Lunar Reference System (LRS)

Updated 21 October 2025

Lunar Reference System is defined by its origin (lunocenter), orientation (principal axes), and scale normalized within GR to ensure precise lunar geodesy.
It is realized using Lunar Laser Ranging, high-precision ephemerides, and rigorous error estimation methods, achieving centimeter-level accuracy.
The LRS supports mission planning and autonomous navigation while evolving with next-generation retroreflectors and advanced measurement technologies.

The Lunar Reference System (LRS) is the foundational spatial and temporal framework for the precise positioning, navigation, and science operations on and around the Moon. Its definition, realization, and continual refinement are driven by decades of lunar laser ranging (LLR), advanced planetary ephemerides, relativistic time scales, and the synthesis of multiple observation and measurement techniques. The LRS underpins all lunar geodesy, mission planning, and timing synchronization, and is realized through international reference frames such as the International Lunar Reference Frame (ILRF), constructed according to principles set forth in IAU and associated scientific resolutions.

1. Definition and Structural Components of the Lunar Reference System

The LRS is defined by three fundamental constituents: origin, orientation, and scale (Sośnica et al., 17 Oct 2025). Its origin is the lunar center of mass (“lunocenter”), established in the Geocentric Celestial Reference System (GCRS). The orientation is fixed by the principal axes (PA) of the lunar inertia tensor, ensuring a co-rotating frame attached to the Moon’s surface and naturally reflecting physical librations. The scale is normalized within the framework of General Relativity (GR) in line with IAU 2024 resolutions, guaranteeing internal consistency with planetary and lunar ephemerides.

The LRS may be formulated in different frame conventions. The Principal Axis system (PA) is the standard for realization, but the Mean Earth/Rotation Axis (ME) system is also employed, defined by three static Euler rotations relating the PA to ME via gravity field coefficients and dissipative models (Fienga et al., 16 Sep 2024).

Reference Frame Transformation Table

Frame	Reference Axes	Transformation Formula
PA	Inertia tensor diagonal	PA = R₍z₎(φ) * R₍x₎(θ) * R₍z₎(ψ) * LCRF
ME	Mean Earth and pole	ME = R₍z₎(φ_ME) * R₍x₎(θ_ME) * R₍z₎(ψ_ME) * LCRF

2. Methodologies for Realization: Lunar Laser Ranging and Ephemeris Solutions

The LRS is realized through an internationally coordinated reference frame, the ILRF, using Lunar Laser Ranging data, high-precision numerical ephemerides (INPOP21a, DE430, EPM2021), and variance component estimation (VCE) (Sośnica et al., 17 Oct 2025). Key to this realization are:

Retroreflector Networks: Fiducial markers (Apollo, Lunokhod arrays) define physical tie points on the lunar surface. Next-generation arrays (LLRRA-21) provide ranging accuracy improvements by factors of 10–100, narrowing geodetic grid errors to meter or sub-meter levels (Burns et al., 2012).
Ephemeris Combination: Position (origin) and rotation (precession, nutation, proper rotation) state vectors from multiple ephemerides are combined across epochs (every 0.75 days) with normalized VCE weights (e.g., 0.451 for EPM2021, 0.381 for INPOP21a, 0.168 for DE430).
Transformation and Error Estimation: Helmert transformations (7-parameter, with translation, scale, and rotation) translate retroreflector coordinates between ILRF, PA frames, and ME frames, yielding mean transformation errors as low as 3–5 cm between ILRF and other reference frames (Sośnica et al., 17 Oct 2025).

3. Advanced Measurement Techniques and Technologies

LLR is the core measurement method, utilizing the round‐trip light travel time from Earth-based lasers to lunar retroreflectors. The primary ranging equation is:

$R = \frac{c\,\Delta t}{2}$

where $R$ is the Earth–Moon distance, $c$ is the speed of light, and $\Delta t$ is the measured round‐trip time delay (Burns et al., 2012). Advanced methodologies include:

Differential LLR: Rapid switching between widely separated reflectors enables cancellation of common-mode errors and tens-of-micrometer precision (Turyshev, 5 Feb 2025).
Infrared Detection: IR laser links densify observations across lunar phases, improving statistical uncertainty per normal point and ensuring spatially homogeneous LRS realization (Courde et al., 2017).
Next-generation Retroreflectors: Single, large hollow CCRs with tailored optical coatings and dihedral geometry enable instantaneous range precisions of ~1 mm and strong signal return, minimizing pulse spreading caused by lunar librations (Turyshev et al., 2012).
VLBI Beacons: Techniques such as COMPASS use ultra-wideband beacons and differential VLBI for sub-meter orbit determination, enhancing the tie between lunar geodetic frames and the ICRF (Eubanks, 2020).

4. Relativistic Time Scales and Temporal Reference

Consistent with IAU 2000 and 2024 resolutions, the LRS incorporates fully relativistic coordinate time scales (Kopeikin et al., 5 Jul 2024, Bourgoin et al., 29 Jul 2025, Liu et al., 21 Jul 2025, Turyshev, 29 Jul 2025). Lunar Coordinate Time (TCL) is defined at the Moon’s center of mass (Lunicentric Celestial Reference System, LCRS) and is related to Barycentric Coordinate Time (TCB) by:

$\frac{dTCL}{dTCB} = 1 + \frac{1}{c^2}\alpha_L + \frac{1}{c^4}\beta_L + O(c^{-5})$

where $\alpha_L = -\frac{1}{2}v_L^2 - \sum_{A\neq L}\frac{GM_A}{r_{LA}}$ and $\beta_L$ includes smaller, higher-order effects (Fienga et al., 16 Sep 2024). Multiple definitions for lunar time are reviewed (Bourgoin et al., 29 Jul 2025):

TL = TCL: No additional scaling, direct adoption of TCL as the lunar reference time.
Selenoid-aligned: TL is scaled to match clock rate on a lunar equipotential.
TT-aligned: TL forced to agree with Terrestrial Time, aiding interplanetary synchronization.

The recommended approach is TL ≡ TCL, avoiding scaling ambiguities. High-fidelity frameworks retain all contributions above 5 × 10⁻¹⁸ fractional threshold, achieving sub-picosecond clock synchronization and centimeter-level navigation (Turyshev, 29 Jul 2025).

5. Error Analysis, Uncertainties, and Kinematic Corrections

The composite accuracy of the ILRF is determined through rigorous error budget analysis:

Mean Error (2010–2030): 17.6 cm overall; 15.3 cm from origin; 8.6 cm from orientation (Sośnica et al., 17 Oct 2025).
Sources of Error: Along-track origin error is driven by retroreflector network geometry and its correlation with scale. LLR post-fit residuals are at the level of 2–3 cm for best-performing stations.
Transformation Errors: Errors in transforming between ILRF and other frames (e.g., ME) are at 3–5 cm (7-parameter Helmert) and up to 14 cm for simplified approaches.
Kinematic Corrections: Empirical terms (e.g., $A_v = A_1\cos(l') + A_2\cos(2l-2D) + A_3\cos(2F-2l)$ ) optimized to fit periodicities (annual, semilunar, 3-year Delaunay arguments) further minimize residuals to a few cm.

The LRS directly enables high-precision lunar positioning, navigation, and timing (PNT) operations (Pöhlmann et al., 14 Aug 2025, Gong et al., 4 Apr 2025). Hybrid lunar PNT systems combine satellite signals with cooperative local ranging and reference station anchoring, attaining sub-meter precision even with only two visible satellites. Realistic error models, including temporal correlations (Gauss–Markov processes), are integrated into navigation filters, allowing robust uncertainty quantification and reliable autonomous navigation.

For surface operations, low lunar orbits targeted for coverage (e.g., 120 km altitude, 90° inclination) enable Doppler-based 3D geolocation with single-satellite algorithms, validated by simulation for sub–10 m errors (Gong et al., 4 Apr 2025). Multiple techniques, including differential corrections, VLBI beacons, and lunar altimeter data, reinforce both the spatial and temporal realization of the LRS (Fienga et al., 16 Sep 2024, Eubanks, 2020).

7. Prospects for Improvement and Future Developments

Current realization errors of the ILRF are dominated by retroreflector geometry and along-track uncertainties. Simulations and prospective deployments (e.g., Artemis III at the lunar south pole, new retroreflectors, altimeter data from MoonLight) indicate that adding pole stations and new precision ranging devices could reduce uncertainties in position and orientation by tens of percent and, in some parameters (e.g., Love numbers, low-degree gravity coefficients), by up to ~80% (Viswanathan et al., 2020, Fienga et al., 16 Sep 2024). This suggests a trajectory toward robust, high-resolution lunar geodesy, better constraints on lunar interior properties, and more reliable navigation infrastructure for a wide range of exploration scenarios.

Summary Table: ILRF Mean Errors (2010–2030)

Component	Mean Error (cm)	Principal Source
Origin	15.3	Along-track, retroreflector net
Orientation	8.6	Euler angle combination
Transformation (ILRF↔PA)	3.0	Ephemeris tie
Transformation (ILRF↔ME)	5.0	Euler libration/inclination

References to Key Papers

LLRRA-21 and improved ranging accuracy: (Burns et al., 2012)
Advanced CCR design, thermal, optical, mechanical optimization: (Turyshev et al., 2012)
ILRF definition, VCE methodology, error budgets, transformations: (Sośnica et al., 17 Oct 2025)
Relativistic temporal frameworks and lunar time scales: (Kopeikin et al., 5 Jul 2024, Bourgoin et al., 29 Jul 2025, Liu et al., 21 Jul 2025, Turyshev, 29 Jul 2025)
Hybrid lunar PNT systems and error modeling: (Pöhlmann et al., 14 Aug 2025)
Single-satellite geolocation and Doppler algorithms: (Gong et al., 4 Apr 2025)
VLBI beacon techniques, lunar navigation: (Eubanks, 2020, Fienga et al., 16 Sep 2024, Viswanathan et al., 2020)
LLR and lunar orientation parameter refinement: (Pavlov, 2019, Viswanathan et al., 2020)

The LRS and its current realization in the ILRF provide the technical and conceptual infrastructure required for precise lunar geodesy, navigation, science operations, and future mission planning in cislunar space. Continuing improvements in measurement techniques, error modeling, network geometry, and time synchronization will further reduce uncertainties and enhance the robustness and interoperability of lunar geodetic products.