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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,

$\tau = \int \kappa(x)dx$

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.