Modes of Star Formation in Finite Molecular Clouds
  We analytically investigate the modes of gravity-induced star formation
possible in idealized finite molecular clouds where global collapse competes
against both local Jeans instabilities and discontinuous edge instabilities. We
examine these timescales for collapse in spheres, discs, and cylinders, with
emphasis on the structure, size, and degree of internal perturbations required
in order for local collapse to occur before global collapse. We find that
internal, local collapse is more effective for the lower dimensional objects.
Spheres and discs, if unsupported against global collapse, must either contain
strong perturbations or must be unrealistically large in order for small
density perturbations to collapse significantly faster than the entire cloud.
We find, on the other hand, that filamentary geometry is the most favorable
situation for the smallest perturbations to grow before global collapse
overwhelms them and that filaments containing only a few Jeans masses and weak
density perturbations can readily fragment. These idealized solutions are
compared with simulations of star-forming regions in an attempt to delineate
the role of global, local, and edge instabilities in determining the
fragmentation properties of molecular clouds. The combined results are also
discussed in the context of recent observations of Galactic molecular clouds.
