    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).    