Abstract
The determination of preliminary orbits of celestial bodies is of interest to observational astronomy in terms of discovering new bodies or identifying them with already known ones. The solution of this problem requires methods that are not limited both by the eccentricity of the orbit and by the time intervals between observations. This article considers the Cauchy–Kuryshev–Perov geometric method for determining a preliminary orbit. It is shown how to determine an orbit that does not lie in the observer’s plane of motion within the two-body problem, based only on geometric constructions, and using five angular observations. This method makes it possible to reduce the problem of determining a preliminary orbit to the algebraic system of equations relative to two dimensionless variables with a finite number of solutions. The method is suitable for determining both elliptical and hyperbolic orbits. It has no restrictions on the length of the orbital arc of the observed body and is not limited by the number of complete revolutions around the attractive center between observations. All possible combinations of positions of the body in the orbit are divided into 64 variants and described by the corresponding systems of equations. This article presents an algorithm for finding solutions to the problem without having prior information about the desired orbit. The solutions are sought in a bounded region in which triangulations are performed with triangles ranked according to the search conditions, thus eliminating the consideration of most of them at the initial stage. The solutions of the system are found by the Nelder–Mead method through the search for minima of the target function. The obtained orbits are compared by means of a representation of observations, and the best one is selected. An example of determining the orbit of the comet 2I/Borisov is given.
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To cite this work: Kuznetsov V.B. “Determination of a Preliminary Orbit by the Cauchy–Kuryshev–Perov Geometrical Method,” Vestnik of St. Petersburg University. Mathematics. Mechanics. Astronomy, 2021, vol. 8(66), no. 4, pp. 716–727. (In Russian.) https://doi.org/10.21638/spbu01.2021.417.
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Translated by O. Pismenov
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Kuznetsov, V.B. Determination of a Preliminary Orbit by the Cauchy–Kuryshev–Perov Geometric Method. Vestnik St.Petersb. Univ.Math. 54, 452–460 (2021). https://doi.org/10.1134/S1063454121040117
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DOI: https://doi.org/10.1134/S1063454121040117