Few-Body Systems

, 59:43 | Cite as

Pulsating Zero Velocity Surfaces and Fractal Basin of Oblate Infinitesimal in the Elliptic Restricted Three Body Problem

  • Ashutosh Narayan
  • Anindita Chakraborty
  • Akanksha Dewangan
Article
  • 21 Downloads

Abstract

The pulsating surfaces of zero velocity of the elliptic restricted three body problem when both the primaries are luminous oblate spheroids is investigated considering the effect of the oblateness of the infinitesimal. It is observed that if the third participating body in the restricted problem has sufficiently high value of the oblateness factor, the shape of the pulsating zero velocity surfaces changes for certain values of true anomaly. The projection of the zero velocity surfaces on xy- and xz-plane is also studied along with the low-velocity sub-regions in the respective planes. Employing the multivariate Newton–Raphson iterative scheme, the basins of attraction of the equilibrium points on the xy- and xz-plane is determined. Simulation technique is applied to demonstrate the influence of the oblateness of the infinitesimal on the basins of attraction.

Mathematics Subject Classification

70F07 70E50 

Notes

Acknowledgements

The financial assistance from Chattisgarh Council of Science and Technology is duly acknowledged with gratitude (Endt. No. 2260/CCOST/MRP/2015).

References

  1. 1.
    M.C. Asique, U. Prasad, M.R. Hassan, M.S. Suraj, On the photogravitational R4BP when the third primary is a triaxial rigid body. Astrophys. Space Sci. 361, 379 (2016)ADSMathSciNetCrossRefGoogle Scholar
  2. 2.
    S.A. Astakhov, D. Farrelly, Capture and escape in the elliptic restricted three-body problem. Mon. Not. R. Astron. Soc. 354, 971 (2004).  https://doi.org/10.1111/j.1365-2966.2004.08280.x ADSCrossRefGoogle Scholar
  3. 3.
    A.N. Baltagiannis, K.E. Papadakis, Equilibrium points and their stability in the restricted four-body problem. Int. J. Bifurc. Chaos 21, 2179–2193 (2011)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Y.A. Chernikov, The photo-gravitational restricted three body problem. Sov. Astron. Astron. J. 14, 176 (1970)ADSGoogle Scholar
  5. 5.
    S. Campagnola, M. Lo, P. Newton, Subregions of motion and elliptic halo orbits in the elliptic restricted three-body problem. Adv. Astronaut. Sci. 130, 1541–1556 (2008)Google Scholar
  6. 6.
    G. Contopoulos, Integrals of motion in plane elliptic restricted three-body problems for orbits with small eccentricity near primaries. Astron. J. 72, 669 (1967)ADSCrossRefGoogle Scholar
  7. 7.
    C.N. Douskos, Collinear equilibrium points of Hills problem with radiation and oblateness and their fractal basins of attraction. Astrophys. Space Sci. 326, 263–271 (2010)ADSCrossRefMATHGoogle Scholar
  8. 8.
    C.N. Douskos, V. Kalantonis, P. Markellos, E. Perdios, On Sitnikovlike motions generating new kinds of 3D periodic orbits in the R3BP with prolate primaries. Astrophys. Space Sci. 337, 99–106 (2012)ADSCrossRefMATHGoogle Scholar
  9. 9.
    C.N. Douskos, V.V. Markellos, Out-of-plane equilibrium points in the restricted three-body problem with oblateness. Astron. Astrophys. 446, 357–360 (2006)ADSCrossRefGoogle Scholar
  10. 10.
    A. Elipe, S. Ferrer, On the equilibrium solution in the circular planar restricted three rigid bodies problem. Celest. Mech. 37, 59 (1985)ADSCrossRefGoogle Scholar
  11. 11.
    S.M. El-Shaboury, M.A. El-Tantawy, Eulerian libration points of restricted problem of three oblate spheroids. Earth Moon Planets 63, 23–28 (1978)ADSCrossRefMATHGoogle Scholar
  12. 12.
    D.P. Hamilton, J.A. Burns, Orbital stability zones about asteroids II. The destabilizing effects of eccentric orbits and of solar radiation. ICARUS 96, 43–64 (1992).  https://doi.org/10.1016/0019-1035(92)90005-R ADSCrossRefGoogle Scholar
  13. 13.
    G.W. Hill, Researches in the field of Lunar theory. Am. J. Math. 1, 129 (1878)CrossRefGoogle Scholar
  14. 14.
    C.G.J. Jacobi, Sur le mouvement d’un point et sur un cas particulier du probleme des trois corps. C. R. lAcad. Sci. 59, 1836 (1836)Google Scholar
  15. 15.
    T.J. Kalvouridis, On some new aspects of the photo-gravitational Copenhagen problem. Astrophys. Space Sci. 317, 107–117 (2008)ADSCrossRefMATHGoogle Scholar
  16. 16.
    T.J. Kalvouridis, M.C. Gousidou-Koutita, Basins of attraction in the Copenhagen problem where the primaries are magnetic dipoles. Appl. Math. 3, 541–548 (2012)CrossRefGoogle Scholar
  17. 17.
    A. Kumar, J.P. Sharma, B. Ishwar, Linear stability of triangular equilibrium points in the photogravitational restricted three body problem with Poynting–Robertson drag when both primaries are oblate spheroid. J. Dyn. Syst. Geom. Theor. 5, 193–202 (2007)MathSciNetMATHGoogle Scholar
  18. 18.
    R. Kumari, B.S. Kushvah, Stability regions of equilibrium points in restricted four-body problem with oblateness effects. Astrophys. Space Sci. 349, 693–704 (2014)ADSCrossRefGoogle Scholar
  19. 19.
    A.L. Kunitsyn, The stability of triangular liberation points in the photo-gravitational three-body problem. Prikl. Mat. Mekh. 65, 788 (2000)MATHGoogle Scholar
  20. 20.
    A.L. Kunitsyn, The stability of collinear liberation points in the photo-gravitational three-body problem. J. Appl. Math. Mech. 65, 703 (2001)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Z. Mako, F. Szenkovits, Capture in the circular and elliptic restricted three body problem. Celet. Mech. Dyn. Astron. 90, 51–58 (2004)ADSMathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Z. Mako, F. Szenkovits, Pulsating zero velocity surfaces and capture in elliptic restricted three body problem. PADEU 15, 21 (2005)Google Scholar
  23. 23.
    S.W. McCuskey, Introduction to Celestial Mechanics (Addison-Wesley, New York, 1963)MATHGoogle Scholar
  24. 24.
    A. Narayan, C. Ramesh, Effects of photogravitation and oblateness on the triangular Lagrangian points in elliptic restricted three-body problem. Int. J. Pure Appl. Math. 68, 201–224 (2011)MathSciNetGoogle Scholar
  25. 25.
    A. Narayan, C. Ramesh, Stability of triangular equilibrium points in elliptic restricted three-body problem under the effects of photogravitation and oblateness of the primaries. Int. J. Pure Appl. Math. 70, 735–754 (2011)Google Scholar
  26. 26.
    A. Narayan, A. Shrivatav, Pulsating different curves of zero velocity around triangular equilibrium points in elliptical restricted three-body problem. J. Math. (2013).  https://doi.org/10.1155/2013/936859 MathSciNetMATHGoogle Scholar
  27. 27.
    P. Oberti, A. Vienne, An upgraded theory for Helene, Telesto, and Calypso. Astron. Astrophys. 397, 353 (2003)ADSCrossRefGoogle Scholar
  28. 28.
    M.W. Ovenden, A.E. Roy, On the use of the Jacobi integral of the restricted three-body problem. Mon. Not. R. Astron. Soc. 123, 1 (1961)ADSMathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    D.W. Schuerman, The restricted three body problem including radiation pressure. Astrophs. J. 238, 337–342 (1980)ADSMathSciNetCrossRefGoogle Scholar
  30. 30.
    J. Singh, S. Haruna, Equilibrium points and stability under effect of radiation and perturbing forces in the restricted problem of three oblate bodies. Astrophys. Space Sci. 349, 107 (2014).  https://doi.org/10.1007/s10509-013-1627-7 ADSCrossRefGoogle Scholar
  31. 31.
    J. Singh, A. Umar, On ’out of plane’ equilibrium points in the elliptic restricted three body problem with radiating and oblate primaries. Astrophys. Space Sci. 344, 13–19 (2013a)ADSCrossRefMATHGoogle Scholar
  32. 32.
    J. Singh, A. Umar, Collinear equilibrium points in the Elliptic R3BP with oblateness and radiation. Adv. Space Res. 52, 1489–1496 (2013b)ADSCrossRefGoogle Scholar
  33. 33.
    J. Singh, A. Umar, Application of binary pulsars to axisymmetric bodies in the Elliptic R3BP. Astrophys. Space Sci. 348, 393 (2013c)ADSCrossRefGoogle Scholar
  34. 34.
    M.S. Suraj, M.C. Asique, U. Prasad, M.R. Hassan, K. Shalini, Fractal basins of attraction in the restricted four-body problem when the primaries are triaxial rigid bodies. Astrophys. Space Sci. 362, 211 (2017)ADSMathSciNetCrossRefGoogle Scholar
  35. 35.
    V. Szebehely, Theory of Orbits, The Restricted Problem of Three Bodies (Academic Press, London, 1967)MATHGoogle Scholar
  36. 36.
    V. Szebehely, G.E.O. Giacaglia, On the elliptic restricted problem of three bodies. Astron. J. 69, 230–235 (1964)ADSMathSciNetCrossRefGoogle Scholar
  37. 37.
    E.E. Zotos, Fractal basins of attraction in the planar circular restricted three-body problem with oblateness and radiation pressure. Astrophys. Space Sci. 361, 181 (2016)ADSMathSciNetCrossRefGoogle Scholar
  38. 38.
    E.E. Zotos, Revealing the basins of convergence in the planar equilateral restricted four-body problem. Astrophys. Space Sci. 362, 2 (2017)ADSMathSciNetCrossRefGoogle Scholar
  39. 39.
    E.E. Zotos, Determining the Newton–Raphson basins of attraction in the electromagnetic Copenhagen problem. Int. J. Nonlinear Mech. 90, 111–123 (2017)ADSCrossRefGoogle Scholar
  40. 40.
    E.E. Zotos, Basins of convergence of equilibrium points in the pseudo-Newtonian planar circular restricted three-body problem. Astrophys. Space Sci. 362, 195 (2017).  https://doi.org/10.1007/s10509-017-3172-2 ADSMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Austria, part of Springer Nature 2018

Authors and Affiliations

  • Ashutosh Narayan
    • 1
  • Anindita Chakraborty
    • 1
  • Akanksha Dewangan
    • 1
  1. 1.Department of MathematicsBhilai Institute of TechnologyDurgIndia

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