World Library  
Flag as Inappropriate
Email this Article

Optical depth

Article Id: WHEBN0000038993
Reproduction Date:

Title: Optical depth  
Author: World Heritage Encyclopedia
Language: English
Subject: Beer–Lambert law, Diazenylium, Photosphere, Assured Clear Distance Ahead, Visibility
Collection: Scattering, Absorption and Radiative Transfer (Optics), Stellar Physics, Visibility
Publisher: World Heritage Encyclopedia

Optical depth

Optical depth, or optical thickness, is a measure of transparency. Optical depth is defined as the negative natural logarithm of the fraction of radiation (e.g., light) that is not scattered or absorbed on a path. Hence optical depth is dimensionless, and in particular is not a length, though it is a monotonically increasing function of path length, and approaches zero as the path length approaches zero.


  • Formulations 1
    • Calculation from fundamental principles 1.1
    • Atmospheric science 1.2
    • Stellar physics 1.3
  • See also 2
  • References 3
  • External links 4


The optical depth expresses the quantity of light removed from a beam by scattering or absorption during its path through a medium. If I_0 is the intensity of radiation at the source and I is the observed intensity after a given path, then the optical depth \tau is defined by the following equation:[1]

I / I_0 = e^{-\tau},


\tau = -\ln{I / I_0} = -\ln{T},

where T is the transmittance.

Calculation from fundamental principles

In atomic physics, the optical depth of a cloud of atoms can be calculated from the quantum-mechanical properties of the atoms. It is given by

\tau = \frac{d^2 \nu N} {2 c \hbar \epsilon_0 A \gamma},

where d denotes the transition dipole moment, γ the natural linewidth of the transition, ν the frequency, N the number of atoms, and A the cross-section of the beam.

Atmospheric science

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. Since τ refers to a vertical path, the optical depth for a slant path is τ′ = m τ, where m is called the relative airmass, and for a plane-parallel atmosphere it is determined as m = sec θ, where θ is the zenith angle corresponding to the given path. Therefore

I / I_0 = e^{-m \tau}.

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.

Stellar physics

Another example occurs in astronomy, where 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 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.

See also


  1. ^ Kitchin, Christopher Robert (1987). Stars, Nebulae and the Interstellar Medium: Observational Physics and Astrophysics. CRC Press. 
  • David R. Brooks (August 2006). "Monitoring Solar Radiation and Its Transmission Through the Atmosphere". Drexel University. Retrieved 2013-08-28. 

External links

  • optical depth equations
This article was sourced from Creative Commons Attribution-ShareAlike License; additional terms may apply. World Heritage Encyclopedia content is assembled from numerous content providers, Open Access Publishing, and in compliance with The Fair Access to Science and Technology Research Act (FASTR), Wikimedia Foundation, Inc., Public Library of Science, The Encyclopedia of Life, Open Book Publishers (OBP), PubMed, U.S. National Library of Medicine, National Center for Biotechnology Information, U.S. National Library of Medicine, National Institutes of Health (NIH), U.S. Department of Health & Human Services, and, which sources content from all federal, state, local, tribal, and territorial government publication portals (.gov, .mil, .edu). Funding for and content contributors is made possible from the U.S. Congress, E-Government Act of 2002.
Crowd sourced content that is contributed to World Heritage Encyclopedia is peer reviewed and edited by our editorial staff to ensure quality scholarly research articles.
By using this site, you agree to the Terms of Use and Privacy Policy. World Heritage Encyclopedia™ is a registered trademark of the World Public Library Association, a non-profit organization.

Copyright © World Library Foundation. All rights reserved. eBooks from Project Gutenberg are sponsored by the World Library Foundation,
a 501c(4) Member's Support Non-Profit Organization, and is NOT affiliated with any governmental agency or department.