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1.8: Diffuse Reflection and Transmission

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    A scattering layer of finite optical thickness t may be used to model e.g. a planetary ring. If we use the Lommel-Seeliger model, then the reflected radiance of such a layer may be determined by changing the upper limit of the integral in equation (20) so that

    \[L_{r}=\frac{\varpi_{0} \mathbf{F}}{4 \pi \mu} \times \int_{0}^{t} \exp \left[-\tau\left(\frac{1}{\mu_{0}}+\frac{1}{\mu}\right)\right] d \tau\]

    resulting in

    \[L_{r}=\frac{w_{0}}{4 \pi} \frac{1}{\mu+\mu_{0}} \times \left[1-\exp \left\{-t\left(\frac{1}{\mu_{0}}+\frac{1}{\mu}\right)\right\}\right] \mu_{0} \mathbf{F}\]

    For the transmitted radiance, it is readily shown that

    \[d L_{t}=\frac{\varpi_{0} \mathbf{F} e^{-\tau / \mu_{0}}}{4 \pi \mu} e^{-(t-\tau) / \mu} d \tau\]

    and in the special case \(μ = μ_0\), integration results in

    \[L_{t}=\frac{\varpi_{0} \mathbf{F} t}{4 \pi \mu_{0}} e^{-t / \mu_{0}}\]

    and otherwise

    \[L_{t}=\frac{\varpi_{0} \mathbf{F}}{4 \pi} \frac{\mu_{0}}{\mu-\mu_{0}}\left[e^{-t / \mu}-e^{-t / \mu_{0}}\right]\]

    In all cases the values of μ and μ0 are positive; some authors even explicitly put in absolute value symbols to emphasise this point!

    This page titled 1.8: Diffuse Reflection and Transmission was last modified on Wed, 30 Dec 2020 21:12:11 GMT and is shared under a CC BY-NC 4.0 license and was authored, remixed, and/or curated by Max Fairbairn & Jeremy Tatum via source content that was edited to the style and standards of the LibreTexts platform.