Quaternion Methods and Regular Models of Celestial Mechanics and Space Flight Mechanics: Local Regularization of the Singularities of the Equations of the Perturbed Spatial Restricted Three-Body Problem Generated by Gravitational Forces

Abstract

The problem of local regularization of differential equations of a perturbed spatial restricted three-body problem is studied: elimination of singularities (dividing by zero) generated by gravity forces of differential equations of perturbed spatial motion of a material point M, which has a negligibly small mass, in the vicinity of two gravitating points M₀ and M₁ by writing equations of motion in rotating coordinate systems, the use of new regular variables and the regularizing transformation of time. Various systems of regular quaternion differential equations (RQDE) for this problem are obtained. The following groups of variables act as variables in these equations: (1) four-dimensional Kustaanheimo–Stiefel variables, Keplerian energies and time t, (2) distances from the point M to the points M0 and M1, modules of the vectors of the moment of velocities of the point M with respect to the points M₀ and M₁, Keplerian energy, time t and Euler (Rodrigues–Hamilton) parameters characterizing the orientations of the orbital coordinate systems in the inertial coordinate system; (3) two-dimensional Levi-Civita variables describing the motion of the point M in ideal coordinate systems, Keplerian energies, time t and Euler parameters characterizing the orientations of ideal coordinate systems in the inertial coordinate system and being osculating elements (slowly changing variables) for the motion of the point M in the neighborhood gravitating point M₀ or M₁, respectively. To construct the RQDE, the equations of the perturbed spatial restricted three-body problem, written either in nonholonomic (azimuthally free), or in orbital, or in ideal coordinate systems, were used as initial ones; “fictitious” times τ₀ and τ₁ are used as new independent variables (i.e., regularizing differential transformations of the Sundmann time are used) or angular variables φ₀ and φ₁, which are traditionally used in the study of orbital motion as part of polar coordinates. To match the two independent variables used in the vicinity of the gravitating points M₀ and M₁, additional differential equations are used.

The obtained various locally regular quaternion differential equations of the perturbed spatial restricted three-body problem make it possible to develop regular analytical and numerical methods for studying the motion of a body of negligibly small mass in the vicinity of two other bodies with finite masses, and also make it possible to construct regular algorithms for the numerical integration of these equations. The equations can be effectively used to study the orbital motion of celestial and cosmic bodies and spacecraft, to predict their motion, as well as to solve problems of controlling the orbital motion of spacecraft and solving problems of inertial navigation in space.