Lunar Reference Timescale

TL;DR

The paper develops a relativistic framework for a lunar reference timescale by extending the IAU’s post-Newtonian reference-system formalism to the Moon, defining a Lunar Celestial Reference System (LCRS) with Lunar Coordinate Time (TCL) and exploring how TCL relates to Earth-based timescales via various intermediate systems (BCRS/TCB, GCRS/TCG, LCRS/TCL). It analyzes the rates of proper time for clocks on the Moon and Earth, the impact of lunar gravity, tides, and topography, and the feasibility of a rescaled TCL (TL) as a practical realization, including potential alignments with TT or TCL itself. The authors propose three TL options and discuss their implications for clock steering, scaling of distances and masses, and traceability to UTC, advocating TL = TCL as the natural, minimally complex choice while acknowledging alternatives and the need for broadcast of relativistic corrections in lunar PNT services. They also address the practical realization and real-time traceability of TL through UTC(k) links, emphasizing future autonomous lunar clocks and the IAU’s role in standardizing the lunar timescale. Overall, the work provides a comprehensive, covariance-consistent pathway to define and realize a lunar reference timescale that can be extended to Mars and other planets, with clear guidance for navigation, timing, and mission planning in a relativistic regime.

Abstract

Setting up a relativistic lunar reference frame is of a prime importance in the context of future exploration missions to the Moon. If the procedure for building a consistent reference frame within the framework of the general theory of relativity is well established (cf. resolutions B.3 of IAU 2000), there is still some freedom in the choice of the coordinate timescale to be adopted as reference in the lunar region. In this paper, we review the orders of magnitude of the relativistic effects resulting from (i) the gravitational redshift of a clock on the lunar surface and (ii) the time transformations between a clock on the surface of the Moon and a clock on the surface of the Earth. We then discuss possible options for a lunar reference timescale with their advantages and drawbacks, taking note that the solution which is adopted for the Moon shall then be reemployed for Mars and other planets. Finally, we propose possible realizations of the lunar reference timescale as well as its traceability to UTC.

Figures (13)

• Figure 1: Map of the relative frequency difference between the proper time of a clock at rest on the lunar surface and $(a)$ TCL [i.e., $\mathcal{T}$ in Eq. \ref{['eq:dpptimedtclsimpfinal']}], $(b)$ a rescaled TCL [i.e., $\mathcal{T}^\dag$ in Eq. \ref{['eq:tau-TL']}]. The clock is assumed at radial position $\mathcal{R}_\mathrm{l}=R_{\mathrm{topo}}(\varphi_\mathrm{l},\theta_\mathrm{l})$ [cf. Eq. \ref{['eq:Rtopo']}]. The origin of the longitude represents lunar prime meridian.
• Figure 2: Coordinate time differences $[\mathcal{T} - t]_{(t_\mathrm{l},\bm{x}_\mathrm{l})}-[T - t]_{(t_\mathrm{e},\bm{x}_\mathrm{e})}$ over the year 2024 with $\bm{x}_\mathrm{l}$ at a lunar latitude and longitude ($0^\circ$,$0^\circ$) and $\bm{x}_\mathrm{e}$ at a Earth latitude and longitude ($0^\circ$,$0^\circ$).
• Figure 3: Top: scalar product $\bm{v}_\mathrm{E} \cdot (\bm{x}-\bm{x}_\mathrm{E})$ from Eq. \ref{['eq:TCB-TCG']} over one month in 2024, evaluated at Earth latitudes and longitudes ($0^\circ$,$0^\circ$) in green, ($90^\circ$,$0^\circ$) in blue and ($0^\circ$,$120^\circ$) in orange. Bottom: scalar product $\bm{v}_\mathrm{L} \cdot (\bm{x}-\bm{x}_\mathrm{L})$ from Eq. \ref{['eq:TCB-TCL']} over one month in 2024, evaluated at Moon latitudes and longitudes ($0^\circ$,$0^\circ$) in green, ($90^\circ$,$0^\circ$) in blue and ($0^\circ$,$90^\circ$) in orange.
• Figure 4: Top: Time differences $[T - \mathcal{T}]_{(\mathcal{T}_\mathrm{l},\bm{\mathcal{X}}_\mathrm{l})}$ over the year 2024, with $\bm{\mathcal{X}}_\mathrm{l}$ at a lunar latitude and longitude ($0^\circ$,$0^\circ$), after removing the secular term of $1.5 \textnormal{~$\mu$s} / \mathrm{day}$. Bottom: Scalar product $\bm{V}_\mathrm{L} \cdot (\bm{X}_\mathrm{l}-\bm{X}_\mathrm{L})$ from Eq. \ref{['eq:Kop']} over two months in 2024, evaluated at Moon latitudes and longitudes ($0^\circ$,$0^\circ$) in green, ($90^\circ$,$0^\circ$) in blue, and ($0^\circ$,$90^\circ$) in orange.
• Figure 5: Top: Time differences $[T - \mathcal{T}]_{(\mathcal{T}_\mathrm{e},\bm{\mathcal{X}}_\mathrm{e})}$ over the year 2024, with the Earth clock $\bm{\mathcal{X}}_\mathrm{e}$ at latitude and longitude ($0^\circ$,$0^\circ$), after removing the secular term of $1.5 \textnormal{~$\mu$s} / \mathrm{day}$. Bottom: Scalar product $\bm{\mathcal{V}}_\mathrm{E} \cdot (\bm{\mathcal{X}}_\mathrm{e}-\bm{\mathcal{X}}_\mathrm{E})$ from Eq. \ref{['eq:Kop']} over one month in 2024, evaluated at Earth latitudes and longitudes ($0^\circ$,$0^\circ$) in green, ($90^\circ$,$0^\circ$) in blue and ($0^\circ$,$120^\circ$) in orange.