Local stability of a gravitating filament: a dispersion relation
  Filamentary structures are ubiquitous in astrophysics and are observed at
various scales. On a cosmological scale, matter is usually distributed along
filaments, and filaments are also typical features of the interstellar medium.
Within a cosmic filament, matter can contract and form galaxies, whereas an
interstellar gas filament can clump into a series of bead-like structures which
can then turn into stars. To investigate the growth of such instabilities, we
derive a local dispersion relation for an idealized self-gravitating filament,
and study some of its properties. Our idealized picture consists of an infinite
self-gravitating and rotating cylinder with pressure and density related by a
polytropic equation of state. We assume no specific density distribution, treat
matter as a fluid, and use hydrodynamics to derive the linearized equations
that govern the local perturbations. We obtain a dispersion relation for
axisymmetric perturbations and study its properties in the (k_R, k_z) phase
space, where k_R and k_z are respectively the radial and longitudinal
wavenumbers. While the boundary between the stable and unstable regimes is
symmetrical in k_R and k_z and analogous to the Jeans criterion, the most
unstable mode displays an asymmetry that could constrain the shape of the
structures that form within the filament. Here the results are applied to a
fiducial interstellar filament, but could be extended for more astrophysical
systems such as cosmological filaments and tidal tails.
