# Fried parameter

The Fried parameter[1] or Fried's coherence length (commonly designated as ${\displaystyle r_{0}}$) is a measure of the quality of optical transmission through the atmosphere due to random inhomogeneities in the atmosphere's 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 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 rms wavefront aberration due to passage through the atmosphere is equal to 1 radian. For a telescope with an aperture, ${\displaystyle D}$, the smallest spot that can be observed is given by the telescopes Point spread function (PSF). Atmospheric turbulence increases the diameter of the smallest spot by a factor approximately ${\displaystyle D/r_{0}}$ (for long exposures[2]). As such, imaging from telescopes with apertures much smaller than ${\displaystyle r_{0}}$ is less affected by atmospheric seeing than diffraction due to the telescope's small aperture. However, the imaging resolution of telescopes with apertures much larger than ${\displaystyle r_{0}}$ (thus including all professional telescopes) will be limited by the turbulent atmosphere, preventing the instruments from approaching the diffraction limit.

Although not explicitly written in his article, the Fried parameter at wavelength ${\displaystyle \lambda }$ can be expressed[3] in terms of the so-called atmospheric turbulence strength ${\displaystyle C_{n}^{2}}$ (which is actually a function of temperature fluctuations as well as turbulence) along ${\displaystyle z'}$ the path of the starlight :

${\displaystyle r_{0}=\left[0.423\,k^{2}\,\int _{\mathrm {Path} }C_{n}^{2}(z')\,dz'\right]^{-3/5}}$

where ${\displaystyle k=2\pi /\lambda }$ 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 angle ${\displaystyle \zeta }$, the line of sight passes through an air column which is ${\displaystyle \sec \zeta }$ times longer, producing a greater disturbance in the wavefront quality. This results in a smaller ${\displaystyle r_{0}}$, so that in terms of the vertical path z, the operative Fried parameter ${\displaystyle r_{0}}$ is reduced according to:

${\displaystyle r_{0}=\left[0.423\,k^{2}\,\sec \zeta \int _{\mathrm {Vertical} }C_{n}^{2}(z)\,dz\right]^{-3/5}=(\cos \zeta )^{3/5}\ r_{0}^{(vertical)}.}$

At locations selected for observatories, typical values for ${\displaystyle r_{0}}$ range from 10 cm for average seeing to 20 cm under excellent seeing conditions. The angular resolution is then limited to about ${\displaystyle \lambda /r_{0}}$ due to the effect of the atmosphere, whereas the resolution due to diffraction by a circular aperture of diameter ${\displaystyle D}$ is generally given as ${\displaystyle 1.22\lambda /D}$. Since professional telescopes have diameters ${\displaystyle D\gg r_{0}}$, they can only obtain an image resolution approaching their diffraction limits by employing adaptive optics.

Because ${\displaystyle r_{0}}$ is a function of wavelength, varying as ${\displaystyle \lambda ^{6/5}}$, its value is only meaningful in relation to a specified wavelength. When not stated explicitly, the wavelength is typically understood to be ${\displaystyle \lambda =0.5\mu m}$.