Crocco's theorem

2014-01-01T15:36:48Z

145.94.181.131:

In [[adaptive optics]], the '''Greenwood frequency'''<ref>{{cite journal|last=Greenwood|first=Darryl P.|title=Bandwidth specification for adaptive optics systems|journal=Journal of the Optical Society of America|year=1977|month=March|volume=67|issue=3|pages=390–303|doi=10.1364/JOSA.67.000390|url=http://www.astro.uu.nl/~werkhvn/study/Y5_07_08/master/papers/1977JOSA...67..390G.pdf|bibcode=1977JOSA...67..390G}}</ref> is the frequency or [[Bandwidth (signal processing)|bandwidth]] required for optimal correction with an adaptive optics system. It depends on the [[transverse]] windspeed and the turbulence strength in the atmosphere. This can be easily understood since if the turbulence moves over the telescope opening faster, the speed at which the [[wavefront]] needs to be corrected is higher, and vice-versa. There are various ways to define the Greenwood frequency, but all the definitions attempt to represent the frequency at which the turbulence distortion of the image is changing. The [[Multiplicative inverse|reciprocal]] of the Greenwood frequency is sometimes known as the Greenwood or atmospheric time constant (τ<sub>0</sub>). Since the distortions are approximately constant over a period less than this time constant, adapting the optical system at a faster rate yields negligible benefits; conversely, adaptive system performance degrades significantly as the response speed decreases below the Greenwood value, since that means that the distortions are changing faster than the system can adapt. Greenwood frequencies in common applications typically run from tens of [[hertz]] up to hundreds or even a few kilohertz, but unusual atmospheric conditions or unusual optical equipment can give very different values.

One expression for the Greenwood frequency is given by

<math>
f_{\mathrm G} = 2.31 \, \lambda^{-6/5} \left[ \sec \zeta \int_{\mathrm{Path}} C_n^2(z) \, v_{\mathrm{Wind}}(z)^{5/3} \, dz \right]^{3/5}
</math><ref>{{cite book|last=Tyson|first=Robert K.|title=Field guide to adaptive optics|year=2004|publisher=SPIE Press|coauthors=Frazier, Benjamin W.|url=http://www.slac.stanford.edu/spires/find/books/www?key=343315|page=14|isbn=0-8194-5319-6}}</ref>

With <math>\zeta</math> the [[zenith]] angle, <math>v_{\mathrm{Wind}}(z)</math> the windspeed as function of height and <math>C_n^2(z)</math> the so-called atmospheric turbulence constant structure function, a measure of the turbulence strength as function of height.

== See also ==
* [[Astronomical seeing]]
* [[Fried parameter]]
* [[Adaptive optics]]

==References==
{{reflist}}

[[Category:Astronomical imaging]]
[[Category:Turbulence]]

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Crocco's theorem