The space-ground laser time transfer experiment using the laser ranging and laser timing payload equipped on the lunar orbiter scheduled to be launched in 2023 by the Chinese Academy of Sciences aims to evaluate the performance of the on-board atomic clock and study the high-accuracy time comparison technology and autonomous navigation technology in Earth-Moon space to support China’s manned lunar exploration project and other interstellar missions in the future. Compared with dedicated time transfer experiments such as Time Transfer by Laser Link (T2L2) and Atomic Clock Ensemble in Space (ACES), whose model is studied in the local geocentric frame of reference, the model for laser time transfer in Earth-Moon space needs to be represented in the solar-system barycentric space-time frame of reference using the Barycentric Coordinate Time (TCB) or Barycentric Dynamical Time (TDB) as the reference time scale. To ensure the smooth progress of the high-accuracy time comparison mission, a measurement model for laser time transfer in Earth-Moon space with an uncertainty of order 1 ns is developed.

The proposed model is represented in the Barycentric Celestial Reference System (BCRS) using TCB as the reference time scale. The numerical time ephemeris of the earth, TE405, which is calculated using NASA’s Jet Propulsion Laboratory (JPL) development ephemerides, was employed to achieve a transformation with a 0.1 ns level accuracy between the Geocentric Coordinate Time (TCG) and TCB. Based on the IAU resolution B1.5 (2000), the nominal trajectory of the lunar spacecraft was used to calculate the influence of the gravitational fields of celestial bodies in the solar system on the rate of the on-board atomic clock. The round-trip light time model of laser pulses with an accuracy of 10 ps level was deduced. We built the model of Shapiro time delay with an accuracy of better than 10 ps, and established center of mass correction model, geometric position correction model, system delay model. Finally, based on the engineering background of the lunar orbiter scheduled to be launched in 2023, factors such as the stochastic and deterministic clock errors, satellite visibility, echo rate setting, and discontinuity of observations were all considered when performing the data simulation using the proposed measurement model for laser time transfer.

The data generated by the proposed measurement model are paired to identify the triplets, analyzed and processed to estimate the time offset and frequency offset of the on-board clock. Random white noise is then superimposed onto the nominal trajectory of the satellite to generate the position of the satellite, which is not constrained by the dynamic model and is used for data processing, resulting in a 1σ error of 3300 ns in the calculation of the one-way light time. This drowns out the deterministic error of the on-board clock by half an hour (0.5 ns/s×1800 s=900 ns). Thus, during data preprocessing, instead of using the traditional observed minus computed (O-C) residuals method to remove the trend caused by the orbit, the effective echoes are directly extracted from the observations (Fig.9, Fig.10). Such a processing method results in a lower accuracy of frequency offset estimation (the estimation error is approximately 15%) and larger root mean square error (approximately 40 ns). The dynamic model will be introduced into the proposed data processing procedure to constrain the variation in the kinematic parameters, which will certainly reduce these errors. Moreover, the standard deviation of the time comparison is at the ns-level (Fig.11, Fig.12), indicating that the precision of our measurement model reaches this level.

In this study, a calculation error for the relativistic shift of approximately 1.1×10^-16 was obtained by neglecting the 1/c⁴ term and influence of the gravitational field of celestial bodies, other than that of the sun, earth, and moon. Consequently, the accuracy of time transformation between the proper time and TCB is better than 3.5 ns within one year, and the calculation error of the Shapiro delay is less than 10 ps when only considering the gravitational fields of the sun and earth. Moreover, the light time model of the round-trip is deduced and the model error caused by neglecting the 1/c⁴ term is less than 20 ps. The clock error time histories are also generated from Allan variance or Hadamard variance profiles using two different methods, the power-law spectral density model and Kalman filter state function. Although the calculation time for the former is equal to approximately 10% of the latter, the noise characteristics generated by the latter are closer to the given index. The feasibility of the ns-level uncertainty of the proposed measurement model for use in the high-accuracy laser time comparison missions in Earth-Moon space is also verified. The proposed time transformations, clock error model, and light time model can also be used in other scenarios such as in the development of a high-precision and accurate measurement model for microwave time transfer in Earth-Moon space, and in laser asynchronous transponders in Earth-Moon space.