Distant orbits are orbits found in the restricted three-body problem. A spacecraft following those orbits is in fact orbiting the major primary of the system while in formation with the smaller primary. Thus, a viewer on the smaller primary will see the spacecraft orbiting around it. In a Star-Planet system those orbits are known as Distant Prograde Orbits (DPO) and Distant Retrograde Orbits (DRO), which indicate the direction of the orbit around the Earth, the same as the Earth is rotating around the Sun for the former and opposite it for the latter. These orbits are of interest because of their distance from Earth and from its disturbances, such as radiation and magnetic fields, while at the same time sustaining a periodic orbit and confining the spacecraft to a closed area.

The fact that the restricted three-body problem cannot be solved analytically, makes computation of even the simplest orbit a time consuming numerical work. However, through the use of variation of parameters and averaging methods we were able to obtain a closed-form analytical approximation to the DROs. This enables a much simpler preliminary design for space missions that are to fly on distant orbits.

In this work we use the instability of DPOs to design a transfer trajectory from Low Earth Orbits (LEO) to DPOs using the orbit stable and unstable manifolds. This method enables us to find a transfer trajectory for which the amount of fuel needed will be smaller than trajectories utilizing the theory of the Two-Body Problem.