Combined effect of Stokes drag, oblateness and radiation pressure on the existence and stability of equilibrium points in the restricted four-body problem

  • Jagadish Singh
  • Solomon Okpanachi OmaleEmail author
Original Article


This paper studies numerically the existence of collinear and non-collinear equilibrium points and their linear stability in the frame work of photogravitational circular restricted four-body problem with Stokes drag acting as a dissipative force and considering the first primary as a radiating body and the second primary as an oblate spheroid. The mass of the fourth body is assumed to be infinitesimal and does not affect the motion of the three primaries which are always at the vertices of an equilateral triangle (Lagrangian configuration). It is found that at constant dissipative force, and a simultaneous increase in both radiation pressure and oblateness coefficients, the curves that define the path of the motion of the infinitesimal body are found to shrink due to shifts in the positions of equilibrium points. All the collinear and non-collinear equilibrium points are found to be linearly unstable under the combined effect of radiation pressure, oblateness and Stokes drag. The energy integral is seen to be time dependent due to the presence of the drag force. More so the dynamic property of the system is investigated with the help of Lyapunov characteristic exponents (LCEs). It is found that the system is chaotic as the trajectories locally diverge from each other and the equilibrium points are chaotic attractors. We justified the relevance of the model in astronomy by applying it to a stellar system (Gliese 667C) and another found that both the existence and stability of the equilibrium points of any restricted few body system greatly depend of the value of the mass parameter.


Restricted four-body problem Stokes drag Radiation pressure Oblateness Stability LCEs 


  1. Abduraheem, 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) ADSCrossRefGoogle Scholar
  2. Abouelmagd, E.I., El-Shaboury, S.M.: Periodic orbits under combined effects of oblateness and radiation in the restricted problem of three bodies. Astrophys. Space Sci. 341, 331–341 (2012) ADSCrossRefGoogle Scholar
  3. Ansari, A.: The photogravitational circular restricted four-body problem with variable masses. J. Eng. Appl. Sci. 3(2) (2016) Google Scholar
  4. Baltagiannis, A.N., Papadakis, K.E.: Equilibrium points and their stability in the restricted four-body problem. Int. J. Bifurc. Chaos 21, 2179–2193 (2011) MathSciNetCrossRefGoogle Scholar
  5. Benettin, G., Galgani, L., Giorgilli, A., Strelcyn, J.M.: Lyapunov exponents for smooth dynamical systems and for Hamiltonian systems—a method for computing all of them. I—Theory. II—Numerical application. Meccanica 15, 19–30 (1980) ADSzbMATHGoogle Scholar
  6. Douskos, C.N.: Equilibrium points of the restricted three-body problem with equal prolate and radiating primaries, and their stability. Astrophys. Space Sci. 333, 79–87 (2011). ADSCrossRefzbMATHGoogle Scholar
  7. Dubeibe, F.L., Bermudez-Almanza, L.D.: Optimal conditions for the numerical calculation of the largest Lyapunov exponent for systems of ordinary differential equations. Int. J. Mod. Phys. C (2013). CrossRefGoogle Scholar
  8. Froeschle, C.: The Lyapunov characteristic exponents—applications to celestial mechanics. Celest. Mech. 34, 95–115 (1984). ADSMathSciNetCrossRefzbMATHGoogle Scholar
  9. Geisel, C.D.: Spacecraft orbit design in the circular restricted three-body problem using higher dimensional Poincare maps. Dissertations. Open Access, 109 (2013) Google Scholar
  10. Hadjidemetriou, J.D.: The restricted planetary 4-body problem. Celest. Mech. 21, 63–71 (1980). ADSMathSciNetCrossRefzbMATHGoogle Scholar
  11. Jain, M., Aggarwal, R.: Restricted three-body problem with Stokes drag effect. Int. J. Astron. Astrophys. 5, 95–105 (2015). CrossRefGoogle Scholar
  12. Kalvouridis, T.J., Arribas, M., Ellipe, A.: Parametric evolution of periodic orbits in the restricted four-body problem with radiation pressure. Planet. Space Sci. 55, 475–493 (2007). ADSCrossRefGoogle Scholar
  13. Khanna, M., Bhatnagar, K.B.: Existence and stability of libration points in the restricted three body problem when the smaller primary is a triaxial rigid body and the bigger one an oblate spheroid. Indian J. Pure Appl. Math. 30, 721–733 (1999) ADSzbMATHGoogle Scholar
  14. 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). arXiv:1212.2368 ADSCrossRefzbMATHGoogle Scholar
  15. Kumari, R., Kushvah, B.S.: Stability regions of equilibrium points in restricted four-body problem with oblateness effects. Astrophys. Space Sci. 349, 693–704 (2014). ADSCrossRefGoogle Scholar
  16. Kushvah, B.S., Sharma, J.P., Ishwar, B.: Nonlinear stability in the generalized photogravitational restricted three-body problem with Poynting–Robertson drag. Astrophys. Space Sci. 312, 279–293 (2007). ADSCrossRefzbMATHGoogle Scholar
  17. Machuy, A.L., Prado, A.F., Stuchi, T.J.: Adv. Space Res. 40, 118–124 (2007) ADSCrossRefGoogle Scholar
  18. Michalodimitrakis, M.: The circular restricted four-body problem. Astrophys. Space Sci. 75, 289–305 (1981). ADSCrossRefGoogle Scholar
  19. Murray, C.D., Dermott, S.F.: Solar System Dynamics. Cambridge University Press, Cambridge (1999) zbMATHGoogle Scholar
  20. Oseledec, V.: A multiplicative ergodic theorem, Lyapunov characteristic numbers for dynamical systems. Trans. Mosc. Math. Soc. 19, 197–231 (1968) MathSciNetGoogle Scholar
  21. Papadouris, J.P., Papadakis, K.E.: Equilibrium points in the photogravitational restricted four-body problem. Astrophys. Space Sci. 344, 21–38 (2013). ADSCrossRefzbMATHGoogle Scholar
  22. Robutel, P., Gabern, F.: The resonant structure of Jupiter’s Trojan asteroids—I. Long term stability and diffusion. Mon. Not. R. Astron. Soc. 372, 1463–1482 (2006) ADSCrossRefGoogle Scholar
  23. Sandri, M.: Numerical calculation of Lyapunov exponents. Math. J. (1995).
  24. Scheurman, D.: The restricted three-body problem including radiation pressure. Astrophys. J. 238, 337–342 (1980). ADSMathSciNetCrossRefGoogle Scholar
  25. Schwarz, R., Suli, A., Dvorac, R., Pilat-Lohinger, E.: Stability of Trojan planets in multi-planetary systems. Celest. Mech. Dyn. Astron. 104, 69–84 (2009a) ADSCrossRefGoogle Scholar
  26. Schwarz, R., Suli, A., Dvorac, R.: Dynamics of possible Trojan planets in binary systems. Mon. Not. R. Astron. Soc. 398, 2085–2090 (2009b) ADSCrossRefGoogle Scholar
  27. Sharma, R.K., Rao, P.V.S.: Collinear equilibria and their characteristic exponents in the restricted three-body problem when the primaries are oblate spheroids. Celest. Mech. 12, 189–201 (1975). ADSCrossRefzbMATHGoogle Scholar
  28. Simmons, J., McDonald, A., Brown, J.: The restricted 3-body problem with radiation pressure. Celest. Mech. 35, 145–187 (1985) ADSMathSciNetCrossRefGoogle Scholar
  29. Singh, J.: Combined effects of oblateness and radiation on the nonlinear stability of L4 in the restricted three-body problem. Astron. J. 137, 3286 (2009) ADSCrossRefGoogle Scholar
  30. Singh, J., Aguda, E.V.: Equilibrium points in the restricted four-body problem with radiation pressure. Few-Body Syst. 57, 83–91 (2016). ADSCrossRefGoogle Scholar
  31. Singh, J., Emmanuel, A.B.: Stability of triangular points in the photogravitational CR3BP with Poynting–Robertson drag and a smaller triaxial primary. Astrophys. Space Sci. 353(1), 97–103 (2014) ADSCrossRefGoogle Scholar
  32. Singh, J., Tujadeen, O.A.: Poynting–Robertson (P–R) drag and oblateness effects on motion around the triangular points in the photogravitational R3BP. Astrophys. Space Sci. 350, 119–126 (2014). ADSCrossRefGoogle Scholar
  33. 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). ADSCrossRefGoogle Scholar
  34. Subbarao, P.V., Sharma, R.K.: A note on the stability of the triangular points of equilibrium in the restricted three-body problem. Astron. Astrophys. 43, 381–383 (1975) ADSzbMATHGoogle Scholar
  35. Xuetang, Z., Lizhong, Y.: Chin. Phys. Lett. 10, 16 (1993) Google Scholar

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© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of Physical SciencesAhmadu Bello UniversityZariaNigeria
  2. 2.Engineering and Space Systems Department, National Space Research and Development Agency (NASRDA)Obasanjo Space CentreAbujaNigeria

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