Aspect Ratio Dependence of the Free-Fall Time for Non-Spherical Symmetries

We investigate the collapse of non-spherical substructures, such as sheets and filaments, which are ubiquitous in molecular clouds. Such non-spherical substructures collapse homologously in their interiors but are influenced by an edge effect that causes their edges to be preferentially accelerated. We analytically compute the homologous collapse timescales of the interiors of uniform-density, self-gravitating filaments and find that the homologous collapse timescale scales linearly with the aspect ratio. The characteristic timescale for an edge driven collapse mode in a filament, however, is shown to have a square root dependence on the aspect ratio. For both filaments and circular sheets, we find that selective edge acceleration becomes more important with increasing aspect ratio. In general, we find that lower dimensional objects and objects with larger aspect ratios have longer collapse timescales. We show that estimates for star formation rates, based upon gas densities, can be overestimated by an order of magnitude if the geometry of a cloud is not taken into account.