On the Generalized Kepler Problem under the Effect of Outer Third-Body Perturbation

Abstract

This study presents generalized solutions to the Kepler problem involving the gravitational perturbation of a third body external to a two-body system. The spatial case of the non-coplanar motion of perturbing and perturbed bodies is considered. The expression for the relative position of the perturbing body relative to that of the perturbed body is expanded in Legendre polynomials. This expansion allows us to obtain the equation of motion and the energy of the system in terms of the orbital elements (I, Ω, ω, ν) for both the perturbed body and the perturbing body. By applying the Sundman transformation to regularize the dynamics of the problem, we eliminate a singularity and find two regularized equations of motion. Solutions to these equations are obtained by employing the families of the Battin and Stumpff functions. From these solutions, we derive formulas that are universally applicable and free from indeterminacy. This includes the expression for the radial distance, the generalized Kepler’s equation, the closed-form expressions for the coefficients f and g, along with their derivative forms, and the expressions for the orbital state vectors.