TARGET PLANE CONFIDENCE BOUNDARIES

**Andrea Milani
Dipartimento di Matematica, Università di Pisa
Via Buonarroti 2
56127 PISA, ITALY
E-mail: **

**Revised version, February 23, 1999
Manuscript pages: 25; Figures: 10.
Running title: Target Plane Confidence Boundaries
Keywords: Asteroids, Dynamics; Comets, Dynamics; Impact Processes**

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 XF*_{11}, 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.

- 1. Introduction
- 2. Projections on the modified target plane
- 3. Application I: The 1997 XF
_{11}scare - 4. Application II: a fictitious impactor
- 5. Conclusions
- 6. How to obtain the software
- Bibliography
- About this document ...