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Physicsofplanetaryatmospheres
Lecture2:Introductionto
RadiativeEnergyTransportPartI
EquationofRadiativeTransport• Radiationpropagationhasdirection.Thus,weneedto
followaray.• Photonsinaraycarryenergy(intensity)andinteract
withmatter.Astheresultintensitycandecrease(extinction)orincrease(emissivity).
• Forsmallgeometricalstepalongtheraywecanwritethechangeofintensityas:
• Decreaseofintensityisproportionaltointensity.• Stimulatedemissionisalsoproportionaltointensityand
assuchistreatedas“negative”extinction.
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Extinction• Twoprocessescanremoveenergyfromabeam:absorptionandscattering.
• Absorptiondestroysthephoton.Photonisconvertedtointernalenergy(excitation)oftheabsorber(atom,molecule,dustparticle)thatisthenreleasedininelasticcollisionorcascadeemission.
• Scatteringdoesnotdestroyaphotonbutredirectsit.Absorptionfollowedbyre-emissioninvolvingthesametwoenergylevelscanbetreatedasscatteringorabsorption.
Extinctioncoefficient
• Extinctionscaleswiththeamount(density)ofmaterialalongtheray.
• Itislogicaltoseparatethispart,notconnectedtoradiation.
• Itcanbedoneinseveralways:whereismassextinction,ismassdensityisabsorptioncross-sectionandnisabsorbernumberdensity.
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Opticalpath(ordepth)
• Variousdefinitionsofextinctionconnectinthepreviousslidedefinenewquantity–opticalpath:
• Thefirstthingwecandoisre-writingtheRTequation:where:isthesourcefunction.
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Opticalpath
• τ-xrelationmeasurestransparencyofmatter(surfaceforτ>>1).
• Forincomingradiationitshowshowdeepradiationwillpenetratedbeforeitisfullyabsorbed.
• Foremissionofmatteritshowsthedepthoflayerfromwhichtheemissionoriginates.
Slangandtraditions• Opticalpathismeasuredalongtheray.• In1Dmedium(allchangescanbedescribedasfunctionofasinglevariable)wecanuseopticaldepth.
• Inthiscase,opticalpathalongtherayis:
• andRTequationchangesto(notesignchange):
τ
θ
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Examples:emission
Example:absorption
Emissivity
• Emissivityistheprocessofkineticenergyconversiontoradiation.
• Itcangoascollisionalexcitation(inelasticcollision)followedbyradiativede-excitation
• Oritcanbescatteringfromotherrays.
SolvingRT• Noemissivity(sourcefunctioniszero)
• Surfaceintensity(streamingout,μ>0)exponentiallygrowsupthestreamwithe-foldinglengthτ/μ=1.
• TheformalsolutionofRTequationis:
• Homework:derivetheformalsolution.
I = I0 · e⌧/µ<latexit sha1_base64="gUHb/KC91g9wlf7mUE0CScKdrJY=">AAACAHicbZC7SgNBFIbPxluMt6ighc1gEKziro02QtDGdAmYC2RjnJ2dxCGzF2bOCmFJ46vYWChi62PY2fgsTi6FJv4w8PGfczhzfi+WQqNtf1mZhcWl5ZXsam5tfWNzK7+9U9dRohivsUhGqulRzaUIeQ0FSt6MFaeBJ3nD61+N6o0HrrSIwhscxLwd0F4ouoJRNFYnv1++KHdsl/kREn6bukiTEzdIhp18wS7aY5F5cKZQKO1Vv+8AoNLJf7p+xJKAh8gk1brl2DG2U6pQMMmHOTfRPKasT3u8ZTCkAdftdHzAkBwZxyfdSJkXIhm7vydSGmg9CDzTGVC817O1kflfrZVg97ydijBOkIdssqibSIIRGaVBfKE4QzkwQJkS5q+E3VNFGZrMciYEZ/bkeaifFh3DVZPGJUyUhQM4hGNw4AxKcA0VqAGDITzBC7xaj9az9Wa9T1oz1nRmF/7I+vgBViOXxA==</latexit><latexit sha1_base64="YMbkBb6Y1HFgdTZhjDojZ688mXI=">AAACAHicbZC7SgNBFIZn4y3GW1TQwmYwCFZx10YbIcTGdAmYC2TjMjs7SQZnd5aZs0JY0vgqNhYGsbX2CexsfBYnl0ITfxj4+M85nDm/Hwuuwba/rMzS8srqWnY9t7G5tb2T391raJkoyupUCqlaPtFM8IjVgYNgrVgxEvqCNf3763G9+cCU5jK6hUHMOiHpRbzLKQFjefnDylXFs10aSMDsLnWBJGdumAy9fMEu2hPhRXBmUCgd1L75qPxR9fKfbiBpErIIqCBatx07hk5KFHAq2DDnJprFhN6THmsbjEjIdCedHDDEJ8YJcFcq8yLAE/f3REpCrQehbzpDAn09Xxub/9XaCXQvOymP4gRYRKeLuonAIPE4DRxwxSiIgQFCFTd/xbRPFKFgMsuZEJz5kxehcV50DNdMGmU0VRYdoWN0ihx0gUroBlVRHVE0RE/oBY2sR+vZerXepq0Zazazj/7Iev8Bp2SZgA==</latexit><latexit sha1_base64="YMbkBb6Y1HFgdTZhjDojZ688mXI=">AAACAHicbZC7SgNBFIZn4y3GW1TQwmYwCFZx10YbIcTGdAmYC2TjMjs7SQZnd5aZs0JY0vgqNhYGsbX2CexsfBYnl0ITfxj4+M85nDm/Hwuuwba/rMzS8srqWnY9t7G5tb2T391raJkoyupUCqlaPtFM8IjVgYNgrVgxEvqCNf3763G9+cCU5jK6hUHMOiHpRbzLKQFjefnDylXFs10aSMDsLnWBJGdumAy9fMEu2hPhRXBmUCgd1L75qPxR9fKfbiBpErIIqCBatx07hk5KFHAq2DDnJprFhN6THmsbjEjIdCedHDDEJ8YJcFcq8yLAE/f3REpCrQehbzpDAn09Xxub/9XaCXQvOymP4gRYRKeLuonAIPE4DRxwxSiIgQFCFTd/xbRPFKFgMsuZEJz5kxehcV50DNdMGmU0VRYdoWN0ihx0gUroBlVRHVE0RE/oBY2sR+vZerXepq0Zazazj/7Iev8Bp2SZgA==</latexit><latexit sha1_base64="/4gjRwZLSzg4P2OYNocZdMVPbP8=">AAACAHicbZC7TsMwFIYdrqXcAgwMLBYVElNJWGBBqmChW5HoRWpC5DhOa9W5yD5BqqIsvAoLAwix8hhsvA1umwFafsnSp/+co+Pz+6ngCizr21haXlldW69sVDe3tnd2zb39jkoySVmbJiKRPZ8oJnjM2sBBsF4qGYl8wbr+6GZS7z4yqXgS38M4ZW5EBjEPOSWgLc88bF41PcuhQQKYPeQOkOzMibLCM2tW3ZoKL4JdQg2VannmlxMkNItYDFQQpfq2lYKbEwmcClZUnUyxlNARGbC+xphETLn59IACn2gnwGEi9YsBT93fEzmJlBpHvu6MCAzVfG1i/lfrZxBeujmP0wxYTGeLwkxgSPAkDRxwySiIsQZCJdd/xXRIJKGgM6vqEOz5kxehc163Nd9ZtcZ1GUcFHaFjdIpsdIEa6Ba1UBtRVKBn9IrejCfjxXg3PmatS0Y5c4D+yPj8Acd8ld8=</latexit>
I1 = I0 · e�(⌧1�⌧0)/µ +1
µ
Z ⌧1
⌧0
S(t) · e�(t�⌧0)/µdt<latexit sha1_base64="w7ERW8d1vW8QF/Hu6QRt3X0CksU=">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</latexit><latexit sha1_base64="v4zut456lI117UzIT91GEcpAqhk=">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</latexit><latexit sha1_base64="v4zut456lI117UzIT91GEcpAqhk=">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</latexit><latexit sha1_base64="NAHC+A6As41tYbV7ga+70KNfKkA=">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</latexit>
AbsorptionandScattering
• Lookingcloseratextinctionwecanseparatethecross-sectionintothetrueabsorptionandscatteringpart:
• Wedefinetherelativecontributionofscatteringas:
• Forlightreflectedfromasurfacethisisalbedo.
�e = �a + �s<latexit sha1_base64="k5h2Qr4irUsAWtoYw3PZbHP/9BY=">AAACA3icbZDLSsNAFIZPvNZ6iwoudBMsgiCUxI1uhKIbly3YC7QhTqaTdujMJMxMhBIKbnwVNy4UcetLuHPjszhts9DWHwY+/nMOZ84fJowq7bpf1sLi0vLKamGtuL6xubVt7+w2VJxKTOo4ZrFshUgRRgWpa6oZaSWSIB4y0gwH1+N6855IRWNxq4cJ8TnqCRpRjLSxAvugo2iPo4Bc5oBOc1CBXXLL7kTOPHg5lCr7te87AKgG9menG+OUE6ExQ0q1PTfRfoakppiRUbGTKpIgPEA90jYoECfKzyY3jJxj43SdKJbmCe1M3N8TGeJKDXloOjnSfTVbG5v/1dqpji78jIok1UTg6aIoZY6OnXEgTpdKgjUbGkBYUvNXB/eRRFib2IomBG/25HlonJU9wzWTxhVMVYBDOIIT8OAcKnADVagDhgd4ghd4tR6tZ+vNep+2Llj5zB78kfXxA4FamZU=</latexit><latexit sha1_base64="Nxg6Zbtlea6uf0bgRJmT6MRNhp0=">AAACA3icbZC7SgNBFIZnvcZ4WxUstFkMgiCEXRtthBAbywTMBZJlOTuZTYbMzC4zs0JYAja+io2IIrZ2PoGdjc/i5FJo4g8DH/85hzPnDxNGlXbdL2thcWl5ZTW3ll/f2Nzatnd26ypOJSY1HLNYNkNQhFFBappqRpqJJMBDRhph/2pUb9wSqWgsbvQgIT6HrqARxaCNFdgHbUW7HAJyOQU4nYIK7IJbdMdy5sGbQqG0X/2mT+WPSmB/tjsxTjkRGjNQquW5ifYzkJpiRob5dqpIArgPXdIyKIAT5WfjG4bOsXE6ThRL84R2xu7viQy4UgMemk4OuqdmayPzv1or1dGFn1GRpJoIPFkUpczRsTMKxOlQSbBmAwOAJTV/dXAPJGBtYsubELzZk+ehflb0DFdNGmU0UQ4doiN0gjx0jkroGlVQDWF0hx7QM3qx7q1H69V6m7QuWNOZPfRH1vsP0pubUQ==</latexit><latexit sha1_base64="Nxg6Zbtlea6uf0bgRJmT6MRNhp0=">AAACA3icbZC7SgNBFIZnvcZ4WxUstFkMgiCEXRtthBAbywTMBZJlOTuZTYbMzC4zs0JYAja+io2IIrZ2PoGdjc/i5FJo4g8DH/85hzPnDxNGlXbdL2thcWl5ZTW3ll/f2Nzatnd26ypOJSY1HLNYNkNQhFFBappqRpqJJMBDRhph/2pUb9wSqWgsbvQgIT6HrqARxaCNFdgHbUW7HAJyOQU4nYIK7IJbdMdy5sGbQqG0X/2mT+WPSmB/tjsxTjkRGjNQquW5ifYzkJpiRob5dqpIArgPXdIyKIAT5WfjG4bOsXE6ThRL84R2xu7viQy4UgMemk4OuqdmayPzv1or1dGFn1GRpJoIPFkUpczRsTMKxOlQSbBmAwOAJTV/dXAPJGBtYsubELzZk+ehflb0DFdNGmU0UQ4doiN0gjx0jkroGlVQDWF0hx7QM3qx7q1H69V6m7QuWNOZPfRH1vsP0pubUQ==</latexit><latexit sha1_base64="VqLbRv+6CfRpWbyLbHvwPtXNN4A=">AAACA3icbZDNSgMxFIUz9a/Wv1F3ugkWQRDKjBvdCEU3LivYWmiH4U6aaUOTzJBkhDIU3Pgqblwo4taXcOfbmLaz0NYDgY9z7+XmnijlTBvP+3ZKS8srq2vl9crG5tb2jru719JJpghtkoQnqh2BppxJ2jTMcNpOFQURcXofDa8n9fsHqjRL5J0ZpTQQ0JcsZgSMtUL3oKtZX0BILwuA0wJ06Fa9mjcVXgS/gCoq1Ajdr24vIZmg0hAOWnd8LzVBDsowwum40s00TYEMoU87FiUIqoN8esMYH1unh+NE2ScNnrq/J3IQWo9EZDsFmIGer03M/2qdzMQXQc5kmhkqyWxRnHFsEjwJBPeYosTwkQUgitm/YjIABcTY2Co2BH/+5EVondV8y7detX5VxFFGh+gInSAfnaM6ukEN1EQEPaJn9IrenCfnxXl3PmatJaeY2Ud/5Hz+APKzl7A=</latexit>
!0 =�s
�e⌘ �s
�a + �s<latexit sha1_base64="dDIU75UiG0P5f6Og6e/GDPatxlo=">AAACM3icdZBNS8NAEIYnflu/qh69BEUQBElE0IsiCiKeFKwKTQmT7aQu7iZxd1Moof/Fn+DFP+JBEA+KePU/uG1V/BxYeHjfGWbnjTLBtfG8e6evf2BwaHhktDQ2PjE5VZ6eOdFprhhVWCpSdRahJsETqhhuBJ1lilBGgk6ji92Of9okpXmaHJtWRjWJjYTHnKGxUlg+CFJJDQy9zSBWyIpA84bEULc/iNoBXea8+Y+Ny59SWF7wVrxuub/Bf4eF7a29tSsAOAzLt0E9ZbmkxDCBWld9LzO1ApXhTFC7FOSaMmQX2KCqxQQl6VrRvbntLlql7sapsi8xblf9OlGg1LolI9sp0Zzrn15H/Mur5ibeqBU8yXJDCestinPhmtTtBOjWuSJmRMsCMsXtX112jjYcY2Mu2RD8nyf/hpPVFd/ykU1jB3o1AnMwD0vgwzpswz4cQgUYXMMdPMKTc+M8OM/OS6+1z3mfmYVv5by+AdJnrvA=</latexit><latexit sha1_base64="zHVv2NyFxi0q77bRfSO6DtcPVGo=">AAACM3icdZDLSsNAFIYn3q23qks3QREEoSQi6EbxAiKuKlgVmhJOpift0JkkzkyEEvocvoYb8T1cCOJCEbe+g9OLoq0eGPj4/3M4c/4g4Uxpx3myhoZHRsfGJyZzU9Mzs3P5+YVzFaeSYonGPJaXASjkLMKSZprjZSIRRMDxImgctv2La5SKxdGZbiZYEVCLWMgoaCP5+RMvFlgD39nxQgk08xSrCfBV64uw5eFVyq7/sWH9W/LzK07B6ZQ9CG4PVvZ2jzZv7vdPin7+wavGNBUYacpBqbLrJLqSgdSMcmzlvFRhArQBNSwbjECgqmSdm1v2qlGqdhhL8yJtd9SfExkIpZoiMJ0CdF31e23xL6+c6nC7krEoSTVGtLsoTLmtY7sdoF1lEqnmTQNAJTN/tWkdTDjaxJwzIbj9Jw/C+UbBNXxq0jgg3ZogS2SZrBGXbJE9ckyKpEQouSWP5IW8WnfWs/VmvXdbh6zezCL5VdbHJ3UGsCk=</latexit><latexit sha1_base64="zHVv2NyFxi0q77bRfSO6DtcPVGo=">AAACM3icdZDLSsNAFIYn3q23qks3QREEoSQi6EbxAiKuKlgVmhJOpift0JkkzkyEEvocvoYb8T1cCOJCEbe+g9OLoq0eGPj4/3M4c/4g4Uxpx3myhoZHRsfGJyZzU9Mzs3P5+YVzFaeSYonGPJaXASjkLMKSZprjZSIRRMDxImgctv2La5SKxdGZbiZYEVCLWMgoaCP5+RMvFlgD39nxQgk08xSrCfBV64uw5eFVyq7/sWH9W/LzK07B6ZQ9CG4PVvZ2jzZv7vdPin7+wavGNBUYacpBqbLrJLqSgdSMcmzlvFRhArQBNSwbjECgqmSdm1v2qlGqdhhL8yJtd9SfExkIpZoiMJ0CdF31e23xL6+c6nC7krEoSTVGtLsoTLmtY7sdoF1lEqnmTQNAJTN/tWkdTDjaxJwzIbj9Jw/C+UbBNXxq0jgg3ZogS2SZrBGXbJE9ckyKpEQouSWP5IW8WnfWs/VmvXdbh6zezCL5VdbHJ3UGsCk=</latexit><latexit sha1_base64="Ak4tienjvxjGYsyY7g+sY2UhIqg=">AAACM3icdZBNS8NAEIY39bt+VT16CRZBEEriRS9C0Yt4UrC20JQw2U7apbtJ3N0USuh/8uIf8SCIB0W8+h/c1ija6sDCw/vOMDtvkHCmtOM8WoWZ2bn5hcWl4vLK6tp6aWPzWsWppFijMY9lIwCFnEVY00xzbCQSQQQc60HvdOTX+ygVi6MrPUiwJaATsZBR0EbyS+deLLADvnPshRJo5inWEeCr4Rfh0MOblPX/sWH/W/JLZafijMueBjeHMsnrwi/de+2YpgIjTTko1XSdRLcykJpRjsOilypMgPagg02DEQhUrWx889DeNUrbDmNpXqTtsfpzIgOh1EAEplOA7qpJbyT+5TVTHR61MhYlqcaIfi4KU27r2B4FaLeZRKr5wABQycxfbdoFE442MRdNCO7kydNwfVBxDV865epJHsci2SY7ZI+45JBUyRm5IDVCyS15IM/kxbqznqxX6+2ztWDlM1vkV1nvH6uPrVg=</latexit>
Splittingthesourcefunction
• Usingthedefinitionofwecanseparatethesourcefunctionintothermalemissionandscatteringparts:whereJ/4πisthemeanintensityinagivenpointandBisthePlanckfunction.
• ThecorrespondingRTequationwillbe:
!0<latexit sha1_base64="Q3zXwzbCAIlA7FvnZZdsdrFMvYk=">AAAB73icbZC7SgNBFIbPeo3xFhVsbAaDYBV2bbQM2lgmYC6QLHF2cjYZMjO7zswKIeQlbCwUsfV17Gx8FieXQhN/GPj4zznMOX+UCm6s7395K6tr6xubua389s7u3n7h4LBukkwzrLFEJLoZUYOCK6xZbgU2U41URgIb0eBmUm88ojY8UXd2mGIoaU/xmDNqndVsJxJ7tON3CkW/5E9FliGYQ7F8XP2+B4BKp/DZ7iYsk6gsE9SYVuCnNhxRbTkTOM63M4MpZQPaw5ZDRSWacDTdd0zOnNMlcaLdU5ZM3d8TIyqNGcrIdUpq+2axNjH/q7UyG1+FI67SzKJis4/iTBCbkMnxpMs1MiuGDijT3O1KWJ9qyqyLKO9CCBZPXob6RSlwXHVpXMNMOTiBUziHAC6hDLdQgRowEPAEL/DqPXjP3pv3Pmtd8eYzR/BH3scPRWyRnw==</latexit><latexit sha1_base64="b9Nob4v0vZNySWN+up+SfQ8qBAM=">AAAB73icbZC7SgNBFIZnvcZ4iwo2NoNBsAq7NlqG2FgmYC6QLGF2cjYZMpd1ZlYIS17CxkIRWwvfwiews/FZnFwKTfxh4OM/5zDn/FHCmbG+/+WtrK6tb2zmtvLbO7t7+4WDw4ZRqaZQp4or3YqIAc4k1C2zHFqJBiIiDs1oeD2pN+9BG6bkrR0lEArSlyxmlFhntTpKQJ90/W6h6Jf8qfAyBHMolo9r3+y98lHtFj47PUVTAdJSToxpB35iw4xoyyiHcb6TGkgIHZI+tB1KIsCE2XTfMT5zTg/HSrsnLZ66vycyIowZich1CmIHZrE2Mf+rtVMbX4UZk0lqQdLZR3HKsVV4cjzuMQ3U8pEDQjVzu2I6IJpQ6yLKuxCCxZOXoXFRChzXXBoVNFMOnaBTdI4CdInK6AZVUR1RxNEDekLP3p336L14r7PWFW8+c4T+yHv7AZatk1s=</latexit><latexit sha1_base64="b9Nob4v0vZNySWN+up+SfQ8qBAM=">AAAB73icbZC7SgNBFIZnvcZ4iwo2NoNBsAq7NlqG2FgmYC6QLGF2cjYZMpd1ZlYIS17CxkIRWwvfwiews/FZnFwKTfxh4OM/5zDn/FHCmbG+/+WtrK6tb2zmtvLbO7t7+4WDw4ZRqaZQp4or3YqIAc4k1C2zHFqJBiIiDs1oeD2pN+9BG6bkrR0lEArSlyxmlFhntTpKQJ90/W6h6Jf8qfAyBHMolo9r3+y98lHtFj47PUVTAdJSToxpB35iw4xoyyiHcb6TGkgIHZI+tB1KIsCE2XTfMT5zTg/HSrsnLZ66vycyIowZich1CmIHZrE2Mf+rtVMbX4UZk0lqQdLZR3HKsVV4cjzuMQ3U8pEDQjVzu2I6IJpQ6yLKuxCCxZOXoXFRChzXXBoVNFMOnaBTdI4CdInK6AZVUR1RxNEDekLP3p336L14r7PWFW8+c4T+yHv7AZatk1s=</latexit><latexit sha1_base64="IOc3LRMiyg33YvueR+rjXW74kB8=">AAAB73icbZBNSwMxEIZn/az1q+rRS7AInsquFz0WvXisYD+gXUo2nW1Dk+yaZIWy9E948aCIV/+ON/+NabsHbX0h8PDODJl5o1RwY33/21tb39jc2i7tlHf39g8OK0fHLZNkmmGTJSLRnYgaFFxh03IrsJNqpDIS2I7Gt7N6+wm14Yl6sJMUQ0mHisecUeusTi+ROKR9v1+p+jV/LrIKQQFVKNToV756g4RlEpVlghrTDfzUhjnVljOB03IvM5hSNqZD7DpUVKIJ8/m+U3LunAGJE+2esmTu/p7IqTRmIiPXKakdmeXazPyv1s1sfB3mXKWZRcUWH8WZIDYhs+PJgGtkVkwcUKa525WwEdWUWRdR2YUQLJ+8Cq3LWuD43q/Wb4o4SnAKZ3ABAVxBHe6gAU1gIOAZXuHNe/RevHfvY9G65hUzJ/BH3ucPtsWPug==</latexit>
S = !0 ·J
4⇡+ (1� !0) ·B
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µdI�d⌧
= I� � !0 ·J�4⇡
� (1� !0) ·B�<latexit sha1_base64="2Bwlpou39EjjbMgRQjSCq46y6KQ=">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</latexit><latexit sha1_base64="DthEA9g7tk8D80h6AyArOm5zb90=">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</latexit><latexit sha1_base64="DthEA9g7tk8D80h6AyArOm5zb90=">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</latexit><latexit sha1_base64="+5os1BqYOH6kWcU4F9UqUJfKYMk=">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</latexit>
Transmissionfunction(1)• ScatteringmakesRTimpossibletosolvedirectly.
• AssumingnoscatteringandconstantPlanckfunctionwecansolveitforaplanetaryatmosphere.
• LetI2andI1beintensitiesatthebottomandthetopoftheatmospheres.ThenintegratingoverμofrewrittenRTequation:
µI2 · e�(⌧2�⌧1)/µ � µI1 = �B
Z ⌧2
⌧1
e�(t�⌧1)/µdt<latexit sha1_base64="2zZjbSnNSJnyOCQSbc3fqAboRP0=">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</latexit><latexit sha1_base64="CllbKI7E0EhJNDW4OM47XVcRLC0=">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</latexit><latexit sha1_base64="CllbKI7E0EhJNDW4OM47XVcRLC0=">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</latexit><latexit sha1_base64="jRJ0+HYe1ZGboZAMXqbGp7X6YXE=">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</latexit>
Transmissionfunction(2)
… fortheupanddowndirectionsgives:wherearrowsindicatethedirectionofweusedthedefinitionofflux:Transmissionfunctionisdefinedas:
F"1 = F"2T + ⇡B(1� T )
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F =
Z 1
0Iµdµ
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T ⌘ µ
Z 1
0µe�(⌧2�⌧1)/µdµ
<latexit sha1_base64="EYAzlwyvplmAw84BNEkq8SerFn8=">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</latexit><latexit sha1_base64="iUc6B85ngshISSYLjyXTrdCcTWg=">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</latexit><latexit sha1_base64="iUc6B85ngshISSYLjyXTrdCcTWg=">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</latexit><latexit sha1_base64="fl452ZUzUDIVACkgaA4OgaHvu2U=">AAACHHicbZA9T8MwEIYdPkv5KjCyWFRIMFCSMsBYwcJYpBaQmhI57pVaOE6wz5WqqD+Ehb/CwgBCLAxI/Bvc0oGvk+x79N6d7HvjTAqDvv/hTU3PzM7NFxaKi0vLK6ultfVzk1rNoclTmerLmBmQQkETBUq4zDSwJJZwEd+cjOoXfdBGpKqBgwzaCbtWois4QydFpYNGCLdW9MPEhkJh5F8F1DGFq3xvJ0Rmo+reOAW7+04fdtwVlcp+xR8H/QvBBMpkEvWo9BZ2Um4TUMglM6YV+Bm2c6ZRcAnDYmgNZIzfsGtoOVQsAdPOx8sN6bZTOrSbancU0rH6fSJniTGDJHadCcOe+V0bif/VWha7R+1cqMwiKP71UNdKiikdOUU7QgNHOXDAuBbur5T3mGYcnZ9FZ0Lwe+W/cF6tBI7P/HLteGJHgWySLbJDAnJIauSU1EmTcHJHHsgTefbuvUfvxXv9ap3yJjMb5Ed4759de6Di</latexit>
Transmissionfunction(3)
• IfTis0theatmosphereistotallyopaque.Theonlylightcomingoutisgeneratedfromthethermalenergy.
• IfTis1theatmosphereistotallytransparent.Whatcomesinstreamsoutandnothermalemissionisadded.
• Thisistrueinbothdirections.• Ofcourse,weignoredscatteringandchangeoftemperaturealongtheway.
Anoteaboutunits• Intensityistheamountofenergyperunitspectralbinthatcomesfromaunitsolidangleandcrossesunitareaperunittimealongagivenray.
• Thusunitsareergcm-2st-1s-1Å-1orergcm-2st-1s-1Hz-1.
• Emissivityhasthesameunitsasintensity.• Fluxisthetotalenergy(fromalldirections)thatcrossesunitarea,sonosolidangle.
• Absorptionisincm-1andisindependentofspectralbinunits.
• Opticalpathisunitless.
Specialcase:instantaneoustransitiontolargeopticaldepth
• Surfacesofsolidobjects.• GeometricalalbedoAgisthecomparisonofthefractionoflightreflectedononedirection(totheobserver)ascomparedwithflatsurfaceofthesameshapethatscatterslightisotropically(Lambertdisk).
• Reflectedlightfromaplanetisalsoaffectedbythegeometry(phases).ThecorrespondingSphericalalbedoAsisrelatedtotheAgthroughphasefunction:As=AgxPg.
Moreaboutalbedo
• Geometricandsphericalalbedoarewavelengthdependent.
• MeanorBondalbedoisatruemeasureoftotalfractionofreflectedenergy.Itiscomputedas:
AB =
RI?�Asd�RI?�d�
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WhatwillbeneededtosolveRT?
• TosolvetheRTequationweneedmodelatmosphere:T,P,partialnumberdensityofallabsorbers,somethingthatwecanmeasureinthesolarsystembutnotelsewhere.
• Let’sassumewehaveaguessforverticalT-Pstructureandchemicalcomposition.
• Thenforeachlayerweneedtocomputethenumberofparticlescontributingtoabsorptionandscatteringacrossthewholewavelengthrange.
SolvingRT
• OncewehaveopacitiesweneedarobustmethodforsolvingRT(accountingforscattering).
• Assumingenergybalance(whatcomesinfromthehoststarisradiatedout,albeitatdifferentwavelengths)wecanadjusttemperaturestructure.
• NewtemperaturewillrequiresolvingtheequationofhydrostaticequilibriumtogetP.ThisisdoneiterativelyasTaffectspartialpressuresthataffectmeandensity.
• ThisonlyworksifweignoreatmosphericdynamicsJ• Thentheconvergenceisachievedifthetotalradiativefluxthroughanylayeriszero!
Homework
• 2.8.3.Greenhouseeffect• 2.8.6.Transitradiusandopticaldepth
