Stability analysis of triangular equilibrium points in restricted three-body problem under effects of circumbinary disc, radiation and drag forces

  • Jagadish Singh
  • Tajudeen Oluwafemi AmudaEmail author


This paper examines the linear stability analysis around triangular equilibrium points of a test body in the gravitational field of a low-mass post-AGB binary system, enclosed by circumbinary disc and radiating with effective Poynting–Robertson (P–R) drag force. The equations of motion are derived and positions of triangular equilibrium points are located. These points are determined by; the circumbinary disc, radiation and P–R drag. In particular, for our numerical computations of triangular equilibrium points and the linear stability analysis, we have taken a pulsating star, IRAS 11472-0800 as the first primary, with a young white dwarf star; G29-38 as the second primary. We observe that the disc does not change the positions of the triangular points significantly, except on the y-axis. However, radiation, P–R drag and the mass parameter \(\mu \) contribute effectively in shifting the location of the triangular points. Regarding the stability analysis, it is seen that these points under the combined effects of radiation, P–R drag and the disc, are unstable in the linear sense due to at least a complex root having a positive real part. In order to discern the effects of the parameters on the stability outcome, we consider the range of the mass parameter to be in the region of the Routhonian critical mass (0.038520). It is seen that in the absence of radiation and the presence of the disc, when the mass parameter is less than the critical mass, all the roots are pure imaginary and the triangular point is stable. However, when \(\mu =0.038521\), the four roots are complex, but become pure imaginary quantities when the disc is present. This proofs that the disc is a stabilizing force. On introducing the radiation force, all earlier purely imaginary roots became complex roots in the entire range of the mass parameter. Hence, the component of the radiation force is strongly a destabilizing force and induces instability at the triangular points making it an unstable equilibrium point.


Celestial mechanics R3BP radiation disc 


  1. AbdulRaheem, A., Singh, J. 2006, Astron. J. 131, 1880–1885ADSCrossRefGoogle Scholar
  2. Das, M. K., et al. 2009, JAA, 30, 177–185ADSGoogle Scholar
  3. Greaves, J. S., Holland, W. S., Moriarty-Schieven, G., Jenness, T., Zuckerman, B., McCarthy, C., Dent, W. R. F., Webb, R. A., Butner, H. M., Gear, W. K., Walker, H. J. 1998, ApJ, 506, 133–137ADSCrossRefGoogle Scholar
  4. Jiang, I. G., Yeh, L. C. 2004, AJ, 128, 923ADSCrossRefGoogle Scholar
  5. Jiang, I. G., Yeh, L. C. 2006, Ap&SS, 305, 341ADSCrossRefGoogle Scholar
  6. Kishor, R., Kushvah, B. S. 2013, MNRAS, 436, 1741ADSCrossRefGoogle Scholar
  7. Kuchner, M. J., Koresko, C. D., Brown, M. E. 1998, ApJ, 508, 81ADSCrossRefGoogle Scholar
  8. Kushvah, B. S. 2008, Ap&SS, 318, 41ADSCrossRefGoogle Scholar
  9. McCook, G. P., Sion, E. M. 1987, ApJS, 65, 603ADSCrossRefGoogle Scholar
  10. Miyamoto, M., Nagai, R. 1975, Astron. Soc. Jpn. Publ. 27, 533ADSGoogle Scholar
  11. Murray, C. D. 1994, Icarus 122, 465ADSCrossRefGoogle Scholar
  12. Olivier, E. A., Whitelock, P., Marang, F. 2001, MNRAS, 326, 490ADSCrossRefGoogle Scholar
  13. Ragos, O., Zafiropoulos, F. A. 1995, A&A, 300, 568ADSGoogle Scholar
  14. Reach, William T., et al. 2005, ApJ, 635, 161ADSCrossRefGoogle Scholar
  15. Schuerman, D. W. 1980, ApJ, 238, 337ADSMathSciNetCrossRefGoogle Scholar
  16. Simmons, J. F. L., McDonald, A. J. C., Brown, J. C. 1985, Celest. Mech., 35, 145ADSCrossRefGoogle Scholar
  17. Singh, J., Amuda, T. O. 2014, AP&SS, 350, 119ADSCrossRefGoogle Scholar
  18. Singh, J., Amuda, T. O. 2017, J. Dyn. Syst. Geom. Theor., 15, 177MathSciNetGoogle Scholar
  19. Singh, J., Leke, O. 2010, Ap&SS, 326, 305ADSCrossRefGoogle Scholar
  20. Singh, J., Leke, O. 2014a, Ap&SS, 350,143ADSCrossRefGoogle Scholar
  21. Singh, J., Leke, O. 2014b, Adv. Space Res. 54 1659ADSCrossRefGoogle Scholar
  22. Singh, J., Taura, J. J. 2014, AP&SS, 349, 681ADSCrossRefGoogle Scholar
  23. Szebehely, V. G. 1967, Theory of Orbits: The Restricted Problem of Three Bodies, Academic press, New YorkzbMATHGoogle Scholar
  24. Trilling, D. E., Stansberry’J, A., Stapelfeldt, K. R., Rieke, G. H., Su, K. Y. L., Gray, R. O., Corbally, C. J., Bryden, G., Chen, C. H., Boden, A., Beichman, C. A., 2007, ApJ 658, 1289ADSCrossRefGoogle Scholar
  25. Van Winckel, H., et al. 2009, A&A, 505, 1221ADSCrossRefGoogle Scholar
  26. Van Winckel, H., Hrivnak, B., Gorlova, N., Gielen, C., Lu, W. 2012, A&A, 542, 53CrossRefGoogle Scholar
  27. Vassiliadis, E., Wood, P. R., 1994, ApJ, 92, 125ADSGoogle Scholar
  28. Yeh, L. C., Jiang, I. G., 2006, AP&SS, 306, 189ADSCrossRefGoogle Scholar
  29. Ziyad A. Alhussain 2018, J. Taibah Univ. Sci., 12, 455CrossRefGoogle Scholar
  30. Zotos, Euaggelos E. 2016, Ap&SS, 361, 181ADSCrossRefGoogle Scholar
  31. Zuckerman, B., Becklin, E. E. 1987, Nature 330, 138ADSCrossRefGoogle Scholar

Copyright information

© Indian Academy of Sciences 2019

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of Physical SciencesAhmadu Bello UniversityZariaNigeria

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