Stellar Dynamics -- Theory of spiral density waves Dynamics of Galaxies Françoise COMBES
Stellar Dynamics -- Theory of spiral density waves
Dynamics of Galaxies
Françoise COMBES
2
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
3
NGC 1232 (VLT image)SAB(rs)c
NGC 2997 (VLT)SA(s)c
4
NGC 1365 (VLT)(R')SB(s)b
Messier 83 (VLT)NGC 5236SAB(s)c
5
Hubble Sequence (tuning fork)
Sequence of mass, of concentration
Gas Fraction
6
The interstellar medium
• 90% H, 10% He
• 3 Phases: neutral, molecular, ionised
H
He
Poussière
10-405 107
10103 - 105105 - 1061 – 5 109
10 000103 - 104100 - 1000
100 - 10000.1 – 103 109HI
HII
H2
Dust
Mass Cloud TDensity
Msun Msun (K)cm-3
Orion
7
The HI gas - Radial Extensions
8
Extension of galaxies in HI
HI
M83: optical
Spiral of the Milky Way type (109 M in HI): M83
Exploration of dark halos
HI Radius2-4 times the optical radius
HI the only component which doesnot fall exponentially with R
(may be also diffuse UV?)
9
The HI gas- Deformations (warps)
Bottema 1996
10
HI rotation curves
Sofue & Rubin 2001
11
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 crossingtime 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
12
Stability -- Toomre CriterionJeans 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 ω2 = σ2k2 - 4 π G ρ0 (σ velocity dispersion)
Jeans length λJ = σ / (G ρ0)1/2 = σ tff
The scales > λJ are unstable
13
Stability due to the rotation
The rotation stabilises the large scalesIn other words, tidal forces destroy all structuresLarger 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 = σ/ σcrit > 1
Q Toomre parameter
14
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
15
Epicycles
Perturbations of the circular trajectoryr = R +xθ= Ωt + y Ω2 = 1/R dU/dr
Developpment in polar coordinates, and linearisation two harmonic oscillatorsd2x/dt2 + κ2 (x-x0) = 0
κ2 = R d Ω2 /dR + 4 Ω2
κ = 2 Ω for a rotation curve Ω = csteκ = (2)1/2 Ω for a flat rotation curve V= cste
16
a) Epicyclic Approximationb) epicyle is run in the retrograde sensec) special case κ = 2 Ω d) corotation
Examples of values of κ always comprised between Ω & 2 Ω
17
Lindblad Resonances
There always exists a referential frame, where there is a rationnalratio 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 rotationcurve, therefore to the mass distribution in galaxies
Is the ratio 2/1, or -2/1
Resonance of corotation: when Ω = Ωb
18
Representation of resonantorbits in the rotatingframe
ILR: Ωb = Ω - κ/2
OLR: Ωb = Ω + κ/2
Corotation: Ωb = Ω There can exist 0, 1 or 2ILRs,
Always a CR, OLR
19
Kinematical waves
The winding problem shows that it cannot be alwaysthe same stars in the same spiral armsGalaxies do not rotate like solid bodies
The concept of density waves is well represented by the schemaof 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
20
Precession of orbits ofelliptical shape at rate Ω - κ/2
This quantity is almost constant all over theinner Galaxy
21
Orbits aligned in abarred configuration
If the quasi-resonantorbits are alignedin a given configuration
Since the precessionrate is almostconstant
There is little deformation
The self-gravity modifies the precessing rates, and made them constantTherefore the density waves, taking into account self-gravity, may explain the formation of spiral arms
22
Flocculent Spirals
There exist also other kinds of spirals, very irregular, formedfrom spiral pieces, which are not sustained density waves
They do not extend all over the galaxy (cf NGC 2841)
Gerola & Seiden 1978
23
Dispersion relation for wavesLet 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
24
Frequencyν = 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 beginsto dominate λcrit = 4π2 Gμ/κ
There exists a forbiden zone, if Q > 1 (disk too hot to allow the developpment of waves) around corotation
25
Geometrical shape of the waves can bedetermined from the dispersion relation
The wave length is ~Q (short)or ~1/Q, for the long waves
a) long branchb) 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 reflexionat the centre and the outer boundaries, or at resonances, Or also at the Q barrier
26
The main amplification occurs at Corotation, when waves aretransmitted and reflected
Waves have energy of different sign on each side of Corotation
The transmission of a waveof negative energy amplifiesthe wave of positive energywhich is reflected
-> Group velocity of paquetsA-B short leadingC-D long leading, openingILR (E) --> long trailingreflected at CR inshort trailing
27
Swing Amplification
Processus of amplification,when the leading paquettransforms in trailing
•Differential Rotation •self-gravity•Epicyclic motions
All three contribute to thisamplification
28
Winding change sign whenwaves cross the centre
A, B, C trailing A', B', C' leading
Group velocity AA'=BB'=CC'=cste
Principle of amplificationof "swing"a) leading, opens in b)c & d) trailing
Gray color = armx= radial, y=tangentialToomre 1981
29
Two fundamental parametres for the swingQ , 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
30
Wave damping
The gas has a strong answer to the excitation, given its lowvelocity dispersion
very non-linear, and dissipative
Analogy of pendulae
Shock waves
31
Shock waves at the entranceof spiral armsContrast of 5-10Compression which triggersstar formation
Large variations of velocity at thecrossing of spiral arms
"Streaming" motions characteristicdiagnostics of density waves
Roberts 1969
32
Wave Generation
The problem of the persistence of spiral arms is notcompletely solved by density wavesSince waves are damped
Is still required a mechanism of generation and maintenanceIn fact, spiral waves are not long-lived in galaxiesIn presence of gas, they can form and reform continuously
Waves transfer angular momentum from the centre to the outer partsThey are thus the essential engine for matter accretionThe sense depends on the wave nature: trailing/leading
Predominance of trailing waves
33
Torques exerted by the spirals
Spiral waves in fact are not very tightly woundThe potential is not local
The density of stars is not in phase with the potential
Potential __________Density +++++Gas ***
Density in advance Inside corotation
Stars only Stars + gas+ bar
34
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 rin the plane of the target
35
Tidal forces are the differential over the plane of thetarget galaxy of gravity forces from the companion
Ftid ~ GMd/D3
V = -GM (r2 + D2 - 2rD cosθ) -1/2
Principle of tidal forcesLet us consider the referential frame fixed with O
The forces on the point P are the attraction of M (companion)- inertial force (attraction fromM on O)
36
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) +...
37
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
38
Conclusions (spirals)
Spiral galaxies are crossed by spiral density wave paquetswhich 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
39
Experimental tests
Can we find orderingalong the orbits of the various SF tracers?
Cross-correlation in polarcoordinates have beendone
No clear answer
Foyle et al 2011
Simulations byDobbs & Pringle 2010
40
Star formation triggered by arms
Different ages of starclusters
Foyle et al 2011
The SF processes are not assimple
There are multiple pattern speedsHarmonics of spirals
+ Flocculence triggered byInstabilities on each arm, etc..
41
Elliptical GalaxiesElliptical 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 individuallyvery broad (> 200km/s)
One has to do a deconvolution: correlation with templatesAs a function of type and stellar populations
Stellar spectra• Absorption lines
LOSVD
star
galaxy
Calcium triplet
V [km/s]
[ang]Deconvolution: G = S* LOSVD
LOSVD : Line Of Sight Velocity Distribution
43
Rotation of Ellipticals
Small E MB> -20.5: filledLarge E MB<-20.5 emptyBulges = crosses
from Davies et al (1983)
Solid line: relation for oblate rotators with isotropic dispersion(Binney 1978)
44
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
45
King ProfilesF(E) = 0 E> Eo
F(E) = (22)-1.5 o [ exp(Eo-E)/2 -1] E < Eo
C=log(rt/rc)
rt =tidal radiusrc= core radius
46
Deformations of Ellipticals
The various profiles correspond to the tidal deformation of ellipticalgalaxies
T1: isolated galaxiesT3: near neighbors
Depart from a de Vaucouleurs distribution
from Kormendy 1982
47
Triaxiality of ellipticals
Tests on observations show that elliptical galaxies aretriaxialWith triaxiality and variation of ellipticity with radius , There exists then isophote rotation
No intrinsic deformation!
48
Ellipticals & Early-types
Some galaxies are difficult to classify, between lenticularsand ellipticals. Most of E-galaxies have a stellar disk
49
Anisotropy of velocities= 1 –
r, -, 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
Radius
50
Comparison with data forN821 (green), N3379(violet)N4494 (brown), N4697 (blue)
Young stars arein yellow contours
51
SAURON Fast and slow rotators
FR have high and rising R
SR have flat or decreasing R
Emsellem et al 2007
52
SAURON Integral field spectroscopy
Emsellem et al 2007
53
Faber-Jackson relation for E-gal
Ziegler et al 2005
54
Tully-Fisher relation for spirals
Relation between maximum velocityand luminosityV corrected from inclinationMuch less scatter in I or K-band(no extinction)
Correlation with VflatBetter than Vmax
Uma clusterVerheijen 2001
55
McGaugh et al (2000) Baryonic Tully-Fisher
Tully-Fisher relationfor gaseous galaxiesworks much better inadding gas mass
Relation Mbaryons
with Rotational V
Mb ~ Vc4
56
Fundamental plane for E-gal
First found by Djorgovski et al 1987
57
Scaling relations
• Tully-Fisher: Mbaryons ~ v4
• Faber-Jackson: L ~ 4
• Fundamental Plane: