  Fragmentation of protoplanetary disks due to gravitational instabilities is a
candidate of a formation mechanism of binary stars, brown dwarfs, and gaseous
giant planets. The condition for the fragmentation has been thought that the
disk cooling timescale is comparable to its dynamical timescale. However, some
numerical simulations suggest that the fragmentation does not occur even if the
cooling time is small enough, or the fragmentation can occur even when the
cooling is inefficient. To reveal a realistic condition for fragmentation of
self-gravitating disks, we perform two-dimensional numerical simulations that
take into account the effect of the irradiation of the central star and
radiation cooling of the disk, and precisely investigate the structure of the
spiral arms formed in the protoplanetary disks. We show that the Toomre Q
parameter in the spiral arms is an essential parameter for fragmentation. The
spiral arms fragment only when Q < 0.6 in the spiral arms. We have further
confirmed that this fragmentation condition observed in the numerical
simulations can be obtained from the linear stability analysis for the
self-gravitating spiral arms. These results indicate that the process of
fragmentation of protoplanetary disks is divided into two stages: formation of
the spiral arms in the disks; and fragmentation of the spiral arm. Our work
reduces the condition for the fragmentation of the protoplanetary disks to the
condition of the formation of the spiral arm that satisfies Q < 0.6.
