On the fragmentation of differentially rotating clouds

We show from general considerations that if the angular velocity Ω of a cloud which is unstable to collapse is initially such that |$\Omega\tilde\omega$| decreases with increase in |$\tilde\omega$| (the distance from the rotation axis), intense rings can form. We have simulated the collapse of a cloud numerically both with Ω constant and with |$\Omega\tilde\omega$| decreasing with |$\tilde\omega$|⁠. The numerical results confirm the general theory and show, in addition, that the ring rapidly fragments. The evolution of clouds which initially rotate differentially is markedly different from that of clouds which initially rotate uniformly. The numerical method we use is a particle method (SPH) which incorporates an artificial viscosity to simulate shock transitions. We describe tests of the ability of various forms of artificial viscosity to simulate shocks and to transport angular momentum correctly. These tests show that some standard forms of artificial viscosity give poor results for shock tube phenomena. Furthermore the tests show that unless the viscosity acts only on the component of linear momentum parallel to the rotation axis the errors from spurious angular transport become unacceptably large.