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Celestial Mechanics and Dynamical Astronomy

, Volume 120, Issue 1, pp 19–38 | Cite as

Dynamics of Kepler problem with linear drag

  • Alessandro Margheri
  • Rafael Ortega
  • Carlota Rebelo
Original Article

Abstract

We study the dynamics of Kepler problem with linear drag. We prove that motions with nonzero angular momentum have no collisions and travel from infinity to the singularity. In the process, the energy takes all real values and the angular velocity becomes unbounded. We also prove that there are two types of linear motions: capture–collision and ejection–collision. The behaviour of solutions at collisions is the same as in the conservative case. Proofs are obtained using the geometric theory of ordinary differential equations and two regularizations for the singularity of Kepler problem equation. The first, already considered in Diacu (Celest Mech Dyn Astron 75:1–15, 1999), is mainly used for the study of the linear motions. The second, the well known Levi-Civita transformation, allows to complete the study of the asymptotic values of the energy and to prove the existence of collision solutions with arbitrary energy.

Keywords

Kepler problem Linear drag Collision Levi-Civita transformation 

Notes

Acknowledgments

Rafael Ortega was supported by project MTM2011-23652, Spain. Alessandro Margheri and Carlota Rebelo were supported by Fundação para a Ciência e Tecnologia, PEst, OE/MAT/UI0209/2011 and project PTDC/MAT/113383/2009.

References

  1. Breiter, S., Jackson, A.: Unified analytical solutions to two-body problems with drag. Mon. Not. R. Astron. Soc. 299, 237–243 (1998)CrossRefADSGoogle Scholar
  2. Celletti, A., Stefanelli, L., Lega, E., Froeschlé, C.: Some results on the global dynamics of the regularized restricted three-body problem with dissipation. Celest. Mech. Dyn. Astron. 109(3), 265–284 (2011)CrossRefzbMATHADSGoogle Scholar
  3. Corne, J., Rouche, N.: Attractivity of closed sets proved by using a family of lyapunov functions. J. Diff. Eq. 13, 231–246 (1973)CrossRefzbMATHMathSciNetADSGoogle Scholar
  4. Danby, J.M.A.: Fundamentals of Celestial Mechanics. The Macmillan Company, New York (1962)Google Scholar
  5. Diacu, F.: Two body problems with drag or thrust: qualitative results. Celest. Mech. Dyn. Astron. 75, 1–15 (1999)CrossRefzbMATHMathSciNetADSGoogle Scholar
  6. Goursat, E.: Les transformations isogonales en mécanique. C. R. Acad. Sci. Paris CVIII, 446–448 (1889)Google Scholar
  7. Jacobi, C.G.J.: Jacobi’s Lectures on Dynamics, vol. 51 of Texts and Readings in Mathematics. revised edn, Hindustan Book Agency, New Delhi. Delivered at the University of Königsberg in the winter semester 1842–1843 and according to the notes prepared by C. W. Brockardt, Edited by A. Clebsch, Translated from the original German by K. Balagangadharan, Translation edited by Biswarup Banerjee (2009)Google Scholar
  8. Leach, P.G.L.: The first integrals and orbit equation for the Kepler problem with drag. J. Phys. A 20(8), 1997–2002 (1987)CrossRefzbMATHMathSciNetADSGoogle Scholar
  9. Margheri, A., Ortega, R., Rebelo, C.: Some analytical results about periodic orbits in the restricted three body problem with dissipation. Celest. Mech. Dyn. Astron. 113, 279–290 (2012)CrossRefzbMATHMathSciNetADSGoogle Scholar
  10. Mavraganis, A.G., Michalakis, D.G.: The two-body problem with drag and radiation pressure. Celest. Mech. Dyn. Astron. 58(4), 393–403 (1994)CrossRefMathSciNetADSGoogle Scholar
  11. Mittleman, D., Jezewski, D.: An analytic solution to the classical two-body problem with drag. Celest. Mech. 28(4), 401–413 (1982)CrossRefzbMATHMathSciNetADSGoogle Scholar
  12. Ortega, R.: Linear motions in a periodically forced Kepler problem. Port. Math. 68(2), 149–176 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  13. Sperling, H.J.: The collision singularity in a perturbed two-body problem. Celest. Mech. 1, 213–221 (1969/1970)Google Scholar
  14. Zhang, Z.F., Ding, T.R., Huang, W.Z., Dong, Z. X.: Qualitative Theory of differential Equations, Vol. 101 of Translations of Mathematical Monographs. American Mathematical Society, Providence, RI. Translated from the Chinese by Anthony Wing Kwok Leung (1992)Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  • Alessandro Margheri
    • 1
  • Rafael Ortega
    • 2
  • Carlota Rebelo
    • 1
  1. 1.Centro de Matemática e Aplicações FundamentaisFaculdade de Ciências Universidade de LisboaLisboaPortugal
  2. 2.Departamento de Matemática AplicadaUniversidad de GranadaGranadaSpain

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