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A. Milani, G.F. Gronchi, Z. Knezevic, M.E. Sansaturio (Department of Mathematics, University of Pisa)
When the observation of a new asteroid are not enough to compute an orbit we can represent them with an attributable (two angles and their time derivatives). The undetermined range and range rate span an admissible region of solar system orbits, which can be represented by a set of Virtual Asteroids (VAs) selected by an optimal triangulation (see the presentation by G. Gronchi).
The four coordinates of the attributable are the result of a fit and have a covariance matrix. Thus the predictions of future observations have a quasi-product structure (admissible region times confidence ellipsoid), approximated by a triangulation with a confidence ellipsoid for each node.
If we have >2 observations we can also estimate the geodetic curvature and the acceleration of the observed path on the celestial sphere. If both are significantly measured they constrain the range and the range rate and may allow to reduce the size of the admissible region.
To compute a a preliminary orbit starting from two attributables, for each VA (selected in the admissible region of the first arc) we consider the prediction at the time of the second and its covariance matrix, and we compare them with the attributable of the second arc with its covariance. By using the identification penalty (as in the algorithms for orbit identification) we can select as a preliminary orbit the VAs which fits together both arcs in the 8-dimensional space.
Two attributables may not be enough to compute an orbit with convergent differential corrections. The preliminary orbit is used in a constrained differential correction, providing solutions along the Line Of Variations, to be used as second generation VAs to predict the observations at the time of a third arc. In general the identification with a third arc ensures a well determined orbit.
The author(s) of this abstract have provided an email address for comments about the abstract: milani@dm.unipi.it
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Bulletin of the American Astronomical Society, 36 #2
© 2004. The American Astronomical Soceity.