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METO 621 LESSON 8 Thermal emission from a surface • Let ˆ ) cos d I ( c be the emitted energy from a flat surface of temperature Ts , within the solid angle din the direction .A blackbody would emit B (T )cosd.The spectral directional s emittance is defined as ˆ ˆ I () cos d Ie () ˆ ( , , TS ) B (TS ) cos d B (TS ) e 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 2psteradians relative to a blackbody is defined as the flux or bulk emittance ( ,2p , TS ) ˆ) d cos Ie ( d cos B (T 1 p ˆ ,T ) d cos ( , S S ) ˆ , T ) B (T ) d cos ( , S S p B (TS ) 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: ˆ ˆ I ( ' ) cos ' d ' I a a (' ) ˆ ( ,' , TS ) ˆ ' ) cos ' d ' I ( ˆ ') I ( 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 ˆ ˆ d ' cos ' ( ,' , TS )I (' ) ( ,2p , TS ) 1 • F Kirchoff showed that for an opaque surface ˆ ,T ) (, ˆ ,T ) (, S S • That is, a good absorber is also a good emitter, and vice-versa Surface reflection : the BRDF ˆ ). I ( ˆ ) cos d '. I ( Consider a downward beam with intensity The energy incident on a flat surface is Let the intensity of the reflected light around the ˆ within a solid angle d be dI then direction r ˆ dI ) r ( ˆ ˆ ( ,', ) ˆ I (' )cos ' d ' ˆ ', ˆ ) is the bidirectional relfectance where ( , distribution function, or BRDF. BRDF ˆ, The total reflected intensity in the direction from all beams is ˆ ) d I ( ˆ ) d ' cos ' ( , ˆ ', ˆ ) I ( ˆ ') I ( r r If a reflecting surface has a BRDF which is independen t of both the incidence and observatio n directions , then it is called a Lambert surface. ˆ ', ˆ ) ( ), and In this case ( , L ˆ ' ) ( ) F Ir L ( ) d ' cos 'I ( L Surface reflectance - BRDF Collimated incidence Collimated Incidence - Lambert Surface • If the incident light is direct sunlight then I () F S ( 0 ) F S (cos cos 0 ) ( 0 ) The incident flux is given by Hence F F S cos 0 Ir L cos 0 F S For a collimated beam the intensity reflected from a Lambert surface is proportional to the cosine of the angle of incidence. Collimated Incidence - Specular reflectance • Here the reflected intensity is directed along the angle of reflection only. • Hence ’and ’+p • Spectral reflection function S(,) ˆ ) ( )F S(cos cos )( p ) Ir ( S 0 0 • and the reflected flux: Fr S (0 )F S cos 0 Absorption and Scattering in Planetary Media • Kirchoff’s Law for volume absorption and Emission ( ,T) ( ,T) k( ) The volume emittance is proportional to the absorption coefficient 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 ˆ ' )dAdtd d d E I ( 4 Differential Equation of Radiative Transfer • The radiative energy scattered in all directions is ds d E' 4 • 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 ˆ ', ˆ ) d /4p p( 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 , ˆ ˆ p( ' , ) ' ˆ 4 d E ( ) dV dt d d d ' I (' ) 4p 4p • The emission coefficient for scattering is jSC d4 E d ' ˆ ˆ ˆ ') ( ) p(', )I ( dV dt d d 4 p 4p Differential Equation of Radiative Transfer • The source function for scattering is thus SC j ( ) d ' ˆ ˆ SC ˆ ˆ ') S ( rˆ, ) p(', ) I ( k( ) k( ) 4 p 4 p • 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 dI a( ) ˆ ', ˆ )I I 1 a( )B (T ) d ' p ( d s 4p 4p