Stability and formation of the fractal

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Stability and formation of the fractal

• SAAS-FEE Lecture 4
• Françoise COMBES

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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 ?J s
tff s/(2p G?)1/2 in 2D (disk) S h ? and h
s2 / ( 2p G S ) gt ?J s2 / ( 2p G S )
h Large-scale Stabilisation by rotational
shear Tidal forces Ftid d(O2 R)/dR ?R ?2
?R Internal gravity forces of the condensation
?R (G S p ?R2)/ ?R2 Ftid gt Lcrit G S /
?2 Lcrit ?J gt scrit p G S / ? Q
s/ scrit gt 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 contributions Unstable if (k 2p/?)
(2p G k Ss)/ (?2 k2 ss2) (2p G k Sg)/ (?2
k2 sg2) gt 1 For low values of k (large ?),
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 Qg is not sufficient
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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
lt?gt of the spherical distribution is 3 M(R)/(4 p
R3) Ftid -d(O2 R)/dR ?R, O2 R
GM(R)/R2 Ftid 4 p G (2/3 lt ?gt - ?) ?R
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-? profile, with ? gt1 (Das Jog 1999) Ftid 4
p G (2/(3- ? )- 1)? ?R
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Can this play a role in the formation of dense
nuclear gaseous disks in starburst
galaxies? High H2 volumic density predicted
In ULIRGs, the tidal field may become compressive
inside 200 pc (Virial equilibrium, in
presence of compressive force)
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Stability and vertical structure
Reduction factor taking into account the
thickness Romeo 1992
stable
Combined Q as a function of Qs Qg and the gas
fraction e Jog, 1996
unstable
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Disks are marginally stable
What is the Q parameter at large-scale? Exponenti
al 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) ? ?o sech2 (z/zo)
zo sz2 /(2pG S) correspond to
observations The derived Q values are about
constant over the stellar disk 2-3 Bottema
(1993) Q sr ?/ S
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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 sg?/ S gas unstable if S gt S crit
Critical density reached for the ultimate HII
regions radius Here, only HI gas No local
correlation
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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 n1.4
(Kennicutt 98) S SFR
S g 1.4 Another formulation works as well
S SFR S g O or S g/tdyn SFR
gas density/tff ? 1.5 or cloud-cloud
collisions in ? 2 (Scoville 00) may explain the
Tully-Fisher relation (Silk 97, Tan 00) L R2S
SFR R2S g O Virial V2 S R LV3
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Slope n1.4 Normal 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-scale
• if gas too cold and unstable, gravitational
  instabilities develop
• and heat the medium until marginal stability is
  reached
• Qg for stability might not be 1, but 2 or 3
  according to the stellar
• disk properties (Qs) 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?

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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 tff t KH 3/2
NkT/L with L 4p f R2 sT4 M 4 10-3 T1/4
µ-9/4 f-1/2 Mo (Rees 1976)
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Mass 10-3 Mo density 1010 cm-3 radius 20
AU N(H2) 1025 cm-2 tff 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)
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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 gt either coalescence, or shredding
and increase of ?V 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
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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, R2, or exponential
flaring gt Star formation and gravitational
stability not the same criterion
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NGC 2915 ATCA HI Regular rotation
Bar spiral
Q gt 5 no instability
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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 barspiral? Either more gas in
the disk? Or a tumbling triaxial halo (Bureau et
al 99)
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HI surface density required in the disk to
explain the instabilities a) X 3 swing
optimisation b) Q2
Scaled by 47.7
c) observed HI surface density
X ?/?crit ?crit 4p2 G S / ?2 X ? r/s Q
Ratio of a) to b) Scrit for star formation
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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)
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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
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
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N-body simulations, periodic Tree-code
collisional scheme self-gravity
dissipation Variable time-steps dt
dr3/2 Initial tiny fluctuations Zeldovich
approximation Pv(k) ka-2 P?(k) ka
Scheme to stop the dissipation and
fragmentation at the smallest scale (20AU)
117000 particules
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Two different schemes for dissipation super-elasti
c collisions at small scale to inject energy at
this level
Schema of the shear simulations
Fractal D as a function of scale Various cruves,
as a function of time
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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
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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
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Clump mass spectra for two values of a at
different evolution times unit (time) is
tff/10 At t5 slope -0.380.03 a-1 slope
-0.180.03, a-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
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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
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Clumping in the z direction
Smaller D when more dissipation
3D with a thin plane necessary when
clumping couples the 3rd dimension
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 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) sr gt sf gt sz
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Conclusions
Gaseous disks, and in particular the H2 gas, are
not in equilibrium or marginally gt 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