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

Onishi et al. (2002)

Taurus Molecular Cloud

5 pc

0.25 km/s

0.2 km/ssc 0.6 km/s

velocity dispersionsound speed

distance = 140 pc

protostar

T Tauri star

Goodman et al. (1990) – polarization map of Taurus region

Goldsmith et al. (2005) – high resolution 12CO map of Taurus (FCRAO)

Correlation of Magnetic Field with Gas Structure

Velocity dispersion-Size relations

[Size scale of the cloud]

[Vel

ocity

dis

pers

ion]

1/ 2

1/ 2

S

Self-gravitational equilibrium withturbulence for an ensemble of clouds.

Solomon et al. (1987)

Heyer and Brunt (2004)

Internal cloud correlations (open circles) and global correlations (filled circles).

Magnetic Field Strength Data

.4

,

loslos,

2/1los

Bv

B

Av

v

since averaged over all possible viewing angles.

,los

0.91v

Av

Correlation previously noted by Myers & Goodman (1988); also

Bertoldi & McKee (1992), Mouschovias & Psaltis (1995).

0.45v

Av

From Basu (2000), based on Zeeman data compiled by Crutcher (1999). BB 5.0los

Best fit

sub-Alfvénic motions.

magneticforce

gravity

MHD wave pressure

Turbulence

Magnetic field line

CloudCloud

Magnetized Interstellar Cloud Schematic Picture

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:

ambipolar diffusion:

turbulence:

• How is the stellar mass accumulated?

Nature of disk accretion?

How does the accretion terminate?

2max 0 02 sc G

max 0

max ? depends on power spectrum

depends on B field, ionization

3D Cloud Model with Protostellar Formation and Feedback

Li & Nakamura (2006) – 128 x 128 x 128 simulation

1.5 pc

tg=6 x 105 yr

Magnetic field line

High resolution global 1.5D MHD simulation

A sinusoidal driving force is input into the molecular cloud.

Self-gravity

Magnetic field line

Driving force

z

Molecular cloud

Hot medium

criticalM

Our simulationbox

Most of the previous simulationsmodel a local region.

Periodic boundary box

If we want to study the global structureof the cloud, this is NOT a good setting for the problem.

Low density andhot gas

Molecular cloud

Kudoh & Basu (2003)

3D Periodic Box Simulations

Stone, Gammie, & Ostriker (1998).

Similar results from Mac Low et al. (1998) and many subsequent studies.

Main Results:

Turbulent dissipation rate

,03

0 turb

= fixed mean density of the periodic box

= one-dimensional velocity dispersion

turbulent driving scaleEnergy dissipation time scale

,0 dt

i.e., the crossing time across the driving scale.

A local region of a molecular cloud.

Problems with Application of Local Simulations

Consider a cloud of size L

If driving scale L0

and

Ltt crossd 0

then can such rapidly decaying turbulence be maintained by equally strong stirring?

Basu & Murali (2001) – “Luminosity problem” if L0Model prediction of CO luminosity would far exceed that actually observed.

Resolution: Characteristic scale for dissipation is L for each cloud, not some inner scale

L

Dissipation time = global crossing time. Is that OK?

-1/26

3 -37.5 10 yr

10 cmd cross

L nt t

Compare to estimated lifetime

for molecular clouds.

(3 5)d crosst t would be better!

yr 103 7lifet

z

v

zv

tz

z

zy

yz

zz g

z

BB

z

P

z

vv

t

v

4

11

),(

4

1 txF

z

BB

z

vv

t

v yz

yz

y

)( zyyzy BvBv

zt

B

m

kTP

Gz

g z 4

0

z

Tv

t

Tz

1-D Magnetohydrodynamic (MHD) equations

(mass)

(z-momentum)

(y-momentum)

(magnetic field)

(self-gravity)

(gas)

(isothermality)

Ideal MHD

Movie: Time evolution of density and wave component of the magnetic field

(z)0.25pc

34

00

cm10

m

n

Interface between cold cloudand hot low-density gas

Kudoh & Basu (2003)

Note: 1D simulation allows exceptional resolution (50 points per scale length). grid z = 0.001 pc

Snapshot of density

0.25pc

Shock waves

3400 cm10

mn

The density structure is complicated and has many shock waves.

Kudoh & Basu (2003)

Time averaged densityTime averaged quantities and are for Lagrangian particles.

Initial condition

Time averaged density

The scale height is about 3 times larger than that of the initial condition.

3400 cm10

mn

0.25pc

The time averaged density shows a smooth distribution.

t

tz

year105.7 6Input constant amplitude disturbance during this period.

The density plots at various times are stacked with time increasing upward.

Turbulent driving amplitude increases linearly with time between t=0 and t=10t0.

Driving is terminated at t =40 t0.

aDensity Evolution

yr 105.2 5

000

scHt

Large scale oscillations survive longest after internal driving discontinued.

Dark Cloud Barnard 68

Lada et al. (2003)

Thermal linewidths and near-equilibrium, but evidence for large scale oscillatory motion.

Angle-averaged profile can be fit by a thermally supported Bonnor-Ebert sphere model.

Alves, Lada, & Lada (2001)

Data from an Ensemble of Cloud Models

2/1Z

Velocity dispersion () vs. Scale of the clouds

Consistent with observations

Time-averaged gravitational equilibrium

Filled circles = half-mass position, open circles = full-mass position for a variety of driving amplitudes.

Best fit to data is for = 1 ≈ 0.5 VA) Cloud self-regulation => highly super-Alfvenic motions not possible!

4A

BV

2 208 sc B

Kudoh & Basu (2006)

Result: Power spectrum of a time snap shot

0kH

3/5k

Power spectrum as a function of wave number (k) at t =30t0.

3/5k

Note that there is significant power on scales larger than the driving scale ( ). This is different from power spectra in uniform media.

0kH

0H

Po

we

r sp

ectr

um o

f By

Po

we

r sp

ectr

um o

f vy

yB yv

Kudoh & Basu (2006)

drivingsource

Dissipation time of energy

Magnetic energy

Kinetic energy (vertical)

Kinetic energy (lateral)

The sum of all

The time we stop driving force

Dissipation timeyear105.210 6

0 ttd

dtteE /

Note that the energy in transversemodes remains much greater thanthat in generated longitudinal modes.

A few crossing timesof the expanded cloud.

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?

Magnetic field line

MHD simulation: 2-dimensional

Self-gravity

Magnetic field line

Driving force

z

Molecular cloud

Hot medium

1D simulationbox

Low density andhot gas

Molecular cloud

Structure of the z-direction is integrated intothe plane 2D approximation.

2D simulationbox

Indebetouw & Zweibel (2000)Basu & Ciolek (2004)Li & Nakamura (2004)

Gravitational collapse occurs.

Dense cloud

Two-Fluid Thin-Disk MHD Equations

ˆ ˆ( : ,

ˆ ˆ, .)

p

p x y

Note x yx y

v v x v y etc

,

, 2, ,

,

, ,

2 2

1/ 2

0

2 2

0

2 2

,2 2

1.4 ,

2,

np n n p

n n p z pp n n p n p s p n n p z p z

zp z i p

z pnii p n p z p z

n

nn s n ext mag

n

i nni i n

i in

p p

t

B Zc B B

tB

Bt

B ZB B

Z c G P P

m mn Kn

w

GFT

k

v

v Bv v g

v

Bv v

g

2 2

2 2

1,

n

x y

p p z

x y

FTk

FT FT Bk k

B

(some higher order terms dropped)Magnetic thin-disk approximation.

Basu & Ciolek (2004)

(mass)

(momentum)

(vertical magnetic field)

(ion-neutral drift)

(vertical equilibrium)

(ionization balance)

(planar gravity)

(planar magnetic field)

Supercritical (B = ½ Bcritical) Transcritical (B = Bcritical)

• 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

Columndensity

MHD Model of Gravitational InstabilityBasu & Ciolek (2004) - Two-dimensional grid (128 x 128), normal to mean B field. Small random perturbations added to initially uniform state.

Comparison of models to observations

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,

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.

Taurus

Onishi et al. (2002)

Still an early stage of comparison between theory and observation.

2max 0(2 4) sc G

Zoom in to simulate the collapse of a nonaxisymmetric supercritical core

Basu & Ciolek (2004)

A self-consistent model of core collapse leading to protostar and disk formation

Vorobyov & Basu (2005)

A disk that forms naturally from the collapse of the core. Previous models have usually studied isolated disks.

0.5 pc

100 AU

Disk Formation and Protostellar Accretion

Vorobyov & Basu (2005)

Ideal MHD 2-D (r,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.

Spiral Structure and Episodic Accretion

Vorobyov & Basu (2005)

FU Ori events

integrated gravitational torque

smooth mode burst mode

2D logarithmically spaced grid follows collapse to late accretion phase.

Hartmann (1998) – empirical inference, based on ideas advocated by Kenyon et al. (1990).

YSO Accretion History

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.

Diffuse, flocculent spiral structure during the quiescent phase between bursts.

Vorobyov & Basu (2006)

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?