Research on third-order analytical solution determination of halo orbits around equilibrium point of asteroid

Though the existence of periodic orbits has been proved, it's a challenging task for us to find the periodic solution with a certain accuracy in a given dynamics system. A method to determine accurate periodic orbit (also called halo orbit) surround the equilibrium points of asteroids is presented. Firstly, to expend the dynamics model and equation of motion, the right end in equation of motion expanded into third-order power series form. Then the nonlinear equations of motion can be extended to quasi-linear differential equations. Secondly, to solve the extended equations of motion by using Lindstedt-Poincaré method, the periodic solution and its frequency are expanded into third-order power series. After substituting the two power series into the quasi-linear differential equations, linear equations of motion with three different orders are got. Solving the differential equations successively and eliminating the secular teams in the solutions, then, the third-order analytical solution for halo orbits is obtained. Finally, the accurate halo orbit in true gravitational field by using differential correction to amend the analytical solution is got.