Three-dimensional hydrodynamic simulations of collapsing prolate clouds

We present the results of collapse calculations for elongated clouds performed using the numerical method of smoothed particle hydrodynamics (SPH). The clouds considered are isothermal, prolate spheroids with different axial ratios (a/b). Results are obtained for different values of |$a/b \enspace \text and \enspace \bar{m}_{L},$|⁠, the mean mass per unit length. It is found that initially uniform clouds undergo fragmentation when the collapse is preferentially down on to the major axis, due to the intrinsic instability of a linear configuration. This occurs when the value of |$\bar{m}_{L}$| is sufficiently large. A criterion for elongated clouds to undergo linear collapse is derived using the tensor virial theorem, and it is found that the numerically obtained value of |$\bar{m}_{L}$| for which fragmentation occurs corresponds closely to that expected from analytical considerations. The addition of small density perturbations simply results in clouds that fragment more easily, particularly for cases in which a/b is close to unity.

Previous calculations, presented by other authors for the case of finite cylinders, show that clouds with cylindrical geometries are highly unstable to the formation of two fragments that occur at the ends of the cylinder. We find that collapsing, prolate spheroids show qualitatively different behaviour, with no preferred tendency to form fragments at the ends of the cloud. Instead fragmentation appears to occur more readily towards the centre of the cloud where the local mass per unit length is greatest.

Our implementation of SPH employs spatially variable smoothing lengths, h. In order to obtain a Hamiltonian system, we incorporate terms involving the spatial variability of h in the particle equations of motion, not included in previous implementations. We find that inclusion of these ∇h terms results in much improved energy conservation, but has little effect on the qualitative outcome of the calculations presented here.