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

Abstract

The restricted circular three-body problem is investigated when two of the massive bodies, which are treated as point masses, move in specified circular orbits in a single plane while third body of small mass is assumed to be spherically symmetric and deformable and its centre of mass moves in the 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 affecting the evolution of its motion. This energy dissipation leads to the evolution of its orbit and angular velocity of rotation. Since it is assumed that the masses of the two bodies (in the case of the solar system, these could be the Sun and Jupiter) relete as one to μ (μ ⪡ 1), the evolution of the motion of the deformable body develops in two stages. During the first, “fast” stage of evolution, its orbit tends towards circular with its centre in the massive body with mass equal to unity, and the rotation is identical to the orbital rotation (a state of gravitational stabilization, 1:1 resonance). In this case, the body turns out to be deformed (oblate with respect to its poles and stretched along the radius which joins this body of small mass to the massive [1, 2]. In the second, “slow” stage of evolution, the effect of the body with mass μ is taken into consideration, which leads to the evolution of the circular orbit of the deformable body.