# Atmospheric phenomena caused by scattering

Lord Rayleigh

In the article radiation in participating media, scattering was discussed for its effect on the intensity passing through a participating medium. Most discussion was for isotropic scattering. Scattering of solar radiation does occur in the Earth's atmosphere. Even on a clear day, solar radiation is scattered from very small aerosol particles and even from molecules in the atmosphere.

When the parameter $\varsigma = \pi D/\lambda {\rm{ }}$ is small, the scattering is a limiting case of Mie scattering called Rayleigh scattering, after Lord Rayleigh (right) who derived the scattering relations using dimensional analysis. In Rayleigh scattering, the wavelength dependent scattering coefficient σsλ is found to be proportional to 1 / λ4. The complete relation for nonabsorbing particles in air depends on the particle diameter D, refractive index of the scattering particles n, the number density of particles N (particles/m3) and $\varsigma$ through

${\sigma _{s,\lambda }} = \frac{8}{3}\frac{{\pi {D^2}}}{4}\frac{{{\varsigma ^4}}}{N}{\left( {\frac{{{n^2} - 1}}{{{n^2} + 2}}} \right)^2}\qquad \qquad(1)$
Figure 1: Atmospheric scattering causing a blue sky.

The wavelength dependence enters through $\varsigma$. Remembering that the visible portion of the spectrum stretches from the blue portion at about λ = 0.4εm to red at about 0.7 εm (see electromagnetic waves and thermal radiation), we expect the atmosphere to scatter much more of the blue portion than the red in the visible portion of the spectrum. Thus after multiple scatterings over the long mean free paths present in the clear atmosphere (Fig. 1), what an observer on the ground sees when looking at the sky is multiply scattered solar radiation strongly biased to the shorter wavelengths… a blue sky!

Figure 2: Atmospheric scattering resulting in a red sunset.
Figure 3: Phase function for Rayleigh scattering.

Similar reasoning is used to examine what happens to solar radiation in the atmosphere near sunset. Figure 2 indicates that near sunset, solar radiation takes a much longer path through the atmosphere, and more scattering occurs. The blue portion of the spectrum is lost through Rayleigh scattering along the path. What is left when viewing the sun is the remaining portion of the visible spectrum, strongly skewed toward the longer wavelengths (red). The effect is even stronger because the refractive index change of the atmosphere with

altitude curves the sunlight around the Earth's curvature, making the pathlength even longer. The phase function Φ(θ) for Rayleigh scattering is given by the smooth function

$\Phi (\theta ) = \frac{3}{4}\left( {1 + {{\cos }^2}\theta } \right)\qquad \qquad(2)$

which has a lobed shape with scattering strongly into the forward direction, an equal amount backscattered, and small scattering into side directions relative to the direction of travel of the intensity (Fig. 3).

If you look at angles near the sun from Earth around noon, the sky near the sun appears nearly white rather than blue. This is because you are observing solar radiation that has been near-forward scattered once or twice into angles near the solar direction, so the blue-scattering effects have not yet become dominant. You are observing forward scattered (white) light from across the visible spectrum.

## References

Faghri, A., Zhang, Y., and Howell, J. R., 2010, Advanced Heat and Mass Transfer, Global Digital Press, Columbia, MO.