HaL-13k: NRHO for Cis-Lunar Missions

• HaL-13k is a defined near-rectilinear halo orbit with an out-of-plane amplitude of about 13,000 km, derived from the CR3BP framework.
• The orbit is generated through differential correction and single-shooting methods, then propagated in a high-fidelity inertial J2000 model including detailed lunar, Earth, and solar effects.
• Stability analysis shows mixed oscillatory and unstable modes, requiring periodic station-keeping maneuvers (approximately 1 m/s every 12–18 days) to maintain accuracy.

Searching arXiv for the cited HaL-13k source and closely related NRHO/HALO material. arXiv.search {"query":"(Granier et al., 2024) HALO High-Precision Orbit Propagation Tool mission design cis-lunar domain NRHO", "max_results": 5} arXiv.search returned 5 results:

1. (Granier et al., 2024) — "HALO: A High-Precision Orbit Propagation Tool for Mission Design in the Cis-Lunar Domain"
2. (Lahabi et al., 2017) — "Distant retrograde orbits as viable states for future space infrastructure"
3. (Wang et al., 2024) — "The Gateway Program and Cislunar Space"
4. (Xu et al., 2021) — "Near Rectilinear Halo Orbit in the Circular Restricted Three-Body Problem"
5. (Cuspinera et al., 2019) — "Dynamical Analysis and Station-Keeping of Near Rectilinear Halo Orbits" Searching specifically for “HaL-13k”. arXiv.search {"query":"\"HaL-13k\" OR \"HAL-13k\" NRHO", "max_results": 10} arXiv.search returned 1 result:
6. (Granier et al., 2024) — "HALO: A High-Precision Orbit Propagation Tool for Mission Design in the Cis-Lunar Domain" HaL-13k denotes, within HALO, the Near-Rectilinear Halo Orbit (NRHO) with out-of-plane amplitude $A_z \simeq 13{,}000$ km. In the cited exposition, it is treated as a specific cis-lunar periodic orbit obtained from a Circular Restricted Three-Body Problem (CR3BP) construction and then propagated with a high-fidelity inertial model in HALO, an open-source mission design tool for the cis-lunar domain. The same exposition uses HaL-13k to summarize orbit generation, correction, stability characterization, station-keeping, and validation against NASA JPL Horizons ephemerides (Granier et al., 2024).

1. Definition and dynamical setting

By “HaL-13k” the source denotes the NRHO with out-of-plane amplitude $A_z \simeq 13{,}000$ km in HALO. In nondimensional units,

$$z_0 = A_z / L \;\approx\; 13{,}000 / 384{,}700 \;\approx\; 0.0338.$$

The underlying analytical model is the standard nondimensional CR3BP in a frame rotating with the mean motion of the Moon about the Earth. The Earth and Moon masses are $m_1$ and $m_2$, with total mass $m_1+m_2=1$, mass ratio

$\mu=\frac{m_2}{m_1+m_2}\approx0.0121505856,$

distance unit $L=R_{\rm EM}\approx3.846\times10^5\,\rm km$, and time unit chosen such that the mean motion $n=1$. Coordinates $(x,y,z)$ are expressed in the rotating frame with origin at the Earth–Moon barycenter, $A_z \simeq 13{,}000$0-axis toward the Moon, $A_z \simeq 13{,}000$1-axis in direction of motion, and $A_z \simeq 13{,}000$2-axis out-of-plane (Granier et al., 2024).

This definition is technically important because HaL-13k is not introduced as a generic halo orbit, nor as a separate propagator, but as a particular NRHO variant embedded in HALO’s modeling stack. A common confusion is to conflate the HALO software name with the orbital family name “halo orbit”; the source distinguishes these by using HALO for the tool and HaL-13k for the orbit instance.

2. CR3BP formulation and periodic-orbit construction

The CR3BP formulation uses the distances to the two primaries

$A_z \simeq 13{,}000$3

and the pseudo-potential

$A_z \simeq 13{,}000$4

The equations of motion in the rotating frame are \begin{align} \ddot x - 2\,\dot y &= x -(1-\mu)\,\frac{x+\mu}{r_1^3} -\mu\,\frac{x-(1-\mu)}{r_2^3}, \ \ddot y + 2\,\dot x &= y -(1-\mu)\,\frac{y}{r_1^3} -\mu\,\frac{y}{r_2^3}, \ \ddot z &= -(1-\mu)\,\frac{z}{r_1^3} -\mu\,\frac{z}{r_2^3}. \end{align} Motion is constrained by the Jacobi integral

$A_z \simeq 13{,}000$5

which remains constant along a solution (Granier et al., 2024).

HALO computes a halo-orbit initial guess through single-shooting differential correction. One seeks a periodic solution satisfying $A_z \simeq 13{,}000$6 together with a symmetry condition such as $A_z \simeq 13{,}000$7. The procedure defines an initial state $A_z \simeq 13{,}000$8, integrates the CR3BP dynamics over one period, forms a residual

$A_z \simeq 13{,}000$9

linearizes the Poincaré map with monodromy sensitivity matrix

$$z_0 = A_z / L \;\approx\; 13{,}000 / 384{,}700 \;\approx\; 0.0338.$$0

and solves

$$z_0 = A_z / L \;\approx\; 13{,}000 / 384{,}700 \;\approx\; 0.0338.$$1

until $$z_0 = A_z / L \;\approx\; 13{,}000 / 384{,}700 \;\approx\; 0.0338.$$2 is below the desired tolerance. For a symmetric halo, the residual is written as

$$z_0 = A_z / L \;\approx\; 13{,}000 / 384{,}700 \;\approx\; 0.0338.$$3

This places HaL-13k within a standard periodic-orbit workflow: a CR3BP solution is first corrected in the rotating frame and then transferred to the higher-fidelity propagation environment. This suggests that the orbit is intended to serve not merely as a numerical trajectory sample, but as a reproducible design point within a broader mission-analysis pipeline.

3. Nominal HaL-13k parameters

HALO’s final corrected CR3BP solution for HaL-13k, at the crossing $$z_0 = A_z / L \;\approx\; 13{,}000 / 384{,}700 \;\approx\; 0.0338.$$4 heading south, is

$$z_0 = A_z / L \;\approx\; 13{,}000 / 384{,}700 \;\approx\; 0.0338.$$5

typically with

$$z_0 = A_z / L \;\approx\; 13{,}000 / 384{,}700 \;\approx\; 0.0338.$$6

The physical periapse relative to Moon center is $$z_0 = A_z / L \;\approx\; 13{,}000 / 384{,}700 \;\approx\; 0.0338.$$7 km, and the apoapse is $$z_0 = A_z / L \;\approx\; 13{,}000 / 384{,}700 \;\approx\; 0.0338.$$8 km. The Jacobi constant is

$$z_0 = A_z / L \;\approx\; 13{,}000 / 384{,}700 \;\approx\; 0.0338.$$9

The synodic period in dimensional form is reported as

$m_1$0

via

$m_1$1

Quantity	Value
Out-of-plane amplitude $m_1$2	$m_1$3 km
$m_1$4	$m_1$5
$m_1$6	$m_1$7
$m_1$8	$m_1$9
Periapse $m_2$0	$m_2$1 km
Apoapse $m_2$2	$m_2$3 km
Jacobi constant $m_2$4	$m_2$5
Synodic period $m_2$6	$m_2$7 days

These values identify HaL-13k as a concrete NRHO representative rather than an abstract family description. A plausible implication is that the designation “13k” is operationally tied to the out-of-plane amplitude, while the remaining orbital invariants and state components specify a particular corrected member of that family.

4. High-fidelity propagation in HALO

HALO propagates the full equations of motion in an inertial J2000 frame. The acceleration model sums lunar gravity via a spherical-harmonic expansion up to degree/order 350 using the GRAIL field, Earth gravity up to degree/order 100 using EGM2008, third-body point masses including Sun, Jupiter, and Venus with perturbation form

$m_2$8

solar radiation pressure with eclipses, Earth albedo, and a general-relativistic correction (Granier et al., 2024).

After testing ODE45, ODE78, ODE89, and the variable-order Adams–Bashforth–Moulton method ODE113, HALO adopts MATLAB’s ODE113. The relative tolerance is $m_2$9 for LLO/ELFO and $m_1+m_2=1$0 for NRHO/DRO, while the absolute tolerance is fixed at $m_1+m_2=1$1. The propagation workflow loads SPICE kernels, sets the integrator, converts the spacecraft state to Earth–Moon–Sun geometry via SPICE, evaluates the component accelerations, and integrates

$m_1+m_2=1$2

Component	Specification
Frame	Inertial J2000
Lunar gravity	Degree/order 350 spherical harmonics
Earth gravity	Degree/order 100 spherical harmonics
Third bodies	Sun, Jupiter, Venus…
Radiation/environment	SRP with eclipses, Earth albedo
Relativity	General-relativistic correction
Integrator	MATLAB ODE113
NRHO relative tolerance	$m_1+m_2=1$3
Absolute tolerance	$m_1+m_2=1$4

The role of this model is methodological as much as numerical. HaL-13k is first generated from CR3BP structure, but its operational assessment is then carried out in a force model that is explicitly higher fidelity than the rotating-frame construction. This distinguishes orbit design from orbit validation.

5. Stability structure and station-keeping

Once the periodic solution $m_1+m_2=1$5 is obtained, HALO integrates the variational equations over one period to form the monodromy matrix $m_1+m_2=1$6. Its eigenvalues are reported to come in four unit-modulus pairs $m_1+m_2=1$7 and one real-unstable pair $m_1+m_2=1$8. For HaL-13k, HALO reports

$m_1+m_2=1$9

The real pair implies an instability exponent

$\mu=\frac{m_2}{m_1+m_2}\approx0.0121505856,$0

so an initial perturbation grows by a factor $\mu=\frac{m_2}{m_1+m_2}\approx0.0121505856,$1 per revolution (Granier et al., 2024).

The recommended station-keeping policy is a $\mu=\frac{m_2}{m_1+m_2}\approx0.0121505856,$2 m/s correction every $\mu=\frac{m_2}{m_1+m_2}\approx0.0121505856,$3–$\mu=\frac{m_2}{m_1+m_2}\approx0.0121505856,$4 revolutions, approximately $\mu=\frac{m_2}{m_1+m_2}\approx0.0121505856,$5–$\mu=\frac{m_2}{m_1+m_2}\approx0.0121505856,$6 days, with burns timed near periapse to minimize $\mu=\frac{m_2}{m_1+m_2}\approx0.0121505856,$7. In the source exposition, these values are presented as practical consequences of the monodromy analysis rather than as a separate optimization study.

A common misconception is that NRHOs are neutrally stable because much of the monodromy spectrum lies on the unit circle. The reported real-unstable pair contradicts that simplification: HaL-13k has oscillatory modes together with a genuine unstable direction, and the stated correction cadence follows from that mixed spectral structure.

6. Validation, perturbation hierarchy, and use in cis-lunar analysis

For validation, HALO propagates HaL-13k from its CR3BP-derived initial state through the full high-fidelity model using ODE113 with $\mu=\frac{m_2}{m_1+m_2}\approx0.0121505856,$8, and compares the resulting inertial-frame trajectory against NASA JPL Horizons ephemerides. Over one period of approximately $\mu=\frac{m_2}{m_1+m_2}\approx0.0121505856,$9 days, the reported errors are a position RMS error of approximately $L=R_{\rm EM}\approx3.846\times10^5\,\rm km$0 km and a velocity RMS error of approximately $L=R_{\rm EM}\approx3.846\times10^5\,\rm km$1 m/s, with CPU time of approximately $L=R_{\rm EM}\approx3.846\times10^5\,\rm km$2 s on a mid-range i5 laptop (Granier et al., 2024).

Sensitivity runs obtained by removing individual perturbations indicate that, for HaL-13k, the dominant perturbations are Earth gravity and third-body Sun terms. Solar radiation pressure, albedo, and high-order lunar harmonics each affect the $L=R_{\rm EM}\approx3.846\times10^5\,\rm km$3–$L=R_{\rm EM}\approx3.846\times10^5\,\rm km$4 m level over one period. These results delimit the perturbation hierarchy for this specific NRHO variant: some force-model components dominate short-horizon behavior, whereas others remain secondary at the one-period scale.

This suggests two complementary interpretations. First, HaL-13k is sufficiently structured in CR3BP to admit systematic differential correction and monodromy analysis. Second, its high-fidelity propagation remains sensitive enough to require nontrivial environmental modeling and regular station-keeping. In that sense, it occupies a characteristic middle ground of cis-lunar mission design: analytically tractable enough for dynamical systems methods, yet operationally rich enough to require full-force-model validation. The detailed derivations and code references are stated to be available through the open-source HALO repository associated with the source paper.