The Jeans condition and collapsing molecular cloud cores: Filaments or binaries?

The 1997 and 1998 studies by Truelove and colleagues introduced the Jeans condition as a necessary condition for avoiding artificial fragmentation during protostellar collapse calculations. They found that when the Jeans condition was properly satisfied with their adaptive mesh refinement (AMR) code, an isothermal cloud with an initial Gaussian density profile collapsed to form a thin filament rather than the binary or quadruple protostar systems found in previous calculations. Using a completely different self-gravitational hydrodynamics code introduced by Boss & Myhill in 1992 (B&M), we present here calculations that reproduce the filamentary solution first obtained by Truelove et al. in 1997. The filamentary solution only emerged with very high spatial resolution with the B&M code, with effectively 12,500 radial grid points (R-12500). Reproducing the filamentary collapse solution appears to be an excellent means for testing the reliability of self-gravitational hydrodynamics codes, whether grid-based or particle-based. We then show that in the more physically realistic case of an identical initial cloud with nonisothermal heating (calculated in the Eddington approximation with code B&M), thermal retardation of the collapse permits the Gaussian cloud to fragment into a binary protostar system at the same maximum density where the isothermal collapse yields a thin filament. However, the binary clumps soon thereafter evolve into a central clump surrounded by spiral arms containing two more clumps. A roughly similar evolution is obtained using the AMR code with a barotropic equation of state--formation of a transient binary, followed by decay of the binary to form a central object surrounded by spiral arms, though in this case the spiral arms do not form clumps. When the same barotropic equation of state is used with the B&M code, the agreement with the initial phases of the AMR calculation is quite good, showing that these two codes yield mutually consistent results. However, the B&M barotropic result differs significantly from the B&M Eddington result at the same maximum density, demonstrating the importance of detailed radiative transfer effects. Finally, we confirm that even in the case of isothermal collapse, an initially uniform density sphere can collapse and fragment into a binary system, in agreement with the 1998 results of Truelove et al. Fragmentation of molecular cloud cores thus appears to remain as a likely explanation of the formation of binary stars, but the sensitivity of these calculations to the numerical resolution and to the thermodynamical treatment demonstrates the need for considerable caution in computing and interpreting three-dimensional protostellar collapse calculations.