Stability of the equilibrium points in the circular restricted four body problem with oblate primary and variable mass

Authors

  • Abdullah Abduljabar Ansari Majmaah University

DOI:

https://doi.org/10.14419/ijaa.v4i1.5831

Published:

2016-03-08

Keywords:

Asymptotically Stable, Circular Restricted Four Body Problem, Isosceles Triangular Configuration, Oblateness Factor, Variable Mass.

Abstract

This paper investigates the liberation points and stability of the restricted four body problem with one of the primaries as oblate body and the infinitesimal body is taken as variable mass. Due to oblateness, the equilateral triangular configuration is no longer exists and becomes an isosceles triangular configuration. Moreover, we have found seven equilibrium points out of which three are asymptotically stable (dark black in the tables) and rest four are unstable.

References

[1] Abouelmagd, EI, Mostafa, A (2015) Out of plane equilibrium points locations and the forbidden movement regions in the restricted three-body problem with variable mass. Astrophys. Space Sci. 357:58, http://dx.doi.org/10.1007/s10509-015-2294-7.

[2] Baltagiannis, AN, Papadakis, KE (2011) Equilibrium points and their stability in the restricted four-body problem. International Journal of Bifurcation and Chaos 21, 2179, http://dx.doi.org/10.1142/s0218127411029707.

[3] Bhatnagar, KB (1971) Periodic orbits of collision in the plane circular problem of four bodies. Indian Journal of Pure and Applied Math. Vol-2, No. - 4.

[4] Douskos, CN, Markellos, VV (2006) Out-of-plane equilibrium points in the restricted three body problem with oblateness. Astron. Astrophys. 446, 357–360. http://dx.doi.org/10.1051/0004-6361:20053828.

[5] Jeans, JH (1928) Astronomy and Cosmogony. Cambridge University Press, Cambridge.

[6] Khanna, M, Bhatnagar, KB (1999) Existance and stability of libration points in the restricted three body problem when the smaller primary is a triaxial rigid body and the bigger one and oblate spheroid. Indian Journal of Pure and Applied Math. 30(7), 721.

[7] McCuskey, SW (1963) Introduction to Celestial Mechanics. Addison-Wesley, Publishing Company, Inc, USA.

[8] Md. Chand Asique, et al. (a) (2015) On the R4BP when third primary is an oblate spheroid. Astrophys Space Sci. 357, 82 http://dx.doi.org/10.1007/s10509-015-2235-5.

[9] Md. Chand Asique, et al. (b) (2015) on the photogravitational R4BP when third primary is an oblate/ prolate spheroid. Astrophys. Space Sci. 360, 13 http://dx.doi.org/10.1007/s10509-015-2522-1.

[10] Meshcherskii, I.V. (1949) Studies on the mechanics of bodies of variable mass. GITTL, Moscow.

[11] Mittal, A, et al. (2009) 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. http://dx.doi.org/10.1007/s10509-008-9942-0.

[12] Moulton, FR (1900) on a class of particular solutions of the problem of four bodies. Trans. Am. Math. Soc. 1, 17-29. http://dx.doi.org/10.1090/S0002-9947-1900-1500520-3.

[13] Shrivastava, AK, Ishwar, B (1983) Equations of motion of the restricted problem of three bodies with variable mass. Celest. Mech. 30, 323-328. http://dx.doi.org/10.1007/BF01232197.

[14] Singh, J, Ishwar, B (1984) Effect of perturbations on the location of equilibrium points in the restricted problem of three bodies with variable mass. Celest. Mech. 32 (4), 297-305. http://dx.doi.org/10.1007/BF01229086.

[15] Singh, J, Ishwar, B (1985) Effect of perturbations on the stability of triangular points in the restricted problem of three bodies with variable mass. Celest. Mech. 35, 201-207. http://dx.doi.org/10.1007/BF01227652.

[16] Singh, J (2003) Photogravitational restricted three body problems with variable mass. Indian Journal of Pure and Applied Math. 32 (2) 335-341.

[17] Singh, J & Leke (2010) Stability of photogravitational restricted three body problem with variable mass, Astrophysics and Space Sci. 326 (2), 305-314 http://dx.doi.org/10.1007/s10509-009-0253-x.

[18] Szebehely, V (1967) Theory of orbits: The restricted Problem of Three Bodies. Academic Press, New York.

[19] Zhang, M J, Zhao, C Y & Xiong, Y Q (2012) on the triangular libration points in photo-gravitational restricted three body problem with variable mass. Astrophysics Space Sci. 337, 107-113, http://dx.doi.org/10.1007/s10509-011-0821-8.

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