METO 621

LESSON 8

Thermal emission from a surface• Let

€

Iνc+ ( ˆ Ω )cosθ dω be the emitted

energy from a flat surface of temperature Ts , within the solid angle d in the direction A blackbody would emit B(Ts)cosd The spectral directional emittance is defined as

)(

)ˆ(

cos)(

cos)ˆ(),ˆ,(

S

e

S

eS TB

I

dTB

dIT

ε

=

=++

Thermal emission from a surface

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

),ˆ,(cos1

)(

)(),ˆ,(cos

)(cos

)ˆ(cos),2,(

S

S

SS

S

e

S

Td

TB

TBTd

TBd

IdT

=

=

≡

∫

∫∫∫

+

+

+

+

+

ε

ε

ε

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

)'ˆ(

)'ˆ(

''cos)'ˆ(

''cos)'ˆ(),'ˆ,(

=

=−−

−

−

−

α

II

dIdI

T aaS

Absorption by a surface• Similar to emission, we can define a flux

absorptance

• Kirchoff showed that for an opaque surface

€

α(ν ,− ˆ Ω ,TS ) = ε(ν , ˆ Ω ,TS )

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

−−

−∫ −=−

αα

F

ITd

TS

S1

)'ˆ(),'ˆ,('cos'

),2,(

Surface reflection : the BRDF

€

Consider a downward beam with intensity Iν− ( ˆ Ω ).

The energy incident on a flat surface is Iν−( ˆ Ω )cosθ dω' .

Let the intensity of the reflected light around the

direction ˆ Ω within a solid angle dω be dIνr− then

ρ (ν ,− ˆ Ω ', ˆ Ω )=dIνr

− ( ˆ Ω )

Iν ( ˆ Ω ')cosθ 'dω'

where ρ (ν ,− ˆ Ω ', ˆ Ω ) is the bidirectional relfectance

distribution function, or BRDF.

BRDF

FIdI

IdIdI

LLr

L

rr

)()'ˆ('cos')(

and ,)()ˆ,'ˆ,( case In this

surface.Lambert a called isit then ,directions

n observatio and incidence both the oft independen

is which BRDF a has surface reflecting a If

)'ˆ()ˆ,'ˆ,('cos')ˆ()ˆ(

is beams all from

,ˆdirection in theintensity reflected totalThe

ρρ

ρρ

ρ

==

=−

−==

−

−

+

−

−+

−

+

∫

∫∫

Surface reflectance - BRDF

Collimated incidence

Collimated Incidence - Lambert Surface

• If the incident light is direct sunlight then

€

I−() Ω ) = F Sδ(

) Ω −

) Ω 0) = F S (cosθ − cosθ0)δ(φ −ϕ 0)

The incident flux is given by F− = F S cosθ0

Hence Ir+ = ρ L cosθ0F

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 ’+• Spectral reflection function ρS(

€

Ir+( ˆ Ω ) = ρ S (θ)F Sδ(cosθ0 − cosθ)δ(φ − φ0 + π[ ])

• and the reflected flux:

€

Fr+ = ρ S (θ0)F S cosθ0

Absorption and Scattering in Planetary Media

• Kirchoff’s Law for volume absorption and Emission

€

ε (ν ,T) =α (ν ,T)

k(ν )

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

dddtdAIEd )'ˆ(4 Ω=+

Differential Equation of Radiative Transfer

€

σ ds d4 E '

• 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

€

p( ˆ Ω ', ˆ Ω ) dω /4π

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

€

d4 E = σ (ν ) dV dt dν dω dω'p( ˆ Ω ', ˆ Ω )

4π4π

∫ Iν' ( ˆ Ω ')

• The emission coefficient for scattering is

€

jνSC ≡

d4 E

dV dt dν dω= σ (ν )

dω'

4π4 π

∫ p( ˆ Ω ', ˆ Ω )Iν ( ˆ Ω ')

Differential Equation of Radiative Transfer

• The source function for scattering is thus

€

SνSC ( ˆ r , ˆ Ω ) =

jνSC

k(ν )=

σ (ν )

k(ν )

dω'

4π4 π

∫ p( ˆ Ω ', ˆ Ω ) Iν ( ˆ Ω ')

• The quantity σk( is called the single-scattering albedo and given the symbol a(

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

Differential Equation of Radiative Transfer

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

[ ] ∫ +−+−=

τ 4

)ˆ,'ˆ('4

)()()(1 Ipd

aTBaI

d

dI

s