On the motion around the centre of mass of a viscously elastic sphere in the restricted elliptic three-body problem

Abstract

Al~traet. In the restricted three-body problem we consider the motion of a viscously elastic sphere (planet) with its centre of mass moving in a conditionally-periodic orbit. The approximate equations describing the rotational motion of the sphere in terms of the Andoyer variables are obtained by the method of the separation of motion and averaging; the evolution of the motion is also analysed. Formulation of the ProblemConsider the restricted elliptic problem of three bodies two of which are mass points M 1 and M 2 with masses m, and m2, respectively, which are moving one relative to the other in an elliptic orbit with the radius ofwhere a, e and ~ are the major semiaxis, eccentricity and true anomaly of the orbit, respectively, and the third body M is a passively gravitating deformed sphere. In the inertial system of coordinates O~1~2~3, in which the point O is the barycentre of the two attracting points M, and M2, the axis O~3 is perpendicular to the plane of the orbit and the axis O~ is the line of apsides of the elliptic motion of the attracting points. The radius vectors of the points M~ and M 2 are (see Figure 1):Here xi is the coordinate of the attracting centre Mi in the rotating coordinate system Oxyz, in which the axis Oz coincides with the axis 0¢3 of the inertial system and the axis Ox passes through the points M 1 and M2; G ° is a unit vector along the axis O~. The angular speed of rotation around the axis Oz is defined by the relation d~ COo(1 + e cos ~ )2 f(ml + m2) dt = (1 --e2) 3/2 ' 032 ----a 3 'where o9 0 is the average speed of revolution of the mass points M, and M2, and f is a gravitational constant.