Editing Optical depth

{{Short description|Physics concept}}
{{Other uses|Optical depth (astrophysics)}}

[[File:Aerosol Optical Depth (haze) at Geronimo Creek Observatory, Texas (1990-2016).jpg|thumb|Aerosol Optical Depth (AOD) at 830 nm measured with the same LED sun photometer from 1990 to 2016 at Geronimo Creek Observatory, Texas. Measurements made at or near solar noon when the Sun is not obstructed by clouds. Peaks indicate smoke, dust and smog. Saharan dust events are measured each summer.]]

In [[physics]], '''optical depth''' or '''optical thickness''' is the [[natural logarithm]] of the ratio of incident to ''transmitted'' [[radiant power]] through a material.
Thus, the larger the optical depth, the smaller the amount of transmitted radiant power through the material. 
'''Spectral optical depth''' or '''spectral optical thickness''' is the natural logarithm of the ratio of incident to transmitted [[Radiant power|spectral radiant power]] through a material.<ref name=GoldBook>{{GoldBookRef|title=Absorbance|file=A00028|accessdate=2015-03-15}}</ref> Optical depth is [[Dimensionless quantity|dimensionless]], and in particular is not a length, though it is a [[Monotonic function | monotonically]] increasing function of [[optical path length]], and approaches zero as the path length approaches zero. The use of the term "optical density" for optical depth is discouraged.<ref name=GoldBook />

In [[chemistry]], a closely related quantity called "[[absorbance]]" or "decadic absorbance" is used instead of optical depth: the [[common logarithm]] of the ratio of incident to transmitted radiant power through a material. It is the optical depth divided by {{math| ''log''<sub>''e''</sub>(10)}}, because of the different logarithm bases used.

== Mathematical definitions ==
=== Optical depth ===
The optical depth of a material, denoted <math display="inline">\tau</math>, is given by:<ref>{{cite book|author=Christopher Robert Kitchin|year=1987|title=Stars, Nebulae and the Interstellar Medium: Observational Physics and Astrophysics|publisher=[[CRC Press]]}}</ref><math display="block">\tau = \ln\!\left(\frac{\Phi_\mathrm{e}^\mathrm{i}}{\Phi_\mathrm{e}^\mathrm{t}}\right) = -\ln T</math>where
* <math display="inline">\Phi_\mathrm{e}^\mathrm{i}</math> is the [[radiant flux]] received by that material;
* <math display="inline">\Phi_\mathrm{e}^\mathrm{t}</math> is the [[radiant flux]] transmitted by that material;
* <math display="inline">T</math> is the [[transmittance]] of that material.

The absorbance <math display="inline">A</math> is related to optical depth by:<math display="block">\tau = A \ln{10}</math>

=== Spectral optical depth ===
The spectral optical depth in frequency (denoted <math>\tau_\nu</math>) or in wavelength (<math>\tau_\lambda</math>) of a material is given by:<ref name=GoldBook />
<math display="block">\tau_\nu = \ln\!\left(\frac{\Phi_{\mathrm{e},\nu}^\mathrm{i}}{\Phi_{\mathrm{e},\nu}^\mathrm{t}}\right) = -\ln T_\nu</math><math display="block">\tau_\lambda = \ln\!\left(\frac{\Phi_{\mathrm{e},\lambda}^\mathrm{i}}{\Phi_{\mathrm{e},\lambda}^\mathrm{t}}\right) = -\ln T_\lambda,</math>
where
* <math>\Phi_{\mathrm{e},\nu}^\mathrm{t}</math> is the [[Radiant flux|spectral radiant flux in frequency]] transmitted by that material;
* <math>\Phi_{\mathrm{e},\nu}^\mathrm{i}</math> is the spectral radiant flux in frequency received by that material;
* <math>T_\nu</math> is the [[Transmittance|spectral transmittance in frequency]] of that material;
* <math>\Phi_{\mathrm{e},\lambda}^\mathrm{t}</math> is the [[Radiant flux|spectral radiant flux in wavelength]] transmitted by that material;
* <math>\Phi_{\mathrm{e},\lambda}^\mathrm{i}</math> is the spectral radiant flux in wavelength received by that material;
* <math>T_\lambda</math> is the [[Transmittance|spectral transmittance in wavelength]] of that material.

Spectral absorbance is related to spectral optical depth by:
<math display="block">\tau_\nu = A_\nu \ln 10,</math><math display="block">\tau_\lambda =A_\lambda \ln 10,</math>
where
* <math>A_\nu</math> is the spectral absorbance in frequency;
* <math>A_\lambda</math> is the spectral absorbance in wavelength.

== Relationship with attenuation ==

=== Attenuation ===
{{Main article|Attenuation}}
Optical depth measures the attenuation of the transmitted radiant power in a material. Attenuation can be caused by absorption, but also reflection, scattering, and other physical processes. Optical depth of a material is approximately equal to its [[attenuation]] when both the absorbance is much less than 1 and the emittance of that material (not to be confused with [[radiant exitance]] or [[emissivity]]) is much less than the optical depth:
<math display="block">\Phi_\mathrm{e}^\mathrm{t} + \Phi_\mathrm{e}^\mathrm{att} = \Phi_\mathrm{e}^\mathrm{i} + \Phi_\mathrm{e}^\mathrm{e},</math><math display="block">T + ATT = 1 + E,</math>
where
* Φ<sub>e</sub><sup>t</sup> is the radiant power transmitted by that material;
* Φ<sub>e</sub><sup>att</sup> is the radiant power attenuated by that material;
* Φ<sub>e</sub><sup>i</sup> is the radiant power received by that material;
* Φ<sub>e</sub><sup>e</sup> is the radiant power emitted by that material;
* ''T'' = Φ<sub>e</sub><sup>t</sup>/Φ<sub>e</sub><sup>i</sup> is the transmittance of that material;
* ''ATT'' = Φ<sub>e</sub><sup>att</sup>/Φ<sub>e</sub><sup>i</sup> is the attenuation of that material;
* ''E'' = Φ<sub>e</sub><sup>e</sup>/Φ<sub>e</sub><sup>i</sup> is the emittance of that material,
and according to the [[Beer–Lambert law]],
<math display="block">T = e^{-\tau},</math>so:<math display="block">ATT = 1 - e^{-\tau} + E \approx \tau + E \approx \tau,\quad \text{if}\ \tau \ll 1\ \text{and}\ E \ll \tau.</math>

=== Attenuation coefficient ===
Optical depth of a material is also related to its [[attenuation coefficient]] by:<math display="block">\tau = \int_0^l \alpha(z)\, \mathrm{d}z,</math>where
* ''l'' is the thickness of that material through which the light travels;
* ''α''(''z'') is the attenuation coefficient or Napierian attenuation coefficient of that material at ''z'',
and if ''α''(''z'') is uniform along the path, the attenuation is said to be a linear attenuation and the relation becomes: <math display="block">\tau = \alpha l</math>

Sometimes the relation is given using the [[Cross section (physics)|attenuation cross section]] of the material, that is its attenuation coefficient divided by its [[number density]]:<math display="block">\tau = \int_0^l \sigma n(z)\, \mathrm{d}z,</math> where
* ''σ'' is the attenuation cross section of that material;
* ''n''(''z'') is the number density of that material at ''z'',
and if <math>n</math> is uniform along the path, i.e., <math>n(z)\equiv N</math>, the relation becomes:<math display="block">\tau = \sigma Nl</math>

== 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]])]]

== See also ==
* [[Air mass (astronomy)]]
* [[Absorptance]]
* [[Actinometer]]
* [[Aerosol]]
* [[Angstrom exponent]]
* [[Attenuation coefficient]]
* [[Beer–Lambert law]]
* [[Pyranometer]]
* [[Radiative transfer]]
* [[Sun photometer]]
* [[Transparency and translucency]]

== References ==
{{reflist}}

== External links ==
* [http://scienceworld.wolfram.com/physics/OpticalDepth.html Optical depth equations]

[[Category:Optical quantities]]
[[Category:Scattering, absorption and radiative transfer (optics)]]
[[Category:Spectroscopy]]
[[Category:Visibility]]