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| {{about|the bending instability in galaxies|the firehose instability in magnetized plasmas|Instability}}
| | My name is Carissa and I am studying Japanese Studies and Gender and Women's Studies at Kobenhavn V / Denmark.<br><br>Feel free to visit my homepage :: [http://www.ezramedicalcare.com/EZRA_04_2/4814437 великий пост читать] |
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| [[Image:FirehoseNbody.png|thumb|right|300px|Fig. 1. The firehose instability in an [[N-body simulation]] of a [[prolate]] [[elliptical galaxy]]. Time progresses top–down, from upper left to lower right. Initially, the long-to-short axis ratio of the galaxy is 10:1. After the instability has run its course, the axis ratio is approximately 3:1. Note the boxy shape of the final galaxy, similar to the shapes of [[barred spiral galaxy|bars]] observed in many [[spiral galaxy|spiral galaxies]].
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| ]]
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| The '''firehose instability''' (or '''hose-pipe instability''') is a dynamical [[instability]] of thin or elongated [[galaxies]]. The instability causes the galaxy to buckle or bend in a direction perpendicular to its long axis. After the instability has run its course, the galaxy is less elongated (i.e. rounder) than before. 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.
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| The firehose instability is probably responsible for the fact that [[elliptical galaxies]] and [[dark matter halo]]es never have axis ratios more extreme than about 3:1, since this is roughly the axis ratio at which the instability sets in.<ref name="FP84">{{Citation
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| | last1 = Fridman | first1 = A. M.
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| | last2 = Polyachenko | first2 = V. L.
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| | title = Physics of Gravitating Systems. II — Nonlinear collective processes: Nonlinear waves, solitons, collisionless shocks, turbulence. Astrophysical applications
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| | publisher = [[Springer Science+Business Media|Springer]]
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| | location = Berlin
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| | year = 1984
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| | isbn = 978-0-387-13103-0}}</ref> It may also play a role in the formation of [[barred spiral galaxy|barred spiral galaxies]], by causing the bar to thicken in the direction perpendicular to the galaxy disk.<ref name="Raha91">{{Citation
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| | last1 = Raha | first1 = N.
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| | last2 = Sellwood | first2 = J. A.
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| | last3 = James | first3 = R. A.
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| | last4 = Kahn | first4 = F. A.
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| | title = A dynamical instability of bars in disk galaxies
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| | journal = Nature
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| | volume = 352
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| | pages = 411–412
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| | year = 1991
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| | doi = 10.1038/352411a0|bibcode = 1991Natur.352..411R | issue=6334}}
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| </ref>
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| The firehose instability derives its name from a similar instability in magnetized [[Plasma (physics)|plasmas]].<ref name="Parker58">{{citation
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| |author=Parker, E. N.
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| |author-link=Eugene Parker
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| |title=Dynamical Instability in an Anisotropic Ionized Gas of Low Density
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| |journal=Physical Review
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| |volume=109
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| |year=1958
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| |pages=1874–1876
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| |doi=10.1103/PhysRev.109.1874
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| |bibcode = 1958PhRv..109.1874P }}</ref> However, from a dynamical point of view, a better analogy is with the [[Kelvin–Helmholtz instability]],<ref name="Toomre66">{{citation
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| | author = Toomre, A.
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| | author-link=Alar Toomre
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| | journal=Notes from the Geophysical Fluid Dynamics Summer Study Program, Woods Hole Oceanographic Inst.
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| | title=A Kelvin–Helmholtz Instability
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| | year=1966
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| | pages=111–114
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| }}</ref> or with beads sliding along an oscillating string.<ref>In spite of its name, the firehose instability is not related dynamically to the oscillatory motion of a hose spewing water from its nozzle.</ref>
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| ==Stability analysis: sheets and wires==
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| The firehose instability can be analyzed exactly in the case of an infinitely thin, self-gravitating sheet of stars.<ref name="Toomre66"/> If the sheet experiences a small displacement <math>h(x,t)</math> in the <math>z</math> direction, the vertical acceleration for stars of <math>x</math> velocity <math>u</math> as they move around the bend is
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| :<math>
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| a_z = \left({\partial\over\partial t} + u{\partial\over\partial x}\right)^2h =
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| {\partial^2h\over\partial t^2} + 2u {\partial^2h\over\partial t \partial x} + u^2 {\partial^2h\over\partial x^2},\,
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| </math>
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| provided the bend is small enough that the horizontal velocity is unaffected. Averaged over all stars at <math>x</math>, this acceleration must equal the gravitational restoring force per unit mass <math>F_x</math>. In a frame chosen such that the mean streaming motions are zero, this relation becomes
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| :<math>
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| {\partial^2h\over\partial t^2} + \sigma_u^2 {\partial^2h\over\partial x^2} - F_z(x,t) =0,\,
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| </math>
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| where <math>\sigma_u</math> is the horizontal velocity dispersion in that frame.
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| For a perturbation of the form
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| :<math>
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| h(x,t)=H\exp\left[i\left(kx-\omega t\right)\right]
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| </math>
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| the gravitational restoring force is
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| :<math>
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| F_z(x,t) = -G\Sigma\int_{-\infty}^\infty dy' \int_{-\infty}^{\infty}
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| {\left[ h(x,t) - h(x',t)\right]\over \left[(x-x')^2+(y-y')^2\right]^{3/2}}dx'
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| = -2\pi G\Sigma k h(x,t)
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| </math>
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| where <math>\Sigma</math> is the surface mass density. The [[dispersion relation]] for a thin self-gravitating sheet is then<ref name="Toomre66"/>
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| :<math>
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| \omega^2 = 2\pi G\Sigma k - \sigma_u^2 k^2.
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| </math>
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| The first term, which arises from the perturbed gravity, is stabilizing, while the second term, due to the [[centrifugal force]] that the stars exert on the sheet, is destabilizing.
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| For sufficiently long wavelengths:
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| :<math>
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| \lambda =2\pi/k > \lambda_J = \sigma_u^2/G\Sigma
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| </math>
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| the gravitational restoring force dominates, and the sheet is stable; while at short wavelengths the sheet is unstable. The firehose instability is precisely complementary, in this sense, to the [[Jeans instability]] in the plane, which is ''stabilized'' at short wavelengths, <math>\lambda < \lambda_J</math>.<ref name="KMC71">{{Citation
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| | last1 = Kulsrud | first1 = R. M.
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| | last2 = Mark | first2 = J. W. K.
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| | last3 = Caruso | first3 = A.
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| | title=The Hose-Pipe Instability in Stellar Systems
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| | journal=Astrophysics and Space Science
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| | volume=14
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| | year=1971
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| | pages=52–55
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| | doi=10.1007/BF00649194
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| | postscript = .
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| | bibcode=1971Ap&SS..14...52K
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| }}</ref>
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| [[Image:FirehoseEvenModes.png|thumb|left|250px|Fig. 2. Unstable eigenmodes of a one-dimensional (prolate) galaxy. Growth rates are given at the left.
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| ]]
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| 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.<ref name="MH91">{{Citation
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| | last1 = Merritt | first1 = D.
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| | author1-link = David Merritt
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| | last2 = Hernquist | first2 = L.
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| | title=Stability of Nonrotating Stellar Systems
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| | journal=The Astrophysical Journal
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| | volume=376
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| | year=1991
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| | pages=439–457
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| | doi=10.1086/170293
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| | postscript = .
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| | bibcode=1991ApJ...376..439M
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| }}</ref> This is a simple model of a ([[prolate]]) elliptical galaxy. Some unstable [[eigenmode]]s are shown in Figure 2 at left.
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| ==Stability analysis: finite-thickness galaxies==
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| 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 <math>\kappa_z</math>; like any oscillator, the phase of the star's response to the imposed bending depends entirely on whether the forcing frequency <math>ku</math> is greater than or less than its natural frequency. If <math>ku>\kappa_z</math> for most stars, the overall density response to the perturbation will produce a gravitational potential opposite to that imposed by the bend and the disurbance will be damped.<ref name="MS94">{{citation
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| | last1 = Merritt | first1 = D.
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| | author1-link = David Merritt
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| | last2 = Sellwood | first2 = J.
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| | title=Bending Instabilities of Stellar Systems
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| | journal=The Astrophysical Journal
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| | volume=425
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| | year=1994
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| | pages=551–567
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| | doi=10.1086/174005
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| | bibcode=1994ApJ...425..551M
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| }}</ref> These arguments imply that a sufficiently thick galaxy (with low <math>\kappa_z</math>) will be stable to bending at all wavelengths, both short and long.
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| 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.<ref name="Toomre66"/><ref>{{cite journal
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| | author = Araki, S.
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| | title = A Theoretical Study of the Stability of Disk Galaxies and Planetary Rings. PhD Thesis, MIT
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| | year = 1985
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| | oclc = 13915550}}
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| </ref> 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 <ref name="FP84"/> that the critical axis ratio for stability of homogeneous (constant-density) [[Oblate spheroid|oblate]] and prolate spheroids was roughly 3:1, not 15:1 as implied by the infinite slab, and Merritt & Hernquist<ref name="MH91"/> found a similar result in an [[N-body simulation|N-body]] study of inhomogeneous prolate spheroids (Fig. 1).
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| The discrepancy was resolved in 1994.<ref name="MS94"/> 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. In these more realistic models, a typical star feels a vertical forcing frequency from a long-wavelength bend that is roughly twice the frequency <math>\Omega_z</math> of its unperturbed orbital motion along the long axis. Stability to global bending modes then requires that this forcing frequency be greater than <math>\Omega_z</math>, the frequency of orbital motion parallel to the short axis. The resulting (approximate) condition
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| :<math> | |
| 2\Omega_x > \Omega_z\,
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| </math>
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| predicts stability for homogeneous prolate spheroids rounder than 2.94:1, in excellent agreement with the normal-mode calculations of Fridman & Polyachenko<ref name="FP84"/> and with N-body simulations of homogeneous oblate<ref name="Jessop97">{{Citation
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| | last1 = Jessop | first1 = C. M.
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| | last2 = Duncan | first2 = M. J.
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| | last3 = Levison | first3 = H. F.
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| | title = Bending Instabilities in Homogenous Oblate Spheroidal Galaxy Models
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| | journal = The Astrophysical Journal
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| | volume = 489
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| | pages = 49–62
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| | year = 1997
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| | doi = 10.1086/304751 | bibcode=1997ApJ...489...49J}}
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| </ref> and inhomogeneous prolate <ref name=MH91/> galaxies.
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| The situation for [[spiral galaxy|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.<ref name="SM94">{{Citation
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| | last1 = Sellwood | first1 = J.
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| | last2 = Merritt | first2 = D.
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| | author2-link = David Merritt
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| | title = Instabilities of counterrotating stellar disks
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| | journal = The Astrophysical Journal
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| | volume = 425
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| | pages = 530–550
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| | year = 1994
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| | doi = 10.1086/174004
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| | bibcode=1994ApJ...425..530S}}
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| </ref> If the stellar velocities are azimuthally biased, the orbits are approximately circular and so othe dominant modes are angular (corrugation) modes, <math>\delta z \propto e^{im\phi}</math>. The approximate condition for stability becomes
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| :<math>
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| m\Omega>\kappa_z\,
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| </math>
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| with <math>\Omega</math> the circular orbital frequency.
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| ==Importance==
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| The firehose instability is believed to play an important role in determining the structure of both [[spiral galaxy|spiral]] and [[elliptical galaxy|elliptical]] galaxies and of [[dark matter halo]]es.
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| * As noted by [[Edwin Hubble]] and others, elliptical galaxies are rarely if ever observed to be more elongated than [[Galaxy morphological classification|E6 or E7]], corresponding to a maximum axis ratio of about 3:1. The firehose instability is probably responsible for this fact, since an elliptical galaxy that formed with an initially more elongated shape would be unstable to bending modes, causing it to become rounder.
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| * Simulated [[dark matter halo]]es, like elliptical galaxies, never have elongations greater than about 3:1. This is probably also a consequence of the firehose instability.<ref name="Bett07">{{citation
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| |author=Bett, P. ''et al.''
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| |title=The spin and shape of dark matter haloes in the Millennium simulation of a Λ cold dark matter universe
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| |journal=Monthly Notices of the Royal Astronomical Society
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| |volume=376
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| |year=2007
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| |pages=215–232
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| |doi=10.1111/j.1365-2966.2007.11432.x
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| |bibcode=2007MNRAS.376..215B
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| |arxiv = astro-ph/0608607 }}</ref>
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| * N-body simulations reveal that the bars of [[barred spiral galaxies]] often "puff up" spontaneously, converting the initially thin bar into a [[bulge (astronomy)|bulge]] or [[Star count|thick disk]] subsystem.<ref name="Combes90">{{citation
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| |author=Combes, F. ''et al.''
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| |title=Box and peanut shapes generated by stellar bars
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| |journal=Astronomy and Astrophysics
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| |volume=233
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| |year=1990
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| |pages=82–95
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| |bibcode=1990A&A...233...82C
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| }}</ref> The bending instability is sometimes violent enough to weaken the bar.<ref name="Raha91"/> Bulges formed in this way are very "boxy" in appearance, similar to what is often observed.<ref name="Combes90"/>
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| * The firehose instability may play a role in the formation of [[galactic warp]]s.<ref name="RP04">{{citation
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| |last1=Revaz | first1=Y.
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| |last2=Pfenniger | first2=D.
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| |title=Bending instabilities at the origin of persistent warps: A new constraint on dark matter halos
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| |journal=Astronomy and Astrophysics
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| |volume=425
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| |year=2004
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| |pages=67–76
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| |doi=10.1051/0004-6361:20041386
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| |bibcode=2004A&A...425...67R
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| |arxiv = astro-ph/0406339 }}</ref>
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| ==See also==
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| *[[Stellar dynamics]]
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| ==References==
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| {{reflist|2}}
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| [[Category:Astrophysics]]
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| [[Category:Stability theory]]
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