week2 - 1 Monday, September 26: Radiative Transfer As light...

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1 Monday, September 26: Radiative Trans- fer As light travels through the universe, things happen to it. By interacting with charged particles, photons can gain energy; they can lose energy; they can change their direction of motion. Photons can also be absorbed by opaque lumps of matter, such as dust particles. The photons can also be joined by new photons emitted by the medium through which they travel. It’s very useful to have a shorthand description of what happens to the speciFc intensity I ν of light as it propagates through the universe. This de- scription is given by the equation of radiative transfer . The radiative transfer equation is a key equation for the study of stellar structure. (How does light get from the center of the Sun to its photosphere? Gamma rays are emit- ted by fusion reactions in the Sun’s core, but the photons that escape from the photosphere are largely at near-infrared, visible, and near-ultraviolet fre- quencies. Obviously, as photons travel through the Sun, their mean energy must be decreased.) The radiative transfer equation is also a key equation for the study of the interstellar medium. (How does the light emitted by a star’s photosphere di±er from the light we observe at our telescope? The di±erence in absorption at di±erent wavelengths can tell us about the composition of the dust and gas of the interstellar medium.) To derive the equation of radiative transfer, let’s start by setting up two identical transparent windows, each of area dA , separated by a short distance ds , as shown in ²igure 1. After passing through the Frst window with spe- ciFc intensity I ν , the light passes through the second window with speciFc intensity I 0 ν . If the space between the two windows is totally empty, then speciFc intensity is conserved: I 0 ν = I ν . (1) This result is derived in the textbook; it results in a straightforward way from Euclidean geometry. Let me just note one of its implications. If you move further away from an extended light source (like the Sun, for instance), the power emitted per unit solid angle remains constant. However, the solid angle subtended by the light source decreases as one over the square of the distance. Thus, the light source’s ³ux, integrated over its angular area, decreases as one over the square of the distance to the light source. The constancy of speciFc intensity implies the inverse square law of ³ux, and vice versa. 1
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Figure 1: Geometry of radiative transfer Now, suppose the space between the two windows isn’t empty. It may contain hot, ionized gas (a bit of the Sun’s interior, or at a much lower density, the coronal gas of the interstellar medium). It may contain cool molecular gas (a bit of the Earth’s atmosphere, or at a much lower density, the molecular clouds of the interstellar medium). For that matter, it may contain a slab of lead. Each of these materials will scatter and absorb photons in di±erent ways. They will also produce photons in di±erent ways.
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This note was uploaded on 07/17/2008 for the course ASTRO 822 taught by Professor Ryden during the Fall '05 term at Ohio State.

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week2 - 1 Monday, September 26: Radiative Transfer As light...

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