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Equation of Transfer(Mihalas Chapter 2)

Interaction of Radiation & MatterTransfer EquationFormal Solution

Eddington-Barbier Relation: Limb DarkeningSimple Examples

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Scattering of Light• Photon interacts with a scattering center and

emerges in a new direction with a possibly slightly different energy

• Photon interacts with atom: e- transition from low to high to low states with emission of photon given by a redistribution function

• Photon with free electron: Thomson scattering• Photon with free charged particle:

Compton scattering (high energy)• Resonance with bound atom: Rayleigh scattering

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Absorption of Light• Photon captured, energy goes into gas thermal energy• Photoionization (bound-free absorption) (or inverse:

radiative recombination): related to thermal velocity distribution of gas

• Photon absorbed by free e- moving in the field of an ion (free-free absorption) (or inverse: bremsstrahlung)

• Photoexcitation (bound-bound absorption) followed by collisional energy loss (collisional de-excitation) (or inverse)

• Photoexcitation, subsequent collisional ionization• Not always clear: need statistical rates for all states

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Extinction Coefficent

• Opacity Χ• Amount of energy absorbed from beam

E = Χ I dS ds dω dν dt• Χ = (absorption cross section)(#density)

cm2 cm-3

• 1/χ is photon mean free path (cm)• Χ = κ+σ, absorption + scattering parts

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Emission Coefficient

• Emissivity η• Radiant energy added to the beam

E = η dS ds dω dν dt(units of erg cm-3 sr-1 Hz-1 sec-1)

• In local thermodynamic equilibrium (LTE) energy emitted = energy absorbedηt = κ I = κν Bν Kirchhoff-Planck relation

• Scattering part of emission coefficient isηs = σ J

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Transfer Equation

• Compare energy entering and leaving element of material

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Transfer Equation

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Transfer Equation

• Usually assume:(1) time independent

(2) 1-D atmosphere

• Transfer equation:

It

0

dz ds ds cos

Iz

I

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Optical Depth

• Optical depth scale defined by

• Increase inwards into atmosphere

d d z

τ

z

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Source Function

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Boundary Conditions

• Finite slab: specifyI- from outer spaceI+ from lower level

• Semi-infinite: I- = 0

lim /

I e 0

0T

0

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Formal Solution

• Use integrating factor• Thus

• In TE:

e /

( )/ / //

I e eI I

ee I

I

II S I e S e / /1

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Integrate for Formal Solution

I e I e S e d t

I I e S e d t

I I e S e d t

t

t

1 2

1 2

1 2

1 2

2

1

2 1

2

1

1

2 1

1

2

1

1

1

1

/ / /

/ /

/ /

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Formal Solution: Semi-infinite case

• Emergent intensity at τ = 0 is

• Intensity is Laplace transform of source function

1 2

0

0

I S t e d tt / /

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Eddington-Barbier Relation

S S S

I S S t e d t

f g f g f g

f e f e g t

e d t e dx e

t e d t t e e d t

I S S

t

t t

t x x

t t t

0 1

0 10

0 0 0

00

0 1

1

/

/ /

/

/ / /

/

, ,

/

/ /

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Application: Stellar Limb Darkening

• Limb darkening of Sun and stars shows how S varies from τ = 0 to 1, and thus how T(τ) varies (since S ≈ B(T), Planck function)

μ=1 at center; μ=0 at edge

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Simple Examples (Rutten p. 16)

• Suppose S = constant, τ1 = 0, μ = 1 and find radiation leaving gas slab

• Small optical depth• Large optical depth

I I e S e d t

I I e S e

t

1 20

2

0 11

1

2

2

2 2

, /

I I S 2 2 21I S

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Simple Examples (Rutten p. 16)

Optically thick case

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Simple Examples (Rutten p. 16)

Optically thin case

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Simple Examples (Rutten p. 16)

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Simple Examples (Rutten p. 16)

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Simple Examples (Rutten p. 16)

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Simple Examples (Rutten p. 16)