Radiation Transfer
with Scattering Process
Yoshiaki Kato (NAOJ)‣ Radiation Transfer Equation with Scattering Process‣ Line Profile Formation‣ Scattering Processes✓ Complete “Frequency” Re-Distribution (CRD)✓ Partial “Frequency” Re-Distribution (PRD)
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Friday, September 14, 12

Radiative Transfer Equations
with Scattering Process“Classical” Radiative Transfer EquationI ν (r, t; n) : specific intensityχν (r, t; n) = κν + σν : extinction coefficient

Source Functionεν (r, t; n) : emissivity

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Friday, September 14, 12

Derivation of RT Equations #0
Description of the Radiation Field• Intensity

I (x, y, z, t; θ, ϕ, ν)

dE = I dt dA dΩ dνdAdΩ(x, y, z)(θ, ϕ)• Photon distribution function: fRh3 ν 2I (x, t; n, ν) = chν 3 fR (x, t; n, ν)
cfR: 6-dimensional phase space

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Derivation of RT Equations #1Relation to Boltzmann Transport Equation• Boltzmann Transport Equation !
∂fR∂fR∂fR+v·+F ·=∂t∂x∂pDfRDt"coll• Radiative Transfer Equation

1 ∂I+ n · ∇I = χ (S − I)c ∂t= η − χIη (x, t; n, ν) : emissivityχ (x, t; n, ν) : opacity per unit length
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Derivation of RT Equations #2Classical, macroscopic, and phenomenological derivation
３３４

ＦＯＵＮＤＡＴＩＯＮＳＯＦＲＡＤＩＡＴＩＯＮＨＹＤＲＯＤＹＮＡＭＩＣＳ

工（ｘ＋△ｘ，ｔ＋△ｔ；ｎ，ｙ）

工（輿，ｔ；ｎ，・）

皿ｇ．７６．１Ｐｅｎｃｉｌｏｆｒａｄｉａｔｉｏｎｐａｓｓｉｎｇｔｈｒｏｕｇｈａｍａｔｅｒｉａｌｅｌｅｍｅｎｔ．

[I (x + ∆x, t + ∆t; n, ν) − I (x, t; n, ν)] dSdtdωdν!
"1 ∂I (x, t; n, ν) ∂I (x, t; n, ν)+=dsdSdtdωdνc∂t∂sＢｅｃａｕｓｅｓｉｓａｃｏｏｒｄｉｎａｔｅ­ｉｎｄｅｐｅｎｄｅｎｔｐａｔｈｌｅｎｇｔｈ，（７６．３）ａｌ）ｐｌｉｅｓｉｎａｒｂｉｔ­Ｉ丿ａｒｙｃｏｏｒｄｉｎａｔｅｓｙｓｔｅｍｓ，Ｉ：）１ʼｏｖｉｄｅｄｗｅｕｓｅａｎａｌ）１）ｒｏｐｒｉａｔｅｅｘｐｒｅｓｓｉｏｎｔｏ
ｅｖａｌｕａｔｅ（／ｓ）．

Ｔｈｅｄｅｒｉｖａｔｉｏｎｊｕｓｔｇｉｖｅｎｏｆｔｈｅｔｒａｎｓｆｅｒｅｑｕａｔｉｏｎｉｓｄａｓｓｉｃａｌ，ｍａｃｒｏｓ­

ｃｏｐｉｃ，ａｎｄｐｈｅｎｏｍｅｎｏｌｏｇｉｃａｌｉｎｃｈａｒａｃｔｅｒ．ｌｔｏｍｉｔｓｒｅｆｅｒｅｎｃｅｔｏｓｕｃｈ

ｉｍｐｏｒｔａｎｔｐｈｅｎｏｍｅｎａｉｌｓｐｏｌａｒｉｚａｔｉｏｎ，ｄｉｓｐｅｒｓｉｏｎ，ｃｏｈｅｒｅｎｃｅ，ｉｎｔｅｒｆｅｒｅｎｃｅ，ａｎｄｑｕａｎｔｕｍｅｆｅｃｔｓ，ｎｏｎｅｏｆｗｈｉｃｈａｒｅｃｏｒｒｅｃｔｌｙｄｅｓｃｒｉｂｅｄｂｙ（７６．３）．Ａｎ= [η (x, t; n, ν) − χ (x, t; n, ν) I (x, t; n, ν)] dsdSdtdωdν
ｅｘｃｅｌｌｅｎｔｄｉｓｃｕｓｓｉｏｎｏｆｔｈｅａｐｐｒｏ）ｃｉｍａｔｉｏｎｓｉｎｈｅｒｅｎｔｉｎ，ａｎｄｔｈｅｖａｌｉｄｉｔｙｏｆ，

ｔｈｅｃｌａｓｓｉｃａ１１ʻａｄｉａｔｉｖｅｔｌａｎｓｆｅｒ（５ｑｕａｔｉｏｎｉｓｇｉｖｅｎｉｎ（Ｐ３，４７­４９）．Ｇｏｏｄ

ｄｉｓｃｕｓｓｉｏｎｓｏｆｔｈｅｔｒａｎｓｆｅｒｅｑｕａｔｉｏｎｆｒｏｍｔｈｅｐｏｉｎｔｏｆｖｉｅｗｏｆｑｕａｎｔｕｍｆｉｅｌｄ
ｔｈｅｏｒｙａｒｅがｖｅｎｉｎ（ＨＩ），（Ｌ：１．），（Ｌ２λ（Ｌ３），（０１）．

1 ∂I (x, t; n, ν) ∂I (x, t; n, ν)+= η (x, t; n, ν) − χ (x, t; n, ν) I (x, t; n, ν)c∂t∂sＴｈｅｍａｔｈｅｍａｔｉｃａｌｅｘｐｒｅｓｓｉｏｎｆｏｒ（ ／ ｓ）ｄｅｐｅｎｄｓｏｎｇｅｏｍｅｔｒｙ．ｌｎＣａｒ­

ｔｅｓｉａｎｃｏｏｒｄｉｎａｔｅｓ

ま゜郎だ十②訂十（だ）だ゜ｎ，£＋ｎｙ；ｊ７十ｎ，£，（７６．４）

5ｗｈｅｒｅ（ｎｘ，馬。ｎｚ）ａｒｅｃｏｍｐｏｎｅｎｔｓｏｆｔｈｅｕｎｉｔｖｅｃｔｏｒｎａｌｏｎｇｔｈｅｄｉｒｅｃｔｉｏｎｏｆ

Friday, September 14, 12

ｐｒｏｐａｇａｔｉｏｎ．ＴｈｅｔｒａｎｓｆｅｒｅｑｕａｔｉｏｎｉｓｔｈｅｎDerivation of RT Equations #3
Schematics of RT Equation with Scatteringn!

Scattering

Scattering

Absorption

EmissionnχS−χIx!

1 ∂I+ n · ∇I = χabs B + χscac ∂t"#I (n! ) φ (n, n! ) dΩ − (χabs + χsca ) I

1 ∂I+ n · ∇I = χ (S − I)c ∂t6Friday, September 14, 12

Radiative Transfer Equations
Opacity and Level populations

Opacity!

2πem0 c"Φij (ν) : the spectral line profile function
ni fij Φij (ν)! "2!
"γν22ν ν02π 2πe2
πe2 γ1σ (ν) =fs →fs#$2222γm0 c (ν 2 − ν 2 ) + ν 2
m0 c 4π (ν − ν0 ) + (γ/4π)
02π

κ (νij ) =Rate Equations (Spontaneous + Induced + Collisional rates)!"## ! "dninj Aji + Bji Uνij + Cji
Aij + Bij Uνij + Cij += −nidtj!=ij!=iAij , Bij : radiative processes

Cij : collisional processes

Uνij radiation energy density in the range between hνij = εi − εj7
Friday, September 14, 12Line Profile Formation
Overview• Gaussian Profile!

1(ν − ν0 )
Φ (ν) = √exp −2σ 22πσ

• Lorentz Profile
Φ (ν) =

∆νL /2π2(ν − ν0 ) + (∆νL /2)

2! +∞"• The Voigt Profile
Φ (ν) =2"−∞2#∆νL exp −∆νd (∆ν)$%√
22π π∆νD (ν − ν0 − ∆ν) + (∆νL /2)
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Gaussian & Lorentz Profile
Doppler, Resonance, Collision, Natural Broadening!

1(ν − ν0 )
Φ (ν) = √exp −2σ 22πσ2"22σ 2 = ∆νD/ (4 ln 2)#$1/2!
"2ν02kTk∆νD =+V2ln 2cMΦ (ν) =

Gaussian profile∆νD = 2∆νL /2π2(ν − ν0 ) + (∆νL /2)∆νL = (2πτ )−1Lorentz profile∆νL = 22= [2π (τN + τR + τC )]−1Frequency [1/s]τN = 1/AijτC : the mean collision time
9Friday, September 14, 124mνmnνmn
τR =
≈
[s]
27N e fa
6 × 10 N fa

The Voigt Profile

Gaussian Profile + Lorentz Profile
Φ (ν) =! +∞−∞"22/∆νD2#∆νL exp −∆νd (∆ν)$%√
22π π∆νD (ν − ν0 − ∆ν) + (∆νL /2)√2 ln 2
Φ(ν) = √H (a, b)π∆νD√
a = ln 2∆νL / (2∆νD )√b = 2 ln 2 (ν − ν0 ) /∆νD" 2#! ∞exp −y dyaH (a, b) =π −∞ (b − y)2 + a2!

H (a, b) ≈ exp −b2"λLyα = 1215.67 Å

∆νL≈ 0.001∆νDTgas = 4000 Kν − ν0∆νDa+√ 2πb10

Friday, September 14, 12

A21 = 6.26 × 108 s−1

Scattering Processes
Line ProfilenΦ! (ν)!

Φ(ν)
Scattering

Scattering−χInχSεAbsorption

Emissionx!

1 ∂I+ n · ∇I = χabs B + χscac ∂t"#I (n! ) φ (n, n! ) dΩ − (χabs + χsca ) I

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Timescales

Collision time .vs. Transition time

• Collision time for neutral Hydrogen atoms
tcoll ≈ 10

12−3=1.58
×10
T −1/2 n−1H!

T
4000 [K]"−1/2 !

nH1013 [1/cc]• Transition time of Hydrogen atom (2p→1s)11−9=1.597
×10[sec]t2p→1s ==8−1A216.26 × 10 [s ]tcoll ! t2p→1s
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Frequency Redistribution

Complete Re-Distribution (CRD) and Partial Re-Distribution (PRD)• Atom 1 ν → Atom 2 ν• Complete Re-Distribution (CRD) in the core12Φ! (ν)

Φ(ν)ν1 νj ν2νj ν2• Partial Re-Distribution (PRD) in the wings!

Φ(ν)

Φ (ν)
νj ν1

νj

ν2

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ν2

Emergent Intensity Profile

Complete Re-Distribution (CRD) and Partial Re-Distribution (PRD)• Complete Re-Distribution (CRD)!

I (νj )I (ν2 ) = ΦCRD (ν2 )j(PRD)• Partial Re-Distribution!

ΦPRD (νj ) I (νj )I (ν2 ) =j14
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To be continued,,,‣ Methods for Solving Radiation Transfer Equations✓ Mesh-based Radiation Transfer Solver✓ Monte-Carlo Radiation Transfer Solver‣ Test Calculation of the Solar Model Atmosphere✓ Static Plane-Parallel Model Atmosphere✓ RMHD Model Atmosphere

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