Solar Radiation in the Atmosphere - Short-Wave Solar Radiation in the Earth's Atmosphere - page 32

function (1.50), compute the integral over the azimuth and level items with
the equal numbers m in the left-hand and right-hand parts of the equation.
Only the items with equal numbers m will be nonzero in the product of the
series (the phase function by the radiance) due to the orthogonality of the
trigonometric functions:

π 2

ϕ cos m 2 ϕ d ϕ = 0for m 1 = cos m 1 m 2 ,

0

π = for m 1 m 2 ,

π

2

ϕ cos m 2 ϕ d ϕ = sin m 1 0.

0

Finally, obtain:

1

= ω 0 (

τ )

B m ( p m ( µ ) I m ( µ ) d µ

τ µ µ 0 ) τ µ τ µ , , , , , ,

2

(1.66)

−1

+ ω 0 (

τ )

Sp m ( τ µ µ 0 )exp(− τ|µ 0 ). , ,

4

Further from (1.51) the following equation is derived:

τ µ µ 0 )

dI m (

, ,

µ = − I m ( τ µ µ 0 )+ B m ( τ µ µ 0 ) , , , , , (1.67)

τ d

with boundary conditions:

I m (0, µ µ 0 ) = µ 0 > 0and , 0, for

(1.68)

I m ( τ 0 , µ µ 0 ) = µ 0 < 0. , 0for

The following relations are correct for it:

τ

µ 0 )exp − τ

d

τ 1

µ

−

I m ( τ µ µ 0 ) = B m ( τ , µ τ µ , , , > 0,

µ

0

(1.69)

τ 0

µ 0 )exp − τ

d

τ

− 1
µ

−

I m ( B m (

τ , τ

τ µ µ 0 ) = µ µ

, , , < 0.

µ

τ