The effect of pressure confinement on the maximum mass of rapidly accreting supermassive stars
We derive an extension of Chandrasekhar's pulsational stability criterion for relativistic stars with a pressure boundary condition at the surface. We then apply the new criterion to hylotropes, a model describing rapidly accreting supermassive stars with a core-envelope structure. We find that confinement typically stabilises stars, delaying the onset of the general relativistic instability when the external pressure at the surface is roughly $\sim10^{-3}$ of the stars' hydrostatic pressure scale. Beyond this value, the critical mass at marginal stability increases rapidly, exceeding the value of $\sim 10^5$ M$_{\odot}$ for helium burning, isolated $n=3$ polytropes by orders of magnitude. We discuss how the required pressure boundary conditions may be produced by the combined ram and magnetic pressure supplied by the accretion flow as it assembles the supermassive star, and provide a fitting formula for the final mass before collapse. As a rough estimate assuming equipartition between magnetic and thermal pressures in a standard $α$-disk, we find that the critical mass doubles at accretion rates of order $\dot M\sim100\,{\rm M_\odot}\,{\rm yr}^{-1}$, again increasing rapidly after this value. Our results show under what conditions, potentially realized in a subset of high redshift halos, non-rotating SMS may collapse into black holes beyond the typical scale of heavy seeds.