1. On the Property of Collapsing Primordial Cloud Core Tsuribe, T. (Osaka University) 2003/09/03-04 at Niigata Univ.

2. Abel, Bryan & Norman 2002 Cosmological Formation of the First Star Fully H 2 molecular cloud core of 1M sun at n=10 8.

3. Stable Core Formation M=0.005M sun Omukai & Nishi (1998) Decreasing M J Runaway Collapse Gravitational Collapse of Primordial Cloud Core Fragmentation again? In ３ D only n<10 8 In １ D T&I 2001 Fragmentation (Bromm, Coppi, & Larson 2002) This work Currently available numerical results are … n=10 3-4, T=a few 100K M J =10 3 M sun No H 2

4. Adiabatic cloud (gamma>4/3) : Yes. Cold dust cloud : No. (Lin, Mestel, Shu 1966) (Collapsing) Isothermal cloud : No. (Hanawa & Matusmoto 1999) Even without rotation, … How about cooling primordial cloud ? Question : Does a collapsing cloud keep its spherically symmetric shape during collapse with cooling ?

5. (1) Some results of numerical calculations (isothermal, polytrope) (2) Analytical explanation (NEW!) Today I will concentrate on …

6. Numerical Study

7. Model & Assumptions ＊ (unperturbed) Initial Density Profile : Spherical & Centrally Concentrated (similar to BE sphere but Fp ＝ alpha Fg ー＞ Rapid convergence to self-similar solution) ＊ Non Spherical Perturbation ： Shrink in x, z direction Expand in y direction Initial Aspect Ratio = 1.1 ＊ Initial Velocity Distribution ： None or with Rotation ＊ Rigid Rotation ＊ Differential Rotation (Fc = beta Fg) ＊ Equation of State : Polytrope : gamma = 1.0, 1.03, 1.06, 1.10, 1.13 * Gravity is dominated by gas itself (no dark matter)

8. Examples of Numerical Calculations

9. An example of numerical results (1) Isothermal (gamma=1.0) Density Filament Formation Fragmentation?

10. An example of numerical results (2) Polytrope (gamma=1.1) Density (almost) Sperical Single Core Results are sensitive to slight difference of the equation of state!

11. Cases with No Rotation Results by Detailed Analysis :

12. Gamma= 1.0 1.03 1.06 1.1 1.13 Log(Aspect Ratio-1) Ｎ＝３２０, ０００ Log (Density) Fragmentation Fp/Fg=0.25 Dependence on Polytropic Index

13. Dependence on Initial Conditions 0.5 0.4 0.3 0.2 Gamma=1.1 Fp/Fg Delta=0.3

14. Dependence on Initial Conditions Gamma=1.1 Delta=1.0 Fragment 0.5 0.4 0.3 0.2 Fp/Fg

15. Cases with Rotation Results by Detailed Analysis :

16. N=1,000,000 Numerical Result: Gamma=1.0 1.1 Fragmentation Log(Aspect Ratio-1) Rigid Rotation t_rot =1/omega = 2.3 t_ff

17. Analytic Study

18. Non-spherical Model (uniform) : time r/z Lin, Mestel & Shu 1965

19. Spherical Model: Non-homologous Self-similar Runaway Omukai&Nishi 1998 Larson 1969

20. Non-spherical Perturbation on Self-similar Solution : Matsumoto & Hanawa 1999 Hanawa & Matsumoto 1999, 2000 Linear Stability Analysis on Larson-Penston solution Numerical Simulation

21. This work : Numerical & Analytic result possibly inconsistent. Analytic Result does not contain the convergence property to self-similar flow from general initial condition. * Physical Reason of growth is not clear enough. Points to be resolved: New analytic model for Non-spherical Runaway Collapse

22. Model Description : Basic Equation for the Core Log r Log density velocity Core Runaway Collapsing Core Non-spherical Gravity Pressure Effect

23. Critical value of f f = Pressure Gradient Force Gravitational Force Analytic solution for f<0.624 Results

24. Polytropic collapse with gamma=1.1 f = 0.8 Pressure effect supresses non-spherical elongation Single round core formation in the center (without rotation). Isothermal Collapse Larson-Penston solution f = 3/5 n = 1/6 Unstable for elongation Non-spherical gravity effect dominates pressure effect Filament Formation in high density limit

25. Summary ： * Results of current 3D cosmological calculations of primordial cloud collapse are available only for n<10 8 although a stable core of the first star forms at n=10 24 (in 1D results). During runaway collapse of fragments in 10 4 < n < 10 20, growth of non-sphericity is supressed in gamma=1.1 cloud, different from isothermal clouds. Fragmentation take place only for Fp/Fg 1. Analytic model for non-spherical runaway collapse core model is constructed. Critical ratio of pressure gradient to gravity and growth rate are derived analytically. Isothermal case  unstable and n=1/6 gamma = 1.1  stable, consistent with 3D results.