The Jeans Condition: A New Constraint on Spatial Resolution in Simulations of Isothermal Self-gravitational Hydrodynamics

We demonstrate with a new three-dimensional adaptive mesh refinement code that perturbations arising from discretization of the equations of self-gravitational hydrodynamics can grow into fragments in multiple-grid simulations, a process we term "artificial fragmentation." We present star formation calculations of isothermal collapse of dense molecular cloud cores. In simulation of a Gaussian-profile cloud free of applied perturbations, we find artificial fragmentation can be avoided across the isothermal density regime by ensuring the ratio of cell size to Jeans length, which we call the Jeans number, J ≡ Δx/λJ, is kept below 0.25. We refer to the constraint that λJ be resolved as the Jeans condition. When an m=2 perturbation is included, we again find it necessary to keep J≤0.25 to achieve a converged morphology. Collapse to a filamentary singularity occurs without fragmentation of the filament, in agreement with the predictions of Inutsuka & Miyama. Simulation beyond the time of this singularity requires an arresting agent to slow the runaway density growth. Physically, the breakdown of isothermality due to the buildup of opacity acts as this agent, but many published calculations have instead used artificial viscosity for this purpose. Because artificial viscosity is resolution dependent, such calculations produce resolution-dependent results. In the context of the perturbed Gaussian cloud, we show that use of artificial viscosity to arrest collapse results in significant violation of the Jeans condition. We also show that if the applied perturbation is removed from such a calculation, numerical fluctuations grow to produce substantial fragments not unlike those found when the perturbation is included. These findings indicate that calculations that employ artificial viscosity to halt collapse are susceptible to contamination by artificial fragmentation. The Jeans condition has important implications for numerical studies of isothermal self-gravitational hydrodynamics problems insofar as it is a necessary but not, in general, sufficient condition for convergence.