Low-Energy Transfer

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

Definition and Criteria

A low-energy transfer is a collective term for Earth–Moon transfer trajectories that consume less energy than a Hohmann transfer at the price of a longer transfer time (Xu 2010, Zheng & Zhao 2023). Three low-energy criteria coexist in the literature; state which is meant when citing:

Cost criterion: total ΔV below that of a Hohmann transfer between the same endpoints. The magnitude depends on the endpoints: a WSB transfer saves about 18% over Hohmann (Belbruno & Miller 1993), and typical savings run 80–270 m/s (the example relayed by Xu 2010 is about 150 m/s; Sousa-Silva 2018 about 80 m/s). Reference lower bound: Sweetser estimated the global minimum ΔV in the CR3BP from a 167 km Earth circular orbit to a 100 km lunar circular orbit at about 3.72 km/s; low-energy transfers including solar perturbation can go below this CR3BP theoretical bound (Anderson & Parker 2012).
Geometric criterion: instantaneous eccentricity below 1 throughout the transfer: the spacecraft remains in a two-body-bound (elliptic) state relative to Earth/Moon at all times; a Hohmann transfer arrives near the Moon at hyperbolic speed and can never be ballistically captured (Xu 2010, Xu 2013).
Arrival-end energy criterion: the two-body (Keplerian) energy relative to the Moon turns from positive to negative on approach (Topputo 2013, Parker & Anderson 2014); see Ballistic Capture.

In time, low-energy transfers typically take 2.5–5 months (Belbruno & Miller 1993, Parker & Anderson 2014), versus about 3 days for a Hohmann.

Three Technical Routes

A low-energy transfer is not a single technique but three interlocking routes:

The invariant-manifold route (inside the Earth–Moon system): Conley 1968 established the orbit classification for the neck region near a libration point (transit / non-transit / asymptotic / periodic orbits, with manifolds as the separatrices); Koon and colleagues patched the manifold tubes of the Sun–Earth and Earth–Moon CR3BPs on Poincaré sections, and trajectories continuous in both position and velocity are low-energy transfers (Xu 2010 calls this the solution to a theoretical problem that had troubled celestial mechanicians for over a decade). Interior transfers purely within the Earth–Moon system need no solar perturbation and can be defined inside the Earth–Moon three-body model (Topputo 2013).
The WSB/exterior route: a lunar flyby sends the spacecraft out to Earth's weak stability boundary (about 4 times the Earth–Moon distance), where the region's dynamical sensitivity allows a near-zero-maneuver junction to a ballistic-capture orbit on the Moon's WSB (Belbruno & Miller 1993). See Weak Stability Boundary Transfer Trajectory.
Ballistic capture is the arrival mechanism, not an independent route: both the WSB transfer and the manifold route end in ballistic capture: targeting the region inside the stable-manifold tube of an Earth–Moon L2 Lyapunov orbit constructs an orbit ballistically captured by the Moon (Ross 2022).

Belbruno 2010 unifies the three routes: within a certain energy range, the points of the secondary primary's WSB are exactly the points on the stable manifolds of the L1/L2 Lyapunov orbits where the radial velocity is zero and the Keplerian energy relative to the secondary is negative.

Classifications

By trajectory geometry (Topputo 2013): exterior transfers reach an apogee of about 4 Earth–Moon distances, and the apogee must lie in the second or fourth quadrant of the Earth-centered frame (x-axis pointing anti-Sun); interior transfers stay mostly inside the lunar orbit and can exploit resonances to shorten the time.
By the libration point used (Zheng & Zhao 2023): the L1 manifolds give inner-capture-type transfers (always inside the Earth–Moon system, shorter); the L2 manifolds give outer-capture-type transfers (= the WSB route, requiring solar gravity assist, longer).
By the object transited (Xu 2010): LL1 transit (minimum-energy transfer), LL1-halo transit, LL2 transit (inner/outer side: outer is the WSB), LL2-halo transit.
By mission profile (Parker & Anderson 2014): direct (3–6 days), direct-staging (2–10 weeks), direct to Earth–Moon L1 (1–5 weeks), low-thrust (months), low-energy (2.5–4 months).

Design Methods and Numbers

Two-impulse fully ballistic solutions: the global optimum is 82.6 days, 3769 m/s (Topputo 2013); an exterior transfer with one lunar flyby takes 162 days, 3775 m/s (Campana 2024).
Three/four-impulse schemes (Qiao 2024, in the EML1 halo→GEO direction): the three-impulse energy optimum is 1.55 km/s over 40 days; the four-impulse 1.47 km/s over 48 days: turning at the unstable manifold's minimum-velocity point and braking at its maximum, with one maneuver at each end of the Lambert arc.
A single stable manifold directly connecting near-Earth and near-Moon orbits: total ΔV about 3921 m/s over 62 days (Zheng & Zhao 2023).
Cross-system manifold patching by bilevel optimization: the minimum patching impulse is about 12.6 m/s (Li 2024).

Mission Applications

Earth–Moon system: Hiten (1991, first WSB/ballistic-capture transfer), GRAIL (2011, first mission with a low-energy transfer as the main leg, performing lunar orbit insertion directly), ARTEMIS (2010, low-energy transfer to libration point orbits; using a nearly ballistic transfer including statistical corrections and small deterministic maneuvers). Sun–Earth reference: Genesis; SMART-1 combined low thrust with the WSB concept.

Terminology Variants

Term	Meaning	Source
Low-energy transfer (LET)	Same as low-energy transfer; literature abbreviation LET	Fantino 2010
Low-energy lunar transfer	2.5–4-month Earth–Moon transfer exploiting solar gravity	Parker & Anderson 2014
Exterior transfer	Low-energy transfer with apogee about 4 Earth–Moon distances	Topputo 2013
Outer-capture / inner-capture type	Classified by the libration point used (L2/L1)	Zheng & Zhao 2023
Three/four-impulse low-energy transfer	Scheme with maneuvers at manifold velocity extrema plus Lambert arcs	Qiao 2024

References

Conley, 1968, Low energy transit orbits in the restricted three-body problem
Belbruno & Miller, 1993, Sun-perturbed Earth-to-Moon transfers with ballistic capture
Belbruno, Gidea & Topputo, 2010, Weak stability boundary and invariant manifolds
Fantino et al., 2010, A note on libration point orbits, temporary capture and low-energy transfers
Xu, 2010, Onset conditions and trajectory construction of Earth–Moon low-energy transfers
Xu et al., 2013, On the construction of low-energy cislunar and trans-lunar transfers based on the libration points
Topputo, 2013, On optimal two-impulse Earth–Moon transfers in a four-body model
Anderson & Parker, 2012, Survey of ballistic transfers to the lunar surface
Anderson & Parker, 2013, Comparison of low-energy lunar transfer trajectories to invariant manifolds
Parker & Anderson, 2014, Low-Energy Lunar Trajectory Design
Sousa-Silva et al., 2018, Fast Earth–Moon transfers with ballistic capture
Ross et al., 2022, Dynamical Systems, the Three-Body Problem, and Space Mission Design
Zheng & Zhao, 2023, Earth–Moon transfer method based on the stable manifolds of large-amplitude Lyapunov orbits
Li et al., 2024, Low-energy Earth–Moon transfer trajectory design based on weak stability boundary theory
Qiao & Yang, 2024, Design and optimization of low-energy transfer trajectories at Earth–Moon L1
Campana et al., 2024, Low-energy Earth–Moon transfers via theory of functional connections and homotopy
Grossi et al., 2024, High-efficiency exterior low-energy transfer study