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We explore the role of the modified Eddington limit due to rapid rotation(the so-called $\Omega\Gamma-$limit) in the formation of Population III stars.We performed one-dimensional stellar evolution simulations of mass-accretingzero-metallicity protostars at a very high rate ($\dot{M} \sim10^{-3}~\mathrm{M_\odot~yr^{-1}}$) and dealt with stellar rotation as aseparate post-process. The protostar would reach the Keplerian rotation verysoon after the onset of mass accretion, but mass accretion would continue asstellar angular momentum is transferred outward to the accretion disk byviscous stress. The protostar envelope expands rapidly when the stellar massreaches $5\sim7~\mathrm{M_\odot}$ and the Eddington factor sharply increases.This makes the protostar rotate critically at a rate that is significantlybelow the Keplerian value (i.e., the $\Omega\Gamma-$limit). The resultantpositive gradient of the angluar velocity in the boundary layer between theprotostar and the Keplerian disk prohibits angular momentum transport from thestar to the disk, and consequently further rapid mass accretion. This wouldprevent the protostar from growing significantly beyond $20 -40~\mathrm{M_\odot}$. Another important consequence of the $\Omega\Gamma-$limitis that the protostar can remain fairly compact ($R \lesssim50~\mathrm{R_\odot}$) and avoid a fluffy structure ($R \gtrsim500~\mathrm{R_\odot}$) that is usually found with a very high mass accretionrate. This effect would make the protostar less prone to binary interactionsduring the protostar phase. Although our analysis is based on Pop III protostarmodels, this role of the $\Omega\Gamma-$limit would be universal in theformation process of massive stars, regardless of metallicity.

On the Role of the $\Omega\Gamma$ Limit in the Formation of Population III Massive Stars