Maximally accreting supermassive stars: a fundamental limit imposed by hydrostatic equilibrium

Major mergers of gas-rich galaxies provide promising conditions for the formation of supermassive black holes (SMBHs; $\gtrsim10^5$ M$_\odot$) by direct collapse because they can trigger mass inflows as high as $10^4-10^5$ M$_\odot$ yr$^{-1}$ on sub-parsec scales. However, the channel of SMBH formation in this case, either dark collapse (direct collapse without prior stellar phase) or supermassive star (SMS; $\gtrsim10^4$ M$_\odot$), remains unknown. Here, we investigate the limit in accretion rate up to which stars can maintain hydrostatic equilibrium. We compute hydrostatic models of SMSs accreting at $1-1000$ M$_\odot$ yr$^{-1}$, and estimate the departures from equilibrium a posteriori by taking into account the finite speed of sound. We find that stars accreting above the atomic cooling limit ($\gtrsim10$ M$_\odot$ yr$^{-1}$) can only maintain hydrostatic equilibrium once they are supermassive. In this case, they evolve adiabatically with a hylotropic structure, that is, entropy is locally conserved and scales with the square root of the mass coordinate. Our results imply that stars can only become supermassive by accretion at the rates of atomically cooled haloes ($\sim0.1-10$ M$_\odot$ yr$^{-1}$). Once they are supermassive, larger rates are possible.