Four Methods for Determining Intermediate Perturbed Orbits from Three Observations: A Comparison

Theoretical and numerical comparison of four methods for determining the orbit of a small celestial body from three measurements of its angular coordinates at three time moments is provided. The methods are intended for constructing intermediate orbits considering most of perturbations in motion of the examined body. Two methods are based on the solutions of the differential equations of motion and on their second derivatives in the form of series in terms of powers of time intervals (the Herrick-Gibbs approach), and the two others are based on the solutions for some intermediate perturbed motions in the closed form, without their representation in the form of series (the approach of the author). A dependence of methodic errors on the length of the reference time interval determined by the moments of observation beginning and ending is investigated. By way of examples, results of calculation of the orbit of the Apophis asteroid are presented.