent bodies. The local properties of a distance function between orbits are of primary importance for the problems of the search for families of related celestial bodies, because orbits of family members cluster together in a small region of the orbit space. Several metrics on the set of Keplerian orbits $$\mathbb{H}$$ and its quotient sets are considered in this paper. For each of these metrics we solve the question: is there a normed vector space that is locally isometric to the orbit metric space? In two of the considered cases, the answer turns out to be positive: the quotient space of $$\mathbb{H}$$ by the equivalence relation neglecting the magnitude of the pericenter argument of the orbit can be isometrically embedded into $${{\mathbb{R}}^{4}}$$ . The embedding into $${{\mathbb{R}}^{3}}$$ also exists for the quotient space by the pair of elements: the longitude of ascending node and the pericenter argument. It is shown in this paper that for other metrics, the answer to the stated question is negative. The possibility of an isometric embedding of the orbit space or its part into Euclidean space is useful in application to the aforementioned problems of celestial mechanics. The isometric map helps to define the mean of the orbit family in a natural way: the arithmetic mean of images corresponds to the orbit with the minimum square deviation of distances from the orbits of the family.