Celestial Mechanics and Dynamical Astronomy

, Volume 94, Issue 2, pp 173–195

Canonical Elements for Öpik Theory


    • Department of MathematicsUniversity of Pisa

DOI: 10.1007/s10569-005-4820-3

Cite this article as:
Tommei, G. Celestial Mech Dyn Astr (2006) 94: 173. doi:10.1007/s10569-005-4820-3


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 elementshyperbolic collision orbitsÖpik theory

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© Springer 2006