### Effects of Triaxiality and Radiation Pressure on Existence of Resonance Stability of Triangular Equilibrium Points in Elliptical Restricted Three-Body Problem

#### Abstract

This paper deals with the effects of radiation and triaxiality on linear stability of the triangular librations points for small value of eccentricity ‘\(e\)’ in presence of resonance. The study is carried out for various values of radiation pressure and Triaxiality parameter of both Primaries. It is found that the parametric resonance is only possible at the resonance frequency \(\omega_2 =1/2\) in elliptical case.

#### Keywords

#### Full Text:

PDF#### References

M. K. Ammar, The effect of solar radiation pressure on the Lagrangian points in the elliptical restricted three body problem, Astrophysics and Space Science 313 (2008), 393 – 408, DOI: 10.1007/s10509-007-9709-2.

A. Bennet, Characteristic exponents of the five equilibrium solutions in the Elliptically Restricted Problem, Icarus 4 (1965), 177 – 187, DOI: 10.1016/0019-1035(65)90060-6.

J. M. A. Danby, Stability of the triangular points in the elliptical restricted problem of three bodies, Astronomical Journal 69 (1964), 166 – 174, DOI: 10.1086/109254.

E. A. Grebnikov, On the stability of the Lagrangian triangle solutions of the restricted elliptic three body problem, Soviet Astronomy 8(3) (1964), 567 – 578.

J. Gyorgyey, On the non-linear motions around the elliptical restricted problem of three bodies, Celestial Mechanics and Dynamical Astronomy 36(3) (1985), 281 – 285, DOI: 10.1007/BF01230741.

S. Kumar and B. Ishwar, Location of collinear equilibrium points in the generalized photogravitational elliptic restricted three body problem, Int. J. Eng. Sci. Tech. 3(2) (2011), 157 – 162, DOI: 10.4314/ijest.v3i2.68143.

V. Kumar and R. K. Choudhary, Non-linear stability of the triangular libration points for the photogravitational elliptic restricted problem of three bodies, Celestial Mechanics and Dynamical Astronomy 48(4) (1990), 299 – 317, DOI: 10.1007/BF00049387.

A. P. Markeev, Libration points in celestial mechanics and cosmic dynamics, Izdatel’stvo Nauka (1978), 312 p. (in Russian).

A. P. Markeev, On one special case of parametric resonance in problems of celestial mechanics, Astronomy Letters 31(5) (2005), 350 – 356.

V. V. Markellos, E. Perdios and P. Labrapoulou, Linear stability of the triangular equilibrium points in the photogravitational elliptic restricted three body problem, Astrophysics and Space Science 194 (1992), 207 – 214, DOI: 10.1007/BF00643991.

F. R. Moulton, On the stability of direct and retrograde orbits, Astron. Soc. 75 (1914), 40 – 57, DOI: 10.1093/mnras/75.2.40.

A. Narayan and C. R. Kumar, 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(2) (2011), 201 – 224.

A. Narayan and N. Singh, Motion and stability of triangular equilibrium points in elliptical restricted three body problem under the radiating primaries, Astrophys. Space Sci. 352(1) (2014), 57 – 70, DOI: 10.1007/s10509-014-1903-1.

A. Narayan and T. Usha, Stability of triangular equilibrium points in the elliptic restricted problem of three bodies with radiation and triaxial primaries, Astrophysics and Space Science 351(1) (2014), 135 – 142, DOI: 10.1007/s10509-014-1818-x.

A. Narayan and A. Shrivastava, Existence of resonance stability of triangular equilibrium points in circular case of the planar elliptical restricted three-body problem under the oblate and radiating primaries around the binary system, Advances in Astronomy 2014 (2014) 1 – 17, DOI: 10.1155/2014/287174.

A. Narayan and N. Singh, Resonance stability of triangular equilibrium points in elliptical restricted three body problem under the radiating primaries, Astrophysics and Space Science 353(2) (2014), 441 – 455, DOI: 10.1007/s10509-014-2085-6.

A. Narayan, K. K. Pandey, S. K. Shrivastava, Effects of radiation and triaxiality of triangular equilibrium points in elliptical restricted three body problem, International Journal of Advanced Astronomy 3(2) (2015), 97 – 106, DOI: 10.14419/ijaa.v3i2.5302.

A. Narayan, K. K. Pandey and S. K. Shrivastava, Characteristics exponents of the triangular solutions in elliptical restricted three-body problem under radiating and triaxial primaries, International Journal of Advanced Astronomy 5(1) (2017), 12 – 18, DOI: 10.14419/ijaa.v5.i3.7114.

J. Singh and U. Aishetu, Motion in the photogravitational elliptic restricted three body problem under an oblate primary, Astronomical Journal 143(5) (2012), 109 (22 pages), DOI: 10.1088/0004-6256/143/5/109.

J. Singh and U. Aishetu, On the stability of triangular points in the elliptic restricted three bodies under the radiating and oblate Primaries, Astrophysics and Space Science 341 (2012), 349 – 358, DOI: 10.1007/s10509-012-1109-3.

V. Szebebely, Stability of the points of equilibrium in the restricted problem, Astron. J. 72 (1967), 7 – 9, DOI: 10.1086/110195.

T. Usha, A. Narayan and B. Ishwar, Effects of radiation and triaxiality of primaries on triangular equilibrium points in elliptic restricted three body problem, Astrophysics and Space Science 349(1) (2014), 151 – 164, DOI: 10.1007/s10509-013-1655-3.

T. Usha and A. Narayan, Linear stability and resonance of triangular equilibrium points in elliptical restricted three-body problem with radiating primary and triaxial secondary, International Journal of Advanced Astronomy 4(2) (2016), 82 – 89, DOI: 10.14419/ijaa.v4i2.6536.

A. S. Zimvoschikov and V. N. Thkai, Instability of libration points and resonance phenomena in the photogravitational elliptical restricted three body problem, Solar System Research 38(2) (2004), 155 – 163, DOI: 10.1023/B:SOLS.0000022826.31475.97.

DOI: http://dx.doi.org/10.26713%2Fjims.v11i3-4.897

eISSN 0975-5748; pISSN 0974-875X