Analytical propagation solution for planet-displaced orbit in the presence of third-body perturbations

Planet-displaced orbits (PDOs) play an important role in space missions such as solar observation, gravitational wave detection, and near-Earth asteroid detection. To propagate the PDOs accurately and efficiently, this paper develops an analytical solution considering the Solar central gravitational force and the time-varying third-body perturbation of the corresponding planet. First, an approximated third-body perturbation model is established based on the planet displacement angle (PDA), which is found to be the core variable affecting the evolution of the orbit. The model can describe both secular and periodic terms of the third-body perturbation accurately. Then, based on the established third body perturbation model, a two-step procedure is developed to iteratively derive the analytical orbit propagation solution of the PDO via the Picard iteration method. The analytical solution is successfully applied to propagate the orbit in an Earth-trailing orbit case: the Laser Interferometer Space Antenna (LISA). Simulation shows that the analytical orbit propagation solution can accurately predict the orbit in both the long-time and short-time cases. The relative error is less than 0.1% in 10 years. The proposed analytical solution can be potentially useful in designing and optimizing PDOs.