Download Supercritical clouds

Supercritical clouds
• Rapid contraction.
• Fragmentation into subregions
– Also supercritical if size R ≥ clump height Z.
• Compression of trapped magnetic flux.
• High star-forming efficiency:
– Need 20 to 50% efficiency to form bound cluster.
– 20 to 50% of 270 MSun pc–2 in OB stars gives luminosity
density 104 to 105 LSun pc–2.
– cf. Trapezium region in Orion.
AS 4002
Star Formation & Plasma Astrophysics
Subcritical clouds
• No way of initiating collapse if supporting
magnetic flux is frozen-in.
• Solution: only (molecular) ions are tied to
field lines.
• Ambipolar diffusion: field lines can slip
through the neutrals, allowing supercritical
cores to form.
• Long diffusion timescale means inefficient
star formation.
AS 4002
Star Formation & Plasma Astrophysics
Heating & cooling
• Main heating agent: cosmic-ray ionization
occurs at rate
  1  10 17 s -1
• Main cooling agent: optically thin emission from
dust, CO etc
• Equilibrium temperature ~ 10 to 15 K
• Cores become optically thick at n ≥ 1016 m–3
– Heat trapped => dense cores warmer
– Higher T => greater critical mass => more massive stars?
AS 4002
Star Formation & Plasma Astrophysics
Ionization balance
• Two-body recombinations of charged particles
and on charged grains give ion-neutral balance:
-15
-3/2
1/2
i  C1/2
;
C

9.5
10
m
kg
n
Weak function of gas temperature; value
given for T ~ 10 to 30 K, metal depletion 0.1
• Typically:
 i  30,  n  2.3
• Hence for n ~ 1010 m–3, get ionization fraction
1/2


ni C  n
 
  1.2  10 7
n  i nmH 
AS 4002
Star Formation & Plasma Astrophysics
Drag force of neutrals
• Ions tied to field:
Cyclotron
frequency
eB
  n
mi
Mean ion-neutral collision frequency
= no of collisions per sec felt by ion
Hence ion completes many cycles around fieldline
between collisions
NB:
  3.5  10 m kg s
2
 r i  r n  ud 3
10
~
AS 4002
mi  mn
3
-1 -1
m kg -1s-1
Geometric value
for collisions with
ud > 10 km s-1
Star Formation & Plasma Astrophysics
• Collision between ions and neutrals ->
exchange of momentum -> neutrals exert a
frictional (or drag) force on the ions.
• This force is balanced by the Lorentz force.
jB 
1
0
  B  B  i n  u i  u n 
Hence the ion-neutral drift velocity ud
ud  ui  un  
AS 4002
1
 n i  0
  B  B
Star Formation & Plasma Astrophysics
Ambipolar diffusion
• Induction equation for the ion fluid
B
    B  ui   0
t
• Substitute to get nonlinear diffusion equation
for neutrals:
 B

B
   B  un     
 B    B


t
 0i n 

AS 4002
Star Formation & Plasma Astrophysics
Effective diffusion coefficient
• Express RHS in terms of a diffusion coeff:
D~
u2At ni ,
where
u2A

B2
 0 n
Alfvén speed
in combined
medium
and t ni 
1
i 
Collision time
for neutral in
sea of ions
• Diffusion and dynamical timescales for field
with length scale R:
2
t AD
AS 4002
R
R
R
~
~
and t dyn ~
D
ud
uA
Star Formation & Plasma Astrophysics
Typical timescales
• Cloud core of mass 1 MSun, radius 0.1 pc and
critical flux density has:
3/2
GM
R
R
uA2 ~
 tdyn ~
~
R
uA GM 1/2
• This gives a dynamical timescale of a few
times 105 y, and an ambipolar diffusion time an
order of magnitude or so greater.
• From detailed analysis in slab geometry (see
Shu 1987, Ann Rev A & A 25, 23):
t AD
C
~
1/2  8
t dyn 22 G 
AS 4002
Star Formation & Plasma Astrophysics