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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, \(\int \kappa(x)dx\), 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.