Download Supersonic turbulence?

Supersonic turbulence?
• If CO linewidths interpreted as turbulence,
velocities approach virial values:
Molecular
line FWHM
in km/sec
2GM 1/2
u ~ 

 R 
Cloud radius
• Problem: specific energy ~ u2 in macroscopic
motions, length scale l gets converted to heat
(cascading eddies, radiative shocks) in time ~ l/u.
• Need u ~ virial speed and l << R for support.
• But this means dissipation time l/u << crossing time
R/u, so need constant replenishment (winds??)
• Ordered magnetic field => turbulence not dominant.
AS 4002
Star Formation & Plasma Astrophysics
Magnetic support
• A fancier form of the scalar virial theorem:
1 d2I
2

3Mc
s 
2
2 dt
 B 
dV
  
2 0 
V
2
Volume integrated
magnetic energy

B2 
1
r.dS   (r.B)B.dS
  p 
 2  0 
0 S
S
Surface term due to
external gas + magnetic
pressure at boundary
AS 4002
Surface term
due to mag. tension
Star Formation & Plasma Astrophysics
Contracting spherical cloud
• Uniform external field B0
outside R0.
• Uniformly compressed
field inside R:
R B 
2
2
R0 B0
B0


B
Total flux threading cloud
R0
R
• Radial field between R0
and R:
R02
B(r,  )  B0 2 cos 
r
AS 4002
Star Formation & Plasma Astrophysics
Magnetic volume term inside R0
 B 
4 3 B
 20 dV  3 R 2 0
V
2
2
 R0
 
0 R
B(r, )2
20
Energy of uniform field
inside inner sphere
Energy of radial field between
surfaces R and R0
.2r sin  .r drd
2 2 3 2 2 4 1 1 

B R 
B0 R0


3 0
3 0
R R0 

AS 4002
Star Formation & Plasma Astrophysics
Magnetic surface terms at R0
B2
S
(r.B)B.dS  
r.dS

0 S
20
S
1
• Can compute surface integral over R0 directly,
but beware of sign errors!
• More simply, note that when R= R0 , field is
uniform and so exerts no forces and makes no
net contribution to Virial Theorem:
2 2 3 2 2 4  1
1 
B0 R0 
B0 R0

S0

3 0
30
R0 R0 

• Hence deduce that volume and surface terms
add to zero when R= R0 , so the sum of the two
surface terms must be:
2
S
AS 4002
3 0
B02 R03
Star Formation & Plasma Astrophysics
Total magnetic virial contribution
• Sum of volume terms:
2 2 3 2 2 4 1 1 
BR 
B0 R0


3 0
30
R R0 

• Sum of surface terms:
2 2 3

B0 R0
3 0
• Total (volume + surface) for R < R0 is:
4 2 4 1 1 
B0 R0


3 0
R R0 

AS 4002
Star Formation & Plasma Astrophysics
Magnetic support
• Add surface and volume terms, and balance
against gravitational binding energy for
uniform sphere:
2


4 2 4 1 1
3GM
B0 R0



3 0
5R
R R0 

R 90 GM 2
 1

R0
20 2
ie magnetic energy
is sufficient to support
cloud at radius R.
• Set R=0 to get minimum flux/mass needed to
prevent collapse:
 3(  0 ) G

M
20
1/2
AS 4002
1/2
Star Formation & Plasma Astrophysics
Alfvén speed in critical-mass clouds
• For clouds near critical state, can rearrange
condition for support to get:
B2
2
3 GM
3
GM

 Mu2A 
 0 5 R
5 R
i.e. Alfvén speed is automatically
of the magnitude needed for
virial equilibrium.
• May explain non-thermal velocities as Alfvénwave motions.
• Nonlinear Alfvén waves may provide support
against collapse along field into sheets.
AS 4002
Star Formation & Plasma Astrophysics
Reality check
• Critical mass for clouds with typical observed R,
B (Zeeman splitting of OH lines):
Mcr  750
R B
2
G1 / 2
2
B  R 

 
 1.66  10 MSun
3nT 2pc
3
• For constant B, get column density:
Mcr
 R  constant,
2
R
 u A   1/2  R1/2
AS 4002
Possible explanation
of observed power-law
relation between CO
linewidth and clump radius
Star Formation & Plasma Astrophysics
Cloud contraction and fragmentation
• Critical surface density and visual extinction
for flattened critical-mass clouds:
Mcl
MSun
2  270
2
R
pc
 B 


3nT 
 B 
 AV  (4 to 5 mag) + 2.5log 

3nT 
(typical
gas/dust
ratios)
• cf. Observed values:
– Taurus-Auriga: AV ~ 1-2 mag in envelope, ~ 10 mag in
cores with T ~ 10 to 11 K. Subcritical, low efficiency
formation of low-mass stars in unbound cluster
–  Oph: AV ~ 6 mag in envelope, ~ 102 mag in densest
cores with T ~ 30 to 35 K. Supercritical, high efficiency
formation of low-mass stars + a few B stars in bound
cluster
AS 4002
Star Formation & Plasma Astrophysics