# Metrics in Keplerian orbits quotient spaces

## Abstract

Quotient spaces of Keplerian orbits are important instruments for the modelling of orbit samples of celestial bodies on a large time span. We suppose that variations of the orbital eccentricities, inclinations and semi-major axes remain sufficiently small, while arbitrary perturbations are allowed for the arguments of pericentres or longitudes of the nodes, or both. The distance between orbits or their images in quotient spaces serves as a numerical criterion for such problems of Celestial Mechanics as search for common origin of meteoroid streams, comets, and asteroids, asteroid families identification, and others. In this paper, we consider quotient sets of the non-rectilinear Keplerian orbits space \(\mathbb H\). Their elements are identified irrespective of the values of pericentre arguments or node longitudes. We prove that distance functions on the quotient sets, introduced in Kholshevnikov et al. (Mon Not R Astron Soc 462:2275–2283, 2016), satisfy metric space axioms and discuss theoretical and practical importance of this result. Isometric embeddings of the quotient spaces into \(\mathbb R^n\), and a space of compact subsets of \(\mathbb H\) with Hausdorff metric are constructed. The Euclidean representations of the orbits spaces find its applications in a problem of orbit averaging and computational algorithms specific to Euclidean space. We also explore completions of \(\mathbb H\) and its quotient spaces with respect to corresponding metrics and establish a relation between elements of the extended spaces and rectilinear trajectories. Distance between an orbit and subsets of elliptic and hyperbolic orbits is calculated. This quantity provides an upper bound for the metric value in a problem of close orbits identification. Finally the invariance of the equivalence relations in \(\mathbb H\) under coordinates change is discussed.

## Keywords

Metric space Space of Keplerian orbits Embedding in Euclidean space Hausdorff distance Isometric embedding Quotient space## Notes

### Acknowledgements

We are grateful to Professor K. V. Kholshevnikov for the statement of the problem and important remarks. We also thank unknown reviewers and the editor for detailed analysis and valuable suggestions. This work is supported by the Saint Petersburg State University, research Grant 6.37.341.2015.

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