  Fragmentation of protostellar disks 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
disks 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 disk structures and their
evolution in the framework of the standard steady disks. Overall, with higher
metallicity, more efficient cooling results in the lower accretion rate and
lower temperature inside the disk: at zero metallicity, the accretion rate is ~
1e-3Msun/yr and the disk temperature is ~ 1000 K, while at solar metallicity, ~
1e-6Msun/yr and 10 K. Despite the large difference in these values, the zero-
and solar-metallicity disks have similar stability properties: the Toomre
parameter for the gravitational stability, which can be written using the ratio
of temperatures in the disk and in the envelope as Q ~ (T_disk/T_env)^3/2, is >
1, i.e., marginally stable. At intermediate metallicities of 1e-5--1e-3Zsun,
however, the disks are found to be strongly unstable with Q ~ 0.1--1 since dust
cooling, which is effective only in the disks due to their high density (> 1e10
cm^-3), makes the temperature in the disks lower than that in the envelopes.
This indicates that masses of the individual stars formed as a result of the
protostellar disk fragmentation can be significantly smaller than their parent
core in this metallicity range. The typical stellar mass in this case would be
a few Msun, which is consistent with the observationally suggested mass-scale
of extremely metal-poor stars.
