Motion and rotation of celestial bodies in the post-Newtonian approximation

Consistent post-Newtonian description of motion and precession in a system ofN extended slowly rotating bodies is developed in the framework of the post-Newtonian approximation scheme (PNA). The solution of Einstein equations for the metric in the local reference system related to a body of the system is obtained. This metric is used to derive the equations of motion and precession of the considered body on the basis of some relativistic generalization of the model of rigid body. These equations are solved in order to find the first order corrections to nutation theory and to the osculating orbital elements of the body. Another important application of such local metric, concerning the motion of a test particle (e.g., artificial satellite) orbiting the body, is also investigated in this paper.