Temporal properties of the atmosphere

In order to investigate the temporal properties of the atmosphere during these observations and help determine the optimum exposure time for the Lucky Exposures method it is of interest to look at the typical variation in the flux at a point in the image plane speckle pattern from one short exposure to the next. The statistics of the temporal fluctuations at one fixed point in the image plane should be representative of the fluctuations at any other point in the image plane.

In order to minimise the effects of long-timescale drift in the location of the stellar PSF on the measurements, the pixel located at the centroid of the long exposure average image of the run on $\epsilon$ Aquilae was selected (i.e. the centroid of Figure 3.14d). The average image for this run had an unusually compact PSF, showing no evidence for substantial drift in the location of the speckle pattern during the $30$ $s$ run. The stellar flux from the selected pixel was recorded in each short exposure, producing a one-dimensional dataset characterising the temporal fluctuations in the PSF. The temporal power spectrum of this dataset is shown as curve A in Figure 3.15. The power spectrum shows a peak at a frequency of $16$ $Hz$. This is in close agreement with the first harmonic of mechanical oscillation for the telescope structure, and movies made from the raw speckle images clearly show motion consistent with such oscillations. The frequency was confirmed to be $16$ $Hz$ by measuring the position of the brightest speckle in each short exposure and then looking at the temporal power spectrum of this time series.

Figure 3.15: Temporal power spectra generated from flux measurements at a fixed point in the image plane. The curves have been smoothed by binning together adjacent spatial frequencies. A shows the result for data taken on the star $\epsilon$ Aquilae at the NOT. B shows results from an example atmospheric simulation, in this case with two wind blown Taylor screens, one having $75\%$ of the turbulence strength and taking $420$ $ms$ to cross the telescope diameter, and one having $25\%$ of the turbulence strength and taking $63$ $ms$ to cross the aperture. Model 2 was used for the mirror aberrations. The simulation used for curve C was the same as that for B but with a diffraction-limited mirror (model 3). For curve D all the turbulence was in a single layer taking $420$ $ms$ to cross the telescope mirror with model 2 for the aberrations in the mirror. Curve E is for the same case as D but with a diffraction-limited mirror. The lines are offset vertically for clarity.

The effect of telescope oscillation on the temporal fluctuations at a fixed point in the image plane can be seen by splitting into partial derivatives the derivative of the flux $I$ with respect to time at a fixed point in the telescope image plane. It is simplest to work in a coordinate frame which is fixed with respect to the speckle pattern, and study the effect of moving the optical detector at a velocity $\mathbf{v}$ relative to the speckle pattern. In these coordinates, the time derivative of the flux is:

$$\frac{\mbox{d}I}{\mbox{d}t}=\left.\frac{\partial I}{\partial... ...t\left.\frac{\partial I}{\partial \mathbf{r}}\right \vert _{t}$$ (3.3)

where $\mathbf{r}$ is the position of the optical detector in the speckle pattern, and $\mathbf{v}$ is the velocity of the detector with respect to the speckle pattern. The motion of the detector with respect to the speckle pattern resulting from telescope oscillation thus produces a coupling between the spatial variations of the flux in the speckle pattern and temporal fluctuations measured at a fixed point in the image plane.

It is the total differential from Equation 3.3 which limits the exposure time we can use during observations at the NOT. For our data it is the telescope oscillation which provides the dominant contribution on short timescales. If the amplitude of the telescope oscillation could somehow be reduced however, the ultimate limit to the exposure time would be set by the partial derivative $\left.\frac{\partial I}{\partial t}\right \vert _{r}$. This term represents the component of the time variation in the flux which is introduced directly by changes in the speckle pattern. It is of interest to try to measure the timescale associated with this, as it would be applicable to other telescopes operating under similar atmospheric conditions.

Curve A in Figure 3.16 shows the temporal autocorrelation of the same dataset from $\epsilon$ Aquilae as Figure 3.15 (it represents the Fourier transform of curve A in Figure 3.15). The curve has been normalised so that it ranges from unity at zero time difference to a mean value of zero for time differences between $\sim200$ $ms$ and $\sim2000$ $ms$. The $16$ $Hz$ oscillatory component is clearly visible. This oscillation is largely responsible for the initial decorrelation in the measured flux as a function of time. The telescope oscillation will only reduce the temporal correlation, so the true autocorrelation function corresponding to the atmosphere would lie above curve A for all time differences. The effect of photon-shot noise was negligible in these observations due to the high flux in each individual exposure. The frame rate ($185$ $Hz$) was sufficiently high that the sharp peak in curve A around zero time difference is relatively well sampled in this dataset (the peak does not simply correspond to a single high value at zero time difference, but contains several data points).

Figure 3.16: Curve A) shows the temporal autocorrelation of the flux at a fixed point in the image plane for the $\epsilon$ Aquilae data. Curve B is based on the same data, but the oscillation in the curve has been artificially suppressed as described in the text. C is a fit to curve B based on Equation 2.2 (the model of Aime et al. (1986)).

Curve B in this Figure is a function extrapolated from the measured curve by dividing it by a decaying sinusoid having the same period as the telescope oscillation. The amplitude and decay time of the sinusoid were chosen so as to minimise the residual component at $16$ $Hz$. This curve is intended to represent a possible shape for the temporal autocorrelation in the absence of telescope oscillation.

Curve C shows a fit to curve B of the form of Equation 2.2 (based on the model of temporal fluctuations by Aime et al. (1986)). The broad peak produced by this model does not seem consistent with the sharp peak seen in the experimental data.

The sharp peak seen in curves A and B of Figure 3.16 could be reproduced qualitatively in numerical simulations if multiple Taylor screens were used with a scatter of different wind velocities. One example of a simulation which gave a better fit to the shape of the experimentally measured temporal autocorrelation is shown as curve C in Figure 3.17. The atmospheric model for this simulation consisted of two Taylor screens moving at constant velocities. Both layers moved in the same direction but with different speeds. One layer, containing $75\%$ of the the turbulence took $420$ $ms$ to cross the diameter of the telescope aperture. The other contained $25\%$ of the turbulence, but took only $63$ $ms$ to cross the telescope aperture. Curves A and B from Figure 3.16 are also reproduced as curves A and B in Figure 3.17 for comparison. Temporal power spectra generated using this model of the atmosphere are plotted as curves B and C in Figure 3.15. It is clear that they provide a much better fit to the experimentally measured data in curve A than the single layer atmospheric models shown in curves D and E.

Figure 3.17: Curve A) shows the temporal autocorrelation of $\epsilon$ Aquilae data and B shows the same data with the oscillation artificially suppressed, as in Figure 3.16. Curve C shows an example of one simulation which fitted the data of curve B. The model atmosphere had two layers: one with $75$% of the turbulence strength taking $420$ $ms$ to move across the diameter of the aperture, and one with $25$% of the turbulence taking $63$ $ms$ to move across the aperture. This curve is the Fourier transform of the power spectrum shown in curve B of Figure 3.15

It is clear from the temporal autocorrelation plots of Figures 3.16 and 3.17 that there are (at least) two timescales associated with the decorrelation of the speckle pattern: the half-period of the telescope oscillation and the timescale for the decorrelation of the atmosphere. The decorrelation timescale $\tau _{e}$ (as defined in Chapter 2.2.1) which results from the combination of these two effects is $22$ $ms$. Using a simple fit to the oscillatory component (used to produce curve B in both Figures) the decorrelation timescale for the atmosphere alone was calculated to be $65$ $ms$.

If the atmosphere had a single, boiling-free layer then Equation 2.14 could be used to obtain the wind velocity. Taking $r_{0}=0.37$ $m$ (consistent with the seeing disk, and with the Strehl ratios in Figure 3.10), a wind velocity of $17$ $m$ $s^{-1}$ is obtained. This is significantly larger than the wind velocity near ground level of $5$ $m$ $s^{-1}$ (from Figure 3.7), but would not be implausible if the turbulence were situated at high altitude.

If the atmosphere had multiple layers travelling at different velocities, and the timescale for decorrelation of the wavefronts was shorter than the wind crossing timescale of the telescope aperture, then the dispersion in the wind velocities $\Delta v$ could be calculated using Equation 2.12. The value obtained for $\tau_{e}=65$ $ms$ is $\Delta v=5.7$ $m$ $s^{-1}$ (again taking $r_{0}=0.37$ $m$). This level of dispersion in wind velocities between atmospheric layers seems consistent with the wind velocity of $5$ $m$ $s^{-1}$ measured near to the ground.

Both of these atmospheric configurations are plausible. The second is perhaps more likely given that the strongest turbulence is most commonly found at relatively low altitudes, where small wind speeds were observed.