Abstract
Three-dimensional motions in the Chermnykh restricted three-body problem are studied. Specifically, families of three-dimensional periodic orbits are determined through bifurcations of the family of straight-line periodic oscillations of the problem which exists for equal masses of the primaries. These rectilinear oscillations are perpendicular to the plane of the primaries and give rise to an infinite number of families consisting entirely of periodic orbits which belong to the three-dimensional space except their respective one-dimensional bifurcations as well as their planar terminations. Many of the computed branch families are continued in all mass range that they exist.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Arnold, V.I.: Mathematical Methods of Classical Mechanics (trans. from the Russian by Vogtmann, K. and Weinstein, A.). Springer, New York (1987).
Beevi, A.S., Sharma, R.K.: Oblateness effect of Saturn on periodic orbits in the Saturn-Titan restricted three-body problem. Astrophys. Space Sci. 340, 245–261 (2012)
Belbruno, E., Llibre, J., Ollé, M.: On the families of periodic orbits which bifurcate from the circular Sitnikov motions. Celest. Mech. Dyn. Astron. 60, 99–129 (1994)
Chermnykh, S.V.: On the stability of libration points in a certain gravitational field. Vestn. Leningr. Univ. 2, 10–13 (1987)
Das, M.K., Narang, P., Mahajan, S., Yuasa, M.: Effect of radiation on the stability of a retrograde particle orbit in different stellar systems. Planet. Space Sci. 57, 836–845 (2009)
Dutt, P., Sharma, R.K.: Evolution of periodic orbits near the Lagrangian point L 2. Adv. Space Res. 47, 1894–1904 (2011)
Dvorak, R.: Numerical results to the Sitnikov problem. Celest. Mech. Dyn. Astron. 56, 71–80 (1993)
Elipe, A., Lara, M.: Periodic orbits in the restricted three-body problem with radiation pressure. Celest. Mech. Dyn. Astron. 68, 1–11 (1997)
Gómez, G., Jorba, À., Simó, C., Masdemont, J.: Dynamics and Mission Design Near Libration Points. Advanced Methods for Collinear Points, vol. III. World Scientific, Singapore (2001)
Goździewski, K., Maciejewski, A.J.: Nonlinear stability of the Lagrangian libration points in the Chermnykh problem. Celest. Mech. Dyn. Astron. 70, 41–58 (1998)
Hénon, M.: Vertical stability of periodic orbits in the restricted problem: I. Equal masses. Astron. Astrophys. 28, 415–426 (1973)
Howell, K.C., Kakoi, M.: Transfers between the Earth-Moon and Sun-Earth systems using manifolds and transit orbits. Acta Astronaut. 59, 367–380 (2006)
Jiang, I.G., Yeh, L.C.: On the Chermnykh-like problems: I. The mass parameter μ=0.5. Astrophys. Space Sci. 305, 341–348 (2006)
Kalantonis, V.S., Perdios, E.A., Perdiou, A.E.: The Sitnikov family and the associated families of 3D periodic orbits in the photogravitational RTBP with oblateness. Astrophys. Space Sci. 315, 323–334 (2008)
Kovács, T., Érdi, B.: Transient chaos in the Sitnikov problem. Celest. Mech. Dyn. Astron. 105, 289–304 (2009)
Kushvah, B.S.: Linear stability of equilibrium points in the generalized photogravitational Chermnykh’s problem. Astrophys. Space Sci. 318, 41–50 (2008)
Kushvah, B.S.: Linearization of the Hamiltonian in the generalized photogravitational Chermnykh’s problem. Astrophys. Space Sci. 323, 57–63 (2009)
Markellos, V.V.: Numerical investigation of the planar restricted three-body problem: II. Regions of stability for retrograde satellites of Jupiter as determined by periodic orbits of the second generation. Celest. Mech. 10, 87–134 (1974)
Markellos, V.V.: Bifurcations of plane with three-dimensional asymmetric periodic orbits in the restricted three-body problem. Mon. Not. R. Astron. Soc. 180, 103–116 (1977)
Murison, M.A.: The fractal dynamics of satellite capture in the circular restricted three-body problem. Astron. J. 98, 2346–2359 (1989)
Oberti, P., Vienne, A.: An upgraded theory for Helene, Telesto and Calypso. Astron. Astrophys. 397, 353–359 (2003)
Papadakis, K.E.: Numerical exploration of Chermnykh’s problem. Astrophys. Space Sci. 299, 67–81 (2005)
Perdios, E.A.: The manifolds of families of 3D periodic orbits associated to Sitnikov motions in the restricted three-body problem. Celest. Mech. Dyn. Astron. 99, 85–104 (2007)
Perdios, E.A., Markellos, V.V.: Stability and bifurcations of Sitnikov motions. Celest. Mech. 42, 187–200 (1988)
Perdios, E.A., Ragos, O.: Asymptotic and periodic motion around collinear equilibria in Chermnykh’s problem. Astron. Astrophys. 414, 361–371 (2004)
Sidorenko, V.V.: On the circular Sitnikov problem: the alternation of stability and instability in the family of vertical motions. Celest. Mech. Dyn. Astron. 109, 367–384 (2011)
Simmons, J.F.L., McDonald, A.J.C., Brown, J.C.: The restricted 3-body problem with radiation pressure. Celest. Mech. 35, 145–187 (1985)
Schuerman, D.W.: The restricted three-body problem including radiation pressure. Astrophys. J. 238, 337–342 (1980)
Sitnikov, K.: Existence of oscillating motions for the three-body problem. Dokl. Akad. Nauk SSSR 133, 303–306 (1960)
Subba Rao, P.V., Sharma, R.K.: Effect of oblateness on the non-linear stability of L 4 in the restricted three-body problem. Celest. Mech. Dyn. Astron. 65, 291–312 (1997)
Szebehely, V.G.: Theory of Orbits: the Restricted Problem of Three Bodies. Academic Press, New York (1967)
Acknowledgements
One of the authors (A.A. Nikaki) acknowledges financial support under a University of Patras “K. Karatheodory” research grant.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Perdiou, A.E., Nikaki, A.A. & Perdios, E.A. Periodic motions in the spatial Chermnykh restricted three-body problem. Astrophys Space Sci 345, 57–66 (2013). https://doi.org/10.1007/s10509-013-1368-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10509-013-1368-7