Stellar Dynamics -- Theory of spiral density waves

1. 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. Messier 83 (VLT) NGC 5236 SAB(s)c NGC 1365 (VLT) (R')SB(s)b

5. Hubble Sequence (tuning fork) Sequence of mass, of concentration Gas Fraction

6. 3 109 0.1 – 10 100 - 1000 100 - 1000 103 - 104 10 000 1 – 5 109 105 - 106 103 - 105 10 5 107 10-40 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

7. The HI gas - Radial Extensions

8. M83: optical Extension of galaxies in HI 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

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 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

12. 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 ω2 = σ2k2 - 4 π G ρ0 (σ velocity dispersion) Jeans length λJ = σ / (G ρ0)1/2 = σ tff The scales l > λJ are unstable

13. 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 / κQ = 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 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

16. 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

17. 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

18. 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

19. 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

20. Precession of orbits of elliptical shape at rate Ω - κ/2 This quantity is almost constant all over the inner Galaxy

21. 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

22. 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

23. 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

24. 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 developpment of waves) around corotation

25. 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

26. 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

27. Swing Amplification • Processus of amplification, • when the leading paquet • transforms in trailing • Differential Rotation • self-gravity • Epicyclic motions • All three contribute to this • amplification

28. 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

29. 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

30. 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

31. 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

32. 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

33. 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

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 r in the plane of the target

35. 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)

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 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

39. 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

40. 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..

41. 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

42. Stellar spectra galaxy • Absorption lines star Calcium triplet Deconvolution: G = S*  LOSVD LOSVD : Line Of Sight Velocity Distribution l [ang] LOSVD V [km/s]

43. 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)

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 Profiles F(E) = 0 E> Eo F(E) = (2p s2)-1.5ro [ exp(Eo-E)/s2 -1] E < Eo C=log(rt/rc) rt =tidal radius rc= core radius

46. 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

47. 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!

48. Ellipticals & Early-types Some galaxies are difficult to classify, between lenticulars and ellipticals. Most of E-galaxies have a stellar disk

49. Radius 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

50. Young stars are in yellow contours Comparison with data for N821 (green),N3379(violet) N4494 (brown),N4697 (blue)