Pulsational instability of supergiant protostars: Do they grow supermassive by accretion?

Supermassive stars (SMSs; M_* ≥ 10⁵M_ȯ) and their remnant black holes are promising progenitors for supermassive black holes (SMBHs) observed in the early universe at z ≥ 7. It has been postulated that SMSs form through very rapid mass accretion on to a protostar at a high rate exceeding 0.01M_ȯ yr^-1. According to recent studies, such rapidly accreting protostars evolve into 'supergiant protostars', i.e. protostars consisting of a bloated envelope and a contracting core, similar to giant star. However, like massive stars as well as giant stars, both of which are known to be pulsationally unstable, supergiant protostars may also be unstable to launch strong pulsation-driven outflows. If this is the case, the stellar growth via accretion will be hindered by the mass-loss. We here study the pulsational stability of the supergiant protostars in the mass range M_*≤ (10³M_ȯ through the method of the linear perturbation analysis. We find that the supergiant protostars with M_* ≥ 600M_ȯ and very high accretion rate ̇ M_acc ≥ 1.0M_ȯ yr^-1 are unstable due to the κ mechanism. The pulsation is excited in the He+ ionization layer in the envelope. Even under a conservative assumption that all the pulsation energy is converted into the kinetic energy of the outflows, the massloss rate is ~10^-3M_ȯ yr^-1, which is lower than the accretion rate by more than two orders of magnitude. We thus conclude that the supergiant protostars should grow stably via rapid accretion at least in the mass range we studied. As long as the rapid accretion is maintained in the later stage, protostars will become SMSs, which eventually produce seeds for the high-z SMBHs.