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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
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
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