Editing Optical depth (section)

===Atmospheric sciences===
{{See also|Beer–Lambert law}}
In [[atmospheric sciences]], one often refers to the optical depth of the atmosphere as corresponding to the vertical path from Earth's surface to outer space; at other times the optical path is from the observer's altitude to outer space. The optical depth for a slant path is {{nobreak|1=''τ'' = ''mτ''′}}, where ''τ′'' refers to a vertical path, ''m'' is called the [[airmass|relative airmass]], and for a plane-parallel atmosphere it is determined as {{nobreak|1=''m'' = sec ''θ''}} where ''θ'' is the [[zenith angle]] corresponding to the given path. Therefore,<math display="block">T = e^{-\tau} = e^{-m\tau'}</math>The optical depth of the atmosphere can be divided into several components, ascribed to [[Rayleigh scattering]], [[aerosols]], and gaseous [[absorption (electromagnetic radiation)|absorption]]. The optical depth of the atmosphere can be measured with a [[Sun photometer]].

The optical depth with respect to the height within the atmosphere is given by<ref name=":0" />
<math display="block">\tau(z) = k_\text{a}w_1\rho_0H e^{-z/H}</math>
and it follows that the total atmospheric optical depth is given by<ref name=":0" />
<math display="block">\tau(0) = k_\text{a}w_1\rho_0H</math>

In both equations:
* ''k''<sub>a</sub> is the absorption coefficient
* ''w''<sub>1</sub> is the mixing ratio
* ''ρ''<sub>0</sub> is the density of air at sea level
* ''H'' is the [[scale height]] of the atmosphere
* ''z'' is the height in question

The optical depth of a plane parallel cloud layer is given by<ref name=":0">{{Cite book|title=A first course in atmospheric radiation|last=Petty|first=Grant W.|year=2006|publisher=Sundog Pub|isbn=9780972903318|oclc=932561283}}</ref><math display="block">\tau = Q_\text{e} \left[\frac{9\pi L^2 H N}{16\rho_l^2}\right]^{1/3}</math>where:
* ''Q''<sub>e</sub> is the extinction efficiency
* ''L'' is the [[liquid water path]]
* ''H'' is the geometrical thickness
* ''N'' is the concentration of droplets
* ''ρ''<sub>l</sub> is the density of liquid water

So, with a fixed depth and total liquid water path, <math display="inline">\tau \propto N^{1/3}</math>.<ref name=":0" />