Regular quaternion equations orbital motion in the earth’s gravitational field in KS-variables and their modifications. Reduction of dimensionality, first integrals of equations

Regular quaternion differential equations of the perturbed orbital motion of a cosmic body (in particular, a spacecraft, an asteroid) in the Earth’s gravitational field are considered, which take into account zonal, tesseral and sectorial harmonics of the field. These equations, unlike classical equations, are regular (do not contain special points such as singularity (division by zero)) for perturbed orbital motion in the central gravitational field of the Earth. In these equations, the main variables are four-dimensional Kustaanheim–Stiefel variables (KS-variables) or four-dimensional variables proposed by the author of the article, in which the equations of orbital motion have a simpler and symmetric structure compared to equations in KS-variables. Additional variables in the equations are orbital energy and time. The new independent variable is related to time by a differential relation containing the distance from the cosmic body to the Earth’s center of mass (the Sundman differential time transformation is used). Regular equations of perturbed orbital motion in quaternion osculating (slowly changing) variables are proposed. The equations are convenient for using methods of nonlinear mechanics and high-precision numerical calculations, in particular, for forecasting and correcting the orbital motion of spacecraft. In the case of orbital motion in the Earth’s gravitational field, the description of which takes into account the central and zonal harmonics of the field, the first integrals of the equations of orbital motion of the eighth order are given, changes of variables and transformations of these equations are considered, which made it possible to obtain closed systems of differential equations of the sixth order for the study of orbital motion, as well as systems of differential equations of the fourth and third orders, including a system of differential equations of the third order with respect to the distance from the cosmic body to the center of mass of the Earth and the sine of geocentric latitude, as well as a system of two integro-differential equations of the first order with respect to these two variables.