A TECHNIQUE TO DERIVE IMPROVED PROPER MOTIONS FOR KEPLER OBJECTS OF INTEREST^*

The Kepler telescope trails the Earth in a Sun-centered orbit. Details on the photometric performance and focal plane array can be found in Borucki et al. (2010) and Caldwell et al. (2010a). The following explorations restrict themselves to the so-called long-cadence data where each subsection containing a star of interest in the array is read out once every 30 minutes. These subsections of the Kepler field of view (FOV; hereafter, postage stamps) range from 4×5 pixels for fainter stars to larger than 8×8 pixels for brighter stars. Kepler pixels are a little less than 4'' on a side. Targets observed with long cadence generate approximately 4700 postage stamps per star per quarter.

We obtained our Kepler data from the Space Telescope Science Institute Multimission Archive (MAST). These data include both pipelined positions (the *_llc.fits files, where "*" is a global replacement marker) and postage stamp image data (the *_lpd-targ.fits files). The Kepler Archive Manual (Fraquelli & Thompson 2012) greatly assisted us with any access issues.

Positions used in this paper are generated from the Kepler postage stamp image data using a simple first-moment centering algorithm. We calculate

$$\begin{eqnarray} &&{MOM}\_{CENTR}\_X = \sum i*z/\sum i, \end{eqnarray} \tag{ 1 }$$

$$\begin{eqnarray} &&{MOM}\_{CENTR}\_Y = \sum j*z/\sum j \protect, \protect \end{eqnarray} \tag{ 2 }$$

where z = flux(i, j) are the flux count values for each Kepler pixel within the postage stamp. To generate positions from the optimal aperture, any z value not in the optimal aperture is set to zero. To reduce the computational load and to smooth out high-frequency positional variations, normal points (NPs) are formed by averaging the x and y positions for a specified time span. The tests and results reported herein are based on nine day NPs. We also only use data within an optimal aperture for each star defined by the Kepler team. We provide an example of an optimal aperture for a star with Kepler identification number KID = 7031732 in Figure 3. There are positional corrections tabulated in the MAST data products, e.g., POS_CORR1. These corrections report the size of the differential velocity aberration, pointing drift, and thermal effects applicable to the region of sky recorded in the file. These corrections are applied to our derived centroids. The final positions used in the test are corrected using the MAST position correction values, e.g., XY_CORR = MOM_CENTR_XY - POS_CORR_XY. We calculate the standard deviation of each NP along each axis for each star. The average standard deviation of these NP is typically on the order of 1 mas, demonstrating exceptional astrometric stability within each postage stamp. However, this small, formal random error is a significant underestimation of the total star-to-star astrometric quality, as we will see below in Section 4.1.

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We explored utilizing PSF fitting methods to extract positions. That approach did not resolve the issue of poor astrometric performance over multiple quarters (see Section 4.3 below). PSF fitting is computationally intensive and complicated given the Kepler FOV crowded stellar field and the significant PSF variations over that field (Bryson et al. 2010).

These tests use 95 stars located in Channel 21, Season 0, Quarter 10 that are identified as red giants in the Kepler Input Catalog (KIC). This initial filtering by star type potentially minimizes any effects of proper motion over the span of one or even two quarters. The NP generator, run on each star selected from test Channel 21, effectively reduces the number of discrete data sets per star from on the order of 4500 down to 9. We assign the 9 NPs for each of the 95 stars in the test to the 9 "plates." Each plate now contains 95 stars whose epochs are separated in time by approximately 9 days. Using the positions and positional errors generated by the NP code (now organized as 9 "plates" each containing 95 star positions and associated errors), we determine scale, rotation, and offset "plate constants" relative to an arbitrarily adopted constraint epoch (the so-called "master plate") for each observation set (the positions generated for each star within each "plate" at each of the nine NP epochs). The solved equations of condition are

$$\begin{equation} \xi = Ax + By + C, \end{equation} \tag{ 3 }$$

$$\begin{equation} \eta = -Bx + Ay + F \protect, \protect \end{equation} \tag{ 4 }$$

where $x$ and $y$ are the measured NP coordinates from the Kepler postage stamps. A and B are scale and rotation plate constants, C and F are offsets. For this test spanning only a single 90 day quarter, we ignore proper motions. When modeling these positions, in order to approach a near-unity χ²/DOF (DOF = degrees of freedom), the input positional errors, standard deviations from the NP averaging process, had to be increased by a factor of four. The final catalog of (ξ, η) positions have average 〈σ_ξ〉 = 0.31 millipixel and 〈σ_η〉 = 0.64 millipixel (1.23 and 2.54 mas), which is seemingly quite encouraging if one's goal is precision astrometry with Kepler.

However, the results of this modeling, shown in Figure 4, exhibit large systematic effects that are well correlated with time. The constraint epoch for this reduction is JD−24400000 = 15785.2, that is, the middle epoch of the nine plotted. We tentatively blame the typically larger y residuals (y is larger than x within each epoch) in Figure 4 with charge-transfer smearing along the CCD column readout direction (Kozhurina-Platais et al. 2008; Quintana et al. 2010). Our ultimate goal is to tease out stellar positional behavior similarly correlated with time: proper motion. We find stars 9 (=KID 6363534) and 27 (=KID 6606001) to exhibit some of the largest and most strikingly systematic residual patterns. Figure 5 provides an explanation for the behavior of star 27 (a close companion that perturbed the simple first-moment centering algorithm), and presents a puzzle regarding star 9. This star has no bright companions, yet it is a poorly behaved component of our astrometric reference frame. We suspect that this is due to the CCD channel-to-channel cross-talk discussed in Caldwell et al. (2010b). Four CCDs share common readout electronics. A bright star on one CCD can affect the measured charge in another.

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To determine whether or not there might be unmodeled—but possibly correctable—systematic effects at the 10 millipixel level, we plotted the reference frame x and y residuals against a number of parameters. These included the following: x, y position within the channel; radial distance from the channel center; reference star magnitude and color; and epoch of observation. We saw no obvious trends, other than an expected increase in positional uncertainty with reference star magnitude. Models with separate x and y scales (six parameters; in Equation (2) above, where −B and A are replaced by D and E) or color terms (eight parameters) provided no improvement in χ²/DOF.

Given that the pipelined positions available in the _llc.fits files are also first-moment centroids calculated from the optimal apertures, we developed the capability to generate these independently as a further test of the Kepler astrometry. The code used to generate the positions whose residuals are plotted in Figure 4 can also produce first-moment centroid data using the entire postage stamp (e.g., all the flux values in the middle panel in Figure 3). By comparing the positions extracted from the entire postage stamp against the optimal subset of the postage stamp for Channel 21, the average absolute value residual is reduced by 30% when using the optimal apertures. However, tests carried out on Channel 41 (Season 0, Quarter 10) near the Kepler FOV center result in a much less significant improvement, only 12%. This can be explained by considering the degradation in the PSF from the center to the edge of the entire Kepler FOV (Bryson et al. 2010; Tenenbaum & Jenkins 2010).

4.2.1. Same Stars—Multiple Seasons and Quarters

To test whether or not the peculiar fan pattern in the residuals against time seen in Figure 4 is channel-specific, we carried out similar tests for Channels 41–44, i.e., the four central CCDs in the Kepler focal plane. We sampled Quarters 3 through 14 using the same ∼75 stars in each channel. The results of this four parameter modeling are provided in Figure 6. Every quarter exhibits time-dependent residual behavior. The patterns often repeat within the same season. For example, in Season 1, the x residuals for star 5 (=KID 8949862) start out large and positive and move down to large and negative over ∼90 days. However, star 5 is relatively well behaved for any other season. A comparison of Figure 6 with Figure 5 in Barclay (2011)⁶ is convincing evidence that astrometry quality and primary mirror temperature changes are correlated. Stable temperatures at any level yield better astrometry (smaller residuals).

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4.2.2. An External Check of Single-channel, Single-season Data

We carried out a four parameter modeling of 127 randomly chosen stars (a mix of dwarfs and giants according to the KIC) in Channel 26 from Season 3, Quarter 5 and found residual patterns similar to that for Season 3, Quarter 5, Channel 44 shown in Figure 6. We then extracted a subset of 10 stars, 5 with relatively well-behaved x residuals (stars 4–62) and 5 with x residuals that are not constant with time (stars 67–106 in Figure 7). We list these in Table 2.

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Table 2. Companion Test Stars

#a	KID	KEPMAG	K	FluxFracb
4	5698236	15.637	14.243	0.886
5	5698325	12.264	10.747	0.920
10	5698466	13.113	11.561	0.921
34	5783576	14.135	12.656	0.874
62	5784222	15.475	13.885	0.881
67	5784291	13.148	11.074	0.936
77	5869153	15.596	13.537	0.706
96	5869586	15.466	13.672	0.886
103	5869826	15.768	13.473	0.774
106	5870047	11.747	6.328	0.962

Notes. ^aNumbering in Figure 7. ^bFraction of target flux in the Kepler project-defined optimum aperture.

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To explore the hypothesis that astrophysical effects (i.e., companions undetectable at the resolution of the Kepler detectors) cause the observed residual behavior, these 10 stars were observed with the Keck NIRC2-AO system (Wizinowich et al. 2004; Johansson et al. 2008) on the nights of 2013 August 19–21 UT with the NIRC2 instrument on Keck II. The targets themselves were used as natural guide stars and observations were performed in the K' filter, or the Br-γ filter if the star was too bright for the broader K' filter. The native seeing on the three nights (before adaptive optics (AO)) was approximately 06 at 2 μm. The NIRC2 instrument was in the narrow field mode with a pixel scale of approximately 0009942 pixel⁻¹ and a FOV of approximately 101 on a side. Each data set was collected with a three point dither pattern, avoiding the lower left quadrant of the NIRC2 array, with five images per dither position, each shifted 1'' from the previous. Each frame was dark subtracted and flat fielded. The sky frames were constructed for each target from the target frames themselves by median filtering and coadding the 15 dithered frames. Individual exposure times varied depending on the brightness of the target but were typically 10–30 s per frame. Data reduction was performed using a custom set of IDL routines.

To estimate companion detection limits as a function of distance from the selected targets, we utilized PSF planting and cross correlation (Tanner et al. 2010). The science target was extracted from the image, sky subtracted, and normalized. Then it was added at random positions around the image such that an equal number of plant positions occur in each radius bin of 01 around the science target. The flux within each planted PSF is scaled by a random value ranging from 10⁻³ to 10³ times the original number of counts in the star. The counts were determined through aperture photometry with a radius of 02 and a sky annulus of 02–03. Once added to the image, the threshold for detection was established by cross-correlating the planted star with the normalized PSF. A scaled and random PSF plant was considered to be detected if the cross-correlation value was above 0.5—a value determined with a "by eye" assessment. The location and flux of those PSFs that were detected were recorded over 5000 PSF plants. In each radius bin, the PSF with the smallest flux was used in the resulting plot of detected minimum magnitude difference (ΔK_s) versus distance from the science target.

We found no companion candidates in these images within a radius of 12. Figure 8 contains the contrast curves for star 4 (constant residuals) and star 67 (time-varying residuals), along with a fit to the normalized average contrast curves for stars 4–62 (〈Good〉) and 67–106 (〈Bad〉). To fit the average contrast curves, we employed an exponential function (y = K0 + K1*exp(− (x − x0)/K2)) with offset x0. The similarity of the average contrast curves removes small angular separation, fainter companions as the cause of the behavior displayed in Figure 7. Finally, the average Season 3 crowding, contamination, and flux fraction parameters (see Fraquelli & Thompson 2012 for parameter details) of the two groups differed little.

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4.2.3. Lessons Learned

With as rich a data set as produced by Kepler, our approach is to exercise extensive editing to establish the best astrometric reference frame: a reference frame with χ²/DOF ∼ 1 and Gaussian distribution of residuals. If we model only epochs 3–7 in Figure 4, as shown in Figure 9, we generate a final catalog of (ξ, η) positions with average 〈σ_ξ〉 = 0.22 millipixel and 〈σ_η〉 = 0.46 millipixel (0.87 and 1.83 mas). The residuals are Gaussian (Figure 10) and naturally larger than the average catalog errors because of the effective averaging to produce a catalog. Again, the significantly larger residuals along the y axis are likely due to CCD read-out issues (Kozhurina-Platais et al. 2008; Quintana et al. 2010).

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Again restricting our test to include only stars identified as red giants to minimize proper-motion effects, we now run a plate overlap model for the same set of stars appearing on two different Channels (21, 37) for Quarters 10 and 11, respectively. Given that the average absolute value UCAC4 proper motion for this suite of test stars is 7.5 mas yr⁻¹ (less than 2 millipixel yr⁻¹), the roughly 180 day span of these data should exhibit very little scatter due to unmodeled motions. A four parameter model (with the constraint plate chosen to be from Channel 21) yielded a final catalog with 〈σ_ξ〉 = 2.60 millipixel and 〈σ_η〉 = 6.6 millipixel (10.33 and 26.23 mas), which is significantly poorer astrometric performance than for a single channel and quarter (Section 4.1). A six parameter model with separate scales along x and y yielded only a 0.8% reduction in the large value of the reduced χ²/DOF.

The Kepler telescope has a Schmidt–Cassegrain design. An effective astrometric model for such a telescope, used successfully in the past on Palomar Schmidt photographic plates, is introduced in Abbot et al. (1975) and used in, e.g., Benedict et al. (1978). That model,

$$\begin{eqnarray} \xi &=& Ax^{\prime } + By^{\prime } + Cx^{\prime }y^{\prime } + Dx^{\prime 2} + Ey^{\prime 2} \nonumber\\ && +Fx^{\prime }(x^{\prime 2}+y^{\prime 2}) + G \protect, \protect \end{eqnarray} \tag{ 5 }$$

$$\begin{eqnarray} \eta &=& A^{\prime }x^{\prime } + B^{\prime }y^{\prime } + C^{\prime }x^{\prime }y^{\prime } + D^{\prime }x^{\prime 2} + E^{\prime }y^{\prime 2} \nonumber\\ && +F^{\prime }y^{\prime }(x^{\prime 2}+y^{\prime 2}) + G^{\prime } \protect, \protect \end{eqnarray} \tag{ 6 }$$

when applied to the Channels 21 and 37 data, provided a 9.1% reduction in reduced χ²/DOF, but a final catalog with errors almost exactly as found for the four and six parameter models. The run of residuals with time is shown in Figure 11. The residuals from this modeling are on average eight times larger than for Channel 21 alone. That the residuals remain large even with a Schmidt model demonstrates that the astrometric effects are not due to the Schmidt nature of the Kepler Telescope. The residuals as a function of position within the Channel 37 CCD show large variations on extremely small spatial scales (Figure 12). We have yet to identify the source of these high-frequency spatial defects, but cannot yet rule out the individual field flatteners atop each module containing four channels (Tenenbaum & Jenkins 2010). This inter-channel behavior effectively prohibits the measurement of precise parallaxes using only Kepler data.

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To fully populate the H_K, (J − K)₀ plane of an RPM diagram, we extract Channel 21 long-cadence image data (*lpd-targ*) for stars with 14 > KEPMAG > 11.6 (Quarters 6, 10, and 14, and all Season 0):

• 1.  
  Approximately 100 stars classified as red giants;
• 2.  
  Approximately 100 stars with T_eff > 6500 K;
• 3.  
  Approximately 100 stars with 6200 > T_eff > 5100 K;
• 4.  
  Approximately 30 stars with T_eff < 5000 K and Total_PM ⩾024yr⁻¹ with any KEPMAG value; and
• 5.  
  All KOIs found in Channel 21, e.g., Planetary_candidate and Exoplanet_host_star condition flag objects. These also have unrestricted KEPMAG.

We extract positions and generate nine day average NPs using only the Kepler team defined optimal apertures for each star. When including these data in our modeling with the ground-based catalogs, we re-origin the Kepler x, y coordinate values to (0,0) at the center of Channel 21.

To place our relative astrometry onto a right ascension (R.A.), declination (decl.) system, we extract J2000 positions and proper motions from the UCAC4 (Zacharias et al. 2013) and PPMXL (Roeser et al. 2010) catalogs. The catalog positions scale the Kepler astrometry and provide an approximately 12 yr baseline for proper-motion determination. Additionally, the catalog proper motions with associated errors are entered into the modeling as quasi-Bayesian priors. These values are not entered as hardwired quantities known to infinite precision. The χ² minimization is allowed to adjust the parameter values suggested by these data values within limits defined by the data input errors.

The input positional errors average 19 mas for the UCAC4 and 63 mas for the PPMXL. The average per-axis proper-motion errors are 2.6 mas yr⁻¹ for UCAC4 and 3.9 mas yr⁻¹ for PPMXL. A comparison of the two catalogs yields an average per-star absolute value proper-motion disagreement of 5.1 mas yr⁻¹, indicating that there is room for improvement. The R.A. and decl. positions from the two catalogs are used to calculate ξ, η standard coordinates transformed from radians to arcseconds (van de Kamp 1967) using the center of Channel 21 as the tangent point.

Using the central five epochs of positions from Quarters 6, 10, and 14 from Kepler Channel 21 (the editing of each quarter can be illustrated by comparing Figure 4 to Figure 9), spanning 2.14 yr, with standard coordinates from PPMXL and UCAC4, and proper-motion priors from the latter two catalogs, we determine "plate constants" relative to the UCAC4 catalog (this catalog having smaller formal positional errors). The constraint epoch is thus 2000.0. Our reference frame after pruning out astrometrically misbehaving objects contains 226 stars. For this model, we include only those stars with a restricted magnitude range, 14 > KEPMAG > 11.6 (samples 1–3 discussed in Section 5.1 above). The average magnitude for this magnitude-selected reference frame is 〈KEPMAG〉 = 13.3.

Again, we employ GaussFit (Jefferys et al. 1988) to minimize χ². The solved condition equations for the Channel 21 field are now

$$\begin{eqnarray} \xi &=& Ax^{\prime } + By^{\prime } + Cx^{\prime }y^{\prime } + Dx^{\prime 2} + Ey^{\prime 2} \nonumber\\ && +Fx^{\prime }(x^{\prime 2}+y^{\prime 2}) + G - \mu _x^{\prime } \Delta t, \end{eqnarray} \tag{ 7 }$$

$$\begin{eqnarray} \eta &=& A^{\prime }x^{\prime } + B^{\prime }y^{\prime } + C^{\prime }x^{\prime }y^{\prime } + D^{\prime }x^{\prime 2} + E^{\prime }y^{\prime 2} \nonumber\\ && +F^{\prime }y^{\prime }(x^{\prime 2}+y^{\prime 2}) + G^{\prime } - \mu _y^{\prime }\Delta t \protect, \protect \end{eqnarray} \tag{ 8 }$$

where $x^{\prime }=x -500$ and $y^{\prime }=y-500$ are the re-origined measured coordinates from Kepler and the standard coordinates from UCAC4 and PPMXL; μ_x and μ_y are proper motions; and Δt is the epoch difference from the mean epoch.

Based on the resulting astrometric parameters, we form a plate scale of

$$\begin{equation} S=(BA^{\prime }-AB^{\prime })^{1/2}, \end{equation} \tag{ 9 }$$

and for the 15 epochs (five for each of the three quarters) of Kepler observations find 〈S〉 = 397664 ± 0000009 pixel⁻¹, which is close to the nominal Kepler plate scale (van Cleve & Caldwell 2009) and an indication that the Kepler telescope plate scale as sampled in Channel 21 was quite constant. The scale factor of the PPMXL catalog relative to the UCAC4 catalog was 1.000012.

Using the UCAC4 catalog as the constraint plate to achieve a χ²/DOF ∼ 1, the Kepler NP data errors (NP standard deviations) had to be increased by a factor of 16. Histograms of the Kepler NP residuals were characterized by σ_x = 3.6 mas, σ_y = 6.4 mas. The average absolute value residual for Kepler was 4.8 mas, 24.3 mas for UCAC4, and 61.6 mas for PPMXL. The resulting 226 star reference frame "catalog" in ξ and η standard coordinates was determined with average positional errors of 〈σ_{ξ, η}〉 = 8.6 mas, a 55% improvement in relative position over the UCAC4 catalog. The average proper-motion error for the stars comprising the reference frame is 0.8 mas yr⁻¹.

Again, to determine whether or not there might be unmodeled—but possibly correctable—systematic effects, we plotted the reference frame x and y residuals against a number of parameters. These included the x, y position within the channel (Figure 13), the radial distance from the channel center, the reference star magnitude and color, and the epoch of observation. We saw no obvious trends other than an expected increase in positional uncertainty with reference star magnitude. Plots of x and y residual versus pixel phase also indicated no trends. We calculate the pixel phase, ϕ_x = x − int(x + 0.5), where int returns the integer part of the (for example) x coordinate.

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To ensure that the typically fainter (and hence less valuable contributors to the astrometric reference frame) stars do not affect our astrometric modeling of the Channel 21 CCD, an identical model in Section 5.3 is re-run, adding NP positions for the K and M stars (sample 4) and KOIs (sample 5) from Section 5.1, holding the Equations (7) and (8) coefficients A–G' to the values determined in Section 5.3. Note that we do solve for the positions and proper motions. This yields final catalog positions and proper motions for 301 stars representing all the categories listed in Section 5.1. The inclusion of fainter stars results in a "catalog" with ξ and η standard coordinates average relative positional errors 〈σ_{ξ, η}〉 = 11.1 mas, and an average proper-motion error for all stars of 1.0 mas yr⁻¹. The decrease in proper-motion precision relative to that found for the reference-frame-only stars is driven by the inclusion of the typically fainter K, M, and KOI stars with average 〈KEPMAG〉 = 15.1. As shown in Section 5.7 below, the centroids of fainter stars have lower signal to noise and, if included, would degrade our astrometric reference frame.

We present final KOI proper motions and errors in Table 3. These are, in a sense, absolute proper motions because of the use of prior information. To reiterate, we treated the UCAC4 and PPMXL proper-motion priors as observations with corresponding errors. The Table 3 KOI proper-motion parameters (and those for the entire set of reference stars modeled above) were adjusted by various amounts depending on the data input errors to arrive at a final result that minimized χ².

Our NP generation process (Section 3.2) also produces an average magnitude. In the case of nine day NPs, all of the measured flux values in each optimum aperture are averaged over the nine day interval and converted to a magnitude with an arbitrary zero point through m_f = 25.768–2.5*log₁₀(〈flux〉). Because we restricted this test to a single channel, no background correction is applied. The standard deviation for the 15 average m_f magnitudes is plotted against reference star ID number in Figure 14. Referencing Section 5.1, stars 1–99 are classified as red giants in the KIC (sample 1), stars 100–199 are hotter stars (sample 2), stars 201–299 are intermediate temperature stars (sample 3), stars 300–350 are selected to be more likely K and M dwarfs (sample 4), and stars 400–452 are the KOIs found in Channel 21 (sample 5). Note that all ID numbers are not present in the plot due to the editing process (Section 5.3) that produces the final astrometric reference frame.

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The highest maximum variability with a nine day cadence is found in our sample of suspected giants, which was not an unexpected result (Bastien et al. 2013). The KOIs seem to have photometric variability characteristics most similar to the K, M group. We note trends toward smaller variation with increasing number within each group (as defined in Section 5.1). This may be a function of photometric noise characteristics having positional dependence within Channel 21. The selection process populating each group and allocating a running number within each group always assigned the lowest numbers closer to (x, y)=(0, 1000) and the highest closer to (x, y) = (1000,0).

Figure 15 contains an average absolute value Kepler x residual for each reference star and KOI (numbering from Table 3) as a function of m_f. The residuals are calculated from the Section 5.3 modeling results. We choose the x residual as a potential diagnostic given that the y residuals are generally systematically larger (see Figures 10 and 13). The trend line is a quadratic fit to the x residuals for the reference stars only (samples 1–3 in Section 5.1). The planet-hosting KOIs over-plotted with large font show no extreme astrometric behavior, all lying within ±3σ of the relation. However, several KOI hosting unconfirmed planetary companions exhibit astrometric peculiarities. Both KOI 426 and 452 were inspected in 2MASS and Palomar Sky Survey images and showed no nearby stellar companions or image structure indicative of close stellar companions. In addition, KOI 452 is the most photometrically variable (Figure 14) host candidate star. Unfortunately, given the random eruption of astrometric peculiarity (see Figure 5), astrometry alone cannot serve as a reliable indicator of astrophysically interesting behavior.

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We now have the proper motions required to generate H_K(0) values for an RPM (Section 2). K-band magnitudes, J − K colors, and interstellar extinction values, A_V, E(B − V), were extracted from the online Kepler target database at MAST. We assumed (Schlegel et al. 1998) extinction-corrected by K(0) = K − A_K and (J − K)₀ = (J − K)−E(J − K), with A_K = A_V/9 and E(J − K)= 0.53*E(B − V). The left-hand RPM in Figure 16 shows H_K(0) and (J − K)₀ for all stars except the KOIs and exhibits a distribution of points that appears to have a main sequence and an ascending sub-giant branch. The average H_K(0) error is 0.43 mag, but is dependent on the value of μ_vec with an increased error toward bright values of H_K(0).

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The scatter in the left of Figure 16 can be due to several causes. These causes include proper-motion accuracy, random motions of stars, and systematic motions of stars. The H_K(0) average error bar in the figure indicates a ±0.4 mag of scatter due to measurement errors. The amount due to random stellar motions is unknown. Any particular star could have a large radial component to its random motion and be erroneously placed among the giant stars with typically lower than average proper motions. Those two effects increase the random scatter in an RPM. That systematic motions can corrupt an RPM is illustrated in Benedict et al.'s (2011) Figure 3. There, the RR Lyr variables all lie below and blueward of the broad main sequence. These Pop II giant stars have anomalously large proper motions, causing their erroneous placement in the RPM.

Comparing the HST-derived RPM (Figure 1, right) with the left-hand RPM in Figure 16 yields a systematic difference. What we identify as the locus of main-sequence stars from the right of Figure 1 appears to be substantially offset toward more negative H_K(0) values by ΔH_K(0) = −1.0. As a check, we produced an RPM by averaging the measured proper motions from the UCAC4 and PPMXL catalogs and obtain the same offset. Averaging the proper motions from the two catalogs, the typical H_K(0) error is 1.58 mag, which is a factor of three larger than when we include Kepler astrometry. There is also a far greater increase in error for brighter values of H_K(0).

To bring the main-sequence stars into coincidence with the HST main sequence would require decreasing the average μ_vec proper motions of the final Table 3 proper motions by ∼10 mas yr⁻¹. This proper-motion offset is likely not from systematic effects on the proper motion due to the space velocity of the Sun. The Kepler field is very near the solar apex at R.A. ≃ 287°, decl. ≃ +37° (Vityazev & Tsvetkov 2013). With most of the vector of solar motion in the radial direction, stars near the solar apex will exhibit very little systematic proper motion due to solar motion. From Vityazev & Tsvetkov (2013), the average transverse velocity of stars in the solar neighborhood toward the Galactic center is $\bar{U} = 9$ km s⁻¹and the average transverse velocity perpendicular to the Galactic plane is $\bar{W}=6$ km s⁻¹ for a systematic total velocity of V_t = 10.8 km s⁻¹. Our sample of F–G dwarfs has 〈K〉 = 11.85 mag with 〈M_K〉 ≃ 3 (Cox 2000), and hence an average distance of 500 pc. The expected proper motion can be estimated from

$$\begin{equation} \mu _{\rm vec} = \pi V_t/4.74 \protect. \protect \end{equation} \tag{ 10 }$$

This yields μ_vec = 4.6 mas yr⁻¹, which could explain some but not all of the positive H_K(0) offset in Figure 16.

However, as mentioned above in Section 2, a systematic effect of Galactic rotation on stellar velocities does exist. Our HST-derived RPM main sequence is composed of stars belonging to the Hyades and Pleiades star clusters. They have a Galactic longitude of ℓ ∼ 175°. The Kepler field has ℓ ∼ 74°. The velocity difference due to Galactic rotation is ∼30 km s⁻¹, translating to a proper-motion difference, Δμ_vec = 12.7 mas yr⁻¹, close to the correction needed above.

We seek luminosity class differentiation, which is a relative determination within an RPM. We ascribe the need for a correction to bring the Channel 21 RPM into closer agreement with the right side of Figure 1 to a mix of the random and systematic proper motions just identified. We add the ΔH_K(0) = −1.0 and replot this time including the KOI, similarly corrected for offset (right-hand side of Figure 16). The identification numbers in this plot, KIC numbers, and KOI numbers are collected in Table 4.