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The circular restricted eight-body problem

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Abstract

We study the motion of infinitesimal mass in the vicinity of the dominant primaries under the Newtonian law of gravitation in the restricted eight-body problem. The proposed problem is a particular case of n + 1-body problem studied by Kalvouridis (Astrophys. Space Sci 260: 309 325, 1999). We consider six peripheral primaries P1, P2, …, P6, each of mass m, revolve in a circular orbit of radius a with an angular velocity ω about their common center of mass. The primaries Pi (i = 1, 2, …, 6) are revolve in a way such that P1, P3, P5 and P2, P4, P6 always form equilateral triangles of side l and have a common circumcenter where the seventh more massive primary P0 of mass m0 rests. The primaries form a symmetric configuration with respect to the origin at any instant of time. This is observed that there exist 18 equilibrium points out of which four equilibrium points are on x-axis, two on y-axis and rest are in orbital plane of the primaries. All the equilibrium points lie on the concentric circles C1, C2 and C3 centered at origin and there exists exactly six equilibrium points on each circle. The equilibrium points on circle C2 are stable for the critical mass parameter β0 while the equilibrium points on circles C1 and C3 are unstable for all values of mass parameter β. The regions of motion for infinitesimal mass are also analyzed in this paper.

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References

  1. Ansari, A.A.: The circular restricted four-body problem with triaxial primaries and variable mass. Appl. Appl. Math. 13, 818–838 (2018)

    MathSciNet  MATH  Google Scholar 

  2. Baltagiannis, A.N., Papadakis, K.E.: Equilibrium points and their stability in the restricted four-body problem. Int. J. Bifurc. Chaos 21(8), 2179–2193 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bang, D., Elmabsout, B.: Restricted N+1-body problem: existence and stability of relative equilibria. Celest. Mech. Dyn. Astron. 89, 305–318 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bhatnagar, K.B., Hallan, P.P.: Effect of perturbed potentials on the stability of libration points in the restricted problem. Celest. Mech. 20, 95–103 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  5. Casasayas, J., Nunes, A.: A restricted charged four-body problem. Celest. Mech. Dyn. Astron. 47, 245–266 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  6. Celli, M.: The central configurations of four masses x, -x, y, -y. J. Differ. Eq. 235, 668–682 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  7. Chen, J., Yang, M.: Central configurations of the 5-body problem with four infinitesimal particles. Few-Body Syst. 61, 26 (2020)

    Article  Google Scholar 

  8. Chernikov, Y.A.: The photogravitational restricted three body problem. Soviet Astronomy-AJ 14, 176–181 (1970)

    MathSciNet  Google Scholar 

  9. Choudhary, R.K.: Libration points in the generalized elliptic restricted three body problem. Celest. Mech. 16, 411–419 (1977)

    Article  Google Scholar 

  10. Cid, R., Ferrer, S., Caballero, J.A.: Asymptotic solutions of the restricted problem near equilateral Lagrangian points. Celest. Mech. Dyn. Astron. 35, 189–200 (1985)

    Article  MATH  Google Scholar 

  11. Corbera, M., Cors, J.M., Llibre, J.: On the central configurations of the planar 1+3 body problem. Celest. Mech. Dyn. Astron. 109, 27–43 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  12. Danby, J.M.A.: Stability of the triangular points in the elliptic restricted problem of three bodies. Astron. J. 69(2), 165–172 (1964)

    Article  MathSciNet  Google Scholar 

  13. Davis, C., Geyer, S., Johnson, W., Xie, Z.: Inverse problem of central configurations in the collinear 5-body problem. J. Math. Phys. 59, 052902 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  14. Douskos, C.N.: Collinear equilibrium points of Hill’s problem with radiation and oblateness and their fractal basins of attraction. Astrophys. Space Sci. 326, 263–271 (2010)

    Article  MATH  Google Scholar 

  15. Ershkov, S.V., Leshchenko, D.: Solving procedure for 3D motions near libration points in CR3BP. Astrophys. Space Sci. 364, 207 (2019)

    Article  MathSciNet  Google Scholar 

  16. Ershkov, S., Leshchenko, D., Prosviryakov, Y.: Semi-analytical approach in BiER4BP for exploring the stable positioning of the elements of a Dyson sphere. Symmetry 15, 326 (2023)

    Article  Google Scholar 

  17. Gao, C., Yuan, J., Sun, C.: Equilibrium points and zero velocity surfaces in the axisymmetric five-body problem. Astrophys. Space Sci. 362, 72 (2017)

    Article  MathSciNet  Google Scholar 

  18. Idrisi, M.J., Taqvi, Z.A.: Restricted three-body problem when one of the primaries is an ellipsoid. Astrophys. Space Sci. 348, 41–56 (2013)

    Article  Google Scholar 

  19. Idrisi, M.J.: Existence and stability of the non-collinear libration points in the restricted three body problem when both the primaries are ellipsoid. Astrophys. Space Sci. 350, 133–141 (2014)

    Article  Google Scholar 

  20. Idrisi, M.J.: A study of libration points in modified CR3BP under albedo effect when smaller primary is an ellipsoid. J. Astronaut. Sci. 64, 379–398 (2017)

    Article  Google Scholar 

  21. Idrisi, M.J., Ullah, M.S.: Non-collinear libration points in ER3BP with albedo effect and oblateness. J. Astrophys. Astron. 39, 28 (2018)

    Article  Google Scholar 

  22. Idrisi, M.J., Ullah, M.S.: A study of albedo effects on libration points in the elliptic restricted three-body problem. J. Astronaut. Sci. 67, 863–879 (2020a)

    Article  Google Scholar 

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

    Article  Google Scholar 

  24. Idrisi, M.J., Ullah, M.S., Sikkandhar, A.: Effect of perturbations in coriolis and centrifugal forces on equilibrium points in the restricted six-body problem. J. Astronaut. Sci. 68, 4–25 (2021)

    Article  Google Scholar 

  25. Idrisi, M.J., Ullah, M.S.: Out-of-plane equilibrium points in the elliptic restricted three-body problem under albedo effect. New Astron. 89, 101629 (2021)

    Article  Google Scholar 

  26. Idrisi, M.J., Ullah, M.S.: Motion around out-of-plane equilibrium points in the frame of restricted six-body problem under radiation pressure. Few-Body Syst. 63, 50 (2022)

    Article  Google Scholar 

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

    Article  MATH  Google Scholar 

  28. Kumari, R., Kushvah, B.S.: Equilibrium points and zero velocity surfaces in the restricted four-body problem with solar wind drag. Astrophys. Space Sci. 344, 347–359 (2013)

    Article  MATH  Google Scholar 

  29. Kumari, R., Pal, A.K., Bairwa, L.K.: Periodic solution of circular Sitnikov restricted four-body problem using multiple scales method. Arch. Appl. Mech. 92, 3847–3860 (2022)

    Article  Google Scholar 

  30. Marchesin, M.: Stability of a rhomboidal configuration with a central body. Astrophys. Space Sci. 362, 1–13 (2017)

    Article  MathSciNet  Google Scholar 

  31. Markellos, V.V., Papadakis, K.E., Perdios, E.A.: Non-linear stability zones around triangular equilibria in the plane circular restricted three-body problem with oblateness. Astrophys. Space Sci. 245, 157–164 (1996)

    Article  MATH  Google Scholar 

  32. Michalodimitrakis, M.: The circular restricted four-body problem. Astrophys. Space Sci. 75, 289–305 (1981)

    Article  Google Scholar 

  33. Pacella, F.: Central configurations of the N-body problem via equivariant Morse theory. Arch. Rat. Mech. Anal. 97, 59–74 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  34. Papadouris, J.P., Papadakis, K.E.: Equilibrium points in the photogravitational restricted four-body problem. Astrophys. Space Sci. 344, 21–38 (2013)

    Article  MATH  Google Scholar 

  35. Pappalarado, C.M., Lettieri, A., Guida, D.: Stability analysis of rigid multibody mechanical systems with holonomic and nonholonomic constraints. Arch. Appl. Mech. 90, 1961–2005 (2020)

    Article  Google Scholar 

  36. Qiu-Dong, W.: The global solution of the N-body problem. Celest. Mech. Dyn. Astron. 50, 73–88 (1990)

    Article  MathSciNet  Google Scholar 

  37. Roberts, G.E.: Linear stability of the elliptic lagrangian triangle solutions in the three-body problem. J. Differ. Eq. 182(1), 191–218 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  38. Roy, A.E., Steves, B.A.: Some special restricted four-body problems–II. From caledonia to copenhagen. Planetary Space Sci. 46, 1475–1486 (1998)

    Article  Google Scholar 

  39. Siddique, M.A.R., Kashif, A.R., Shoaib, M., Hussain, S.: Stability analysis of the rhomboidal restricted six-body problem. Adv. Astronomy 6, 1–15 (2021)

    Article  Google Scholar 

  40. Siddique, M.A.R., Kashif, A.R.: The restricted six-body problem with stable equilibrium points and a rhomboidal configuration. Adv. Astronomy 2022, 8100523 (2022)

    Article  Google Scholar 

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

    MATH  Google Scholar 

  42. Turyshev, S.G., Toth, V.T.: New perturbative method for solving the gravitational N-body problem in the general theory of relativity. Int. J. Modern Phys. 24, 1550039 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  43. Ullah, M.S., Idrisi, M.J., Kumar, V.: Elliptic Sitnikov five-body problem under radiation pressure. New Astron. 80, 101398 (2020)

    Article  Google Scholar 

  44. Ullah, M.S., Idrisi, M.J., Sharma, B.K., Kaur, C.: Sitnikov five-body problem with combined effects of radiation pressure and oblateness. New Astron. 87, 101574 (2021)

    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, Worku Tenna & Andualem Derebe

  2. Department of Mathematics, Zakir Husain Delhi College, New Delhi, 110002, India

    M. Shahbaz Ullah

  3. Department of Physics, College of Natural and Computational Science, Mizan-Tepi University, Tepi, Ethiopia

    Getachew Mulu

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  1. M. Javed Idrisi
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Idrisi, M.J., Ullah, M.S., Mulu, G. et al. The circular restricted eight-body problem. Arch Appl Mech 93, 2191–2207 (2023). https://doi.org/10.1007/s00419-023-02379-3

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