The Radiative Transfer Equationgriffith/PTYS517/RT1-equation.pdf · The radiative transfer equation...

TheRadiativeTransferEquation

•  Assumethataplanetradiateslikeablackbodyataconstanttemperature,Teff,wherethesolarenergyabsorbedbytheplanetequalsthatemi;ed.

•  Thistemperatureiscalledtheplanet’seffec>vetemperature,Teff

•  Howwouldyouderiveit?

Effectivetemperatureofaplanet

SolidAngle:Ratioofanareaonaspheredividedbytheradiussquared

Unitlessunit:steradian

Thisis1steradian

SolidAngle:De>inition

Largestsolidangle:

~12.6

•  Definition: solid angle

Theradiativetransferequation&somesimplesolutions

Unit: steradian

SolidanglesonEarth

•  Intensity

Groupsofphotons

ThepointPissurroundedbyasmallelementofareadπs,perpendiculartothedirec>onoftheunitvectors.Fromeachpointondπsaconeofsolidangledωisdrawnaboutthesvector.Thebundleofrays,origina>ngondπs,andcontainedwithindω,transportsin>medtandinthewavelengthrangeλtoλ+dλ,theenergy

Here,dΩ=dω=solidangle

Whatifyouwantameasureoftheinsolationthatheatsthesurfaceofaplanet?

•  Intensity

Groupsofphotons

WhatistheSun’sintensity?Whatisitsflux?

Whatifyouwantedtoknowtheflux?Thatisthetotalnumberofphotonsabsorbedperunitsurfaceareapersecondperunitwavelength?

•  Intensity

•  Flux

Groupsofphotons

Direc>onofintensitybeamDirec>onofFlux

WhatistheSun’sintensity?Whatisitsflux?

=cos(θ)θ

Intensity:

MeanIntensityoverallangles:

MeanIntensity:

•  Photonmovesadistancedsina>medt=ds/c

•  Thends=cdt

•  Since:

• 

•  Writeenergytransportas:

•  Edensity/steradian:

•  Energydensityisthen

Morede>initions

dsA

volume

Where:

Absorption,scattering&emissionDefineabsorp>oncoefficientofaabsorp>vegas(sayCH4):overasmallwavelengthregionasThiscoefficientexpressesthedecreaseinintensityperunitareaoveradistancedswherethemolecularnumberdensityofCH4is

Writetheexpressionforthedecreaseinintensityintermsoftheintensity,thedensityandthedistance.

dsρ

CH4

Absorption,scattering&emissionDefineabsorp>oncoefficientofaabsorp>vegas(sayCH4):overasmallwavelengthregionaspermoleculeThiscoefficientexpressesthedecreaseinintensityperunitareaoveradistancedswherethemolecularnumberdensityofCH4is

Writetheexpressionforthedecreaseinintensityintermsoftheintensity,thedensityandthedistance

dsρ

CH4

•  Ex>nc>on:aninterac>onbetweenlight&ma;erthatdecreasestheintensity

•  Emissionaddstointensity:

•  Equa>onofTransfer:

Absorption,scattering&emission

Canbeduetolightsca;eredbackintobeam!

•  Equa>onofTransfer:

•  Defini>onofSourceFunc>on:

•  Op>calDepth:

•  SimplifiedEqua>on:

TheSourceFunction

ρdsisthecolumnabundance

IntegratedColumnAbundances

•  Equa>onofTransfer:

GeneralSolution

Howwouldyousolvethisequa>on,i.e.gettheintensitytermsononesideoftheequa>on?

•  Equa>onofTransfer:

•  Mul>plybothsidesbyeτ

•  GeneralSolu>on:

GeneralSolution

Whatdoesthismean?

FoundSimplestSolution?

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FoundSimplestSolution

NoSourcefunc>on

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Whatisitcalled?

Essai d’optique sur la gradation de la lumière (1729)

August Beer 1825-1863

Johann Lambert (1728-1777)

•  e-τ(s)αprobabilitythatthephotonmakesittoadistances.•  Normalizetosumofalltheprobabili>esthatitgoesanydistancetogettheprobabilitythatthepar>cleisabsorbedbetweens&s+ds:

Meanfreepathofaphoton

Theexpected(ormean)valueofsisthen

Meanfreepath:averagedistancebeforebeingabsorbedorsca;ered

Bouguer’slawisthen:I=I0e-s/<s>

Beer’slaworLambert’sLaw,butreallyBouguer’slaw…

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Beer’slaworLambert’sLaw,butreallyBouguer’slaw…

26

De>initionofTransmission&Absorption

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Transmission: T = e-τ

Absorption: A = 1 - e-τ

•  observedspectralradiance(intensity)dividedbythesolarirradiance(flux).

De>initionofI/F

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ApplyBeer’slawtoanatmosphere

ρ1 τ1

ρ2 τ2

ρ3 τ3

ρ4 τ4

ρ5 τ5

ρ6 τ6

ρ7 τ7

IλI1λ

I1λ = Iλe-τ

Airmass

Airmass=1/cos(θ)

Whereθistheanglefromzenith

PercentageAbsorption

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UseBeer’slawtocalculatethetransmissionThroughEarth’satmosphere

TransmissionofEarth’satmosphere

Photometricbands(optical)

H2O&OHlines

TransmissionofEarth’satmosphere

Photometricbands(IR)

TransmissionthoughEarth’sAtmosphere:UVtomicrowave