Establishing a reliable determination of the final mass for rapidly accreting supermassive stars

Context. The formation of supermassive black holes possibly takes place via direct collapse, with a supermassive star (SMS) as its progenitor. In this scenario, the SMS accretes at > 0.1 M⊙ yr−1 until it collapses into a massive black hole seed as a result of the general-relativistic (GR) instability. However, the exact mass at which the collapse occurs is not known, as existing numerical simulations give us a divergent range of results. Aims. Here, we address this problem analytically, which allows for reliable ab initio determination of the onset point of the GR instability, for given hydrostatic structures. Methods. We applied the relativistic equation of radial pulsations in its general form to the hydrostatic GENE