High Precision Photometry for K2 Campaign 1

High Precision Photometry for K2 Campaign 1

C. X. Huang11affiliation: Department of Astrophysical Sciences, Princeton University, Princeton, NJ 08544; email: chelsea@astro.princeton.edu , K. Penev11affiliation: Department of Astrophysical Sciences, Princeton University, Princeton, NJ 08544; email: chelsea@astro.princeton.edu , J. D. Hartman11affiliation: Department of Astrophysical Sciences, Princeton University, Princeton, NJ 08544; email: chelsea@astro.princeton.edu , G. Á. Bakos11affiliation: Department of Astrophysical Sciences, Princeton University, Princeton, NJ 08544; email: chelsea@astro.princeton.edu 22affiliation: Sloan Fellow 33affiliation: Packard Fellow , W. Bhatti11affiliation: Department of Astrophysical Sciences, Princeton University, Princeton, NJ 08544; email: chelsea@astro.princeton.edu , I. Domsa11affiliation: Department of Astrophysical Sciences, Princeton University, Princeton, NJ 08544; email: chelsea@astro.princeton.edu , M. de Val-Borro11affiliation: Department of Astrophysical Sciences, Princeton University, Princeton, NJ 08544; email: chelsea@astro.princeton.edu

The two reaction wheel K2 mission promises and has delivered new discoveries in the stellar and exoplanet fields. However, due to the loss of accurate pointing, it also brings new challenges for the data reduction processes. In this paper, we describe a new reduction pipeline for extracting high precision photometry from the K2 dataset, and present public light curves for the K2 Campaign 1 target pixel dataset. Key to our reduction is the derivation of global astrometric solutions from the target stamps, from which accurate centroids are passed on for high precision photometry extraction. We extract target light curves for sources from a combined UCAC4 and EPIC catalogue – this includes not only primary targets of the K2 campaign 1, but also any other stars that happen to fall on the pixel stamps. We provide the raw light curves, and the products of various detrending processes aimed at removing different types of systematics. Our astrometric solutions achieve a median residual of . For bright stars, our best 6.5 hour precision for raw light curves is parts per million (ppm). For our detrended light curves, the best 6.5 hour precision achieved is ppm. We show that our detrended light curves have fewer systematic effects (or trends, or red-noise) than light curves produced by other groups from the same observations. Example light curves of transiting planets and a Cepheid variable candidate, are also presented. We make all light curves public, including the raw and de-trended photometry, at http://k2.hatsurveys.org.

Subject headings:
K2, astrometry, photometry
44affiliationtext: Hungarian Astronomical Association, Budapest, Hungary

1. Introduction

The Kepler spacecraft ended its primary mission after the failure of two reaction wheels. The K2 mission uses the Kepler spacecraft to perform 80-day observations of selected fields in the ecliptic plane. This brings new opportunities to study transiting planets around different stellar populations compared to the original Kepler field, such as clusters of young and pre-main sequence stars (Howell et al., 2014).

K2 uses the remaining two reaction wheels, and solar radiation pressure, to maintain close to constant pointing of the spacecraft over the 80-day per-field observations. Currently, observations are performed with 21 modules, each module consisting of 4 CCD channels, yielding 76 channels (2 modules failed). Due to the limited bandwidth, only postage stamps containing proposed targets are downloaded. These postage stamps are typically 2525 pixels in size (depending on the brightness of the targets and campaigns). These make up only less than 10 percent of the entire field of view (FOV). The majority of stamps are observed at 30 minutes cadence. Typically, two Full Field Images (FFIs) are downloaded for the beginning and the end of campaign.

However, the two reaction wheel mode also brings in new challenges for the data reduction processes. The spacecraft pointing is less stable compared to the primary mission, leading to a potential decrease in the photometric precision. Although the disturbance from the solar pressure is mostly controlled by the two reaction wheels and the thruster firing (every 2 days), there is still a low frequency motion remaining, resulting in the targets drifting across the field of view. The extracted aperture photometry light curves are dominated by the systematics induced by this drift pattern. Vanderburg & Johnson (2014) (hereafter VA14) minimized this drift systematic by decorrelating the light curves with the motion of the spacecraft. They achieved a photometric precision that is within a factor of two of the original Kepler photometry. Various other teams also developed their own tools to reduce the K2 data. Aigrain et al. (2015) used aperture photometry and a semi-parametric Gaussian process model to extract photometry from the K2 engineering data. Lund et al. (2015) presented K2P, a pipeline specifically designed for astrometric analyses. Foreman-Mackey et al. (2015) and Angus et al. (2015) proposed a method to analysis the K2 data without a general detrending process.

There is, however, room for further improvements. VA14 reduction achieved the highest precision among all the past works, but only derived photometry for the proposed Kepler targets (not all targets falling on silicon), and are also known to have remaining systematic variations affected by the spacecraft roll (Angus et al., 2015). Aigrain et al. (2015) and Lund et al. (2015) derive photometry for all of the targets on silicon, but achieved slightly lower precision than VA14, especially for the bright stars. Here we present a new reduction of the K2 data drawing on techniques used in analysing data from ground-based surveys (Bakos et al., 2010, e.g.).

We approach the K2 pixel file reduction with the following steps: 1) improved astrometry for source centroiding and flux extraction; 2) photometric extraction for all the stars observed on the K2 postage stamps; 3) removal of first order systematics via a modified External Parameter Decorrelation (EPD) procedure (broadly similar to VA14); 4) further reduction of the shared systematic trends via an implementation of the Trend Filtering Algorithm (TFA) and semi-periodic stellar oscillations via cosine-filtering. The global astrometry step is key to this process – it minimizes the effect of spacecraft drift on the aperture photometry, and allows us to accurately model the spacecraft motion for further detrending.

In this paper, we describe our K2 photometry pipeline and the high precision light curves from the reduction of K2 Campaign 1. We introduce our effort of deriving accurate astrometry for the K2 observations, making use of the K2 FFIs, and present a revised K2 Campaign 1 target list in §2. In §3, we present our aperture photometry method. In §4, we revisit our detrending techniques and present our light curves at different detrending stages. In §5, we compare our photometry with that of other studies.

2. Astrometry

2.1. Background

The first step of our reduction is to derive an accurate astrometric solution of the K2 data. Despite its large pixel scale (″) and PSF FWHM ( ″), the original Kepler mission turned out to be a great tool for accurate astrometry itself because of its extremely high SNR photometry and stable pointing. Monet et al. (2010) reported a preliminary astrometric solution precision, from the first few months of Kepler data, to be 0.001 pixel, nominal 4 mas. This high astrometric precision, and high stability of the centroid position, enabled the high photometric precision of the Kepler primary mission.

Unlike the original Kepler Mission, the K2 stars typically drift across the CCD plane at a speed of 1-3 of a pixel every 30 min. Since the 30 min K2 frame is composed of 270 short exposures of 6 s each, the final PSF is inevitably distorted, and neighbouring stars tend to become blended. Therefore, it is difficult to determine accurate centroids from source extraction alone. Thus, we use an external catalogue, namely the fourth United States Naval Observatory (USNO) CCD Astrograph Catalogue, UCAC4 (Zacharias et al., 2013), which has an astrometric precision of 15-100 mas, to derive good astrometric solutions for the K2 frames.

A good astrometric solution does not only benefit the photometric precision, but also enable us to make maximal use of the K2 observations. The K2 campaigns observe targets proposed by the community, and each target was then assigned a stamp of size 20-50 pixels across. This stamp size is much bigger than the original Kepler stamp size. In addition to the target, many other sources are observed in a typical K2 stamp. We provide position information and reduced light curves for all of the stars observed in the K2 stamps. We anticipate an improved planet yield from this approach, due to the larger number of sources available, and the availability of the light curves of neighbouring stars, useful in blend analyses.

We first derive a general astrometric solution using the Kepler FFIs and our custom developed astrometry software used for HATNet. This general solution is then used as an initial guess for the remaining stamp observations. We stitch all the K2 stamps together into a “Sparse FFI” (SFFI), with the unobserved regions masked. We fit for an astrometric solution to the SFFI, which is assumed to be a low order polynomial distortion from the FFI astrometric solution.

2.2. Astrometry Standard Catalogue

We use the UCAC4 catalogue (Zacharias et al., 2013) as our astrometry standard for deriving the astrometric solution. It contains over 113 million objects, and is complete down to magnitude .

The precision of coordinates provided by UCAC4 is mas. UCAC4 catalogue also contains proper motion of million stars, with errors around 1 to 10 . Both the coordinates and proper motions are measured on the International Celestial Reference System (ICRS) at a mean epoch of 2000. We linearly corrected the coordinates based on the proper motions to epoch 2014. UCAC4 also contains Two Micron All-Sky Survey ((2MASS; Skrutskie et al., 2006)) photometry for around 110 million stars, and AAVSO Photometric All-Sky Survey (APASS) five-band (BVgri) photometry for over 51 million stars. For the stars in 2MASS but without APASS photometry, their band photometry are estimated using the 2MASS magnitudes. BV band magnitudes are adopted from the Tycho-2 catalogue where available, otherwise also estimated from 2MASS. We use the B and V magnitude to estimate the magnitude of stars in the Kepler band when needed.

2.3. Astrometry on the Full Frame Image

The Kepler FFIs are divided into subimages by readout channels. There are 84 subimages for each FFI. Two of the CCD modules (8 channels altogether) failed during the Kepler main mission. The remaining 76 subimages were used to create the images from the 38 working CCDs (following the Kepler Instrument Handbook). We use fistar (Pál, 2012) for source extraction. The uncertainties of the source extractor is about 0.07 pixels. This is estimated by comparing the extracted source positions on the two different FFIs taken from the Campaign 1. The relative shift and rotation between the two FFIs were taken into account by fitting a low order polynomial to the two extracted source lists. We also experimented with other source extractors such as sextractor (Bertin & Arnouts, 1996), and all gave similar uncertainties.

The astrometric solution is provided by anmatch, a software routinely produces arcsecond precision astrometric solutions for the HATNet/HATSouth observations. anmatch first uses the engine of astrometry.net (Lang et al., 2010) for a low order solution (3rd order Simple Imaging Polynomial [SIP] tweak) to obtain an initial guess, then fits a 3rd order polynomial on a bigger matched list between the extracted source and the catalogue source to obtain the final solution.

The histogram of the residuals from the astrometric solution, for all 38 CCDs in the K2 Campaign 1 FFIs, is plotted in black in Figure 1. The astrometric solution residual is defined as the distance between the projected pixel coordinates of catalogue sources and the corresponding coordinates for the same stars from our source extractor. The median of the astrometric solution residual for these raw frames is pixel for K2 Campaign 1. 111We expect a factor of two difference between the estimated uncertainties from the source extractor, and our astrometric residual, given the different methods by which these two uncertainties are calculated. We estimated the uncertainty of the source extractor by compare the .rms. difference of the source positions on two frames (allowing a spacial transformation), while the uncertainty of the astrometric residual is estimated by the median of residual between the extracted position and the projected solution. Given the Kepler CCD has a plate scale of 3.98, our astrometric solution residual corresponds to 0.127.

Figure 1.— The residual of the astrometric solution for all channels of the K2 Campaign 1 field FFIs (black) and SFFIs (red and hatched). This is computed by comparing the distance between the projected catalogue coordinates and the detected source coordinates on the CCDs.

2.4. Astrometry for All the Stamps

The SFFIs are very sparse. For a single channel, typically more than of the pixels are not downloaded. It is impossible to solve for the astrometric solution of these SFFIs via a direct catalogue matching. We use the FFI astrometric solution as an initial guess to overcome this problem.

We first use fistar to extract the sources from the SFFIs. We then project the UCAC4 catalogue on to the SFFIs with the astrometric solutions obtained in §2.3. We use an iterative point matching algorithm allowing a field centre shift from the FFI to SFFI to match the extracted sources and the projected coordinates of catalogue stars. We solve for the distortion between these matched pairs using a second order polynomial to obtain the final solution. In Figure 2, we show the corresponding region of FFI and SFFI from the same CCD channel (K2 Campaign 1, module 13, channel 41). This region consists of three stamps in the SFFI observations. We marked out the detected source by cyan circles, and the projected sources from catalogue by red circles. The original K2 targets are marked out in the black circles. Some stamps consist of multiple stars. We also show that in the top rightmost stamp in Figure 2, the projected catalogue indicates that there are additional sources blended in the primary source’s PSF, which was originally missed by the source extractor, and the light of which would be measured together with that of the primary source.

We compute our astrometric residuals as per §2.3. The astrometric residuals on the SFFIs for K2 Campaign 1 are shown in red in Figure 1. The median astrometric residual is around 0.034 pixels (0.135), comparable with what we achieved on the FFIs.

Figure 2.— An 85 56 pixel region of the FFI frame (left) and its corresponding SFFI frame (right). The detected sources are marked by cyan circles, and the projected sources from UCAC4 catalogue by red circles. The original K2 targets are marked by black big circles centred on centroids as determined from our astrometric solution. The white region in the SFFI image were not observed, and are masked out.

2.5. A revised K2 target catalogue

We projected the UCAC4 catalogue on the K2 Campaign 1 SFFIs using the astrometric solution we obtained in §2.4. Stars with centroids within 3 pixels from the stamp edges were excluded. We also included those stars in the original Ecliptic Plane Input Catalogue (EPIC, Huber & Bryson (2015)) but not in the UCAC4 catalogue. The EPIC catalogue is a combination of the Hipparcos catalogue (van Leeuwen, 2007), Tycho-2 catalogue (Høg et al., 2000), UCAC4 catalogue, 2MASS, and SDSS DR9 (Ahn et al., 2012) for the selected K2 target stars. There may be systematic offsets between the coordinates from the above catalogue, but they are relatively small ( 10 mas), and can be ignored when combining these catalogues. We estimated the B and V magnitudes of stars not in the EPIC catalogue as per the Kepler Instrument Handbook. Altogether, we found 14778 stars from the UCAC4 catalogue, and an additional 7939 stars from the EPIC catalogue only, in K2 Campaign 1. This combined set of K2 target catalogue (22717 stars in total) is larger by 5 than the total in the original EPIC catalogue (21647 stars in total). This increase will be more pronounced for other, more crowded K2 fields. Part of the final K2 Campaign 1 target list catalogue is shown in Table 1. We provide the centroid positions on the corresponding postage stamp, for each target.

2.6. The refined motion for each module

VA14 pointed out that the K2 photometry is strongly correlated with the centroid positions of the stars. They also found that the centroid position of individual stars, as determined by their weighted light centres, are often not good enough. As such, they chose the centroid motion of a bright star to represent all the stars observed in the same campaign. Taking advantage of our derived astrometric solution, we find that by combining many stars observed on the same module, we can achieve even better constrained , motion tracks. We define the , motion derived for the centre pixel position of SFFI modules as the refined motion for each module. As an example, we show in Figure 3 the relative centroid drift of module 4, a module in the corner of the focal plane. We did not derive a rolling motion for the entire spacecraft to avoid correcting for additional rotations between modules. We notice that although the spacecraft attempted to correct its roll drift every 12 hours, the drifting segments can last longer. The drifting segments are defined as a time series of smooth , motion without significant outliers. We identify each drifting segment, and the outliers in between segments, by applying a 1-d edge detection method (Sobel operator) on the motion of each module. A example of the edge detection is show for module 4, in Figure 3, with the red dashed lines separating each segment.

Figure 3.— A segment of relative centroid drift from module 4, between cadence 1000 and 1500. The dashed red lines separate the drifting segments we identified via edge detection.

3. Photometry

For each target, we use fiphot (Pál, 2012) to extract photometry in 36 circular apertures around the derived centroids. The flux from the sources are estimated by summing up all the pixels within an aperture and weighting edge pixels by the fraction of which lie within the circular aperture. The background flux is measured by taking the median, with iterative outlier rejection, in an annulus of pixels around the aperture, then multiplying it by the area of the aperture, and subtracting it from the flux. The aperture sizes range from 2.5 — 5 pixels, and are chosen so as to optimize the photometric precision for a wide range of magnitudes. For apertures with sizes smaller (larger) than 4.5 pixels, the background annulus has inner radii of 5 (6) pixels and outer radii of 11 (12) pixels. They are designed to optimize the photometric precision for stars in different magnitude bins. We note that for saturated stars (KepMag ), our photometric method cannot capture all the leaked electrons in the stamps, therefore leading to degraded photometry. For the saturated stars, the fixed aperture approach taken by VA14 remains the best way to extract optimized photometry for now.

4. Light Curves and Detrending

We present the raw aperture photometry light curves (described in section §3), and apply a three-step detrending process on our light curves. The detrending methodology is adapted from the HATNet pipeline, as well as the Kepler light curve detrending pipeline described in Huang et al. (2013). Each step of detrending is aimed to correct different aspects of the noise in the light curves. Users are able to query light curves detrended up to an intermediate step to suit their own purpose 222http://k2.hatsurveys.org/. In this section, we will first describe the properties of our detrending methods, and then demonstrate the light curve products from each detrending step. At the end of this section, we will compare our light curves with those from other works.

4.1. Detrending

We refer to our light curves from the aperture photometry as RAWLC. Our detrending pipeline applied to these RAWLC can be divided into the following three steps:

(1) External Parameter Decorrelation (EPD);

(2) Trend filtering (TFA);

(3) Cosine filtering (COS).

To correct for the photometric variations due to the motion of the spacecraft, we performed EPD on the RAWLC (Bakos et al., 2010). We follow a similar methodology as described in VA14 to deal with the thrust fire events of the spacecraft. Instead of correcting for the drift effect due to a 6 hour roll, we make use of the drifting pattern of each module we identified from §2.6. We first reject the data points that fall in between any drifting segments. We then divide the data into 6 segments before detrending, as defined in Table 2. These segments are designed to separate large amplitude flux offset in the data, and allowing each segments to be represented by low order smooth functions. For each segment, we iteratively fit a 3rd order B-spline through the median magnitude of each drifting segment with 3- outlier rejection until the fit converges. This long term trend represented by the B-spline is then removed.

We then fit for the variation due to spacecraft drift as per the following:

in which, , represent the relative , drift of the module on which the target sits. The fitted , trend is then removed from the original RAWLC, and the B-spline long term trend added back in. This preserves the long-term trend while minimizing the effects of short-term spacecraft motion. The light curves at this stage is called EPDLC.

The shared systematics between the stars are then corrected using an adaptation of the Trend Filtering Algorithm (TFA) designed for Kepler. The idea of TFA is to select a set of template light curves, that is representative of all the systematic variations present in the data. Each target light curve is then corrected based on a linear filter that identifies the shared trends between the target and the template light curves. We found that using only template stars observed in the same channel as the target provided the best results. Since the number of stars observed in each channel in K2 Campaign 1 is quite small, we use all the stars but the target as templates in the TFA procedure. The TFA filtered light curves are denoted as TFALC.

The last step is to filter all the low frequency variabilities (mostly due to intrinsic stellar variability) using a set of cosine and sine functions. This method was implemented by Huang et al. (2013) for the independent search of planetary candidates in the original Kepler data. We aim to keep all periodicities at or below the protected timescale of the transit undisturbed, while minimizing any other variations following Kipping et al. (2013). The cosine function detrending is applied to the EPDLC light curves, the resulting light curves are called the COSLC. The cosine function detrending process is independent of the TFA process above. Due to its purpose, astrophysical signals such as stellar pulsations are no longer preserved in the COSLC.

4.2. Light Curve Products

We provide two types of measurements about the precision of our light curves. We use the point to point median scatter around the median (MAD) to represent the overall variability in the light curves  which is used as an estimation of noise level in our transit search algorithm. We also report the 6.5 hour precision as per VA14, which characterizes the noise of light curves at a time scale relevant to the transit duration of an earth analog. Figure 4 shows the MAD of our light curves. The best precision of RAWLC, EPDLC, COSLC and TFALC for the bright stars are (120 ppm), (60 ppm), (50 ppm) and (50 ppm), respectively. Figure 5 shows the estimated 6.5 hour precision of our light curves. The best precision of RAWLC for the bright stars are (20 ppm), and (15 ppm) for all the other three types of light curves. We also overlaid the estimation of the bottom envelop of the original Kepler 6.5 Hour precision based on Jenkins et al. (2010).

We show that the EPD process always improves both the short time scale (6.5 hours) and the long time scale (whole campaign) precision compared to the RAWLC. We find a greater improvement for the bright stars compared to the faint stars. The COS filtering process improves the precision in both time scales compared to EPDLC. For most stars, the TFALC has a similar or marginally worse precision compared with the COSLC, but the TFA process tends to preserve the intrinsic variabilities of the stars. We note that for a small fraction of the stars, the COSLC and TFALC can have a worse 6.5 hour precision compared to the EPDLC. This is because the cosine filter and TFA algorithm both use linear least square method aiming to minimize the overall point-to-point scattering in the light curves, which is sometime achieved at the cost of increasing noise at specific time scale. We compare the noise properties of the light curves in Figure 6 by showing the ratio of per point root mean square (RMS) and MAD versus magnitude. If the noise is composited with pure white noise, this ratio should be . The TFALC have the most white noise compare to other detrending stages.

Figure 4.— The point to point median standard deviation around the median (MAD) versus Kepler magnitude of all the light curve products at different detrending stages. From left to right, we show the RAWLC  the EPDLC, the COSLC  the TFALC and the VALC. The dashed red line is the fitted function for the magnitude versus the median MAD in the magnitude bin. The solid horizontal line indicates a scatter of (100 ppm). The vertical scale is logarithmic, and is the same for each panel.
Figure 5.— The 6.5 hour precision versus Kepler magnitude of all the light curve products at different detrending stages. From left to right, we show the RAWLC  the EPDLC, the COSLC  the TFALC and the VALC. The solid black horizontal line indicates a scatter of (100 ppm). The vertical scale are the same for each panel, linear and in units of parts per million (ppm). The dashed red line is the fitted function for the magnitude versus the median 6.5 hour precision in the magnitude bin. The dashed blue line is the fitted function indicate the bottom envelope of the original Kepler 6.5 Hour precision based on Jenkins et al. (2010).
Figure 6.— Ratio of point to point RMS and MAD versus Kepler magnitude. From left to right, we show the RAWLC  the EPDLC, the COSLC and the TFALC. The red horizontal line indicates the value of , which should be the value or their ratio for pure white noise.
Figure 7.— Photometry precision of our light curves compared to VA14 K2 Campaign 1 light curves. Top panel: 6.5 hour precision ratio between our light curves and VA14 light curves versus Kepler magnitude. Bottom panel: Per point MAD ratio between our light curves and VA14 light curves versus Kepler magnitude. From left to right, we show the RAWLC  the EPDLC, the COSLC and the TFALC. The red horizontal line indicates the value of 1. We note the vertical scale in the two panels are different, and both in log scale.

5. Comparison with other works

Many teams have developed methods to improve K2 photometry. We summarize the different approaches according to their photometry and detrending methods.

5.1. Photometry Methods of previous works

All the teams use aperture photometry method to extract the light curves from K2 data. However, they differ in the details of aperture choice and centroid measurements:

  • Fixed Mask method:

    The fixed mask method is such that the flux of the target is summed up over pixels within a fixed predetermined mask, while the pixels are accounted in a binary way. VA14 used a combination of approximate circular aperture and fitted apertures using the Kepler Pixel Responding Function (Bryson et al., 2010). Foreman-Mackey et al. (2015) and Angus et al. (2015) used approximate circular but binary apertures (do not include pixels partially) and present the photometry from the best apertures. Lund et al. (2015) used the density-based spatial clustering of applications with noise routine (DBSCAN, Ester et al. (1996)) to chose their pixel mask (aperture).

  • Moving Circular Aperture method:

    Aigrain et al. (2015) used 6 circular apertures to extract photometry. The apertures are soft-edged, in the sense that pixels straddling the edge of the aperture contribute partially to the flux.

  • Centroids from astrometric solution:

    Aigrain et al. (2015) used centroids derived from their own astrometric solution with the 2MASS all-sky point-source catalogue.

  • Centroids from WCS header:

    Foreman-Mackey et al. (2015) and Angus et al. (2015) used centroids from the WCS header of the K2 target pixel files. Only one WCS solution is given for the entire time series of each star.

  • Centroids from weighted centre of flux:

    VA14 and Lund et al. (2015) used the weighted centre of flux as the centroids of the stars. We note, in the subsequent detrending, VA14 used the centroid of star EPIC 201611708 instead of the centroids of individual stars.

In this work we used 36 moving circular apertures, with the centroids of apertures determined by precise astrometric solutions, to determine the photometry of each star.

5.2. Detrending Methods of previous work

There are three different types of “detrending” methods used by other authors.

  • Decorrelation:

    VA14 , Armstrong et al. (2014) and Lund et al. (2015) used a self-flat-field method to decorrelate the aperture photometry from centroid position of the image. There are, however, subtle differences between these studies. VA14 used the centroids from a representative star, Armstrong et al. (2014) seemed to use centroids for individual stars, while Lund et al. (2015) used the weighted light centroids derived for individual stars. VA14 used a 1-d decorrelation along the trajectory of the drift, Armstrong et al. (2014) used 2-d centroid surface to decorrelate with the flux, and Lund et al. (2015) used both the 1-d and 2-d approach in their pipeline.

  • Gaussian Process:

    Aigrain et al. (2015) and Crossfield et al. (2015) used Gaussian process model, with the rolling angle as the input variable to detrend the light curves. They assume the systematics can be modelled as a function form of the rolling angle, and that function’s form can vary from star to star.

  • Not Detrending:

    Foreman-Mackey et al. (2015) and Angus et al. (2015) choose to not detrend their light curve prior to the search of signal, but instead, they simultaneously fit for the systematics and the signal of interest.

In this work, we applied three stages of detrending. In the first stage, we applied a similar method as the decorrelation detrend in VA14. We additionally applied TFA and COS filtering to further filter the data. TFA is aimed to correct for shared systematics between the stars observed on the same channel, while preserving the stellar variability. The COS filtering method aimed to correct for any variability in the light curves  and is optimized for searching for transit signals.

5.3. Centroids Determination

We took a similar approach as Aigrain et al. (2015) in the determination of centroids by deriving an astrometric solution for each image. We made use of a more precise catalogue, UCAC4 instead of 2MASS, and the Full Frame Image as a better initial guess, and achieved higher precision in our astrometric solution. Aigrain et al. (2015) reported a typical root mean square of the astrometric solution of 0.4″, or approximately 0.1 pixel, times larger than our typical astrometric residuals.

5.4. Photometric Precision

To date, only VA14 have released their detrended K2 Campaign 1 light curves, therefore we will focus on comparing our photometry precision with their work.

We show in Figure 7 the precision ratio between our light curves and the VA14 light curves, for the same stars at both times scales. The EPDLC, COSLC and TFALC from this work have comparable precision compare to VA14 light curves on the 6.5 hour time scale, and a smaller point-to-point scatter over the entire observation length.

Aigrain et al. (2015) presented their (similar to MAD) and 6.5 hour Combined Differential Photometric Precision (CDPP) for K2 engineering data photometry. Their best precision for is 300 ppm for the bright stars, and 60 ppm for 6.5 hour CDPP. In this work, we achieved higher precision on both time scales (50 ppm and 15 ppm, respectively). Although, we caution that the noise characteristic for K2 engineering data and K2 Campaign 1 data could be different.

5.5. Power Spectrum

Previous works (Lund et al., 2015; Angus et al., 2015) noted the detrended light curves from a 1D decorrelation may still have residual spikes around the harmonics of   in their power spectra. These residuals may be largely due to aliasing of the low frequency power, induced by data gaps from rejected points during thruster firing. These harmonics are less obvious in our light curves.

We computed the Discrete Fourier Transformation (DFT) power spectrum using vartools (Hartman et al., 2008) for the star EPIC 201183188, following Angus et al. (2015). We find our EPDLC does not show the frequencies corresponding to the 6 hour roll in the DFT power spectrum present in the VA14 light curves. Figure 8 compares our EPDLC for EPIC 201183188 with the VA14 reduction, and Figure 9 compares the DFT power spectrum of our EPDLC with that of the VA14 light curve. The offset around epoch BJD 2456015 in the VALC could be blamed for contributing to noise peaks corresponding to the space craft rolling frequency.

We further investigate the noise characteristic of the light curves by computing the median of DFT power spectra of 1661 stars in the magnitude range of 10–12. To eliminate the influence from the long term trend, we first filter out the strongest low frequency peak in the light curves, and then recompute the DFT power spectra of each star before taking the median. We show this median spectrum in Figure 10. The peaks related to the rolling frequency are still present, but have less power compared to the median DFT spectrum computed with the same set of stars from VA14. When the signal from stellar variability is strong, such as in the case of EPIC 201183188, the systematic noise peaks are negligible.

Figure 8.— The EPDLC for EPIC 201183188 from this work (black), and VA14 (red). The DFT power spectrum of this star is shown in Figure 9.
Figure 9.— The DFT power spectrum for EPIC 201183188 from this work (top panel) and VA14 (bottom panel). Note the vertical scales are different in the two panels. The dash lines indicate the corresponding frequency associated with the spacecraft roll motion. For more discussion, see Angus et al. (2015).
Figure 10.— The median DFT power spectrum of 1661 stars in the magnitude range 10-12. Red dashed lines show the result from the VA14 light curves. Black solid lines show the result from the EPDLC of this work. In the small window of this figure, we show the zoom-in of these spectra around the space craft roll frequency .

5.6. Example Light Curves

To demonstrate the products of our photometric extraction and detrending in the context of stellar variability and transit searches, we show example light curves for stars with known variability and transiting signals.

EPIC 201711881, is a Cepheid candidate discovered by the ASAS project (Schmidt et al., 2009). The original discovery paper reported a period of 2.7353 days. The period we detected is day, consistent with (Schmidt et al., 2009). The RAWLC and EPDLC  folded with the Cepheid period, are shown in Figure 11. Our RAWLC  without any detrending, already shows a clean periodic signal. In addition, our EPDLC preserves the amplitude of the pulsation after removing the systematics. We also find a periodic eclipsing signal in the light curve with twice the pulsation period. This eclipsing signal can be visually seen, even in the RAWLC. The COSLC, phase folded with the eclipse period, is also plotted in the bottom panel of Figure 11. It seems that this system is more likely to be an eclipsing binary system with large out-of-transit variation rather than a real Cepheid. To further characterize this signal, the COSLC from the general detrending pipeline is not good enough. The COSLC we show in Figure 11 have been reconstructed after the discovery of the eclipsing signal. Only the out-of-eclipse part have been filtered by the cosine filters in order to preserve the shape and amplitude of the eclipsing signal.

WASP-85b: WASP-85 (EPIC 201852715), was observed in K2 Campaign 1, in module 15, channel 49. 1182 other stars were observed in the same module. We show in Figure 12 the light curves of WASP-85 at different detrending stages from this work. In the second panel from the top, we also overlaid the VA14 detrended light curve. WASP-85b has a period of 2.65 days, and known depth of . We show the WASP-85 light curve folded with the detected period and epoch in phase space for both the COSLC and TFALC in the bottom of Figure 12.

K2-3: K2-3 (EPIC 201367065), was observed in K2 Campaign 1, in module 12, channel 40. The host star is an M dwarf, with three transiting super-earths discovered by Crossfield et al. (2015). We show the light curves for K2-3 in Figure 13. The transits of the biggest planet (1 mmag) is visible in our RAWLC, and the transits of all three planets are visible in all the other light curves. We also show the phase folded COSLC and TFALC for all three planets in the bottom panel of Figure 13.

EPIC 201613023: This star was identified as a transiting planet candidate system by Foreman-Mackey et al. (2015). We show our light curves in Figure 14. The transit signal has depth of 400 ppm. Individual transits from the planets are visible in the EPDLC, TFALC and COSLC. The phase folded COSLC and TFALC with the detected epoch and period are shown in the bottom panel.

Figure 11.— The light curves of EPIC 201711881, a Cepheid Candidate discovered by Schmidt et al. (2009). From the top to bottom, we show the RAWLC (A), EPDLC (B), TFALC (C) and COSLC (D). In panel (E) and (F), we show RAWLC and EPDLC folded with the period of the pulsation period (2.735 day) in the phase space. The last panel we show COSLC (G) folded with twice the period of the pulsation. The eclipse events can be seen at phase 0. The COSLC in panel (D) and (E) have been reconstructed after the discovery of the eclipsing signal. Only the out-of-eclipse part have been filtered by the cosine filters in order to preserve the signal. The red light curve in figure (B) and (F) is from VA14. We note that the vertical scales are different for each panel.
Figure 12.— The light curves of WASP85 (black). From the top to bottom, we show the RAWLC, EPDLC, TFALC and COSLC. The bottom most panel, we show COSLC and TFALC of WASP85 folded with the epoch and period of WASP-85b (Brown et al., 2014). The red light curve in the second panel is from VA14. We note that the vertical scales in different panels are different.
Figure 13.— Same as Figure 12, but for K2-3 (Crossfield et al., 2015). In the bottom panel, we show all the three planets detected in this system folded with their own periods and epochs. K2-3 b (P 10 day, R2 ), c (P 25 day, R1.7 ), d (P 44 day, R1.5  ) are presented in red,blue and black curves, respectively. We note that the vertical scales in different panels are different.
Figure 14.— Same as Figure 12, but for EPIC 201613023 (Foreman-Mackey et al., 2015). In the bottom panel we show the detected planet candidate folded with its period (8.28 days). The depth of the transit is 400 ppm. We note that the vertical scales in different panels are different.

6. Conclusion and Discussion

In this article, we present our effort to extract high precision photometry from K2 Campaign 1 data. Our method has three distinct advantages:

  • Making use of accurate astrometric solution (0.127 or 0.034 pixels) from the FFIs for aperture centroiding;

  • Providing photometry for all sources on the stamps, not only for the proposed targets from the input catalogue;

  • Presenting light curves with very low systematic variations.

Our extracted light curves are of high precision at both the long (entire campaign) and short (6.5 hours) time scales, even for the raw light curves without any detrending. Light curves derived from all 36 photometric apertures at all four detrending stages are provided for the public at http://k2.hatsurveys.org.


We would like to thank the referee for their helpful comments. We also thank A.V for his thoughtful suggestions. G.Á.B. and X.H acknowledge funding from the Packard Foundation. This work was also supported by NASA grant NNX13AJ15G. K.P. acknowledges support from NASA grant NNX13AQ62G. The K2 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 NNX09AF08G and by other grants and contracts. This paper includes data collected by the Kepler telescope. Funding for the K2 Mission is provided by the NASA Science Mission directorate.


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UCAC4ID RA DEC J H K B V g r i x() bbthe and coordinate of star at time 0, time 0 is defined as cadence 1 (BJD-2455895.528) for K2.0, and cadence 102 (BJD-2455975.178) for K2.1 y() bbthe and coordinate of star at time 0, time 0 is defined as cadence 1 (BJD-2455895.528) for K2.0, and cadence 102 (BJD-2455975.178) for K2.1 channel ccthe channel number of which the star is observed K2ID ddthe given K2 ID of the star flagaaThe photometry flag indicator: A-APASS photometry; T-Tycho photometry; e-estimated with 2MASS photometry
UCAC4-555-033290 101.764743 20.947725 11.705 11.192 11.067 11.927 11.661 11.756 11.696 11.726 41.098906 885.251787 24 202071861 AAeeeAAAe
UCAC4-555-033327 101.789413 20.952493 14.487 12.700 12.157 15.417 14.481 14.920 14.144 13.857 42.030470 863.957681 24 202071849 AAeeeAAAe
UCAC4-555-033328 101.789718 20.941804 15.207 14.082 13.808 15.967 15.332 15.603 15.144 14.984 32.436217 865.242115 24 202071849 eeeeeeeee
UCAC4-555-033330 101.790216 20.942716 16.218 14.420 13.828 17.923 16.773 17.021 16.130 15.738 33.183423 864.694662 24 202071849 eeeeeeeee
UCAC4-555-033335 101.795353 20.945481 11.554 11.266 11.224 11.589 11.466 11.478 11.559 11.677 34.961871 860.006732 24 202071849 AAeeeAAAe
UCAC4-555-033336 101.795597 20.950077 16.008 14.560 14.051 17.444 16.468 16.771 16.030 15.720 39.036805 859.140643 24 202071849 eeeeeeeee
UCAC4-555-033338 101.799223 20.951700 15.611 14.331 14.073 15.935 15.416 15.625 15.249 15.125 39.999106 855.879192 24 202071849 eeeeeeeee
UCAC4-555-033418 101.876901 20.980552 14.091 13.383 13.252 14.427 14.080 14.193 14.071 14.044 55.344496 786.888096 24 202068459 AAeeeAAAe
UCAC4-555-033420 101.876908 20.972664 12.859 11.979 11.702 13.485 12.948 13.181 12.839 12.712 48.289514 788.011635 24 202068459 AAeeeAAAe
UCAC4-555-033425 101.881285 20.975126 16.224 13.317 12.709 17.317 16.082 16.193 15.201 14.762 49.903488 784.005099 24 202068459 eeeeeeeee
Table 1K2 target list
Segment No start Cadence end Cadence
1 0 454
2 455 1005
3 1006 1989
4 2050 2314
5 2315 2997
6 2998 4020
Table 2Light Curve Segments used in EPD
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