Let us assume that the orbit with initial conditions *X*_{i} undergoes a
close approach to the Earth at a time
;
the *Modified
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
differentiable map

which can be explicitly computed by orbit propagation from epoch

By this linear map, the ellipsoid of Eq. (1) is mapped onto the target plane as an elliptical disk with equations [Milani and Valsecchi 1999]

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 *t*_{0}increases, 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 *t*_{0}, 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 *T*_{min} 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 *d*_{min} of *T*_{min} 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 *F*_{T} is replaced by its
linearization *DF*_{T}. 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.