Varying the fraction of exposures selected

In order to assess the performance of the Lucky Exposures method with different exposure selection parameters, I analysed the $\zeta$ Boötis data several times. In each case the images of the two binary components appear very similar, suggesting that the field is isoplanatic with little variation in the imaging PSF as a function of position.

Some of the results from the $\zeta$ Boötis data are summarised in Figure 3.21. Figure 3.21a shows an image generated from the short exposures have the highest $1$% of Strehl ratios as measured on the left-hand component of $\zeta$ Boötis. The stellar images appear almost diffraction limited, with the first Airy ring clearly visible. The diffuse halo surrounding the stars is very faint and barely visible in the image. Figure 3.21b shows the result when the process is repeated using the right-hand star as the reference for measuring Strehl ratio and the position of the brightest speckle. Figures 3.21a and 3.21b are almost indistinguishable to the eye, emphasising the high degree of isoplanatism and the good signal-to-noise in the images. The Strehl ratio for the reference star in each case is $0.19$.

Figure 3.21: Comparison of images generated using different exposure selection approaches in $\zeta$ Boötis. In the three left-hand panels (a), c), e)), the left-hand binary component has been used as the reference star, while the right-hand component was used for the images in the right-hand panels. The properties of the six images are as follows: a) & b) best $1\%$ of exposures selected, Strehl ratio of reference star is $0.190$ in both images; c) & d) best $10\%$ of exposures selected, Strehl ratio of reference star is $0.136$ in both images; e) all exposures shifted and added, Strehl ratio of reference star is $0.0782$; and f) all exposures shifted and added, Strehl ratio of reference star is $0.0783$. In each case the Strehl ratio for the binary companion star is found to be a factor of $0.985\pm 0.005$ times lower than for the reference star.

Figures 3.21c and 3.21d show images generated in a similar way but using the short exposures which have the highest $10\%$ of Strehl ratios. A diffuse halo is clearly visible around both stars slightly reducing the Strehl ratio for the reference star images to $0.14$.

Figures 3.21e and 3.21f show images generated in a similar way but using all of the short exposures regardless of Strehl ratio. The diffuse halos are much more prominent around the stars reducing the Strehl ratio for the reference star images to $0.078$. These represent the conventional shift-and-add images from the same data.

For all six images shown in Figure 3.21, the Strehl ratio for the binary companion was found to be only $98.5\%\pm0.5\%$ as high as that of the reference star. This indicates a small level of decorrelation between the shapes of the stellar images for the two binary components as recorded on the detector. It is likely that the decorrelation comes partially from noise sources such as detector readout noise, photon shot noise, and in particular the pixellation of the stellar image on the detector. Both the exposure selection step and the image re-centring have a tendency to coherently add the noise components in the image of the reference star to give an artificially high Strehl ratio for this binary component. This effect is described in detail by Nieto & Thouvenot (1991) for the photon-shot noise component. The noise contribution is not expected to show strong correlation between the separate binary components, so the Strehl ratio for the binary companion should not be systematically affected in this way.

It is clear from Figures 3.21a to 3.21f that the imaging PSF degrades gradually as the fraction of exposures selected is increased. The gradual nature of this change may be extremely useful in astronomical programs as the performance of the Lucky Exposures method can be adjusted according to the scientific needs. If an astronomical target is too faint to give good signal to noise using only the best $1\%$ of exposures, the astronomer can choose to use a larger fraction of exposures at the expense of a small degradation in the image quality. If the observational data are stored in a suitable manner, the fraction of exposures selected can be adjusted after the observations have been completed (during the data reduction) in order to give the highest quality science results.