Effects of Mass Variation in the Collinear Perturbed Moulton-Copenhagen Configuration

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Abdullah A. Ansari
Brijendra Yadav

Abstract

The main idea of this paper is to investigate the motion properties of the smallest body under the gravitational forces of the three collinear spherical primaries. Here we place the three primaries on the same line where the masses of two primary bodies are taken equal and third primary body is having the solar radiation effect. The effects of Coriolis and centrifugal forces on the system is considered. Therefore this system is recognized as collinear perturbed Moulton-Copenhagen configuration. After determining the equations of motion and quasi-Jacobi integral, we numerically illustrate the locations of equilibrium points (in-plane and out-of-plane), regions of motion, Poincaré surfaces of section, basins of attraction and periodic orbits. And then the examination of stability for the equilibrium points lie either in-plane or out-of-plane are examined.

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References

  1. K.B. Bhatnagar, Periodic orbits of collision in the plane circular problem of four bodies, Indian J. Pure Appl. Math. 2 (1970), 583–596.
  2. C. Simo, Relative equilibrium solutions in the four body problem, Celest. Mech. 18 (1978), 165-184. https://doi.org/10.1007/bf01228714.
  3. M. Michalodimitrakis, The circular restricted four-body problem, Astrophys. Space Sci. 75 (1981), 289-305. https://doi.org/10.1007/bf00648643.
  4. R.F. Arenstorf, Central configurations of four bodies with one inferior mass, Celest. Mech. 28 (1982), 9-15. https://doi.org/10.1007/bf01230655.
  5. J.I. Palmore, Collinear relative equilibria of the planarN-body problem, Celest. Mech. 28 (1982), 17-24. https://doi.org/10.1007/bf01230656.
  6. J.F.L. Simmons, A.J.C. McDonald, J.C. Brown, The restricted 3-body problem with radiation pressure, Celest. Mech. 35 (1985), 145-187. https://doi.org/10.1007/bf01227667.
  7. C.G. Zagouras, Periodic motion around the triangular equilibrium points of the photogravitational restricted problem of three bodies, Celest. Mech. Dyn. Astr. 51 (1991), 331-348. https://doi.org/10.1007/bf00052926.
  8. E.S.G. Leandro, On the central configurations of the planar restricted four-body problem, J. Differ. Equ. 226 (2006), 323-351. https://doi.org/10.1016/j.jde.2005.10.015.
  9. K.E. Papadakis, Asymptotic orbits in the restricted four-body problem, Planet. Space Sci. 55 (2007), 1368-1379. https://doi.org/10.1016/j.pss.2007.02.005.
  10. T.J. Kalvouridis, M. Arribas, A. Elipe, Parametric evolution of periodic orbits in the restricted four-body problem with radiation pressure, Planet. Space Sci. 55 (2007), 475-493. https://doi.org/10.1016/j.pss.2006.07.005.
  11. A.N. Baltagiannis, K.E. Papadakis, Equilibrium points and their stability in the restricted four-body problem, Int. J. Bifurcation Chaos. 21 (2011), 2179-2193. https://doi.org/10.1142/s0218127411029707.
  12. J.P. Papadouris, K.E. Papadakis, Equilibrium points in the photogravitational restricted four-body problem, Astrophys. Space Sci. 344 (2012), 21-38. https://doi.org/10.1007/s10509-012-1319-8.
  13. J. Singh, A.E. Vincent, Equilibrium points in the restricted four-body problem with radiation pressure, Few-Body Syst. 57 (2015), 83-91. https://doi.org/10.1007/s00601-015-1030-8.
  14. M. Arribas, A. Abad, A. Elipe, et al. Equilibria of the symmetric collinear restricted four-body problem with radiation pressure, Astrophys. Space Sci. 361 (2016), 84. https://doi.org/10.1007/s10509-016-2671-x.
  15. M. Arribas, A. Abad, A. Elipe, et al. Out-of-plane equilibria in the symmetric collinear restricted four-body problem with radiation pressure, Astrophys. Space Sci. 361 (2016), 270. https://doi.org/10.1007/s10509-016-2858-1.
  16. M. Palacios, M. Arribas, A. Abad, A. Elipe, Symmetric periodic orbits in the Moulton-Copenhagen problem, Celest. Mech. Dyn. Astr. 131 (2019), 16. https://doi.org/10.1007/s10569-019-9893-5.
  17. J. Llibre, D. Paşca, C. Valls, The circular restricted 4-body problem with three equal primaries in the collinear central configuration of the 3-body problem, Celest. Mech. Dyn. Astr. 133 (2021), 53. https://doi.org/10.1007/s10569-021-10052-6.
  18. J. Singh, B. Ishwar, Effect of perturbations on the stability of triangular points. In the restricted problem of three bodies with variable mass, Celest. Mech. 35 (1985), 201-207. https://doi.org/10.1007/bf01227652.
  19. A.A. Ansari, S.N. Prasad, Generalized elliptic restricted four-body problem with variable mass, Astron. Lett. 46 (2020), 275-288. https://doi.org/10.1134/s1063773720040015.
  20. A.A. Ansari, K.R. Meena, S.N. Prasad, Perturbed six-body configuration with variable mass, Romanian Astron. J. 30 (2020), 135-152.
  21. M.J. Zhang, C.Y. Zhao, Y.Q. Xiong, On the triangular libration points in photogravitational restricted threebody problem with variable mass, Astrophys. Space Sci. 337 (2011), 107-113. https://doi.org/10.1007/s10509-011-0821-8.
  22. A.A. Ansari, Effect of albedo on the motion of the infinitesimal body in circular restricted three-body problem with variable masses, Italian J. Pure Appl. Math. 38 (2017), 581-600.
  23. A.A. Ansari, The circular restricted four-body problem with triaxial primaries and variable infinitesimal mass, Appl. Appl. Math. 13 (2018), 818-838.
  24. A.A. Ansari, R. Kellil, S.K. Sahdev, Dynamical behavior of an infinitesimal variable mass body in the elliptical hill problem, Romanian Astron. J. 32 (2022), 15-33.
  25. E.I. Abouelmagd, A. Mostafa, Out of plane equilibrium points locations and the forbidden movement regions in the restricted three-body problem with variable mass, Astrophys. Space Sci. 357 (2015), 58. https://doi.org/10.1007/s10509-015-2294-7.
  26. J.H. Jeans, Astronomy and cosmogony, Cambridge University Press, Cambridge, (1928).
  27. I.V. Meshcherskii, Works on the mechanics of bodies of variable mass, GITTL, Moscow, (1949).
  28. L.G. Luk’yanov, On the restricted circular conservative three-body problem with variable masses, Astron. Lett. 35 (2009), 349-359. https://doi.org/10.1134/s1063773709050107.
  29. F. Bouaziz, A.A. Ansari, Perturbed Hill’s problem with variable mass, Astron. Nachr. 342 (2021), 666-674. https://doi.org/10.1002/asna.202113870.