Abstract
A new analytic expression for the position of the infinitesimal body in the elliptic Sitnikov problem is presented. This solution is valid for small bounded oscillations in cases of moderate primary eccentricities. We first linearize the problem and obtain solution to this Hill's type equation. After that the lowest order nonlinear force is added to the problem. The final solution to the equation with nonlinear force included is obtained through first the use of a Courant and Snyder transformation followed by the Lindstedt–Poincaré perturbation method and again an application of Courant and Snyder transformation. The solution thus obtained is compared with existing solutions, and satisfactory agreement is found.
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References
Arnold, V. I.: 1963, 'Small denominators and problems of stability of motion in classical and celestial mechanics', Usp. Mat. Nauk. 18, 91-192.
Belbruno, E., Llibre, J. and Olle, M.: 1994, 'On the families of periodic orbits which bifurcate from the circular Sitnikov motions', Celest. Mech. Dyn. Astr. 60, 99-129.
Brouwer, D. and Clemence, G.: 1961, Methods of Celestial Mechanics, Academic Press, New York.
Corbera, M. and Llibre, J.: 2000, 'Periodic orbits of the Sitnikov problem via a Poincare map', Celest. Mech. Dyn. Astr. 77, 273-303.
Corbera, M. and Llibre, J.: 2001, 'On symmetric periodic orbits of the elliptic Sitnikov problem via the analytical continuation method', Contemp. Math. (in press).
Courant, E. D., Livingston, M. S. and Snyder, H. S.: 1952, 'The strong-focusing synchrotron-a new high energy accelerator', Phys. Rev. 88, 1190-1196.
Dvorak, R.: 1993, 'Numerical results to the Sitnikov problem', Celest. Mech. Dyn. Astr. 56, 71-80.
Faruque, S. B.: 2002, 'Axial oscillation of a planetoid in restricted three body problem: the circular Sitnikov problem', B. Astron. Soc. India 30, 895-909.
Hagel, J.: 1992, 'A new analytic approach to the Sitnikov problem', Celest. Mech. Dyn. Astr. 53, 267-292.
Jalali, M. A. and Pourtakdoust, S. H.: 1997, 'Regular and chaotic solutions of the Sitnikov problem near the 3/2 commensurability', Celest. Mech. Dyn. Astr. 68, 151-162.
Jose, J. V. and Saletan, E. J.: 1998, Classical Dynamics, Cambridge University Press, Cambridge.
Liu, J. and Sun, Y.: 1990, 'On the Sitnikov problem', Celest. Mech. Dyn. Astr. 49, 285-302.
MacMillan, W. D.: 1913, 'An integrable case in the restricted problem of three bodies', Astron. J. 27, 11-13.
Nayfeh, A. H.: 2000, Perturbation Methods, John Wiley & Sons, Inc., New York.
Olle, M. and Pacha, J. R.: 1999, 'The 3D elliptic restricted three-body problem: periodic orbits which bifurcate from limiting restricted problems', Astron. Astrophys. 351, 1149-1164.
Perdios, E. and Markellos, V. V.: 1988, 'Stability and bifurcations of Sitnikov motions', Celest.Mech. Dyn. Astr. 42, 187-200.
Sitnikov, K. A.: 1960, 'Existence of oscillating motion for the three-body problem', Dokl. Akad. Nauk. USSR 133, 303-306.
Whittaker, E. T.: 1988, A treatise on the analytical dynamics, Cambridge University Press, Cambridge.
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Faruque, S.B. Solution of the Sitnikov Problem. Celestial Mechanics and Dynamical Astronomy 87, 353–369 (2003). https://doi.org/10.1023/B:CELE.0000006721.86255.3e
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DOI: https://doi.org/10.1023/B:CELE.0000006721.86255.3e