Skip to main content

Dynamics of Kepler problem with linear drag


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.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5


  1. Besides the classical works by Levi-Civita it is interesting to mention Goursat’s paper (1889). The authors thank Dr. Lei Zhao for calling their attention to this paper.

  2. It can be proved that \({\mathcal {M}}\) is a connected manifold of dimension four that is not compact and has the same type of homotopy of a 3-sphere \(\mathbb {S}^3\).


  • Breiter, S., Jackson, A.: Unified analytical solutions to two-body problems with drag. Mon. Not. R. Astron. Soc. 299, 237–243 (1998)

    Article  ADS  Google Scholar 

  • 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)

    Article  MATH  ADS  Google Scholar 

  • Corne, J., Rouche, N.: Attractivity of closed sets proved by using a family of lyapunov functions. J. Diff. Eq. 13, 231–246 (1973)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  • Danby, J.M.A.: Fundamentals of Celestial Mechanics. The Macmillan Company, New York (1962)

    Google Scholar 

  • Diacu, F.: Two body problems with drag or thrust: qualitative results. Celest. Mech. Dyn. Astron. 75, 1–15 (1999)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  • Goursat, E.: Les transformations isogonales en mécanique. C. R. Acad. Sci. Paris CVIII, 446–448 (1889)

  • 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)

  • Leach, P.G.L.: The first integrals and orbit equation for the Kepler problem with drag. J. Phys. A 20(8), 1997–2002 (1987)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  • 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)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  • Mavraganis, A.G., Michalakis, D.G.: The two-body problem with drag and radiation pressure. Celest. Mech. Dyn. Astron. 58(4), 393–403 (1994)

    Article  MathSciNet  ADS  Google Scholar 

  • Mittleman, D., Jezewski, D.: An analytic solution to the classical two-body problem with drag. Celest. Mech. 28(4), 401–413 (1982)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  • Ortega, R.: Linear motions in a periodically forced Kepler problem. Port. Math. 68(2), 149–176 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  • Sperling, H.J.: The collision singularity in a perturbed two-body problem. Celest. Mech. 1, 213–221 (1969/1970)

  • 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)

Download references


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.

Author information

Authors and Affiliations


Rights and permissions

Reprints and Permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Margheri, A., Ortega, R. & Rebelo, C. Dynamics of Kepler problem with linear drag. Celest Mech Dyn Astr 120, 19–38 (2014).

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: