The Asteroid Identification Problem Iii: Target Plane Confidence Boundaries

The nominal orbit solution for an asteroid/comet resulting from a least squares t to astrometric observations is surrounded by a region containing solutions equally compatible with the data, the con dence region. If the observed arc is not too short, and for an epoch close to the observations, the con dence region in the 6-dimensional space of orbital elements is well approximated by an ellipsoid. This uncertainty of the orbital elements maps to a position uncertainty at close approach, which can be represented on a Modi ed Target Plane (MTP), a modi cation of the one used by  Opik. The MTP is orthogonal to the geocentric velocity at the closest approach point along the nominal orbit. In the linear approximation, the con dence ellipsoids are mapped on the MTP into concentric ellipses, computed by solving the variational equation. For an object observed at only one opposition, however, if the close approach is expected after many revolutions, the ellipses on the MTP become extremely elongated, therefore the linear approximation may fail, and the con dence boundaries on the MTP, by de nition the nonlinear images of the con dence ellipsoids, may not be well approximated by the ellipses. In theory the Monte Carlo method by [Muinonen and Bowell 1993] can be used to compute the nonlinear con dence boundaries, but in practice the computational load is very heavy. We propose a new method to compute semilinear con dence boundaries on the MTP, based on the theory developed by [Milani 1999] to e ciently compute con dence boundaries for predicted observations. This method is a reasonable compromise between reliability and computational load, and can be used for real time risk assessment. These arguments can be applied to whatever small body approaching any planet, but in the case of a Potentially Hazardous Object (PHO), either an asteroid or a comet whose orbit comes very close to that of the Earth, the application is most important. We apply this technique to discuss the recent case of asteroid 1997 XF11, which, on the basis of the observations available up to March 11, 1998, appeared to be on an orbit with a near miss of the Earth in 2028. Although the least squares solution had a close approach at 1/8 of the lunar distance, the linear con dence regions corresponding to acceptable size of the residuals are very elongated ellipses which do not include collision; this computation was reported by Chodas and Yeomans. In this paper, we compute the semilinear con dence boundaries, and nd that they agree with the results of the Monte Carlo method, but di er in a signi cant way from the linear ellipses, although the di erences occur only far from the Earth. The use of the 1990 pre-discovery observations has con rmed the impossibility of an impact in 2028 and reduces the semilinear con dence regions to subsets of the regions computed with less data, as expected. The con dence regions computed using the linear approximation, on the other hand, do not reduce to subsets of the regions computed with less data. We also discuss a simulated example [Bowell and Muinonen 1992] of an Earth impacting asteroid. In this hypothetical case the semilinear con dence boundary has a completely di erent shape from the linear ellipse, and indeed for orbits determined with only few weeks of observational data the semilinear con dence boundary correctly includes possible collisions, while the linear one does not. Free software is available now, allowing everyone to compute target plane con dence boundaries as in this paper; in case a new asteroid with worrisome close approaches is discovered, our method allows to quickly perform an accurate risk assessment.