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M51, the Whirlpool Galaxy
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
• If the arm structure rotates differentially, then the pitch must diminish,
and in a tipical time scale of 10^8 years the arms will become tightly
wound.
• but the proportion of spiral with tightly wound arms is small and
galaxies are typically 10^10 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
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.
An important application of it is to determine
whether a given disk is locally stable to
axisymmetric perturbations (m=0)
This study lead to the famous
Toomre stability criterium:
Q  ( ) /(3.36G)  1
Where σ is the radial velocity dispersion and Σ
The surface density
(for our Galaxy: Q=1.7)
and to to the critical lenght:
  4 G / 
2
(the longest wavelenght taht could be unstable: it
provides a useful yardstick
for Jeans-type instabilities of all kinds)
2
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.
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 Toomre’s locally
criterium shows that the system is
unstable against very large-scale
modes.
Uniformly rotating disk of 100000 stars
Moving under a purely radial gravitational
field
Non axisymmetric evolution of the disk
• These experiments show that:
• Toomre criterium is sufficient against global
instability in axysymmetric modes
• (m=0)
• But is not sufficient against non axysymmetric
modes (m=1,2)
• Hohl noticed that Q >2.5 stabilizes the disk
against bar instability
(too high value for real galaxies)
Resonances
LIR and OLR
Two important disk parameters:
Q and J
• Toomre stability parameter:
Q   /(3.36G)
And the parameter J
J  2m (2 /  )(d ln  / d ln r )1 / 2
where   G / r
parameter
1/ 2
is the selfgravity
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.
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
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
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
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.
• Toomre’s mechanism suggests three
different ways in which the bar mode can
be stabilized
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 Ω decreasing outward. Both types of
motion occur in a direction opposite to Ω 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
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
• 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.
• 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
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.
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
S moderate S violent
Superposition of a
Bar mode onto its axysim
etric density distribution
Responses of a V=const. disk of stars to transient
gravity forces from the imposed masses
The top
tow shows
the excess
densities
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
Bertin and Lin (1996):
Numerical work on bar models
• Orbit families in frozen bar-like potentials
(Lindblad resonances, Lagrangian points..)
• Orbital structure of a bar formed in an Nbody simulation (2d and 3d)
• Origin of bars (global instability followed
by Nbody simulations)
• Controlling bar instability
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
Qui dovrebbe stare l’immagine che mi
deve scannerizzare Giuseppe
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
• 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.
Controlling the bar instability
using 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?
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)
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
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
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