chap radtrans
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Radiative Transfer:
Interpretingthe observed light
?
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References:
A standard book on radiative processes inastrophysics is: Rybicki & Lightman RadiativeProcesses in Astrophysics Wiley-Interscience
For radiative transfer in particular there aresome excellent lecture notes on-line by RobRutten Radiative transfer in stellaratmosphereshttp://www.astro.uu.nl/~rutten/Course_notes.html
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Radiation as a messenger
I,in I,out
Spectra
van Kempen et al. (2010)
Images
Hubble ImageOne image is wortha 1000 words...
One spectrum is wortha 1000 images...
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Radiative quantities
Basic radiation quantity: intensity
I ( , ) erg
s cm 2 Hz ster
Definition of mean intensity
J ( ) 14
I ( , )d 4
ergs cm 2 Hz ster
Definition of flux
r
F ( ) I ( , )r
d 4
ergs cm 2 Hz
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Thermal radiationPlanck function:
In dense isothermal medium, the radiation field is in thermodynamicequilibrium. The intensity of such an equilibrium radiation field is:
I
B (T )
2h 3 /c 2
[exp( h /kT ) 1](Planck function)
Wien Rayleigh-Jeans
In Rayleigh-Jeans limit (h
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Thermal radiation
Blackbody emission:
An opaque surface of a given temperature emits a fluxaccording to the following formula:
F B (T )
Integrated over all frequencies (i.e. total emitted energy):
F F d 0 B (T )d 0
If you work this out you get:
F T 4 5.67 10 5 erg/cm 2/K 4 /s
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Radiative transferIn vaccuum: intensity is constant along a ray
Example: a star
A B
F A
r
B
2
r A
2 F
B
A r B
2
r A2
B
F I
I const
Non-vacuum: emission and absorption change intensity:dI ds
S I
Emission Extinction
(s is path length)
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Radiative transfer
dI ds
(S I )
Radiative transfer equation again:
Over length scales larger than 1/ intensity I tends toapproach source function S.
Photon mean free path: l free, 1
Optical depth of acloud of size L:
Ll free, L
In case of local thermodynamic
equilibrium: S is Planck function:S B (T )
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Rad. trans. through a cloud of fixed T
I,bg I,out
Tcloud
cloud
Tbg =6000 K
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Rad. trans. through a cloud of fixed T
I,bg I,out
Tcloud
cloud
Tbg =6000 K
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Rad. trans. through a cloud of fixed T
I,bg I,out
Tcloud
cloud
Tbg =6000 K
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Rad. trans. through a cloud of fixed T
I,bg I,out
Tcloud
cloud
Tbg =6000 K
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Rad. trans. through a cloud of fixed T
I,bg I,out
Tcloud
cloud
Tbg =6000 K
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Rad. trans. through a cloud of fixed T
I,bg I,out
Tcloud
cloud
Tbg =6000 K
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Rad. trans. through a cloud of fixed T
I,bg I,out
Tcloud
cloud
Tbg =6000 K
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Rad. trans. through a cloud of fixed T
I,bg I,out
Tcloud
cloud
Tbg =6000 K
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Formal radiative transfer solution
Observed flux from single-temperature slab:
I obs I
0e (1 e ) B (T )
B (T )for 1
I 0 0and
dI ds
(S I )
Radiative transfer equation again:
L
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Emission vs. absorption lines
Line Profile:
K e 2 / 2
line
line1
c
2kT
(for thermal broadning)
line
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Emission vs. absorption lines
I,bg I,out
Tcloud
cloud
Tbg =6000 K
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Emission vs. absorption lines
I,bg I,out
Tcloud
cloud
Tbg =6000 K
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Emission vs. absorption lines
I,bg I,out
Tcloud
cloud
Tbg =6000 K
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Emission vs. absorption lines
I,bg I,out
Tcloud
cloud
Tbg =6000 K
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Emission vs. absorption lines
I,bg I,out
Tcloud
cloud
Tbg =6000 K
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Emission vs. absorption lines
I,bg I,out
Tcloud
cloud
Tbg =6000 K
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Emission vs. absorption lines
I,bg I,out
Tcloud
cloud
Tbg =6000 K
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Emission vs. absorption lines
I,bg I,out
Tcloud
cloud
Tbg =6000 K
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Emission vs. absorption lines
Hot surface layer 1
1
Flux
Cool surface layer
Flux
I obs I
0e (1 e ) B (T )
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Example: The Suns photosphere
Spectrum of the sun:
Fraunhofer lines = absorption lines
What do we learn?
Temperature of thegas goes downtoward the suns surface!
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Example: The Suns corona
X-ray spectrum of the sun using CORONAS-F
Sylwester, Sylwester & Phillips (2010)
What do we learn?
There must be veryhot plasma hoveringabove the suns surface! And thisplasma is opticallythin!
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Suns temperature structure
Model by Fedun, Shelyag, Erdelyi (2011)
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Example: Protoplanetary Disks
Spitzer Spectra of T Tauri disks by Furlan et al. (2006)
What do we learn?The surface layersof the disk must bewarm compared tothe interior!
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How a disk gets a warm surface layer
Literature: Chiang & Goldreich (1997), DAlessio et al. (1998), Dullemond & Dominik (2004)
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Lines of atoms and molecules
4
3
Example:a fictive 6-level atom.
21
56 E6
E5 E4
E3 E2 E1=0
E n e r g y
The energies
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Lines of atoms and molecules
4
3
Example:a fictive 6-level atom.
21
56 g6=2
g5=1g4=1
g3=3g2=1g1=4
E n e r g y
Level degeneracies
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Lines of atoms and molecules
4
3
Example:a fictive 6-level atom.
21
56 E6
E5 E4
E3 E2 E1=0
E n e r g y
Polulating the levels
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Lines of atoms and molecules
4
3
Example:a fictive 6-level atom.
21
56 E6
E5 E4
E3 E2 E1=0
E n e r g y
Spontaneous
radiative decay(= line emission)
[sec -1]Einstein A-coefficient (radiative decay rate):
A4,3
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Lines of atoms and molecules
4
3
Example:a fictive 6-level atom.
21
56 E6
E5 E4
E3 E2 E1=0
E n e r g y
Line absorption
Einstein B-coefficient (radiative absorption coefficient): B
3,4 B3,4 J 3,4 [sec-1]
J 3,4 14 I ( , ) 3,4 ( ) d d
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Lines of atoms and molecules
4
3
Example:a fictive 6-level atom.
21
56 E6
E5 E4
E3 E2 E1=0
E n e r g y
Stimulated emission
Einstein B-coefficient (stimulated emission coefficient): B
4,3 B4,3 J 3,4 [sec-1]
J 3,4 14 I ( , ) 3,4 ( ) d d
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Lines of atoms and molecules
4
3
Example:a fictive 6-level atom.
21
56 E6
E5 E4
E3 E2 E1=0
E n e r g y
Einstein relations:
B4,3 A4,3c 2
2h 3 B4,3
g 3 g
4
B3,4
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Lines of atoms and molecules
4
3
Example:a fictive 6-level atom.
21
56 E6
E5 E4
E3 E2 E1=0
E n e r g y
Spontaneous
radiative decay(= line emission)can be from anypair of levels,provided the transitionobeys selection rules
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Lines of atoms and molecules
4
3
Example:a fictive 6-level atom.
21
56 E6
E5 E4
E3 E2 E1=0
E n e r g y
Ecollision Collisional excitation
Our atom
free electron
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Example: Protoplanetary Disks
Carr & Najita 2008
What do we learn?
Organic moleculesexist already duringthe epoch of planetformation. Modelsof chemistry can tellus why. Models ofrad. trans. tell us
Tgas and gas .
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Lines of atoms and molecules
Partition function:(usually available on databaseson the web in tabulated form)
How to determine the absolute populations?
Z (T ) g ie E i / kT
i
If we know total number of atoms: N
...then we can compute the nr ofatoms N i in each level i : N i
N Z (T ) g ie
E i / kT
Note: Works only under LTE conditions (high enough density)
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Using multiple lines for finding T gas
van Kempen et al. (2010)
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Using excitation diagrams to infer T gas
Martin-Zaidi et al. 2008
What do we learn?
There are clearlytwo componentswith different gastemperatures: Onewith T=56 K andone with T=373 K.
0 1000 2000 3000 4000Energy [K]
l o g
( N / g )
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Lines of atoms and molecules
Radiative transfer in lines:
j h 4
n i Aik ik ( )
h 4
(n k Bki n i Bik ) ik ( )
dI ds j
I
extinction stimulated emission
( ) 1
exp ( 0 )
2
2
...where the lineprofile function is:
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Beware of non-LTE! In this lecture we focused on LTE conditions ,
where the level populations can be derivedfrom the temperature using the partitionfunction.
In astrophysics we often encounter non-LTEconditions when the densities are very low
(like in the interstellar medium). Then linetransfer becomes much more complex ,because then the populations must be
computed together with the rad. trans.
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Using doppler shift to probe motion
( ) 1
exp ( 0 )2
2
Line profile withoutdoppler shift:
Line profile withdoppler shift:
( , ) 1
exp ( 0 0
u
/c)2
2
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Example: Position-velocity diagramsMotion of neutral hydrogen gas in the Milky Way
Kalberla et al. 2008
l l h l
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Example: Velocity channel maps
From: Alyssa Goodman (CfA Harvard), the COMPLETE survey
Viewing the Omega Nebula (M17) at different velocity channels
b d
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Continuum emission/extinction by dust
Atoms in dust grains do not produce lines.They produce continuum + broad features.
From lectureEwine vanDishoeck
CO ice
CO ice+gas
CO gas
solid CO 2 CO gas
CO gas+ice
CO ice
D i i E l ili
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Dust opacities. Example: silicateOpacity of amorphous silicate
E l B68 l l l d
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Example: B68 molecular cloud
Credit: European Southern Observatory
E l Th l d i i M51
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Example: Thermal dust emission M51
Made with theHerschel SpaceTelescope:
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Using radiative transfer modelsto interpret observational data
F d d li M d l fi i
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I,in I,out
?
Modelcloud
Radiative transfer program
Forward modeling: Model fitting
van Kempen et al. (2010)
F d d li M d l fi i
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I,in I,out
?
Model cloud
Radiative transfer program
Forward modeling: Model fitting
van Kempen et al. (2010)
F d d li M d l fitti
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I,in I,out
?
Model cloud
Radiative transfer program
Got it!
Forward modeling: Model fitting
van Kempen et al. (2010)
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A t t d fitti g
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Automated fitting
Then we need a procedure to scan model-parameter space:
Brute force method
Pontoppidan et al. 2007
2-contours
A t t d fitti g
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Automated fitting
Then we need a procedure to scan model-parameter space:
Brute force method
Pontoppidan et al. 2007
2-contours
But strongdegeneracy
Best fit
Automated fitting
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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
...
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Some useful radiative transfer codes
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Some useful radiative transfer codes...
Infrared and submillimeter lines: RADEXhttp://www.sron.rug.nl/~vdtak/radex/radex.php RATRANhttp://www.strw.leidenuniv.nl/~michiel/ratran/ SIMLINEhttp://hera.ph1.uni-koeln.de/~ossk/Myself/simline.html
Stellar atmosphere codes: TLUSTYhttp://nova.astro.umd.edu/ PHOENIXhttp://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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