Solar Radiation in the Atmosphere - Short-Wave Solar Radiation in the Earth's Atmosphere - page 30

because any function within the interval could be expanded to the series over
Legendre Polynomials. The following is deduced for the phase function:

∞

χ = χ

x ( ) x i P i ( )

=

i 0

(1.60)

1

2 i +1
2

= χ ) P i ( χ ) d χ . x i x (

−1

From the normalizing condition of the phase function (1.18) and fromequality
P 0 = = 1italwaysfollows x 0 1.Thefirstcoefficientoftheexpansion x 1 is of an

important physical sense:

1

3
2

= χ χ χ = x 1 x ( ) d 3 g . (1.61)

−1

From the phase function interpretation as a probability density of the scat-
tering to the certain angle it follows that value g = | 3is themeancosineof

scattering angle . It determines the elongation of the phase function, namely,
as g is closer to unity then the phase function is more extended to the forward
direction and weaker extended to the backscatter direction. In the context of
parameter g the Henyey-Greenstein approximation (1.31) is appropriate. It is
easy testing that its mean cosine is just equal to the parameter of the approx-
imation and it is specified with the same sign g (but it is not otherwise, the
using of sign g for the mean cosine does not imply the Henyey-Greenstein ap-
proximation is obligatory). Other expansion items of the Henyey-Greenstein
function over Legendre Polynomials are also simply expressed through its pa-
rameter: x i

x 1

(2 i +1) g i . This very reason determines the wide application of
the Henyey-Greenstein function but not an accuracy of the real phase function
approximation.
Practically the series is to break at the certain item with number N .The
value N was shown in the study by Dave (1970) to reach hundreds and even
thousands to approximate the phase function with the necessary accuracy.
It is not appropriate for expansion (1.60) using for the radiance calculation
even with modern computers. It is the essential problem of the application of
the described methodology. We would like to point out that for the molecu-
lar scattering determined by (1.25) the phase function is much more simple
( N

=

=

2):

δ

)+ 1−

χ = χ χ x m ( ) P 0 ( P 2 ( ).

δ 2+

χ The phase function cosine intransfer equation(1.47) (and inall consequences

from it) is a function of directions of incident and scattered radiation (1.46).