The Thermal Regulation of Gravitational Instabilities in Protoplanetary Disks. II. Extended Simulations with Varied Cooling Rates

Abstract

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, Rdisk = 40 AU, Mdisk = 0.07 M☉, M* = 0.5 M☉, and Qmin = 1.5. Sustained cooling, with tcool = 2 ORPs (outer rotation periods; 1 ORP ≈ 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 Σ(r), and a roughly steady ≈ 5 × 10-7 M☉ yr-1. The transport is driven by global low-order spiral modes. We restart the calculation at 11.2 ORPs with tcool = 1 and ORPs. The latter case is also run at high azimuthal resolution. We find that shorter cooling times lead to increased -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 tcool. Fragmentation occurs only in the high-resolution tcool = 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.