Regular Quaternion Models of Perturbed Orbital Motion of a Rigid Body in the Earth's Gravitational Field

In this paper, we propose regular quaternion models of perturbed orbital motion of a rigid body. They do not have the features inherent in classical models when a body moves in a Newtonian gravitational field and, in the general case, when a body moves in a central force field with the potential in the form of a polynomial of negative degrees of distance to the center of attraction of the fourth-order. We also propose regularized quaternion models of the orbital motion of the body in the Earth's gravitational field with allowance for not only the central (Newtonian), but also zonal, tesseral, and sectorial harmonics of the gravitational field potential that take the Earth's nonsphericity into account. In these models, the negative degrees of distance to the attraction center are reduced by several orders of magnitude in terms describing the effect of the zonal, tesseral, and sectorial harmonics of the Earth's gravitational field potential on the orbital motion of a rigid body. The main variables are the Euler parameters, the distance from the center of mass of the body to the center of attraction, the total energy of the orbital motion of the body, and the square of the absolute value for the vector of the orbital velocity moment of the body (or projection of this vector). These models use a new independent variable associated with time by a differential relation containing the square of the distance from the center of mass of the body to the center of attraction. In the case of the orbital motion of the body in the Earth's gravitational field, in the description of which only its central and zonal harmonics are taken into account, we found the first integrals of the equations of orbital motion and proposed substitutions of variables and transformations of these equations. For studying the motion of the body, this allowed us to obtain closed systems of differential equations of lower dimension, in particular, a system of third-order equations for distance, the sine of the geocentric latitude, and the square of the absolute value of the vector of the orbital angular momentum.