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

Abstract

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.

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References

  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)

    ADS  Article  Google 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)

    ADS  Article  Google Scholar 

  3. Ansari, A.: The photogravitational circular restricted four-body problem with variable masses. J. Eng. Appl. Sci. 3(2) (2016)

  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)

    MathSciNet  Article  Google 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)

    ADS  MATH  Google 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). https://doi.org/10.1007/s10509-009-0213-5

    ADS  Article  MATH  Google 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). https://doi.org/10.1142/S0129183114500247

    Article  Google Scholar 

  8. Froeschle, C.: The Lyapunov characteristic exponents—applications to celestial mechanics. Celest. Mech. 34, 95–115 (1984). https://doi.org/10.1007/BF01235793

    ADS  MathSciNet  Article  MATH  Google 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)

  10. Hadjidemetriou, J.D.: The restricted planetary 4-body problem. Celest. Mech. 21, 63–71 (1980). https://doi.org/10.1007/BF01230248

    ADS  MathSciNet  Article  MATH  Google Scholar 

  11. https://en.wikipedia.org/wiki/Gliese_570

  12. Jain, M., Aggarwal, R.: Restricted three-body problem with Stokes drag effect. Int. J. Astron. Astrophys. 5, 95–105 (2015). https://doi.org/10.4236/ijaa.2015.52013

    Article  Google Scholar 

  13. 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). https://doi.org/10.1016/j.pss.2006.07.0005

    ADS  Article  Google Scholar 

  14. 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)

    ADS  MATH  Google Scholar 

  15. 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). https://doi.org/10.1007/s10509-012-1340-y. arXiv:1212.2368

    ADS  Article  MATH  Google Scholar 

  16. 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). https://doi.org/10.1007/s10509-013-1689-6

    ADS  Article  Google Scholar 

  17. 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). https://doi.org/10.1007/s10509-014-2023-7

    ADS  Article  MATH  Google Scholar 

  18. Machuy, A.L., Prado, A.F., Stuchi, T.J.: Adv. Space Res. 40, 118–124 (2007)

    ADS  Article  Google Scholar 

  19. Michalodimitrakis, M.: The circular restricted four-body problem. Astrophys. Space Sci. 75, 289–305 (1981). https://doi.org/10.1007/BF00648643

    ADS  Article  Google Scholar 

  20. Murray, C.D., Dermott, S.F.: Solar System Dynamics. Cambridge University Press, Cambridge (1999)

    Google Scholar 

  21. Oseledec, V.: A multiplicative ergodic theorem, Lyapunov characteristic numbers for dynamical systems. Trans. Mosc. Math. Soc. 19, 197–231 (1968)

    MathSciNet  Google Scholar 

  22. Papadouris, J.P., Papadakis, K.E.: Equilibrium points in the photogravitational restricted four-body problem. Astrophys. Space Sci. 344, 21–38 (2013). https://doi.org/10.1007/s10509-012-1319-8

    ADS  Article  MATH  Google Scholar 

  23. 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)

    ADS  Article  Google Scholar 

  24. Sandri, M.: Numerical calculation of Lyapunov exponents. Math. J. (1995). www.msandri.it/docs/lce.m

  25. Scheurman, D.: The restricted three-body problem including radiation pressure. Astrophys. J. 238, 337–342 (1980). https://doi.org/10.1086/157989

    ADS  MathSciNet  Article  Google Scholar 

  26. 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)

    ADS  Article  Google Scholar 

  27. Schwarz, R., Suli, A., Dvorac, R.: Dynamics of possible Trojan planets in binary systems. Mon. Not. R. Astron. Soc. 398, 2085–2090 (2009b)

    ADS  Article  Google Scholar 

  28. 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). https://doi.org/10.1007/BF01230211

    ADS  Article  MATH  Google Scholar 

  29. Simmons, J., McDonald, A., Brown, J.: The restricted 3-body problem with radiation pressure. Celest. Mech. 35, 145–187 (1985)

    ADS  MathSciNet  Article  Google Scholar 

  30. 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)

    ADS  Article  Google Scholar 

  31. Singh, J., Aguda, E.V.: Equilibrium points in the restricted four-body problem with radiation pressure. Few-Body Syst. 57, 83–91 (2016). https://doi.org/10.1007/s00601-015-1030-8

    ADS  Article  Google Scholar 

  32. 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)

    ADS  Article  Google Scholar 

  33. 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). https://doi.org/10.1007/s10509-013-1707-8

    ADS  Article  Google Scholar 

  34. 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). https://doi.org/10.1007/s00601-015-1019-3

    ADS  Article  Google Scholar 

  35. 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)

    ADS  MATH  Google Scholar 

  36. Xuetang, Z., Lizhong, Y.: Chin. Phys. Lett. 10, 16 (1993)

    Google Scholar 

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Correspondence to Solomon Okpanachi Omale.

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Singh, J., Omale, S.O. Combined effect of Stokes drag, oblateness and radiation pressure on the existence and stability of equilibrium points in the restricted four-body problem. Astrophys Space Sci 364, 6 (2019). https://doi.org/10.1007/s10509-019-3494-3

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Keywords

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