Blackbody radiation. Main radiation laws. Sun as an energy source. Solar spectrum and solar constant.

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1 Lecture 3. lackbody radiation. Main radiation law. Sun a an energy ource. Solar pectrum and olar contant. Objective:. Concept of a blackbody, thermodynamical equilibrium, and local thermodynamical equilibrium.. Main radiation law: Planck function. Stefan-oltzmann law. Wien diplacement law. Kirchhoff law. 3. Sun a an energy ource. Required reading: L0:.,.4.3, Appendix A; Additional reading: Harder, J. W., J. M. Fontenla, P. Pilewkie, E. C. Richard, and T. N. Wood (009), Trend in olar pectral irradiance variability in the viible and infrared, Geophy. Re. Lett., 36, L0780, doi:0.09/008gl Concept of a blackbody and thermodynamical equilibrium. Thermodynamical equilibrium decribe the tate of matter and radiation inide an iolated contant-temperature encloure. lackbody i a perfect aborber (emitter) of radiation. Propertie of blackbody radiation: Radiation emitted by a blackbody i iotropic, homogeneou and unpolarized; lackbody radiation at a given wavelength depend only on the temperature; Any two blackbodie at the ame temperature emit preciely the ame radiation; A blackbody emit more radiation than any other type of an object at the ame temperature; NOTE: The atmophere i not trictly in the thermodynamic equilibrium becaue it temperature and preure are function of poition. Therefore, it i uually ubdivided into mall ubytem each of which i effectively iothermal and iobaric referred to a Local Thermodynamical Equilibrium (LTE).

2 . Main radiation law. Planck function (T) give the intenity (or radiance) emitted by a blackbody having temperature T. NOTE: Ditribution of blackbody radiation a a function of wavelength, known a the Planck law, cannot be predicted uing claical phyic. Thi derivation require quantum mechanic. See L0, Appendix A, for the derivation of the Plank function. Plank function can be expreed in wavelength, frequency, or wavenumber domain a hc ( T ) = 5 (exp( hc / k T) ) [3.] ~ 3 hν ~ ν ( T) = (exp( ~ [3.] c hν / k T) ) 3 hν c ν ( T) = [3.3] exp( hνc / k T) where i the wavelength; ν ~ i the frequency; ν i the wavenumber; h i the Plank contant; k i the oltzmann contant (k =.3806x0-3 J K - ); c i the peed of light; and T i the abolute temperature of a blackbody. NOTE: The relation between ~ ν ( T ); ν ( T ) and (T ) are derived uing that T d ~ ~ ( ) ν ( T ) dν = ( T ) d ν ν =, and that = c / ~ ν = / ν => ( T ) = ( ) and ν ( T ) = ( T ) c ~ ν T

3 Figure 3. Planck function on log-log plot for everal temperature. Aymptotic behavior of Planck function: If (or ~ ν -> 0) (known a Rayleigh Jean ditribution): k Tc ( T ) = 4 [3.4] k ~ Tν ~ ν = [3.5] c NOTE: Thi longwave limit ha a direct application to paive microwave remote ening. If -> 0 (or ~ ν ): hc ( T ) = exp( hc / k T ) 5 [3.6] ~ 3 hν = exp( h ~ ν / k ) [3.7] c ~ ν T 3

4 Figure 3. Plank function and it aymptotic behavior. Stefan-oltzmann law: The total power (energy per unit time) emitted by a blackbody, per unit urface area of the blackbody, varie a the fourth power of the temperature. F = π (T) = σ b T 4 [3.8] where σ b i the Stefan-oltzmann contant (σ b = 5.67x0-8 W m - K -4 ), F i the radiant flux [W m - ], and T i blackbody temperature [K]; and 0 ( T ) = ( T ) d 4

5 Wien diplacement law: The wavelength at which the blackbody emiion pectrum i mot intene varie inverely with the blackbody temperature. The contant of proportionality i Wien contant (897 K µm): m = 897 / T [3.9] where m i the wavelength (in micrometer, µm) at which the peak emiion intenity occur, and T i the temperature of the blackbody (in degree Kelvin, K). NOTE: Thi law i derived from d /d = 0 NOTE: Eay to remember tatement of the Wien diplacement law: the hotter the object the horter the wavelength of the maximum intenity emitted Kirchhoff law: The emiivity, ε, of a medium i equal to the aborptivity, Α, of thi medium under thermodynamic equilibrium: ε = Α [3.0] where ε i defined a the ratio of the emitted intenity to the Planck function; Α i defined a the ratio of the aborbed intenity to the Planck function. For a blackbody: ε = Α = For a non-blackbody: ε = Α < For a gray body (i.e., no dependency on the wavelength): ε = Α < NOTE: The Kirchhoff law applie to gae, liquid and olid if they in TE or LTE. For atmopheric radiation tranfer application, one need to ditinguih between the emiivity of the urface (e.g., variou type of land, ice, etc.) and the emiivity of an atmopheric volume (coniting of gae, aerool, and/or cloud). 5

6 Emiivity of an atmopheric volume: Aborption and thermal emiion of the atmophere volume i iotropic. Kirchhoff law applied to volume thermal emiion give j = β ( ) [3.], thermal a, T where β a, i the aborption coefficient of the atmopheric volume and j i the thermal emiion coefficient which relate to the ource function J (introduced in Lecture ) a J = (j, thermal + j, cattering ) / β e, and β e, i the extinction coefficient of the atmopheric volume. Recall the elementary olution of the radiative tranfer equation (Eq.[.3], Lecture ): I ( ) = I (0) exp( τ ( ;0)) + exp( τ ( ; )) J β e, 0 d For a non-cattering medium in the thermodynamical equilibrium: J = (T), where (T ) Alo, for the non-cattering medium, we have β i Plank function. = β = k ρ, where k a, i the e, a, a, ma aborption coefficient and ρ i the denity (ee Lecture ). Thu, the olution of the equation radiative tranfer for thi cae can be expreed a I ( ) = I (0)exp( τ ( ;0)) + exp( τ ( ; )) ( T( )) ka, ρd [3.] 0 NOTE: The optical depth in Eq.[3.] i due to aborption only, o τ ( ; ) = β a, ( d = ) k a, ρd 6

4 A) Meaurement of olar contant from five independent pace-baed radiometer ince 978 (top) have been combined to produce the compoite olar

7 4. Sun a an energy ource. Solar contant i total radiation emitted by the Sun reaching the top of the Earth atmophere. Not a contant, but it varie a a function of everal parameter (including un activity, un pot, ditance between Sun and Earth). A Figure 3.4 A) Meaurement of olar contant from five independent pace-baed radiometer ince 978 (top) have been combined to produce the compoite olar irradiance (bottom) over two decade. They how that the Sun output fluctuate during each -year unpot cycle, changing by about 0. percent between maximum (980 and 990) and minimum (987 and 997) in 7

magnetic activity. The larger number of unpot near the peak in the -year cycle i accompanied by a rie in magnetic activity that create an increae in olar radiation.

8 magnetic activity. The larger number of unpot near the peak in the -year cycle i accompanied by a rie in magnetic activity that create an increae in olar radiation. The capital letter are acronym for the different atellite radiometer (ee L0, capture for figure.) ) Updated meaurement (from The pectrum of olar radiation i well approximated by the emiion of a blackbody with temperature of about 5800K Figure 3.4 The pectrum of olar radiation (at the top of the atmophere) and a blackbody with T=6000 K. Figure 3.5 Solar radiation at the top of the atmophere, at the urface (for repreentative atmopheric condition) and 0 m under the ocean urface. 8

Blackbody radiation. Main Laws. Brightness temperature. 1. Concepts of a blackbody and thermodynamical equilibrium.

Blackbody radiation. Main Laws. Brightness temperature. 1. Concepts of a blackbody and thermodynamical equilibrium. Lecture 4 lackbody radiation. Main Laws. rightness temperature. Objectives: 1. Concepts of a blackbody, thermodynamical equilibrium, and local thermodynamical equilibrium.. Main laws: lackbody emission: