Gravitational stability and fragmentation condition for discs around
  accreting supermassive stars
  Supermassive stars (SMSs) with mass $\sim10^{5}~\rm{M}_{\odot}$ are promising
candidates for the origin of supermassive black holes observed at redshift
$\gtrsim6$. 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
circum-stellar 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
$10-10^5~\rm{M}_{\odot}$ and the accretion rate
$10^{-3}-1~\rm{M}_{\odot}~\rm{yr}^{-1}$. 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
$10^{-1}~\rm{M}_{\odot}~\rm{yr}^{-1}$ 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 appears to
be marginally satisfied according to numerical simulations, suggesting that
disc fragmentation can be common during SMS formation.
