Lunar Time Ephemeris $ exttt{LTE440}$: definitions, algorithm and performance

Robotic and human activities in the cislunar space are expected to rapidly increase in the future. Modeling, jointly analysis and sharing of time measurements made in the vicinity of the Moon might indispensably demand calculating a lunar time scale and transforming it into other time scales. For users, we present a ready-to-use software package of Lunar Time Ephemeris $\texttt{LTE440}$ that can calculate the Lunar Coordinate Time (TCL) and its relations with the Barycentric Coordinate Time (TCB) and the Barycentric Dynamical Time (TDB). According to the International Astronomical Union Resolutions on relativistic time scales, we numerically calculate the relativistic time-dilation integral in the transformation between TCL and TCB/TDB with the JPL ephemeris DE440 including the gravitational contributions from the Sun, all planets, the main belt asteroids and the Kuiper belt objects, and export data files in the SPICE format. At a conservative estimate, $\texttt{LTE440}$ 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 secular drifts between the coordinate times in $\texttt{LTE440}$ are respectively estimated as $\langle \mathrm{d},\mathrm{TCL}/\mathrm{d},\mathrm{TCB}\rangle=1-1.482,536,216,7\times10^{-8}$ and $\langle \mathrm{d},\mathrm{TCL}/\mathrm{d},\mathrm{TDB}\rangle=1+6.798,355,24\times10^{-10}$. Its most significant periodic variations are an annual term with amplitude of 1.65 ms and a monthly term with amplitude of 126 $μ$s. $\texttt{LTE440}$ might satisfy most of current needs and is publicly available.

💡 Research Summary

The paper presents LTE440, a lunar time ephemeris designed to provide a precise transformation between the newly defined Lunar Coordinate Time (TCL) and the widely used Barycentric Coordinate Time (TCB) and Barycentric Dynamical Time (TDB). Motivated by the anticipated surge in robotic and crewed activities in cislunar space, the authors argue that a dedicated lunar time scale is essential for modeling, joint analysis, and sharing of time measurements made near the Moon.

Following the International Astronomical Union (IAU) 2024 resolutions on relativistic time scales, the authors derive the exact relativistic time‑dilation integral that links TCL to TCB (and, via the linear TCB–TDB relation, to TDB). The integral incorporates the Moon’s barycentric position and velocity, the Newtonian potential of all external bodies (Sun, planets, main‑belt asteroids, Kuiper‑belt objects), non‑spherical contributions, and higher‑order coupling terms up to order c⁻⁴. Equations (1)–(4) define the potentials w₀ᴹ, wˡᴹ, and Δᴹ, while equations (5)–(12) rewrite the transformation in TDB‑compatible form, yielding the core function L_TE(TDB) (Eq. 7) and the position‑dependent correction L_PD(TDB, x) (Eq. 12).

The numerical implementation uses the JPL DE440 planetary and lunar ephemeris, which provides the most up‑to‑date positions, velocities, and masses for the Sun, eight planets, the main‑belt asteroids, and Kuiper‑belt objects. The time‑dilation integral is evaluated with a 10th‑order Romberg scheme on half‑day (12 h) intervals. Because the integrand consists of a constant secular drift plus periodic terms, the authors adopt an iterative drift‑removal technique: an initial constant drift estimate is subtracted, the resulting curve is fitted with a linear function, the drift is refined, and the process repeats until convergence (a method inspired by Fukushima 1995 and Irwin & Fukushima 1999). After the drift is isolated, the remaining periodic components are integrated, and the final ephemeris is reconstructed by adding back the converged secular term.

Performance is assessed by comparing LTE440 with two auxiliary ephemerides built on DE430 (LTE430) and DE441 (LTE441). The secular drift rates are reported as ⟨dTCL/dTCB⟩ = 1 − 1.482 536 216 7 × 10⁻⁸ and ⟨dTCL/dTDB⟩ = 1 + 6.798 355 24 × 10⁻¹⁰. The most prominent periodic variations are an annual term of 1.65 ms amplitude and a monthly term of 126 µs amplitude. The authors claim a conservative absolute accuracy better than 0.15 ns before the year 2050 and a numerical precision of 1 ps over the entire time span of the ephemeris.

To facilitate immediate use, the authors provide the ephemeris data in SPICE kernel format and release a software package (available on GitHub) that implements functions for computing TCL − TCB, TCL − TDB, and TCL for any given TDB epoch. The package also includes utilities for converting between TCL and terrestrial time scales (TT) by combining LTE440’s TCL − TDB with the TT − TDB offsets already supplied by DE440, INPOP21a, and EPM2021.

The paper discusses scientific and operational implications. For lunar‑based VLBI, the sub‑microsecond timing precision enabled by LTE440 would allow Earth‑Moon baselines to achieve astrometric accuracies comparable to, or better than, ground‑based VLBI. The secular drift and periodic terms are essential for correcting phase errors in long‑baseline interferometry, precise orbit determination of lunar orbiters, and for tracing the proper time of lunar clocks back to Coordinated Universal Time (UTC). The authors note that a lunar reference time (LRT) with an uncertainty of ≤10 ns per day translates to a required ephemeris derivative error of 10⁻¹³; LTE440’s reported systematic errors (≈0.1 ns) comfortably meet this requirement.

Limitations are acknowledged. The stated accuracy is validated only up to 2050; beyond that, the propagation of DE440’s orbital uncertainties may degrade performance. The half‑day integration step, while sufficient for the claimed picosecond precision, may not be optimal for future missions demanding femtosecond‑level timing. Additionally, the ephemeris does not yet incorporate possible time‑varying lunar interior mass distributions or solar activity‑induced gravitational potential changes, which could become relevant for ultra‑high‑precision applications.

In summary, LTE440 delivers a high‑precision, publicly available lunar time ephemeris that bridges TCL with the standard barycentric time scales, quantifies secular drifts and dominant periodic terms, and provides ready‑to‑use software tools. It constitutes a critical infrastructure component for upcoming lunar exploration, lunar‑Earth VLBI, and any scientific endeavor requiring nanosecond‑level synchronization across the Earth‑Moon system.