Characteristics exponents of the triangular solution in the elliptical restricted three body problem under the radiation and oblateness of primaries

20150204 https://doi.org/10.14419/ijaa.v3i1.4074 
Elliptical Restricted Three Body Problem, Stability, Radiation, Oblateness, Binary System. 
This paper studies effects of the oblateness and radiation of both the primaries on the stability of the infinitesimal motion about triangular equilibrium points (L4,5) in the elliptical restricted three body problem (ER3BP) around the binary system We have exploited analytical method for determining of characteristics exponent to the variational equations with periodic coefficients, developed by Bennet (! 965b), which is based on the Floquet's theory. The stability of the infinitesimal motion about the triangular points under the effects of radiation and oblateness of both the primaries around the binary systems Achird, Luyten7268, Kruger 60, Alpha Centauri AB and Xi Bootis, has been studied. The stability of infinitesimal around the triangular points has been studied based on the analytical and numerical exploration is simulated by drawing transition curves bounding the region of stability in the (Î¼e) plane. The region of stability changed with variations in eccentricity, oblateness and radiation pressures. It is observed that the equilibrium points stable in the shaded portion of the transition curve, whereas unstable outside the region of the transition curves.

References
[1] Ammar M. K (20008). â€œThe effect of solar radiation pressure on the Lagrangian points in the elliptic restricted three body problemsâ€ Astrophys, space sci. Vol. 313, pp.393408, http://dx.doi.org/10.1007/s105090079709z.
[2] Bennett, A. (1965). Characteristic Exponent of the five equilibrium solutions in the elliptically restricted problems. Icarus. 4:177187. http://dx.doi.org/10.1016/00191035(65)900606.
[3] Conxita Pinyol, (1995) â€œEjection collision orbits with the more massive primary in the planar elliptic restricted three body problemâ€ Celest  Mech and Dyn Astro. Vol.61,pp.315331,
[4] Danby J.M.A.(1964) â€œStability of the triangular points in elliptic restricted problem of three bodiesâ€,The Astronomical Journal, vol. â€“ 69, pp. 165172,
[5] Erdi, B., Dajka, E.F., Nagy, I. and Rajnai, R. (2009). A parametric study of stability and resonances around L4 in the elliptical restricted three body problem. Celestial Mechanics and Dynamical Astronomy. DOI 1007/1056900991972.
[6] Gyorgrey J., (1985). â€œOn the nonlinear stability of motionâ€™s around L5 in the elliptic restricted problem of the three bodiesâ€, Celestial Mech., Dyn.Astro., vol. 36, no.3, pp.281285
[7] Khasan S.N. (1990) â€œThree dimensional periodic solutions to the radiational Hill problemâ€ Cosmic researchâ€ Vol.34, no.5, pp.299317.
[8] Khasan S.N, (1996)â€œLiberation Solutions to the radiational restricted three body problemâ€, Cosmic researchâ€ vol.34, no.2, pp.146151
[9] Kumar V. and Choudhary R.K. (1990) â€œNonlinear stability of the triangular libration points for the photogravitational elliptic restricted problem of three bodiesâ€, celestial Mech. and Dyn. Astro. Vol. 48, no. 4, pp. 299317,
[10] Markeev A.P., (1978) â€œLibration points in celestial mechanics and cosmodynamicsâ€, Nauk Moscow,
[11] Markeev A.P. (2005) â€œOne special case of parametric resonance in problem of celestial mechanicsâ€ Astronomy letter vol 31, No. 5, pp. 300356, 2005. http://dx.doi.org/10.1134/1.1922534.
[12] Markellos V.V., Papadakis K.E. and Perdios E.A. (1996). â€œNonlinear stability zones around triangular equilibria in the plane circular restricted three body problem with oblatenessâ€ Astrophysics and space science, vol.245.issue 1, pp 157164). http://dx.doi.org/10.1007/BF00637811.
[13] Markellos V.V., Perdios, E. and Labropoulou P. (1992). â€œLinear Stability of the triangular equilibrium points in the radiational elliptical restricted problemâ€ J: Astrophysics and Space Science,: pp. 207213.
[14] Meire R. (1981) â€œThe stability of the triangular points in the elliptical restricted problem.â€ Celestial Mechanics, Vol23, pp. 8995. http://dx.doi.org/10.1007/BF01228547.
[15] Narayan A. and Shrivastava Amit , (2012). â€œEffects of oblateness and radiation of primaries on the equilibrium points in the ellipted restricted three body problems.â€ International Journal of Mathematical Science, Vol. 32, issue10, pp. 330345,
[16] Narayan A. and Shrivastava Amit: â€œExistence of Resonance Stability of Triangular Equilibrium Points in Circular Case of the Planar Elliptical Restricted ThreeBody Problem under the Oblate and Radiating Primaries around the Binary Systemâ€ Advances in Astronomy; doi.org/10.1155/2014/287174(2014).
[17] Narayan A. and Singh N. (2014): â€œMotion and stability of triangular equilibrium points in elliptical restricted three body problem under the radiating primariesâ€ Astrophysics Space sci. DOI10.1007/s.105090141903I.
[18] Narayan A. and Usha T (2014): â€œEffects of radiation and triaxiality of primaries on triangular equilibrium points in elliptic restricted three body problemâ€ Astrophysics Space sci. DOI10.1007/s.105090141818x.
[19] Sandoor Zsoft and B. Erdi, (2003) â€œSympletic mapping for the Trojantype motion in the elliptic restricted three body problemâ€, Celest Mech and Dyn. Astro. vol. 86, pp.301319, 2003.
[20] Schauner T. (1971) â€œDie Bevegung in der Nach der Dreieckspu mkte des elliptischen eingeschrinkten Dreikorpen problemsâ€. Celest. Mech.3, pp. 189196, 1971. http://dx.doi.org/10.1007/BF01228032.
[21] Selaru D and CucuDumitrescu C (1995). â€œInfinitesimal orbit around Lagrange points in the elliptic restricted three body problemâ€, Celest. Mech. Dyn. Astron. Vol. 61, no. 4, pp. 333346, 1995. http://dx.doi.org/10.1007/BF00049514.
[22] Singh Jagdish and Umar Aishetu(2012) â€œOn the stability of triangular points in the elliptical R3BP under the radiating and oblate primaries.â€ Astrophys Space Science DOI 10.1007/s1050901211093. http://dx.doi.org/10.1007/s1050901211093.
[23] Zimvoschikov A.S. And Thakai V.N. (2004) â€œInstability of libration points and resonance phenomena in the photogravitaional in the elliptical restricted three body problem.â€Solar system Research, 38(2), 1554. http://dx.doi.org/10.1023/B:SOLS.0000022826.31475.

How to Cite
Narayan, A., Shrivastava, A., & Ishwar, B. (2015). Characteristics exponents of the triangular solution in the elliptical restricted three body problem under the radiation and oblateness of primaries. International Journal of Advanced Astronomy, 3(1), 816. https://doi.org/10.14419/ijaa.v3i1.4074