Heterogeneous primaries in CR4BP

  • Authors

    • Abdullah A. Ansari ICAIR
    • Ashraf Ali Majmaah University
    • Kumari Shalini University of Delhi
    • Mehtab Alam ICAIR
    2019-10-02
    https://doi.org/10.14419/ijaa.v7i2.29648
  • Equilateral Triangle, Equilibrium Points, Heterogeneous Primaries, R4BP, Stability.
  • Abstract

    This paper investigates the motion of the massless body moving under the influence of the gravitational forces of the three equal heterogeneous oblate spheroids placed at Lagrangian configuration. After determining the equations of motion and the Jacobian constant of the massless body, we have illustrated the numerical work (Stationary points, zero-velocity curves, regions of motion, Poincare surfaces of section and basins of attraction). And then we have checked the linear stability of the stationary points and found that all the stationary points are unstable.

     

  • References

    1. [1] Abdullah (2014), Periodic orbits around Lagrangian points of the circular restricted four-body problem. Invertis Journal of Science and Technology. 7(1), 29-38.

      [2] Ansari AA (2016a), Stability of the equilibrium points in the photo-gravitational circular restricted four-body problem with the effect of perturbations and variable mass. Science International (Lahore). 28, 859- 866.

      [3] Ansari AA (2016b), Stability of the equilibrium points in the circular restricted four-body problem with oblate primary and variable mass. International Journal of Advanced Astronomy. 4(1), 14-19. https://doi.org/10.14419/ijaa.v4i1.5831.

      [4] Ansari AA (2016c), the photo-gravitational circular restricted four-body problem with variable masses. Journal of Engineering and Applied Sciences. 3(2), 30-38.

      [5] Ansari AA, Kellil R, Alhussain Z (2017), The effect of perturbations on the circular restricted four-body problem with variable mass. Journal of Mathematics and Computer Science. 17(3), 365-377. https://doi.org/10.22436/jmcs.017.03.03.

      [6] Ansari AA, Alhussain Z, Prasad S (2018), The circular restricted three-body problem when both the primaries are heterogeneous spheroid of three layers and infinitesimal body varies it's mass. Journal of Astrophysics and Astronomy. 39, 57. https://doi.org/10.1007/s12036-018-9540-7.

      [7] Arribas M, Abad A, Elipe A, Palacios M (2016), Out-of-plane equilibria in the symmetric collinear restricted four-body problem with radiation pressure. Astrophys. Space Sci. 361, 270. https://doi.org/10.1007/s10509-016-2858-1.

      [8] Asique MC, et al. (2015a), On the R4BP when third primary is an oblate spheroid. Astrophys. Space Sci. 357, 82(1), https://doi.org/10.1007/s10509-015-2235-5.

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

      [10] Asique MC, et al. (2016), on the photo-gravitational R4BP when the third primary is a tri-axial rigid body. Astrophys. Space Sci. 361, 379, https://doi.org/10.1007/s10509-016-2959-x.

      [11] Asique MC, et al. (2017), On the R4BP when Third Primary is an Ellipsoid. Journal of Astronaut. Sci. 64, 231-250, https://doi.org/10.1007/s40295-016-0104-2.

      [12] Baltagiannis A, Papadakis KE (2011a), Equilibrium points and their stability in the restricted four body problem. International Journal of Bifurcation and Chaos. 21(8), 2179-2193, https://doi.org/10.1142/S0218127411029707.

      [13] Baltagiannis AN, Papadakis KE (2011b), Families of periodic orbits in the restricted four-body problem. Astrophys. Space Sci. 336, 357–367. https://doi.org/10.1007/s10509-011-0778-7.

      [14] Ceccaroni M, Biggs J (2012), Low-thrust propulsion in a coplanar circular restricted four body problem. Celest. Mech. Dyn. Astron. 112,191–219. https://doi.org/10.1007/s10569-011-9391-x.

      [15] Esteban EP, Vazquez S (2001), Rotating stratified heterogeneous oblate spheriod in Newtonian Physics, Celestial Mechanics and Dynamical Astronomy. 81, 299 – 312. https://doi.org/10.1023/A:1013292529030.

      [16] Idrisi J and Shalini K (2016), Non-collinear libration points in CR3BP when less massive primary is an heterogeneous oblate body with N-layers. International Journal of Advanced Astronomy. 4(1), 39-42. https://doi.org/10.14419/ijaa.v4i1.5928.

      [17] Kalvouridis TJ, Arribas M, Elipe A (2006a), Dynamical properties of the restricted four-body problem with radiation pressure. Mechanics research communications. 33, 811-817. https://doi.org/10.1016/j.mechrescom.2006.01.008.

      [18] Kalvouridis T, Arribas M, Elipe A (2006b), the photo-gravitational restricted four-body problem: an exploration of its dynamical properties. AIP Conf. Proc., 848, 637. https://doi.org/10.1063/1.2348041.

      [19] Kumari R, Kushvah BS (2013), Equilibrium points and zero velocity surfaces in the restricted four-body problem with solar wind drag. Astrophys. Space Sci. 344, 347-359. https://doi.org/10.1007/s10509-012-1340-y.

      [20] Kumari R, Kushvah BS (2014), Stability regions of equilibrium points in the restricted four-body problem with oblateness effects. Astrophys. Space Sci. 349, 693-704. https://doi.org/10.1007/s10509-013-1689-6.

      [21] Papadakis KE (2007), Asymptotic orbits in the restricted four-body problem. Planet. Space Sci., 55, 1368. https://doi.org/10.1016/j.pss.2007.02.005.

      [22] Papadakis KE (2016a), Families of three-dimensional periodic solutions in the circular restricted four-body problem. Astrophys. Space Sci. 361, 129. https://doi.org/10.1007/s10509-016-2713-4.

      [23] Papadakis KE (2016b), Families of asymmetric periodic solutions in the restricted four-body problem. Astrophys. Space Sci. 361, 377. https://doi.org/10.1007/s10509-016-2965-z.

      [24] Papadouris JP, Papadakis KE (2013), Equilibrium points in the photo-gravitational restricted four-body problem. Astrophys. Space Sci. 344, 21-38. https://doi.org/10.1007/s10509-012-1319-8.

      [25] Papadouris JP, Papadakis KE (2014), Periodic solutions in the photo-gravitational restricted four-body problem. MNRAS. 442, 1628-1639. https://doi.org/10.1093/mnras/stu981.

      [26] Shalini K, et al. (2017), the non-linear stability of L4 in the R3BP when the smaller primary is a heterogeneous spheroid. J. Astronaut. Sci., 64 (1), 18–49. https://doi.org/10.1007/s40295-016-0093-1.

      [27] Singh J, Vincent AE (2015a), Out-of-plane equilibrium points in the photo-gravitational restricted four-body problem. Astrophys. Space Sci. 359, 38. https://doi.org/10.1007/s10509-015-2487-0.

      [28] Singh J, Vincent AE (2015b), Effect of perturbations in the Coriolis and centrifugal forces on the stability of equilibrium points in the restricted four-body problem. Few-Body Syst. 56, 713–723. https://doi.org/10.1007/s00601-015-1019-3.

      [29] Singh J, Vincent AE (2016a), Equilibrium points in the restricted four-body problem with radiation pressure. Few-Body Syst. 57, 83-91. https://doi.org/10.1007/s00601-015-1030-8.

      [30] Singh J, Vincent AE (2016b), Out-of-plane equilibrium points in the photo-gravitational restricted four-body problem with oblateness. British Journal of Mathematics and Computer Science. 19(5), 1-15. https://doi.org/10.9734/BJMCS/2016/29698.

      [31] Suraj MS, Hassan MR, Chand MA (2014), the photo gravitational R3BP when the primaries are heterogeneous spheroid with three layers. J. Astronaut. Sci. 61 (2), 133. https://doi.org/10.1007/s40295-014-0026-9.

      [32] Suraj MS, Aggarwal R, Arora M (2017), On the restricted four-body problem with the effect of small perturbations in the Coriolis and centrifugal forces. Astrophys. Space Sci. 362, 159. https://doi.org/10.1007/s10509-017-3123-y.

      [33] Suraj MS, et al. (2017), Fractal basins of attraction in the restricted four-body problem when the primaries are tri-axial rigid bodies. Astrophys. Space Sci. 362: 211. https://doi.org/10.1007/s10509-017-3188-7.

      [34] Zotos EE (2017), revealing the basins of convergence in the planar equilateral restricted four-body problem. Astrophys. Space Sci. 362(2). https://doi.org/10.1007/s10509-017-3172-2.

  • Downloads

  • How to Cite

    A. Ansari, A., Ali, A., Shalini, K., & Alam, M. (2019). Heterogeneous primaries in CR4BP. International Journal of Advanced Astronomy, 7(2), 49-56. https://doi.org/10.14419/ijaa.v7i2.29648

    Received date: 2019-07-16

    Accepted date: 2019-09-14

    Published date: 2019-10-02