Constructing an Initial Confidence Region Far from the Observational ARC and Estimating the Probability of an Asteroid Impact with the Earth

A method for reducing nonlinearity in the problem of asteroid orbit determination is considered. The method is based on the use of variables that take into account the stretching of the initial confidence region mainly along the trajectory when the initial epoch is located far from the observational arc. It is shown that in this case, the nonlinearity of the inverse problem is revealed only along the largest axis of the confidence ellipsoid, which is directed almost along the trajectory of the object, while the other axes are not deformed. This allows us to introduce new variables in which one (largest) axis is curved and approximated using polynomials, while the remaining axes remain the same as in the confidence ellipsoid. The confidence region in the new variables is an ellipsoid, which makes it possible to fill it with a cloud of random points according to the law of multidimensional normal distribution and, thus, significantly increase their number. In addition, it is shown that with a significant distance of the initial epoch from the observational arc, the coordinates and velocities noticeably correlate, which simplifies the approximation of the point cloud using an ellipsoid. The method was used to estimate the probability of a collision with the Earth (and the Moon) of potentially dangerous asteroids 2024 YR4, 2023 DO and 2018 CB in their upcoming closest approach.