Dynamics of a dumbbell satellite under the zonal harmonic effect of an oblate body

• The dynamics of a dumbbell satellite moving in a gravity field generated by an oblate body is studied.
• The effect of the zonal harmonic parameter is consider.
• It is proved that the mass center of the system is periodic.
• Equations of motion in the satellite approximation are deduced.
• Equations of motion are reduced to reduced to Beletsky’s equation.

The aim of the present paper is to study the dynamics of a dumbbell satellite moving in a gravity field generated by an oblate body considering the effect of the zonal harmonic parameter. We prove that the pass trajectory of the mass center of the system is periodic and different from the classical one when the effect of the zonal harmonic parameter is non zero. Moreover, we complete the classical theory showing that the equations of motion in the satellite approximation can be reduced to Beletsky’s equation when the zonal harmonic parameter is zero. The main tool for proving these results is the Lindstedt–Poincare’s technique.