Supermassive stars (SMSs) with mass ∼105 M are promising candidates for the origin of supermassive black holes observed at redshift z 6. They are supposed to form as a result of rapid accretion of primordial gas, although it can be obstructed by the time variation caused by circumstellar disc fragmentation due to gravitational instability. To assess the occurrence of fragmentation, we study the structure of marginally gravitationally unstable accretion discs, by using a steady one-dimensional thin disc model with detailed treatment of chemical and thermal processes. Motivated by two SMS formation scenarios, i.e. those with strong ultraviolet radiation background or with large velocity difference between the baryon and the dark matter, we consider two types of flows, i.e. atomic and molecular flows, respectively, for a wide range of the central stellar mass 10-105 M and the accretion rate 10−3-1 M yr−1. In the case of a mostly atomic gas flowing to the disc outer boundary, the fragmentation condition is expressed as the accretion rate being higher than the critical value of 10−1 M yr−1 regardless of the central stellar mass. On the other hand, in the case of molecular flows, there is a critical disc radius outside of which the disc becomes unstable. Those conditions appear to be marginally satisfied according to numerical simulations, suggesting that disc fragmentation can be common during SMS formation.
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