Two birds with one stone: simultaneous realization of both Lunar Coordinate Time and lunar geoid time by a single orbital clock

TL;DR

This work addresses the lunar reference time problem by contrasting Lunar Coordinate Time ($T_{CL}$) with selenoid proper time and proposes a time_aligned_orbit to bridge them. In this orbit, the clock reading can match the selenoid proper time $\tau_s$ and be mapped to $T_{CL}$ via a linear scaling tied to $L_L$. Theory shows that equating $L_L$ and $L_P$ yields a mean semi-major axis near $1.5\,R_M$ and, for a particular inclination, allows $\tau_s=\tau_p$, with a simple relation to UTC. Simulations including perturbations show desynchronization up to $190$ ns/year and frequency offsets around $6\times 10^{-15}$, which can be reduced to $13$ ns and $4\times 10^{-16}$ when mean-element deviations are corrected. Overall, the work presents a viable, scalable approach to realize both LRT options with a single orbital clock and points to applicability beyond the Earth–Moon system.

Abstract

Context. Among options for definition of the lunar reference time, the option taking Lunar Coordinate Time (O1) has its simplicity but cannot be realized by any clock without steering, while another option adopting the lunar geoid (selenoid) proper time (O2) has its convenience for users on the lunar surface but would bring a new scaling of spatial coordinates and mass parameter of the Moon. Aims. We propose a ''time aligned orbit'' that the readings of an ideal clock in this orbit could equal to the selenoid proper time in O2 and these readings could be converted to Lunar Coordinate Time in O1 by a known linear transformation. Methods. We show that there exist the time aligned orbit around the Moon with its semi-major axis of about 1.5 lunar radius slightly depending on its inclination. We conduct a set of numerical simulations to assess to what extent a clock on these orbits could realize O2 in a more realistic lunar environment. Results. We find that the proper time in our simulations would desynchronize from the selenoid proper time up to 190 ns after a year with a frequency offset of 6E-15, which is solely 3.75% of the frequency difference in O2 caused by the lunar surface topography. These numbers might be further reduced to 13 ns and 4E-16, if we could account for the deviation of the mean orbits in our simulations from the nominal ones. Conclusions. One might simultaneously realize O1 and O2 by deployment of a single clock in the time aligned orbit. This approach also has its scalability for other terrestrial planets beyond the Earth-Moon system.

Figures (1)

• Figure 1: Left panel: the desynchronization $\Delta^*=\tau_{\mathrm{p}}^*-\tau_s$ and their frequency offsets $\Delta f^*$ of four numerically simulated time aligned orbit s with different initial inclinations. Right panel: the corrected $\Delta^*_{\mathrm{c}}=\tau^*_{\mathrm{p,c}}-\tau_s$ and $\Delta f^*_{\mathrm{c}}$ for the same four simulated orbits by accounting for the deviation of the mean orbital elements in our simulations from the ones required by the time aligned orbit s.