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

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1 Lecture 4 lackbody radiation. Main Laws. rightness temperature. Objectives: 1. Concepts of a blackbody, thermodynamical equilibrium, and local thermodynamical equilibrium.. Main laws: lackbody emission: Planck function. Stefan-oltzmann law. Wien s displacement law. Kirchhoff s law. 3. rightness temperature. 4. Sun as an energy source. Solar constant. Required reading: G.5 1. Concepts of a blackbody and thermodynamical equilibrium. lackbody is a body whose absorbs all radiation incident upon it. Thermodynamical equilibrium describes the state of matter and radiation inside an isolated constant-temperature enclosure. lackbody radiation is the radiative field inside a cavity in thermodynamic equilibrium. T =const I (T) lackbody cavity 1

3 Plank function can be expressed in wavelength, frequency, or wavenumber domains as hc ( T ) = 5 (exp( hc / k T) 1) [4.1] ~ 3 hν ( T) = c (exp( h ~ ν / k ~ ν [4.] T) 1) 3 hν c ν ( T) = [4.3] exp( hνc / k T) 1 where is the wavelength; ν ~ is the frequency; ν is the wavenumber; h is the Plank s constant; k is the oltzmann s constant (k = 1.38x10-3 J K -1 ); c is the velocity of light; and T is the absolute temperature (in K) of a blackbody. NOTE: The relations between ~ ν ( T ); ν ( T ) and (T ) are derived using that I d ~ ν = I dν = I ν ν d ~, and that = c / ~ ν = 1/ ν ~ ν ( T ) = ( T ) and ν ( T ) = ( T ) c Asymptotic behavior of Planck function: If (or ~ ν -> 0) (known as Rayleigh Jeans distributions): kc ( T ) = T 4 [4.4a] k ~ ν ~ ν ( T ) = T [4.4b] c 3

4 NOTE: Rayleigh Jeans distributions has a direct application to passive microwave remote sensing. For large wavelengths, the emission is directly proportional to T. If -> 0 (or ~ ν ): ( T ) hc = exp( hc / k T ) [4.5a] 5 ~ 3 hν = exp( h ~ ν / k ) [4.5b] c ~ ν T Figure 4.1 Planck function on log-log plot for several temperatures. 4

5 Stefan-oltzmann law. The Stefan-oltzmann law states that the total power (energy per unit time) emitted by a blackbody, per unit surface area of the blackbody, varies as the fourth power of the temperature. F = π (T) = σ b T 4 [4.6] where σ b is the Stefan-oltzmann constant (σ b = 5.671x10-8 W m - K -4 ), F is energy flux [W m - ], and T is blackbody temperature (in degrees Kelvin, K); and (T) = 0 ( T ) d Wien s displacement law. The Wien s displacement law states that the wavelength at which the blackbody emission spectrum is most intense varies inversely with the blackbody s temperature. The constant of proportionality is Wien s constant (897 K µm): m = 897 / T [4.7] where m is the wavelength (in micrometers, µm) at which the peak emission intensity occurs, and T is the temperature of the blackbody (in degrees Kelvin, K). NOTE: This law is simply derived from d /d = 0 NOTE: Easy to remember statement of the Wien s displacement law: the hotter the object the shorter the wavelengths of the maximum intensity emitted 5

6 Kirchhoff s law. The Kirchhoff s law states that the emissivity, ε, of a medium is equal to the absorptivity, Α, of this medium under thermodynamic equilibrium: ε = Α [4.8] where ε is defined as the ratio of the emitting intensity to the Planck function; Α is defined as the ratio of the absorbed intensity to the Planck function. For a blackbody: ε = Α = 1 For a gray body: ε = Α < 1 (i.e., no dependency on the wavelength) For a non-blackbody: ε = Α < 1 NOTE: Kirchhoff s law applies to gases, liquids and solids if they in TE or LTE. NOTE: In remote sensing applications, one needs to distinguish between the emissivity of the surface (e.g., various types of lands, ice, ocean etc., see Lecture 5) and the emissivity of an atmospheric volume (consisting of gases, aerosols, and/or clouds, see Lecture 7). 3. rightness temperature. rightness temperature, T b, is defined as the temperature of a blackbody that emits the same intensity as measured. rightness temperature is found by inverting the Planck function. For instance, from Eq.[4.1]: C T b = [4.9] C 1 ln[ 1 + ] 5 I where I is the measured intensity, and C 1 = x10 8 W m - sr -1 µm 4 ; C = x10 4 K µm 6

7 For a blackbody: brightness temperature = kinetic temperature ( T b = T) For natural materials: T b 4 = ε T 4 (ε is the broadband emissivity) NOTE: In the microwave region, the Rayleigh Jeans distributions gives T b = ε T. However, ε is a complex function of several parameters (see Lecture 5) 4. Sun as an energy source. Solar constant. Solar flux reaching the earth is a function of time determined by 1) the orbital characteristics of the earth and the sun (i.e., eccentricity; obliquity, and periodic precession) ) the sun properties (e.g., solar surface activity). NOTE: a) Sun is a gaseous sphere consisting of hydrogen, helium, iron, silicon, etc. Solar energy: nuclear fusion (conversion of four hydrogen atoms to one helium atom) b) Temperature of sun s photosphere is about 5800 K. b) Sunspots are cooler regions of the sun (with T = 4000K). Period between sunspot maxima is about 11 years (called 11-year-cycle). Solar constant, S 0, is defined as total flux of solar energy, reaching the top of the atmosphere, per unit surface normal to the solar beam at the mean distance between the sun and the earth. The sun emits about F sun = 6.x10 7 W/m. On the basis of energy conservation law F sun 4 π r sun = S 0 4 π d 0 where r sun is the radius of the sun (6.96x10 5 km), and d 0 is the mean distance between the sun and the earth (1.5x10 8 km). Hence, S 0 = F sun (r sun /d 0 ) [4.10] Mean measured value S 0 = 1366 W m - with the measurement uncertainty +/ 3W m -. 7

8 Actual solar flux at the top of the atmosphere at a given time is F o d o = S0 cos( θ o ) [4.11] d where d o is the mean distance from the center of the sun to the earth and d is the actual distance on a given day (depends of the earth orbit). Solar zenith angle cos (θ 0 ): cos θ 0 = sin( φ ) sin( δ ) + cos( φ ) cos( δ ) cos( h) [4.1] where φ is the latitude; δ is the solar inclination angle (varies throughout the seasons) h is the local hour angle of the sun (h=0 at local solar noon). Satellite measurements of solar constant: Nimbus 7 (Earth Radiation udget) Solar Maximum Mission: Active Cavity Radiometer Irradiance Monitor I (ACRIM I) Earth Radiation udget Satellite (ERS) Solar Monitor Measurements Upper Atmosphere Research Satellite (UARS) ACRIM II Measurements

9 New satellite measurements of solar constant: Solar Radiation and Climate Experiment (SORCE), 003: Total Irradiance Monitor (TIM) measures total (full spectrum) solar irradiance with absolute accuracy of 0.1 % Spectral Irradiance Monitor (SIM) measures the solar radiation spectrum from 00 nm to 500 nm, with spectral resolution of 1 nm and absolute accuracy of % Solar Stellar Irradiance Comparison Experiment (SOLSTICE) measures solar UV spectrum from 5 to 440 nm with 3% absolute accuracy and 1 nm resolution NPOESS program, 006: will use both TIM and SIM 9

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