In recent years, atomic clocks have made spectacular progress, with ground-based optical lattice atomic clocks reaching stabilities of less than 10^-18 and accuracies of 1×10^-18. High-performance atomic-clock satellite-satellite, satellite-ground, and ground-ground interconnections can be achieved by establishing time-frequency transfer links. This advancement provides insights into various critical technological and scientific domains, including global satellite navigation systems, deep space exploration, verification of general relativity, measurement of gravitational waves, gravity field assessment of the earth, and fundamental physical constant measurements. This year, the Chinese Academy of Sciences plans to launch a lunar orbiting spacecraft equipped with a laser time-frequency transfer payload to assess the performance of onboard hydrogen atomic clocks and to conduct a comparison of clocks in remote observatories. For high-precision time-frequency transfer data processing, establishing a computational model that meets the requirements of the mission within the framework of general relativity is necessary.

Based on the existing relativistic theory for time and frequency transfer, in this study, we derived a relativistic model of one- and two-way satellite-ground laser time-frequency transfer on a distant retrograde orbit (DRO), which can be directly used in the data processing of the DRO laser time-frequency transfer. Using the simulated DRO orbit, we analyzed the magnitudes and distribution patterns of various error correction terms in the laser time-frequency transfer. These terms include light-time correction, atmospheric refraction, relativistic rate shifts, and position correction between the detector and reflector. In addition, Monte Carlo methods were employed to simulate and compute the uncertainties of link corrections, considering the DRO orbit and attitude determinations and the probe payload calibration parameters. The effects of these uncertainties on the stability and accuracy of the time-frequency transfer measurements were investigated for both one- and two-way satellite-ground modes and for one-way laser time-frequency transfer in the common-view mode.

In the two-way mode, the dependence on the satellite orbit accuracy is relatively weak. With a radial distance error in orbit determination of 10 m, we anticipate that the accuracy of link error correction will exceed 1.5 ps, with corresponding link stability (modified Allan deviation) of better than 2×10^-17@10000 s. However, the comparative performance of the one-way mode is highly dependent on the accuracy of the satellite orbit determination. The measurement clock error is coupled with the orbit errors. In scenario 1, the link correction accuracy is expected to be within 38 ns, with corresponding link stabilities of approximately 8×10^-15@1000 s and 7×10^-14@10000 s. This may affect the assessment of the long-term stability of onboard hydrogen clocks.

Correction of the detector–reflector positioning relationship is a major factor that affects the laser time-frequency transfer in low-earth-orbit satellites. In lunar laser time-frequency transfer, assuming ground measurement errors of 1 cm and attitude measurement errors of 360 arcsec, the correction uncertainty is approximately 1×10^-13. Both one- and two-way modes require correction for atmospheric refraction, with the one-way mode possibly requiring more accurate meteorological parameters than the two-way mode.

When two observation stations each use one-way mode to conduct measurements on a DRO probe and achieve one-way laser time-frequency transfer in the common-view mode, the link is expected to achieve an accuracy of better than 1.7 ns and a stability of 1×10^-14@1000 s.

The computational model developed in this research enables high-precision processing of one-way and two-way laser time-frequency transfer measurement data between the earth and the moon, thereby providing a theoretical foundation for the performance evaluation of atomic clocks and time synchronization in deep space exploration missions.