Optical depth

Optical depth

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Optical depth, or optical thickness, is a measure of transparency. Optical depth is defined by the negative logarithm
Logarithm
The logarithm of a number is the exponent by which another fixed value, the base, has to be raised to produce that number. For example, the logarithm of 1000 to base 10 is 3, because 1000 is 10 to the power 3: More generally, if x = by, then y is the logarithm of x to base b, and is written...

 of the fraction of radiation
Radiation
In physics, radiation is a process in which energetic particles or energetic waves travel through a medium or space. There are two distinct types of radiation; ionizing and non-ionizing...

 (e.g., light
Light
Light or visible light is electromagnetic radiation that is visible to the human eye, and is responsible for the sense of sight. Visible light has wavelength in a range from about 380 nanometres to about 740 nm, with a frequency range of about 405 THz to 790 THz...

) that is not scattered or absorbed on a path. Hence optical depth is dimensionless
Dimensionless quantity
In dimensional analysis, a dimensionless quantity or quantity of dimension one is a quantity without an associated physical dimension. It is thus a "pure" number, and as such always has a dimension of 1. Dimensionless quantities are widely used in mathematics, physics, engineering, economics, and...

, and particularly is not expressed in units of length; however the optical depth to an object will typically approach zero as the spatial distance to the object decreases.

Formulations


The optical depth expresses the quantity of light removed from a beam by scattering
Scattering
Scattering is a general physical process where some forms of radiation, such as light, sound, or moving particles, are forced to deviate from a straight trajectory by one or more localized non-uniformities in the medium through which they pass. In conventional use, this also includes deviation of...

 or absorption
Absorption (electromagnetic radiation)
In physics, absorption of electromagnetic radiation is the way by which the energy of a photon is taken up by matter, typically the electrons of an atom. Thus, the electromagnetic energy is transformed to other forms of energy for example, to heat. The absorption of light during wave propagation is...

 during its path through a medium
Transmission medium
A transmission medium is a material substance that can propagate energy waves...

.
If is the intensity
Intensity (physics)
In physics, intensity is a measure of the energy flux, averaged over the period of the wave. The word "intensity" here is not synonymous with "strength", "amplitude", or "level", as it sometimes is in colloquial speech...

 of radiation at the source and
is the observed intensity after a given path, then optical depth is defined by the
following equation:

Calculation from fundamental principles


In atomic physics
Atomic physics
Atomic physics is the field of physics that studies atoms as an isolated system of electrons and an atomic nucleus. It is primarily concerned with the arrangement of electrons around the nucleus and...

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



where denotes the transition dipole moment, the natural linewidth of the transition, the frequency, the number of atoms, and the cross-section of the beam.

Atmospheric Science


In atmospheric sciences
Atmospheric sciences
Atmospheric sciences is an umbrella term for the study of the atmosphere, its processes, the effects other systems have on the atmosphere, and the effects of the atmosphere on these other systems. Meteorology includes atmospheric chemistry and atmospheric physics with a major focus on weather...

, 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 refers to a vertical path, the optical depth for a slant path is
, where is called the relative airmass
Airmass
In astronomy, air mass is the optical path length through Earth’s atmosphere for light from a celestial source. As it passes through the atmosphere, light is attenuated by scattering and absorption; the more atmosphere through which it passes, the greater the attenuation. Consequently, celestial...

, and for a plane-parallel
atmosphere it is determined as , where is the
zenith angle
Celestial coordinate system
In astronomy, a celestial coordinate system is a coordinate system for mapping positions on the celestial sphere.There are different celestial coordinate systems each using a system of spherical coordinates projected on the celestial sphere, in analogy to the geographic coordinate system used on...

 corresponding to the given path.
Therefore


The optical depth of the atmosphere can be divided into several components, ascribed to Rayleigh scattering
Rayleigh scattering
Rayleigh scattering, named after the British physicist Lord Rayleigh, is the elastic scattering of light or other electromagnetic radiation by particles much smaller than the wavelength of the light. The particles may be individual atoms or molecules. It can occur when light travels through...

,
aerosols, and gaseous absorption
Absorption (electromagnetic radiation)
In physics, absorption of electromagnetic radiation is the way by which the energy of a photon is taken up by matter, typically the electrons of an atom. Thus, the electromagnetic energy is transformed to other forms of energy for example, to heat. The absorption of light during wave propagation is...

.
The optical depth of the atmosphere can be measured with a sun photometer
Sun photometer
A sun photometer is a type of photometer conceived in such a way that it points at the Sun.Recent sun photometers are automated instruments incorporating a Sun-tracking unit, an appropriate opticalsystem, a spectrally filtering device, a photodetector, and a...

.

Stellar Physics


Another example occurs in astronomy
Astronomy
Astronomy is a natural science that deals with the study of celestial objects and phenomena that originate outside the atmosphere of Earth...

 where the photosphere
Photosphere
The photosphere of an astronomical object is the region from which externally received light originates. The term itself is derived from Ancient Greek roots, φῶς, φωτός/phos, photos meaning "light" and σφαῖρα/sphaira meaning "sphere", in reference to the fact that it is a spheric surface perceived...

 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 (wavelength
Wavelength
In physics, the wavelength of a sinusoidal wave is the spatial period of the wave—the distance over which the wave's shape repeats.It is usually determined by considering the distance between consecutive corresponding points of the same phase, such as crests, troughs, or zero crossings, and is a...

s) 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.

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