Tsuribe, T. (Osaka U.) Cloud Fragmentation via filament formation 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) Contents:

Formation Process of Astronomical Objects in CDM Cosmology Turn Around Non-homologous Collapse Cosmic Expansion Linear Growth Nonlinear Growth Tidal Interaction Infall to Dark Matter Potential Shock Formation Cooling ? No Stable Oscillation Yes Collapse Fragment? ? Stellar Cluster? Massive Black Hole? Massive star? Low mass star?

Density Fluctuations Cloud 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 project is to construct a simple but more accurate theory for fragmentation in the collapsing cloud cores as a useful tool for astrophysical applications Ultimately… origin of IMF t H > t cool t ff > t cool M > M J Sufficient ? CRITERION? singlebinarymultiple 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 fragment spontaneously 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)fragmentation of self-gravitating run-away collapsing cloud core, Physical property of non-spherical gravitational collapse is a key. Elongation & Filament Formation? Fragmentation? Ring formation? Disk formation? … this talk c.f., Omukai-san’s talk Collapsing cloud 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^-1 Fp = 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 finite size core is also investigated. In a infinite length filament, since density G P Isothermal With increasing T P G 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 coordinate Equations in self-similar frame Lai (2001)

Perturbations Unperturbed state Larson-Penston type self-similar Solution (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= RandomGaussian

Fragmentation Fragmentation Sites (by linear growth + thresholds + Monte Carlro) Solved range

Z=10^-5 Axis ratio1:2

Z=10^-5 Axis 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 range of 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=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 = … 1: 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): Initial state (n=10) log n Log(p/rho) Local T minimum n=1000 The case 1:2pi bounced The case 1:5 bounced The case with 1:4 fragmented The case with 1:3 fragmented The case with 1:2 without fragmentation Local T Maximum 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)