Analysis of Long-term Systematic Errors in Kepler K2

Despite being repurposed as the "K2" mission (Howell et al. 2014), Kepler is a potentially powerful facility for characterizing the variability of faint active galactic nuclei (AGNs), as K2 light curves probe timescales on the order of hours to weeks. Multiple systematic errors plague K2 light curves, including electronic interference, thermally driven focus change, differential velocity aberration (DVA), and arc-drift (e.g., Van Cleve et al. 2016). Removal of systematic effects is complicated by the fact that K2 has limited on-board solid-state storage, making it difficult to establish calibrations that are necessary for standard photometric reduction techniques. Various algorithms (e.g., Vanderburg & Johnson 2014; Armstrong et al. 2016; Luger et al. 2017) have been devised to detrend K2 light curves and remove systematic effects.

While investigating "detrended" K2 light-curves of AGNs, we noticed that there are residual systematic errors that affect our ability to perform an accurate investigation of AGNs. AGNs are stochastic by nature and should have uncorrelated light curves, yet we observe persistent patterns between independent objects. Recent analysis of AGN light curves by Aranzana et al. (2018) provides a good illustration of the problem: e.g., the light curves of objects 201207010, 201157230, 201167738, 201185828, 201189418 are impossibly similar. Further, many of the K2 light curves of faint AGNs with similar patterns are phase shifted from one another, which severely complicates the determination (and application) of a simple correction.

Figure 1 examines long-term systematic effects of interest herein. Using all available pixels, at each epoch we sort by flux into percentile bins and compute the median flux for each bin. We plot median light curves (rows in Figure 1) after "whitening" and differencing neighboring rows to highlight changes. Crucially, patterns we observed in AGNs within the same CCD are broadly replicated in Figure 1.

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We highlight three features in Figure 1. First, in every module, the faint pixel "light curves" are found to exhibit a broad shift roughly halfway through the campaign, consistent with behavior seen in other work (e.g., Foreman-Mackey et al. 2015; Armstrong et al. 2016; Van Cleve et al. 2016). Second, bright pixels (with most of the flux from the intended targets) show a trend that is opposite to that of the faint "background" pixels. Third, there are discrete features (spanning ∼10 days) in some modules that appear smoothly phase shifted as a function of pixel brightness. Figure 1 illustrates Campaign 8; however, these trends are ubiquitous in the campaigns that we have inspected.

We argue that it is possible to understand the broad trend (systematically brighter/fainter with time) in terms of a "differential focus error." When a module is out of focus, bright objects spread more of their light into other pixels in the aperture. Thus, bright and faint pixels will behave in opposite ways. In this interpretation, modules like 14 remain near focus, modules like 6 (purple-orange trend) start out of focus and move towards better focus, and modules like 12 (orange-purple) start in focus and move out of focus. Objects are brighter when observed in focus not only because more of the light is in the aperture, but also because more "background" gets subtracted out when the focus is poor. This behavior explains the pattern seen in the first three "eigenvector light curves" of Foreman-Mackey et al. (2015) and the first/second half issue noted in Armstrong et al. (2016). While the roll angle direction noted by Aigrain et al. (2016) may have some effect, campaigns noted to have more than 2 "seasons" by Armstrong et al. (2016) and Luger et al. (2017) do not have trends different than those seen in Figure 1. The long-term trends appear to be module dependent in a manner that is inconsistent with being due to DVA (c.f. Aranzana et al. 2018). Proper point-spread-function photometry with great care taken in defining the background and a global flat-field image will likely be needed to correct for this effect.

We do not have an explanation for the more discrete features or their phase shifting; however, we can confirm that they are module-dependent and are persistent from campaign to campaign. The effect is not limited to channels known to be affected by "rolling band" and is unlikely to be due to "ghosting" (as suggested by Aranzana et al. 2018)—unless they are solar ghosts (or rather trends in amplifier temperature) that occur at equivalent Sun-K2 geometries in different campaigns.

Our attempt at correction is most similar to the pixel level decorrelation in Luger et al. (2017) except that we seek to use pixels unassociated with the object in question (but with similar illumination levels). However, this process is only well defined for faint pixels, and AGNs are not the ideal objects for testing the method. Thus, the purpose of this Research Note is to inform other K2 investigators of these problems with faint objects, which may persist at relevant levels for bright objects, and to seek input from the community on ways to test and improve upon recipes for light curve correction.