2.2 Scanning for close approaches

Given a catalog of virtual asteroids $X_i,\; i=1, \ldots ,1201$ we 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 X_i is in general different from the list for a different, but nearby one, say $X_{i \pm 1}$. This arises from the MTP stretching $\Gamma_{\mathrm{MTP}}$ [Milani et al. 1999] which separates the solutions on the MTP at a given encounter by a distance

$$d=\Gamma_{\mathrm{MTP}}\;\left\vert\sigma(X_i)-\sigma(X_j)\right\vert\ .$$

For example, if $\Gamma_{\mathrm{MTP}} = 40\; \mathrm{AU}/\sigma$, then two consecutive solutions with $\Delta \sigma=0.005$ 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 $\Gamma_{\mathrm{MTP}} > 40 \mathrm{AU}/\sigma$ may not be found during the propagation if $\Delta \sigma \geq 0.005$.

The sequence of close approaches to a given planet, such as the Earth, for a given solution X_i, 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 X_i 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.