The thermal regulation of gravitational instabilities in protoplanetary disks.: II.: Extended simulations with varied cooling rates

In order to investigate mass transport and planet formation through gravitational instabilities (GIs), we have extended our three-dimensional hydrodynamic simulations of protoplanetary disks from a previous paper. Our goal is to determine the asymptotic behavior of GIs and how it is affected by different constant cooling times. Initially, R-disk 40 AU, M-disk 0.07 M-circle dot, M-* = 0.5 M-circle dot, and Q(min) 1.5. Sustained cooling, with t(cool) = 2 ORPs (outer rotation periods; 1 ORP approximate to 250 yr), drives the disk to instability in about 4 ORPs. This calculation is followed for 23.5 ORPs. After 12 ORPs, the disk settles into a quasi-steady state with sustained nonlinear instabilities, an average Q = 1.44 over the outer disk, a well-defined power law Sigma(r), and a roughly steady (M)over dot approximate to 5 x 10(-7) M-circle dot yr(-1). The transport is driven by global low-order spiral modes. We restart the calculation at 11.2 ORPs with t(cool) 1 and 1/4 ORPs. The latter case is also run at high azimuthal resolution. We find that shorter cooling times lead to increased (M)over dot-values, denser and thinner spiral structures, and more violent dynamic behavior. The asymptotic total internal energy and the azimuthally averaged Q(r) are insensitive to t(cool). Fragmentation occurs only in the high-resolution t(cool) 1/4 ORP case; however, none of the fragments survive for even a quarter of an orbit. Ringlike density enhancements appear and grow near the boundary between GI-active and GI-inactive regions. We discuss the possible implications of these rings for gas giant planet formation.