Magnetic Fields and Protostellar Cores

Shantanu Basu

University of Western Ontario

YLU Meeting, La Thuile, Italy, March 24, 2004

Magnetic Field Strength Data

Crutcher (1999) and Basu (2000)

?2/1nB

constant?4

BvA

A better correlation2/1nB v

Av v

Best fit slope = 0.47

Best fit slope = 1.00

1-D velocity dispersion

Magnetic Field Strength DataTwo separate correlations

12/12

GB

Best fit => .14.3los

(1)

However, los2 BB

5.1

(2)

Dimensionless mass-to-flux ratio

21

2

2 vcG

e.g., Myers & Goodman (1988)

Pressure of self-gravity Turbulent pressure

Magnetic Field Strength Data

.21

,8

1

2/12/1

1

cv

cB

A

v

v

Using Blos, best fit implies

i.e., Alfvenic motions in molecular clouds?

,los

0.91v

Av

e.g., Myers & Goodman (1988), Bertoldi & McKee (1992), Mouschovias & Psaltis (1995).

(3)

0.45v

Av

Basu (2000)

self-gravity

perturbation

Molecular cloud

Magnetic field line

Schematic picture of our simulation

A sinusoidal perturbation is input into the molecular cloud.

Magnetic field line

Low-density andhot medium

Simulationbox

z

Molecular cloud

Hot medium

Kudoh & Basu (2003)

Basic MHD equations in 1.5 dimensions

2

0

18

14

0

0

4

z

yz zz z

y y yz z

z

yy z z y

z

vt z

Bv v Pv gt z z zv v B

v Bt z z

T Tvt zB

v B v Bt z

g Gz

kTPm

mass continuity

z-momentum

y-momentum

isothermality

magnetic induction

self-gravity (Poisson’s eqn.)

ideal gas law

A Model for Turbulent Molecular CloudsNumerical solution of MHD equations in 1-D.Start with Spitzer 1-D equilibrium state

• Cloud has a moving boundary

• Density stratification due to gravity

• Add nonlinear forcing near z = 0 => nonzero

200, 0 sech ,

ˆ( , 0) .z

z t z HH

B z t B z

.,, zyy vvBKudoh & Basu (2003)

Molecular cloud

Hot medium

A Model for MHD Turbulence in Molecular Clouds

Kudoh & Basu (2003)

Highlights: Cloud expands due to turbulent pressure, achieves “steady state” between t = 10 and t = 40; later contracts when forcing discontinued at t = 40. Outer cloud undergoes largest amplitude oscillations.

Resolution: 50 points per length H0 .

in this model.

20 0 0 030 , , 1s sa c H c H

Parameters:

Snapshots of density

0.25pc

Shock waves

3400 cm10

mn

The density structure is complicated and has many shock waves.

Time averaged density Time averaged quantities and are for Lagrangian particles.

Initial condition

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

4 300 10 cmn

m

0.25pc

The time averaged density shows a smooth distribution.

t

tz

A Model for MHD Turbulence

Transverse standing wave => boundary is a node for By, antinode for vy.

sub-Alfvenic motions

Results for an ensemble of clouds with different turbulent driving strengths:

.50,40,30,20,10 02

0 Hca s

Solid circles => half-mass position

Open circles => edge of cloud

1/ 2Z

0.5 Av

Correlations of Global Properties

Ideal MHD Turbulence in a Stratified Cloud

• Clouds are in a time-averaged balance between turbulent support and gravity.

• Inner cloud obeys equipartition of transverse wave energy,

• Transverse modes dominate,

• Outer low density part of cloud undergoes large longitudinal oscillations, and exhibits transverse (Alfvenic) standing wave modes.

• Correlations and naturally satisfied.

221 .

8 2y

y

Bv

2 2.y zv v

0.5 Av 1/ 2Z

MHD Model of Gravitational Instability

Courtesy of Nakamura & Hanawa (1997)

Complementary to previous model. Solve for dynamics in plane perpendicular to mean magnetic field. No driven turbulence. Ion-neutral friction allowed => non-ideal MHD.

Basu & Ciolek (2004)

A sub-region of a cloud in which turbulence has largely dissipated.

Two-Fluid 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

tB Zc B B

tB Bt

B Z B B

Z c G P

m m n Knw

GFTk 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.

MHD Model of Gravitational InstabilityBasu & Ciolek (2004)

Small perturbations added to periodic initially uniform state.3 3

0 ,01, 3 10 cm .nn

Column density Mass-to-flux ratio

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

Triaxial but more nearly oblate cores.

. 0.57 pcT m

MHD Model of Gravitational Instability

0 1 Infall motions are subsonic. Maximum

0.5 .sc

e.g., observations of L1544, Tafalla et al. (1998)

Note merger of column density into background, e.g., mid-infrared maps of Bacmann et al. (2000).

Horizontal slice through a core.

MHD Model of Gravitational Instability

0 2 supercritical cloud. All other parameters identical.Supersonic infall in cores and extended near-sonic infall.Observationally distinguishable!

6,max .0 10 at 4.2 10 yr.n n t Basu & Ciolek (2004)

Two-Fluid Non-ideal MHD Gravitational Instability

• Ambipolar diffusion leads naturally to a non-uniform distribution of mass-to-flux ratio. Stars form preferentially in the most supercritical regions.

• Supercritical cores and subcritical envelopes created simultaneously by flux redistribution if

• Initially critical model => subsonic infall. Initially significantly supercritical model => supersonic infall.

• Neutral speeds typically greater than ion speeds – gravitationally driven motions.

• Core densities merge into background near-uniform value.

0 1.

MHD Model of Gravitational Instability

0 0.5 Subcritical sheet

The coefficient ofChandrasekhar-Fermi formula

Surface of the cloud

A

yy

Vv

BB ||||

0

=1 (for linear wave)

=0.23

<1 at the surface of the cloud0.25pc

By is small near the surface but vy is not – a standing wave effect!

Dissipation time of energy

Magnetic energy

Kinetic energy (vertical)

Kinetic energy (lateral)

The sum of the all

The time we stop driving force

Dissipation timeyear100.28 6

0 ttd

dtteE /

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