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.

$$\tau = Na(\lambda)\\$$

		 N 		: 		 column density


a($\lambda)$
:		 line absorption coefficient

$$a(\lambda) = a_o\/ \phi_{\lambda}$$

		$\phi_{\lambda}$
: 		 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

$$a_{kl} = f_{lk}\ \ \frac{g_{l}}{g_{k}}\ \ \frac{1}{\lambda^{2}_{lk}}\ \ \frac {8\pi^{2}e^{2}}{m_{e}c}$$

		flk = upward oscillator strength