3.3 The contribution of the 1990 observations

On March 12, 1998, four pre-discovery observations of 1997 XF₁₁were found on a film exposed by Helin, Lawrence and Roman on March 22 and 23, 1990 at the small Palomar Schmidt telescope, and preserved in the JPL archives. We may wonder why this detection of an Apollo asteroid went unnoticed in 1990, to the point that the positions were not even measured and astrometrically reduced. The answer is that the proper motion of 1997 XF₁₁ at that time was only -0.33degrees per day in right ascension, and 0.14 in declination; these values did not attract the attention of the blinker searching for NEOs, although in hindsight it could be argued that these are somewhat strange values for a main belt asteroid, especially considering that they were taken at $\simeq 21^\circ$ from opposition.

  

Figure: Prediction for one of the March 22, 1990 precovery observation, based upon the orbital solution for 1997 XF₁₁ fitted to the 1997-98 data. The curve is the semilinear confidence boundary (in this case, the linear ellipse would not be significantly different) for $\sigma =3$; the cross just outside the boundary curve is the actual precovery observation.

In any case, when the orbit of 1997 XF₁₁ was propagated back to 1990, it was possible to detect on the archived film and to attribute the 4 observations; the residuals with respect to the nominal orbit were as high as -7 arc minutes in right ascension and 2 arc minutes in declination, but this was within the range of uncertainty to be expected when extrapolating the orbit by more than 7 years; Figure 5 shows the $\sigma =3$ semilinear confidence boundary, as defined in Paper I, for one of these precovery observations. The same figure also shows the real observation (cross); note that it is slightly outside the $\sigma =3$ boundary, but still inside the $\sigma=4$ boundary. This could be used as an additional argument in favour of a slightly more prudent normalisation.

  

Figure: Residuals (arc seconds) in right ascension (top) and in declination (bottom) for the orbital fit of 1997 XF₁₁ by using the 1997-98 together with the 1990 observations; here only the residuals of the 1997-98 observations are shown, with the same conventions of the Figure 2. Changes in the systematic trends, present in the residuals of each observatory, are apparent.

By adding these 4 observations, a new fit to 97 observations is obtained, and because of the 30 times longer time span the confidence region in the space of orbital elements shrinks a great deal; the conditioning number of the covariance matrix becomes $153\,000$, and the largest eigenvalue becomes $2.4\times 10^{-10}$; roughly speaking, the orbital elements are better determined by an order of magnitude. The residuals of this more accurate solution are plotted in Figure 6 for the 1997-98 period; the residuals of the 1990 observations are $\leq 1$ arc second. The change in the systematic observatory dependent residuals is apparent by comparing with Figure 2: for example, the declination residuals decrease by about 0.6 arc seconds in december 1997, and increase by about 0.4 arc seconds in March 1998. Nevertheless, the overall RMS is increased only to 0.55 arc seconds, and there are no additional outliers. When propagated to the close approach time in 2028, this nominal 1990-98 orbit has a closest approach distance of $0.006\,45$AU. The next question is: where is the more accurate solution including the 1990 data, with its much smaller confidence boundary, with respect to the confidence boundary of the previous solution? If the word confidence has some meaning, it must be inside! Now we have computed two confidence boundaries, the linear one and the semilinear one; as we have seen, they are in part superimposed (near the Earth) and in part separate (near the tips of the ellipse). If the ``true'' solution, more rigorously the solution based upon more information, intersects the MTP where the two confidence regions are disjoint, then one of the two must be wrong.

  

Figure: Enlarged view of the Modified Target Plane, showing the semilinear confidence boundaries for both the solution with 1997-98 data only (dotted line) and the complete solution including the 1990 data (continous line); the curves for the latter are almost entirely included in the ones for the former, for the $\sigma =3$ value used in this plot. The linear confidence boundary of the 1997-98 solution, to the contrary, is in this region off by about one and half Earth radii from the real confidence region, and fails to contain the actual orbit, as determined with additional information.

To perform this comparison in an accurate way, we need to map the solution including the 1990 data on the same MTP on which we have traced the confidence boundary of the 1997-98 solution; we must not use the MTP defined by the closest approach point of the nominal 1990-98 solution, because the two planes are not the same, and even a small difference in the two MTP normal vectors would result in a displacement which would invalidate the comparison. The intercept point of the nominal 1990-98 orbit on the MTP of the 1997-98 orbit is at a geocentric distance of $0.006\,55$ AU (it is not the closest approach point, but the difference is small), the linear confidence ellipse is of course much smaller, for $\sigma =3$ it has semiaxes of 377 and $88\,000$ km.

In Figure 4 we have also plotted the confidence boundary (for $\sigma =3$) of the 1990-98 solution, by plotting the intersection points of the orbits forming the semilinear boundary with the MTP of the nominal 1997-98 solution; the plot shows that the semilinear confidence boundary is by far the winner. Thus, not only can the linear confidence boundary deviate by a large amount from the semilinear one, but this difference might result in an abuse of confidence, in that the ``true'' solution may well be outside the ellipse.

To better assess the reliability of the semilinear confidence boundary, we have prepared an enlarged view of the portion of the MTP containing the confidence region of the 1990-98 solution (Figure 7). The Figure shows in an even more evident fashion the inaccuracy of the linear confidence ellipse, but the enlargement allows us to see that the $\sigma =3$ semilinear confidence boundary for the 1990-98 solution pokes out of the $\sigma =3$ boundary of the 1997-98 solution; the same happens for the close approach manifold representation (the two are not very different, because the close approaches of the orbits in this window are all shallow). This could indicate that we should have used a more prudent normalisation, or equivalently, a value of the $\sigma$ parameter around 4.

We believe that this indicates that the semilinear confidence boundary method works in a way which can be considered reliable, but the normalisation problem requires further study to come out with a reliable algorithm to select weights. In most cases, including 1997 XF₁₁, the difference between the $\sigma =3$ and the $\sigma=4$ boundary does not matter as far as the possibility of an impact is concerned; however, it is always possible to contrive an example in which the Earth would be between the $\sigma=4$ and the $\sigma =3$ boundary. Even the difference between the semilinear boundary and the fully nonlinear one may be important in some cases, including the marginal ones with a semilinear boundary very near the Earth.