ATSC 5003 Atmospheric Radiation Lab
Rayleigh Scattering, phase function, and skylight
See Liou, section 3.3
In the formulation of Rayleigh scattering the following is written to describe the scattered E&M
wave, E, from a small spherical particle, E = Eo exp[-ik(r-ct)] • k2 • α/r • sin(γ), where k is the
circular wave number = 2 π / λ, α the polarizability of the medium, r the distance from the
scatterer, and γ the angle of the scattered wave. This relation holds if the particle size is << λ.
Mathematical exercises:
1) Write E as two perpendicular
components, Er, perpendicular to
the plane of scattering, and El in
the plane of scattering, thereby
defining γr (=γ1 in figure) and γl
(=γ2 in figure) in terms of the
scattering angle θ. The plane of
scattering is the plane defined by the incident and scattered wave.
2)
Using the fact that scattered intensity I is proportional to │E│2, show that for sunlight,
which is unpolarized, I = Io (α/r)2 (2 π/λ)4 (1+cos2θ)/2.
3) A phase function, P(θ), describes the distribution of scattered radiation as a function of
angle from the incident radiation, with the normalization that ∫4π P(θ)/(4 π) dΩ = 1. Show
that the phase function for Rayleigh scattering is P(θ) = ¾ (1 + cos2θ) and thus that
Rayleigh scattering intensity can be written as I(θ) = Io (α/r)2 128 π5/(3 λ4) P(θ)/(4 π)
4) The degree of linear polarization is defined as LP = -Q/I, for the Stokes vectors Q = Il –
Ir, and I = Il + Ir. Show that for Rayleigh scattering LP = sin2θ / (1 + cos2θ).
For the following graphical exercises we are only interested in either the angular or wavelength
dependence, so we can assume that Io, α, and r = 1.0
5) Prepare a graph to show the variation of scattered intensity from unpolarized sunlight as a
function of wavelength in the visible. Show this relationship for several angles, e.g. 0,
π/4, π/2, 3π/4, and π. Include on this page a second graph showing the scattered intensity
of the visible spectrum relative to the scattered intensity in the violet at 380 nm. Based on
this relationship explain why the sky is blue and sunsets are red.
6) Graph using a polar plot (see /polar keyword in the IDL call to plot), Il, Ir and I, thus
showing the scattering pattern for Rayleigh scattering by a small object such as a
molecule. If this scattering pattern were extended to three dimensions what would it be
called?
7) Graph the degree of polarization as a function of angle from forward scatter. Where do
you find the greatest polarization? How could you test this result with a simple
observation?