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This paper aims to obtain an analytic approximation to the evolution ofcircular orbits governed by the Earth's $J_2$ and the luni-solar gravitationalperturbations. Assuming that the lunar orbital plane coincides with theecliptic plane, Allan and Cook (1964) derived an analytic solution to theorbital plane evolution of circular orbits. Using their result as anintermediate solution, we establish an approximate analytic model with lunarorbital inclination and its node regression be taken into account. Finally, anapproximate analytic expression is derived, which is accurate compared to thenumerical results except for the resonant cases when the period of thereference orbit approximately equals the integer multiples (especially 1 or 2times) of lunar node regression period.

Analytic model for the long-term evolution of circular Earth satellite orbits including lunar node regression