Multiwavelength Photometry Derived from Monochromatic Kepler Data

During the prime mission (2009–2013), Kepler provided stable, precise time-series measurements of ≈200,000 targets, each with high-precision photometry lasting up to 4 yr. These data capture time-series variability at extreme precision; Kepler achieved precision better than 30 ppm for Kepler band magnitudes less than 12.5 over 6.5 hr (Gilliland et al. 2011). Kepler is capable of observing only at a single wavelength. The center of the Kepler bandpass is ≈640 nm, and it has 50% relative response between 450 and 825 nm. More information on the characteristics of the Kepler telescope can be found in the Kepler Instrument Handbook (Van Cleve & Caldwell 2016), hereafter referred to as KIH. Kepler observations are split into quarters, where each target is observed continuously for three months. The spacecraft rotates each season, such that the target falls on a new detector module each season (every fourth quarter the star falls on the same module). During the prime mission the pointing was stable, with pointing better than an arcsecond over the duration of a quarter. This small, long-term drift is caused by velocity aberration (see Jenkins et al. 2010a) and focus change due to changes in spacecraft temperature. These factors cause the position of a target to vary slowly over the course of an observation, characteristically moving ≲1 pixel over the course of a single quarter. These drifts, coupled with the intrapixel sensitivity variations (see KIH), cause the flux in each pixel to vary slowly over time.

The data volume from Kepler was high, owing to its short observing cadence (≈30 minutes in long-cadence mode, ≈1 minute in short cadence mode). In order to reduce the data volume, the Kepler telescope downlinked data in small image cutouts, centered around targets of interest. Data were delivered in target pixel files (TPFs), which consist of stacks of images typically ≈5 × 5 pixels across, with one image per 30 minutes cadence. An example of a frame from a TPF is shown in Figure 1. TPFs are then converted into light curves by the Kepler Pipeline (Jenkins et al. 2010b) using simple aperture photometry (SAP). Pixels inside apertures specified by the Kepler Pipeline are summed in order to create a single flux time series of each target.

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When flux from a target goes through the optical system of the telescope, it is spread from a point source into a broad point-spread function (PSF). The PSF for Kepler is wider than a single pixel, and so light from the source is registered on multiple pixels (for an example of the theoretical PSF of Kepler see Figure 2). The image that is recorded on the detector is known as the pixel response function (PRF), which is the true image combined with the detector sensitivity. For the bright target in Figure 1 the PRF covers tens of pixels.

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During the prime mission, systematics in the light curves produced by the mission (such as focus change and velocity aberration) were corrected by the Kepler pipeline (see Kepler Data Processing Handbook, Jenkins 2017). The pipeline approach was to use cotrending basis vectors (CBVs) to remove common trends in the light-curve sample (see Smith et al. 2012). In the mission's second phase, known as K2 (2013–2018) (Howell et al. 2014), the observations were subject to increased motion noise, owing to the spacecraft pointing in a two-reaction wheel mode. Intrapixel sensitivity variations cause the K2 time series to have variability at a level of ≈0.1%, and were not explicitly corrected by the Kepler pipeline. The CBVs employed by the pipeline removed common systematics between targets, but owing to the unique sensitivity variations of each pixel, this approach is not able to completely remove the systematic motion noise. Several groups produced techniques to mitigate the motion noise in the light curves produced by the K2 mission (e.g., see Vanderburg & Johnson 2014; Aigrain et al. 2015; Luger et al. 2016), employing techniques to correct the pixel data local to the source. These works are able to achieve light curves with a precision within ≈1–4× that of the prime mission. Since any wavelength dependence in Kepler is predicted to be weak, it is paramount to correct all other known systematics in the Kepler data, such as motion over the intrapixel flat field. In this work, we will use similar techniques to those used to correct K2 systematics in Luger et al. (2016) to model systematics from data from the Kepler prime mission at the pixel level, leaving any wavelength dependence intact.

1.1.1. Saturation

There are two known effects that cause a wavelength dependence in the PRF of the Kepler telescope: (1) chromatic aberration from the optical system; (2) wavelength dependence of the intrapixel sensitivity. Together, these two effects combine to make the PRF of the instrument weakly color-dependent. Below we discuss each of these effects.

1.2.1. PSF Wavelength Dependence

1.2.2. Wavelength Dependence of Intrapixel Sensitivity

1.2.3. The PRF

In this section we describe the data-driven model we use to model each pixel light curve, first in a general sense and then in the specific case of Kepler data in Section 2.2.

In general, we have a time-series measurement of a source, which we label f , which contains both astrophysical time series and unwanted instrument systematics. In this general section, this could be any flux time-series data; f could be a single-pixel time series or a SAP light curve. Each data point in the time series has some associated error σ _f . K _f is the covariance matrix for the data.

We label the true, astrophysical flux time series of the source as s . s is a vector with n time points, and our goal is to extract the most reliable estimate of s . We have measurements of the source from a telescope, which recorded the source brightness. For this data set, an acceptable model would describe the underlying true astrophysical signal s , and some multiplicative factors and/or additive factors that take into account systematic trends from the telescope. For example, systematic motion over the intrapixel sensitivity function would cause flux to be altered by a multiplicative factor. An unrelated background source would cause an additive offset in the measured flux. The model for our measured signal f is given as

$$\begin{eqnarray}&&{{\boldsymbol{f}}}_{{\boldsymbol{m}}}={\boldsymbol{S}}\cdot {\boldsymbol{X}}\cdot {\boldsymbol{w}}\end{eqnarray} \tag{ 1 }$$

where f _m is the model flux, S is a diagonal matrix where the diagonal values are s , X is a design matrix with predictors for systematics, and w is a vector of weights for these systematic components.

The design matrix X is a matrix made up of m column vectors, each of length n (the number of time points in the time series f ). Each vector in X is some predictor of a systematic, such as spacecraft temperature, motion, measured background scattered light etc. These vectors encode our understanding of the systematics. An example of the structure of X is

$$\begin{eqnarray}{\boldsymbol{X}}=\left(\begin{array}{ccccc}{x}_{\mathrm{1,1}} & {x}_{\mathrm{1,1}} & ... & {x}_{1,m-1} & 1\\ {x}_{\mathrm{2,1}} & {x}_{\mathrm{2,2}} & ... & {x}_{2,m-1} & 1\\ ... & ... & ... & ... & ...\\ {x}_{n,1} & {x}_{n,2} & ... & {x}_{n,m-1} & 1\end{array}\right).\end{eqnarray} \tag{ 2 }$$

Note that in the example matrix, the final column of X is a row of ones. This column allows us to fit an additive offset in the time series (i.e., to fit the mean level).

In order to calculate the best fitting model, if we know the underlying astrophysical signal s , we must solve for the weights w . In this general case, we assume we know or have a reasonable estimate of s . To solve for w and find the best fitting model m we can use linear regression, employing the following relations (under the assumption that our errors are Gaussian):

$$\begin{eqnarray}&&{{\boldsymbol{K}}}_{{\boldsymbol{w}}}^{-1}={({\boldsymbol{S}}\cdot {\boldsymbol{X}})}^{\top }\cdot {{\boldsymbol{K}}}_{{\boldsymbol{f}}}^{-1}\cdot ({\boldsymbol{S}}\cdot {\boldsymbol{X}})\end{eqnarray} \tag{ 3 }$$

$$\begin{eqnarray}&&\hat{{\boldsymbol{w}}}={{\boldsymbol{K}}}_{{\boldsymbol{w}}}^{-1}\cdot \left({({\boldsymbol{S}}\cdot {\boldsymbol{X}})}^{\top }\cdot {{\boldsymbol{K}}}_{{\boldsymbol{f}}}^{-1}\cdot {\boldsymbol{f}}\right)\end{eqnarray} \tag{ 4 }$$

where $\hat{{\boldsymbol{w}}}$ is a vector of length m of the mean of the best fitting coefficients and K _w is the covariance matrix of w with size m × m. These relations are derived in Luger et al. (2017). By rearranging Equation (1) we can now build a systematics model and find our best estimate of s , the true astrophysical signal.

In the more realistic case, we have several flux time series for the source, because the flux has been recorded on several pixels due to the broad PRF. We describe the flux time series for the ith pixel at time t as f _i,t . We will assume there are n points in time, m vectors in our design matrix, and l pixel time series in our TPF. In this work, we assume there is no covariance either between data points in time or between pixels, and our matrix K _f is a simple diagonal matrix, where the diagonal values are fixed at ${\sigma }_{{{\boldsymbol{f}}}_{i}}$ (the error on each flux measurement). This assumption implies that there are no correlations between either pixels or time points in the data set, and is adopted here for simplicity. A more robust analysis might include a K _f with off-diagonal terms to include covariance in the time series (encoding covariance between "frames") and covariance between pixels (encoding prior knowledge of the "true" PRF shape). In this first proof of concept, we treat each flux measurement as independent.

In this case, our "best guess" for the true underlying signal s is the SAP light curve. This light curve is the sum of all pixels within the aperture predetermined by the Kepler pipeline. Our assumption here is that each pixel should contain the same signal as the "total" brightness from the source. Said another way, the s measured in each pixel should be identical (for now, ignoring wavelength dependences). The SAP flux of a target is simply

$$\begin{eqnarray}&&{\boldsymbol{s}}=\displaystyle \sum _{i=0}^{l^{\prime} }{{\boldsymbol{f}}}_{i,t}\end{eqnarray} \tag{ 5 }$$

where ${\sum }_{i=0}^{l^{\prime} }$ denotes summation of the pixel time series inside the optimum aperture specified by the Kepler Pipeline. (Note that $l^{\prime}$ denotes the pixels inside the aperture, and there are a total of l pixels in the TPF). If there is no PRF shape variability in our data, the SAP light curve should be identically reflected in every pixel. Because we know there are some unrelated shape and position changes due to focus change, intrapixel sensitivity variations, and velocity aberration, we allow for common systematics between pixels with variable weights using the design matrix X and weights w .

We build the design matrix X using the following components:

• 1.  
  The first two cotrending basis vectors (CBVs) from the Kepler Pipeline. These are components built by the Kepler pipeline (Jenkins et al. 2010b) and reflect common variability across all targets in a single Kepler channel. In our case, they are mostly predictive of focus change. Focus change causes a shape change in the PRF that is common across the entire channel, and has a greater magnitude at the start of quarters (where the spacecraft is changing temperature due to a new pointing). We include these components here, and they are considered a nuisance parameter. If these components are omitted, the PRF shape change due to focus change would not be fit well in our model. We find empirically in our work that the first two CBVs are the most predictive of focus change and sufficiently capture this variability.
• 2.  
  The pixel column and row positions of the PSF as estimated by the Kepler Pipeline. These are provided by the TPF in the POS_CORR1 and POS_CORR2 arguments respectively, found in the pipeline processed fits files. We include a 2D, second-order polynomial in vectors of row ( r ) and column ( c ) position of the target. These vectors are { r , c , r ², c ², r . c , r ². c , r . c ², ( r . c )²}. This allows us to take into account the motion over the subpixel flat field. For Kepler, the motion takes a simple form and changes gradually, and so a low-order 2D polynomial is a reasonable model. For the data taken during the K2 mission, this assumption does not hold, and a different model for centroid position is required (see Section 5.3.1). This polynomial is analogous to the "arclength" used in Vanderburg & Johnson (2014) to model K2 systematics.
• 3.  
  A higher-order polynomial of flux and position. As shown in other works on the sensitivity of the Kepler pixels, the drop-off in sensitivity toward the edge of the pixel is not linear (e.g., see Vorobiev et al. 2018). As such, we cannot model the systematics linearly with s , and we must include higher-order terms. To capture this nonlinearity, we include columns containing a first-order polynomial in position c and r , multiplied by s . The vectors included in the design matrix for this component are { s . r , s . c , s . c . r , s }.
• 4.  
  A two-day B-spline. In order to capture time variability on short and long timescales, such as from the known Kepler rolling band (short term) and velocity aberration (long term), we include a third-order basis spline in our matrix that has knots every two days, this allows us to flexibly capture variability. In this work, where we are concerned with variability due to eclipsing binaries that occurs on timescales of hours, a two-day B-spline is acceptable. However, in other applications, a longer baseline may be more appropriate.
• 5.  
  A column of ones. This allows the mean of the time series to be fit by the model. This column allows each pixel to have a different mean. In practice, this enables an offset in each pixel due to background flux from neighboring contaminant stars. This term takes into account the missing offsets from the other polynomial terms given above.

An example of the design matrix is

$$\begin{eqnarray}{\boldsymbol{X}}=\left(\begin{array}{cccccc}{\mathrm{CBV}}_{\mathrm{0,0}}, & {\mathrm{CBV}}_{\mathrm{1,0}}, & {r}_{0}, & {c}_{0}, & ... & 1\\ {\mathrm{CBV}}_{\mathrm{0,1}}, & {\mathrm{CBV}}_{\mathrm{1,1}}, & {r}_{1}, & {c}_{1}, & ... & 1\\ {\mathrm{CBV}}_{\mathrm{0,2}}, & {\mathrm{CBV}}_{\mathrm{1,2}}, & {r}_{2}, & {c}_{2}, & ... & 1\\ ... & ... & ... & ... & ... & ...\\ {\mathrm{CBV}}_{0,N}, & {\mathrm{CBV}}_{1,N}, & {r}_{N}, & {c}_{N}, & ... & 1\end{array}\right).\end{eqnarray} \tag{ 6 }$$

By utilizing Equations (3) and (4) we can now find the weights in each pixel w _i by using the relations

$$\begin{eqnarray}&&{{\boldsymbol{K}}}_{{{\boldsymbol{w}}}_{i}}^{-1}={({\boldsymbol{S}}\cdot {\boldsymbol{X}})}^{\top }\cdot {{\boldsymbol{K}}}_{{{\boldsymbol{f}}}_{i}}^{-1}\cdot ({\boldsymbol{S}}\cdot {\boldsymbol{X}})\end{eqnarray} \tag{ 7 }$$

$$\begin{eqnarray}&&\hat{{{\boldsymbol{w}}}_{i}}={{\boldsymbol{K}}}_{{{\boldsymbol{w}}}_{i}}^{-1}\cdot \left({({\boldsymbol{S}}\cdot {\boldsymbol{X}})}^{\top }\cdot {{\boldsymbol{K}}}_{{{\boldsymbol{f}}}_{i}}^{-1}\cdot {{\boldsymbol{f}}}_{i}\right)\end{eqnarray} \tag{ 8 }$$

where ${\hat{{\boldsymbol{w}}}}_{i}$ are the best fitting weights in pixel i, and is a vector with length m, with one coefficient per weight.

Our full model f _m (given by Equation (1)) is a matrix of size n × l, where the model has been computed for each time t in each independent pixel i. The resultant model is a stack of "images" of our data-driven model, including instrument systematics, under the assumption that the astrophysical signal s is the same in each pixel, i.e., a model of the PRF, where the PRF shape may vary only in a way correlated with the CBV components or with the two-day basis-spline components. An example of the data and the model are shown for KIC 2708156 in Figure 3. Using this model, we can now analyze the residuals to see whether the PRF shape change does indeed remain constant.

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In this work we fit each pixel time series individually, as shown in Equations (7) and (8). We are able to do this only when the signal-to-noise ratio in each pixel time series is high enough to measure the intrinsic astrophysical variability (i.e., for bright targets with significant variability). This is the case for bright eclipsing binaries observed with Kepler, but for fainter targets with lower amplitude variability this may not modeled well by the model discussed here. Further investigation is needed to find the limit of the method described here.

Finding w _i by jointly fitting multiple pixels and multiple targets, taking into account covariances between pixels, would enable us to find a model for the full detector, creating a data-driven PRF model in absolute terms. Such a model, given an input flux value for a given time and pixel ( f _i,t ), would be able to return the estimated flux for a given pixel based on the full context of the detector, accounting for all common systematics. However, to create such a model would require the incorporation of further information into the model, including the true underlying image (i.e., a sky catalog of the star field, including exact, subpixel positions of every target). Such a model is not necessarily easily described by simple linear combinations of polynomials and would be computationally intensive to generate. Generating and testing such a model is beyond the scope of this initial work, but would in theory enable this model to be expanded to the full Kepler sample.

To demonstrate the detrending power of the model we have described, we apply it to the data set on transiting exoplanet Kepler-1b (also known as TrES-2b) (O'Donovan et al. 2006). Kepler-1b is an ideal test case for this model; the deep and short-period transits, as well as the bright host star, make the astrophysical signal in each pixel strong. In addition, a transiting planet is expected to cause no wavelength-dependent variability in the target flux. Even in the case of the hot planet of Kepler-1b, any emission from the planet would peak well outside the Kepler bandpass in the infrared, and changes in limb darkening are anticipated to be small. Figure 4 shows the model fit to a single pixel of Quarter 13 data of Kepler-1b. Our model has an exceptional fit to the data, providing an accurate fit at better than 200 ppm in most cadences. Our model is similarly accurate in each pixel in the data set.

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The upper left panel of Figure 5 shows the SAP flux for Kepler-1b during Quarter 13 of Kepler, folded at the orbital period of Kepler-1b. Our pixel-level model is also summed over every pixel to create a model SAP flux light curve (blue). In the lower panel, the reduced chi-squared statistic is shown for each cadence, showing that our model has good agreement at all phases of the light curve (reduced chi-squared value of ∼1). There are no significant excesses or deviations from the model during transit, and crucially there is no correlation between the residuals and the transit signal. A single quarter has been chosen for demonstration, but our model is similarly effective in all quarters.

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The Kepler-1b data set is not expected to have any wavelength dependence, since exoplanets are not bright enough to have significant emission in the Kepler bandpass. To find targets where wavelength dependence is both expected and predictable, we can look to eclipsing binaries. Eclipsing binaries are highly variable, and their astrophysical signal can easily be identified using only a single pixel of the broad PRF. Eclipsing binaries also have a predictable color change; if the two stars have different temperatures, they will have different apparent colors in the Kepler bandpass. As one star eclipses the other, the color of the target will change, as each star's contribution is downweighted when it is eclipsed. KIC 2708156 is an eclipsing binary, which was observed by Kepler during the prime mission (parameters available in Table 1). The right panel of Figure 5 shows our model fit to Quarter 13 data on KIC 2708156, and shows that our model has a significantly poorer fit during the eclipses of this target. The true SAP flux is shown for KIC 2708156, as well as the modeled SAP flux. The reduced chi-squared of the model fit is shown in the lower panel, summed up over all pixels in the aperture. The reduced chi-squared shows poor agreement between the model and the data, and crucially shows poor agreement at certain phases of the eclipsing binary. The model fits much more poorly during secondary and primary eclipses.

Table 1. Table of Parameters of the Eclipsing Binaries Used in This Work. The Kepler Magnitudes and Effective Temperatures have been Derived from Matson et al. (2017). Here Subscript of 1 Denotes the Primary Body, and Subscript of 2 Denotes the Secondary. The Radius, Inclination and Amplitude Values were Derived in this Work by Fitting an Eclipsing Binary Model Using starry (Luger et al. 2019)

KIC ID	Kepler Mag	${{\rm{T}}}_{{eff}1}$ a	${{\rm{T}}}_{{eff}2}$ a	Period	t0	Radius1	Radius2	Inclination°	Amplitude1	Amplitude2
		K	K	days		${R}_{\odot }$	${R}_{\odot }$
2708156	10.672	11061	5671	1.891269	170.508742	$2.27939{\pm }_{-0.00011}^{0.00011}$	$2.08991{\pm }_{-0.00003}^{0.00003}$	$82.61361{\pm }_{-0.00045}^{0.00045}$	$0.90881{\pm }_{-0.00001}^{0.00001}$	$0.09119{\pm }_{-0.00001}^{0.00001}$
3241619	12.524	5715	4285	1.703347	169.942399	$1.06836{\pm }_{-0.00021}^{0.00021}$	$1.02367{\pm }_{-0.00018}^{0.00017}$	$85.17719{\pm }_{-0.00136}^{0.00139}$	$0.76921{\pm }_{-0.00014}^{0.00014}$	$0.23079{\pm }_{-0.00014}^{0.00014}$
4574310	13.242	7153	4077	1.306220	169.993896	$1.49917{\pm }_{-0.00043}^{0.00044}$	$1.58066{\pm }_{-0.00062}^{0.00059}$	$78.49502{\pm }_{-0.00672}^{0.00689}$	$0.80368{\pm }_{-0.00017}^{0.00017}$	$0.19632{\pm }_{-0.00017}^{0.00017}$
4665989	13.016	7559	6846	2.248068	170.972872	$1.84602{\pm }_{-0.00036}^{0.00037}$	$1.36237{\pm }_{-0.00086}^{0.00083}$	$81.94781{\pm }_{-0.00561}^{0.00567}$	$0.74771{\pm }_{-0.00030}^{0.00031}$	$0.25229{\pm }_{-0.00031}^{0.00030}$
4848423	11.825	6239	6176	3.003646	633.146586	$1.50426{\pm }_{-0.00023}^{0.00023}$	$1.38874{\pm }_{-0.00025}^{0.00026}$	$87.57999{\pm }_{-0.00069}^{0.00068}$	$0.54403{\pm }_{-0.00017}^{0.00016}$	$0.45597{\pm }_{-0.00016}^{0.00017}$
5444392	11.378	5965	5726	1.519529	170.538898	$1.59339{\pm }_{-0.00013}^{0.00013}$	$1.66385{\pm }_{-0.00009}^{0.00009}$	$84.27161{\pm }_{-0.00062}^{0.00062}$	$0.51747{\pm }_{-0.00007}^{0.00007}$	$0.48253{\pm }_{-0.00007}^{0.00007}$
5513861	11.638	6479	6411	1.510208	170.329100	$1.64494{\pm }_{-0.00040}^{0.00040}$	$1.49045{\pm }_{-0.00049}^{0.00047}$	$78.72670{\pm }_{-0.00234}^{0.00242}$	$0.56207{\pm }_{-0.00026}^{0.00027}$	$0.43793{\pm }_{-0.00027}^{0.00026}$
5738698	11.941	6792	6773	4.808774	171.679679	$1.69030{\pm }_{-0.00052}^{0.00053}$	$1.53567{\pm }_{-0.00067}^{0.00067}$	$86.19971{\pm }_{-0.00154}^{0.00158}$	$0.55585{\pm }_{-0.00036}^{0.00037}$	$0.44415{\pm }_{-0.00037}^{0.00036}$
6206751	12.142	6965	4885	1.245343	170.824532	$1.50527{\pm }_{-0.00164}^{0.00160}$	$1.58397{\pm }_{-0.00308}^{0.00309}$	$71.73854{\pm }_{-0.02101}^{0.02087}$	$0.77538{\pm }_{-0.00115}^{0.00115}$	$0.22462{\pm }_{-0.00115}^{0.00115}$
8262223	12.146	9128	6849	1.613015	170.460863	$1.36443{\pm }_{-0.00310}^{0.00301}$	$1.82295{\pm }_{-0.00338}^{0.00345}$	$72.16185{\pm }_{-0.00712}^{0.00714}$	$0.58017{\pm }_{-0.00229}^{0.00226}$	$0.41983{\pm }_{-0.00226}^{0.00229}$
8553788	12.691	8045	5328	1.606164	170.180393	$1.95352{\pm }_{-0.00604}^{0.00600}$	$1.61936{\pm }_{-0.01344}^{0.01344}$	$70.23075{\pm }_{-0.09297}^{0.09245}$	$0.91970{\pm }_{-0.00184}^{0.00182}$	$0.08030{\pm }_{-0.00182}^{0.00184}$
8823397	13.249	8540	5724	1.506504	170.654968	$1.69962{\pm }_{-0.00016}^{0.00017}$	$1.25994{\pm }_{-0.00027}^{0.00027}$	$84.49796{\pm }_{-0.00570}^{0.00578}$	$0.85919{\pm }_{-0.00006}^{0.00006}$	$0.14081{\pm }_{-0.00006}^{0.00006}$
9159301	12.146	7959	4209	3.044772	172.019893	$2.85734{\pm }_{-0.00003}^{0.00003}$	$1.99571{\pm }_{-0.00003}^{0.00003}$	$85.84383{\pm }_{-0.00047}^{0.00049}$	$0.95374{\pm }_{-0.00001}^{0.00001}$	$0.04626{\pm }_{-0.00001}^{0.00001}$
9357275	12.186	7545	5580	1.588298	170.303369	$1.55017{\pm }_{-0.00277}^{0.00279}$	$1.60682{\pm }_{-0.00490}^{0.00485}$	$72.45522{\pm }_{-0.02677}^{0.02667}$	$0.80551{\pm }_{-0.00171}^{0.00173}$	$0.19449{\pm }_{-0.00173}^{0.00171}$
9592855	12.216	7290	7087	1.219325	907.436255	$1.59043{\pm }_{-0.04540}^{0.04555}$	$1.63729{\pm }_{-0.04631}^{0.04601}$	$68.48705{\pm }_{-0.02219}^{0.02090}$	$0.49956{\pm }_{-0.03212}^{0.03222}$	$0.50044{\pm }_{-0.03222}^{0.03212}$
9602595	11.882	9679	5705	3.556513	172.643854	$2.59819{\pm }_{-0.00017}^{0.00017}$	$3.09062{\pm }_{-0.00006}^{0.00006}$	$81.85038{\pm }_{-0.00044}^{0.00045}$	$0.88844{\pm }_{-0.00002}^{0.00002}$	$0.11156{\pm }_{-0.00002}^{0.00002}$
9851944	11.249	7026	6902	2.163902	170.471968	$1.81528{\pm }_{-0.00102}^{0.00099}$	$2.72157{\pm }_{-0.00084}^{0.00086}$	$73.91465{\pm }_{-0.00370}^{0.00374}$	$0.31987{\pm }_{-0.00040}^{0.00039}$	$0.68013{\pm }_{-0.00039}^{0.00040}$
9899416	10.028	11056	6278	1.332564	170.693626	$2.24077{\pm }_{-0.00122}^{0.00113}$	$1.73692{\pm }_{-0.00101}^{0.00103}$	$87.46126{\pm }_{-0.13032}^{0.13031}$	$0.88206{\pm }_{-0.00006}^{0.00006}$	$0.11794{\pm }_{-0.00006}^{0.00006}$
10156064	10.367	7424	6268	4.855936	170.017881	$1.36860{\pm }_{-0.00157}^{0.00150}$	$1.79987{\pm }_{-0.00161}^{0.00170}$	$81.72808{\pm }_{-0.00122}^{0.00118}$	$0.55287{\pm }_{-0.00117}^{0.00111}$	$0.44713{\pm }_{-0.00111}^{0.00117}$
10191056	10.811	6588	6455	2.427495	170.581538	$1.84616{\pm }_{-0.00072}^{0.00073}$	$1.63960{\pm }_{-0.00083}^{0.00080}$	$78.92767{\pm }_{-0.00193}^{0.00189}$	$0.56878{\pm }_{-0.00045}^{0.00045}$	$0.43122{\pm }_{-0.00045}^{0.00045}$
10486425	12.465	7018	5847	5.274818	173.633819	$1.68109{\pm }_{-0.00132}^{0.00128}$	$0.93051{\pm }_{-0.00426}^{0.00432}$	$83.97970{\pm }_{-0.01582}^{0.01546}$	$0.93073{\pm }_{-0.00075}^{0.00072}$	$0.06927{\pm }_{-0.00072}^{0.00075}$
10619109	11.704	7028	3903	2.045166	171.224091	$1.78472{\pm }_{-0.00175}^{0.00177}$	$2.45157{\pm }_{-0.00357}^{0.00352}$	$70.46071{\pm }_{-0.01975}^{0.01999}$	$0.84881{\pm }_{-0.00066}^{0.00067}$	$0.15119{\pm }_{-0.00067}^{0.00066}$
10661783	9.586	7764	6001	1.231363	169.978747	$0.86273{\pm }_{-0.00001}^{0.00001}$	$2.79316{\pm }_{-0.00002}^{0.00002}$	$72.36971{\pm }_{-0.00030}^{0.00030}$	$0.18305{\pm }_{-0.00000}^{0.00000}$	$0.81695{\pm }_{-0.00000}^{0.00000}$
10686876	11.727	7944	5842	2.618415	170.700071	$0.57829{\pm }_{-0.00006}^{0.00006}$	$2.80106{\pm }_{-0.00014}^{0.00014}$	$76.28074{\pm }_{-0.00118}^{0.00115}$	$0.21066{\pm }_{-0.00002}^{0.00002}$	$0.78934{\pm }_{-0.00002}^{0.00002}$
10736223	13.621	7797	5069	1.105094	170.119937	$1.49741{\pm }_{-0.00008}^{0.00008}$	$1.29056{\pm }_{-0.00004}^{0.00004}$	$88.43092{\pm }_{-0.00205}^{0.00207}$	$0.88487{\pm }_{-0.00002}^{0.00002}$	$0.11513{\pm }_{-0.00002}^{0.00002}$
10858720	10.971	7282	7223	0.952378	170.430133	$1.54288{\pm }_{-0.00001}^{0.00001}$	$1.36804{\pm }_{-0.00001}^{0.00001}$	$88.33953{\pm }_{-0.00047}^{0.00047}$	$0.57041{\pm }_{-0.00000}^{0.00000}$	$0.42959{\pm }_{-0.00000}^{0.00000}$

Note.

^a Effective temperatures are from Matson et al. (2017).

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To show these residuals more clearly, Figure 6 shows the average residual for each pixel for KIC 2708156 during (1) primary eclipse, (2) secondary eclipse, (3) out of eclipse in Quarter 13. These images can be thought of as the residual between our PRF model and the data in pixel space. Given that we expect KIC 2708156 to have a color dependence, and that the PRF should be wavelength-dependent, we would expect that the PRF shape will change during primary and secondary eclipses.

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In Figure 6, we see that (1) the PRF is, on average, broader during secondary eclipse, (2) the PRF is, on average, narrower during primary eclipse, (3) the PRF shapes during primary and secondary eclipses are the inverse of each other, and (4) the shape change during secondary eclipse is qualitatively similar to the theoretical PSF shape shown in Figure 2. Together, these facts imply that we are seeing a PSF shape change due to color variability. Outside of eclipse we see that there is still a slight residual, and in Figure 6 we see this is due to there being slight variations in the phase curve of the eclipsing binary in each pixel, implying that there may be color variability in this phase curve.

Figure 7 shows the residuals, per pixel, as a function of phase in the eclipsing binary. Each pixel where a significant residual was detected during secondary eclipse is shown. While the residuals are small (≲200 ppm) they are evidently correlated with the phase of the eclipsing binary. We see there is a clear change in eclipse depth, and that the ingress and egress of the primary eclipse change significantly in each pixel.

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Matson et al. (2017) provide temperature estimates for a small sample of bright eclipsing binaries. According to Matson et al. (2017), the primary of the KIC 2708156 system has an effective temperature of 11,061 K, and the secondary has an effective temperature of 5671 K. These two different temperatures will result in each eclipse depth (both primary and secondary) having a strong color dependence. In the case of KIC 2708156, we expect the target will become redder during primary eclipse, as the hotter star is eclipsed by the cooler star. Similarly, we expect the target will become bluer during secondary eclipse. Given that the target will cycle between redder and bluer phases, it is compelling that the primary and secondary PRF residuals shown in Figures 6 and 7 are the inverse of each other.

To further demonstrate the PRF shape, we show the residuals in each Kepler quarter. The Kepler spacecraft rotated 90° every quarter, and so each target falls on a different channel of the detector every season. Every fourth quarter, the target will fall on the same channel, in almost the same place. However, because of temperature changes and velocity aberration, the target center may have moved ≲1 pixel. Because KIC 2708156 falls on a different channel each season, we would expect the PRF to be different each season (owing to the unique PSF shape and detector systematics of each channel). However, during quarters that fall on the same channel, we would expect the PRF, and thus the residual PRF, to be largely the same. Figure 8 shows the residual PRF during secondary eclipse (the central panel from Figure 6) for every quarter of Kepler data on KIC 2708156. The color scaling is fixed between quarters to show the same level of variability. We see that all quarters that fall on the same channel have extremely similar residual PRF shapes during secondary eclipse, further suggesting that the effect is real, and intrinsic to the target PRF. The color map has been chosen in this figure to reflect the true color change, i.e., blue pixels indicate that the secondary eclipse depth residual became positive, indicating a bluer pixel light curve, and red pixels indicate that it became negative, indicating a redder pixel light curve. Figure 6 shows that pixels invert residuals during primary and secondary eclipses, as the eclipsing binary changes color. We also see that the PRF shape is also qualitatively similar to Figure 2; there is a small central cluster of pixels (red) that are inverted compared to the broader halo of pixels (blue).

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In the above sections, we have shown the PRF shape for a single eclipsing binary is highly correlated with the phase of the binary. There exists a known systematic effect that may cause the PRF shape to change in a way that is correlated with the astrophysical signal, known as the "brighter–fatter" effect. This systematic effect is well documented (e.g., Antilogus et al. 2014; Guyonnet et al. 2015; Lage et al. 2017), and causes the shape of PSFs to get broader as sources become brighter, due to increased electrostatic repulsion as a pixel accumulates charge.

In the above sections, we have purposefully chosen a bright eclipsing binary with large eclipse depth. We have included terms to account for nonlinearity in each pixel, which should in theory entirely account for the brighter–fatter effect (as we are treating each pixel independently). However, the flux of the target is changing significantly, and we must verify that the shape change we have identified is not due to the brighter–fatter effect. To do this, we show the reduced chi-squared of the model for KIC 2708156 as a function of flux in Figure 9 (it is also shown in Figure 5 as a function of time).

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In Figure 9, the primary and secondary eclipses have been highlighted in red and blue respectively. In reduced chi-squared, the primary and secondary eclipses clearly follow two tracks, with the secondary having a much higher reduced chi-squared than the primary at a flux of 0.9. Crucially, there are several instances where the fluxes of the primary and secondary eclipses are the same (normalized flux ≈0.85–1), but the reduced chi-squared is significantly higher for the secondary eclipse. Despite having the same measured flux on the detector, the model residual is more significant during secondary eclipse; the pixel-level model fits more poorly during secondary eclipse, indicating a more significant shape change. Given that the measured flux is the same, the brighter–fatter effect is not able to account for this difference in PRF shape between primary and secondary eclipses, and we can discount this as the source of the PRF shape change.

In this work we use a small literature sample of eclipsing binaries with temperature estimates for each component from Matson et al. (2017). From that sample, we exclude systems with nonzero eccentricities, obvious eclipse timing variations, and uncertain temperatures. Our reduced sample of 27 eclipsing binaries is given in Table 1 in the Appendix. All targets in this sample are bright Kepler eclipsing binaries with short periods. We select this small sample only as a proof of concept for the wavelength-dependent PRF, but in theory this effect would apply to all bright eclipsing binaries observed with Kepler.

For each of the eclipsing binaries in Table 1 we fit the SAP light curve to find the best-fit orbital parameters (including radii and impact parameter). We use the analytic light-curve modeling code starry ⁴ (Luger et al. 2019) to find the best-fit solution. In order to model phase-curve variation, we use simple models from Faigler & Mazeh (2011), given as

$$\begin{eqnarray}&&\begin{array}{l}{a}_{\mathrm{re}}\cos (2\pi \phi +\pi )+{a}_{\mathrm{el}}\cos (4\pi \phi +\pi )\\ \quad +\,{a}_{\mathrm{do}}\sin (2\pi \phi +\pi )+c\end{array}\end{eqnarray} \tag{ 9 }$$

where ϕ is the phase of the eclipsing binary, a_re, a_el, and a_do are free parameters for the weights of the reflected, ellipsoidal, and Doppler beaming contributions respectively, and c is a constant. Using literature values of effective temperatures from Matson et al. (2017), and assuming a blackbody emission for each component, we then predict the time series of the eclipsing binary at 400, 500, 700, and 800 nm. Only the luminosity of each star is varied at each wavelength point, and any change in the radius of the star is assumed to be negligible.

Using this grid of wavelength-dependent light curves, we can now test whether the Kepler data on the sample of eclipsing binaries show consistent evidence of color variability, in the same way as KIC 2708156. To test this, we first build pixel model light curves for every target, for every pixel and quarter, using the same methods as discussed in Section 2.2. We then build the residuals of the true Kepler data and our data-driven pixel time-series model.

For targets where there is a strong wavelength dependence, we expect the secondary eclipse depth residual to be large. But, since each pixel may have a different effective wavelength, the residual eclipse depth varies between pixels (see the right panel of Figure 7). We define the "measured eclipse depth variation" in this data set as the standard deviation of the residual eclipse depths, measured in each pixel for observation of a single quarter, in pixels where a residual is detected at ≥3σ significance. This is the standard deviation of the values in the left panel of Figure 7. We take this measurement once per quarter. In this section we choose the standard deviation as our metric for the change in eclipse depth, which allows us to combine the residuals in all pixels into a single, simple metric. (In Section 4 we use a more sophisticated approach to extract wavelength information.)

Each eclipsing binary has a unique expected color variability; binaries containing stars with similar temperatures will vary less in wavelength than stars with extreme temperatures, and binaries with deeper eclipses will have a larger relative change. To understand our sample, we need a single numeric value to assess the magnitude of the expected, theoretical wavelength dependence. We define the theoretical eclipse depth variation as the difference between the time-series models of eclipsing binaries at 500 and at 700 nm, averaged during secondary eclipse. We choose wavelengths that are slightly bluer and slightly redder than the center of the Kepler bandpass. This provides a single, quantitative estimate of the magnitude of color variability anticipated for each target, based on the temperatures and orbital parameters of the star components. These wavelength values have been chosen somewhat arbitrarily to construct this simple numeric estimator, in order to demonstrate the consistent wavelength dependence in our sample of eclipsing binaries.

Figure 10 shows the correlation between the theoretical eclipse depth variation and the measured eclipse depth variation, for each target. We plot the average measured eclipse depth variation across all Kepler quarters, with uncertainties given by the standard deviation of the measurement across all quarters. As shown in Figure 10, we find a statistically significant correlation between the measured variation in eclipse depth and the predicted theoretical variation across our sample. Using a Pearson correlation test, we find there is a (∼3σ) correlation between our theoretical prediction of the color variability of our sample and the measured PRF shape change. This demonstrates that the wavelength-dependent effects shown in Section 2.5 are not simply coincidence, but true evidence of the wavelength dependence of Kepler observations.

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It is beyond the scope of this initial work to fully model the PRF shape of Kepler. Fully modeling the PRF would require an investigation in the entire data set, and would provide a PRF model as a function of magnitude, spatial position on the detector, and wavelength. However, using our sample, we can provide a coarse wavelength calibration, and estimate the range of effective wavelengths for each pixel in our sample.

To provide a wavelength calibration we will fit the residuals of the Kepler data less our PRF model (Section 2) with our wavelength-dependent time-series model of eclipsing binaries, for every pixel and every quarter. We use a grid of pre-computed eclipsing binary models for each target, evaluated at 300–900 nm with a grid spacing of 100 nm. We then interpolate between these grid points to find the best fitting wavelength model. We additionally fit phase-curve variations using Equation (9), to allow for wavelength-dependent phase-curve variation (we consider this to be a nuisance parameter in this work). Using this model we estimate the effective wavelength of each pixel. An example of this fit is shown in Figure 11 for two example pixels of KIC 2708156. In Figure 11 it is clear that at phase 0 there is a significant, sharp change in the eclipse depth. We attribute this to wavelength-dependent changes in limb darkening. (This effect is more clearly shown in Figure 14.) Our model here is approximate, and could be improved by including physically motivated wavelength-dependent limb darkening and phase-curve variation parameters. We therefore mask out this part of the phase curve when fitting our eclipsing binary model.

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Figure 12 shows a histogram of best-fit wavelengths to every pixel in our sample, (i.e., every pixel, in every quarter, for every target). Using a simple chi-squared test, we remove pixels where the eclipsing binary model is a poorer fit than a flat line. A value of 0 indicates that there is no significant difference between the SAP flux light curve and the pixel light curve. We find the distribution of wavelengths to be approximately Gaussian, with a standard deviation of 40 nm. Figure 12 shows that, for bright eclipsing binaries observed by Kepler, a broad range of wavelengths are accessible.

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The model for the PRF we have described here is limited. In particular, we have ignored all correlations, both in time between cadences and between pixels. In reality, (1) the systematics in consecutive cadences are highly correlated, and (2) pixels that fall close to each other in the PRF are highly correlated. In addition, observations that fall on the same channel (and therefore similar pixels) of the Kepler detector should be correlated, because they experience similar systematics and exhibit similar PRF shapes. Second, instrument effects, such as undershoot, column-dependent smear, and row-dependent bias could cause significant correlation between pixels (see Van Cleve & Caldwell 2016), which we do not account for in this work for simplicity. By including all these sources of correlations, it may be possible to build a more predictive model, applicable to targets beyond bright eclipsing binaries.

Our data-driven approach fits a model to each pixel, and each new target requires a unique model. As such, using our current model, it is not possible to approximate the effective wavelength of an arbitrary pixel on the detector. By building a fully calibrated PRF model based on all the Kepler data it is theoretically possible to estimate the effective wavelength of each pixel. We leave this as further work. Using the work presented here, and a large enough sample of eclipsing binaries, it may be possible to create a model for CAP apertures, and use those apertures to build coarse color photometry from Kepler.

This work has shown that there is a color dependence in the Kepler PRF, and each pixel has a different effective wavelength. We have not explored the limits of this effect in terms of magnitude, and have restricted our study to only bright targets with extreme and predictable color variation. Further work is required to study the chromaticity of the Kepler PRF, and find the magnitude limit of the effect across the Kepler sample.

Finally, the Kepler bandpass is in fact wide, spanning 400 nm (see KIH). While we have identified the approximate effective wavelength of pixels in our sample, this is an oversimplification. In fact, each pixel should be thought of as having a unique bandpass, which is the compound effect of all of the optical systems in the telescope, and which is different to that of the average Kepler bandpass. Further investigation is needed to understand the differences between our simplistic approach and a thorough analysis that takes into account the possibility of a variable bandpass in each pixel.

Despite the considerations discussed in Section 5.1, and despite the limited wavelength range of the chromaticity effect, there are several ways that the community can capitalize on this coarse multiwavelength photometry. In this section we discuss some of the ways that this effect could be employed to find new insight using the archival Kepler data, including gaining a better understanding of eclipsing binaries across multiple wavelengths, vetting exoplanet candidates, identifying and characterizing flares, and studying pulsators. There are numerous further uses for this coarse wavelength photometry that go beyond what is discussed here, including better understanding of reddening in young stars, the color dependence of supernovae, or wavelength-dependent asteroseismology measurements. This coarse multiwavelength photometry, in conjunction with Kepler's extreme precision, can be used both for inference and for classification of objects. For example, by incorporating pixel-level modeling into machine-learning classification algorithms (e.g., Shallue & Vanderburg 2018) it may be possible to more accurately classify true transit events.

5.2.1. Classification: Exoplanet Vetting

Exoplanet candidates require vetting, in order to establish whether they are true planets or false positives (for example from blended background eclipsing binaries). The Kepler pipeline, and tools such as the Robovetter (Coughlin 2017), employed some pixel-level vetting of exoplanet candidates to aid in distinguishing true planet candidates from false positives for the Kepler mission. This vetting included analyzing centroid offsets to identify blended targets (see Twicken et al. 2020; Thompson et al. 2018). The model presented here can be used in conjunction with existing tools to vet exoplanet candidates and provide additional vetting statistics at the pixel level.

Unlike eclipsing binaries, exoplanet systems do not usually exhibit strong color dependence at visible wavelengths. Exoplanets usually do not emit in the visible, unless they have extremely high effective temperatures. As such, for true planets, we expect no changes in transit depth due to color dependence to be detectable using the techniques described in this work. We suggest that, using the model described in this work, it is possible to vet planet candidates by searching for significant shape variation in the PRF during transit. Significant PRF shape changes would indicate that the candidate planet is in fact a grazing eclipsing binary with a significant color variation. Using chromaticity of the Kepler PRF to vet planet candidates can help distinguish between grazing eclipsing binaries and grazing planets, which are otherwise difficult to distinguish (as both exhibit V-shaped transits).

Figure 15 shows an example of how the Kepler PRF chromaticity can be used to vet planet transit candidates by comparing an eclipsing binary signal to a true planet candidate. Figure 15 shows the folded light curves of true planet Kepler-1b and eclipsing binary KIC 8703887, which shows an eclipse depth consistent with a large planet. These targets have been chosen as a demonstration owing to their similar transit/eclipse depths, similar durations, and similar apparent magnitudes. In Figure 15 we show the reduced chi-squared metric of our PRF model fit for each cadence, which shows a significant disagreement during the eclipse of KIC 8703887. Kepler-1b shows no significant disagreement in reduced chi-squared during transit, showing that it is not a false positive eclipsing binary. Using the techniques presented in this work, we can identify planet candidate targets where there is significant PRF shape variation during transit, and discount them as false positives. We note that a transit/eclipse with high signal-to-noise ratio would be required to use this technique.

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There are some caveats when using this method to vet exoplanet transit signals in the Kepler data set. First, it is important to understand the magnitude of the PRF shape change due to wavelength-dependent variability as a function of Kepler magnitude and location on the detector. In this work we do not identify the magnitude limit of this effect. Second, it is important to account for astrophysical effects, such as emission for hot planets and wavelength-dependent limb darkening. Accounting for such effects is beyond the scope of this work, but is enabled by the full Kepler data set.

5.2.2. Characterization: Flares

Stellar flares have been studied extensively in Kepler data (see, e.g., Davenport 2016; Yang & Liu 2019). However, flare studies with Kepler have been adversely affected by two key factors: (1) Kepler's monochromatic bandpass makes estimates of flare energy degenerate, because there is no way to constrain the flare temperature, (2) the majority of Kepler data were taken in long-cadence mode, where each exposure is 30 minutes, which can make identification of short-term flares on timescales ≲1 hr difficult. We propose that the chromaticity of the Kepler PRF can help alleviate both of these problems.

Figure 16 shows an example light curve of a flaring Kepler target, with the reduced chi-squared statistic of our systematics model. Flares are a significantly poorer fit than the stellar variability, showing that there is a change in the PRF shape during flares. We attribute this to a wavelength change during flares: because the flares are much hotter than the stellar photosphere we anticipate the target flux to be bluer during flares. Using the PRF shape change, it is possible to intercompare flares within a single target and investigate whether flares are all the same temperature (peak wavelength) or whether they vary.

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Short-term flares on timescales of ≲1 hr are frequently discarded by flare searches because they are indistinguishable from the signal of cosmic rays. Using the model described here it would be possible to identify true flares by using the shape of the PRF and to reject cadences where the PRF shape is more consistent with a cosmic ray than with the target PRF or the PRF shape change due to color variability. Changes in shape due to flares should be centered on the main target, and should have a shape similar to a PSF, whereas cosmic rays can appear in any pixel, and are usually either single-pixel events or sharp, bright lines. Using simple image statistics (e.g., centroiding) it would be possible to distinguish these processes. This pixel-level shape modeling could be capitalized on by machine-learning algorithms, alongside time-variable flux information, in order to better identify short-term flares in the Kepler long-cadence data set.

If a fully calibrated model for PRF shape were available, the wavelength dependence of the Kepler PRF could be used to extract temperature information on flares in the Kepler data set. Currently we do not know the parameter space that would be probed by a fully calibrated model (e.g., completeness in Kepler magnitude, flare brightness, flare temperature etc.), but in theory it will be possible to at least place limits on the flare temperature, and to intercompare different flare energies within a single target.

5.2.3. Characterization: Pulsators

Pulsating stars such as RR Lyrae are rarely studied at the precision and cadence of the Kepler spacecraft while also obtaining multiwavelength photometry. The chromatic PRF of Kepler provides a unique opportunity to study an enormous catalog of pulsators with a 4 yr long observational baseline and high-precision photometry, and to investigate the wavelength dependence of their pulsations. Figure 17 shows an example light curve of an RR Lyrae star observed by Kepler, with the reduced chi-squared statistic for our pixel-level model fit at each phase of the pulsation. The fit is significantly poorer during the rapid rise of the pulsation, indicating that there may be a significant shape change in the PRF, and therefore a color dependence, during the pulsation. Better characterization of this color change across the sample of pulsators identified in Kepler, alongside Gaia distances, may allow more robust characterization of the temperatures of these pulsators.

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5.2.4. Characterization: Eclipsing Binaries

The effect discussed in this work is not specific to only the Kepler spacecraft. The chromaticity of the Kepler PRF is due to wavelength dependence in the optical system, namely in the PSF as it is transmitted through the optical path of the telescope, and in the detector as the PSF is recorded. Other spacecraft will have analogous effects due to wavelength dependence in their optical systems. In particular, we highlight that the TESS spacecraft is anticipated to have a similar wavelength dependence, owing to its use of lenses and large detector pixels. In fact, the magnitude of the effect for the TESS spacecraft is anticipated to be greater than that for Kepler. We encourage the investigation of other detectors to identify wavelength-dependent PSFs, which could unlock serendipitous wavelength-dependent photometry from other NASA time-series missions.

5.3.1. Modifying the Model for K2