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Homework Statement
An optically thin cloud at temperature T radiates power P_{\nu} per unit volume. Find an expression for the cloud's brightness I_{\nu} as a function of distance from the centre of the cloud in the case where:
(a) the cloud is a cube of side d
(b) the cloud is a sphere of radius R
(c) How would your answers change if the cloud were optically thick?
Homework Equations
First of all, terminology varies wildly, but astronomical terminology is being used here i.e. "brightness" refers to the radiometric unit that is a measure of the rate (with time) at which energy arrives per unit of a given perpendular area per frequency band and from a given direction (i.e. per unit solid angle subtended presumably by the source) as measured in \textrm{W} \cdot \textrm{m}^{-2} \cdot \textrm{Hz}^{-1} \cdot \textrm{sr}^{-1}
The relevant equation given is the Radiative Transfer Equation, which describes how this "brightness" varies with distance from the centre of the source, one term being a loss due to absorption, and the other term being a gain due to radiation:
\frac{dI_{\nu}(s)}{ds} = -\alpha_{\nu}(s)I_{\nu}(s) + j_{\nu} [/itex]
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where \alpha_{\nu}(s) is the absorption coefficient, and j_{\nu} is the rate of change of brightness with distance due to emission i.e. the energy radiated per unit time, per unit volume, per unit solid angle. <br />
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The optical depth, \tau_{\nu}(s) is defined by:<br />
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<div style="text-align: center">\tau_{\nu}(s) = \int_{s_0}^s \alpha_{\nu}(s^\prime)\, ds^\prime<br />​</div><br />
Optically thin means tau << 1<br />
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<h2>The Attempt at a Solution</h2><br />
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I wasn't 100% sure how to proceed, but my first thought was that maybe the relationship between \j_{\nu} and P_{\nu} just depends on the shape of the cloud. Furthermore, we're given an equation that (from what I understand), is true under any circumstances, so I set about trying to solve the ODE using the method of integrating factors:<br />
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let, <br />
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<div style="text-align: center">\phi(s) = \exp{(\int_{s_0}^s \alpha_{\nu}(s^\prime)\, ds^\prime)} = e^{\tau_{\nu}(s)}​</div><br />
then, <br />
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<div style="text-align: center">e^{\tau_{\nu}}\frac{dI_{\nu}}{ds} + e^{\tau_{\nu}}\alpha_{\nu}I_{\nu} = e^{\tau_{\nu}}j_{\nu} <br />
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e^{\tau_{\nu}}\frac{dI_{\nu}}{ds} + \frac{d}{ds}\left(e^{\tau_{\nu}}\right)I_{\nu} = e^{\tau_{\nu}}j_{\nu} <br />
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\frac{d}{ds}\left(e^{\tau_{\nu}}I_{\nu}\right) = e^{\tau_{\nu}}j_{\nu} <br />
<br />
e^{\tau_{\nu}}I_{\nu}= \int e^{\tau_{\nu}}j_{\nu} \, ds​</div><br />
Now this is where I am stuck (i.e. I don't know what to do with this, or whether I'm on the right track.
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