5. Conclusions

The study of the 1997 XF₁₁ scare, and even more the one of the 1992 KP₄ simulation, show that it is not possible to just rely on the experience of the usual linear analysis. The events such as extremely close approaches and impacts are rare, nevertheless one has to be ready to analyse them in a quantitative way. From this the following conclusions can be drawn.

In Paper I the theory of semilinear prediction was introduced, which can be used on any target space, provided the mapping is differentiable. If the target space is 2-dimensional , the semilinear confidence boundary curves can be computed using only a few tens to a few hundreds orbit propagations. In this paper we have applied the theory to close approach analysis by using a Modified Target Plane (MTP) as the target space. Other choices for target plane are possible.

To exclude the possibility of an impact -conversely to decide that it cannot be excluded- we recommend the following procedure:

1.

find the least squares solution for the existing observations, discard the outliers and select weights;

2.

propagate the orbit for the time span to be monitored, and detect the close approaches; note that the most dangerous close approaches are not necessarily the ones with the smallest nominal miss distance;

3.

having selected a close approach and its MTP, compute the linear confidence boundary; analyse the size and orientation of the confidence ellipse;

4.

if the linear ellipse gets close to the Earth, especially when this occurs far from the nominal orbit, use the semilinear approximation;

5.

if the semilinear approximation indicates the possibility of an impact, and also in borderline cases (e.g. if the Earth is touched by the boundaries corresponding to values of $\sigma$ between 4 and 5), resort to a fully nonlinear exploration by using the Monte Carlo method [Muinonen and Bowell 1993].

When subjected to the above procedure, a potential impactor might look like the simulated case of 1992 KP₄. The confidence boundaries on the MTP might have wild shapes, not easy to understand; they should be analysed in some non singular coordinate system, such as the one provided by the Öpik plane.

An impact by asteroid 1997 XF₁₁ in 2028 was not compatible with the observations from the discovery to March 4, 1998, unless it could be admitted that the RMS observation error was about 4 arc seconds.

The algorithms described in this paper are implemented as part of the free software package OrbFit, which can be obtained as described in the Appendix.