Near-Rectilinear Halo Orbit

Author: Tianjiang Says

Website: https://cislunarspace.cn

Definition

A Near-Rectilinear Halo Orbit (NRHO) is a sub-class of Halo orbits near the Earth-Moon collinear libration points $L_1$ or $L_2$. In the synodic reference frame, when the out-of-plane amplitude $A_z$ of a Halo orbit is much larger than the in-plane amplitude $A_y$, the orbit shape transitions from the classic "cashew-shaped" Halo orbit to an approximately linear reciprocating motion -- i.e., the NRHO. In other words, the NRHO corresponds to the extreme members of the Halo orbit family with large $A_z/A_y$ ratios.

Earth-Moon L1 northern family and L2 southern family Halo orbits, with the extreme configuration being the NRHO

Geometric Characteristics

• Extremely low perilune altitude: typically below 100 km
• Apolune: located near the Earth-Moon $L_2$ point
• Orbital plane symmetric about the $xOz$ plane: with southern and northern families as two branches
• Overall presents an approximately linear reciprocating motion

Resonance Relationships

Similar to DROs, NRHOs also exhibit resonance relationships with the Moon's orbital period. When the orbital period $T$ and the Moon's orbital period $T_M$ satisfy $T/T_M \approx n/m$, it is referred to as an $m:n$ synodic resonant NRHO.

Dynamic Symmetry

Unlike DROs which exhibit symmetry about the $x$-axis, NRHOs display mirror symmetry about the $xOz$ plane. When an NRHO crosses the $xOz$ plane, the velocity components satisfy the conditions: $\dot{x}$ and $\dot{z}$ change sign while $\dot{y}$ remains unchanged. This symmetry provides natural shooting conditions for differential correction: select an initial point on the $xOz$ plane, retaining only $z_0$ and $\dot{y}_0$ as free variables, integrate for half a period, and verify the $xOz$ plane crossing conditions -- enabling iterative convergence to a periodic orbit.

Stability Characteristics

NRHO stability analysis requires attention to:

• Floquet multipliers: eigenvalues of the monodromy matrix characterizing the amplification/attenuation characteristics of orbital perturbations in each direction
• Perilune distance $r_p$: excessively small $r_p$ may risk lunar surface impact, while excessively large $r_p$ weakens communications and exploration advantages
• Coupling with invariant manifolds in the libration point region: a unique dynamic characteristic distinguishing NRHOs from DROs

Engineering Applications

NRHOs have become a popular candidate orbit for current cislunar space missions:

• China's Chang'e-4 relay satellite "Queqiao": successfully operating in an Earth-Moon $L_2$ point Halo orbit, providing communications relay services for lunar far-side exploration
• NASA "Gateway" space station: planned deployment in the $L_2$ southern family 9:2 resonant NRHO
• Cislunar space situation awareness: NRHOs, with their unique orbital position, are well-suited for deploying relay communications and observation platforms

Related Concepts

• Distant Retrograde Orbit (DRO)
• Earth-Moon L1/L2 Halo Orbits (EML1/EML2 Halo)
• Starshade
• Birkhoff-Gustavson Normal Form
• Central Manifold
• Action-Angle Variables
• Orbit Identification
• Halo orbit
• Circular Restricted Three-Body Problem (CR3BP)
• Libration point (Lagrangian point)
• Ephemeris model
• Invariant manifold

References

• Zimovan E M. Rectilinear halo orbits and their applications in cislunar space[D]. Purdue University, 2017.
• Williams J, Whitley R. Targeting cislunar rectilinear halo orbits for spacecraft missions[C]. 2017.
• Wu Weiren. Chang'e-4 Lunar Far-Side Soft Landing Mission Design[J]. 2017.
• 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.