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Tsuribe, T. (Osaka U.)
Cloud Fragmentation via filament formation
IntroductionBasic Aspects of Cloud FragmentationApplication 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)
Contents:
2009/01/14-16 @Tsukuba
Formation Process of Astronomical Objectsin CDM Cosmology
Turn Around
Non-homologous Collapse
Cosmic Expansion
Linear Growth
Nonlinear GrowthTidal
Interaction
Infall to Dark Matter PotentialShock Formation
Cooling ? No Stable Oscillation
Yes Collapse Fragment? ?
Stellar Cluster?
Massive Black Hole?
Massive star?Low mass star?
Density FluctuationsCloud Core
Fragments
Core Formation
Stars
Fragmentation
Runaway Collapse
Accretion / Merging
Feedback …UV, SNe, etc.
?When fragmentation stops?
Simple criterion?
Possibility of Subfragmentation?
Purpose of this projectis to construct a simple but more accurate theory for fragmentation in the collapsing cloud cores as a usefultool for astrophysical applications
Ultimately… origin of IMF
tH > tcool
tff > tcool
M > MJ
Sufficient ?
CRITERION?
single binary multiple
Simple arguments..
…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
Fastest growing mode
Finite size is spontaneously chosen!
Filamentary clouds also fragmentspontaneously into a finite size object.
Linear analysis of gravitational instability 3: Filament-like cloud case
Fastest growing mode
In this talk, in order to understand the possibility of (sub)fragmentationof self-gravitating run-away collapsing cloud core, Physical property of non-spherical gravitational collapse is a key.
Elongation & FilamentFormation? Fragmentation?
Ring formation?
Disk formation?
… this talk c.f., Omukai-san’s talk
Collapsingcloud core
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)
Fg = GM/R … R^-1Fp = cs^2 rho/R … R^-1 (for isothermal),
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 finitesize core is also investigated.
In a infinite length filament, since
density
GP
Isothermal
With increasing TPG
density
Elongation of cloud core
If non-spherical perturbation is given to a spherical fragment …
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?
Elongation
Hanawa&Matsumoto (2000)
Non-spherical elongation of a self-similar collapse solution
Zooming coordinateEquations in self-similar frame
Lai (2001)
Perturbations
Unperturbed stateLarson-Penston type self-similarSolution (various gamma)
Eigen value for bar-mode
Elongation evolves as rho^n
Linear growth rate
grow
decay
Unstable for isothermal
Stable for gamma>1.1
Effect of the dust cooling for elongation
Thermal evolution
Dust cooling
Gamma~1.1
Results: Linear Elongation Rate
Elongation by dust cooling
Fragmentation
Fragmentation Sites
(by linear growth + thresholds + Monte Carlro) mass function
Dependence on Metalicity of Mass function
Initial amplitude= Random Gaussian
Fragmentation
Fragmentation Sites
(by linear growth + thresholds + Monte Carlro)
Solved range
Z=10^-5Axis ratio1:2
Z=10^-5Axis ratio1:1.32
Effect of Sudden heating + Dust cooling
Fragmentation
Fragmentation Sites
(by linear growth + thresholds + Monte Carlro)
Solved range
Low metallicity Case (dust cooling)Effect of 3-body H2 formation heating
3body H2 formation heating
Dust cooling
[M/H]=-4.5
[M/H]=-5.5
Without rotation
With rotation [M/H]=-4.5
[M/H]=-5.5
Rule of thumb
For filament fragmentation, elongation > 30 is required.
Fragmented
Not fragmented
Axis Ratio-1
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 rangeof isothermal evolution is generated by CMB effect
(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)
Thermal evolution (from 1zone result)
n
T
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=10Nsph=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 timeIn 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_sFree-fall time = 1/sqrt( G rho )by using alpha0=1 for the spherical state, the condition2 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-sphericalrho^0.5 for cylindrical shape
Condition for filament formation before the first minimumtemperature … at n=1e3Since 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: 1.628For smaller than this value, cloud is expected to not to beFilamentally 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 beforen=10^3 and collapse does not halted in the early state.
There is another condition, Sound crossing time in short axis < free-fall timeRarefaction wave reach the center of axisCentral region of the filament becomes equilibriumCentral bounceThis 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 maximum1:3 … fragmented1:4 … fragmented (just after local T minimum)
Results (so far):
Initial state(n=10)
log n
Log(p/rho)
Local T minimumn=1000
The case1:2pibounced
The case1:5 bounced
The case with1:4 fragmented
The case with 1:3 fragmented
The case with1:2 without fragmentation
Local TMaximum n = 1e6
Z=0.01Zsun
Initial state(n=10)
log n
The cases with 1:4,3,2 forming spindle
With the effect of isothermal temperature floor:
Fragmentation is not prominent during isothermal stage
•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.
Discussion (preliminary)
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