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From Clouds to Cores to
Protostars and Disks
New Insights from Numerical Simulations
Shantanu Basu
The University of Western Ontario
Collaborators:
Glenn E. Ciolek (RPI), Takahiro Kudoh (NAO, Japan),
Eduard I. Vorobyov (UWO)
University of Massachusetts, April 13, 2006
Taurus Molecular Cloud
sound speed
cs  0.2 km/s
5 pc
distance = 140 pc
velocity dispersion
  0.6 km/s
protostar
T Tauri star
  0.25 km/s
Onishi et al. (2002)
Correlation of Magnetic Field with Gas Structure
Goodman et al. (1990) –
polarization map of Taurus
region
Goldsmith et al. (2005)
– high resolution 12CO
map of Taurus
(FCRAO)
Velocity dispersion-Size relations
Heyer and Brunt (2004)
[Velocity dispersion]
Solomon et al. (1987)
[Size scale of the cloud]
 S

1/ 2
1/ 2
Self-gravitational
equilibrium with
turbulence for an
ensemble of
clouds.
Internal cloud
correlations (open
circles) and global
correlations (filled
circles).
Magnetic Field Strength Data
Blos   v  1/ 2 ,
  v  v A,los  Blos
Best fit
v
vA
From Basu (2000), based on Zeeman
data compiled by Crutcher (1999).
Correlation previously noted by
Myers & Goodman (1988); also
Bertoldi & McKee (1992),
Mouschovias & Psaltis (1995).
v
vA,los
 0.45
4 .
 0.91 
sub-Alfvénic
motions.
since Blos  0.5 B averaged
over all possible viewing
angles.
Magnetized Interstellar Cloud Schematic Picture
Magnetic
field line
MHD wave pressure
Cloud
Cloud
magnetic
gravity
force
Turbulence
Protostellar-driven turbulence
Quillen et al. (2006)
NGC 1333
Green circles = outflow
driven cavity locations,
from velocity channel map
Diamonds = Herbig-Haro
objects
Triangles = compact
submm sources
Stars = protostars
Key Questions
• How is interstellar molecular cloud turbulence driven?
external: galactic shear, supernova shells?
internal: outflows, OB stars?
• How is turbulence maintained?
dissipation time < lifetime of cloud?
• What controls fragmentation of molecular clouds?
gravity:
max  0  2 cs2 G 0
ambipolar diffusion:
turbulence:
max  ?
max  0
depends on B field, ionization
depends on power spectrum
• How is the stellar mass accumulated?
Nature of disk accretion?
How does the accretion terminate?
3D Cloud Model with Protostellar Formation and Feedback
Li & Nakamura (2006) – 128 x 128 x 128 simulation
tg=6 x 105 yr
1.5 pc
High resolution global 1.5D MHD simulation
Most of the previous
simulations
Magnetic
field line
Magnetic field line
model a local region.
Low density and
Hot medium
hot gas
Our simulation
box
z
Periodic boundary box
Molecular cloud
Kudoh & Basu (2003)
Molecular
cloud
Self-gravity
If we want to study the global structure
Driving force
of the cloud, this is NOT a good setting
the problem.
Afor
sinusoidal
driving force is input into
M
the molecular cloud.
 critical

3D Periodic Box Simulations
A local region of a molecular cloud.
Main Results:
Turbulent dissipation rate
turb   0 3 0 ,
0 = fixed mean density of the periodic
box
 = one-dimensional velocity dispersion
0  turbulent driving scale
Energy dissipation time scale
td  0  ,
i.e., the crossing time across the
driving scale.
Stone, Gammie, & Ostriker (1998).
Similar results from Mac Low et al.
(1998) and many subsequent
studies.
Problems with Application of Local Simulations
Consider a cloud of size L
If driving scale
and
0  L
0
L
td 
 tcross 


0
then can such rapidly decaying turbulence be
maintained by equally strong stirring?
Basu & Murali (2001) –
“Luminosity problem” if 0  L
L
Model prediction of CO luminosity
would far exceed that actually
observed.
Resolution: Characteristic scale for
dissipation is L for each cloud, not some
inner scale 0.
Dissipation time = global crossing time. Is that OK?
td  tcross
n


  7.5 10  3
-3 

10
cm


L
6
-1/2
yr
Compare to estimated lifetime
tlife  3 10 yr
7
for molecular clouds.
 td  (3  5)  tcross
would be better!
1-D Magnetohydrodynamic (MHD) equations


v
 vz
  z
t
z
z
(mass)
Ideal MHD
By
vz
vz
1 P
1
 vz


By
 g z (z-momentum)
t
z
 z 4
z
v y
v y
By F ( x, t )
1
(y-momentum)
 vz

Bz

t
z 4
z

By

 (vz By  v y Bz )
t
z
g z
 4G
z
kT
P
m
T
T
 vz
0
t
z
(magnetic field)
(self-gravity)
(gas)
(isothermality)
Movie: Time evolution of density and
wave component of the magnetic field
n0 
0
m
 10 4 cm 3
Interface between cold cloud
and hot low-density gas
Note: 1D
simulation
allows
exceptional
resolution (50
points per
scale length).
0.25pc
grid Dz = 0.001 pc
(z)
Kudoh & Basu (2003)
Snapshot of density
Kudoh & Basu (2003)
n0 
0
m
 10 4 cm 3
Shock waves
0.25pc
The density structure is complicated and has many shock waves.
Time averaged density
n0 
0
m
Time averaged quantities  t and
z t are for Lagrangian particles.
 10 4 cm 3
Time averaged density
The scale height is about
3 times larger than that
Initial condition
of the initial condition.
0.25pc
The time averaged density shows a smooth distribution.
Density
Evolution
The density
plots at
various times
are stacked
with time
increasing
upward.
Large scale
oscillations survive
longest after
internal driving
discontinued.
t 0  H 0 cs 0
a
 2.5 105 yr
Driving is terminated
at t =40 t0.
7.5 106 year
Input constant
amplitude
disturbance
during this period.
Turbulent driving
amplitude increases
linearly with time
between t=0 and t=10t0.
Dark Cloud Barnard 68
Alves, Lada, & Lada (2001)
Angle-averaged profile
can be fit by a thermally
supported Bonnor-Ebert
sphere model.
Lada et al. (2003)
Thermal linewidths and near-equilibrium, but
evidence for large scale oscillatory motion.
Data from an Ensemble of Cloud Models
Velocity dispersion () vs. Scale of the clouds
  Z 1/ 2
Time-averaged
gravitational
equilibrium
Consistent with
observations
Filled circles = half-mass position, open
circles = full-mass position for a variety of
driving amplitudes.
b  80cs2 B 2
B
  VA 
4
Best fit to data is for b = 1 ( ≈ 0.5 VA).
Cloud self-regulation => highly
super-Alfvenic motions not possible!
Kudoh & Basu (2006)
Result: Power spectrum of a time snap shot
k 5 / 3
driving
source
By
Power spectrum of vy
Power spectrum of By
Power spectrum as a function of wave number (k) at t =30t0.
k 5 / 3
vy
kH0
kH0
Note that there is significant power on scales larger than
the driving scale (  H 0 ). This is different from power
spectra in uniform media.
Kudoh & Basu (2006)
Dissipation time of energy
The sum of all
Kinetic energy
(lateral)
Dissipation time
t d  10t0  2.5 106 year
Ee
 t / t d A few
crossing times
of the expanded
cloud.
Magnetic energy
Kinetic energy
(vertical)
Note that the energy in transverse
modes remains much greater than
The time we stop driving force
that in generated longitudinal modes.
Key Conclusions of High-resolution
1.5D Turbulence Model
• Turbulence from local sources quickly propagate to fill the cloud.
• Outer regions of low density have high Alfvén speed  leads to large
amplitude motions and generation of long wavelength modes in outer
cloud.
•   Z1/2 and   VA relations naturally satisfied by time-averaged
quantities. Highly super-Alfvénic motions not possible.
• Power spectrum contains most power on the largest scales, in spite of
driving on a smaller inner scale (unlike periodic models).
• Large scale oscillations survive longest after internal turbulence
dissipates.
• Dissipation time is a few crossing times of the expanded cloud, less
than but within reach of estimated cloud lifetimes. It is longer than in
periodic 3D models. But, will this result survive in a 3D global model?
MHD simulation: 2-dimensional
Magnetic
field lineinto
Structure of the z-direction
is integrated
the plane  2D approximation.
Magnetic field line
Low density and
hot gas
2D
simulation
box
Hot medium
1D simulation
box
Molecular cloud
z
Molecular
cloud
Dense
cloud
Self-gravity
Gravitational collapse
occurs.
Driving
force
Indebetouw & Zweibel (2000)
Basu & Ciolek (2004)
Li & Nakamura (2004)
Two-Fluid Thin-Disk MHD Equations
Magnetic thin-disk approximation.
 n
(mass)
(some higher order terms dropped)
  p  ( n v n, p   0
t
 ( n v n, p 
Bz B p Z
2
  p  (  n v n , p v n , p   cs  p  n   n g p 

Bz  p Bz (momentum)
t
2
2
Bz


(vertical magnetic field)
  p  ( Bz v i , p   0
( Note:  p  xˆ 
yˆ ,
t
x
y
v p  vx xˆ  v y yˆ , etc.)

 ni  Bz B p Z
(ion-neutral
drift)
vi, p  v n, p 

Bz  p Bz 

 n  2
2

Z
n

,  n cs2  G n2  Pext  Pmag
2n
2
 ni  1.4
mi  mn
, ni  Kn1/n 2
i  w in
g p   p , FT (   
B p   p  , FT (   
(ionization balance)
2 G
k x2  k y2
1
k k
2
x
(vertical equilibrium)
2
y
FT (  n 
FT ( Bz 
(planar gravity)
Basu &
Ciolek (2004)
(planar magnetic field)
MHD Model of Gravitational Instability
Basu & Ciolek (2004) - Two-dimensional grid (128 x 128), normal to mean B field.
Small random perturbations added to initially uniform state.
Supercritical (B = ½ Bcritical)
Transcritical (B = Bcritical)
Column
density
• Gravitational collapse
happens quickly:
sound crossing time
~ 106 year
• Infall velocity supersonic on ~
0.1 pc scales
• Gravity wins again, but slowly:
magnetic diffusion time
~ 107 year
• Infall velocity is subsonic
• Core spacing is larger
Comparison of models to observations
Taurus
Onishi et al.
(2002)
Still an early stage of comparison
between theory and observation.
Is core formation driven by
gravity, ambipolar diffusion,
or turbulence? Or what
combination of these?
Gravity (i.e., highly supercritical fragmentation)  accounts for YSO
spacings in dense regions, max  (2  4) cs2 G 0
Ambipolar diffusion (i.e., transcritical or subcritical fragmentation)
 accounts for large-scale subsonic infall and possibly for the low
star formation efficiency (SFE). Transcritical fragmentation may be
related to “isolated star formation”.
Turbulence  is strong in cloud common envelope, and may also
enforce low SFE if not dissipated quickly.
A self-consistent model of core collapse
leading to protostar and disk formation
A disk that forms naturally
from the collapse of the
core. Previous models
Zoom in to simulate
have usually studied
the collapse of a
isolated disks.
nonaxisymmetric
supercritical core
0.5 pc
Basu & Ciolek (2004)
100 AU
Vorobyov & Basu (2005)
Disk Formation
and Protostellar
Accretion
Ideal MHD 2-D (r,q 
simulation of rotating
supercritical core.
Logarithmically
spaced grid. Finest
spacing (0.3 AU)
near center. Sink
cell introduced after
protostar formation.
Disk evolution
driven by infalling
envelope.
Vorobyov & Basu (2005)
Spiral Structure and Episodic Accretion
2D logarithmically spaced grid follows collapse
to late accretion phase.
FU Ori events
smooth
mode
burst mode
  integrated gravitational torque
Vorobyov & Basu (2005)
YSO Accretion History
Hartmann (1998) –
empirical inference,
based on ideas
advocated by Kenyon
et al. (1990).
Vorobyov & Basu (2006) –
theoretical calculation of disk
formation and evolution
Spiral Structure
Sharp spiral structure and
embedded clumps (denoted by
arrows) just before a burst occurs.
Vorobyov & Basu (2006)
Diffuse, flocculent spiral
structure during the quiescent
phase between bursts.
Summary: From Clouds to Cores to Disks
• One-dimensional simulations of turbulence:
- largest (supersonic) speeds in outermost parts of stratified cloud
- significant power generated on largest scales even with driving on
smaller scales, due to stratification effect
- dissipation time is related to cloud size, not internal driving scale:
provides a way out of “luminosity problem” if this result holds for the
large cloud complexes.
• Two-dimensional simulations of magnetically-regulated fragmentation:
- infall speed subsonic or supersonic depending on magnetic field
strength. Core spacing can also depend on B field.
- transcritical fragmentation may be important for understanding
isolated low mass star formation and low SFE.
• Detailed collapse of nonaxisymmetric rotating cores:
- newly discovered “burst mode” of accretion
- envelope accretion onto disk  disk instability  clump formation
 clumps driven onto protostar  repeats until envelope accretion
declines sufficiently
- explains FU Ori bursts, low disk luminosity. Protoplanet formation?