Solar Radiation in the Atmosphere - Short-Wave Solar Radiation in the Earth's Atmosphere - page 14

γ further the phase functions depending only upon the scattering angle with

the normalization relation (1.18). Thus, we obtain the following relation for
energy scattered along direction γ

4

γ = γ Ω dE d ( ) x ( ) E 0 d dl . (1.21)

π

γ This relation may be accepted as a definition of phase function x ( )ofthe

elementary volume of the medium (however, owing to the definition formality
it is often used the definition of the phase function as a probability density of
radiation scattering to angle γ ).

Let us link the characteristics of the interaction between radiation and
a separate particle with the elementary volume. Let every particle interact with
radiation independently of others. Then extinction energy of the elementary
volume is equal to a sum of extinction energies of all particles in the volume.
Suppose firstly that all particles are similar; they have an extinction cross-
section C e and their number concentration (number of particle in the unit
volume) is equal to n . The particle number in the elementary volume is ndV .
Substituting the sum of extinction energies to the extinction coefficient def-
inition (1.19) in accordance with (1.12) and accounting the definition of the
irradiance (1.3) we obtain the following:

λ ndVC e F 0 d dt

α = = nC e .

λ F 0 dSd dtdl

Thus, the volume extinction coefficient is equal to the product of particle
number concentration by the extinction cross-section of one particle. 5
If there are extenuating particles of M kinds with concentrations n i and
cross-sections C e , i in the elementary volume of the medium then it is valid:
dE e = i = 1 n i dVC e , i F 0 d λ dt . Analogously considering the energies of scatter-

ing, absorption and directional scattering, we are obtaining the formulas,
which link the volume coefficients and cross-sections of the interaction:

M M

α = σ =

n i C e , i , n i C s , i ,

= =

i 1 i 1

(1.22)

M M

κ = σ γ = γ

n i C a , i , x ( ) n i C s , i x i ( ).

= = i 1 i 1

We would like to point out that the separate items in (1.22) make sense of the
volume coefficients of the interaction for the separate kinds of particles. There-
fore, highly important for practical problems are the “rules of summarizing”
following from (1.22). These rules allow us to derive separately the coefficients

5 Just by this reason, the term “volume” and not “linear” is used for the coefficient. It is defined by
numerical concentration in the unit volume of the air.