On the Property of Collapsing Primordial Cloud Core

Tsuribe, T. (Osaka University)

2003/09/03-04 at Niigata Univ.

Abel, Bryan & Norman 2002

Cosmological Formation of the First Star

Fully H2 molecular cloud core of 1Msun at n=108.

Stable Core FormationM=0.005Msun

Omukai & Nishi (1998)

1.1KP

Decreasing MJ

Runaway Collapse

Gravitational Collapse of Primordial Cloud Core

Fragmentation again?

In ３ D only n<108

In １D

T&I 2001

Fragmentation

(Bromm, Coppi, & Larson 2002)

This work

Currently available numerical results are …

n=103-4, T=a few 100K MJ=103Msun

No H2

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 ?

(1) Some results of numerical calculations (isothermal, polytrope)

(2) Analytical explanation (NEW!)

Today I will concentrate on …

Numerical Study

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)

Examples of

Numerical Calculations

An example of numerical results (1)

Isothermal (gamma=1.0)

Density

Filament Formation

Fragmentation?

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!

Cases with No Rotation

Results by Detailed Analysis :

Gamma= 1.0 1.031.06

1.1

1.13Log(Aspect Ratio-1)

Ｎ＝３２０ , ０００

Log (Density)

Fragmentation Fp/Fg=0.25

Dependence on Polytropic Index

Dependence on Initial Conditions

0.5 0.4

0.3

0.2

Gamma=1.1

Fp/Fg

Delta=0.3

Dependence on Initial Conditions Gamma=1.1

Delta=1.0

Fragment

0.5

0.4

0.3

0.2Fp/Fg

Cases with Rotation

Results by Detailed Analysis :

N=1,000,000Numerical Result:

Gamma=1.0

1.1

Fragmentation

Log(Aspect Ratio-1)

Rigid Rotation 　t_rot =1/omega = 2.3 t_ff

Analytic Study

Non-spherical Model (uniform) :

time

r/z

Lin, Mestel & Shu 1965

Spherical Model: Non-homologousSelf-similar Runaway

Omukai&Nishi 1998

Larson 1969

Non-spherical Perturbation on Self-similar Solution :

Matsumoto & Hanawa 1999Hanawa & Matsumoto 1999, 2000

Linear Stability Analysis on Larson-Penston solution

354.00

20

)(

)(

tt

tt

bar

c

177.0 bar

Numerical Simulation

6/1 bar

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

Model Description : Basic Equation for the Core

Log r

Log

density

velocity

Core

Runaway Collapsing CoreNon-spherical GravityPressure Effect z

z

r

z

B

A

z

c

z

BzG

dt

zd

cAzG

dt

d

s

s

20

20

0

2/32

2/1222

2/32

2/122

20

200

2

2

2

20

200

2

2

)1(

)1()1ln(3)(

)1(

)1()1ln(

2

3)(

)(

3

4

)(

3

4

Critical value of f

f = Pressure Gradient Force

Gravitational Force

n

Analytic solution

01

5/11318 2

f

fnn

11

5/1811

12

1

f

fn

for f<0.624

Results

Polytropic collapse with gamma=1.1

f = 0.8

Pressure effect supresses non-spherical elongation

Single round core formationin the center (without rotation).

Isothermal Collapse

Larson-Penston solution

f = 3/5

n = 1/6

Unstable for elongation

Non-spherical gravity effectdominates pressure effect

Filament Formation in high density limit

Summary ：

* Results of current 3D cosmological calculations of primordial cloud collapse are available only for n<108 although a stable core of the first star forms at n=1024 (in 1D results).

• During runaway collapse of fragments in 104 < n < 1020, growth of non-sphericity is supressed in gamm

a=1.1 cloud, different from isothermal clouds. Fragmentation take place only for Fp/Fg<0.2, Delta>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.