Download Contraction of a Magnetized Rotating Cloud

Fragmentation and
Evolution of the First Core
Kohji Tomisaka
and
Kazuya Saigo (NAOJ)
consensus of scenarios of
Star Formation of ~M8 stars
Molecular Cloud
Cloud Cores
Dynamical Contraction
of Cores
Starless Cores
Prestellar Cores
Protostars
Main-sequence Stars
Pre-main-sequence Stars
(T Tau. Stars) 10 7 yr
10 5 yr
Protostellar
Cores
IR Sources
1010 yr
Hydrogen Nuclear Fusion
Accretion Energy
Opt, Near IR Sources
Gravitational Energy of
Stars
Gas (B = 0,
Spherical Collapse
 = 0) contracting under the self-gravity
(1) isothermal g= 1 r<rA=10-13 g cm-3
run-away collapse
first collapse (dynamical collapse)
central density increases greatly
in a finite time-scale.
(2) adiabatic g= 7/5 rA <r< rB= 5.6 10-8 g
cm-3
first core: a nearly hydrostatic
core forms at the center
(outflow and fragmentation)
(3) H2 dissociation g= 1.1 rB <r< rC= 2.0
10-3g cm-3
second collapse: dynamical
collapse begins at the center of the
1st core.
Larson (1969)
Temp-density relation of gas @
cloud center. (Tohline 1982)
First Collapse
rC
First
γ= 1Collapse rB rC
γ= 1
rB
rA
Second Collapse
γ= 1.10
rA
log nc (cm-3)
cf. Masunaga & Inutsuka (2000)
Runaway Collapse
1013 s
First core
Isothermal spherical
collapse shows:
1013 s
log r
(1)Convergence to
a power-law structure
1013 s
r (r )  r 2
1013 s
(2)Divergence 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
How about a rotating magnetized cloud?
1. In case with B and , a contracting disk is made in the
runaway collapse phase. As a consequence,
(a) A flat first core is born.
(b) Outflow is driven by the effect of twisted B-field and a
rotating disk.
(c) B-field transfers the angular momentum from the
contracting disk to the envelope.
2. Star formation process is controlled by the rotation speed
of the first core.
Evolution is classified into three types.
(a) A slow rotator evolves similarly to the B==0 cloud.
(b) An intermediate rotator forms a bar and spiral arms,
which transfer the angular momentum from the center.
Owing to this, collapse proceeds.
(c) A bar or spiral arms of fast rotator finally fragments.
This leads to the binary formation.
Numerical Method
Initial Condition

B
periodic
boundary
To achieve high resolution near
the center of the cloud.
Nested grid
L=1
L=2
L=3
(1) We assume a hydrostatic balance
between pressure, Lorentz force, centrifugal
force and gravity.
(2) We added both axisymmetric and nonaxisymmetric r perturbations.
The coarsest grid
r

Nested 4-times finer grid
r

a=1,=0.5
First Core Phase Evolution
Nested 28-times finer grid

r
(1)Just after the central
density exceeds rA
(first core formation),
outflow begins to blow.
(2) In this case, gas is
accelerated by the
magnetocentrifugal wind
mechanism.
(3) 10% of gas in mass
is ejected with almost all
the angular momentum.
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 with
having 99.9% of angular momentum, j* would
be reduced to 10-3 jcl.
Inflow  star
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
 

 System
 2 R  of10day
Orbital Angular Momentum
a Binary
16
1/ 2
jbin
 Rbin 
 4  10 

100AU


19
jcl j*
jcl  jbin
1/2
 M 
2 -1
cm
s


M 
Specific Angular Momentum of a Parent Gas
F
I
R IF 
cm s
G
J
G
J
H0.1pc KH4kms pc K
2
jcl 5 10
21
2 -1
-1
-1
Centrifugal Radius
j
j

  M 
Rc =
 0.06pc 

21
2 -1  
GM
5

10
cm
s
M

 

2
Tomisaka 2000 ApJL 528 L41--L44
2
1
Rc R*
Rc  Rbin
Molecular Outflow
H13 CO+
L1551 IRS5
Saito, Kawabe,
Kitamura&Sunada
1996
12
CO J 1  0
Optical Jets
105 AU
1400AU = 10"
Snell, Loren, &Plambeck 1980
Evolution of a Rotating First Core
Saigo & Tomisaka (2006, ApJ, 645, 381-394)
Saigo, Matsumoto, Tomisaka (2006, in prep.)
I have showed that B-field
controls the angular momentum of Temp-density relation of IS gas.
(Tohline 1982)
the first core.
Fragmentation develops quickly inJ core First Collapse
a hydrostatic state (first core)
rC
γ= 1
rB
rather than in a contracting
circumstance (runaway phase)
Second Collapse
Fragmentation in a first core
rA
γ= 1.10
brings binary or multiple stars.
binaries are more popular than
single stars.
We study the fragmentation of Masunaga & Inutsuka (2000)
the 1st core using a nested grid
hydrocode.
Hydrostatic Equilibrium
Hydrostatic Axisymmetric Configuration for Barotropic Gas
 r r , 0, 0  P  r = 0,
2
 = 4 G r
 K1 r 7 / 5
p
1.1
K
r
 2
( r  rdis )
( r  rdis )
Angular 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  


Self-consistent Field Method (SCF)
M
,J
r
Hachisu(1986), Tohline, Durisen
core the
coreevolution
c of first core
– to understand
 rc , M core   J core
dissociation
density
Examples of Hydrostatic Configuration for
Rotating Barotropes
J
M
Mass and simultaneous angular momentum
accretions drives an evolution like this:
Three models have the same central density rc=4rdiss, 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 ( = 0 )
Below
r < rdis,
mass increases with central density r.c
rdis = 5 108 gcm 3
– Mass is prop. to Jeans mass
MJ  T
3/ 2
r
1/ 2
r
1/10
(Chandrasekhar 1949)
– Mass accretion drives the core
from lower-left to upper-right.
Above r  a few rdis mass
decreases due to soft EOS.
– Further accretion destabilizes
the cloud and drives dynamical contraction (2nd collapse).
 Maximum mass of the 1st core is 0.01 M8.
Hydrodynamical Simulation
run-away (1st collapse) 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
hydrostatic sphere
Rotation w=tff=0~0.3
w = t ff = 0  0.3
increase the BE density
4
-3
10
H
cm
by 1.1~8 times
2
10K
– Perturbations m=2 and m=3
dr/r=10%
Numerical method
– HD nested grid
– barotropic EOS
Increase the mass by 1.1-8 times
Add m=2 and m=3 perturbation
Non-rotating model
1. Unless the cloud is
much more massive
than the B-E mass, the
first core evolves to
follow a path expected
by quasi-hydrostatic
evolution.
2. Maximum mass of a first
core is small ~0.01M8.
3. Quasi-static evolution
gives a good agreement
with HD result.
M core
1032
max.mass
M = 8M BE
10
31
M = (1.1 3)M BE
1012 1010 108 106 r c
Mass-Density Relation (  0 )
rotation rate of parent cloud
j
w=
M

2c s / G

w<0.015
– similar to non-rot. case.
– second collapse
w>0.015
– Mass increases much
rdis
well below rc
rdis = 5 108 gcm 3
Rotating Cloud (w=0.05)
First, the 1st core
increases its mass
(upwardly in Mcl-rc plane).
–
–
follows a hydrostatic
evolution path.
Shape: round spherical
disk.
Then, the first core begins
to contract (rightward in the
plane)
–
–
This phase, spiral arms
appear.
J is transferred outwardly.
Core+disk continues to
contract.
M core
0.03M
rc
Comparison with previous
simulations
Bate (1998)
– SPH simulation
– w=0.08
– spiral  transfer J
Matsumoto, Hanawa
(2003)
– Nested Grid Eulerian
Hydrodynamics
– w=0.03
spiral
– w=0.05
spiral
fragmentation
Matsumoto, Hanawa 2003
Bate 1998
w = 0.08
w = 0.03
w = 0.05
Nonaxisymmetric instability
Rotational-to-gravitational energy
ratio: T/|W|
– A polytropic disk with T/|W|>0.27
(g=5/3) is dynamically unstable under a
wide range of conditions (g=5/3:
Pickett et al. 1996; g=7/5, 9/5, 5/3
Imamura et al. 2000 )
rdis = 5 108 gcm 3
w = 0.03
0.1
0.05
T/|W| increases with mass
accretion.
After T/|W| exceeds the critical
value,
– nonaxisymmetric instability grows.
– Angular momentum is transferred
outwardly.
– This may stabilize the disk again.
Fragmentation
In a fast rotating cloud,
fragmentation (more
than 2 fragments) is
observed in the 1st
core.
This occurs after nonaxisymmetric
instability is triggered.
Typical Rotation Rate
NH3 cores (n~3 104cm-3) Goodman et al
(1993)
-6
1
 (0.3  2) 10
rad
yr
1/ 2
 3 
5
 ff = 
2

10
yr

 32G r 
w 0.06  0.4
N2H+ cores (~ 2 105cm-3) Caselli et al.
(2002)
 (0.5  6) 10-6 rad yr 1
1/ 2
 3 
4
 ff = 
8

10
yr

 32G r 
w 0.04  0.5
Luminosity of the First Core
3D simulation
Lbol
w =0
dEtot
d
L=
=  W  Ekin  Eth 
dt
dt
Quasi-static evolution
second collapse
second collapse
0.03
0.05
0.1
Lifetime of 1st core is
not short !
L is a decreasing
function
of w.
5
1
M ~ 2 10 M yr
L ~ (3  5) 102 L
t(yr)
Absolute value L and timescale are obtained
after M is given. L  M t  M 1
Mass Accretion Rate
Mass accretion
rate is between
the LP solution
and a SH disk
solution.
Much higher than
that expected for
SIS.
M = 1.1M BE
(Whitworth & Summers 1985)
M = (2  4) M yr 1
Summary
The evolution of a 1st core is well described with the
quasi-static evolution.
Slow (or no) rotation model exhibits the second collapse
(w<0.015).
– Maximum mass of the 1st core ~0.02 M8 (w=0.015).
Rotating cloud with w>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 w>0.1, we found fragmentation of
the 1st core.