METO 621 LESSON 8

Thermal emission from a surface Let be the emitted energy from a flat surface of temperature T s, within the solid angle d  in the direction  A blackbody would emit B (T s )cos  d  The spectral directional emittance is defined as

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

Flux emittance The energy emitted into 2  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 -  emphasizes 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

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  ’  and  ’+  Spectral reflection function  S (  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 d  centered about the direction  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  The emission coefficient for scattering is

Differential Equation of Radiative Transfer The source function for scattering is thus The quantity  k(  is called the single-scattering albedo and given the symbol a(  If thermal emission is involved, (1-a) is the volume emittance 

Differential Equation of Radiative Transfer The complete time-independent radiative transfer equation which includes both multiple scattering and absorption is