Optical depth
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 spectral radiant power through a material. Optical depth is dimensionless, and in particular is not a length, though it is a 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. 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, that is the optical depth divided by ln 10. Mathematical definitions. Optical depth. Optical depth of a material, denoted formula_1, is given by: where The absorbance formula_6 is related to optical depth by: Spectral optical depth. Spectral optical depth in frequency and spectral optical depth in wavelength of a material, denoted formula_8 and formula_9 respectively, are given by: where  Spectral absorbance is related to spectral optical depth by: where Relationship with attenuation. 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: where and according to the Beer–Lambert law, so: Attenuation coefficient. Optical depth of a material is  also related to its attenuation coefficient by: where and if "α"("z") is uniform along the path, the attenuation is said to be a linear attenuation and the relation becomes: Sometimes the relation is given using the attenuation cross section of the material, that is its attenuation coefficient divided by its number density: where and if formula_29 is uniform along the path, i.e., formula_30, the relation becomes: 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 where Atmospheric sciences. 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 , where "τ′" refers to a vertical path, "m" is called the relative airmass, and for a plane-parallel atmosphere it is determined as where "θ" is the zenith angle corresponding to the given path. Therefore, The optical depth of the atmosphere can be divided into several components, ascribed to Rayleigh scattering, aerosols , and gaseous 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 formula_34 and it follows that the total atmospheric optical depth is given by formula_35. In both equations: The optical depth of a plane parallel cloud layer is given by formula_36 where: So, with a fixed depth and total liquid water path, formula_37. Astronomy. 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. Note that the optical depth of a given medium will be different for different colors (wavelengths) 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.