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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?
21
FoundSimplestSolution
NoSourcefunc>on
22
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…
25
Beer’slaworLambert’sLaw,butreallyBouguer’slaw…
26
De>initionofTransmission&Absorption
27
Transmission: T = e-τ
Absorption: A = 1 - e-τ
• observedspectralradiance(intensity)dividedbythesolarirradiance(flux).
De>initionofI/F
29
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
31
UseBeer’slawtocalculatethetransmissionThroughEarth’satmosphere
TransmissionofEarth’satmosphere
Photometricbands(optical)
H2O&OHlines
TransmissionofEarth’satmosphere
Photometricbands(IR)
TransmissionthoughEarth’sAtmosphere:UVtomicrowave