The free-fall time of finite sheets and filaments

Abstract

Molecular clouds often exhibit filamentary or sheet-like shapes. We compute the free-fall time (τff) for finite, uniform, self-gravitating circular sheets and filamentary clouds of small but finite thickness, so that their volume density ρ can still be defined. We find that, for thin sheets, the free-fall time is larger than that of a uniform sphere with the same volume density by a factor proportional to √A, where the aspect ratio A is given by A = R/h, R being the sheet's radius and h is its thickness. For filamentary clouds, the aspect ratio is defined as √A, where L is the filament's half-length and is its (small) radius, and the modification factor is more complicated, although in the limit of large A it again reduces to nearly . We propose that our result for filamentary shapes naturally explains the ubiquitous configuration of clumps fed by filaments observed in the densest structures of molecular clouds. Also, the longer free-fall times for non-spherical geometries in general may contribute toward partially alleviating the "star formation conundrum," namely, the star formation rate in the Galaxy appears to be proceeding in a timescale much larger than the total molecular mass in the Galaxy divided by its typical free-fall time. If molecular clouds are in general formed by thin sheets and long filaments, then their relevant free-fall time may have been systematically underestimated, possibly by factors of up to one order of magnitude. © 2012 The American Astronomical Society. All rights reserved.