The Gravitational Instability of Adiabatic Filaments
  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 $k_{\rm crit} \simeq few$, where $k_{\rm
crit}$ is measured relative to the radius of the cylinder. Below this critical
wavenumber perturbations grow as $\propto e^{\sigma_{\rm u}\tau}$, where $\tau$
is time relative to the sound crossing time across the diameter of the
cylinder, and we derive the growth rate $\sigma_{\rm u}$ as a function of
wavenumber. We find that there is a maximum growth rate $\sigma_{\rm max} \sim
1$ that occurs at a specific wavenumber $k_{\rm max} \sim 1$, and we derive the
growth rate $\sigma_{\rm max}$ and the wavenumbers $k_{\rm max}$ and $k_{\rm
crit}$ 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.
