Lunar Reference Timescale

• Lunar Reference Timescale is the temporal component of a Moon-centered relativistic reference system that enables navigation, geodesy, and high-precision timing in cislunar space.
• It is formally defined through Lunar Coordinate Time (TCL) within the Lunar Celestial Reference System, utilizing post-Newtonian transformations to relate coordinate and proper time.
• Numerical implementations like LTE440 employ high-order integration and polynomial approximations to achieve sub-nanosecond precision over extended time spans.

Searching arXiv for papers on lunar reference timescales, TCL, and lunar time ephemerides. arxiv_search(query="lunar reference timescale TCL lunar coordinate time LTE440", max_results=10) A lunar reference timescale is the temporal component of a Moon-centered relativistic reference-system architecture for navigation, geodesy, timing, and interoperability in cislunar space. In the current literature, its formal basis is Lunar Coordinate Time, $TCL$, the coordinate time associated with the Lunar Celestial Reference System (LCRS) recommended by IAU 2024 Resolution II, while the central practical question is whether lunar operations should use unscaled $TCL$ directly or a scaled lunar time tied either to a lunar equipotential or to terrestrial practice (Kopeikin et al., 2024, Defraigne et al., 4 Nov 2025).

1. Coordinate-time framework and conceptual scope

The lunar reference timescale is embedded in the same relativistic hierarchy that already structures solar-system and terrestrial timing. In that hierarchy, TCB is the coordinate time of the Barycentric Celestial Reference System (BCRS), TCG is the coordinate time of the Geocentric Celestial Reference System (GCRS), and TCL is the coordinate time of the Lunar Celestial Reference System (LCRS). The Moon-centered system is treated as the luni-centric counterpart of the geocentric system, with origin at the Moon’s center of mass and a local coordinate time defined by the same post-Newtonian logic used for Earth (Fienga et al., 2024, Lu et al., 23 Sep 2025).

Within this framework, the literature draws a strict distinction between coordinate time and proper time. $TCL$ is a coordinate time; it is not the proper time of a clock on the lunar surface, in lunar orbit, or elsewhere in cislunar space. A real clock measures proper time $\tau$ along its world line, and any lunar timing architecture must therefore model the transformation between $\tau$ and the relevant coordinate times rather than treating a physical clock as if it directly realized $TCL$ (Lu et al., 23 Sep 2025, Jin et al., 1 Jul 2026).

This distinction is operationally consequential. Lunar orbit determination, navigation message generation, inter-satellite synchronization, relay operations, surface dating and localization, and interoperability with Earth systems all require a common Moon-centered temporal reference. The literature therefore treats a lunar reference timescale not as a single “Moon clock,” but as a relativistic reference quantity attached to a Moon-centered frame and linked to physical clocks through modeled transformations (Fienga et al., 2024, Jin et al., 1 Jul 2026).

2. Relativistic definition of $TCL$

The defining relation used in current lunar-time work is the Moon-substituted analogue of the standard local-to-barycentric coordinate-time transformation. In one common formulation,

$$\begin{aligned} TCL-TCB \ & \xlongequal{TCB} - \frac{1}{c^2}\int_{t_0}^t\left(\frac{v_M^2}{2}+w_\mathrm{0M}+w_{lM}\right)\mathrm{d}t - \frac{1}{c^2}\boldsymbol{v}_M\cdot(\boldsymbol{x}-\boldsymbol{x}_{M}) \ & \quad +\frac{1}{c^4}\int_{t_0}^t\left(-\frac{v_M^4}8-\frac{3}{2}v_M^2w_\mathrm{0M} + 4\boldsymbol{v}_M\cdot\boldsymbol{w}_{M} +\frac12w_\mathrm{0M}^2 +\Delta_M\right)\mathrm{d}t \ & \quad - \frac{1}{c^4}\left(3w_\mathrm{0M}+\frac{v_M^2}2\right)\boldsymbol{v}_M\cdot(\boldsymbol{x}-\boldsymbol{x}_{M}), \end{aligned}$$

where $t$ is TCB, $\boldsymbol{x}_M,\boldsymbol{v}_M$ are the Moon’s barycentric position and velocity, $TCL$0 is the external Newtonian potential at the lunar center, $TCL$1 is the nonspherical external potential, $TCL$2 is the vector potential, and $TCL$3 collects higher-order nonlinear terms (Lu et al., 23 Sep 2025).

A closely related Earth–Moon formulation gives the difference between $TCL$4 and $TCL$5 in terms of relative Earth–Moon kinetic and potential terms plus the leading solar tidal contribution. In one explicit expression,

$TCL$6

with $TCL$7 and $TCL$8 (Kopeikin et al., 2024).

The physical content of these formulas is standard post-Newtonian clock theory. The leading terms are the Moon’s barycentric kinetic energy, the external gravitational potential at the Moon, and position-dependent simultaneity terms within the lunar frame; higher-order terms introduce vector potentials, nonlinear couplings, and nonspherical contributions. The literature repeatedly emphasizes that, by the equivalence principle, the common solar-system free fall of the Earth–Moon system cancels from the local Earth–Moon comparison, so the remaining external effects enter chiefly through tidal terms (Kopeikin et al., 2024, Seyffert, 10 Sep 2025).

A separate line of work argues for an Earth–Moon intermediate frame and treats direct coordinate-time-to-coordinate-time transformations as conceptually suspect, but that stance is presented as differing from standard IAU practice rather than replacing it (Liu et al., 21 Jul 2025).

3. Competing definitions of the operational lunar reference time

Although $TCL$9 is the formal coordinate time of the LCRS, the literature distinguishes between that formal definition and the practical lunar reference timescale that would be broadcast, realized, or steered operationally. Three options recur.

The first is to use $TCL$0 directly. In the common parameterization

$TCL$1

this corresponds to $TCL$2. Its principal attraction is conceptual and computational simplicity: the lunar reference timescale is simply the coordinate time of the lunar relativistic reference frame, with no new scaling of spatial coordinates or mass parameters (Defraigne et al., 4 Nov 2025, Bourgoin et al., 29 Jul 2025).

The second is to define a lunar-surface-scaled time. One version introduces a lunar analogue of TT tied to a reference lunar equipotential $TCL$3, with

$TCL$4

so that a clock on the adopted selenoid has, on average, no offset from the scaled time (Bourgoin et al., 29 Jul 2025). A related construction in the cislunar GNSS literature uses

$TCL$5

with

$TCL$6

and $TCL$7 (Iiyama et al., 23 Jan 2026). Earlier terrestrial-style derivations similarly define a scaled lunar time by

$TCL$8

with

$TCL$9

as the lunar analogue of $\tau$0 (Turyshev et al., 2024).

The third option is to define a scaled lunar time with no secular drift relative to TT/UTC. In the Moonlight formulation,

$\tau$1

with

$\tau$2

so that the differences between $\tau$3 and $\tau$4 are only periodic (Fienga et al., 2024).

Later trade-off analyses evaluate these options by two explicit criteria: scaling and need for clock steering. Those analyses conclude that the best practical lunar reference time is simply $\tau$5 itself, without defining a new scaled lunar timescale. The main argument is that any new scaling factor would require corresponding scaling of spatial coordinates and mass parameters, would introduce additional numerical conventions and software complexity, and would offer only narrow practical benefit except for very high-performance surface-clock applications (Defraigne et al., 4 Nov 2025, Bourgoin et al., 29 Jul 2025).

4. Quantitative behavior of lunar and Earth–Moon clock relations

Several numerical scales recur across the literature. For a clock at rest on the lunar surface relative to $\tau$6, the dominant proper-time offset is about

$\tau$7

which corresponds to about

$\tau$8

The variation over the surface due to topography is within

$\tau$9

and one analysis quotes a total low-to-high variation of about

$\tau$0

A specifically lunar feature is the permanent Earth-raised tidal component; inserted into the proper-time formula, this constant surface-location-dependent term contributes at the level of

$\tau$1

These values show that a free-running surface clock does not, in general, realize $\tau$2 directly, and that surface location enters at the $\tau$3–$\tau$4 level depending on the effect considered (Defraigne et al., 4 Nov 2025, Seyffert, 10 Sep 2025).

For Earth–Moon surface-clock comparison, the dominant secular effect is much larger. One first-principles estimate gives

$\tau$5

for a clock near the Moon’s equator relative to one near the Earth’s equator (Ashby et al., 2024). A terrestrial-style relativistic transformation derives a secular drift of

$\tau$6

and periodic terms with the largest amplitude of about

$\tau$7

at the mean anomalistic period (Turyshev et al., 2024). A separate Earth–Moon frequency-comparison model decomposes the comparison as

$\tau$8

and finds that gravity potential differences impact observations at the $\tau$9 level, while the coordinate time ratio enters at $TCL$0 (Zhang et al., 19 Jun 2025).

For the coordinate times themselves, several papers quote a leading Earth–Moon coordinate-time rate offset of about

$TCL$1

for $TCL$2 relative to $TCL$3, together with periodic terms of order $TCL$4 (Seyffert, 10 Sep 2025, Kopeikin et al., 2024). One review stresses that there is no universal scalar function “$TCL$5” valid independently of clock location; one can only determine the difference between one clock realizing $TCL$6 on the Moon and one clock realizing TT on Earth, and this also depends on the frame used for light-time computation (Defraigne et al., 4 Nov 2025).

5. Numerical realization: time ephemerides and LTE440

The decisive step from formal definition to usable product is the construction of a lunar time ephemeris. The best-developed example is LTE440, a numerical realization of the transformation between $TCL$7 and barycentric time scales (Lu et al., 24 Jun 2025, Lu et al., 23 Sep 2025).

LTE440 adopts the operational convention that the transformation between $TCL$8 and $TCL$9 is given by the TCG–TDB relation, with the Earth-related quantities replaced by the Moon-related quantities. It numerically evaluates the relativistic time-dilation integral using JPL DE440 state data, including gravitational contributions from the Sun, all planets, Pluto, 343 main-belt asteroids, and 30 Kuiper belt objects and a ring (Lu et al., 23 Sep 2025).

The numerical scheme is explicit. The time integral is computed with a 10th-order Romberg scheme using half-day integration intervals. The resulting periodic residual, after removal of the secular drift, is represented by 13th-degree Chebyshev polynomials on 4-day granules using the Newhall method. Validation is performed by reproducing TT–TDB at the geocenter from DE440, with agreement better than 1 ps over the whole DE440 span (Lu et al., 23 Sep 2025).

The package architecture separates the transformation into a secular drift and a periodic part. In the public release, lte440.bsp stores the periodic part of $TCL$0, and lte440.tpc stores the coefficient of the secular drift together with metadata. The BSP is a binary SPK kernel, its time argument is TDB, its unit is seconds, the $TCL$1 quantity is assigned NAIF ID 1000000005, and the secular drift coefficient is stored under BODY1000000005_RATE (Lu et al., 24 Jun 2025).

Quantitatively, LTE440 estimates

$TCL$2

and

$TCL$3

The dominant periodic terms are an annual term with amplitude

$TCL$4

and period

$TCL$5

and a monthly term with amplitude

$TCL$6

and period

$TCL$7

The paper gives accuracy better than 0.15 ns before 2050 and numerical precision at the level of 1 ps over its entire time span; because it is derived from DE440, the effective span is 1550–2650 (Lu et al., 23 Sep 2025).

6. Realization, traceability, and unresolved issues

The literature is explicit that $TCL$8 is not itself a practical clock standard. A realizable lunar timescale must be produced by clocks on or near the Moon and steered or reduced to the coordinate-time framework. One recent synthesis states that multi-CRS consistency relies on documented transformation chains rather than a single master clock and distinguishes formal $TCL$9 from an operational lunar scale LTC, “the lunar counterpart of UTC, realised by atomic clocks on or near the Moon, steered to TCL, and disseminated to users” (Jin et al., 1 Jul 2026).

In estimation architectures for lunar navigation constellations, this distinction already appears explicitly. One cislunar GNSS framework assumes an operational LT for broadcast use while estimating spacecraft clock states with respect to TCL, and writes the terrestrial-to-lunar transformation chain as

$$\begin{aligned} TCL-TCB \ & \xlongequal{TCB} - \frac{1}{c^2}\int_{t_0}^t\left(\frac{v_M^2}{2}+w_\mathrm{0M}+w_{lM}\right)\mathrm{d}t - \frac{1}{c^2}\boldsymbol{v}_M\cdot(\boldsymbol{x}-\boldsymbol{x}_{M}) \ & \quad +\frac{1}{c^4}\int_{t_0}^t\left(-\frac{v_M^4}8-\frac{3}{2}v_M^2w_\mathrm{0M} + 4\boldsymbol{v}_M\cdot\boldsymbol{w}_{M} +\frac12w_\mathrm{0M}^2 +\Delta_M\right)\mathrm{d}t \ & \quad - \frac{1}{c^4}\left(3w_\mathrm{0M}+\frac{v_M^2}2\right)\boldsymbol{v}_M\cdot(\boldsymbol{x}-\boldsymbol{x}_{M}), \end{aligned}$$0

with

$$\begin{aligned} TCL-TCB \ & \xlongequal{TCB} - \frac{1}{c^2}\int_{t_0}^t\left(\frac{v_M^2}{2}+w_\mathrm{0M}+w_{lM}\right)\mathrm{d}t - \frac{1}{c^2}\boldsymbol{v}_M\cdot(\boldsymbol{x}-\boldsymbol{x}_{M}) \ & \quad +\frac{1}{c^4}\int_{t_0}^t\left(-\frac{v_M^4}8-\frac{3}{2}v_M^2w_\mathrm{0M} + 4\boldsymbol{v}_M\cdot\boldsymbol{w}_{M} +\frac12w_\mathrm{0M}^2 +\Delta_M\right)\mathrm{d}t \ & \quad - \frac{1}{c^4}\left(3w_\mathrm{0M}+\frac{v_M^2}2\right)\boldsymbol{v}_M\cdot(\boldsymbol{x}-\boldsymbol{x}_{M}), \end{aligned}$$1

and

$$\begin{aligned} TCL-TCB \ & \xlongequal{TCB} - \frac{1}{c^2}\int_{t_0}^t\left(\frac{v_M^2}{2}+w_\mathrm{0M}+w_{lM}\right)\mathrm{d}t - \frac{1}{c^2}\boldsymbol{v}_M\cdot(\boldsymbol{x}-\boldsymbol{x}_{M}) \ & \quad +\frac{1}{c^4}\int_{t_0}^t\left(-\frac{v_M^4}8-\frac{3}{2}v_M^2w_\mathrm{0M} + 4\boldsymbol{v}_M\cdot\boldsymbol{w}_{M} +\frac12w_\mathrm{0M}^2 +\Delta_M\right)\mathrm{d}t \ & \quad - \frac{1}{c^4}\left(3w_\mathrm{0M}+\frac{v_M^2}2\right)\boldsymbol{v}_M\cdot(\boldsymbol{x}-\boldsymbol{x}_{M}), \end{aligned}$$2

This makes the intended operational logic explicit: terrestrial GNSS system times are mapped through terrestrial coordinate time into lunar coordinate time and then, if needed, into a broadcast lunar scale (Iiyama et al., 23 Jan 2026).

The same literature also stresses traceability to terrestrial time. Low-accuracy human operations will likely continue to use UTC for ordinary coordination, but high-accuracy PNT requires full relativistic time transfer rather than a fixed Earth–Moon offset (Defraigne et al., 4 Nov 2025). One dedicated study formulates traceability by correcting a measured lunar-clock-to-UTC offset with modeled proper-time, coordinate-time, and light-time terms, thereby relating the lunar clock error to whichever lunar reference option has been adopted (Bourgoin et al., 29 Jul 2025).

Several unresolved issues remain. There is still no internationally agreed lunar operational timescale analogous to UTC or TAI. A lunar $$\begin{aligned} TCL-TCB \ & \xlongequal{TCB} - \frac{1}{c^2}\int_{t_0}^t\left(\frac{v_M^2}{2}+w_\mathrm{0M}+w_{lM}\right)\mathrm{d}t - \frac{1}{c^2}\boldsymbol{v}_M\cdot(\boldsymbol{x}-\boldsymbol{x}_{M}) \ & \quad +\frac{1}{c^4}\int_{t_0}^t\left(-\frac{v_M^4}8-\frac{3}{2}v_M^2w_\mathrm{0M} + 4\boldsymbol{v}_M\cdot\boldsymbol{w}_{M} +\frac12w_\mathrm{0M}^2 +\Delta_M\right)\mathrm{d}t \ & \quad - \frac{1}{c^4}\left(3w_\mathrm{0M}+\frac{v_M^2}2\right)\boldsymbol{v}_M\cdot(\boldsymbol{x}-\boldsymbol{x}_{M}), \end{aligned}$$3 or selenoid convention has not been adopted internationally, which complicates any TT-like lunar surface scaling (Defraigne et al., 4 Nov 2025, Seyffert, 10 Sep 2025). There is also no mature analytical lunar time series comparable to the mature Earth time-ephemeris literature; numerical realization currently dominates (Lu et al., 23 Sep 2025).

An emerging realization concept seeks to bridge theory and practice without choosing irreversibly between unscaled $$\begin{aligned} TCL-TCB \ & \xlongequal{TCB} - \frac{1}{c^2}\int_{t_0}^t\left(\frac{v_M^2}{2}+w_\mathrm{0M}+w_{lM}\right)\mathrm{d}t - \frac{1}{c^2}\boldsymbol{v}_M\cdot(\boldsymbol{x}-\boldsymbol{x}_{M}) \ & \quad +\frac{1}{c^4}\int_{t_0}^t\left(-\frac{v_M^4}8-\frac{3}{2}v_M^2w_\mathrm{0M} + 4\boldsymbol{v}_M\cdot\boldsymbol{w}_{M} +\frac12w_\mathrm{0M}^2 +\Delta_M\right)\mathrm{d}t \ & \quad - \frac{1}{c^4}\left(3w_\mathrm{0M}+\frac{v_M^2}2\right)\boldsymbol{v}_M\cdot(\boldsymbol{x}-\boldsymbol{x}_{M}), \end{aligned}$$4 and a selenoid-based scale. The proposed “time aligned orbit” is a lunar orbit with mean semi-major axis about 1.5 lunar radii such that the proper time of an ideal orbital clock equals the proper time rate associated with the selenoid, while $$\begin{aligned} TCL-TCB \ & \xlongequal{TCB} - \frac{1}{c^2}\int_{t_0}^t\left(\frac{v_M^2}{2}+w_\mathrm{0M}+w_{lM}\right)\mathrm{d}t - \frac{1}{c^2}\boldsymbol{v}_M\cdot(\boldsymbol{x}-\boldsymbol{x}_{M}) \ & \quad +\frac{1}{c^4}\int_{t_0}^t\left(-\frac{v_M^4}8-\frac{3}{2}v_M^2w_\mathrm{0M} + 4\boldsymbol{v}_M\cdot\boldsymbol{w}_{M} +\frac12w_\mathrm{0M}^2 +\Delta_M\right)\mathrm{d}t \ & \quad - \frac{1}{c^4}\left(3w_\mathrm{0M}+\frac{v_M^2}2\right)\boldsymbol{v}_M\cdot(\boldsymbol{x}-\boldsymbol{x}_{M}), \end{aligned}$$5 is recovered by a known linear transformation. In realistic simulations including solar-system perturbations and lunar harmonics up to degree 100, the orbital proper time desynchronizes from the selenoid proper time by up to 190 ns after a year with a frequency offset of $$\begin{aligned} TCL-TCB \ & \xlongequal{TCB} - \frac{1}{c^2}\int_{t_0}^t\left(\frac{v_M^2}{2}+w_\mathrm{0M}+w_{lM}\right)\mathrm{d}t - \frac{1}{c^2}\boldsymbol{v}_M\cdot(\boldsymbol{x}-\boldsymbol{x}_{M}) \ & \quad +\frac{1}{c^4}\int_{t_0}^t\left(-\frac{v_M^4}8-\frac{3}{2}v_M^2w_\mathrm{0M} + 4\boldsymbol{v}_M\cdot\boldsymbol{w}_{M} +\frac12w_\mathrm{0M}^2 +\Delta_M\right)\mathrm{d}t \ & \quad - \frac{1}{c^4}\left(3w_\mathrm{0M}+\frac{v_M^2}2\right)\boldsymbol{v}_M\cdot(\boldsymbol{x}-\boldsymbol{x}_{M}), \end{aligned}$$6; after correcting for mean-orbit deviations, those numbers become 13 ns and $$\begin{aligned} TCL-TCB \ & \xlongequal{TCB} - \frac{1}{c^2}\int_{t_0}^t\left(\frac{v_M^2}{2}+w_\mathrm{0M}+w_{lM}\right)\mathrm{d}t - \frac{1}{c^2}\boldsymbol{v}_M\cdot(\boldsymbol{x}-\boldsymbol{x}_{M}) \ & \quad +\frac{1}{c^4}\int_{t_0}^t\left(-\frac{v_M^4}8-\frac{3}{2}v_M^2w_\mathrm{0M} + 4\boldsymbol{v}_M\cdot\boldsymbol{w}_{M} +\frac12w_\mathrm{0M}^2 +\Delta_M\right)\mathrm{d}t \ & \quad - \frac{1}{c^4}\left(3w_\mathrm{0M}+\frac{v_M^2}2\right)\boldsymbol{v}_M\cdot(\boldsymbol{x}-\boldsymbol{x}_{M}), \end{aligned}$$7 (Yang et al., 28 Dec 2025).

The overall direction of the field is therefore clear even though standardization is incomplete. The formal backbone is $$\begin{aligned} TCL-TCB \ & \xlongequal{TCB} - \frac{1}{c^2}\int_{t_0}^t\left(\frac{v_M^2}{2}+w_\mathrm{0M}+w_{lM}\right)\mathrm{d}t - \frac{1}{c^2}\boldsymbol{v}_M\cdot(\boldsymbol{x}-\boldsymbol{x}_{M}) \ & \quad +\frac{1}{c^4}\int_{t_0}^t\left(-\frac{v_M^4}8-\frac{3}{2}v_M^2w_\mathrm{0M} + 4\boldsymbol{v}_M\cdot\boldsymbol{w}_{M} +\frac12w_\mathrm{0M}^2 +\Delta_M\right)\mathrm{d}t \ & \quad - \frac{1}{c^4}\left(3w_\mathrm{0M}+\frac{v_M^2}2\right)\boldsymbol{v}_M\cdot(\boldsymbol{x}-\boldsymbol{x}_{M}), \end{aligned}$$8 in the LCRS; the operational problem is how to realize, steer, disseminate, and trace a practical lunar timescale without introducing unnecessary scaling complexity or losing interoperability with terrestrial and barycentric timing conventions (Defraigne et al., 4 Nov 2025, Jin et al., 1 Jul 2026).