Difference between revisions of "Radiative Transfer Equation"

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\usepackage{eufrak}
 
\usepackage{eufrak}
 
\begin{document}
 
\begin{document}
\section*{ Fundamental Equation of Radiative Transfer}
+
\section{ The Fundamental Equation of Radiative Transfer}
  
The fundamental equation of transfer is governed by emission and extinction.  
+
The fundamental equation of radiative transfer is governed by emission and extinction.  
 
Extinction is brought about by absorption (which changes photon energy) or
 
Extinction is brought about by absorption (which changes photon energy) or
by scattering (which does not).  Examples of scattering are Thomson scattering
+
by scattering (which does not).  Examples of scattering are Thomson scattering of light
 
off of cold electrons, Rayleigh scattering in the atmosphere, and Line
 
off of cold electrons, Rayleigh scattering in the atmosphere, and Line
 
scattering (reemission in a different direction).  
 
scattering (reemission in a different direction).  
 
An example of absorption is photoionization (where a photon ionizes an atom,  
 
An example of absorption is photoionization (where a photon ionizes an atom,  
say by knocking off an electron).\par
+
say by knocking off an electron).
 +
 
 +
\subsection{Absorption}
  
\begin{itemize}
 
\item  Absorption:\par
 
 
Let's say radiation $I_\nu$ passes through a region $ds$ of  
 
Let's say radiation $I_\nu$ passes through a region $ds$ of  
 
absorption/scattering on its way to us. Then:
 
absorption/scattering on its way to us. Then:
$$dI_\nu=-\alpha_\nu I_\nu ds$$
+
\begin{equation}
 +
dI_\nu=-\alpha_\nu I_\nu ds
 +
\end{equation}
 
where $\alpha_\nu$ is the {\it extinction} coefficient (units of $cm^{-1}$).
 
where $\alpha_\nu$ is the {\it extinction} coefficient (units of $cm^{-1}$).
 
We may compute $\alpha_\nu$ a couple different ways:
 
We may compute $\alpha_\nu$ a couple different ways:
$$\alpha_\nu=\overbrace{n}^{\#\ density}
+
\begin{equation}
 +
\alpha_\nu=\overbrace{n}^{\#\ density}
 
\overbrace{\sigma_\nu}^{cross\ section}
 
\overbrace{\sigma_\nu}^{cross\ section}
=\overbrace{\rho}^{mass\ density}\overbrace{\kappa_\nu}^{opacity}$$
+
=\overbrace{\rho}^{mass\ density}\overbrace{\kappa_\nu}^{opacity}
 +
\end{equation}
 
Solving for intensity:
 
Solving for intensity:
$$\begin{aligned}I_\nu(s)&=I_\nu(0)e^{-n\sigma_\nu s}\\  
+
\begin{eqnarray}
&=I_\nu(0)e^{-\tau_\nu}\\ \end{aligned}$$
+
I_\nu(s)&=I_\nu(0)e^{-n\sigma_\nu s}\\  
 +
&=I_\nu(0)e^{-\tau_\nu}\\
 +
\end{eqnarray}
 
where $\tau_\nu$ is the {\it optical depth} at $\nu$.
 
where $\tau_\nu$ is the {\it optical depth} at $\nu$.
 +
 
Optical depth is often computed as:
 
Optical depth is often computed as:
$$\tau_\nu=n\sigma_\nu s=N\sigma_\nu$$
+
\begin{equation}
 +
\tau_\nu=n\sigma_\nu s=N\sigma_\nu
 +
\end{equation}
 
where $N$, the {\it column density}, is in $cm^{-2}$ and is the \# of  
 
where $N$, the {\it column density}, is in $cm^{-2}$ and is the \# of  
 
extinguishers per unit area.
 
extinguishers per unit area.
 
Similarly,  
 
Similarly,  
$$\tau_\nu=\rho\kappa_\nu s=\Sigma\kappa_\nu$$
+
\begin{equation}
where $\Sigma$ is the mass surface density.
+
\tau_\nu=\rho\kappa_\nu s=\Sigma\kappa_\nu
 +
\end{equation}
 +
where $\Sigma$ is the mass surface density and $\kappa_\nu$ is the density-weighted extinction coefficient.
 
$$\tau_\nu\begin{cases}\ll 1 &optically\ thin\\
 
$$\tau_\nu\begin{cases}\ll 1 &optically\ thin\\
 
\gg 1 &optically\ thick\end{cases}$$
 
\gg 1 &optically\ thick\end{cases}$$
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$$I_\nu(s)=I_\nu(0)e^{-\alpha_\nu s}$$
 
$$I_\nu(s)=I_\nu(0)e^{-\alpha_\nu s}$$
  
\item Emission:\par
+
 
 +
\subsection{Emission}
 +
 
 
If $j_\nu $ is the emissivity, then the contribution of the emissivity of
 
If $j_\nu $ is the emissivity, then the contribution of the emissivity of
 
a medium to the flux is:
 
a medium to the flux is:
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It is often convenient to express this in terms of optical depth.   
 
It is often convenient to express this in terms of optical depth.   
 
Dividing by $\alpha_\nu$ and recognizing $d\tau_\nu = ds \alpha_\nu$:
 
Dividing by $\alpha_\nu$ and recognizing $d\tau_\nu = ds \alpha_\nu$:
$$\begin{aligned}{dI\nu\over d\tau_\nu}&={j_\nu\over\alpha_\nu}-I_\nu\\  
+
\begin{eqnarray}
 +
{dI\nu\over d\tau_\nu}&={j_\nu\over\alpha_\nu}-I_\nu\\  
 
&=S_\nu-I_\nu\\
 
&=S_\nu-I_\nu\\
\end{aligned}$$
+
\end{eqnarray}
 
where $S_\nu$ is a ``source function''. In general,
 
where $S_\nu$ is a ``source function''. In general,
 
$$S_\nu\eval{scattering}\propto\int{I_\nu d\Omega}$$
 
$$S_\nu\eval{scattering}\propto\int{I_\nu d\Omega}$$

Revision as of 13:09, 25 August 2016

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Reference Material

1 The Fundamental Equation of Radiative Transfer

The fundamental equation of radiative transfer is governed by emission and extinction. Extinction is brought about by absorption (which changes photon energy) or by scattering (which does not). Examples of scattering are Thomson scattering of light off of cold electrons, Rayleigh scattering in the atmosphere, and Line scattering (reemission in a different direction). An example of absorption is photoionization (where a photon ionizes an atom, say by knocking off an electron).

1.1 Absorption

Let’s say radiation passes through a region of absorption/scattering on its way to us. Then:

where is the extinction coefficient (units of ). We may compute a couple different ways:

Solving for intensity:

where is the optical depth at .

Optical depth is often computed as:

where , the column density, is in and is the # of extinguishers per unit area. Similarly,

where is the mass surface density and is the density-weighted extinction coefficient.

The Mean Free Path is given by: . Thus:

That is, the optical depth is the number of mean-free-paths deep a medium is. For Poisson processes, the probability of absorption is given by:

Therefore:

1.2 Emission

If is the emissivity, then the contribution of the emissivity of a medium to the flux is:

Emission and Extinction together:

(Fundamental Equation of Transfer)

It is often convenient to express this in terms of optical depth. Dividing by and recognizing :

where is a “source function”. In general,

There is a formal solution for . Let’s define and . Then:

If is constant with , then:

That second term on the righthand side can be approximated as for , since self-absorption is negligible. Similarly, for , it may be approximated as . The source function is everything. It has both the absorption and emission coefficients embedded in it.