7. Conclusions and future work

We have defined and implemented a new, numerically stable and efficient algorithm to compute the secular evolution of planet crossing orbits. The output of this computation for all known Near Earth Asteroids is now available online. This is an important step forward, since the collision singularity previously prevented this computation. However, we are well aware that this has to be considered just a first step. The following are the main limitations of this theory, and our future work will try to remove them.

First, the averaging with respect to the fast angle variables $\ell, \ell'$ is applicable provided there are no mean motion resonances. We should detect this condition and either remove the corresponding proper elements, which are not reliable, from the catalog, or use a different definition and a different algorithm, possibly along the lines of ([Milani and Baccili 1998]).

Second, this theory is of order zero in the eccentricities and inclinations of the perturbing planet; an extension to at least order one in these small quantities would be necessary. This would significantly improve the accuracy, hence the stability in time of the proper elements, and also improve the secular frequencies and allow to better detect the orbits affected by secular resonances.

Third, the very occurrence of crossings, the encounter conditions, and especially the times of the node crossings are significantly affected by the eccentricity of the target planet; even the Earth's current eccentricity of $0.017$ is not negligible.

Fourth, the proper elements should be computed starting from mean elements, that is after removing the short periodic perturbations.