Averaged equations of motion in a central field in the presence of a constant in absolute value disturbing acceleration

Abstract

The motion ofapoint of zero massdue to the attraction to a centralbody S and thedisturbing acceleration P is considered. We suppose that P is small in comparison with the main acceleration caused by the gravitation of S. The second assumption is the constancy of the vector P in one of the two standard in celestial mechanics frames of references with the common origin S but different behaviour of their axes: the main inertial one O with fixed axes and the accompanying one O1 with rotating axes directed along the radius-vector, along the transversal (perpendicular to the radius-vector in the plane of the osculating orbittowardsthedirectionof motion),and thebinormal(directed along theangularmomentum vector).Averaging transformis applied toEuler-like equationsin osculating elements.Namely,formulaefor transition from osculating elements to the mean ones, and the equations of motion in the mean elements are derived in the first approximation. The result is expressed in closed form, without expansions in powers of the eccentricity or the inclination. The equations of motion are conservative if P is constant in the system O. The same assertion is true in the system O1 if the acceleration P is a central one only.

Keywords:

variation of osculating elements, averaging transform, solution in a closed form