Oscillations and stability of polytropic filaments

We study the oscillations and stability of self-gravitating cylindrically symmetric fluid systems and collisionless systems. This is done by studying small perturbations to the equilibrium system and finding the normal modes, using methods similar to those used in astroseismology. We find that there is a single sequence of purely radial modes that become unstable if the adiabatic exponent is less than 1. Non-radial modes can be divided into p modes, which are stable and pressure driven, and g modes, which are gravity driven. The g modes become unstable if the adiabatic exponent is greater than the polytrope index. These modes are analogous to the modes of a spherical star, but their behaviour is somewhat different because a cylindrical geometry has less symmetry than a spherical geometry. This implies that perturbations are classified by a radial quantum number, an azimuthal quantum number and wavelength in the z direction, which can become arbitrarily large. We find that decreasing this wavelength increases the frequency of stable modes and increases the growth rate of unstable modes. We use variational arguments to demonstrate that filaments of collisionless matter with ergodic distribution functions are stable to purely radial perturbations, and that filaments with ergodic power-law distribution functions are stable to all perturbations.