1. 1 Magnetic fields in star forming regions: theory Daniele Galli INAF-Osservatorio di Arcetri Italy

2. 2 Outline Zeeman effect and polarization Models of magnetized clouds: Magnetic braking Equilibrium Stability Quasistatic evolution Dynamical collapse

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

4. 4 Basic observational techniques: Zeeman effect and polarization

5. 5 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

6. 6 Zeeman measurements in molecular clouds (cm -3 )  G) B  1/2 è Crutcher (1999)

7. 7 Summary of Zeeman measurements HI gas molecular clouds OH masers H 2 O masers SiO masers Vallée (1997)

8. 8 Polarization (Weintraub et al. 2000)

9. 9 Polarization map of background starlight in the Milky Way Mathewson & Ford (1970) Taurus Orion Ophiuchus

10. 10 The magnetic field in M51 optical polarization (Scarrott et al. 1987) radio synchrotron polarization (Beck et al. 1987)

11. 11 Optical polarization map of Taurus 5 pc Moneti et al. (1984), Heyer et al. (1987)

12. 12 Weintraub et al. (2000)

13. 13 (Weintraub et al. 2000)

14. 14 Hourglass field geometry in OMC-1? Schleuning (1998)

15. 15 Barnard 1 at 850  m Matthews & Wilson (2002)

16. 16 Submillimiter polarization in cloud cores L183L1544 Ward-Thompson et al. (2000)

17. 17 Models of magnetized clouds: I. Equilibrium

18. 18 force balance no 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

19. 19 Axially symmetric magnetostatic models Li & Shu (1996), Galli et al. (1999) Shu et al. (2000), Galli et al. (2001) 2-D 3-D

20. 20 line-of-sight

21. 21 Gonçalves, Galli, & Walmsley (2004)

22. 22 Models of magnetized clouds: II. Stability

23. 23 The magnetic virial theorem the magnetic critical mass the critical mass-to-flux ratio Chandrasekhar & Fermi (1953), Mestel & Spitzer (1956), Strittmatter (1966)

24. 24 The role of the magnetic critical mass stable unstable Boyle’s law

25. 25 Cloud supported by thermal pressure: M cr =M J, the Jeans mass Cloud supported by magnetic fields: M cr =M  In general, M cr = M J +M  to within 5% (McKee 1989) For T=10 K, n=10 5 cm -3, R=0.1 pc, B=10  G: M J = M  = 1 M  Summary of stability conditions

26. 26 R RR mass M magnetic flux  m 

27. 27 RR R

28. 28 Bourke et al. (2001)  = 0.1 The magnetic mass-to-flux ratio: observations 10 1 1  = 0.1

29. 29 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

30. 30 Models of magnetized clouds: II. Quasistatic evolution

31. 31 Ionisation fraction in molecular clouds Bergin et al. (1999)

32. 32 Field-plasma coupling gyration frequency  = qB/mc collision time with neutrals  =1/ n example: n=10 4 cm -3, B=10  G (  electrons =10 7, (  ions =10 3 >>1 magnetic field well coupled to the plasma

33. 33 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: F ni =  in n i n n in (v i -v n ) The field slips through the neutrals at a velocity v drift = v i -v n that depends on the field strength and the ionisation fraction (Mestel & Spitzer 1956)

34. 34 Diffusion of the magnetic field t ad (  ) in  <(  ) in v drift

35. 35 Timescale of magnetic flux loss at n=10 4 cm -3, x e =10 -7, M/  =(M/  ) cr,, L=0.1 pc ambipolar diffusion timescale: Ohmic dissipation timescale: 1-10 Myr 10 15 yr

36. 36 Density distribution and magnetic fieldlines Desch & Mouschovias (2001) 7.1 Myr 15.2308 Myr 15.23195 Myr 15.17 Myr

37. 37 Evolution of the central density Desch & Mouschovias (2001) t0t0 t1t1 t2t2

38. 38 The velocity and mass-to-flux radial profiles Desch & Mouschovias (2001) t0t0 t0t0 t2t2 t1t1 t2t2 t1t1 subcritical supercritical supersonic subsonic

39. 39 Core evolution by ambipolar diffusion Fiedler & Mouschovias (1992,1993) R=0.75 pc M=10 M  n=10 3 -10 7 cm -3 v max =0.4 km s -1

40. 40 Models of magnetized clouds: II. Collapse

41. 41 The equations of magnetohydrodynamics equation of continuity equation of momentum induction equation no monopoles Poisson’s equation

42. 42 t = 5.7 x 10 4 yr Galli & Shu (1993) t = 0 Singular isothermal sphere with uniform magnetic field

43. 43 t = 1.1 x 10 5 yr

44. 44 t = 1.7 x 10 5 yr

45. 45 Magnetic reconnection Mestel & Strittmatter (1966)

46. 46 Magnetic braking

47. 47 The angular momentum problem 1M of ISM (n = 1 cm -3,  = 10 -15 rad/s): J/M = 10 22 cm 2 /s 1M dense core (n = 10 4 cm -3,  =1 km s -1 /pc): J/M = 10 21 cm 2 /s 1M wide binary (T = 100 yr): J/M = 10 20 cm 2 /s Solar system: J/M = 10 18 cm 2 /s   

48. 48 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: t b  R/(2   v A )  

49. 49 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.)

50. 50 Magnetic braking: observations Ohashi et al. (1997) J/M R  const. J/M  const. Solar system wide binary

51. 51 Conclusions Zeeman effect and polarization Models of magnetized clouds: Magnetic braking Equilibrium Stability Quasistatic evolution Dynamical collapse