Magnetic fields in star forming regions: theory
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Magnetic fields in star forming regions:
theory
Daniele Galli
INAF-Osservatorio di Arcetri
Italy
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Outline
• Zeeman effect and polarization• Models of magnetized clouds:
• Magnetic braking
• Equilibrium• Stability• Quasistatic evolution• Dynamical collapse
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Pieter Zeeman (1865 – 1943)
ApJ, 5, 332 (1897)
2 citations (source: ADS)
1 Nobel prize
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Basic observational techniques: Zeeman effect and polarization
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The Zeeman effect in OH toward Orion B
Bourke et al. (2001)
OH line profile
Stokes V spectrum
(RCP-LCP)
Zeeman << line in molecular clouds
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Zeeman measurements in molecular clouds
(cm-3)
G
)B 1/2
Crutcher (1999)
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Summary of Zeeman measurements
HI gas
molecular clouds
OH masers
H2O masers
SiO masers
Vallée (1997)
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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 mMatthews & Wilson (2002)
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Submillimiter polarization in cloud cores
L183L1544
Ward-Thompson et al. (2000)
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Models of magnetized clouds: I. Equilibrium
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force balanceno monopoles
Poisson’s equation
known solutions:• axisymmetric: Mouschovias, Nakano, Tomisaka, etc.• cylindrical: Chandraskhar & Fermi, etc.• helical: Fiege & Pudritz, etc.
System of 5 quasi-linear PDEs in 5 unknowns
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Axially symmetric magnetostatic models
Li & Shu (1996), Galli et al. (1999)
Shu et al. (2000), Galli et al. (2001)
2-D 3-D
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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
un
stab
le
Boyle’s law
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• Cloud supported by thermal pressure: Mcr=MJ, the Jeans mass
• Cloud supported by magnetic fields: Mcr=M
• In general, Mcr= MJ+Mto within 5% (McKee 1989)
• For T=10 K, n=105 cm-3, R=0.1 pc, B=10 G: MJ= M= 1 M
Summary of stability conditions
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R
R
mass M
magnetic flux
m
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R
R
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= 0.1
The magnetic mass-to-flux ratio: observations
101
101
= 0.1
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The magnetic flux problem
• Molecular clouds: /M = (/M)cr
• Magnetic stars with 1-30 kG fields: /M = 10-5 – 10-3 (/M)cr
• Ordinary stars (e.g. the Sun): /M = 10-8 (/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 = qB/mc
• collision time with neutrals =1/ n <vrel>
• example: n=104 cm-3, B=10 G
(electrons=107, (ions=103 >>1
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: Fni=in ni nn <vrel>in (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
tad
()in
<()in
vdrift
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Timescale of magnetic flux loss
at n=104 cm-3, xe=10-7, M/=(M/)cr,, L=0.1 pc
• ambipolar diffusion timescale:
• Ohmic dissipation timescale:
1-10 Myr
1015 yr
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Density distribution and magnetic fieldlines
Desch & Mouschovias (2001)
7.1 Myr
15.2308 Myr 15.23195 Myr
15.17 Myr
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Evolution of the central density
Desch & Mouschovias (2001)
t0 t1 t2
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The velocity and mass-to-flux radial profiles
Desch & Mouschovias (2001)
t0
t0
t2 t1
t2 t1
subcritical
supercritical
supersonic
subsonic
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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
• Poisson’s equation
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t = 5.7x104 yr
Galli & Shu (1993)
t = 0
Singular isothermal sphere with uniform magnetic field
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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, = 10-15 rad/s): J/M = 1022 cm2/s
• 1M dense core (n = 104 cm-3, =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
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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:tb
R/(2vA)
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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 (const.)
• in supercritical collapse, specific angular momentum is conserved (J/M=const.)
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Magnetic braking: observations
Ohashi et al. (1997)
J/M
R
const
.J/Mconst.
Solar system
wide binary
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Conclusions
• Zeeman effect and polarization• Models of magnetized clouds:
• Magnetic braking
• Equilibrium• Stability• Quasistatic evolution• Dynamical collapse