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scratch3

-              v12
+              v13
 (where the absorption coefficient reduces the intensity of the ray by [[latex($dI_\nu=-\alpha_\nu I_\nu ds$)]])
 This implies that the optical depth is,
+The optical depth is then defined by,
 [[latex($\tau_R\approx4.3~\frac{g}{cm^2}$)]]
+[[latex($\tau_\nu (s)=\int^s_{s_0} \alpha_\nu(s')ds'$)]]
+When [[latex($\tau>1$)]] (integrated along a typical path through the medium), the material is said to be optically thick, and when [[latex($\tau<1$)]], 'optically thin'.
+Recall that optical depth, opacity, and mean free path are related by,
+The mean optical depth of an absorbing material can be shown to =1, and so in terms of the mean free path ([[latex($l_\nu$)]]) we have:
+[[latex($$)]]
+[[latex($\bar{\tau_\nu}=\alpha_\nu l_\nu = 1$)]]
+or
+[[latex($l_\nu=\frac{1}{\rho \kappa}$)]]