In this work, we suggest a simpler model to study the motion around irregularly elongated asteroids. This new model includes a variable density for the elongated asteroid. More precisely, we consider the motion of an infinitesimal mass attracted by the gravitational force induced by a body modeled as a non-homogeneous straight segment. We consider two situations: the segment can rotate uniformly with angular velocity \(\omega \ne 0\) , or it is fixed (i.e., \(\omega =0\) ). The aim of this paper is to prove the existence of different family of periodic solutions for this problem, such as those obtained by the averaging method for Hamiltonian or as the continuation method of Poincaré by using discrete symmetries. We prove the existence of several families of periodic solutions as a continuation of circular orbits of the (fixed or rotating) Kepler problem. We also obtain periodic solutions as a continuation of circular solutions of the Coriolis problem. The analysis of the linear stability of some periodic solutions is considered. The families of periodic orbits studied in this work constitute an example of the variety of orbits that can be followed by a particle orbiting an elongated asteroid from an analytic point of view. This helps us understand the dynamics around these bodies.
