Magnetic fields in star forming regions: theory

1
Magnetic fields in star
forming regions theory

• Daniele Galli
• INAF-Osservatorio di Arcetri
• Italy

2
Outline

• Zeeman effect and polarization
• Models of magnetized clouds
• Magnetic braking

• Equilibrium
• Stability
• Quasistatic evolution
• Dynamical collapse

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ApJ, 5, 332 (1897)
2 citations (source ADS) 1 Nobel prize
Pieter Zeeman (1865 1943)
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Basic observational techniques Zeeman effect
and polarization
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The Zeeman effect in OH toward Orion B
OH line profile
Stokes V spectrum (RCP-LCP) DnZeeman ltlt Dnline
in molecular clouds
Bourke et al. (2001)
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Zeeman measurements in molecular clouds
(mG)
B r1/2
è
(cm-3)
Crutcher (1999)
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Summary of Zeeman measurements
H2O masers
OH masers
molecular clouds
SiO masers
HI gas
Vallée (1997)
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Polarization
(Weintraub et al. 2000)
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Polarization map of background starlight
in the Milky Way
Ophiuchus
Taurus
Orion
Mathewson Ford (1970)
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The magnetic field in M51
optical polarization (Scarrott et al. 1987)
radio synchrotron polarization (Beck et al.
1987)
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Optical polarization map of Taurus
5 pc
Moneti et al. (1984), Heyer et al. (1987)
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Weintraub et al. (2000)
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(Weintraub et al. 2000)
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Hourglass field geometry in OMC-1?
Schleuning (1998)
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Barnard 1 at 850 mm
Matthews Wilson (2002)
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Submillimiter polarization in cloud cores
L183
L1544
Ward-Thompson et al. (2000)
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Models of magnetized clouds I. Equilibrium
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force balance
no monopoles
Poissons equation
System of 5 quasi-linear PDEs in 5 unknowns

• known solutions
• axisymmetric Mouschovias, Nakano, Tomisaka, etc.
• cylindrical Chandraskhar Fermi, etc.
• helical Fiege Pudritz, etc.

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Axially symmetric magnetostatic models
3-D
2-D
Li Shu (1996), Galli et
al. (1999)
Shu et al. (2000), Galli
et al. (2001)
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line-of-sight
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Gonçalves, Galli, Walmsley (2004)
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Models of magnetized clouds II. Stability
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The magnetic virial theorem
the magnetic critical mass
the critical mass-to-flux ratio
Chandrasekhar Fermi (1953), Mestel Spitzer
(1956), Strittmatter (1966)
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The role of the magnetic critical mass
stable
unstable
Boyles law
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Summary of stability conditions

• Cloud supported by thermal pressure
  McrMJ, the Jeans mass
• Cloud supported by magnetic fields
  McrMF
• In general, Mcr MJMF to within 5
  (McKee 1989)
• For T10 K, n105 cm-3, R0.1 pc, B10 mG
  MJ MF 1 M8

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R
mass M
magnetic flux F
m
eR
f
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R
eR
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The magnetic mass-to-flux ratio observations
M/F 0.1
1
10
M/F 0.1
1
10
Bourke et al. (2001)
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The magnetic flux problem

• Molecular clouds
  F/M (F/M)cr
• Magnetic stars with 1-30 kG fields
  F/M 10-5 10-3 (F/M)cr
• Ordinary stars (e.g. the Sun)
  F/M 10-8 (F/M)cr

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Models of magnetized clouds II. Quasistatic
evolution
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Ionisation fraction in molecular clouds
Bergin et al. (1999)
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Field-plasma coupling

• gyration frequency w qB/mc
• collision time with neutrals t 1/ n ltsvrelgt
• example n104 cm-3, B10 mG
• (wt)electrons107, (wt)ions103 gtgt1
• magnetic field well coupled to the plasma

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Effects of the field on the neutrals

• The field acts on neutrals indirectly only
  through collisions between neutral and charged
  particles
• frictional force on the neutrals
  Fnimin ni nn ltsvrelgtin (vi-vn)
• The field slips through the neutrals at a
  velocity
• vdrift vi-vn that depends on the field
  strength and the ionisation fraction (Mestel
  Spitzer 1956)

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Diffusion of the magnetic field
vdrift
(F/M)in
tad
F/Mlt(F/M)in
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Timescale of magnetic flux loss

• at n104 cm-3, xe10-7, M/F(M/F)cr,, L0.1 pc
• ambipolar diffusion timescale
• Ohmic dissipation timescale

1-10 Myr
1015 yr
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Density distribution and magnetic fieldlines
15.17 Myr
7.1 Myr
15.23195 Myr
15.2308 Myr
Desch Mouschovias (2001)
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Evolution of the central density
t0
t1
t2
Desch Mouschovias (2001)
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The velocity and mass-to-flux radial profiles
supercritical
subsonic
t0
t2
t1
t2
t1
t0
subcritical
supersonic
Desch Mouschovias (2001)
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Core evolution by ambipolar diffusion
R0.75 pc
M10 M8
n103-107 cm-3
vmax0.4 km s-1
Fiedler Mouschovias (1992,1993)
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Models of magnetized clouds II. Collapse
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The equations of magnetohydrodynamics

• equation of continuity

• equation of momentum

• induction equation

• no monopoles

• Poissons equation

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t 5.7x104 yr
t 0
Singular isothermal sphere with uniform
magnetic field
Galli Shu (1993)
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t 1.1x105 yr
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t 1.7x105 yr
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Magnetic reconnection
Mestel Strittmatter (1966)
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Magnetic braking
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The angular momentum problem

• 1M of ISM (n 1 cm-3, W 10-15 rad/s)
  J/M 1022 cm2/s
• 1M dense core (n 104 cm-3, W 1 km s-1/pc)
  J/M 1021 cm2/s
• 1M wide binary (T 100 yr)
  J/M 1020 cm2/s
• Solar system
  J/M 1018 cm2/s

8
8
8
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Magnetic Braking

• Magnetic fields can redistribute angular momentum
  away from a collapsing region
• Outgoing torsional Alfvèn waves must couple with
  mass equal to mass in collapsing region
  (Mouschovias Paleologou 1979, 1980)
• Timescale for magnetic braking

r0
r
tb r R/(2r0vA)
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• MHD waves transport angular momentum from the
  core to the envelope
• magnetic braking timescale shorter than ambipolar
  diffusion, but longer than free-fall
• during ambipolar diffusion stage, core corotates
  with envelope (Wconst.)
• in supercritical collapse, specific angular
  momentum is conserved (J/Mconst.)

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Magnetic braking observations
J/M
W const.
J/M const.
wide binary
Solar system
R
Ohashi et al. (1997)
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Conclusions

• Zeeman effect and polarization
• Models of magnetized clouds
• Magnetic braking

• Equilibrium
• Stability
• Quasistatic evolution
• Dynamical collapse