Supermassive primordial stars are suspected to be the progenitors of the most massive quasars at z \sim 6 .
Previous studies of such stars were either unable to resolve hydrodynamical timescales or considered stars in isolation , not in the extreme accretion flows in which they actually form .
Therefore , they could not self-consistently predict their final masses at collapse , or those of the resulting supermassive black hole seeds , but rather invoked comparison to simple polytropic models .
Here , we systematically examine the birth , evolution and collapse of accreting non-rotating supermassive stars under accretion rates of 0.01 - 10 { { \mathrm { M } _ { \odot } } { \mathrm { yr } } ^ { -1 } } using the stellar evolution code Kepler .
Our approach includes post-Newtonian corrections to the stellar structure and an adaptive nuclear network , and can transition to following the hydrodynamic evolution of supermassive stars after they encounter the general relativistic instability .
We find that this instability triggers the collapse of the star at masses of 150 , 000 - 330 , 000 { \mathrm { M } _ { \odot } } for accretion rates of 0.1 - 10 { { \mathrm { M } _ { \odot } } { \mathrm { yr } } ^ { -1 } } , and that the final mass of the star scales roughly logarithmically with the rate .
The structure of the star , and thus its stability against collapse , is sensitive to the treatment of convection , and the heat content of the outer accreted envelope .
Comparison with other codes suggests differences here may lead to small deviations in the evolutionary state of the star as a function of time , that worsen with accretion rate .
Since the general relativistic instability leads to the immediate death of these stars , our models place an upper limit on the masses of the first quasars at birth .