The Journal of the Astronautical Sciences

, Volume 64, Issue 3, pp 231–250 | Cite as

On the R4BP when Third Primary is an Ellipsoid

  • Md Chand Asique
  • Umakant Prasad
  • M. R. Hassan
  • Md Sanam Suraj
Article

Abstract

The present paper deals with the restricted four-body problem (R4BP), when the third primary placed at the triangular libration point of the restricted three-body problem is an ellipsoid. The third primary m3 is not influencing the motion of the dominating primaries m1 and m2. We have studied the motion of m4, moving under the influence of the three primaries mi, i = 1, 2, 3, but the motion of the primaries is not being influenced by infinitesimal mass m4. Further, we have developed the equations of motion of infinitesimal mass m4 which involves elliptic integrals and shows the existence and locations of equilibrium points. We have also discussed the zero velocity curves(ZVCs) for various value of Jacobian constant.

Keywords

R4BP Ellipsoid Libration points ZVCs 

References

  1. 1.
    Alvarez-Ramírez, M., Barrabés, E.: Transport orbits in an equilateral restricted four-body problem. Celest. Mech. Dyn. Astr. 121, 191–210 (2015)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Alvarez-Ramírez, M., Skea, J.E.F., Stuchi, T.J.: Nonlinear stability analysis in a equilateral restricted four-body problem. Astrophys. Space Sci. 358, 3 (2015)CrossRefGoogle Scholar
  3. 3.
    Amit, M., Aggarwal, R., Suraj, M.S., Bisht, V.S.: Stability of libration points in the restricted four-body problem with variable mass. Astrophys. Space Sci 361, 329 (2016). doi:10.1007/s10509-016-2901-2 MathSciNetCrossRefGoogle Scholar
  4. 4.
    Arribas, M., Abad, A., Elipe1, A., Palacios, M.: Equilibria of the symmetric collinear restricted four-body problem with radiation pressure. Astrophys. Space Sci. 384, 361 (2016)Google Scholar
  5. 5.
    Baltagiannis, A.N., Papadakis, K.E.: Equilibrium points and their stability in the restricted four-body poblem. Int. J. Bifurc. Chaos. 21, 2179–2193 (2011a)CrossRefMATHGoogle Scholar
  6. 6.
    Baltagiannis, A.N., Papadakis, K.E.: Periodic solutions in the Sun-Jupiter-Trojan Asteroid-Spacecraft system. Planet. Space Sci. 75, 148—157 (2013)CrossRefGoogle Scholar
  7. 7.
    Burgaos-Garcia, J., Gidea, M.: Hill’s approximation in a restricted four-body problem. Celest. Mech. Dyn. Astr. 122, 117–141 (2015)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Burgos-García, J., Delgado, J.: Periodic orbits in the restricted four-body problem with two equal masses. Astrophys. Space Sci. 345, 247–263 (2013)CrossRefMATHGoogle Scholar
  9. 9.
    Chernikov, Y.A.: The photogravitational restricted three-body problem. Sov. Astron. AJ 14, 176 (1970)MathSciNetGoogle Scholar
  10. 10.
    Ceccaroni, M., Biggs, J.: Extension of low-thrust propulsion to the Autonomous Coplanar Circular Restricted Four Body Problem with application to future Trojan Asteroid missions. In: 61st International Astronautical Congress, IAC-10-1.1.3, Prague (2010)Google Scholar
  11. 11.
    Ceccaroni, M., Biggs, J.: Low-thrust propulsion in a Coplanar Circular Restricted Four Body Problem. Celest. Mech. Dyn. Astr. 112, 191–219 (2012)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Chand, M.A., Umakant, P., Hassan, M.R., Suraj, M.S.: On the R4BP when third primary is an oblate spheroid. Astrophys. Space. Sci. 357, 87 (2015a). doi:10.1007/s10509-015-2235-5 CrossRefGoogle Scholar
  13. 13.
    Chand, M.A., Umakant, P., Hassan, M.R., Suraj, M.S.: On the photogravitational R4BP when third primary is an oblate/prolate spheroid. Astrophys. Space. Sci. 360, 313 (2015b). doi:10.1007/s10509-015-2522-1 Google Scholar
  14. 14.
    Chand, M.A., Umakant, P., Hassan, M.R., Suraj, M.S.: On the photogravitational R4BP when the third primary is a triaxial rigid body. Astrophys. Space Sci. 361, 379 (2016). doi:10.1007/s10509-016-2959-x MathSciNetCrossRefGoogle Scholar
  15. 15.
    Croustalloudi and Kalvouridis, T.J.: The restricted 2+2 body problem : Parametric variation of the equilibrium states of the minor bodies and their attracting regions. ISRN Astronomy and Astrophysics, Volume 2013 Article ID 281849 (2013)Google Scholar
  16. 16.
    Douskos, C., Kalantonis, V., Markellos, P., Perdios, E.: On Sitnikov-like motionsgenerating new kinds of 3D periodic orbits in the R3BP with prolate primaries. Astrophys. Space Sci. 337, 99–106 (2012)CrossRefMATHGoogle Scholar
  17. 17.
    Idrisi, M.J., Taqvi, Z.A.: Restricted three-body problem when one of the primaries is an ellipsoid. Astrophys. Space Sci. 348(1), 41–56 (2013)CrossRefGoogle Scholar
  18. 18.
    Idrisi, M.J., Taqvi, Z.A.: Eistence and stability of the non-collinear libration points in restricted three-body problem when both the primaries are ellipsoid. Astrophys. Space Sci. 350(1), 133–141 (2014)CrossRefGoogle Scholar
  19. 19.
    Idrisi, M.J., Taqvi, Z.A.: Eistence and stability of libration points in CR3BP when the smallar primary is an oblate spheroid. Astrophys. Space Sci. 354(1), 311–325 (2014)Google Scholar
  20. 20.
    Kalvouridis, T., Arribas, M., Elipe, A.: Dynamical properties of the restricted four-body problem with radiation pressure. Mech. Res. Commun. 33, 811 (2006a)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Kalvouridis, T., Arribas, M., Elipe, A.: The photo-gravitational restricted four-body problem: an exploration of its dynamical properties. AIP Conf. Proc. 848, 637 (2006b)CrossRefMATHGoogle Scholar
  22. 22.
    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)CrossRefMATHGoogle Scholar
  23. 23.
    Moulton, F.R.: On a class of particular solutions of the problem of four bodies. Trans. Am. Math. Soc. 1, 17 (1900)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Papadouris, J.P., Papadakis, K.E.: Equilibrium points in the photogravitational restricted four-body problem. Astrophys. Space Sci. 344, 21–38 (2013)CrossRefMATHGoogle Scholar
  25. 25.
    Papadakis, K.E.: Families of three dimensional periodic solutions in the circular restricted four-body problem. Astrophys. Space Sci. 361, 129 (2016)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Singh, J., Vincent, A.E.: Out-of-plane equilibrium points in the photogravitational restricted four-body problem. Astrophys. Space Sci. 359, 38 (2015)CrossRefGoogle Scholar
  27. 27.
    Singh, J., Vincent, A.E.: Equilibrium Points in the Restricted Four-Body Problem with Radiation Pressure. Few-Body Syst. doi:10.1007/s00601-015-1030-8
  28. 28.
    Singh, J., Vincent, A.E.: 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 (2015). doi:10.1007/s00601-015-1019-3 CrossRefGoogle Scholar
  29. 29.
    Suraj, M.S., Hassan, M.R.: Sitnikov restricted four-body problem with radiation pressure. Astrophys. Space Sci. 349(2), 705–716 (2014)CrossRefGoogle Scholar
  30. 30.
    Suraj, M.S., Hassan, M.R., Chand, M.A.: The Photo-Gravitational R3BP when the Primaries are Heterogeneous Spheroid with Three Layers. J. Astronaut. Sci. 61 (2014)Google Scholar
  31. 31.
    Zotos, E.E.: Escape and collision dynamics in the planar equilateral restricted four- body problem. Int. J. Non Linear Mech. 86, 66–82 (2016)CrossRefGoogle Scholar

Copyright information

© American Astronautical Society 2016

Authors and Affiliations

  • Md Chand Asique
    • 1
  • Umakant Prasad
    • 2
  • M. R. Hassan
    • 3
  • Md Sanam Suraj
    • 4
  1. 1.Research Scholar, Department of PhysicsTilka Manjhi Bhagalpur UniversityBhagalpurIndia
  2. 2.Department of Physics, TNB College, BhagalpurTilka Manjhi Bhagalpur UniversityBiharIndia
  3. 3.P.G.Department of Mathematics, S M College BhagalpurBhagalpurIndia
  4. 4.Department of Mathematics, SGTB Khalsa CollegeUniversity of Delhi North CampusNew DelhiIndia

Personalised recommendations