Approximate analytic periodic solution of circular Sitnikov restricted four-body problem

Abstract In this technical note, a high-precision approximate analytic periodic solution of the circular Sitnikov restricted four-body problem is constructed. In order to improve the accuracy, the governing differential equation is first converted into another form without using Taylor expansion. Then the approximate solution is established by combining the harmonic balance method and Newton’s method. Compared with the fourth-order approximate analytic solution obtained by using the Lindstedt-Poincaré method, only one Newton iteration can obtain higher precision, wider amplitude range, and shorter approximate analytic period and periodic solution expressions.