# Rayleigh Scattering: Phase Function

**Definitions of Angles**

See Figure 11. The angle between the incident flux and the
local normal is given by and µ_{0} cos().
The angle between the local normal and the observer is given by
and µ cos(). The phase angle,
, is the angle between the incident and emission through the
origin. The azimuthal angle, , is the projection of the
solar incidence and emission directions onto the local horizon. Using
spherical trigonometry cosine laws (*e.g.*, see CRC pg. 146) we can
easily obtain the value of ) given µ_{0}, µ, and . For solar scattering to an observer above the atmosphere the
angles are related by:

cos() = [(1-µ_{0}^{2})(1-µ^{2})] · cos() -/+
µ µ_{0} (S = +, T = -)

This is the method employed by Chandrasekar, Liou, and others. We could also write the equations in terms of the scattering angles. The scattering angle between incidence and emission, , which is related to the phase function, is given by = - . The scattering angle in the horizontal plane, is related to the azimuthal angle, and is given by = - . This definition is used by Hansen, Tomasko, and Danielson. It is the method employed within the VIAMP programs. Note the sign change between the two methods.

cos() = [ (1-µ_{0}^{2})(1-µ^{2})] · cos() ±
µ µ_{0} (T = +, S = -)

The Rayleigh phase function is given in terms of scattering angles or phase angles by

*P*() = (3/4) (1 + cos

_{ray}^{2}()) = (3/4)(1 + cos

^{2}()) =

*P*()

_{ray}
The cos^{2} term can be expressed in terms of
µ, µ_{0}, and as

cos^{2}() = (1-µ_{0}^{2})(1-µ^{2}) · cos^{2}()
± 2µµ_{0} [ (1-µ_{0}^{2})(1-µ^{2}) ] · cos()
+ (µ^{2} µ_{0}^{2})

cos(2) = 2 cos^{2}() - 1 or,

^{2}() = 1/2 cos(2) + 1/2

cos^{2}() = 1/2 (1-µ_{0}^{2})(1-µ^{2}) · cos(2)
±
2 µ µ_{0} [ (1-µ_{0}^{2})(1-µ^{2}) ] · cos()
+
µ^{2} µ_{0}^{2} + 1/2 (1-µ_{0}^{2})(1-µ^{2})

So the Rayleigh phase function can be written in terms of 3 Fourier moments:

*P _{ray}*() =

*P*+

_{1}*P*cos() +

_{2}*P*cos(2)

_{3}

*P _{0}* = 3/4 [ 1 + µ

^{2}µ

_{0}

^{2}+ 1/2 (1-µ

_{0}

^{2})(1-µ

^{2}) ] = 3/8 ( 3 + 3 µ

^{2}µ

_{0}

^{2}-µ

_{0}

^{2}-µ

^{2})

*P _{1}* = ± 3/2 ( µ µ

_{0}[(1-µ

_{0}

^{2})(1-µ

^{2})] ) (T = +, S = -)

*P _{2}* = 3/8 (1-µ

_{0}

^{2})(1-µ

^{2})