Effect of Atmospheric Turbulence on the Long Exposure PSF

It is important to be able to determine the telescope MTF when atmospheric perturbations are present. In other words, we want to compute the MTF of a telescope looking through the turbulent atmosphere. In the astronomical case, the exposure time is usually much longer than the atmospheric correlation time; thus, the relevant MTF is the time-averaged MTF. Assuming ergodicity conditions, we will substitute the time average with a statistic average. For this computation, we will have to use the phase structure function defined already. The results reported in this section have been obtained by Fried in 1966 [11]. The expression for the MTF in the case of an integration time much longer then the correlation time of refraction index fluctuations is given by

(MTF(/)) ot^j W (r + d)W (r)dr^ = J P (r + d)P (r)dr ^ , where (and) represent the expectation value of the given quantity a. Now, considering the central limit theorem the above expression can be rewritten as

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