Fractal structure of H2 gas

SAAS-FEE Lecture 3

Françoise COMBES

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Definition of a fractal

Astrophysical fractals: only an approximationonly between two limiting scalesdifferent from pure mathematical fractals than are infinite

Random fractals (statistical self-similarity)

No characteristic scaleHaussdorff dimension D

M ( r ) ~ rD

non integer dimension D ~1.7

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Taurus Molecular Cloudat 100pc from the Sun

IRAS 100μ emissionfrom heated dust

Self-similar structure(except for resolution!)

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FCRAO CO survey of the 2nd quadrant (Mark Heyer et al)

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Interstellar Clouds

Very irregular and fragmented structure

From size 100pc, the Giant Molecular Clouds or GMCs

down to 10 AU, structures observed in VLBI(HI in absorption in front of quasars, Diamond et al 89, Davis et al96, Faison et al 98)TSAS tiny scale atomic structures (Heiles 97)

Also in front of pulsars (Frail et al 94)Extreme Scattering events "ESE" detected in QSO monitoringFiedler et al (87, 94, 97)

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Interpretation of the origin of these ESEne ~103 cm-3 electrons required to scatter

How do these clouds hold together? Long controversyover pressureor self-gravitating (Pfenniger & Combes 1994)Walker & Wardle (98) evaporation model

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Clouds mapped in HI 21cm absorption

3C138

10 AU

VLBA, or VLBI(Davis et al 96, Faison et al 98)

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Fractal Structure -- Scaling laws

Self-similarity -- Larson relations (1981)

Size-linewidth relations ΔV ~ Rq 0.3 < q < 0.5

Virialisation of almost all scales ΔV2 ~ M/R(debated at small scale, where there is no good mass tracer)

Size-Mass M ( r ) ~ rD 1.6 < D < 2

Density decreases as ~R-α 1 < α < 1.4

Hierarchical structure, or treedegree of imbrication not well known (Houlahan & Scalo 92)

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Hierarchical orrandomized?

Evidence is found moreof a hierarchical model(Houlahan & Scalo 92)

with 0.04 filling factorin Taurus

efficiency of fragmentation0.4

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Filamentary structure

Taurus-Auriga 100μ Cold matter = I100μ -I60μ/0.15

Abergel et al1994

Aspect ratio close to 10 or larger, 30% massAbrupt transition (shocks? Not photodissociation)Confinement?Scaling in mass with star-forming activity

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Scaling relations

Size-linewidth from galacticsurveysIf virial assumed, the second law is derivedJustification to derive H2 massConversion ratio based on virial

Slope size-line width between 0.3 and 0.5

Mass-to-size, between 1.6 and 2, larger if small-scales are notvirialised

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High Latitude CloudsHeithausen 96, Magnani et al 85

compared to GMC (Solomon et al 87)

CO is not a good tracerof H2 mass(slope different from 1)

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Area-contour length relation

P ~A D2/2 D2 = 1.36 (CO, IRAS, HI)Bazell & Désert 1988, Vogelaar & Walker 1994

Projection of a fractalDp=D if D < 2, Dp = 2 if D > 2 (Falconer 1990)

Bazell & Désert 88

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Observational BiasesObservations of an optically thick line COradiative transfer in a fractal?

Mass spectrum dN/ dm ~ mγ, with γ ~ -1.5(Heithausen et al 1996, 98)

This power law is not related to the fractal dimension, but to thegeometry of the fractal, its hierachical character, etc..

If the fractal is entirely hierachical, the power-law expected is 1

since m N(m) = cste or n(M) dlogm ~m-1 dlogm

(and all the mass is included in the smallest fragments)

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Formation by recursive Jeans fragmentation?

A simple way to form a hierarchical fractal

ML = N ML-1

rLD = NrL-1

D

α = rL-1/rL= N-1/D

cf Pfenniger & Combes 1994

D=2.2

D=1.8

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Projected mass log scale (15 mag)

N=10, L=9

Filling factor in surface

is a strong function of D

less than 1% at D=1.7

Pfenniger & Combes 1994

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Computation of surfacefilling factors

DM/DΣcumulated M (< Σ)and Area (> Σ)

D=3 (solid)D=2.5 (dot)D=2 (short dash)D=1.5 (long dash)

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Ratios (log) of the cumulated mass in various D models

D=3/D=2.5 dots D=3/D=2 dash D=3/D=1.5 long-dash

Case A density law truncated 1/r2 in each clumpCase B less abruptly truncated 1/r2 (1 +r/rL)5

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Turbulence

Very low viscosity => Reynolds number Re = v d /ν ~ 109 >> Rc

Advection term dominates viscous term

But only incompressible turbulence is well known,here compressible and supersonic

Energy transfer v2 / (r/v)

==> Kolmogorov relation v ~ r1/3

Energy cascade, injected at large scales, dissipated at small scales

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Intermittence: a current feature of fluid turbulenceNon-gaussian lines, large wings

2D turbulence (section pictures) appear to yield the same dimensions for area-contour length fractal D2 = 1.36 (Sreenivasan & Méneveau 1986)

Chaos: sensitivity to initial conditionsnon-linearity, fluctuations at all scales

Two-gaussian fits in thesevelocity profilesFalgarone & Phillips 90

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Numerical simulations of turbulence

Magnetic field important at small scale, traced by orientation of thedust, and polarisation of lightB sometimes //, sometimes perpendicular to the filaments

Intensity traced by Zeeman splitting, B ~20 μG

Hydrodynamical, MHD, 2D essentiallyNot yet enough dynamical range

Self-similar statistics?

Vasquez-Semadeni et al 1994: without pressure, without self-gravity, only 2 hierarchical levels

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Δv ~ r1/2 ρ ~ r-1 ==> Δv ~ ρ -1/2 Since dP/d ρ = Δv2 , then dP/d ρ = ρ -1

equation of "logatrope" P ~ log ρ

This equation has not been retrieved in numerical tests (Vasquez-Semadeni et al 1998), instead a polytrope with γ = 2

But: depends on equilibrium states of clouds (McLaughlin & Pudritz 1996)

Are Larson relations retrieved in 2D hydro simulations(self-gravity + MHD) ?No (Vasquez-Semadeni et al 1997)Problems: diffusion, hyper-viscosity, clouds of only a few pixels

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2D turbulent simulation800x800, with star formation70 MyrRatio 1000 between max andmin densities(Vazquez-Semadeni et al 97)

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Chemical fluctuations: can thicken the picture

Simulations from Rousseau et al (1998)minimum level of modelisation: time dependent, radiative transfer+chemistry, frozen turbulent velocity spectrum

chemical coupling, no characteristic scale

Thermal instability may play a role in triggering star formation,but appears as secondary, with respect to stellar forcing, self-gravity,magnetic fields (Vazquez-Semadeni et al 00, Nomura & Kamaya 01)

Y=log(Ab) vs T in several cellsline is the asymptoticsteady behaviour

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Self-gravity

In turbulent simulations with Self-g, most power is on small scalesdue to gravitational collapse (Ossenkopf et al (2001)

Already Hoyle (1953) proposes recursive fragmentation, in anisothermal regime

Principle: Jeans mass decreases faster than the cloud fragmentstff ~ ρ-1/2 MJ ~ ρ-1/2

Properties of an isothermal self-gravitating gas: negative specific heat==> gravothermal catastrophy (Lynden-Bell & Wood 1978)

Time of collapse ~time of first free-fall α = rL-1/rL= N-1/D

τ ~ 1 + 1/k + 1/k2 + … = k/(k-1) k = (5 fragments)1/4 ~ 1.5

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Relation between dimensionlesstemperature (kT rb/GM)

and dimensionless Energy =Erb/GM2

for a mass M of isothermal gas in aspherical container of radius rb

The curve spirals inwards to the pointcorresponding to the singular isothermal sphere

Gravothermal Catastrophy

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Turbulence in a gravitational field, more than pressure

Statistical equilibrium between collapse, fragmentation, coalescence

collisions are favored because of the fractaldistribution (Pfenniger & Combes 1994)

Simulations: big numerical problems alsoexample: artificial fragmentation (Truelove et al 97)

Must always have cell < Jeans size•artificial viscosity•softening necessary

•error to have minimum size of pressure forces smallerthan gravitational softening

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Klessen et al (1997, 98, 00): self-gravity + hydro, periodic boundary conditions

Initially: gaussian perturbations

ZEUS-3D, or SPHimpossible to prevent densecores to formor unrealistic short-scale driving

There can be local collapseeven in global stability

Can explain isolated versuscluster star formation

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Fraction of mass in dense coresvs time, with different driving(cores are replaced by a sink particle)

Magnetic field cannot prevent collapse

Supersonic turbulence, will globally support a molecular cloudbut will allow local collapse

Fluctuations in local turbulent flow are highly transient

To maintain the stability of clouds, required short-scale drivingStellar driving?

When globally unstable ==> stellar clusters

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Klessen et al (98)

Gas clumps (thin line)Protostellar cores (thick)

vertical: limit with N=5105

dN/ dm ~ mγ, with γ ~ -1.5

At the end figure, 60% of themass in cores

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Molecular Fractal and IMF

Several theories for the IMF, at least 4 types of models

(1) Formation of a single star, then variation of parameters to getseveral star masses (Larson 73, Silk 77, Zinnecker 84)

Random variations with time and positions

(2) Clustered star formation, and protostars interaction (Bastien 81,

Murray & Lin 96) simplistic interactions, collisions, coalescenceAdvantage: most stars are born in clusters

(3) Clustered star formation, with competitive accretion (Larson 78,Tohline 80, Myers 00)

Assumes nearly uniform reservoir of gasBut dense clumps are observed (Motte et al 98, Belloche et al 00), with a small filling factor and they don't favor accretion

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IMF slopes in different clustersfrom Scalo (1998)Salpeter -1.35 = dashMW solidLMC open

Models from randomsampling(Elmegreen 1999)

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(4) Observations favor stars forming in pre-formed clumpsTherefore the IMF does not give any highlight in star formationbut only in cloud formation!

The IMF has two characteristic masses, and 3 slopes

=> why the characteristic mass of 0.3-1 Mo?By accretion and collapse of smaller clumps

Stars grow until a self-limiting mass, a little larger than deuteriumburning limit (stellar winds) (Larson 82, Shu et al 87)

It is not the smallest fragments in clouds that determine the smallest star mass!

Are the smallest clouds self-gravitating?(Elmegreen & Falgarone 96, Walker & Wardle 98)

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The smallest mass depends on temperature and local pressure(Bonnert-Ebert)M ~P1.5 n-2

Note a certain chaos between initial cloud masses and the finalstar mass, due to a series of dynamical events, unpredictable(accretion, fragmentation, turbulence, self-gravity..)

Another model for a limiting smallest mass,opacity-limited clumps, of the order of 10-3 Mo

Then accretion was to operate to reach the first stellar mass(and may be form brown dwarfs in between)Rees 1976, Yoshii & Saio 85

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Pure hierarchy of clouds n(M)dlogm ~m-1 dlogmthis contains equal mass per logarithmic interval. Clouds are hierarchical, but it is difficult to know whether the hierarchyis pure

The probability to have a random mass m is

m-1 dlogm or m-2 dm

This is what is observed, n(m) in m-1.85

(Heithausen et al 98)

Stellar clusters also obey m-1 dlogm

Stars also would have this kind of law, except that they compete with gas

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Once a star is formed at a certain level of the hierarchy, the higher scale has no longer any gas to form other stars

At a higher level, a star that normally would have contained much higher mass, because comprising a lot of subclumps, has now a masslower than expected from the cloud hierarchy

==> steepening the slope of the mass spectrum, during the processof star formationClusters don't compete for mass, since they have the total mass of their components => not true for stars

Another parameter: time-scale of star formation tff ~ρ-1/2 ~ r1/2

==> again steepens the IMF

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Star formation is a dynamical process, and the fractal re-adjusts itsstructure during the process

The two factors for steepening can explain the Salpeter law

m-1.35 dlogm

instead of that of the pure hierarchy law(simulations by Elmegreen 97, 99)Then the intermediate to high m of the IMF comes from scales largerby a factor 10 (from the m ~rD, with D=2)

The low mass end (flat IMF) comes from fragmentation

Models done from random sampling in hierarchical clouds arereasonably corresponding to observations

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Ns = ƒ nc P(ε) dlog εε = Ms/Mc= Mstar/Mclump

P(ε) dlog ε =cste

minimum mass?

Physical mechanism unknown

for instance formation of browndwarfs in accretion disk of stars

Slope 1.3

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Is there a physical limit for the upper mass?

-- Eddington limit (if optically thin, Norbert & Maeder 00)-- accretion (Zinnecker et al 86)

If the IMF is prolonged Mup = 7000 MoBut observed is 120-150 Mo (30 Dor)

Why? Problem of oscillations, instabilities (but stars, not proto-stars)

Timing? After the smaller stars have formed, then a large fraction of the cloud is used up, and this leads to cloud destruction, before awhole GMC can form a big star

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Fractal structure in Galactic Star Fields

HST images of 10 galaxies (archive) are gaussian smoothed atvarious scales from 10kpc to kpc along the spiral arms(Elmegreen & Elmegreen 2001)

==> gives the fractal dimension D = 2.3

about the same as that of interstellar cloudsPassive tracer?

The densest structures are like the Pleiades, at the bottom ofthe hierarchy

If all stars form in the densest clouds, the fractal structure is then onlydue to the hierarchy of their position

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Observations of NGC 2207

n(S) dlog(S) ~ S-D dlog(S), with D=1.12

But projection effects, and overlap in counting?

Models with n(R) dlog(R) ~ R-2.3 dlog(R) depending essentially on the center positions and not on the shape of the clouds (spectrum of cloud sizes)

Size-luminosity relation for star clustersL ~ R2.3 (Elmegreen et al 2001)

Number of HII regions n(L) ~ L-2 dL (Kennicutt et al 89, Oey & Clarke 98)

Models are smoothed by photoshop by 2, 4, 8, 16, 32, 64 pixelsand counted!

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NGC 2207 galaxy

6 levels of smoothingfor this star-forming region

(nb of pixels)

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Number of objects foundaccording to the smoothing level (Elmegreen & Elmegreen 2001)

+ several galaxies

dash = slope 1 observations best fitted byD=2.3

Models withD=2.3

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They show that approximately the projection has fractal dimensionone less than the 3D object D=1.12 obs means D ~2 in 3D space

Stars form in the densest part of the fractal structure of the cloudsThey keep the global fractal structure at least for a few tdyn of the largest structures

tdyn is short at small scales, where the stars have time to disperse, but at large scale the fractal survives (Kroupa 2000)

τ ~ r1/2

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Conclusions

Molecular clouds have a fractal structure, over 9 orders in massand 6 orders in scale

Fractal dimension around 2, filamentary geometry

Highly Hierarchical

Turbulence and self-gravity are the key factors

Formation of stars, and IMF, results from this structure