# Effects of radiation and triaxiality of triangular equilibrium points in elliptical restricted three body problem

## DOI:

https://doi.org/10.14419/ijaa.v3i2.5302## Published:

2015-10-19## Keywords:

ER3BP, Dynamical System, Libration Points, Stablity## Abstract

This paper studies effects of the triaxiality and radiation pressure of both the primaries on the stability of the infinitesimal motion about triangular equilibrium points in the elliptical restricted three body problem(ER3BP), assuming that the bigger and the smaller primaries are triaxial and the source of radiation as well. It is observed that the motion around these points is stable under certain condition with respect to the radiation pressure and oblate triaxiality. The critical mass ratio depends on the radiation pressure, triaxiality, semi -major axis and eccentricity of the orbits. It is further analyzed that an increase in any of these parameters has destabilizing effects on the orbits of the infinitesimal.

## References

**[1] **Ammar MK (2008), The effect of solar radiation pressure on the Lagrangian points in the elliptical restricted three body problem, Astrophysics and Space Science, 313, 393-408. http://dx.doi.org/10.1007/s10509-007-9709-z.

**[2] **Bennet A (1965), Characteristic exponents of the five equilibrium solutions in the Elliptically Restricted Problem, ICARUS 4, 177-187. http://dx.doi.org/10.1016/0019-1035(65)90060-6.

**[3] **Danby JMA (1964), Stability of the triangular points in the elliptical restricted problem of three bodies, Astronomical Journal, 69,166-174.

**[4] **Eilpe A (1985), Ferrer,S. On the equilibrium solution in the circular planar restricted three rigid bodies, Celestial Mechanics, 37, 59. http://dx.doi.org/10.1007/BF01230341.

**[5] **F A Abd El-Salam (2015), Stability of Triangular equilibrium points in the elliptic restricted three body problem with oblate and triaxial Primaries, AstrophysSpaceSci, 357:15. http://dx.doi.org/10.1007/s10509-015-2308-5.

**[6] **Grebnikov E A (1964), On the stability of the Lagrangian triangle solutions of the restricted elliptic three body problem, Soviet Astronomy,8, No.3, 567-578.

**[7] **Grebnikov E A (1986), the methods of Averaging Applications, NAUKA, Moscow Revised.

**[8] **Gyorgyey J (1985), on the non-linear motions around the elliptical restricted problem of three bodies, Celestial Mechanics and Dynamical Astronomy, 36, No.3, 281-285. http://dx.doi.org/10.1007/BF01230741.

**[9] **Ishwar B &Elipe A (2001), Secular Solutions at Triangular Equilibrium Point in the Generalized Photogravitational Restricted Three Body Problem, Astrophysics and Space Science, 277, No.3, 437-446. http://dx.doi.org/10.1023/A:1012528929233.

**[10] **Khanna M &Bhatnagar K B (1998), Existence and Stability of the Libration points in the restricted three body problem when the smaller primary is a triaxial rigid body, Indian Journal of Pure and Applied Mathematics,29(10), 1011-1023.

**[11] **Kumar V & Choudhary R K (1990), Non-linear stability of the triangular libration points for the photogravitational elliptic restricted problem of three bodies, Celestial Mechanics and Dynamical Astronomy, 48, No.4, 299-317.http://dx.doi.org/10.1007/BF00049387.

**[12] **Kumar S &Ishwar B (2009), Solutions of Generalized Photogravitational Elliptical Restricted Three Body Problems, AIP Conf.Proc. 1146,456.http://dx.doi.org/10.1063/1.3183564.

**[13] **Markeev A P (1978), Libration points in celestial mechanics and cosmic dynamics, NAUKA Moscow, 312.

**[14] **Markellos V V, Perdios E &Labrapoulou P (1992), linear stability of the triangular equilibrium points in the photogravitational elliptic restricted three body problem, Astrophysics and Space Science, 194, 207-214.http://dx.doi.org/10.1007/BF00643991.

**[15] **McCusky S W (1963), Introduction to celestial Mechanics, Addison Wesley.

**[16] **Narayan A & Kumar C R (2011), Effects of photogravitational and oblateness on the Triangular Lagrangian points in the Elliptical Restricted three body problem, Indian Journal of Pure and Applied Mathematics,68, No.2, 201-224.

**[17] **Narayan A &Usha T (2014), Stability of triangular equilibrium points in the elliptic restricted problem of three bodies with radiation and triaxial primaries, Astrophysics and Space Science, 351(1), 135-142.http://dx.doi.org/10.1007/s10509-014-1818-x.

**[18] **Singh, J &Aishetu U (2012a), Motion in the photogravitational elliptic restricted three body problem under an oblate primary, Astronomical Journal, 143, No.5, 109.http://dx.doi.org/10.1088/0004-6256/143/5/109.

**[19] **Singh J &Aishetu U (2012b), On the stability of triangular points in the elliptic restricted three bodies under the radiating and oblate Primaries, Astrophysics and Space Science ,341, 349-358.http://dx.doi.org/10.1007/s10509-012-1109-3.

**[20] **Subbarao P V & Sharma R K (1975), A Note on the Stability of the Triangular points of Equilibrium in the Restricted-Three body problem, Astronomy and Astrophysics, 43,381-383.

**[21] **Szebehely V (1975), Stability of the points of equilibrium in restricted problem, Astronomical Journal, 72,7-9.http://dx.doi.org/10.1086/110195.

**[22] **Szebehely V (1967), Theory of Orbits, Academic press, New-York.

**[23] **Usha T, Narayan A &Ishwar B (2014), Effects of radiation and triaxiality of primaries on triangular equilibrium points in elliptic restricted three body problem, Astrophysics and Space Science, 349(1), 151-164.http://dx.doi.org/10.1007/s10509-013-1655-3.

**[24] **Zimvoschikova S &Thkai V N (2004), Instability of libration points and resonance phenomena in the photogravitational elliptical restricted three body problem, Solar system research, 38(2), 155-163.