Potential generated by a massive inhomogeneous straight segment

The discoveries of binary asteroids have opened an important new field of research concerning the calculation of potential generated by irregular bodies. Some of them have an elongated shape. A simple model to describe the motion of a test particle in that kind of potential requires consideration of a finite homogeneous straight segment. We construct this model by adding an inhomogeneous distribution of mass. To be consistent with the geometrical shape of the asteroid, we explore a parabolic profile of the density. We establish the closed analytical form of the potential generated by this inhomogeneous massive segment. The study of the dynamical behavior is fulfilled by the use of Lagrangian formulation, which allows us to calculate some two and three dimensional orbits.