THE COLLAPSE OF CYLINDRICAL ISOTHERMAL AND POLYTROPIC CLOUDS WITH ROTATION

Previous investigations of the effects of rotation on star formation have considered only a very limited number of initial cloud shapes. We present two-dimensional numerical hydrodynamic calculations to study the effects of the geometry on the process of gravitational collapse of a rotating axisymmetric cloud. Both a polytropic and an isothermal equation of state, specified by the polytropic constant-gamma, are incorporated. The initial conditions of a collapse are specified by the Jeans number (J), the ratio of the gravitational potential energy to the thermal energy, and beta, the ratio of rotational energy to gravitational potential energy. The geometry of the cloud is specified by the ratio of its length to its diameter (L/D). The axis of rotation is parallel to the symmetry axis. The shape of the cloud undergoing collapse is at least as important as the initial values of J and beta in determining the outcome of the evolution. In fact, the shape of the cloud is dominant in all cases except where gamma is large (gamma > 4/3). The critical Jeans number, defined as the minimum Jeans number for which collapse will proceed, is modified when rotation is included. Without rotation, clouds differing from a uniform geometry (L/D not-equal 1) collapse more readily, while for the rotating case oblate (L/D < 1) clouds require a larger initial Jeans number than do either clouds with L/D = 1 or prolate (L/D > 1) clouds. The collapse of elongated clouds (L/D > 1) leads to a spindle, in the absence of rotation, if J0 > J(spindle) is satisfied. With rotation, a spindle is not formed, even for large initial Jeans numbers J0. Instead, collapse leads to a hollow cylinder. The critical value of J0, say J(hc), at which the hollow cylinder is formed in the presence of rotation is greater than J(spindle) and increases with beta. For smaller J0, J0 < J(hc), two toroids are formed when rotation is included, compared with the nonrotating case, where two condensations are formed on the axis of symmetry. The possibility of the toroids' fragmentation would lead to the formation of a multiple protostellar system.