Strehl ratios obtained for $\zeta$ Boötis

Figure 3.22 shows a scatter diagram comparing the Strehl ratios for each binary companion in the individual short exposures. There is a strong linear correlation between the Strehl ratios measured for the two stars, with a linear regression correlation coefficient of $r^{2}=0.975$ for a straight line through the origin. The gradient of $1.001$ for the best fit line gives a good consistency check on the magnitude difference between the two binary companions (calculated as $\delta m_{(810nm)}=0.045\pm0.03$ from the images in Figure 3.21, and used for calculating the Strehl ratios). The scatter in the points about the best fit line indicates the error in the Strehl ratio measurements from individual short exposures. The RMS difference between the Strehl ratios measured for the two stars was $0.0043$. If the random error component of the Strehl ratio measurements is equal for each of the stars, this would imply an RMS random error of $0.003$ on each Strehl ratio measurement.

Figure 3.22: Comparison of measured Strehl ratios for the two components in $\zeta$ Boötis. The regression coefficient for a straight line through origin is $r^{2}=0.975$. Only exposures of $\zeta$ Boötis where both stars are more than 16 pixels from edge of usable detector area were utilised, using data from both runs on this target.

Figure 3.22 compares the Strehl ratios of the brightest speckle in the PSF obtained from each of the stars in individual exposures. This comparison is not sensitive to variations in the relative positions of the brightest speckles for the two stars. In observations of distant ground-based artificial light sources through a turbulent medium, Englander et al. (1983) found relative motions in the position of the brightest speckle in Lucky Exposures for light sources which were separated in the object plane but within the isoplanatic patch. A qualitative discussion of this effect for astronomical observations is also found in Dantowitz et al. (2000); Dantowitz (1998). If such a variation occurs in the relative positions of the brightest speckles for the two components of $\zeta$ Boötis, this will lead to blurring of the image of the companion star when the short exposures are re-centred and co-added based on measurements of the reference star.

In order to investigate the magnitude of this effect, the short exposures were sorted by Strehl ratio into groups which each contained $1\%$ of the total number of exposures. The exposures in each group were then re-centred and co-added based on the measured positions of the brightest speckle for the reference star. This gave a single averaged PSF for the exposures in that group. The Strehl ratios for binary component b are plotted against the Strehl ratios for the reference star (component a) in Figure 3.23 for each of the summed images generated in this way. There is extremely good correlation between the Strehl ratios for the two stars, as emphasised by Figure 3.24. In this Figure, the Strehl ratio for component b has been divided by the Strehl ratio for component a for each of the summed images. The Strehl ratios for component b are typically only $0.5\%$ lower than those for the reference star although there is a more significant difference for the poorest exposures. It is clear from these Figures that there must be very good correlation between the positions of the brightest speckle for the two stars, and that measurements of the position of the brightest speckle using a reference star can reliably be used for re-centring images of another object in the field.

Figure 3.23: Exposures of $\zeta$ Boötis were binned into $100$ equal groups according to the Strehl ratio measured for binary component a (the reference star). The exposures in each group were then re-centred and co-added according to the position of the brightest speckle in the image of the reference star. The Strehl ratios measured in the shift-and-add images for the two binary components are plotted in the figure. The regression coefficient for a straight line through the origin is $r^{2}=0.99985$. Only exposures from both runs on $\zeta$ Boötis where both stars are more than 8 pixels from edge of usable detector area were used for this analysis.

Figure 3.24: The same data as for Figure 3.23, but the Strehl ratios measured for component b have been divided by the Strehl ratio for component a in each of the summed images.

The precise shape of the PSF obtained for the reference star and also that for other objects in the vicinity of a reference star is of interest in determining the applicability of the Lucky Exposures method for astronomical programs. The extent of the wings of the PSF determines the area of sky around bright stars which will be ``polluted'' by photon shot noise from starlight. If the image of the reference star is sufficiently similar to the PSF obtained for other objects in the field it can be used for deconvolving the astronomical image. For this reason I undertook an investigation of the faint wings of the PSF, and the differences between the PSF obtained for the reference star and that for the binary companion.

Figure 3.25 shows the best $1\%$ of exposures of $\zeta$ Boötis using the brighter (left-hand) component as a reference for Strehl ratio measurements. Figure 3.25a shows a linear greyscale ranging from zero to the maximum flux in the image. Figure 3.25b shows the same image with a stretched linear greyscale ranging from zero to one-tenth of the maximum flux. In order to investigate the level of similarity between the PSFs for the two binary companions, I subtracted the image of the right-hand star from the image of the left-hand star. A copy of the image shown in Figure 3.25 was multiplied by the intensity difference between the two stars, shifted by the separation of the stars and subtracted from the original image. This eliminated most of the flux from the left-hand star, as shown in Figure 3.26. The small residual component visible in the greyscale-stretched version of this image shown in Figure 3.26b is largely due to a small error in the measured separation of the stars due to the finite pixel size used. There is no clear evidence for anisoplanatism between the two binary components.

Figure 3.25: The best $1\%$ of exposures of $\zeta$ Boötis processed using the Lucky Exposures method and plotted with two different greyscales. Both greyscales are linear below saturation. The upper left star (component a) was used as the reference star.

The remaining binary component in Figure 3.26 represents a good measure of the PSF for imaging in the vicinity of a reference star using the Lucky Exposures method. The compact image core and steeply decaying wings around the star in this figure indicate that high resolution, high dynamic range imaging will be possible using the Lucky Exposures technique.

Figure 3.26: The reference star was suppressed from the image shown in Figure 3.25 by subtracting the image of binary component b from the reference star image (component a). This was done by taking a copy of the image shown in Figure 3.25, scaling it by the magnitude difference for the binary, shifting it by the binary separation, and subtracting it from the original image. Both greyscales are linear below saturation.

The subtracted image in Figure 3.26 allowed investigation of the faint wings of the PSF for component b without strong effects from the contribution of the reference star (component a). Figure 3.27 shows profiles through the image in Figure 3.26. Curve X in Figure 3.27a shows a single cross-section along a line perpendicular to the separation vector between the two stars, passing through binary component b. Curve Y shows the flux averaged around the circumference of a circle centred on the star, plotted as a function of the circle radius (i.e. a radial profile). At large distances from the core, the flux in the PSF drops off exponentially in both of these curves (with an $e$-folding distance of $0.17$ $as$). This is highlighted in the logarithmic plots of the same curves shown in Figure 3.27b. The kink at $\sim 0.78$ $as$ in the radial profile plot corresponds to the location of the reference star, indicating that it was not fully subtracted from the images. For comparison the profile of a diffraction-limited PSF sampled with the same pixel scale is shown in both figures as curve Z.

Figure 3.27: Cross sections through the b component of $\zeta$ Boötis plotted on a linear scale in the left-hand panel and on a logarithmic scale in the right-hand panel. Curve X shows the variation of flux along a line perpendicular to the separation between the binary components in Figure 3.25. Curve Y shows a radially averaged profile of the b component after subtraction of the a component (based on Figure 3.26). Curve Z shows the simulated profile in the absence of atmospheric turbulence (but with the same pixel sampling and a $32\times 32$ grid of sub-pixel position offsets for star, resampled, re-centred and co-added in the usual way).

If a large number of selected exposures are co-added, the speckle patterns in the wings of the PSF will average out to give a smooth halo. If the flux in this halo follows the exponentially decaying radial distribution shown by curves X and Y of Figure 3.27, then the halo flux could be removed using deconvolution with a simple axisymmetric, exponentially decaying model for the PSF. This would only leave a small residual component from the photon shot noise and small deviations of the PSF halo from the model. It is clear from the rapid decay of the curves in Figure 3.27 that very high dynamic range imaging should be possible, even within relatively crowded fields.