A Technique for Extracting Highly Precise Photometry for the Two-Wheeled Kepler Mission

Unlike Kepler data, K2 data is dominated by noise due to the spacecraft's pointing jitter. Our approach is to extract aperture photometry and image centroid position information (as a proxy for the spacecraft's motion) from the K2 pixel-level data and correct the photometry using our knowledge of the spacecraft's pointing.

Kepler collected the data we use in our analysis during a 9 day test of the new two-wheeled operation mode conducted between 2014 February 4 and 2014 February 13. During those nine days, Kepler pointed to a field centered at R.A. = -1.35°, decl. = -2.15°, and monitored about 2000 targets with 30 minute "long cadence" exposures and 17 targets with 1 minute "short cadence" exposures. After the first 2.5 days of the test, Kepler achieved fine guiding control; that is, the spacecraft locked onto the center of its field, its pointing stabilized, and Kepler began collecting high quality data. We focus primarily on the long cadence data collected after Kepler achieved fine guiding control.

We downloaded target pixel files for all K2 engineering targets from the Mikulski Archive for Space Telescopes (MAST) and measured raw photometry for each star in the dataset. Due to spacecraft datalink bandwidth limitations, Kepler does not download a full-frame image for every photometric timestep. Instead, small subimages, or "postage stamps," are available for each target. Due to the increased uncertainty in the spacecraft's pointing, the K2 Engineering Test postage stamps are 50 by 50 pixel squares, much larger than the Kepler postage stamps. For each of these postage stamps, we start by defining an aperture mask in two ways: extracting a circle of pixels around the target star, and extracting a region defined by flux from the telescope's pixel response function (PRF). We analyze the photometry from each method separately and choose the one that provides the best photometric precision as measured using the procedure outlined in § 3.1. All of our apertures are stationary; the movement of the star on the Kepler detector is small enough (typically within one pixel) that moving apertures are not required. The first mask we define is a circular aperture centered on the target star, with a radius determined by the Kepler bandpass magnitude (K_p ) of the target star. After optimizing the mask radius for robustness by trial and error, we arrived at a mask radius function that varies between 13 pixels for the brightest targets (K_p  = 9) to 6 pixels for stars fainter than K_p  = 13.5. The circular masks are intentionally large and conservative in order to prevent flux from spilling out of the aperture as the star drifts. The large circular masks typically give better photometric precision for saturated stars and stars with close companions.

In addition to defining large circular apertures around the target stars, we also define more aggressively fitted apertures using the Kepler PRF (Bryson et al. 2010). We fit the PRF downloaded from the MAST to one long-cadence image of the target star, using a Levenberg–Marquardt least squares minimization algorithm (Markwardt 2009). We fit for four free parameters: horizontal and vertical centroid positions, an amplitude, and a background offset. For stars with K_p  < 13, we define our aperture mask as those pixels for which the PRF model flux is greater than 0.005% of the total PRF model flux. For stars with K_p  > 13, we define our aperture mask as pixels where the PRF model flux is greater than 0.1% the total PRF model flux. The flux levels that define the apertures were determined through trial and error to maximize photometric precision. This procedure yields a tightly shaped aperture about the stellar image on the detector. The smaller apertures limit background light contamination and thus improve photometric precision for faint, background-limited stars, while decreasing the robustness of aperture photometry for bright stars, particularly for saturated stars.

After defining the apertures, which remain stationary over the length of the K2 observations, we extract photometry for each K2 image using the FLUX tag from the Kepler FITS files data structure. First, for each image we estimate the background flux by taking the median value of the background pixels that lie outside of our aperture. We have experimented with estimating the background flux using a robust (outlier-rejected) mean calculation, but found no improvement over the median. We then subtract the background flux from the image and sum the pixels within the aperture. After extracting photometry for all K2 datapoints for a particular star, we divide by the median measured brightness to roughly continuum-normalize our photometry.

We then estimate the position of the star on the detector for each K2 datapoint using two methods. First, we calculate the centroid using a "center of flux" calculation given by

where x_c and y_c are the centroid positions in the horizontal and vertical dimensions of the K2 image, respectively; x_i and y_i are the individual horizontal and vertical positions of pixels; and f_i is the flux in each individual pixel.

We also estimate the position of the star on the detector by fitting a multivariate Gaussian to the image with a Levenberg–Marquardt minimization routine. We fit a Gaussian model to the star instead of the Kepler PRF for computational speed. We allow the amplitude of the Gaussian, an offset, widths in the horizontal and vertical directions, a cross term that rotates an elongated Gaussian, and horizontal and vertical positions to vary. Specifically, the model is given by:

where

A is the flux at the center of the star on the detector, x and y are the horizontal and vertical positions on the detector, respectively, σ_x and σ_y are the widths in the horizontal and vertical directions, respectively, B is the amplitude of a cross term that rotates the elongated PSF along an arbitrary axis, and D is an offset to account for the background.

For each K2 target, we automatically choose between the "center of flux" and Gaussian centroids based on which exhibit smaller root mean square (rms) residuals when fitted by a fifth-order polynomial. In most cases, the Gaussian centroids (eq. [2]) are more precise than the "center of flux" centroids (eq. [1]). But in some cases, particularly for saturated stars exhibiting bleed trails, a Gaussian model does not describe the shape of the stellar image and the "center of flux" measurements are better estimators of the centroid position.

Extracting raw photometry and centroid positions from the K2 images results in flux and position time series characterized by jagged features occurring several times a day. Figure 1 shows an example of a "raw" light curve and image centroid positions for a 10th magnitude K2 target. The position of the star on the detector drifts on time scales of hours and then is quickly corrected back to its starting point when the spacecraft thrusters fire. The changes in position introduce significant artifacts into the raw K2 photometry, which must be corrected in order to achieve the photometric precision characteristic of the original Kepler Mission.

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After extracting raw photometry, we begin by excluding certain data from further analysis. In particular, we exclude data from the first 2.5 days of the Two-Wheel Concept Engineering Test, which were taken before the spacecraft achieved adequate fine pointing control. These data exhibit larger variations in the image centroid position, and the photometric precision is worse than after fine guiding was established. We also exclude points labeled by the Kepler pipeline as having poor quality, where the QUALITY tag of the Kepler FITS file data structure was not 0. This cut excludes data with a variety of anomalies, including cosmic rays, pointing adjustments, and detector glitches. Typically, between 2% and 4% of datapoints are excluded at this stage.

Finally, we exclude points taken while the spacecraft's thrusters were firing. The behavior of the Kepler spacecraft during K2 data collection is to acquire the field using thrusters and stabilize the spacecraft's pointing with Kepler's two remaining reaction wheels, while balancing the spacecraft against solar radiant flux in an unstable equilibrium. After approximately 6 hr of drifting off equilibrium due to the Solar wind, Kepler fires its thrusters to bring the spacecraft closer to the equilibrium point. The result of this is that the spacecraft drifts only along its roll axis, and the movement of the spacecraft is one dimensional in an arc about Kepler's boresight, or the center of its field. We observed that during thruster fires K2 photometry exhibits discontinuous behavior. Excluding these points yields improved photometric precision.

We identify the points during which Kepler's thrusters were firing through analysis of the image centroids. Because the spacecraft moves almost solely in the roll direction, the position of the spacecraft follows a one-dimensional curve on the detector to good approximation. We automatically detect the direction of motion using a procedure similar to that of a Principal Component Analysis (PCA). We take the image centroids measured both from the center-of-flux analysis and the Gaussian fits and separately calculate the covariance matrix between horizontal and vertical centroid positions and its eigenvectors. Because the eigenvectors of the covariance matrix are the basis in which the covariance matrix is diagonal (that is, the covariance is zero), the eigenvectors lie parallel and perpendicular to the direction of motion of the star. Figure 2 shows an example of the path of the star on the detector, as well as the measured direction of movement.

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We then rotate the image centroids into new coordinates with the x^' axis along the direction of the strongest correlation (that is, the eigenvector associated with the largest eigenvalue). We fit a fifth-order polynomial to both sets of the transformed centroids, while excluding 5–σ outliers from the fit. Artifacts from the centroiding algorithm (like the wiggle seen in Fig. 2) complicate the star's measured path along the detector to the point where a high order polynomial fit is required. At this point, we choose between the center-of-flux centroids and the Gaussian-fit centroids and proceed with the set of centroid measurements that provide the lower rms residuals to the polynomial fit.

We calculate the arclength along the polynomial curve, where arclength, s, is defined as:

where is the transformed x coordinate of the point with the smallest x^' position, and is the best-fit polynomial function. We then differentiate arclength with respect to time to calculate ds/dt. When Kepler's thrusters fire there are sharp changes in the position of the spacecraft, so ds/dt is significantly different from the typical drifting behavior. We identify the points during which the spacecraft fires its thrusters as 5–σ outliers from the distribution of ds/dt points and exclude them from further calculations.

After excluding points with poor photometric performance, we used a "self-flat-fielding" (SFF) approach to remove photometric variability caused by the motion of the spacecraft. We start by isolating the short-period (timescales τ ≲ 24 hr) variability of the light curve, which we assume is dominated by the 6-hr spacecraft pointing jitter for most stars. We estimate the low frequency variability in the light curve by iteratively fitting a basis spline (B-spline) with breakpoints every 1.5 days in the light curve. We iteratively fit the B-spline, identify and exclude 3–σ outlier points from the fit, and refit the B-spline, repeating this process until convergence (typically less than five iterations). This process is illustrated in Figure 3. We found this approach to be a good balance between the desire to retain enough flexibility needed to describe the stellar variation and avoiding over-fitting to noise caused by pointing jitter. We then isolate high frequencies by dividing our raw aperture photometric time series by the final B-spline fit.

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The short-timescale variations remaining in the light curve are a combination of astrophysical variability (such as eclipse/transit events, flares, oscillations or flicker) and noise due to the spacecraft's pointing jitter, whether caused by uneven pixel sensitivity (e.g., Ballard et al. 2010) or different amounts of flux falling outside of the aperture (e.g., Stevenson et al. 2012). We separate astrophysical variability from pointing jitter by removing the noise component that correlates with the position of the star on the detector, as parametrized by arclength, s (eq. [4]). Figure 4 shows a plot of the high-pass filtered photometry versus the arclength. The dependence of measured flux on the image position is evident.

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We divide the data points into 15 bins in arclength and calculate the mean within each bin with 3σ outlier exclusion. The outlier exclusion serves to reduce the influence of short-timescale astrophysical noise on the correction. We then perform a linear interpolation between the mean of each bin. The result is a "correction" that can be applied to raw aperture photometry to remove the effects of pointing jitter. We chose a piecewise linear function to model the flux variation because it was simpler than other alternatives (like a spline or polynomial fit), and produced light curves of the same quality. Finally, we applied the correction by dividing the raw aperture photometry time series by the piecewise linear fit. The resulting light curve gives a much better estimate of the low-frequency variations than the raw aperture photometry, so we recalculated the low-frequency component with an iterative B-spline fit to the corrected lightcurve, which we removed from the uncorrected lightcurve, and used to recalculate the SFF correction. This process typically converged after one or two iterations.

We remove variation due to centroid position changes in only one dimension, specifically along the arclength of the polynomial fit. The spacecraft's pointing was repeatable enough over the 6.5 days of the fine pointing test to not require removal of variation transverse to the direction of drift on the detector. However, our technique is easily generalizable to removing photometric artifacts due to image centroid variations in two dimensions. This might be important for future K2 data releases for longer campaigns, particularly if the boresight of the spacecraft drifts over the course of 75 days of observations, and the path of the spacecraft's pointing jitter cannot be approximated as one-dimensional.

Figure 5 shows the result of the SFF correction and data exclusion on the K2 photometry. For the target star EPIC 60021426, the SFF correction decreases the scatter measured on 6-hr timescales by a factor of 5.

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We applied our SFF reduction technique to all targets observed during the K2 Two-Wheel Concept Engineering Test to assess its performance. The total runtime was about 5 hr on a laptop computer. We find that for almost all targets, our reduction improves the photometric precision over raw aperture photometry. Exceptions to this rule include stars with rapid astrophysical variation such as Cepheid variable stars, contact binaries, and giant stars with large-amplitude oscillation modes.

We assess the quality of our K2 light curves by estimating their photometric precision over a 6-hr window, a metric similar to the Combined Differential Photometric Precision (CDPP) metric used by the Kepler pipeline. We perform this measurement using a method inspired by the Kepler Guest Observer tools routine kepstddev ³ . For each target, we calculate the standard deviation within a running bin of 13 long-cadence measurements in length, divide by , and report the median value of the running standard deviation as the target's photometric precision. For each star, we calculate the photometric precision of the light curves produced from both the large circular apertures and the small PRF-defined apertures, as described in § 2.1, and retain the smaller of the two estimates.

We find that photometric precision of the K2 sample compares favorably to the photometric precision of the Kepler mission. To ensure a differential comparison, we downloaded light curves for all Kepler targets observed in Quarter 10 from the MAST and estimate photometric precision of the PDCSAP_FLUX light curve using the same procedure as we used for the K2 engineering data. Figure 6 shows the measured photometric precision for dwarf stars observed by K2 and Kepler as a function of Kepler magnitude. We select the dwarfs observed by Kepler by taking all stars with log g≥4 according to the Kepler Input Catalog (KIC; Brown et al. 2011), and we selected dwarfs observed by K2 as those stars whose target list was designated as cool_star or GKM_dwarf according to the K2 engineering test target list ⁴ . We find that in the best cases, the photometric precision of K2 approaches that of Kepler, but the typical precision of a K2 target is consistently worse by roughly a factor of 1.3–2. We also note that there is more scatter in the photometric precision of K2 targets than in Kepler targets, and investigate this further in § 3.2.

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We summarize the results of our SFF photometric analysis in comparison to Kepler's original precision in Table 1. Raw K2 photometry exhibits worse photometric precision than the original Kepler mission by at least a factor of four, and is comparatively worse for brighter target stars. The SFF technique presented in this paper restores K2 photometry to within a factor of two of the precision of Kepler photometry when comparing the median photometric precision of each data set.

Figure 6 shows evidence for large scatter in the photometric precision of K2 targets at a given magnitude, so we investigated factors beyond the brightness of a target that affect its photometric precision. We found a correlation between a star's position on the Kepler detector and its measured photometric precision. We calculated the angular distance of each star from the center of the Kepler boresight, held during the Engineering Test at R.A. = -1.3507626°, decl. = -2.1523890° (T. Barclay 2014, private communication). Figure 7 shows our measured 6-hr photometric precision of the K2 data as a function of angular distance of the target from the center of the boresight.

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We divided the K2 engineering sample of bright stars into 15 bins in radial distance and found the median of each bin. For the bright stars (K_p  < 13), we noticed a slight linear trend of worsening photometric precision as a function of angular distance from the boresight. We fit a line to the medians of the radial bins (with errors estimated by taking the standard deviation of points within each bin with 3-σ outlier exclusion) using a Markov Chain Monte Carlo algorithm with an affine invariant ensemble sampler (adapted for IDL from the algorithm of Goodman & Weare 2010; Foreman-Mackey et al. 2013). Any polynomial model more complex than a line resulted in high order polynomial terms consistent with zero and was not justified by the Bayesian Information Criterion (BIC, Schwarz et al. 1978), a test which balances model complexity with improvements in the fit. We measured a slight radial increase in 6-hr photometric scatter of 0.57 ± 0.15 ppm/degree. Our best-fit polynomial was:

where P_6,bright is the 6 hr photometric precision for bright stars in ppm and r is the distance from the boresight in degrees. The worsening photometric precision as a function of distance from the boresight is likely because stars farther from the center of the field move a larger distance on the detector as the spacecraft's roll drifts. This introduces larger and more complex photometric variations farther from the center of the spacecraft's field of view, which are more difficult to correct and result in larger photometric scatter.

The median photometric precision of faint stars (K_p  > 14) also shows a dependence on the angular distance from the center of the boresight. We investigated this dependence by dividing the K2 sample of faint stars into 15 bins in radial distance and noticed that the trend in radius was nonmonotonic—photometric scatter increases both very close to the boresight and very far away. We fit a second order polynomial to the median values of the radial bins (using the same technique and model complexity tests as before) and found that photometric precision is best at a distance of 3.62 ± 0.11° from the boresight. Our best-fit polynomial was:

where P_6,faint is the 6 hr photometric precision for bright stars in ppm and r is the distance from the boresight in degrees. A similar nonmonotonic dependence of precision on distance from the boresight was noticed by Christiansen et al. (2012), who attributed the effect to the quality of the telescope's focus, which is best between 3–6° from the center of the field. We speculate that the focus quality affects faint stars more than bright stars because they are closer to background-limited photometric precision. The degraded pointing stability of the two-wheeled Kepler spacecraft forces the use of larger photometric apertures, which increases the amount of scattered background light in the aperture, adding to photometric uncertainties.

The SFF correction primarily removes red noise (that is, low-frequency noise) introduced to K2 data by pointing drift and correctional thruster fires. We investigated the noise power spectrum of K2 data, in comparison to that of Kepler data, to understand the extent to which red noise is introduced into K2 data. We selected two stars observed by K2, the 10th magnitude EPIC 60021426 and the 14th magnitude EPIC 60029819, with photometric precisions close to the median for K2 targets of their brightness and after removing long-period stellar variability we calculated their power spectra via Fast Fourier Transform (FFT). We then selected two stars observed by Kepler of similar brightnesses, KIC 009579208 (K_p  = 10.3) and KIC 012066509 (K_p  = 14.7) with photometric precisions close to the median for Kepler targets of their brightness. We examined a baseline of photometry of exactly the same length as that of the K2 data and calculated the noise power spectra for the Kepler targets in the same way. Inspection of the power spectra showed no significant differences between the noise properties of Kepler and K2 targets, other than that the K2 data has consistently higher levels of noise at all frequencies. Over the short time baseline we tested, we do not see obvious excesses of noise at low frequencies corresponding to the interval between K2 thruster fires.

We tested the ability of our reduction technique to recover exoplanet transits by injecting artificial signals directly into the K2 pixel-level data and passing the modified data through our reduction pipeline. Because our technique uses the star's flux as a flat-field, it is possible that flux decrements, like transit events, could bias the flux correction and suppress or distort transit events. We calculated a model transit light curve based on transit parameters from the NASA Exoplanet Archive (Akeson et al. 2013), using a Mandel & Agol (2002) model, as implemented by Eastman et al. (2013), with limb darkening parameters from Claret & Bloemen (2011). We then scaled our PRF model from § 2.1 by the flux decrement from the transit light curve model and subtracted this scaled PRF from each K2 image. After the injection, we performed the data reduction in exactly the same manner as described in § 2.

We injected transit models of several different exoplanets into two K2 target stars, one bright (K_p  = 10.3) and one faint (K_p  = 14.7), with 6-hr photometric precisions close to the median K2 target for their brightness. The recovered light curves are shown in Figure 8, along with the injected signals. Visual inspection shows that the recovered light curves show minimal distortion of the transit signal. To quantify this, we fit the recovered light curves with a Mandel & Agol (2002) model (once again as implemented by Eastman et al. [2013]) and compared our recovered parameters to our inputs. We fitted model transit light curves to the data using a Markov Chain Monte Carlo algorithm with an affine invariant ensemble sampler while holding most parameters fixed.

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For the two deep transits (Kepler 4-b and KOI 843.01), we held all parameters fixed except for the orbital period, the ratio of the planet's radius to that of its star (R_p /R_⋆), the ratio of the stellar radius to the orbital semimajor axis (a/R_⋆), and a constant baseline flux offset. Both of the injected signals had only two transit events, so fitting for the orbital period is equivalent to fitting for one of the two transit times, thus testing our ability to recover Transit Timing Variations (TTVs). For the shallow transit (Kepler 10-b), we froze the orbital period because of the smaller signal, but still fit for a/R_⋆ and R_p /R_⋆. We found in each of the three cases shown in Figure 8, the recovered best-fit R_p /R_⋆, a/R_⋆, flux offset, and, where applicable, orbital periods were within the 1–σ uncertainties of the input parameters. Evidently, at the level of precision attainable with 6.5 days of K2 data, our SFF technique does not distort or suppress transit light curves. That being said, it is possible that with more data, transit parameter determinations will be precise enough to notice a bias. In that case, it will be necessary to incorporate the SFF technique into light curve fitting, in order to ensure that the SFF corrections are not biased by transit events.

Our technique is designed to be generally applicable to most or all K2 targets and to be fast enough to be run on a personal computer. Because the algorithm is meant to be generally applicable to the K2 sample, there is room for photometric improvement on individual stars by "hand-tweaking" at various steps in the algorithm. This process can improve the photometric precision of individual stars when objects of interest are revealed using the general-purpose pipeline.

The ability to adjust the reduction algorithm by hand on individual interesting targets is enabled by the fact that the technique presented in this paper relies only on data collected from one single K2 postage stamp. On the other hand, the technique could presumably be made more efficient and reliable by using data from multiple, independent K2 apertures. In particular, the motion of the spacecraft can be estimated from any number of bright but unsaturated stars to higher precision than is possible for the fainter targets in the K2 sample.

Finally, our technique was designed to be computationally efficient to run multiple times on the entire K2 dataset. As presently constructed, processing one K2 light curve takes ≃10 s per 9-day light curve on a laptop computer. The majority of the computer time is spent fitting the Kepler PRF to define the aperture and the Gaussian model to each image to estimate the centroid. It is possible that more computationally expensive operations, like extracting photometry by fitting the Kepler PRF to each image, could yield better photometry but the computational expense would be high.

4.2.1. Exoplanet Detection

4.2.2. Flicker

We acknowledge the tremendous effort of the K2 team and Ball Aerospace to make the K2 mission a success. We thank the anonymous reviewer for many insightful comments and suggestions. We thank Tom Barclay for helpful conversations and for providing the precise boresight coordinates for the Kepler Two-Wheel Concept Engineering Test. We are grateful to Fabienne Bastien for a useful conversation regarding flicker measurements with K2 data. We thank Eric Agol, Suzanne Aigrain, Juliette Becker, Christian Clanton, Daniel Foreman–Mackey, Heather Knutson, Benjamin Montet, and Jonathan Swift for their helpful comments on the manuscript. Some/all of the data presented in this paper were obtained from the Mikulski Archive for Space Telescopes (MAST). STScI is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS5-26555. Support for MAST for non-HST data is provided by the NASA Office of Space Science via grant NNX13AC07G and by other grants and contracts. This paper includes data collected by the Kepler mission. Funding for the Kepler mission is provided by the NASA Science Mission directorate. This research has made use of NASA's Astrophysics Data System and the NASA Exoplanet Archive, which is operated by the California Institute of Technology, under contract with the National Aeronautics and Space Administration under the Exoplanet Exploration Program. A.V. is supported by the NSF Graduate Research Fellowship, Grant No. DGE 1144152. J.A.J is supported by generous grants from the David and Lucile Packard Foundation and the Alfred P. Sloan Foundation.