stellar dynamics -- theory of spiral density waves dynamics of galaxies françoise combes

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Stellar Dynamics -- Theory of spiral density waves Dynamics of Galaxies Françoise COMBES

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Page 1: Stellar Dynamics -- Theory of spiral density waves Dynamics of Galaxies Françoise COMBES

Stellar Dynamics -- Theory of spiral density waves

Dynamics of Galaxies

Françoise COMBES

Page 2: 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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NGC 1365 (VLT)(R')SB(s)b

Messier 83 (VLT)NGC 5236SAB(s)c

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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

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

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The HI gas - Radial Extensions

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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?)

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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 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

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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

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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

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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 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

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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

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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

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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

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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

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Precession of orbits ofelliptical shape at rate Ω - κ/2

This quantity is almost constant all over theinner Galaxy

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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

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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

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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

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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

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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

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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

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Swing Amplification

Processus of amplification,when the leading paquettransforms in trailing

•Differential Rotation •self-gravity•Epicyclic motions

All three contribute to thisamplification

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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

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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

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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

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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

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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

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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

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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 rin the plane of the target

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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)

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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 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

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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

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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..

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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

Page 42: Stellar Dynamics -- Theory of spiral density waves Dynamics of Galaxies Françoise COMBES

Stellar spectra• Absorption lines

LOSVD

star

galaxy

Calcium triplet

V [km/s]

[ang]Deconvolution: G = S* LOSVD

LOSVD : Line Of Sight Velocity Distribution

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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)

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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 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

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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

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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!

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Ellipticals & Early-types

Some galaxies are difficult to classify, between lenticularsand ellipticals. Most of E-galaxies have a stellar disk

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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

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Comparison with data forN821 (green), N3379(violet)N4494 (brown), N4697 (blue)

Young stars arein yellow contours

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SAURON Fast and slow rotators

FR have high and rising R

SR have flat or decreasing R

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 velocityand luminosityV corrected from inclinationMuch less scatter in I or K-band(no extinction)

Correlation with VflatBetter than Vmax

Uma clusterVerheijen 2001

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McGaugh et al (2000) Baryonic Tully-Fisher

Tully-Fisher relationfor gaseous galaxiesworks much better inadding gas mass

Relation Mbaryons

with Rotational V

Mb ~ Vc4

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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 ~ 4

• Fundamental Plane: