Lecture Notes on Radiation Transport for Spectroscopy• Summed over populations and radiative transitions: • In NLTE, the source function has a complex dependence on plasma parameters

LLNL-PRES-668311 This work was performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344. Lawrence Livermore National Security, LLC

Lecture Notes on Radiation Transport for Spectroscopy

ICTP-IAEA School on Atomic Processes in Plasmas

27 February – 3 March 2017 Trieste, Italy

Outline •  Definitions, assumptions and terminology

•  Equilibrium limit •  Radiation transport equation

•  Characteristic form & formal solution •  Material radiative properties

•  Absorption, emission, scattering •  Coupled systems

•  LTE / non-LTE •  Line radiation

•  Line shapes •  Redistribution •  Material motion

•  Solution methods •  Transport operators •  2-level & multi-level atoms •  Escape factors

Radiation Transport

Basic assumptions

Classical / Semi-classical description –

•  Radiation field described by either specific intensity Iν or the photon distribution function fν

•  Unpolarized radiation

•  Neglect index of refraction effects (n≈1, ω>>ωp) è Photons travel in straight lines

•  Neglect true scattering (mostly)

•  Static material (for now) è Single reference frame

Macroscopic - specific intensity Iν •  energy per (area x solid angle x time) within

the frequency range

•  dE = energy crossing area dA within

Microscopic - photon distribution function

Radiation Description

dE = Iv

r ,Ω,t( ) n • Ω( )dAdΩdv dt

dE = hv( )

spins∑ fν

!r , !p,t( ) d 3!xd 3 !ph3

Iv = 2 hv3

c2 fν!r , !p,t( )

(ν ,ν + dν )

(dΩdv dt)

p = hv

cΩ

0th moment = energy density x c/4π

1st moment = flux x 1 /4π

2nd moment = pressure tensor x c /4π

For isotropic radiation, Kν is diagonal with equal elements:

In this case, radiation looks like an ideal gas with = 4/3

Jν =

14π

Iν dΩ∫

Hν =

14π

nIν dΩ∫

Kν =

14π

nnIν dΩ∫

Kν =

13

Jν I (P = 13

E)

Angular Moments

γ

Thermal Equilibrium

Intensity: Planck function Distribution function: Bose-Einstein Energy density For Te=Tr and nH = 1023 cm-3 :

Bv = 2 hv3

c2

1

ehv

kTr −1 fν = 1

ehv

kTr −1

Erad Tr( ) = 4π

cBv Tr( )

0

∞

∫ dv = aTr4

Erad = Ematter ⇔ T ≈ 300eV

LTE(LocalThermodynamicEquilibrium)-par8cleshavethermaldistribu8ons(Te,Ti)photondistribu8oncanbearbitrary

•  Boltzmann equation for the photon distribution function:

•  The LHS describes the flow of radiation in phase space •  The RHS describes absorption and emission

•  Absorption & emission coefficients depend on atomic physics •  Photon # is not conserved (except for scattering)

•  Photon mean free path

Radiation Transport Equation

= absorption coefficient (fraction of energy absorbed per unit length)

= emissivity (energy emitted per unit time, volume, frequency, solid angle)

1c∂ Iv

∂ t+!Ω i∇Iv = −α v Iv +ηv

1c∂ fν∂ t

+!Ω i∇fν =

∂ fν∂ t

⎛⎝⎜

⎞⎠⎟ coll

α v

ηv

Iv = 2hν hν

c⎛⎝⎜

⎞⎠⎟

2

fν

λν = 1/αν

Characteristic Form

Define the source function Sν and optical depth τν: Sν = ην/αν = Bν in LTE dτν = αν ds Along a characteristic, the radiation transport equation becomes

This solution is useful when material properties are fixed, e.g. postprocessing for diagnostics Important features: •  Explicit non-local relationship between Iν and Sν •  Escaping radiation comes from depth τν ≈ 1 •  Implicit Sν(Iν) dependence comes from radiation / material coupling

dIv

dτ v

= − Iv + Sv ⇒ Iv τ v( ) = Iv 0( )e−τ v + e− τ v− ′τν( )Sv ′τν( )0

τ v∫ d ′τν

Self-consistently determining Sν and Iν is the hard part of radiation transport

Limiting Cases

Optically-thin τν<<1 (viewed in emission):

Iν reflects spectral characteristics of emission, independent of absorption Optically-thick τν>>1:

Iν reflects spectral characteristics of Sν (over ~last optical depth) Negligible emission (viewed in absorption):

Iν / Iν(0) reflects spectral characteristics of absorption

Iv = Sv0

τ v∫ d ′τν = ηv0

dx

∫ d ′x →ηvdx

Iv τ v( ) = e− τ v− ′τν( )Sv ′τν( )

0

τ v∫ d ′τν → Sv τν( )

Iv τ v( ) = Iv 0( )e−τ v , τν = α v0

dx

∫ d ′x →α vdx

10-4

10-3

10-2

10-1

100

optic

al d

epth

2000150010005000photon energy (eV)

2500

2000

1500

1000

500

0

flux

(cgs

)


Example – Flux from a uniform sphere

Kr @ T = 200 eV, ρ = 0.01 g/cc, LTE

Δr=0.001cm

2500

2000

1500

1000

500

0

flux

(cgs

)


0.01

0.1

1

10

100

optic

al d

epth



Kr @ T = 200 eV, ρ = 0.01 g/cc, LTE

Δr=0.1cm

2500

2000

1500

1000

500

0

flux

(cgs

)


100

101

102

103

104

optic

al d

epth



Kr @ T = 200 eV, ρ = 0.01 g/cc, LTE

Δr=10.0cm

Absorption / emission coefficients !!"#$%#&&!'($)*$&&+),)$)-.&

Plasma Physics Division

The simulated and experimental K-shell spectra reveal plasma conditions in the emitting region are Te~3.5 keV and ne ~ 3x1022 cm-3.

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Macroscopic description – energy changes •  Energy removed from radiation passing through material of area dA,

depth ds, over time dt

•  Energy emitted by material

Microscopic description – radiative transitions •  Absorption and emission coefficients are

constructed from atomic populations yi and cross sections σij:

dE = −αν Iν dAdsdΩdν dt

dE = ην dAdsdΩdν dt

αν = σν , iji< j∑ (yi −

gigjyj ) , ην =

2hν 3

c2 σν , iji< j∑ gi

gjyj

Bound-bound Transitions

Probability (per unit time) of •  Spontaneous emission: A21 •  Absorption: B12 •  Stimulated emission: B21

A21, B12, B21 are Einstein coefficients

2 g2

1

n2

g1 n1

Two energy level system

g1B21 = g2B12 , A21 =2hv0

3

c2B21

J

φ(ν )

0

∞

∫ dν =1

Line profile measures probability of absorption φ v⎛⎝ ⎞

⎠

Transition rate from level 1 to level 2

R12 = B12J , J = Jv0

∞

∫ φ v( )dv

ΔE = hv0

(assuming that the linewidth << ΔE)

J

αν = n1

πe2

mcf12φ v( ) 1−

g1n2

g2n1

⎡

⎣⎢

⎤

⎦⎥

ην =2hν 3

c2

⎛⎝⎜

⎞⎠⎟

n2

πe2

mcf12φ v( )

Absorption and emission coefficients:

absorp8on–s8mulatedemissionspontaneousemission

σ v( ) = hvo

4πB21φ v( ) = πe2

mcf12φ v( )

Cross section for absorption

Oscillator strength f12 relates the quantum mechanical result to the classical treatment of a harmonic oscillator

•  Strong transitions have f ~1

•  Sum rule (# of bound electrons) fij = Zj∑

For now we are assuming that absorption and emission have the same line profile

f12 = oscillator strength

πe2

mc=0.02654 cm2 /s

Bound-free absorption

Free-free absorption

σ bf = 7.91⋅10−18 ngbf

ν0

ν⎛⎝⎜

⎞⎠⎟

3

1− e−hν

kTe⎛⎝⎜

⎞⎠⎟

cm2

Absorption cross section from state of principal quantum number n and charge Z

Absorption cross section per ion of charge Z

Gauntfactor hv0 = thresholdenergy

σ ff = 3.69 ⋅108 Z 2

ν 3 Te

g ff ne 1− e−hν

kTe⎛⎝⎜

⎞⎠⎟

cm2

The term accounts for stimulated emission 1− e−hν

kTe

Gauntfactor

Scattering

Interaction in which the photon energy is (mostly) conserved (i.e. not converted to kinetic energy)

Examples: Scattering by bound electrons – Rayleigh scattering - important in atmospheric radiation transport Scattering by free electrons – Thomson / Compton scattering

Note: frequency shift from scattering is ~

Doppler shift from electron velocity is ~ For most laboratory plasmas, these types of scattering are negligible Note: X-ray Thomson scattering (off ion acoustic waves and plasma oscillations) can be a powerful diagnostic for multiple plasma parameters (Te, Ti, ne)

σ T =

8π3

e2

mec2

⎛⎝⎜

⎞⎠⎟

2

= 6.65x10−25 cm2 hν mec2

hν /mec2

2kT /mec2

Radiation transport equation with scattering (and frequency changes):

The redistribution function R describes the scattering of photons (ν,Ω) è (ν’,Ω’)

Neglecting frequency changes, this simplifies to Scattering contributes to both absorption and emission terms (and may be

denoted separately or included in αν and ην)

1c∂ Iv

∂ t+Ω i∇Iv = −(α v +σ v )Iv +ηv +σ v

d ′Ω4π∫ Iv ( ′

Ω )g

Ω i ′Ω( )

→−(α v +σ v )Iv +ηv +σ v Jv

1c∂ Iv

∂ t+Ω i∇Iv = −α v Iv +ηv +

σ v d ′ν d ′Ω4π

−R( ′ν , ′Ω ;ν ,Ω) ν′ν

Iv (1+ ′fν )+ R(ν ,Ω; ′ν , ′Ω )I ′v (1+ fν )⎡

⎣⎢

⎤

⎦⎥∫

0

∞

∫s8mulatedscaGering

forisotropicscaGering

Effective scattering

Photons also “scatter” by e.g. resonant absorption / emission Upper level 2 can decay a) radiatively A21 b) collisionally neC21 The fraction of photons are destroyed / thermalized The fraction (1-ε) of photons are “scattered”

è energy changes only slightly (mostly Doppler shifts) è undergo many “scatterings” before being thermalized

Note: is the condition for a strongly non-LTE transition

and is easily satisfied for low density or high ΔE !

2 g2

1

n2

g1 n1

Two energy level system

ε ≈ neC21 / A21

ε 1

Population distribution

LTE: Saha-Boltzmann equation

•  Excited states follow a Boltzmann distribution

•  Ionization stages obey the Saha equation

NLTE: Collisional-radiative model •  Calculate populations by integrating rate equations

yiyj

= gigje−(εi−ε j )/Te

Nq

Nq+1

= 12neUq

Uq+1

h2

2πmeTe

⎛⎝⎜

⎞⎠⎟

3/2

e−(ε0q+1−ε0

q )/Te

ε i = energyof state iTe = electron temperature

Nq = yii∈q∑ e−(εi−ε0

q )/Te number densityof chargestateq

Uq = gii∈q∑ e−(εi−ε0

q )/Te partition functionof chargestateq

dydt

= Ay Aij = Cij + Rij + (other)ij

Rij = σ ij (ν )J(ν ) dνhν

, Rji =∫ σ ij (ν ) J(ν )+ 2hν 3

c2

⎡

⎣⎢

⎤

⎦⎥ e

−hν kT∫dνhν

Cij = ne vσ ij (v) f (v)dv∫

•  Summed over populations and radiative transitions:

•  In NLTE, the source function has a complex dependence on plasma parameters and on the radiation spectrum:

•  In LTE, absorption and emission spectra are complex but the source function obeys Kirchoff’s law:

Summary of absorption / emission coefficients

Sν = Sν (ne ,Te; yi(Jν ))

αν = yiπe2

mc2fij φij v( ) 1− gi y jg j yi

⎛

⎝⎜

⎞

⎠⎟ + σ ij

bf (ν ) + neσ ijff (ν )⎡⎣ ⎤⎦ (1− e

−hν /kTe )⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪ij∑

ην =2hν 3

c2y j

πe2

mc2gig jfij φij v( )+ σ ij

bf (ν ) + neσ ijff (ν )⎡⎣ ⎤⎦e

−hν /kTe⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪ij∑

Sν = Bν (Te )

Opacity & mean opacities

opacity = absorption coefficient / mass density

κν =αν ρ

1κ R

=dν

0

∞

∫ 1κν

dBν

dT

dν0

∞

∫ dBν

dT

κ P =dν

0

∞

∫ κνBν

dν0

∞

∫ Bν

Rosseland mean opacity: emphasizes transmission includes scattering appropriate for average flux

Planck mean opacity:

emphasizes absorption no scattering appropriate for energy exchange

0.1

1

10

100

1000

abso

rptio

n co

effic

ient

(1/

cm)

300025002000150010005000photon energy (eV)

dB/dT

Kr @ T = 200 eV, ρ = 0.01 g/cc, LTE

10-12

10-11

10-10

10-9

10-8

10-7

10-6

10-5

10-4

abso

rptio

n co

effic

ient

(cm

-1)

0.1 1 10 100photon energy (eV)

total bound-free free-free n=1

n=2n=3

Example – Hydrogen (Te= 2 eV, ne=1014 cm-3)

absorption coefficient

10-11

10-10

10-9

10-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

abso

rptio

n co

effic

ient

(cm

-1)


LTE NLTE

Hydrogen, again (Te= 2 eV, ne=1014 cm-3)

emissivity sourcefunc8on

10-1

100

101

102

103

104

105

106

107

108

emiss

ivity

(er

g/cm

3 /s/e

V/st

er)


6x1011

5

4

3

2

1

0

sour

ce fu

nctio

n (e

rg/c

m2 /s

/eV/

ster

)


LTE NLTE

10-4

10-3

10-2

10-1

100

optic

al d

epth


2500

2000

1500

1000

500

0

flux

(cgs

)


LTE NLTE

Flux from a uniform sphere revisited

Kr @ T = 200 eV, ρ = 0.01 g/cc, LTE / NLTE

Δr=0.001cm

2500

2000

1500

1000

500

0

flux

(cgs

)


LTE NLTE

0.01

0.1

1

10

100

optic

al d

epth




Δr=0.1cm

100

101

102

103

104

optic

al d

epth


2500

2000

1500

1000

500

0

flux

(cgs

)


LTE NLTE



Δr=10.0cm

Flux from a uniform sphere - Summary

200

150

100

50

0T r

, eff

1.00.80.60.40.20.0r/∆r

LTE NLTE

∆r = 10. cm

∆r = 0.1 cm

∆r = 0.001 cm

§  “Black-body” emission requires large optical depths

§  Large optical depths è high radiation fields èLTE conditions

§  Boundaries introduce non-uniformities through radiation fields

§  The radiation field will also change the material temperature

Conditions do not remain uniform in the

presence of radiation transport and boundaries

1.0x10-9

0.8

0.6

0.4

0.2

0.0

spec

ific

inte

nsity

(cg

s)

-0.002 -0.001 0.000 0.001 0.002photon energy w.r.t. line center (eV)

90º 10º

10-5

10-4

10-3

10-2

10-1

100

101

opti

cal depth

10.20210.20010.19810.196photon energy (eV)

Ly-α emission from a uniform plasma •  Te = 1 eV, ne = 1014 cm-3 •  Moderate optical depth τ ~ 5 •  Viewing angles 90o and 10o

show optical depth broadening

6

5

4

3

2

1

0

opti

cal depth

-0.002 -0.001 0.000 0.001 0.002photon energy w.r.t. line center (eV)

90o10o

1 cm

Example – Hydrogen Ly-α

3.0x10-8

2.5

2.0

1.5

1.0

0.5

0.0

n=2

frac

tiona

l pop

ulat

ion

1.00.80.60.40.20.0

position (cm)

self-consistent uniform

6x10-9

5

4

3

2

1

0

spec

ific

inte

nsity

(cg

s)

-0.002 -0.001 0.000 0.001 0.002

photon energy w.r.t. line center (eV)

self-consistent 90º 10º

uniform

90º 10º

Example – Hydrogen Ly-α Ly-α emission from a plasma with uniform temperature and density

•  Te= 1 eV, ne= 1014 cm-3 •  Self-consistent solution displays effects of

–  Radiation trapping / pumping –  Non-uniformity due to boundaries

90o10o

1 cm

Coupled systems – or – What does “Radiation Transport” mean?

The system of equations and emphasis varies with the application

For laboratory plasmas, these two sets are most useful -

LTE / energy transport : •  Coupled to energy balance

•  Indirect radiation-material coupling through energy/temperature

•  Collisions couple all frequencies locally, independent of Jν

•  Solution methods concentrate on non-local aspects

dEm

dt= 4π αν (Jν − Sν )∫ dν

αν =αν (Te ) , Sν = Bν (Te )

NLTE / spectroscopy : •  Coupled to rate equations

•  Direct coupling of radiation to material •  Collisions couple frequencies over

narrow band (line profiles) •  Solution methods concentrate on local

material-radiation coupling •  Non-local aspects are less critical

dydt

= Ay , A ij =A ij (Te , Jν )

Sν =2hν3

c2 Sij , Sij ≈ a + b Jij

Line Profiles

Line profiles are determined by multiple effects: •  Natural broadening (A12) - Lorentzian •  Collisional broadening (ne, Te) - Lorentzian •  Doppler broadening (Ti) - Gaussian •  Stark effect (plasma microfields) - complex

-5.0 -3.0 -1.0 1.0 3.0 5.0

φ, li

ne p

rofil

e fa

ctor

Gaussian, Voigt and Lorentzian Profiles

Gaussian

Lorentzian

Voigt, a=0.001Voigt, a=0.01Voigt, a=0.1

(ν-νo)/∆

10

10−1

10−2

10−3

10−4

10−5

φ(ν ) = 1ΔνD π

H (a, x)

a = Γ4πΔνD

ΔνD = ν0c

2kTimi

H (a, x) = aπ

e− y2

(x − y)2 + a2−∞

∞

∫ dy

φ(ν ) = Γ 4π 2

ν −ν0( )2 + Γ 4π( )2

Γ =destruc8onrate

Redistribution

The emission profile is determined by multiple effects: •  collisional excitation è natural line profile (Lorentzian) •  photo excitation + coherent scattering •  photo excitation + elastic scattering è ~absorption profile •  Doppler broadening This is described by the redistribution function

Complete redistribution (CRD): Doppler broadening is only slightly different from CRD, while coherent scattering gives A good approximation for partial redistribution (PRD) is often where f (<<1 for X-rays) is the ratio of elastic scattering and de-excitation rates, includes coherent scattering and Doppler broadening

ψν = φν

ψν

R(ν , ′ν )

R(ν , ′ν )

0

∞

∫ dv = φ( ′ν ) , ψ (ν ) = R(ν , ′ν )0

∞

∫ J ( ′ν )d ′v φ( ′ν )0

∞

∫ J ( ′ν )d ′v

R(ν , ′ν ) = φ(ν )δ (ν − ′ν )

R(ν , ′ν ) = (1− f )φ( ′ν )φ(ν )+ f RII (ν , ′ν )

RII

6x10-9

5

4

3

2

1

0

spec

ific

inte

nsity

(cg

s)

-0.002 -0.001 0.000 0.001 0.002


PRD 90º 10º

CRD

90º 10º

uniform

90º 10º

3.5x10-8

3.0

2.5

2.0

1.5

1.0

0.5

0.0

n=2

frac

tiona

l pop

ulat

ion

1.00.80.60.40.20.0

position (cm)

PRD CRD uniform

Hydrogen Ly-α w/ Partial Redistribution

90o10o

1 cm

Ly-α emission from a plasma with uniform temperature and density •  Te= 1 eV, ne= 1014 cm-3 •  Optical depth τ ~ 5 •  Voigt parameter a ~ 0.0003

Material Velocity

The discussion so far applies in the reference frame of the material

Doppler shifts matter for line radiation when v/c ~ ΔE/E

If velocity gradients are present, either a)  Transform the RTE into the co-moving frame - or – b)  Transform material properties into the laboratory frame

Option (a) is complicated (particularly when v/c è 1) - see the references by Castor and Mihalas for discussions

Option (b) is relatively simple, but makes the absorption and emission coefficients direction-dependent

Al sphere w/ uniform expansion velocity

10-5

10-4

10-3

10-2

10-1

100

101

102

optic

al d

epth

240022002000180016001400photon energy (eV)

0.1

1

10

100

area

l flux

(er

g/s/

Hz/s

ter)

240022002000180016001400photon energy (eV)

v/c = 0 v/c = 0.003 v/c = 0.01 v/c = 0.03

optical depth areal flux

Te=500eVρ=10-3g/cm3Router=1.0cmRinner=0.9cm

0.1

1

10

100

area

l flux

(er

g/s/

Hz/s

ter)


v/c = 0 v/c = 0.003 v/c = 0.01 v/c = 0.03

Al sphere w/ uniform expansion velocity

He-α Ly-α

2-Level Atom

Rate equation for two levels in steady state: Absorption / Emission: Source Function:

2 g2

1

n2

g1 n1

Two energy level systemn1(B12J12 +C12 ) = n2 (A21 + B21J12 +C21)

J12 = Jv0

∞

∫ φ12 v( )dv , C12 =g2g1

e−hν0 kTC21ΔE = hv0

αν =hν4π

n1B12 − n2B21( )φ12 (ν ) , ην =hν4π

n2A21φ12 (ν )

Sν =n2A21

n1B12 − n2B21= (1− ε )J12 + εBν , ε

1− ε=C21

A211− e−hν0 kT( )

Sν is independent of frequency and linear in - solution methods exploit this dependence

J

A Popular Solution Technique

For a single line, the system of equations can be expressed as where the lambda operator and the source function depend on the populations through the absorption and emission coefficients. A numerical solution for uniquely specifies the entire system. Since depends on through the populations, the full system is non-linear and requires iterating solutions of the rate equations and radiation transport equation. The dependence of on is usually weak, so convergence is rapid.

dIv

dτ v

= − Iv + Sv ⇒ Iv =λνSv (J )

Sv = aν + bν J , J = 14π

dΩ∫ Iv0

∞

∫ φ v( )dv

λν

J

J

J

λν

λν

Solution technique for the linear source function: This linear system can (in principle) be solved directly for and in 1D this is very efficient The (angle,frequency)-integrated operator includes factors, so amplifies (radiation trapping) Efficient solution methods approximate key parts of

NLTE – local frequency scattering LTE (linear in ΔTe) – non-local coupling

Iv =λν aν + bν J⎡⎣ ⎤⎦

⇒ J = 14π

dΩ∫λν aν + bν J⎡⎣ ⎤⎦0

∞

∫ φ v( )dv

⇒ J = 1−Λ⎡⎣ ⎤⎦

−1 14π

dΩ∫λνaν0

∞

∫ φ v( )dv

Λ = 1

4πdΩ∫

λνbν0

∞

∫ φ v( )dv

J

J Λ

1−Λ⎡⎣ ⎤⎦

−11− e−τν

Λ

For multiple lines, the source function for each line can be put in the two-level form – ETLA (Extended Two Level Atom) – or the full source function can be expressed in the following manner: Solve for each individually (if coupling between lines is not important) or simultaneously. (Complete) linearization – expand in terms of and solve as before. Partial redistribution usually converges at a slightly slower rate.

J

Sv =ην

c + ηνℓ(Jℓ )φν

ℓ

ℓ∑

ανc + αν

ℓ(Jℓ )φνℓ

ℓ∑

Multi-Level Atom

Sν ΔJℓ

Sν =Sν (Jℓ

0 )+ ∂Sν∂Jℓ

Jℓ − Jℓ0( )

ℓ∑

Numerical methods need to fulfill 2 requirements

1.  Accurate formal transport solution which is •  conservative, •  non-negative •  2nd order (spatial) accuracy (diffusion limit as τ >> 1) •  causal (+ efficient) Many options are available – each has advantages and disadvantages

2. Method to converge solution of coupled implicit equations •  Multiple methods fall into a few classes

–  Full nonlinear system solution –  Accelerated transport solution –  Incorporate transport information into other physics

•  Optimized methods are available for specific regimes, but no single method works well across all regimes

Method #1 – Source (or lambda) iteration

Advantages – Simple to implement Independent of formal transport method

Disadvantages –

Can require many iterations: # iterations ~ τ2 False convergence is a problem for τ >> 1

1.  Evaluate source function 2.  Formal solution of radiation

transport equation 3.  Use intensities to evaluate

temperature / populations

iteratetoconvergence

Hydrogen Lyman-α revisited •  Source iteration (green curves) approaches self-consistent solution slowly •  Linearization achieves convergence in 1 iteration

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3.0x10-8

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Method #2 – Monte Carlo Formal solution method – 1)  Sample emission distribution in (space, frequency, direction) to create “photons” 2)  Track “photons” until they escape or are destroyed

Advantages – Works well for complicated geometries Not overly constrained by discretizations è does details very well Disadvantages – Statistical noise improves slowly with # of particles Expense increases with optical depth Iterative evaluation of coupled system is not possible / advisable Semi-implicit nature requires careful timestep control Convergence – A procedure that transforms absorption/emission events into effective scatterings (Implicit Monte Carlo) provides a semi-implicit solution Symbolic IMC provides a fully-implicit solution at the cost of a solving a single mesh-wide nonlinear equation

Method #3 – Discrete Ordinates (SN) Formal solution method –

Discretization in angle converts integro-differential equations into a set of coupled differential equations (provides lambda operator)

Advantages – Handles regions with τ<<1 and τ>>1 equally well Modern spatial discretizations achieve the diffusion limit Deterministic method can be iterated to convergence Disadvantages – Ray effects due to preferred directions angular profiles become inaccurate well before angular integrals Required # of angles in 2D/3D can become enormous Discretization in 7 dimensions requires large computational resources Convergence –

Effective solution algorithms exist for both LTE and NLTE versions [6] – e.g. LTE – synthetic grey transport (or diffusion)

NLTE – complete linearization (provides linear source function) + accelerated lambda iteration in 2D/3D (Note: this applies to all deterministic methods)

Method #4 – Escape factors Escape factor pe is used to eliminate radiation field from net radiative rate Equivalent to incorporating a (partial) lambda operator into the rate equations è combines the formal solution + convergence method Advantages – Very fast – no transport equation solution required Can be combined with other physics with no (or minimal) changes Disadvantages – Details of transport solution are absent Escape factors depend on line profiles, system geometry Iterative improvement is possible, but usually not worthwhile

yjBjiJ − yiBij J = yjAji pe

Escape factor is built off the single flight escape probability Iron’s theorem – this gives the correct rate on average

(spatial average weighted by emission) Note that the optical depth depends on the line profile, plus continuum Asymptotic expressions for large optical depth:

Gaussian profile Voigt profile Evaluating pe can be complicated by overlapping lines, Doppler shifts, etc. Many variations and extensions exist in a large literature

pe =

14π

dΩ∫ e−τν0

∞

∫ φ v( )dv

1− J S = pe

e−τν

pe ≈1 4τ ln τ π( ) pe ≈ 13 a τ

References

Stellar Atmospheres, 2nd edition, D. Mihalas, Freeman, 1978. Theory of Stellar Atmospheres: An Introduction to Astrophysical Non-

equilibrium Quantitative Spectroscopic Analysis, I. Hubeny and D. Mihalas, Princeton, 2014.

Radiation Hydrodynamics, J. Castor, Cambridge University Press, 2004. Kinetics Theory of Particles and Photons (Theoretical Foundations of Non-

LTE Plasma Spectroscopy), J. Oxenius, Springer-Verlag, 1986. Radiation Trapping in Atomic Vapours, A.F. Molisch and B.P. Oehry, Oxford

University Press, 1998. Radiation Processes in Plasmas, G. Bekefi, John Wiley and Sons, 1966. Computational Methods of Neutron Transport, E.E. Lewis and W.F. Miller, Jr.,

John Wiley and Sons, 1984.

Documents

Lecture Notes on Radiation Transport for Spectroscopy• Summed over populations and radiative transitions: • In NLTE, the source function has a complex dependence on plasma parameters