Editing Optical depth (section)

== 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>