Convergence Studies of Mass Transport in Disks With Gravitational Instabilities. Ii. The Radiative Cooling Case

Abstract

We conduct a convergence study of a protoplanetary disk subject to gravitational instabilities (GIs) at a time of approximate balance between heating produced by the GIs and radiative cooling governed by realistic dust opacities. We examine cooling times, characterize GI-driven spiral waves and their resultant gravitational torques, and evaluate how accurately mass transport can be represented by an α-disk formulation. Four simulations, identical except for azimuthal resolution, are conducted with a grid-based three-dimensional hydrodynamics code. There are two regions in which behaviors differ as resolution increases. The inner region, which contains 75% of the disk mass and is optically thick, has long cooling times and is well converged in terms of various measures of structure and mass transport for the three highest resolutions. The longest cooling times coincide with radii where the Toomre Q has its minimum value. Torques are dominated in this region by two-and three-armed spirals. The effective α arising from gravitational stresses is typically a few ×10 −3 and is only roughly consistent with local balance of heating and cooling when time-averaged over many dynamic times and a wide range of radii. On the other hand, the outer disk region, which is mostly optically thin, has relatively short cooling times and does not show convergence as resolution increases. Treatment of unstable disks with optical depths near unity with realistic radiative transport is a difficult numerical problem requiring further study. We discuss possible implications of our results for numerical convergence of fragmentation criteria in disk simulations.

Physical Processes in Protoplanetary Disks

“…The transport could be largely non-local, driven for example by large-scale structures in the density field (such as bars) or by waves that transport energy and angular momentum a significant distance before dissipating [47]. There is no precise criterion for when self-gravitating transport can be described using a local theory, but numerical simulations indicate that this is a reasonable approximation for low-mass disks with M disk /M * ≈ 0.1 [256,103,136,387]. Transport in more massive disks, such as might be present during the Class 0 and Class I phases of star formation, cannot be described locally (for multiple reasons, e.g.…”

Gravitoturbulence and the excitation of small-scale parametric instability in astrophysical discs

“…Previous grid-based global simulations can compete on vertical resolution, but suffer from poor azimuthal resolution especially (e.g. Michael et al 2012;Steiman-Cameron et al 2013). The radial boundary conditions in global simulations impose further complications that muddy their interpretation.…”

Transport and Accretion in Planet-Forming Disks

“…Pioneering simulations (Fromang, 2005) need to be revisited at higher numerical resolution; (2) thermodynamically self-consistent disks with realistic opacities and external irradiation. While much work has already been done (Boss, 2007b;Cai et al, 2008;Steiman-Cameron et al, 2013) tracking cooling inside proto-fragments requires very high spatial resolution.…”