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M51, the Whirlpool Galaxy
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Lord Rosse discovered the spiral structure in M51
in 1850

• The explanation of this beautiful form has been
  one of the outstanding problems in astronomy.
• Jeans tried to identify the arms with pieces of
  material that would be shed equatorially as a
  uniformly rotating centrally-condensed mass
  slowly shrank.
• Lindblad attempted to give an explanation of arms
  in term of orbits and then in terms of
  self-gravitating perturbations of a stellar
  system

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• If the arm structure rotates differentially, then
  the pitch must diminish, and in a tipical time
  scale of 108 years the arms will become tightly
  wound.
• but the proportion of spiral with tightly wound
  arms is small and galaxies are typically 1010
  years olds.
• We deduce that
• The spiral structure rotates nearly
  uniformly although the material rotates
  differentially.
• The most promising theory to explain this
  property is the density wave theory

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The density wave analysis is a complicate
procedure There is an important limit in which
this Analysis is much simpler the tightly wound
or WKB approximation (tightly wound the radial
wavelenght is much less than the radius) (WKB
Wentzel-Kramers-Brillouin) In this framework is
possible to deduce the dispersion relations for
stellar disks they establish the relation
between wavenumber and frequency for a
traveling wave as it propagates across the
disk.
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An important application of it is to determine
whether a given disk is locally stable to
axisymmetric perturbations (m0) This study lead
to the famous Toomre stability criterium Where
s is the radial velocity dispersion and S The
surface density (for our Galaxy Q1.7) and to
to the critical lenght (the longest wavelenght
taht could be unstable it provides a useful
yardstick for Jeans-type instabilities of all
kinds)
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Unfortunately WKB analysis does not give a
complete picture of disk dynamics, because it
does not apply to loosely wound
structures. There are no analytic method that
can determine the stability of a general galactic
disk to arbitrary perturbations.
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NUMERICAL WORK ON DISK STABILITY

• One of the earliest studies was carried on by
  Hohl (1971)
• The evolution of a rotating disk of stars, with
  an initial velocity dispersion given by Toomres
  locally criterium shows that the system is
  unstable against very large-scale modes.

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Uniformly rotating disk of 100000 stars Moving
under a purely radial gravitational field
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Non axisymmetric evolution of the disk
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• These experiments show that
• Toomre criterium is sufficient against global
  instability in axysymmetric modes
• (m0)
• But is not sufficient against non axysymmetric
  modes (m1,2)
• Hohl noticed that Q gt2.5 stabilizes the disk
• against bar instability
• (too high value for real galaxies)

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ResonancesLIR and OLR
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Two important disk parametersQ and J

• Toomre stability parameter
• And the parameter J
• where is the
  selfgravity
• parameter

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Swing amplification (Julian and Toomre ,
Goldreich and Lynden Bell , 1965)

• Toomre argues that the bar instability was driven
  by a positive feedback to the swing amplification
  mechanism
• Remarkably, the most features of global
  instability can be understood by augmenting the
  WKB dispersion relations with the swing
  amplification.

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A mode is a standing wave, by definition, But
Toomre showed that the bar instability is more
easily understood in terms of a propagating
wave packet A leading spiral disturbance
originating near the disk center propagates
outward toward corotation, where the swing
amplification causes it to shear into a trailing
wave of much greater amplitude
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The transfer of angular momentum toward the
outer regions is accompanied by the
amplification of the incoming wave
OVERREFLECTION. Overreflection operates inside
the corotation radius as a resonant
cavity. Overreflection can occur in two
different forms In the regime of low J ,
operating on trailing wave and in the regime of
Higher J, in which overreflection operates
Converting a leading wave into a pair of
trailing waves. This later form of
overreflection corresponds to the so called
swing amplification
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Two different regimes for galaxies

• Light disks (low J), in which all the relevant
• cycle can be all trailing, and gives rise to
  self excited normal (unbarred) galaxies (Spirals
  are generally trailing)
• Heavy disks (high J) the relevant cycle is
  based on a leading and a trailing wave and
  generates barred spiral modes (two blobs
  structures inside the CR , due to the
  superposition of the leading with the trailing
  wave

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The bar mode is simply the standing wave
resulting from an endless wave train
propagating trough this cycle

• Wave action is conserved by an outwardingly
  propagating wave beyond corotation.
• Toomres mechanism suggests three different ways
  in which the bar mode can be stabilized

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Swing amplification, according to Toomre is a
strong cooperative effect that inhibits
interarm travel It results from a three fold
conspiracy Shear, shaking and self
gravity Shear flow and epicyclic vibrations
share the same sense in any normal disk having
angular speed O decreasing outward. Both types
of motion occur in a direction opposite to O
itself It is precisely this agreement that makes
it possible for a wide-open pattern of epiciclic
vibrations to resonate with the shear flow. The
only extra-need is for stellar communication and
this bring us to self gravity
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These three ingredients suggest three different
ways in which the bar mode can be stabilized

• The first is to embed the whole disk in a
  massive unresponsive halo (decrease self
• gravity)
• This solution is effective only if sufficient
  mass lies interior to corotation radius

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• The second way is to raise the level of random
  motion in the disk (heat the disk, inhibit
  collective behaviour)
• The third is to breack the feedback loop
  inserting an inner Lindblad resonance between
  corotation and disk center.

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• Some combination of these three mechanism (e.g.
  a massive, dense bulge ,a responsive dark halo)
  is presumably responsive for the stability of
  most galaxies.
• In spite of the encouraging results of the modal
  description in the interpretation of spiral
  structures in galaxies, we are at only the
  beginning in our understanding of galaxy
  evolution.
• This is largely due to our general lack of tools
  to describe the nonlinear evolution of a
  dynamical sistem , even when at the linear level
  its dynamics is dominated by a few modes

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Dynamical classification of spiral morfologies

• An extensive survey of realistic models of galaxy
  disks has shown that the morphology types of the
  global spiral modes that can be generated in a
  disk match the general morphological categories
  that are found along the Hubble sequence.

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Depending on the parameter regime of a given
galaxy disk, the dominant mode may be of the A
Or B type. Different excitation mechanisms
operate for the two Classes of modes. Moreover a
mode rely on a combined support of gas And
stars .
SB0
SB
Superposition of a Bar mode onto its axysim etric
density distribution
S moderate
S violent
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Responses of a Vconst. disk of stars to
transient gravity forces from the imposed masses
The top tow shows the excess densities
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These transient imposed forces (1 of the
galacto centric force on particle A and 0.25 on
particle B)
soon yield an evolving Spiral pattern of
impressive severity among the disk stars
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Bertin and Lin (1996)
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Numerical work on bar models

• Orbit families in frozen bar-like potentials
  (Lindblad resonances, Lagrangian points..)
• Orbital structure of a bar formed in an N-body
  simulation (2d and 3d)
• Origin of bars (global instability followed by
  Nbody simulations)
• Controlling bar instability

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Numerical work on the dissipative component

• Gas behaviour inside a frozen potential or in
  a Nbody-SPH simulation
• Gravitational coupling between stellar bar and
  interstellar medium.
• Star formation in SPH bar simulations
• Coupling between stars and ISM via STF

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Qui dovrebbe stare limmagine che mi deve
scannerizzare Giuseppe
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Bar forming modes

• The type of behaviour illustrated is typical of
  almost every two dimensional simulation for which
  the underlying model is unstable to global
  bisymmetric distorsions As the instability
  runs, the transient features in the surrounding
  disk fade and the only non axisymmetric feature
  to survive is the steadly tumbling bar

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• The bar ends just inside corotation
• The axis ratio of the bar depends upon the degree
  o random motions in the original disks the
  cooler the initial disk the narrower the
  resulting bar.
• When the initial bar is short, it continues to
  interact with the outer disk through spiral
  activity the trailing spirals remove angular
  momentum from particles at their inner end. This
  enable more stars to be trapped into the bar,
  increasing its length and lowering its pattern
  speed.
• These changes in both bar length and pattern
  speed conspire to keep co-rotation just beyon the
  end of the bar.

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Controlling the bar instabilityusing Dark
Matter haloes

• Spherical halo triaxial halo
• Non rotating halo spinning halo
• Analytical passive halo live
  halo
• Static halo dynamical halo
• HOW THE BAR INSTABILITY IS REACTING TO SUCH MORE
  REALISTIC HALOES MODELS?

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What happens if we include gas in the disk?

• Dissipation triggers significant gas fueling of
  the central regiones once the bar has formed
• This leads to a high central mass concentration
  wihich is in the end responsible for the
  destruction of the bar.
• (as soon as the mass accreted by the central
  regions represents a non negligible part of the
  galaxy mass (1-2) a strong ILR appears)

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What happens if we include star formation in
the disk?

• Stronger bars tends to form inside more massive
  non relaxed haloes
• If star formation is included, it seems to
  favour bar formation, lenghtening the bar
  lifetime (SF works against strong mass
  concentration in the center of the disk)
• Since stronger bursts of star formation are
  triggered in more massive and concentrated
  haloes, stronber bars develop in more
  concentrated massive haloes

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Thus the star formation, which depend on local
conditions Is however governed by the local
dynamics of the galaxy. And Vice-versa , local
Star Formation can modify the global Dynamics,
resulting in a highly non linear feed back
mechanism. Moreover SF efficiency, IMF, cooling
function can tehmselves Be dependent on the
metallicity. Since clearly this metallicity
is Related to the previous SF hustory, this add
another feed back Mechanism. All this seems to
suggest that global self regulated non stationary
Processes could take place in disk galaxies
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Controlling bar instability through
Cosmological DM haloes

• We adopt fully cosmological DM haloes, inside a
  real cosmological scenario, to imbed our stellar
  disk.
• We therefore can investigate the role of the
  infall, the influence of the matter outside the
  system. The cosmological expansion and so
  on.
• The aim is to get the disk evolution as a
  redshift function