In this manuscript, the existence of periodic orbits of collision of the first kind has been discussed on the model of Autonomous Four-body Problem by the method of analytic continuation given by Giacaglia [1] and Bhatnagar [2] [3]. For the existence of periodic orbits, Duboshin’s criterion [4] has been satisfied and it has been confirmed by analyzing the Poincare surfaces of section (PSS) [5]. Also it has been shown that the case of collision given by Levi-Civita [6] [7] is conserved by the method analytic continuation. In all sections of this manuscript, equilateral triangular configuration given by Ceccaroni and Biggs [8] has been considered. In this model, third primary of L4 inferior mass (in comparison of the other primaries) is placed at the equilibrium point of the R3BP.
References
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[2]
Bhatnagar, K.B. (1969) Periodic Orbits of Collision in the Plane Elliptic Restricted Problem of Three Bodies. National Institute of Science India, 35A, 829-844.
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Bhatnagar, K.B. (1971) Periodic Orbits of Collision in the Plane Circular Problem of Four Bodies. Indian Journal of Pure and Applied Mathematics, 2, 583-596.
Poincare, H. (1905) Lecons de Mécanique Céleste. Gauthier-Villars, Paris, 1.
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Levi-Civita, T. (1903) Traiettorie singolari ed urti nel problema ristreto deri teri corpi. Annali di Mathematica Pura ed Applicata, 9, 1-32.
[7]
Levi-Civita, T. (1906) Sur la résolution qualitative du probleme restrient des trios corps. Acta Mathematica, 30, 305-327. https://doi.org/10.1007/BF02418577
[8]
Ceccaroni, M. and Biggs, J. (2012) Low-Thrust Propulsion in a Coplanar Circular Restricted Four body Problem. Celestial Mechanics and Dynamical Astronomy, 112, 191-219. https://doi.org/10.1007/s10569-011-9391-x
[9]
Choudhry, R.K. (1966) Existence of Periodic Orbits of the Third Kind in the Elliptic Restricted Problem of the Three Bodies and the Stability of the Generating Solution. Proceedings of the National Academy of Sciences India Section A, 36, 249-264.
