The Free-Fall time of finite Sheets and Filaments

TL;DR

This paper calculates the free-fall time for finite, self-gravitating sheets and filaments, revealing that their collapse times are longer than spherical models, which may help explain observed star formation rates.

Contribution

It provides new formulas for free-fall times of finite sheets and filaments, accounting for aspect ratios, and links these results to star formation observations.

Findings

Free-fall time for thin sheets exceeds spherical case by factor ~√A.

Filament free-fall times also increase with aspect ratio, approaching √A.

Longer free-fall times may explain slower star formation rates in galaxies.

Abstract

Molecular clouds often exhibit filamentary or sheet-like shapes. We compute the free-fall time ($\tff$) 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 $A=R/h$, $R$ being the sheet's radius and $h$ is its thickness. For filamentary clouds, the aspect ratio is defined as $A=L/\calR$, where $L$ is the filament's half length and $\calR$ 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.