Quaternion Equations of Disturbed Motion of an Artificial Earth Satellite

Аннотация

The quaternion equations of disturbed motion of an artificial Earth satellite (AES) in Earth’s gravitational field (with regard to its zonal, tesseral, and sectorial harmonics) are obtained in four-dimensional Kustaanheimo–Stiefel variables and in modified four-dimensional variables. In the latter variables the equations of satellite motion have a simpler and more symmetric structure, as compared to the equations of motion in the Kustaanheimo–Stiefel variables. The obtained equations are linear for undisturbed Keplerian motion and, in general, have the form of equations of motion of a disturbed four-dimensional oscillator. These equations, in contrast to classical equations, are regular (they do not contain specific points of singularity) for satellite motion in the central gravitational field of Earth under an effect of disturbing forces, the description of which does not contain negative, higher-than-first powers of the satellite distance to Earth’s center. In these equations, the basic variables are Kustaanheimo–Stiefel variables, or modified four-dimensional variables proposed by the author, as well as satellite motion energy and time. The new independent variable is associated with time by the differential relation containing the satellite distance to Earth’s center of mass. The equations are convenient for applying nonlinear mechanics methods and for high-precision numerical calculations. In the case of artificial satellite motion in Earth’s gravitational field, the description of which does not take into account the tesseral and sectorial harmonics, but takes into account its zonal harmonics, the first integrals of the equations of satellite motion are found, and the replacements of variables and transformations of these equations are proposed, which allowed one to get, for studying satellite motion, closed systems of differential equations of lower dimension, in particular, the system of equations of fourth and third orders.