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]

1•10!8

2•10!8

3•10!8

4•10!8

5•10!8

!2 s!1

6300 [K]

! " "" "! # $

0 2000 4000 6000 8000 10000Wavelength [nm]

1•10!8

2•10!8

3•10!8

4•10!8

5•10!8

!2 s!1

] 5770 [K]

6300 [K]

! " "" "! # $

0 2000 4000 6000 8000 10000Wavelength [nm]

1•10!8

2•10!8

3•10!8

4•10!8

5•10!8

!2 s!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

0 2 4 6 8x [arcsec]

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

!2 s!1

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

1•10!8

2•10!8

3•10!8

4•10!8

!2 s!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

!I" d! =

0I" sin % d% d'

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

Basic Radiative Transfer: Flux

Flow of radiative energy through a surface.Flux:

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

I" cos % d! =

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

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

Flux in radial direction:

F"(z) =

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

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

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

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

! F+" (z)" F"

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" =

jc" + jl"!c" + !l

=Sc" + ("Sl

1 + (", (" ! !l

Basic Radiative Transfer: Transport Equation

Transport along a ray:

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

= j" " !"I"

dI"!"ds

=dI"d)"

= S" " I"

Optical length and thickness:

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

)"(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)

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

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"

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

10 sin !

r/R = sin %

Absorption lines in the solar spectrum

429.0 429.5 430.0 430.5 431.0 431.5 432.0wavelength [nm]

Why do we get spectral lines in absoption?

Optical depth unity in the Na i D2 line

0 1000 2000 3000 4000 5000x [km]

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]

Why do we get spectral lines in emission?

Why do we get spectral lines in emission and absorption?

,*- )*)+,

Eddington–Barbier is an approximation!!

T [103 K]

6 8 10

0 1 2 3 4 5x [Mm]

µ = 0.93

Eddington–Barbier is an approximation!!

0 1 2 3 4 5x [Mm]

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

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

Di!erences in spectral lines

393.0 393.2 393.4 393.6 393.8 394.0Wavelength [nm]

2.0•10!9

4.0•10!9

6.0•10!9

8.0•10!9

1.0•10!8

1.2•10!8

!2 s!1

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 !"

H" Opacity

0 1 2 3 4Wavelength [µm]

2•10!29

4•10!29

6•10!29

8•10!29

1•10!28

% & [m

Hydrogen H! opacity T = 6000

H! b ! fH! f ! f

"Ebf = 0.754 eV Back

Bound–Free Cross Sections of Hydrogen

Why Radiative Transfer? Introduction to Solar...

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