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Magnetic fields in star forming regions: theory

Magnetic fields in star forming regions: theory. Daniele Galli INAF-Osservatorio di Arcetri Italy. Outline. Zeeman effect and polarization Models of magnetized clouds: Magnetic braking. Equilibrium Stability Quasistatic evolution Dynamical collapse.

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Magnetic fields in star forming regions: theory

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  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

  3. ApJ, 5, 332 (1897) 2 citations (source: ADS) 1 Nobel prize Pieter Zeeman(1865 – 1943)

  4. Basic observational techniques: Zeeman effect and polarization

  5. The Zeeman effect in OH toward Orion B OH line profile Stokes V spectrum (RCP-LCP) DnZeeman << Dnline in molecular clouds Bourke et al. (2001)

  6. Zeeman measurements in molecular clouds (mG) Br1/2 è (cm-3) Crutcher (1999)

  7. Summary ofZeeman measurements H2O masers OH masers molecular clouds SiO masers HI gas Vallée (1997)

  8. (Weintraub et al. 2000)

  9. Hourglass field geometry in OMC-1? Schleuning (1998)

  10. Barnard 1 at 850 mm Matthews & Wilson (2002)

  11. Submillimiter polarization in cloud cores L1544 L183 Ward-Thompson et al. (2000)

  12. Models of magnetized clouds: I. Equilibrium

  13. force balance no monopoles known solutions: • axisymmetric: Mouschovias, Nakano, Tomisaka, etc. • cylindrical: Chandraskhar & Fermi, etc. • helical: Fiege & Pudritz, etc. Poisson’s equation System of 5 quasi-linear PDEs in 5 unknowns

  14. Axially symmetric magnetostatic models 3-D 2-D Shu et al. (2000), Galli et al. (2001) Li & Shu (1996), Galli et al. (1999)

  15. line-of-sight

  16. Gonçalves, Galli, & Walmsley (2004)

  17. Models of magnetized clouds: II. Stability

  18. The magnetic virial theorem the magnetic critical mass the critical mass-to-flux ratio Chandrasekhar & Fermi (1953), Mestel & Spitzer (1956), Strittmatter (1966)

  19. The role of the magnetic critical mass unstable stable Boyle’s law

  20. Summary of stability conditions • Cloud supported by thermal pressure: Mcr=MJ, the Jeans mass • Cloud supported by magnetic fields: Mcr=MF • In general, Mcr= MJ+MFto within 5% (McKee 1989) • For T=10 K, n=105 cm-3, R=0.1 pc, B=10 mG: MJ= MF= 1 M8

  21. R mass M magnetic flux F m eR f

  22. R eR

  23. The magnetic mass-to-flux ratio: observations M/F = 0.1 1 10 M/F = 0.1 1 10 Bourke et al. (2001)

  24. 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

  25. Models of magnetized clouds: II. Quasistatic evolution

  26. Ionisation fraction in molecular clouds Bergin et al. (1999)

  27. Field-plasma coupling • gyration frequency w = qB/mc • collision time with neutrals t =1/ n <svrel> • example: n=104 cm-3, B=10 mG (wt)electrons=107, (wt)ions=103 >>1 magnetic field well coupled to the plasma

  28. 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=min ni nn <svrel>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)

  29. Diffusion of the magnetic field vdrift (F/M)in tad F/M<(F/M)in

  30. Timescale of magnetic flux loss at n=104 cm-3, xe=10-7, M/F=(M/F)cr,, L=0.1 pc • ambipolar diffusion timescale: • Ohmic dissipation timescale: 1-10 Myr 1015 yr

  31. Density distribution and magnetic fieldlines 15.17 Myr 7.1 Myr 15.23195 Myr 15.2308 Myr Desch & Mouschovias (2001)

  32. Evolution of the central density t0 t1 t2 Desch & Mouschovias (2001)

  33. The velocity and mass-to-flux radial profiles subsonic supercritical t2 t1 t0 t2 t1 t0 subcritical supersonic Desch & Mouschovias (2001)

  34. Models of magnetized clouds: II. Collapse

  35. The equations of magnetohydrodynamics • equation of continuity • equation of momentum • induction equation • no monopoles • Poisson’s equation

  36. t = 5.7x104 yr t = 0 Singular isothermal sphere with uniform magnetic field Galli & Shu (1993)

  37. t = 1.1x105 yr

  38. t = 1.7x105 yr

  39. Magnetic reconnection Mestel & Strittmatter (1966)

  40. Magnetic braking

  41. 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

  42. 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)

  43. 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 (W=const.) • in supercritical collapse, specific angular momentum is conserved (J/M=const.)

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

  45. Conclusions • Zeeman effect and polarization • Models of magnetized clouds: • Magnetic braking • Equilibrium • Stability • Quasistatic evolution • Dynamical collapse

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