[PPT] - Radiation Transfer with Scattering Process Yoshiaki Kato (NAOJ) PowerPoint Presentation

SLIDE 1

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)

1

Friday, September 14, 12

SLIDE 2

Radiative Transfer Equations

with Scattering Process

“Classical” Radiative Transfer Equation Source Function Iν (r, t; n) : specific intensity χν (r, t; n) = κν + σν : extinction coefficient εν (r, t; n) : emissivity

2

Friday, September 14, 12

SLIDE 3

Derivation of RT Equations #0

Description of the Radiation Field

Intensity
Photon distribution function: fR

3

I (x, y, z, t; θ, ϕ, ν) dE = I dt dA dΩ dν

dA (x, y, z) dΩ (θ, ϕ)

fR: 6-dimensional phase space

I (x, t; n, ν) = chν h3ν2 c3 fR (x, t; n, ν)

Friday, September 14, 12

SLIDE 4

Derivation of RT Equations #1

Relation to Boltzmann Transport Equation

Boltzmann Transport Equation
Radiative Transfer Equation

4

∂fR ∂t + v · ∂fR ∂x + F · ∂fR ∂p = DfR Dt

coll

1 c ∂I ∂t + n · ∇I = χ (S − I) χ (x, t; n, ν) : opacity per unit length = η − χI η (x, t; n, ν) : emissivity

Friday, September 14, 12

SLIDE 5

Derivation of RT Equations #2

Classical, macroscopic, and phenomenological derivation

5

３３４

工（輿，ｔ；ｎ，・）

ＦＯＵＮＤＡＴＩＯＮＳＯＦＲＡＤＩＡＴＩＯＮＨＹＤＲＯＤＹＮＡＭＩＣＳ

工（ｘ＋△ｘ，ｔ＋△ｔ；ｎ，ｙ）皿ｇ．７６．１Ｐｅｎｃｉｌｏｆｒａｄｉａｔｉｏｎｐａｓｓｉｎｇｔｈｒｏｕｇｈａｍａｔｅｒｉａｌｅｌｅｍｅｎｔ．

Ｂｅｃａｕｓｅｓｉｓａｃｏｏｒｄｉｎａｔｅ

ｉ

ｎｄｅｐｅｎｄｅｎｔｐａｔｈｌｅｎｇｔｈ，（７６．３）ａｌ）ｐｌｉｅｓｉｎａｒｂｉｔ

Ｉ

丿ａｒｙｃｏｏｒｄｉｎａｔｅｓｙｓｔｅｍｓ，Ｉ：）１ ’ ｏｖｉｄｅｄｗｅｕｓｅａｎａｌ）１）ｒｏｐｒｉａｔｅｅｘｐｒｅｓｓｉｏｎｔｏｅｖａｌｕａｔｅ（ ∂ ／ ∂ ｓ）．Ｔｈｅｄｅｒｉｖａｔｉｏｎｊｕｓｔｇｉｖｅｎｏｆｔｈｅｔｒａｎｓｆｅｒｅｑｕａｔｉｏｎｉｓｄａｓｓｉｃａｌ，ｍａｃｒｏｓ

ｃ

ｏｐｉｃ，ａｎｄｐｈｅｎｏｍｅｎｏｌｏｇｉｃａｌｉｎｃｈａｒａｃｔｅｒ．ｌｔｏｍｉｔｓｒｅｆｅｒｅｎｃｅｔｏｓｕｃｈｉｍｐｏｒｔａｎｔｐｈｅｎｏｍｅｎａｉｌｓｐｏｌａｒｉｚａｔｉｏｎ，ｄｉｓｐｅｒｓｉｏｎ，ｃｏｈｅｒｅｎｃｅ，ｉｎｔｅｒｆｅｒｅｎｃｅ，ａｎｄｑｕａｎｔｕｍｅｆｅｃｔｓ，ｎｏｎｅｏｆｗｈｉｃｈａｒｅｃｏｒｒｅｃｔｌｙｄｅｓｃｒｉｂｅｄｂｙ（７６．３）．Ａｎｅｘｃｅｌｌｅｎｔｄｉｓｃｕｓｓｉｏｎｏｆｔｈｅａｐｐｒｏ）ｃｉｍａｔｉｏｎｓｉｎｈｅｒｅｎｔｉｎ，ａｎｄｔｈｅｖａｌｉｄｉｔｙｏｆ，ｔｈｅｃｌａｓｓｉｃａ１１ ‘ ａｄｉａｔｉｖｅｔｌ ° ａｎｓｆｅｒ（５ｑｕａｔｉｏｎｉｓｇｉｖｅｎｉｎ（Ｐ３，４７

４

９）．Ｇｏｏｄｄｉｓｃｕｓｓｉｏｎｓｏｆｔｈｅｔｒａｎｓｆｅｒｅｑｕａｔｉｏｎｆｒｏｍｔｈｅｐｏｉｎｔｏｆｖｉｅｗｏｆｑｕａｎｔｕｍｆｉｅｌｄｔｈｅｏｒｙａｒｅがｖｅｎｉｎ（ＨＩ），（Ｌ：１．），（Ｌ２ λ （Ｌ３），（０１）．Ｔｈｅｍａｔｈｅｍａｔｉｃａｌｅｘｐｒｅｓｓｉｏｎｆｏｒ（∂／∂ｓ）ｄｅｐｅｎｄｓｏｎｇｅｏｍｅｔｒｙ．ｌｎＣａｒ- ｔｅｓｉａｎｃｏｏｒｄｉｎａｔｅｓま゜郎だ十 ② 訂十（だ）だ゜ｎ， £ ＋ｎｙ；ｊ７十ｎ， £ ，（７６．４）ｗｈｅｒｅ（ｎｘ，馬。ｎｚ）ａｒｅｃｏｍｐｏｎｅｎｔｓｏｆｔｈｅｕｎｉｔｖｅｃｔｏｒｎａｌｏｎｇｔｈｅｄｉｒｅｃｔｉｏｎｏｆｐｒｏｐａｇａｔｉｏｎ．Ｔｈｅｔｒａｎｓｆｅｒｅｑｕａｔｉｏｎｉｓｔｈｅｎ（贈尋）（ ∂ ／ ∂ １）十（ｎ・ ▽ ）］ｆ（ｘ，１；ｎ，ｐ）＝・ｒｌ（ｘ，１；ｎ，ｐ）

χ

（ｘ，ｔ；ｎ，ｐ）１（ｘ，ｔ；ｎ，ｙ）。（７６．５）Ｆｏｒａｏｎｅ

ｄ

ｉｍｅｎｓｉｏｎａｌｐｌａｎａｒａｔｍｏｓｐｈｅｒｅ，（７６．５りｅｄｕｃｅｓｔｏ（嶺三）（ ∂ ／ ∂ １）＋１１．（ｉ）／ ∂ ｚ）］１（ｚ，１；１１，１・）＝７１（ｚ，１；１１，１・）

χ

（ｚ，１；１４。ｌｚ）ｆ（４１； μ ，ｌｚ）。（７６．６）ａｎｄｆｏｒｓｔａｔｉｃｍｅｄｉａｏｒｓｔｅａｄｙｎｏｗｓｔｈｅｔｉｍｅｄｅｒｉｖａｔｉｖｅｃａｎｂｅｄｒｏｐｐｅｄ，ｙｉｅｌｄｉｎｇ μ ［ ∂ １（ｚ； μ ，ｌｚ）／ ∂ ｚ］＝・ｙｌ（ｚ； μ ，ｐ）

λ

・（ｚ； μ ，ｌｊ）１（ｚ； μ ，ｐ）。（７６．７）ｌｆｔｈｅｏｐａｃｉｔｙａｎｄｅｎｌｉｓｓｉｖｉｔｙａｒｅｇｉｖｅｎ，（ ’ ７６．７）ｉｓａｎｏｒｄｉｎａｒｙｄｉｆ（５１ ’ ｅｎｔｉａｌｅｑｕａｔｉｏｎ，ｗｈｉｌｅ（７６．６）ｉｓａｐａｒｔｉａｌｄｉｆｉｅｒｅｎｔｉａｌｅ・４ｕａｔｉｏｎ．ｌｆｓｃａｔｔｅｒｉｎｇｔｅｒｍｓ

[I (x + ∆x, t + ∆t; n, ν) − I (x, t; n, ν)] dSdtdωdν = 1 c ∂I (x, t; n, ν) ∂t + ∂I (x, t; n, ν) ∂s

dsdSdtdωdν

1 c ∂I (x, t; n, ν) ∂t + ∂I (x, t; n, ν) ∂s = η (x, t; n, ν) − χ (x, t; n, ν) I (x, t; n, ν) = [η (x, t; n, ν) − χ (x, t; n, ν) I (x, t; n, ν)] dsdSdtdωdν

Friday, September 14, 12

SLIDE 6

Derivation of RT Equations #3

Schematics of RT Equation with Scattering

6

−χI n n′

Scattering Absorption Emission Scattering

x χS

1 c ∂I ∂t + n · ∇I =

χabsB + χsca
I (n′) φ (n, n′) dΩ
− (χabs + χsca) I

1 c ∂I ∂t + n · ∇I = χ (S − I)

Friday, September 14, 12

SLIDE 7

Radiative Transfer Equations

Opacity and Level populations Opacity Rate Equations (Spontaneous + Induced + Collisional rates)

7

Aij, Bij : radiative processes Cij : collisional processes dni dt = −ni

j=i
Aij + BijUνij + Cij
+
j=i

nj

Aji + BjiUνij + Cji
Uνij radiation energy density in the range between hνij = εi − εj

Φij(ν) : the spectral line profile function κ (νij) = πe2 m0c

nifijΦij (ν)

σ (ν) = πe2 m0c 2ν2

ν ν0

2

γ 2π2

(ν2 − ν2

0)2 + ν2 γ 2π

2 fs → πe2 m0c γ 4π

1

(ν − ν0)2 + (γ/4π)2

fs

Friday, September 14, 12

SLIDE 8

Line Profile Formation

Overview

Gaussian Profile
Lorentz Profile
The

Voigt Profile

8

Φ (ν) = +∞

−∞

∆νL exp

−∆ν2/∆ν2

D

d (∆ν)

2π√π∆νD

(ν − ν0 − ∆ν)2 + (∆νL/2)2

Φ (ν) = ∆νL/2π (ν − ν0)2 + (∆νL/2)2 Φ (ν) = 1 √ 2πσ exp

−(ν − ν0)2

2σ2

Friday, September 14, 12

SLIDE 9

Gaussian & Lorentz Profile

Doppler, Resonance, Collision, Natural Broadening

9

Φ (ν) = ∆νL/2π (ν − ν0)2 + (∆νL/2)2 ∆νL = (2πτ)−1 = [2π (τN + τR + τC)]−1 τN = 1/Aij τC : the mean collision time τR = 4mνmn Ne2fa ≈ νmn 6 × 107Nfa [s] 2σ2 = ∆ν2

D/ (4 ln 2)

∆νD = 2ν0 c

ln 2

2kTk M + V 2 1/2

Gaussian profile ∆νD = 2 ∆νL = 2 Lorentz profile

Frequency [1/s]

Φ (ν) = 1 √ 2πσ exp

−(ν − ν0)2

2σ2

Friday, September 14, 12

SLIDE 10

The Voigt Profile

Gaussian Profile + Lorentz Profile

10

Φ(ν) = 2 √ ln 2 √π∆νD H (a, b) a = √ ln 2∆νL/ (2∆νD) b = 2 √ ln 2 (ν − ν0) /∆νD H (a, b) = a π ∞

−∞

exp

−y2

dy (b − y)2 + a2 H (a, b) ≈ exp

−b2

+ a √πb2

Φ (ν) = +∞

−∞

∆νL exp

−∆ν2/∆ν2

D

d (∆ν)

2π√π∆νD

(ν − ν0 − ∆ν)2 + (∆νL/2)2

ν − ν0 ∆νD ∆νL ∆νD ≈ 0.001 λLyα = 1215.67 ˚ A Tgas = 4000 K A21 = 6.26 × 108 s−1

Friday, September 14, 12

SLIDE 11

Scattering Processes

Line Profile

11

ε −χI n n′

Scattering Absorption Emission Scattering

x χS

1 c ∂I ∂t + n · ∇I =

χabsB + χsca
I (n′) φ (n, n′) dΩ
− (χabs + χsca) I

Φ′(ν) Φ(ν)

Friday, September 14, 12

SLIDE 12

Timescales

Collision time .vs. Transition time

12

Collision time for neutral Hydrogen atoms
Transition time of Hydrogen atom (2p→1s)

tcoll ≈ 1012T −1/2n−1

H = 1.58 × 10−3

T

4000 [K] −1/2 nH 1013 [1/cc] −1 [sec] t2p→1s = 1 A21 = 1 6.26 × 108 [s−1] = 1.597 × 10−9 [sec]

tcoll t2p→1s

Friday, September 14, 12

SLIDE 13

Atom 1 → Atom 2
Complete Re-Distribution (CRD) in the core
Partial Re-Distribution (PRD) in the wings

Frequency Redistribution

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

13 ν1 ν2 ν2

Φ′(ν) Φ(ν)

ν1 ν2 ν2

Φ′(ν) Φ(ν) ν1 ν2

νj νj νj νj

Friday, September 14, 12

SLIDE 14

Complete Re-Distribution (CRD)
Partial Re-Distribution (PRD)

Emergent Intensity Profile

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

14

I (ν2) = ΦCRD (ν2)

j

I (νj) I (ν2) =

j

ΦPRD (νj) I (νj)

Friday, September 14, 12

SLIDE 15

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

15

Friday, September 14, 12