Qi Liu, Lingda Kong, Can Güngör, Lorenzo Ducci, Long Ji, Wei Wang, Xiaohang Dai, Andrea Santangelo

We present the results of the spectral and timing analyses of the accreting X-ray pulsar, 2S 1417-624, during the 2018 and 2021 outbursts with Insight-HXMT. We find that the pulse profiles in all energy bands exhibit clear double-peaked structures at low flux states. In the 1-10 keV band, the pulse profiles evolve from double to triple peaks at a flux level of $\sim$4.1$\ \times \ 10^{-9}$ erg cm$^{-2}$ s$^{-1}$, and from triple to quadruple peaks at $\sim$6.4$\ \times \ 10^{-9}$ erg cm$^{-2}$ s$^{-1}$. In the 10-30 keV and 30-100 keV bands, the pulse profiles become narrower at the first transition flux level, followed by a stark transition to quadruple-peaked and triple-peaked structures around the second flux level, respectively. The change of the pulse profile {during the second transition} reveals the transition of the emission pattern from the sub-critical (pencil beam) to the supercritical (fan beam) regime. By performing the binary orbital fitting of the observed spin periods, we provide new measurements of the orbital parameters from the 2021 outburst. Applying different accretion torque models and using the critical luminosity inferred from the pulse profile transitions, we derive a self-consistent distance of {2S 1417-624} in the range of approximately 12.0-15.0~kpc, based on the magnetic field strength derived from the cyclotron resonance scattering feature (CRSF). From the estimated distance of 13 kpc and Gaia's distance of 7.4 kpc, we can infer the observed transition luminosity of \((1.0-1.4) \times 10^{38} \, \mathrm{erg \, s^{-1}}\) and \((3.0-5.0) \times 10^{37} \, \mathrm{erg \, s^{-1}}\), respectively, and compare them with theoretical models. The spectral continuum parameters and the hardness ratio also show significant transitions around the second transition, strongly supporting a change in the accretion regime.