2. Projections on the modified target plane

Let the initial conditions of the asteroid/comet under consideration be described by the 6-dimensional vector X of orbital elements, and let X^* be the solution best fitting the available observations; let C_X be the normal matrix, and $\Gamma_X=C_X^{-1}$ the covariance matrix of this fit. X^* is a minimum for a function Q proportional to the sum of squares of the residuals (Q is often called target function; to avoid confusion, in this paper it will be called cost function). The expansion of the cost function in the neighbourhood of X^* is

$$Q(X)=Q(X^*) +{\displaystyle 2 \over \displaystyle m} \,(X-X^*)^T\, C_X\,(X-X^*)+ \ldots= Q(X^*)+\Delta Q(X)$$

where the dots contain higher order terms, and m is the number of residuals. A confidence ellipsoid is a region where the quadratic part of the penalty $\Delta Q$ is less than a given constant: we shall indicate with $Z_X(\sigma)$ the ellipsoid defined by the inequality:

$$(X-X^*)^T \, C_X\,(X-X^*)\leq\sigma^2 \ .$$

If the confidence ellipsoid under consideration is small enough, the higher order terms in $\Delta Q(X)$ are also small, and the confidence ellipsoid itself is a good approximation of the region in the space of orbital elements where the penalty $\Delta Q$ is less than $\sigma^2$, where the alternative solutions still compatible with the observations are contained.

If the normal and covariance matrices have a large conditioning number (the ratio between the largest and the smallest eigenvalue), as is often the case for an asteroid/comet observed only over a short arc, the quadratic approximation is poor; in this case, however, the analysis of potential close encounters would be of little significance anyway, because of the excessive uncertainties in the predicted positions at times far from the observations. Thus we shall restrict our discussion to reasonably well observed orbits, as is now becoming the rule for newly discovered Potentially Hazardous Objects (PHO); especially among older discoveries there are objects observed only over very short arcs, for which the algorithms described in this paper may not give satisfactory results. This ``reasonable goodness'' hypothesis can be easily checked, in each individual case, by inspection of the residuals and of the eigenvalues of the covariance matrix: the distribution of the residuals, after outlier removal, must not be very different from the central portion of a Gaussian, and the square root of the largest eigenvalue of the covariance matrix must be small, say less than 10^-4.

To describe the totality of the close approaches compatible with the given set of observations we should, as a matter of principle, compute all the orbits contained in some confidence ellipsoid (for a suitable value of $\sigma$) and monitor the portion of the orbits in the vicinity of the concerned planets. Since it is not conceivable to compute an infinite number of orbits, a Monte Carlo scheme has been proposed by [Muinonen and Bowell 1993] to explicitly compute the statistical distribution of the predicted outcomes of the compatible orbits: a probability distribution in the orbital elements space, which can be derived from the Gaussian theory [Gauss 1809], is randomly sampled and each orbit in the sample individually computed until the close approach under study takes place [Muinonen 1999]. This method works, but in practice the computational load is very heavy; our challenge is to obtain a description of the totality of the close approaches, practically equivalent to the one obtained by the Monte Carlo method, by computing only a very limited number of orbits.