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The '''Fried parameter'''<ref>{{cite journal|last=Fried|first=D. L.|title=Optical Resolution Through a Randomly Inhomogeneous Medium for Very Long and Very Short Exposures|journal=Journal of the Optical Society of America|year=1966|month=October|volume=56|issue=10|pages=1372–1379|doi=10.1364/JOSA.56.001372|url=http://www.opticsinfobase.org/abstract.cfm?id=53167|bibcode=1966JOSA...56.1372F}}</ref> or '''Fried's coherence length''' (commonly designated as <math>r_0</math>) is a measure of the quality of optical transmission through the [[atmosphere]] due to random inhomogeneities in its refractive index. In practice, such inhomogeneities are primarily due to tiny variations in temperature (and thus density) on smaller spatial scales resulting from random turbulent mixing of larger temperature variations on larger spatial scales as first [[Turbulence#Kolmogorov's theory of 1941|described]] by [[Kolmogorov]]. The Fried parameter has units of length and is typically expressed in centimeters. It is defined as the diameter of a circular area over which the [[root mean square|rms]] [[wavefront]] aberration due to passage through the atmosphere is equal to 1 [[radian]]. As such, imaging from telescopes with apertures much smaller than <math>r_0</math> is less affected by atmospheric [[Astronomical seeing|seeing]] than [[diffraction]] due to the telescope's small aperture. However the imaging resolution of telescopes with apertures much larger than <math>r_0</math> (thus including all professional telescopes) will be limited by the turbulent atmosphere, preventing the instrument from approaching the [[diffraction limit]].
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Although not explicitly written in his article, the Fried parameter at wavelength <math>\lambda</math> can be expressed<ref>{{cite book|last=Hardy|first=Johw W.|title=Adaptive optics for astronomical telescopes|year=1998|publisher=Oxford University Press|url=http://openlibrary.org/works/OL2633873W/Adaptive_optics_for_astronomical_telescopes|page=92|isbn=0-19-509019-5}}</ref> in terms of the so-called  [[Astronomical seeing#The profile|atmospheric turbulence strength]] <math>C_n^2</math> (which is actually a function of temperature fluctuations as well as turbulence) along <math>z'</math> the path of the starlight :
 
<math>
r_0 = \left [ 0.423 \, k^2 \,  \int_{\mathrm{Path}} C_n^2(z') \, dz' \right ]^{-3/5}
</math>
 
where <math>k= 2 \pi / \lambda </math> is the [[wavenumber]]. If not specified, a reference to the Fried Parameter in astronomy is understood to refer to a path in the vertical direction. When observing at a [[Zenith#Relevance and use|zenith angle]] <math>\zeta</math>, the line of sight passes through an air column which is <math>\sec \zeta</math> times longer, producing a greater disturbance in the wavefront quality. This results in a smaller <math>r_0</math>, so that in terms of the ''vertical'' path ''z'' the operative Fried parameter <math>r_0</math> is reduced according to:
 
<math>
r_0 = \left [ 0.423 \, k^2 \, \sec \zeta \int_{\mathrm{Vertical}} C_n^2(z) \, dz \right ]^{-3/5} = (\cos \zeta)^{3/5} \ r_0^{(vertical)}.
</math>
 
At locations selected for observatories, typical values for <math>r_0</math> range from 10&nbsp;cm for average seeing to 20&nbsp;cm under excellent seeing conditions. The angular resolution is then limited to about <math>\lambda / r_0</math> due to the effect of the atmosphere, whereas the [[Angular resolution|resolution due to diffraction]] by a circular aperture of diameter <math>D</math> is generally given as <math>1.22 \lambda / D</math>. Since professional telescopes have diameters <math>D \gg r_0</math>, they can only obtain an image resolution approaching their diffraction limits by employing [[adaptive optics]].
Because <math>r_0</math> is a function of wavelength, varying as <math>\lambda ^{6/5}</math>, its value is only meaningful in relation to a specified wavelength. When not stated explicitely, the wavelength is typically understood to be <math>\lambda = 0.5 \mu m</math>.
 
== See also ==
* [[Astronomical seeing]]
* [[Adaptive optics]]
* [[Greenwood frequency]]
 
== References ==
{{reflist}}
 
[[Category:Astronomical imaging]]

Latest revision as of 17:34, 10 November 2014

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