Filamentary structures , or long and narrow streams of material , arise in many areas of astronomy .
Here we investigate the stability of such filaments by performing an eigenmode analysis of adiabatic and polytropic fluid cylinders , which are the cylindrical analog of spherical polytropes .
We show that these cylinders are gravitationally unstable to perturbations along the axis of the cylinder below a critical wavenumber k _ { crit } \simeq few , where k _ { crit } is measured relative to the radius of the cylinder .
Below this critical wavenumber perturbations grow as \propto e ^ { \sigma _ { u } \tau } , where \tau is time relative to the sound crossing time across the diameter of the cylinder , and we derive the growth rate \sigma _ { u } as a function of wavenumber .
We find that there is a maximum growth rate \sigma _ { max } \sim 1 that occurs at a specific wavenumber k _ { max } \sim 1 , and we derive the growth rate \sigma _ { max } and the wavenumbers k _ { max } and k _ { crit } for a range of adiabatic indices .
To the extent that filamentary structures can be approximated as adiabatic and fluid-like , our results imply that these filaments are unstable without the need to appeal to magnetic fields or external media .
Further , the objects that condense out of the instability of such filaments are separated by a preferred length scale , form over a preferred timescale , and possess a preferred mass scale .