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The discovery of pulsating ultra-luminous X-ray sources (PULX) suggests thatneutron stars are presumably common within the ultra-luminous X-ray source(ULX) population though the majority of the population members are currentlylacking pulsations. These systems are likely to host neutron stars accretingmass at super-Eddington (super-critical) rates from their massive companion inhigh-mass X-ray binaries. Taking into account the spherization of the accretionflow in the super-critical regime, the beaming of X-ray emission, and thereduction of the scattering cross-section in a strong magnetic field, we inferthe ranges for the neutron-star surface magnetic dipole field strengths,beaming fractions, and fastness parameters in the PULX M82 X-2, ULX NGC 5907,ULX NGC 7793 P13, NGC 300 ULX1, M51 ULX-7, NGC 1313 X-2, and Swift J0243.6+6124from a set of conditions based on a variety of combinations of different spinand luminosity states. Using the observed spin-up rates under the criticalluminosity condition, we estimate the surface-field strengths in the $\sim10^{11}-10^{13}\,{\rm G}$ range for all PULX. In general, the results of ouranalysis under the subcritical luminosity condition indicate surface-fieldstrengths in the $\sim 10^{11}-10^{15}\,{\rm G}$ range. We argue that the PULXdo not require magnetar-strength surface dipole fields if beaming is taken intoaccount; yet the fields are strong enough for the neutron stars in ULX tomagnetically channel the accretion flow in super-critical accretion disks.

On the Magnetic Fields, Beaming Fractions, and Fastness Parameters of Pulsating Ultra-Luminous X-Ray Sources