The space of Keplerian orbits and a family of its quotient spaces
| Autor: | Konstantin V. Kholshevnikov, D. V. Milanov, Anastasia S. Shchepalova |
|---|---|
| Rok vydání: | 2021 |
| Předmět: |
Longitude of the ascending node
Triangle inequality General Mathematics General Physics and Astronomy Function (mathematics) Quotient space (topology) Type (model theory) Space (mathematics) Combinatorics Metric space Physics::Space Physics Astrophysics::Earth and Planetary Astrophysics Quotient Mathematics |
| Zdroj: | Vestnik of Saint Petersburg University. Mathematics. Mechanics. Astronomy. 8:359-369 |
| ISSN: | 2587-5884 1025-3106 |
| DOI: | 10.21638/spbu01.2021.215 |
| Popis: | The distance functions on the set of Keplerian orbits play an important role in solving the problems of searching for the parent bodies of meteoroid streams. A special kind of function is the distances in the quotient spaces of orbits. Three metrics of this type were developed earlier. These metrics allow disregarding the longitude of the ascending node or the argument of pericenter, or both. Here, we introduce one more quotient space, where two orbits are considered to be identical if they differ only in their longitudes of nodes and arguments of pericenters, but have the same sum of these elements (the longitude of pericenter). Function $${{\varrho }_{6}}$$ is defined to calculate the distance between two equivalence classes of orbits. The algorithm for calculation of $${{\varrho }_{6}}$$ from given elements of the orbits is provided along with a reference to the corresponding program written in the C++ programming language. Unfortunately, function $${{\varrho }_{6}}$$ is not a full-fledged metric. We have proven that it satisfies the first two axioms of a metric space; however, it does not satisfy the third axiom: the triangle inequality does not hold, at least in the case of large eccentricities. However, there are two important cases when the triangle axiom is satisfied: one of three orbits is circular and the longitudes of pericenters of all three orbits coincide. Perhaps, the inequality holds for all elliptic orbits, but this is a matter of further research. |
| Databáze: | OpenAIRE |
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