Criteria for fragmentation of rotating isothermal clouds revisited

The collapse of rotating isothermal clouds is investigated by three-dimensi onal self-gravitating hydrodynamical calculations. The criterion that predi cts the outcome after the collapse is presented for the initially uniform-d ensity rigid-rotating sphere. It is shown that the central flatness, that i s, the axial ratio of the isodensity contour in the central region, is a go od indicator for the fate of the cloud. If the central flatness is greater than the critical value similar to 4 pi, a collapsing cloud with moderate p erturbations is unstable for fragmentation, while if the central flatness i s smaller than the critical value, it does not fragment at least before adi abatic core formation. The relation between the central flatness and the in itial value of the ratio of the thermal (alpha(0)) and rotational energy (b eta(0)) to the gravitational energy is also presented. Warm clouds (alpha(0 ) greater than or equal to 0.5) are not expected to fragment before adiabat ic core formation almost independent of the initial rotation (beta(0)) and the properties of the initial perturbation.