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Magnetic Fields and Protostellar Cores
Shantanu Basu
University of Western Ontario
YLU Meeting, La Thuile, Italy, March 24, 2004
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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
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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
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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)
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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)
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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
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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
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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:
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Snapshots of density
0.25pc
Shock waves
3400 cm10
mn
The density structure is complicated and has many shock waves.
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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
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A Model for MHD Turbulence
Transverse standing wave => boundary is a node for By, antinode for vy.
sub-Alfvenic motions
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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
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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
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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.
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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.
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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
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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.
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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)
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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.
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MHD Model of Gravitational Instability
0 0.5 Subcritical sheet
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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!
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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.