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 on-board the Insight-Hard X-ray Modulation Telescope (Insight-HXMT). These 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 on-board crystal oscillator, especially for ME that 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. As the ephemeris of the Crab pulsar given by Jodrell Bank observatory has systematic errors around 40 {\mu}s (Rots et al. 2014), we use the quasi-simultaneous observations of the X-ray Timing Instrument (XTI) on-board the Neutron star Interior Composition Explorer (NICER) to produce the Crab ephemerides and to verify the timing system of Insight-HXMT. The energy-dependent ToAs' offsets relative to the NICER measurements including physical and instrumental origins are about 24.7{\mu}s, 10.1{\mu}s and 864.7{\mu}s, and the systematic errors of the timing system are determined as 12.1{\mu}s, 8.6{\mu}s, and 15.8{\mu}s, for HE, ME and LE respectively.