Core Formation due to Magnetic Fields, Ambipolar Diffusion, and Turbulence
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Core Formation due to Magnetic Fields, Ambipolar Diffusion, and Turbulence
Shantanu BasuThe University of Western Ontario
Collaborators: Glenn Ciolek (RPI), Wolfgang Dapp (UWO), Takahiro Kudoh (NAOJ), James Wurster (UWO)
The Cosmic AgitatorUniversity of Kentucky
March 29, 2008
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Onishi et al. (2002)
Taurus Molecular Cloud
5 pc
0.25 km/s
0.2 km/ssc 0.6 km/s velocity dispersion
sound speed
distance = 140 pc
= T Tauri star
= protostar
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Large Scale Molecular Gas Structure of Taurus
Goldsmith et al. (2008)
12CO emission
Striations of gas emission consistent with magnetically-dominated envelope.
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Layer Instability
Consider a layer of surface density .Linear perturbation analysis yields dispersion relation
2
.scrit
cH
G
Moreover, there is a preferred fragmentation scale.
2
max
22 scH
G
at which the growth time is a minimum.
H is the vertical scale height of the layer.
2zg G
2 2 2 2 .c k G ks Gravitational instability if
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Effect of Magnetic Field
Critical magnetic field if
228
2
GB
magnetic pressure
self-gravitational pressure
Where magnetic flux-freezing applies:
Subcritical cloud 12/12
GB No fragmentation occurs
Supercritical cloud 12/12
GB Fragmentation occurs
B
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Ambipolar Diffusion
In a weakly ionized gas, the mean velocity of neutral atoms or molecules will not generally equal the mean velocity of ions and electrons.
Neutrals do not feel the Lorentz force directly, but only through collisions arising from a drift relative to ions.
, , Lorentz
1/21.4 ,
nii p n p
n
i nni i n
i in
F
m mn Kn
w
v v
neutral-ion collision time
Even SUBCRITICAL clouds can undergo fragmentation instability due to ambipolar diffusion, i.e. ion-neutral slip.
ion density vs. neutral density relation, primarily due to cosmic ray ionization
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Magnetic field line
MHD simulation: 2-dimensional
Low density andhot gas
Molecular cloud
Integrate thorugh structure of the z-direction near the midplane 2D approximation.
2D simulationbox
Gravitational fragmentation occurs.
Dense core
Kudoh & Basu (2003,2006) – dense midplane of stratified turbulent cloud has transonic/subsonic motions.
1D simulationbox (Kudoh & Basu)
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Modes of Fragmentation
Gravitational Fragmentation
Turbulent Fragmentation
dynamic (supercritical mass-to-flux ratio)
quasistatic ambipolar-diffusion (subcritical mass-to-flux ratio)
(Linear perturbations)
We can test all of these scenarios including the effects of magnetic fields and ambipolar diffusion.
supercritical subcritical
(Highly nonlinear perturbations)
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MHD Model of Gravitational Fragmentation
In all images, but magnetic field strength varies.
Added small (few %) initial random white noise perturbations to column density, magnetic field.
100,max, nn
Thin disk approximation
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Linear Perturbation Analysis for Magnetic Disk with AD
Ciolek & Basu (2006)
g g
flux freezing imperfect coupling
2 ,H H cg s
7 4 310 at 10 cmi
n
CR ionization
1/ 200 ,0
0
2 , neutral-ion collision timeniGB
growth time of instability
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Ciolek & Basu (2006); Basu, Ciolek, & Wurster (2008)
Fragmentation Scales
This curve first derived by Morton & Mouschovias (1991)
Solid lines = linear fragmentation theory. Symbols = result of 2D numerical simulations
rate.growth maximumh with wavelengt, mg
Case of low external pressure on disk
High external pressure case
Converges to 2 for both,
1 and 1.
Hg m
1/ 200
0
2 GB
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MHD Model of Gravitational Instability
0 2
Basu, Ciolek & Wurster (2008)
0 10 0 0.5
Very weak B
Weak StrongB
B
in all images
- t = 20- |v|max=1.1 cs
- moderate elongation- large spacing
- t = 200- |v|max=0.4 cs
-mildest core elongation
- t = 10- |v|max=1.2 cs
- most elongated
Highly supercritical
Critical B
- t = 100- |v|max=0.7 cs
- largest spacing
10 Supercritical
Subcritical
box size ~ 2 pc, time unit ~ 2 x 105 yr if nn,0= 3 x 10-3 cm-3, scales as nn,0-1/2.
100,max, nn 1282 cells in each model
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MHD Model of Gravitational Instability
10 0 2
Basu, Ciolek, & Wurster (2008)
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INITIAL Core Mass Function (Grav. Fragmentation)
Narrow lognormal-like. High-mass slope much steeper than observed CMF/IMF.
“Core” = enclosed region with
.20, nn
Basu, Ciolek, & Wurster (2008)
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Observed Core Mass Fcn and Initial Mass Fcn
Data from Nutter & Ward-Thompson (2007)
Data from Muench, Lada, & Lada (2002)
Lognormal Lognormal/power-law
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Cumulative histogram of 1524 cores from over 400 separate simulations
INITIAL Core Mass Function from Multiple Models
Add results from a range of models from =0.5 to 0 = 2.0.
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Turbulent Fragmentation with B and Ambipolar DiffusionThin disk approximation Li & Nakamura (2004)
(a)-(e) subcritical (0.83model, (f)-(h) supercritical ( = 1.25model.
vk2~ k -4 spectrum – has a large-scale flow
note filamentarity and velocity vectors
time unit = 2 Myr; box width = 3.7 pc
Will this work in 3D?
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Magnetic field line
Molecular clouddense sheet
Low density and hot gasMagnetic field
3D simulationbox
X
y
z
Input large perturbation perpendicular to magnetic field at t=0
MHD simulation: (1+2 =) 3-dimensional
Low density andhot gas
Kudoh, Basu, Ogata, & Yabe (2007), Kudoh & Basu (2008)
Top view
Sideview
MHD with ion-neutral slip
We have a new explicit 3D non-ideal MHD code.
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3D Turbulent Fragmentation with B and AD
Kudoh & Basu (2008)
42 kvk
Nonlinear initial velocity field
rms amplitude
0 0.5
yr102 50 t
Gas density in midplane (z=0)
A vertical slice of gas density
Nonlinear IC
Linear IC
using 64 x 64 x 40 cells
allowed to decay
box width = 2.5 pc
3 Alfv c va s
trans-Alfvenic
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3D Turbulent Fragmentation with B and AD
Kudoh & Basu (2008)
What’s really happening?
.4
Bv Bnt
ni B B Bn
28 2csB
2 2 3/2 2
2 2 2L L Ln n i n
ADB B Bni
is a proxy for .
Early turbulent compression
5/2 quickly as L LAD Then, higher density region evolves with near vertical force balance
1/2 more slowlyAD
Rapid contraction when/where 1.
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Core Mass Spectrum for 2D Turbulent Model
Basu, Ciolek, Dapp, & Wurster (2008)
BROAD tail, ROUGHLY consistent with IMF.
But, final fate still undetermined.
Thin disk (2D) allows many runs in order to compile statistics
0 2
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Conclusions
• Transcritical gravitational fragmentation has a maximum (>> Jeans) fragment scale. Subcritical and supercritical fragmentation both occur at ~ Jeans scale
• Nonlinear gravitational fragmentation yields expected fragment spacing and observationally testable kinematics for different ’s
• Initial CMF is narrow for gravitational fragmentation with any single mass-to-flux ratio, but a variety of ’s can yield an overall broad distribution.
• Model of 3D turbulent fragmentation reveals accelerated fragmentation for subcritical clouds with trans-Alfvenic turbulence.
• Initial CMF from turbulent fragmentation is also broad but clear mapping to IMF via monolithic collapse not assured
A wide range of results are available for comparison to observations.