Advertisement

The Three-Body Interaction Effect on the Families of 3D Periodic Orbits Associated to Sitnikov Motion in the Circular Restricted Three-Body Problem

  • Omiros RagosEmail author
  • Angela E. Perdiou
  • Efstathios A. Perdios
Article
  • 29 Downloads

Abstract

This paper deals with a modified version of the Circular Restricted Three-Body Problem (CR3BP). In this version, the additional effect of a three-body interaction is taken into account. In particular, we examine numerically the result of this interaction on the evolution of the well-known family of Sitnikov motion of CR3BP as well as that on the families of 3D periodic orbits bifurcating from this family.

Keywords

Circular restricted three-body problem Sitnikov motions Body interaction Three dimensional periodic orbits Bifurcation points Numerical continuation 

Notes

Acknowledgments

The authors would like to thank the anonymous reviewers for their helpful and constructive comments that led us to greatly improve the final version of the paper.

References

  1. 1.
    Alfaro, J.M., Chiralt, C.: Invariant rotational curves in Sitnikov’s problem. Celest. Mech. Dyn. Astr. 55, 351–367 (1993)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Belbruno, E., Llibre, J., Ollé, M.: On the families of periodic orbits which bifurcate from the circular Sitnikov motions. Celest. Mech. Dyn. Astr. 60, 99–129 (1994)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Bosanac, N.: Exploring the influence of a three-body interaction added to the gravitational potential function in the circular restricted three-body problem: a numerical frequency analysis. M.Sc. thesis, School of Aeronautics and Astronautics, Purdue University, West Lafayette (2012)Google Scholar
  4. 4.
    Bosanac, N., Howell, K.C., Fischbach, E.: Exploring the Impact of a Three-Body Interaction Added to the Gravitational Potential Function in the Restricted Three-Body Problem. In: 23Rd AAS/AIAA Space Flight Mechanics Meeting, pp. 13–490. AAS, Hawaii (2013)Google Scholar
  5. 5.
    Bosanac, N., Howell, K.C., Fischbach, E.: A Natural Autonomous Force Added in the Restricted Problem and Explored via Stability Analysis and Discrete Variational Mechanics. In: 25Th AAS/AIAA Space Flight Mechanics Meeting, pp. 15–265. AAS, Williamsburg (2015)Google Scholar
  6. 6.
    Bosanac, N.: Leveraging natural dynamical structures to explore multi-body systems. Ph.D Dissertation, Purdue University, West Lafayette (2016)Google Scholar
  7. 7.
    Bosanac, N., Howell, K.C., Fischbach, E.: Leveraging Discrete Variational Mechanics to Explore the Effect of an Autonomous Three-Body Interaction Added to the Restricted Problem. Astrodynamics Network AstroNet-II. Springer (2016)Google Scholar
  8. 8.
    Bountis, T., Papadakis, K.E.: The stability of vertical motion in the N-body circular Sitnikov problem. Celest. Mech. Dyn. Astr. 104, 205–225 (2009)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Bray, T.A., Goudas, C.L.: Three dimensional oscillations about l 1, l 2, l 3. Adv. Astron. Astrophys. 5, 71–130 (1967)CrossRefGoogle Scholar
  10. 10.
    Corbera, M., Llibre, J.: Periodic orbits of the Sitnikov problem via a poincaré map. Celest. Mech. 77, 273–303 (2000)CrossRefGoogle Scholar
  11. 11.
    Douskos, C., Kalantonis, V., Markellos, P.: Effects of resonances on the stability of retrograde satellites. Astrophys. Space Sci. 310, 245–249 (2007)CrossRefGoogle Scholar
  12. 12.
    Douskos, C., Kalantonis, V., Markellos, P., Perdios, E.: On Sitnikov-like motions generating new kinds of 3D periodic orbits in the r3BP with prolate primaries. Astrophys. Space Sci. 337, 99–106 (2012)CrossRefGoogle Scholar
  13. 13.
    Douskos, C.: Effect of three-body interaction on the number and location of equilibrium points of the restricted three-body problem. Astrophys. Space Sci. 356, 251–268 (2015)CrossRefGoogle Scholar
  14. 14.
    Dvorak, R.: Numerical results to the Sitnikov-problem. Celest. Mech. Dyn. Astr. 56, 71–80 (1993)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Faruque, S.B.: Solution of the Sitnikov problem. Celest. Mech. Dyn. Astr. 87, 353–369 (2003)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Hagel, J.: A new analytic approach to the Sitnikov problem. Celest. Mech. Dyn. Astr. 53, 267–292 (1992)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Hagel, J., Llotka, C.: A High order perturbation analysis of the Sitnikov problem. Celest. Mech. Dyn. Astr. 93, 201–228 (2005)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Hénon, M.: Vertical stability of periodic orbits in the restricted problem. I. Equal masses. Astron. Astroph. 28, 415–426 (1973)zbMATHGoogle Scholar
  19. 19.
    Jalali, M.A., Pourtakdoust, S.H.: Regular and chaotic solutions of the Sitnikov problem near the 3/2 commensurability. Celest. Mech. Dyn. Astr. 68, 151–162 (1997)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Kalantonis, V.S., Perdios, E.A., Perdiou, A.E.: The Sitnikov family and the associated families of 3D periodic orbits in the photogravitational RTBP with oblateness. Astrophys. Space Sci. 315(1–4), 323–334 (2008)CrossRefGoogle Scholar
  21. 21.
    Kallrath, J., Dvorak, R., Schlöder, J.: Periodic orbits in the Sitnikov problem. in The Dynamical Behaviour of Our Planetary System (Ramsau 1996), Kluwer, Dordrecht, pp. 323–334 (1997)CrossRefGoogle Scholar
  22. 22.
    Kovács, T., Érdi, B.: The structure of the extended phase space of the Sitnikov problem. Astron. Nachr. 328(8), 801–804 (2007)CrossRefGoogle Scholar
  23. 23.
    Liu, J., Sun, Y.S.: On the Sitnikov problem. Celest. Mech. Dyn. Astr. 49, 285–302 (1990)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Markellos, V.V.: Bifurcations of plane with three-dimensional asymmetric periodic orbits in the restricted three-body problem. Mon. Not. R. Asrt. Soc. 180, 103–116 (1977)CrossRefGoogle Scholar
  25. 25.
    Ollé, M., Pacha, J.R.: The 3D elliptic restricted three-body problem: periodic orbits which bifurcate from limiting restricted problems. Astron. Astroph. 351, 1149–1164 (1999)Google Scholar
  26. 26.
    Perdios, E.A., Markellos, V.V.: Stability and bifurcations of Sitnikov motion. Celest. Mech. 42, 187–200 (1988)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Perdios, E.A.: The manifolds of families of 3D periodic orbits associated to Sitnikov motions in the restricted three-body problem. Celest. Mech. Dyn. Astr. 99 (2), 85–104 (2007)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Perdios, E.A., Kalantonis, V.S.: Self-resonant bifurcations of the Sitnikov family and the appearance of 3D isolas in the restricted three-body problem. Celest. Mech. Dyn. Astr. 113, 377–386 (2012)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Perdios, E.A., Kalantonis, V.S., Douskos, C.N.: Straight-line oscillations generating three-dimensional motions in the photogravitational restricted three-body problem. Astrophys. Space Sci. 314(1–3), 199–208 (2008)CrossRefGoogle Scholar
  30. 30.
    Poincaré, H.: Les Méthodes Nouvelles De La MéCanique CéLeste. 1, (1892); 2, (1893); 3 (1899); Gauthiers-Villars, ParisGoogle Scholar
  31. 31.
    Rahman, M.A., Garain, D.N., Hassan, M.R.: Stability and periodicity in the Sitnikov three-body problem when primaries are oblate spheroids. Astrophys. Space Sci. 357, 64 (2015)CrossRefGoogle Scholar
  32. 32.
    Robin, I.A., Markellos, V.V.: Numerical determination of three-dimensional periodic orbits generated from vertical self-resonant satellite orbits. Celest. Mech. 21, 395–434 (1980)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Seydel, R.: Practical bifurcation and stability analysis. Springer, New York (2010)CrossRefGoogle Scholar
  34. 34.
    Sidorenko, V.V.: On the circular Sitnikov problem: the alternation of stability and instability in the family of vertical motions. Celest. Mech. Dyn. Astr. 109, 367–384 (2011)MathSciNetCrossRefGoogle Scholar
  35. 35.
    Sitnikov, K.: The existence of oscillatory motions in the three-body problem. Dokl. Akad. Nauk. 133, 303–306 (1960)MathSciNetzbMATHGoogle Scholar
  36. 36.
    Szebehely, V.: Theory of orbits: The restricted problem of three bodies. Academic Press, London (1967)Google Scholar
  37. 37.
    Ullah, M.S., Majda, B., Ullah, M.Z., Ullah, M.S.: Series solutions of the Sitnikov restricted N + 1-body problem: elliptic case. Astrophys. Space Sci. 357, 166 (2015)CrossRefGoogle Scholar
  38. 38.
    Ullah, M.S.: Sitnikov problem in the square configuration: elliptic case. Astrophys. Space Sci. 361, 171 (2016)MathSciNetCrossRefGoogle Scholar

Copyright information

© American Astronautical Society 2019

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of PatrasPatrasGreece
  2. 2.Department of Civil EngineeringUniversity of PatrasPatrasGreece
  3. 3.Department of Electrical and Computer EngineeringUniversity of PatrasPatrasGreece

Personalised recommendations