Stellar Dynamics -- Theory of
spiral density waves
Dynamics of Galaxies
Françoise COMBES
Stellar Dynamics in Spirals
Spiral galaxies represent about 2/3 of all galaxies
Origin of spiral structure ?
Winding problem, differential rotation
Theory of density waves, excitation and maintenance
Stellar Dynamics -- Stability
The main part of the mass today in galaxy disks is stellar
(~10% of gas)
Dominant forces: gravity at large scale
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NGC 1232 (VLT image)
SAB(rs)c
NGC 2997 (VLT)
SA(s)c
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Messier 83 (VLT)
NGC 5236
SAB(s)c
NGC 1365 (VLT)
(R')SB(s)b
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Hubble Sequence (tuning fork)
Sequence of mass, of concentration
Gas Fraction
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The interstellar medium
• 90% H, 10% He
H
He
Poussière
• 3 Phases: neutral, molecular, ionised
Mass
Orion
HI
1 – 5 109
Density
100 - 1000
100 - 1000
103 - 104
10 000
105 - 106
103 - 105
10
5 107
Msun
T
0.1 – 10
3 109
HII
H2
Dust
Cloud
10-40
Msun
cm-3
(K)
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The HI gas - Radial Extensions
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Extension of galaxies in HI
M83: optical
Exploration of dark halos
HI
HI Radius
2-4 times the optical radius
HI the only component which does
not fall exponentially with R
(may be also diffuse UV?)
Spiral of the Milky Way type (109 M in HI): M83
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The HI gas- Deformations (warps)
Bottema 1996
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HI rotation curves
Sofue & Rubin 2001
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Stars are a medium without collisions
The more so as the number of particles is larger N ~1011
(paradox)
In the disk (R, h)
Two body encounters, where stars exchange energy
Two-body relaxation time-scale Trel, compared to the crossing
time tc = R/v :
Trel/tc ~ h/R N/(8 log N)
Order of magnitude
tc ~108 y Trel/tc ~ 108
The gravitational potential of a small number of bodies is « grainy »
and scatters particles, while when N>> 1, the potential is smoothed
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Stability -- Toomre Criterion
Jeans Instability
Assume an homogeneous medium (up to infinity, "Jeans Swindle")
ρ = ρ0 + ρ1
ρ1 = α exp [i (kr - ωt)]
Linearising equations  ω(k)
If ω2 <0 , a solution increases exponentially with time
The system is unstable
Fluid P0 = ρ0 σ2
Jeans length
The scales
ω2 = σ2k2 - 4 π G ρ0
(σ velocity dispersion)
λJ = σ / (G ρ0)1/2 = σ tff
l > λJ are unstable
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Stability due to the rotation
The rotation stabilises the large scales
In other words, tidal forces destroy all structures
Larger than a characteristic scale Lcrit
Tidal forces Ftid = d(Ω2 R)/dR ΔR ~ κ2 ΔR
Ω angular frequency of rotation
κ epicyclique frequency (cf further down)
Internal gravity forces of the condensation ΔR
(G Σ π ΔR2)/ ΔR2 = Ftid  Lcrit ~ G Σ / κ2
Lcrit = λJ
 σcrit ~ π G Σ / κ
Q Toomre parameter
Q = σ/ σcrit > 1
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In this expression, we have assumed a galactic disk (2D)
Jeans Criterion λJ = σ tff = σ/(2π Gρ)1/2
Disk of surface density Σ and height h
The isothermal equilibrium of the self-gravitating disk:
P = ρσ2
ΔΦ = 4πGρ
grad P = - ρ grad Φ
d/dz (1/ρ dρ/dz) = -ρ 4πG/σ2
ρ = ρ0 sech2(z/h) = ρ0 / ch2(z/h)
avec h2 = σ2 /2πGρ
Σ = h ρ and h = σ2 / ( 2π G Σ )  λJ = σ2 / ( 2π G Σ ) = h
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Epicycles
Perturbations of the circular trajectory
r = R +x
θ= Ωt + y
Ω2 = 1/R dU/dr
Developpment in polar coordinates, and linearisation
two harmonic oscillators
d2x/dt2 + κ2 (x-x0) = 0
κ2 = R d Ω2 /dR + 4 Ω2
κ = 2 Ω for a rotation curve Ω = cste
κ = (2)1/2 Ω for a flat rotation curve V= cste
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a) Epicyclic Approximation
b) epicyle is run in the
retrograde sense
c) special case κ = 2 Ω
d) corotation
Examples of values of κ
always comprised between Ω & 2 Ω
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Lindblad Resonances
There always exists a referential frame, where there is a rationnal
ratio between epicyclic frequency κ and the frequency of rotation
Ω - Ωb
Then the orbit is closed in this referential frame
The most frequent case, corresponding to the shape of the rotation
curve, therefore to the mass distribution in galaxies
Is the ratio 2/1, or -2/1
Resonance of corotation: when Ω = Ωb
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Representation of resonant
orbits in the rotating
frame
ILR: Ωb = Ω - κ/2
OLR: Ωb = Ω + κ/2
Corotation: Ωb = Ω
There can exist 0, 1 or 2
ILRs,
Always a CR, OLR
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Kinematical waves
The winding problem shows that it cannot be always
the same stars in the same spiral arms
Galaxies do not rotate like solid bodies
The concept of density waves is well represented by the schema
of kinematical waves
The trajectory of a particle can be considered under 2 points of view:
•Either a circle + an epicycle
•Or a resonant closed orbit, plus a precession
The precession rate: Ω - κ/2
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Precession of orbits of
elliptical shape at rate
Ω - κ/2
This quantity is almost
constant all over the
inner Galaxy
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If the quasi-resonant
orbits are aligned
in a given configuration
Since the precession
rate is almost
constant
Orbits aligned in a
barred configuration
There is little deformation
The self-gravity modifies the precessing rates, and made them constant
Therefore the density waves, taking into account self-gravity,
may explain the formation of spiral arms
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Flocculent Spirals
There exist also other kinds of spirals, very irregular, formed
from spiral pieces, which are not sustained density waves
They do not extend all over the galaxy (cf NGC 2841)
Gerola & Seiden 1978
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Dispersion relation for waves
Let us assume a perturbation Σ = Σ0 + Σ1( r ) exp[-im(θ-θo) +iωt]
We linearise the equations, of Poisson, of Boltzman
pitch angle tan (i) = 1/r dr/dθo = 1/(kr)
k = 2π/λ
Assuming also that spiral waves
are tightly wound
pitch angle ~ 0 kr >>1
or
λ << r  WKB
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Frequency
ν = m (Ωp - Ω)/κ
m=2 nbre of arms
ν = 0 Corotation
ILR ν = -1, OLR ν = 1 (Lin & Shu 1964)
relation of dispersion, identical for trailing or leading waves
The critical wave length is the scale where self-gravity begins
to dominate λcrit = 4π2 Gμ/κ
There exists a forbiden zone, if Q > 1 (disk too hot to allow the
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developpment of waves) around corotation
Geometrical shape of the waves can be
determined from the dispersion relation
The wave length is ~Q (short)
or ~1/Q, for the long waves
a) long branch
b) Short branch
In fact the waves travel in wave paquets, with the group
velocity vg = dω/dk
There can be wave amplification, when there is reflexion
at the centre and the outer boundaries, or at resonances,
Or also at the Q barrier
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The main amplification occurs at Corotation, when waves are
transmitted and reflected
Waves have energy of different sign on each side of Corotation
The transmission of a wave
of negative energy amplifies
the wave of positive energy
which is reflected
-> Group velocity of paquets
A-B short leading
C-D long leading, opening
ILR (E) --> long trailing
reflected at CR in
short trailing
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Swing Amplification
Processus of amplification,
when the leading paquet
transforms in trailing
•Differential Rotation
•self-gravity
•Epicyclic motions
All three contribute to this
amplification
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Winding change sign when
waves cross the centre
A, B, C trailing
 A', B', C' leading
Group velocity AA'=BB'=CC'=cste
Principle of amplification
of "swing"
a) leading, opens in b)
c & d) trailing
Gray color = arm
x= radial, y=tangential
Toomre 1981
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Two fundamental parametres for the swing
Q , but also X = λ/sini / λ crit
Amplification is weaker for a hot system (high Q)
X optimum = 2, from 3 and above  no efficiency
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Wave damping
The gas has a strong answer to the excitation, given its low
velocity dispersion
 very non-linear, and dissipative
Analogy of pendulae
 Shock waves
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Shock waves at the entrance
of spiral arms
Contrast of 5-10
Compression which triggers
star formation
Large variations of velocity at the
crossing of spiral arms
"Streaming" motions characteristic
diagnostics of density waves
Roberts 1969
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Wave Generation
The problem of the persistence of spiral arms is not
completely solved by density waves
Since waves are damped
Is still required a mechanism of generation and maintenance
In fact, spiral waves are not long-lived in galaxies
In presence of gas, they can form and reform continuously
Waves transfer angular momentum from the centre to the outer parts
They are thus the essential engine for matter accretion
The sense depends on the wave nature: trailing/leading
Predominance of trailing waves
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Torques exerted by the spirals
Spiral waves in fact are not very tightly wound
The potential is not local
The density of stars is not in phase with the potential
Stars only
Stars + gas
+ bar
Potential __________
Density +++++
Gas ***
Density in advance
Inside corotation
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Spiral waves and tides
Tidal forces are bisymetrical
in cos 2θ
Already m=2 spiral arms can easily
form in numerical simulations
Restricted 3-body
(Toomre & Toomre 1972)
But this cannot explain M51 and
All other galaxies in interaction
Tidal forces increase with r
in the plane of the target
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Tidal forces are the differential over the plane of the
target galaxy of gravity forces from the companion
Ftid ~ GMd/D3
V = -GM (r2 + D2 - 2rD cosθ) -1/2
Principle of tidal forces
Let us consider the referential
frame fixed with O
The forces on the point P are
the attraction of M (companion)
- inertial force (attraction from
M on O)
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Inertial force -Gmu/D2
u unit vector along OM
Vtot = -GM (r2 + D2 - 2rD cosθ) -1/2
+ GM/D2 rcosθ + cste
After developpment
V = -GM r2/D3 (1/4 +3/4 cos2θ) +...
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Tidal forces in the perpendicular direction
Fz = D sini GM [(r2 + D2 - 2rD cosθ cosi) -3/2 - D-3]
= 3/2 GMr/D3 sin2i cosθ
perturbation m=1
 warp of the plane
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Conclusions (spirals)
Spiral galaxies are crossed by spiral density wave paquets
which are not permanent
Between two episodes, disks can develop flocculent spirals,
generated by the contagious propagation of star formation
Spiral waves transform deeply the galaxies:
•Heat old stars, transfer angular momentum
•Trigger bursts of star formation
•and the accretion & concentration of matter towards the centre
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Experimental tests
Can we find ordering
along the orbits of the
various SF tracers?
Cross-correlation in polar
coordinates have been
done
No clear answer
Foyle et al 2011
Simulations by
Dobbs & Pringle 2010
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Star formation triggered by arms
Different ages of star
clusters
Foyle et al 2011
The SF processes are not as
simple
There are multiple pattern speeds
Harmonics of spirals
+ Flocculence triggered by
Instabilities on each arm, etc..
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Elliptical Galaxies
Elliptical galaxies are not supported by rotation
(Illingworth et al 1978)
But by an anisotropic velocity dispersion
Certainly this must be due to their formation mode: mergers?
Very difficult to measure the rotation of elliptical galaxies
Stellar spectra (absorption lines) are individually
very broad (> 200km/s)
One has to do a deconvolution: correlation with templates
As a function of type and stellar populations
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Stellar spectra
galaxy
• Absorption lines
star
Calcium triplet
Deconvolution:
G = S*  LOSVD
LOSVD :
Line Of Sight
Velocity Distribution
l [ang]
LOSVD
V [km/s]
Rotation of Ellipticals
Small E MB> -20.5: filled
Large E MB<-20.5 empty
Bulges = crosses
from Davies et al (1983)
Solid line: relation for
oblate rotators with
isotropic dispersion
(Binney 1978)
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Density Profiles
The profile of de Vaucouleurs in r1/4 log(I/Ie)= -3.33 (r/re1/4 -1)
The profile of Hubble I/Io = [r/a+1]-2
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King Profiles
F(E) = 0 E> Eo
F(E) = (2p s2)-1.5 ro [ exp(Eo-E)/s2 -1] E < Eo
C=log(rt/rc)
rt =tidal radius
rc= core radius
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Deformations of Ellipticals
The various profiles correspond to
the tidal deformation of elliptical
galaxies
T1: isolated galaxies
T3: near neighbors
Depart from a de Vaucouleurs
distribution
from Kormendy 1982
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Triaxiality of ellipticals
Tests on observations show that elliptical galaxies are
triaxial
With triaxiality and variation of ellipticity with radius ,
 There exists then isophote rotation
No intrinsic
deformation!
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Ellipticals & Early-types
Some galaxies are difficult to classify, between lenticulars
and ellipticals. Most of E-galaxies have a stellar disk
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Anisotropy of velocities
= 1 – s2q/s2r,
-, 0, 1

 circular, isotropic and radial orbits
When galaxy form by mergers,
orbits in the outer parts are
strongly radial, which could explain
the low projected dispersion
(Dekel et al 2005)
Radius
The observation of the velocity profile is somewhat degenerate
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Young stars are
in yellow contours
Comparison with data for
N821 (green), N3379(violet)
N4494 (brown), N4697 (blue)
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SAURON Fast and slow rotators
FR have high
and rising lR
SR have flat
or decreasing lR
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Emsellem et al 2007
SAURON Integral field spectroscopy
Emsellem et al 2007
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Faber-Jackson relation for E-gal
Ziegler et al 2005
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Tully-Fisher relation for spirals
Relation between maximum velocity
and luminosity
DV corrected from inclination
Much less scatter in I or K-band
(no extinction)
Correlation with Vflat
Better than Vmax
Uma cluster
Verheijen 2001
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Tully-Fisher relation
for gaseous galaxies
works much better in
adding gas mass
Relation Mbaryons
with Rotational V
Mb ~ Vc4
McGaugh et al (2000)  Baryonic Tully-Fisher
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Fundamental plane for E-gal
First found by Djorgovski et al 1987
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Scaling relations
• Tully-Fisher: Mbaryons ~ v4
• Faber-Jackson: L ~ s4
• Fundamental Plane:
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