introduction to solar radiative transfer i · introduction to solar radiative transfer: ......

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)


Bibliography

• Rutten: Radiative Transfer in Stellear Atmospheres(http://www.astro.uu.nl/˜rutten/Astronomy lecture.html)


http://www.astro.uu.nl/~rutten/Astronomy_lecture.html

Bibliography


• Rybicki and Lightman: Radiative Processes in Astrophysics



Bibliography



• Gray: Observation and Analysis of Stellar Photospheres



Bibliography




• Mihalas: Stellar Atmospheres



Bibliography





• Shu: The Physics of Astrophysics. I. Radiation



Bibliography





• Shu: The Physics of Astrophysics. I. Radiation

• Allen: Astrophysical Quantities



Short History

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


Short History


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


Short History



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


Short History




• 1842 Becquerel photographs spectra, discovers UV lines beyond thevisible


Short History




• 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.


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.






• 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.


Solar line spectrum


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 ]


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


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


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:

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)



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)


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


Basic Radiative Transfer: Local Changes

Source function:

Sν = jν/αν (7)


Stotν =

∑jν/∑

αν (8)

Stotν =

jcν + jlναcν + αlν

=Scν + ηνS

lν

1 + ην, ην ≡ αlν/αcν (9)


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


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)


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


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)


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)


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)



Sν 0

1

2

0 1 2 3 40

−τνe

θ I

τ ν

ν

−τνeSν

ντ



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


Basic Radiative Transfer: Limb Darkening

Sν a

b

h0

I ν

0

a

b

10 sin θ

θ

a

b


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


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


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


Spherical Coordinates

θ ∆ϕ

r ∆θ

r sin

ϕ

z

y

x

θ

Back


introduction to solar radiative transfer i · introduction to solar radiative transfer: ......

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