A semi-analytical approach using the single and double averaged methods and the Lidov–Kozai mechanism

Abstract

An analysis of the orbital motion of artificial satellites around Mercury is presented taking into account its non-sphericity (J₂, J₃, C₂₂) and the perturbation of the third body. The disturbing potential due to the third body is developed in circular and inclined orbit. The double-averaged method should be used with caution in some situations where the averaging is applied at different timescales. In this work, a study is presented considering this observation for orbits around Mercury. When the mean anomaly of the Sun is eliminated, the idea is that all effects whose periods below 88 days are neglected. As the rotation of Mercury is about 58.6 days, this means that the perturbation due to the C₂₂ term must also be neglected. However, since the C₂₂ term is important and should be taken into account, then terms longer than 58.6 days should also be preserved. In other words, keeping the C₂₂ term with a period of 58.6 days means that the solar terms with the longest period (88 days) will be maintained here. The single-averaged method is applied to eliminate only the mean anomaly of the spacecraft. A comparison between the single and double averaged models is presented. We found that for the case of Mercury the two models are in agreement, but the single-averaged model is more realistic because it keeps more terms in the disturbing potential. Several types of resonances can be analyzed starting of the single-averaged potential. Considering our single-averaged disturbing potential, the terms due to Lidov–Kozai resonance were isolated to make a qualitative analysis considering the libration and circulation regions in the diagram, eccentricity versus argument of the pericenter.