Central Field Motion with Perturbing Acceleration Varying According to the Inverse Square Law in the Reference Frame Associated with the Radius Vector

The motion of a point with zero mass under the action of attraction to the central body and perturbing acceleration , inversely proportional to the square of the distance to , is considered. It is assumed that is small in absolute value compared to the main acceleration, caused by the attraction of the central body. Further, the vector components are constant in the reference frame used in astronomy, with the origin in the central body and the axes directed along the radius vector, the transversal (perpendicular to the radius vector in the osculating plane in the direction of motion), and the binormal (directed along the area vector). Earlier, we performed an averaging transformation of Euler-type equations of motion in osculating elements and obtained mean element evolutionary differential equations of motion in the first approximation in a small parameter. This article is devoted to solving the averaged equations, which are integrated completely. Moreover, the quadratures were expressed via elementary functions. The solution found has singularities at zero eccentricity and in the absence of the transverse acceleration. These and some other special cases are considered separately. There are at least two applications of the problem considered which are: an asteroid’s motion with allowance for the Yarkovsky–Radzievsky effect and a spacecraft’s motion with a solar sail. In both cases, the perturbation is inversely proportional to the squared distance from the Sun.