Transcript METO 621
METO 621
LESSON 8
Thermal emission from a surface
ˆ )cos d
• Let 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(Ts)cosd.The spectral directional
emittance is defined as
ˆ
ˆ
I () cosd Ie ()
ˆ
( , , TS )
B (TS ) cosd 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
( ,2p , TS )
•
ˆ
ˆ
d ' cos ' ( ,' , TS )I (' )
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 ).
Consider a downward beam with intensitIy (
ˆ )cos d '.
T he energy incident on a flat surfaceI is(
Let the intensit y of the reflected light around the
ˆ within a solid angled be dI t hen
direction
r
ˆ
dI
)
r (
ˆ
ˆ
( ,', ) ˆ
I (')cos ' d '
ˆ ',
ˆ ) is the bidirectional relfect ance
where ( ,
distribution function,or BRDF.
BRDF
ˆ,
T he t ot alreflect edint ensit yin t hedirect ion
from all beams is
ˆ ) d I (
ˆ ) d ' cos ' ( ,
ˆ ',
ˆ ) I (
ˆ ')
Ir (
r
If a reflect ingsurface has a BRDF which is
independent of bot h t heincidenceand observat ion
direct ions, t henit is called a Lambertsurface.
ˆ ',
ˆ ) ( ), and
In t hiscase ( ,
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 )
T he incident flux is given F
by F S cos0
Hence
Ir L cos 0 F S
For a collimat ed beam t he intensity reflect ed from
a Lambert surface is proport ional to t he 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 cos0
Absorption and Scattering in Planetary
Media
• Kirchoff’s Law for volume absorption and
Emission
( ,T)
( ,T)
k( )
T he volume emit t ance is proport ional t o t
absorpt ion 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 dtd 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 4p
• 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