Given a catalog of virtual asteroids
propagate all the orbits from the initial epoch t0 until some
target epoch t1 and record all the close approaches to the
terrestrial planets within a distance of 0.1 AU and taking place
between t0 and t1. Note that the list of such approaches for
one solution Xi is in general different from the list for a
different, but nearby one, say
This arises from the MTP
[Milani et al. 1999] which separates the
solutions on the MTP at a given encounter by a distance
For example, if , then two consecutive solutions with are separated by d=0.2 AU on the MTP, thus they cannot both pass within 0.1 AU of the Earth at that encounter. This fact dictates the resolution limit: a close approach with may not be found during the propagation if .
The sequence of close approaches to a given planet, such as the Earth, for a given solution Xi, is controlled by the complex interplay of resonances and orbital changes resulting from close approaches, as in the case of 1999 AN10 discussed in [Milani et al. 1999]. It is possible a posteriori to explain the occurrence of a given close approach for one solution Xi as the result of a sequence of resonant and/or non-resonant returns. However, the cascade of returns is often so complex that it generates several possibilities of close approach every year; some of these possibilities have such a large stretching, hence such small probabilities of occurring, that they are not worth considering.
For this reason we have adopted a procedure which works the other way round: we use the scan of 1201 orbits to detect close approaches, in this way selecting the returns with a significant probability, then by inspecting the sequence of returns of a given solution we can easily identify the resonance mechanism which has allowed the returns to take place.