# Kepler Bonus: Aperture Photometry Light Curves of EXBA Sources

Jorge Martínez-Palomera Bay Area Environmental Research Institute, P.O. Box
25, Moffett Field, CA 94035, USA.  NASA Ames Research Center, Moffett Field,
CA, USA  Christina Hedges Bay Area Environmental Research Institute, P.O. Box
25, Moffett Field, CA 94035, USA.  NASA Ames Research Center, Moffett Field,
CA, USA  Joseph E. Rodriguez Department of Physics and Astronomy, Michigan
State University, East Lansing, MI 48824, USA  Geert Barentsen Bay Area
Environmental Research Institute, P.O. Box 25, Moffett Field, CA 94035, USA.
NASA Ames Research Center, Moffett Field, CA, USA  Jessie Dotson NASA Ames
Research Center, Moffett Field, CA, USA

###### Abstract

NASA’s Kepler mission observed background regions across its field of view for
more than three consecutive years using custom designed super apertures (EXBA
masks). Since these apertures were designed to capture a region of the sky
rather than single targets, the Kepler Science Data Processing pipeline
produced Target Pixel Files, but did not produce light curves for the sources
within these background regions. In this work we produce light curves for
$9,327$ sources observed in the EXBA masks. These light curves are generated
using aperture photometry estimated from the instrument’s Pixel Response
Function (PRF) profile computed from Kepler’s full-frame images. The PRF
models enable the creation of apertures that follow the characteristic shapes
of the PSF in the image and the computation of flux completeness and
contamination metrics. The light curves are available at MAST as a High Level
Science Product (kbonus-apexba). Alongside this dataset, we present kepler-
apertures, a Python library to compute PRF models and use them to perform
aperture photometry on Kepler-like data. Using light curves from the EXBA
masks we found an exoplanet candidate around Gaia EDR3
2077240046296834304consistent with a large planet companion with a $0.81R_{J}$
radius. Additionally, we report a catalog of 69 eclipsing binaries. We
encourage the community to exploit this new dataset to perform in depth time
domain analysis, such as eclipsing binaries demographic and others.

Time Domain Astronomy – Light Curves – Exoplanets – Eclipsing Binaries –
Astronomy databases

††facilities: Kepler††software: astropy (Astropy Collaboration et al., 2013),
lightkurve (Lightkurve Collaboration et al., 2018), scipy (Virtanen et al.,
2020), psfmachine (Hedges & Martínez-Palomera, 2021), kepler-apertures
(Martínez-Palomera, 2021), numpy (Harris et al., 2020), exoplanet (Foreman-
Mackey et al., 2021), pymc3 (Salvatier et al., 2016), contaminante (Hedges et
al., 2021b),

##  1 Introduction

NASA’s Kepler mission delivered to the community one of the finest time series
datasets ever produced. Kepler observed more than $400,000$ target stars
(Borucki et al., 2010) Kepler found more than $4,000$ exoplanet candidates
(Thompson et al., 2018), observed numerous supernovae from earliest stages of
explosion (Olling et al., 2015; Garnavich et al., 2016; Li et al., 2019), and
more than $2,900$ eclipsing binary systems (Kirk et al., 2016), to name some
examples. The Kepler mission mainly downlinked image data around selected
targets (KIC, Brown et al., 2011). These target cutouts are centered on the
object of interest and usually cover a few pixels around the source. Kepler
observations are organized into seventeen 93-days periods. These observation
periods are named quarters. The Kepler Science Data Processing Pipeline
(Jenkins et al., 2010), produced two science products. First, the Kepler
pipeline produced Target Pixel Files (TPFs), which are cutout images of each
observed target. A TPF contains all observed cadences for the target in a
single quarter as well as information on which pixels fall into the aperture
of each target star. Secondly, the Light Curve Files (LCFs), which are flux
time series of the target of interest. The LCFs contain two types of light
curves, one is Simple Aperture Photometry (SAP), where the flux is measured
within an aperture selected to minimize contamination from background sources.
The second type are Presearch Data Conditioned Simple Aperture Photometry
(PDCSAP) light curves (Smith et al., 2012) that corrects the SAP light curves
for systematics of the instrument.

Kepler’s mission also downlinked single cadence observations of the entire
Kepler field each month. These Full Frame Images (FFIs) contain more than 1
million sources per image. They were observed mainly for calibration and
diagnostic purposes (Van Cleve & Caldwell, 2016). Although the FFIs covered
the entire focal plane of the instrument, the time resolution is not
comparable to the main targets observed with a cadence of 1 or 30 minutes.
Montet et al. (2017) used Kepler’s FFIs to study long-term variability of Sun-
like stars due to magnetic cycles. Additionally, two types of custom aperture
targets were observed using the 30 min cadence mode. First, aperture masks
covering two open clusters, NGC 6819 (Stello et al., 2010; Balona et al.,
2013; Brewer et al., 2013) and NGC 6791 (Stello et al., 2011; Corsaro et al.,
2012; McClure et al., 2021). The second custom masks are the background super
apertures, or EXBA masks. The EXBA masks cover relatively dark regions of the
sky at a common location on all CCD channels, and were observed continuously
between quarters 5 and 17. The scientific motivation to collect these images
was to obtain an unbiased characterization of the background Eclipsing Binary
(EB) rate in the Kepler field (Barentsen et al., 2018).

The Kepler pipeline relied upon aperture photometry to create the flux time
series of the targets. There are multiple ways to define the shape and the
size of a photometric aperture. The simplest approach is defining circular
apertures that enclose the source flux. A circular aperture would work well
for isolated sources and where the instrument’s Pixel Response Function (PRF)
profile is symmetric (typically found in Kepler at the center of the focal
plane). But a circular aperture will lose considerable flux where the PRF
shape becomes distorted (typically when moving away from the center of the
image) Aperture photometry can be improved by allowing for elliptical or non-
circular shapes, e.g. isophote apertures. Regardless of shape simple apertures
become sub-optimal in crowded regions, where source contamination is the
dominant factor. Bryson et al. (2010) shows that Kepler’s PRF profiles are
distorted at the edges of the focal plane, with elongated shapes in multiple
directions. Therefore, the simple aperture methods will perform poorly. The
Kepler pipeline performed photometry by computing optimized apertures for
every target source and providing metrics that characterize the completeness
of the flux, and amount of contamination within the aperture.

The EXBA masks were chosen to have an unbiased representation of the Kepler
field. The number of Gaia sources (brighter than $20$th magnitude in the $G$
band) per channel mask fluctuates from less populated regions with a few
dozens of objects to more crowded ones with a couple of hundred stars. In this
work, we implement aperture photometry by creating custom apertures that
follow the PRF profile of the image. This PRF models are computed using
Kepler’s FFIs and evaluated on every EXBA source. Until now, the EXBA masks
have not been systematically analyzed and the Kepler pipeline did not produce
LCFs for observed sources. In this work, we produce light curves for more than
$9,300$ sources detected in the EXBA masks. We follow the procedures
introduced by the Linearized Field Deblending (LFD) photometry method (Hedges
et al., 2021a) to obtain a detailed, yet fast to compute and evaluate, model
of the PRF profile for all the channels in the focal plane of Kepler.
Evaluating the PRF models on the EXBA sources provide a model of the object,
which is used to define the photometric aperture and compute flux metrics. All
the light curves produced in this work are publicly available to the community
as FITS Light Curve Files and can be accessed throughout the Mikulski Archive
for Space Telescopes (MAST) archive111Kepler Bonus, APEXBA,
10.17909/t9-d5wy-e535. We present kepler-
apertures222https://github.com/jorgemarpa/kepler-apertures/releases/tag/v0.1.0
(Martínez-Palomera, 2021) a the set of Python tools to compute and use PRF
models to perform aperture photometry on Kepler-like data. As an example of
the opportunities that this new light-curve dataset provides, we perform a
search of exoplanet transiting signals using the BLS periodogram method. We
identify a candidate around the source Gaia EDR3 2077240046296834304that shows
a transiting signal consistent with a large planet or sub-stellar companion.
Our search also provide a catalog of 69 EBs observed in the EXBA masks.

This article is structured as follows. Section 3 details the characteristics
of the data used for this work as well as the steps followed to compute the
PRF models, aperture photometry, flux metrics, and light curves. In Section 4
we present our results: high-level science products consisting of light curve
files, and the search for transiting signals which resulted in an exoplanet
candidate and a catalog of EBs. In Section 5 we discuss the opportunities that
this new unexplored dataset provides to the community, as well as its
limitations. Finally, Section 6 summarizes this work.

##  2 Kepler Observations

The Kepler mission observed an specific area of the sky centered at
R.A.$=19^{h}22^{m}40^{s}$ and Dec$=+44^{\circ}30^{\prime}00^{\prime\prime}$
(J2000) between 2009 and 2013. The observations were split into quarters and
taken in two cadence mode, a short 1-minute cadence and a long 30-minute
cadence. The telescope camera is an array of 21 modules with two CCDs each. A
CCD has two read-out channels adding to a total of 84 CCD channels. Kepler
telescope has an aperture of 0.95 meters, a 105 square deg field-of-view, and
the instrument has a pixel scale of 3.98 arcsec per pixel.

Typically three FFIs were downlinked during quarter 5 and 17, one at the
beginning, middle, and end of the quarter. A single FFI channel has
$1100\times 1024$ pixels, adding to a total of 95 mega pixels for the entire
focal plane. Each FFI has an exposure time of 27 minutes. Kepler’s FFIs can be
downloaded from the MAST archive333https://archive.stsci.edu/missions-and-
data/kepler/kepler-bulk-downloads.

The EXBA images were observed continuously between quarters 5 and 17 of
Kepler’s prime mission using the long cadence setup. These data consist in 4
custom mask of $9\times 60$ pixels per channel, each of these 4 tiles are
located next to each other in the CCD producing a $36\times 60$ pixel image
per channel. Kepler EXBA images can be downloaded from the MAST
archive444https://archive.stsci.edu/kepler/, by searching with the EXBA
keyword in the Investigation ID field. An example of the tiled EXBA mask for
channel 48 is shown in Figure 1. Valid data exist for 80 out of 84 channels.
The four channel are missing because of failure of circuitry powering in
module 3 of the camera (Van Cleve et al., 2016).

Figure 1: Example full EXBA mask (four tiles) for channel 48 observed during
quarter 5. The image shown here is 36 x 60 pixels and the flux values averaged
across cadences. 123 sources obtained from the Gaia EDR3 catalog down to
$G\leq 20$ magnitude are marked with red circles (see Section 3.2). The
brightest visible star is $G=13.7$ magnitudes. Our pipeline filters out all
contaminating sources within $2\arcsec$ to fit the PRF models. Sources outside
the coverage of the mask (up to $4\arcsec$) are allowed and extracted for
photometry.

##  3 Data Processing

To obtain light curves based on aperture photometry that meet similar
standards as those provided by Kepler’s pipeline in the main target LCFs, we
follow these steps:

  1. 1.

Create a detailed model of the PRF profile for each channel and quarter
combination using Kepler’s FFIs.

  2. 2.

Use Gaia catalogs to obtain accurate positions of the sources observed in the
EXBA mask with a limiting magnitude of $G=20$.

  3. 3.

Obtain a flux-normalized model of each source by evaluating the PRF model on
all Gaia sources observed in the EXBA mask.

  4. 4.

Define multiple aperture masks according to the flux normalized source models
and compute flux metrics that characterize them.

  5. 5.

Finally, compute multi-aperture photometry for every source, then compile and
save the flux time series in FITS files following a similar structure as the
LCFs.

The following sections describe these steps in detail.

###  3.1 Modeling the PRF Profile

Kepler’s FFIs contain more than 1.1 million sources in total, on average $\sim
12,000$ sources can be detected in each channel. The large amount of sources
available enables the creation of detailed PRF models for every channel.

Figure 2: PRF models across the focal plane computed from Kepler’s FFIs as
described in 3.1. The models are based on data observed during quarter 5. Each
cell in the figure grid represents a single CCD channel (number in red). In a
cell, each data point represents the pixel position from the reference source
in Cartesian coordinates, while the color represents the pixel flux value
normalized by the total flux of the corresponding reference source. As
discussed in Hedges et al. (2021a), the PRF profiles show steep gradients and
are elongated, particularly as we move away from the center of the focal
plane. Note that in this figure the CCD channels have the axis origin at each
cell’s lower-left corner, then the orientations of the plotted PRFs do not
reflect the axis origin of the CCD channels on Kepler’s focal plane which are
arranged to be invariant under rotation. This is only aesthetics and does not
affect the use of our models.

To compute PRF models of each channel, we use the PRF scene modeling described
in the LFD photometry method (Hedges et al., 2021a). This LFD photometry
method uses Gaia catalogs to find and fix the coordinate position of sources
in the image and models the scene PRF profile by fitting a linear model with
spline basis to the data in polar coordinates. The PRF modeling done by the
LFD method assumes the following: it relies upon the accurate astrometry done
by Gaia catalogs; the PRF shapes do not vary significantly over the spatial
scale of the detector channel; and the PRF shapes do not change in time within
each quarter.

Figure 2 shows the PRF models for all 80 channels with available FFI data
during quarter 5. At the center of the focal plane, the PRF shape is fairly
circular, smooth, and shows angular symmetry. As we move away from the center,
the PRF profile starts gradually showing elongation in both radial and
tangential axis, this effect becomes extreme in the channels at the border.
These characteristics of the data are conveniently captured by the model
thanks to the use of polar coordinates, where the radial symmetry benefits the
fit of a smoother model.

In Hedges et al. (2021a), the PRF profile is modeled using $~{}\sim 500$ Gaia
sources and around $7,000$ pixel data points. The LFD method only uses pixel
where flux from the sources is detected and removes all background pixels.
Here, in contrast, thanks to the large number of sources ($\sim 12,000$) in
each FFIs channel image, the number of pixels reach a larger value $\sim
100,000$, allowing for more dense and robust representation of the PRF.
Additionally, Hedges et al. (2021a) models the drift in position of all
sources due to velocity aberration, this correction is only necessary when
doing LFD photometry on all cadences of a TPF stack to obtain light curves. In
this work, we only follow the LFD method to obtain the PRF models from the
FFIs and fitting the scene motion it is not only limited by the low number of
FFI cadences but also out of the scope of this use-case. Computing the PRF
models from Kepler FFIs can be computationally expensive, fortunately, the
convenience use of scipy’s sparse matrices (Virtanen et al., 2020), keeps the
memory use low ($\sim 450MB$ for a single PRF model) and computing time in the
order of $100$ seconds when using a modern personal laptop5552020 MacbookPro,
16 GB RAM, 2 GHz Quad-Core Intel Core i5 processor.. Evaluation of a PRF model
is faster, for the same machine it takes $\sim 1$ milliseconds per source.

Our pipeline produce a total of $1,080$ PRF models, one for every channel and
quarter combination where the EXBA masks were collected. These PRF models can
be later evaluated for a grid of positions (in Cartesian coordinates) to
obtain a flux normalized model for either a single source or a set of sources
in any Kepler-like data, e.g. TPFs, FFI, and SuperStamps666Cody et al. (2018)
created SuperStamp FITS files for K2 observation of open clusters (M35, M67,
Ruprecht 147, and NGC 6530) and the Galactic bulge. files. Modeling the PRFs
for every quarter capture the natural time degradation in sensitivity of the
CCDs. The later is a factor that could have not been taken into account in
Bryson et al. (2010) works due to the data availability, at the time of their
work, only commissioning data was available. Furthermore, our PRF models use a
small number of linear components which can be evaluated faster than grid-
based models and are forced to be smooth with no edge effects. We direct the
reader to Hedges et al. (2021a) for further discussion.

###  3.2 The EXBA Sources

We use Gaia EDR3 (Gaia Collaboration et al., 2021) as source catalog. This
catalog provides high precision astrometry and source completeness, $0.5$ mas
at $G=20$ and, complete between $G=12$ and $G=17$, respectively (Fabricius et
al., 2021). Although Gaia $G$ band (Jordi et al., 2010) does not exactly
matches Kepler’s response curve (Van Cleve & Caldwell, 2016), it present an
appropriate proxy for Kepler magnitudes and can be used as a observed source
catalog. We query the Gaia catalog with a limiting magnitude of $G=20$ and
account for the source proper motions between each Kepler’s quarter and the
Gaia EDR3 epoch. We flag contaminated sources ($<5\%$) in crowded regions
where the separation between sources is $<2\arcsec$ (0.5 of a pixel), and we
allow for sources outside the limits of the EXBA mask up to $4\arcsec$ (1
pixel). Only non-flagged sources will be extracted, but contaminants are
considered when computing flux metrics (see following section for details).
After this, a total of $9,327$ (non-flagged) sources are found in the sky area
covered by the EXBA mask ($\sim 0.2$ sq. deg.) across all quarters. A CCD
channel covers the same sky area every four quarters due to the rotation of
the spacecraft between quarters. This effect, combined with the mask alignment
in the rotational symmetric channels, leads to sources near the borders of the
mask (within $20\arcsec$) to fall outside of it in certain quarters. As a
result, $45\%$ of the sources have observations in all 13 quarters while
$13\%$ of the sources have data in either 6, 7, or 11 quarters (see Figure 3
for details).

Figure 3: The number of EXBA sources with the total number of observed
quarters. $46.5\%$ of the total sources have observations in all 13 quarters
that Kepler used the EXBA masks. The rest have 12 or fewer quarters available.

Figure 4 shows the Color Magnitude Diagram (CMD) for both, main Kepler Targets
and the EXBA sources. The object photometry comes from Gaia EDR3 catalog and
their distances computed by Bailer-Jones et al. (2021). The EXBA sources
follow the distribution of Kepler’s main targets except for the horizontal and
asymptotic branches. A large fraction of the EXBA sources are located in the
main sequence and binary sequence regions while a small fraction are evolved
stars. The EXBA masks were chosen to be representative of the background field
in comparison with the selection bias of Kepler’s mission that specifically
targeted evolved stars (Batalha et al., 2010).

Figure 4: Color Magnitude Diagram of the EXBA sources (9,327 sources in blue)
in comparison with the main targets (213,393 sources) of Kepler’s mission
(density plot in black). The red marker represents the location of Gaia EDR3
2077240046296834304(see Section 4.2.3) that falls in the binary sequence.

###  3.3 Aperture Photometry

With the detailed PRF models computed from the FFI data for every channel and
quarter combination we are able to evaluate them for every source in the EXBA
images. We evaluate the PRF model on the Cartesian grid of pixel coordinates
of the image data and with a maximum distance up to six pixels from the center
of the source (obtained from the Gaia astrometry). The PRF model evaluation
leads to a normalized (to sum 1) flux model of the source. In this work we
evaluate the PRF model to obtain a normalized model of the sources aiming to
capture the characteristic shape of the PRF, in contrast with the LFD
photometry method where the evaluation aims to obtain the total flux of the
source.

Following _Kepler_ ’s crowding and flux loss (CROWDSAP and FLFRCSAP
respectively, Kinemuchi et al., 2012) metrics that the pipeline calculated for
the photometric aperture of every primary target, we compute two equivalent
metrics. Per definition, FLFRCSAP is the fraction of target flux contained in
the photometric aperture over the total target flux. While CROWDSAP is the
ratio of target flux relative to the total flux within the photometric
aperture including contaminating sources. Therefore a CROWDSAP value of 1
means no contamination and a FLFRCSAP value of 1 means the aperture contains
the entire flux of the source. Evaluating the PRF model for all sources and
given their photometric aperture, we can compute both crowding and
completeness metrics for every source.

We use isophotes to define the boundaries of the photometric apertures. The
boundary of each isophote for a given source is defined by the flux value at a
certain percentile of the normalized flux distribution. The seven percentiles
are $0,15,30,45,60,75$ and $90$. This leads to seven different apertures that
follow the characteristic PRF shape and enclose different levels of flux
completeness and contamination. Per definition, larger percentile values lead
to smaller apertures and therefore less contamination (larger CROWDSAP values)
but less flux completeness (smaller FLFRCSAP values). Additionally, we include
an extra aperture in which the percentile that defines the isophote value is
computed by simultaneously optimizing both metrics, FLFRCSAP and CROWDSAP.
This scalar optimization is performed with scipy’s Brent’s
method777https://docs.scipy.org/doc/scipy/reference/generated/scipy.optimize.minimize_scalar.html.
After the target value is reached for either of the metrics, a softening
term888We redefine the updated metric value as
$metric^{\prime}=target+leakFactor(metric-target)$ where the $target$ is the
desired target value for the metric (e.g. $0.9$), $metric$ is the current
value, and $leakFactor$ is a user defined value (e.g. $0.001$). is applied to
allow the optimization to focus in the second metric until both reach their
respective target scores, instead of driving a single metric to the optimal
point. We optimize the aperture size by targeting for a CROWDSAP $>0.8$ and a
FLFRCSAP $>0.5$.

Providing multiple aperture photometry allows the users to choose the most
suitable one depending on their scientific goal. Each aperture photometry has
its associated FLFRCSAP and CROWDSAP metrics. Later, as indicated in Kepler
Archive Manual (Thompson et al., 2016), crowding corrected fluxes can be
computed by applying the following equation:

$F_{corrected}=F-med(F)(1-\rm{CROWDSAP})$ (1)

where $F$ is the original flux as provided in the light curve files, CROWDSAP
is the crowding metric, and $med(F)$ is the median value of the uncorrected
fluxes. Finally, both metrics can be used to stitch and adjust the light
curves from different quarters to account for aperture variations due to
changes in the PRF profile.

##  4 Results

###  4.1 Data Products

In this work we produce two high-level data products. First, a data set of
light curves with multiple aperture photometry for $9,327$ sources observed by
Kepler in the EXBA masks. We defined the multi-apertures as seven different
isophote levels of the PRF model evaluated at every source. Secondly, a set of
Python tools that allow the use of our precomputed PRF models to perform
aperture photometry in any Kepler-like data, such as TPFs and the EXBA data.

The light curves provided in this work are stored at the MAST archive999Kepler
Bonus, APEXBA, 10.17909/t9-d5wy-e535 and can be accessed similarly as the main
Kepler’s LCFs. Each FITS file contains light curves of a single EXBA source
observed during a quarter. The FITS files are multi-extension, with a table
containing the flux time series from multi-aperture photometry, and 8 extra
image extension containing the aperture mask used for every photometric
aperture. There are a total of $90,224$ LCFs, which accounts for the number of
observed sources and the number of quarters each source was observed. Each
file header contain relevant keywords such as Gaia’s source information, EXBA
origin file, and the completeness and contamination metrics for all the
photometric apertures. Alongside the LCFs, a catalog with all observed Gaia
sources for each channel and quarter is also available. Appendix A details the
structure and content of each EXBA LCF.

Accompanying this dataset of light curves we release kepler-
apertures101010https://github.com/jorgemarpa/kepler-
apertures/releases/tag/v0.1.0, a Python library that allows to reproduce the
results presented in this work. The PRF models computed from Kepler’s FFIs are
included in the library. Through using the this PRF models users can build
aperture masks and compute flux metrics for any type of Kepler’s prime mission
data. kepler-apertures also provides the functionalities to compute new PRF
models from FFI data by using the LFD method.

###  4.2 Finding Transiting Signals

As planned, the Kepler pipeline only produced TPFs for the EXBA mask and no
light curves were extracted. Thus, the EXBA data has never been systematically
analyzed to search for exoplanet candidates, or any other time domain studies.
We follow a standard approach to search for transiting signals.

####  4.2.1 Detrending

Before analyzing each light curve for scientific purposes, we process them to
correct for instrumental effects. Not only that, numerous stars show stellar
activity due to rotation or pulsation, these signals need to be removed to
isolate small variations of the incoming flux originated from transit events.
Removing these types of signals is often refer as detrending.

We use the Cotrending Basis Vectors (CBVs, Smith et al., 2012) to account for
the systematic signals introduced by the instrument. CBVs are a series of 16
basis vectors that capture most of the known correlated features in a
reference ensemble of flux time series. The Kepler pipeline computed these
basis vectors for every quarter and output channel combination. As shown in
Smith et al. (2012), the first 8 CBV components capture most of the systematic
errors, therefore for this work we only use the first eight basis vectors. To
capture variability due to stellar activity, such as rotation or pulsation, we
use a set of basis splines. Similar to Vanderburg & Johnson (2014) we found
that the number of knots for the spline basis equivalent to a spacing of 1 day
allows fitting variability in short time scales without losing transit-like
signals. Combining the aforementioned set of basis vectors we defined a design
matrix that is fitted using linear regression to the data and obtain the
corrected time series. We perform this detrending correction for all the
source light curves on a per quarter basis. As an example, Figure 5 shows the
flux time series of Gaia EDR3 2077240046296834304. The light curve corrected
only with the CBVs (upper panel) shows clear stellar variability in the scale
of days. The light curve after full detrending (middle panel) preserves the
transiting signal. Although the fully corrected light curve still shows
residual stellar variability (as seen in the 1 day binned light curve shown in
black), the amplitude is significantly smaller that the transiting signal.

####  4.2.2 Box Least Square Periodogram

We perform a search for periodic transiting signals using the Box Least Square
Periodogram (BLS, Kovács et al., 2002). The BLS method searches for periodic
variability by fitting the data with an upside-down top hat periodic model.
The model is parameterized by the period, transit duration, and the time at
mid-transit.

Due to the structure of our light curves, which are separated by quarters, we
compute the BLS periodograms in two time regimes. First, on a per quarter
basis search using a grid periods between 0.2 and 30 days. The grid consist in
$60,000$ values evenly spaced in frequency. The upper limit allows for
detecting at least three transits within the $\sim 90$ days time window of
each quarter. Secondly, we search for signals with periods ranging from 60 to
360 days, for this we use the object light curves from all available quarters.
The BLS periodogram return us the parameters of the model that best fits the
data.

To remove periodic signals originated from high-amplitude eclipsing binary
systems and other types of periodic pulsators we filtered the BLS parameters
results. This filtering isolates transiting signal potentially originated from
small companions. We only consider sources with the following BLS results: 1)
transit depth lower than $10\%$ in normalized flux units, 2) a signal-to-noise
ratio of the periodogram maximum power larger than $20$, and 3) a periodogram
maximum power larger than $20$ after standardizing the periodogram by the mean
and standard deviation of the ensemble of periodograms.

Figure 5: Time-based light curve (top two panels) and phase-folded light curve
(bottom panel) of Gaia EDR3 2077240046296834304. Blue markers are the flux
measurements, while black markers are the binned light curves using a time bin
size of 1 day for the time light curve. The top and middle panels show the
CBV-only corrected and fully detrended (CBVs and splines) light curves both
respectively, for observations during quarter 7. The phase-folded light curve
contains data from all quarters as described in Section 4.2.3. The transit
u-like shape and depth ($\sim 0.5\%$) evidence the possibility of a large
planet candidate or a sub-stellar companion. The parameters that characterize
the best transit model are listed in Table 1.

After a visual inspection of less than two hundred light curves after the
aforementioned filtering, one source stands out. Gaia EDR3
2077240046296834304(here and after G…304) exhibits the characteristic transit
shape and depth of a large planet or a sub-stellar companion. Figure 5 shows
both time-based (quarter 7) and phase folded light curve of G…304.

####  4.2.3 Gaia EDR3 2077240046296834304

G…304 was observed in the EXBA masks during quarters 7, 8, 11, 12, 15, and 16,
and it is located outside the coverage of the mask (within 1 pixel from the
mask border) according to Gaia astrometry. Moreover, it was also observed as a
background source in the main TPF of KIC 7214804 continuously between quarters
2 and 17. As shown in Figure 6, Gaia astrometry (solid red dot in the figure)
locates G…304 inside the TPF, but sufficient flux is contained in the EXBA
data to detect the transit events.

Figure 6: Image in pixel space of the EXBA data ($\rm{Pixel\ Column}\geq 556$)
and the KIC 7214804 TPF ($\rm{Pixel\ Column}<556$) where the host candidate
Gaia EDR3 2077240046296834304is located. The flux data is an average of
cadences observed during quarter 7. Red circles show Gaia EDR3 sources, while
the solid circle locates the host candidate. The red squares illustrate the
aperture mask used for this work.

Because the EXBA masks did not capture G…304 during quarters 2 through 6, 9,
10, 13, 14, and 17, we decided to work only with quarters that have both EXBA
and TPF data. Combining the EXBA and TPF pixel data we are able to create a
”stitched” pixel image which improves flux completeness leading to a more
accurate photometry. Not using all available quarters does not impact the
number of observed transits in a significant amount and allows to obtain
accurate model fitting. We compute a new photometric aperture from the
combined image that maximizes the transiting signal using the
contaminante111111https://github.com/christinahedges/contaminante (Hedges et
al., 2021b) Python package. Given a period value and transit reference time,
contaminante creates an aperture mask by finding which pixels contain the
transiting signal at high significance, marginalizing over known instrument
systematics.

We defined the orbital parameters of G…304 using the BLS algorithm at a finer
period grid. We found a transit period of $4.99667$ days, a transit reference
time of $2455468.058$ Julian days, a duration of $0.091$ days, and a
normalized transit depth of $0.005$. An odd-even transit depth test (Wu et
al., 2010) helps to separate transit signals originated from eclipsing
binaries. We perform this test with the best BLS model parameters resulting in
a test statistic value of $0.00235$ and significance level of $96.17\%$
indicating the odd and even transits are statistically consistent. The
transiting signal is a viable case for a transiting planet (see Figure 5,
lower panel), with a transit-like shape and a transit depth consistent with a
large planet or sub-stellar companion. G…304 light curve also evidences
significant variability due to stellar activity in the scale of days.

Unfortunately, Gaia DR2 and EDR3 do not provide estimates of the stellar
radius, luminosity, or line-of-sight extinction for G…304, only an estimate of
the effective temperature ($T_{eff}=4,358$ K). To constrain the host star
parameters and understand the evolutionary state of G…304, we perform a
spectral energy distribution (SED) analysis of the available Gaia ($G$,
$B_{P}$, and $R_{P}$, (Gaia Collaboration et al., 2018)), 2MASS
($J_{2\rm{M}},H_{2\rm{M}}$, and $K_{2\rm{M}}$ (Cutri et al., 2003)), and WISE
(W1 and W2, Cutri & et al., 2012) photometry using the publicly available
exoplanet fitting suite, EXOFASTv2 (Eastman et al., 2013, 2019). We included
the MESA Isochrones and Stellar Tracks (MIST) stellar evolution models (Paxton
et al., 2011, 2013, 2015; Choi et al., 2016; Dotter, 2016) within the fit to
precisely estimate the host star parameters. A Gaussian prior on the parallax
from Gaia DR2 of 1.05010$\pm$0.22882 mas (corrected for the -30 $\mu$as offset
reported by Lindegren et al. (2018) was enforced on the fit and we also
include an upper limit of 0.6677 mag on the line-of-sight extinction (AV) from
Schlegel et al. (1998) & Schlafly & Finkbeiner (2011). Lastly, we assume a
solar metallicity prior of 0.0$\pm$0.25 dex. Our analysis showed that G…304 is
a K-dwarf star with a mass of $0.749M_{\odot}$, a stellar radius of
$0.720R_{\odot}$, effective temperature of $4,410$ K consistent with the value
provided by Gaia DR2, and a density of $2.83g/cm^{3}$. Table 1 summarizes
these results.

We model the light curve of G…304 using the exoplanet library (Foreman-Mackey
et al., 2021) and pymc3 (Salvatier et al., 2016). These tools allow to set
priors for the stellar parameters, such as the stellar radius (R∗), density
($\rho_{*}$) and Limb Darkening, and for the orbital parameters such as period
($P$), transit reference time ($t_{0}$), impact parameter ($b$) and the ratio
between the planet and the star (R${}_{p}/$ R∗). In a similar manner as
Sandford & Kipping (2017) and including the stellar parameter results from the
SED fitting for this system, we adopt the following priors:

  * •

$t_{0}$: a uniform prior bounded by $t_{0\rm{,BLS}}-0.5$ days and
$t_{0\rm{,BLS}}+0.5$, where $t_{0\rm{,BLS}}$ is the transit time at maximum
power from the BLS periodogram.

  * •

$P$: a uniform prior from $0.9P_{\rm{BLS}}$ to $1.1P_{\rm{BLS}}$ days, where
$P_{\rm{BLS}}$ is the period at maximum power from the BLS periodogram.

  * •

$b$: a uniform prior from 0 to 2.

  * •

R${}_{p}/$ R∗: a uniform prior from 0 to 1.

  * •

quadratic Limb Darkening coefficients, $u_{1}$ and $u_{2}$, following the
uninformative prior defined by Kipping (2013).

  * •

$\rho_{*}$: reparameterized as $log_{10}(\rho_{*}[kg/m^{3}])$ and a uniform
prior in log space between 0 and 3.5, which is the stellar density from the
SED fitting in units of $[kg/m^{3}]$.

  * •

R∗: a uniform prior between 0 and $1.1\rm{R}_{*,\rm{SED}}$.

G…304 locates close to the binary sequence in the HR diagram (red marker in
Figure 4) and has a large value of Gaia’s Renormalized Unit Weigh Error (RUWE
=
4.2)121212https://gea.esac.esa.int/archive/documentation/GDR2/Gaia_archive/chap_datamodel/sec_dm_main_tables/ssec_dm_ruwe.html
suggesting that the target could be an unresolved binary system. Considering
the relation between the RUWE and the photocentre perturbation derived by
Belokurov et al. (2020), G…304 has a centroid perturbation of $\sim 2$ mas
($\sim 2$ AU at 970 pc distance). Considering a physically possible binary
systems with a K-dwarf primary (as the SED fitting suggested) and a K-dwarf
(or a smaller M-dwarf) secondary with a 5-day period orbit and an estimated
$0.065$ AU semi-major axis (assuming non-eccentric Keplerian orbit) can not
account for a 0.5% transit depth and the measured astrometric scatter.
Therefore, the transits must be produced by a third body orbiting one of the
stars in the binary system. We added a dilution term to our light curve model
that accounts for the flux provided by a companion following the
parametrization suggested by Espinoza et al. (2019):

$\mathcal{M}(t)=[\mathcal{T}(t)D+(1-D)]\left(\frac{1}{1+DM}\right)$ (2)

where $\mathcal{T}(t)$ is the transit model defined by the stellar and orbital
parameters, $D$ is the dilution term, and $M$ is the out-of-transit target
flux.

The results of the model fitting are presented in Table 1 while the joint
posterior distributions are shown in Figure 7. The inferred parameters for the
radius $0.81^{+0.54}_{-0.54}R_{J}$ of the candidate (with a maximum radius of
$2.2R_{J}$ from the posterior distribution), indicate the transits are
produced by a substellar companion orbiting at $0.034$ AU. Subtracting the
most representative model to the data shows no remaining signal, as shown in
Figure 8. The inferred stellar density ($2.60g/cm^{3}$) is in agreement
(within the errors) with the value obtained from the SED fitting. The mean
value in the stellar parameters inferred from the model fitting ($R_{*}=0.4\pm
0.26R_{\odot}$ and $M_{*}=0.116^{+0.54}_{-0.11}M_{\odot}$) differ with those
find from the SED fitting ($R_{*,\rm{SED}}=0.720^{+0.048}_{-0.046}R_{\odot}$
and $M_{*,\rm{SED}}=0.749^{+0.048}_{-0.049}M_{\odot}$) due to the inclusion of
the dilution term. During the SED fitting, we treated the target as a single
star. Including the dilution term also affected the width of the posterior
distributions, enlarging the derived uncertainties for stellar radius and
mass.

Figure 7: Posterior distributions of inferred stellar and transit parameters for the transiting signal observed around Gaia EDR3 2077240046296834304. The model fitting was performed using the exoplanet Python package following the priors described in Section 4.2.3. The posteriors suggest the eclipse signal is produced by a large planet with a radius $<2R_{J}$.  Table 1: Best model parameters from the SED fitting (first block) and the transiting signal fitting using the exoplanet library (second and third block) around Gaia EDR3 2077240046296834304. The last block details the host Gaia EDR3 values.  Parameter | Description | Value
---|---|---
$R_{*,\rm{SED}}$ | Radius [$R_{\odot}$] | $0.720^{+0.048}_{-0.046}$
$M_{*,\rm{SED}}$ | Mass [$M_{\odot}$] | $0.749^{+0.048}_{-0.049}$
$L_{*}$ | Luminosity [$L_{\odot}$] | $0.175^{+0.044}_{-0.034}$
$\rho_{*,\rm{SED}}$ | Stellar density [$g/cm^{3}$] | $2.83^{+0.48}_{-0.40}$
$log(g)$ | Surface gravity [$cm/s^{2}$] | $4.598\pm 0.04$
$T_{\rm{eff,SED}}$ | Effective Temperature, SED [K] | $4410^{+150}_{-140}$
$[$Fe/H$]$ | Metallicity (dex) | $0.32^{+0.15}_{-0.18}$
$R_{*}$ | Radius [$R_{\odot}$] | $0.400^{+0.26}_{-0.26}$
$M_{*}$ | Mass [$M_{\odot}$] | $0.116^{+0.39}_{-0.11}$
$\rho_{*}$ | Stellar density [$g/cm^{3}$] | $2.60^{+0.23}_{-0.38}$
$u_{1}$ | Limb Darkening Coefficient 1 | $0.65^{+0.13}_{-0.15}$
$u_{2}$ | Limb Darkening Coefficient 2 | $0.13^{+0.24}_{-0.28}$
$D$ | Dilution | $0.09^{+0.03}_{-0.02}$
$R_{P}$ | Radius [$R_{Jup}$] | $0.81^{+0.54}_{-0.54}$
$P$ | Period [days] | $4.99671123\pm 4.66\times 10^{-6}$
$t_{0}$ | Transit Mid Point [BKJD] | $635.0541\pm 0.0005$
$t_{14}$ | Duration [hours] | $0.1\pm 1\times 10^{-6}$
$b$ | Impact parameter | $0.2^{+0.17}_{-014}$
$a$ | Semi-Major Axis [AU] | $0.032^{+0.02}_{-0.02}$
$a/R_{*}$ | Semi-Major Axis / R∗ | $17.48\pm 1\times 10^{-5}$
$\alpha_{\rm{J2000}}$ | R.A. | $295.859489\pm 0.000132$
$\delta_{\rm{J2000}}$ | Decl. | $42.713179\pm 0.000142$
$\mu_{\alpha}$ . | Proper Motion RA [mas/yr] | $-1.99\pm 0.17$
$\mu_{\delta}$ . | Proper Motion Dec [mas/yr] | $-9.40\pm 0.20$
$\pi$ | Parallax [mas/yr] | $1.03\pm 0.15$
$T_{\rm{eff,Gaia}}$ | Effective Temperature, Gaia DR2 [K] | $4358.5^{+244.7}_{-351.5}$
G | Gaia EDR3 G magnitude | $16.5922$
$R_{\rm{P}}$ | Gaia EDR3 $R_{\rm{P}}$ magnitude | $15.6498$
$B_{\rm{P}}$ | Gaia EDR3 $B_{\rm{P}}$ magnitude | $17.2516$
Figure 8: Transit model fitted using the exoplanet tool. The upper panel shows
the light curve data (black markers), the median model (solid blue line), and
the 3rd and 97th percentile (in orange) from the posterior distribution.
Residuals are shown in the lower panel with no clear evidence of a remaining
signal. The transit shape and depth are typical of a large planet companion.

###  4.3 Eclipsing Binaries

The EXBA masks were designed and observed to estimate the unbiased occurrence
fraction of EBs in the Kepler field. Although in this work we do not pursue a
thorough and complete search of this type of variables, during the exoplanet
search previously describe we found many EBs. Here we present a catalog of EBs
and their eclipsing parameters derived from the BLS periodogram. Table 2 lists
69 EB systems found in the EXBA masks. The model parameters period, transit
depth, reference time, and duration are derived from the best BLS fitting. We
perform a visual inspection to remove clear contaminated sources and a cross-
reference search with the literature (Simbad search). The two types of EBs are
present in this catalog. One of them are contact systems (EWs, see Figure 9)
where both components are in contact and sharing a common envelope. EWs have
light curves in which both minima are of similar depth and their typical
periods are shorter than a day showing large-amplitude variability. The second
type are detached systems (EAs, see Figure 9), where both components are of
different spectral type. Therefore, both the primary and secondary minima have
different depths, and the beginning and end of the eclipses are typically very
well defined. EAs are the major source of contamination when searching for
exoplanets using the transiting method.

Figure 9: Examples of Eclipsing Binaries found in Kepler’s EXBA masks.
Detached systems (EAs) show well defined primary and secondary eclipses and
are the major source of false positive when looking for exoplanet candidates.
Contact systems (EWs) show continuous eclipses and typically have shorter
periods than detached EBs.

Table 2: Catalog of 69 Eclipsing Binary systems found in the EXBA masks. This
EB catalog represents an incomplete search for this type of variables and is
only presented as an illustration of the opportunities that this data
presents. These EBs are a result of a search for periodic transiting signal
limited to periods between 0.2 and 100 days and with a standardized
periodogram peak value larger than 20, see Section 4.2.2 for further details.

Gaia ID | Period | $BKJD_{0}$ | Duration | Depth
---|---|---|---|---
| days | days | days | $\%$
Gaia EDR3 2051774184464360064 | 0.816420 | 443.893508 | 0.099 | 1.030747
Gaia EDR3 2051871182004752512 | 0.531408 | 443.762933 | 0.450 | 2.848130
Gaia EDR3 2052499488484831616 | 0.523529 | 630.445491 | 0.450 | 6.528912
Gaia EDR3 2052503164977145088 | 0.551766 | 443.795923 | 0.450 | 0.104680
Gaia EDR3 2052503164977145472 | 0.551766 | 443.795923 | 0.450 | 12.661261
Gaia EDR3 2052524429346526464 | 0.969631 | 540.414513 | 0.099 | 9.448619
Gaia EDR3 2053614118393506176 | 0.532076 | 443.725578 | 0.099 | 18.842998
Gaia EDR3 2053614182812468480 | 0.532076 | 443.725578 | 0.099 | 0.997837
Gaia EDR3 2073608428123884416 | 0.707867 | 443.657609 | 0.150 | 0.096724
Gaia EDR3 2073608462478158848 | 2.006032 | 445.105109 | 0.099 | 0.198396
Gaia EDR3 2073608462483635968 | 2.006032 | 445.105109 | 0.099 | 0.205073
Gaia EDR3 2073682438989499392 | 0.707867 | 443.660681 | 0.126 | 0.056299
Gaia EDR3 2073682438989966592 | 0.707867 | 443.660681 | 0.126 | 0.042618
Gaia EDR3 2073682640842559360 | 0.707867 | 630.538708 | 0.126 | 5.968305
Gaia EDR3 2073682645147919616 | 0.707867 | 443.840681 | 0.450 | 7.558699
Gaia EDR3 2075404789599593088 | 3.034506 | 444.869691 | 0.330 | 2.204107
Gaia EDR3 2075405133185880704 | 0.541367 | 443.837691 | 0.450 | 0.821595
Gaia EDR3 2075405133185894656 | 0.541367 | 443.837691 | 0.450 | 0.230565
Gaia EDR3 2075406541944219520 | 0.630683 | 443.765691 | 0.450 | 0.072754
Gaia EDR3 2076208292075451136 | 0.646481 | 443.890268 | 0.399 | 0.829846
Gaia EDR3 2076209357217808384 | 0.618034 | 443.722268 | 0.399 | 5.536100
Gaia EDR3 2076209456001789056 | 6.810999 | 445.073768 | 0.150 | 1.027946
Gaia EDR3 2076505018468886784 | 0.542118 | 443.714849 | 0.450 | 0.707253
Gaia EDR3 2076789723264283008 | 13.362883 | 446.483764 | 0.450 | 1.503108
Gaia EDR3 2076789727566520192 | 3.524663 | 445.685764 | 0.450 | 0.800765
Gaia EDR3 2076836727387736576 | 3.554231 | 542.873966 | 0.201 | 2.813851
Gaia EDR3 2076836864825475456 | 0.707867 | 443.663764 | 0.126 | 1.174416
Gaia EDR3 2077245814431169792 | 0.637804 | 630.807001 | 0.051 | 32.182456
Gaia EDR3 2077246024892437120 | 12.062975 | 445.235872 | 0.450 | 0.831434
Gaia EDR3 2077810555389602176 | 0.604800 | 539.753960 | 0.399 | 2.754031
Gaia EDR3 2078117181696007296 | 0.983511 | 444.397464 | 0.099 | 11.280642
Gaia EDR3 2078734140854317440 | 0.525966 | 735.642432 | 0.450 | 6.108658
Gaia EDR3 2078735858841858816 | 2.414770 | 445.490808 | 0.174 | 1.672115
Gaia EDR3 2079019292324040704 | 0.628500 | 443.743214 | 0.051 | 1.849365
Gaia EDR3 2079399006085883008 | 11.236515 | 446.155307 | 0.099 | 1.319522
Gaia EDR3 2080297753770786944 | 2.170456 | 443.921954 | 0.126 | 1.089285
Gaia EDR3 2080297753770788864 | 0.827853 | 444.160454 | 0.051 | 4.862135
Gaia EDR3 2082091568328182528 | 0.701906 | 443.740152 | 0.051 | 2.486457
Gaia EDR3 2086516449796423936 | 2.632309 | 443.777988 | 0.174 | 18.155153
Gaia EDR3 2086516454099067776 | 2.632309 | 443.780988 | 0.126 | 0.191745
Gaia EDR3 2086592354758573568 | 6.359473 | 444.454549 | 0.399 | 1.276964
Gaia EDR3 2100240347774313728 | 1.672736 | 443.540000 | 0.099 | 10.979652
Gaia EDR3 2100490555389337984 | 0.580361 | 443.824932 | 0.051 | 11.126328
Gaia EDR3 2101033920292227584 | 3.734361 | 447.016742 | 0.051 | 6.638635
Gaia EDR3 2101136415390982528 | 0.752610 | 443.745168 | 0.450 | 1.506718
Gaia EDR3 2101136419694167040 | 0.752610 | 443.754168 | 0.450 | 1.984499
Gaia EDR3 2101752936479788544 | 0.552366 | 443.755662 | 0.051 | 9.413652
Gaia EDR3 2101753516292218624 | 0.530687 | 443.817162 | 0.450 | 7.978087
Gaia EDR3 2101753520591885568 | 0.530687 | 443.817162 | 0.450 | 7.094880
Gaia EDR3 2101753619371695872 | 0.885682 | 444.205662 | 0.099 | 4.122006
Gaia EDR3 2102900959754861440 | 1.481648 | 443.734874 | 0.099 | 3.322106
Gaia EDR3 2103827298302304128 | 0.533752 | 539.746877 | 0.450 | 1.101425
Gaia EDR3 2104814389164263168 | 0.595171 | 443.855354 | 0.051 | 4.316045
Gaia EDR3 2106327622103110784 | 20.544483 | 463.879921 | 0.450 | 5.975260
Gaia EDR3 2107265956496766208 | 0.604584 | 443.898714 | 0.450 | 5.087825
Gaia EDR3 2107266536313762816 | 1.640355 | 444.341214 | 0.099 | 12.684281
Gaia EDR3 2126083788772171648 | 0.608346 | 539.695481 | 0.450 | 0.021125
Gaia EDR3 2126085369316699648 | 0.601366 | 539.876981 | 0.399 | 2.233859
Gaia EDR3 2126085369320144640 | 0.841974 | 539.699981 | 0.051 | 3.655029
Gaia EDR3 2126955323535081728 | 0.608346 | 539.695419 | 0.450 | 16.312343
Gaia EDR3 2127956149696086272 | 0.601936 | 539.759749 | 0.399 | 1.934922
Gaia EDR3 2127957734541969152 | 0.658113 | 443.884613 | 0.051 | 10.867783
Gaia EDR3 2128160285200885888 | 0.562013 | 443.844143 | 0.450 | 5.224848
Gaia EDR3 2128267968619535872 | 0.515723 | 443.959706 | 0.051 | 0.231262
Gaia EDR3 2128268587094813056 | 0.585238 | 443.715206 | 0.450 | 1.411582
Gaia EDR3 2129514093249096832 | 0.875163 | 444.168417 | 0.450 | 1.554462
Gaia EDR3 2130912843839236224 | 25.235131 | 460.013006 | 0.249 | 3.355718
Gaia EDR3 2133602451140200192 | 0.985036 | 444.216754 | 0.174 | 2.357031
Gaia EDR3 2134814804448660608 | 0.707178 | 444.006163 | 0.150 | 10.898769

Note. — The $BKJD_{0}$, duration, and depth are the results from the best BLS
model and account for the primary eclipse. The $BKJD_{0}$ correspond to the
Barycentric Kepler Julian Day ($BKJD=BJD-2,454,833.0$). These parameters were
obtained from fitting the BLS model with a broad period and duration grid,
therefore are not necessarily the most accurate results.

##  5 Discussion

In this section, we will discuss the relevant assumptions of our work as well
as its limitations.

The use of custom apertures that follow the PRF shape becomes important when
its profile are elongated and a significant fraction of the flux falls in the
wings of the profile. Elongated PRF profiles are particularly true for Kepler
as we move away from the center of the focal plane (see Figure 2). Although
these aperture masks are more complex than standard circular-like apertures,
photometry in crowded regions where multiple sources are located within pixel
distance is sub-optimal. Under these circumstances, PSF/PRF photometry is
recommended, such as the LFD photometry method. Nonetheless, for isolated
sources, custom apertures represent a fast and accurate method to extract
their photometry.

As discussed in Hedges et al. (2021a), fitting the PRF model follows a series
of assumptions. Two of them are relevant to our work: the PRF does not change
with time during each quarter and is uniform across each channel. Both
assumptions are reasonable up to some limit, the PRF shape can suffer changes
in time due to focus recalibration of the instrument and spatially within a
channel.

The time dependency can be mostly corrected using the CBVs as they capture the
instrument systematics. Another approach is to introduce a time dependency to
the PRF model, although this can not be done using Kepler’s FFIs due to their
limited observing cadence. This strategy is feasible if using Kepler’s
clusters apertures (e.g. NGC 6819), where the number of sources is high enough
to provide a smooth PRF model ($\sim 2,000$ sources and $\sim 40,000$ pixel
data points) and the observed cadence is in long mode ($\sim 4400$ cadences).

The PRF profile can slightly change across a single channel, these variations
are averaged when estimating the shape model for the full CCD. As the typical
scale of a PRF model is $<4$ pixels and its variations within the channel
happen at a smaller scale (1 to 2 pixels), the effect in the photometric
apertures defined from the PRF profile is minimal, especially for the small
size of the EXBA masks (36 x 60 pixels) compared to the FFIs ($1,024$ x
$1,100$ pixels). If needed, the spatial change of the PRF profile can be model
by including a spatial dependency to the PRF model or by dividing the channel
image into a grid where each cell contains a dedicated model of the PRF.

The catalog of light curves for $9,327$ sources presented in this work is
based on a Gaia search done at a limiting magnitude of $G=20$. Thus, fainter
sources not only do not have time series but also are not considered for
contamination metrics. This sets the CROWDSAP metric reported in this work as
an upper limit, but the flux contribution from sources fainter than 20th
magnitude is minimal in Kepler observations. The Gaia search could be relaxed
in future works to allow fainter sources.

##  6 Summary

In this work, we extracted source light curves from Kepler’s EXBA masks for
the first time. The EXBA masks are a set of custom masks observed by Kepler
during its prime mission between quarters 5 and 17, which has never been
analyzed before. We created light curve files for $9,327$ Gaia sources
observed in the EXBA masks. In our processing pipeline, we used the method
described in the LFD photometry (Hedges et al., 2021a) to model the PRF
profile of all channels using Kepler’s full frame images. We capitalized on
the large number of sources observed in these FFIs ($\sim 10,000$ per channel)
to obtained detailed PRF models. We evaluated the PRF models for every source
in the EXBA masks to define photometric aperture. These aperture masks follow
the characteristics shapes of the PRF and enabled to compute flux completeness
and crowdedness metrics. The light curves files produced in this work are
available for public access through the MAST archive. The PRF profile models
built from the FFIs are a sub-product of our work. The Python library kepler-
apertures enables user to utilize our precomputed PRF models (included in the
package) to perform aperture photometry on Kepler like data.

As an example of what can be accomplished with these high-level science
products, we present two scientific cases. First, we find evidence of an
exoplanet candidate around Gaia EDR3 2077240046296834304. The results of the
transit model fitting suggest that the object producing this transit has a
size smaller than $2.2R_{J}$. Further investigations need to be conducted to
constrain the stellar properties of the host as well as the parameters of the
transiting objects in order to confirm its planetary nature. As a second case,
we presented a catalog of Eclipsing Binaries found during the search for
transiting signals. The EXBA masks were designed with the primary purpose of
estimating the fraction of background EBs systems in the Kepler field. EBs are
the major source of false positives when searching for exoplanets. Although in
this work we did not attempt a complete search of EBs, we found a significant
number (69) of them and compute its basic parameters using the BLS
periodogram. A more thorough search and analysis of the EXBA light curves to
estimate the occurrence fraction of EBs is left to the community.

This work presents the tools to perform aperture photometry of Kepler data in
a similar fashion as the Kepler pipeline and provides to the community a new
dataset that has not been systematically explored yet. This methodology can be
applied systematically to all the background sources found in the Kepler
targets, which are estimated to contain more than $129,000$ sources. Although
a large fraction of them are blended sources and a fully PSF photometry
approach is more suitable (e.g. LFD photometry), numerous of these sources are
isolated, and fast aperture photometry is the optimal approach. Even more, the
K2 mission shares similar data characteristics as Kepler’s prime mission and
the application of these tools extends naturally.

This paper includes data collected by the Kepler mission and obtained from the
MAST data archive at the Space Telescope Science Institute (STScI). Funding
for the Kepler mission is provided by the NASA Science Mission Directorate.
STScI is operated by the Association of Universities for Research in
Astronomy, Inc., under NASA contract NAS 5–26555. This work has made use of
data from the European Space Agency (ESA) mission Gaia
(https://www.cosmos.esa.int/gaia), processed by the Gaia Data Processing and
Analysis Consortium (DPAC,
https://www.cosmos.esa.int/web/gaia/dpac/consortium). Funding for the DPAC has
been provided by national institutions, in particular the institutions
participating in the Gaia Multilateral Agreement. This research made use of
exoplanet (Foreman-Mackey et al., 2021) and its dependencies(Astropy
Collaboration et al., 2013, 2018; Kipping, 2013; The Theano Development Team
et al., 2016; Salvatier et al., 2016; Luger et al., 2019; Kumar et al., 2019;
Agol et al., 2020; Foreman-Mackey et al., 2021). Funding for this work for JMP
and CH is provided by grant number 80NSSC20K0874, through NASA ROSES.

##  Appendix A EXBA Light Curve Files

Each EXBA Light Curve File (LCF) contains the light curve of a single source
and the aperture masks used for the photometry of a single quarter. Each LCF
is named under the following pattern hlsp_kbonus-
apexba_kepler_kepler_<gaia_source_id>-q<quarter>_kepler_v1.0_lc.fits where the
fields are the Gaia EDR3 source id and the quarter number, respectively. Each
LCF is a multi-extension FITS file that contains the following:

  * •

The primary header unit. This extension contain the main metadata necessary to
identify the object.

  * •

The LIGHTCURVE extension. A binary table header unit containing the flux time
series information. The columns of the table are detailed in Table 3.

  * •

The APERTURE extensions. Eight image header units containing the aperture
masks used for the eight different apertures as described in Section 3.3:
first an optimized aperture (0) and then 7 apertures with decreasing size as
percentile increase. Each aperture mask matches the shape of the 4 tiled EXBA
masks (36 x 60 pixels) for a given channel.

Accompanying the EXBA LCFs for a given quarter and channel combination, there
are the hlsp_kbonus-apexba_kepler_kepler_exba-mask-
ch<channel>-q<quarter>_kepler_v1.0_sup.fits files. These are support files
used for plotting purposes. Each file contains three extensions:

  * •

The primary header unit containing basic information of the EXBA mask such as
center R.A. and Dec coordinates and pixel reference position.

  * •

An image header unit with an averaged image across cadences of the tiled EXBA
mask (36 x 60 pixels).

  * •

A binary table unit containing the Gaia EDR3 catalog of sources observed in
the EXBA mask.

Table 3: Description of the columns available in the Binary Table header unit containing the flux time series of sources in every EXBA Light Curve File.  Column | Field | Format | Units | Description
---|---|---|---|---
1 | Time | float32 | BJD - 2454833 | Time value.
2 | Cadenceno | int64 | - | Cadence number.
3 | Flux | float32 | $e^{-}/s$ | Flux from the optimized aperture.
4 | Flux0 | float32 | $e^{-}/s$ | Flux from the 1st aperture at percentile 0.
5 | Flux15 | float32 | $e^{-}/s$ | Flux from the 2nd aperture at percentile 15.
6 | Flux30 | float32 | $e^{-}/s$ | Flux from the 3rd aperture at percentile 30.
7 | Flux45 | float32 | $e^{-}/s$ | Flux from the 4th aperture at percentile 45.
8 | Flux60 | float32 | $e^{-}/s$ | Flux from the 5th aperture at percentile 60.
9 | Flux75 | float32 | $e^{-}/s$ | Flux from the 6th aperture at percentile 75.
10 | Flux90 | float32 | $e^{-}/s$ | Flux from the 7th aperture at percentile 90.
11 | Flux_err | float32 | $e^{-}/s$ | Flux error from the optimized aperture.
12 | Flux_err0 | float32 | $e^{-}/s$ | Flux error from the 1st aperture at percentile 0.
13 | Flux_err15 | float32 | $e^{-}/s$ | Flux error from the 2nd aperture at percentile 15.
14 | Flux_err30 | float32 | $e^{-}/s$ | Flux error from the 3rd aperture at percentile 30.
15 | Flux_err45 | float32 | $e^{-}/s$ | Flux error from the 4th aperture at percentile 45.
16 | Flux_err60 | float32 | $e^{-}/s$ | Flux error from the 5th aperture at percentile 60.
17 | Flux_err75 | float32 | $e^{-}/s$ | Flux error from the 6th aperture at percentile 75.
18 | Flux_err90 | float32 | $e^{-}/s$ | Flux error from the 7th aperture at percentile 90.
19 | Quality | int64 | - | Quality flag from the TPF.
20 | SAP_Quality | int64 | - | Quality flag introduced by lightkurve.KeplerLightCurve.

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