Download tsuribe

Cloud Fragmentation via filament formation
Tsuribe, T. (Osaka U.)
Contents:
Introduction
Basic Aspects of Cloud Fragmentation
Application to the Metal deficient Star Formation
based on Omukai,TT,Schineider,Ferrara 2005, ApJ
TT&Omukai 2006, ApJL
TT&Omukai 2008, ApJL
(+if possible, some new preliminary results)
2009/01/14-16 @Tsukuba
Formation Process of Astronomical Objects
in CDM Cosmology
Cosmic Expansion
Linear Growth
Tidal
Nonlinear Growth Interaction
Stellar Cluster?
Massive Black Hole?
Turn Around
Non-homologous Collapse
Massive star?
Low mass star?
Infall to Dark Matter Potential
Shock Formation
Cooling ?
No
Stable Oscillation
Yes
Collapse
Fragment?
?
Possibility of Subfragmentation?
Density Fluctuations
Cloud Core
?
Fragmentation
When fragmentation stops?
Fragments
Simple criterion?
Runaway Collapse
Core Formation
Accretion ／ Merging
Stars
Feedback …UV, SNe, etc.
Purpose of this project
is to construct a simple but more accurate theory for
fragmentation in the collapsing cloud cores as a useful
tool for astrophysical applications
Simple arguments..
CRITERION?
single
binary
multiple
Ultimately… origin of IMF
tH > tcool
tff > tcool
M > MJ
Sufficient ?
…No (for me)
Linear analysis of gravitational
instability 1: Uniform cloud case
Dispersion relation:
Sound wave
Growing mode
Fastest growing mode
(no fragmentation)
Linear analysis of gravitational
instability 2: Sheet-like cloud case
Finite size is
spontaneously chosen!
Fastest growing mode
Linear analysis of gravitational instability 3:
Filament-like cloud case
Fastest growing mode
Filamentary clouds also fragment
spontaneously into a finite size object.
In this talk,
in order to understand the possibility of (sub)fragmentation
of self-gravitating run-away collapsing cloud core,
Physical property of non-spherical gravitational collapse
is a key.
Collapsing
cloud core
Elongation & Filament
Formation? Fragmentation?
… this talk
Disk formation?
Ring formation?
c.f., Omukai-san’s talk
In primordial star formation,
infinite length filament is investigated by e.g.,
Uehara,Susa,Nishi,Yamada&Nakamura(1996)
Uehara&Inutsuka(2000)
Nakamura&Umemura (1999,2001,2002)
Isothermal
G
P
density
In a infinite length filament, since
With increasing T
P
Fg = GM/R
… R^-1
G
Fp = cs^2 rho/R … R^-1 (for isothermal),
density
isothermal evolution has a special meaning.
… Break down of isothermality is sometimes interpreted as a
site of fragmentation
In this work, the formation process of filament from the finite
size core is also investigated.
Elongation of cloud core
If non-spherical perturbation is given to a spherical fragment …
Elongation
Unstable  It will elongate to form sheet or filament
 Possibly fragment again
Stable
 It keeps spherical shape
 It will form massive object without fragmentation
Condition of elongation instability? Condition for fragmentation?
Non-spherical elongation of
a self-similar collapse solution
Hanawa&Matsumoto (2000)
Lai (2001)
Zooming coordinate
Equations in self-similar frame
Unperturbed state
Larson-Penston type self-similar
Solution (various gamma)
Perturbations
Elongation evolves as rho^n
Linear growth rate
grow
Unstable for
isothermal
decay
Eigen value for bar-mode
Stable for
gamma>1.1
Effect of the dust cooling for elongation
Thermal evolution
Gamma~1.1
Dust cooling
Results： Linear Elongation Rate
Elongation by
dust cooling
Fragmentation Sites
(by linear growth + thresholds + Monte Carlro)  mass function
Fragmentation
Dependence on Metalicity of
Mass function
Initial amplitude= Random
Gaussian
Fragmentation Sites
(by linear growth + thresholds + Monte Carlro)
Solved range
Fragmentation
Z=10^-5
Axis ratio1:2
Z=10^-5
Axis ratio1:1.32
Effect of
Sudden heating
+
Dust cooling
Fragmentation Sites
(by linear growth + thresholds + Monte Carlro)
Solved range
Fragmentation
Low metallicity Case (dust cooling)
Effect of 3-body H2 formation heating
3body H2 formation
heating
Dust cooling
Without rotation
[M/H]
=-4.5
[M/H]
=-5.5
With rotation
[M/H]=-4.5
[M/H]=-5.5
Rule of thumb
Fragmented
Axis Ratio-1
Not
fragmented
For filament fragmentation, elongation > 30 is required.
Summary 1:
(1) Filament fragmentation is one mode of fragmentatation which
can generate small mass objects
(2) Starting from a finite-size-cloud core with moderate initial
elongation, elongation is supressed in the case with gamma>1.1
(3) Dust cooling in metal deficient clouds as low as 10^-5~10^-6
Zsun provides the possible thermal evolution in which filament
fragmentation works, provided that moderate elongation ~1:2
exists at the onset of dust cooling.
(4) If the cloud is suffered from sudden heating process before dust
cooling, axis ratio becomes close to unity and filament
fragmentation can not be expected even with dust cooling.
(5) With the rotation, elongation become larger but the effect is
limited.
Effect of
isothermal temperature floor
by CMB
(Preliminary results)
Thermal evolution under CMB
Wide density range
of isothermal evolution is generated by CMB effect
Thermal evolution
(from 1zone result)
T
n
(1) Z=0.01Zsun, redshift=0. T peak is because of line cooling reach LTE and
rate becomes small and heating due to H2 formation (red)
(2) Isothermalized temperature floor is inserted between two local minimum
(simple model : green)
(3) With CMB effect (redshift=20) (blue)
Model:
(1) Prepare uniform sphere with |Eg|=|Eth|
(2) Elongate it to with keeping mass and density to
Axis ratio = 1:2 pi, 1:5, 1:4, 1:3, 1:2
(3) Follow the gravitational collapse
Initial density n=10
Nsph=10^6
Result : final density so far (n=4e6)
(1) Bounce -> No collapse 1:2pi, 1:5
(2) Collapse -> filament formation -> fragmentation 1:4,1:3
(3) Collapse -> filament formation -> Jeans Condition 1:2
(4) Collapse -> almost spherical (not calculated) 1:1.01 etc.
(1) cases with bounce and no collapse: (axis ratio=1:2pi,1:5)
2 Sound crossing time
In short axis direction < free fall time
Pressure force prevent from collapsing
For the axis ratio f, short axis becomes A=(1/f)^(1/3)R,
where R is radius of spherical state.
Sound crossing in the short axis = A/c_s
Free-fall time = 1/sqrt( G rho )
by using alpha0=1 for the spherical state, the condition
2 A/c_s < 1/sqrt(G rho) gives axis ratio < critical value
(4) Cases with Non-filamentary collapse
Axis Ratio Growth Rate
rho^0.354 for quasi-spherical
rho^0.5 for cylindrical shape
Condition for filament formation before the first minimum
temperature … at n=1e3
Since n0=10, n/n0=1e2, therefore even initial cylindrical
Shape is assumed, we need at least
Initial axis ratio > 2 pi/sqrt(1e2) = 2 pi/10 = 0.628
… 1: 1.628
For smaller than this value, cloud is expected to not to be
Filamentally shape enough to fragment.
(2),(3) Collapse & Filament Formation
Initial Axis ratio = 1:4, 1:3, and 1:2
In these cases, growth rate of axis ratio is rho^0.5.
2Sound crossing time is larger than free-fall time.
Therefore, axis ratio becomes larger than 2 pi before
n=10^3 and collapse does not halted in the early state.
There is another condition,
Sound crossing time in short axis < free-fall time
Rarefaction wave reach the center of axis
Central region of the filament becomes equilibrium
Central bounce
This condition seems to be between the cases with 1:3 and 1:2
Case with Z=0.01Zsun with local T maximum
Density
Fragmentation is seen during temperature increasing phase
Case with Z=0.01Zsun without local T maximum
Density
Fragmentation is not seen with the isothermal temperature floor
1:2 … no central bounce  further filament collapse
 no fragmentation, spindle formation
 fragmentation later
1:3,1:4 … central bounce and equilibrium filamentary core
 dynamical time become larger than free-fall time
 fragmentation can be expected here.
Numerical Result:
1:2 … no fragmentation before T local maximum
1:3 … fragmented
1:4 … fragmented (just after local T minimum)
Results (so far): Z=0.01Zsun
Log(p/rho)
Initial state
(n=10)
The case
1:2pi
bounced
The case
1:5 bounced
The case with
1:4 fragmented
The case with
1:2 without
fragmentation
The case with 1:3
fragmented
Local T minimum
n=1000
Local T
Maximum
n = 1e6
log n
With the effect of isothermal temperature floor:
Initial state
(n=10)
log n
The cases with 1:4,3,2
forming spindle
Fragmentation is not prominent during isothermal stage
Discussion (preliminary)
•For a cloud with dust, filament fragmentation may be effective
for clouds with moderate initial elongation
•Once filament is formed, fragmentation can be possible at the
continuous density range where T is weakly increasing (not only
just after the temperature minimum).
•Fragmentation density (i.e. mass) of above mode depends on the
degree of initial elongation.
•Once filament fragmentation takes place, in temperature
increasing phase, each fragment tend to have highly spherical
shape.  Further subfragmentation via filament fragmentation
may be rare event (still under investigation) but disk
fragmentation is not excluded.
•In the density range with the isothermalized EOS, perturnation
growth is not prominent within the time scale of filament
collapse of the whole system indicating smaller mass
fragmentation in later stage.