Исследование движения пробных частиц в гравитационном поле аксиально симметричного центрального тела в классической физике

The article deals with an axially symmetric body and examines its internal and external gravitational field within the framework of classical theory of gravity. The Maclaurin spheroid is used as a deformed body to represent objects with homogeneous density and rigid rotation, hence, axially symmetric bodies. The gravitational potential is derived from the Poisson equation for the external and internal fields, satisfying the boundary conditions at the center, on the surface of the body, and at infinity. The Poisson equation is solved analytically and exactly, applying the Green's function and the expansion into spherical harmonics (spherical functions). Moreover, the matching on the surface of the body for small deformations is shown as an example. In addition, the quadrupole moment of a deformed central object is considered and its influence on the motion of test bodies (particles) in the field of a given object is investigated. The quasi-Kepler problem is solved numerically in the Wolfram Mathematica program. It was shown that numerical calculations correspond to the analytical solution of the quasi-Kepler problem in the equatorial plane of the orbit. The perihelion shift of the planets of the solar system were also analyzed.The article pursues scientific, methodological and academic goals and is intended for a wide audience of students, graduates and doctoral students in the specialties of physics, mechanics and astronomy.