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A revised condition for self-gravitational fragmentation of protoplanetary disks

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.