3.2 Linear and semilinear confidence boundaries

We now proceed to compute the differential, at the nominal orbit, of the map onto the MTP, with the formulas of Section 2, and we compute the confidence ellipse $K_{lin}(\sigma)$, for $\sigma =3$. The ellipse has very uneven semiaxes, the short one being only $\simeq 1\,300$ km, while the long one is $\simeq 1\,440\,000$ km long; this is not unexpected, since the conditioning number of the covariance matrix of the orbital elements grows at least quadratically with time (see Paper I, Section 4.1); intuitively, the orbital error accumulates along track as an effect of the original uncertainty in the semimajor axis. The good news is, the direction of the major axis of the confidence ellipse is not directed towards the origin in the MTP, where the Earth is; it is directed away from it by an angle $\beta=41^\circ$. Since the ellipse is very narrow, almost a segment, the minimum distance from the linear analysis is essentially the nominal minimum distance d_n times $\sin \beta$, in this case $0.000\,19$ AU. If the linear approximation is applicable, then $d_{n}\,\sin\beta$ is all what matters, while both d_n and $\beta$ can change for solutions using different data sets. If $d_{n}\,\sin\beta$ is safely above the surface of the Earth, this is enough to conclude that the risk of impact is essentially zero ¹.

  

Figure: On the Modified Target Plane (scales in AU; $\eta$ on the abscissa, $\zeta$ on the ordinate axis) the semilinear and the linear confidence boundaries are quite close in the portion of the plane near the the nominal solution, and therefore near the Earth; they diverge significantly near the edge of the figure.

To see whether this conclusion holds, even when taking into account the nonlinearity of the map onto the MTP, we have computed the semilinear confidence boundary $K_N(\sigma)$, for the same $\sigma =3$. Figure 3 shows a square window on the MTP with a side of about $300\,000$ km; the Earth is the circle around the center, drawn at the same scale. The linear and semilinear confidence boundaries are almost the same near the nominal solution (which is along the $\eta$ axis, towards the right in the Figure); the semilinear boundary begins to bend with respect to the major axis of the ellipse, until at a distance of about $100\,000$ km the two confidence regions do not overlap at all. Near the Earth, however, the two boundaries are essentially the same: this implies that the conclusion on the essentially zero risk of impact is valid, in this case, also within a nonlinear model.

  

Figure: Global view of the confidence boundaries on the MTP (for $\sigma =3$). Both the linear and the semilinear confidence boundaries (dotted lines) are extremely elongated, to the point that their thickness is not visible in this plot; towards the two tips, the difference between them is significant. The confidence boundary of the solution which uses also the 1990 precovery observations bounds a much smaller region, which is either inside, or very close to, the semilinear confidence region of the solution with 1997-98 data only, but well outside the linear confidence ellipse computed with the same data.

To decide whether the applicability of the linear approximation is a general result, rather than an accidental property of this example, we have plotted the entire confidence boundaries K_lin(3) and K_N(3) in Figure 4; to this purpose, we have used a window on the MTP with a side of about $2\,400\,000$ km. This makes clear that the distance of the 2028 closest approach, as assessed by using only the 1997-98 data (until March 4, 1998), could well be up to 4 times the Earth-Moon distance. The plot also shows that the linear ellipse and the semilinear confidence boundary can be far apart, near the two extremities, by more than the diameter of the Earth. In other words, the fact that the linear and the semilinear method give the same answer to the question about the possibility of an impact depends upon where the Earth is, with respect to the nominal solution and with respect to the tips of the confidence ellipse.

Thus Figure 4 allows us to conclude: (1) that the impact risk by 1997 XF₁₁ in 2028 is essentially zero, even using the 1997-98 observations only; (2) that the close approach of October 2028 is not constrained, by the 1997-98 observations, to be at a distance closer than the Moon; (3) that the linear approximation, successful as it is in this case, can not be relied upon in every case. All three conclusions have been also obtained by the Monte Carlo method [Muinonen and Bowell 1993]².