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Introduction to Solar Radiative Transfer:I Basic Radiative Transfer
Han UitenbroekNational Solar Observatory/Sacramento Peak
Sunspot NM, USA
Summerschool Sunspot, Jun 18 2008
Overview
• I Basic Radiative Transfer (Intensity, emission, absorption, sourcefunction, optical depth)
Han Uitenbroek, NSO/SP Introduction to Solar Radiative Transfer I y w ww wy p 7
Overview
• I Basic Radiative Transfer (Intensity, emission, absorption, sourcefunction, optical depth)
• II Detailed Radiative Processes (Spectral lines, radiative transitions,collisions, continuum processes, Saha-Boltzmann, polarization)
Han Uitenbroek, NSO/SP Introduction to Solar Radiative Transfer I y w ww wy p 7
Bibliography
• Rutten: Radiative Transfer in Stellear Atmospheres(http://www.astro.uu.nl/˜rutten/Astronomy lecture.html)
Han Uitenbroek, NSO/SP Introduction to Solar Radiative Transfer I y w ww wy p 7
Bibliography
• Rutten: Radiative Transfer in Stellear Atmospheres(http://www.astro.uu.nl/˜rutten/Astronomy lecture.html)
• Rybicki and Lightman: Radiative Processes in Astrophysics
Han Uitenbroek, NSO/SP Introduction to Solar Radiative Transfer I y w ww wy p 7
Bibliography
• Rutten: Radiative Transfer in Stellear Atmospheres(http://www.astro.uu.nl/˜rutten/Astronomy lecture.html)
• Rybicki and Lightman: Radiative Processes in Astrophysics
• Gray: Observation and Analysis of Stellar Photospheres
Han Uitenbroek, NSO/SP Introduction to Solar Radiative Transfer I y w ww wy p 7
Bibliography
• Rutten: Radiative Transfer in Stellear Atmospheres(http://www.astro.uu.nl/˜rutten/Astronomy lecture.html)
• Rybicki and Lightman: Radiative Processes in Astrophysics
• Gray: Observation and Analysis of Stellar Photospheres
• Mihalas: Stellar Atmospheres
Han Uitenbroek, NSO/SP Introduction to Solar Radiative Transfer I y w ww wy p 7
Bibliography
• Rutten: Radiative Transfer in Stellear Atmospheres(http://www.astro.uu.nl/˜rutten/Astronomy lecture.html)
• Rybicki and Lightman: Radiative Processes in Astrophysics
• Gray: Observation and Analysis of Stellar Photospheres
• Mihalas: Stellar Atmospheres
• Shu: The Physics of Astrophysics. I. Radiation
Han Uitenbroek, NSO/SP Introduction to Solar Radiative Transfer I y w ww wy p 7
Bibliography
• Rutten: Radiative Transfer in Stellear Atmospheres(http://www.astro.uu.nl/˜rutten/Astronomy lecture.html)
• Rybicki and Lightman: Radiative Processes in Astrophysics
• Gray: Observation and Analysis of Stellar Photospheres
• Mihalas: Stellar Atmospheres
• Shu: The Physics of Astrophysics. I. Radiation
• Allen: Astrophysical Quantities
Han Uitenbroek, NSO/SP Introduction to Solar Radiative Transfer I y w ww wy p 7
Short History
• 1802 Wollaston First to observe dark gaps in spectrum: spectral lines
Han Uitenbroek, NSO/SP Introduction to Solar Radiative Transfer I y w ww wy p 7
Short History
• 1802 Wollaston First to observe dark gaps in spectrum: spectral lines
• 1814 Fraunhofer rediscovers lines. Assigns names e.g.,C (Hα), D (Na i), G (CH molecules), F (Hβ), and H (Ca ii)
Han Uitenbroek, NSO/SP Introduction to Solar Radiative Transfer I y w ww wy p 7
Short History
• 1802 Wollaston First to observe dark gaps in spectrum: spectral lines
• 1814 Fraunhofer rediscovers lines. Assigns names e.g.,C (Hα), D (Na i), G (CH molecules), F (Hβ), and H (Ca ii)
• 1823 Herschel realized spectra contain information on composition ofsource from flame spectra
Han Uitenbroek, NSO/SP Introduction to Solar Radiative Transfer I y w ww wy p 7
Short History
• 1802 Wollaston First to observe dark gaps in spectrum: spectral lines
• 1814 Fraunhofer rediscovers lines. Assigns names e.g.,C (Hα), D (Na i), G (CH molecules), F (Hβ), and H (Ca ii)
• 1823 Herschel realized spectra contain information on composition ofsource from flame spectra
• 1842 Becquerel photographs spectra, discovers UV lines beyond thevisible
Han Uitenbroek, NSO/SP Introduction to Solar Radiative Transfer I y w ww wy p 7
Short History
• 1802 Wollaston First to observe dark gaps in spectrum: spectral lines
• 1814 Fraunhofer rediscovers lines. Assigns names e.g.,C (Hα), D (Na i), G (CH molecules), F (Hβ), and H (Ca ii)
• 1823 Herschel realized spectra contain information on composition ofsource from flame spectra
• 1842 Becquerel photographs spectra, discovers UV lines beyond thevisible
• 1858 Bunsen and Kirchhoff discover wavelength correspondencebetween bright flame emission and dark solar absorption lines. Start ofquantitative spectroscopy.
Han Uitenbroek, NSO/SP Introduction to Solar Radiative Transfer I y w ww wy p 7
Why Radiative Transfer?
• Information on astronomical objects in general has to be recovered fromthe electromagnetic radiation the object emits and/or reflects.
Han Uitenbroek, NSO/SP Introduction to Solar Radiative Transfer I y w ww wy p 7
Why Radiative Transfer?
• Information on astronomical objects in general has to be recovered fromthe electromagnetic radiation the object emits and/or reflects.
• Spectroscopic observation (remote sensing) and analysis is the onlyavailable method to determine the physical conditions of theseastronomical objects.
Han Uitenbroek, NSO/SP Introduction to Solar Radiative Transfer I y w ww wy p 7
Why Radiative Transfer?
• Information on astronomical objects in general has to be recovered fromthe electromagnetic radiation the object emits and/or reflects.
• Spectroscopic observation (remote sensing) and analysis is the onlyavailable method to determine the physical conditions of theseastronomical objects.
• To analyze spectroscopic data meaningfully we need to understand howthis physical information is encoded in the radiation RadiativeTransfer.
Han Uitenbroek, NSO/SP Introduction to Solar Radiative Transfer I y w ww wy p 7
Why Radiative Transfer?
• Information on astronomical objects in general has to be recovered fromthe electromagnetic radiation the object emits and/or reflects.
• Spectroscopic observation (remote sensing) and analysis is the onlyavailable method to determine the physical conditions of theseastronomical objects.
• To analyze spectroscopic data meaningfully we need to understand howthis physical information is encoded in the radiation RadiativeTransfer.
• Observational techniques need to be applied to obtain the relevantproperties of the radiation signal.
Han Uitenbroek, NSO/SP Introduction to Solar Radiative Transfer I y w ww wy p 7
Why Radiative Transfer?
• Information on astronomical objects in general has to be recovered fromthe electromagnetic radiation the object emits and/or reflects.
• Spectroscopic observation (remote sensing) and analysis is the onlyavailable method to determine the physical conditions of theseastronomical objects.
• To analyze spectroscopic data meaningfully we need to understand howthis physical information is encoded in the radiation RadiativeTransfer.
• Observational techniques need to be applied to obtain the relevantproperties of the radiation signal.
• We need to understand how the radiative signal is modified as it travelsfrom the object to our instruments.
Han Uitenbroek, NSO/SP Introduction to Solar Radiative Transfer I y w ww wy p 7
Solar line spectrum
Han Uitenbroek, NSO/SP Introduction to Solar Radiative Transfer I y w ww wy p 7
Differences in Spectral Lines
391 392 393 394 395 396 397 398 399Wavelength [nm]
0
5.0•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 H
z−1 s
r−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 H
z−1 s
r−1 ]
Han Uitenbroek, NSO/SP Introduction to Solar Radiative Transfer I y w ww wy p 7
Molecular 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 y w ww wy p 7
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 y w ww wy p 7
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
Han Uitenbroek, NSO/SP Introduction to Solar Radiative Transfer I y w ww wy p 7
Basic Radiative Transfer: Radiation Field
Specific Intensity:
dEν ≡ Iν(~r,~l, t) dtdA dν dΩ (1)
= Iν(x, y, z, θ, ϕ, t) cos θ dtdA dν dΩ
Units: J s−1 m−2 Hz−1 ster−1
Mean intensity:
Jν(~r, t) ≡1
4π
∫Iν dΩ =
14π
∫ 2π
0
∫ π
0
Iν sin θ dθ dϕ (2)
Units: J s−1 m−2 Hz−1 ster−1
Han Uitenbroek, NSO/SP Introduction to Solar Radiative Transfer I y w ww wy p 7
Basic Radiative Transfer: Radiation Field
Flux:
Fν(~r, ~n, t) ≡∫Iν cos θ dΩ =
∫ 2π
0
∫ π
0
Iν cos θ sin θ dθ dϕ (3)
Units: J s−1 m−2 Hz−1
Flux in radial direction:
Fν(z) =∫ 2π
0
∫ π2
0
Iν cos θ sin θ dθ dϕ+∫ 2π
0
∫ π
π2
Iν cos θ sin θ dθ dϕ (4)
=∫ 2π
0
∫ π2
0
Iν cos θ sin θ dθ dϕ−∫ 2π
0
∫ π2
0
Iν(π − θ) cos θ sin θ dθ dϕ
≡ F+ν (z)−F−ν (z)
Han Uitenbroek, NSO/SP Introduction to Solar Radiative Transfer I y w ww wy p 7
Basic Radiative Transfer: Local Changes
Emission:
dEν ≡ jν dV dtdν dΩ (5)
dIν = jν(s)ds
Units: J m−3 s−1 Hz−1 ster−1
Extinction:
dIν ≡ −ανIνds (6)
Units: m−1
Han Uitenbroek, NSO/SP Introduction to Solar Radiative Transfer I y w ww wy p 7
Basic Radiative Transfer: Local Changes
Source function:
Sν = jν/αν (7)
Units: J s−1 m−2 Hz−1 ster−1
Stotν =
∑jν/∑
αν (8)
Stotν =
jcν + jlναcν + αlν
=Scν + ηνS
lν
1 + ην, ην ≡ αlν/αcν (9)
Han Uitenbroek, NSO/SP Introduction to Solar Radiative Transfer I y w ww wy p 7
Basic Radiative Transfer: Transport Equation
Transport along a ray:
dIν(s) = Iν(s+ ds)− Iν(s) = jν(s)ds− αν(s)Iν(s)ds (10)
dIνds
= jν − ανIν
dIνανds
= Sν − Iν
Optical length and thickness:
dτν ≡ αν(s)ds (11)
τν(D) =∫ D
0
αν(s)ds
Han Uitenbroek, NSO/SP Introduction to Solar Radiative Transfer I y w ww wy p 7
Basic Radiative Transfer: Transport Equation
Transport along a ray:
dIνdτν
= Sν − Iν (12)
Iν(τν) = Iν(0)e−τν +∫ τν
0
Sν(t)e−(τν−t)dt
Homogeneous medium:
Iν(D) = Iν(0)e−τν(D) + Sν
(1− e−τν(D)
)(13)
Optically thick: Iν(D) ≈ SνOptically thin: Iν(D) ≈ Iν(0) + [Sν − Iν(0)] τν(D)
Han Uitenbroek, NSO/SP Introduction to Solar Radiative Transfer I y w ww wy p 7
Basic Radiative Transfer: Homogeneous Medium
ν (D) >> 1
ν0
I ν
S ν
0ν ν
0
I ν
S ν
0ν
τν (D) < 1
I (0) = 0ν
ν0
I ν
S ν
I ν(0)0
ν
I (0) < Sν ν
τν (D) < 1
ν0
I ν(0)
S νI ν
τ
0ν
τν (D) > 10
τν (D) < 1
I (0) < Sν ν
ν0
S ν
I ν(0)I ν
0ν
τν (D) > 10
τν (D) < 1
ν νI (0) > S
ν0
S ν
I ν(0)I ν
0ν
τν (D) < 1
I (0) > Sν ν
Han Uitenbroek, NSO/SP Introduction to Solar Radiative Transfer I y w ww wy p 7
Basic Radiative Transfer: Through an Atmosphere
Optical depth:
dτµν = ανds ≡ −ανdz|µ|
(14)
τν(z1) =∫ z1
zsurf
−ανdz =∫ zsurf
z1
ανdz
Standard plane parallel transport equation:
µdIνdτν
= Iν − Sν (15)
Han Uitenbroek, NSO/SP Introduction to Solar Radiative Transfer I y w ww wy p 7
Basic Radiative Transfer: Through an Atmosphere
Formal solution in upward direction:
I+ν (τν, µ) =
∫ ∞τν
Sν(t)e−(t−τν)/µdt/µ, µ > 0 (16)
Formal solution in downward direction:
I−ν (τν, µ) = −∫ τν
0
Sν(t)e−(t−τν)/µdt/µ, µ < 0 (17)
Han Uitenbroek, NSO/SP Introduction to Solar Radiative Transfer I y w ww wy p 7
Basic Radiative Transfer: Eddington–Barbier
Emergent intensity at the surface:
I+ν (τν = 0, µ) =
∫ ∞0
Sν(t)e−t/µdt/µ (18)
Substitute power series:
Sν(τν) =N∑n=0
anτnν (19)
I+ν (τν = 0, µ) = a0 + a1µ+ 2a2µ
2 + . . .+ n!aNµN
Eddington–Barbier relation:
I+ν (τν = 0, µ) ≈ Sν(τν = µ) (20)
Han Uitenbroek, NSO/SP Introduction to Solar Radiative Transfer I y w ww wy p 7
Basic Radiative Transfer: Eddington–Barbier
Sν 0
1
2
0 1 2 3 40
−τνe
θ I
τ ν
ν
−τνeSν
ντ
Han Uitenbroek, NSO/SP Introduction to Solar Radiative Transfer I y w ww wy p 7
Basic Radiative Transfer: Eddington–Barbier
T [103 K]
6 8 10
−0.2
0.0
0.2
0.4
y [
Mm
]
0 1 2 3 4 5x [Mm]
µ = 0.93
Han Uitenbroek, NSO/SP Introduction to Solar Radiative Transfer I y w ww wy p 7
Basic Radiative Transfer: Limb Darkening
Sν a
b
h0
I ν
0
a
b
10 sin θ
θ
a
b
Han Uitenbroek, NSO/SP Introduction to Solar Radiative Transfer I y w ww wy p 7
End Part I
Han Uitenbroek, NSO/SP Introduction to Solar Radiative Transfer I y w ww wy p 7
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 y w ww wy p 7
Differences 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 H
z−1 s
r−1 ]
Back
Han Uitenbroek, NSO/SP Introduction to Solar Radiative Transfer I y w ww wy p 7
Invariance of Specific Intensity along Rays
Specific Intensity has been defined in such a way as to be independent ofthe source and the observer.
dEν = Iν cos θ dt dAdν dΩ = I ′ν cos θ′ dtdA′ dν dΩ′
dΩ = dA′ cos θ′/R2
dΩ′ = dA cos θ/R2
Iν = I ′ν
Back
Han Uitenbroek, NSO/SP Introduction to Solar Radiative Transfer I y w ww wy p 7
Spherical Coordinates
θ ∆ϕ
r ∆θ
r sin
ϕ
z
y
x
θ
Back
Han Uitenbroek, NSO/SP Introduction to Solar Radiative Transfer I y w ww wy p 7