3.4 Risk assessment

The question still pending, after some reasonable confidence boundary (for either $\sigma =3$ or $\sigma=4$, as discussed above) on the MTP has been computed, is whether an impact is possible, given the available observations. If the 1990 precovery observations are taken into account, this possibility can be excluded without complicated arguments. For the solution using the 1997-98 data only, we have computed the confidence boundaries, in both the linear and the semilinear approximations, corresponding to $\sigma =60$; as shown in Figure 8, only for values of the $\sigma$ parameter this high does an impact become possible, in both approximations.

  

Figure: The linear (dotted lines) and semilinear (continous lines) confidence boundaries on the MTP are plotted for $\sigma =60$, taking into account only the 1997-98 observations; they both intersect the Earth surface. This value of $\sigma$ corresponds to unacceptably high residuals.

From the discussion of the previous subsections it should be clear that we are not fully convinced that it is possible to give a quantitatively rigorous estimate of the probability of an impact for the 2028 encounter of 1997 XF₁₁ with the Earth. Even excluding the 1990 precovery observations, the probability is a very small number; to compute it, one needs to use the very extreme tails of an assumed Gaussian distribution. The fact is, the observation errors do not follow a Gaussian distribution, so much so that whenever a residual is above 3 times the RMS of 0.5 arc seconds we discard it. If a single residual were of the order of 60 times the assumed weight of 1 arc second, no one on Earth would include it in an orbital solution, but rather think that the observation belongs to another asteroid.

If, on the other hand, all the observations conspired to allow an impact in 2028, then the RMS value of the residuals would become no less than $\simeq 4$ arc seconds; this is so far along the tail of a Gaussian distribution with RMS 0.5 that the formal probability would be a very small number, but this would stretch the application of Gaussian statistics far beyond the region where it correctly represents the observation errors. Rather than using an extremely small number to describe the probability of such a solution, we would rather describe the situation in the following way. Accepting an impact solution as compatible with the 1997-98 observations would be equivalent to believing that there has not been any progress in astrometry since the second half of the eighteenth century, when typical observation errors were $\simeq 4$ arc seconds.

We find this argument, based on the size of the residuals required to allow an Earth impact, more convincing than a statement of a tiny impact probability computed from an inappropriate application of Gaussian statistics.