Stability of the Moons Orbits in Solar System in the Restricted Three-Body Problem - Sergey V. Ershkov (Advances in Astronomy)
We consider the equations of motion of three-body problem in a Lagrange form (which means a consideration of relative motions of 3 bodies in regard to each other). Analyzing such a system of equations, we consider in detail the case of moon’s motion of negligible mass m3 around the 2nd of two giant-bodies m1, m2 (which are rotating around their common centre of masses on Kepler’s trajectories), the mass of which is assumed to be less than the mass of central body. Under assumptions of R3BP, we obtain the equations of motion which describe the relative mutual motion of the centre of mass of 2nd giant-body m2 (planet) and the centre of mass of 3rd body (moon) with additional effective mass ξ·m2 placed in that centre of mass ξ·m2+m3, where ξ is the dimensionless dynamical parameter. They should be rotating around their common centre of masses on Kepler’s elliptic orbits. For negligible effective mass ξ·m2+m3 it gives the equations of motion which should describe a quasi-elliptic orbit of 3rd body (moon) around the 2nd body m2 (planet) for most of the moons of the planets in Solar System.