Regular Quaternion Equations of the Spatial Hill Problem in Kustaanheimo–Stiefel Variables and Quaternion Osculating Elements

Regular quaternion equations of the spatial Hill problem (a variant of the limited three-body problem (Sun, Earth, Moon (or another low-mass moving cosmic body under study)) are obtained, when the distance between two bodies with finite masses is considered very large, in four-dimensional Kustaanheimo-Stiefel variables (KS-variables) within the framework of the elliptical and circular spatial bounded three-body problem, as well as the regular quaternion equations of the planar Hill problem in two-dimensional Levi-Civita variables. In these equations, the variables are KS-variables or Levi-Civita variables and the energy of relative motion of the body under study, or a variable that converts for the circular Hill problem into a constant of motion of this body (the Jacobi integration constant), as well as the planetocentric distance of the Sun and real time associated with a new independent variable by the Sundman differential transformation of time or other more complex differential ratio. These equations are supplemented by the equation of the Earth’s orbit in polar coordinates and the equation for the true anomaly characterizing the Earth’s position in the orbit. The first integral of the obtained equations in KS-variables in the case of a circular problem is established. Another first partial integral in the general case is a bilinear relation connecting KS-variables and their first derivatives. Three new forms of regular equations of the spatial Hill problem in quaternion osculating elements (slowly changing quaternion variables) are proposed. The proposed regular quaternion equations have an oscillatory form or the form of equations with slowly changing variables, which makes it possible to effectively use analytical and numerical methods of oscillation theory and methods of nonlinear mechanics in the study of the Hill problem.