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We perform both timing and spectral analyses using the archival X-ray datataken with Swift, XMM-Newton, NICER, and NuSTAR from 2016 to 2020 to study anultra-luminous pulsar, NGC 7793 P13, that showed a long period ofsuper-Eddington accretion. We use the Rayleigh test to investigate thepulsation at different epochs, and confirm the variation of the pulse profilewith the finite Gaussian mixture modelling and two-sample Kuiper test. Takinginto account the periodic variation of the spin periods caused by the orbitalDoppler effect, we further determine an orbital period of ~65 d and show thatno significant correlation can be detected between the orbital phase and thepulsed fraction. The pulsed spectrum of NGC7793 P13 in 0.5-20keV can be simplydescribed using a power-law with a high energy exponential cutoff, while thebroad-band phase-averaged spectrum of the same energy range requires twoadditional components to account for the contribution of a thermal accretiondisk and the Comptonization photons scattered into the hard X-rays. We findthat NGC 7793 P13 stayed in the hard ultra-luminous state and the pulsedspectrum was relatively soft when the source was faint in the end of 2019.Moreover, an absorption feature close to 1.3 keV is marginally detected fromthe pulsed spectra and it is possibly associated with a cyclotron resonantscattering feature.

Investigation of the timing and spectral properties of an Ultra-luminous X-ray pulsar NGC 7793 P13