The asteroid identification problem - II. Target plane confidence boundaries

The nominal orbit solution for an asteroid/comet resulting from a least squ ares fit to astrometric observations is surrounded by a region containing s olutions equally compatible with the data, the confidence region. If the ob served are is not too short, and for an epoch close to the observations, th e confidence region in the six-dimensional space of orbital elements is wel l approximated by an ellipsoid. This uncertainty of the orbital elements ma ps to a position uncertainty at close approach, which can be represented on a Modified Target plane (MTP), a modification 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 confidence ellip soids are mapped on the MTP into concentric ellipses, computed by solving t he variational equation. For an object observed at only one opposition, how ever, if the close approach is expected after many revolutions, the ellipse s on the MTP become extremely elongated, therefore the linear approximation may fail! and the confidence boundaries on the MTP, by definition the nonl inear images of the confidence ellipsoids, may not be well approximated by the ellipses. In theory the Monte Carlo method by Muinonen and Bowell (1993 , Icarus 104, 255-279) can be used to compute the nonlinear confidence boun daries, but in practice the computational load is very heavy. We propose a new method to compute semilinear confidence boundaries on the MTP, based on the theory developed by Milani (1999, Icarus 137, 269-292) to efficiently compute confidence boundaries far predicted observations. This method is a reasonable compromise between reliability and computational load, and can b e used for real time risk assessment. These arguments can be applied to any small body approaching any planet, but in the case of a potentially hazard ous object (PHO), either an asteroid or a comet whose orbit comes very clos e to that of the Earth, the application is most important. We apply this te chnique to discuss the recent case of asteroid 1997 XF11, which, on the bas is of the observations available up to March ii, 1998, appeared to be on an orbit with a near miss of the Earth in 2028. Although the least squares so lution had a close approach at 1/8 of the lunar distance, the linear confid ence regions corresponding to acceptable size of the residuals are very elo ngated ellipses which do not include collision; this computation was report ed by Chodas and Yeomans. In this paper, we compute the semilinear confiden ce boundaries and find that they agree with the results of the Monte Carlo method, but differ in a significant way from the linear ellipses, although the differences occur only far from the Earth. The use of the 1930 pre-disc overy observations has confirmed the impossibility of an impact in 2028 and reduces the semilinear confidence regions to subsets of the regions comput ed with less data, as expected. The confidence regions computed using the l inear approximation, on the other hand, do not reduce to subsets of the reg ions computed with less data. We also discuss a simulated example (Bowell a nd Muinonen 1992, Bull. Am. Astron. Soc. 24, 965) of an Earth-impacting ast eroid. In this hypothetical case the semilinear confidence boundary has a c ompletely different shape from the linear ellipse, and indeed for orbits de termined with only few weeks of observational data the semilinear confidenc e boundary correctly includes possible collisions, while the linear one doe s not. Free software is available now, allowing everyone to compute target plane c onfidence boundaries as in this paper; in case a new asteroid with worrisom e close approaches is discovered, our method allows to quickly perform an a ccurate risk assessment. (C) 1999 Academic Press.