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Optical Depth

After having crossed an absorbing cloud, the intensity of a source, I $_{o}(\lambda)$, is received by the observer as I($\lambda$) = I $_{o}(\lambda)e^{-\tau}$, where $\tau$ is the optical depth of the cloud.

Let's connect $\tau$ to the physical parameters.

\begin{displaymath}\tau = Na(\lambda)\\

		 N 		: 		 column density

a($\lambda)$ : line absorption coefficient

\begin{displaymath}a(\lambda) = a_o\/ \phi_{\lambda}

: 		 broadening function

ao = $\frac {\lambda^{4}}{8\pi c}$ $\frac {g_{k}}{g_{l}} a_{kl}$
l : lower level of the atomic transition
k : upper level of the atomic transition
$\lambda_{lk}$ : rest wavelength of the transition
gl : statistical weight of the lower level
gk : statistical weight of the upper level
akl : spontaneous transition probability

\begin{displaymath}a_{kl} = f_{lk}\ \ \frac{g_{l}}{g_{k}}\ \ \frac{1}{\lambda^{2}_{lk}}\ \ \frac

		flk = upward oscillator strength

Petra Nass