The circular Sitnikov problem is revisited, using matched asymptotic expansions. In the case of large oscillation periods, approximate analytical expressions for the period and the orbit of the third body are found. The results are compared with those described in the literature and show that the movement of the third body can be well described by two analytical solutions, the inner and outer solutions.
Marco Rosales-Vera. "Matched Asymptotic Expansions to the Circular Sitnikov Problem with Long Period." J. Appl. Math. 2012 1 - 15, 2012. https://doi.org/10.1155/2012/479093