Detailed Analysis of the Dynamics of Two Point Masses Under Gravitational Interaction

The dynamics of two point masses interacting in a gravitational field has been the object of several scientific works. However, the complete explicit solution of the two-body problem is, to the best of our knowledge, not always available in the scientific literature. In this work, we describe the dynamics of a two-body system with that of an equivalent single-body with a reduced mass. Then, we solve the specific problems for elliptical, circular and parabolic trajectories, starting from different initial conditions. Through detailed analytical calculations, we write the Cartesian equations of the trajectories and the equations of motion both in the reference system of the centre of mass and in the original reference system. The proposed methodology is a simple but rigorous way to analyse the two-body dynamics under gravitational interactions, and can be applied also to more complex cases, such as the motion in a perturbed Newtonian potential and/or precession problems. The treatment presented in this work is particularly suitable to undergraduate students.