Calibration of the Timing Performance of GECAM-C

As a new member of the Gravitational-wave high-energy Electromagnetic Counterpart All-sky Monitor (GECAM) after GECAM-A and GECAM-B, GECAM-C (originally called HEBS), which was launched on board the SATech-01 satellite on 2022 July 27, aims to monitor and localize X-ray and gamma-ray transients from similar to 6 keV to 6 MeV. GECAM-C utilizes a similar design to GECAM but operates in a more complex orbital environment. In this work, we utilize the secondary particles simultaneously produced by the cosmic-ray events on orbit and recorded by multiple detectors to calibrate the relative timing accuracy between all detectors of GECAM-C. We find the result is 0.1 mu s, which is the highest time resolution among all GRB detectors ever flown and very helpful in timing analyses such as minimum variable timescale and spectral lags, as well as in time delay localization. Besides this, we calibrate the absolute time accuracy using the one-year Crab Pulsar data observed by GECAM-C and Fermi's Gamma-ray Burst Monitor, as well as GECAM-C and GECAM-B. The results are 2.02 +/- 2.26 mu s and 5.82 +/- 3.59 mu s, respectively. Finally, we investigate the spectral lag between the different energy bands of the Crab Pulsar observed by GECAM and the Gamma-ray Burst Monitor, which is similar to -0.2 mu s keV-1.