1. Stability and formation of the fractal SAAS-FEE Lecture 4 Françoise COMBES

2. 2 Stability of the molecular disk Usual homogeneous disk: the Toomre criterion Collaboration between the pressure at small-scale and the rotation at large-scale Small-scale: Jeans criterion λ J = σ t ff = σ/(2π Gρ) 1/2 in 2D (disk) Σ = h ρ and h = σ 2 / ( 2π G Σ ) ==> λ J = σ 2 / ( 2π G Σ ) = h Large-scale: Stabilisation by rotational shear Tidal forces F tid = d(Ω 2 R)/dR ΔR ~ κ 2 ΔR Internal gravity forces of the condensation ΔR (G Σ π ΔR 2 )/ ΔR 2 = F tid ==> L crit ~ G Σ / κ 2 L crit = λ J ==> σ crit ~ π G Σ / κ Q = σ/ σ crit > 1

3. 3 Stability when several components Disk of stars and gas, each one stabilises or de-stabilises the other Approximate estimation of the two contributions Unstable if (k = 2π/λ) (2π G k Σ s )/ (κ 2 + k 2 σ s 2 ) + (2π G k Σ g )/ (κ 2 + k 2 σ g 2 ) > 1 For low values of k (large λ), the stellar component dominates the instability; at small scale, the gas dominates by its low dispersion For maintaining instabilities, gas is required, since it dissipates Stars may be unstable only transiently, since the component heats up and becomes stable (self-regulation) For star formation at large-scale Q g is not sufficient

4. 4 The tidal field is not always disruptive it can also be compressive, in the center of galaxies, when there is a flat core If the mean density of the spherical distribution is 3 M(R)/(4 π R 3 ) F tid = -d(Ω 2 R)/dR ΔR, Ω 2 R = GM(R)/R 2 F tid = 4 π G (2/3 - ρ) ΔR If flat density inside a certain radius (core), the gas will be compressed while the tides are always disruptive, in case of a power law density r -γ profile, with γ >1 (Das & Jog 1999) F tid = 4 π G (2/(3- γ )- 1)ρ ΔR

5. 5 Can this play a role in the formation of dense nuclear gaseous disks in starburst galaxies? High H 2 volumic density predicted In ULIRGs, the tidal field may become compressive inside 200 pc (Virial equilibrium, in presence of compressive force)

6. 6 Stability and vertical structure Combined Q as a function of Q s & Q g and the gas fraction ε Jog, 1996 stable unstable Reduction factor taking into account the thickness Romeo 1992

7. 7 Disks are marginally stable What is the Q parameter at large-scale? Exponential disks of stars ~ exp(-r/h), and exponentially decreasing velocity dispersion ~ exp(-r/2h), accounting for constant scale-height (van der Kruit & Searle 81, 82) self-gravitating isothermal slab (1st approx) ρ = ρo sech 2 (z/zo) zo = σ z 2 /(2πG Σ) correspond to observations The derived Q values are about constant over the stellar disk ~2-3 Bottema (1993) Q ~ σ r κ/ Σ

8. 8 Either M/L is assumed constant or the thickness of the planes as a function of luminosity Final Q ( R ) Bottema 93

9. 9 Critical gas surface density Often used to justify star formation (Kennicutt 89) Q g ~ σ g κ/ Σ gas unstable if Σ > Σ crit Critical density reached for the ultimate HII regions radius Here, only HI gas No local correlation

10. 10 Star formation rate For normal disks as for starburst galaxies, the star formation rate appears to be proportional to gas density But average on large-scale, the whole disk Global Schmidt law, with a power n=1.4 (Kennicutt 98) Σ SFR ~ Σ g 1.4 Another formulation works as well Σ SFR ~ Σ g Ω or Σ g /t dyn SFR ~gas density/t ff ~ ρ 1.5 or cloud-cloud collisions in ρ 2 (Scoville 00) may explain the Tully-Fisher relation (Silk 97, Tan 00) L ~ R 2 Σ SFR ~ R 2 Σ g Ω Virial V 2 ~ Σ R L~V 3

11. 11 Slope n=1.4 Normal galaxies (filled circles) starburst (squares) nuclei (open circles) Slope 1

12. 12 Problems with the use of Q g Disks are self-regulated, on a dynamical time-scale if gas too cold and unstable, gravitational instabilities develop and heat the medium until marginal stability is reached Q g for stability might not be 1, but 2 or 3 according to the stellar disk properties (Q s ) or the thickness, etc.. Difficult to measure the total gas, especially the CO/H2 conversion ratio not known within a factor 2 Time delay for the feed-back? Instabilities: formation of structures, or stars?

13. 13 Small-scale stability Always a puzzle Free fall time of small observed clumps is much less than 1 Myr Pressure support is necessary Magnetic field cannot halt the collapse For an isothermal gas, fragmentation cannot be stopped until the fragments are so dense that they become optically thick, and shift in the adiabatic regime Without external perturbations, the smallest fragment when this occurs is about 10 -3 Mo t ff ~t KH = 3/2 NkT/L with L = 4π f R 2 σT 4 M = 4 10 -3 T 1/4 μ -9/4 f -1/2 Mo (Rees 1976)

14. 14 Mass ~ 10 -3 Mo density ~10 10 cm -3 radius ~ 20 AU N(H 2 ) ~ 10 25 cm -2 t ff ~ 1000 yr But the pressure support ensures that the life-time is much longer If in a fractal, collisions lead to coalescence, heating, and to a statistical equilibrium (Pfenniger & Combes 94)

15. 15 Observations: dense cores with isolated star formation dense cores with clustered star-formation dense cores without any star formation The triggering of star formation could be due to un-balanced time-scales Pertubation is a non-linear increase of velocity dispersion, due for instance to galaxy encounters These trigger collisions => either coalescence, or shredding and increase of ΔV If there is a time-delay between the formation time of massive clouds leading to SF, and the SF feed-back, then a starburst is triggered Modelisation with many parameters (cooling of the gas, fresh supply of gas, etc..) limit cycles appear, chaotic behaviour

16. 16 Gas in the outer parts Observationnally, the gas in the outer parts is stable with respect to star formation, although not to gravitational perturbations Examples of HI-21cm maps, with clumpy structure, and spiral structure at large-scale (cf M101, NGC 2915, etc..) Similar conditions in LSB Volumic density? Flaring? Linear, R 2, or exponential flaring ==> Star formation and gravitational stability: not the same criterion

17. 17 NGC 2915 ATCA HI Regular rotation Bar +spiral Q > 5 no instability

18. 18 Determination of the bar pattern speed Method of Tremaine-Weinberg, based on the hypothesis of conservation of the matter along an orbit Measurement of the velocity and density profiles The bar is quite slow, its corotation is at 1.7Rb NGC 2915 isolated, what is the trigger of the bar+spiral? Either more gas in the disk? Or a tumbling triaxial halo (Bureau et al 99)

19. 19 HI surface density required in the disk to explain the instabilities a) X = 3 swing optimisation b) Q=2 c) observed HI surface density X = λ/λ crit λ crit = 4π 2 G Σ / κ 2 X ~ κ r/σ Q Ratio of a) to b) Σcrit for star formation Scaled by 47.7

20. 20 If the dark matter is placed in the disk, it solves the problem of creating the observed instabilities (bar + spiral) But then, it also would mean that the disk in unstable to star formation Why no stars? Another criterium taking into account volumic density? Warped distribution of the HI in NGC 2915 Dark halo could be triaxial, and tumbling very slowly? (Bureau et al 1999)

21. 21 Formation of the structures How to form and stabilize the hierarchical structure of the H 2 gas? Effect of self-gravity: at large scale, structures virialised without contestation Recursive fragmentation should occur Can form self-similar structure (field theory, renormalization group) N-body simulations (Semelin & Combes 00, Huber & Pfenniger 01) Unlike previous simulations, to form the dense cores (Klessen et al) there is a schematical process to change to adiabatic regime at low scale + taking into account galactic shear

22. 22 N-body simulations, periodic Tree-code + collisional scheme self-gravity +dissipation Variable time-steps dt ~dr 3/2 Initial tiny fluctuations Zeldovich approximation P v (k) ~ k α-2 P ρ (k) ~ k α Scheme to stop the dissipation and fragmentation at the smallest scale (20AU) 117000 particules

23. 23 Two different schemes for dissipation super-elastic collisions at small scale to inject energy at this level Fractal D as a function of scale Various cruves, as a function of time Schema of the shear simulations

24. 24 Results of the shear simulations the only way to maintain the fractal structure is to re-inject energy at large scale The natural way is from the galactic shear Structure at small and large scale subsist statistically Constantly the shear destroys the small clumps formed again and again Filaments continuously form at large scale

25. 25 Fractal dimension computed at different epoch in the shear simulation Independent of initial conditions Several examples of extreme distributions and their Dimension D D independent of r is neither sufficient nor necessary

26. 26 Clump mass spectra for two values of α at different evolution times unit (time) is tff/10 At t=5 slope -0.38+0.03 α=-1 slope -0.18+0.03, α=-2 In summary: the galactic rotation is the best source of energy to maintain the fractal structure Contrary to initial collapse (in cosmological simulations) a quasi- steady state could be obtain independent of initial conditions

27. 27 Galaxy plane simulations, Huber & Pfenniger (01) 2D simulations with varying gas dissipation, FFT code, periodic weak, middle and strong Different structures (more clumpy when strong) velocity dispersion increase Middle dissipation

28. 28 Smaller D when more dissipation 3D with a thin plane necessary when clumping couples the 3rd dimension Clumping in the z direction Top: flat V( r ) Bottom: V( r ) ~ r 1/2

29. 29 Strongly depends on differential rotation and dissipation The structure shifts from filamentary to clumpy, when the dissipation increases, and when the shear decreases The dynamical range of the simulations until now is too small to probe a true fractal structure and the Larson relations, for example Problem of boundary conditions Ellipsoid of velocity has the right shape, compared to observations (Huber & Pfenniger 2001) σ r > σ φ > σ z

30. 30 Conclusions Gaseous disks, and in particular the H 2 gas, are not in equilibrium or marginally ==> unstable at all scales, spiral structure, filaments, clumpy hierarchical structure To explain this fractal, self-gravity is required, together to injection of energy at large-scale (and may be small scale) The galactic rotation is the main source of energy, and it takes Gyr for the gas in a galactic disk to flow slowly to the center (faster in the case of perturbations) The criterium for gravitational instabilities, for cloud and structure formation is different than for star formation