Regularization Techniques for PSF-Matching Kernels. I. Choice of Kernel Basis

We review current methods for building PSF-matching kernels for the purposes of image subtraction or coaddition. Such methods use a linear decomposition of the kernel on a series of basis functions. The correct choice of these basis functions is fundamental to the efficiency and effectiveness of the matching - the chosen bases should represent the underlying signal using a reasonably small number of shapes, and/or have a minimum number of user-adjustable tuning parameters. We examine methods whose bases comprise multiple Gauss-Hermite polynomials, as well as a form free basis composed of delta-functions. Kernels derived from delta-functions are unsurprisingly shown to be more expressive; they are able to take more general shapes and perform better in situations where sum-of-Gaussian methods are known to fail. However, due to its many degrees of freedom (the maximum number allowed by the kernel size) this basis tends to overfit the problem, and yields noisy kernels having large variance. We introduce a new technique to regularize these delta-function kernel solutions, which bridges the gap between the generality of delta-function kernels, and the compactness of sum-of-Gaussian kernels. Through this regularization we are able to create general kernel solutions that represent the intrinsic shape of the PSF-matching kernel with only one degree of freedom, the strength of the regularization lambda. The role of lambda is effectively to exchange variance in the resulting difference image with variance in the kernel itself. We examine considerations in choosing the value of lambda, including statistical risk estimators and the ability of the solution to predict solutions for adjacent areas. Both of these suggest moderate strengths of lambda between 0.1 and 1.0, although this optimization is likely dataset dependent.

Periodic Variability of Low-Mass Stars in SDSS Stripe 82

We present a catalog of periodic stellar variability in the Stripe 82 region of the Sloan Digital Sky Survey (SDSS). After aggregating and recalibrating catalog-level data from the survey, we ran a period-finding algorithm (Supersmoother) on all point-source lightcurves. We used color selection to identify systems that are likely to contain low-mass stars, in particular M dwarfs and white dwarfs. In total, we found 207 candidates, the vast majority of which appear to be in eclipsing binary systems. The catalog described in this paper includes 42 candidate M dwarf / white dwarf pairs, 4 white-dwarf pairs, 59 systems whose colors indicate they are composed of 2 M dwarfs and whose lightcurve shapes suggest they are in detached eclipsing binaries, and 28 M dwarf systems whose lightcurve shapes suggest they are in contact binaries. We find no detached systems with periods longer than 3 days, thus the majority of our sources are likely to have experienced orbital spin-up and enhanced magnetic activity. Indeed, twenty-six of twenty-seven M dwarf systems that we have spectra for show signs of chromospheric magnetic activity, far higher than the 24% seen in field stars of the same spectral type. We also find binaries composed of stars that bracket the expected boundary between partially and fully convective interiors, which will allow the measurement of the stellar mass-radius relationship across this transition. The majority of our contact systems have short orbital periods, with small variance (0.02 days) in the sample near the observed cutoff of 0.22 days. The accumulation of these stars at short orbital period suggests that the process of angular momentum loss, leading to period evolution, becomes less efficient at short periods. (Abridged)

Two More Candidate AM Canum Venaticorum (AM CVn) Binaries from the Sloan Digital Sky Survey

AM CVn systems are a select group of ultracompact binaries with the shortest orbital periods of any known binary subclass; mass-transfer is likely from a low-mass (partially-)degenerate secondary onto a white dwarf primary, driven by gravitational radiation. In the past few years, the Sloan Digital Sky Survey (SDSS) has provided five new AM CVns. Here we report on two further candidates selected from more recent SDSS data. SDSS J1208+3550 is similar to the earlier SDSS discoveries, recognized as an AM CVn via its distinctive spectrum which is dominated by helium emission. From the expanded SDSS Data Release 6 (DR6) spectroscopic area, we provide an updated surface density estimate for such AM CVns of order 10^{-3.1} to 10^{-2.5} per deg^2 for 15<g<20.5. In addition, we present another new candidate AM CVn, SDSS J2047+0008, that was discovered in the course of followup of SDSS-II supernova candidates. It shows nova-like outbursts in multi-epoch imaging data; in contrast to the other SDSS AM CVn discoveries, its (outburst) spectrum is dominated by helium absorption lines, reminiscent of KL Dra and 2003aw. The variability selection of SDSS J2047+0008 from the 300 deg^2 of SDSS Stripe 82 presages further AM CVn discoveries in future deep, multicolor, and time-domain surveys such as LSST. The new additions bring the total SDSS yield to seven AM CVns thus far, a substantial contribution to this rare subclass, versus the dozen previously known.