Download The formation of stars and planets

The formation of stars and planets
Day 2, Topic 1:
Giant Molecular Clouds
and
Gravitational Stability
Lecture by: C.P. Dullemond
Giant Molecular Clouds
• Typical characteristics of GMCs:
–
–
–
–
–
–
–
–
Mass
= 104...106 M
Distance to nearest GMC
= 140 pc (Taurus)
Typical size
= 5..100 pc
Size on the sky of near GMCs
= 5..20 x full moon
Average temperature (in cold parts) = 20...30 K
Typical density
= 103...106 molec/cm3
Typical (estimated) life time
= ~107 year
Star formation efficiency
= ~1%..10%
Giant Molecular Clouds
• Composition of material:
– 99% gas, 1% solid sub-micron particles (‘dust’)
(by
mass)
– Gas: 0.9 H2/H, 0.1 He, 10-4 CO, 10-5 other molecules
(by number)
– Dust: Mostly silicates + carbonaceous (< m in size)
• Properties of the gas:
– Gas mostly in molecular form: hydrogen in H2, carbon
in CO, oxygen in O (O2?), nitrogen in N2(?).
– At the edges of molecular clouds: transition to atomic
species. “Photo-Dissociation Regions” (PDRs).
– H2 cannot be easily observed. Therefore CO often
used as tracer.
Giant Molecular Clouds
Nearby well-studied GMCs:
• Taurus (dist ≈ 140 pc, size ≈ 30 pc, mass ≈104 M): Only low
mass stars (~105), quiet slow star formation, mostly isolated
star formation.
• Ophiuchus (dist ≈ 140 pc, size ≈ 6 pc, mass ≈ 104 M): Low
mass stars (~78), strongly clustered in western core (stellar
density 50 stars/pc), high star formation efficiency
• Orion (dist ≈ 400 pc, size ≈ 60 pc, mass ≈ 106 M): Cluster of
O-stars at center, strongly ionized GMC, O-stars strongly
affect the low-mass star formation
• Chamaeleon...
• Serpens...
Orion GMC
From: CfA Harvard, Millimeter Wave Group
Orion Nebula (part
of Orion GMC)
Giant Molecular Clouds
Structure of GMCs: two descriptions
• Clump picture: hierarchical structure
– Clouds (≥ 10 pc)
– Clumps (~1 pc)
• Precursors of stellar clusters
– Cores (~0.1 pc)
• High density regions which form individual stars or binaries
• Fractal picture: clouds are scale-free
V  AD / 2
D  1.4
fractal dimension

Clump mass spectrum
Orion B: First GMC systematically surveyed for dense gas
and embedded YSOs by E. Lada 1990
Survey of gas
clumps
Clumps in range
M = 8..500 M
dN
dN
1.60.6

MM
ddM
ln M
MdN
0.4
M
d ln M
Most of mass in
massive clumps
Core mass spectrum
Most clumps don’t form stars. But if they do, they form many.
Core mass spectrum is more interesting for predicting the
stellar masses of the newborn stars.
Deep 1.3 mm
continuum map of 
Ophiuchi (140 pc)
at 0.01 pc (=2000
AU) resolution.
Motte et al. 1998
Core mass spectrum
Result of survey:
dN
 M 0.6
d ln M
dN
 M (1.11.5)
d ln M
for M < 0.5 M

for M > 0.5 M

Motte et al. 1998
Core mass spectrum
Similar to stellar IMF (Initial Mass Function)
Stellar IMF:
Meyer et al. PP IV
Salpeter (1955) IMF:
dN
 M 1.35
d ln M

Jeans mass
• Given a homogeneous medium of density 0
• Do linear perturbation analysis to see if there exist
unstable wave modes:
  0  1
Continuity equation:


Euler equation:

Poisson’s equation:
v  v1
  1
1
 0  v1  0
t

v1
P1
 1 
t
0
 21  4 G 1
Jeans mass
1
 0  v1  0
t
 2 1
v1
  0 
0
2
t
t
Take derivative to t:
v1
P
 1  1
t
0

 21  4 G 1
kT
P1 
1
 mH


 2 1
P1 
 0  1 
 0
2
t
0 

 2 1
kT 2
 4  G  0 1 
 1  0
2
t
m H
Jeans mass
Equation to solve:
 1
kT 2
 4  G  0 1 
 1  0
2
t
m H
2
Try a plane wave:
 2 x

1  expi 
  t 

  

Obtain dispersion relation:

2


2  kT 
2
    
 4 G 0
    m H 
Jeans mass
2   kT 
    
 4 G 0
    m H 
2
2
For  larger than:

  kT 
J  

 m H G0 
1/ 2
Jean’s length
the wave grows exponentially.
This is true for all waves (in all directions) with >J. This
defines maximum stable mass: a sphere with diameter J:

  kT 
M J  0

6  m H G0 

3/2
Jean’s mass
Problem of star formation efficiency
Gas in the galaxy should be wildly gravitationally unstable.
It should convert all its mass into stars on a free-fall time
scale:
3
3.4 107
tff 

year
32G 
n
For interstellar medium (ISM):
n  17 cm-3
t ff  8 10 6 year
9
~
2
10
M sun
Total amount of molecular gas in the Galaxy:

~ 250 M /year
Expected star formation rate:
Observed star formation rate:
sun
~ 3 Msun /year

Something slows star formation down...
Magnetic field support
In presence of B-field, the stability analysis changes.
Magnetic fields can provide support against gravity.
Replace Jeans mass with critical mass, defined as:
 B  R 
M
3
M cr  0.12 1/2 10 M sun 
 
G
30 G 2pc 
2
Magnetic field support
Consider an initially stable cloud. We now compress it. The
density thereby increases, but the mass of the cloud stays
constant.
Jeans mass decreases:
MJ 
1

If no magnetic fields: there will come a time when M>MJ
and the cloud will collapse.
But Mcr stays constant (magnetic
flux freezing)
So if B-field is strong enough to support a cloud, no
compression will cause it to collapse.
Ambipolar diffusion
But magnetic flux freezing is not perfect. Only the (few)
electrons and ions are stuck to the field lines. The neutral
molecules do not feel the B-field. They may slowly diffuse
through the ‘fixed’ background of ions and electrons.
Friction between ions and neutrals:
mi m n
f  ni nn v
vi  vn   i n vi  vn 
mi  m n
The drift velocity is inversely proportional to the friction:
vd  vi  vn 
1
4   i n
  B  B
1
fL 
  B  B
4
Ambipolar diffusion
Slowly a cloud (supported by B-field) will expell the field,
and contract, until it can no longer support itself, and will
collapse.
Simulations:
See later...
Lizano & Shu (1989)
HII
Regions
Remember:
HII Regions
Strong UV flux from O star ionizes GMC.
Simple model: constant density, spherically symmetric.
HII region
(‘Strömgren sphere’)
O star
Ionization, heating, recombination...
Thermalization of electron to
local gas temperature. This
heats the gas to high
temperatures
Continuum (free
electron)
Excited
states
(bound
electron)
Ground state
Recombination to the
ground state
produces a photon
that immediately
ionizes another atom.
Ionization, heating, recombination...
Thermalization of electron to
local gas temperature. This
heats the gas to very high
temperatures
Continuum (free
electron)
Excited
states
(bound
electron)
Ground state
Strömgren sphere
From: Osterbrock “Astrophysics of Gaseous Nebulae and AGN”
Ionization balance:
 4 J

N H0 
a (H 0 ) d  N e N p  (H 0,T)
 0 h
Mean intensity of ionizing radiation:


L
4 J 
4 r 2
Approximation:
 L
 0 h a (H0 )d  LN a 0 (H0 )

The ionization balance then becomes:

LN
N H0 a 0
 N e N p 
 (H 0,T)
2
4 r
 0  3.29 1015 Hz
0  912 Å
L
LN  
d
 0 h



Strömgren sphere
Express NH0, Ne and Np as:
N H0   N H

N e  N p  (1  )N H
LN
2
 a 0

(1

)
N H  (H 0,T)
2
4 r

Approximate ionization cross section of atomic hydrogen:

a 0  6 1018 cm2
Approximate recombination coefficient:
 (H 0 )  4 1013 cm3 /s

Strömgren sphere
Example:
O6 star with T=40,000 K:
LN  5 10 48 photons/s
Hydrogen density of 10 atoms / cm3
At r = 5 pc we get  = 4x10-4, i.e. nearly complete ionization!
Conclusion: Unless LN drops really low (or one is very far
away from the star),  will be near 0, i.e. virtually complete
ionization.
Strömgren sphere
Effect of extinction:
Inside sphere: virtually complete ionization. Recombination
rate per volume element is:
N H2  (H 0 ,T)
Need continuous re-ionization to compensate for
recombination. This `eats away’ stellar photons (extinction):
 dLN
 4 r 2 N H2  (H 0,T)
dr
4 3 2
LN (r)  LN (0) 
r N H  (H 0 ,T)
3


Strömgren sphere
Outer radius of Strömgren
sphere: where all photons are
used up, i.e. where LN(r)=0.
 3
LN (0) 
rs  

2
4 N H  (H 0,T) 
1/ 3
Strömgren sphere
Abundance of neutral hydrogen :
ionized
Very sharp transition to neutral.
neutral
Expansion
Ionized material inside the HII region is very hot (~104 K).
Therefore pressure is about thousand times higher than in
the neutral surrounding medium.
The sphere expands and drives a strong shock through
the medium.
Champagne flows (‘blisters’)
When shell reaches end of Molecular Cloud, it bursts out
with high velocity outflow. Similarity to uncorking a bottle
of champagne, hence the name “Champagne Flows”.
Orion Nebula (rotated 90 deg)