1. Fragmentation and Evolution of the First Core
Kohji Tomisaka
and
Kazuya Saigo (NAOJ)

2. Star Formation of ~M8 stars
consensus of scenarios of
Star Formation of ~M8 stars
Dynamical Contraction
of Cores
Molecular Cloud
Cloud Cores
Starless Cores
Prestellar Cores
Main-sequence Stars
Protostars
Pre-main-sequence Stars
(T Tau. Stars)
Protostellar
Cores
IR Sources
Accretion Energy
Hydrogen Nuclear Fusion
Opt, Near IR Sources
Gravitational Energy of Stars

3. Spherical Collapse Gas ( ) contracting under the self-gravity
(1) isothermal g= 1 r<rA= 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= 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= g cm-3 　　
second collapse: dynamical collapse begins at the center of the 1st core.
Larson (1969)
Temp-density relation of cloud center. (Tohline 1982)
First Collapse
γ= 1 rC
First Collapse
γ= 1 rB rC rB
The first core begins to contract again, after the central temperature becomes sufficient to dissociate the hydrogen molecule.
Because the dissociation process absorbs thermal energy, the effective equation state becomes soft and gamma is estimated as 1.1.
This phase is called 2nd collapse.
First Core
g=7/5
Second Collapse
γ= 1.10
Second Collapse
γ= 1.10 rA rA
log nc (cm-3)
cf. Masunaga & Inutsuka (2000)

4. Runaway Collapse First core Isothermal spherical collapse shows:
(1)Convergence to
a power-law structure
(2)Divergence of central
density in a finite time.
(3)Only a central part
contracts. 半径
This is called
“runaway collapse.”
Larson 1969, MNRAS, 145,271

5. How about a rotating magnetized cloud?
1. In case with B and W, 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=W=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.
I will talk about two topics.
The first one is for aligned rotator B//Omega.
The central issue will be: whether the magnetic field promotes the fragmentation or suppresses it?
The second topics is on the evolution of non-aligned rotator.
Is the outflow ejected parallel to B or J?
How different is the direction of local B-field from that of global one?

6. Initial Condition Numerical Method B W periodic boundary
To achieve high resolution near the center of the cloud. W B
Nested grid
L=1
L=2
L=3
periodic
boundary
(1) We assume a hydrostatic balance between pressure, Lorentz force, centrifugal force and gravity.
(2) We added both axisymmetric and nonaxisymmetric r perturbations.

7. Nested 4-times finer grid W r W r
The coarsest grid
Nested 4-times finer grid W r W r
a=1,W=0.5

8. First Core Phase Evolution
Nested 28-times finer grid W r
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.

9. 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.
Outflow
Disk
B-Fields
Mass
Inflow ® star
Outflow
Ang.Mom.

10. Angular Momentum Problem
Specific Angular Momentum of a New-born Star
Orbital Angular Momentum of a Binary System
Specific Angular Momentum of a Parent Gas
Centrifugal Radius
Tomisaka 2000 ApJL 528 L41--L44

11. Angular Momentum Transfer
High-density region is formed by gases with small j.
Run-away Collapse
Specific Angular Momentum
of gas <M
Initial
Core Formation
Magnetic torque brings the angular momentum from the disk to the outflow.
Outflow brings the angular momentum.
Accretion Stage
X-axis is the mass measured from the center and Y-axis is specific angular momentum.
I plotted the distribution for three epochs, initial or low-density stage, core formation epoch, and 7000 yr after the core formation.
In the run-away collapse stage, the specific angular momentum of the central, say, 0.1 M sol decreases to 1/3.
However, it is too insufficient to solve the angular momentum problem.
How about the accretion stage? In 7000 yr more than 0.1 solar mass is accreted on the core. J of this gas reduces 3-4 orders of magnitudes.
Magnetic torque seems to bring the angular momentum from the disk to the outflow. And outflow brings the angular momentum away.
7000 yr after
Core Formation
Accumulated Mass

12. L1551 IRS5 Molecular Outflow Optical Jets Snell, Loren, &Plambeck 1980
Saito, Kawabe, Kitamura&Sunada 1996
L1551 IRS5
Snell, Loren, &Plambeck 1980
Optical Jets

13. Evolution of a Rotating First Core
Saigo & Tomisaka (2006, ApJ, 645, )
Saigo, Matsumoto, Tomisaka (2006, in prep.)
I have showed that B-field controls the angular momentum of the first core.
Fragmentation develops quickly in a hydrostatic state (first core) rather than in a contracting circumstance (runaway phase)
Fragmentation in a first core brings binary or multiple stars.
binaries are more popular than single stars.
We study the fragmentation of the 1st core using a nested grid hydrocode.
Temp-density relation of IS gas.
(Tohline 1982)
Masunaga & Inutsuka (2000)
First Collapse
γ= 1 rC rB
First Core
γ= 7/5
Second Collapse
γ= 1.10 rA

14. Hydrostatic Equilibrium
Hydrostatic Axisymmetric Configuration for Barotropic Gas
Angular Momentum Distribution
same as a uniform-density sphere with rigid-body rotation
total mass and total ang. mom.
Self-consistent Field Method (SCF) Hachisu(1986), Tohline, Durisen
to understand the evolution of first core
dissociation
density

15. Examples of Hydrostatic Configuration for Rotating Barotropes
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).

16. Mass-Density Relation ( )
Below , mass increases with central density .
Mass is prop. to Jeans mass
(Chandrasekhar 1949)
Mass accretion drives the core
from lower-left to upper-right.
Above 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.

17. 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
Rotation w=Wtff=0~0.3
increase the BE density
by 1.1~8 times
Perturbations m=2 and m=3
dr/r=10%
Numerical method
HD nested grid
barotropic EOS
hydrostatic sphere
Increase the mass by times
Add m=2 and m=3 perturbation

18. Non-rotating cloud density rc<rdis,Mcl increases with rc.
Jeans mass
Evolution is driven by accretion.
At rc>2rdis, soft EOS makes Mcl decrease.
dynamical contraction (2nd collapse)
maximum mass of a 1st core is equal to ~0.01 M8.

19. Non-rotating model
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.
Maximum mass of a first core is small ~0.01M8.
Quasi-static evolution gives a good agreement with HD result.
max.mass

20. Mass-Density Relation ( )
rotation rate of parent cloud
w<0.015
similar to non-rot. case.
second collapse
w>0.015
Mass increases much
well below

21. 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.

22. 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
fragmentation

23. 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 )
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.

24. Rotating Cloud (w=0.05) Evolution depends on the initial mass. M~MBE:
Follow quasihydrostatic evolution
M>>MBE:
evolution depends on its initial condition.

25. Fragmentation
In a fast rotating cloud, fragmentation (more than 2 fragments) is observed in the 1st core.
This occurs after non-axisymmetric instability is triggered.

26. Typical Rotation Rate NH3 cores (n~3 104cm-3) Goodman et al (1993)
N2H+ cores (~ 2 105cm-3) Caselli et al. (2002)

27. Luminosity of the First Core
3D simulation
Quasi-static evolution
second collapse
second collapse
Lbol
Lifetime of 1st core is
not short !
t(yr)
L is a decreasing function of w.
Absolute value L and timescale are obtained
after is given.

28. Mass Accretion Rate
Mass accretion rate is between the LP solution and a SH disk solution.
Much higher than that expected for SIS.
(Whitworth & Summers 1985)

30. 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.