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• 1. Radio Science using Interferometric Arrays
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1.3. Radiative transport

In this chapter we briefly review the simplest form of a mathematical treatment of the the propagation of electromagnetic radiation through a medium, in which radiation gets absorbed, but also generated. While the concept is very simple, it is also very fundamental and should not miss in a brief overview of astronomial quantities.

In the previous chapter, we have discussed that the Intensity is independent of the distance to a source as long as no emission is generated or absorbed. On its way to the observer along a path s, a radiative bundle may gain in intensity or loose in intensity. The loss of intensity is generally proportional to the intensity itself (imagine an absorption probability for radiation). In a medium, along an infinitesimal path length $ds$, $I_\nu$ looses a fraction $ -\kappa_\nu(s)\,I_\nu \,ds$. $\kappa_\nu(s)$ is called the linear absorption coefficient. Intensity will generally be generated independently of the incoming radiation, such that along the infinitesimal path length the intensity will increase by a constant amount $\varepsilon_\nu \,ds$, where $\varepsilon_\nu $ the so-called emission coefficient. $\kappa_\nu$ and $\varepsilon_\nu$ may generally be functions of $I$. In the following we will assume that this is not the case. From above, we can write down the equation of radiative transfer

$$ \begin{align} dI_\nu \,&= -\kappa_\nu\,I_\nu\,ds+\varepsilon_\nu\,ds\\ &\Leftrightarrow\\ \frac{dI_\nu}{ds} \,&= -\kappa_\nu\,I_\nu+\varepsilon_\nu\qquad .\\ \end{align} $$

The equation of radiative transfer has simple solutions for $\kappa_\nu\,=\,0$ or $\varepsilon_\nu\,=\,0$:

$$ \begin{align} \kappa_\nu\,&=\,0 &\Rightarrow\quad& I_\nu(s)\,=\,I_\nu(s_0)+\int_{s_0}^{s}\varepsilon_\nu(t)\,dt\\ \varepsilon_\nu\,&=\,0 &\Rightarrow\quad& I_\nu(s)\,=\,I_\nu(s_0)\,e^{-\int_{s_0}^{s}\kappa_\nu(t)\,dt}\qquad .\\ \end{align} $$

To proceed to the general solution of the equation of radiative transfer, one defines the optical depth $\tau_\nu$ via

$$ \begin{align} d\tau_\nu\,&\underset{def}{=}\,-\kappa_\nu\,dt\\ &\Rightarrow\\ \tau_\nu\,&=\,\int_s^{s_0}\kappa_\nu(t)\,dt\qquad .\\ \end{align} $$

The observer is at the position $s_0$, such that $\tau_\nu\,=\,0$ at the position of the observer. With that, one basically introduces a more relevant quantity than the actual path length as a parameter. Substituting via the chain rule

$$ \begin{align} \frac{dI_\nu}{ds} \,&= \,\frac{dI_\nu}{d\tau_\nu}\frac{d\tau_\nu}{ds}\\ &=\, -\kappa_\nu\frac{dI_\nu}{d\tau_\nu}\\ &=\, -\kappa_\nu\,I_\nu+\varepsilon_\nu\qquad .\\ \end{align} $$

The figure below shows the radiative transport process in a pictorial form.

By defining the source function or efficiency $s_\nu$ (unfortunately often the same symbol $S_\nu$ as for the flux density is commonly used, which we avoid here)

$$ \begin{align} s_\nu\,&\underset{def}{=}\,\frac{\varepsilon_\nu}{\tau_\nu}\\ \end{align} $$

we get

$$ \begin{align} \frac{dI_\nu}{d\tau_\nu}\,=\,I_\nu-s_\nu\qquad , \end{align} $$

which can be solved via multiplication with $e^{-\tau_\nu}$ and integration

$$ \begin{align} \frac{d\left(I_\nu \,e^{-\tau_\nu}\right)}{d\tau_\nu}\,&=\,\frac{dI_\nu}{d\tau_\nu}\,e^{-\tau_\nu}-I_\nu\,e^{-\tau_\nu}\\ &=\,-s_\nu\,e^{-\tau_nu}\\ &\Rightarrow\\ \int_0^{\tau_\nu(s)} \frac{d\left(I_\nu \,e^{-\tau_\nu}\right)}{d\tau_\nu}\,d\tau_\nu\,&=\,I_\nu(\tau_\nu(s))\,e^{-\tau_nu(s)}-I_\nu(0)e^{0}\\ &=\,I_\nu(\tau_\nu(s))\,e^{-\tau_nu(s)}-I_\nu(0)\\ \\&=\,-\int_0^{\tau_\nu(s)}s_\nu\,e^{-\tau_\nu}\,d\tau_\nu\\ &\Leftrightarrow\\ I_\nu(0)\,&=\, I_\nu(s_0)\\ &=\, I_\nu\left(\tau_\nu(s)\right)\,e^{-\tau_\nu(s)}+\int_0^{\tau_\nu(s)}s_\nu\,e^{-\tau_\nu}\,d\tau_\nu \qquad . \end{align} $$

In a local thermodynamical equilibrium (LTE) at local temperature $T$, the emissivity equals the absorbed radiation (Kirchhoff's Law), or

$$ \begin{align} \varepsilon_\nu\,&\underset{LTE}{=}\,\kappa_\nu\,B(T)\\ s_\nu\,&=\,B(T) \qquad , \end{align} $$

where $B(T)$ is the radiation emitted by a black body of temperature $T$ (see section 1.5.1 ➞ ).

local thermodynamical equilibrium the emissivity equals the absorbed radiation, and hence, starting at any position s,

$$ \begin{align} I_\nu(0)\,&=\, I_\nu(s_0)\\ &=I_\nu\left(\tau_\nu(s)\right)\,e^{-\tau_\nu(s)}+\int_0^{\tau(s)}s_\nu\,e^{-\tau_\nu}\,d\tau_\nu\\ &=I_\nu\left(\tau_\nu(s)\right)\,e^{-\tau_\nu(s)}+\int_0^{\tau(s)}B_\nu(T)\,e^{-\tau_\nu}\,d\tau_\nu \qquad . \end{align} $$

If the temperature $T$ is constant, $B_\nu(T)$ is constant, and

$$ \begin{align} I_\nu(0)\,&=\, I_\nu(s_0)\\ &=I_\nu\left(\tau_\nu(s)\right)\,B_\nu(T)\,e^{-\tau_\nu(s)}+\int_0^{\tau(s)}B_\nu(T)\,e^{-\tau_\nu}\,d\tau_\nu\\ &=I_\nu\left(\tau_\nu(s)\right)\,B_\nu(T)\,e^{-\tau_\nu(s)}+B_\nu(T)\left(1-e^{-\tau_\nu(s_0)}\right) \qquad. \end{align} $$

It is easy to see that for an opaque source

$$ \begin{align} \tau(s) \,&=\, \infty\\ &\Rightarrow\\ I_\nu(s_0) \,&=\, B_\nu(T)\qquad . \end{align} $$

Generally, if the source function is constant (substitute $B_\nu$ for $S_\nu$ for LTE)

$$ \begin{align} s_\nu \,&=\, s_\nu^0 = const.\\ &\Rightarrow\\ I_\nu(0)\,&=\, I_\nu(s_0)\\ &=I_\nu\left(\tau_\nu(s)\right)\,s_\nu\,e^{-\tau_\nu(s)}+\int_0^{\tau(s)}s_\nu\,e^{-\tau_\nu}\,d\tau_\nu\\ &=I_\nu\left(\tau_\nu(s)\right)\,s_\nu\,e^{-\tau_\nu(s)}+s_\nu\left(1-e^{-\tau_\nu(s_0)}\right) \qquad. \end{align} $$

From this, it is easy to see that if all emission comes from a slab of material, which is optically thin

$$ \begin{align} s_\nu \,&=\, s_\nu^0 = const.\\ I_\nu(\tau_\nu(s)) \,&=\, 0\\ \tau_\nu(s) \,&\ll\, 1\\ &\Rightarrow\\ I_\nu(0)\,&=\, I_\nu(s_0)\,=\, s_\nu^0\,\tau_\nu(s)\qquad ,\\ \end{align} $$

which is proportional to $\kappa_\nu$ if also $\kappa_\nu$ is constant and that for a background source in the optically thin case

$$ \begin{align} s_\nu \,&=\, s_\nu^0 = const.\\ I_\nu(\tau_\nu(s)) \,&\neq\, 0\\ \tau_\nu(s) \,&\ll\, 1\\ &\Rightarrow\\ I_\nu(0)\,&=\, I_\nu(s_0)\,=\, I_\nu(\tau(s))-\tau_\nu(s)(I_\nu(\tau(s))-s_\nu^0)\qquad .\\ \end{align} $$

If in that case the background intensity $I_\nu(\tau(s))$ is larger than the source function $s_\nu^0$, the emission gets reduced (absorption), if it is lower than the source function, we witness an increase in brightness (emission case). As the blackbody radiation is always larger for a larger temperature, we can say that absorption occurs if a background source of a certain temperature is obscured by an optically thin slab of gas of lower temperature, and that additional emission arises from the slab of gas if it has a higher temperature than the background source.

• Next: 1.4 Radio regime