Editing Optical depth (section)

== 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.