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

Context. There are three options for defining the lunar reference time. Option O1, using Lunar Coordinate Time, has the advantage of simplicity, while options O2 – using the lunar geoid (selenoid) time – and O3 – using an average alignment with Terrestrial Time – have the advantage of convenience for users with instruments on the lunar surface and those using Earth navigation satellite signals, respectively. Clock steering must be performed for all three options. O2 and O3 provide new scalings of spatial coordinates and mass parameters in the Solar System. Aims. We propose a ‘time-aligned orbit’ in which the readings of an ideal clock in this orbit are equal to the selenoid time in O2; these readings can be converted to Lunar Coordinate Time in O1 via a known linear transformation. Methods. We show that there exists a time-aligned orbit around the Moon with a semi-major axis of about 1.5 lunar radii that slightly depends on its inclination with respect to the equator of the Moon. We conducted a set of numerical simulations to assess to what extent a clock on these orbits could be used in O2 in a more realistic lunar environment. Results. The proper time in our simulations de-synchronizes from the selenoid time by up to 190 ns after a year with a frequency offset of 6 × 10−15, which is only 3.75% of the frequency difference in O2 caused by the lunar surface topography. This could be further reduced to 13 ns and 4 × 10−16 if we are able to account for the deviation of the mean orbits in our simulations from the nominal ones. Conclusions. One can simultaneously realize and use options O1 and O2 by deploying a single clock in the time-aligned orbit. This approach is scalable to other terrestrial planets beyond the Earth–Moon system.