Firehose instability
The instability causes the galaxy to buckle or bend in a direction perpendicular to its long axis.
Any sufficiently thin stellar system, in which some component of the internal velocity is in the form of random or counter-streaming motions (as opposed to rotation), is subject to the instability.
[3] However, from a dynamical point of view, a better analogy is with the Kelvin–Helmholtz instability,[4] or with beads sliding along an oscillating string.
[5] The firehose instability can be analyzed exactly in the case of an infinitely thin, self-gravitating sheet of stars.
[6] A similar analysis can be carried out for a galaxy that is idealized as a one-dimensional wire, with density that varies along the axis.
At wavelengths shorter than the actual vertical thickness of a galaxy, the bending is stabilized.
The reason is that stars in a finite-thickness galaxy oscillate vertically with an unperturbed frequency
; like any oscillator, the phase of the star's response to the imposed bending depends entirely on whether the forcing frequency
for most stars, the overall density response to the perturbation will produce a gravitational potential opposite to that imposed by the bend and the disturbance will be damped.
Analysis of the linear normal modes of a finite-thickness slab shows that bending is indeed stabilized when the ratio of vertical to horizontal velocity dispersions exceeds about 0.3.
[4][9] Since the elongation of a stellar system with this anisotropy is approximately 15:1 — much more extreme than observed in real galaxies — bending instabilities were believed for many years to be of little importance.
However, Fridman & Polyachenko showed [1] that the critical axis ratio for stability of homogeneous (constant-density) oblate and prolate spheroids was roughly 3:1, not 15:1 as implied by the infinite slab, and Merritt & Hernquist[7] found a similar result in an N-body study of inhomogeneous prolate spheroids (Fig.
[8] The gravitational restoring force from a bend is substantially weaker in finite or inhomogeneous galaxies than in infinite sheets and slabs, since there is less matter at large distances to contribute to the restoring force.
As a result, the long-wavelength modes are not stabilized by gravity, as implied by the dispersion relation derived above.
Stability to global bending modes then requires that this forcing frequency be greater than
The resulting (approximate) condition predicts stability for homogeneous prolate spheroids rounder than 2.94:1, in excellent agreement with the normal-mode calculations of Fridman & Polyachenko[1] and with N-body simulations of homogeneous oblate[10] and inhomogeneous prolate [7] galaxies.
The situation for disk galaxies is more complicated, since the shapes of the dominant modes depend on whether the internal velocities are azimuthally or radially biased.
In oblate galaxies with radially-elongated velocity ellipsoids, arguments similar to those given above suggest that an axis ratio of roughly 3:1 is again close to critical, in agreement with N-body simulations for thickened disks.
The firehose instability is believed to play an important role in determining the structure of both spiral and elliptical galaxies and of dark matter haloes.
