Lunar Coordinate Time (TCL) Fundamentals

• Lunar Coordinate Time (TCL) is the relativistic coordinate time for the Lunar Celestial Reference System (LCRS), distinct from proper time and defined by IAU 2024.
• TCL uses methods analogous to Earth’s TCG, incorporating post-Newtonian corrections and transformations with TCB and TDB for high-precision temporal modeling.
• Realized through tools like the LTE440 ephemeris, TCL underpins high-accuracy lunar navigation, VLBI, and time transfer applications essential for lunar metrology.

Lunar Coordinate Time (TCL) is the coordinate time of the Lunar Celestial Reference System (LCRS), established in IAU 2024 Resolution II as the lunar analogue of Geocentric Coordinate Time (TCG) in the Geocentric Celestial Reference System. It is intended for the modeling, joint analysis, and sharing of time measurements made in the vicinity of the Moon, and it is distinct both from Barycentric Coordinate Time (TCB) and Barycentric Dynamical Time (TDB), and from the proper time measured by any physical clock. In the current literature, TCL is treated as the relativistic temporal backbone for lunar and cislunar navigation, tracking, communications, synchronization, Earth–Moon VLBI, and prospective lunar reference-time realizations (Lu et al., 23 Sep 2025, Defraigne et al., 4 Nov 2025).

1. Definition, standards context, and conceptual status

IAU 2024 Resolution II identifies TCL as the coordinate time associated with the LCRS, and the literature repeatedly states that the LCRS and TCL can be constructed with the same techniques used to construct the GCRS and TCG. In that sense, TCL is not an ad hoc operational clock reading; it belongs to the same relativistic reference-system architecture as the established BCRS/TCB and GCRS/TCG hierarchy. Several later syntheses argue that the IAU 2000 framework already implies a local GCRS-like reference system and a TCG-like coordinate time for each major Solar-System body, with the lunar case denoted LCRS/TCL (Lu et al., 23 Sep 2025, Klioner, 17 Apr 2026).

The conceptual distinction between coordinate time and proper time is fundamental. TCL is a coordinate assigned to events in the lunar local relativistic frame, whereas a real clock on the lunar surface, in lunar orbit, or at a libration region measures proper time along its own worldline. The literature is explicit that TCL is not given by any real clock, even at the origin of the lunar frame. It is therefore misleading to identify TCL with the direct reading of a lunar surface clock, or with a civil timescale analogous to UTC. Future operational scales such as a Lunar Reference Time, Lunar Time, LTC, or TL are treated as separate questions (Defraigne et al., 4 Nov 2025, Bourgoin et al., 29 Jul 2025).

The spatial scope of TCL is also framed as local rather than universal. One synthesis states that the LCRS is clearly useful within roughly $\sim 10$ lunar radii from the lunar center, and recommends the BCRS and TCB outside the immediate lunar vicinity. This suggests that TCL is primarily the natural coordinate time for lunar local dynamics, lunar geodesy, spacecraft motion near the Moon, and reference-system consistency, rather than a substitute for barycentric timing throughout all of cislunar space (Klioner, 17 Apr 2026).

2. Relativistic formulation and transformation structure

The defining transformation is constructed by direct analogy with the standard TCB–TCG relation, replacing Earth-centered quantities with Moon-centered quantities. In first post-Newtonian form, one published expression is

$$[TCL -TCB]_{(TCB,\mathbf{x})} = -\frac{1}{c^2} \left\{\int_{t_0}^{TCB}\left[\frac{v_L^2(t')}{2}+w_{L}(\mathbf{x}_L(t'))\right] d t' + \mathbf{v}_L (TCB) \cdot [\mathbf{x}-\mathbf{x}_L(TCB)] \right\}+\mathcal{O}(c^{-4}) ,$$

where $c$ is the speed of light, $TCB$ is barycentric coordinate time, $\mathbf{x}_L,\mathbf{v}_L$ are the Moon’s barycentric position and velocity, and $w_L$ is the Newtonian potential at the Moon center due to all solar-system bodies except the Moon (Defraigne et al., 4 Nov 2025).

The practical TDB-based implementation used in LTE440 starts from the IAU 2006 definition

$TDB=TCB-B(TCB-T_0)+TDB_0,$

and then evaluates the lunar analogue of the geocentric relativistic transformation with TDB as the independent variable. The explicit $TDB-TCL$ formula in the LTE440 documentation includes the Moon’s barycentric velocity $v_M$, the external scalar potential $w_{0M}$, the external oblateness term $$[TCL -TCB]_{(TCB,\mathbf{x})} = -\frac{1}{c^2} \left\{\int_{t_0}^{TCB}\left[\frac{v_L^2(t')}{2}+w_{L}(\mathbf{x}_L(t'))\right] d t' + \mathbf{v}_L (TCB) \cdot [\mathbf{x}-\mathbf{x}_L(TCB)] \right\}+\mathcal{O}(c^{-4}) ,$$0, the vector potential $$[TCL -TCB]_{(TCB,\mathbf{x})} = -\frac{1}{c^2} \left\{\int_{t_0}^{TCB}\left[\frac{v_L^2(t')}{2}+w_{L}(\mathbf{x}_L(t'))\right] d t' + \mathbf{v}_L (TCB) \cdot [\mathbf{x}-\mathbf{x}_L(TCB)] \right\}+\mathcal{O}(c^{-4}) ,$$1, and the post-Newtonian correction $$[TCL -TCB]_{(TCB,\mathbf{x})} = -\frac{1}{c^2} \left\{\int_{t_0}^{TCB}\left[\frac{v_L^2(t')}{2}+w_{L}(\mathbf{x}_L(t'))\right] d t' + \mathbf{v}_L (TCB) \cdot [\mathbf{x}-\mathbf{x}_L(TCB)] \right\}+\mathcal{O}(c^{-4}) ,$$2, with the time argument taken as TDB and all right-hand-side variables TDB-compatible (Lu et al., 24 Jun 2025).

In the LTE440 model, the external scalar potential includes the Sun, all planets, 343 main-belt asteroids, 30 Kuiper belt objects, and the Kuiper belt ring, matching the DE440 dynamical model. The external oblateness term $$[TCL -TCB]_{(TCB,\mathbf{x})} = -\frac{1}{c^2} \left\{\int_{t_0}^{TCB}\left[\frac{v_L^2(t')}{2}+w_{L}(\mathbf{x}_L(t'))\right] d t' + \mathbf{v}_L (TCB) \cdot [\mathbf{x}-\mathbf{x}_L(TCB)] \right\}+\mathcal{O}(c^{-4}) ,$$3 includes only the Sun and Earth. The model is explicitly described as closely following the TT−TDB model in DE440, but transferred to the Moon through Moon-centered replacement of Earth-centered terms (Lu et al., 23 Sep 2025, Lu et al., 24 Jun 2025).

The literature also fixes a standard epoch convention. One paper gives the reference epoch as 1977 January 1, $$[TCL -TCB]_{(TCB,\mathbf{x})} = -\frac{1}{c^2} \left\{\int_{t_0}^{TCB}\left[\frac{v_L^2(t')}{2}+w_{L}(\mathbf{x}_L(t'))\right] d t' + \mathbf{v}_L (TCB) \cdot [\mathbf{x}-\mathbf{x}_L(TCB)] \right\}+\mathcal{O}(c^{-4}) ,$$4 TCB, equivalent to 1977 January 1, $$[TCL -TCB]_{(TCB,\mathbf{x})} = -\frac{1}{c^2} \left\{\int_{t_0}^{TCB}\left[\frac{v_L^2(t')}{2}+w_{L}(\mathbf{x}_L(t'))\right] d t' + \mathbf{v}_L (TCB) \cdot [\mathbf{x}-\mathbf{x}_L(TCB)] \right\}+\mathcal{O}(c^{-4}) ,$$5 TAI. Another formulation uses the same IAU convention that $$[TCL -TCB]_{(TCB,\mathbf{x})} = -\frac{1}{c^2} \left\{\int_{t_0}^{TCB}\left[\frac{v_L^2(t')}{2}+w_{L}(\mathbf{x}_L(t'))\right] d t' + \mathbf{v}_L (TCB) \cdot [\mathbf{x}-\mathbf{x}_L(TCB)] \right\}+\mathcal{O}(c^{-4}) ,$$6 at 1977 January 1, 0h 0m 0s TAI at the geocenter, and extends it to local $$[TCL -TCB]_{(TCB,\mathbf{x})} = -\frac{1}{c^2} \left\{\int_{t_0}^{TCB}\left[\frac{v_L^2(t')}{2}+w_{L}(\mathbf{x}_L(t'))\right] d t' + \mathbf{v}_L (TCB) \cdot [\mathbf{x}-\mathbf{x}_L(TCB)] \right\}+\mathcal{O}(c^{-4}) ,$$7 times including TCL (Defraigne et al., 4 Nov 2025, Klioner, 17 Apr 2026).

3. Coordinate time versus proper time near the Moon

A central proper-time relation for a clock on or around the Moon is

$$[TCL -TCB]_{(TCB,\mathbf{x})} = -\frac{1}{c^2} \left\{\int_{t_0}^{TCB}\left[\frac{v_L^2(t')}{2}+w_{L}(\mathbf{x}_L(t'))\right] d t' + \mathbf{v}_L (TCB) \cdot [\mathbf{x}-\mathbf{x}_L(TCB)] \right\}+\mathcal{O}(c^{-4}) ,$$8

with

$$[TCL -TCB]_{(TCB,\mathbf{x})} = -\frac{1}{c^2} \left\{\int_{t_0}^{TCB}\left[\frac{v_L^2(t')}{2}+w_{L}(\mathbf{x}_L(t'))\right] d t' + \mathbf{v}_L (TCB) \cdot [\mathbf{x}-\mathbf{x}_L(TCB)] \right\}+\mathcal{O}(c^{-4}) ,$$9

where $c$0 is the lunar Newtonian potential, $c$1 is the tidal potential, and $c$2 is the inertial potential caused by the non-geodesic acceleration of the Moon’s center of mass. The same paper states that, as for Earth, the inertial potential is negligible (Defraigne et al., 4 Nov 2025).

The lunar case differs from the terrestrial one in an important respect: because of the Moon’s 1:1 spin-orbit resonance with Earth, one part of the Earth-raised tide is permanent. Inserted into the proper-time equation, this constant tide contributes at about

$c$3

in relative frequency, depending on location on the lunar surface. This is one of the reasons the literature insists that surface clocks, orbiters, and clocks at Earth–Moon libration regions do not realize a common proper time that can replace TCL (Defraigne et al., 4 Nov 2025).

Using lunar gravity from GRAIL and topography from LOLA, one trade-off study estimates that the dominant effect for a clock at rest on the lunar surface is the lunar monopole term

$c$4

which leads to a surface-clock relative frequency offset of about

$c$5

with respect to TCL, corresponding to about

$c$6

Across the lunar surface, spatial variations due mainly to topography are within about

$c$7

in relative frequency (Defraigne et al., 4 Nov 2025).

A separate Earth–Moon clock-comparison study uses the same hierarchy of local proper-time and coordinate-time transformations and concludes that gravity potential differences affect Earth–Moon comparisons at the $c$8 level, while the coordinate-time ratio term enters at the $c$9 level. On the lunar side, the static GRAIL potential contributes $TCB$0, while lunar rotational and tidal terms are at the $TCB$1 to $TCB$2 level depending on the component retained (Zhang et al., 19 Jun 2025).

4. Numerical characterization and the LTE440 ephemeris

The principal numerical implementation currently described in the literature is the Lunar Time Ephemeris $TCB$3, a ready-to-use software package that calculates TCL and its relations with TCB and TDB and exports data files in SPICE format. LTE440 consists of two files: lte440.bsp, which stores the periodic part of $TCB$4 as a binary SPK kernel with TDB as the time argument, and lte440.tpc, which stores the coefficient of secular drift of $TCB$5 as a text PCK kernel. The assigned NAIF ID for $TCB$6 is 1000000005, with dummy center body 1000000000 and secular rate coefficient BODY1000000005_RATE (Lu et al., 23 Sep 2025, Lu et al., 24 Jun 2025).

The numerical method is based on direct evaluation of the integral terms using JPL DE440/DE441 source ephemerides, 10th-order Romberg integration with 0.5-day step size, and 13th-degree Chebyshev polynomials on 4-day intervals. The secular trend is removed before fitting so that the BSP stores only the oscillatory part, and users must add the secular term from the TPC file. The implementation was validated by applying the same machinery to TT−TDB in DE440, obtaining agreement within 1 picosecond (Lu et al., 24 Jun 2025).

The published performance claims are stringent. One paper states that, at a conservative estimate, $TCB$7 has an accuracy better than 0.15 ns before 2050 and a numerical precision at the level of 1 ps over its entire time span. The user manual characterizes the numerical precision as at the level of several picoseconds. These statements refer to the numerical time ephemeris and its representation, not necessarily to all model-systematic uncertainties (Lu et al., 23 Sep 2025, Lu et al., 24 Jun 2025).

The dominant structure of $TCB$8 is not numerical noise but a secular term plus large periodic modulations. The most significant periodic terms are an annual term with amplitude of about $TCB$9 microseconds and a monthly term with amplitude of about $\mathbf{x}_L,\mathbf{v}_L$0 microseconds. One FFT analysis identifies 13 periodic terms with amplitudes greater than 1 microsecond; the leading two have amplitude $\mathbf{x}_L,\mathbf{v}_L$1 with period $\mathbf{x}_L,\mathbf{v}_L$2 day, and amplitude $\mathbf{x}_L,\mathbf{v}_L$3 with period $\mathbf{x}_L,\mathbf{v}_L$4 day (Lu et al., 23 Sep 2025, Lu et al., 24 Jun 2025).

The secular rates reported for LTE440 are not entirely uniform across the supplied descriptions. The abstract of “Lunar Time Ephemeris $\mathbf{x}_L,\mathbf{v}_L$5: definitions, algorithm and performance” gives

$\mathbf{x}_L,\mathbf{v}_L$6

and

$\mathbf{x}_L,\mathbf{v}_L$7

whereas the user manual states that the paper explicitly gives

$\mathbf{x}_L,\mathbf{v}_L$8

and remarks that the minus sign is “physically and textually important” (Lu et al., 23 Sep 2025, Lu et al., 24 Jun 2025).

5. Scaled lunar times, operational options, and the main debate

A major controversy in the 2024–2026 literature concerns whether TCL should be used directly as the practical lunar reference time, or whether it should be replaced by a scaled derivative analogous to TT or TDB. One paper parameterizes candidate lunar reference times as

$\mathbf{x}_L,\mathbf{v}_L$9

with three options: direct use of TCL with $w_L$0; a scaled lunar time matching, on average, proper time on a lunar surface equipotential with $w_L$1; and a scaled lunar time with no secular drift relative to TT with $w_L$2 (Defraigne et al., 4 Nov 2025).

The strongest argument against new scaled lunar times is scaling complexity. The literature emphasizes that scaling coordinate time forces corresponding scaling of spatial coordinates, mass parameters, and distances used in ephemerides and ranging. One trade-off analysis quantifies the effect on the Earth–Moon distance as about 1 cm for the surface-matched option and about 30 cm for the TT-drift-free option, and notes that Lunar Laser Ranging would then require an additional third scaling on top of the existing TT-compatible and TDB-compatible values (Defraigne et al., 4 Nov 2025).

The corresponding steering argument is more limited. For direct TCL, an ideal surface clock differs by $w_L$3, or about $w_L$4. For a surface-matched scaled lunar time, the residual difference can be as low as $w_L$5 and later summarized as not more than $w_L$6. For the TT-drift-free option, the surface clock differs by about $w_L$7, or about $w_L$8, which the same paper treats as operationally unattractive for onboard lunar clocks (Defraigne et al., 4 Nov 2025).

An earlier JPL-led formulation had explicitly introduced a scaled lunar surface time $w_L$9 from coordinate time $TDB=TCB-B(TCB-T_0)+TDB_0,$0 through

$TDB=TCB-B(TCB-T_0)+TDB_0,$1

with

$TDB=TCB-B(TCB-T_0)+TDB_0,$2

together with the metric-preserving scalings

$TDB=TCB-B(TCB-T_0)+TDB_0,$3

That work treats $TDB=TCB-B(TCB-T_0)+TDB_0,$4 as the lunar analogue of TT and explicitly distinguishes it from TCL (Turyshev et al., 2024).

Later papers reject that move. One concludes that “there is no need to define a new lunar timescale based on scaling TCL” and that the preferred practical solution is simply to use TCL itself as the lunar reference time. Another argues more broadly that scaling a local lunar coordinate time is “unnecessary, unreasonable and even risky.” A third operational study likewise recommends $TDB=TCB-B(TCB-T_0)+TDB_0,$5 as “the less constraining and natural solution” (Defraigne et al., 4 Nov 2025, Klioner, 17 Apr 2026, Bourgoin et al., 29 Jul 2025).

6. Realization, applications, and operational architectures

The operational motivations for TCL are explicit and immediate: modeling and analyzing measurements near the Moon, transforming lunar-region times to and from barycentric and Earth-based times, supporting navigation, tracking, communications, synchronization, tracing the proper time of a Moon-based clock back to UTC, and enabling Earth–Moon VLBI and future cislunar high-precision astrometry (Lu et al., 23 Sep 2025, Defraigne et al., 4 Nov 2025).

The proposed realization strategy is hierarchical. One paper recommends using TCL as the lunar coordinate reference time for high-accuracy PNT, disseminating it through lunar navigation systems or a ground station—initially on Earth and later potentially on the Moon. For lower-accuracy operational coordination with Earth, the same paper expects UTC to remain the common timescale and states that a simple secular correction relative to UTC may be adequate at the microsecond level. For higher-accuracy applications, the exact relation between TCL and UTC must be computed through relativistic time transfer, accounting for the actual clock positions on Earth and Moon (Defraigne et al., 4 Nov 2025).

A more specialized realization proposal is the “time aligned orbit.” In that construction, a clock in a suitable lunar orbit realizes the proper time of a chosen lunar geoid or selenoid while remaining linearly convertible to TCL through

$TDB=TCB-B(TCB-T_0)+TDB_0,$6

The paper states that such an orbit exists around the Moon with semi-major axis of about 1.5 lunar radius slightly depending on inclination. Numerical simulations report desynchronization from selenoid proper time up to 190 ns after a year with a frequency offset of $TDB=TCB-B(TCB-T_0)+TDB_0,$7, reducible to 13 ns and $TDB=TCB-B(TCB-T_0)+TDB_0,$8 if the deviation of the mean simulated orbit from the nominal one is accounted for (Yang et al., 28 Dec 2025).

The broader lunar PNT literature also emphasizes that timing architecture and frame realization cannot be separated. One observability study states that practical lunar timing services reduce to a joint determination of lunar infrastructure geometry and clock states, tied to an Earth or inertial frame, but also insists that such clock-offset estimation is not itself a definition or realization of TCL. This distinction mirrors the general literature: TCL is the relativistic coordinate-time definition of the LCRS, whereas operational services must still solve the engineering problems of dissemination, steering, time transfer, orbit determination, and traceability (Baweja, 29 Jun 2026).

In summary, TCL occupies a precise but nontrivial position in contemporary lunar metrology. It is the relativistically defined coordinate time of the lunar local reference system; it is not the proper time of any generic lunar clock; it is numerically realizable through products such as LTE440; and it remains the focal point of an ongoing standards debate over whether the Moon should use direct coordinate time or a new scaled derivative. The dominant trend in the current literature is toward retaining TCL itself as the primary lunar reference-time backbone, while treating physical clock ensembles, UTC traceability, and dissemination mechanisms as downstream realization problems (Lu et al., 23 Sep 2025, Defraigne et al., 4 Nov 2025, Klioner, 17 Apr 2026).