Stellar Dynamics -- Theory of spiral density waves

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Stellar Dynamics -- Theory of spiral density
waves

• Dynamics of Galaxies
• Françoise COMBES

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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
• 3 Phases neutral, molecular, ionised

Mass
Cloud
T
Density
HI
Orion
HII
H2
Dust
Msun
Msun
(K)
cm-3
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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 Radius 2-4 times
the optical radius HI the only component which
does not fall exponentially with R (may be also
diffuse UV?)
HI
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 Ngtgt 1, the potential is
smoothed
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Stability -- Toomre Criterion
Jeans Instability Assume an homogeneous medium
(up to infinity, "Jeans Swindle") ? ?0 ?1
?1 a exp i (kr - ?t) Linearising
equations ? ?(k) If ?2 lt0 , a solution increases
exponentially with time
The system is unstable
Fluid P0 ?0 s2 ?2 s2k2 - 4
p G ?0 (s velocity dispersion) Jeans
length ?J s / (G ?0)1/2 s tff The
scales l gt ?J 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(O2 R)/dR ?R
?2 ?R O angular frequency of rotation ?
epicyclique frequency (cf further down) Internal
gravity forces of the condensation ?R (G S p
?R2)/ ?R2 Ftid ? Lcrit G S / ?2 Lcrit
?J ? scrit p G S / ? Q s/
scrit gt 1 Q Toomre parameter
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In this expression, we have assumed a galactic
disk (2D) Jeans Criterion ?J s tff s/(2p
G?)1/2 Disk of surface density S and height
h The isothermal equilibrium of the
self-gravitating disk P ?s2 ?F
4pG? grad P - ? grad F d/dz (1/?
d?/dz) -? 4pG/s2 ? ?0
sech2(z/h) ?0 / ch2(z/h)
avec h2 s2 /2pG? S h ? and h s2
/ ( 2p G S ) ? ?J s2 / ( 2p G S ) h
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Epicycles
Perturbations of the circular trajectory r R
x ? Ot y O2 1/R dU/dr
Developpment in polar coordinates, and
linearisation ?two harmonic oscillators d2x/dt2
?2 (x-x0) 0 ?2
R d O2 /dR 4 O2 ? 2 O for a
rotation curve O cste ? (2)1/2 O 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 ? 2 O
d) corotation
Examples of values of ? always comprised
between O 2 O
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Lindblad Resonances
There always exists a referential frame, where
there is a rationnal ratio between epicyclic
frequency ? and the frequency of rotation O -
Ob 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 O Ob
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Representation of resonant orbits in the
rotating frame ILR Ob O - ?/2 OLR Ob O
?/2 Corotation Ob O 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 O - ?/2

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Precession of orbits of elliptical shape at rate
O - ?/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 S S0 S1( r )
exp-im(?-?o) i?t We linearise the equations,
of Poisson, of Boltzman pitch angle tan (i)
1/r dr/d?o 1/(kr) k 2p/? Assuming also
that spiral waves are tightly wound pitch
angle 0 kr gtgt1 or ? ltlt r ? WKB
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Frequency ? m (Op - O)/? m2 nbre of arms ?
0 Corotation
ILR ? -1, OLR ? 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 ?crit 4p2 Gµ/? There exists a
forbiden zone, if Q gt 1 (disk too hot to allow
the developpment of waves) around corotation
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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 d?/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 -gt Group velocity of paquets A-B short
leading C-D long leading, opening ILR (E) --gt
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, ytangential Toomre 1981
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Two fundamental parametres for the swing Q , but
also X ?/sini / ? 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
2? Already m2 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
cos?) -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 cos?) -1/2 GM/D2
rcos? cste After developpment V -GM
r2/D3 (1/4 3/4 cos2?) ...
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Tidal forces in the perpendicular direction
Fz D sini GM (r2 D2 - 2rD cos? cosi) -3/2 -
D-3 3/2 GMr/D3 sin2i cos?
perturbation m1 ? 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 (gt
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
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Rotation of Ellipticals
Small E MBgt -20.5 filled Large E MBlt-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/a1-2
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King Profiles
F(E) 0 Egt Eo F(E) (2p s2)-1.5 ro
exp(Eo-E)/s2 -1 E lt Eo
Clog(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, -µ, 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)
• The observation of the velocity profile is
  somewhat degenerate

b
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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
Emsellem et al 2007
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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