# Radiative Transfer - Emitting Ring

1. Jan 26, 2010

### astrop

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

2. Relevant equations

$$dE = I_{\nu} dA cos\theta d\nu d\omega dt$$

Emission:
$$\frac{dI_{\nu}}{ds}cos\theta = j_{\nu} - k_{\nu}I_{\nu}$$

3. The attempt at a solution
I'm assuming that the center of the ring is at a distance D>>$$r_{out}$$. The angle $$\theta$$ 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:
$$\frac{dI_{\nu}}{ds}cos\theta = j_{\nu}$$

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?