On the other hand planet-crossing orbits have been extensively studied by numerical integrations and these studies have led to the conclusion that secular resonances are the main dynamical phenomenon controlling their long term evolution (see Froschlé et al. 1995) . These resonances have been detected empirically as librations of some critical arguments in the numerical integration outputs; they have also been studied analytically, but only for low eccentricity, non-crossing orbits (Michel and Froschlè 1997). To predict where in the phase space these secular resonances occur, however, we should have a definition of proper frequencies for planet-crossing orbits, together with an explicit algorithm to compute them.

We have introduced a generalized definition of averaged solution, which has a rigorous meaning also for planet-crossing orbits, applicable in the approximation neglecting the eccentricities and the mutual inclinations of the planetary orbits, but without limitations on the eccentricities and the inclinations of the asteroids or comets. This allows us to define proper frequencies of precession of the perihelia and nodes, and also proper elements for planet crossing orbits with arbitrary semimajor axis, eccentricity and inclination. This computation can now be performed without neglecting the crossed planets as source of secular perturbations, as it was previously the case.

A systematic exploration of the phase space by the corresponding semianalytic theory would allow to identify the secular resonance surfaces in the planet-crossing region. This need to be followed by numerical integration tests to verify the accuracy and even the applicability of this theory, which is expected to be extremely non uniform, depending upon the geometry of the node crossing and upon the proper frequencies themselves.

2000-05-16