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### METO 621

LESSON 8

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

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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