The temporal power spectrum of intensity fluctuations

Aime et al. (1986) showed that experimentally measured temporal power spectra of photometric measurements in the image plane of a telescope can be well fitted at high frequencies by negative exponential functions of the form:

$$P\left (f \right ) = A e^{\left (-a \left \vert f \right \vert \right )}$$ (2.1)

In many of their observations there is excess power at low frequencies, attributed to long-timescale motion of the image centroid (this excess power in the power spectrum is sometimes fitted empirically by adding another exponential term to Equation 2.1).

Equation 2.1 can be used to predict the form of the high frequency component of the temporal autocorrelation of stellar speckle patterns. After normalisation as described in Chapter 2.2.1, the temporal autocorrelation $C\left ( t \right)$ corresponding to Equation 2.1 has the form:

$$C\left ( t \right ) = \frac{a^{2}}{a^{2}+t^{2}}$$ (2.2)

The coherence timescale $\tau _{e}$ for the case described by Equation 2.1 will be:

$$\tau_{e}=a \sqrt{e-1}$$ (2.3)

Many measurements of the atmospheric coherence time $\tau _{e}$ for speckle imaging have been made at a variety of observatory sites. At $500$ $nm$ wavelength the measured timescales are usually found to be a few milliseconds or tens of milliseconds (Scaddan & Walker, 1978; Karo & Schneiderman, 1978; Parry et al. , 1979; Dainty et al. , 1981; Vernin & Muñoz-Tuñón, 1994; Roddier et al. , 1990; Marks et al. , 1999; Lohmann & Weigelt, 1979) although Aime et al. (1981) report timescales as long as a few hundred milliseconds under good conditions.

It will now be of interest to compare these experimental results and empirical analysis with atmospheric simulations.