Metrics in the space of orbits and their application to searching for celestial objects of common origin

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Finding a common origin for various celestial bodies, especially the relations between meteoroid streams, comets and asteroids (possibly extinct comets), remains one of the important problems in Solar system astronomy. Different criteria, starting with one by Southworth and Hawkins, have been used for this purpose. Ideally, they must represent some kind of metric in a five-dimensional space of orbits. Unfortunately, they are not ideal. The majority of the criteria have been examined by us. It turns out that they all represent pseudometrics for which the triangle axiom is not fulfilled. Besides this, they are inapplicable if at least one of the orbits is circular. We propose metrics free of all the aforementioned drawbacks. In addition, the metric properties of three factor-spaces, where orbits are identified irrespective of the values of node longitudes, pericentre arguments or both, are examined. The results are applied to the problem of searching for minor bodies of the Solar system with a common origin. The relationship between comet 96P/Machholz 1 and asteroid 2003EH1, as well as that between comet 2P/Encke and asteroid 2004TG10, has been proved. Using all criteria considered and the new metrics leads to practically identical results. This is explained by the fact that only close and essentially non-circular orbits were analysed. Besides this, the measure of orbit triples for which the triangle axiom failed is likely small, though this problem has not been examined yet.