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 …
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)
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!
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
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
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.