Investigating the Kepler Target Selection Function Using Gaia DR2

The Stars Kepler Missed: Investigating the Kepler Target Selection Function Using Gaia DR2

Linnea Wolniewicza, Travis Bergerb, Daniel Huberb

aUniversity of Colorado, Boulder, Libby Dr, Boulder, CO 80302, USA bInstitute for Astronomy, University of Hawai‘i, 2680 Woodlawn Drive, Honolulu, HI 96822, USA

Abstract

A critical piece of information to exploit Kepler data is the biases of the Kepler selection function. The Kepler Space Telescope selected nearly 200,000 targets for observation from a sample of over a million stars positioned over the Kepler CCDs with minimal information of their evolutionary state, binarity, or proper motions. Planetary occurrence rates have assumed Kepler’s selection of the stars in its ﬁeld of view to be complete and unbiased with respect to stellar multiplicity and kinematics. We identify the stars positioned on the Kepler CCDs that were not chosen for observation. We use Gaia DR2 to identify stellar parameters for both the observed and non-observed samples. We compare this non-observed sample to the observed sample and identify biases in the Kepler target selection function. We ﬁnd that the selection is complete for main sequence stars, including solar analogs, with Kp < 14 mag but drops to 80% at Kp = 15 mag and 60% at Kp = 16 mag. We use the Gaia Re-normalized Unit Weight Error (RUWE) to show that Kepler was biased against stellar multiplicity. We furthermore use the Gaia proper motions to show that the Kepler selection function was unbiased with respect to kinematics. Keywords: Gaia DR2, Kepler Input Catalog

1. Introduction influence of a binary companion. Dressing and Charbon- neau (2013) found that for the M dwarf stars in the Kepler The discovery of life on other planets is a pursuit that sample, the Earth-sized (0.5 R⊕ - 1.4 R⊕) planetary oc- has long inspired many astronomers, and is fueled by the currence rate is 0.51, all with orbital periods less than 50 search for Earth-like planets. Today, the complex detec- days. All of these analyses have caused the Kepler stars tion of planets around stars is known as extra-solar planet to become one of best-characterized star samples (Berger (exoplanet) science. Exoplanet scientists are constantly et al., 2020). detecting and investigating planets beyond our solar sys- A critical piece of information to exploit Kepler data is tem and categorizing them by size and composition. This the process with which targets were selected. For exam- important science would not be possible without the use of ple, planet occurrence rates studies have so far assumed high-resolution telescopes. For this very purpose of discov- that the Kepler target selection function is unbiased with ering Earth-like planets, NASA launched the Kepler space respect to stellar multiplicity and galacto-centric stellar telescope in March of 2009 - the original exoplanet hunter. velocities (Kraus et al., 2016; McTier and Kipping, 2019). The Kepler mission (Borucki et al., 2010), officially re- Yet Kepler was forced to select targets, as only 200,000 tired in 2018, has left behind a legacy dataset for stellar stars could be observed over the course of the mission. astrophysics and exoplanet science. Most planets observed Many of the underlying assumptions and biases of the Ke- around Kepler host-stars have been found to have sizes be- pler selection function remains unexplored despite its pro- tween Earth and Neptune, a distribution absent in our own lific use in astrophysics research today. solar system (Howard et al., 2012). The uniquely-sized The Kepler Input Catalog (KIC) contains physical and planets found by Howard et al. (2012) have orbital peri- photometric data for sources in the Kepler field of view ods less than 50 days. Kraus et al. (2016) found that there (Brown et al., 2011). The goal of the KIC was to distin- was a deficit in the number of expected binary compan- guish cool dwarf stars from red giants, to which they found ions among planet host stars from the Kepler data, leading themselves to be 98% effective (Brown et al., 2011). Ke- to the conclusion that a fifth of all solar-type stars in the pler selected the optimal targets for observation using the Milky Way are disallowed from hosting planets due to the KIC, with a goal of selecting solar type stars that could host terrestrial-sized planets. The KIC used broadband photometry to infer T values for 85% of their stars. Few Email address: [email protected] (Linnea Wolniewicz) eff

Preprint submitted to Icarus August 3, 2020 log(g) values were known, as broadband photometry is in- As a first step, we cross-matched the KIC with Gaia DR2 sensitive to log(g). The highest priority stars were those to obtain Gaia information for each star in the KIC. To where it would be possible to detect an Earth-sized planet do this, we used the Centre de Donnéesastronomiques de in the habitable zone (HZ), and for which high precision Strasbourg (CDS) cross-match 1. This service is provided radial velocities could be determined (1 - 3 m s−1). The by the Universitéde Strasbourg and joins any VizieR data, next criterion of the selection process was to include stars in this case Gaia DR2, with a private data table based on that were brighter than the 14th magnitude in the Kepler the right ascension (RA) and declination (Dec) of the stars. passband (Kp). The following highest priority targets were This cross-match created duplicates of Kepler ID’s where the brightest stars where terrestrial-sized planets would be multiple stars were located at the same RA and Dec. We detectable in the HZ, even if they were as faint as 15 or 16 removed duplicates by selecting only the Kepler and Gaia magnitudes. Finally, the criterion for detectable planets ID’s associated with the most similar magnitudes in the in the HZ was relaxed, allowing stars that would benefit Kepler passband Kp and Gaia passband (G) for analysis. from additional S/N by observing more transits to be ob- This was done with use of the pandas library in Python served. This created a list of 261,363 stars brighter than (pandas development team, 2020; Wes McKinney, 2010). 16th magnitude in the Kepler passband, which was reduced to less than 200,000 stars due to mission constraints We then queried our data with the Gaia Re-normalized (Batalha et al., 2010). unit weight error (RUWE) values through TOPCAT, as RUWE values can be used to make determinations of The recent Gaia second data release (DR2) provides a stellar multiplicity in Gaia sources. The unit weight er- unique opportunity to look back at the Kepler data and ror (UWE) values are a representation of the normalized better understand its selection function (Gaia Collabora- chi-squared values resulting from the fitting of Gaia DR2 tion et al., 2018). Gaia DR2 has provided astrometry, vari- sources to single-star point spread functions (PSFs). The able star classifications, and astrophysical parameter esti- RUWE value corresponds to a PSF fitting corrected for mates for a total of 1,692,919,135 sources (Lindegren et al., color-dependent biases. RUWE values center around 1.0, 2018). Gaia DR2 is a much expanded and improved astro- but can be large if the fit is not good or there is more metric and photometric data set compared to its predeces- noise than expected. A large RUWE value, such as 1.2 or sor, Gaia DR1, and has been shown to be near complete higher, indicates a multi-stellar system where the presence for G magnitudes between 7 and 20 (Boubert and Everall, of stellar companions increases the noise. 2020). The sources analyzed in Gaia were pulled from 52 billion detections, and reduced to 1.7 billion to reduce spu- rious sources as well as to analyze only those sources with 2.2. Filtering the Data a minimum of 5 detections by the Gaia spacecraft (Linde- gren et al., 2018). With the use of this powerful tool, it 52.5 is possible to look back at the stars chosen by Kepler and garner an understanding of their biases, or lack thereof. 50.0

The goal of this paper is to use the comprehensive Gaia 47.5 DR2 (Gaia Collaboration et al., 2018) to look back at the 45.0 stars the Kepler mission observed, or intended to, and understand the biases of the data and its selection. Many 42.5 conclusions drawn from the Kepler mission data assume Declination (Dec) 40.0 Kepler was unbiased to parameters such as stellar multiplicity (Kraus et al., 2016). Thus, characterizing the biases 37.5 behind the stars that the Kepler mission chose to observe is important. 300 295 290 285 280 Right Ascension (RA)

Figure 1: Spatial extent of the Kepler field. Turquoise points are stars whose light did not fall on the CCDs, while red points are 2. Methodology stars whose light did fall on the CCDs for all four seasons and are 8 arcseconds away from the CCD edges. 2.1. Catalog Cross-matching In this paper we analyze only the stars that fall on the Kepler CCDs that could have been observed. The KIC This paper analyzes the Kepler Input Catalog dataset for contained stars in the Kepler field of view, only a fraction all sources close to the Kepler field of view from the Mikul- of which were located on the Kepler CCDs. Figure 1 shows ski Archive for Space Telescopes (MAST). We analyze only the spatial extent of the Kepler field for stars brighter than the Kepler stars brighter than Kp = 16 mag, as nearly all Kepler targets were below this threshold (Batalha et al., 2010). 1http://cdsxmatch.u-strasbg.fr 2 Kp = 16 mag, consisting of 580,000 stars. Of the 580,000 creasing Kp mag. We can see in this histogram the Kepler stars in the Kepler field of view, only 379,000 stars had selection function detailed in Batalha et al. (2010) at work; light that actually fell upon the Kepler charge-coupled de- at Kp = 14 mag the observed count shows a drop, and vices (red points). The remaining 214,000 stars’ light fell likewise beyond Kp = 16 mag very few stars are observed. in the cracks of the CCDs and were physically unobserv- The stars fainter than Kp = 16 were likely cherry-picked able (blue points). rather than chosen by a general selection function, so they The stars that were possible for Kepler to observe are de- are not indicative of the selection function and its biases. fined as those located more than 8 arcseconds away from It is for this reason that we analyze the stars brighter than the edge of the charge-coupled devices, or 2 Kepler pix- Kp < 16 mag (stars to the left of the dotted grey line) separately from the stars fainter than K > 16 mag and els, for all 4 quarters of the Kepler telescope. As the Ke- p brighter than K < 18 mag in section 5. pler telescope trailed the Earth’s orbit about the Sun, it p made a 90 degree roll about its optical axis every 3 months We calculated revised stellar parameters for all of the to keep its solar array pointed at the Sun and its radia- stars brighter than Kp = 16 mag using the calculations of tor that cools the detectors pointed to deep space. The Berger et al. (2020). We made this decision upon discov- 3 months before rotation define a quarter on the Kepler ering that over 60,000 of our 379,000 stars lacked radii and telescope. Because of pointing discrepancies between rolls luminosity values from Gaia. In addition, Gaia DR2 does stars needed to be on the CCDs and no closer than 2 pixels not take into account interstellar extinction, which affects to the edge in order to be visible in all quarters. our data strongly as many stars in the Kepler field of view Finally, we cross-matched the refined dataset with the Ke- are located in the galactic plane. We used isoclassify based on the grid model from Berger et al. (2020) with pler target list (Batalha et al., 2010), a table possessing all the Gaia parallaxes, G, B, and R passbands and uncer- the Kepler ID’s of stars that were observed. This match was performed on the Kepler ID’s, and all matches of this tainties to derive revised stellar parameters for all 379,000 join represent the stars that Kepler observed. Likewise, all stars in our dataset. This gave us access to temperatures, non-matches are the stars that Kepler did not observe. radii, and luminosities for nearly 70,000 more stars than previously, or 19% of our dataset. Comparisons of our re- 105 sults from the Gaia and revised stellar parameters are de- Total tailed in 5. We subsequently reduce our dataset to 327,849 stars by removing stars with parallax percent uncertainties Observed larger than 20%. 104

3. Full Kepler Sample Count 103 3.1. HR Diagram Figure 3 displays the HR diagrams, based on the revised ef- 102 fective temperature and radii determined by the processes 1.0 Kepler Magnitude of Berger et al. (2020), for 6 subsets of the Kepler data deﬁned by their maximum Kepler magnitude. The color 0.8 corresponds to the percentage of stars that were observed for each eﬀective temperature and radius value. We vary 0.6 the minimum count for each Kp magnitude so that the shapes we see are not dictated by noise. 0.4 Figure 3 shows that for the brighter Kepler magnitudes, such as those below 14, nearly all stars were observed. 0.2 This is to be expected as it corresponds precisely with the Percentage Observed Observed selection function detailed in Batalha et al. (2010), and the 0.0 number of stars fainter than Kp < 14 mag was so few that 12 14 16 18 Kepler could observe everything. As we view the selection Kepler Magnitude function of fainter Kp mag, however, some patterns take Figure 2: (a) Distribution of Kepler magnitude for observed and non- shape in the contouring of the diagram. observed samples. The total number of stars brighter than Kp < 18 The broadband photometry used by the KIC is insensitive mag for stars on the Kepler CCDs is 1,559,884, and the total number of observed stars is 194,859. (b) Percentage observed versus Kepler to log(g) values, and it was later found that KIC surface magnitude. Dotted grey line marks the Kp = 16 mag. gravities were frequently overestimated by up to 0.2 dex, resulting in underestimated radii by up to 50% (Huber Figure 2 shows the distributions of stars observed for in- et al., 2014; Verner et al., 2011). In addition, it wasn’t 3 2 Kepmag < 11 Kepmag < 12 Kepmag < 13 10 1.0 )

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Figure 3: Radii plotted against Effective Temperature of Kepler stars for increasing Kepler magnitudes. The first 6 panels are for cumulative Kepler magnitudes and the last 3 panels are for binned Kepler magnitudes. The color of each bin corresponds to the percentage of points observed by Kepler in that bin. The Sun is shown as the circled dot in each panel. until the Gaia mission that distances to stars were known required the full 4 years of Kepler data to resolve (Bányai to accuracies below 20% (Gaia Collaboration et al., 2018), et al., 2013), and luminous giants are important candi- and so the Kepler mission’s choices of what stars to ob- dates for asteroseismic research. In addition they are very serve were not based off of the fleshed-out graphs shown luminous and so are all in the Kp < 14 mag range where in Figure 3. Instead, they had information on what stars Kepler’s observation was most complete. were dwarfs, giants, and the areas in between. Along the main sequence, the observed fraction remains The observed fraction on the red giant branch is lowest relatively high except for lower values near the solar region for cool stars with low luminosities. We suspect that this and for cool M dwarfs. The observation fraction of the is due to the fact that these stars could be easily distin- subgiant stars is similar to the solar region because Kepler guished from cool dwarfs at the same temperature. Dwarfs selected targets in bands of Teff , as seen in the bottom are far more likely to host planets than red giant stars, panels of Figure 3. and so the giants in this region were dropped from the The main sequence is well observed for bright Kepler mag- target list. The observed fraction on the red giant branch nitudes, but decreases drastically at the K = 16 mag. is highest for the most luminous giants. This is most likely p Interestingly, solar-type stars and cool M dwarfs are the because these large giants have long pulsation periods that

4 least observed stars on the main sequence. Likewise, we see 3.3. Effective Temperature Analysis little difference in color between the solar-type stars and Figure 5 gives a deeper analysis of the observations of stars their subgiant neighbors. These observations likely stem for different effective temperatures. The far left plot is a from the same cause; that with such minimal information histogram of all the stars in our sample, broken into those on the evolutionary states of stars Kepler couldn’t distin- observed and those not observed. As anticipated, the ob- guish the subgiant stars from solar-type main sequence served sample peaks around T , confirming that Kepler stars. As a consequence, many solar-type stars were not did indeed observe more solar-type stars than anything observed because their placement on the main sequence else. The second peak is around the temperature of red was not known, and many subgiant stars were observed giant stars, as another other goal of the Kepler mission was because they were thought to be solar-type main sequence to observe luminous red giant stars for asteroseismology. stars. Looking right, we see that for both the main sequence and subgiants the curves peak around 5800 K. This is because, as discussed previously, it was hard for Kepler to distinguish between main sequence and subgiant stars around the solar temperature. As such, all the stars with tem- 3.2. Evolutionary States peratures close to T were observed without knowledge of their evolutionary states. The shape of the non-observed curves for each evolutionary state resemble the conclusions of Figure 3. For main se- To quantify the percentage of observed stars, we calcu- quence stars with temperatures below or above 5800 K, the late different evolutionary states of the HR diagram using percentage of stars observed is smaller than it is around the 2 evolstate . It defines 3 main evolutionary states, each solar temperature. For subgiant stars the curves are simi- defined the lines in Figure 4 (a). The percentage of stars lar for all Teff . Finally, the red giant non-observed curve Kepler observed for each evolutionary state is then plotted is significantly larger than the observed curve. This sug- in Figure 4 (b). gests that the red giant stars were not as well observed as Figure 4 (b) confirms the conclusion of Figure 3 that Ke- stars of other evolutionary states, a conclusion supported pler observed nearly all the stars brighter than Kp = 14 by the yellow giant branch in Figure 3 and the drop of the mag. The main sequence is the most observed evolution- red line in Figure 4 (b). ary state of Figure 4 (b), dropping to only 60% observed at Kp = 16 mag. The subgiant stars closely resemble the 3.4. Kinematics main sequence stars, for a final observation rate of 50% at McTier and Kipping (2019) compared galacto-centric ve- Kp = 16 mag. This is most likely due to the Kepler selection function’s inability to distinguish subgiant stars from locities of Kepler host stars against other stars in the Ke- solar-type main sequence stars, and it has been shown that pler dataset and found the host stars to be moving sig- the Kepler mission preferentially selected subgiant stars for nificantly slower than the rest of the Kepler dataset, and observation (Huber et al., 2014). concluded that the Kepler selection was biased to observing slower host stars. However, by using Gaia DR2 to At brighter Kepler magnitudes, we see a large number of identify Kepler’s non-observed sample we can understand red giant stars observed for at least one quarter. This is much better the biases of the selection function with re- because they are more luminous and therefore more likely spect to kinematics. We analyze velocities using Gaia DR2 to be observed by astronomers implementing observational proper motion values converted to kms−1 using distance cuts based on apparent magnitude. However, as the goal of values. It should be noted that these velocities do not the mission was to observe solar-type stars many red giants take into account radial components, and are therefore were dropped from the target list after being observed for sky-plane velocities. one quarter, as shown by the plummet of the red dotted line in Figure 4 (b). We reduce our investigation of kinematics to include only Kepler magnitudes between 14 and 16. We analyze only Figure 4 (c) breaks the main sequence into three subsec- the faintest stars of our dataset because for Kepler magni- tions; solar analogs, MS stars cooler than the Sun, and MS tudes brighter than 14, nearly all stars were observed. It stars hotter than the Sun. Interestingly, the stars cooler is only in the fainter than Kp = 14 mag domain where Ke- than the Sun are more observed than the solar analogs at pler was forced to make tough decisions and the selection fainter Kp magnitudes. We suspect the reason behind this function is at work. choice is because the small and cool dwarf stars could easily be distinguished from giants. In addition, M dwarf stars The different distributions of distance of the observed and are likely planetary hosts with terrestrial-sized planets in non-observed samples have a strong effect on the proper the habitable zone, as shown in Dressing and Charbonneau (2013). 2http://https://github.com/danxhuber/evolstate 5 102 Red Giant Branch

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Figure 4: (a): Effective temperature and radius plot of Kp < 16 mag with lines defining different evolutionary states. Red separates the giant and subgiants, magenta the subgiant and main sequence (MS) branch, blue the MS stars cooler and hotter than the Sun, and yellow the solar analogs (5700 K < Teff < 5900 K, 0.9 < R < 1.1). (b) and (c): Percentage observed at each Kp magnitude for different evolutionary states, defined above. Dotted lines represent stars observed for more than 8 quarters of the Kepler mission. motions of the samples, and therefore their sky-plane ve- because the red giants are more luminous and can be seen locities. To account for this we match the non-observed from greater distances. The greater distance red giant sample distance distribution to the observed sample in- stars are likely located in different areas of the galaxy than dividually for each evolutionary state to the best of our the thin disk and therefore orbit differently than we do. ability. The distributions of sky-plane velocity for each The solar analog and main sequence stars are the only evolutionary state is shown in Figure 6. panels of Figure 6 possessing any noticeable differences between the observed and non-observed samples. The non- The red giant distributions of Figure 6 are wider spread

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Figure 5: Histogram of the eﬀective temperatures of Kepler stars for diﬀerent evolutionary states. Solid lines represent the observed sample, and dashed lines represent the non-observed sample.

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Figure 6: Histogram of the quadrature sum of the proper motions in RA and Dec (kms−1) of Kepler stars for different evolutionary states. Solid lines represent the observed sample, and dashed lines represent the non-observed sample. observed samples possess more stars with large sky-plane An RUWE value above 1.2 suggests that the star in ques- velocities, a difference explained the non-observed sample tion has a binary companion. Thus, if the Kepler observed being systematically farther away since it is made up of and non-observed samples have different distributions for fainter Kp. While we try to account for this by matching large RUWE we can conclude that the selection function the distance distributions of the samples, the match is not was biased in some way with respect to stellar multiplicity. perfect and the non-observed sample is still consistently Similarly to the analyses of 3.4 we first reduce our sam- farther away. The effect of distance on sky-plane velocities ple to the stars with 14 < Kp < 16 mag, both because is strong, partly because far away stars appear more static the selection function was most active in this magnitude than close stars, and because the farther away stars are range and because the unit-weight error (UWE) has been located in galactic structures other than the thin disk and shown to be inaccurate for Gaia magnitudes fainter than possess different kinematics. 13th mag (Lindegren, 2018). As a next step we match the non-observed sample distance distribution to the ob- 3.5. Stellar Multiplicity served sample distribution individually for each evolutionary state. The different distance distributions of the two Kraus et al. (2016) used the Kepler data to conclude that samples have a strong effect on the RUWE distribution, a fifth of all solar-type stars in the Milky Way have binary as farther away stars are more likely to have an improper companions, a conclusion based on the assumption that fit and high UWE. The RUWE distributions are shown the Kepler data is unbiased with respect to stellar mul- below in Figure 7. tiplicity. To evaluate this claim we evaluate the RUWE RUWE values greater than 1.2 (shown by the dotted grey values of the stars in the Kepler field of view for both the line in Figure 7) correlate with multiplicity, so the tails observed and non-observed samples. The RUWE value, as of the distributions are where we focus our investigation. discussed in 2, is the Gaia re-normalized unit weight error. 7 Solar Analog Stars Main Sequence Stars Sub Giant Stars Red Giant Stars 101 Observed Observed Observed Observed Un-observed Un-observed Un-observed Un-observed

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Figure 7: Histogram of RUWE values of Kepler stars for diﬀerent evolutionary states. Solid lines represent the observed sample, and dashed lines represent the non-observed sample. The dotted grey line marks the RUWE value beyond which stars likely possess companions.

To quantify the fraction of stars with stellar companions would be an invalid comparison because main sequence for both samples we calculate the percentage of stars in stars span many Teff while subgiants exist in a small range each sample with RUWE values greater than 1.2. This of Teff . This difference could also be attributed to im- number is simply calculated by dividing the number of precise matching of the distance distributions between the stars with RUWE > 1.2 by the total number of stars for samples, leading us to investigate this difference in RUWE each sample. This statistic, its uncertainty, and the sigma further in Section 4. difference between the samples is listed below in Table 1.

Sample: Observed Non-Observed ∆σ 4. Host Star Sample Solar Analog: 5.39 ± 0.4 % 7.25 ± 0.7 % -2.4 Red Giant: 4.51 ± 0.4 % 3.97 ± 0.4 % 0.97 The Kepler data is integral for exoplanet science, and so it Subgiant: 7.47 ± 0.2 % 5.65 ± 0.2 % 7.1 is important to investigate the effects of the Kepler selec- Main Sequence: 9.17 ± 0.1 % 10.5 ± 0.2 % -5.9 tion function on the Kepler host sample. We use the Ke- pler host sample as a basis for our investigations into the Table 1: Percentage of RUWE values greater than 1.2 and their differences between the observed and non-observed sam- Poisson uncertainties. ples RUWE and sky-plane velocity distributions. Our host sample consists of 2,066 stars with either confirmed or can- The differences between the red giant and solar analog didate planets and Kepler magnitudes fainter than 14 and observed and non-observed samples are insignificant and brighter than 16 mag. We randomly select stars from our match within three sigma. In contrast, the main sequence observed and non-observed samples that match the host and subgiant observed and non-observed samples have sample distributions of effective temperature, radius, and close to or over six sigma differences. While this differ- distance. By matching our observed and non-observed samples to the host sample we create two samples that ence is significant it is important to note that the non- observed main sequence stars have a greater fraction of are similar in both evolutionary state and distance. As binaries than the observed main sequence stars, and the such, if any differences arise in our comparisons of RUWE non-observed subgiants have a smaller fraction of binaries and sky-plane velocity values between the matched sam- than the observed subgiants. This inverse relationship be- ples they are due to the Kepler selection function. The tween the samples may be due to Kepler’s observation of comparison between the host-matched observed and non- observed samples is shown in Figure 8. main sequence stars in binary systems that appear as subgiants due to their inflated luminosity values. This would Figure 8 (a) and (b) confirm that the distributions of ef- cause the main sequence observed sample to have a lower fective temperature, radius, and distance are matched be- fraction of binary stars than the non-observed sample, between the observed and non-observed samples. Figure 8 cause the observed binaries in the main sequence have been (c) shows the RUWE values of the observed and non- misidentified as subgiant stars. Similarly, it would cause observed samples, and it can be seen that the non-observed the subgiant observed sample to have a greater fraction of sample is consistently above the observed sample for high binary stars than the non-observed sample since the ob- RUWE values. 8.2 ± 0.4% of the observed sample have served sample include main sequence binary stars as well RUWE values beyond 1.2 and 12.3 ± 0.6% of the non- as subgiant binary stars. We could test this by combining observed sample have RUWE values beyond 1.2. This sig- main sequence and subgiant stars together, however this nificant 5.2 sigma difference suggests that Kepler preferen- 8 1 10 10 3 Observed 10 2 Un-observed

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Figure 8: (a) HR diagram of the observed (blue) and non-observed (orange) samples. (b) Histogram of the distance distributions for the observed and the non-observed samples. (c) Histogram of the RUWE values for the observed and the non-observed samples. (d) Histogram of the sky-plane velocity distributions for the observed and the non-observed samples.

tially selected non-binary stars for observation. Kepler’s 5.2. Brighter than Kp = 18 mag bias against stellar multiplicity could mean that when investigating the Kepler host star sample, it will also be One advantage of using the Gaia DR2 stellar parameters is necessary to investigate a control non-observed sample to that they provide parameters for stars fainter than Kp = affirm your results. Conversely, the distributions of Fig- 16 mag. These faint stars are displayed in Figure 10. ure 8 (d) appear very similar, and the Kolmogorv-Smirnov Figure 10 is colored by the percentage of stars observed test confirms this conclusion with a p-value of 18.7%. This by Kepler, on a scale from 0% to 15%. It appears much p-value, as well as the similarities in the sky-plane veloc- as we would expect it to, as towards the end of the Kepler ity distributions of Figure 8 (d), reinforce our conclusion mission it was discovered that M dwarf stars host many of Section 3.4 that Kepler was unbiased with respect to planets (Dressing and Charbonneau, 2013) and were added proper motions. to the Kepler target list. We see few stars with Kp > 17 mag because of Gaia’s cuts on parallax error as well as its inability to derive physical parameters for faint stars. 5. Comparison with Gaia DR2 5.3. Binary Evaluation In previous sections of this paper we use the revised stellar paramaters calculated with the techniques of Berger The inflation of the main sequence shown in Figure 9 (a) al- et al. (2020) for our analysis, however it is important to lows us to identify a new region of stars known as the main understand the differences between the revised and the sequence binary stars. These are cool dwarf stars that ap- Gaia DR2 parameters. Our revised stellar parameters are pear more luminous on an HR diagram than they actually calculated from isochrone fitting with the Gaia parallaxes, are due to the presence of stellar companions. These stars G, B, and R passbands. They provide distances more ac- are defined as those enclosed by the green lines of Figure curate than parallax−1 and take into account interstellar 11. They provide us with another metric to analyze the Kepler selection function’s was bias with respect to stellar extinction. In contrast, the Gaia DR2 Teff is derived using color analysis and does not account for interstellar extinc- multiplicity. tion. Additionally, many radii and luminosity values are Figure 11 (a) shows the definitions of the main sequence missing from Gaia DR2 (Gaia Collaboration et al., 2018). binary stars, as well as their neighboring main sequence stars, with Teff < 4700 K. The main sequence binary stars have significantly larger RUWE values, as shown in 5.1. Revised Radii Comparison Figure 12. Figure 11 (b) displays the percentage observed for sus- The Gaia DR2 stellar parameters are much less precise pected main sequence binary stars with Teff < 4700 K than the revised values, and the main sequence of Figure and the main sequence stars most similar to them, single 9 (a) appears inflated from noise. Small, cool dwarfs were M dwarfs with Teff < 4700 K. It can be seen that the not given Gaia values because of Gaia’s cuts on Teff and percentage observed for binary and single star systems is luminosity, and are characterized much better by Figure 9 very similar with only a 8% difference at Kp = 16mag. (b). Some banding patterns are evident in Figure 9 (a) be- While this difference is small it reinforces the conclusion cause Teff was calculated using color analysis as opposed of Section 4 that Kepler preferentially selected non-binary to isochrone fitting. stars for observation. 9 Gaia Stellar parameters Revised Stellar Parameters 1.0 102

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Figure 9: HR diagrams for stars brighter than KP = 16 mag. Figure (a) shows the Gaia luminosity and eﬀective temperatures. Figure (b) shows the revised radii and eﬀective temperatures from the process detailed in Berger et al. (2020). The Sun is shown as the circled dot in each panel.

102 Kepmag < 16 16 < Kepmag < 17 17 < Kepmag < 18 0.150

0.125

) 0.100 1

R 10 (

s 0.075 u i d a

R 0.050

0.025 100

0.000 Percentage Observed by Kepler 8000 6000 4000 8000 6000 4000 8000 6000 4000 Effective Temperature (K) Effective Temperature (K) Effective Temperature (K)

Figure 10: Gaia DR2 radii plotted against Teff for increasing Kepler magnitudes beyond 18th magnitude. The color of each bin corresponds to the percentage of points observed by Kepler in that bin. The Sun is shown as the circled dot in each panel. 302,008 stars are plotted in (a), 163,572 stars in (b), and 15,066 stars are plotted in (c).

6. Discussions and Concluding Remarks main sequence binary stars. Our main results are as fol- lows: In this paper we have analyzed the Kepler mission’s target selection function used to decide which stars to observe • We find that the Kepler target selection is complete from the Kepler field of view. We use MAST and Gaia for stars brighter than Kp = 14 mag. The observed DR2 data to identify stars on the Kepler CCDs that were fraction for all main sequence stars drops from 95% to not chosen for observation, and use the python pandas 60% between 14 < Kp < 16 mag. For stars along the library to analyze the observed and non-observed sam- main sequence completeness is best for stars cooler ples. We calculate revised stellar parameters for the stars than the Sun and worst for stars hotter than the Sun, brighter than Kp = 16 mag with the processes detailed in with Solar Analog stars in between. We find that the Berger et al. (2020) and analyze the Kepler missions com- observed fraction for subgiant stars is only lower than pleteness at each evolutionary state. We use the Kepler the main sequence stars by 10%, implying that many host sample to analyze the RUWE and sky-plane velocity subgiant stars selected for observation were believed distributions of the observed and non-observed samples to be main sequence stars. and the biases therein. For the stars fainter than Kp = 16 mag we use the Gaia DR2 stellar parameters to compare • We find that the observed fraction for red giant stars results of the revised parameters and identify suspected drops from 90% at Kp = 14 mag to 45% at Kp = 16 10 Kepmag < 13 Kepmag < 14 102 )

1 R

( 10

s u i d a R

100

Kepmag < 15 Kepmag < 16 102 )

1 R

( 10

s u i d a R

100

8000 7000 6000 5000 4000 3000 8000 7000 6000 5000 4000 3000 Effective Temperature (K) Effective Temperature (K)

1.0 MS Binaries < 4700K MS not Binaries < 4700K 0.9

0.8

0.7 Percentage Observed

12.0 12.5 13.0 13.5 14.0 14.5 15.0 15.5 16.0 Kepler Magnitude

Figure 11: (a): Effective temperature and radius plots of Kp < 13, 14, 15, 16 mag with lines defining different evolutionary states. Red separates the giant and subgiants, magenta the subgiant and main sequence stars, green the suspected main sequence binaries, blue the stars cooler and hotter than the Sun, and yellow the solar analogs. (b): Percentage observed at each Kp magnitude for main sequence and main sequence binary stars with Teff < 4700 K.

mag. Kepler’s selection of red giant stars was biased biased with respect to kinematics. against cool and low luminosity giants, with their observation dropping below 30%. We suspect this low • We find that the distribution of Gaia re-normalized observation rate is due to the KIC’s efficient sepa- unit weight error (RUWE) of the observed and non- ration between giants and dwarfs for temperatures observed samples differ for main sequence and sub- cooler than 5000 K. giant stars. When we account for differences in distance, Teff , and radius by matching our samples to • We observe a difference between the main sequence Kepler host sample we find this difference in RUWE observed and non-observed proper motion distribu- remains, and conclude that Kepler was significantly tions. However, when we match our samples to the biased against stellar multiplicity and preferentially Kepler host sample, accounting for differences in dis- selected non-binary stars for observation. To this tance, Teff , and radius, we find that Kepler was un- end, we also compare the percentage observed for sus- 11 Astronomy, University of Hawaii-Manoa funded through NSF grant 6104374. Linnea Wolniewicz would like to thank the Institute for Astronomy for their kind hospi- 100 tality during the course of this project. This research has made use of the VizieR catalogue access tool, CDS, Strasbourg, France. We obtain our Kepler data from MAST, and our Gaia DR2 data from VizieR. This 1 10 paper has also made use of the open-source TOPCAT ap-

Normalized Count plication, as well as python tools such as pandas, SciPy, MS Binaries < 4700K and Matplotlib. MS not Binaries < 4700K 10 2 0.5 1.0 1.5 2.0 2.5 3.0 RUWE

Figure 12: Histogram of the RUWE values for main sequence and main sequence binary stars with Teff < 4700 K.

pected main sequence binary stars and non-binary main sequence stars and ﬁnd that our conclusion holds. • For the faintest stars we ﬁnd Kepler was biased towards observing cool dwarf stars. The observed M dwarf fraction is 13.57% for 16 < Kp < 17 mag, and falls to 7.53% for 17 < Kp < 18 mag.

This paper has provided a comprehensive evaluation of the biases and successes of the Kepler selection function. The biases evaluated here will prove useful in determinations of exoplanets and their host stars and for conclusions of binarity and kinematics using the Kepler dataset. The Kepler selection function’s bias against stellar multiplicity suggests that future research will require analysis of a control sample of non-observed, non-host stars. In the future it would be useful to explore more tools to determine stellar multiplicity such as the Gaia RUWE and include them in future missions. Understanding the biases behind selection functions of datasets is essential, and it would be beneﬁ- cial to investigate the biases behind Gaia DR2 and other comprehensive observation missions. We look forward to future investigations of the Kepler data to continue the legacy left behind by the original exoplanet hunter.

Acknowledgements

We gratefully acknowledge the Gaia and Kepler missions and the people involved for their hard work that have made this paper possible. We thank Robert Jedicke, Aaron Do, Ben Shappee, Michael Bottom, Victoria Catlett, Jing- wen Zhang, Casey Brinkman, Ashley Chontos, Vanshree Bhalotia, Nicholas Saunders, Larissa Nofi, Jamie Tayar, and Lauren Weiss for their helpful discussions and feed- back. Linnea Wolniewicz acknowledges support from Research Experience for Undergraduate program at the Institute for 12 References A. H. Andrei, E. Anglada Varela, E. Antiche, T. Antoja, B. Ar- cay, T. L. Astraatmadja, N. Bach, S. G. Baker, L. Balaguer- E. Bányai, L. L. Kiss, T. R. Bedding, B. Bellamy, J. M. Benk˝o, Núñez,P. Balm, C. Barache, C. Barata, D. Barbato, F. Bar- A. Bódi,J. R. Callingham, D. Compton, I. Csányi, A. Derekas, blan, P. S. Barklem, D. Barrado, M. Barros, M. A. Barstow, J. Dorval, D. Huber, O. Shrier, A. E. Simon, D. Stello, G. M. S. BartholoméMuñoz,J. L. Bassilana, U. Becciani, M. 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