The uncertainty associated with astrometric observations of an
asteroid limits the precision with which the orbital elements of the
latter can be determined. After computing, by the method of least
squares, the solution that minimizes the sum of the residuals squared
(the nominal solution), one can define a confidence region
in the space
of the orbital elements at some fixed epoch t0.
Using the same notation as [Milani 1999, Sec. 2], the 6-dimensional
ellipsoid (with interior) approximating the confidence region is given
Following [Milani 1999, Sec. 5], we sample the confidence region by computing multiple solutions by moving along the weak direction with a small step ; in this way we sample uniformly a line in the confidence region, the Line Of Variations (LOV). The resolution in the detection of possible impacts is controlled by ; more precisely, the probability of impact above which all the impacts are certainly detected is inversely proportional to [Milani et al. 1999]. We are currently using , that is we sample the line of variations along the weak direction for with 1201 solutions Xi; these multiple solutions are used as virtual asteroids, that is each one of them is taken as representative of a small region in the space of initial conditions. Note that other authors, e.g., [Muinonen 1999] and [Chodas and Yeomans 1996], use different methods to sample the confidence region; these differences are important for the efficiency and the reliability of the computations, but they are not conceptually essential. We use virtual asteroids Xi uniformly spaced along a smooth line in the space because this allows us to exploit the topological properties of a string, e.g., if some continuous function is positive for Xi and negative for Xi+1 we can conclude that it has a zero along the LOV.