Meto 621
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METO 621. LESSON 8. Thermal emission from a surface. be the emitted. energy from a flat surface of temperature T s , within the solid angle d w in the direction W. A blackbody would emit B n (T s )cos q d w. The spectral directional emittance is defined as. Let.

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METO 621

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METO 621


Thermal emission from a surface

be the emitted

energy from a flat surface of temperature Ts , within the solid angle dw in the direction W. A blackbody would emit Bn(Ts)cosqdw. The spectral directional emittance is defined as

  • Let

Thermal emission from a surface

  • In general e depends on the direction of emission, the surface temperature, and the frequency of the radiation. A surface for which eis unity for all directions and frequencies is a blackbody. A hypothetical surface for which e = constant<1 for all frequencies is a graybody.

Flux emittance

  • The energy emitted into 2p steradians relative to a blackbody is defined as the flux or bulk emittance

Absorption by a surface

  • Let a surface be illuminated by a downward intensity I. Then a certain amount of this energy will be absorbed by the surface. We define the spectral directional absorptance as:

  • The minus sign in -Wemphasizes the downward direction of the incident radiation

Absorption by a surface

  • Similar to emission, we can define a flux absorptance

  • Kirchoff showed that for an opaque surface

  • That is, a good absorber is also a good emitter, and vice-versa

Surface reflection : the BRDF


Surface reflectance - BRDF

Collimated incidence

Collimated Incidence - Lambert Surface

  • If the incident light is direct sunlight then

Collimated Incidence - Specular reflectance

  • Here the reflected intensity is directed along the angle of reflection only.

  • Hence q’=q and f=f’+p

  • Spectral reflection function rS(n,q)

  • and the reflected flux:

Absorption and Scattering in Planetary Media

  • Kirchoff’s Law for volume absorption and Emission

Differential equation of Radiative Transfer

  • Consider conservative scattering - no change in frequency.

  • Assume the incident radiation is collimated

  • We now need to look more closely at the secondary ‘emission’ that results from scattering. Remember that from the definition of the intensity that

Differential Equation of Radiative Transfer

  • The radiative energy scattered in all directions is

  • We are interested in that fraction of the scattered energy that is directed into the solid angle dwcentered about the direction W.

  • This fraction is proportional to

Differential Equation of Radiative Transfer

  • If we multiply the scattered energy by this fraction and then integrate over all incoming angles, we get the total scattered energy emerging from the volume element in the direction W,

  • The emission coefficient for scattering is

Differential Equation of Radiative Transfer

  • The source function for scattering is thus

  • The quantity s(n)/k(n) is called the single-scattering albedo and given the symbol a(n).

  • If thermal emission is involved, (1-a) is the volume emittance e.

Differential Equation of Radiative Transfer

  • The complete time-independent radiative transfer equation which includes both multiple scattering and absorption is

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