General-relativistic instability in hylotropic supermassive stars

Context. The formation of supermassive black holes by direct collapse would imply the existence of supermassive stars (SMSs) and their collapse through the general-relativistic (GR) instability into massive black hole seeds. However, the final mass of SMSs is weakly constrained by existing models, in spite of the importance of this value for the consistency of the direct collapse scenario. Aims. We estimate the final masses of spherical SMSs within the whole parameter space that is relevant to these objects. Methods. We built analytical stellar structures (hylotropes) that mimic existing numerical SMS models, accounting for full stellar evolution with rapid accretion. From these hydrostatic structures, we determine ab initio the conditions for GR instability and compare the results with the predictions for full stellar evolution. Results. We show that hylotropic models predict the onset of GR instability with a high level of precision. The mass of the convective core appears as a decisive quantity. The lower it is, the larger the total mass required for GR instability. The typical conditions for GR instability feature a total mass of ≳10 5 M ⊙ with a core mass of ≳10 4 M ⊙ . If the core mass remains below 10 4 M ⊙ , total masses in excess of 10 6 − 10 7 M ⊙ can be reached. Conclusions. Our results confirm that spherical SMSs forming in primordial, atomically cooled haloes collapse at masses below 500 000 M ⊙ . On the other hand, accretion rates in excess of 1000 M ⊙ yr −1 , leading to final stellar masses of ≳10 6 M ⊙ , are required for massive black hole formation in metal-rich gas. Thus, the different channels of direct collapse imply distinct final masses for the progenitor of the black hole seed.