5.4: Optical Depth
The product of linear extinction coefficient and distance, or, more properly, if the extinction coefficient varies with distance, the integral of the extinction coefficient with respect to distance,
\[\tau = \int \kappa(x)dx\]
is the optical depth, or optical thickness, \(\tau\). It is dimensionless. Specific intensity falls off with optical depth as
\[I = I^0 e^{-\tau}.\]
Thus optical depth can also be defined by \(\ln (I^0/I)\). While the optical depth \(\ln (I^0 /I)\) is generally used to describe how opaque a stellar atmosphere or an interstellar cloud is, when describing how opaque a filter is, one generally uses \(\log_{10} (I^0/I)\), which is called the density \(d\) of the filter. Density is \(0.4343\) times optical depth. If a star is hidden behind a cloud of optical depth \(\tau\) it will be dimmed by \(1.086\tau\) magnitudes. If it is hidden behind a filter of density \(d\) it will be dimmed by \(2.5d\) magnitudes. The reader is encouraged to verify these assertions.