The formation of stars and planets

Day 2, Topic 1:

Giant Molecular Cloudsand

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

• 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:

Orion GMC

Orion Nebula (part of Orion GMC)From: CfA Harvard, Millimeter Wave Group

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

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

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 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

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

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

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

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

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

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...

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.

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.

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

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...

HII Regions

Remember:

HII Regions

Strong UV flux from O star ionizes GMC.

Simple model: constant density, spherically symmetric.

O star

HII region (‘Strömgren sphere’)

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.

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

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

∞

∫

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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

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)

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.

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)

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

Strömgren sphereAbundance of neutral hydrogen :

ionized neutral

Very sharp transition to 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)