A simple theory of accretion onto magnetized neutron stars is discussed. Particularly, the maximally possible luminosity is calculated as a function of the neutron stars magnetic field and spin period. It is shown that luminosity can reach values of the order of 10⁴⁰ erg/s and even higher for the magnetar-like magnetic fields (B>10¹⁴ G) and long spin periods (P>1.5 sec). The relative narrowness of an area of feasible neutron stars parameters which are able to provide higher luminosities leads to the conclusion that 10⁴⁰ erg/s is a good estimate for the limiting accretion luminosity of neutron stars. From the facts that this value (i) can be reached by accretion onto the neutron stars, and (ii) coincides the cut-off value observed in the HMXBs luminosity function, and that (iii) HMXBs luminosity function does not show any features at lower luminosities, we are making conclusion that the substantial part of ultra-luminous X-ray sources (ULXs) are accreting neutron stars in binary systems.

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