Radiative Transfer: ?

Radiative Transfer:

Interpreting

the observed light

?

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

I_ν,in I_ν,out

Spectra

van Kempen et al. (2010)

Images

Hubble Image

One image is worth a 1000 words...

One spectrum is worth a 1000 images...

Radiative quantities

Basic radiation quantity: intensity

€

I(Ω,ν ) = erg

s cm²Hz ster

Definition of mean intensity

€

J(ν ) = 1

4π I(Ω,ν )dΩ

4π

∫ ⁼ _{s cm}2^ergHz ster

Definition of flux

€

F r (ν ) = I(Ω,ν ) r

Ω dΩ

4π

∫ ⁼ _{s cm}^erg2Hz

Thermal radiation

Planck function:

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

€

I_ν = B_ν (T) ≡ 2hν ³ /c²

[exp(hν /kT) −1] (Planck function)

Wien Rayleigh-Jeans

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

€

I_ν = B_ν (T) ≡ 2kTν ² c²

€

ν²

Thermal radiation

Blackbody emission:

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

€

F_ν = π B_ν (T)

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

€

F ≡ F_ν dν

0

∫ ∞ ^{= π} ∫₀^{∞Bν (T)dν}

If you work this out you get:

€

F = σ T⁴

€

σ= 5.67 ×10⁻⁵erg/cm²/K⁴/s

Radiative transfer

In vaccuum: intensity is constant along a ray Example: a star

A € B

F_A = r_B² r_A² F_B

€

ΔΩ_A = r_B²

r_A² ΔΩ_B

€

F = IΔΩ

€

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

€

dI_ν

ds = ρ κ_ν (S_ν −I_ν )

Radiative transfer equation again:

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

Photon mean free path:

€

l_free,ν = 1 ρ κ_ν

Optical depth of a cloud of size L:

€

τ_ν = L

l_free,ν = Lρ κ_ν

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

€

S_ν = B_ν (T)

Rad. trans. through a cloud of fixed T

I_ν,bg I_ν,out

T_cloud

τ_cloud

T_bg=6000 K

Rad. trans. through a cloud of fixed T

I_ν,bg I_ν,out

T_cloud

τ_cloud

T_bg=6000 K

Rad. trans. through a cloud of fixed T

I_ν,bg I_ν,out

T_cloud

τ_cloud

T_bg=6000 K

Rad. trans. through a cloud of fixed T

I_ν,bg I_ν,out

T_cloud

τ_cloud

T_bg=6000 K

Rad. trans. through a cloud of fixed T

I_ν,bg I_ν,out

T_cloud

τ_cloud

T_bg=6000 K

Rad. trans. through a cloud of fixed T

I_ν,bg I_ν,out

T_cloud

τ_cloud

T_bg=6000 K

Rad. trans. through a cloud of fixed T

I_ν,bg I_ν,out

T_cloud

τ_cloud

T_bg=6000 K

Rad. trans. through a cloud of fixed T

I_ν,bg I_ν,out

T_cloud

τ_cloud

T_bg=6000 K

Formal radiative transfer solution

Observed flux from single-temperature slab:

€

I_ν^obs = I_ν⁰e^−τν + (1− e^−τν ) B_ν (T)

€

≈τ_ν B_ν (T)

for

€

τ_ν <<1

€

I_ν⁰ = 0

and

€

dI_ν

ds = ρ κ_ν (S_ν −I_ν )

Radiative transfer equation again:

€

τ_ν = Lρκ_ν

Emission vs. absorption lines

Line Profile:

€

ρκ_ν = K e^{−Δν 2 /σ 2}

€

Δν = ν −ν_line

€

σ = ν_line ¹ c

2kT μ

(for thermal broadning)

ρκ_ν

ν σ

ν_line

Emission vs. absorption lines

I_ν,bg I_ν,out

T_cloud

τ_cloud

T_bg=6000 K

Emission vs. absorption lines

I_ν,bg I_ν,out

T_cloud

τ_cloud

T_bg=6000 K

Emission vs. absorption lines

I_ν,bg I_ν,out

T_cloud

τ_cloud

T_bg=6000 K

Emission vs. absorption lines

I_ν,bg I_ν,out

T_cloud

τ_cloud

T_bg=6000 K

Emission vs. absorption lines

I_ν,bg I_ν,out

T_cloud

τ_cloud

T_bg=6000 K

Emission vs. absorption lines

I_ν,bg I_ν,out

T_cloud

τ_cloud

T_bg=6000 K

Emission vs. absorption lines

I_ν,bg I_ν,out

T_cloud

τ_cloud

T_bg=6000 K

Emission vs. absorption lines

I_ν,bg I_ν,out

T_cloud

τ_cloud

T_bg=6000 K

Emission vs. absorption lines

Hot surface layer

€

τ_ν ≤1

€

τ_ν >>1

Flux



Cool surface layer

Flux



€

I_ν^obs = I_ν⁰e^−τν + (1− e^−τν ) B_ν (T)

Example: The Sun’s photosphere

Spectrum of the sun:

Fraunhofer lines = absorption lines

What do we learn?

Temperature of the gas goes down

toward the sun’s surface!

Example: The Sun’s corona

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

What do we learn?

There must be very hot plasma hovering above the sun’s

surface! And this plasma is optically thin!

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 be warm compared to the 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 E₂ E₁=0

Energy

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=1 g₁=4

Energy

Level degeneracies

Lines of atoms and molecules

4 3 Example:

a fictive 6-level atom.

21

56 E₆

E₅ E₄ E₃ E₂ E₁=0

Energy

Polulating the levels

Lines of atoms and molecules

4 3 Example:

a fictive 6-level atom.

21

56 E₆

E₅ E₄ E₃ E₂ E₁=0

Energy γ

Spontaneous radiative decay (= line emission)

[sec^-1]

Einstein A-coefficient (radiative decay rate):

€

A_4,3

Lines of atoms and molecules

4 3 Example:

a fictive 6-level atom.

21

56 E₆

E₅ E₄ E₃ E₂ E₁=0

Energy

Line absorption γ

Einstein B-coefficient (radiative absorption coefficient):

€

B_3,4

€

B_3,4J _3,4 [sec^-1]

€

J _3,4 = ₄¹_π

∫ ∫

Lines of atoms and molecules

4 3 Example:

a fictive 6-level atom.

21

56 E₆

E₅ E₄ E₃ E₂ E₁=0

Energy

Stimulated emission γ

Einstein B-coefficient (stimulated emission coefficient):

€

B_4,3

€

B_4,3J _3,4 [sec^-1]

€

J _3,4 = ₄¹_π

∫ ∫

γ

Lines of atoms and molecules

4 3 Example:

a fictive 6-level atom.

21

56 E₆

E₅ E₄ E₃ E₂ E₁=0

Energy

Einstein relations:

€

B_4,3 = A_4,3 c² 2hν ³

€

B_4,3 = g₃

g₄ B_3,4

Lines of atoms and molecules

4 3 Example:

a fictive 6-level atom.

21

56 E₆

E₅ E₄ E₃ E₂ E₁=0

Energy γ

Spontaneous radiative decay (= line emission) can be from any pair of levels,

provided the transition obeys selection rules

Lines of atoms and molecules

4 3 Example:

a fictive 6-level atom.

21

56 E₆

E₅ E₄ E₃ E₂ E₁=0

Energy

E_collision

Collisional excitation

Our atom

free electron

Lines of atoms and molecules

4 3 Example:

a fictive 6-level atom.

21

56 E₆

E₅ E₄ E₃ E₂ E₁=0

Energy

E_collision

Collisional deexcitation

Our atom

free electron

Example: Protoplanetary Disks

Carr & Najita 2008

What do we learn?

Organic molecules exist already during the epoch of planet formation. Models of chemistry can tell us why. Models of rad. trans. tell us T_gas and ρ_gas.

Lines of atoms and molecules

n_i is population of level nr i 4

3 At high enough densities the

populations of the levels

are thermalized. This is called

“Local Thermodynamic

Equilibrium” (LTE). For LTE the ratio of populations of any two levels is given by:

€

n_i

n_k = g_i

g_k e^{−(Ei −Ek )/kT}

How to determine the absolute populations?

21 56

Lines of atoms and molecules

Partition function:

(usually available on databases on the web in tabulated form)

How to determine the absolute populations?

€

Z(T) = g_ie^{−Ei /kT}

i

∑

If we know total number of atoms: N

...then we can compute the nr of atoms N_i in each level i:

€

N_i = N

Z(T) g_ie^{−Ei /kT}

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

Using multiple lines for finding T_gas

van Kempen et al. (2010)

Using excitation diagrams to infer T_gas

Martin-Zaidi et al. 2008

What do we learn?

There are clearly two components with different gas temperatures: One with T=56 K and one with T=373 K.

0 1000 2000 3000 4000 Energy [K]

log(N/g)

Lines of atoms and molecules

Radiative transfer in lines:

€

j_ν = hν

4π n_i A_ik ϕ _ik (ν )

€

ρκ_ν = hν

4π (n_k B_ki − n_i B_ik)ϕ _ik(ν )

€

dI_ν

ds = j_ν − ρ κ_ν I_ν

extinction stimulated emission

€

ϕ(ν ) = 1

σ π exp −(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 π exp − (ν −ν ₀)² σ ²

⎛

⎝⎜ ⎞

⎠⎟ Line profile without

doppler shift:

Line profile with

doppler shift:

€

ϕ(Ω,ν ) = 1

σ π exp −(ν −ν ₀ −ν ₀r u • r

Ω /c)² σ ²

⎛

⎝⎜ ⎞

⎠⎟

Example: Position-velocity diagrams

Motion 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 dust

Atoms in dust grains do not produce lines.

They produce continuum + broad features.

From lecture Ewine van Dishoeck

CO ice

CO ice+gas

CO gas

solid CO₂ CO gas CO gas+ice

CO ice

Dust opacities. Example: silicate

Opacity of amorphous silicate

Example: B68 molecular cloud

Credit: European Southern Observatory

Example: Thermal dust emission M51

Made with the Herschel Space Telescope:

Using radiative transfer models to interpret observational data

I_ν,in I_ν,out

?

Model cloud Model

cloud

Radiative transfer program

Forward modeling: “Model fitting”

van Kempen et al. (2010)

I_ν,in I_ν,out

?

Model cloud Model cloud

Radiative transfer program

Forward modeling: “Model fitting”

van Kempen et al. (2010)

I_ν,in I_ν,out

?

Model cloud Model cloud

Radiative transfer program

Got it!

Forward modeling: “Model fitting”

van Kempen et al. (2010)

Automated fitting

χ² Error estimate:

€

χ² = (y_i^obs − y_i^model)² σ _i²

i=1

∑N

...where σ_i is the weight (usually taken to be the uncertainty in 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 fitting

Then we need a procedure to scan model-parameter space:

Brute force method

Pontoppidan et al. 2007

χ²-contours

Automated fitting

Then we need a procedure to scan model-parameter space:

Brute force method

Pontoppidan et al. 2007

χ²-contours

But strong degeneracy

“Best fit”

Automated fitting

Then 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