1 equation of transfer (mihalas chapter 2) interaction of radiation matter transfer equation formal...
DESCRIPTION
3 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 statesTRANSCRIPT
![Page 1: 1 Equation of Transfer (Mihalas Chapter 2) Interaction of Radiation Matter Transfer Equation Formal Solution Eddington-Barbier Relation: Limb Darkening](https://reader036.vdocuments.mx/reader036/viewer/2022070611/5a4d1bb57f8b9ab0599ce8bd/html5/thumbnails/1.jpg)
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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
![Page 3: 1 Equation of Transfer (Mihalas Chapter 2) Interaction of Radiation Matter Transfer Equation Formal Solution Eddington-Barbier Relation: Limb Darkening](https://reader036.vdocuments.mx/reader036/viewer/2022070611/5a4d1bb57f8b9ab0599ce8bd/html5/thumbnails/3.jpg)
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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
![Page 4: 1 Equation of Transfer (Mihalas Chapter 2) Interaction of Radiation Matter Transfer Equation Formal Solution Eddington-Barbier Relation: Limb Darkening](https://reader036.vdocuments.mx/reader036/viewer/2022070611/5a4d1bb57f8b9ab0599ce8bd/html5/thumbnails/4.jpg)
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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
![Page 5: 1 Equation of Transfer (Mihalas Chapter 2) Interaction of Radiation Matter Transfer Equation Formal Solution Eddington-Barbier Relation: Limb Darkening](https://reader036.vdocuments.mx/reader036/viewer/2022070611/5a4d1bb57f8b9ab0599ce8bd/html5/thumbnails/5.jpg)
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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
![Page 6: 1 Equation of Transfer (Mihalas Chapter 2) Interaction of Radiation Matter Transfer Equation Formal Solution Eddington-Barbier Relation: Limb Darkening](https://reader036.vdocuments.mx/reader036/viewer/2022070611/5a4d1bb57f8b9ab0599ce8bd/html5/thumbnails/6.jpg)
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Transfer Equation
• Compare energy entering and leaving element of material
![Page 7: 1 Equation of Transfer (Mihalas Chapter 2) Interaction of Radiation Matter Transfer Equation Formal Solution Eddington-Barbier Relation: Limb Darkening](https://reader036.vdocuments.mx/reader036/viewer/2022070611/5a4d1bb57f8b9ab0599ce8bd/html5/thumbnails/7.jpg)
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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
![Page 9: 1 Equation of Transfer (Mihalas Chapter 2) Interaction of Radiation Matter Transfer Equation Formal Solution Eddington-Barbier Relation: Limb Darkening](https://reader036.vdocuments.mx/reader036/viewer/2022070611/5a4d1bb57f8b9ab0599ce8bd/html5/thumbnails/9.jpg)
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Optical Depth
• Optical depth scale defined by
• Increase inwards into atmosphere
d d z
τ
z
![Page 10: 1 Equation of Transfer (Mihalas Chapter 2) Interaction of Radiation Matter Transfer Equation Formal Solution Eddington-Barbier Relation: Limb Darkening](https://reader036.vdocuments.mx/reader036/viewer/2022070611/5a4d1bb57f8b9ab0599ce8bd/html5/thumbnails/10.jpg)
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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 / /
![Page 15: 1 Equation of Transfer (Mihalas Chapter 2) Interaction of Radiation Matter Transfer Equation Formal Solution Eddington-Barbier Relation: Limb Darkening](https://reader036.vdocuments.mx/reader036/viewer/2022070611/5a4d1bb57f8b9ab0599ce8bd/html5/thumbnails/15.jpg)
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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
/
/ /
/
/ / /
/
, ,
/
/ /
![Page 16: 1 Equation of Transfer (Mihalas Chapter 2) Interaction of Radiation Matter Transfer Equation Formal Solution Eddington-Barbier Relation: Limb Darkening](https://reader036.vdocuments.mx/reader036/viewer/2022070611/5a4d1bb57f8b9ab0599ce8bd/html5/thumbnails/16.jpg)
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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
![Page 17: 1 Equation of Transfer (Mihalas Chapter 2) Interaction of Radiation Matter Transfer Equation Formal Solution Eddington-Barbier Relation: Limb Darkening](https://reader036.vdocuments.mx/reader036/viewer/2022070611/5a4d1bb57f8b9ab0599ce8bd/html5/thumbnails/17.jpg)
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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
![Page 20: 1 Equation of Transfer (Mihalas Chapter 2) Interaction of Radiation Matter Transfer Equation Formal Solution Eddington-Barbier Relation: Limb Darkening](https://reader036.vdocuments.mx/reader036/viewer/2022070611/5a4d1bb57f8b9ab0599ce8bd/html5/thumbnails/20.jpg)
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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)