Fractal structure of H

2

gas

SAAS-FEE Lecture 3 Françoise COMBES

Definition of a fractal

Astrophysical fractals:

only an approximation

only between two limiting scales

different from pure mathematical fractals than are infinite Random fractals (statistical self-similarity)

No characteristic scale

Haussdorff dimension D

M ( r ) ~ r D

non integer dimension D ~1.7

2

Taurus Molecular Cloud at 100pc from the Sun IRAS 100μ emission from heated dust Self-similar structure (except for resolution!) 3

FCRAO CO survey of the 2nd quadrant (Mark Heyer et al) 4

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 al 96, 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 monitoring Fiedler et al (87, 94, 97) 5

Interpretation of the origin of these ESE ne ~10 3 cm -3 electrons required to scatter How do these clouds hold together?

over pressure or self-gravitating (Pfenniger & Combes 1994) Walker & Wardle (98) evaporation model Long controversy 6

Clouds mapped in HI 21cm absorption

3C138 10 AU VLBA, or VLBI (Davis et al 96, Faison et al 98) 7

Fractal Structure -- Scaling laws

Self-similarity -- Larson relations (1981) Size-linewidth relations ΔV ~ R q 0.3 < q < 0.5

Virialisation of almost all scales ΔV 2 ~ M/R (debated at small scale, where there is no good mass tracer) Size-Mass

M ( r ) ~ r D

1.6 < D < 2

Density decreases as ~R -α 1 < α < 1.4

Hierarchical structure, or tree degree of imbrication not well known (Houlahan & Scalo 92) 8

Hierarchical or randomized?

Evidence is found more of a hierarchical model (Houlahan & Scalo 92) with 0.04 filling factor in Taurus efficiency of fragmentation 0.4

9

Filamentary structure

Aspect ratio close to 10 or larger, 30% mass Abrupt transition (shocks? Not photodissociation) Confinement?

Scaling in mass with star-forming activity Taurus-Auriga 100μ Cold matter = I100μ -I60μ/0.15

Abergel et al 1994 10

Scaling relations Size-linewidth from galactic surveys If virial assumed, the second law is derived Justification to derive H 2 Conversion ratio based mass 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 not virialised 11

CO is not a good tracer of H 2 mass (slope different from 1) High Latitude Clouds Heithausen 96, Magnani et al 85 compared to GMC (Solomon et al 87) 12

Area-contour length relation P ~A D2/2 D2 = 1.36

(CO, IRAS, HI) Bazell & Désert 1988, Vogelaar & Walker 1994 Projection of a fractal Dp=D if D < 2, Dp = 2 if D > 2 (Falconer 1990) Bazell & Désert 88 13

Observational Biases

Observations of an optically thick line CO radiative transfer in a fractal?

Mass spectrum dN/ dm ~ m γ , with γ ~ -1.5

(Heithausen et al 1996, 98) This power law is not related to the fractal dimension, but to the geometry 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) 14

D=1.8

Formation by recursive Jeans fragmentation?

A simple way to form a hierarchical fractal M L = N M L-1 r L D = Nr L-1 D α = r L-1 /r L = N -1/D cf Pfenniger & Combes 1994 D=2.2

15

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 16

Computation of surface filling factors DM/DΣ cumulated M (< Σ) and Area (> Σ) D=3 (solid) D=2.5 (dot) D=2 (short dash) D=1.5 (long dash) 17

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/r 2 in each clump Case B less abruptly truncated 1/r 2 (1 +r/r L ) 5 18

Turbulence

Very low viscosity => Reynolds number Re = v d /ν ~ 10 9 >> Rc Advection term dominates viscous term But only incompressible turbulence is well known, here

compressible and supersonic

Energy transfer v 2 / (r/v) ==> Kolmogorov relation v ~ r 1/3 Energy cascade, injected at large scales, dissipated at small scales 19

Intermittence: a current feature of fluid turbulence Non-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 conditions non-linearity, fluctuations at all scales Two-gaussian fits in these velocity profiles Falgarone & Phillips 90 20

Numerical simulations of turbulence

Magnetic field important at small scale, traced by orientation of the dust, and polarisation of light B sometimes //, sometimes perpendicular to the filaments Intensity traced by Zeeman splitting , B ~20 μG Hydrodynamical, MHD, 2D essentially Not yet enough dynamical range

Self-similar statistics?

Vasquez-Semadeni et al 1994: without pressure, without self gravity, only 2 hierarchical levels 21

Δv ~ r 1/2 ρ ~ r -1 ==> Δv ~ ρ -1/2 Since dP/d ρ = Δv 2 , 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 22

2D turbulent simulation 800x800, with star formation 70 Myr Ratio 1000 between max and min densities (Vazquez-Semadeni et al 97) 23

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 Y=log(Ab) vs T in several cells line is the asymptotic steady behaviour Thermal instability may play a role in triggering star formation, but appears as secondary, with respect to stellar forcing, self-gravity,

Self-gravity

In turbulent simulations with Self-g, most power is on small scales due to gravitational collapse (Ossenkopf et al (2001) Already Hoyle (1953) proposes recursive fragmentation , in an isothermal regime Principle: Jeans mass decreases faster than the cloud fragments t ff ~ ρ -1/2 M J ~ 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 = r L-1 /r L = N -1/D τ ~ 1 + 1/k + 1/k 2 + … = k/(k-1) k = (5 fragments) 1/4 ~ 1.5

25

Gravothermal Catastrophy

Relation between dimensionless temperature (kT rb/GM) and dimensionless Energy =Erb/GM 2 for a mass M of isothermal gas in a spherical container of radius rb The curve spirals inwards to the point corresponding to the singular isothermal sphere 26

Turbulence in a gravitational field, more than pressure

Statistical equilibrium

between collapse, fragmentation, coalescence collisions are favored because of the fractal distribution (Pfenniger & Combes 1994) Simulations: big numerical problems also example: artificial fragmentation (Truelove et al 97) Must always have cell < Jeans size •artificial viscosity •softening necessary •error to have minimum size of pressure forces smaller than gravitational softening 27

Klessen et al (1997, 98, 00):

self-gravity + hydro, periodic boundary conditions Initially: gaussian perturbations ZEUS-3D, or SPH impossible to prevent dense cores to form or unrealistic short-scale driving There can be local collapse even in global stability Can explain isolated versus cluster star formation 28

Fraction of mass in dense cores vs time, with different driving (cores are replaced by a sink particle) Magnetic field cannot prevent collapse Supersonic turbulence, will globally support a molecular cloud but will allow local collapse Fluctuations in local turbulent flow are highly transient To maintain the stability of clouds, required short-scale driving Stellar driving?

When globally unstable ==> stellar clusters 29

Klessen et al (98) Gas clumps (thin line) Protostellar cores (thick) vertical: limit with N=510 5 dN/ dm ~ m γ , with γ ~ -1.5

At the end figure, 60% of the mass in cores 30

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 get several 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, coalescence Advantage: most stars are born in clusters (3) Clustered star formation, with competitive accretion (Larson 78, Tohline 80, Myers 00) Assumes nearly uniform reservoir of gas But dense clumps are observed (Motte et al 98, Belloche et al 00), with a small filling factor and they don't favor accretion 31

IMF slopes in different clusters from Scalo (1998) Salpeter -1.35 = dash MW solid LMC open Models from random sampling (Elmegreen 1999) 32

(4) Observations favor stars forming in pre-formed clumps Therefore the IMF does not give any highlight in star formation but 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 deuterium burning 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) 33

The smallest mass depends on temperature and local pressure (Bonnert-Ebert) M ~P 1.5

n -2 Note a certain chaos between initial cloud masses and the final star 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 34

Pure hierarchy of clouds n(M)dlogm ~m -1 dlogm this contains equal mass per logarithmic interval. Clouds are hierarchical, but it is difficult to know whether the hierarchy is 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 35

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 mass lower than expected from the cloud hierarchy ==> steepening the slope of the mass spectrum, during the process of star formation Clusters 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 t ff ~ρ -1/2 ~ r 1/2 ==> again steepens the IMF 36

Star formation is a dynamical process, and the fractal re-adjusts its structure 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 larger by a factor 10 (from the m ~r D , with D=2) The low mass end (flat IMF) comes from fragmentation Models done from random sampling in hierarchical clouds are reasonably corresponding to observations 37

Ns = ƒ nc P(ε) dlog ε ε = Ms/Mc= Mstar/Mclump P(ε) dlog ε =cste minimum mass?

Physical mechanism unknown for instance formation of brown dwarfs in accretion disk of stars Slope 1.3

38

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 Mo But 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 a whole GMC can form a big star 39

Fractal structure in Galactic Star Fields

HST images of 10 galaxies (archive) are gaussian smoothed at various 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 clouds Passive tracer?

The densest structures are like the Pleiades, at the bottom of the hierarchy If all stars form in the densest clouds, the fractal structure is then only due to the hierarchy of their position 40

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 clusters L ~ R 2.3

(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 pixels and counted!

41

NGC 2207 galaxy 6 levels of smoothing for this star-forming region (nb of pixels) 42

Models with D=2.3

Number of objects found according to the smoothing level (Elmegreen & Elmegreen 2001) + several galaxies dash = slope 1 observations best fitted by D=2.3

43

They show that approximately the projection has fractal dimension one 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 clouds They keep the global fractal structure at least for a few t dyn largest structures of the t dyn is short at small scales, where the stars have time to disperse, but at large scale the fractal survives (Kroupa 2000) τ ~ r 1/2 44

Conclusions

Molecular clouds have a fractal structure , over 9 orders in mass and 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 45