Effects of the Isothermal Region in Protoplanetary Disks and the Protostar Irradiation on the Disk Instability Model for Giant Planet Formation

We construct an analytical model of gravitationally unstable protoplanetary disks consisting of three regions: the inner region where the internal dissipation dominates the heating, the intermediate region where the central protostar irradiation dominates, and the outer region where background irradiation dominates. We use this analytical model and an evolutionary numerical model of protoplanetary disks to calculate the cooling time and find out the location of the isothermal region. We investigate the effects of the isothermal region on the disk instability model for giant planet formation. We find that the fragmentation region found in previous studies is contained in the isothermal region of a disk. In this case, the cooling time criterion is not applicable for fragmentation. Therefore, the constraint on the disk instability model caused by the cooling time criterion should be relieved. The viability of the disk instability model is improved. When the isothermal region is considered, the inner boundary of the fragmentation region is extended inward to similar to 20 au. We also show that if the contribution of the protostar irradiation to the disk surface temperature can be included in the cooling rate, the fragmentation region defined by the cooling time criterion can be extended inward to similar to 26 au. We find that a disk tends to be isothermal in the region where the cooling time criterion is satisfied. We also find that at the later stage of disk instability, the inner boundary of the fragmentation region is determined by the inner boundary of the gravitationally unstable region.