The formation of stars and planets

Day 1, Topic 2:

Radiation physics

Lecture by: C.P. Dullemond

Astronomical Constants

• CGS units used throughout lecture (cm,erg,s...)• AU = Astronomical Unit

= distance earth - sun = 1.49x1013 cm

• pc = Parsec = 3.26 lightyear = 3.09x1018 cm • ’’ = Arcsec = 4.8x10-6

radian• Definition: 1’’ at 1 pc = 1 AU

• M = Mass of sun = 1.99x1033 gram

• M = Mass of Earth = 5.97x1027 gram

• L = 3.85x1033 erg/s

Radiative transfer

Basic radiation quantity: intensity

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I(Ω,ν ) =erg

s cm2Hz ster

Definition of mean intensity

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J(ν ) =1

4πI(Ω,ν )dΩ

4π∫ =

erg

s cm2Hz

Definition of flux:

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F(ν ) =1

4πI(Ω,ν )ΩdΩ

4π∫ =

erg

s cm2Hz

Radiative transferPlanck function:

In dense isothermal medium, the radiation field is in thermodynamic equilibrium. The intensity of such an equilibrium radiation field is:

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Iν = Bν (T) ≡2hν 3 /c 2

[exp(hν /kT) −1](Planck function)

Wien Rayleigh-Jeans

In Rayleigh-Jeans limit (h<<kT) this becomes a power law:

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Iν = Bν (T) ≡2kTν 2

c 2€

2

Radiative transferBlackbody emission:

An opaque surface of a given temperature emits a flux according to the following formula:

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Fν = π Bν (T)

Integrated over all frequencies (i.e. total emitted energy):

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F ≡ Fν dν0

∞

∫ = π Bν (T)dν0

∞

∫If you work this out you get:

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F =σ T 4

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σ=5.67 ×10−5 erg/cm2/K4 /s

Radiative transfer

In vaccuum: intensity is constant along a ray

Example: a star

Non-vacuum: emission and absorption change intensity:

A B€

FA =rB2

rA2FB

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ΩA =rB2

rA2ΩB

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F = IΩ

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I = const

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dI

ds= ρ κ S − ρ κ I

Emission Extinction

(s is path length)

Radiative transfer

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dIνds

= ρ κ ν (Sν − Iν )

Radiative transfer equation again:

Over length scales larger than 1/ intensity I tends to approach source function S.

Photon mean free path:

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lfree,ν =1

ρ κ ν

Optical depth of a cloud of size L:

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τ =L

lfree,ν= Lρκ ν

In case of local thermodynamic equilibrium: S is Planck function:

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Sν = Bν (T)

Radiative transfer

Radiative transfer

Observed flux from single-temperature slab:

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Iνobs = Iν

0e−τν + (1− e−τν ) Bν (T)

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≈τ Bν (T)

for

€

τ <<1

€

Iν0 = 0and

Radiative transfer

Emission/absorption lines:

Hot surface layer

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τ ≤1

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τ >>1

Flux

Cool surface layer

Flux

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Iνobs = Iν

0e−τν + (1− e−τν ) Bν (T)

Difficulty of dust radiative transfer• If temperature of dust is given (ignoring scattering for the

moment), then radiative transfer is a mere integral along a ray: i.e. easy.

• Problem: dust temperature is affected by radiation, even the radiation it emits itself.

• Therefore: must solve radiative transfer and thermal balance simultaneously.

• Difficulty: each point in cloud can heat (and receive heat from) each other point.

Dust opacities. Example: silicate

Opacity of amorphous olivine (silicate) for different grain sizes

Rotating molecules

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≡Bh J(J +1)€

E rot =J 2

2IClassical case:

Quantum case:

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E rot =h2

2IJ(J +1)

I is the moment of inertia

J is the rotational quantum number:J = 0,1,2,3...

B has the dimension of frequency (Hertz)

Rotating molecules

Dipole radiative transition: JJ-1:

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ΔE = Bh J(J +1) − (J −1)J[ ]

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=2Bh J

Quadrupole radiative transition: JJ-2:

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ΔE = Bh J(J +1) − (J −2)(J −1)[ ]

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=2Bh (2J −1)

Transition energies linear in J

Rotating molecules

Carbon-monoxide (CO):

J = 10 =2.6 mmJ = 21 =1.3 mmJ = 32 =0.87 mm

CO: I = 1.46E-39

Ground based millimeter dishes. CO emission is brightest molecular rotational emission from space. Often used!

Plateau de Bure James Clerck Maxwell Telescope

Rotating molecules

Molecular hydrogen (H2)

H2: I = 4.7E-41 J = 20 =28 mJ = 31 =17 mJ = 42 =12 m

Need space telescopes (atmosphere not transparent). Emission is very weak. Only rarely detected.

Due to symmetry: only quadrupole transitions:

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Δ J = ±2

Spitzer Space Telescope

Rotating molecules

Carbon-monoxide (CO) and Molecular hydrogen (H2)

Rotating molecules

Molecules with 2 or 3 moments of inertia:

H

H

H

NH

H

O

“Symmetric top”: 2 different moments of inertia, e.g. NH3:

“Asymmetric top”: 3 different moments of inertia, e.g. H2O:

These molecules do not have only J, but also additional quantum numbers.

Water is notorious: very strong transitions + the presence of masers (both nasty for radiative transfer codes)

Rotating moleculesSymmetric top: NH3

H

H

H

N

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E rot ≡ Bh J(J +1) + (A − B)hK 2

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A ≅ 2.02 ×1011Hz

B ≅ 3.20 ×1011Hz

Quadrupole ΔK0 transitions are slow (10-9 s) but possible (along backbone)

backbone

Radiative J to J-1 transitions with ΔK=0 are rapid (~10-1...-2 s). But ΔK0 transitions are ‘forbidden’ (do not exist as dipole transitions).

After book by Stahler & Palla

Isotopes

• Molecules with atomic isotopes have slightly different moments of inertia, hence different line positions.

• Molecules with atoms with non-standard isotopes have lower abundance, hence lines are less optically thick.

• Examples: – [12CO]/[13CO] ~ 30...130– [13CO]/[C18O] ~ 10...40

Vibrating molecules: case of H2

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Evib ≡hν 0 v + 12( )

Atomic bonds are flexible: distance between atoms in a molecule can oscillate.

Vibrational frequency for H2:

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0 ≅1.247 ×1014 Hz

Vibrating molecule can also rotate. Sum of rot + vib energy:

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E tot = Evib + E rot

Selection rules for H2-rovib transitions: from v to any v’, but ΔJ=-2,0,2 (quadrupole transitions).Quadrupole transitions are weak: H2 difficult to detect...

Vibrating molecules: case of CO

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Evib = hν 0 v + 12( )

For CO same mechanism as for H2:

Vibrational frequency for CO:

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0 ≅ 6.4 ×1013 Hz

Often v=0,1,2 of importance

Selection rules for CO-rovib transitions: from v to any v’, but ΔJ = -1,0,1.

Δv=-1 is called fundamental 4.7 m

Δv=-2 is called first overtone 2.4 m

Vibrating molecules: case of CO

J to J-1: R-branch transitions: during vibrational transition also downward rotational transition: more energy release. Lines blueward of Q-lines, bluer for higher J.

J to J: Q-branch transitions: Pure vibrational transitions (no change in J). All lines roughly at same position.

Transitions from v to v-1 or v-2 etc:

J to J+1: P-branch transitions: during vibrational transition also upward rotational transition: less energy release. Lines redward of Q-lines, redder for higher J.

R Q P

Vibrating molecules: case of CO

• “Band head”:– Rotational moment of intertia for v=1 slightly larger than

for v=0– Therefore rotational energy levels of v=1 slightly less

than for v=0– R branch (blue branch): distance between lines

decreases for increasing J. Eventually lines reach minimum wavelength (band head) and go back to longer wavelengths.

Calvet et al. 1991

2.294 2.302 [m]

Band head CO first overtone 2-0

Overview of location of molecular lines

Photodissociation of molecules

First and second electronic excited state

Electronic ground state

Ultraviolet photons can excite an atom in the molecule to a higher electronic state.

The decay of this state can release energy in vibrational continuum, which destroys the molecule.

Formation of molecules: H2

Due to low radiative efficiency, H+H cannot form H2 in gas phase (energy cannot be lost). Main formation process is on dust grain surfaces:

Binding energy of H = 0.04 eV due to unpaired e-.

In lattice fault: 0.1 eV, so it stays there.

Once two H meet, they form H2, which has no unpaired e-. So H2 is realeased.

Many other molecules also formed in this way (e.g. H2O, though water stays frozen onto dust grain: ice mantel).