Lunar Coordinate Time (TCL/O1)

• Lunar Coordinate Time (O1) is the universal relativistic time coordinate for the Moon’s inertial system, enabling precision in navigation and scientific event tagging.
• TCL is derived via post-Newtonian transformations that incorporate lunar gravitational potential, kinematic effects, and periodic timing variations.
• Its realization uses ultra-stable clocks, two-way time transfers, and advanced ephemerides to achieve sub-nanosecond accuracy for lunar positioning and timing.

Lunar Coordinate Time (O1)

Lunar Coordinate Time (O1), also universally denoted as TCL per IAU 2024 Resolution II, is the canonical relativistic coordinate time associated with the Lunar Celestial Reference System (LCRS). It extends the methodology of terrestrial time metrology into cislunar and planetary contexts, providing the four-dimensional parametric time variable that defines simultaneity, ephemeris integration, and data timestamping throughout the lunar neighborhood. TCL is a fundamental backbone for high-fidelity position, navigation, and timing (PNT) systems on and around the Moon, analogous to TCG (Geocentric Coordinate Time) for Earth, and is intrinsic to the realization of practical lunar GNSS, scientific analyses, and PNT interoperability across the Earth–Moon system (Yang et al., 28 Dec 2025, &&&1&&&, Bourgoin et al., 29 Jul 2025, Seyffert, 10 Sep 2025, Turyshev et al., 2024, Lu et al., 23 Sep 2025, Lu et al., 24 Jun 2025, Kopeikin et al., 2024, Defraigne et al., 4 Nov 2025, Turyshev, 29 Jul 2025, Fienga et al., 2024, Liu et al., 21 Jul 2025, Turyshev et al., 2012, Zhang et al., 19 Jun 2025).

1. Theoretical Foundation and Definition

TCL (O1) is the time coordinate of the LCRS, a non-rotating, Moon-centered local inertial system prescribed by the IAU [LCRS; IAU 2024 Resolution II]. The metric in LCRS, to O(c⁻⁴), mirrors GCRS construction but replaces all terrestrial reference quantities with lunar analogs. The line element is: $$ds^2 = -c^2\left[1 - \frac{2W(T, \mathbf{X})}{c^2} + \frac{2W(T, \mathbf{X})^2}{c^4}\right]dT^2 - \frac{4}{c}W_i(T, \mathbf{X})\,dT\,dX^i + \left[\delta_{ij} + \frac{2W(T, \mathbf{X})}{c^2}\delta_{ij}\right]dX^i\,dX^j + O(c^{-5})$$ where $$W = U_L(\mathbf{X}) + U_\text{tidal}(T,\mathbf{X}) + W_\text{inertial}(T)$$, with $U_L$ as the lunar monopole, plus all higher multipoles and external tidal potentials (Defraigne et al., 4 Nov 2025, Bourgoin et al., 29 Jul 2025, Kopeikin et al., 2024, Fienga et al., 2024, Turyshev, 29 Jul 2025, Turyshev et al., 2012).

The coordinate time transformation, to first post-Newtonian order, from the Solar System Barycentric Coordinate Time (TCB) to TCL is: $$TCL - TCB = -\frac{1}{c^2}\int_{t_0}^{TCB} \left( \frac{1}{2} v_M^2 + w_{0M} + w_{lM} \right) \,dt + O(c^{-4})$$ where $v_M$ is the Moon’s barycentric velocity, $w_{0M}$ the external (Solar, planetary, asteroid) potential at the selenocenter, and $w_{lM}$ higher-order nonspherical terms (Lu et al., 23 Sep 2025, Lu et al., 24 Jun 2025, Turyshev et al., 2012, Fienga et al., 2024).

TCL is thus a universal, periodicity-free celestial time parameter: it does not coincide with any clock's proper time, nor is it subject to periodic gravitational or kinematic perturbations by construction (Yang et al., 28 Dec 2025, Turyshev, 29 Jul 2025, Turyshev et al., 2012).

2. Relation to Proper Time: Gravitational Redshift and Realization

No unsteered physical clock remains synchronous with TCL. Any real clock—whether surface or orbital—experiences both gravitational redshift and kinematic (special-relativistic) effects. The differential proper-to-coordinate time relation in LCRS, accurate to O(c⁻²), is: $$\frac{d\tau}{dTCL} = 1 - \frac{1}{c^2}[U_M(Y) + \tfrac{1}{2}\dot{Y}^2] + O(c^{-4})$$ For a stationary clock at lunar geoid potential $W_{M0}$, the offset is: $$TCL = (1 + L)\,(\tau_s - \tau_{s0}) + \text{constant} + O(c^{-4})$$ with

$$L = \frac{W_{M0}}{c^2} \simeq \frac{GM_M}{c^2R_M}\left[1+\frac{1}{2}J_2^M+\frac{1}{2}\eta_M\right]$$

where $G, M_M, R_M$ are the lunar gravitational constant, mass, and mean radius, $J_2^M$ the quadrupole, and $\eta_M$ a rotation-related correction (Yang et al., 28 Dec 2025, Bourgoin et al., 29 Jul 2025, Turyshev, 29 Jul 2025, Kopeikin et al., 2024).

On the mean selenoid, $L \simeq 3.14 \times 10^{-11}$, yielding a rate offset of ≈2.7 μs/day between proper time and TCL (Yang et al., 28 Dec 2025, Defraigne et al., 4 Nov 2025, Bourgoin et al., 29 Jul 2025).

3. Transformation to Other Time Scales and Scaling Implications

TCL is integrated into the hierarchy of solar-system time scales, with exact transformations to TCB, TDB, TCG, and ultimately TT, using the same post-Newtonian framework as for GCRS/TCG and BCRS/TCB (Kopeikin et al., 2024, Lu et al., 24 Jun 2025, Lu et al., 23 Sep 2025, Turyshev et al., 2012). The transformation with secular and periodic behavior is: $$TCL - TDB = \text{drift}_{\text{sec}}(t) + \sum_{i=1}^{13} A_i \sin\left(2\pi \frac{t-t_0}{T_i}+\phi_i\right)$$ with $$\langle dTCL/dTDB \rangle = 1 + 6.80 \times 10^{-10}$$ (i.e., TCL accumulates ≈+58.7 μs/day vs. TT) and dominant periodic terms at amplitudes of 1.65 ms (annual), 126 μs (monthly) (Lu et al., 24 Jun 2025, Lu et al., 23 Sep 2025).

The three main time scale options considered for practical GNSS and scientific use are:

• Option 1 (O1/TCL): adopt the coordinate time TCL unscaled; surface clocks will tick slower by $L\sim3 \times 10^{-11}$ (≈2.7 μs/day).
• Option 2: scale TCL so that surface clocks at the mean geoid tick at the rate of the timescale (removes the geoid redshift).
• Option 3: scale TCL so that it matches the terrestrial standard (TT) in rate (i.e., removes secular drift relative to TT, but introduces a large offset in lunar coordinate system scaling) (Bourgoin et al., 29 Jul 2025, Defraigne et al., 4 Nov 2025).

Operational simplicity and minimal proliferation of coordinate scalings are achieved by adopting Option 1: plain TCL (O1) (Defraigne et al., 4 Nov 2025, Bourgoin et al., 29 Jul 2025).

4. Realization Architecture and Metrological Performance

Physical realization of TCL utilizes a combination of ultra-stable clocks in lunar orbiters and at carefully selected surface sites, with time transfer performed by microwave and/or optical links. Key steps:

• High-precision clocks, steered via frequency offsets, are synchronized to traceably realize the common TCL ephemeris (Yang et al., 28 Dec 2025, Fienga et al., 2024, Turyshev, 29 Jul 2025).
• Two-way time transfer and broadcast of ephemerides and clock corrections enable dissemination at the ∼1 ns level of absolute accuracy (Fienga et al., 2024).
• The LTE440 numerical lunar time ephemeris provides TCL–TDB/TCB transformations to a picosecond threshold, incorporating all post-Newtonian corrections, barycentric potentials, and planetary effects (Lu et al., 24 Jun 2025, Lu et al., 23 Sep 2025).
• Systematic uncertainties are sub-ns for clock noise, link error, and ephemeris modeling; total realized error is constrained to ∼0.1 ns over hours and <1 ns absolutely (Fienga et al., 2024, Lu et al., 24 Jun 2025, Lu et al., 23 Sep 2025).

Key elements for implementation are summarized below:

Component	Achievable Accuracy (Typical)	Main Contributor
Clock stability	≤10⁻¹⁴ at 1 s, ≤10⁻¹⁵ at 10³ s	Ultra-stable optical/Rb clocks
Two-way time transfer	Optical ≤0.1 ns, MW ≈1 ns	Link noise, multipath
Barycentric ephemerides	<1 m (Moon center), ≈3 ns equiv	Ephemeris, GRAIL/LLR

These techniques support time-stamping, navigation, and precise event tagging at ∼1 ns/1 m lunar scales (Fienga et al., 2024, Kopeikin et al., 2024, Turyshev, 29 Jul 2025).

5. The Time Aligned Orbit: Simultaneous Realization of O1 and O2

Physical realization of both Lunar Coordinate Time (O1) and lunar geoid (selenoid) proper time (O2) with a single device leverages a "time aligned orbit": a specific class of lunar orbits whose secular redshift matches that of the mean selenoid (Yang et al., 28 Dec 2025). The defining criterion is

$L_P = L$

where $L_P$ is the orbital clock’s secular redshift, matched to the geoid's $L$. For the Moon, this is achieved by quasi-circular orbits with semi-major axis $\bar{a}_p \approx 1.5\,R_M$ and inclination ≈55°, which cancels quadrupole effects to first order. Clocks in such orbits realize, up to O(c⁻⁴), both the geoid proper time and, by a linear scaling, TCL. Simulations indicate desynchronization stays below 190 ns (raw) or 13 ns (corrected) per year, with frequency deviation ≤6 × 10⁻¹⁵, improving to 4 × 10⁻¹⁶ post correction (Yang et al., 28 Dec 2025). This approach is extensible, mutatis mutandis, to Mercury, Venus, Earth, Mars, and other planetary bodies with low relative surface rotation (Yang et al., 28 Dec 2025).

6. Applications: GNSS, Scientific Data Tagging, Navigation and Scaling Trade-offs

TCL is the core timescale for lunar GNSS, scientific measurement campaigns, and the fundamental time parameter for lunar reference frames used in navigation and PNT:

• GNSS satellite clock correction, ephemeris broadcasting, and receiver processing are structured around TCL, replicating terrestrial protocols but with lunar potentials and kinematics (Defraigne et al., 4 Nov 2025, Seyffert, 10 Sep 2025).
• Sagnac and relativistic signal propagation corrections (tidal, rotation, and third-body harmonics) are included at levels <1 ns (Defraigne et al., 4 Nov 2025, Kopeikin et al., 2024, Turyshev, 29 Jul 2025).
• Synchronization to terrestrial TT and UTC is via TCL–TDB–TT chains, with IAU-prescribed offsets and leap second adjustments (Defraigne et al., 4 Nov 2025, Bourgoin et al., 29 Jul 2025, Lu et al., 23 Sep 2025).
• The adoption of unscaled TCL (O1) ensures future extensibility to Mars and other planets, avoiding the proliferation of coordinate scalings and non-SI adjustments (Defraigne et al., 4 Nov 2025, Bourgoin et al., 29 Jul 2025).

7. Numerical Behavior: Secular Drift, Periodic Terms, and Precision

Empirical and analytical work (ephemerides LTE440, JPL DE440/441, INPOP21a) establish the detailed behavior:

• Secular drift: TCL accumulates ≈+58.7 μs/day relative to TT, due to lunar gravitational potential and kinematic effects (Lu et al., 24 Jun 2025, Lu et al., 23 Sep 2025, Turyshev et al., 2024).
• Periodic harmonics: Annual (1.65 ms amplitude), monthly (126 μs), and a spectrum of smaller terms, encoded in the LTE440 and related ephemeris products (Lu et al., 24 Jun 2025, Lu et al., 23 Sep 2025).
• Site/topography dependence: Surface clock redshift varies ±14–15 ns/day peak-to-peak (topography, gravity field modeling), total range ≈28.7 ns/day (Fienga et al., 2024, Seyffert, 10 Sep 2025, Turyshev, 29 Jul 2025).
• Transformation precision: LTE440 and related software provide sub-picosecond-level transformation accuracy over century timescales (Lu et al., 24 Jun 2025, Lu et al., 23 Sep 2025).

• (Yang et al., 28 Dec 2025) Two birds with one stone: simultaneous realization of both Lunar Coordinate Time and lunar geoid time by a single orbital clock (2025)
• (Ashby et al., 2024) A Relativistic Framework to Establish Coordinate Time on the Moon and Beyond (2024)
• (Bourgoin et al., 29 Jul 2025) Lunar Reference Timescale (2025)
• (Seyffert, 10 Sep 2025) Relativistic Time Modeling for Lunar Positioning Navigation and Timing (2025)
• (Turyshev et al., 2024) Relativistic Time Transformations Between the Solar System Barycenter, Earth, and Moon (2024)
• (Lu et al., 23 Sep 2025) Lunar Time Ephemeris LTE440: definitions, algorithm and performance (2025)
• (Lu et al., 24 Jun 2025) Lunar Time Ephemeris LTE440: User Manual (2025)
• (Kopeikin et al., 2024) Lunar Time in General Relativity (2024)
• (Defraigne et al., 4 Nov 2025) Lunar Time (2025)
• (Turyshev, 29 Jul 2025) High-Precision Relativistic Time Scales for Cislunar Navigation (2025)
• (Fienga et al., 2024) Lunar References Systems, Frames and Time-scales in the context of the ESA Programme Moonlight (2024)
• (Liu et al., 21 Jul 2025) Lunar and Terrestrial Time Transformation Based on the Principle of General Relativity (2025)
• (Turyshev et al., 2012) General relativistic observables of the GRAIL mission (2012)
• (Zhang et al., 19 Jun 2025) Frequency Differences between Clocks on the Earth and the Moon (2025)