Solar-Perturbation Lunar Gravity Assist (Forward/Backward LGA)

Author: Tianjiang Shuo
Website: https://cislunarspace.cn

Definition

In the bicircular restricted four-body problem (BCR4BP), solar gravity is a time-periodic perturbation on the Earth-Moon CR3BP. The derivative of the spacecraft's Earth-Moon mechanical energy $E$ with respect to time has a definite sign over most of the orbital plane:

\frac{dE}{dt} = \mu(1-\mu)\,y\left(\frac{1}{r_1^3} - \frac{1}{r_2^3}\right)

(Wang et al. 2025, Eq. 8), where $\mu$ is the Earth-Moon mass ratio and $r_1$ , $r_2$ are the distances to Earth and Moon. Using $x = 0.5 - \mu$ and $y = 0$ as the coordinate axes, the plane is divided into four quadrants. The sign of $dE/dt$ is positive in quadrants II and IV and negative in quadrants I and III (and nearly zero far from any massive body). Wang et al. (2025) call this coordinate system the mechanical energy coordinate system.

Because $E + C/2$ equals (up to a constant) the angular momentum $H$ (Wang et al. 2025, Eq. 10), $C$ and $H$ vary together with $E$ . We therefore label:

Forward lunar gravity assist — flight through quadrants II and IV, where solar perturbation raises $E$ , $C$ , and $H$ and lifts the perigee;
Backward lunar gravity assist — flight through quadrants I and III, where solar perturbation lowers $E$ , $C$ , and $H$ and drops the perigee.

"Forward" and "backward" here refer to the direction of the energy trend under solar perturbation, not to the spacecraft's sense of motion around the Moon (see the separate prograde/retrograde classification). "Assist" is used in the sense of an energy-trend label rather than a literal flyby event.

Why the Sun, not the Moon, is the driver

Inside the Earth-Moon sphere of influence the spacecraft's Keplerian energy with respect to either the Earth or the Moon is conserved to two-body accuracy; gravity-assist energy exchange happens at the encounter. Solar gravity is different: it acts continuously, and in the BCR4BP it is the term that makes $E$ and $C$ non-constant. The Moon's role is to route the spacecraft through a chosen quadrant sequence — a lunar encounter can flip the spacecraft from a backward quadrant into a forward quadrant, switching the sign of $dE/dt$ and thus the energy trend. Low-energy transfer design exploits precisely this routing.

Application notes

Low-energy DRO insertion. For 2:1 DRO insertion via WSB, Wang et al. (2025) identify the low-energy transfer gateway (LEGT) — the energy-and-geometry region a WSB return trajectory must satisfy if it is to admit a low-impulse DRO insertion. About 73.6% of candidate trajectories that pass the LEGT filter are feasible, versus under 1% for a naive grid search, which is the practical payoff of quadrant-aware forward/backward analysis.
Phase partitioning. The transfer is split into three phases — Earth-Moon transfer (where LGA reduces departure energy), Sun-Earth WSB transfer (where forward quadrants do the work), and DRO low-energy capture — each with its own dominant dynamics and its own energy-trend expectation.
Handoff. At the WSB-to-DRO handoff the spacecraft must arrive with $E$ in roughly $[-1/3, -1/5]$ (the energy range of the WSB region) and with a compatible angular-momentum sign. A trajectory that returns through backward quadrants is unsuitable for low-energy DRO insertion and can be filtered out before expensive numerical optimization.

Common confusions

"Forward/backward LGA" is sometimes read as "fore/aft of the Moon" or "prograde/retrograde flyby" — both wrong. The axis is the sign of the solar-perturbation energy trend in the mechanical-energy coordinate system, which is set by which synodic quadrant the spacecraft is in, not by the geometry of the lunar encounter itself.
The energy $E$ here is not the Keplerian two-body energy about the Earth or the Moon; it is the Earth-Moon synodic-frame energy defined by Wang et al. (2025, Eq. 6), which in two-body limits reduces to the usual Keplerian form but in the BCR4BP carries the integrated effect of solar gravity.

References

Wang M., Zhang C., Zhang H., 2025, "Mechanism analysis of the DRO low-energy transfer problem: an energy perspective," Lect. Notes Eng. (defines the mechanical-energy coordinate system, the quadrant division, the LEGT, and the forward/backward classification)
Peng et al., 2024, "Low-Energy Transfers to Lunar Distant Retrograde Orbits from Geostationary Transfer Orbits," J. Spacecraft and Rockets, doi:10.2514/1.A35623