In-orbit timing calibration of the Insight-Hard X-ray Modulation Telescope

We describe the timing system and the timing calibration results of the three payloads onboard the Insight-Hard X-ray Modulation Telescope (Insight-HXMT). The three payloads are the High Energy X-ray telescope (HE, 20-250 keV), the Medium Energy X-ray telescope (ME, 5-30 keV) and the low Energy X-ray telescope (LE, 1-10 keV). We present a method to correct the temperature-dependent period response and the long-term variation of the onboard crystal oscillator, especially for ME which does not carry a temperature-compensated crystal oscillator. The time of arrivals (ToAs) of the Crab pulsar are measured to evaluate the accuracy of the timing system. In the case that the Crab ephemeris of radio Jodrell Bank has 40 {\mu}s systematic error (Rots et al. 2004), we use the data observed by X-ray Timing Instrument (XTI) onboard the Neutron star Interior Composition Explorer (NICER) to produce the Crab ephemerides. The energy-dependent ToAs' offsets relative to NICER measurements including physical and instrumental origins are about 24.7 {\mu}s, 10.1 {\mu}s and 864.7 {\mu}s for HE, ME and LE respectively. The systematic errors of the timing system, which indicates a good timing performance, are determined as 12.1 {\mu}s, 8.6 {\mu}s, and 15.8 {\mu}s for HE, ME and LE respectively.