Collapse and fragmentation of molecular cloud cores .4. Oblate clouds and small cluster formation

Binary and multiple stars are believed to form primarily by the gravitational collapse of dense molecular cloud cores and by the fragmentation of the clouds into systems of protostars. Previous numerical calculations have shown that initially prolate (or cylindrical) cloud cores may collapse and fragment into binary or higher order protostellar systems. Observations of precollapse clouds are consistent with a significant fraction being oblate as well as prolate, yet the collapse of oblate clouds has not been investigated to date with a three-dimensional hydrodynamics code. We present here three-dimensional calculations of the isothermal collapse of oblate cloud cores, with initial axial ratios of 2:1 and uniform or centrally condensed (Gaussian) radial density profiles. The calculations are performed with a spatially and temporally second-order accurate hydrodynamics code with relatively high resolution in the azimuthal direction (N-phi = 128). Oblate clouds with moderate solid-body rotation rates collapse in two steps. First, the clouds collapse along their short axes to form thin pancakes, which are unsupported by centrifugal effects. Second, the pancakes collapse radially inward along their midplanes until centrifugal support is achieved in their inner regions. During the second phase, the surface density of the pancakes becomes so high [sigma(r) > sigma(J)(r) approximate to (c(s)(2) pi)/(Gr)] that the inner regions of the pancakes fragment into multiple protostars. Fragmentation occurs when the initial ratio of thermal to gravitational energy (alpha(i)) is about 0.4 or less. Varying the ratio of rotational to gravitational energy (beta(i)) from 0.14 to 0.018 has relatively little effect, because the highly fragmentable pancakes are formed primarily as a result of the oblate geometry of the initial clouds rather than by rotational flattening. The most dramatic result is that oblate clouds appear to fragment into a larger number of protostars than do prolate clouds with similar values of alpha(i) and beta(i). The models imply that centrally condensed, oblate cloud cores close to virial equilibrium (or alpha approximate to 1/2) should collapse and fragment into small clusters containing on the order of 10 protostars.