Averaged equations of motion for a perturbing acceleration which is constant in various reference frames

摘要

The motion of a point with zero mass under the action of the gravitation of a central body and a perturbing acceleration P that is constant in one of the three most widely used reference frames in astronomy is considered: a main inertial frame and two orbital frames- with its x axis directed along the radius vector and with its x axis directed along the velocity vector. |P| is taken to be small compared to the main acceleration due to the gravitation of the central body. An averaging transformation in a first approximation with respect to a small parameter is applied to the equations of motion in the osculating elements. Closed expressions for both the variable-substitution functions and the right-hand sides of the equations of motion in the averaged elements are obtained. In the and frames, all the functions used are elementary, while elliptical integrals appear in the frame. Expansions of all required quantities in series in powers of the eccentricity e that converge for |e| < 1 are obtained (solutions of the averaged equations will be presented in a future publication). Possible applications of the results are noted: the motion of a spacecraft with a small thrust with a constant magnitude, and the motion of an asteroid on which an engine providing a small thrust with a constant magnitude acts, for example, in order to avert a collision with the Earth.