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<title>Radiative transfer</title>

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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Radiative transfer</span></span>

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<p><b>Radiative transfer</b> (also called <b>radiation transport</b>) is the physical phenomenon of energy transfer in the form of electromagnetic radiation. The propagation of radiation through a medium is affected by <a href="Absorption_(electromagnetic_radiation)" title="Absorption (electromagnetic radiation)">absorption</a>, <a href="Emission_(electromagnetic_radiation)" class="mw-redirect" title="Emission (electromagnetic radiation)">emission</a>, and <a href="Scattering" title="Scattering">scattering</a> processes. The <b>equation of radiative transfer</b> describes these interactions mathematically. Equations of radiative transfer have application in a wide variety of subjects including optics, astrophysics, atmospheric science, and <a href="Remote_sensing" title="Remote sensing">remote sensing</a>. Analytic solutions to the radiative transfer equation (RTE) exist for simple cases but for more realistic media, with complex multiple scattering effects, numerical methods are required.

The present article is largely focused on the condition of <a href="Radiative_equilibrium" title="Radiative equilibrium">radiative equilibrium</a>.<sup id="cite_ref-chandrasekhar_1-0" class="reference"><a href="#cite_note-chandrasekhar-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-lenoble_2-0" class="reference"><a href="#cite_note-lenoble-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup>

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<div class="mw-heading mw-heading2"><h2 id="Definitions">Definitions</h2></div>

<p>The fundamental quantity that describes a field of radiation is called <b><a href="Spectral_radiance" title="Spectral radiance">spectral radiance</a></b> in radiometric terms (in other fields it is often called <b><a href="Specific_radiative_intensity" class="mw-redirect" title="Specific radiative intensity">specific intensity</a></b>). For a very small area element in the radiation field, there can be electromagnetic radiation passing in both senses in every spatial direction through it. In radiometric terms, the passage can be completely characterized by the amount of energy radiated in each of the two senses in each spatial direction, per unit time, per unit area of surface of sourcing passage, per unit <a href="Solid_angle" title="Solid angle">solid angle</a> of reception at a distance, per unit wavelength interval being considered (<a href="Polarization_(waves)" title="Polarization (waves)">polarization</a> will be ignored for the moment).

</p><p>In terms of the spectral radiance, <span class="nowrap"><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{\nu }}">

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<annotation encoding="application/x-tex">{\displaystyle I_{\nu }}</annotation>

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</math></span><img src="./172f1a90292ff1160dabfbc32fd0dcb802fc9f98.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.127ex; height:2.509ex;" alt="{\displaystyle I_{\nu }}" loading="lazy"></span>,</span> the energy flowing across an area element of area <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle da}">

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<annotation encoding="application/x-tex">{\displaystyle da}</annotation>

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</math></span><img src="./4bc9b93f275d94ae7e0e9ed1e8acc754cb751418.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.446ex; height:2.176ex;" alt="{\displaystyle da}" loading="lazy"></span> located at <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {r} }">

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</math></span><img src="./eca0f46511c4c986c48b254073732c0bd98ae0c1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.102ex; height:1.676ex;" alt="{\displaystyle \mathbf {r} }" loading="lazy"></span> in time <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle dt}">

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<annotation encoding="application/x-tex">{\displaystyle dt}</annotation>

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</math></span><img src="./ebee76a835701fd1f26047a09855f2ea36bb08fc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.055ex; height:2.176ex;" alt="{\displaystyle dt}" loading="lazy"></span> in the solid angle <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d\Omega }">

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<annotation encoding="application/x-tex">{\displaystyle d\Omega }</annotation>

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</math></span><img src="./9a77777df83cbdbf96f0eaa6e74ba90eba9ce43b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.894ex; height:2.176ex;" alt="{\displaystyle d\Omega }" loading="lazy"></span> about the direction <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {\mathbf {n} }}}">

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<annotation encoding="application/x-tex">{\displaystyle {\hat {\mathbf {n} }}}</annotation>

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</math></span><img src="./aae87b164ba005e99b51066c46d1eacc7f56564a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.485ex; height:2.343ex;" alt="{\displaystyle {\hat {\mathbf {n} }}}" loading="lazy"></span> in the frequency interval <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu \,}">

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<annotation encoding="application/x-tex">{\displaystyle \nu \,}</annotation>

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</math></span><img src="./32a13ab64f1a72c7827e70aed42c8fcb323bd7bf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.619ex; height:1.676ex;" alt="{\displaystyle \nu \,}" loading="lazy"></span> to <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu +d\nu \,}">

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<annotation encoding="application/x-tex">{\displaystyle \nu +d\nu \,}</annotation>

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</math></span><img src="./3f0853787a777205a4a10977ef8b24da125bb416.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.908ex; height:2.343ex;" alt="{\displaystyle \nu +d\nu \,}" loading="lazy"></span> is

</p><p><span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle dE_{\nu }=I_{\nu }(\mathbf {r} ,{\hat {\mathbf {n} }},t)\cos \theta \ d\nu \,da\,d\Omega \,dt}">

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<annotation encoding="application/x-tex">{\displaystyle dE_{\nu }=I_{\nu }(\mathbf {r} ,{\hat {\mathbf {n} }},t)\cos \theta \ d\nu \,da\,d\Omega \,dt}</annotation>

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</p><p>where <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta }">

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<annotation encoding="application/x-tex">{\displaystyle \theta }</annotation>

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</math></span><img src="./6e5ab2664b422d53eb0c7df3b87e1360d75ad9af.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }" loading="lazy"></span> is the angle that the unit direction vector <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {\mathbf {n} }}}">

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<annotation encoding="application/x-tex">{\displaystyle {\hat {\mathbf {n} }}}</annotation>

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</math></span><img src="./aae87b164ba005e99b51066c46d1eacc7f56564a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.485ex; height:2.343ex;" alt="{\displaystyle {\hat {\mathbf {n} }}}" loading="lazy"></span> makes with a normal to the area element. The units of the spectral radiance are seen to be energy/time/area/solid angle/frequency. In MKS units this would be <span class="nowrap">W·m<sup>−2</sup>·sr<sup>−1</sup>·Hz<sup>−1</sup></span> (watts per square-metre-steradian-hertz).

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<div class="mw-heading mw-heading2"><h2 id="The_equation_of_radiative_transfer">The equation of radiative transfer</h2></div>

<p>The equation of radiative transfer simply says that as a beam of radiation travels, it loses energy to absorption, gains energy by emission processes, and redistributes energy by scattering. The differential form of the equation for radiative transfer is:

</p><p><span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{c}}{\frac {\partial I_{\nu }}{\partial t}}+{\hat {\Omega }}\cdot \nabla I_{\nu }+\left(k_{\nu ,s}+k_{\nu ,a}\right)\rho I_{\nu }=j_{\nu }\rho +{\frac {k_{\nu ,s}\rho }{4\pi }}\int _{\Omega }I_{\nu }\,d\Omega }">

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<annotation encoding="application/x-tex">{\displaystyle {\frac {1}{c}}{\frac {\partial I_{\nu }}{\partial t}}+{\hat {\Omega }}\cdot \nabla I_{\nu }+\left(k_{\nu ,s}+k_{\nu ,a}\right)\rho I_{\nu }=j_{\nu }\rho +{\frac {k_{\nu ,s}\rho }{4\pi }}\int _{\Omega }I_{\nu }\,d\Omega }</annotation>

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</p><p>where <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c}">

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<annotation encoding="application/x-tex">{\displaystyle c}</annotation>

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</math></span><img src="./86a67b81c2de995bd608d5b2df50cd8cd7d92455.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\displaystyle c}" loading="lazy"></span> is the speed of light, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j_{\nu }}">

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<annotation encoding="application/x-tex">{\displaystyle j_{\nu }}</annotation>

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</math></span><img src="./2d7118a0b30c7fbe680079f309ec8d84eebc88a2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:2.088ex; height:2.509ex;" alt="{\displaystyle j_{\nu }}" loading="lazy"></span> is the emission coefficient, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k_{\nu ,s}}">

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<annotation encoding="application/x-tex">{\displaystyle k_{\nu ,s}}</annotation>

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</math></span><img src="./89f255959e9e08f6c12f3edf668ddddf5f4d7615.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.543ex; height:2.843ex;" alt="{\displaystyle k_{\nu ,s}}" loading="lazy"></span> is the scattering opacity, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k_{\nu ,a}}">

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<annotation encoding="application/x-tex">{\displaystyle k_{\nu ,a}}</annotation>

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</math></span><img src="./41b266b28daeeeffb526447a879d308afe919f81.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.642ex; height:2.843ex;" alt="{\displaystyle k_{\nu ,a}}" loading="lazy"></span> is the absorption opacity, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho }">

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<annotation encoding="application/x-tex">{\displaystyle \rho }</annotation>

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</math></span><img src="./1f7d439671d1289b6a816e6af7a304be40608d64.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.202ex; height:2.176ex;" alt="{\displaystyle \rho }" loading="lazy"></span> is the mass density and the <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\frac {1}{4\pi }}k_{\nu ,s}\int _{\Omega }I_{\nu }\,d\Omega }">

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<annotation encoding="application/x-tex">{\textstyle {\frac {1}{4\pi }}k_{\nu ,s}\int _{\Omega }I_{\nu }\,d\Omega }</annotation>

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</math></span><img src="./a5824ee6d98ef7d865e9d324c1aaf26843436041.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:14.841ex; height:3.676ex;" alt="{\textstyle {\frac {1}{4\pi }}k_{\nu ,s}\int _{\Omega }I_{\nu }\,d\Omega }" loading="lazy"></span> term represents radiation scattered from other directions onto a surface.

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<div class="mw-heading mw-heading2"><h2 id="Solutions_to_the_equation_of_radiative_transfer">Solutions to the equation of radiative transfer</h2></div>

<p>Solutions to the equation of radiative transfer form an enormous body of work. The differences however, are essentially due to the various forms for the emission and absorption coefficients. If scattering is ignored, then a general steady state solution in terms of the emission and absorption coefficients may be written:

</p><p><span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{\nu }(s)=I_{\nu }(s_{0})e^{-\tau _{\nu }(s_{0},s)}+\int _{s_{0}}^{s}j_{\nu }(x)e^{-\tau _{\nu }(x,s)}\,dx}">

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<annotation encoding="application/x-tex">{\displaystyle I_{\nu }(s)=I_{\nu }(s_{0})e^{-\tau _{\nu }(s_{0},s)}+\int _{s_{0}}^{s}j_{\nu }(x)e^{-\tau _{\nu }(x,s)}\,dx}</annotation>

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</p><p>where <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau _{\nu }(s_{1},s_{2})}">

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<annotation encoding="application/x-tex">{\displaystyle \tau _{\nu }(s_{1},s_{2})}</annotation>

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</math></span><img src="./c89f97170b271b5cd91a36bd190be3d9fc9be375.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.252ex; height:2.843ex;" alt="{\displaystyle \tau _{\nu }(s_{1},s_{2})}" loading="lazy"></span> is the <a href="Optical_depth" title="Optical depth">optical depth</a> of the medium between positions <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s_{1}}">

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</math></span><img src="./d4b9a7acc0ae8f54da4b7f4eef2c777d44faecd4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.145ex; height:2.009ex;" alt="{\displaystyle s_{2}}" loading="lazy"></span>:

</p><p><span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau _{\nu }(s_{1},s_{2})\ {\stackrel {\mathrm {def} }{=}}\ \int _{s_{1}}^{s_{2}}\alpha _{\nu }(s)\,ds}">

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<div class="mw-heading mw-heading3"><h3 id="Local_thermodynamic_equilibrium">Local thermodynamic equilibrium</h3></div>

<p>A particularly useful simplification of the equation of radiative transfer occurs under the conditions of <a href="Local_thermodynamic_equilibrium" class="mw-redirect" title="Local thermodynamic equilibrium">local thermodynamic equilibrium</a> (LTE). It is important to note that local equilibrium may apply only to a certain subset of particles in the system. For example, LTE is usually applied only to massive particles. In a radiating gas, the photons being emitted and absorbed by the gas do not need to be in a thermodynamic equilibrium with each other or with the massive particles of the gas in order for LTE to exist.

</p><p>In this situation, the absorbing/emitting medium consists of massive particles which are locally in equilibrium with each other, and therefore have a definable temperature (<a href="Zeroth_law_of_thermodynamics" title="Zeroth law of thermodynamics">Zeroth Law of Thermodynamics</a>). The radiation field is not, however in equilibrium and is being entirely driven by the presence of the massive particles. For a medium in LTE, the emission coefficient and absorption coefficient are functions of temperature and density only, and are related by:

</p><p><span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {j_{\nu }}{\alpha _{\nu }}}=B_{\nu }(T)}">

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</p><p>where <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B_{\nu }(T)}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle B_{\nu }(T)}</annotation>

</semantics>

</math></span><img src="./c94ca2424e3bc90a5acceef97eb5918bf11672d4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.313ex; height:2.843ex;" alt="{\displaystyle B_{\nu }(T)}" loading="lazy"></span> is the <a href="Black_body" title="Black body">black body</a> spectral radiance at temperature <span class="texhtml mvar" style="font-style:italic;">T</span>. The solution to the equation of radiative transfer is then:

</p><p><span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{\nu }(s)=I_{\nu }(s_{0})e^{-\tau _{\nu }(s_{0},s)}+\int _{s_{0}}^{s}B_{\nu }(T(x))\,\alpha _{\nu }(x)\,e^{-\tau _{\nu }(x,s)}\,dx}">

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<annotation encoding="application/x-tex">{\displaystyle I_{\nu }(s)=I_{\nu }(s_{0})e^{-\tau _{\nu }(s_{0},s)}+\int _{s_{0}}^{s}B_{\nu }(T(x))\,\alpha _{\nu }(x)\,e^{-\tau _{\nu }(x,s)}\,dx}</annotation>

</semantics>

</math></span></span>

</p><p>Knowing the temperature profile and the density profile of the medium is sufficient to calculate a solution to the equation of radiative transfer.

</p>

<div class="mw-heading mw-heading3"><h3 id="The_Eddington_approximation">The Eddington approximation</h3></div>

<p>The Eddington approximation is distinct from the <a href="Two-stream_approximation" title="Two-stream approximation">two-stream approximation</a>. The two-stream approximation assumes that the intensity is constant with angle in the upward hemisphere, with a different constant value in the downward hemisphere. The Eddington approximation instead assumes that the intensity is a linear function of <span class="nowrap"><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu =\cos \theta }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mu =\cos \theta }</annotation>

</semantics>

</math></span><img src="./60c87dad43a560c1222665a86ee030c874bdcaf9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.089ex; height:2.676ex;" alt="{\displaystyle \mu =\cos \theta }" loading="lazy"></span>,</span> i.e.,

</p><p><span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{\nu }(\mu ,z)=a(z)+\mu b(z)}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle I_{\nu }(\mu ,z)=a(z)+\mu b(z)}</annotation>

</semantics>

</math></span></span>

where <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle z}</annotation>

</semantics>

</math></span><img src="./bf368e72c009decd9b6686ee84a375632e11de98.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}" loading="lazy"></span> is the normal direction to the slab-like medium. Note that expressing angular integrals in terms of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mu }</annotation>

</semantics>

</math></span><img src="./9fd47b2a39f7a7856952afec1f1db72c67af6161.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }" loading="lazy"></span> simplifies things because <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d\mu =-\sin \theta \,d\theta }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle d\mu =-\sin \theta \,d\theta }</annotation>

</semantics>

</math></span><img src="./2315a7240e27e8ec1fbb8412fe1c05b0e49056a4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.938ex; height:2.676ex;" alt="{\displaystyle d\mu =-\sin \theta \,d\theta }" loading="lazy"></span> appears in the <a href="Jacobian_matrix_and_determinant" title="Jacobian matrix and determinant">Jacobian</a> of integrals in <a href="Spherical_coordinate_system" title="Spherical coordinate system">spherical coordinates</a>. The Eddington approximation can be used to obtain the spectral radiance in a "plane-parallel" medium (one in which properties only vary in the perpendicular direction) with isotropic frequency-independent scattering.

</p><p>Extracting the first few moments of the spectral radiance with respect to <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mu }</annotation>

</semantics>

</math></span><img src="./9fd47b2a39f7a7856952afec1f1db72c67af6161.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }" loading="lazy"></span> yields

</p><p><span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}J_{\nu }&={\frac {1}{2}}\int _{-1}^{1}I_{\nu }\,d\mu =a,\\[1ex]H_{\nu }&={\frac {1}{2}}\int _{-1}^{1}\mu I_{\nu }\,d\mu ={\frac {b}{3}},\\[1ex]K_{\nu }&={\frac {1}{2}}\int _{-1}^{1}\mu ^{2}I_{\nu }\,d\mu ={\frac {a}{3}}\end{aligned}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}J_{\nu }&={\frac {1}{2}}\int _{-1}^{1}I_{\nu }\,d\mu =a,\\[1ex]H_{\nu }&={\frac {1}{2}}\int _{-1}^{1}\mu I_{\nu }\,d\mu ={\frac {b}{3}},\\[1ex]K_{\nu }&={\frac {1}{2}}\int _{-1}^{1}\mu ^{2}I_{\nu }\,d\mu ={\frac {a}{3}}\end{aligned}}}</annotation>

</semantics>

</math></span></span>

</p><p>Thus the Eddington approximation is equivalent to setting <span class="nowrap"><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle K_{\nu }={\frac {1}{3}}J_{\nu }}">

<msub>

</mrow>

</msub>

<mfrac>

</mfrac>

</mrow>

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</msub>

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</mrow>

<annotation encoding="application/x-tex">{\textstyle K_{\nu }={\frac {1}{3}}J_{\nu }}</annotation>

</semantics>

</math></span><img src="./c11136a3bad9704f829f0aecc4a235800fd650b2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:10.227ex; height:3.676ex;" alt="{\textstyle K_{\nu }={\frac {1}{3}}J_{\nu }}" loading="lazy"></span>.</span> Higher order versions of the Eddington approximation also exist, and consist of more complicated linear relations of the intensity moments. This extra equation can be used as a closure relation for the truncated system of moments.

</p><p>Note that the first two moments have simple physical meanings. <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J_{\nu }}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle J_{\nu }}</annotation>

</semantics>

</math></span><img src="./93f5eaf64d9da9c637c9e42042719a0e47bdca28.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.394ex; height:2.509ex;" alt="{\displaystyle J_{\nu }}" loading="lazy"></span> is the isotropic intensity at a point, and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H_{\nu }}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle H_{\nu }}</annotation>

</semantics>

</math></span><img src="./36749733a64d7f7cda7d3d3ce6b31d3b2bb062b3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.035ex; height:2.509ex;" alt="{\displaystyle H_{\nu }}" loading="lazy"></span> is the flux through that point in the <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle z}</annotation>

</semantics>

</math></span><img src="./bf368e72c009decd9b6686ee84a375632e11de98.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}" loading="lazy"></span> direction.

</p><p>The radiative transfer through an isotropically scattering medium with scattering coefficient <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{\nu }}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sigma _{\nu }}</annotation>

</semantics>

</math></span><img src="./50360e0a88f92f46ae5c32d659c2c472d8d8cf47.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.431ex; height:2.009ex;" alt="{\displaystyle \sigma _{\nu }}" loading="lazy"></span> at local thermodynamic equilibrium is given by

<mfrac>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mu {\frac {dI_{\nu }}{dz}}=-\alpha _{\nu }\left(I_{\nu }-B_{\nu }\right)+\sigma _{\nu }\left(J_{\nu }-I_{\nu }\right)}</annotation>

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</math></span></span>

</p><p>Integrating over all angles yields

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<annotation encoding="application/x-tex">{\displaystyle {\frac {dH_{\nu }}{dz}}=\alpha _{\nu }\left(B_{\nu }-J_{\nu }\right)}</annotation>

</semantics>

</math></span></span>

Premultiplying by <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mu }</annotation>

</semantics>

</math></span><img src="./9fd47b2a39f7a7856952afec1f1db72c67af6161.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }" loading="lazy"></span>, and then integrating over all angles gives

<mfrac>

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<msub>

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</mrow>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {dK_{\nu }}{dz}}=-\left(\alpha _{\nu }+\sigma _{\nu }\right)H_{\nu }}</annotation>

</semantics>

</math></span></span>

</p><p>Substituting in the closure relation, and differentiating with respect to <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle z}</annotation>

</semantics>

</math></span><img src="./bf368e72c009decd9b6686ee84a375632e11de98.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}" loading="lazy"></span> allows the two above equations to be combined to form the radiative diffusion equation

<mfrac>

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<annotation encoding="application/x-tex">{\displaystyle {\frac {d^{2}J_{\nu }}{dz^{2}}}=3\alpha _{\nu }\left(\alpha _{\nu }+\sigma _{\nu }\right)\left(J_{\nu }-B_{\nu }\right)}</annotation>

</semantics>

</math></span></span>

</p><p>This equation shows how the effective optical depth in scattering-dominated systems may be significantly different from that given by the scattering opacity if the absorptive opacity is small.

</p>

<ul><li><a href="Beer-Lambert_law" class="mw-redirect" title="Beer-Lambert law">Beer-Lambert law</a></li>

<li><a href="Kirchhoff's_law_of_thermal_radiation" title="Kirchhoff's law of thermal radiation">Kirchhoff's law of thermal radiation</a></li>

<li><a href="List_of_atmospheric_radiative_transfer_codes" class="mw-redirect" title="List of atmospheric radiative transfer codes">List of atmospheric radiative transfer codes</a></li>

<li><a href="Optical_depth" title="Optical depth">Optical depth</a></li>

<li><a href="Planck's_law" title="Planck's law">Planck's law</a></li>

<li><a href="Radiative_transfer_equation_and_diffusion_theory_for_photon_transport_in_biological_tissue" title="Radiative transfer equation and diffusion theory for photon transport in biological tissue">Radiative transfer equation and diffusion theory for photon transport in biological tissue</a></li>

<li><a href="Schwarzschild's_equation_for_radiative_transfer" title="Schwarzschild's equation for radiative transfer">Schwarzschild's equation for radiative transfer</a></li>

<li><a href="Vector_radiative_transfer" title="Vector radiative transfer">Vector radiative transfer</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("./mw/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("./mw/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("./mw/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("./mw/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}

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</style><cite id="CITEREFS._Chandrasekhar1960" class="citation book cs1">S. Chandrasekhar (1960). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/radiativetransfe0000chan"><i>Radiative Transfer</i></a></span>. Dover Publications Inc. p. <a rel="nofollow" class="external text" href="https://archive.org/details/radiativetransfe0000chan/page/393">393</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-486-60590-6</bdi>.</cite> <span class="cs1-hidden-error citation-comment"><code class="cs1-code">{{cite book}}</code>: </span><span class="cs1-hidden-error citation-comment">ISBN / Date incompatibility (help)</span></span>

</li>

<cite id="CITEREFJacqueline_Lenoble1985" class="citation book cs1">Jacqueline Lenoble (1985). <i>Radiative Transfer in Scattering and Absorbing Atmospheres: Standard Computational Procedures</i>. A. Deepak Publishing. p. 583. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-12-451451-5</bdi>.</cite></span>

</li>

</ol></div></div>

<div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2></div>

<ul><li><cite id="CITEREFIvan_HubenyDimitri_Mihalas2015" class="citation book cs1">Ivan Hubeny; <a href="Dimitri_Mihalas" title="Dimitri Mihalas">Dimitri Mihalas</a> (2015). <i>Theory of Stellar Atmospheres, An Introduction to Astrophysical Non-equilibrium Quantitative Spectroscopic Analysis</i>. <a href="Princeton_University_Press" title="Princeton University Press">Princeton University Press</a>. p. 944. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9780691163291</bdi>.</cite></li>

<li><cite id="CITEREFSubrahmanyan_Chandrasekhar1960" class="citation book cs1"><a href="S._Chandrasekhar" class="mw-redirect" title="S. Chandrasekhar">Subrahmanyan Chandrasekhar</a> (1960). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/radiativetransfe0000chan"><i>Radiative Transfer</i></a></span>. Dover Publications Inc. p. <a rel="nofollow" class="external text" href="https://archive.org/details/radiativetransfe0000chan/page/393">393</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-486-60590-6</bdi>.</cite> <span class="cs1-hidden-error citation-comment"><code class="cs1-code">{{cite book}}</code>: </span><span class="cs1-hidden-error citation-comment">ISBN / Date incompatibility (help)</span></li>

<li><cite id="CITEREFJacqueline_Lenoble1985" class="citation book cs1">Jacqueline Lenoble (1985). <i>Radiative Transfer in Scattering and Absorbing Atmospheres: Standard Computational Procedures</i>. A. Deepak Publishing. p. 583. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-12-451451-5</bdi>.</cite></li>

<li><cite id="CITEREFGrant_Petty2006" class="citation book cs1">Grant Petty (2006). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20130329033531/http://www.sundogpublishing.com/shop/a-first-course-in-atmospheric-radiation-2nd-ed"><i>A First Course in Atmospheric Radiation (2nd Ed.)</i></a>. Sundog Publishing (Madison, Wisconsin). <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-9729033-1-8</bdi>. Archived from <a rel="nofollow" class="external text" href="http://www.sundogpublishing.com/shop/a-first-course-in-atmospheric-radiation-2nd-ed/">the original</a> on 2013-03-29<span class="reference-accessdate">. Retrieved <span class="nowrap">2012-03-14</span></span>.</cite></li>

<li><cite id="CITEREFDimitri_MihalasBarbara_Weibel-Mihalas1984" class="citation book cs1"><a href="Dimitri_Mihalas" title="Dimitri Mihalas">Dimitri Mihalas</a>; Barbara Weibel-Mihalas (1984). <i>Foundations of Radiation Hydrodynamics</i>. Dover Publications, Inc. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-486-40925-2</bdi>.</cite></li>

<li><cite id="CITEREFGeorge_B._RybickiAlan_P._Lightman1985" class="citation book cs1">George B. Rybicki; Alan P. Lightman (1985). <i>Radiative Processes in Astrophysics</i>. Wiley-Interscience. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-471-82759-7</bdi>.</cite></li>

<li><cite id="CITEREFG._E._ThomasK._Stamnes1999" class="citation book cs1">G. E. Thomas & K. Stamnes (1999). <i>Radiative Transfer in the Atmosphere and Ocean</i>. <a href="Cambridge_University_Press" title="Cambridge University Press">Cambridge University Press</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-521-40124-1</bdi>.</cite></li>

<li><cite id="CITEREFC._Bohren2006" class="citation book cs1"><a href="Craig_Bohren" title="Craig Bohren">C. Bohren</a> (2006). <i>Fundamentals of Atmospheric Radiation: an Introduction with 400 Problems</i>. <a href="John_Wiley_%26_Sons" class="mw-redirect" title="John Wiley & Sons">John Wiley & Sons</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-527-40503-9</bdi>.</cite></li>

<li><cite id="CITEREFR._T._Pierrehumbert2010" class="citation book cs1"><a href="Raymond_Pierrehumbert" title="Raymond Pierrehumbert">R. T. Pierrehumbert</a> (2010). <i>Principles of Planetary Climate</i>. <a href="Cambridge_University_Press" title="Cambridge University Press">Cambridge University Press</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9780521865562</bdi>.</cite></li></ul>

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