Invariant Manifolds

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Invariant Manifolds
Halo orbits are a type of Unstable Periodic Orbit (UPO) and as such, they have what are known as invariant manifolds. Each point on a halo orbit has a characteristic stable and unstable direction. If a spacecraft is perturbed in the unstable direction, the spacecraft will "fall off" of the halo orbit. The set of all the trajectories that fall off a halo orbit can be called the Unstable Invariant Manifold, or the unstable manifold (WU). On the other hand, if a spacecraft is perturbed in the stable direction, the spacecraft will then "fall back" to the halo orbit asymptotically. The set of all the trajectories that arrive on the halo orbit following the stable direction is called the Stable Invariant Manifold, or stable manifold (WS).
Animation
The animation above shows several spacecraft evenly spaced on a halo orbit. Each spacecraft is perturbed in the unstable direction and they all fall off the halo orbit and make it easy to visualize the shape of the unstable manifold of that halo orbit. The main thing to remember here is that only a small perturbation is needed to leave the halo orbit and go somewhere else. This is what enables low-energy transfers.
Animation
The animation above shows several spacecraft following the stable manifold of a halo orbit. The spacecraft arrive onto the halo orbit ballistically, or in other words they asymptotically approach the halo orbit with no insertion maneuver required. The trajectories on the stable manifold would allow spacecraft transfer to halo orbits with no ΔV required on arrival.

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