The determination of a preliminary orbit by Cauchy-Kuryshev-Perov method
DOI:
https://doi.org/10.21638/spbu01.2021.417Abstract
The determination of preliminary orbits of celestial bodies is of interest for observational astronomy, from the point of the view of the discovery of new bodies or identification with already known ones. To solve this problem, the techniques are required that are not limited both by the values of eccentricity of the orbit and by the time intervals between observations. In this paper, we consider the geometric Cauchy-Kuryshev-Perov method for determining the preliminary orbit. It is shown how, within the framework of the two-body problem, proceeding only from geometric constructions, using five angular observations to determine an orbit that does not lie in the plane of the observer’s motion. This method allows us to reduce the problem of determining the preliminary orbit to an algebraic system of equations for two dimensionless variables, with a finite number of solutions. The method is suitable for determining both elliptical and hyperbolic orbits. Moreover, it has no restrictions on the length of the orbital arc of the observed body and is unlimited by the number of complete revolutions around the attracting center between observations. All possible combinations of body positions in orbit are divided into 64 variants and represented by the corresponding systems of equations. This article presents an algorithm for finding solutions to the problem without a priori information about the desired orbit. Solutions are sought in a limited area, in which triangulation is performed with ranking triangles for compliance with the search conditions, which makes it possible to exclude consideration of most of them at the initial stage. Solutions of the system are found by the Nelder-Mead method through the search for the minima of the objective function. The obtained orbits are compared, through the presentation of observations, and the best one is selected. An example of determining the orbit of comet Borisov 2I is given.Keywords:
preliminary orbit determination, geometric method of Cauchy-Kuryshev-Perov, method of Nelder-Mead, algebraic equations, 2I/Borisov
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Литература
1. Кузнецов В.Б. Применение геометрического метода Курышева-Перова для определения орбит астероидов и комет. Труды международной научн. конф. «Околоземная астрономия - 2015». п. Терскол, 31 августа - 5 сентября 2015 г., Москва, 51–57 (2015).
2. Cauchy A. Memoire sur deux formules generals etc. C. R. 25 (1847).
3. Harzer P. Uber eine geometrische Methode z¨ur Bahnbestimmung der Bahnen von Himmelskorper aus funf Beobachtungen. A. N., 184 (1910); Kiel, Publ. der Sternwarte, 12 (1910).
4. Вильев М.А. Исследования по вопросу о числе решенийосновной задачи теоретической астрономии в связи с общим ее положением в настоящее время. В: Труды АО ЛГУ, серия математических наук (астрономия) 5 (27), 81–256 (1938).
5. Курышев В.И., Перов Н.И. О нетрадиционном способе определения элементов орбит неизвестных космических объектов по данным обработки обзорных фотоснимков на ЭВМ. Астрономический журнал 59 (6), 1212–1217 (1982).
6. Кузнецов В.Б. К вопросу об определении предварительнойорбиты небесного тела. Астрономический вестник 53 (6), 456–466 (2019). https://doi.org/10.1134/S0320930X19060057
7. Stumpff K. Himmelsmechanik. Vol. 1. Berlin, Deutscher Verlag (1959).
8. Sarnecki A.J. A projective approach to orbit determination from three sight-lines. Celest. Mech. Dyn. Astr. 66 (4), 425–451 (1996). https://doi.org/10.1007/BF00049380
9. Fong C. Analytical methods for squaring the disc. In: Seoul ICM 2014 (2014).
10. Шалашилин В.И., Кузнецов Е.Б. Метод продолжения решения по параметру и наилучшая параметризация в прикладной математике и механике. Москва, Эдиториал УРСС (1999).
11. Самотохин А.С., Хуторовский З.Н. Метод первоначального определения параметров околоземных орбит по трем угловым измерениям. Препринты ИПМ им. М.В.Келдыша 44 (2014).
12. Nelder J.A., Mead R. A Simplex Method for Function Minimization. The Computer Journal 7 (4), 308–313 (1965). https://doi.org/10.1093/comjnl/7.4.308
13. Химмельблау Д. Прикладное нелинейное программирование, пер. с англ. Москва, Мир (1975).
14. Потемкин В. Язык программирования Fortran (2005). Доступно на: http://fortran-90.pvbk.spb.ru/min.html#FM28 (дата обращения: 10.09.2021).
15. Арушанян О.Б. Научно-образовательный интернет-ресурс НИВЦ МГУ по численному анализу (2006). Доступно на: http://num-anal.srcc.msu.ru/lib_na/cat/mn/mnb6r.htm (дата обращения: 10.09.2021).
16. The International Astronomical Union. Minor Planet Center 2I/Borisov = C/2019 Q4 (Borisov) (2020). Доступно на: https://minorplanetcenter.net/db_search/show_object?utf8=%E2%9C%93&object_id=2I (дата обращения: 10.09.2021).
References
1. Kuznetsov V.B. The using of geometrical method of Kuryshev-Perov for determination of asteroid and comet orbits. Proceedings of the International conference “Near-Earth Astronomy - 2015”, Terskol, 31 August - 5 September 2015, Moscow, 51–57 (2015). (In Russian)
2. Cauchy A. Memoire sur deux formules generals etc. C. R. 25 (1847).
3. Harzer P. Uber eine geometrische Methode z¨ur Bahnbestimmung der Bahnen von Himmelskorper aus funf Beobachtungen. A. N., 184 (1910); Kiel, Publ. der Sternwarte, 12 (1910).
4. Vil’ev М.А. Research on the number of solutions to the main problem of theoretical astronomy in connection with its general position at the present time. In: Trudy AO LGU, seriia matematicheskikh nauk (astronomiia) 5 (27), 81–256 (1938). (In Russian)
5. Kuryshev V.I. Perov N.I. A nontraditional means of determining the orbital elements of unknown cosmic objects from the data of computer processing of survey photographs. Astronomicheskii Zhurnal 59 (6), 1212–1217 (1982) (In Russian) [Engl. transl.: Soviet Astronomy 26 (6), 727–729 (1982)].
6. Kuznetsov V.B. Revisiting the Determination of a Preliminary orbit for a Celestial Body. Astronomicheskii vestnik 53 (6), 456–466 (2019). https://doi.org/10.1134/S0320930X19060057 (In Russian) [Engl. transl.: Solar System Research 53 (6), 462–472 (2019). https://doi.org/10.1134/S0038094619060054].
7. Stumpff K. Himmelsmechanik. Vol. 1. Berlin, Deutscher Verlag (1959).
8. Sarnecki A.J. A projective approach to orbit determination from three sight-lines. Celest. Mech. Dyn. Astr. 66 (4), 425–451 (1996). https://doi.org/10.1007/BF00049380
9. Fong C. Analytical methods for squaring the disc. In: Seoul ICM 2014 (2014). 10. Shalashilin V. I., Kuznetsov E.B. Metod prodolzheniia resheniia po parametru i nailuchshaia parametrizatsiia v prikladnoi matematike i mekhanike. Moscow, Editorial URSS Publ. (In Russian) [Engl. transl.: Shalashilin V. I. Kuznetsov Е.B. Parametric Continuation and Optimal Parametrization in Applied Mathematics and Mechanics. Dordrecht, Springer (2003)].
11. Samotokhin A. S., Khutorovsky Z.N. Method for initial determining Earth satellites orbits on base of three angular observations. Keldysh Institute Preprints 44 (2014). (In Russian)
12. Nelder J.A., Mead R. A Simplex Method for Function Minimization. The Computer Journal 7 (4), 308–313 (1965). https://doi.org/10.1093/comjnl/7.4.308
13. Himmelblau D.M. Applied Nonlinear Programming. Texas, McGraw-Hill Book Company (1972). [Russ. ed.: Himmelblau D.M. Prikladnoe nelineinoe programmirovanie. Moscow, Mir Publ. (1975)].
14. Potemkin V. Programming language Fortran (2005). Available at: http://fortran-90.pvbk.spb.ru/min.html#FM28 (accessed: September 10, 2021). (In Russian)
15. Arushanyan O.B. Scientific and educational Internet resource of the Research Computing Center of Moscow State University on numerical analysis (2006). Available at:http://numanal.srcc.msu.ru/lib_na/cat/mn/mnb6r.htm (accessed: September 10, 2021). (In Russian)
16. The International Astronomical Union. Minor Planet Center 2I/Borisov = C/2019 Q4 (Borisov) (2020). Available at: https://minorplanetcenter.net/db_search/show_object?utf8=%E2%9C%93&object_id=2I (accessed: September 10, 2021). (In Russian)
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Published
2022-01-04
How to Cite
Kuznetsov, V. B. (2022). The determination of a preliminary orbit by Cauchy-Kuryshev-Perov method. Vestnik of Saint Petersburg University. Mathematics. Mechanics. Astronomy, 8(4), 716–727. https://doi.org/10.21638/spbu01.2021.417
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Astronomy
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Articles of "Vestnik of Saint Petersburg University. Mathematics. Mechanics. Astronomy" are open access distributed under the terms of the License Agreement with Saint Petersburg State University, which permits to the authors unrestricted distribution and self-archiving free of charge.
