Model on picometer-level light gravitational delay in the GRACE Follow-On-like missions

Laser interferometry plays a crucial role in laser ranging for high-precision space missions such as GRACE (Gravity Recovery and Climate Experiment) Follow-On-like missions and gravitational wave detectors. For such accuracy of modern space missions, a precise relativistic model of light propagation is required. With the post-Newtonian approximation, we utilize the Synge world function method to study the light propagation in the Earth's gravitational field, deriving the gravitational delays up to order c^‑4. Then, we investigate the influences of gravitational delays in three inter-satellite laser ranging techniques, including one-way ranging, dual one-way ranging, and transponder-based ranging. By combining the parameters of Kepler orbit, the gravitational delays are expanded up to the order of e² (e is the orbital eccentricity). Finally, considering the GRACE Follow-On-like missions, we estimate the gravitational delays to the level of picometer. The results demonstrate some high-order gravitational and coupling effects, such as c^‑4-order gravitational delays and coupling of Shapiro and beat frequency, which may be non-negligible for higher precision laser ranging in the future.