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Depolarization by Air

Rayleigh scattering, as described above, is valid for an ensemble of isotropic spherical particles. However, the molecules which comprise air (principly N2 and O2) are diatomic and hence slightly anisotropic. The polarizability of a molecule will depend on its orientation relative to the direction of the incident light and, in general, is a tensor of rank 2. For diatomic molecules, the polarizability reduces to parallel and perpendicular components.

In practice, molecular anisotropy can be accounted for by considering molecular scattering to be a combination of true Rayleigh and isotropic scattering (Chandrasekhar, 1959) through the use of a depolarization factor. Assuming incident unpolarized radiation, the depolarization factor has a value equal to the ratio of the perpendicular and parallel scattered intensities at right angles, $\rho= ( I_{\perp}/I_{\parallel} ) _{\Theta=\pi/2}$. The depolarization factor is slightly wavelength-dependent and is different for different molecules. The value for air is 0.031 (Hansen and Travis, 1974).

After accounting for depolarization, the Rayleigh scattering matrix takes the form,

 \begin{displaymath}{\b P}(\Theta) = \frac{3}{2} \frac{(1-\rho)}{(1+\rho/2)}
... & \frac{(1-2\rho)}{(1-\rho)}\cos\Theta \\
\end{displaymath} (3.55)

and the Rayleigh scattering cross-section becomes,

\begin{displaymath}\sigma_R = \frac{8 \pi^3}{3} \frac{(m^2-1)^2}{\lambda^4 N^2}
\end{displaymath} (3.56)

As both m and $\rho$ are weak functions of wavelength, the Rayleigh scattering cross-section has a slight departure from the expected inverse-fourth power relationship. An empirical relation for the Rayleigh scattering cross-section, which accounts for molecular anisotropy, is used throughout this study (Nicolet, 1984),

\begin{displaymath}\sigma_R = \frac{4.02 \times 10^{-28}}{\lambda^{4+\chi}}
\end{displaymath} (3.57)


\begin{displaymath}\chi = \left\{ \begin{array}{ll}
0.389 \lambda - 0.3228 + 0.0...
... m} \\
0.04 & \lambda > 0.55~\rm {\mu m}
\end{array} \right.
\end{displaymath} (3.58)

and where $\sigma_R$ is in units of cm2 and $\lambda$ in $\mu $m. While technically not correct, the term `Rayleigh' is retained to designate scattering by the molecules which comprise air.

Assuming initially unpolarized light, the degree of linear polarization upon a Rayleigh scattering event is, by equation (2.42),

\begin{displaymath}LP(\Theta) = \frac{\sin^2\Theta} {\frac{(1+\rho)}{(1-\rho)} +\cos^2\Theta}.
\end{displaymath} (3.59)

Hence, accounting for molecular anisotropy results in a maximum polarization of less than one. For $\rho=0.031$, the maximum polarization is $LP(90^{\circ})=0.94$. Recognition of this fact has led to the discovery of ice crystals, small enough to be true Rayleigh scatterers, in the mesosphere. The measured polarization was too close to unity to be from molecularly scattered light (Witt et al., 1976).

Raman scattering, the inelastic version of Rayleigh scattering, is discussed briefly in Chapter 6 as it impacts the retrieval of atmospheric trace gases.

next up previous
Next: Atmospheric Aerosols Up: Types of Scattering Previous: Rayleigh Scattering
Chris McLinden