Revisiting the Determination of a Preliminary Orbit for a Celestial Body

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This paper revisits the classical problem of determining a preliminary orbit of a celestial body from angular observations. It is proposed to solve this problem using a universal method based on the Gaussian equation in the Shefer form. The resulting solution is free of uncertainties for the radial position vectors of the observed body and can be obtained for all types of Keplerian orbits. The method demonstrates an exact functional dependence of the desired parameters on the initial data in the two-body problem, without any limitations. Noncoplanar orbits are determined using a system of two transcendental equations for two unknowns. The desired parameters can be both dimensional (distances) and dimensionless. In the latter case, limitations can be imposed on the solution search domain. The Lerch transcendental function is used to represent motion with any eccentricity value. The time intervals between observations are unlimited and may exceed the period of revolution for elliptical orbits. Motion in the ecliptic plane is also examined by analyzing three-equation systems for nonrectilinear orbits and two-equation systems for rectilinear ones. The results of determining the orbit of the dwarf planet Ceres are given as an example.