On Hasen's Method for Computing Special Perturbations

摘要

The purpose of this paper was to derive a set of differential equations which describe the motion of a satellite moving in a nearly elliptical orbit around the earth. The derivation presented is a modification of that given by Peter Musen in his paper 'Modified formulae for Hansen's special perturbations.' Hannsen's method describes the motion of the satellite relative to a fixed auxiliary ellipse in the instantaneous orbit plane. At the same time, the position of the ellipse relative to an inertial coordinate system is determined. Two sets of differential equations are derived. The first, which is similar to that given by Musen, uses an auxiliary ellipse which is not permitted to rotate in the orbit plane. It is suitable for orbits with small eccentricity. The second uses an auxiliary ellipse rotating in the plane so that its axes are coincident with the axes of instantaneous ellipse which is associated with the motion. Its set of equations is similar to that of the variation of the elements method. However 'One point which differentiates Hansen's method from all others consists in the addition of the perturbations directly to the mean anomaly of a certain auxiliary ellipse in the plane of the instantaneous orbit instead of to the true anomaly or to the true longitude on the fixed plane.'