Forbidden Zones for Circular Regular Orbits of the Moons in Solar System, R3BP

Abstract

Previously, we have considered the equations of motion of the three-body problem in a Lagrange form (which means a consideration of relative motions of 3-bodies in regard to each other). Analysing such a system of equations, we considered the case of small-body motion of negligible mass m ₃ around the second of two giant-bodies m ₁, m ₂ (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. In the current development, we have derived a key parameter η that determines the character of quasi-circular motion of the small third body m ₃ relative to the second body m ₂ (planet). Namely, by making several approximations in the equations of motion of the three-body problem, such the system could be reduced to the key governing Riccati-type ordinary differential equations. Under assumptions of R3BP (restricted three-body problem), we additionally note that Riccati-type ODEs above should have the invariant form if the key governing (dimensionless) parameter η remains in the range 10⁻² 10⁻³. Such an amazing fact let us evaluate the forbidden zones for Moon’s orbits in the inner solar system or the zones of distances (between Moon and Planet) for which the motion of small body could be predicted to be unstable according to basic features of the solutions of Riccati-type.