Comparing the super-Eddington accretion of SMC X-3 and RX J0209.6-7427 with Swift J0243.6+6124

We study the giant outbursts of SMC X-3 and RX J0209.6-7427 (hereafter RX J0209) to compare their super-Eddington accretion regime with that of Swift J0243.6+6124 (hereafter Swift J0243). The high double-peak profile of SMC X-3 is found to be 0.25 phase offset from that below 2.3 × 10³⁸ erg s^-1, which is similar to Swift J0243 (happened around 0.9 × 10³⁸ erg s^-1). The profile of RX J0209 shows a similar 0.25 phase offset between high double-peak and low double-peak around 1.25 × 10³⁸ erg s^-1. The 0.25 phase offset corresponds to a 90° angle change of the emission beam and strongly supports for a transition from a fan beam to a pencil beam. Their critical luminosities imply a surface magnetic field ~4 × 10¹³ and 2 × 10¹³ G for SMC X-3 and RX J0209, respectively, based on the recently measured cyclotron line of Swift J0243. The spin-up rate and luminosity of SMC X-3 follows a relation of $\dot{\nu }\propto L^{0.94\pm 0.03}$, while that of RX J0209 follows $\dot{\nu }\propto L^{1.00\pm 0.03}$, which are similar to Swift J0243 and consistent with the prediction of a radiation-pressure-dominated disc. These results indicate that accretion columns are indeed formed above Eddington luminosity, and the population of ultraluminous X-ray pulsars likely corresponds to X-ray pulsars of highest magnetic field.