Pitch motion of a gravity gradient satellite in an elliptical orbit is studied. The cell mapping method is employed to find periodic solutions and analyze the global behavior of the system. Stability characteristics of the solutions are established using a point mapping approximation algorithm. The proposed approach does not depend on existence of a small parameter and, therefore, no limitations are imposed on the magnitudes of eccentricity or amplitude of motion. This is in contrast to the perturbation based approaches that require assumptions of small orbital eccentricity and small motion. As a result, stable periodic solutions of twice and three times the orbital period are found for some different eccentricities and inertia parameters. Global behavior and evolution of periodic solutions are also demonstrated using invariant surfaces and bifurcation diagrams.