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<H3><A NAME="SECTION001321100000000000000">
Optical Depth</A>
</H3>
<P>
After having crossed an absorbing cloud, the intensity of a source,
I
<!-- MATH: $_{o}(\lambda)$ -->
<IMG
WIDTH="1" HEIGHT="47" ALIGN="MIDDLE" BORDER="0"
SRC="img336.gif"
ALT="$_{o}(\lambda)$">,
is received by the observer as I(<IMG
WIDTH="42" HEIGHT="56" ALIGN="BOTTOM" BORDER="0"
SRC="img337.gif"
ALT="$\lambda$">)
= I
<!-- MATH: $_{o}(\lambda)e^{-\tau}$ -->
<IMG
WIDTH="32" HEIGHT="46" ALIGN="MIDDLE" BORDER="0"
SRC="img338.gif"
ALT="$_{o}(\lambda)e^{-\tau}$">,
where
<IMG
WIDTH="42" HEIGHT="54" ALIGN="BOTTOM" BORDER="0"
SRC="img339.gif"
ALT="$\tau$">
is the optical depth of the cloud.
<P>
Let's connect <IMG
WIDTH="42" HEIGHT="54" ALIGN="BOTTOM" BORDER="0"
SRC="img340.gif"
ALT="$\tau$">
to the physical parameters.
<P>
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH: \begin{displaymath}
\tau = Na(\lambda)\\
\end{displaymath} -->
<IMG
WIDTH="100" HEIGHT="23"
SRC="img341.gif"
ALT="\begin{displaymath}\tau = Na(\lambda)\\
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P><PRE><TT>
N : column density
<BR>
a(<IMG
WIDTH="52" HEIGHT="74" ALIGN="MIDDLE" BORDER="0"
SRC="img342.gif"
ALT="$\lambda)$">
: line absorption coefficient
</TT></PRE>
<P>
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH: \begin{displaymath}
a(\lambda) = a_o\/ \phi_{\lambda}
\end{displaymath} -->
<IMG
WIDTH="114" HEIGHT="23"
SRC="img343.gif"
ALT="\begin{displaymath}a(\lambda) = a_o\/ \phi_{\lambda}
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P><PRE><TT>
<IMG
WIDTH="54" HEIGHT="72" ALIGN="MIDDLE" BORDER="0"
SRC="img344.gif"
ALT="$\phi_{\lambda}$">
: broadening function
<BR> <I>a</I><SUB><I>o</I></SUB> = <IMG
WIDTH="60" HEIGHT="86" ALIGN="MIDDLE" BORDER="0"
SRC="img345.gif"
ALT="$\frac {\lambda^{4}}{8\pi c}$">
<IMG
WIDTH="10" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
SRC="img346.gif"
ALT="$\frac {g_{k}}{g_{l}} a_{kl}$">
<BR>
l : lower level of the atomic transition
<BR>
k : upper level of the atomic transition
<BR> <IMG
WIDTH="58" HEIGHT="72" ALIGN="MIDDLE" BORDER="0"
SRC="img347.gif"
ALT="$\lambda_{lk}$">
: rest wavelength of the transition
<BR>
g<SUB><I>l</I></SUB> : statistical weight of the lower level
<BR>
g<SUB><I>k</I></SUB> : statistical weight of the upper level
<BR>
a<SUB><I>kl</I></SUB> : spontaneous transition probability
</TT></PRE>
<P>
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH: \begin{displaymath}
a_{kl} = f_{lk}\ \ \frac{g_{l}}{g_{k}}\ \ \frac{1}{\lambda^{2}_{lk}}\ \ \frac
{8\pi^{2}e^{2}}{m_{e}c}
\end{displaymath} -->
<IMG
WIDTH="240" HEIGHT="64"
SRC="img348.gif"
ALT="\begin{displaymath}a_{kl} = f_{lk}\ \ \frac{g_{l}}{g_{k}}\ \ \frac{1}{\lambda^{2}_{lk}}\ \ \frac
{8\pi^{2}e^{2}}{m_{e}c}
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P><PRE><TT>
f<SUB><I>lk</I></SUB> = upward oscillator strength
</TT></PRE>
<BR><HR>
<ADDRESS>
<I>Petra Nass</I>
<BR><I>1999-06-15</I>
</ADDRESS>
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