Astrophysics and Space Science

, Volume 341, Issue 2, pp 331–341 | Cite as

Periodic orbits under combined effects of oblateness and radiation in the restricted problem of three bodies

  • Elbaz I. Abouelmagd
  • S. M. El-Shaboury
Original Article


In this paper, we study the existence of libration points and their linear stability when the three participating bodies are axisymmetric and the primaries are radiating, we found that the collinear points remain unstable, it is further seen that the triangular points are stable for 0<μ<μ c , and unstable for \(\mu_{c}\leq\mu\leq\tfrac{1}{2}\) where \(\mu_{c}\in(0,\tfrac{1}{2})\), it is also observed that for these points the range of stability will decrease. In addition to this we have studied periodic orbits around these points in the range 0<μ<μ c , we found that these orbits are elliptical; the frequencies of long and short orbits of the periodic motion are affected by the terms which involve parameters that characterize the oblateness and radiation repulsive forces. The implication is that the period of long periodic orbits adjusts with the change in its frequency while the period of short periodic orbit will decrease.


Stability Axisymmetric Oblateness Solar radiation pressure Restricted three- body problem 



The authors are grateful for the referees and the editor for their constructive suggestions. Also, the first author greatly thanks Prof. M. Al-Hazmi, Dr. M. Alamin and Dr. Y. Abd Elmaboud for their helping and encouragement (K.A.A. University).


  1. AbdulRaheem, A., Singh, J.: Astrophys. Space Sci. 317, 9 (2008) ADSMATHCrossRefGoogle Scholar
  2. Bhatnagar, K.B., Hallan, P.P.: Celest. Mech. 20, 95 (1979) MathSciNetADSMATHCrossRefGoogle Scholar
  3. Celleti, A.: Stability and Chaos in Celestial Mechanics. Praxis Publishing Ltd., Chichester (2010) CrossRefGoogle Scholar
  4. Elipe, A., Ferrer, S.: Celest. Mech. 37, 59 (1985) ADSCrossRefGoogle Scholar
  5. Elipe, A., Lara, M.: Celest. Mech. 68, 1 (1997) MathSciNetADSMATHCrossRefGoogle Scholar
  6. El-Shaboury, S.M.: Earth Moon Planets 45, 205 (1989) ADSCrossRefGoogle Scholar
  7. El-Shaboury, S.M.: Earth Moon Planets 48, 233 (1990) ADSMATHCrossRefGoogle Scholar
  8. El-Shaboury, S.M., El-Tantawy, M.A.: Earth Moon Planets 63, 23 (1993) ADSMATHCrossRefGoogle Scholar
  9. El-Shaboury, S.M., Shaker, M.O., El-dessoky, A.E, Eltantawy, M.A.: Earth Moon Planets 52, 69 (1991) ADSCrossRefGoogle Scholar
  10. Hénon, M.: Generating Families in the Restricted Three-Body Problem. Springer, New York (1997) MATHGoogle Scholar
  11. Mittal, A., Iqbal, A., Bhatnagar, K.B.: Astrophys. Space Sci. 63, 319 (2009) Google Scholar
  12. Murray, C.D., Dermott, S.F.: Solar System Dynamics. Cambridge University Press, Cambridge (1999) MATHGoogle Scholar
  13. Perdios, E.A.: Astrophys. Space Sci. 278, 403 (2001) ADSMATHGoogle Scholar
  14. Sharma, R.K., Taqvi, Z.A., Bhatngar, K.B.: Indian J. Pure Appl. Math. 32(7), 981 (2001) MathSciNetMATHGoogle Scholar
  15. Simmons, J.F.L., McDonald, A.J.C., Brown, J.C.: Celest. Mech. 35, 145 (1985) MathSciNetADSMATHCrossRefGoogle Scholar
  16. Singh, J.: Astrophys. Space Sci. 332, 331 (2011) ADSMATHCrossRefGoogle Scholar
  17. Singh, J., Ishwar, B.: Bull. Astron. Soc. India 27, 415 (1999) ADSGoogle Scholar
  18. Subbarao, P.V., Sharma, R.K.: Astron. Astrophys. 43, 381 (1975) ADSMATHGoogle Scholar
  19. Szebehely, V.: Theory of Orbits. Academic Press, New York (1967) Google Scholar

Copyright information

© Springer Science+Business Media B.V. 2012

Authors and Affiliations

  1. 1.Mathematics Department, Faculty of Science and ArtsKing Abdulaziz UniversityKhulaisSaudi Arabia
  2. 2.Mathematics Department, Faculty of ScienceAin Shams UniversityCairoEgypt

Personalised recommendations