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Radiative Transfer - Emitting Ring

  1. Jan 26, 2010 #1
    1. The problem statement, all variables and given/known data
    Suppose there is an optically thin emitting ring with inner radius [tex]r_{in}[/tex] and outer radius [tex]r_{out}[/tex] seen edge on. Compute the relative surface brightness of the ring as a function of its projected position in the sky.

    2. Relevant equations

    Radiative transfer:
    [tex]dE = I_{\nu} dA cos\theta d\nu d\omega dt[/tex]

    Emission:
    [tex]\frac{dI_{\nu}}{ds}cos\theta = j_{\nu} - k_{\nu}I_{\nu}[/tex]

    3. The attempt at a solution
    I'm assuming that the center of the ring is at a distance D>>[tex]r_{out}[/tex]. The angle [tex]\theta[/tex] is the angle between the normal to the ground and the line going to the center of the ring. I think I can drop the extinction coefficient so that:
    [tex]\frac{dI_{\nu}}{ds}cos\theta = j_{\nu}[/tex]

    I'm not entirely sure how exactly to proceed. Do I just need to figure out what ds is? This would be the area of the ring covered by some angle from the center of the ring?
     
  2. jcsd
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