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Radiative Transfer Equation
Specific intensity of radiation fields
is defined by
the radiation energy transferred by photons with wavelength
through a unit surface placed at
whose normal is directed to
per unit time per unit wavelength, and per unit steradian.
Average intensity of radiation
is defined as

(2.121) 
where is obtained by averaging for the solid angle.
This is related to the energy density of radiation
as

(2.122) 
If the radiation is absorbed in the displacement as
,
must be proportional to and
as

(2.123) 
where is a coefficient and called volume absorption coefficient.
The dimension of is .
We can rewrite the above to the differential equation as

(2.124) 
where we used mass absorption coefficient which represents
the absorption per mass.
Equation (2.123) is reduced to

(2.125) 
where

(2.126) 
is called the optical depth.
This means that is a measure for absorption as
the intensity decreases at a factor from to .
If the ray runs crossing a volume whose
volume emissivity equal to
the intensity increases

(2.127) 
The volume emissivity
is the energy
emitted by a unit volume at a position
per unit time
per unit solid angle per unit wavelength.
From equations (2.123) and (2.127),
the radiation transfer is written as

(2.128) 
Using the optical denpth
,
this gives

(2.129) 
where is called the source function and is defined as
.
Assuming the specific intensity at the point of ,
that at the point on the same ray is given

(2.130) 
In the case of constant source term const it reduces to

(2.131) 
Equation (2.130) gives

(2.132) 
This indicates that if we see an optically thick cloud
the specific intensity reaches us represents ,
while if we see an transparent cloud , represents
that of background.
Problem
Show that
equations (2.130) and (2.130)
are solutions of equation (2.129).
Next: Einstein's Coefficients
Up: Radiative Transfer
Previous: Radiative Transfer
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Kohji Tomisaka
20070708