Lunar Coordinate Time (LTC)

• Lunar Coordinate Time (TCL/LTC) is a Moon-centered relativistic time scale defined within the LCRS, essential for accurate navigation and time-transfer near the Moon.
• It distinguishes coordinate time from proper time, requiring complex relativistic transformations to correct for gravitational and kinematic effects in both surface and orbital clocks.
• Numerical models like LTE440 achieve sub-nanosecond precision in linking TCL with TDB/TCB, ensuring reliable performance for GNSS-like lunar positioning and timing applications.

Lunar Coordinate Time is the Moon-centered relativistic coordinate time required for high-accuracy modeling of navigation, positioning, timing, orbit determination, and time transfer in the lunar vicinity. In the formal IAU-aligned literature, the standard symbol is TCL (Temps-Coordonnée Lunaire), the coordinate time of the Lunar Celestial Reference System (LCRS); “LTC” is usually an informal alternate label rather than the official notation, although some later discussions reserve LTC for an operational realized lunar scale steered to TCL (Defraigne et al., 4 Nov 2025, Jin et al., 1 Jul 2026).

1. Terminology and conceptual status

The defining distinction in the lunar-time literature is between coordinate time and proper time. Coordinate times are global labels attached to relativistic reference systems: TCB for the solar-system barycentric frame, TCG for the geocentric frame, and TCL for the lunar frame. Proper time is what an actual clock measures along its own worldline. Because the rate of proper time depends on local gravity and motion, no single natural proper time is shared by all clocks on the Moon or in lunar orbit. TCL therefore plays the same structural role for the Moon that TCG plays for Earth: it is the common relativistic time coordinate needed to model clocks, signal propagation, and dynamics consistently in a Moon-centered system (Lu et al., 23 Sep 2025).

In the IAU 2024 framework, the Moon-centered relativistic system is the LCRS, constructed “with the same techniques used to construct the GCRS and TCG,” and its associated coordinate time is TCL. The unit of measurement of TCL is kept consistent with the SI second. The same literature is explicit that TCL “is not given by any real clock”: a real clock on the surface or in orbit realizes proper time, and its relation to TCL must be computed from relativistic transformations (Lu et al., 23 Sep 2025).

The practical motivation is operational rather than terminological. Once lunar PNT targets meter-level or better performance, timing must be controlled at the nanosecond level, and relativity becomes unavoidable. The cited papers therefore treat a lunar coordinate time not as a convenience but as a necessary component of any GNSS-like lunar infrastructure, Earth–Moon VLBI, or interoperable cislunar timing architecture (Defraigne et al., 4 Nov 2025).

2. Relativistic definition in the LCRS

The formal construction of TCL is the Moon-specialized instance of the standard IAU body-centric transformation. For a local reference system tied to body $X$, the generic post-Newtonian relation between barycentric coordinates $(t,x^i)$ and local coordinates $(T,X^a)$ is written as

$T = t - {1 \over c^2} \biggl[ A(t) + v_X^i r_X^i\biggr] + {1 \over c^4} \biggl[B(t) + B^i(t)r_X^i + B^{ij}(t)r_X^i r_X^j + C(t,\ve{x})\biggr ] + O(c^{-5}),$

with $r_X^i=x^i-x_X^i$. For the Moon one sets $X=L$, so $T=\mathrm{TCL}$, $x_X^i=x_L^i$, $v_X^i=v_L^i$, and $r_X^i=r_L^i$ (Klioner, 17 Apr 2026).

A more explicit lunar 1PN/2PN expression is given in the DE440-based time-ephemeris work: $(t,x^i)$0 where the dominant secular pieces are the Moon’s barycentric kinetic term and the external Newtonian potential, especially the Sun’s contribution (Lu et al., 23 Sep 2025).

The comparison between coordinate time and proper time in the lunar environment is organized through the local LCRS relation

$(t,x^i)$1

with

$(t,x^i)$2

so the rate of a lunar clock relative to TCL depends on its LCRS velocity, the lunar Newtonian potential, the tidal potential, and inertial terms (Defraigne et al., 4 Nov 2025).

A specifically lunar feature is the permanent Earth tide. Because the Moon is in a 1:1 spin-orbit resonance with Earth, the lunar tidal potential contains a constant term absent in the terrestrial case, contributing at the level of

$(t,x^i)$3

in relative frequency and varying with location on the lunar surface (Defraigne et al., 4 Nov 2025).

3. Surface clocks, orbital clocks, and realizations of TCL

For clocks at rest on the lunar surface, the dominant offset from TCL is set by the Moon’s monopole potential. Using GRAIL gravity and LOLA topography, one study finds a principal surface effect of about

$(t,x^i)$4

corresponding to about 2.7 $(t,x^i)$5s/day relative to TCL, with surface variations due to topography within

$(t,x^i)$6

A free-running surface clock therefore does not realize TCL automatically, even before one includes finer tidal and kinematic terms (Defraigne et al., 4 Nov 2025).

Surface-clock studies that model topography more explicitly obtain closely related magnitudes. The same thesis literature reports a mean surface drift relative to TCL of

$(t,x^i)$7

with location-dependent variation of about

$(t,x^i)$8

and a total low-to-high terrain spread of about

$(t,x^i)$9

The same work finds that changing lunar orientation parameters contribute only about

$(T,X^a)$0

on average, with periodic variations not exceeding

$(T,X^a)$1

so libration-driven rotational timing effects are negligible for stationary surface clocks at the stated precision level (Seyffert, 10 Sep 2025).

Orbital clocks depart from TCL in a different regime. In the Moonlight-oriented analysis, a circular-like orbiter with $(T,X^a)$2 km and $(T,X^a)$3 has a periodic amplitude of about 1.5 ns and a drift of about $(T,X^a)$4 relative to a surface-related lunar timescale, while an eccentric case with $(T,X^a)$5 km and $(T,X^a)$6 reaches a periodic amplitude of about 27 ns and a drift magnitude of about 1.76 $(T,X^a)$7s/day (Fienga et al., 2024). For Moonlight-like ELFO navigation satellites, another detailed simulation finds that orbiter proper time runs faster than a lunar surface reference by about 1.986–1.987 $(T,X^a)$8s/day, with periodic variations below about 0.1 $(T,X^a)$9s (Seyffert, 10 Sep 2025).

A distinct realization proposal is the time aligned orbit. In that construction, a single ideal clock in a lunar orbit with semi-major axis of about 1.5 lunar radius can directly realize the proper time of a chosen lunar selenoid while remaining convertible to TCL by a known linear transformation. Numerical simulations yield desynchronization from the selenoid proper time up to 190 ns after a year with a frequency offset of $T = t - {1 \over c^2} \biggl[ A(t) + v_X^i r_X^i\biggr] + {1 \over c^4} \biggl[B(t) + B^i(t)r_X^i + B^{ij}(t)r_X^i r_X^j + C(t,\ve{x})\biggr ] + O(c^{-5}),$0, and as low as 13 ns and $T = t - {1 \over c^2} \biggl[ A(t) + v_X^i r_X^i\biggr] + {1 \over c^4} \biggl[B(t) + B^i(t)r_X^i + B^{ij}(t)r_X^i r_X^j + C(t,\ve{x})\biggr ] + O(c^{-5}),$1 when mean-orbit deviations are accounted for (Yang et al., 28 Dec 2025).

4. Relation to terrestrial times and Earth–Moon time transfer

The Earth–Moon problem introduces an additional layer absent from purely terrestrial chronometric geodesy: the relation between the geocentric and selenocentric coordinate times. In one Earth–Moon clock-comparison formulation,

$T = t - {1 \over c^2} \biggl[ A(t) + v_X^i r_X^i\biggr] + {1 \over c^4} \biggl[B(t) + B^i(t)r_X^i + B^{ij}(t)r_X^i r_X^j + C(t,\ve{x})\biggr ] + O(c^{-5}),$2

where $T = t - {1 \over c^2} \biggl[ A(t) + v_X^i r_X^i\biggr] + {1 \over c^4} \biggl[B(t) + B^i(t)r_X^i + B^{ij}(t)r_X^i r_X^j + C(t,\ve{x})\biggr ] + O(c^{-5}),$3 and $T = t - {1 \over c^2} \biggl[ A(t) + v_X^i r_X^i\biggr] + {1 \over c^4} \biggl[B(t) + B^i(t)r_X^i + B^{ij}(t)r_X^i r_X^j + C(t,\ve{x})\biggr ] + O(c^{-5}),$4 are the local Earth and Moon proper-to-coordinate contributions and $T = t - {1 \over c^2} \biggl[ A(t) + v_X^i r_X^i\biggr] + {1 \over c^4} \biggl[B(t) + B^i(t)r_X^i + B^{ij}(t)r_X^i r_X^j + C(t,\ve{x})\biggr ] + O(c^{-5}),$5 is the Earth–Moon coordinate-time ratio. The same study states that gravity-potential differences affect observations at the $T = t - {1 \over c^2} \biggl[ A(t) + v_X^i r_X^i\biggr] + {1 \over c^4} \biggl[B(t) + B^i(t)r_X^i + B^{ij}(t)r_X^i r_X^j + C(t,\ve{x})\biggr ] + O(c^{-5}),$6 level, while the coordinate-time ratio contributes at the $T = t - {1 \over c^2} \biggl[ A(t) + v_X^i r_X^i\biggr] + {1 \over c^4} \biggl[B(t) + B^i(t)r_X^i + B^{ij}(t)r_X^i r_X^j + C(t,\ve{x})\biggr ] + O(c^{-5}),$7 level (Zhang et al., 19 Jun 2025).

Different quoted Earth–Moon offsets correspond to different time pairs. In a Keplerian Earth–Moon model, the rate of a clock near the Moon’s equator relative to one near the Earth’s equator is

$T = t - {1 \over c^2} \biggl[ A(t) + v_X^i r_X^i\biggr] + {1 \over c^4} \biggl[B(t) + B^i(t)r_X^i + B^{ij}(t)r_X^i r_X^j + C(t,\ve{x})\biggr ] + O(c^{-5}),$8

with the $T = t - {1 \over c^2} \biggl[ A(t) + v_X^i r_X^i\biggr] + {1 \over c^4} \biggl[B(t) + B^i(t)r_X^i + B^{ij}(t)r_X^i r_X^j + C(t,\ve{x})\biggr ] + O(c^{-5}),$9 term as the mean secular difference and the $r_X^i=x^i-x_X^i$0 term as the eccentricity-driven modulation (Ashby et al., 2024).

For the scaled lunar-surface time $r_X^i=x^i-x_X^i$1 defined in one post-Newtonian framework, the direct terrestrial comparison is

$r_X^i=x^i-x_X^i$2

with the largest periodic term about $r_X^i=x^i-x_X^i$3 at the mean anomalistic period (Turyshev et al., 2024). In a dedicated TCL–TCG algorithm, the coordinate-time drift itself is about 1.4769 $r_X^i=x^i-x_X^i$4s/day, while a lunar surface clock on the selenoid runs faster than a terrestrial geoid clock by about 56.025 $r_X^i=x^i-x_X^i$5s/day, with a dominant monthly component of about 0.4707 $r_X^i=x^i-x_X^i$6s (Kopeikin et al., 2024).

The literature is also explicit about a common misconception: there is no universal scalar function “TCL minus TT” valid everywhere. The quantity depends on the chosen clocks, their worldlines, and the time-transfer model used between Earth and Moon. For low-accuracy coordination, UTC may remain sufficient if one corrects the secular trend; for high-accuracy lunar navigation, the relevant object is a full relativistic time-transfer model, not a globally unique lunar–Earth offset (Defraigne et al., 4 Nov 2025).

5. Scaled lunar times and the reference-timescale debate

A central post-2024 question is whether lunar operations should use TCL directly or some scaled version analogous to TT or TDB. One trade-off analysis writes the generic scaled form as

$r_X^i=x^i-x_X^i$7

and studies three options. Option (1) sets $r_X^i=x^i-x_X^i$8, so the practical lunar reference is TCL itself. Option (2) uses a mean rate matched to the proper time on a chosen lunar equipotential, with

$r_X^i=x^i-x_X^i$9

Option (3) chooses the scaling so that the difference from terrestrial time has no secular drift, with

$X=L$0

The same paper concludes that TCL should be used directly, because option (2) requires a conventional lunar reference potential $X=L$1 that “does not exist to date,” while option (3) would make a perfect surface clock differ from the adopted scale by about 56 $X=L$2s/day and would impose another layer of scaling on coordinates and mass parameters (Defraigne et al., 4 Nov 2025).

The coordinate-scaling issue is not cosmetic. When time is scaled, spatial coordinates and mass parameters must be scaled as well. For a prospective lunar scaling, the induced change in the Earth–Moon distance is estimated as about 1 cm for the equipotential-matched option and about 30 cm for the no-secular-drift option. A broader review therefore argues that scaling local coordinate times such as TCL is “unnecessary, unreasonable and even risky,” both because additional linear drifts remain unavoidable among TCB, TCG, and a scaled lunar time, and because each scaling propagates into distances, $X=L$3 values, and related constants (Klioner, 17 Apr 2026).

Not all proposals use the same operational symbol. One relativistic timing framework defines TL as the Moon-surface analog of TT, with

$X=L$4

so that TL is a scaled form of TCL appropriate to clocks on or near the lunar surface (Turyshev et al., 2024). By contrast, the Moonlight appendix introduces

$X=L$5

precisely so that differences between TL and TT are only periodic (Fienga et al., 2024). A later lunar-reference-timescale review returns to the unscaled choice and judges the least constraining and most natural solution to be

$X=L$6

(Bourgoin et al., 29 Jul 2025).

6. Numerical ephemerides, software realization, and domain of use

The most explicit numerical realization of TCL presently described in the literature is the Lunar Time Ephemeris LTE440. It computes TCL and its relations with TCB and TDB using JPL DE440, including contributions from the Sun, all 8 planets, the Moon, Pluto, 343 main-belt asteroids, and 30 Kuiper belt objects plus a Kuiper-belt ring. The integration uses a 10th-order Romberg scheme on half-day integration intervals; the resulting detrended series is represented by 13th-degree Chebyshev polynomials on 4-day granules and exported in SPICE format. The reported performance is better than 0.15 ns before 2050 in accuracy and 1 ps in numerical precision over the ephemeris span, with secular rates

$X=L$7

and dominant periodic terms of 1.65 ms annually and 126 $X=L$8s monthly (Lu et al., 23 Sep 2025).

The associated user manual specifies the practical interface. LTE440 is distributed through lte440.bsp for the periodic part of $X=L$9 and lte440.tpc for the secular drift coefficient, with SPICE/CALCEPH access routines such as tclmtdb(jd_tdb) and tclmtcb(jd_tdb). Evaluation is performed at a TDB Julian Date, and the ephemeris returns $T=\mathrm{TCL}$0 or $T=\mathrm{TCL}$1 in seconds (Lu et al., 24 Jun 2025).

The domain of validity of the lunar local frame is not the whole Earth–Moon system. One Solar-System-wide review states that the LCRS remains clearly useful within $T=\mathrm{TCL}$2 lunar radii from the center of the Moon, and explicitly says that the use of the LCRS for the whole cislunar space “cannot be recommended.” Outside the immediate lunar vicinity, the recommended framework is the BCRS with TCB. The same study computed body-centric time ephemerides for all major bodies with INPOP19a, represented them with Chebyshev polynomials, and kept computed numerical errors below 10 ps, but described these products as demonstrators rather than ultimate authoritative standards (Klioner, 17 Apr 2026).

A further terminological split appears in later systems papers: TCL remains the fundamental coordinate time of the LCRS, while LTC can denote a proposed operational lunar scale, “the lunar counterpart of UTC,” realized by atomic clocks on or near the Moon, steered to TCL, and disseminated to users. That usage preserves the distinction between the abstract coordinate time and its realized operational counterpart, even when the acronym “LTC” is employed in mission architecture rather than in the formal IAU symbol set (Jin et al., 1 Jul 2026).