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We study properties of luminous X-ray pulsars using a simplified model of theaccretion column. The maximal possible luminosity is calculated as a functionof the neutron star (NS) magnetic field and spin period. It is shown that theluminosity can reach values of the order of $10^{40}\,{\rm erg/s}$ for themagnetar-like magnetic field ($B\gtrsim 10^{14}\,{\rm G}$) and long spinperiods ($P\gtrsim 1.5\,{\rm s}$). The relative narrowness of an area offeasible NS parameters which are able to provide higher luminosities leads tothe conclusion that $L\simeq 10^{40}\,\,{\rm erg/s}$ is a good estimate for thelimiting accretion luminosity of a NS. Because this luminosity coincides withthe cut-off observed in the high mass X-ray binaries luminosity function whichotherwise does not show any features at lower luminosities, we can concludethat a substantial part of ultra-luminous X-ray sources are accreting neutronstars in binary systems.

On the maximum accretion luminosity of magnetized neutron stars: connecting X-ray pulsars and ultraluminous X-ray sources