Quaternion Regularization of the Equations of the Perturbed Spatial Restricted Three-Body Problem: II

Abstract

A quaternion method for the regularization of differential equations of the perturbed spatial restricted three-body problem is developed. It is closely related, from the methodological point of view, to the quaternion method for the regularization of the differential equations of the perturbed spatial three-body problem in Kustaanheimo–Stiefel variables that was earlier proposed by the author of this article.

Various local and global regular quaternion differential equations of the perturbed spatial restricted three-body problem (both circular and non-circular problem) i.e. equations that are regular in the vicinity of the first or second body of finite mass and equations that are regular at the same time both in the neighborhood of the first and second body of finite mass are obtained. The equations are systems of nonlinear nonstationary differential equations of the tenth or eleventh or nineteenth order with respect to the Kustaanheimo–Stiefel variables, their first derivatives, Kepler or total energies, or variables that are Jacobi integration constants in the case of the unperturbed spatial circular restricted three-body problem, as well as with respect to time and auxiliary time variable. The equations obtained allow one to construct different regular algorithms for integrating the differential equations of the perturbed spatial restricted three-body problem.

This study is an extension of [1, 2].