International Lunar Reference Frame (ILRF)

Updated 21 October 2025

International Lunar Reference Frame is a globally recognized selenocentric system constructed using variance component estimation of LLR data and multiple lunar ephemerides.
The framework achieves sub-decimeter accuracy by synthesizing data from INPOP21a, DE430, and EPM2021 and rigorously applying relativistic corrections.
Future improvements focus on expanding the retroreflector network and integrating VLBI and new laser technologies to enhance lunar navigation and inter-frame consistency.

The International Lunar Reference Frame (ILRF) is the globally agreed-upon, physically realized selenocentric, body-fixed reference frame for the Moon. The ILRF provides the essential foundation for precision positioning, navigation, cartography, and time dissemination on the lunar surface and in cislunar space. Its construction synthesizes advances in lunar laser ranging (LLR), radio-metric astrometry, high-precision dynamical and rotational ephemerides, and state-of-the-art relativistic time scales, establishing continuity with the conventions of terrestrial and celestial geodesy and consistent traceability to Earth-based and barycentric reference systems. This article details the definition, realization, methodologies, error analysis, and future directions for the ILRF, with explicit attention to its mathematical underpinnings and practical operational role.

1. Origins and Conceptual Foundation

The rationale for the ILRF draws direct analogy to the International Terrestrial Reference Frame (ITRF) and the International Celestial Reference Frame (ICRF) (Malkin, 2013). The ILRF defines a lunar body-fixed frame with its origin at the lunar center of mass (the "lunocenter"), its orientation tied to physically meaningful axes, and its scale rigorously constructed within the framework of general relativity (Sośnica et al., 17 Oct 2025). The adoption of a principal axis (PA) system—where the axes are defined by the diagonalization of the lunar inertia tensor and co-rotate with the Moon—is foundational. This choice ensures that the frame is dynamically meaningful, body-fixed, and optimal for selenodesy and onboard navigation.

In terms of mathematical realization, the ILRF uses a 6D state vector for the position and orientation of the Moon,

$\mathbf{X}(t) = \begin{bmatrix} x(t) \ y(t) \ z(t) \ \varphi(t) \ \theta(t) \ \psi(t) \end{bmatrix},$

where $(x, y, z)$ is the lunocenter in an inertial frame and $(\varphi, \theta, \psi)$ are the Euler angles (precession, nutation, proper rotation) relating the PA frame to the lunar inertial frame (Sośnica et al., 17 Oct 2025).

2. Realization of the ILRF: Data and Numerical Methods

The first realization of the ILRF is constructed as a weighted combination of three major lunar ephemerides—INPOP21a, DE430, and EPM2021—each of which integrates lunar orbital and rotational dynamics based on decades of LLR data (Sośnica et al., 17 Oct 2025, Pavlov et al., 2016, Viswanathan et al., 2017). The state vectors from these ephemerides are combined using variance component estimation (VCE), which iteratively calculates normalized weights based on the residuals between each ephemeris solution and the current combined solution: $\mathbf{X}^{k+1}(t) = \sum_{i=1}^{3} w_i^k \mathbf{X}_i(t)$ with weights updated in each iteration until convergence (<1 mm change on the lunar surface is achieved).

The VCE procedure evaluates both positional and rotational (Euler angle) parts of the residuals. Appropriate normalization constants convert angle differences (in radians) into equivalent surface displacements (via the lunar radius). This approach ensures statistically consistent combination, accounting for both internal variance of the ephemeris models and the geometry of the underlying laser retroreflector network.

LLR data are the core observational input for all contributing ephemerides, with corrections for relativistic propagation delays, Earth's precession-nutation, ocean and solid Earth tides, and station deformation (Pavlov et al., 2016, Pavlov, 2019). High-density, homogeneous LLR normal points from modern stations (with mm-cm range precision) are particularly critical to the temporal stability and accuracy of the ILRF realization (Courde et al., 2017, Viswanathan et al., 2020).

3. Error Budget and Limiting Factors

For the era 2010–2030, the mean error in the realized ILRF is 17.6 cm, split into 15.3 cm from the origin and 8.6 cm from orientation. The dominant contribution to origin error arises from the non-optimal spatial distribution of the retroreflector network—primarily near the near-side lunar equator, with only a single station in the southern hemisphere (Sośnica et al., 17 Oct 2025). This network geometry introduces high correlations (up to –0.97) between the X translation parameter and overall scale, a critical degeneracy in the inversion process.

Orientation (Euler angle) determination benefits from physical libration modeling constrained by LLR and improved thermal stability in new retroreflector hardware (Turyshev et al., 2012). Post-fit LLR residuals in the realized ILRF for best-performing stations are at the level of 2–3 cm in one-way range standard deviation; transformation errors from ILRF (PA) to other PA frames are about 3 cm, and to legacy Mean Earth (ME) frames about 5 cm. Errors increase beyond the epoch range covered by direct LLR data; extrapolation to 2052, while supported, is less reliable than in the observational period (Sośnica et al., 17 Oct 2025).

4. Relativistic Reference Frame and Time Scale

The ILRF is constructed in the context of the selenocentric Principal Axis (PA) system, with the time coordinate drawn from a relativistically defined lunar timescale. The consensus is to use Lunar Coordinate Time (TCL) as defined in the Lunar Celestial Reference System (LCRS), based on the IAU convention for relativistic reference frames (Kopeikin et al., 5 Jul 2024, Fienga et al., 16 Sep 2024, Bourgoin et al., 29 Jul 2025, Turyshev, 29 Jul 2025). The transformation between TCL and TCG (Geocentric Coordinate Time) incorporates the contributions from gravitational potential differences, kinematic dilation, and periodic tidal perturbations: $s = u - \frac{1}{c^2}\int_{t_0}^t \left\{ \frac{v^2}{2} + \frac{\mu_E}{r} - \frac{2\mu_L}{r} + (\text{tidal corrections}) \right\} dt - \frac{1}{c^2}(\mathbf{v} \cdot \Delta\mathbf{r}),$ where $s$ is TCL, $u$ is TCG, and the residual terms model the relevant relativistic corrections (Kopeikin et al., 5 Jul 2024).

Several candidate timescales (all affine functions of TCL) are under discussion for the operational lunar time coordinate: direct use of TCL; a frequency-scaled variant matching proper time on an equipotential ("seleniod"), or a version with secular drift with respect to TT removed so only periodic residuals remain (Bourgoin et al., 29 Jul 2025). Each has different implications for navigation and interoperability; direct TCL is simplest and avoids rescaling spatial coordinates, while scaled variants may facilitate synchrony with Earth-based systems.

High-precision lunar time-ephemeris products (e.g., LTE440) now provide transformations between TCL and barycentric timescales (TCB, TDB) with sub-nanosecond accuracy, resolving secular drift rates (e.g., $\langle d\mathrm{TCL}/d\mathrm{TCB}\rangle=1-1.4825362167 \times 10^{-8}$ ) and dominant periodic terms (annual ∼1.65 ms, monthly ∼126 μs), supporting clock synchronization and time transfer at the picosecond level (Lu et al., 23 Sep 2025, Turyshev, 29 Jul 2025).

5. Integration with Terrestrial and Celestial Reference Frames

The ILRF must be tied both to the terrestrial International Terrestrial Reference Frame (ITRF) and to the International Celestial Reference Frame (ICRF). Lessons from the realization of ITRF/ICRF at the mm/μas level highlight the importance of dense, globally-distributed observing stations, global solution strategies (no legacy "defining points"), and sophisticated modeling of non-linear motion (seasonal, tidal, post-seismic effects) in the terrestrial network (Malkin, 2013).

The transformation between ILRF and ITRF (or the lunar selenocentric and terrestrial frames) is conceptually similar to that in Earth-based geodesy: $\mathbf{r}_{\mathrm{ILRF}} = \mathbf{R}_m(t) \cdot \mathbf{r}_{\mathrm{ITRF}} + \mathbf{T}_m,$ where $\mathbf{R}_m(t)$ encapsulates rotation/libration and $\mathbf{T}_m$ translation (Malkin, 2013). For integration with the ICRF and planetary ephemeris frames, the tie is ultimately established through combined LLR, spacecraft VLBI, and time transfer measurements (Pavlov, 2019, Kurdubov et al., 2019), with transformation parameters fitted as part of the global parameter estimation.

Integration of Earth–Moon Very Long Baseline Interferometry (VLBI) is projected to decrease the uncertainties in lunar retroreflector positions and rotational parameters by factors of 4–10, with positional errors dropping from tens of centimeters to millimeter-level (Kurdubov et al., 2019). This approach also enhances the direct tie to the celestial frame and improves fundamental physics tests (e.g., measurement of the PPN parameter γ with increased sensitivity).

6. Technological Advances and Prospects for Improvement

The accuracy, reliability, and sustainability of the ILRF depend on enhancements in both observing hardware and data analysis:

Deployment of next-generation corner-cube retroreflectors (CCR) with large apertures and optimized optical design supports single-pulse 1-mm range precision, improving normal point accuracy and supporting mm-level frame realization [(Turyshev et al., 2012); (Viswanathan et al., 2020)].
Widespread adoption of infrared LLR, as demonstrated at Grasse, densifies observations across lunar phases and balances usage of all retroreflectors, enhancing network geometry and reducing measurement uncertainties (Courde et al., 2017).
Active laser transponders, when deployed alongside CCRs, potentially allow significant expansion of the ground station network and increase photon return via 1/R² scaling in signal strength (Viswanathan et al., 2020).
Differential LLR, rapid inter-target switching, and the application of "differential observables" to reduce systematics (analogous to same-beam interferometry) are expected to reveal previously masked effects (subtle geophysical or interior signatures) and further tighten constraints on lunar orientation, tidal dissipation, and Love numbers (Viswanathan et al., 2020).

Current limitations are primarily due to the geometry of the retroreflector network (concentration on the near-side equator), which induces strong parameter correlations and limits lateral component accuracy; strategic deployment of new CCRs (especially near the poles and far side) is anticipated to significantly improve the realized origin and orientation (Fienga et al., 16 Sep 2024, Sośnica et al., 17 Oct 2025). Orbiter altimetry, if incorporated at the 1-cm accuracy level, offers another independent constraint on lunar geometry and orientation.

Advances in timekeeping and time ephemerides (e.g., LTE440) ensure that the ILRF can operate with traceable time scales compatible with Earth-based and celestial systems, a necessity for lunar navigation, satnav, and metrology applications (Lu et al., 23 Sep 2025).

7. Future Outlook and Operational Synthesis

The ILRF is poised to serve as the global reference for all lunar navigation, science, and infrastructure efforts. For operational readiness, the current realization is validated for 1970–2052, with ongoing improvements tied to the expansion of the reflector network, improved station distribution (notably in the southern hemisphere), and integration of new VLBI facilities and high-precision clocks (Sośnica et al., 17 Oct 2025, Bourgoin et al., 29 Jul 2025).

Coordination with reference time standards (TCL, TL) and traceability to UTC guarantee that lunar and terrestrial operations can be seamlessly integrated, minimizing time-transfer discontinuities. The variance component estimation approach for frame realization ensures that future upgrades—using better ephemerides, more robust network geometry, and advanced ranging techniques—can be readily incorporated without disrupting continuity.

Potential upgrades include new geodetic missions (e.g., NovaMoon, Moonlight), additional distributed CCRs, and networked active laser transponders. Continued alignment with evolving IAU and IERS conventions is essential for interoperability and legal traceability.