In this chapter I describe a number of methods that I have developed for the measurement of the parameters of the close binaries discovered by LuckyCam in chapters 6 and 7. Many of these techniques are specific to the use of data from L3CCDs and/or Lucky Imaging. As the high-speed high-QE photon counting capabilities of L3CCDs are extremely useful for astronomy, and Lucky Imaging shows excellent promise for very extensive astronomical use, these methods are likely to be useful for a range of instruments in the future.
Performing accurate and precise photometry of very close binary systems is particularly challenging, because the PSFs of the binary components are often blended. An extra complication is introduced in Lucky Imaging images – faint guide stars have different PSFs from all other stars in the field. In this section I first describe the faint guide star problem, followed by the development and tests of a general photometry system for LuckyCam images of close binaries.
If the Lucky Imaging guide star is faint, so that photon shot noise in the PSF core becomes significant, the frame selection and alignment can be biased away from the true brightest speckle of the PSF by high excursions of the photon noise. In the final high-resolution images the guide star then has a photon-event-shaped core overlayed on the real PSF. The PSF is also degraded in resolution by mis-alignments.
The guide star shares its underlying PSF with the other stars in the field (within the limits of anisoplanatism), because each star is corrected with the same field motions and frame selections. However, only the guide star suffers from the photon-event-shaped core, as all other stars have photon noise uncorrelated with that of the guide star. Because of the (often very) different guide star PSF, standard PSF fitting photometry cannot include a faint guide star. Since many LuckyCam targets serve as their own guide stars, this would be a serious limitation without corrective measures.
The problem is made worse by the standard Lucky Imaging image construction. Because the convolution section of the frame alignment process effectively blurs the registration on individual photon events, the photon-event-shape overlay on the guide star acquires the shape of a diffraction-limited PSF core. Although the photon events themselves are not convolved by the diffraction-limited PSF during the Drizzle reconstruction, the blurred alignment leads to the final shape of the many summed events to replicate that of the convolution kernel. This is very hard to remove, because the real PSF itself includes a diffraction-limited core component (figure 5.1, top left).
Although in principle a calibrator star could be used to determine the level of this effect, the star would have to be matched to the target’s brightness, because the strength of the effect is photon-noise and seeing dependent. A second calibrator would also have to be found nearby to estimate the non-guide star PSF.
The effect can, however, be limited if PSF convolution and resampling is disable during the Lucky Imaging process. Alignment and selection then proceeds purely on the brightest pixel in the input image. This mode limits Lucky Imaging performance, but constrains any high excursions of photon noise to a single pixel – as each pixel contains photon noise uncorrelated with that of its neighbours. That pixel may then be ignored or re-estimated in subsequent PSF fitting procedures (top two images of figure 5.1). In practise, CTE effects lead to the two pixels to the right of the peak also being affected, and those pixels must also be masked.
As discussed in chapter 3, very close binaries often appear as triple systems because the Lucky Imaging system occasionally shifts lock from one component of the binary to the other. This leads to the secondary appearing in the position of the primary in some fraction of the images. The primary then forms a tertiary image separated from the guide star by the binary separation, but in the opposite direction (clearly visible in figure 5.1).
The three components of the triple image have the following fluxes, with F1 being the flux of the image in the position of the true primary, F2 that in the position of the true secondary, and F3 the flux in the position of the tertiary image:
 | (5.1) |
 | (5.2) |
 | (5.3) |
FP and FS are the true primary and secondary components’ fluxes respectively, and P is the probability of mistakenly using the secondary component as a guide star instead of the primary.
P varies with the flux ratio between the two components and the SNR of the brightest speckles. Equal-flux binaries will swap lock between the two components with P=50%. Bright binaries with high contrast ratios and high SNRs will have a very small P; for an example of this, see the observations of the i=+11 mags Ross 530 (chapter 6), which is a 0.15 arcsec binary but does not suffer from tripling. If the SNR is low, however, even high contrast ratio binaries have high P – see the observations of the possible substellar companion to LSPM J1735+2634 in chapter 6.
For useful photometry, we wish to recover the original primary and secondary fluxes from the above three measurements. The de-tripling procedure described in chapter 3 operates only down to separations of ~0.25 arcsec, requires significant manual experimentation to find the best guide star box positions, and may fail in poor seeing conditions. For these reasons, in the LuckyCam VLM binary survey I chose to measure the fluxes of the three images visible in standard Lucky Imaging images, and solve the above equations simultaneously for the true binary flux ratio.
If we define I12 as F1∕F2 and I13 as F1∕F3 then FR (the true binary flux ratio), is:
 | (5.4) |
This equation is undefined when the tertiary image has no flux (although it has the correct asymptotic behaviour); in that case the binary flux ratio is just the simple flux ratio of the two components. The PSF fitting tests described in the next section include simulated tripling, and the true binary flux was recovered using the above equation.
Because the LuckyCam VLM survey requires resolved photometry for a large number of close binaries, I developed an automated PSF fitting program. PST fits identical PSFs to the components of a LuckyCam image of a binary with the aim of retrieving the binary contrast ratio. Because the LuckyCam VLM binary survey did not include PSF reference stars taken very close in time and space to the targets, I fit the PSF using a parameterised model.
Each PSF is parameterised at a radius r as follows:
 | (5.5) |
The first term is a Moffat point spread function describing the halo of the PSF, where rs is the Moffat scale factor, β varies between 1 and ∞, and FH denotes the proportion of the star’s light contained within the PSF halo. The second term is a 2D Gaussian that models the diffraction-limited core of the PSF. ra is the width of the core and is generally 1-2 pixels. FS is proportional to the total flux from the star. For simplicity, the normalisation of the two PSF terms not included, because identical PSFs are fitted and the system need only produce the flux ratio between the binary components.
Before fitting, the pixels affected by the faint guide star problem described above are rescaled to the average flux of the pixels just above and below them* . All the above PSF parameters are allowed to vary on a per-image basis, as well as the binary position, separation and the two flux ratios (of the primary to the secondary and tertiary images). I use a simulated annealing (Press et al., 1988) algorithm to minimise χ2 summed over all the pixels in the input image. The noise value in each pixel is estimated from the number of photon events within that pixel. An example of the quality of fit is shown in figure 5.1.
To improve the quality of the derived parameters, multiple fits are performed using images of the binary with different PSFs. The wealth of Lucky Imaging frames allows these multiple tests to be performed very easily. By default PST uses 5 per cent non-cumulative selections from a LuckyCam run – that is, the best 5 per cent of frames shifted-and-added into a high-resolution image, followed by the next-best 5 per cent, and so on. Each selection level provides an independent image of the binary, with a unique PSF. 5 separate images ranging from the best 5 per cent of frames to the 20-25 percentiles are used, greatly improving the reliability of the derived binary flux ratios.
I tested the performance of the PSF fitting against simulated binaries of known flux ratio. Real LuckyCam images of single stars from the VLM binary surveys were replicated to form binaries at three flux ratios, in the 5 separate PSFs available for each of the 6 stars tested. For each PSF a large variety of binary separations and position angles were tested. 540 trials were performed in total, with 30 different PSFs covering a wide variety of atmospheric conditions and guide star brightnesses. The trials give two separate results: both the fitting accuracy and precision. The accuracy is found from comparing the measured flux ratio to the simulated input flux ratio; since real PSFs are used the fitting is not perfect. The fitting precision is found from the range of flux ratios derived from different simulated images with different input PSFs.
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The test results are shown in figure 5.2. Down to 0.3 arcsec separation, at a variety of flux ratios, fitting performance is roughly constant at an error of 0.2-0.3 mags. The performance is much poorer at 0.2 arcsec separation, because at this radius the 0.08 arcsec FWHM diffraction limited cores of the two stars become close enough to blend.
Under good conditions speckle imaging and AO measurements achieve precisions of 0.1-0.2 mags down to about 0.2 arcsec separation (see eg. Horch et al. (2004) and references therein). These techniques make use of an unresolved PSF reference star, selected to be very close to the target binary, both in sky position and time of observation. LuckyCam photometry would similarly benefit from the availability of reference PSFs.
The LuckyCam VLM binary survey, which the PSF fitting technique was designed for, is based on a quick look at 50-60 targets per night and did not include nearby PSF reference stars because of time constraints. However, in subsequent reobserving of the detected binaries (to follow the orbital motion of the companion) nearby PSF reference stars will be targeted, and this is likely to very significantly improve the quality of the binary contrast ratio measurements.
Astrometric measurements of very close binaries (<0.2 arcsec) on LuckyCam images is challenging because the crowding of the PSFs, which may be separated by less than a diffraction-limited core width (figure 5.3).
LuckyCam produces a very large data cube of images in each observation, consisting of many thousands of individual frames, each with a different PSF. Although direct Lucky Imaging gives very high resolution images, further processing of the raw data can achieve higher precision determination of the parameters of some targets. In this section I present an example of such processing: speckle imaging to retrieve the separation and position angle of very close binary systems.
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Speckle imaging is briefly described in chapter 1. Essentially, the method relies on the fact that each frame I(x, y) is the result of a convolution of some object brightness distribution O(x,y) with a PSF P(x, y):
 | (5.6) |
If, for example, the object is a binary star (two point sources) the resultant image will have two copies of the PSF, slightly displaced. Short exposure PSFs, consisting of speckles with diffraction-limited sizes, retain information at the diffraction limit after convolution with the object brightness distribution. Speckle interferometry searches for correlations in the image that suggest the observed PSF could be represented as a number of superimposed and displaced PSFs, and in that way recovers the object brightness distribution.
The simplest speckle technique, first suggested in Labeyrie (1970), is to sum the power spectra of many short exposure speckle frames. In the power spectrum, binary stars appears as set of fringes (figure 5.3). The fringes are preserved under addition, unlike the high-resolution components in a simple addition of short-exposure frames. The binary parameters can then be retrieved from fits to the fringes.
The short exposure high SNR frames obtained by LuckyCam meet the requirements of a speckle imager as well as a Lucky imager, and both techniques can operate on the same data.
To obtain the separations and position angles of very close binary stars to high precision I implemented a Fourier Transform speckle imaging system for use on LuckyCam data. The program automatically fits the expected speckle pattern of a close binary to several independent data sets of ~1000 frames targeted at the same star. The independent binary parameter estimates are combined to single values with improved accuracy, and estimates of the random errors are found from the range in parameter fits.
As can be readily seen from the images of the fringes, the position angle of the binary has a 180∘ uncertainty. This is because phase information is lost during image reconstruction. This information can be recovered via more complicated processing techniques (notably the bispectrum speckle masking techniques described in Weigelt (1977); Weigelt & Wirnitzer (1983); Lohmann et al. (1983)). However, when conditions allow Lucky Imaging resolved images of the binary (as for all observations presented here), it is simpler to use those images to resolve the uncertainty (using the detripling procedures if necessary).
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On any instrument, speckle data requires extensive calibration before it can be used for parameter estimation. In particular, the shape of photon events leads to a spatially varying background in Fourier space that can be complicated to remove. In the following, I detail the effects that must be calibrated out from L3CCD speckle images. Figure 5.4 shows the stages in the calibration.
The raw power spectrum, figure 5.4(a), shows clear fringes, as expected for a close binary. In this case, chosen for clarity, the binary separation is 0.31 arcsec, and so several fringe peaks are visible. In most cases where speckle interferometry is necessary to get accurate separation values only one fringe peak (or even only one minimum) is visible.
As for all detectors, noise terms must be added to the imaging equation (eg. Pluzhnik (2005) and references therein):
 | (5.7) |
where the convolving S(x,y) is the photon event shape for the detector (which convolves the shape of all objects in the frame) and N(x,y) is a noise bias floor, which includes pattern noise in the detector. These noise terms are obvious in the power spectrum (figure 5.4). Firstly, and most obviously, there is a vertical bar extending down the center of the power spectrum. This shape corresponds to structures in image space that are single-pixels high but several pixels long – CTE-shaped photon events.
The noise term N(x,y) is introduced by LuckyCam’s electronics, and takes the form of lines and points in the power spectrum (figure 5.4(b)). I remove it by subtracting the power spectrum of a background region of the frames from that of the foreground region. The speckle imaging code also detects those rare cases where the background noise does not subtract well, and masks off those areas as bad pixels.
The remaining two multiplicative terms in the power spectrum are divided out, thus deconvolving the binary power spectrum from their effects. Perhaps the most obvious way to remove the photon-event-shape bar is to divide by a suitably scaled event shape taken from a background region of the same image (figure 5.4(b)). However, as can be seen from the power spectra, the photon event shape is different in the stellar and background regions, because the CTE effects are (weakly) dependent on the amount of charge (light) within the pixels. For this reason, I estimate the shape of the photon events from the topmost and bottommost rows of the PSF power spectrum. These high frequency regions are beyond the frequency limit of the telescope, and so should be dominated by the photon event shape. I then divide the power spectrum by the estimated photon-event power spectrum (corresponding to deconvolving the input image by the photon event shape).
The results of these two background removal procedures are shown in figure 5.4(c). However, the power spectrum does not take the ideal form of a set of cosine fringes, being instead heavily peaked towards the center. This is because the fringe pattern from the binary is multiplied by an envelope function – the power spectra of the two unresolved point source PSFs making up the binary.
Left uncalibrated, the point source power spectrum can “push” a minimum away from the center of the fringe pattern, artificially decreasing the measured binary separation. I remove this effect by dividing the entire power spectrum by a 1-D power spectrum taken though the center of the fringe pattern. The 1-D power spectrum is measured along the central fringe, corresponding to the power spectrum of a single unresolved point source. This deconvolution proceeds under the assumption of circular symmetry of the atmosphere-corrupted power spectrum of the single point source, an assumption that appeared to be reasonable from tests of the speckle patterns from single stars. The final calibrated fringe pattern after the division is shown in figure 5.4(d).
In summary, the procedure developed to make a calibrated speckle power spectrum using data from a LuckyCam-like L3CCD imaging system is:
I developed an iterative procedure that, given a very rough estimate of the binary separation, converges on the true values of the binary separation and position angle. It proceeds as follows:
The procedure could be easily extended to fit more peaks and troughs than just the first minimum, but this is not necessary for the LuckyCam VLM binary survey data – the close binaries that require speckle imaging for parameter determination rather than simple centroid-fitting possess only one minimum within the telescope resolution boundaries.
I found this procedure to be more robust than direct fitting of the fringe pattern in binaries with marginal SNR or very close separations. For 0.1-0.2 arcsec binaries, typical variations in the measured binary parameters with a range of different PSFs are less than 0.01 arcsec in separation and 1-3∘ in position angle. The derived values agree well with (higher-uncertainty) parameters measured from Lucky Imaging data.
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