Long term evolution of the eccentricity in the meo region: Extraction of the most significant terms from the potential expression

The long term evolution of the mean eccentricity of trajectories within the MEO region is driven by the inuence of (mainy) zonal and (but also) tesseral parameters of the Earths gravity field, and the luni-solar attraction. A wide literature now exists to explain the dynamical sources of the eccentricity growth that is observed in many cases, and due to commensurabilities (resonances) between some angles appearing in the potential, linked to the motion of the third bodies and the satellites. Depending on the initial conditions, and with a sensitivity depending upon the width of such commensurabity driving stable or chaotic regions, the two components of the eccentricity vector follow an ellipse through the equations of a harmonic oscillator with two regimes: libration or circulation. We provide in this article the analytic expressions of the equation of motion of the eccen-tricity, under the inuence of satellite' angles-only commensurabilities. This work is led independently from the well known Kaula's development published in 1962 for a potential generated by a third body, This paper is a generalization of the paper by the same authors presented during the 2015 summer conference hosted in Vail, and that was focused only on one of the resonant com-binations of Galileo-like orbits. We extract from the general expression of the third-body potential those terms that are the most significant to explain the temporal variations of the eccentricity, and due to commensurabilities with the satellite angles.