Transcript Stability and formation of the fractal

Stability and formation of the
fractal
SAAS-FEE Lecture 4
Françoise COMBES
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 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
2
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 Σs)/ (κ2 + k2 σs2) + (2π G k Σg)/ (κ2 + k2 σg2) > 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 Qg is not sufficient
3
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 <ρ> of the spherical
distribution 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
4
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)
5
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 ε
Jog, 1996
6
unstable
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 sech2 (z/zo) zo = σz2 /(2πG Σ)
correspond to observations
The derived Q values are about constant over the stellar disk ~2-3
Bottema (1993) Q ~ σr κ/ Σ
7
Either M/L is assumed constant
or the thickness of the planes
as a function of luminosity
Final Q ( R )
Bottema 93
8
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 ultimate
HII regions radius
Here, only HI gas
No local correlation
9
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/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 R
L~V3
10
Slope n=1.4
Normal galaxies (filled circles)
starburst (squares)
nuclei (open circles)
Slope 1
11
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?
12
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 = 4π f R2 σT4
M = 4 10-3 T1/4 μ-9/4 f-1/2 Mo (Rees 1976)
13
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)
14
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
15
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
==> Star formation and gravitational stability:
not the same criterion
16
NGC 2915
ATCA HI
Regular rotation
Bar +spiral
Q>5
no instability
17
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)
18
HI surface density required in the
disk to explain the instabilities
a) X = 3 swing optimisation
b) Q=2
Scaled by 47.7
c) observed HI surface density
X = λ/λcrit
λcrit= 4π2 G Σ / κ2
X ~ κ r/σ Q
19
Ratio of a) to b) Σcrit for star formation
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)
20
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
21
N-body simulations, periodic
Tree-code + collisional scheme
self-gravity +dissipation
Variable time-steps
dt ~dr3/2
Initial tiny fluctuations
Zeldovich approximation
Pv(k) ~ kα-2 Pρ(k) ~ kα
Scheme to stop the dissipation
and fragmentation at the smallest
scale (20AU)
22
117000 particules
Two different schemes for dissipation
super-elastic 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
23
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
24
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
25
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
26
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
27
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
28
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)
σr > σ φ > σ z
29
Conclusions
Gaseous disks, and in particular the H2 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
30
formation is different than for star formation