The Free-Fall time of finite Sheets and Filaments

Molecular clouds often exhibit filamentary or sheet-like shapes. We compute the free-fall time ($\text{tff}$) for finite, uniform, self-gravitating circular sheets and filamentary clouds of small but finite thickness, so that their volume density $\rho$ 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 $\sqrt{A}$, where the aspect ratio $A$ is given by $A=R\mathrm{/}h$, $R$ being the sheet’s radius and $h$ is its thickness. For filamentary clouds, the aspect ratio is defined as $A=L\mathrm{/}\text{calR}$, where $L$ is the filament’s half length and $\text{calR}$ is its (small) radius, and the modification factor is a more complicated, although in the limit of large $A$ it again reduces to nearly $\sqrt{A}$. 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 towards partially alleviating the “star-formation conundrum”, namely, that 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.