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We consider whether the maximum mass of first stars is imposed by theprotostellar spin, i.e., by the so-called $\Omega\Gamma$-limit, which requiresthe sum of the radiation and centrifugal forces at the stellar surface besmaller than the inward pull of the gravity. Once the accreting protostarreaches such a marginal state, the star cannot spin up more and is not allowedto accrete more gas with inward angular momentum flux. So far, however, theeffect of stellar radiation on the structure of the accretion disk has not beenproperly taken into account in discussing the effect of $\Omega\Gamma$-limit onthe first star formation. Here, we obtain a series of the steady accretion-disksolutions considering such effect and find solutions without net angularmomentum influx to the stars with arbitrary rotation rates, in addition tothose with finite angular momentum flows. The accretion of positive angularmomentum flows pushes the star beyond the $\Omega\Gamma$-limit, which isallowed only with the external pressure provided by the circumstellar disk. Onthe other hand, the accretion with no net angular momentum influx does notresult in the spin-up of the star. Thus, the existence of the solution with nonet angular momentum influx indicates that the protostars can keep growing inmass by accretion even after they reach the $\Omega\Gamma$-limit.

Primordial protostars accreting beyond the $\Omega\Gamma$-limit: radiation effect around the star-disk boundary