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

Abstract

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-accreting zero-metallicity protostars at a very high rate ($\dot{M} \sim 10^{-3}~\mathrm{M_\odot~yr^{-1}}$) and dealt with stellar rotation as a separate post-process. The protostar would reach the Keplerian rotation very soon after the onset of mass accretion, but mass accretion would continue as stellar angular momentum is transferred outward to the accretion disk by viscous stress. The protostar envelope expands rapidly when the stellar mass reaches $5\sim7~\mathrm{M_\odot}$ and the Eddington factor sharply increases. This makes the protostar rotate critically at a rate that is significantly below the Keplerian value (i.e., the $\Omega\Gamma-$limit). The resultant positive gradient of the angluar velocity in the boundary layer between the protostar and the Keplerian disk prohibits angular momentum transport from the star to the disk, and consequently further rapid mass accretion. This would prevent the protostar from growing significantly beyond $20 - 40~\mathrm{M_\odot}$. Another important consequence of the $\Omega\Gamma-$limit is that the protostar can remain fairly compact ($R \lesssim 50~\mathrm{R_\odot}$) and avoid a fluffy structure ($R \gtrsim 500~\mathrm{R_\odot}$) that is usually found with a very high mass accretion rate. This effect would make the protostar less prone to binary interactions during the protostar phase. Although our analysis is based on Pop III protostar models, this role of the $\Omega\Gamma-$limit would be universal in the formation process of massive stars, regardless of metallicity.