Celestial Mechanics and Dynamical Astronomy

, Volume 94, Issue 2, pp 173–195 | Cite as

Canonical Elements for Öpik Theory



The purpose of this paper is to find a set of canonical elements to use within the framework of Öpik theory of close encounters of a small body with a planet (Öpik, Interplanetary Encounters, 1976). Since the small body travels along a planetocentric hyperbola during the close approach and Öpik formulas are valid, without approximations, only at collision, we derive a set of canonical elements for hyperbolic collision orbits (eccentricity e → 1+, semi-major axis a fixed) and then we introduce the unperturbed velocity of the small body and the distance covered along the asymptote as a new canonically conjugate pair of orbital elements. An interesting result would be to get a canonical set containing the coordinates in the Target Plane (TP), useful for the analysis of the future encounters: in the last part we prove that this is not possible.


canonical elements hyperbolic collision orbits Öpik theory 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Carusi, A., Valsecchi, G. B., Greenberg, R. 1990‘Planetary close encounters: Geometry of approach and post-encounter orbital parameters’Celest. Mech. Dyn. Astr.49111131ADSGoogle Scholar
  2. Floria, L. 1995‘A simple derivation of the hyperbolic Delaunay variables’Astron. J.110940942ADSGoogle Scholar
  3. Goldstein, H.: 1980, Classical Mechanics, Second Edition, Addison Wesley.Google Scholar
  4. Hori, G. 1961‘The motion of a hyperbolic artificial satellite around the oblate earth’Astron. J.66258263ADSMathSciNetGoogle Scholar
  5. Öpik, E. J. 1976Interplanetary EncountersElsevierAmsterdam-Oxford-New YorkGoogle Scholar
  6. Tremaine, S. 2001‘Canonical elements for collision orbits’Celes. Mech. Dyn. Astr.79231233ADSMATHGoogle Scholar
  7. Valsecchi, G. B., Milani, A., Gronchi, G. F., Chesley, S. R. 2003‘Resonant returns to close approaches: Analytical theory’Astr. Astrophys.40811791196ADSGoogle Scholar

Copyright information

© Springer 2006

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of PisaPisaItaly

Personalised recommendations