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Rayleigh scattering

Rayleigh scattering#

Scattering of light by molecules is well modeled using the Rayleigh approximation (see [Liou, 2002], section 3.3.1 for example).

Phase function#

We use the phase function \([/]\) (units: \(\mathrm{str}^{-1}\)) expression provided by [Hansen and Travis, 1974] (eq. 2.14):

\[p(\theta) = \frac{\Delta}{4 \pi} \left[ \frac{3}{4} \left( 1 + \cos^2 \theta \right) \right] + \frac{1 - \Delta}{4 \pi}\]

where

  • \(\theta\) is the scattering angle \([/]\) (units: \(\mathrm{rad}\)) and

  • \(\Delta\) \([/]\) is given by:

\[\Delta = \frac{1 - \delta}{1 + \delta / 2}\]

where \(\delta\) is the depolarisation factor \([/]\).

We take \(\delta = 0\) for air, which gives \(\Delta = 1\).

Warning

We use the following normalisation rule for the phase function:

\[\int_{0}^{2\pi}\int_{0}^{\pi} p(\theta) \sin\theta \, d\theta \, d\phi = 1\]

where \(\phi\) is the azimuth angle \([/]\) (units: \(\mathrm{rad}\)).

Scattering coefficient#

We use the expression of the scattering coefficient \([L^{-1}]\) for a pure gas provided by [Eberhard, 2010] (eq. 60), and apply it to air:

\[k_{\mathrm s \, \lambda} (n) = \frac{8 \pi^3}{3 \lambda^4} \frac{1}{n} \left( \eta_{\lambda}^2(n) - 1 \right)^2 F_{\lambda}\]

where

  • \(\lambda\) is the wavelength \([L]\) (subscript indicates spectral dependence),

  • \(n\) is the air number density \([L^{-3}]\),

  • \(\eta\) is the air refractive index \([/]\) and

  • \(F_{\lambda}\) is the air King correction factor \([/]\).

Note

Since air is considered as a pure gas, variations of the scattering coefficient with composition are neglected.

The spectral dependence of the air refractive index is computed according to [Peck and Reeder, 1972] (eq. 2), valid in the wavelength range \([230, 1690]\) nm. In the wavelength range [1690, 2400] nm, we assume that this equation remains accurate enough, although we have no data to justify that assumption.

The number density dependence is computed using the simple proportionality rule:

\[\eta(n) = \frac{n}{n_0} \eta_0\]

where

  • \(n_0\) is the air number density under standard conditions of pressure and temperature (101325 Pa, 288.15 K) and

  • \(\eta_0\) is the air refractive index under standard conditions of pressure and temperature.

The King correction factor is computed using the data from [Bates, 1984] (table 1). The data is interpolated within the [200, 1000] nm range and extrapolated using the 1000 nm value for wavelength larger than 1000 nm.

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