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 *t*_{0}.
Using the same notation as [Milani 1999, Sec. 2], the 6-dimensional
ellipsoid (with interior) approximating the confidence region is given
by

where is the change in the orbital elements with respect to the nominal solution, and

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 *X*_{i};
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 *X*_{i} 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 *X*_{i} and negative for *X*_{i+1} we can
conclude that it has a zero along the LOV.