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.
