Contraction of Magnetized
Rotating Clouds andOutflows
Kohji Tomisakawith collaborators: M. N. Machida (Kyoto U.), K.
Saigo (NAOJ), T. Matsumoto (Hosei U.),
Hanawa (Chiba U.)

国立天文台

父島
石垣島

13

Taurus Molecular CloudCO

T Tauri Stars

Protostars

13CO map of Taurus molecular cloud observed by Nagoya 4m radio telescope.

Star Formation in Molecular CloudMolecular Cores

C18 O

C18O integrated intensity map of HLC2 in Taurus molecular cloud.
This shows the molecular cloud consists of many molecular cores.

H13CO

Cores with/without Protostars

IR Source

Starless Core(prestellar core)

Protostellar Core

H13CO integrated intensity map of prestellar (left) and protostellar (right)
cores in Taurus molecular cloud observed by Nobayama 45m radio
telescope

Star Formation of ~M stars
Molecular Cloud

Cloud Cores

Dynamical Contraction of Core

Starless Cores
Protostars
Main-sequence Stars
Pre-main-sequence Stars(T Tau. Stars) ¡10 7 yr
¡10 5 yr

¡1010 yr

IR Sources

Hydrogen Nuclear Fusion

Accretion Energy

Opt, Near IR Sources
Gravitational Energy ofStars

Spherical Collapse
Gas (B = 0, Ω = 0 ) contracting under the self-gravity
Larson (1969)

cm-3

9first core
9outflow
9fragmentation• H2 dissociation γ= 1.1 ρB <ρ< ρC= 2.0

10-3g cm-3

9second collapse

Temp-density relation of IS gas.
(Tohline 1982)
First Collapse
γ= 1

ρB

Fir
γ= st C7/ or
5e

• isothermal γ= 1 ρ<ρA=10-13 g cm-3
9run-away collapse
9first collapse• adiabatic γ= 7/5 ρA <ρ< ρB= 5.6 10-8 g

ρC

se
p
a
l
Col
d
on
Sec 1.10γ=ρA

log nc (cm-3)
cf. Masunaga & Inutsuka (2000)

Runaway Collapse
×1013 sFirst core

Isothermal sphericalcollapse shows:×1013 s(1) Convergence toa power-law structurelog ρ×1013 sρ (r ) ∝ r −2
×1013 s(2) Increase of central
density in a finite time.

×1013 s×1013 s(3) Only a central part
contracts.
半径

log r

This is called“runaway collapse.”

Larson 1969, MNRAS, 145,271

B, Ω ≠ 0
How about a Rotating Magnetized Cloud?
1. In case with B and Ω, a runaway contracting disk ismade. As a consequence,(a) A flat first core is born.
(b) Outflow is driven by a twisted B-field and arotating disk.
(c) B-field transfers the angular momentum from thecontracting disk to the envelope.
3. Star formation process is controlled by the rotation
speed of the first core.
(a) A slow rotator evolves similarly to the B=Ω=0
cloud.
(b) A first core with Ω in a finite range,(c) A fast rotator fragments, which leads to binary
formation.

Numerical Method

Initial ConditionΩB

periodic
boundary

Nestedgrid
grid
Nested

L=1
L=2
L=3

axisymmetric ρ perturbation
nonaxisymmetric ρ perturbation

The coarsest gridρΩ

Nested 4-times finer gridρΩ

Nested 28-times finer gridΩρ(1)Just after the central
density exceeds ρA(first core formation),outflow begins to blow.
(2) In this case, gas isaccelerated by themagnetocentrifugal windmechanism.
(3) 10% of gas in mass
is ejected with almost all
the angular momentum.
ビデオ クリップ

Angular Momentum
Redistribution in Dynamical
Collapse

In outflows driven by magnetic fields:– The angular momentum is transferred
effectively from the disk to the outflow.
– If 10 % of inflowing mass is outflowed withhaving 99.9% of angular momentum, j* would
Outf
be reduced to 10-3 jcl.
l ow
Inflow → star

ow
l
f
t
Ou

Outflow

B-Fields

Mass
Ang.Mom.
DiskAngular Momentum Problem
Specific Angular Momentum of a New-born Star
2

-1

 R*   P 
2 -1
j* ≈ 6 × 10 
cm
s
 

2 R   10day Orbital Angular Momentum of a Binary System
16

1/ 2

 Rbin 
jbin ≈ 4 × 10  100AU 
19

jcl !!j*jcl >> jbin

1/2

 M 
2 -1
cm
s


 M 

Specific Angular Momentum of a Parent Cloud Core
2FI
R IF >j ¡5 10 G JG
cm s
J
H0.1pc KH4kms pc K
21

cl

2 -1

-1

-1

Centrifugal Radiusjj  M Rc =≈ 0.06pc 
21
2 -1  
GM
M
5
10
cm
s
×

  
2

Tomisaka 2000 ApJL 528 L41--L44

2−1

Rc !!R*
Rc >> Rbin

Angular Momentum Problem

Angular Momentum Distribution(1) Mass measured fromthe center(2) Angularmomentum in M( ρ !ρ 1 )

M ( ρ !ρ 1 )

z
z

ρdV

ρ !ρ 1

L( ρ !ρ 1 )ρrvφ dVρ !ρ 1(3) Specific Angularmomentum distributionj (M )L( ρ !ρ 1 )
M ( ρ !ρ 1 )Run-away Collapse
High-density region
is formed by gaseswith small j.

Angular Momentum Transfer
Specific
AngularInitialMomentumof gas <M
Core Formation

Accretion Stage
Magnetic torquebrings the angularmomentum from thedisk to the outflow.

7000 yr afterCore Formation

Outflow brings theangular momentum.
Accumulated Mass

アウトフローの働き
角運動量と質量の分離

ガスの星への降着 アウトフロー
質量
再分配
再分配なし

角運動量
星・惑星系形成時の角運動量過剰問題の解決

Entrainment vs 2 outflows
分子流の角運動量配分は質量配分に比例。
– 角運動量問題を解決しない。

Outf
lo

ow
l
f
t
Ou

B-Fields

分子流の角運動量配分は質量配分より非常に多い。
– 角運動量問題を解決する。

w

Disk

Molecular Outflow

H13 CO +L1551 IRS5

Saito, Kawabe,
Kitamura&Sunada
1996

12

CO J

1 0

Optical Jets

10 5 AU
1400AU = 10 "Snell, Loren, &Plambeck 1980

Binary: To understand Star Formation,
study BINARY FORMATION.
Period distribution of nearby binaries
NumberBinary fraction is high.

orbital period∆K<2mag

if completely surveyed,Ghez et al 1993
bsd ~60%

Gaussian around ~180yr
Duquennoy & Mayor 2001

Binary Fraction

This suggests binary/multiple systems are formed inearly phase.

(3) may depend on the local
stellar density Liu et al. 2003

(1) may depend on the mass ofthe stars¾ Herbig/AeBe 68±11%(SSB) (Baines et al. 06)¾ similar to T Tau

binary fraction relative
to that for field stars
2
Tau
Oph

(2) may be deferent between PMS
and MS.

Trapezium
1

IC348

T Tau
N2024
DM01

MSS

0

1

2

local density
34

1. Binary fraction is a decreasing
function of local stellar density
Ghez et al 1993

3D MHD Simulation of Rotating Magnetized CloudCollapse
Machida, Tomisaka, Matsumoto 04
Model and Numerical Method
Machida, M.,H., Tomisaka, 05
Machida, M. Tomisaka, H. 05
Assume barotropic eq. state.
Machida, M.,H., Tomisaka, 06– mimicing the result of 1D RHD (eg. Masunaga, Inutsuka 2000).

2nd Core

4

Isothermal phase

.
issoc
D
H2

1st Core
ic

3
aba
t

∂ρ

+ ∇ ⋅ ( ρ v ) = 0,∂t ρ  ∂v + v ⋅∇v  = −∇p − ρ∇φ ,   ∂t∂B+ ∇ × ( v × B) = 0,∂t∆φ = 4π G ρ

Adiabatic phase

2
1

adi

 c ρ K ( ρ ≤ ρ crit )p≈
7/5
K ( ρ > ρ crit )
K
ρ

2
s

Ideal MHD

ncrit = 5 × 1010 cm -3

7/5

log T

p = cs2 ρ + cs2 ρ crit ( ρ ρ crit )isothermal

log ρ

Log ρ
Temp-density
relation of IS gas.
(Tohline 1982)

Numerical Method (cont.)
Non-homologous Collapse– Dynamic ranges of size and density scales are huge.
17-3ρ ISM : 102 cm-3 ρ 2ND CORE : 10 cm

LISM : 0.1pc

L2ND CORE : 1011 cm

Nestedgrid
grid
Nested“Nested Grid” Technique– Coarser grid: covers global structure– Finer grid: small-scale structure near the center.
# of cells：128(x)×128(y)×32(z)×17 (level)
equivalent to simulations

L=1

with〜1.5×1020 grids at the center.

New Finer Grid is Generated to Guarantee the Jeans
Condition (Truelove et al. 1997)– To achieve physically correct answer:

L=2
L=3

∆x < λJ / 4 = [(4π G ρ )1/ 2 / cs ]/ 4
– Simulations continues till the “Jeans Condition” is violated at the deepest L=17level of grid (17th Level).

Initial ConditionZ

An isothermal cylindrical cloud– in hydrostatic balance(Stodolkiewicz 1963)
magnetic field line

Ω // B // FilamentB ∝ ρ 1/ 2 ; Ω ∝ ρ 1/ 4
– Perturbations with thewavelength equal to theJeans length is added.

Bonnor-Ebert Sphere with Rigid-body Rotation & Uniform B-FieldX
non-axisymmetricÎ Add
Results
in a Similar Result.
rotateperturbation m=2 as well asaxisymmetric m=0 mode

H=〜106 [AU]
M=〜20 MParameters:– B-field strength and angularrotation speed: Aφ,

B02c / 4π
α= 2cs ρ 0 cω = Ω ( 4π G ρ0 c )

−1/ 2

Y

Initial state
ρ=102 cm-3
L=1

ρ=103 cm-3
L=1

ρ=105 cm-3
L=3

ρ=104 cm-3
L=2

3.Results
Side view
x=0 plane

Pole-on view
z=0 plane(Am2, α, ω)=(0.2, 1, 0.5)

ρ=107 cm-3
L=6

ρ=109 cm-3
L=9

ρ=1010 cm-3
L=11

Model with strong
magnetic field andhigh rotation speed
isothermal phase adiabatic phase

Only after the sufficiently
thin disk is formed, thenon-axisymmetricperturbation can grow

ρ=1011cm-3
L=12

Initial state
ρ=102 cm-3
L=1

ρ=103 cm-3
L=1

ρ=105 cm-3
L=3

ρ=104 cm-3
L=2

Pole-on view
z=0 plane
Model with weakmagnetic field andhigh rotation speed

Side view
x=0 plane

Typical Model(Am2, α, ω)=(0.01, 0.01, 0.5)

ρ=107 cm-3
L=6

ρ=109 cm-3
L=9

ρ=1010 cm-3
L=10

isothermal phase adiabatic phase

ρ=1010cm-3
L=10

Initial state
ρ=102 cm-3
L=1

ρ=103 cm-3
L=1

ρ=104 cm-3
L=2

ρ=105 cm-3
L=3This corresponds
to strong B-field& slow rotation.

Disk -->
Spherical coreCore Model(Am2, α, ω)=(0.01, 1, 0.1)

ρ=107 cm-3
L=6

In this model, the nonaxisymmetry hardly grows.
Core continues to contract.

ρ=109 cm-3
L=9

ρ=1010 cm-3
L=11

isothermal phase adiabatic phase

ρ=1011cm-3
L=12

Bar Fragmentation

Ring Fragmentation

A = 0.2, α = 1, ω = 0.5

A = 0.01, α = 0.01, ω = 0.5

Non-Fragmentation

Non-Fragmentation

A = 0.01, α = 1, ω = 0.1

A = 0.01, α = 0.1, ω = 0.1

B-Ω Flux-Spin Relation

Machida et al. 2005a,b

1/ 2
2
s c

Bc (8p c r ) - Wc (4p Gr c )--Evolutionary Path--

α= 0.01

1/ 2

0.1

C

Wc / r 1/c 2 ｵ Bc / r 1/c 2 ｵ r 1/c 6
Wc / Bc ; const

Magnetic braking
Bc / r c2 / 3 ; const
Wc / r c2 / 3 ]J-lossBc / Wc Z

0.1

1/21/2Ωc /(4πGρΩc/(4πGρc) c)Support deficient.
Î Spherical collapse.
Bc ｵ r c2 / 3 ﾜ Bc R 2 = const
Wc ｵ r c2 / 3 ﾜ Wc R 2 = const

1.0
D

π/4

0.01

A

B

nc=5x102 [cm-3 ]: initial
nc=5x10 4nc=5x106
nc=5x1080.1
Bzc /(8πcs22ρc)1/21/2

Bzc/(8πcs ρc)

1.0

B-Ω Flux-Spin Relation--Evolutionary Path--sliMagnetic brakinggBch/ Wc Ztl

Bc (8p c r ) - Wc (4p Gr c )α= 0.01

1/ 2

0.1

C

1.0
D

0.1

1/21/2Ωc /(4πGρΩc/(4πGρc) c)Support sufficientÎDisk formation
Bc ; const r Z
cWc ; constÆmove left-down
Wc / r 1/c 2 ｵ Bc / r 1/c 2 ｵ r c- 1/ 2
ÎRadial collapse
SupportÎRadialÆmove left-down
sufficient.
collapseÍfrozen-in
Bc / s c ; constÆ
Wc / s c ; constmÍconserve J
1/ 2
s c / r c ; consto Íself-grav. diskÆmove slightly
v
e

1/ 2
2
s c

π/4

0.01

A

B

nc=5x102 [cm-3 ]: initial
nc=5x10 4nc=5x106
nc=5x1080.1
Bzc /(8πcs22ρc)1/21/2

Bzc/(8πcs ρc)

1.0

B-Ω Flux-Spin Relation--Evolutionary Path-In the isothermal run-away collapse,
contraction proceeds self-similarly or solution
converges to a family of self-similar solutions.
All the models converge to a line as
Bc2
W2c
+
=1
2
2
2
(0.36) 8pcs r c (0.2) 4pGr cempirical

There exists a balance between B-field,
centrifugal force, thermal pressure and gravity.

1. We know both the evolutionary path andfragmentation condition εob>3 or Ωc/(4πGρc)1/2>0.2.
2. This gives a diagnosis of a cloud: fragments in the
adiabatic stage or remains single ?
α = 0.001

CΩc / (4πGρc)1/2

Fragmentation
No fragmentation

0.1

1

Vertical Collapse Re
RegionFor the adiabatic
core to fragment,
it must rotate fastenough.
Large symbols
initial states

0.01

D

0.1
nc,0 = 5x106 cm-3
nc,0 = 5x104 cm-3

nc,0 = 5x106 cm-3

Spherical
Sphheri Collapse Region
A

0.01

B

Evolutionarypath

nc = 5x1010 cm-3

initialRing Fragmentation
Bar Fragmentation
Both
No Fragmentation

Ring Fragmentation
Bar Fragmentation
No Fragmentation~~

Small symbolsMagnetic field suppresses the fragmentation~~
Adiabatic core
0

0.01

0.1
Bzc / (8πcs2ρc)1/2

1

To Fragment

W0
G1/ 2
-7
-1
-1 踐 cs> 1/ 2 : 3ｴ 10 yr mG ٛٛٛ顏
B0 2 cs
190ms- 1-1

Prestellar core L1544
Ohashi et al (1999)
vf ; 0.09km s-1 @ r = -15000AU
ﾞ W0 ; 1.3ｴ 10 6 yr -1
vf ; 0.14km s-1 @ r = 7000AU
Williams et al (1999)
ﾞ W0 ; 4.2 ｴ 10- 6 yr -1
B0 ; + 11 ｱ 2mG Zeeman splitting Crutcher & Troland (2000)
!
!
l
W0
-7
-1
-1
a
n
ﾞ
: (1.2 - 3.8)ｴ 10 yr mG
i
rg
B0
a
M

Measurement both Ω and B at the same density
Î future forecast!

Alignment of Outflow and Magnetic Field
IRAS 16293-2422

Wolf et al (2003)

n
g
i
l
a

1800AUngilaOutflow

L1551 IRS5

Magnetic field

n
g
i
l
a

Tamura et al(1995)
1mm radio polarization???
Polarization ofthermal dust
emission map
SCUBA 850µm

nˆ diskMiss-aligned caseJ

B

Bcp ｺ B ﾗnˆ disk Wcp ｺ Ω ﾗnˆ disk

Ωcp/(4πGρc)1/2

Β

Bcp/(8πcs2ρc)1/2

a = 0.01, w = 0.01 Evolution is understood by the spin-flux relation.

Direction of the Disk
Rotation-dominant:– disk ^J

Magnetic-dominant:– disk ^B

Boundary is given
W0
G1/ 2= 0.39B0
cs

J

BDisk, B Field and Rotation in Different Scales (Final state)
Global J

Initial B

x16

x16

Disk oriantation, local B,
and local J change theirdirections according to
the scale.
Matsumoto & T. 2004

Q

li
a
u

v
i
t
ta

e

Summary for Part 1

Magnetic field (B) and angular momentum(J) play cooperatively a role to form e.g.
outflows.
B reduces the power of J to form
fragmentation.
Disk is formed either by J or B dependingon the dominant force: J or B.

Evolution of a Rotating First Core
Saigo & Tomisaka (2006, ApJ, 645, 381-394)
Saigo, Matsumoto, Tomisaka (2006, in prep.)Temp-density relation of IS gas.
I have showed that B-field
controls the angular momentum of (Tohline 1982)

controls the angular momentum of
J core First Collapse
the first core.
ρCγ= 1
Fragmentation develops quickly inρBa hydrostatic state (first core)rather than in a contracting
Second Collapse
ρA
γ= 1.10circumstance (runaway phase)
Fragmentation in a first core may
bring binary or multiple stars.
Åbinaries are more popular than Masunaga & Inutsuka (2000)
single stars.
Ideal MHD should be reconsidered.

Hydrostatic Equilibrium
Hydrostatic Axisymmetric Configuration for Barotropic Gas( ρ rΩ , 0, 0 ) − ∇P − ρ∇ψ = 0,2∆ψ = 4π G ρ K1 ρ 7 / 5 K ( ρ ≤ ρ dis )
p≈
1.1
ρ
K
K ( ρ > ρ dis ) 2Angular Momentum Distribution– same as a uniform-density sphere with rigid-body rotation– total mass M core and total ang. mom. J core
2/3
5  J core    M ( R)  j ( M ( R )) =  1 −  1 − 
2  M core   
M core  dissociation
density

Self-consistent Field Method (SCF) Hachisu(1986), Tohline, Durisen

M core , J core ) → ρ c(– to understand the evolution of first core( ρc , M core ) → J core

Examples of Hydrostatic
ConfigurationΩZ M Z

Three models have the same central density ρc=4ρdiss, but different
angular momenta as 2.25 × 1049 (left), 4.18 × 1049 (middle),
and 9.99 × 1049 g cm2 (right), and masses as 2.77 × 1031 (left),
3.45 × 1031 (middle), and 4.97 × 1031 g (right).

Mass-Density Relation (Ω = 0 )Below ρ <: ρ dis, mass increases with central density ρ c .
ρ dis = 5 × 10−8 gcm −3– Mass is prop. to Jeans mass

M J ∝ T 3/ 2 ρ −1/ 2 ∝ ρ 1/10

(Chandrasekhar 1949)– Mass accretion drives the corefrom lower-left to upper-right.

Above ρ :> a few ρ dis massdecreases due to soft EOS.
– Further accretion destabilizes
the cloud and drives dynamical contraction (2nd collapse).

 Maximum mass of the 1st core is 0.01 M.

Hydrodynamical Simulation
run-away (1st collapse)Æ 1st core
– 1st core grows by mass accretion from contracting
envelope.

Initial Condition– Bonnor-Ebert sphere (+ envelope (R~50,000AU))– nc~104H2cm-3, T=10K– Rotation ω=Ωtff=0~0.3– increase the BE density by 1.1~8 times– Perturbations m=2 and m=3 δρ/ρ=10%

Numerical method– HD nested grid– barotropic EOS

Non-rotating clouddensity ρc<ρdis,Mclincreases with ρc.

ρ dis = 5 × 10−8 gcm −3– Jeans mass
M J ∝ T 3/ 2 ρ −1/ 2 ∝ ρ 1/10
– Evolution is driven byaccretion.

At ρc>2ρdis, soft EOS makes
Mcl decrease.
– dynamical contraction (2ndcollapse)

maximum mass of a 1st
core is equal to ~0.01 M.

ω=0Non-rotating model
1. Unless the cloud ismuch more massive
than the B-E mass, thefirst core evolves tofollow a path expectedby quasi-hydrostatic
evolution.
2. Maximum mass of a firstcore is small ~0.01M.
3. Quasi-static evolution
gives a good agreementwith HD result.

M core1032

max.mass

M = 8M BE
M = (1.1 : 3) M BE

1031
10

−12

10

−10

10−8 10−6 ρ c

Mass-Density Relation (Ω > 0 )rotation rate of parent cloudjω=M( 2c / G )sω<0.015– similar to non-rot. case.
– second collapse

ω>0.015– Mass increases muchwell below ρ c = ρ dis

ρ dis = 5 × 10−8 gcm −3

Rotating Cloud (ω=0.05)
M coreFirst, the 1st core
increases its mass
(upwardly in Mcl-ρc plane).
––follows a hydrostatic
evolution path.
Shape: round spherical
disk.

Then, the first core begins
to contract (rightward in theplane)––

This phase, spiral arms
appear.
J is transferred outwardly.

Core+disk continues tocontract.

0.03M ρcComparison with previous
simulations
Bate (1998)– SPH simulation– ω=0.08– spiral Æ transfer J

Matsumoto, Hanawa(2003)– Nested Grid Eulerian
Hydrodynamics– ω=0.03
spiral– ω=0.05
spiralfragmentation

Matsumoto, Hanawa 2003
Bate 1998
ω = 0.08

ω = 0.03
ω = 0.05

Nonaxisymmetric instability
Rotational-to-gravitational energy
ratio: T/|W|– A polytropic disk with T/|W|>0.27(γ=5/3) is dynamically unstable under awide range of conditions (γ=5/3:
Pickett et al. 1996; γ=7/5, 9/5, 5/3
Imamura et al. 2000 )ρ dis = 5 × 10−8 gcm −3

ω = 0.03

0.1
0.05

T/|W| increases with massaccretion.
After T/|W| exceeds the criticalvalue,– nonaxisymmetric instability grows.
– Angular momentum is transferredoutwardly.
– This may stabilize the disk again.

Rotating Cloud (ω=0.05)
ρ dis = 5 × 10−8 gcm −3

Evolution depends
on the initial mass.
– M~MBE:

1032

M = (1.1 : 3) M BE

M coreFollow
quasihydrostatic
evolution

M = 8M BE– M>>MBE:
evolution depends
on its initial condition.1031

10

−12

10

−10

10

−8

10−6 ρ cFragmentation
In a fast rotating cloud,fragmentation (morethan 2 fragments) isobserved in the 1stcore.
This occurs after nonaxisymmetricinstability is triggered.

Typical Rotation Rate
NH3 cores (n~3 104cm-3) Goodman et al(1993)-6−1Ω ; (0.3 − 2) × 10
rad
yr
1/ 2
 3π 
5
τ ff = 
2
10
;
×
yr

 32G ρ 

ω ; 0.06 − 0.4

N2H+ cores (~ 2 105cm-3) Caselli et al.
(2002)Ω ; (0.5 − 6) × 10-6 rad yr −1
1/ 2
 3π 
4
τ ff = 
8
10
;
×
yr

 32G ρ 

ω ; 0.04 − 0.5

Luminosity of the First Core
3D simulation

Lbol

ω =0

dEtot
d
L=−
= − (W + Ekin + Eth )dt
dt
Quasi-static evolution
second collapse

second collapse

0.03
0.05
Lifetime of 1st core is
0.1 not short !

L is a decreasing
function
of ω.
−5
−1

M& ~ 2 × 10 M  yr
L ~ (3 − 5) × 10−2 Lt(yr)Absolute value L and timescale are obtained& is given. L ∝ M& t ∝ M&−1
after M

Mass Accretion Rate
Mass accretion
rate is between
the LP solution
and a SH disk
solution.
Much higher thanthat expected for
SIS.

M = 1.1M BE(Whitworth & Summers 1985)

M& = (2 − 4) M& yr −1

SummaryThe evolution of a 1st core is well described with thequasi-static evolution.
Slow (or no) rotation model exhibits the second collapse(ω<0.015).
– Maximum mass of the 1st core ~0.02 M (ω=0.015).

Rotating cloud with ω>0.015, the 1st core contracts
slowly.
– After T/|W|>0.27, a dynamical nonaxisymmetric instability grows
and spiral pattern appears.
– Gravitational torque transfers the angular momentum outwardly.
– The 1st core contracts further.

In a rotating cloud with ω>0.1, we found thefragmentation of the 1st core.