The determination of a preliminary orbit by Cauchy-Kuryshev-Perov method

The determination of preliminary orbits of celestial bodies is of interest for observational astronomy, from the point of the view of the discovery of new bodies or identification with already known ones. To solve this problem, the techniques are required that are not limited both by the values of eccentricity of the orbit and by the time intervals between observations. In this paper, we consider the geometric Cauchy-Kuryshev-Perov method for determining the preliminary orbit. It is shown how, within the framework of the two-body problem, proceeding only from geometric constructions, using five angular observations to determine an orbit that does not lie in the plane of the observer's motion. This method allows us to reduce the problem of determining the preliminary orbit to an algebraic system of equations for two dimensionless variables, with a finite number of solutions. The method is suitable for determining both elliptical and hyperbolic orbits. Moreover, it has no restrictions on the length of the orbital arc of the observed body and is unlimited by the number of complete revolutions around the attracting center between observations. All possible combinations of body positions in orbit are divided into 64 variants and represented by the corresponding systems of equations. This article presents an algorithm for finding solutions to the problem without a priori information about the desired orbit. Solutions are sought in a limited area, in which triangulation is performed with ranking triangles for compliance with the search conditions, which makes it possible to exclude consideration of most of them at the initial stage. Solutions of the system are found by the Nelder-Mead method through the search for the minima of the objective function. The obtained orbits are compared, through the presentation of observations, and the best one is selected. An example of determining the orbit of comet Borisov 2I is given.