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

TL;DR

LTE440 provides a numerical lunar time ephemeris package that computes TCL and its relations to TCB and TDB by numerically integrating the relativistic time-dilation integral using DE440 data. The approach achieves sub-nanosecond accuracy by 2050 and picosecond-level precision across the full span, with a clear separation of secular drift and periodic terms via Chebyshev interpolation for SPICE outputs. The authors compare LTE440 with LTE430 and calibrate against LTE441 to quantify long-term drifts, and make the software and data publicly available for lunar time-scale traceability and Moon-based measurements. They also propose cross-validation with other ephemerides (e.g., INPOP21a, EMP2021) to further validate performance and broaden applicability.

Abstract

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.

Figures (3)

• Figure 1: Differences between our reproduced version of TT$-$TDB at the geocenter with the planetary data of DE440 and the inherent version of DE440. Top panel: difference between the derivatives $\delta[\mathrm{d(TT-TDB)}/\mathrm{dTDB}]=[\mathrm{d(TT-TDB)}/\mathrm{dTDB}]_\mathrm{this}-[\mathrm{d(TT-TDB)}/\mathrm{dTDB}]_\mathrm{DE440}$. Bottom panel: difference between the Earth time ephemerides $\delta\mathrm{(TT-TDB)}=\mathrm{(TT-TDB)}_\mathrm{this}-\mathrm{(TT-TDB)}_\mathrm{DE440}$.
• Figure 2: Comparison of LTE$_{430}$ and LTE$_{440}$. (a) Difference between their derivatives: $\delta(\mathrm{dLTE}/\mathrm{dTDB})=\mathrm{dLTE_{430}}/\mathrm{dTDB}-\mathrm{dLTE_{440}}/\mathrm{dTDB}$. (b) Difference between LTE$_{430}$ and LTE$_{440}$: $\delta\mathrm{LTE}=\mathrm{LTE_{430}}-\mathrm{LTE_{440}}$. (c) Detrended difference between LTE$_{430}$ and LTE$_{440}$: $\overline{\delta(\mathrm{LTE})}=\delta(\mathrm{LTE})-\mathrm{linear\ drift}$.
• Figure 3: Detrended differences between mainly contributing sources of LTE$_{430}$ and LTE$_{440}$: (a) the square of Moon's velocity $\overline{\delta(\mathrm{LTE})}_{v^2_\mathrm{M}}$, (b) the gravitational potential of the Sun at the Moon $\overline{\delta(\mathrm{LTE})}_\odot$, and (c) the gravitational potentials of the Kuiper belt object s and ring at the Moon $\overline{ \delta(\mathrm{LTE})}_\mathrm{KBO}$.