The Averaged Equations of Motion in the Presence of an Inverse-Square Perturbing Acceleration

Abstract

The motion of a zero-mass point under the action of a gravitational force toward a central body ${\cal S}$ and a perturbing acceleration P′ whose magnitude is inversely proportional to the square of the distance to ${\cal S}$ is considered. The direction of P′ is constant in one of the three coordinate systems most widely used in astronomy: the main inertial system ${\cal O}$ and two orbiting systems ${{\cal O}_s}$ with their x axes along the radius vector for s = 1 and along the velocity vector for s = 2.The ratio of |P′|to the main acceleration due to the gravitation of the central body is taken to be small. An averaging transformation in a first approximation in a small parameter of the problem is applied to the equations of motion in the osculating elements. Closed expressions are obtained for the right-hand sides of the equations of motion in the mean elements. These are expressed in terms of elementary functions in the systems ${\cal O}$ and ${{\cal O}_1}$; complete elliptical integrals arise in the system ${{\cal O}_2}$. Closed expressions are obtained for the change-of-variable functions. All the functions encountered in the systems ${\cal O}$ and ${{\cal O}_1}$ are elementary functions, apart from those determining the variations of the mean anomaly. The latter is given by an integral of an elementary function, as well as a series in powers of the eccentricity that converges absolutely and uniformly when 0 ⩽ e ⩽ 1. All functions in the system ${{\cal O}_2}$ apart from those determining the variations of the mean anomaly can be expressed in terms of incomplete elliptical integrals. The variations of the mean anomaly are calculated using a Fourier series in the mean anomaly. Integration of the averaged equations of motion will be considered in future papers. Possible applications of this model problem include the motion of an asteroid taking into account the Yarkovsky-Radziewski effect, and the motion of a spacecraft with a solar sail, when the perturbing action is inversely proportional to the square of the distance from the Sun. It stands to reason that determining the components of the vector P′ requires knowledge of the thermal-physical characteristics of the body in question and the parameters of its rotational motion in the former case, and of the orientation of the solar sail in the latter case.