The formation and fragmentation of the ring appearing in the collapse of a rotating cloud

Numerical simulations of the collapse of a slowly rotating cloud have been performed, assuming either isothermality, or a barotropic equation of state that reproduces the expected thermal behaviour of protostellar gas. A ring appears in the late stages of the collapse of a rotating cloud, and we have investigated the effect of differential rotation on the formation and fragmentation of this ring. In the simulations presented here, we have used Godunov-type particle hydrodynamics to avoid the side effects of artificial viscosity in a differentially rotating cloud. The initial state of a cloud is characterized by alpha(0)=T/\Omega\ and beta(0)=R/\Omega\, where T, Omega and are the thermal, gravitational and rotational energies, respectively. If the initial angular velocity, omega, of a cloud is proportional to r(-P), then in the isothermal simulations, a ring forms if P is larger than 0.5, provided beta(o)less than or similar to0.035. In the simulations using a barotropic equation of state, with alpha(o)=0.6 and beta(o)less than or similar to0.035, a ring is always formed, irrespective of whether Pless than or equal to0.5 or P>0.5. However, the mechanism and time of ring formation are different in the two extremes, as are the final configurations. Strong differential rotation (P>0.5) is more effective in inducing fragmentation than solid-body rotation (P = 0), in the sense that fragmentation tends to occur earlier and to produce more fragments when P is larger.