Lunar Synodic Resonance (LSR)

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

Definition

Lunar Synodic Resonance (LSR) is a state in which a spacecraft orbital period $T_{\text{orb}}$ and the lunar synodic period $T_{\text{syn}}$ (~29.5 days, the interval between successive Sun-Earth-Moon conjunctions) satisfy a simple integer ratio:

\frac{T_{\text{orb}}}{T_{\text{syn}}} = \frac{P}{Q}

with positive integers $P$ and $Q$ . The resonance is commonly denoted $M:N$ , meaning the spacecraft completes $M$ revolutions while the Sun-Earth-Moon geometry repeats $N$ times. Because the synodic period is defined by the relative motion of the Sun and Moon, the resonance makes the trajectory predictable with respect to the periodic Sun-Earth-Moon configuration (Zimovan-Spreen et al. 2020; Boudad et al. 2020).

Mathematical and Dynamical Details

In the bicircular restricted four-body problem (BCR4BP), the Sun revolves around the Earth-Moon barycenter at a constant angular rate $\omega_S$ (about $-0.925$ in Earth-Moon non-dimensional units, corresponding to 29.5 days). An $M:N$ synodic resonant periodic orbit must satisfy

T_{4BP} = N \left| \frac{2\pi}{\omega_S} \right|

with $\left| 2\pi / \omega_S \right| \approx 29.5$ days. The corresponding single-revolution orbital period in the CR3BP is

T_{3BP} = \frac{N}{M} \left| \frac{2\pi}{\omega_S} \right|

(Oshima 2022).

Synodic resonant orbits in the BCR4BP are computed by pseudo-arclength continuation from CR3BP solutions: the Sun's mass is gradually increased from zero, with intermediate solutions corresponding to higher-period near-periodic orbits in the CR3BP (Boudad et al. 2020). When $M$ is even, two distinct BCR4BP families ( $T_0$ and $T_{1/2}$ families) can originate from one CR3BP orbit, distinguished by an initial solar phase offset of $\pi$ (Oshima 2022).

Key Examples and Eclipse Avoidance

The principal value of LSR lies in eclipse avoidance. Synodic resonant orbits form repeating lobe/peak geometries in the Sun-Moon and Sun-Earth rotating frames; with careful epoch selection, the Earth and Moon shadow cones pass through the inter-lobe gaps without intersecting the trajectory, achieving long-duration ballistic eclipse-free flight (Zimovan-Spreen et al. 2020).

The following table lists key synodic resonant NRHOs along the $L_2$ halo family in the CR3BP Earth-Moon system:

Resonance	Orbital Period	Perilune Radius	Apolune Radius	Remarks
9:2	~6.53 days	~3150 km	~71000 km	Gateway baseline; 9 revs = 2 synodic periods (~59 days)
4:1	~7.34 days	~5600 km	~75335 km	4 revs = 1 synodic period; wider eclipse margins
3:1	~9.79 days	~15000 km	~84500 km	Simple ratio but higher perilune
5:1	~5.90 days	—	—	Very low perilune

Higher-period orbit families adjacent to NRHOs also host synodic resonant members, e.g. 2:1 P2HO1 (period ~14.76 days), 1:1 P2HO1 (~29.5 days), and 3:2 P2HO1, which preserve eclipse-avoidance geometries analogous to the 4:1 or 9:2 NRHOs (Zimovan-Spreen et al. 2020).

Application Highlights

Eclipse avoidance: in the BCR4BP, the 9:2 NRHO trajectory in the Sun- $B_1$ rotating frame is completely clear of the Earth's penumbra cone, enabling purely ballistic, long-duration eclipse-free flight.
Long-term predictability: the repeating geometry allows reliable estimation of lifetime station-keeping budgets and enables coordinated multi-spacecraft phasing (e.g. two spacecraft on the 9:2 NRHO with a $\pi$ solar-phase offset can fly concurrently without collision) (Boudad et al. 2020).
Navigation and mission planning: the periodic Sun-Earth-Moon geometry simplifies navigation filter design and onboarding epoch selection.

References

Zimovan-Spreen, E. M. et al., 2020, "Near rectilinear halo orbits and nearby higher-period dynamical structures: orbital stability and resonance properties," Acta Astronautica
Boudad, K. D. et al., 2020, "Dynamics of synodic resonant near rectilinear halo orbits in the bicircular four-body problem," Celestial Mechanics and Dynamical Astronomy
Williams, K. E. et al., 2017, "Targeting cislunar near rectilinear halo orbits for human space exploration," AIAA SPACE and Astronautics Forum and Exposition
Oshima, K., 2022, "Multiple families of synodic resonant periodic orbits in the bicircular restricted four–body problem," Advances in Space Research
Lee, K., 2019 (internal NASA report on Gateway NRHO 9:2 synodic resonant orbit analysis)