New Numerical/Analytical Methods for Solving Equations of Orbital Motion

New methods are proposed for solving equations of motion of celestial bodies. The methods are based on the use of superosculating orbits with second- and third-order tangency to the trajectory of the real motion of a body. The construction of these orbits is related to the concept of a fictitious attracting center, whose mass varies in accordance with the first Meshchersky law. In the original reference methods, the perturbed trajectory is represented by a sequence of small arcs of superosculating orbits. The order of accuracy of the reference methods coincides with the order of tangency of the superosculating orbit used in calculations. Using Runge's rule and Richardson's extrapolation scheme leads to the methods of higher order. The efficiency of the new methods in comparison with the numerical integration of equations of motion based on the well-known fourth- and seventh-order Runge–Kutta–Fehlberg methods is illustrated by examples of the calculation of perturbed orbits of some asteroids.