The evolution of the motion of a viscoelastic sphere in the restricted circular three-body problem

The restricted circular three-body problem is investigated when two of the massive bodies, which are treated as point masses, move in specified circul ar orbits in a single plane while the third body of small mass is assumed t o be spherically symmetric and deformable and its centre of mass moves in t he plane of the circular orbits of the first two bodies and rotation around the centre of mass occurs around the normal to the plane of motion of the centre of mass. The energy dissipation accompanying the deformations of the small, spherically symmetric, deformable body is an important factor affec ting the evolution of its motion. This energy dissipation leads to the evol ution of its orbit and angular velocity of rotation. Since it is assumed th at the masses of the two bodies tin the case of the solar system, these cou ld be the Sun and Jupiter) relate as one to mu (mu much less than 1), the e volution of the motion of the deformable body develops in two stages. Durin g the first, "fast" stage of evolution, its orbit tends towards circular wi th its centre in the massive body with mass equal to unity, and the rotatio n is identical to the orbital rotation (a state of gravitational stabilizat ion, 1:1 resonance). In this case, the body turns out to be deformed (oblat e with respect to its poles and stretched along the radius which joins this body of small mass to the massive body [1, 2]. In the second, "slow" stage of evolution, the effect of the body with mass mu is taken into considerat ion, which leads to the evolution of the circular orbit of the deformable b ody. (C) 2001 Elsevier Science Ltd. All rights reserved.