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POINT-SPREAD FUNCTIONS FOR THE EXTREME-ULTRAVIOLET CHANNELS OF SDO/AIA TELESCOPES

We present the stray-light point-spread functions (PSFs) and their inverses we characterized for the Atmospheric Imaging Assembly (AIA) EUV telescopes on board the Solar Dynamics Observatory (SDO) spacecraft. The inverse kernels are approximate inverses under convolution. Convolving the original Level 1 images with them produces images with improved stray-light characteristics. We demonstrate the usefulness of these PSFs by applying them to two specific cases: photometry and differential emission measure (DEM) analysis. The PSFs consist of a narrow Gaussian core, a diffraction component, and a diffuse component represented by the sum of a Gaussian-truncated Lorentzian and a shoulder Gaussian. We determined the diffraction term using the measured geometry of the diffraction pattern identified in flare images and the theoretically computed intensities of the principal maxima of the first few diffraction orders. To determine the diffuse component, we fitted its parameterized model using iterative forward-modeling of the lunar interior in the SDO/AIA images from the 2011 March 4 lunar transit. We find that deconvolution significantly improves the contrast in dark features such as miniature coronal holes, though the effect was marginal in bright features. On a percentage-scattering basis, the PSFs for SDO/AIA are better by a factor of two than that of the EUV telescope on board the Transition Region And Coronal Explorer mission. A preliminary analysis suggests that deconvolution alone does not affect DEM analysis of small coronal loop segments with suitable background subtraction. We include the derived PSFs and their inverses as supplementary digital materials.

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UTILIZATION OF THE WAVEFRONT SENSOR AND SHORT-EXPOSURE IMAGES FOR SIMULTANEOUS ESTIMATION OF QUASI-STATIC ABERRATION AND EXOPLANET INTENSITY

Heretofore, the literature on exoplanet detection with coronagraphic telescope systems has paid little attention to the information content of short exposures and methods of utilizing the measurements of adaptive optics wavefront sensors. This paper provides a framework for the incorporation of the wavefront sensor measurements in the context of observing modes in which the science camera takes millisecond exposures. In this formulation, the wavefront sensor measurements provide a means to jointly estimate the static speckle and the planetary signal. The ability to estimate planetary intensities in as little as a few seconds has the potential to greatly improve the efficiency of exoplanet search surveys. For simplicity, the mathematical development assumes a simple optical system with an idealized Lyot coronagraph. Unlike currently used methods, in which increasing the observation time beyond a certain threshold is useless, this method produces estimates whose error covariances decrease more quickly than inversely proportional to the observation time. This is due to the fact that the estimates of the quasi-static aberrations are informed by a new random (but approximately known) wavefront every millisecond. The method can be extended to include angular (due to diurnal field rotation) and spectral diversity. Numerical experiments are performed with wavefront data from the AEOS Adaptive Optics System sensing at 850 nm. These experiments assume a science camera wavelength {lambda} of 1.1 {mu}, that the measured wavefronts are exact, and a Gaussian approximation of shot-noise. The effects of detector read-out noise and other issues are left to future investigations. A number of static aberrations are introduced, including one with a spatial frequency exactly corresponding the planet location, which was at a distance of Almost-Equal-To 3{lambda}/D from the star. Using only 4 s of simulated observation time, a planetary intensity, of Almost-Equal-To 1 photon ms{sup -1}, and a stellar intensity of Almost-Equal-To 10{sup 5} photons ms{sup -1} (contrast ratio 10{sup 5}), the short-exposure estimation method recovers the amplitudes' static aberrations with 1% accuracy, and the planet brightness with 20% accuracy.

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FOMALHAUT b: INDEPENDENT ANALYSIS OF THE HUBBLE SPACE TELESCOPE PUBLIC ARCHIVE DATA

The nature and even the existence of a putative planet-mass companion ({sup F}omalhaut b{sup )} to Fomalhaut has been debated since 2008. In the present paper, we reanalyze the multi-epoch ACS/STIS/WFC3 Hubble Space Telescope (HST) optical/near-infrared images on which the discovery and some other claims were based. We confirm that the HST images do reveal an object in orbit around Fomalhaut, but the detailed results from our analysis differ in some ways from previous discussions. In particular, we do not confirm flux variability over a two-year interval at 0.6 {mu}m wavelength and we detect Fomalhaut b for the first time at the short wavelength of 0.43 {mu}m. We find that the HST image of Fomalhaut b at 0.8 {mu}m may be extended beyond the point-spread function. We cannot determine from our astrometry if Fomalhaut b will cross or not the dust ring. The optical through mid-infrared spectral energy distribution (SED) of Fomalhaut b cannot be explained as due to direct or scattered radiation from a massive planet. We consider two models to explain the SED: (1) a large circumplanetary disk around an unseen planet and (2) the aftermath of a collision during the past 50-150 yr of two Kuiper-Belt-like objects of radii {approx}50 km.

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Searching for pulsars using image pattern recognition

In the modern era of big data, many fields of astronomy are generating huge volumes of data, the analysis of which can sometimes be the limiting factor in research. Fortunately, computer scientists have developed powerful data-mining techniques that can be applied to various fields. In this paper, we present a novel artificial intelligence (AI) program that identifies pulsars from recent surveys by using image pattern recognition with deep neural nets—the PICS (Pulsar Image-based Classification System) AI. The AI mimics human experts and distinguishes pulsars from noise and interference by looking for patterns from candidate plots. Different from other pulsar selection programs that search for expected patterns, the PICS AI is taught the salient features of different pulsars from a set of human-labeled candidates through machine learning. The training candidates are collected from the Pulsar Arecibo L-band Feed Array (PALFA) survey. The information from each pulsar candidate is synthesized in four diagnostic plots, which consist of image data with up to thousands of pixels. The AI takes these data from each candidate as its input and uses thousands of such candidates to train its ∼9000 neurons. The deep neural networks in this AI system grant it superior ability to recognize various types of pulsars as well as their harmonic signals. The trained AI's performance has been validated with a large set of candidates from a different pulsar survey, the Green Bank North Celestial Cap survey. In this completely independent test, the PICS ranked 264 out of 277 pulsar-related candidates, including all 56 previously known pulsars and 208 of their harmonics, in the top 961 (1%) of 90,008 test candidates, missing only 13 harmonics. The first non-pulsar candidate appears at rank 187, following 45 pulsars and 141 harmonics. In other words, 100% of the pulsars were ranked in the top 1% of all candidates, while 80% were ranked higher than any noise or interference. The performance of this system can be improved over time as more training data are accumulated. This AI system has been integrated into the PALFA survey pipeline and has discovered six new pulsars to date.

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Reducing systematic error in weak lensing cluster surveys

Weak lensing provides an important route toward collecting samples of clusters of galaxies selected by mass. Subtle systematic errors in image reduction can compromise the power of this technique. We use the B-mode signal to quantify this systematic error and to test methods for reducing this error. We show that two procedures are efficient in suppressing systematic error in the B-mode: (1) refinement of the mosaic CCD warping procedure to conform to absolute celestial coordinates and (2) truncation of the smoothing procedure on a scale of 10'. Application of these procedures reduces the systematic error to 20% of its original amplitude. We provide an analytic expression for the distribution of the highest peaks in noise maps that can be used to estimate the fraction of false peaks in the weak-lensing κ-signal-to-noise ratio (S/N) maps as a function of the detection threshold. Based on this analysis, we select a threshold S/N = 4.56 for identifying an uncontaminated set of weak-lensing peaks in two test fields covering a total area of ∼3 deg{sup 2}. Taken together these fields contain seven peaks above the threshold. Among these, six are probable systems of galaxies and one is a superposition. We confirm the reliability of these peaks with dense redshift surveys, X-ray, and imaging observations. The systematic error reduction procedures we apply are general and can be applied to future large-area weak-lensing surveys. Our high-peak analysis suggests that with an S/N threshold of 4.5, there should be only 2.7 spurious weak-lensing peaks even in an area of 1000 deg{sup 2}, where we expect ∼2000 peaks based on our Subaru fields.

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SpArcFiRe: Scalable automated detection of spiral galaxy arm segments

Given an approximately centered image of a spiral galaxy, we describe an entirely automated method that finds, centers, and sizes the galaxy (possibly masking nearby stars and other objects if necessary in order to isolate the galaxy itself) and then automatically extracts structural information about the spiral arms. For each arm segment found, we list the pixels in that segment, allowing image analysis on a per-arm-segment basis. We also perform a least-squares fit of a logarithmic spiral arc to the pixels in that segment, giving per-arc parameters, such as the pitch angle, arm segment length, location, etc. The algorithm takes about one minute per galaxies, and can easily be scaled using parallelism. We have run it on all ∼644,000 Sloan objects that are larger than 40 pixels across and classified as 'galaxies'. We find a very good correlation between our quantitative description of a spiral structure and the qualitative description provided by Galaxy Zoo humans. Our objective, quantitative measures of structure demonstrate the difficulty in defining exactly what constitutes a spiral 'arm', leading us to prefer the term 'arm segment'. We find that pitch angle often varies significantly segment-to-segment in a single spiral galaxy, making it difficult to define the pitch angle for a single galaxy. We demonstrate how our new database of arm segments can be queried to find galaxies satisfying specific quantitative visual criteria. For example, even though our code does not explicitly find rings, a good surrogate is to look for galaxies having one long, low-pitch-angle arm—which is how our code views ring galaxies. SpArcFiRe is available at http://sparcfire.ics.uci.edu.

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Imaging an event horizon: mitigation of scattering toward Sagittarius A*

The image of the emission surrounding the black hole in the center of the Milky Way is predicted to exhibit the imprint of general relativistic (GR) effects, including the existence of a shadow feature and a photon ring of diameter ∼50 μas. Structure on these scales can be resolved by millimeter-wavelength very long baseline interferometry. However, strong-field GR features of interest will be blurred at λ ≥ 1.3 mm due to scattering by interstellar electrons. The scattering properties are well understood over most of the relevant range of baseline lengths, suggesting that the scattering may be (mostly) invertible. We simulate observations of a model image of Sgr A* and demonstrate that the effects of scattering can indeed be mitigated by correcting the visibilities before reconstructing the image. This technique is also applicable to Sgr A* at longer wavelengths.

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Unresolved fine-scale structure in solar coronal loop-tops

New and advanced space-based observing facilities continue to lower the resolution limit and detect solar coronal loops in greater detail. We continue to discover even finer substructures within coronal loop cross-sections, in order to understand the nature of the solar corona. Here, we push this lower limit further to search for the finest coronal loop substructures, through taking advantage of the resolving power of the Swedish 1 m Solar Telescope/CRisp Imaging Spectro-Polarimeter (CRISP), together with co-observations from the Solar Dynamics Observatory/Atmospheric Image Assembly (AIA). High-resolution imaging of the chromospheric Hα 656.28 nm spectral line core and wings can, under certain circumstances, allow one to deduce the topology of the local magnetic environment of the solar atmosphere where its observed. Here, we study post-flare coronal loops, which become filled with evaporated chromosphere that rapidly condenses into chromospheric clumps of plasma (detectable in Hα) known as a coronal rain, to investigate their fine-scale structure. We identify, through analysis of three data sets, large-scale catastrophic cooling in coronal loop-tops and the existence of multi-thermal, multi-stranded substructures. Many cool strands even extend fully intact from loop-top to footpoint. We discover that coronal loop fine-scale strands can appear bunched with as many as eight parallel strands within an AIA coronal loop cross-section. The strand number density versus cross-sectional width distribution, as detected by CRISP within AIA-defined coronal loops, most likely peaks at well below 100 km, and currently, 69% of the substructure strands are statistically unresolved in AIA coronal loops.

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IMFIT: A FAST, FLEXIBLE NEW PROGRAM FOR ASTRONOMICAL IMAGE FITTING

I describe a new, open-source astronomical image-fitting program called IMFIT, specialized for galaxies but potentially useful for other sources, which is fast, flexible, and highly extensible. A key characteristic of the program is an object-oriented design that allows new types of image components (two-dimensional surface-brightness functions) to be easily written and added to the program. Image functions provided with IMFIT include the usual suspects for galaxy decompositions (Sérsic, exponential, Gaussian), along with Core-Sérsic and broken-exponential profiles, elliptical rings, and three components that perform line-of-sight integration through three-dimensional luminosity-density models of disks and rings seen at arbitrary inclinations. Available minimization algorithms include Levenberg-Marquardt, Nelder-Mead simplex, and Differential Evolution, allowing trade-offs between speed and decreased sensitivity to local minima in the fit landscape. Minimization can be done using the standard χ{sup 2} statistic (using either data or model values to estimate per-pixel Gaussian errors, or else user-supplied error images) or Poisson-based maximum-likelihood statistics; the latter approach is particularly appropriate for cases of Poisson data in the low-count regime. I show that fitting low-signal-to-noise ratio galaxy images using χ{sup 2} minimization and individual-pixel Gaussian uncertainties can lead to significant biases in fitted parameter values, which are avoided if a Poisson-based statistic is used; this is true even when Gaussian read noise is present.

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MASS SUBSTRUCTURE IN ABELL 3128

We perform a detailed two-dimensional weak gravitational lensing analysis of the nearby (z = 0.058) galaxy cluster Abell 3128 using deep ugrz imaging from the Dark Energy Camera (DECam). We have designed a pipeline to remove instrumental artifacts from DECam images and stack multiple dithered observations without inducing a spurious ellipticity signal. We develop a new technique to characterize the spatial variation of the point-spread function that enables us to circularize the field to better than 0.5% and thereby extract the intrinsic galaxy ellipticities. By fitting photometric redshifts to sources in the observation, we are able to select a sample of background galaxies for weak-lensing analysis free from low-redshift contaminants. Photometric redshifts are also used to select a high-redshift galaxy subsample with which we successfully isolate the signal from an interloping z = 0.44 cluster. We estimate the total mass of Abell 3128 by fitting the tangential ellipticity of background galaxies with the weak-lensing shear profile of a Navarro–Frenk–White (NFW) halo and also perform NFW fits to substructures detected in the 2D mass maps of the cluster. This study yields one of the highest resolution mass maps of a low-z cluster to date and is the first step in a larger effort to characterize the redshift evolution of mass substructures in clusters.

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MEASURING AN ERUPTIVE PROMINENCE AT LARGE DISTANCES FROM THE SUN. II. APPROACHING 1 AU

The physical properties of eruptive prominences are unknown at large distances from the Sun. They are rarely, if ever, measured by in situ spacecraft and until recently our ability to measure them beyond the fields of view of solar imagers has been severely limited. The data quality of heliospheric imaging has reached a point where some quantitative measurements of prominences are now possible. I present the first such measurements of a bright prominence continually out to distances of around 1 AU from the Sun. This work follows on from the preparatory work presented in an accompanying paper, which showed that that the brightness of a prominence can be safely assumed to arise entirely from Thomson scattering in the STEREO/HI fields of view. Measurements of distance, speed, and mass are provided along with those from its accompanying coronal mass ejection (CME) to demonstrate their geometric, kinematic, and mass relationships. I find that the prominence travels with a slower speed than that of the CME, but its location relative to the CME structure does not conform to the expected location for basic geometric expansion. Further, the mass of the prominence was found to decrease by around an order of magnitude while that of the CME increased by an order of magnitude across the same distance.

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IMPACT OF ATMOSPHERIC CHROMATIC EFFECTS ON WEAK LENSING MEASUREMENTS

Current and future imaging surveys will measure cosmic shear with statistical precision that demands a deeper understanding of potential systematic biases in galaxy shape measurements than has been achieved to date. We use analytic and computational techniques to study the impact on shape measurements of two atmospheric chromatic effects for ground-based surveys such as the Dark Energy Survey and the Large Synoptic Survey Telescope (LSST): (1) atmospheric differential chromatic refraction and (2) wavelength dependence of seeing. We investigate the effects of using the point-spread function (PSF) measured with stars to determine the shapes of galaxies that have different spectral energy distributions than the stars. We find that both chromatic effects lead to significant biases in galaxy shape measurements for current and future surveys, if not corrected. Using simulated galaxy images, we find a form of chromatic “model bias” that arises when fitting a galaxy image with a model that has been convolved with a stellar, instead of galactic, PSF. We show that both forms of atmospheric chromatic biases can be predicted (and corrected) with minimal model bias by applying an ordered set of perturbative PSF-level corrections based on machine-learning techniques applied to six-band photometry. Catalog-level corrections do not address the model bias. We conclude that achieving the ultimate precision for weak lensing from current and future ground-based imaging surveys requires a detailed understanding of the wavelength dependence of the PSF from the atmosphere, and from other sources such as optics and sensors. The source code for this analysis is available at https://github.com/DarkEnergyScienceCollaboration/chroma.

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BEYOND ELLIPSE(S): ACCURATELY MODELING THE ISOPHOTAL STRUCTURE OF GALAXIES WITH ISOFIT AND CMODEL

This work introduces a new fitting formalism for isophotes that enables more accurate modeling of galaxies with non-elliptical shapes, such as disk galaxies viewed edge-on or galaxies with X-shaped/peanut bulges. Within this scheme, the angular parameter that defines quasi-elliptical isophotes is transformed from the commonly used, but inappropriate, polar coordinate to the “eccentric anomaly.” This provides a superior description of deviations from ellipticity, better capturing the true isophotal shape. Furthermore, this makes it possible to accurately recover both the surface brightness profile, using the correct azimuthally averaged isophote, and the two-dimensional model of any galaxy: the hitherto ubiquitous, but artificial, cross-like features in residual images are completely removed. The formalism has been implemented into the Image Reduction and Analysis Facility tasks Ellipse and Bmodel to create the new tasks “Isofit,” and “Cmodel.” The new tools are demonstrated here with application to five galaxies, chosen to be representative case-studies for several areas where this technique makes it possible to gain new scientific insight. Specifically: properly quantifying boxy/disky isophotes via the fourth harmonic order in edge-on galaxies, quantifying X-shaped/peanut bulges, higher-order Fourier moments for modeling bars in disks, and complex isophote shapes. Higher order (n > 4) harmonics now become meaningful and may correlate with structural properties, as boxyness/diskyness is known to do. This work also illustrates how the accurate construction, and subtraction, of a model from a galaxy image facilitates the identification and recovery of over-lapping sources such as globular clusters and the optical counterparts of X-ray sources.

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DETECTING UNSPECIFIED STRUCTURE IN LOW-COUNT IMAGES

Unexpected structure in images of astronomical sources often presents itself upon visual inspection of the image, but such apparent structure may either correspond to true features in the source or be due to noise in the data. This paper presents a method for testing whether inferred structure in an image with Poisson noise represents a significant departure from a baseline (null) model of the image. To infer image structure, we conduct a Bayesian analysis of a full model that uses a multiscale component to allow flexible departures from the posited null model. As a test statistic, we use a tail probability of the posterior distribution under the full model. This choice of test statistic allows us to estimate a computationally efficient upper bound on a p-value that enables us to draw strong conclusions even when there are limited computational resources that can be devoted to simulations under the null model. We demonstrate the statistical performance of our method on simulated images. Applying our method to an X-ray image of the quasar 0730+257, we find significant evidence against the null model of a single point source and uniform background, lending support to the claim of an X-ray jet.

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MORFOMETRYKA—A NEW WAY OF ESTABLISHING MORPHOLOGICAL CLASSIFICATION OF GALAXIES

We present an extended morphometric system to automatically classify galaxies from astronomical images. The new system includes the original and modified versions of the CASGM coefficients (Concentration C{sub 1}, Asymmetry A{sub 3}, and Smoothness S{sub 3}), and the new parameters entropy, H, and spirality σ{sub ψ}. The new parameters A{sub 3}, S{sub 3}, and H are better to discriminate galaxy classes than A{sub 1}, S{sub 1}, and G, respectively. The new parameter σ{sub ψ} captures the amount of non-radial pattern on the image and is almost linearly dependent on T-type. Using a sample of spiral and elliptical galaxies from the Galaxy Zoo project as a training set, we employed the Linear Discriminant Analysis (LDA) technique to classify EFIGI (Baillard et al. 4458 galaxies), Nair and Abraham (14,123 galaxies), and SDSS Legacy (779,235 galaxies) samples. The cross-validation test shows that we can achieve an accuracy of more than 90% with our classification scheme. Therefore, we are able to define a plane in the morphometric parameter space that separates the elliptical and spiral classes with a mismatch between classes smaller than 10%. We use the distance to this plane as a morphometric index (M{sub i}) and we show that it follows the human based T-type index very closely. We calculate morphometric index M{sub i} for ∼780k galaxies from SDSS Legacy Survey–DR7. We discuss how M{sub i} correlates with stellar population parameters obtained using the spectra available from SDSS–DR7.

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IMAGE ANALYSIS FOR COSMOLOGY: RESULTS FROM THE GREAT10 STAR CHALLENGE

We present the results from the first public blind point-spread function (PSF) reconstruction challenge, the GRavitational lEnsing Accuracy Testing 2010 (GREAT10) Star Challenge. Reconstruction of a spatially varying PSF, sparsely sampled by stars, at non-star positions is a critical part in the image analysis for weak lensing where inaccuracies in the modeled ellipticity e and size R {sup 2} can impact the ability to measure the shapes of galaxies. This is of importance because weak lensing is a particularly sensitive probe of dark energy and can be used to map the mass distribution of large scale structure. Participants in the challenge were presented with 27,500 stars over 1300 images subdivided into 26 sets, where in each set a category change was made in the type or spatial variation of the PSF. Thirty submissions were made by nine teams. The best methods reconstructed the PSF with an accuracy of {sigma}(e) Almost-Equal-To 2.5 Multiplication-Sign 10{sup -4} and {sigma}(R {sup 2})/R {sup 2} Almost-Equal-To 7.4 Multiplication-Sign 10{sup -4}. For a fixed pixel scale, narrower PSFs were found to be more difficult to model than larger PSFs, and the PSF reconstruction was severely degraded with the inclusion of an atmospheric turbulence model (although this result is likely to be a strong function of the amplitude of the turbulence power spectrum).

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MILLIONS OF MULTIPLES: DETECTING AND CHARACTERIZING CLOSE-SEPARATION BINARY SYSTEMS IN SYNOPTIC SKY SURVEYS

The direct detection of binary systems in wide-field surveys is limited by the size of the stars' point-spread functions (PSFs). A search for elongated objects can find closer companions, but is limited by the precision to which the PSF shape can be calibrated for individual stars. Based on a technique from weak-lensing analysis, we have developed the BinaryFinder algorithm to search for close binaries by using precision measurements of PSF ellipticity across wide-field survey images. We show that the algorithm is capable of reliably detecting binary systems down to Almost-Equal-To 1/5 of the seeing limit, and can directly measure the systems' position angles, separations, and contrast ratios. To verify the algorithm's performance we evaluated 100,000 objects in Palomar Transient Factory (PTF) wide-field-survey data for signs of binarity, and then used the Robo-AO robotic laser adaptive optics system to verify the parameters of 44 high-confidence targets. We show that BinaryFinder correctly predicts the presence of close companions with a <11% false-positive rate, measures the detected binaries' position angles within 1 Degree-Sign to 4 Degree-Sign (depending on signal-to-noise ratio and separation), and separations within 25%, and weakly constrains their contrast ratios. When applied to the full PTF data set, we estimate that BinaryFinder will discover and characterize {approx}450,000 physically associated binary systems with separations <2 arcsec and magnitudes brighter than m{sub R} = 18. New wide-field synoptic surveys with high sensitivity and sub-arcsecond angular resolution, such as LSST, will allow BinaryFinder to reliably detect millions of very faint binary systems with separations as small as 0.1 arcsec.

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Fringing Analysis and Simulation for the Vera C. Rubin Observatory’s Legacy Survey of Space and Time

Abstract The presence of fringing in astronomical Charge-Coupled Device (CCD) images will have an impact on photometric quality and measurements. Yet its impact on the Vera C. Rubin Observatory’s Legacy Survey of Space and Time (LSST) has not been fully studied. We present a detailed study on fringing for CCDs already implemented on the Rubin Observatory LSST Camera’s focal plane. After making physical measurements and knowing the compositions, we developed a model for the e2v CCDs. We present a method to fit for the internal height variation of the epoxy layer within the sensors based on fringing measurements in a laboratory setting. This method is generic enough that it can be easily modified to work for other CCDs. Using the derived fringing model, we successfully reproduce comparable fringing amplitudes that match the observed levels in images taken by existing telescopes with different optical designs. This model is then used to forecast the expected level of fringing in a single LSST y -band sky background exposure with Rubin telescope optics in the presence of a realistic time-varying sky spectrum. The predicted fringing amplitude in LSST images ranges from 0.04% to 0.2% depending on the location of a CCD on the focal plane. We find that the predicted variation in surface brightness caused by fringing in LSST y -band sky background images is about 0.6 μ Jy arcsec − 2 , which is 40 times larger than the current measurement error. We conclude that it is necessary to include fringing correction in the Rubin’s LSST image processing pipeline.

Astronomy & Astrophysics↗

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