The radiative transfer equation

Most of the work presented in this thesis involves the transportation of energy (as radiation) through a medium, a process which is described by the equation of radiative transfer:

$$% latex2html id marker 679\frac{dI_{\nu}}{ds}=-\alpha_{\nu}I_{\nu}+j_{\nu} \index{radiative transfer!equation}$$ (3.1)

where $\frac{dI_{\nu}}{ds}$ is the rate of change of radiation intensity, $I_\nu$ (at a particular frequency $\nu$), with distance, $s$. The rate at which it changes depends on the absorption coefficient $\alpha_{\nu}$ and the emission coefficient, $j_{\nu}$ (these are defined as the absorption and emission per unit length respectively). In order to provide a formal solution to this it will help to define optical depth as:

$$d\tau_{\nu}=\alpha_{\nu}ds$$

and the source function as:

$$S_{\nu}=\frac{j_\nu}{\alpha_\nu}$$ (3.2)

This then enables equation 3.1 to be re-written as

$$\frac{dI_{\nu}}{d\tau_{\nu}}=-I_{\nu}+S_{\nu}$$ (3.3)

This is now simply a first order differential equation which can be solved by the usual method of multiplying by an integrating factor, in this case $e^{\int^{\tau_{\nu}}_{0}d\tau_{\nu}}=e^{\tau_\nu}$, to give

 

$$\frac{dI_{\nu}}{d\tau_{\nu}}e^{\tau_{\nu}}+I_{\nu}e^{\tau_{\nu}} = S_{\nu} e^{\tau_\nu}$$

$$\frac{d}{d\tau_{\nu}}\left(I_\nu e^{\tau_{\nu}}\right) = S_{\nu} e^{\tau_\nu}$$ (3.4)

Integrating both sides of equation 3.4 from zero to $\tau_{\nu}$ (ie. this represents integrating along a line of sight from the edge of the universe to the observer) then gives

$$\int^{\tau_\nu}_{0} \frac{d}{d\tau_{\nu}}\left(I_\nu e^{\tau_... ...tau_\nu =\int^{\tau_\nu}_{0} S_{\nu}e^{\tau_{\nu}} d\tau_{\nu}$$

If $S_\nu$ is assumed to be constant then this has the solution

$$I_\nu e^{\tau_\nu}-I_{\nu_0}=S_\nu\left[e^{\tau_\nu}-1\right]$$

which can be re-arranged to give

$$I_\nu=S_\nu+e^{-\tau_\nu}\left(I_{\nu_0}-S_\nu \right)$$ (3.5)

where $I_{\nu_0}$ will usually represent the cosmic background radiation. When a telescope looks at a source it sees it against the cosmic background radiation, in other words it will see $I_\nu-I_{\nu_0}$ and thus the observed intensity will be

 

$$I_{\nu_{\rm obs}} = I_\nu-I_{\nu_0}=S_\nu-I_{\nu_0}+e^{-\tau_\nu}\left(I_{\nu_0}-S_\nu \right)$$

$$= (S_\nu-I_{\nu_0})(1-e^{-\tau_\nu})$$ (3.6)

In an analogous way to equation 2.32 a temperature can be associated with this intensity, the observed brightness temperature is the temperature above the background and is called the radiation temperature, $T_R$    

This has of course assumed that $S_\nu$ is not a function of $\tau$, in other words $S_\nu$ does not vary across the cloud. This is an extremely restrictive assumption and in order to model a realistic cloud a way of solving the radiative transfer equation for a variable $S_\nu$ is necessary. Sections 3.3 and 3.5 describe the two main methods that have been used to tackle this problem.