Radiative Transfer (Optically Thin Cloud)

[cepheid]

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:

[tex] \frac{dI_{\nu}(s)}{ds} = -\alpha_{\nu}(s)I_{\nu}(s) + j_{\nu} [/itex]​

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.

The optical depth, $\tau_{\nu}(s)$ is defined by:

$$\tau_{\nu}(s) = \int_{s_0}^s \alpha_{\nu}(s^\prime)\, ds^\prime$$
​

Optically thin means tau << 1

The Attempt at a Solution

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:

let,

$$\phi(s) = \exp{(\int_{s_0}^s \alpha_{\nu}(s^\prime)\, ds^\prime)} = e^{\tau_{\nu}(s)}$$​

then,

$$e^{\tau_{\nu}}\frac{dI_{\nu}}{ds} + e^{\tau_{\nu}}\alpha_{\nu}I_{\nu} = e^{\tau_{\nu}}j_{\nu}$$

$$e^{\tau_{\nu}}\frac{dI_{\nu}}{ds} + \frac{d}{ds}\left(e^{\tau_{\nu}}\right)I_{\nu} = e^{\tau_{\nu}}j_{\nu}$$

$$\frac{d}{ds}\left(e^{\tau_{\nu}}I_{\nu}\right) = e^{\tau_{\nu}}j_{\nu}$$

$$e^{\tau_{\nu}}I_{\nu}= \int e^{\tau_{\nu}}j_{\nu} \, ds$$​

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.

 

[cepheid]

Any thoughts, guys?

 

[JeffKoch]

If it's optically thin, how important is the absorption term in the radiative transfer equation?