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)