Fried parameter

When observing a star through a telescope, the atmosphere distorts the incoming light, making images blurry and causing stars to twinkle. The Fried parameter, or Fried's coherence length, is a quantity that measures the strength of this optical distortion. It is denoted by the symbol $r_{0}$ and has units of length, usually expressed in centimeters.

The Fried parameter can be thought of as the diameter of a "tube" of relatively calm air through the turbulent atmosphere. Within this area, the seeing is good. A telescope with an aperture diameter $D$ that is smaller than $r_{0}$ can achieve a resolution close to its theoretical best (the diffraction limit). However, for telescopes with apertures much larger than $r_{0}$—which includes all modern professional telescopes—the image resolution is limited by the atmosphere, not the telescope's size. The angular resolution of a large telescope without adaptive optics is limited to approximately $\lambda /r_{0}$, where $\lambda$ is the wavelength of the light observed. At good observatory sites, $r_{0}$ is typically 10–20 cm at visible wavelengths. Large ground-based telescopes use adaptive optics to compensate for atmospheric effects and reach the diffraction limit.

Technically, the Fried parameter is defined as the diameter of a circular area over which the rms wavefront aberration is equal to 1 radian.

Mathematical definition

Although not explicitly written in his original article, the Fried parameter at wavelength $\lambda$ can be expressed in terms of the atmospheric turbulence strength $C_{n}^{2}(z')$ (which is a function of temperature and turbulence fluctuations) along the light's path $z'$:$$r_{0}=\left[0.423\,k^{2}\,\int _{\mathrm {Path} }C_{n}^{2}(z')\,dz'\right]^{-3/5}$$where $k=2\pi /\lambda$ is the wavenumber. If not specified, the path is assumed to be in the vertical direction.

When observing a star at a zenith angle $\zeta$, the light travels through a longer column of atmosphere by a factor of $\sec \zeta$. This increases the disturbance, resulting in a smaller $r_{0}$:$$r_{0}=(\cos \zeta )^{3/5}\ r_{0}^{\text{(vertical)}}.$$Because $r_{0}$ varies with wavelength as $\lambda ^{6/5}$, its value is only meaningful when the observation wavelength is specified. If not stated, it is typically assumed to be $\lambda =0.5\,\mathrm {\mu m}$ (in the visible spectrum).

See also

• Astronomical seeing
• Adaptive optics
• Greenwood frequency