The Jeans Mass Constraint and the Fragmentation of Molecular Cloud Cores

The collapse and fragmentation of molecular cloud cores into binary and multiple protostar systems is a demanding computational problem, in part because of the large range of length scales involved. Truelove et al. proposed that the computational cell size Δx must be smaller than 1/4 of the Jeans length, λ_J, if artificial (numerical) fragmentation is to be avoided. For a uniform Cartesian grid, Truelove et al.'s Jeans condition is equivalent to ensuring that the mass of a cell never exceeds ~1/64 of a Jeans mass. It is shown here that for a nonuniform spherical grid, artificial fragmentation can be avoided provided that the cell size of a cube with approximately the same volume as the spherical coordinate cell [Δx=(Δx_rΔx_θΔx_ϕ)^1/3] is less than λ_J/4 (i.e., that the mass inside each cell is much less than a Jeans mass), even if one of the three cell lengths (Δx_r, Δx_θ, or Δx_ϕ) exceeds λ_J/4. For a nonuniform grid, resolving a small fraction of a Jeans mass is less restrictive than resolving 1/4 of a Jeans length in each coordinate direction; resolving all three Jeans lengths is desirable but not necessary in order to avoid artificial fragmentation. The well-resolved collapse of an initially Gaussian-profile cloud is then followed with both isothermal and nonisothermal (Eddington approximation radiative transfer) thermodynamics and is shown to lead to fragmentation into a binary protostar system in both cases. The Jeans mass constraint appears to be a valuable indicator of physically realistic fragmentation.