Computation of Three-Dimensional Periodic Orbits in the Sun-Earth System

From the last few decades, the space science community has shown considerable interest in missions which take place in the vicinity of the Lagrangian points in the restricted three-body problem (RTBP) of the Sun-Earth and the Earth-Moon systems [1]. Designing trajectories for these missions is a challenging task due to inadequacy of the conic approximations. The RTBP deals the situation where one of the three bodies has a negligible mass, and moves under the gravitational influence of two other bodies [2-9]. In the RTBP, the circular restricted three-body problem (CRTBP) is a special case where two massive bodies move in the circular motion around their common centre of mass [10-14]. The collinear Lagrangian point orbits have paid a lot of attentions for the mission design and transfer of trajectories [15-22]. When the frequencies of two oscillations are commensurable, the motion becomes periodic and such an orbit in the three-dimensional space is called halo [23]. Lyapunov orbits are the two-dimensional planar periodic orbits. These planar periodic orbits are not suitable for space applications since they do not allow the out-ofplane motion, e.g., a spacecraft placed in the Sun-Earth 2 L point must have an out-of-plane amplitude in order to avoid the solar exclusion zone (dangerous for the downlink); a space telescope around the Sun-Earth 2 L point must avoid the eclipses and hence requires a three-dimensional periodic orbit. Since the RTBP does not have any analytic solution, the periodic orbits are difficult to obtain because the problem is highly nonlinear and small changes in the initial conditions break the periodicity [24]. Farquhar [23] was the first person who introduced analytic computation of the halo orbit in his PhD thesis. In 1980, [25] introduced a thirdorder analytic approximation of the halo orbits near the collinear libration points in the classical CRTBP for the Sun-Earth system. Thurman & Worfolk [1], and Koon et al. [26] found the halo orbits for the CRTBP with the Sun-Earth system in the absence of any perturbative force using Richardson method [25] up to thirdorder. Breakwell & Brown [27], and Howell [28] numerically obtained the halo orbits in the classical CRTBP Earth-Moon system using the single step differential correction scheme. Numerous applications of the halo orbits in the scientific mission design can be seen such as investigations concerning solar exploration and helio-spheric effects on planetary environments using the spacecraft placed in these orbits at different phases. ISEE-3 was the first mission in a halo orbit of the Sun-Earth system around