Reading about the upcoming (as of the 01/10/2022) NASA mission "Artemis 1" triggered my curiosity to take a closer look at the mission profile and the properties of the chosen Distant Retrograde Orbit (DRO) around the Moon.
Artemis 1 is a planned uncrewed Moon-orbiting mission, the first spaceflight in NASA's Artemis Moon exploration program and the first flight of the agency's powerful Space Launch System (SLS). The goal of the mission is to certify the Orion spacecraft and the Space Launch System for crewed flights beginning with Artemis 2. The mission will lift off from Launch Complex 39B at the Kennedy Space Center aboard the SLS rocket and the Orion spacecraft will be launched on a mission of between 26 and 42 days.
The sequence of events in the mission is as follows: after reaching Earth-orbit and performing a Trans-Lunar Injection (TLI) to bring the spacecraft in a trajectory towards the Moon, the mission will deploy ten CubeSat satellites and the Orion spacecraft will enter a distant retrograde orbit (DRO) for either six or eighteen days (which corresponds to half or one and a half revolutions along the DRO). The Orion spacecraft will then return and re-enter the Earth's atmosphere, protected by its heat shield, and splash down in the Pacific Ocean.
Distant Retrograde Orbits (DROs)
DROs are specific orbits that occur within two-body systems like the Moon-Earth system and arise as a solution of the three-body problem. As the name suggests, they are pretty far from the moon (hence "distant") - as a reference the one chosen for the Artemis 1 mission is about 70,000 km from the surface of the moon - and also rotate in the opposite direction relative to the motion of the Moon (hence "retrograde"). Technically, a probe that is on a DRO is still orbiting around the Earth in a ellipsoid-like trajectory, but relative to the Moon (in a rotating frame with the same period as the Moon's rotation around the earth), a retrograde orbit is obtained. This is illustrated in the short video underneath, where a DRO with period of ~12 days and average distance from the moon of around 70,000 km is propagated in time:
As it is a solution of the circuular restricted three-body problem, the orbit that is used for the Artemis 1 mission is in the Earth-Moon plane, meaning that it's flat, around the Moon’s equator. The DRO family is predominantly comprised of stable members that travel clockwise around the Moon and since a stable orbit can potentially require little propellant for maintenance, DROs have been considered as viable candidates for long-term storage orbits and are a great candidate for the first Artemis 1 demonstrator mission. Moreover, DROs have been considered for asteroid retrieval mission concepts, such as the NASA Asteroid Redirect Robotic Mission concept. Initially, DROs were proposed for observational purposes in the Sun-Earth system due to their favorable deep space environment and periodic, predictable behavior (see Demeyer et al, Stramacchia et al.)
The DRO orbits are generally not periodic in an inertial frame, but the family possesses some resonant members. Example of a 3:2 and 4:3 resonant orbits (with periods of ~18.3 and ~20.5 days approximately) for the Earth-Moon system are shown in the figures underneath in a rotational frame:
... and in an inertial frame:
An interesting analysis of the DROs along with other orbits in the Earth-Moon system is given in Whitley et al., where an extensive comparison is shown with the main criteria being the flexibility that an orbit offers in terms of access from Earth, access to other destinations, and spacecraft design impacts. Establishing a viable staging orbit in cis-lunar space is a key step in the human exploration journey beyond Low Earth and DROs are one promising candidate family of orbits for this goal. According to the same study, the Near Rectilinear Orbit (NRO) appears to be the most favorable orbit to meet the multiple, sometimes
competing, constraints and requirements.
Table from Whitley et al.
Transfer towards DROs
Getting to DROs from low-earth orbit (LEO) has been the subject of multiple studies. As I find the specific transfer problem to be quite interesting, I wanted to calculate the transfer trajectory to different points along a DRO. The chosen target orbit is similar to the one that Artemis 1 is aiming for (~70,000km distance to the Moon, ~12 days period).
I am showing a direct transfer and a powered Moon-flyby transfer. For both cases, the constraints and boundary conditions are identical:
Solving the circular restricted three-body problem (CR3BP)
Starting from a 300km circular LEO
Avoiding collision with the Moon's surface
Minimizing the DeltaV (sum of flyby and DRO insertion maneuver, excluding the TLI DeltaV)
Direct transfer
For the direct transfer, selected optimal trajectories are shown in the figure below. In all plots, x=0 corresponds to the barycenter of the Earth-Moon system, which is within the Earth, at ~75% of the planet's radius.
Note that the rotation of the DRO is clockwise. It can be seen that for the first 3 plotted trajectories, the change in direction upon arrival at the DRO is quite large, which is connected to a high fuel expenditure and DeltaV.
Looking at the evolution of the transfer time and fuel requirements as a function of the location along the DRO, one sees that there is a clear minimum of the DRO-insertion burn DeltaV for the point on the orbit that is the furthest from the earth (corresponding to 180 degrees in the diagram below). The DeltaV needed to depart from the LEO however, does not seem to be singnificantly affected by the location of the point on the PO.
The fuel-optimal trajectory for the direct insertion into the DRO then looks like the following:
Moon-flyby transfer
Leveraging a flyby close to the Moon to reduce the DeltaV is also the option that has been chosen for the Artemis 1 mission.
Selected optimal trajectories for different DRO insertion points can be seen in the following diagram. Only locations between 0 and 140 degrees along the DRO were investigated here, as they are most relevant for Artemis 1.
The corresponding DeltaVs and transfer times are also plotted here. As expected, trageting the Moon for a flyby before the final DRO insertion is benefitial in terms of the total fuel consumption, but does so in an expense of transfer time.
Artemis 1 mission
In order to recreate an example of what the Artemis 1 trajectory may look like, I chose a DRO insertion point at 90 degrees from the Earth-Moon line (in the rotational frame). As an altitude for the flyby 100km were chosen, while the time of flight on the DRO after insertion was defined to be 18 days (1.5 revolutions on the DRO). For the final splashdown, no constraint on the entry angle was introduced, which of course would need to be taken care of to ensure that the angle of attack of Orion is reasonable within the Earth's atmosphere.
The objective to be minimized was the sum of the DeltaVs for
Moon flyby 1
Insertion into the DRO
Departure from the DRO
Moon flyby 2
Again, the TLI DeltaV was not included in the objective function for the optimization.
In the simulation scenario above, the spacecraft arrives at the location of the first lunar flyby after an initial transfer of ~6 days, . Upon the completion of the flyby, it enters a ~3 day travel towards the DRO where it stays for 1.5 revolutions, i.e ~18 days, before the DRO departure burn is carried out. The next 6 days correspond to the departure leg trajectory from the DRO, and after that the final flyby burn is carried out to send the Orion spacecraft towards thr earth.
No mid-course corrections in the outbound and return transits have been included.