Why Radiative Transfer? Introduction to Solar...
Transcript of 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]
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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
] 5770 [K]
6300 [K]
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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
] 5770 [K]6000 [K]6300 [K]
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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
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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
Han Uitenbroek, NSO/SP Introduction to Solar Radiative Transfer I ! " "" "! # $
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.
Han Uitenbroek, NSO/SP Introduction to Solar Radiative Transfer I ! " "" "! # $
The Sun
Han Uitenbroek, NSO/SP Introduction to Solar Radiative Transfer I ! " "" "! # $
Solar atmosphere is also very strongly time dependent
Courtesy: Mats Carlsson
Han Uitenbroek, NSO/SP Introduction to Solar Radiative Transfer I ! " "" "! # $
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
]
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The Visible Solar Spectrum
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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
]
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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
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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
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Basic Radiative Transfer: Mean Intensity
Angle-averaged Mean intensity:
J!(#r, t) !1
4&
!I" d! =
1
4&
! 2#
0
! #
0I" sin % d% d'
Units: J s"1 m"2 nm"1 ster"1
Unlike the Specific Intensity theAngle-averaged Mean Intensity is notconserved with distance
! "#
r "!
r sin
#
z
y
x
!
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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)
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Basic Radiative Transfer: Absorption
Absorption !":
I"(s+ ds) = I"(s) + dI" = I" " !"I"ds
Units: m"1
! !"#"$!
$%
$&
$'
(
Han Uitenbroek, NSO/SP Introduction to Solar Radiative Transfer I ! " "" "! # $
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
! !"#"$!
$%
$&
$'
(
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Basic Radiative Transfer: Source Function
Source function:
S" ! j"/!"
Units: J s"1 m"2 nm"1 ster"1
For multiple proceses active at the same wavelength:
Stot" =
"j"/
"!"
Stot" =
jc" + jl"!c" + !l
"
=Sc" + ("Sl
"
1 + (", (" ! !l
"/!c"
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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
Han Uitenbroek, NSO/SP Introduction to Solar Radiative Transfer I ! " "" "! # $
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)
Han Uitenbroek, NSO/SP Introduction to Solar Radiative Transfer I ! " "" "! # $
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"
!
"#$%!%&%'
(
Han Uitenbroek, NSO/SP Introduction to Solar Radiative Transfer I ! " "" "! # $
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)
Han Uitenbroek, NSO/SP Introduction to Solar Radiative Transfer I ! " "" "! # $
Eddingtomn-Barbier approximation
!"# !"$
%& '%()('&
'("(*+!"$,
Han Uitenbroek, NSO/SP Introduction to Solar Radiative Transfer I ! " "" "! # $
Basic Radiative Transfer: Limb Darkening
S$ a
b
h0
I $
0
a
b
10 sin !
!
ab
r/R = sin %
Han Uitenbroek, NSO/SP Introduction to Solar Radiative Transfer I ! " "" "! # $
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
Han Uitenbroek, NSO/SP Introduction to Solar Radiative Transfer I ! " "" "! # $
Why do we get spectral lines in absoption?
!1 !0
!1 !0
I! !
S
"!
total
!0
!1
0 0
01
h
h0
!
!
!#
Han Uitenbroek, NSO/SP Introduction to Solar Radiative Transfer I ! " "" "! # $
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]
Han Uitenbroek, NSO/SP Introduction to Solar Radiative Transfer I ! " "" "! # $
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
Han Uitenbroek, NSO/SP Introduction to Solar Radiative Transfer I ! " "" "! # $
Why do we get spectral lines in emission?
!1 !0
!1 !0
I! !
S
"!
!0
!1
0 0
01
h
h
total
0!
!
#!
Han Uitenbroek, NSO/SP Introduction to Solar Radiative Transfer I ! " "" "! # $
Why do we get spectral lines in emission and absorption?
!
!!
"!
"!
#!
#!$ %
!!
&!
&!
&!
!'(!
!
)*)+,
!
!
!!
,*- )*)+,
!
#
#
#!,*-
Han Uitenbroek, NSO/SP Introduction to Solar Radiative Transfer I ! " "" "! # $
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
Han Uitenbroek, NSO/SP Introduction to Solar Radiative Transfer I ! " "" "! # $
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
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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
Han Uitenbroek, NSO/SP Introduction to Solar Radiative Transfer I ! " "" "! # $
There is a lot of information in spectral lines
Uitenbroek & Tritschler, IBIS DST
Han Uitenbroek, NSO/SP Introduction to Solar Radiative Transfer I ! " "" "! # $
End Part I
Han Uitenbroek, NSO/SP Introduction to Solar Radiative Transfer 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
Han Uitenbroek, NSO/SP Introduction to Solar Radiative Transfer I ! " "" "! # $
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
Han Uitenbroek, NSO/SP Introduction to Solar Radiative Transfer I ! " "" "! # $
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
Han Uitenbroek, NSO/SP Introduction to Solar Radiative Transfer I ! " "" "! # $
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
Han Uitenbroek, NSO/SP Introduction to Solar Radiative Transfer I ! " "" "! # $
Bound–Free Cross Sections of Hydrogen
Back
Han Uitenbroek, NSO/SP Introduction to Solar Radiative Transfer I ! " "" "! # $