Let us assume that the orbit with initial conditions Xi undergoes a
close approach to the Earth at a time
Target Plane (MTP) for that encounter is the plane perpendicular to
the geocentric velocity at closest approach. Let this plane be
and the points on it be
then there is a
By computing the eigenvalues of , we find that there is again a weak direction corresponding to the long axis of the MTP ellipse. As the time elapsed from t0increases, the confidence region becomes longer and longer in the phase space, and simultaneously thinner and thinner (this follows from Liouville's theorem, by which the phase space 6-dimensional volume is invariant). Thus, when a close approach takes place decades after the initial epoch t0, the two eigenvalues of have a very large ratio. An orbital solution that moves the MTP intersection along the weak direction results in a negligible change in the value of the target function, that is a negligible increase in the RMS of the observation residuals. On the contrary, a comparatively small change in the orthogonal direction, along the minor axis of the ellipse, would result in a significant increase of the residual RMS.
Thus the points on the MTP which can be reached with a negligible increase in the RMS are the points of the straight line which is the eigenspace of the larger eigenvalue ; let Tmin be the point along that line that is closest to the center of the Earth, provided it is not too far along the line (with respect to ). The distance dmin of Tmin provides an estimate of the closest approach distance possible within the confidence ellipsoid; this estimate involves two approximations. First, the width of the ellipse is neglected; second, the nonlinear map FT is replaced by its linearization DFT. Both approximations can be removed, e.g., by the method of semilinear confidence boundaries described in [Milani and Valsecchi 1999]. However, for the purpose of finding VIs another approach is more efficient.