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Unformatted text preview: 6. Moment equations for radiation Reading: Shu, Vol.I, Ch.2 6.1 Conservation equations Instead of solving the full radiation transport equation one may be content with knowing the moments of the radiation field, much like hydrodynamics is often preferred over kinetic theory. We already know the zeroth moment of the specific intensity, the mean intensity of differential energy density. I I ν d Ω = c u ν = 4 π J ν (6 . 1) Rather than velovity moments we would now be interested in angular moments such as the net flux of radiation. For that purpose we must recall that Eq.5.1, though written as a scalar expression, tacitly involves the direction of the radiation, ~e k . The intensity I ν refers to the energy transfer through an area element with unit normal ~e k , i.e. perpendicular to the direction of radiation. As observers, however, we might be interested in the energy transport rate through a fixed area element with unit normal ~e n . The net flux through the area element d ~ A n would then be F ν,~ e n = I d Ω I ν ~e n · ~e k = Z 2 π dφ Z 1- 1 d cos θ n cos θ n I ν ( φ, θ n ) (6 . 2) using ~e n as the polar axis of the coordinate system.as the polar axis of the coordinate system....
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