Application of a new semi-analytical method to periodic motion due to the J22 tesseral harmonic

We revisit the issue of constructing the first-order periodic solution that incorporates the J22 tesseral harmonic and developing a new semi-analytical solution that may apply to any orbital eccentricity in [0,1）. In our work, the solution is expressed in a finite compact form composed of several definite integrals with varying integration intervals constrained in [0,Tr], in which the traditional Hansen coefficients are no longer involved. Numerical experiments are also given and compared with the traditional series expansion method, and the results show that the derived solution is capable of dealing with highly eccentric orbits. Therefore, the solution given can provide a new technique to analyze the perturbation characteristics arising from the J22 harmonic.