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Optical depth (attenuation of radiation)

Optical depth is a dimensionless measure of how much light or other radiation is absorbed or scattered along a path; it controls transmission, distinguishes optically thin/thick media, and appears in radiative transfer.

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Overview

Optical depth is a dimensionless quantity that characterizes the attenuation of radiation as it travels through a medium. Practically it answers the question: what fraction of a beam is removed by absorption and scattering over a given path? In many contexts the transmitted fraction follows an exponential law: transmission ≈ e^{-τ}, where τ (tau) is the optical depth. Small τ indicates an optically thin medium; large τ corresponds to an optically thick medium. For introductory summaries of radiative concepts see radiative transfer resources.

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Aerosol Optical Depth (haze) at Geronimo Creek Observatory, Texas (1990-2016)
en.wikipedia.org · CC BY-SA 4.0

Definition and physical meaning

Physically, τ is the integral along the path of an extinction coefficient that combines absorption and scattering. In words: τ = integral over path of local extinction per unit length. It is therefore sensitive to the concentration of particles or molecules, their cross sections, and the path length itself. Optical depth does not carry units. Related useful concepts are the mean free path (typical distance between interactions) and the single-scattering albedo (fraction of extinction due to scattering rather than absorption).

Examples and intuition

Everyday examples help build intuition. In fog or haze the optical depth between you and a distant object increases with distance and particle density until the object disappears. A thin window or a slight tint has τ much less than 1; dense clouds or thick smoke can have τ well above 1 so little direct light passes. For visual demonstrations and imagery consult visual examples.

Applications

  • Astronomy: optical depth describes how starlight is dimmed by interstellar dust and gas, and determines whether an emission region is observable directly or only via scattered/thermal radiation. See astronomical uses.
  • Atmospheric science and climate: τ governs Earth's radiative balance by setting how much solar and terrestrial radiation is absorbed or scattered by gases, aerosols, and clouds. See climate and remote sensing applications.
  • Medical and laboratory optics: concepts similar to optical depth appear in tissue imaging, spectroscopy, and material characterization where light attenuation encodes composition.

Measurement, interpretation and notable points

Measuring optical depth often involves comparing incoming and transmitted intensities, inferring τ from the logarithm of their ratio. In complex media multiple scattering invalidates simple single-beam interpretations and full radiative transfer modeling is required. Optical depth is wavelength dependent: a medium can be optically thin at one wavelength and thick at another. The idea extends beyond visible light and is used for radio, infrared, and X-ray regimes. For further technical reading and datasets see more references.

Optical thickness of the atmosphere

Determination

The optical thickness τ \tau the atmosphere enters as extinction coefficient into the transmissivity Tthe atmosphere. This is calculated for a given wavelength according to Lambert-Beer's law to:

{\displaystyle T={\frac {I}{I_{0}}}=e^{-\tau \cdot m}}

with

  • the intensity Iof the solar radiation in the considered wavelength on the ground
  • of the exatmospheric solar radiation I_{0}(solar constant)
  • the atmospheric mass {\displaystyle m=1/\cos \Theta _{z}}, i.e. the distance through the atmosphere as a multiple of the shortest possible distance at zenith insolation ( \Theta _{z}is the solar zenith angle).

Due to the atmospheric mass, the transmissivity depends on the position of the sun, i.e., it changes during the day, even if atmospheric conditions remain constant. In contrast, the optical thickness of the atmosphere does not depend on the position of the sun; it can be measured with a photometer.

Components

The optical thickness of the atmosphere is composed additively:

{\displaystyle \tau =\tau _{\text{Gas}}+\tau _{R}+\tau _{A}}

Thereby describe

  • the gas optical thickness τ {\displaystyle \tau _{\text{Gas}}}the absorption at the atmospheric gases (mainly ozone, oxygen and water vapor), but only in the wavelength ranges λ \lambda of the absorption bands of the gases. The optical thickness of the atmospheric gases (except water vapor) is quasi-constant and can be taken from tables.
  • the Rayleigh optical thickness τ \tau _{R}(\lambda )=0{,}008735\cdot \lambda ^{{-4{,}085}}the extinction caused by Rayleigh scattering of air molecules.
  • the aerosol optical thickness τ \tau _{A}the Mie scattering from larger particles (aerosols). It can be determined from the other components (measured or looked up):

{\displaystyle \Leftrightarrow \tau _{A}=\tau -\tau _{R}-\tau _{\text{Gas}}}

For a more detailed breakdown, see Lambert-Beer's Law, Remote Sensing (Atmosphere).

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