Astronomisches Rechen-Institut Heidelberg

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ARI-Heidelberg Mitteilungen Serie B, No. 33

Author(s): Giffen, R.
Title: A Study of Commensurable Motion in the Asteroid Belt
Source: Astron. Astrophys. 23, 387-403
Year: 1973
Abstract: Schubart's averaging method for studying the secular effects of commensurable motion is applied to the 2/1 Hecuba Gap and the 3/2 Hilda Group in the asteroid belt. The development of this method, which is based on the planar, elliptic restricted three-body problem, and the assumptions used to arrive at this model are discussed. Although a study of this commensurable motion in the circular restricted problem shows no significant differences between these two resonance cases, a study of periodic solutions to the averaged elliptic restricted problem indicates significant differences in the behavior of the motion for both cases. The existence of a non-classical integral for the averaged elliptical problem permits the use of invariant curves in the study of these two resonances, analogous to Henon's study of the circular problem. Here again, apparent differences between the Hilda and Hecuba cases exist. These results indicate that resonant motion is possible over a wide range of eccentricities in the Hilda case. In the Hecuba case, however, a very complex motion is indicated for small eccentricities and long term resonant motion appears possible only for large values of the eccentricity. These results compare favorably with observations.
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