THE ASTEROID IDENTIFICATION PROBLEM II:
TARGET PLANE CONFIDENCE BOUNDARIES

Andrea Milani
Dipartimento di Matematica, Università di Pisa
Via Buonarroti 2
56127 PISA, ITALY
E-mail: milani@dm.unipi.it


Giovanni B. Valsecchi
IAS-Planetologia
Area di ricerca CNR
Via Fosso del Cavaliere
00133 ROMA, ITALY
E-mail: giovanni@ias.rm.cnr.it

Revised version, February 23, 1999
Manuscript pages: 25; Figures: 10.

Running title: Target Plane Confidence Boundaries

Keywords: Asteroids, Dynamics; Comets, Dynamics; Impact Processes

Abstract:

The nominal orbit solution for an asteroid/comet resulting from a least squares fit to astrometric observations is surrounded by a region containing solutions equally compatible with the data, the confidence region. If the observed arc is not too short, and for an epoch close to the observations, the confidence 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 Modified Target Plane (MTP), a modification of the one used by Öpik. The MTP is orthogonal to the geocentric velocity at the closest approach point along the nominal orbit. In the linear approximation, the confidence 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 confidence boundaries on the MTP, by definition the nonlinear images of the confidence 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 confidence boundaries, 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] to efficiently compute confidence 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 confidence 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 confidence 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 1990 pre-discovery observations has confirmed the impossibility of an impact in 2028 and reduces the semilinear confidence regions to subsets of the regions computed with less data, as expected. The confidence 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 confidence boundary has a completely different shape from the linear ellipse, and indeed for orbits determined with only few weeks of observational data the semilinear confidence boundary correctly includes possible collisions, while the linear one does not. Free software is available now, allowing everyone to compute target plane confidence 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.