1.3: Diffuse Reflection and Transmission

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Page ID: 7491

Max Fairbairn & Jeremy Tatum
University of Victoria

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The fundamental problem of planetary photometry is the diffuse reflection and transmission of a plane parallel beam of radiation by a scattering medium, which we would understand as a planetary atmosphere and/or surface or a planetary layer such as the rings of Saturn. Such media may be idealised as locally plane parallel strata in which physical properties are uniform throughout a given layer. In such cases we may use a hybrid Cartesian and spherical frame of reference in which the Oxy plane is the surface and z-axis points in the direction of the surface normal. Directions are then specified by the polar and azimuthal angles \(ϑ\) and \(φ\) (“curly theta” and “curly phi”) respectively. Further, with problems of this kind, rather than working in actual physical distances it is preferable to work in terms of normal optical thickness t, measured downwards from z = 0, such that \(dτ = -κρ dz\). Radiation that has traversed a path of optical thickness t is attenuated by a factor of e^-t.

Using the direction cosine \(m = \cos ϑ\) the standard form of the equation of transfer for plane parallel media is

\[\mu \frac{d L(\tau, \mu, \varphi)}{d \tau}=L(\tau, \mu, \varphi)-\Im(\tau, \mu, \varphi)\]

For a scattering medium, the only contribution to the source function is the scattering of that radiation which has been incident on the medium from external sources, so that, by totalling the contributions impinging on level τ from all directions, the source function is

\[\mathfrak{J}(\tau, \mu, \varphi)=\frac{1}{4 \pi} \int_{-1}^{1} \int_{0}^{2 \pi} p\left(\mu, \varphi ; \mu^{\prime}, \varphi^{\prime}\right) L\left(\tau, \mu^{\prime}, \varphi^{\prime}\right) d \varphi^{\prime} d \mu^{\prime}\]

where p is the normalised phase function which determines the angular distribution of the scattering. A convenient way to think of p is that \( \frac{p}{4 \pi} d \omega\) is the probability that a photon travelling in the direction (μ', φ') would be scattered into an elemental solid angle dw in the direction (μ, φ).

Radiation traversing a normal optical thickness dt in the direction (m,j) will be attenuated by the amount δL = Lδτ/μ. Of this amount, a fraction can be attributed to that caused by scattering alone – this fraction is called the single scattering albedo ϖ₀. It then follows that the phase function p must be normalised according to

\[\int_{4 \pi} \frac{p}{4 \pi} d \omega=0 \leq \varpi_{0} \leq 1\]

and if p is a constant, then p = ϖ₀ and the scattering is isotropic.