Inferring the origins of the pulsed gamma-ray emission from the Crab pulsar with 10-year Fermi LAT data

Context: The Crab pulsar is a bright $\gamma$-ray source detected at photon energies up to $\sim$1 TeV. Its phase-averaged and phase-resolved $\gamma$-ray spectra both exhibit (sub-)exponential cutoffs while those above 10 GeV apparently follow simple power-laws. Aims: We re-visit the $\gamma$-ray properties of the Crab pulsar with 10-year \emph{Fermi} Large Area Telescope (LAT) data in the range of 60 MeV--500 GeV. With the phase-resolved spectra, we investigate the origins and mechanisms responsible for the emissions. Methods: The phaseograms are reconstructed for different energy bands and further analysed using a wavelet decomposition. The phase-resolved energy spectra are combined with the observations of ground-based instruments like MAGIC and VERITAS to achieve a larger energy converage. We fit power-law models to the overlapping energy spectra from 10 GeV to $\sim$1 TeV. We include in the fit a relative cross-calibration of energy scales between air-shower based gamma-ray telescopes with the orbital pair-production telescope of the Fermi mission. Results: We confirm the energy-dependence of the $\gamma$-ray pulse shape, and equivalently, the phase-dependence of the spectral shape for the Crab pulsar. A relatively sharp exponential cutoff at a relatively high energy of $\sim$8 GeV is observed for the bridge-phase emission. The $E>$10 GeV spectrum observed for the second pulse peak is harder than those for other phases. Conclusions: In view of the diversity of phase-resolved spectral shapes of the Crab pulsar, we tentatively propose a multi-origin scenario where the polar-cap, outer-gap and relativistic-wind regions are involved.