Quaternion Equations of Disturbed Motion of an Artificial Earth Satellite

Quaternion methods and models of regular celestial mechanics and astrodynamics

“…The variable γ in the potential of the Earth's gravitational field can be expressed from the KS variables by the relations (36),…”

“…These equations were convenient to study the motion of an Earth artificial satellite, since many forces acting on the satellite, and primarily the Earth's gravity force (the potential of the Earth's gravitational field), depend on relative coordinates of the satellite location (on its geographical coordinates, latitude, and longitude), instead of its absolute coordinates (in the coordinate system moving translationally in relation to the inertial coordinate frame). 7.2 Applications of quaternion regular models in the theory of orbital motion of a solid body (spacecraft and artificial satellite) in the Earth's gravitational field Chelnokov [36] derived the regularized quaternion equations of the perturbed motion of an artificial satellite in the Earth's gravitational field, taking into account its zonal, tesseral, and sectorial harmonics in four-dimensional KS variables and in modified four-dimensional variables, for which the motion equations of a satellite have a simpler and more symmetrical structure compared with those in the KS variables. These equations use energy of motion of a satellite and time as additional variables.…”

“…Different from the models proposed in Ref. [36], Chelnokov [57] proposed regularized quaternion models of orbital motion of a solid body in the Earth's gravitational field, which was defined considering not only central (Newtonian), but also zonal, tesseral, and sectorial harmonics of the gravitational field potential, which take into account nonsphericity of the Earth. In these models, the negative powers of distance to the center of attraction were lower by several orders in terms that define the effect of zonal, tesseral, and sectorial harmonics of the Earth's gravitational field potential on the orbital motion of a solid body.…”

“…The convenience and efficiency of using the regularized quaternion models of orbital motion of a solid body, derived by Chelnokov [36,57] , for analytical research of motion of a body, are exemplified by the motion of a body in the Earth's gravitational field, which is defined taking into account the central and zonal harmonics of the gravitational field potential. The first integrals are found for the derived orbital motion equations.…”

“…Various aspects of quaternion regularization of differential equations of the perturbed spatial two-body problem by using the KS variables have been studied by Velte [1] , Vivarelli [2][3][4] , Shagov [5] , Deprit et al [6] , Vrbik [7][8] , Waldvogel [9][10] , Saha [11] , Zhao [12] , Roa et al [13][14] , Roa and Pelaez [15] , Breiter and Langner [16][17][18] , Ferrer and Crespo [19] , and also by the author of this paper, Chelnokov [20][21][22][23][24][25][26][27][28][29][30][31][32][33][34][35][36] .…”