Scat nr2 QScatrN rdr660

The asymmetry factor, g, can be computed from

where a* denotes the complex conjugate of a. We recall that the complex conjugate of a complex number is obtained by replacing i by —i, wherever it occurs (implicitly or explicitly), and that the product of a complex number and its conjugate is equal to the square of its magnitude.

In Fig. 6.5, we present the variation of Qscat as a function of the size parameter x, for n = 1.4. We see that if the scattering particle has no absorption (n' = 0), Qscat has a sinusoidal form with damping, arising from the interference of light that is refracted through the particle and diffracted around the particle. The irregularities superimposed on the form, correspond to edge phenomena around the scattering sphere. In this case, Qext = Qscat and for large x, Qscat ^ 2. As the absorption by the particle increases (increasing values of the imaginary part n' of the refractive index), the damping becomes stronger until there is

Flg. 6.5. Qscat versus size parameter Flg. 6.6. Qext, Qscat and Qabs versus x, for n = 1.4, for four values of the phase-lag parameter 2x\rn — 1|.

the imaginary part of the refractive index.

saturation beyond x = 10. For large particle sizes and small wavelengths, x becomes very large and the propagation of the EM wave can be described in terms of geometric optics. In Fig. 6.6, we demonstrate the effect of absorption on Qscat and Qabs, for the case of m = 1.2 — 0.3«, as functions of the phaselag parameter 2x|m — 1| describing the phase lag between the wave passing through the sphere and outside the sphere. As the phase-lag parameter increases Qscat ^ Qabs ^ 1. In Fig. 6.7 we show the various Q values for the case of a factor ten less absorption, n = 0.03. In Fig. 6.8 we show the variation of the asymmetry factor g for n = 1.33 for different absorption as described by n .In the case of zero absorption n = 0 we see that g approaches about 0.87 at large particle sizes.

In Fig. 6.9 we show the shape of the phase function with scattering angle 0, for different particle sizes for a refractive index m = 1.371 — 0.272i, which is typical for water at 3 ^m. In the natural atmosphere the size of the particles varies within a large range of values. Usually, a Gamma function distribution is used to describe the probability of finding a particle of specific size r p{r) = (6-63)

where a = (r — rmin)/3, rmin is the smallest radius considered, y is the shape parameter of the distribution that defines the relative population of particles smaller than the most probable size, ¡3 is the scale parameter that controls the width of the distribution and r is the Gamma function. As an example, Fig. 12.9 represents the resulting average phase function distribution over the range of values covered by the Gamma function distribution shown in Fig. 6.11.

Fig . 6.7. The variation of Qext, Qscat and Qabs versus the phase-lag parameter 2x\rn — 1|, for the case of small absorption.

Flg. 6.8. The variation of the asymmetry factor g for n = 1.33 for different absorption as described by n .

Flg. 6.9. The shape of the phase function for different particle sizes, x, for a refractive index m typical of water at 3 ¡m.

Flg. 6.10. Phase function with refractive index m = 1.55 — 0.01i, for a Gamma distribution in sizes (shown below). Phase functions for the different particle sizes of the distribution are also shown.

6.3.3 Rayleigh scattering

Rayleigh scattering by atmospheric molecules is important for wavelengths up to 1.0 ym, as the Rayleigh scattering cross-section decreases rapidly with wave-

Radius (nm)

FlG. 6.11. A Gamma function distribution of particle sizes. The shape parameter is Y and the scale parameter is ¡3.

length. The real part of the refractive index of an atmosphere consisting of scattering molecules that are far apart is given by n = 1 + 2naN, (6.64)

where N is the number density of the molecules and a is called the polarizability of the molecule (see van de Hulst). The Rayleigh scattering cross-section for a molecule is given by or = —k4a2d, (6.65)

where k = 2n/A, and A is the wavelength in vacuum. The original result of Rayleigh was for spherical top molecules, so there is a small but significant correction 5 for nonspherical molecules. The refractive indices of molecular gases at STP are tabulated, for example, in Allen (1976) and have the form

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