From Clouds to Cores: Magnetic Field Effects on the Structure of

Molecular Gas

Shantanu Basu

University of Western Ontario, Canada

Collaborators:

Takahiro Kudoh (UWO)

Carol Jones (UWO)

Glenn Ciolek (RPI)

John Dubinski (Toronto)

Star Formation 2002, Taiwan, June 12, 2002

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

As found by Myers & Goodman (1988)

Pressure of self-gravity Turbulent pressure

Magnetic Field Strength Data

.2

1

,8

1

2/12/1

1

cv

cB

A

v

v

Using Blos, best fit implies

i.e., Alfvenic motions in molecular clouds?

.45.029.0

29.091.0 loslos,

A

v

A

v

v

v3.14

1.57

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

A Model for Turbulent Molecular Clouds

Kudoh & Basu (2002)

Numerical 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

Highlights: Cloud expands due to turbulent pressure, achieves “steady state”; later contracts as forcing discontinued.

.ˆ)0,(

,sech0, 20

zBtzB

HzH

tz

z

z/H z/H

t / t cross,0

.,, zyy vvB

A Model for Turbulent Molecular Clouds

Turbulent driving =>

,0 stAv cv

.4

vz

A

A

Bv

v

until rises and dropsBut

At half-mass position z1/2, simulation yields

.1

5.0

withcloud a forA

v

v

Kudoh & Basu (2002)

Av

toty

z

Dense Cores

Tafalla et al. (1999)The specific case of L1544:

Tafalla et al. (1999) & Williams et al. (1999) => spectra imply

-1infall s km 1.0v over a range of scales r ~ 0.02 – 0.1 pc.

Apparent starless core with low turbulence.

A Model for L1544

• Magnetic field model with ambipolar diffusion. Contraction of supercritical core => infall speed ~ 0.1 km/s for r ~ 0.01 – 0.1 pc.

• Oblate model flattening + observed elongation => implied inclination angle = 74o.

• Supercritcal core ( ~ 2) and = 74o => estimate Blos.

Ciolek & Basu (2000)

Zeeman Data for L1544

• Ciolek & Basu (2000) predict Blos = 16 G within r = 0.06 pc.

• Crutcher & Troland (2000) measure Blos = 11 +/- 2 G within r = 0.06 pc.

• CT (2000) measurement not sensitive to inner region where Blos > 25 G , therefore “measured field consistent with the model prediction”.

Polarization Data for L1544

Ward-Thompson et al. (2000) –

Polarized submillimeter emission

• Angular offset = 29o +/- 6o between apparent minor axis and apparent B direction.

• Inconsistent with pure oblate model.

Magnetic Field Projection for Triaxial Cores

Basu (2000)

Projected B field (dashed) and density contours (solid) for triaxial body ( = 0.3,0.6) seen from three viewing angles.

Probability distribution function for offset angle .

Intrinsic Shapes of Cores

Can invert observations of projected axis ratios to get an intrinsic shape distribution.

Recent Data:

• Lee & Myers (1999) - compile projected b/a for 406 cores (optical selection).

• Jijina, Myers, & Adams (1999) - b/a for 264 cores (NH3 maps).

Analysis:

• Jones, Basu, & Dubinski (2001); Jones & Basu (2002) - invert these and other distributions to obtain intrinsic shapes.

• Previously, Ryden (1996) – several catalogs of 19 – 89 each.

Intrinsic Shapes of Cores

Data for NH3 cores, Jijina et al. (1999)

Two key features of all observed distributions:

(1) Significant decline towards p = 1.

(2) Broad peak near p = 0.5 – 0.65.

Property (1) incompatible with pure oblate or even pure prolate objects.

Triaxiality required for a better fit.

Jones, Basu, & Dubinski (2001)

Intrinsic Shapes of Cores

Jones, Basu, & Dubinski (2001)

A uniform distribution of triaxial axis ratios =c/a, =b/a does not provide a good fit.

Assuming triaxial ellipsoids (Binney 1985), any distribution of ratios =c/a, =b/a yields a distribution of observed axis ratios p for a large number of random viewing angles.

Instead, assume Gaussian distributions of and and find distributions of projected axis ratio p.

Intrinsic Shapes of Cores

Gaussians with centers at width = 0.1. Best fit using analysis =>

inverse values

Best fit is much smaller than => cores primarily flattened in one direction => triaxial but nearly oblate.Broad peak favors near-oblate triaxial objects.

Jones, Basu, & Dubinski (2001)

Shapes of Cores: The Bottom Line

Near-oblate triaxial distributions provide best fit in all cases!

Jones & Basu (2002)

Triaxial Cores

Galli et al. (2001) – nonaxisymmetric equilibrium- L1544 overlay

Origin

• Effect of large scale turbulence?

• Triaxial equilibria?

• Gravitationally driven effect?

Implication

• Leads naturally to multiple star formation?

Nakamura & Li (2002) – nonaxisymmetric collapse

Summary

• Magnetic field strength data imply an ensemble of turbulent clouds satisfying

• We have performed global 1-D ideal MHD simulations of turbulent molecular clouds. Stay tuned for Kudoh’s talk!

• Results agree with above relation; <vA> adjusts to value slightly greater than v, which itself is related to the driving force.

• Oblate MHD supercritical core model for L1544 makes reasonable prediction for Blos.

• Angular offset of B direction with apparent minor axis => triaxiality.

• Detailed analysis of large cloud core data sets show that cores indeed triaxial but nearly oblate.

. Av v