Magnetic Fields in
Star and Planet Formation
Frank H. Shu
UCSD Physics Department
Stars to Planets -- University of Florida
12 April 2007
Outline and Logic of Talk
• Cloud core characterized by dimensionless massto-flux ratio   2 G1/2 M /   1 to 2 collapses
to form star + disk.
• Some loss of flux during collapse results in 0  4
for system. Most of the mass ends up in the star;
almost all of the flux, in the disk. Therefore,
known mass of star implies calculable flux in disk.
• Self-contained theory of MRI for viscous/resistive
spreading yields disk radius R() needed to
contain flux trapped in disk as function of age t.
3 1/2
• Predictions for ( ), Bz ( ), and   f (GM * /  )
with f < 1 (sub-Keplerian rotation) of disk.
• Implications for disk- and X-winds, funnel-flows,
and planetsimal formation.
Catastrophic Magnetic Braking if
Fields Are Perfectly Frozen
Allen, Li, & Shu (2003) – Initial rotation in range specified by Goodman et al. (1993).
Some braking is needed, but frozen-in value is far too much (no Keplerian disk forms).
Ohmic Dissipation of Split Monopole Yields
Central Region of Low Uniform Field
rOhm 
2
2GM *
,
 sm
4G 5 / 2 M *
Bc 

.
2
4
15sm
30 r Ohm
3
For   2  10 20 cm 2 s-1,
split monopole  2, and M *  1 M⊙,
 sm
4G 5 / 2 M *
Bc 

2
15sm 4
30 rOhm
3
rOhm  8.4 AU, and Bc  1 gauss.
Contours of constant FL/FG
Magnetic field lines
Shu, Galli, Lizano, & Cai (2006)
NGC 1333 IRS 4A
Best fit:
split monopole  1 to 2
ROhm  50 AU
  4  10 20 cm 2 /s
for d  300 pc.
Likely value in star
plus disk: 0  4.
Girart et al. (2006); Gonzalez et al. (2007)
Mean-Field MHD of Accretion
Disks in Star Formation
Model equations:
 1
  u   0,
t 
B  Bz
1
2
 
 2
2

B 
Bz
B 



r
GM

2

G
K 0   (r,t)rdr  ,
*


0



  2
1   2
 

  u
[ ] 
 


,
 t
 
  
 
Bz 1 
1     

 Bz u  
B ,

t  
   z0  
where
B  


0
K 0 ( ) 
rdr
r
K 0   Bz (r,t) 2


1
2
—
 1 

z0  A and a 2 
Lubow, Papaloizou, & Pringle (1994);
Shu & Li (1997)
1   cos 
2
 2 cos 

3/2
d
1  B   2 2 GM * 
GM * 2 
A .

   
 A 
2  Bz



Shu, Galli, Lizano, Glassgold, & Diamond (2007)
MRI Turbulence in Magnetized
Accretion Disks
No previous MRI simulation is both global
• Turbulent viscosity:
and has a nonzero net magnetic flux. Reason
 u B ~  B 
why MRI simulations systematically give too
  B ~
small aviscosity compared to astrophysical
systems (cf. King, Livio, & Pringle 2007).

 with  u ~ 

 
 B with  B ~ B  .
 
 B  B (B  )2  
Maxwell stress:
~
.
4
4  
Cf. modeled viscous stress: 





.

2z0

(B  )2 z0
 identify   F
where F is "form factor."
2
Bz 2 z0 Shakura-Sunyaev

In steady state, B  I l Bz    D
2 viscosity with
where D  I l 2 F is an order unity parameter. magnetic pressure
• Turbulent resistivity:
replacing
gas pressure.
 3z0 

 in quasi-steady state.

 2I  
l
Four Astronomical Models
• Steady-state
solution:
1/2
 GM 
  f  3* 
  
,
1/ 4
 GM * M&*2 
  3  ,
f  Il 
M&*

.


1  f 2  3 DA  (GM * )1/2
 2f 
Bz  
 3DA 
1/2
A( )  0.1( / 100 AU)1/4  I l  1.742
M D (R )  M&*t age where

0
0 = 4
Bz 2 d  2 G1/2 M * / 0
 0.5444  M *
1 f 2  
 0 2  M D (R )
T Tau
LMP
FU Ori HMP
M* /M¤
0.5
0.5
M*
M ¤/yr
1x10-8
2x10-6
2x10-4
1x10-4
tage/yr
3x106
1x105
100
1x105
D
10-2.5
1
1
1
0.03
0.20
0.02
10
f
0.658
0.957
0.386
0.957
R/AU
298
318
16.5
1,520
JD (M¤
5.12
51.4
0.473
39,700
0.5
25
.
• Models:
R
Object
MD /M¤
AU km/s)
binary?
binary?
Magnetic Fields and
Surface Densities
Both LMP and TT have ~ 1 G field at 3 AU,
compatible with chondrules in meteorites.
LMP and HMP have several to tens of mG
fields at 100 &1000 AU (check with masers)
FU Ori has kG at 0.05 AU, compatible with
measurement by Donati et al. (2005). These
authors also find rotation to be sub-Keplerian
by factor of 2 to 3, compatible with f = 0.386.
Surface density of neither LMP nor TT looks
like minimum solar nebula. The 
profile inferred from solids probably results
from recycling of hot rocks near the protosun
3/ 2
(Stardust sample-return mission).
Implications for
X-wind/Funnel Flow/Planetesimal Formation
• Inward press of disk field
resisted by outward press of
squeezed magnetosphere.
• Change of magnetically
coupled layers from subKepler to super-Kepler with

change in sign of B Bz , i.e
with change from outward
bend to inward bend of field
lines. (Inner edge helps.)
• X-winds are better focused and
faster with f < 1/3.
• Planetesimals probably form
only in dead zones
Shu, Lizano, Galli, & Cai (2007)