Stability and formation of the fractal

SAAS-FEE Lecture 4

Françoise COMBES

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Stability of the molecular diskUsual homogeneous disk: the Toomre criterion

Collaboration between the pressure at small-scale and the rotationat large-scale

Small-scale: Jeans criterion λJ = σ tff = σ/(2π Gρ)1/2

in 2D (disk) Σ = h ρ and h = σ2 / ( 2π G Σ ) ==> λJ = σ2 / ( 2π G Σ ) = h

Large-scale: Stabilisation by rotational shear

Tidal forces Ftid = d(Ω2 R)/dR ΔR ~ κ2 ΔR 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

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Stability when several components

Disk of stars and gas, each one stabilises or de-stabilises the other

Approximate estimation of the two contributionsUnstable if (k = 2π/λ)

(2π G k Σs)/ (κ2 + k2 σs2) + (2π G k Σg)/ (κ2 + k2 σg

2) > 1

For low values of k (large λ), the stellar component dominates theinstability; at small scale, the gas dominates by its low dispersion

For maintaining instabilities, gas is required, since it dissipatesStars may be unstable only transiently, since the componentheats up and becomes stable (self-regulation)

For star formation at large-scale Qg is not sufficient

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The tidal field is not always disruptiveit can also be compressive, in the centerof galaxies, when there is a flat core

If the mean density <ρ> of the sphericaldistribution is 3 M(R)/(4 π R3)

Ftid = -d(Ω2 R)/dR ΔR, Ω2 R = GM(R)/R2

Ftid= 4 π G (2/3 < ρ> - 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)

Ftid= 4 π G (2/(3- γ )- 1)ρ ΔR

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Can this play a role in the formationof dense nuclear gaseous disksin starburst galaxies?

High H2 volumic density predicted

In ULIRGs, the tidal field maybecome compressive inside200 pc

(Virial equilibrium, in presenceof compressive force)

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Stability and vertical structure

Combined Q as afunction of Qs & Qg

and the gas fraction ε

Jog, 1996

stable

unstable

Reduction factortaking into accountthe thicknessRomeo 1992

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Disks are marginally stable

What is the Q parameter at large-scale?

Exponential disks of stars ~ exp(-r/h), and exponentially decreasingvelocity dispersion ~ exp(-r/2h),accounting for constant scale-height (van der Kruit & Searle 81, 82)

self-gravitating isothermal slab (1st approx)ρ = ρo sech2 (z/zo) zo = σz

2 /(2πG Σ)correspond to observations

The derived Q values are about constant over the stellar disk ~2-3

Bottema (1993) Q ~ σr κ/ Σ

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Either M/L is assumed constant

or the thickness of the planes

as a function of luminosity

Final Q ( R )

Bottema 93

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Critical gas surface density

Often used to justify star formation (Kennicutt 89)Qg ~ σgκ/ Σ

gas unstable if Σ > Σ crit

Critical density reached for the ultimateHII regions radius

Here, only HI gas

No local correlation

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Star formation rateFor normal disks as for starburst galaxies, the star formation rateappears to be proportional to gas densityBut 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/tdyn SFR ~gas density/tff ~ ρ 1.5

or cloud-cloud collisions in ρ 2 (Scoville 00)

may explain the Tully-Fisher relation (Silk 97, Tan 00)L ~ R2Σ SFR ~ R2Σ g Ω Virial V2

~ Σ RL~V3

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

Slope 1

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Problems with the use of Qg

•Disks are self-regulated, on a dynamical time-scaleif gas too cold and unstable, gravitational instabilities developand heat the medium until marginal stability is reached

•Qg for stability might not be 1, but 2 or 3 according to the stellardisk properties (Qs) or the thickness, etc..

•Difficult to measure the total gas, especially the CO/H2 conversionratio not known within a factor 2

•Time delay for the feed-back?

•Instabilities: formation of structures, or stars?

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Small-scale stability

Always a puzzleFree fall time of small observed clumps is much less than 1 Myr

Pressure support is necessaryMagnetic field cannot halt the collapse

For an isothermal gas, fragmentation cannot be stoppeduntil the fragments are so dense that they become opticallythick, and shift in the adiabatic regime

Without external perturbations, the smallest fragment when thisoccurs is about 10-3 Motff ~t KH = 3/2 NkT/L with L = 4π f R2 σT4

M = 4 10-3 T1/4 μ-9/4 f-1/2 Mo (Rees 1976)

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Mass ~ 10-3 Modensity ~1010 cm-3

radius ~ 20 AU

N(H2) ~ 1025 cm-2

tff ~ 1000 yr

But the pressure supportensures that the life-timeis much longer

If in a fractal, collisionslead to coalescence, heating, and to a statistical equilibrium(Pfenniger & Combes 94)

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Observations: dense cores with isolated star formationdense cores with clustered star-formationdense cores without any star formation

The triggering of star formation could be due to un-balancedtime-scalesPertubation is a non-linear increase of velocity dispersion, due forinstance to galaxy encountersThese trigger collisions => either coalescence, or shredding andincrease of ΔVIf 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 supplyof gas, etc..) limit cycles appear, chaotic behaviour

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Gas in the outer parts

Observationnally, the gas in the outer parts is stable with respectto star formation, although not to gravitational perturbations

Examples of HI-21cm maps, with clumpy structure, andspiral structure at large-scale (cf M101, NGC 2915, etc..)

Similar conditions in LSB

Volumic density? Flaring? Linear, R2, or exponential flaring

==> Star formation and gravitational stability: not the same criterion

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NGC 2915ATCA HI

Regular rotation

Bar +spiral

Q > 5no instability

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Determination of the bar pattern speed

Method of Tremaine-Weinberg, based on the hypothesis of conservationof 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)

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HI surface density required in thedisk to explain the instabilitiesa) X = 3 swing optimisationb) Q=2

c) observed HI surface density

X = λ/λcrit

λcrit= 4π2 G Σ / κ2

X ~ κ r/σ Q

Ratio of a) to b) Σcrit for star formation

Scaled by 47.7

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If the dark matter is placed in the disk, it solves the problem ofcreating the observed instabilities (bar + spiral)

But then, it also would mean that the disk in unstable to star formationWhy 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)

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Formation of the structures

How to form and stabilize the hierarchical structure of the H2 gas?

Effect of self-gravity: at large scale, structures virialised withoutcontestationRecursive 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 atlow scale + taking into account galactic shear

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N-body simulations, periodicTree-code + collisional scheme

self-gravity +dissipationVariable time-stepsdt ~dr3/2

Initial tiny fluctuationsZeldovich approximation

Pv(k) ~ kα-2 Pρ(k) ~ kα

Scheme to stop the dissipationand fragmentation at the smallestscale (20AU)

117000 particules

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Two different schemes for dissipationsuper-elastic collisions at small scaleto inject energy at this level

Fractal D as a function of scaleVarious cruves, as a function oftime

Schema of the shear simulations

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Results of the shear simulations

the only way to maintain the fractalstructure is to re-inject energyat large scale

The natural way is from the galacticshear

Structure at small and large scalesubsist statistically

Constantly the shear destroys the small clumps formed again and again

Filaments continuously form atlarge scale

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Fractal dimension computed atdifferent epoch in the shear simulation

Independent of initial conditions

Several examples of extremedistributions and their Dimension DD independent of r is neither sufficientnor necessary

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Clump mass spectra for two values of α at different evolutiontimesunit (time) is tff/10

At t=5 slope -0.38+0.03 α=-1slope -0.18+0.03, α=-2

In summary: the galactic rotation is the best source of energyto maintain the fractal structure

Contrary to initial collapse (in cosmological simulations)a quasi- steady state could be obtainindependent of initial conditions

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Galaxy plane simulations, Huber & Pfenniger (01)

2D simulations with varying gas dissipation, FFT code, periodicweak, middle and strong

Different structures (more clumpy when strong)velocity dispersion increase

Middle dissipation

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Smaller D when moredissipation

3D with a thin planenecessary when clumpingcouples the 3rd dimension

Clumping in the z direction

Top: flat V( r ) Bottom: V( r ) ~ r 1/2

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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 smallto probe a true fractal structureand the Larson relations, for example

Problem of boundary conditions

Ellipsoid of velocity has the right shape, comparedto observations (Huber & Pfenniger 2001)σr > σφ > σz

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Conclusions

Gaseous disks, and in particular the H2 gas, are not in equilibriumor marginally

==> unstable at all scales, spiral structure, filaments, clumpyhierarchical structure

To explain this fractal, self-gravity is required, together to injectionof energy at large-scale (and may be small scale)

The galactic rotation is the main source of energy, and it takesGyr 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 structureformation is different than for star formation