?
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
Images
One image is worth
a 1000 words...
Hubble Image
One spectrum is worth
a 1000 images...
Spectra
van Kempen et al. (2010)

Radiative quantities
Basic radiation quantity: intensity
erg
I(, ) 
s cm2 Hz st er
Definition of mean intensity
1
J( ) 
4
erg
 4  I(, )d  s cm2 Hz st er
Definition of flux
r
F ( ) 
r
 4  I(, ) d 
erg
s cm2 Hz
Thermal radiation
Planck function:
In dense isothermal medium, the radiation field is in thermodynamic
equilibrium. The intensity of such an equilibrium radiation field is:
2h 3 /c 2
I  B (T) 
[exp(h /kT) 1]

(Planck function)
In Rayleigh-Jeans limit (h<<kT)
this becomes a power law:
2
2kT  2
I  B (T) 
c2
Wien
Rayleigh-Jeans

Thermal radiation
Blackbody emission:
An opaque surface of a given temperature emits a flux
according to the following formula:
F   B (T)
Integrated over all frequencies (i.e. total emitted energy):
 F 


0
F d  


0
B (T)d
If you work this out you get:

F  T4
  5.67105 erg/cm2/K 4 /s
Radiative transfer
In vaccuum: intensity is constant along a ray
rB2
FA  2 FB
rA
Example: a star
rB2
A  2 B
rA
F  I 
A

B


I  const
Non-vacuum: emission and absorption change intensity:
dI

  S   I
ds
Emission
(s is path length)
Extinction
Radiative transfer
Radiative transfer equation again:
dI
  (S I )
ds
Over length scales larger than 1/ intensity I tends to
approach source function S.
lfree, 
Photon
 mean free path:
Optical depth of a
cloud of size L:
 

In case of local thermodynamic
equilibrium: S is Planck function:
1
 
L
lfree,
 L 
S  B (T)
Rad. trans. through a cloud of fixed T
Tcloud
Iν,bg
Tbg=6000 K
τcloud
Iν,out
Rad. trans. through a cloud of fixed T
Tcloud
Iν,bg
Tbg=6000 K
τcloud
Iν,out
Rad. trans. through a cloud of fixed T
Tcloud
Iν,bg
Tbg=6000 K
τcloud
Iν,out
Rad. trans. through a cloud of fixed T
Tcloud
Iν,bg
Tbg=6000 K
τcloud
Iν,out
Rad. trans. through a cloud of fixed T
Tcloud
Iν,bg
Tbg=6000 K
τcloud
Iν,out
Rad. trans. through a cloud of fixed T
Tcloud
Iν,bg
Tbg=6000 K
τcloud
Iν,out
Rad. trans. through a cloud of fixed T
Tcloud
Iν,bg
Tbg=6000 K
τcloud
Iν,out
Rad. trans. through a cloud of fixed T
Tcloud
Iν,bg
Tbg=6000 K
τcloud
Iν,out

Formal radiative transfer solution
Radiative transfer equation again:
dI
  (S I )
ds
Observed flux from single-temperature slab:
Iobs I0e  (1 e ) B (T)
   B (T)
for

   1

and
I0  0
   L 

Emission vs. absorption lines
Line Profile:
     line
 2 / 2
  K e

ρκν

σ
νline
ν
1 2kT
   line
c 
(for thermal broadning)
Emission vs. absorption lines
Tcloud
Iν,bg
Tbg=6000 K
τcloud
Iν,out
Emission vs. absorption lines
Tcloud
Iν,bg
Tbg=6000 K
τcloud
Iν,out
Emission vs. absorption lines
Tcloud
Iν,bg
Tbg=6000 K
τcloud
Iν,out
Emission vs. absorption lines
Tcloud
Iν,bg
Tbg=6000 K
τcloud
Iν,out
Emission vs. absorption lines
Tcloud
Iν,bg
Tbg=6000 K
τcloud
Iν,out
Emission vs. absorption lines
Tcloud
Iν,bg
Tbg=6000 K
τcloud
Iν,out
Emission vs. absorption lines
Tcloud
Iν,bg
Tbg=6000 K
τcloud
Iν,out
Emission vs. absorption lines
Tcloud
Iν,bg
Tbg=6000 K
τcloud
Iν,out
Emission vs. absorption lines
  1
Hot surface layer
Cool surface layer
   1
Flux
Flux

0 
I  I e
obs


 (1 e
) B (T)
Example: The Sun’s photosphere
What do we learn?
Temperature of the
gas goes down
toward the sun’s
surface!
Spectrum of the sun:
Fraunhofer lines = absorption lines
Example: The Sun’s corona
What do we learn?
There must be very
hot plasma hovering
above the sun’s
surface! And this
plasma is optically
thin!
X-ray spectrum of the sun using CORONAS-F
Sylwester, Sylwester & Phillips (2010)
Sun’s temperature structure
Model by Fedun, Shelyag, Erdelyi (2011)
Example: Protoplanetary Disks
What do we learn?
The surface layers
of the disk must be
warm compared to
the interior!
Spitzer Spectra of T Tauri disks by Furlan et al. (2006)
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
The energies
Energy
Example:
a fictive 6-level atom.
6
5
E6
E5
4
E4
3
E3
2
1
E2
E1=0
Lines of atoms and molecules
Level degeneracies
Energy
Example:
a fictive 6-level atom.
6
5
g6=2
g5=1
4
g4=1
3
g3=3
2
1
g2=1
g1=4
Lines of atoms and molecules
Polulating the levels
Energy
Example:
a fictive 6-level atom.
6
5
E6
E5
4
E4
3
E3
2
1
E2
E1=0
Lines of atoms and molecules
Spontaneous
radiative decay
(= line emission)
Energy
Example:
a fictive 6-level atom.
6
5
E6
E5
4
E4
3
E3
2
1
E2
E1=0
Einstein A-coefficient (radiative decay rate):
A4,3 [sec-1]
γ
Lines of atoms and molecules
Line absorption
Energy
Example:
a fictive 6-level atom.
6
5
E6
E5
4
E4
γ
3
E3
2
1
E2
E1=0
Einstein B-coefficient (radiative absorption coefficient):
B3,4 J3,4 [sec-1]
B3,4
J3,4 
1
4
  I(, )
3,4
( )d d
Lines of atoms and molecules
Stimulated emission
Energy
Example:
a fictive 6-level atom.
6
5
E6
E5
4
E4
γ
γ
3
E3
2
1
E2
E1=0
Einstein B-coefficient (stimulated emission coefficient):
B4,3 J3,4 [sec-1]
B4,3
J3,4 
1
4
  I(, )
3,4
( )d d
Lines of atoms and molecules
Energy
Example:
a fictive 6-level atom.
6
5
E6
E5
4
E4
3
E3
2
1
E2
E1=0
Einstein relations:
2
B4,3
c
 A4,3
3
2h
g3
B4,3  B3,4
g4
Lines of atoms and molecules
Spontaneous
radiative decay
(= line emission)
can be from any
pair of levels,
provided the transition
obeys selection rules
Energy
Example:
a fictive 6-level atom.
6
5
E6
E5
4
E4
γ
3
E3
2
1
E2
E1=0
Lines of atoms and molecules
Collisional excitation
Energy
Example:
a fictive 6-level atom.
6
5
E6
E5
4
E4
Ecollision
3
E3
2
1
E2
E1=0
Our atom
free electron
Lines of atoms and molecules
Collisional deexcitation
Energy
Example:
a fictive 6-level atom.
6
5
E6
E5
4
E4
Ecollision
3
E3
2
1
E2
E1=0
Our atom
free electron
Example: Protoplanetary Disks
What do we learn?
Carr & Najita 2008
Organic molecules
exist already during
the epoch of planet
formation. Models
of chemistry can tell
us why. Models of
rad. trans. tell us
Tgas and ρgas.
Lines of atoms and molecules
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 gi (E i E k )/ kT
 e
n k gk
6
5
4
3
2
1
ni is population of level nr i
How to determine the absolute populations?
Lines of atoms and molecules
How to determine the absolute populations?
Z(T)   gie
E i / kT
Partition function:
(usually available on databases
on the web in tabulated form)
i
If we know total number of atoms: N

...then we can compute the nr of
atoms Ni in each level i:
N
E i / kT
Ni 
gie
Z(T)
Note: Works only under LTE conditions (high enough density)
Using multiple lines for finding Tgas
van Kempen et al. (2010)
log(N/g)
Using excitation diagrams to infer Tgas
What do we learn?
0
1000
2000
Energy [K]
Martin-Zaidi et al. 2008
3000
4000
There are clearly
two components
with different gas
temperatures: One
with T=56 K and
one with T=373 K.
Lines of atoms and molecules
Radiative transfer in lines:
h
j 
ni Aik ik ( )
4
h
 
(nk Bki  ni Bik )ik ( )
4
extinction
dI
 j   I
ds
stimulated emission

...where the line
profile function is:
 (   0 )2 
( ) 
exp

2

 


1
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
Line profile without
doppler shift:

Line profile with
doppler shift:

 (   0 )2 
( ) 
exp

2

 


1
r r

1
(   0   0u  / c)2 
(, ) 
exp

2

 


Example: Position-velocity diagrams
Motion of neutral hydrogen gas in the Milky Way
Kalberla et al. 2008
Example: Velocity channel maps
Viewing the Omega Nebula (M17) at different velocity channels
From: Alyssa Goodman (CfA Harvard), the COMPLETE survey
Continuum emission/extinction by dust
Atoms in dust grains do not produce lines.
They produce continuum + broad features.
CO ice
CO ice
CO ice+gas
CO gas+ice
CO gas
solid CO2
CO gas
From lecture
Ewine van
Dishoeck
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
Forward modeling: “Model fitting”
?
Iν,out
Iν,in
van Kempen et al. (2010)
Radiative transfer program
Model
cloud
Forward modeling: “Model fitting”
?
Iν,out
Iν,in
van Kempen et al. (2010)
Radiative transfer program
Model cloud
Forward modeling: “Model fitting”
?
Iν,out
Iν,in
van Kempen et al. (2010)
Got it!
Radiative transfer program
Model cloud
Automated fitting
First we need a “goodness of fit indicator”
χ2
Error estimate:
N
2  
i1
(y iobs  y imodel ) 2
 i2
...where σi 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).
“Least squares fitting”
Automated fitting
Then we need a procedure to scan model-parameter space:
Brute force method
χ2-contours
Pontoppidan et al. 2007
Automated fitting
Then we need a procedure to scan model-parameter space:
Brute force method
χ2-contours
“Best fit”
But strong
degeneracy
Pontoppidan et al. 2007
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