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.
