At the initial creation of the Asteroid Redirect Mission concept, the intent was to place the asteroid into LEO (low-Earth orbit) after rendezvous and capture. However, there are many instabilities and perturbations that exist in such an orbit. The current orbit trajectory of choice is one known as the lunar Distant Retrograde Orbit (DRO). It is an orbit that exists due to third-body effects, and is designed via the circular restricted three body problem. For this particular orbit, it operates by incorporating the effects from both the Earth and the Moon (treated as one mass) and the Sun. A brief explanation of the characteristics of a Distant Retrograde Orbit is given below.
Distant Retrograde Orbits
The most appealing aspect of a DRO is its stability, as defined by the Lyapunov stability. The definition of Lyapunov stability is: “an equilibrium point is locally stable if all solutions which start near the point remain near that point for all time” (Lyapunov). This stability allows an object to be an a “quarantine orbit”, while experiencing only slight, periodic variations in its orbit over time.
With previous research done by M. Henon, distant retrograde orbits are classified under the family f of simple-periodic symmetrical orbits. These orbits have the following characteristics:
Retrograde satellites around a second body
The second body's mass is negligible relative to the main body
Stable over long periods of time (hundreds of years)
Stability extends at very large distances away from the second body several millions of kilometers (encompasses the entire range of Jacobi integral)
This can be seen in comparison to other orbit families, such as a, which is unstable. As seen in the figure below, this a family orbit ultimately converges towards it's equilibrium point, and isn't able to keep the object in orbit at a constant orbital radius.
As a source of clarification, the axes ξ and η in the figure a bove are scaled versions of the typical coordinates x and y, respectively.
An example of a large amplitude distant retrograde orbit, that was being analyzed for the use of a future mission concept around Europa, is given below.
The left plot shows the orbits as seen in an inertial frame, while the right one depicts the orbit as seen from the L1 rotating frame. The semimajor axis of this orbit was assumed to be approximately 35,000 km. Due to the retrograde then prograde motion of the orbit in the inertial frame, a satellite's motion cannot be described by Keplerian orbital elements, making the problem more difficult to analyze. The relative size of a Distant Retrograde orbit around the Moon is given in the image below.
Currently, research has not revealed a spacecraft that has utilized a Distant Retrograde Orbit in its operation, despite 40-50 years of research going into this topic. The proposed JIMO mission, before it was cancelled in 2005, was looking into using a DRO for its stability and low-energy transfer qualities. DROs have also been of interest to potential spacecraft in creation of an initial warning system for large coronal mass ejections heading towards Earth. However, neither of these missions came to fruition. Therefore, there is not mission experience in flying a spacecraft to and into such an orbit.