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In this paper we present an analytical theory with numerical simulations to study the orbital motion of lunar artificial satellites. We consider the problem of an artificial satellite perturbed by the non-uniform distribution of mass of the Moon and by a third-body in elliptical orbit (Earth is considered). Legendre polynomials are expanded in powers of the eccentricity up to the degree four and are used for the disturbing potential due to the third-body. We show a new approximated equation to compute the critical semi-major axis for the orbit of the satellite. Lie-Hori perturbation method up to the second-order is applied to eliminate the terms of short-period of the disturbing potential. Coupling terms are analyzed. Emphasis is given to the case of frozen orbits and critical inclination. Numerical simulations for hypothetical lunar artificial satellites are performed, considering that the perturbations are acting together or one at a time.

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The eccentric Kozai-Lidov mechanism, based on the secular theory, has been proposed as a mechanism that plays an important role in producing orbits that switch from prograde to retrograde. In the present work we study the secular dynamics of a triple system composed of a Sun-like central star and a Jupiter-like planet, which are under the gravitational influence of another perturbing star (brown dwarf). The perturbation potential is developed in closed form up to the fifth order in a small parameter (α = a 1 /a 2), where a 1 is the semimajor axis of the extrasolar planet and a 2 is the semimajor axis of the perturbing star. To eliminate the short-period terms of the perturbation potential, the double-average method is applied. In this work we do not eliminate the nodes, a standard method in the literature, before deriving the equations of motion. The main goal is to study the effects of the higher-order terms of the expansion of the perturbing force due to the third body in the orbital evolution of the planet. In particular, we investigate the inclination and the shape (eccentricity) of these orbits. We show the importance of the higher-order terms in changing the inversion times of the