2.1 Confidence region and multiple solutions

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 ${\cal X}$ 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 by

$$\Delta X\cdot C_X\,\Delta X\leq\sigma^2$$ (1)

where $\Delta X$ is the change in the orbital elements $X\in{\cal X}$with respect to the nominal solution, and C_X is the normal matrix. The inverse of the normal matrix is the covariance matrix $\Gamma_X=C_X^{-1}$. The largest eigenvalue of $\Gamma_X$ is associated to a weak direction along which the orbital elements are poorly constrained by the available observations.

Following [Milani 1999, Sec. 5], we sample the confidence region by computing multiple solutions by moving along the weak direction with a small step $\Delta \sigma$; 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 $\Delta \sigma$; more precisely, the probability of impact above which all the impacts are certainly detected is inversely proportional to $\Delta \sigma$ [Milani et al. 1999]. We are currently using $\Delta \sigma=0.005$, that is we sample the line of variations along the weak direction for $\vert\sigma\vert \leq 3$ 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 ${\cal X}$ 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.