Editing Optical depth (section)

== Applications ==

=== Atomic physics ===
In [[atomic physics]], the spectral optical depth of a cloud of atoms can be calculated from the quantum-mechanical properties of the atoms. It is given by<math display="block">\tau_\nu = \frac{d^2 n\nu} {2\mathrm{c} \hbar \varepsilon_0 \sigma \gamma} </math>where
* ''d'' is the [[transition dipole moment]];
* ''n'' is the number of atoms;
* ''ν'' is the frequency of the beam;
* ''c'' is the [[speed of light]];
* ''ħ'' is the [[reduced Planck constant]];
* ''ε''<sub>0</sub> is the [[vacuum permittivity]];
* ''σ'' is the cross section of the beam;
* ''γ'' is the [[natural linewidth]] of the transition.

===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" />

=== Astronomy ===
{{Main article|Optical depth (astrophysics)}}
In [[astronomy]], the [[photosphere]] of a star is defined as the surface where its optical depth is 2/3. This means that each photon emitted at the photosphere suffers an average of less than one scattering before it reaches the observer. At the temperature at optical depth 2/3, the energy emitted by the star (the original derivation is for the Sun) matches the observed total energy emitted.{{citation needed|date=November 2014}}{{clarify|reason=See talk page|date=April 2015}}

Note that the optical depth of a given medium will be different for different colors ([[wavelength]]s) of light.

For [[planetary rings]], the optical depth is the (negative logarithm of the) proportion of light blocked by the ring when it lies between the source and the observer. This is usually obtained by observation of stellar occultations.

[[File:PIA22737-Mars-2018DustStorm-MCS-MRO-Animation-20181030.webm|thumb|center|600x600px|[[Atmosphere of Mars|Mars dust storm]] – optical depth tau – May to September 2018<br />([[Mars Climate Sounder]]; [[Mars Reconnaissance Orbiter]])<br />(1:38; animation; 30 October 2018; [[:File:PIA22737-Mars-2018DustStorm-MCS-MRO-Animation-20181030.webm|file description]])]]