Distant Retrograde Orbit

Author: Tianjiang Says

Website: https://cislunarspace.cn

Definition

A Distant Retrograde Orbit (DRO) is a stable periodic orbit around the Moon in the Circular Restricted Three-Body Problem (CRTBP). In the synodic reference frame, a DRO moves in the opposite direction to the Moon's orbit around Earth, hence the term "retrograde" orbit.

DRO orbit shape in the Earth-Moon synodic reference frame

Geometric configuration of DRO in the barycentric rotating reference frame

Geometric Characteristics

In the synodic reference frame, a planar DRO appears as an approximately elliptical closed curve with the Moon as its geometric center. Its main parameters are:

• $x$-direction amplitude $A_x$: The distance from the intersection of the orbit with the Earth-Moon line to the Moon, which is the primary parameter describing the DRO configuration
  • When $A_x$ is small, the DRO is close to the Moon, approximating a circular lunar orbit
  • As $A_x$ increases, the DRO moves farther from the Moon, and the orbit shape evolves from nearly circular to an ellipse with significant eccentricity
• $z$-direction amplitude $A_z$: Introducing a $z$-direction amplitude yields a three-dimensional non-planar DRO, which exhibits both retrograde motion within the $xOy$ plane and periodic oscillation in the $z$ direction

Resonance Relationships

DROs exhibit resonance relationships with the Moon's orbital period. When the DRO's orbital period $T$ and the Moon's orbital period $T_M$ satisfy $T/T_M \approx n/m$ (where $n$ and $m$ are positive integers), it is referred to as an $m:n$ resonant DRO.

For example, a 2:1 resonant DRO has an orbital period approximately half that of the Moon's orbital period -- meaning the spacecraft completes two orbits around the Moon for every one orbit the Moon completes around Earth.

Dynamic Symmetry

In the CRTBP, DROs exhibit dynamic symmetry about the $x$-axis: when the orbit crosses the $x$-axis, there is only a $y$-direction velocity component $\dot{y}_0$, while the $y$-direction position and the $x$ and $z$ direction velocities are all zero. This symmetry means that one only needs to select an initial point on the $x$-axis, use $\dot{y}_0$ and period $T$ as free variables, integrate for half a period, and verify whether the orbit returns to the $x$-axis -- enabling iterative convergence to a closed periodic orbit.

Behavior in Ephemeris Models

In perturbative environments such as ephemeris models, where celestial body positions change over time, DROs no longer maintain strict periodicity and evolve into quasi-periodic orbits that wind within a limited region. Their trajectories do not close, but the overall shape remains stable.

Application Value

With excellent long-term stability (requiring no or only minimal orbital maneuvers to maintain) and advantageous orbital position, DROs have become the preferred mission orbit for cislunar space infrastructure. Application scenarios include:

• Situation awareness constellation deployment
• Cislunar space navigation system networking
• Deep space relay communications
• Material storage and strategic station-keeping

NASA's Lunar Reconnaissance Orbiter (LRO) mission has validated the application value of DROs in lunar exploration. Recent research has shown that non-planar DROs with $z$-direction amplitude can avoid solar eclipses, further improving observer effectiveness.

Related Concepts

• Near-Rectilinear Halo Orbit (NRHO)
• Earth-Moon L1/L2 Halo Orbits (EML1/EML2 Halo)
• Circular Restricted Three-Body Problem (CR3BP)
• Ephemeris Model
• Starshade
• Birkhoff-Gustavson Normal Form
• Poincaré Section
• Resonance orbit
• Quasi-periodic orbit

References

• Whitley R, Martinez R. Options for staging orbits in cislunar space[C]. 2016.
• Broucke R. Periodic orbits in the restricted three-body problem with Earth-Moon masses[R]. 1968.
• Chen Yuju. DRO Orbit Design and Control Research for Cislunar Space Situation Awareness[D]. 2024.
• Genszler G, Savransky D, Soto G J. Surveying orbits in cislunar space for telescope-starshade observatories[J]. 2026.
• Qiao C, Long X, Yang L, et al. Orbital parameter characterization and objects cataloging for Earth-Moon collinear libration points[J]. Chinese Journal of Aeronautics, 2025. doi: 10.1016/j.cja.2025.103869.