Astrophysics and Space Science

, Volume 326, Issue 2, pp 305–314 | Cite as

Stability of the photogravitational restricted three-body problem with variable masses

  • Jagadish Singh
  • Oni Leke
Original Article


This paper investigates the stability of equilibrium points in the restricted three-body problem, in which the masses of the luminous primaries vary isotropically in accordance with the unified Meshcherskii law, and their motion takes place within the framework of the Gylden–Meshcherskii problem. For the autonomized system, it is found that collinear and coplanar points are unstable, while the triangular points are conditionally stable. It is also observed that, in the triangular case, the presence of a constant κ, of a particular integral of the Gylden–Meshcherskii problem, makes the destabilizing tendency of the radiation pressures strong. The stability of equilibrium points varying with time is tested using the Lyapunov Characteristic Numbers (LCN). It is seen that the range of stability or instability depends on the parameter κ. The motion around the equilibrium points L i (i=1,2,…,7) for the restricted three-body problem with variable masses is in general unstable.


Celestial mechanics Variable masses 


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  1. AbdulRaheem, A., Singh, J.: Astron. J. 131, 1880 (2006) CrossRefADSGoogle Scholar
  2. Bekov, A.A.: Astron. Z. 65, 202 (1988) zbMATHADSGoogle Scholar
  3. Chetaev, N.G.: Stability of Motion. GITTL, Moscow (1955) Google Scholar
  4. Gasanov, S.A.: Astron. Lett. 34, 179 (2008) ADSGoogle Scholar
  5. Gel’fgat, B.E.: Modern Problems of Celestial Mechanics and Astrodynamics. Nauka, Moscow (1973), p. 7 Google Scholar
  6. Gylden, H.: Astron. Nachr. 109, 1 (1884) CrossRefADSGoogle Scholar
  7. Jeans, J.H.: Astronomy and Cosmogony. Cambridge University Press, Cambridge (1928) zbMATHGoogle Scholar
  8. Lu, T.-W.: Publ. Purple Mt. Obs. 9, 290 (1990) ADSGoogle Scholar
  9. Luk’yanov, L.G.: Astron. Z. 66, 180 (1989) zbMATHADSGoogle Scholar
  10. Luk’yanov, L.G.: Astron. Z. 67, 167 (1990) MathSciNetADSGoogle Scholar
  11. Lyapunov, A.M.: A General Problem of Stability of Motion. Acad. Sci. USSR, Moscow (1956) Google Scholar
  12. Maklin, J.G.: Theory of the Stability of Motion. Editorial URSS, Moscow (2004) Google Scholar
  13. Meshcherskii, I.V.: Studies on the Mechanics of Bodies of Variable Mass. GITTL, Moscow (1949) Google Scholar
  14. Meshcherskii, I.V.: Works on the Mechanics of Bodies of Variable Mass. GITTL, Moscow (1952) Google Scholar
  15. Minglibaev, M.J.: Questions of Celestial Mechanics and Stellar Dynamics. Nauka Kazakh SSR, Alma-Ata (1990), p. 36 Google Scholar
  16. Orlov, A.A.: Astron. J. Acad. Sci. Moscow 16, 52 (1939) Google Scholar
  17. Poincaré, H.: Leçons sur les Hypothésis Cosmogoniques (1911) Google Scholar
  18. Radzievskii, V.V.: Astron. J. 27, 249 (1950) MathSciNetGoogle Scholar
  19. Radzievskii, V.V.: Astron. Z. 30, 225 (1953) MathSciNetGoogle Scholar
  20. Singh, J.: Indian J. Pure Appl. Math. 32, 335 (2003) Google Scholar
  21. Singh, J.: Astrophys. Space Sci. 321, 127 (2009) zbMATHCrossRefADSGoogle Scholar
  22. Singh, J., Ishwar, B.: Celest. Mech. 32, 297 (1984) zbMATHCrossRefMathSciNetADSGoogle Scholar
  23. Singh, J., Ishwar, B.: Celest. Mech. 35, 201 (1985) zbMATHCrossRefADSGoogle Scholar
  24. Singh, J., Ishwar, B.: Bull. Astron. Soc. India 27, 415 (1999) ADSGoogle Scholar
  25. Szebehely, V.: Theory of Orbits: The Restricted Problem of Three Bodies. Academic Press, San Diego (1967a) Google Scholar
  26. Szebehely, V.: Astron. J. 72, 7 (1967b) CrossRefMathSciNetADSGoogle Scholar
  27. Tran, T.L.: Agriensis. Sect. Math. 26, 87 (1999) zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2010

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of ScienceAhmadu Bello UniversityZariaNigeria

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