the formation of stars and planets day 2, topic 1: giant molecular clouds and gravitational...
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![Page 1: The formation of stars and planets Day 2, Topic 1: Giant Molecular Clouds and Gravitational Stability Lecture by: C.P. Dullemond](https://reader036.vdocuments.us/reader036/viewer/2022062421/56649d375503460f94a1060f/html5/thumbnails/1.jpg)
The formation of stars and planets
Day 2, Topic 1:
Giant Molecular Cloudsand
Gravitational Stability
Lecture by: C.P. Dullemond
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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%
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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.
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Giant Molecular Clouds
• 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...
Nearby well-studied GMCs:
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Orion GMC
Orion Nebula (part of Orion GMC)From: CfA Harvard, Millimeter Wave Group
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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
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V ∝ AD / 2
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D ≈1.4
fractal dimension
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Clump mass spectrumOrion B: First GMC systematically surveyed for dense gas and embedded YSOs by E. Lada 1990
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dN
dM∝M−1.6
Clumps in range M = 8..500 M
Most of mass in massive clumps
Survey of gas clumps
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dN
d lnM∝M−0.6
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MdN
d lnM∝M 0.4
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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
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Core mass spectrumResult of survey:
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dN
d lnM∝M−0.6
for M < 0.5 M
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dN
d lnM∝M−(1.1−1.5)
for M > 0.5 M
Motte et al. 1998
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Core mass spectrum
Stellar IMF: Meyer et al. PP IV
Similar to stellar IMF (Initial Mass Function)
Salpeter (1955) IMF:
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dN
d lnM∝M−1.35
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Jeans mass
• Given a homogeneous medium of density 0
• Do linear perturbation analysis to see if there exist unstable wave modes:
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=0 + ρ1
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v = v1
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φ=φ1
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∂1
∂t+ ρ 0∇ ⋅v1 = 0Continuity equation:
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∂v1
∂t= −∇φ1 −
∇P1
ρ 0
Euler equation:
Poisson’s equation:
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∇2φ1 = 4π Gρ1
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Jeans mass
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∂2ρ1
∂t 2+ ρ 0∇ ⋅
∂v1
∂t= 0Take derivative to t:
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∂1
∂t+ ρ 0∇ ⋅v1 = 0
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∂v1
∂t= −∇φ1 −
∇P1
ρ 0
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∂2ρ1
∂t 2= ρ 0∇ ⋅ ∇φ1 +
∇P1
ρ 0
⎛
⎝ ⎜
⎞
⎠ ⎟= 0
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∇2φ1 = 4π Gρ1
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∂2ρ1
∂t 2= 4π Gρ 0ρ1 +
kT
μmH
∇ 2ρ1 = 0
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P1 =kT
μ mH
ρ1
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Jeans mass
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∂2ρ1
∂t 2= 4π Gρ 0ρ1 +
kT
μmH
∇ 2ρ1 = 0
Equation to solve:
Try a plane wave:
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1 = exp i2π x
λ−ω t
⎛
⎝ ⎜
⎞
⎠ ⎟
⎡
⎣ ⎢
⎤
⎦ ⎥
Obtain dispersion relation:
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ω 2 =2π
λ
⎛
⎝ ⎜
⎞
⎠ ⎟2
kT
μmH
⎛
⎝ ⎜
⎞
⎠ ⎟− 4π Gρ 0
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Jeans mass
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ω 2 =2π
λ
⎛
⎝ ⎜
⎞
⎠ ⎟2
kT
μmH
⎛
⎝ ⎜
⎞
⎠ ⎟− 4π Gρ 0
For larger than:
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λJ =π kT
μmHGρ 0
⎛
⎝ ⎜
⎞
⎠ ⎟
1/ 2
Jean’s length
the wave grows exponentially.
This is true for all waves (in all directions) with λ>λJ. This defines maximum stable mass: a sphere with diameter λJ:
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MJ =π
6ρ 0
π kT
μmHGρ 0
⎛
⎝ ⎜
⎞
⎠ ⎟
3 / 2
Jean’s mass
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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:
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tff =3π
32Gρ=
3.4 ×107
nyear
For interstellar medium (ISM):
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n ≈17 cm-3
Total amount of molecular gas in the Galaxy:
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~ 2 ×109Msun
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tff = 8 ×106 year
Expected star formation rate:
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~ 250 Msun /year
Observed star formation rate:
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~ 3 Msun /yearSomething slows star formation down...
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Magnetic field support
Replace Jeans mass with critical mass, defined as:
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Mcr = 0.12ΦM
G1/2≈103Msun
B
30μG
⎛
⎝ ⎜
⎞
⎠ ⎟R
2pc
⎛
⎝ ⎜
⎞
⎠ ⎟
2
In presence of B-field, the stability analysis changes.Magnetic fields can provide support against gravity.
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Magnetic field supportConsider an initially stable cloud. We now compress it. The density thereby increases, but the mass of the cloud stays constant.
Jeans mass decreases:
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MJ ∝1
ρ
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.
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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:
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f = ninn σvmimn
mi + mn
vi − vn( ) ≡ ρ iρ nγ vi − vn( )
The drift velocity is inversely proportional to the friction:
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vd ≡ vi − vn =1
4π γ ρ iρ n
∇ ×B( ) ×B
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fL =1
4π∇ ×B( ) ×B
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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: Lizano & Shu (1989)
See later...
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HII Regions
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Remember:
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HII Regions
Strong UV flux from O star ionizes GMC.
Simple model: constant density, spherically symmetric.
O star
HII region (‘Strömgren sphere’)
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Ionization, heating, recombination...
Continuum (free electron)
Excited states (bound electron)
Ground state
Thermalization of electron to local gas temperature. This
heats the gas to high temperatures
Recombination to the ground state
produces a photon that immediately
ionizes another atom.
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Ionization, heating, recombination...
Continuum (free electron)
Excited states (bound electron)
Ground state
Thermalization of electron to local gas temperature. This heats the gas to very high
temperatures
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Strömgren sphere
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NH0
4π Jνhν
aν (H0)ν 0
∞
∫ dν = NeN p α (H0,T)
Ionization balance:
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4π Jν =Lν
4π r2
Mean intensity of ionizing radiation:
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LN ≡Lν
hνdν
ν 0
∞
∫
€
Lν
hνaν (H0)dν
ν 0
∞
∫ ≈ LN aν 0(H0)
Approximation:
The ionization balance then becomes:
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NH0aν 0
LN
4π r2= NeN p α (H0,T)
From: Osterbrock “Astrophysics of Gaseous Nebulae and AGN”
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ν 0 = 3.29 ×1015 Hz
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λ0 = 912Å
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Strömgren sphere
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aν 0≈ 6 ×10−18 cm2
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α(H0) ≈ 4 ×10−13 cm3 /s
Approximate ionization cross section of atomic hydrogen:
Approximate recombination coefficient:
Express NH0, Ne and Np as:
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NH0= ξ NH
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Ne = N p = (1−ξ )NH
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ξaν 0
LN
4π r2= (1−ξ )2NHα (H0,T)
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Strömgren sphere
Example:
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LN ≈ 5 ×1048 photons/s
O6 star with T=40,000 K:
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), ξ will be near 0, i.e. virtually complete ionization.
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Strömgren sphere
Effect of extinction:
Inside sphere: virtually complete ionization. Recombination rate per volume element is:
Need continuous re-ionization to compensate for recombination. This `eats away’ stellar photons (extinction):
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NH2 α (H0,T)
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dLN
dr= −4π r2 NH
2 α (H0,T)
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LN (r) = LN (0) −4π
3r3 NH
2 α (H0,T)
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Strömgren sphere
Outer radius of Strömgren sphere: where all photons are used up, i.e. where LN(r)=0.
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rs =3
4π
LN (0)
NH2 α (H0,T)
⎛
⎝ ⎜
⎞
⎠ ⎟
1/ 3
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Strömgren sphereAbundance of neutral hydrogen :
ionized neutral
Very sharp transition to neutral.
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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.
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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)