Dynamics of tethered satellites in the vicinity of the Lagrangian point \(L_{2}\) of the Earth–Moon system

Original Article
First Online:
Received:
Accepted:

Abstract

This paper analyzes the dynamical evolution of satellites formed by two masses connected by a cable—tethered satellites. We derive the Lagrangian equations of motion in the neighborhood of the collinear equilibrium points, especially for the \(L_{2} \), of the restricted problem of three bodies. The rigid body configuration is expanded in Legendre polynomials up to fourth degree. We present some numerical simulations of the influence of the parameters such as cable length, mass ratio and initial conditions in the behavior of the tethered satellites. The equation for the collinear equilibrium point is derived and numerically solved. The evolution of the equilibria with the variation of the cable length as a parameter is studied. We also present a discussion of the linear stability around these equilibria. Based on this analysis calculate some unstable Lyapunov orbits associated to these equilibrium points. We found periodic orbits in which the tether travels parallel to itself without involving the angular motion. The numerical applications are focused on the Earth–Moon system. However, the general character of the equations allows applications to the \(L_{1}\) equilibrium and obviously to systems other than the Earth–Moon.

Keywords

Tethered satellites Restricted three body problem Earth–Moon system Equilibrium points Linear stability Lyapunov periodic orbits 
This is a preview of subscription content, log in to check access.

Notes

Acknowledgements

The authors would like to thank the anonymous referees for valuable comments.

References

  1. Colombo, G.: The stabilization of artificial satellites at the inferior conjunction point of the Earth–Moon system. Smithsonian Astrophysical Observatory Special Report Number 80 (1961) Google Scholar
  2. Farquhar, R.W.: The flight of ISEE-3/ICE: origins, mission history and a legacy. J. Astronaut. Sci. 49(1), 91–106 (2001) MathSciNetGoogle Scholar
  3. Misra, A.K.: Dynamics of a tethered system near the Earth–Moon Lagrangian point \(L_{2}\). In: Proceedings of the International Symposium on the Systems and Structures in Space, pp. 291–300. Cambridge University Press, England (2002) Google Scholar
  4. Peláez, J., Sanjurjo, R.M., Lucas, F.R., Lara, M.: Dynamics and stability of tethered satellites at Lagrangian points. Final Report to ESA (2008) Google Scholar
  5. Simó, C.: On the analytical and numerical approximation of invariant manifolds. In: Benest, D., Froeschlé, C. (eds.) Modern Methods in Celestial Mechanics, pp. 285–329 (1971). Editions Frontiéres Google Scholar
  6. Wong, B., Patil, R., Misra, A.: Attitude dynamics of rigid bodies in the vicinity of the Lagrangian points. J. Guid. Control Dyn. 31, 252–256 (2008) ADSCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2017

Authors and Affiliations

  1. 1.Observatório do ValongoUniversidade Federal do Rio de JaneiroRio de JaneiroBrazil
  2. 2.Instituto de FísicaUniversidade Federal do Rio de JaneiroRio de JaneiroBrazil

Personalised recommendations