High-order analytical solutions around triangular libration points in the circular restricted three-body problem

Abstract

High-order analytical solutions around the triangular libration points in the circular restricted three-body problem (CRTBP) are constructed in this paper. It is known that the linearized equations of motion in CRTBP have three pairs of imaginary characteristic roots, for all values of the mass parameter μ less than μc (Routh’s critical value) except for a set of values. That is, the motion around the triangular libration points is linearly stable. To describe the general dynamics explicitly, the solutions around triangular libration points are expanded as power series of three amplitude parameters: α corresponding to the long periodic motion, β corresponding to the short periodic motion and γ corresponding to the vertical periodic motion. The solutions are constructed up to arbitrary order by means of the Lindstedt–Poincare method. Using the series expansions constructed, the long, short, vertical and quasi-periodic orbits around triangular libration points can all be parametrized. To check the validity of series expansions, numerically integrated orbits with the same initial states as analytical orbits are computed, and the practical convergence of analytical solutions is considered in detail for the Earth–Moon, Sun–Jupiter and Sun–Earth systems.