Retrograde

Author: CislunarSpace

Website: https://cislunarspace.cn

Definition

Retrograde refers to the motion state where a spacecraft's direction of travel is opposite to the central body's rotation or orbital direction. In the Earth-Moon rotating reference frame, a DRO appears as clockwise motion around the Moon, making it a retrograde orbit. The opposite is prograde orbit, where the motion direction is the same as the central body's motion.

Key Elements

Meaning of Retrograde in the Earth-Moon System

In the Earth-Moon rotating reference frame (with the Earth-Moon line as the xx-axis, rotating at the Moon's orbital rate):

  • Retrograde orbit: The spacecraft's motion around the Moon is opposite to the Moon's orbital direction around Earth, appearing as clockwise motion in the rotating frame
  • Prograde orbit: The spacecraft's motion around the Moon is the same as the Moon's orbital direction around Earth, appearing as counterclockwise motion in the rotating frame

The "retrograde" in DRO (Distant Retrograde Orbit) comes from this -- in the synodic reference frame, a spacecraft on a DRO moves in the opposite direction to the Moon.

Impulsive Advantages of Retrograde Orbits

Research by Wei et al. (2026) shows that during transfers from LEO to DRO, retrograde orbits have significant impulsive advantages over prograde orbits. The physical basis for this advantage includes:

  • Coriolis force effect: In the rotating frame, the Coriolis force for retrograde motion opposes the direction of travel, acting like a "brake" that helps the spacecraft gain more energy during the gravity assist
  • Potential energy surface structure: In the Jacobi constant equipotential surfaces of the Earth-Moon rotating frame, retrograde orbits correspond to more favorable energy channels in phase space
  • Gravity assist geometry: Retrograde-direction lunar gravity assist produces greater orbital energy change, reducing the total ΔV\Delta V required to enter DRO

Specifically, when transferring from LEO to m:nm:n resonant DRO via lunar gravity assist, the retrograde transfer scheme typically has lower total impulsive cost than the prograde scheme.

Mathematical Description of Retrograde and Prograde

In the synodic reference frame, the angular momentum hh of a spacecraft's motion around the Moon can be positive or negative:

  • Retrograde: h<0h < 0 (angular momentum vector opposite to the zz-axis of the rotating frame)
  • Prograde: h>0h > 0 (angular momentum vector aligned with the zz-axis of the rotating frame)

The prograde/retrograde nature of an orbit can be determined from the cross product of velocity and position vectors:

h=r×v\mathbf{h} = \mathbf{r} \times \mathbf{v}

Application Value

Retrograde motion characteristics have core value in cislunar mission design:

  • DRO mission design: DRO is inherently a retrograde orbit; understanding retrograde characteristics is fundamental to DRO transfer orbit design
  • Fuel optimization: Using the impulsive advantage of retrograde orbits can significantly reduce fuel consumption for cislunar transfers
  • Orbital stability: DRO's long-term stability is partly due to its retrograde nature; retrograde orbits have a wider stability region in the CRTBP framework
  • Constellation deployment: In DRO constellation design, retrograde characteristics affect inter-satellite relative motion and coverage properties

References

  • Wei Z et al. Research on lunar gravity-assist injection into cislunar distant retrograde orbit families[J]. 2026.
  • 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.