Abstract
The problem of finding critical points of the distance function between two Keplerian elliptic orbits is reduced to the determination of all real roots of a trigonometric polynomial of degree 8. The coefficients of the polynomial are rational functions of orbital parameters. Using computer algebra methods we show that a polynomial of a smaller degree with such properties does not exist. This fact shows that our result cannot be improved and it allows us to construct an optimal algorithm to find the minimal distance between two Keplerian orbits.
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Vasiliev, N. N.: 'Determining of critical points of distance function between points of two Keplerian orbits', Bull. Inst. Theor. Astron. 14(5) (1978), Leningrad, 266-268 (in Russian).
Buchberger, B.: 'Gröbner bases: an algorithmic method in polynomial ideal theory', In: N. K. Bose (ed.) Progress Directions and Open Problems in Multidimensional Systems Theory, D. Reidel, 1985, pp. 184-232.
Char, B. W., Geddes, K. O., Gonnet, G. H., Leong, B. L., Monagan, M. B. and Watt, S. M.: Maple V Library Reference Manual, Springer, New York, 1993.
Lusternik, L. A. and Schnierelman, L. G.: 'Topological methods in variational problems and applications to differential geometry of surfaces', Uspechi Math. Nauk 2(1) (1947), 166-217 (in Russian).
Kopiev, A. N. and Leskov, E. I.: Private Communication.
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Kholshevnikov, K.V., Vassiliev, N.N. On the Distance Function Between Two Keplerian Elliptic Orbits. Celestial Mechanics and Dynamical Astronomy 75, 75–83 (1999). https://doi.org/10.1023/A:1008312521428
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DOI: https://doi.org/10.1023/A:1008312521428