On the 3/1 planar and circular resonant problem

The simplest model of a resonant problem of second order is the planar and circular case. Simplification like this is very old and for 3/1 resonance, several authors have studied this problem with different purposes. In this work, we test this model for the available asteroids, by applying Hori's perturbation method. Explicit solutions of the intermediate orbit are obtained. In the plane of two constants of the problem, all types of motion are described. By testing the model, it is shown that, in general, one can confirm results of numerical integrations indicating libration for a few number of asteroids and circulation for most of them. However, agreement in numerical values for amplitude and period of librations seems to be not possible mainly if Jupiter's eccentricity is neglected. On the other hand, even though there might be some physical reasons determining that only asteroids with high eccentricity may librate, it is shown that, from mathematical point of view, libration may occur even in the case of small eccentricities provided that some relations are satisfied.