Multidimensional Models of Magnetically Regulated

Star Formation

Shantanu Basu

University of Western Ontario

Collaborators:

Glenn E. Ciolek (RPI), Takahiro Kudoh (NAO, Japan), Eduard I. Vorobyov (UWO)

Submillimeter Astronomy, CfA, June 15, 2005

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

magneticforce

gravity

MHD wave pressure

Turbulence

Magnetic field line

CloudCloud

Magnetized Interstellar Cloud Schematic Picture

Magnetic field line

MHD simulation: 2-dimensional

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 leads to cores.

Dense core

Two-Fluid 2-D 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

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

n

i nni i n

i in

p p

x y

t

B Zc B B

tB

Bt

B ZB B

Z c G P

m mn Kn

w

GFT

k k

v

v Bv v g

v

Bv v

g

2

2 2

1,

n

p p z

x y

FT

FT FT Bk k

B

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

Basu & Ciolek (2004)

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

Column density Mass-to-flux ratio

7,max .0 10 at 3.2 10 yr.n n t

. 0.57 pcT m

0 1 Initially critical mass-to-flux ratio balance between gravity and magnetic restoring forces. But neutrals slip past ions/magnetic field.

likely low SFE

MHD Model of Gravitational Instability

0 1 Infall motions are subsonic.

Maximum

0.5 .scSimilar to infall speeds in cores where measured, e.g., Tafalla et al. (1998), Williams et al. (1999), Lee et al. (1999, 2001, 2004)

Horizontal slice through a core.

0.1 pc

MHD Model of Gravitational Instability

0 2

,max .0 10n n Basu & Ciolek (2005)

0 10

0 0.5

Negligible B

WeakStrong

B

B

in all images

- intermediate time scale ~ 4 Myr- supersonic infall- moderate elongation- large spacing

- longest time scale ~ 50 Myr- subsonic infall- mildest elongation- small spacing

- shortest time scale ~ 2 Myr- supersonic infall- greatest elongation- smallest spacing

MHD Models of Gravitational Instability

Relate to observed maps?

Taurus, C18O (Nanten telescope)

Further effects necessary?

- core spacing

- core masses, shapes

- polarization patterns

- magnitude of infall motions

- turbulent motions (Li’s talk)

- 3D, non-periodic important for turbulence

- microphysics (ionization, heating/cooling)

Magnetic field line

MHD simulation: 1-dimensional

Self-gravity

Magnetic field line

Driving force

z

Molecular cloud

Hot medium

1D simulationbox

Low density andhot gas

Molecular cloud

2D simulationbox

Kudoh & Basu (2003)

A model for turbulent motions

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

year105.7 6

Input 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

Kudoh & Basu (2003)

Linewidth-Size Relation from Ensemble of Cloud Models

Kudoh & Basu (2002) Most power concentrated on largest scales. Large scale oscillations survive longest after internal driving discontinued.

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.

Linewidth-size relation

Power spectrum of a time snap shot

0kH

3/5k

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

3/5k

Note that there is significant power on scales larger than the driving scale ( ).

0kH

0H

Po

we

r sp

ectr

um o

f By

Po

we

r sp

ectr

um o

f vy

yB yv

Kudoh & Basu (2005)

drivingsource

Magnetic field line

MHD simulation: 2-dimensional

1D simulationbox

Low density andhot gas

Molecular cloud

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

2D simulationbox

Back to 2-D model. What happens deep within collapsing cores?

Dense core

Zoom in to simulate the collapse of an intially slightly nonaxisymmetric supercritical core

Basu & Ciolek (2004)

Core to Protostar + Disk

Vorobyov & Basu (2005)

Disk Formation and Protostellar Accretion

Vorobyov & Basu (2005)

Ideal MHD 2-D (r,simulation of rotating supercritical core.

See poster (#76, downstairs) on this subject!

Logaritmically spaced grid; inner zone width 0.3 AU.

Spiral Structure and Episodic Accretion

Vorobyov & Basu (2005)

FU Ori events

integrated gravitational torque

Spiral arms create a strong centrifugal disbalance bursts of mass accretion; 0.01 to 0.05 solar masses are accreted.

Summary• Two-dimensional simulations of magnetically-regulated fragmentation:

- core properties depend on magnetic field strength

- infall speeds subsonic for critical and subcritical cases;

for star formation.

- maximum infall speeds supersonic for supercritical case;

for star formation.

• One-dimensional simulations of turbulence:

- stratified cloud has largest (supersonic) speeds in outermost parts

- significant power generated on largest scales even with driving on smaller scales.

• Collapse of nonaxisymmetric rotating cores:

- leads to centrifugally balanced disk spiral structure burst of enhanced accretion spiral structure regenerated …. cycle

continues due to continued mass infall from envelope.

yr 107

yr 106

(poster Vorobyov & Basu)