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Effect of Perturbations in Coriolis and Centrifugal Forces on Libration Points in the Restricted Six-Body Problem

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Abstract

This paper deals with the existence and stability of libration points in linear sense in central square configuration of restricted six-body problem when small perturbations υ and ρ are given to the coriolis and the centrifugal forces, respectively. This is observed that there exist twelve libration points out of which four libration points are collinear and eight are non-collinear and all libration points lie on the concentric circles C1, C2 and C3 centered at origin. This is also observed that the eight libration points are on the axes and four are off the axes, i.e., L1, L2, L3 and L4 are on x-axis, L5, L6, L7 and L8 are on y-axis and rest are off the axes. Also, the locations of all the libration points are only influenced by a small change in the centrifugal force. It is further observed that the stability of libration points lying on circles C1 and C3 is not influenced by the perturbations and they remain unstable for all values of mass parameter μ while the libration points lying on circle C2 are influenced by the perturbations and stable for the critical mass parameter μc.

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References

  1. AbdulRaheem, A., Singh, J.: Combined effects of perturbations, radiation and oblateness on the stability of equilibrium points in the restricted three-body problem. Astron. J. 131, 1880–1885 (2006)

    Article  Google Scholar 

  2. Aggarwal, R., Mittal, A., Suraj, M.S., Bisht, V.: The effect of small perturbations in the coriolis and centrifugal forces on the existence of libration points in the restricted four-body problem with variable mass. Astronomical Notes, https://doi.org/10.1002/asna.201813411, (2018)

  3. Ansari, A.A., Alhussain, Z.A., Kellil, R.: The effect of perturbations on the circular restricted four-body problem with variable masses. J. Math. Comput. Sci. 17, 365–377 (2017)

    Article  Google Scholar 

  4. Bhatnagar, K.B., Hallan, P.P.: Effect of perturbations in coriolis and centrifugal forces on the stability of libration points in the restricted problem. Celest. Mech. 18, 105–112 (1978)

    Article  MathSciNet  Google Scholar 

  5. Bhatnagar, K.B., Hallan, P.P.: The effect of perturbations in coriolis and centrifugal forces on the nonlinear stability of equilibrium points in the restricted problem of three bodies. Celestial Mech. 30, 97–114 (1983)

    Article  MathSciNet  Google Scholar 

  6. Hallan, P.P., Neelam, R.: Effect of perturbations in coriolis and centrifugal forces on the location and stability of the equilibrium point in the Robe’s circular restricted three body problem. Planet. Space Sci. 49(9), 957–960 (2001)

    Article  Google Scholar 

  7. Idrisi, M.J., Ullah, M.S.: Central-body square configuration of restricted-six body problem. New Astron. 79, 101381 (2020)

    Article  Google Scholar 

  8. Kalvouridis, T.J.: A planar case of the n + 1 body problem: The ‘Ring’ problem. Astrophys. Space Sci. 260, 309–325 (1999)

    Article  Google Scholar 

  9. Shrivastava, A.K., Garain, D.: Effect of perturbation on the location of libration points in the robe restricted problem of three bodies. Celest. Mech. Dyn. Astron. 51(1), 67–73 (1991)

    Article  Google Scholar 

  10. Singh, J.: Effect of perturbations on the non-linear stability of triangular points in the restricted three-body problem variable mass. Astrophys. Space Sci. 321(2), 127–135 (2009)

    Article  Google Scholar 

  11. Singh, J., Omale, A.J.: Effect of perturbations in coriolis and centrifugal forces on the locations and stability of libration points in the Robe’s circular restricted three body problem under oblate-triaxial primaries. Adv. Space Res. 55(1), 297–302 (2015)

    Article  Google Scholar 

  12. Suraj, M.S., Aggarwal, R., Arora, M.: On the restricted four-body problem with the effect of small perturbations in coriolis and centrifugal forces. Astrophys. Space Sci. 362, 159 (2017)

    Article  MathSciNet  Google Scholar 

  13. Suraj, M.S., Sachan, P., Mittal, A., Aggarwal, R.: The effect of small perturbations in the Coriolis and centrifugal forces in the axisymmetric restricted five-body problem. Astrophys. Space Sci. 364, 44 (2019)

    Article  MathSciNet  Google Scholar 

  14. Szebehely, V.: Theory of orbits, the restricted problem of three bodies. Academic Press, New York and London (1967)

    MATH  Google Scholar 

  15. Szebehely, V., Peters, C.F.: Complete solution of a general problem of three bodies. Astron. J. 72(7), 876–883 (1967)

    Article  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, College of Natural and Computational Science, Mizan-Tepi University, Tepi, Ethiopia

    M. Javed Idrisi

  2. Department of Mathematics, T. M. Bhagalpur University, Bhagalpur, Bihar, 812007, India

    M. Shahbaz Ullah

  3. Department of Civil Engineering, College of Engineering and Technology, Mizan-Tepi University, Tepi, Ethiopia

    A. Sikkandhar

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  1. M. Javed Idrisi
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  2. M. Shahbaz Ullah
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  3. A. Sikkandhar
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Correspondence to M. Javed Idrisi.

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Idrisi, M.J., Ullah, M.S. & Sikkandhar, A. Effect of Perturbations in Coriolis and Centrifugal Forces on Libration Points in the Restricted Six-Body Problem. J Astronaut Sci 68, 4–25 (2021). https://doi.org/10.1007/s40295-021-00254-3

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