why radiative transfer? introduction to solar...

Introduction to Solar Radiative TransferI Basic Radiative Transfer

Han UitenbroekNational Solar Observatory/Sacramento Peak

Sunspot NM

George Ellery Hale CGEP, CU BoulderLecture 13, Mar 5 2013

Why Radiative Transfer?

• In general we cannot visit the astronomical objects we are interested in,and thus cannot take in-situ measurements

• Instead, to determine the object’s properties, we have to rely on theinformation carried to us by the electromagnetic radiation emittedand/or reflected by the object.

• Multi-wavelength (spectroscopic) observations and analysis are the onlyavailable means to determine the physical conditions of astronomicalobjects.

• To analyze spectroscopic data meaningfully we need to understand howphysical information is encoded in the radiation (Radiative Transfer).

• We need to understand how the radiative signal is modified as it travelsto our instruments and is detected with them.

! " "" "! # $

The Solar Spectrum and Surface Temperature

0 2000 4000 6000 8000 10000Wavelength [nm]

0

1•10!8

2•10!8

3•10!8

4•10!8

5•10!8

Inte

nsity

[J m

!2 s!1

Hz!1

sr!1

]

6300 [K]

! " "" "! # $


0 2000 4000 6000 8000 10000Wavelength [nm]

0

1•10!8

2•10!8

3•10!8

4•10!8

5•10!8

Inte

nsity

[J m

!2 s!1

Hz!1

sr!1

] 5770 [K]

6300 [K]

! " "" "! # $


0 2000 4000 6000 8000 10000Wavelength [nm]

0

1•10!8

2•10!8

3•10!8

4•10!8

5•10!8

Inte

nsity

[J m

!2 s!1

Hz!1

sr!1

] 5770 [K]6000 [K]6300 [K]

! " "" "! # $

Overview

• I Basic Radiative TransferIntensity, emission, absorption, source function, optical depth, transferequation, line formation

• II Detailed Radiative ProcessesSpectral lines, radiative transitions, collisions, polarization, Non-LTEradiative transfer

• III Observations of Solar RadiationSolar telescopes, spectroscopy, polarimetry

Han Uitenbroek, NSO/SP Introduction to Solar Radiative Transfer I ! " "" "! # $

Bibliography

• Rutten: Radiative Transfer in Stellar Atmospheres(http://esoads.eso.org/abs/2003rtsa.book.....R)

• Rybicki and Lightman: Radiative Processes in Astrophysics

• Mihalas: Stellar Atmospheres

• Shu: The Physics of Astrophysics. I. Radiation

• Gray: Observation and Analysis of Stellar Photospheres

• del Toro Iniesta: Introduction to Spectropolarimetry

• Allen: Astrophysical Quantities


Short History

• 1802 Wollaston First to observe dark gaps in spectrum: spectral lines

• 1814 Fraunhofer rediscovers lines. Assigns names e.g.,C (H!), D (Na i), G (CH molecules), F (H"), and H (Ca ii)

• 1823 Herschel realized spectra contain information on composition ofsource from flame spectra

• 1842 Becquerel photographs spectra, discovers lines in the UV, beyondthe visible

• 1858 Bunsen and Kirchho! discover wavelength correspondencebetween bright flame emission and dark solar absorption lines. Start ofquantitative spectroscopy.


The Sun


Solar atmosphere is also very strongly time dependent

Courtesy: Mats Carlsson


A vertical cross section through a 3-D ConvectionSimulation

6

8

10

T [1

03 K]

0 2 4 6 8x [arcsec]

!200

0

200

400

z [km

]

0 2 4 6 8x [arcsec]

0

2

4

6

8

x [ar

csec

]


The Visible Solar Spectrum


Solar Spectrum in the Blue and Red

391 392 393 394 395 396 397 398 399Wavelength [nm]

05.0•10!9

1.0•10!8

1.5•10!8

2.0•10!8

2.5•10!8

Inte

nsity

[J m

!2 s!1

Hz!1

sr!1

]

588 589 590 591 592 593 594 595 596Wavelength [nm]

0

1•10!8

2•10!8

3•10!8

4•10!8

Inte

nsity

[J m

!2 s!1

Hz!1

sr!1

]


Spatially Resolved Spectral Lines

Intensity

0.5 0.6 0.7 0.8 0.9 1.0

Spectrum

0.2 0.4 0.6 0.8

Polarization

!1.0 !0.5 0.0 0.5


Basic Radiative Transfer: Radiation Field

!"

#

!$

%

&

Specific intensity I! is the radiative energy that flows, at the location #r,per second, per wavelength interval, and per solid angle, in the direction #lthrough the surface area dA! perpendicular to #l. Intensity is conserved withdistance in the absence of emission and absorption or scattering processes.

Specific Intensity:

dErad" ! I"(#r,#l, t) dt dA

! d$ d! = I"(#r,#l, t) dt cos % dA d$ d!

Units: J s"1 m"2 nm"1 ster"1


Basic Radiative Transfer: Mean Intensity

Angle-averaged Mean intensity:

J!(#r, t) !1

4&

!I" d! =

1

4&

! 2#

0

! #

0I" sin % d% d'


Unlike the Specific Intensity theAngle-averaged Mean Intensity is notconserved with distance

! "#

r "!

r sin

#

z

y

x

!


Basic Radiative Transfer: Flux

Flow of radiative energy through a surface.Flux:

F"(#r,#n, t) !!

I" cos % d! =

! 2#

0

! #

0I" cos % sin % d% d' (1)

Units: J s"1 m"2 nm"1

Flux in radial direction:

F"(z) =

! 2#

0

! #2

0I" cos % sin % d% d'+

! 2#

0

! #

#2

I" cos % sin % d% d' (2)

=

! 2#

0

! #2

0I" cos % sin % d% d'"

! 2#

0

! #2

0I"(& " %) cos % sin % d% d'

! F+" (z)" F"

" (z)


Basic Radiative Transfer: Absorption

Absorption !":

I"(s+ ds) = I"(s) + dI" = I" " !"I"ds

Units: m"1

! !"#"$!

$%

$&

$'

(


Basic Radiative Transfer: Emission

Emission j":

I"(s+ ds) = I"(s) + dI" = I" + j"(s)ds

Units: J m"3 s"1 nm"1 ster"1

! !"#"$!

$%

$&

$'

(


Basic Radiative Transfer: Source Function

Source function:

S" ! j"/!"


For multiple proceses active at the same wavelength:

Stot" =

"j"/

"!"

Stot" =

jc" + jl"!c" + !l

"

=Sc" + ("Sl

"

1 + (", (" ! !l

"/!c"


Basic Radiative Transfer: Transport Equation

Transport along a ray:

dI"(s) = I"(s+ ds)" I"(s) = j"(s)ds" !"(s)I"(s)ds (3)

dI"ds

= j" " !"I"

dI"!"ds

=dI"d)"

= S" " I"

Optical length and thickness:

d)" ! !"(s)ds (4)

)"(D) =

! D

0!"(s)ds


Basic Radiative Transfer: Transport Equation

Transport along a ray:

dI"d)"

= S" " I" (5)

I"()") = I"(0)e"$" +

! $"

0S"(t)e

"($""t)dt

Homogeneous medium:

I"(D) = I"(0)e"$"(D) + S"

#1" e"$"(D)

$(6)

Optically thick: I"(D) # S"

Optically thin: I"(D) # I"(0) + [S" " I"(0)] )"(D)


Basic Radiative Transfer: Through an Atmosphere

Optical path:

d)µ" = !"ds ! "!"dz

µ

Standard plane parallel transport equation:

dI"d)µ"

= µdI"d)"

= I""S"

!

"#$%!%&%'

(


Basic Radiative Transfer: Eddington–Barbier

Emergent intensity at the surface:

I+" ()" = 0, µ) =

! #

0S"(t)e

"t/µdt/µ

Substitute power series:

S"()") =N"

n=0

an)n" (using :

! #

0e"ttndt =!n)

I+" ()" = 0, µ) = a0 + a1µ+2a2µ2 + . . .+ n!aNµN

Eddington–Barbier relation:

I+" ()" = 0, µ) # S"()" = µ)


Eddingtomn-Barbier approximation

!"# !"$

%& '%()('&

'("(*+!"$,


Basic Radiative Transfer: Limb Darkening

S$ a

b

h0

I $

0

a

b

10 sin !

!

ab

r/R = sin %


Absorption lines in the solar spectrum

429.0 429.5 430.0 430.5 431.0 431.5 432.0wavelength [nm]

0.0

0.2

0.4

0.6

0.8

1.0

Inte

nsity


Why do we get spectral lines in absoption?

!1 !0

!1 !0

I! !

S

"!

total

!0

!1

0 0

01

h

h0

!

!

!#


Optical depth unity in the Na i D2 line

10!9

10!8

10!7

Sou

rce

Func

tion

[J m

!2 s

!1 H

z!1 s

r!1 ]

0 1000 2000 3000 4000 5000x [km]

!200

0

200

400

600

800

z [k

m]

589.002 [nm]

588.80 588.90 589.00 589.10 589.20 589.30$[nm]


In the UltraViolet the Spectral Lines are in Emission

126 128 130 132 134Wavelength [nm]

0.0

0.2

0.4

0.6

0.8

1.0

Inte

nsity


Why do we get spectral lines in emission?

!1 !0

!1 !0

I! !

S

"!

!0

!1

0 0

01

h

h

total

0!

!

#!


Why do we get spectral lines in emission and absorption?

!

!!

"!

"!

#!

#!$ %

!!

&!

&!

&!

!'(!

!

)*)+,

!

!

!!

,*- )*)+,

!

#

#

#!,*-


Eddington–Barbier is an approximation!!

T [103 K]

6 8 10

!0.2

0.0

0.2

0.4

y [M

m]

0 1 2 3 4 5x [Mm]

µ = 0.93


Eddington–Barbier is an approximation!!

6

8

10

12

T [1

03 K

]

0 1 2 3 4 5x [Mm]

!0.4

!0.2

0.0

0.2

0.4

z [M

m]

0 1 2 3 4 5x [Mm]

5.8

6.0

6.2

6.4

6.6

6.8

7.0

radi

atio

n te

mpe

ratu

re [1

03 K

]

Trad Tform


Continuum processes

Outside spectral lines the solar plasma has significant opacity in so calledcontinuum processes. They are called this way because their opacity variesvery slowly with wavelength.

• Atomic Bound–free and free–free transitions

• H" bound–free and free-free

• Thomson scattering

!Te = Ne*e = Ne

8&

3m4ec

2

q4e(4&+0)2

• Rayleigh scattering

!R(,) = *efij,4/(,2

ij " ,2)2


There is a lot of information in spectral lines

Uitenbroek & Tritschler, IBIS DST


End Part I


Molecular Oxygen in the Earth Atmosphere

Intensity

0.4 0.6 0.8

630.0 630.1 630.2 630.3 630.4Wavelength [nm]

Fe IFe I

O2 O2

Back


Di!erences in spectral lines

393.0 393.2 393.4 393.6 393.8 394.0Wavelength [nm]

0

2.0•10!9

4.0•10!9

6.0•10!9

8.0•10!9

1.0•10!8

1.2•10!8

Inte

nsity

[J m

!2 s!1

Hz!1

sr!1

]

Back


Invariance of Specific Intensity along Rays

Specific Intensity has been defined in such a way as to be independent ofthe source and the observer.

dE" = I" cos % dt dA d$ d! = I !" cos %! dt dA! d$ d!!

d! = dA! cos %!/R2

d!! = dA cos %/R2

I" = I !"

Back


H" Opacity

0 1 2 3 4Wavelength [µm]

0

2•10!29

4•10!29

6•10!29

8•10!29

1•10!28

% & [m

5 /J]

Hydrogen H! opacity T = 6000

H! b ! fH! f ! f

"Ebf = 0.754 eV Back


Bound–Free Cross Sections of Hydrogen

Back


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