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5.4: Optical Depth

Last updated

Mar 5, 2022
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- 5.3: Scattering, Extinction and Opacity
- 5.5: The Equation of Transfer

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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.

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