Chap Radtrans

8/12/2019 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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I,bg I,out

Tcloud

cloud

Tbg =6000 K


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I,bg I,out

Tcloud

cloud

Tbg =6000 K


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I,bg I,out

Tcloud

cloud

Tbg =6000 K


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I,bg I,out

Tcloud

cloud

Tbg =6000 K


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I,bg I,out

Tcloud

cloud

Tbg =6000 K


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I,bg I,out

Tcloud

cloud

Tbg =6000 K


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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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I,bg I,out

Tcloud

cloud

Tbg =6000 K


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I,bg I,out

Tcloud

cloud

Tbg =6000 K


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I,bg I,out

Tcloud

cloud

Tbg =6000 K


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I,bg I,out

Tcloud

cloud

Tbg =6000 K


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I,bg I,out

Tcloud

cloud

Tbg =6000 K


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I,bg I,out

Tcloud

cloud

Tbg =6000 K


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I,bg I,out

Tcloud

cloud

Tbg =6000 K


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I,bg I,out

Tcloud

cloud

Tbg =6000 K


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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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4

3


21

56 g6=2

g5=1g4=1

g3=3g2=1g1=4

E n e r g y

Level degeneracies


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4

3


21

56 E6

E5 E4

E3 E2 E1=0

E n e r g y

Polulating the levels


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4

3


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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4

3


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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4

3


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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4

3


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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4

3


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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4

3


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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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



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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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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


F d d li M d l fi i


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I,in I,out

?

Model cloud




F d d li M d l fitti


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I,in I,out

?

Model cloud


Got it!




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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


Brute force method

Pontoppidan et al. 2007

2-contours

But strongdegeneracy

Best fit

Automated fitting


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Automated fitting


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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Chap Radtrans