Abstract
The orbits about Lagrangian equilibrium points are important for scientific investigations. Since, a number of space missions have been completed and some are being proposed by various space agencies. In light of this, we consider a more realistic model in which a disk, with power-law density profile, is rotating around the common center of mass of the system. Then, we analyze the periodic motion in the neighborhood of Lagrangian equilibrium points for the value of mass parameter \(0<\mu\leq\frac{1}{2}\). Periodic orbits of the infinitesimal mass in the vicinity of equilibrium are studied analytically and numerically. In spite of the periodic orbits, we have found some other kind of orbits like hyperbolic, asymptotic etc. The effects of radiation factor as well as oblateness coefficients on the motion of infinitesimal mass in the neighborhood of equilibrium points are also examined. The stability criteria of the orbits is examined with the help of Poincaré surfaces of section (PSS) and found that stability regions depend on the Jacobi constant as well as other parameters.
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References
Broucke, R.: Periodic orbits in the restricted three-body problem with Earth-Moon masses. JPL technical report, Jet Propulsion Laboratory, California Institute of Technology (1968). http://books.google.co.in/books?id=ZB8HGQAACAAJ
Chermnykh, S.V.: Stability of libration points in a gravitational field. Leningr. Univ. Vestn. Mat. Mekh. Astron. 73–77 (1987)
Elipe, A., Lara, M.: Periodic orbits in the restricted three body problem with radiation pressure. Celest. Mech. Dyn. Astron. 68, 1–11 (1997). doi:10.1023/A:1008233828923
Goździewski, K.: Nonlinear stability of the Lagrangian libration points in the Chermnykh problem. Celest. Mech. Dyn. Astron. 70, 41–58 (1998). doi:10.1023/A:1008250207046
Goździewski, K., Maciejewski, A.J.: Unrestricted planar problem of a symmetric body and a point mass. Triangular libration points and their stability. Celest. Mech. Dyn. Astron. 75, 251–285 (1999)
Henon, M.: New families of periodic orbits in Hill’s problem of three bodies. Celest. Mech. Dyn. Astron. 85, 223–246 (2003). doi:10.1023/A:1022518422926
Henon, M.: Families of asymmetric periodic orbits in Hill’s problem of three bodies. Celest. Mech. Dyn. Astron. 93, 87–100 (2005). doi:10.1007/s10569-005-3641-8
Jiang, I.G., Ip, W.H.: The planetary system of Upsilon Andromedae. Astron. Astrophys. 367, 943–948 (2001). arXiv:astro-ph/0008174. doi:10.1051/0004-6361:20000468
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). arXiv:astro-ph/0610735. doi:10.1007/s10509-006-9065-4
Kushvah, B.S.: Linear stability of equilibrium points in the generalized photogravitational Chermnykh’s problem. Astrophys. Space Sci. 318, 41–50 (2008). arXiv:0806.1132. doi:10.1007/s10509-008-9898-0
Kushvah, B.S., Kishor, R., Dolas, U.: Existence of equilibrium points and their linear stability in the generalized photogravitational Chermnykh-like problem with power-law profile. Astrophys. Space Sci. 337, 115–127 (2012). arXiv:1107.5390. doi:10.1007/s10509-011-0857-9
Markellos, V.V.: Numerical investigation of the planar restricted three-body problem. Celest. Mech. Dyn. Astron. 10, 87–134 (1974). doi:10.1007/BF01261880
McCuskey, S.: Introduction to Celestial Mechanics. Addison-Wesley Series in Aerospace Science. Addison-Wesley, Reading (1963). http://books.google.co.in/books?id=e_ZQAAAAMAAJ
Meyer, K.R., Schmidt, D.S.: Periodic orbits near l 4 for mass ratios near the critical mass ratio of routh. Celest. Mech. Dyn. Astron. 4, 99–109 (1971). doi:10.1007/BF01230325
Mittal, A., Ahmad, I., Bhatnagar, K.: Periodic orbits generated by Lagrangian solutions of the restricted three body problem when one of the primaries is an oblate body. Astrophys. Space Sci. 319, 63–73 (2009). doi:10.1007/s10509-008-9942-0
Murray, C., Dermott, S.: Solar System Dynamics. Cambridge University Press, Cambridge (2000). http://books.google.co.in/books?id=aU6vcy5L8GAC
Papadakis, K.E.: Motion around the triangular equilibrium points of the restricted three-body problem under angular velocity variation. Astrophys. Space Sci. 299, 129–148 (2005a). doi:10.1007/s10509-005-5158-8
Papadakis, K.E.: Numerical exploration of Chermnykh’s problem. Astrophys. Space Sci. 299, 67–81 (2005b). doi:10.1007/s10509-005-3070-x
Perdios, E.A.: Critical symmetric periodic orbits in the photogravitational restricted three-body problem. Astrophys. Space Sci. 286, 501–513 (2003). doi:10.1023/A:1026328832021
Plummer, H.C.: On periodic orbits in the neighbourhood of centres of libration. Mon. Not. R. Astron. Soc. 62, 6 (1901)
Poincaré, H.: Les Methodes Nouvelles de la Mecanique Celeste. Gauthier-Villars, Paris (1892–1899)
Ragos, O., Zagouras, C.G., Perdios, E.: Periodic motion around stable collinear equilibrium point in the photogravitational restricted problem of three bodies. Astrophys. Space Sci. 182, 313–336 (1991). doi:10.1007/BF00645009
Ragos, O., Papadakis, K.E., Zagouras, C.G.: Stability regions and quasi-periodic motion in the vicinity of triangular equilibrium points. Celest. Mech. Dyn. Astron. 67, 251–274 (1997). doi:10.1023/A:1008271126547
Rivera, E.J., Lissauer, J.J.: Stability analysis of the planetary system orbiting υ Andromedae. Astrophys. J. 530, 454–463 (2000). doi:10.1086/308345
Safiya Beevi, A., Sharma, R.K.: Analysis of periodic orbits in the Saturn-Titan system using the method of Poincare section surfaces. Astrophys. Space Sci. 333, 37–48 (2011). doi:10.1007/s10509-011-0630-0
Strand, M.P., Reinhardt, W.P.: Semiclassical quantization of the low lying electronic states of H2 +. J. Chem. Phys. 70, 3812–3827 (1979). doi:10.1063/1.437932
Szebehely, V.: Theory of Orbits. The Restricted Problem of Three Bodies. Academic Press, New York (1967)
Winter, O.C.: The stability evolution of a family of simply periodic lunar orbits. Planet. Space Sci. 48, 23–28 (2000). doi:10.1016/S0032-0633(99)00082-3
Winter, O.C., Murray, C.D.: Resonance and chaos. I. First-order interior resonances. Astron. Astrophys. 319, 290–304 (1997)
Wolfram, S.: The Mathematica Book. Wolfram Media, Champaigh (2003). http://books.google.co.in/books?id=dyK0hmFkNpAC
Yeh, L.C., Jiang, I.G.: On the Chermnykh-like problems: II. The equilibrium points. Astrophys. Space Sci. 306, 189–200 (2006). arXiv:astro-ph/0610767. doi:10.1007/s10509-006-9170-4
Acknowledgements
We are thankful to the Department of Science and Technology, Govt. of India for providing financial support through the SERC-Fast Track Scheme for Young Scientist [SR/FTP/PS-121/2009]. We are also thankful to IUCAA Pune for partial support to visit library and to use computing facility.
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Kishor, R., Kushvah, B.S. Periodic orbits in the generalized photogravitational Chermnykh-like problem with power-law profile. Astrophys Space Sci 344, 333–346 (2013). https://doi.org/10.1007/s10509-012-1334-9
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DOI: https://doi.org/10.1007/s10509-012-1334-9