Abstract
An algorithm for studying the families of symmetric periodic orbits using their generating solutions, whose structure was presented in the first part of this paper [1], is described. The algorithm is essentially based on symmetry of the generating solution and on its initial approximation. More than 20 new families of symmetric periodic solutions of the Hill’s problem have been found and investigated with the use of this algorithm. The families including trajectories for orbital injection into the vicinity of collinear libration points L 1,2 are described.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Batkhin, A.B., Symmetric periodic solutions of the Hill’s problem. I, Kosm. Issled., 2013, vol. 51, no. 4, pp. 308–322. (Cosmic Research, pp. 275–288).
Batkhin, A.B., Symmetric periodic solutions of the Hill’s problem, Preprint of Keldysh Inst. of Applied Math., Russ. Acad. Sci., Moscow, 2012, no. 33. http://www.keldysh.ru/papers/2012/prep2012_52.pdf.)
Szebehely, V., Theory of Orbits. The Restricted Problem of Three Bodies, New York and London: Academic Press, 1967. Translated under the title Teoriya orbit: ogranichennaya zadacha trekh tel, Moscow: Nauka, 1982.
Batkhin, A.B., Symbolic dynamics and generating solutions of planar periodic orbits in the Hill’s problem, Preprint of Keldysh Inst. of Applied Math., Russ. Acad. Sci., Moscow, 2011, no. 34. http://www.keldysh.ru/papers/2011/source/prep2011_34.pdf)
Hénon, M., New families of periodic orbits in Hill’s problem of three bodies, Celestial Mechanics and Dynamical Astronomy, 2003, vol. 85, pp. 223–246.
Hénon, M., Numerical exploration of the restricted problem. V. Hill’s case: Periodic orbits and their stability, Astron. & Astrophys., 1969, vol. 1, pp. 223–238.
Hénon, M., Generating Families in the Restricted Three-Body Problem, vol. 52 of Lecture Notes in Physics, Berlin, Heidelberg, New York: Springer, 1997.
Perko, L., Families of symmetric periodic solutions of Hill’s problem. II: Second species periodic solutions for C ≪ −1, Amer. J. Math., 1981, vol. 104, no. 2, pp. 353–397.
Hénon, M., Numerical exploration of the restricted problem. Hill’s case: non-periodic orbits, Astron. & Astrophys., 1970, no. 9, pp. 24–36.
Parker, T.S. and Chua, L.O., Practical Numerical Algorithms for Chaotic Systems, Springer, 1989.
Batkhin, A.B. and Batkhina, N.V., The Hill’s Problem, Volgograd: Volgograd. Nauch. Izd., 2009.
Hénon, M., Families of asymmetric periodic orbits in Hill’s problem of three bodies, Celest. Mech. Dyn. Astr., 2005, vol. 93, pp. 87–100.
Bryuno, A.D., Ogranichennaya zadacha trekh tel: Ploskie periodicheskie orbity (Restricted Three Body Problem: Planar Periodic Orbits), Moscow: Nauka, 1990.
Breakwell, J.V. and Perko, L.M., Second-order matching in the restricted three-body problem (small μ), Celestial Mechanics, 1974, vol. 9, pp. 437–450.
Perko, L., Families of symmetric periodic solutions of Hill’s problem. I: First species periodic solutions for C≪ −1, Amer. J. Math., 1981, vol. 104, no. 2, pp. 321–352.
Simó, C. and Stuchi, T.J., Central stable/unstable manifolds and the destruction of KAM tori in the planar Hill’s problem, Physica D, 2000, vol. 140, pp. 1–32.
Tsirogiannis, G.A., Perdios, E.A., and Markellos, V.V., Improved grid search method: an efficient tool for global computation of periodic orbits, application to Hill’s problem, Celest. Mech. Dyn. Astr., 2009, no. 103, pp. 49–78.
Bruno, A.D., Zero-multiple and retrograde periodic solutions of the restricted three body problem, Preprint of Keldysh Inst. of Applied Math., Russ. Acad. Sci., 1996, Moscow, 1996, no. 93.
Bruno, A.D. and Varin, V.P., Periodic solutions of the restricted three body problem at small μ, Prikl. Mat. Mekh., 2007, vol. 71, no. 6, pp. 1034–1066.
Bruno, A.D., Periodic solutions to a Hamiltonian system, Kosm. Issled., 2006, vol. 44, no. 3, pp. 258–271. (Cosmic Research, pp. 245–257).
Batkhin, A.B. and Batkhina, N.V., New families of periodic solutions in the Hill’s problem, in Trudy XXXV akademicheskikh chtenii po kosmonavtike (Proceed. of 35th Academic Readings in Astronautics), Moscow: Komissiya RAN, 2011, pp. 122–123.
Benest, D., Libration effects for retrograde satellites in the restricted three-body problem. I: Circular plane Hill’s case, Celestial Mechanics, 1976, no. 13, pp. 203–215.
Kogan, A.Yu., Distant satellite orbits in the restricted three body problem, Kosm. Issled., 1988, vol. 26, no. 6, pp. 813–818. (Cosmic Research, p. 705–710).
Boyarskii, M.N. and Sheikhet, A.I., One-impulse transfer from the satellite orbit to a conventionally periodic trajectory around the collinear libration point of the Sun-Earth system, Kosm. Issled., 1987, vol. 25, no. 1, p. 152.
Lidov, M.L., Lyakhova, V.A., and Teslenko, N.M., One-impulse transfer from the satellite orbit to a conventionally periodic orbit in the vicinity of L 2 point of the Earth-Sun system and related problems, Kosm. Issl., 1987, vol. 25, no. 2, pp. 163–185.
Kreisman, B.B., Single-impulse transfers from orbits of artificial satellites to orbits around L 1 or L 2 libration points, Kosm. Issled., 2011, vol. 49, no. 4, pp. 335–344. (Cosmic Research, pp. 325–334).
Batkhin, A.B., The use of generating solutions for constructing transfers to the vicinity of a collinear libration point, in Trudy XXXIV akademicheskikh chtenii po kosmonavtike (Proc. of 34th Academic Readings in Astronautics), Moscow: Komissiya RAN, 2010, pp. 122–123.
Author information
Authors and Affiliations
Additional information
Original Russian Text © A.B. Batkhin, 2013, published in Kosmicheskie Issledovaniya, 2013, Vol. 51, No. 6, pp. 497–510.
Rights and permissions
About this article
Cite this article
Batkhin, A.B. Symmetric periodic solutions of the Hill’s problem. II. Cosmic Res 51, 452–464 (2013). https://doi.org/10.1134/S0010952513050018
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0010952513050018