Gravitational instability in protostellar discs at low metallicities

Fragmentation of protostellar discs controls the growth of protostars and plays a key role in determining the final mass of newborn stars. In this paper, we investigate the structure and gravitational stability of the protostellar discs in the full metallicity range between zero and the solar value. Using the mass-accretion rates evaluated from the thermal evolution in the preceding collapse phase of the pre-stellar cores, we calculate disc structures and their evolution in the framework of the standard steady discs. Overall, with higher metallicity, more efficient cooling results in the lower accretion rate and lower temperature inside the disc: at zero metallicity, the accretion rate is ̃10^-3M_⊙yr^-1 and the disc temperature is ̃1000 K, while at solar metallicity, ̃10-6M_⊙ yr^-1 and ̃10 K. Despite the large difference in these values, the zero- and solar-metallicity discs have similar stability properties: the Toomre parameter for the gravitational stability,which can be written using the ratio of temperatures in the disc and in the envelope as QT ̃ (T_disc/T_env)^3/2, is ≤ 1, i.e. marginally stable. At intermediate metallicities of 10^-5 to 10^-3 M_⊙, however, the discs are found to be strongly unstable with QT ̃0.1-1 since dust cooling, which is effective only in the discs due to their high density (≤ 10¹⁰ cm^-3), makes the temperature in the discs lower than that in the envelopes. This indicates that masses of the individual stars formed as a result of the protostellar disc fragmentation can be significantly smaller than their parent core in this metallicity range. The typical stellar mass in this case would be a few M_⊙, which is consistent with the observationally suggested mass-scale of extremely metal-poor stars.