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