Recent observations of star-forming regions suggest that binary and multiple young stars are the rule rather than the exception and implicate fragmentation as the likely mechanism for their formation. Most numerical hydrodynamic calculations of fragmentation have neglected the possibly deleterious effects of magnetic fields, despite ample evidence for the importance of magnetic support of precollapse clouds. We present here the first numerical hydrodynamic survey of the collapse and fragmentation of initially magnetically supported clouds that takes into account several magnetic field effects in an approximate manner. The models are calculated with a three-dimensional, finite differences code that solves the equations of hydrodynamics, gravitation, and radiative transfer in the Eddington and diffusion approximations. Magnetic field effects are included through two simple approximations: magnetic pressure is added to the gas pressure, and magnetic tension is approximated by gravity dilution once collapse is well underway. Ambipolar diffusion of the magnetic field leading to cloud collapse is treated approximately as well. Models are calculated for a variety of initial cloud density profiles, shapes, and rotation rates. We find that in spite of the inclusion of magnetic field effects, dense cloud cores are capable of fragmenting into binary and multiple protostar systems. Initially prolate clouds tend to fragment into binary protostars, while initially oblate clouds tend to fragment into multiple protostar systems containing a small number (of the order of 4) of fragments. The latter are likely to be subject to rapid orbital evolution, with close encounters possibly leading to the ejection of fragments. Contrary to expectation, magnetic tension effects appear to enhance fragmentation, allowing lower mass fragments to form than would otherwise be possible, because magnetic tension helps to prevent a central density singularity from forming and producing a dominant single object. Magnetically supported dense cloud cores thus seem to be capable of collapsing and fragmenting into sufficient numbers of binary and multiple protostar systems to be compatible with observations of the relative rarity of single protostars.
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