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