Determination of Variation Lines and Modeling of Initial Trajectory Dispersion in the Presence of Strong Nonlinearity in Orbit Improvement Problems

A method for determining the lines of variation (LOV) in the initial confidence region under strong nonlinearity in the orbit improvement problem is developed, based on finding the maximum modulus of the normal vector to the level surfaces of the objective function using the least-squares method. The method was tested for three asteroids using their observations on a short arc of the orbit, namely, the points of two variation lines corresponding to the directions of the greatest deformation of the initial confidence region were determined. Approximation of variation lines was performed using third-degree polynomials. Using the obtained approximations, new variables were introduced in which the initial confidence region is almost ellipsoidal, i.e., nonlinearity is nearly absent. The modeling of the initial probability dispersion of trajectories is performed in the space of new variables. The resulting dispersion can be further used to identify collision orbits and estimate the probability of asteroid collisions with Earth.