Radiative Transfer:

Interpreting the observed light

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References:

• A standard book on radiative processes in astrophysics is: Rybicki & Lightman “Radiative Processes in Astrophysics” Wiley-Interscience

• For radiative transfer in particular there are some excellent lecture notes on-line by Rob Rutten “Radiative transfer in stellar atmospheres” http://www.astro.uu.nl/~rutten/Course_notes.html

Radiation as a messenger

Iν,in Iν,out

Spectra

van Kempen et al. (2010)

Images

Hubble ImageOne image is wortha 1000 words...

One spectrum is wortha 1000 images...

Radiative quantities

Basic radiation quantity: intensity

€

I(Ω,ν ) =erg

s cm2Hz ster

Definition of mean intensity

€

J(ν ) =1

4πI(Ω,ν )dΩ

4 π∫ =

erg

s cm2Hz ster

Definition of flux

€

rF (ν ) = I(Ω,ν )

r Ω dΩ

4 π∫ =

erg

s cm2Hz

Thermal radiationPlanck function:

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

€

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:

€

Iν = Bν (T) ≡2kTν 2

c 2

€

2

Thermal radiationBlackbody emission:

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

€

Fν = π Bν (T)

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

€

F ≡ Fν dν0

∞

∫ = π Bν (T)dν0

∞

∫

If you work this out you get:

€

F = σ T 4

€

σ=5.67 ×10−5 erg/cm2/K4 /s

Radiative transferIn vaccuum: intensity is constant along a ray

Example: a star

A B€

FA =rB

2

rA2FB

€

ΔΩA =rB

2

rA2

ΔΩB

€

F = I ΔΩ

€

I = const

Non-vacuum: emission and absorption change intensity:

€

dI

ds= ρ κ S − ρ κ I

Emission Extinction

(s is path length)

Radiative transfer

€

dIν

ds= 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:

€

lfree,ν =1

ρ κ ν

Optical depth of a cloud of size L:

€

τ =L

lfree,ν

= Lρ κ ν

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

€

Sν = Bν (T)

Rad. trans. through a cloud of fixed T

Iν,bg Iν,out

Tcloud

τcloud

Tbg=6000 K

Rad. trans. through a cloud of fixed T

Iν,bg Iν,out

Tcloud

τcloud

Tbg=6000 K

Rad. trans. through a cloud of fixed T

Iν,bg Iν,out

Tcloud

τcloud

Tbg=6000 K

Rad. trans. through a cloud of fixed T

Iν,bg Iν,out

Tcloud

τcloud

Tbg=6000 K

Rad. trans. through a cloud of fixed T

Iν,bg Iν,out

Tcloud

τcloud

Tbg=6000 K

Rad. trans. through a cloud of fixed T

Iν,bg Iν,out

Tcloud

τcloud

Tbg=6000 K

Rad. trans. through a cloud of fixed T

Iν,bg Iν,out

Tcloud

τcloud

Tbg=6000 K

Rad. trans. through a cloud of fixed T

Iν,bg Iν,out

Tcloud

τcloud

Tbg=6000 K

Formal radiative transfer solution

Observed flux from single-temperature slab:

€

Iνobs = Iν

0e−τν + (1− e−τν ) Bν (T)

€

≈τ Bν (T)for

€

τ <<1

€

Iν0 = 0and

€

dIν

ds= ρ κ ν (Sν − Iν )

Radiative transfer equation again:

€

τ =Lρκ ν

Emission vs. absorption lines

Line Profile:

€

=K e−Δν 2 /σ 2

€

Δ = − line

€

σ = line

1

c

2kT

μ(for thermal broadning)ρκν

ν

σ

νline

Emission vs. absorption lines

Iν,bg Iν,out

Tcloud

τcloud

Tbg=6000 K

Emission vs. absorption lines

Iν,bg Iν,out

Tcloud

τcloud

Tbg=6000 K

Emission vs. absorption lines

Iν,bg Iν,out

Tcloud

τcloud

Tbg=6000 K

Emission vs. absorption lines

Iν,bg Iν,out

Tcloud

τcloud

Tbg=6000 K

Emission vs. absorption lines

Iν,bg Iν,out

Tcloud

τcloud

Tbg=6000 K

Emission vs. absorption lines

Iν,bg Iν,out

Tcloud

τcloud

Tbg=6000 K

Emission vs. absorption lines

Iν,bg Iν,out

Tcloud

τcloud

Tbg=6000 K

Emission vs. absorption lines

Iν,bg Iν,out

Tcloud

τcloud

Tbg=6000 K

Emission vs. absorption lines

Hot surface layer

€

τ ≤1

€

τ >>1

Flux

Cool surface layer

Flux

€

Iνobs = Iν

0e−τν + (1− e−τν ) Bν (T)

Example: The Sun’s photosphere

Spectrum of the sun:Fraunhofer lines = absorption lines

What do we learn?

Temperature of thegas goes down toward the sun’ssurface!

Example: The Sun’s corona

X-ray spectrum of the sun using CORONAS-FSylwester, Sylwester & Phillips (2010)

What do we learn?

There must be veryhot plasma hoveringabove the sun’ssurface! And thisplasma is opticallythin!

Sun’s temperature structure

Model by Fedun, Shelyag, Erdelyi (2011)

Example: Protoplanetary Disks

Spitzer Spectra of T Tauri disks by Furlan et al. (2006)

What do we learn?

The surface layers of the disk must bewarm compared tothe interior!

How a disk gets a warm surface layer

Literature: Chiang & Goldreich (1997), D’Alessio et al. (1998), Dullemond & Dominik (2004)

Lines of atoms and molecules

4

3

Example: a fictive 6-level atom.

21

56 E6

E5

E4

E3

E2E1=0

Ene

rgy

The energies

Lines of atoms and molecules

4

3

Example: a fictive 6-level atom.

21

56 g6=2

g5=1

g4=1

g3=3

g2=1g1=4

Ene

rgy

Level degeneracies

Lines of atoms and molecules

4

3

Example: a fictive 6-level atom.

21

56 E6

E5

E4

E3

E2E1=0

Ene

rgy

Polulating the levels

Lines of atoms and molecules

4

3

Example: a fictive 6-level atom.

21

56 E6

E5

E4

E3

E2E1=0

Ene

rgy γ

Spontaneous radiative decay(= line emission)

[sec-1]Einstein A-coefficient (radiative decay rate):

€

A4,3

Lines of atoms and molecules

4

3

Example: a fictive 6-level atom.

21

56 E6

E5

E4

E3

E2E1=0

Ene

rgy

γLine absorption

Einstein B-coefficient (radiative absorption coefficient):

€

B3,4

€

B3,4J 3,4 [sec-1]

€

J 3,4 = 14 π I(Ω,ν )ϕ 3,4 (ν ) dΩ∫∫ dν

Lines of atoms and molecules

4

3

Example: a fictive 6-level atom.

21

56 E6

E5

E4

E3

E2E1=0

Ene

rgy

γStimulated emission

Einstein B-coefficient (stimulated emission coefficient):

€

B4,3

€

B4,3J 3,4 [sec-1]

€

J 3,4 = 14 π I(Ω,ν )ϕ 3,4 (ν ) dΩ∫∫ dν

γ

Lines of atoms and molecules

4

3

Example: a fictive 6-level atom.

21

56 E6

E5

E4

E3

E2E1=0

Ene

rgy

Einstein relations:

€

B4,3 = A4,3

c 2

2hν 3

€

B4,3 =g3

g4

B3,4

Lines of atoms and molecules

4

3

Example: a fictive 6-level atom.

21

56 E6

E5

E4

E3

E2E1=0

Ene

rgy γ

Spontaneous radiative decay(= line emission)can be from anypair of levels,provided the transitionobeys selection rules

Lines of atoms and molecules

4

3

Example: a fictive 6-level atom.

21

56 E6

E5

E4

E3

E2E1=0

Ene

rgy

EcollisionCollisional excitation

Our atom

free electron

Lines of atoms and molecules

4

3

Example: a fictive 6-level atom.

21

56 E6

E5

E4

E3

E2E1=0

Ene

rgy

EcollisionCollisional deexcitation

Our atom

free electron

Example: Protoplanetary Disks

Carr & Najita 2008

What do we learn?

Organic moleculesexist already duringthe epoch of planet formation. Modelsof chemistry can tellus why. Models ofrad. trans. tell us Tgas and ρgas.

Lines of atoms and molecules

ni is population of level nr i

4

3

At high enough densities thepopulations of the levelsare thermalized. This is called“Local Thermodynamic Equilibrium” (LTE). For LTEthe ratio of populations of any two levels is given by:

€

ni

nk

=gi

gk

e−(E i −Ek ) / kT

How to determine the absolute populations?

21

56

Lines of atoms and molecules

Partition function:(usually available on databaseson the web in tabulated form)

How to determine the absolute populations?

€

Z(T) = gie−E i / kT

i

∑

If we know total number of atoms: N

...then we can compute the nr ofatoms Ni in each level i:

€

N i =N

Z(T)gie

−E i / kT

Note: Works only under LTE conditions (high enough density)

Using multiple lines for finding Tgas

van Kempen et al. (2010)

Using excitation diagrams to infer Tgas

Martin-Zaidi et al. 2008

What do we learn?

There are clearlytwo componentswith different gastemperatures: Onewith T=56 K and one with T=373 K.

0 1000 2000 3000 4000Energy [K]

log(

N/g

)

Lines of atoms and molecules

Radiative transfer in lines:

€

jν =hν

4πni Aik ϕ ik (ν )

€

=hν

4π(nk Bki − ni Bik )ϕ ik (ν )

€

dIν

ds= jν − ρ κ ν Iν

extinction stimulated emission

€

ϕ (ν ) =1

σ πexp −

(ν −ν 0)2

σ 2

⎛

⎝ ⎜

⎞

⎠ ⎟...where the line

profile function is:

Beware of non-LTE!• In this lecture we focused on LTE conditions,

where the level populations can be derived from the temperature using the partition function.

• In astrophysics we often encounter non-LTE conditions when the densities are very low (like in the interstellar medium). Then line transfer becomes much more complex, because then the populations must be computed together with the rad. trans.

Using doppler shift to probe motion

€

ϕ (ν ) =1

σ πexp −

(ν −ν 0)2

σ 2

⎛

⎝ ⎜

⎞

⎠ ⎟

Line profile withoutdoppler shift:

Line profile withdoppler shift:

€

ϕ (Ω,ν ) =1

σ πexp −

(ν −ν 0 −ν 0

r u •

r Ω /c)2

σ 2

⎛

⎝ ⎜

⎞

⎠ ⎟

Example: Position-velocity diagramsMotion of neutral hydrogen gas in the Milky Way

Kalberla et al. 2008

Example: Velocity channel maps

From: Alyssa Goodman (CfA Harvard), the COMPLETE survey

Viewing the Omega Nebula (M17) at different velocity channels

Continuum emission/extinction by dustAtoms in dust grains do not produce lines. They produce continuum + broad features.

From lectureEwine van Dishoeck

CO ice

CO ice+gas

CO gas

solid CO2CO gas

CO gas+ice

CO ice

Dust opacities. Example: silicateOpacity of amorphous silicate

Example: B68 molecular cloud

Credit: European Southern Observatory

Example: Thermal dust emission M51

Made with theHerschel SpaceTelescope:

Using radiative transfer models to interpret observational data

Iν,in Iν,out

?

Model cloudModel cloud

Radiative transfer program

Forward modeling: “Model fitting”

van Kempen et al. (2010)

Iν,in Iν,out

?

Model cloudModel cloud

Radiative transfer program

Forward modeling: “Model fitting”

van Kempen et al. (2010)

Iν,in Iν,out

?

Model cloudModel cloud

Radiative transfer program

Got it!

Forward modeling: “Model fitting”

van Kempen et al. (2010)

Automated fitting

χ2 Error estimate:

€

χ 2 =(y i

obs − y imodel)2

σ i2

i=1

N

∑

...where σi is the weight (usually taken to be the uncertaintyin the observation, but can also denote the “unimportance”of this measuring value compared to others).

First we need a “goodness of fit indicator”

“Least squares fitting”

Automated fittingThen we need a procedure to scan model-parameter space:

Brute force method

Pontoppidan et al. 2007

χ2-contours

Automated fittingThen we need a procedure to scan model-parameter space:

Brute force method

Pontoppidan et al. 2007

χ2-contours

But strong degeneracy

“Best fit”

Automated fittingThen we need a procedure to scan model-parameter space:

For large parameter spaces, better use one of these:

• Simulated annealing• Amoeba• MCMC• Genetic algorithms• ...

Some useful radiative transfer codes...

• Optical/UV of the interstellar medium:– CLOUDY http://www.nublado.org/

– Meudon PDR code http://pdr.obspm.fr/PDRcode.html

– MOCASSIN http://www.usm.uni-muenchen.de/people/ercolano/

• Dust emission, absorption, scattering:– DUSTY http://www.pa.uky.edu/~moshe/dusty/

– MC3D http://www.astrophysik.uni-kiel.de/~star/Classes/MC3D.html

– RADMC-3D http://www.ita.uni-heidelberg.de/~dullemond/software/radmc-3d/

Some useful radiative transfer codes...

• Infrared and submillimeter lines:– RADEX http://www.sron.rug.nl/~vdtak/radex/radex.php

– RATRAN http://www.strw.leidenuniv.nl/~michiel/ratran/

– SIMLINE http://hera.ph1.uni-koeln.de/~ossk/Myself/simline.html

• Stellar atmosphere codes:– TLUSTY http://nova.astro.umd.edu/ – PHOENIX http://www.hs.uni-hamburg.de/EN/For/ThA/phoenix/index.html

– More codes on: http://en.wikipedia.org/wiki/Model_photosphere