Existence and uniqueness of solution to the system of integral equations in the planar Earth, Sun and satellite system

Abstract

This manuscript delves into the exploration of the existence and uniqueness of the $n$th level approximation within the context of the inertial restricted three-body problem. In this scenario, two massive celestial bodies, namely Earth and the Sun, are held stationary along a straight line, while a less massive object serves as an artificial satellite. Within the manuscript, we have uncovered solutions expressed in terms of quadratures, infinite series, and transcendental functions. Our investigation employs a system of integral equations to address the challenge posed by these two immobile centers. The process initiates with the derivation of the equations of motion for the Earth, Sun, and satellite system, all considered within the inertial coordinate system. Subsequently, we formulate the $n$th level approximation and present its solution for the linear integral equations system. We also meticulously determine the conditions necessary for the solution to converge. Additionally, we engage in an in-depth discussion regarding the existence of such a solution. Moreover, the manuscript firmly establishes the uniqueness of this solution, assuring its singularity. Furthermore, we undertake a rigorous analysis to quantify the error associated with the $n$th level approximated solution.