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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

28Bourke et al. (2001)

= 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