Space technology in binary asteroid system: third-order analytical solutions of periodic orbits around collinear equilibrium points

Periodic motion in binary asteroid systems is fundamental to studying system dynamical behaviors and designing exploration missions. In this paper, we focus on the periodic orbits around collinear equilibrium points of the system. Third-order analytical solutions for three types of periodic motion (i.e., planar Lyapunov, vertical Lyapunov, and Halo orbits) are sought. Considering the mass distribution, we model the system as two triaxial ellipsoids, so the gravitational field consists of their ellipsoid potentials. Elliptic integrals analytically express high-ordered partial derivatives of the ellipsoid potential in preparation for the procedure. After utilizing the Taylor series to linearize the system's effective potential, we employ the Lindstedt-Poincaré method to construct the third-order analytical solutions. Features of three periodic motions help determine the unknown coefficients during the recursive procedure with elaborate manipulations. A good agreement is found between the analytical solution and numerical results for periodic orbits, except for the Halo case. We develop an extra scaling technique to correct the deviation, whose scaling factor is formulated by system parameters through a polynomial fitting procedure. The numerical simulation shows that the Halo approximated solution is also sufficiently accurate, covering binary asteroids within various parameter ranges.