Forbidden zones for circular regular orbits of the Moons in Solar system, R3BP

TL;DR

This paper derives a key parameter in the restricted three-body problem that predicts forbidden zones for stable Moon orbits in the Solar system based on Riccati-type differential equations.

Contribution

It introduces a new parameter and analytical approach to identify unstable orbital zones for moons using Riccati equations in the R3BP context.

Findings

Identifies a parameter range (0.001-0.01) where Riccati equations remain invariant.

Defines forbidden zones for Moon orbits in the inner Solar system.

Provides a method to predict orbital stability based on differential equations.

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). Analyzing such a system of equations, we considered the case of small-body motion of negligible mass around the 2-nd of two giant-bodies (which are rotating around their common centre of masses on Kepler 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 that determines the character of quasi-circular motion of the small 3-rd body relative to the 2-nd body (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 remains in the range 0.001-0.01. Such an amazing fact let us evaluate the forbidden zones for Moons orbits in the inner Solar system or the zones of the meanings of distances (between Moon-Planet) for which the motion of small body could be predicted to be unstable according to basic features of the solutions of Riccati-type.