Celestial Mechanics and Dynamical Astronomy

, Volume 127, Issue 1, pp 35–48 | Cite as

First integrals for the Kepler problem with linear drag

  • Alessandro Margheri
  • Rafael Ortega
  • Carlota Rebelo
Original Article
  • 233 Downloads

Abstract

In this work we consider the Kepler problem with linear drag, and prove the existence of a continuous vector-valued first integral, obtained taking the limit as \(t\rightarrow +\infty \) of the Runge–Lenz vector. The norm of this first integral can be interpreted as an asymptotic eccentricity \(e_{\infty }\) with \(0\le e_{\infty } \le 1\). The orbits satisfying \(e_{\infty } <1\) approach the singularity by an elliptic spiral and the corresponding solutions \(x(t)=r(t)e^{i\theta (t)}\) have a norm r(t) that goes to zero like a negative exponential and an argument \(\theta (t)\) that goes to infinity like a positive exponential. In particular, the difference between consecutive times of passage through the pericenter, say \(T_{n+1} -T_n\), goes to zero as \(\frac{1}{n}\).

Keywords

Kepler problem Linear drag First integral Conformally symplectic Global dynamics Runge-Lenz-type integral 

Notes

Acknowledgments

Alessandro Margheri: Supported by Fundação para a Ciência e Tecnologia, UID/MAT/04561/2013. Rafael Ortega: Supported by project MTM2014–52232–P, Spain. Carlota Rebelo: Supported by Fundação para a Ciência e Tecnologia, UID/MAT/04561/2013.

References

  1. Ai, C.: Effect of tidal dissipation on the motion of celestial bodies. Ph.D. thesis, The Pennsylvania State University, pp. 51, ISBN: 978-1303-05123-4 (2012)Google Scholar
  2. Brouwer, D., Hori, G.I.: Theoretical evaluation of atmospheric drag effects in the motion of an artificial satellite. Astron. J. 66, 193–225 (1961)ADSMathSciNetCrossRefGoogle Scholar
  3. Breiter, S., Jackson, A.: Unified analytical solutions to two-body problems with drag. Mon. Not. R. Astron. Soc. 299, 237–243 (1998)ADSCrossRefGoogle Scholar
  4. Celletti, A.: Stability and Chaos in Celestial Mechanics. Springer, Berlin (2010). (published in association with Praxis Publ. Ltd, Chichester)CrossRefMATHGoogle Scholar
  5. Calleja, R.C., Celletti, A., De la Llave, R.: A KAM theory for conformally symplectic systems: efficient algorithms and their validation. J. Differ. Equ. 255, 978–1049 (2013)ADSMathSciNetCrossRefMATHGoogle Scholar
  6. Corne, J.L., Rouche, N.: Attractivity of closed sets proved by using a family of Lyapunov functions. J. Differ. Equ. 13, 231–246 (1973)ADSCrossRefMATHGoogle Scholar
  7. Danby, J.M.A.: Fundamentals of Celestial Mechanics. The Macmillan Company, New York (1962)Google Scholar
  8. Diacu, F.: Two body problems with drag or thrust: qualitative results. Celest. Mech. Dyn. Astron. 75, 1–15 (1999)ADSMathSciNetCrossRefMATHGoogle Scholar
  9. Gorringe, V.M., Leach, P.G.L.: Hamilton-like vectors for a class of Kepler problem with a force proportional to the velocity. Celest. Mech. 41, 125–130 (1988)ADSMathSciNetCrossRefMATHGoogle Scholar
  10. Goldstein, H.: Classical Mechanics. Addison-Wesley, Reading (1980)MATHGoogle Scholar
  11. Jacobi, C.G.J.: Jacobi’s Lectures on Dynamics, Texts and Readings in Mathematics, vol. 51. Hindustan Book Agency, New Delhi (2009)Google Scholar
  12. Leach, P.G.L.: The first integrals and orbit equation for the Kepler problem with drag. J. Phys. A 20, 1997–2002 (1987)ADSMathSciNetCrossRefMATHGoogle Scholar
  13. Margheri, A., Ortega, R., Rebelo, C.: Dynamics of Kepler problem with linear drag. Celest. Mech. Dynam. Astron. 120, 19–38 (2014)ADSMathSciNetCrossRefMATHGoogle Scholar
  14. Mavraganis, A.G., Michalakis, D.G.: The two-body problem with drag and radiation pressure. Celest. Mech. 4, 393–403 (1994)ADSMathSciNetCrossRefGoogle Scholar
  15. Moser, J.: Integrals Via Asymptotics; the Störmer Problem (1963) (unpublished). http://www.math.harvard.edu/~knill/diplom/lit/Moser1963
  16. Moser, J., Zehnder, E.J.: Notes on dynamical systems. In: Courant Lecture Notes, vol. 12. AMS (2005)Google Scholar
  17. Ryabov, Y., Yankovsky, G.: An Elementary Survey of Celestial Mechanics. Dover Publications Inc, New York (1961)Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  • Alessandro Margheri
    • 1
  • Rafael Ortega
    • 2
  • Carlota Rebelo
    • 1
  1. 1.Fac. Ciências da Univ. de Lisboa e, Centro de MatemáticaAplicações Fundamentais e Investigação OperacionalLisbonPortugal
  2. 2.Departamento de Matemática AplicadaUniversidad de GranadaGranadaSpain

Personalised recommendations