A Technique to Derive Improved Proper Motions for Kepler

Objects of Interest1

G. Fritz Benedict2,

Angelle M. Tanner3, Phillip A. Cargile4, and David R. Ciardi5

ABSTRACT

We outline an approach yielding proper motions with higher precision than

exists in present catalogs for a sample of stars in the Kepler field. To increase

proper motion precision we combine first moment centroids of Kepler pixel data

from a single Season with existing catalog positions and proper motions. We

use this astrometry to produce improved reduced proper motion diagrams, anal-

ogous to a Hertzsprung-Russell diagram, for stars identified as Kepler Objects of

Interest. The more precise the relative proper motions, the better the discrimi-

nation between stellar luminosity classes. With UCAC4 and PPMXL epoch 2000

positions (and proper motions from those catalogs as quasi-bayesian priors) as-

trometry for a single test Channel (21) and Season (0) spanning two years yields

proper motions with an average per-coordinate proper motion error of 1.0 mas

yr−1, over a factor of three better than existing catalogs. We apply a mapping

between a reduced proper motion diagram and an HR diagram, both constructed

using HST parallaxes and proper motions, to estimate Kepler Object of Inter-

est K-band absolute magnitudes. The techniques discussed apply to any future

small-field astrometry as well as the rest of the Kepler field.

Subject headings: astrometry — stars: distances — stars: proper motions —

stars: exoplanet hosts

2McDonald Observatory, University of Texas, Austin, TX 78712

3Department of Physics and Astronomy, Mississippi State University, Starkville, MS 39762

4Department of Physics and Astronomy, Vanderbilt University, Nashville, TN 37235

5NASA Exoplanet Science Institute, Caltech, Pasadena, CA 91125

1Based on observations made with the NASA Kepler Telescope

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1. Introduction

Astrometric precision, ε, in the absence of systematic error is proportional to N−1/2,

where N is the number of observations (van Altena 2013). Theoretically, averaging the

existing vast quantity of Kepler data might permit one to approach HST /FGS astrometric

precision, 1 millisecond of arc per observation. While Kepler was never designed to be an

astrometric instrument, and despite significant astrometric systematics and the challenge of

fat pixels (3.′′9757/pixel), one can reach a particular goal; higher precision proper motions

for Kepler Objects of Interest (KOIs) from Kepler data. Additionally, these techniques

may find utility when future astrometric users of, for example, the Large Synoptic Survey

Telescope (Ivezic et al. 2008) require the highest possible astrometric precision for targets of

interest contained on a single CCD in the focal plane. Finally, proper motion measures from

any Kepler extended mission might benefit from the application of these techniques.

In transit work it is useful to know the luminosity class of a host star when estimating the

size of the companion. This requires a distance, ideally provided by a measurement of par-

allax. With simple centroiding unaware of point spread function (PSF) structure the season

to season Kepler astrometry required for parallaxes presently yields positions with average

errors exceeding 100 milliseconds of arc (mas), insufficient for parallaxes (Section 4.3). Yet

distance is a desirable piece of information. Reduced proper motion (RPM) diagrams may

provide an alternative distance estimate. The concept is simple: proper motion becomes a

proxy for distance (Stromberg 1939; Gould & Morgan 2003; Gould 2004). Statistically, the

nearer any star is to us, the more likely it is to have a larger proper motion. The RPM dia-

gram consists of the proper motion converted to a magnitude-like parameter plotted against

color. The RPM diagram is thus analogous to a Hertzsprung-Russell (HR) diagram. While

some nearby stars might have low proper motions, typically giant and dwarf stars are sep-

arable. The more precise the proper motions, the better the discrimination between stellar

luminosity classes.

In the following sections we describe our approach yielding improved placement within

an RPM for any Kepler target of interest. We outline in Section 2 the utility of RPM dia-

grams, including a calibration to absolute magnitude derived from HST astrometry; discuss

in Section 3 Kepler data acquisition and reduction; present the results of a number of tests

providing insight into the many difficulties associated with Kepler astrometry (Section 4);

describe the modeling and the proper motion results for our selected Kepler test field, (Sec-

tion 5); compare our improved RPM with that previously derived from existing astrometric

catalogs (Section 5.8); and discuss the astrophysical ramifications of our estimated absolute

magnitudes for the over 60 KOI’s in our test field (Section 6). We summarize our findings

in Section 7.

– 3 –

2. A Calibrated RPM Diagram

In past HST astrometric investigations (e.g. Benedict et al. 2011, McArthur et al.

2010) the RPM was used to confirm the spectrophotometric stellar spectral types and lumi-

nosity classes of reference stars. Their estimated parallaxes are entered into the modeling

as observations with associated errors. To minimize absorption effects HST astrometric in-

vestigations use for the magnitude-like parameter HK(0) = K(0) + 5log(µ), and for a color,

(J−K)0, where K magnitudes and (J−K) colors (from 2MASS, Skrutskie et al. 2006) have

been corrected for interstellar extinction. For all of our RPMs we use the vector length

proper motion, µ = (µ2RA + µ2

DEC)1/2.

Compared to Hipparcos, HST has produced only a small number of parallaxes and

proper motions (Benedict et al. 1999, 2000a, 2000b, 2002, 2006, 2007, 2009, 2011; Harrison

et al. 2013; McArthur et al. 2001, 2010, 2011, 2013; Roelofs et al. 2007), but with higher

precision. Parallax and proper motion results for forty-two stars with HST proper motion

and parallax measures are collected in Table 1. Average parallax errors are 0.2 mas. Average

proper motion errors are 0.4 mas yr−1. Compared in Figure 1 are an HR diagram and an

RPM diagram constructed with HST parallax and proper motion results for the targets

listed in Table 1. Conspicuously absent from the RPM diagram are the RR Lyr results from

Benedict et al. (2011) with their large proper motions due to their Halo Pop II identification.

Lines plotted on the HR diagram show predicted loci for 10 Gyr age solar metallicity stars

and 3Gyr age metal-poor stars from Dartmouth Stellar Evolution models (Dotter et al.

2008). These ages and metallicities encompass the majority of what might be expected from

a random sampling of stars in our Galaxy.

Even though these targets are scattered all over the sky and range from PN central

stars to Cepheid variables, the similarity between the HR and RPM diagrams is striking. In

Figure 2 we plot the extinction corrected K-band absolute magnitude derived from HST par-

allaxes against the magnitude-like parameter HK(0). The resulting scatter, 0.7 mag RMS,

suggests that precise proper motion is a good enough proxy for distance to allow the assign-

ment of luminosity class.

Note that while this calibration is produced with proper motions sampling much of the

celestial sphere, the HR diagram and RPM main sequences are defined almost exclusively

by stars belonging to the Hyades and Pleiades clusters. This may become an issue when we

attempt to apply the calibration to a small piece of the sky in a different location. Both

the calibration sources and a random Kepler field have systematic proper motions due to

galactic rotation (c.f. van Leeuwen 2007, section 6.1.5), possibly requiring some correction.

– 4 –

3. Kepler Observations and Data Reduction

The primary mission of the Kepler spacecraft is high-precision photometry with which

to discover transiting planets. Kepler rotates about the boresight once each 90 days to

maximize solar panel illumination. Each such pointing is identified by a Season number; 0–

3. Each 90 day span is identified by a Quarter number; 1–17. The CCDs in the Kepler focal

plane are identified by Channel number; 1-84. Our goal is to produce an astrometric reference

frame across a given Kepler channel containing KOIs of interest, the end product being KOI

proper motions with which to populate an RPM.

3.1. Star Data

The Kepler telescope trails the Earth in 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 FOV (hereafter, postage stamps) range from 4x5

pixels for fainter stars to larger than 8x8 pixels for brighter stars. Kepler pixels are a little

less than 4 arc sec on a side. Targets observed with long-cadence generate approximately

4700 postage stamps per star per Quarter.

We obtain 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.

3.2. Positions from Kepler image data

Positions used in this paper are generated from the Kepler postage stamp image data,

using a simple first-moment centering algorithm. We calculate

MOM CENTR X =∑

i ∗ z/∑

i (1)

MOM CENTR Y =∑

j ∗ z/∑

j (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

– 5 –

set to zero. To reduce the computational load and to smooth out high frequency positional

variations, normal points (NP) 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 normal points. We

also use only 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 report the size of the differential velocity aberration

(DVA), pointing drift, and thermal effects applicable to the region of sky recorded in the file.

These corrections are applied to our derived centroids. Final positions used in the test are

corrected using the MAST position correction values, e.g., XY CORR = MOM CENTR XY

- POS CORR XY. The standard deviation of each normal point along each axis for each

star is calculated. The average standard deviation of these NP is typically on order one 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 shall see below in Section 4.1.

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).

4. Kepler Astrometric Tests

These tests highlight several systematic errors and motivate our simple strategy for

dealing with them. For all astrometric modeling we employ GaussFit (Jefferys et al. 1988)

to minimize χ2.

4.1. Single Channel, Single Season, Single Quarter

These tests use 95 stars located in Channel 21, Season 0, Quarter 10 and 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

normal point generator, run on each star selected from test Channel 21, effectively reduces

the number of discrete data sets per star from on order 4500 down to nine. We assign the

nine normal points for each of the 95 stars in the test to the nine ‘plates’. Each plate now

contains 95 stars whose epochs are now separated in time by approximately nine days. With

– 6 –

the positions and positional errors generated by the normal point code (now organized as nine

‘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 normal point epochs). The solved equations of condition are:

ξ = Ax+By + C (3)

η = −Bx+ Ay + F (4)

where x and y are the measured normal point 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 one 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 normal point averaging process, had to be increased by

a factor of four. The final catalog of (ξ, η) positions have average < σξ >= 0.31 millipixel

and < ση >= 0.64 millipixel (1.23 and 2.54 mas), 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,

well-correlated with time. The constraint epoch for this reduction is JD-24400000=15785.2,

the middle epoch of the nine plotted. We tentatively blame the typically larger y residuals

(y 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. Among the largest and most strikingly systematic residual patterns we find stars 9

(=KID 6363534) and 27 (=KID 6606001). 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. It has no bright companions, yet is a poorly behaved

component of our astrometric reference frame. We suspect 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 measured charge in another.

To determine if 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 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 (6 parameters; in Equation 2 above, where

-B and A are replaced by D and E) or color terms (8 parameters) provided no improvement

– 7 –

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 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).

Comparing 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 field of view center result in a much less significant improvement, only

12%. This can be explained by considering the degradation in the point spread function

(PSF) from the center to the edge of the entire Kepler field of view (Bryson et al. 2010;

Tenenbaum & Jenkins 2010).

4.2. Other Single Channel Tests

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, 42, 43, and 44,

the central four CCDs in the Kepler focal plane. We sampled Quarter 3 through Quarter

14, using the same ∼75 stars in each channel. The results of this 4 parameter modeling are

given in Figure 6. Every Quarter exhibits time-depends 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 negative over ∼ 90

days. Yet star 5 is relatively well-behaved for any other Season. A comparison of Figure 6

with figure 5 in Barclay (2011)2 is convincing evidence that astrometry quality and primary

mirror temperature changes are correlated. Stable temperatures at any level yield better

astrometry (smaller residuals).

2http://archive.stsci.edu/kepler/release notes/release notes12/

DataRelease 12 2011113017.pdf

– 8 –

4.2.2. An External Check of Single Channel, Single Season Data

We carried out a four parameter modeling of 127 stars randomly chosen (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 ten stars, five with relatively well-behaved x residuals (stars 4–62), five

with x residuals not constant with time (stars 67–106 in Figure 7). We list these in Table 2.

To explore the hypothesis that astrophysical effects (i.e., companions undetectable at the

resolution of the Kepler detectors) cause the observed residual behavior, these ten stars were

observed with the Keck NIRC2-AO system (Wizinowich et al. 2004; Johansson et al. 2008)

on the nights of 19-21 August 2013 UT with the NIRC2 instrument on Keck II. The targets

themselves were used as natural guide stars and observations were made 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 AO) was approximately 0.′′6 at 2µm. The NIRC2 instrument was in the

narrow field mode with a pixel scale of approximately 0.′′009942 pixel−1 and a field of view

of approximately 10.′′1 on a side. Each dataset was collected with a 3-point dither pattern,

avoiding the lower left quadrant of the NIRC2 array, with 5 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 typically were 10 - 30 seconds per frame. Data reduction

was performed with 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 0.′′1 around the science target. The flux within each planted PSF is

scaled by a random value which ranges 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 0.′′2

and a sky annulus of 0.′′2-0.′′3. 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 detected if the cross correlation value was above 0.5 - a

value determined by a ”by eye” assessment. The location and flux of those PSFs which 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 (∆Ks) versus

distance from the science target.

We found no companion candidates in these images within a radius of 1.′′2. Figure 8

– 9 –

contains the contrast curves for star 4 (constant residuals) and star 67 (time-varying resid-

uals), along with a fit to 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 Figure 7 be-

havior. Finally, the average Season 3 crowding, contamination, and flux fraction parameters

(see Fraquelli and Thompson 2012 for parameter details) of the two groups differed little.

4.2.3. Lessons Learned

With as rich a set of data 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 < σξ >= 0.22 millipixel

and < ση >= 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).

4.3. Two Channels, Two Contiguous Seasons

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 < σξ >= 2.60 millipixel and < ση >= 6.6

millipixel (10.33 and 26.23 mas), 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 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,

– 10 –

is introduced in Abbot et al. (1975) and used in e.g. Benedict et al. (1978). That model,

ξ = Ax′ +By′ + Cx′y′ +Dx′2 + Ey′2

+Fx′(x′2 + y′2) +G(5)

η = A′x′ +B′y′ + C ′x′y′ +D′x′2 + E ′y′2

+F ′y′(x′2 + y′2) +G′(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 residual 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.

5. Astrometry of a Kepler Test Field

Our ultimate goal is to produce an RPM diagram including KOIs, permitting an esti-

mate of their luminosity class. This may be feasible by restricting astrometry to a single

Channel and Season. Essentially we may be able to ignore the deficiencies demonstrated in

Figures 11 and 12 because each star in any given Season will be observed by the same pixels,

and the starlight is passing through the exact same region of the field flattener. The seven-

teen available Quarters provide 3–4 same-Season observation sets for any Kepler Channel.

Examination of Figure 6 supports our selection of Season 0 as one of the more stable. Tests

similar to those carried out in Section 4.1 yielded very poor results for Quarter 2, hence it

is unused here.

5.1. Populating an RPM Diagram

To fully populate the HK , (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, all Season 0:

– 11 –

1. ∼ 100 stars classified as red giants,

2. ∼ 100 stars with Teff > 6500K ,

3. ∼ 100 stars with 6200 > Teff > 5100K ,

4. ∼ 30 stars with Teff <5000K and Total PM ≥ 0.′′24yr−1 with any KEPMAG value,

and

5. all KOIs found in Channel 21; e.g., Planetary candidate and Exoplanet host star con-

dition flag objects. These, too, have unrestricted KEPMAG.

We extract positions and generate 9-day average normal points using only the Kepler team

defined optimal apertures for each star. When including these data in our modeling with

the ground-based catalogs, the Kepler x,y values are re-origined to the center of Channel

21.

5.2. Reference Star Priors

To place our relative astrometry onto a Right Ascension, Declination 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

approximately a 12 year 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 room for improvement. The RA and Dec

positions from the two catalogs are used to calculate ξ, η standard coordinates transformed

from radians to seconds of arc (van de Kamp 1967), using the center of Channel 21 as the

tangent point.

– 12 –

5.3. The Proper Motion Astrometric Model

With the central five epochs of positions from Quarter 6, Quarter 10, and Quarter 14

from Kepler Channel 21 (the editing of each Quarter illustrated by comparing Figure 4 to

Figure 9), spanning 2.14 years, standard coordinates from PPMXL, and UCAC4, and proper

motion priors from the latter two catalogs we determine “plate constants” relative to the

UCAC4 catalog (it 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 equations

of condition for the Channel 21 field are now:

ξ = Ax′ +By′ + Cx′y′ +Dx′2 + Ey′2

+Fx′(x′2 + y′2) +G− µ′x∆t(7)

η = A′x′ +B′y′ + C ′x′y′ +D′x′2 + E ′y′2

+F ′y′(x′2 + y′2) +G′ − µ′y∆t(8)

where x ′ = x − 500 and y ′ = y − 500 are the re-origined measured coordinates from

Kepler and the standard coordinates from UCAC4 and PPMXL; µx and µy are proper

motions; and ∆t is the epoch difference from the mean epoch.

From the resulting astrometric parameters we form a plate scale

S = (BA′ − AB′)1/2 (9)

and find for the 15 epochs (five for each of the three Quarters) of Kepler observations 〈S〉=3.′′97664±0.′′000009 pixel−1, 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.

5.4. Assessing the Reference Frame

Using the UCAC4 catalog as the constraint plate, to achieve a χ2/DOF∼1 theKepler nor-

mal point data errors (normal point standard deviations) had to be increased by a factor of

– 13 –

sixteen. Histograms of the Kepler normal point residuals were characterized by σx = 3.6

mas, σy = 6.4 mas. The average absolute value residual for Kepler was 4.8 mas; for UCAC4,

24.3 mas; for PPMXL, 61.6 mas. The resulting 226 star reference frame ‘catalog’ in ξ and η

standard coordinates was determined with average positional errors 〈σξ,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 if there might be unmodeled - but possibly correctable - systematic

effects, we plotted reference frame x and y residuals against a number of parameters. These

included x, y position within the channel (Figure 13); 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.

Plots of x and y residual versus pixel phase also indicated no trends. We calculate pixel

phase; φx = x − int(x + 0.5), where int returns the integer part of the (for example) x

coordinate.

5.5. Applying the Reference Frame

To insure that the typically fainter (hence less valuable contributors to the astrometric

reference frame) stars do not affect our astrometric modeling of the Channel 21 CCD, the

identical model in Section 5.3 is re-run adding normal point positions for the K and M stars

(sample 4) and KOI’s (sample 5) from Section 5.1, holding the equations 7–8 coefficients A–

G’ to the values determined in Section 5.3. Note that we do solve for 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 〈σξ,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.

– 14 –

5.6. Reference Star Photometric Stability

Our normal point generation process (Section 3.2) also produces an average magnitude.

In the case of nine-day normal points, all 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 mf = 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 mf

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 KOI’s

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) producing the final astrometric reference frame.

Highest maximum variability with a nine-day cadence is found in our sample of suspected

giants, 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 the trends to 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 nearer (x,y)=(0,1000), the highest nearer

(x,y)=(1000,0).

5.7. Astrometry as a Diagnostic

Figure 15 contains an average absolute value Kepler x residual for each reference star

and KOI (numbering from Table 3) as a function of mf . 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 in general are systematically larger (c.f. 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 KOI over-plotted with large font show no extreme astrometric behavior, all

lying within ±3-σ of the relation. However several KOI hosting unconfirmed planetary com-

panions 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 struc-

ture 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 as-

trometric peculiarity (c.f. Figure 5), astrometry alone cannot serve as a reliable indicator of

astrophysically interesting behavior.

– 15 –

5.8. RPM Diagrams; Pre- and Post-Kepler

We now have the proper motions required to generate HK(0) values for an RPM (Sec-

tion 2). K-band magnitudes, J−K colors and interstellar extinction values, AV , E(B − V )

were extracted from the online Kepler target data base at the MAST. We assumed (Schlegel

et al. 1998)extinction-corrected by K(0) = K-AK , (J −K)0 = (J −K)-E(J −K), with AK

= AV /9 and E(J −K)= 0.53*E(B−V ). The left-hand RPM in Figure 16 shows HK(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 HK(0) error is 0.43

mag, but is dependent on the value of µvec with an increased error towards bright values of

HK(0).

The scatter in Figure 16, left, can be due to several causes. These include proper motion

accuracy, random motions of stars, and systematic motions of stars. The HK(0) average error

bar in the figure indicates ±0.4 magnitude of scatter due to measurement error. 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 amongst 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.

(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 Fig-

ure 16 yields a systematic difference. What we identify as the locus of main sequence stars

from Figure 1, right, appears to be substantially offset towards more negative HK(0) values

by ∆HK(0) = -1.0. As a check we produced an RPM by averaging measured proper motions

from the UCAC4 and PPMXL catalogs and obtain the same offset. Averaging the proper

motions from the two catalogs, the typical HK(0) error is 1.58 mag, a factor of three larger

than when we include Kepler astrometry. There is also a far greater increase in error for

brighter values of HK(0).

To bring the main sequence stars into coincidence with the HST main sequence would

require decreasing the average µvec 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 proper

motion due to the space velocity of the Sun. The Kepler field is very near the Solar Apex

at RA ' 287◦, DEC ' +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 towards the Galactic center is U = 9 km s−1 and

– 16 –

the average transverse velocity perpendicular to the Galactic plane W = 6 km s−1 for a

systematic total velocity Vt = 10.8 km s−1. Our sample of F–G dwarfs has 〈K〉 = 11.85

mag with 〈MK〉 ' 3 (Cox 2000), hence an average distance of 500pc. The expected proper

motion can be estimated from

µvec = πVt/4.74 (10)

This yields µvec = 4.6 mas yr−1, which could explain some but not all the positive HK(0)

offset in Figure 16.

However as mentioned in Section 2 above, 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, ` ∼ 175◦.

The Kepler field has ` ∼ 74◦. The velocity difference due to Galactic rotation is ∼ 30km

s−1, translating to a proper motion difference, ∆µvec = 12.7 mas yr−1, close to the correction

needed above.

We seek luminosity class differentiation, a relative determination within an RPM. We

ascribe the need for a correction to bring the Channel 21 RPM into closer agreement with

Figure 1, right, to a mix of the random and systematic proper motions just identified. We

add the ∆HK(0) = -1.0 and replot, this time including the KOI, similarly corrected for offset

(right-hand side of Figure 16). Identification number in this plot, KIC numbers, and KOI

numbers are collected in Table 4.

6. Estimated Absolute Magnitudes for Channel 21 KOI

Table 4 contains the HK(0) values (corrected by ∆HK(0) = -1.0) derived from the Table 3

KOI proper motions and K(0) magnitudes. The listed K-band absolute magnitudes, MK(0)

are obtained using the Figure 2 calibration; MK(0) = 1.51±0.12+(0.90±0.02)×HK(0). An

MK(0), (J − K)0 HR diagram is shown in Figure 17. An HR diagram constructed using

HK(0) from the UCAC4, PPMXL average proper motions has a distribution similar to

Figure 17, but with significantly increased scatter. Nine stars in Channel 21 (Season 0)

host confirmed planetary systems. Our number 409 is Kepler-100, hosting three confirmed

planets (Marcy et al. 2014), and our number 441 is Kepler-28, hosting two confirmed planets

(Steffen et al. 2012). Recently Rowe et al. (2014) have statistically confirmed a number

of multi planet systems. All exoplanet host stars and associated companions found in the

Channel 21 field are listed in Table 5. Most of their positions in the Figure 17 HR diagram

lie on or close to a solar-metallicity 10Gyr old main sequence (Dartmouth Stellar Evolution

model, Dotter et al. 2008). Our number 435 = KOI-1359 has a photometrically-determined

(hence, low-precision) lowest metallicity in Table 5, consistent with sub-dwarf location on an

– 17 –

HR diagram, as indicated in Figure 17. The other eight hosts of confirmed planetary systems

have locations in the HR diagram consistent with a main sequence dwarf classification.

Inspection of Figure 17 suggests that a significant number of our astrometric reference

stars (and some of the KOI) appear to be sub-giants. As with any apparent magnitude

limited survey, the stars observed with Kepler will have a Malmquist-like bias, i.e., the

survey will be biased towards the inclusion of the most luminous objects in the field as a

result of the greater volume being surveyed for these intrinsically brighter objects (Malmquist

1922). Therefore, within the Kepler field there is a significant bias towards observing stars

that are more massive and/or more evolved.

The Kepler target selection attempted to mitigate this bias by selecting stars identified

as main-sequence solar-type dwarfs based on their KIC values (Batalha et al. 2010). However,

significant uncertainty in KIC surface gravities make this selection process suspect. Brown

et al. (2011) concluded that uncertainties in KIC log(g) are ∼0.4 dex, and are unreliable

for distinguishing giants/main-sequence stars for Teff & 5400 K. Consequently, a significant

fraction of Kepler target stars are expected to be F and G spectral type subgiant stars

(Farmer et al. 2013; Gaidos & Mann 2013).

Finally, to argue for the added value of carrying out this astrometry, Figure 18 plots

a theoretical HR diagram (log g vs. log T) for the astrometric reference stars (grey dots)

and the Table 4 KOI. The log g and T values are from the Kepler Target Search results

tabulated at the STScI MAST3. There are very few stars in the expected locus of F–G sub

giants. Cargile et al. (in preparation) have re-measured Teff and log g for 850 KOIs using

Keck HiRes archived material and find a sub-giant fraction amongst Kepler targets similar

to that shown in Figure 17. Again, we include on this plot a range of metallicities and ages

from the Dartmouth evolution models.

7. Summary

1. Astrometry carried out on Kepler data yields significant systematics in position. These

systematics correlate with time.

2. Astrometric performance correlates with Kepler telescope temperature variations.

Larger variations result in poorer astrometry.

3. Astrometric modeling with a previously successful Schmidt model of more than one

3http://archive.stsci.edu/kepler/kepler fov/search.php

– 18 –

Kepler Season fails to produce astrometric precision allowing for the measurement of

stellar parallax.

4. Combining Kepler astrometry for a single Season and Channel and three Quarters

with existing catalog positions and proper motions, extends the time baseline to over

12 years. This provides a mapping of the lower-spatial frequency distortions over a

Channel, and improves the precision of measured proper motions to 1.0 mas yr−1, over

a factor of three better than UCAC4 and PPMXL.

5. Applying that astrometric model, Kepler measurements yield absolute proper motions

for a number of KOIs with an average proper motion vector error σµ = 2.3 mas yr−1,

or σµ/µ = 19.4%. In contrast, averaging the UCAC4 and PPMXL catalog proper

motions provide σµ = 5.0 mas yr−1, or σµ/µ = 43.0%.

6. An RPM diagram constructed from the proper motions determined by our method,

when compared to one based on HST proper motions, shows a systematic offset. Much

of the offset can be attributed to the effect of Galactic rotation on proper motions.

7. The corrected RPM parameter, HK(0), transformed to MK(0) through an HK(0) -

MK(0) relation derived from HST proper motions and parallaxes, yields MK(0) for fifty

KOIs, including nine stars with confirmed planetary companions, 8 now confirmed as

dwarfs, one a possible sub-dwarf. Six KOIs are identified as giants or sub-giants.

The next significant improvement in KOI proper motions will come from the space-

based, all-sky astrometry mission Gaia (Lindegren et al. 2008) with ∼ 20 microsecond of arc

precision proper motions and parallaxes for the brighter KOI’s. With parallax there will be

no need for RPM diagrams. Final Gaia results are expected early in the next decade.

This paper includes data collected by the Kepler mission. Funding for Kepler is pro-

vided by the NASA Science Mission directorate. All of the Kepler data presented in this

paper were obtained from the Mikulski Archive for Space Telescopes (MAST) at the Space

Telescope Science Institute. STScI is operated by the Association of Universities for Research

in Astronomy, Inc., under NASA contract NAS5-26555. Support for MAST for non-HST

data is provided by the NASA Office of Space Science via grant NNX13AC07G and by

other grants and contracts. Direct support for this work was provided to GFB by NASA

through grant NNX13AC22G. Direct support for this work was provided to AMT by NASA

through grant NNX12AF76G. PAC acknowledges NSF Astronomy and Astrophysics grant

AST-1109612. This publication makes use of data products from the Two Micron All Sky

– 19 –

Survey, which is a joint project of the University of Massachusetts and the Infrared Process-

ing and Analysis Center/California Institute of Technology, funded by NASA and the NSF.

This research has made use of the SIMBAD and Vizier databases and Aladin, operated at

CDS, Strasbourg, France; the NASA/IPAC Extragalactic Database (NED) which is oper-

ated by JPL, California Institute of Technology, under contract with the NASA; and NASA’s

Astrophysics Data System Abstract Service. This research has made use of the NASA Exo-

planet Archive, which is operated by the California Institute of Technology, under contract

with the National Aeronautics and Space Administration under the Exoplanet Exploration

Program. Some of the data presented herein were obtained at the W.M. Keck Observatory,

which is operated as a scientific partnership among the California Institute of Technology,

the University of California and the National Aeronautics and Space Administration. The

Observatory was made possible by the generous financial support of the W.M. Keck Foun-

dation. The authors wish to recognize and acknowledge the very significant cultural role and

reverence that the summit of Mauna Kea has always had within the indigenous Hawaiian

community. We are most fortunate to have the opportunity to conduct observations from

this mountain. GFB thanks Bill Jefferys, Tom Harrison, and Barbara McArthur who over

many years contributed to the techniques reported in this paper. GFB and AMT thank Dave

Monet for several stimulating discussions that should have warned us off from this project,

but didn’t. GFB thanks Debra Winegarten for her able assistance, allowing progress on this

project. We thank an anonymous referee for a thorough, careful, and useful review which

materially improved the final paper.

– 20 –

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Table 1. HST MK(0) and HK(0)

# ID m-M MK(0) SpT µTa K0 (J-K)0 HK(0) Refb

1 HD 213307 7.19 -0.86 B7IV 21.82±0.42 6.32 -0.12 -1.98±0.05 B02

2 υ AND 0.66 2.20 F8 IV-V 419.26 0.14 2.86 0.32 0.97 0.03 M10

3 HD 136118 3.59 2.00 F9V 126.31 1.20 5.60 0.34 1.11 0.03 Mr10

4 HD 33636 2.24 3.32 G0V 220.90 0.40 5.56 0.34 2.28 0.03 Ba07

5 HD 38529 3.00 1.22 G4IV 162.31 0.11 4.22 0.68 0.27 0.03 B10

6 vA 472 3.32 3.69 G5 V 104.69 0.21 7.00 0.50 2.10 0.03 M11

7 55 Cnc 0.49 3.49 G8V 539.24 1.18 3.98 0.70 2.64 0.03 SIMBAD

8 δ Cep 7.19 -4.91 F5Iab: 17.40 0.70 2.28 0.52 -6.51 0.09 B07

9 vA 645 3.79 4.11 K0V 101.81 0.76 7.90 0.77 2.93 0.03 M11

10 HD 128311 1.09 3.99 K1.5V 323.57 0.35 5.08 0.53 2.63 0.03 M13

11 γ Cep 0.67 0.37 K1IV 189.20 0.50 1.04 0.62 -2.58 0.03 B13

12 vA 627 3.31 3.86 K2 V 110.28 0.05 7.17 0.56 2.38 0.03 M11

13 ε Eri -2.47 4.24 K2V 976.54 0.10 1.78 0.45 1.72 0.03 B06

14 vA 310 3.43 4.09 K5 V 114.44 0.27 7.52 0.63 2.82 0.03 M11

15 vA 548 3.39 4.13 K5 V 105.74 0.01 7.52 0.71 2.64 0.03 M11

16 vA 622 3.09 5.13 K7V 107.28 0.05 8.22 0.84 3.38 0.03 M11

17 vA 383 3.35 5.01 M1V 102.60 0.32 8.36 0.91 3.42 0.03 M11

18 Feige 24 4.17 6.38 M2V/WD 71.10 0.60 10.55 0.69 4.81 0.03 B00a

19 GJ 791.2 -0.26 7.57 M4.5V 678.80 0.40 7.31 0.92 6.47 0.03 B00b

20 Barnard -3.68 8.21 M4Ve 10370.00 0.30 4.52 0.72 9.60 0.03 B99

21 Proxima -4.43 8.81 M5Ve 3851.70 0.10 4.38 0.97 7.31 0.03 B99

22 TV Col 7.84 4.84 WD 27.72 0.22 12.68 0.49 4.89 0.03 M01

23 DeHt5 7.69 7.84 WD 21.93 0.12 15.53 -0.07 7.24 0.03 B09

24 N7293 6.67 7.87 WD 38.99 0.24 14.54 -0.23 7.49 0.03 B09

25 N6853 8.04 2.54 WD 8.70 0.11 10.58 1.13 0.27 0.04 B09

26 A31 8.97 6.69 WD 10.49 0.13 15.66 0.25 5.77 0.04 B09

27 V603 Aql 7.20 4.12 CNe 15.71 0.19 11.32 0.31 2.30 0.04 H13

28 DQ Her 8.06 5.00 CNe 13.47 0.32 13.06 0.46 3.70 0.06 H13

29 RR Pic 8.71 3.54 CNe 5.21 0.36 12.25 0.18 0.83 0.15 H13

30 HP Lib 6.47 7.35 WD 33.59 1.54 13.82 -0.12 6.45 0.10 R07

31 CR Boo 7.64 8.59 WD 38.80 1.78 16.23 -1.52 9.17 0.10 R07

– 24 –

Table 1—Continued

# ID m-M MK(0) SpT µTa K0 (J-K)0 HK(0) Refb

32 V803 Cen 7.70 6.12 WD 9.94 2.98 13.82 -0.10 3.81 0.65 R07

33 ` Car 8.56 -7.55 G3Ib 15.20 0.50 0.99 0.55 -8.10 0.08 B07

34 ζ Gem 7.81 -5.73 G0Ibv 6.20 0.50 2.13 0.23 -8.91 0.18 B07

35 β Dor 7.50 -5.62 F6Ia 12.70 0.80 2.06 0.48 -7.42 0.14 B07

36 FF Aql 7.79 -4.39 F6Ib 7.90 0.80 3.45 0.40 -7.06 0.22 B07

37 RT Aur 8.15 -4.25 F8Ibv 15.00 0.40 3.90 0.28 -5.22 0.06 B07

38 κ Pav 6.29 -3.52 F5Ib-II: 18.10 0.10 2.71 0.62 -6.00 0.03 B11

39 VY Pyx 6.00 -0.26 F4III 31.80 0.20 5.63 0.33 -1.86 0.03 B11

40 P3179 5.65 3.02 G0V: 50.36 0.40 8.67 0.35 2.18 0.03 S05

41 P3063 5.65 4.68 K6V: 45.30 0.50 10.33 0.82 3.61 0.04 S05

42 P3030 5.65 4.97 K9V: 43.20 0.50 10.62 0.83 3.79 0.04 S05

aµT = (µ2RA + µ2

DEC)1/2 in mas yr−1

bB99=Benedict et al. (1999), B00a=Benedict et al. (2000a), B00b=Benedict et al. (2000b),

B02=Benedict et al. (2002), B06=Benedict et al. (2006), B07=Benedict et al. (2007), B09=Benedict

et al. (2009), B11=Benedict et al. (2011), Ba07=Bean et al. (2007), H13=Harrison et al.

(2013), Mr10=Martioli et al. (2010), M01=McArthur et al. (2001), M11=McArthur et al. (2011),

R07=Roelofs et al. (2007), S05=Soderblom et al. (2005)

– 25 –

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

aNumbering in Figure 7

bFraction of target flux in the Kepler project-defined optimum aper-

ture.

– 26 –

Table 3. Channel 21 KOI Proper Motions (µ)

ID KID KOI mf µRAa µDEC

b µT

401 6362874 1128 13.51 -0.0063±0.0008 -0.0307±0.0008 0.0314±0.0011

402 6364215 2404 15.66 0.0005 0.0015 -0.0056 0.0019 0.0056 0.0024

403 6364582 3456 12.99 0.0022 0.0011 0.0029 0.0005 0.0036 0.0012

404 6441738 1246 14.90 -0.0038 0.0013 0.0158 0.0012 0.0163 0.0017

405 6442340 664 13.48 0.0090 0.0009 -0.0103 0.0011 0.0137 0.0015

406 6442377 176 13.43 0.0079 0.0009 0.0096 0.0012 0.0124 0.0015

407 6520519 4749 15.61 0.0026 0.0017 -0.0061 0.0019 0.0067 0.0025

408 6520753 4504 11.20 0.0046 0.0087 -0.0535 0.0061 0.0537 0.0106

409 6521045 41 15.20 0.0205 0.0003 -0.0275 0.0006 0.0343 0.0007

410 6522242 855 15.98 0.0025 0.0037 0.0128 0.0029 0.0130 0.0047

411 6523058 4549 13.16 0.0041 0.0019 0.0024 0.0017 0.0048 0.0025

412 6523351 3117 11.38 0.0052 0.0006 -0.0021 0.0006 0.0056 0.0009

413 6603043 368 15.90 -0.0029 0.0005 -0.0018 0.0004 0.0034 0.0006

414 6604328 1736 13.80 0.0033 0.0026 0.0026 0.0021 0.0042 0.0034

415 6605493 2559 15.55 -0.0052 0.0011 -0.0076 0.0012 0.0092 0.0016

416 6606438 2860 13.42 0.0044 0.0025 0.0051 0.0020 0.0068 0.0032

419 6607447 1242 13.75 0.0087 0.0009 0.0002 0.0014 0.0087 0.0017

420 6607644 4159 14.50 -0.0070 0.0017 0.0395 0.0018 0.0402 0.0025

421 6690082 1240 14.47 -0.0027 0.0010 -0.0130 0.0012 0.0133 0.0015

422 6690171 3320 15.95 0.0060 0.0021 -0.0027 0.0015 0.0066 0.0026

423 6690836 2699 15.23 -0.0081 0.0031 -0.0026 0.0019 0.0085 0.0036

424 6691169 4890 15.77 -0.0006 0.0016 -0.0105 0.0021 0.0106 0.0026

425 6693640 1245 14.20 0.0098 0.0012 0.0059 0.0013 0.0115 0.0018

426 6773862 1868 15.22 -0.0089 0.0024 -0.0014 0.0020 0.0091 0.0031

427 6774537 2146 15.33 -0.0023 0.0014 0.0026 0.0013 0.0035 0.0019

428 6774880 2062 15.00 0.0024 0.0019 -0.0019 0.0016 0.0031 0.0025

429 6776401 1847 14.81 -0.0031 0.0014 -0.0295 0.0015 0.0297 0.0020

430 6779260 2678 11.80 0.0000 0.0006 0.0063 0.0004 0.0063 0.0007

431 6779726 3375 15.70 0.0011 0.0031 -0.0084 0.0024 0.0085 0.0040

432 6862721 1982 15.77 0.0026 0.0017 0.0051 0.0022 0.0057 0.0027

433 6863998 867 15.22 0.0076 0.0014 0.0054 0.0012 0.0093 0.0019

– 27 –

Table 3—Continued

ID KID KOI mf µRAa µDEC

b µT

434 6945786 3136 15.74 0.0088 0.0018 0.0010 0.0019 0.0089 0.0026

435 6946199 1359 15.23 0.0291 0.0032 -0.0082 0.0023 0.0303 0.0040

436 6947164 3531 14.62 -0.0002 0.0009 0.0010 0.0013 0.0011 0.0016

437 6947668 3455 15.80 -0.0027 0.0015 -0.0046 0.0017 0.0054 0.0023

438 6948054 869 15.60 0.0112 0.0023 0.0097 0.0020 0.0148 0.0031

439 6948480 2975 15.31 -0.0030 0.0013 -0.0004 0.0013 0.0030 0.0018

440 6949061 1960 14.13 0.0050 0.0012 -0.0022 0.0010 0.0055 0.0015

441 6949607 870 15.04 -0.0003 0.0017 0.0216 0.0019 0.0216 0.0026

442 6949898 3031 15.27 -0.0020 0.0021 -0.0021 0.0015 0.0029 0.0026

443 7031517 871 15.22 0.0066 0.0021 -0.0072 0.0014 0.0097 0.0025

444 7032421 1747 14.79 0.0088 0.0020 0.0160 0.0026 0.0182 0.0033

445 7033233 2339 15.13 0.0102 0.0016 0.0072 0.0017 0.0125 0.0023

446 7033671 670 13.77 0.0034 0.0008 -0.0076 0.0007 0.0083 0.0011

447 7115291 3357 15.19 0.0030 0.0018 -0.0014 0.0013 0.0034 0.0023

448 7115785 672 14.00 -0.0078 0.0011 -0.0117 0.0012 0.0141 0.0016

449 7118364 873 15.02 0.0072 0.0018 -0.0040 0.0015 0.0083 0.0023

450 7199060 4152 12.97 -0.0047 0.0010 -0.0028 0.0005 0.0054 0.0011

451 7199397 75 10.78 -0.0019 0.0007 0.0265 0.0006 0.0266 0.0009

452 7199906 1739 15.13 0.0029 0.0014 0.0041 0.0018 0.0050 0.0022

aProper motions in seconds of arc per year.

bCorrected for cosδ declination.

– 28 –

Table 4. Channel 21 KOI Absolute Magnitude

ID KID KOI K0 (J-K)0 HK(0)a MKb Statusc

401 6362874 1128 11.75 0.41 3.23±0.08 4.4±0.1 PC

402 6364215 2404 14.02 0.14 1.73 0.96 3.1 0.4 PC

403 6364582 3456 11.38 0.38 -1.76 0.71 -0.1 0.9 PC

404 6441738 1246 13.34 0.31 3.39 0.23 4.6 0.1 PC

405 6442340 664 11.96 0.27 1.65 0.23 3.0 0.1 Kepler-206b,c,d

406 6442377 176 12.18 0.24 1.65 0.26 3.0 0.1 PC

407 6520519 4749 13.77 0.39 1.90 0.81 3.2 0.3 PC

408 6520753 4504 13.94 0.43 6.59 0.43 7.4 0.1 PC

409 6521045 41 9.76 0.29 1.43 0.05 2.8 0.1 Kepler-100b,c,d

410 6522242 855 13.27 0.43 2.83 0.78 4.1 0.2 PC

411 6523058 4549 14.24 0.32 1.64 1.14 3.0 0.4 PC

412 6523351 3117 11.47 0.40 -0.80 0.34 0.8 1.7 PC

413 6603043 368 11.03 -0.05 -2.22 0.39 -0.5 0.3 PC

414 6604328 1736 14.24 0.38 1.37 1.73 2.7 0.7 PC

415 6605493 2559 12.30 0.27 1.12 0.38 2.5 0.2 PC

416 6606438 2860 14.04 0.23 2.19 1.03 3.5 0.3 PC

419 6607447 1242 12.43 0.20 1.14 0.42 2.5 0.2 PC

420 6607644 4159 12.60 0.47 4.62 0.14 5.7 0.1 PC

421 6690082 1240 12.81 0.39 2.41 0.26 3.7 0.1 PC

422 6690171 3320 13.79 0.53 1.88 0.85 3.2 0.3 EB; PC

423 6690836 2699 13.27 0.46 1.91 0.92 3.2 0.3 PC

424 6691169 4890 14.41 0.18 3.53 0.54 4.7 0.1 PC

425 6693640 1245 12.79 0.24 2.12 0.33 3.4 0.1 PC

426 6773862 1868 12.27 0.82 1.12 0.70 2.5 0.4 PC

427 6774537 2146 13.13 0.56 -0.15 1.16 1.4 1.3 PC

428 6774880 2062 13.45 0.26 -0.12 1.75 1.4 2.0 PC

429 6776401 1847 12.88 0.45 4.23 0.15 5.3 0.1 PC

430 6779260 2678 10.07 0.41 -1.96 0.23 -0.3 0.3 PC

431 6779726 3375 14.00 0.28 2.65 1.01 3.9 0.3 PC

432 6862721 1982 13.97 0.31 1.76 1.03 3.1 0.4 PC

433 6863998 867 13.11 0.55 1.94 0.44 3.3 0.2 PC

– 29 –

Table 4—Continued

ID KID KOI K0 (J-K)0 HK(0)a MKb Statusc

434 6945786 3136 13.52 0.61 2.24 0.63 3.5 0.2 PC

435 6946199 1359 13.48 0.40 4.88 0.28 5.9 0.1 R14

436 6947164 3531 13.06 0.34 -2.39 2.66 -0.6 1.8 EB; PC

437 6947668 3455 13.93 0.39 1.56 0.92 2.9 0.4 PC

438 6948054 869 13.59 0.41 3.45 0.45 4.6 0.1 Kepler-245b,c,d

439 6948480 2975 13.69 0.31 0.12 1.28 1.6 1.2 PC

440 6949061 1960 12.64 0.29 0.30 0.62 1.8 0.5 Kepler-343b,c

441 6949607 870 12.68 0.56 3.35 0.26 4.5 0.1 Kepler-28b,c

442 6949898 3031 13.61 0.28 0.02 1.83 1.5 2.1 PC

443 7031517 871 13.60 0.40 2.53 0.56 3.8 0.2 PC

444 7032421 1747 13.10 0.39 3.43 0.39 4.6 0.1 R14

445 7033233 2339 12.78 0.61 2.27 0.40 3.6 0.1 R14

446 7033671 670 12.15 0.33 0.72 0.28 2.2 0.2 PC

447 7115291 3357 13.49 0.32 0.11 1.45 1.6 1.3 EB; PC

448 7115785 672 12.32 0.37 2.04 0.25 3.3 0.1 Kepler-209b,c

449 7118364 873 13.34 0.37 1.91 0.62 3.2 0.2 PC

450 7199060 4152 11.84 0.10 -0.49 0.43 1.1 0.8 PC

451 7199397 75 9.37 0.29 0.50 0.08 2.0 0.1 PC

452 7199906 1739 13.43 0.36 1.01 0.93 2.4 0.5 PC

aHK(0) = K(0) + 5log(µT) with ∆HK(0)=-1.0 correction.

bMK(0) = 1.51±0.12 + (0.90±0.02)×HK(0) from the Figure 2 calibration.

cPC = Planetary candidate, EB = Eclipsing Binary, FP = False Positive, Kepler = designated

exoplanets have been confirmed, R14 = statistical multi-exoplanet confirmation (Rowe et al. 2014).

– 30 –

Tab

le5.

Chan

nel

21,

Sea

son

0P

lanet

ary

Syst

ems

IDK

IDK

OI

Pla

net

aP

(day

s)re

f.b

Teff

R�

log

g[F

e/H

](J

-K) 0

MK

(0)

405

6442

340

664.

0120

6c13

.137

5R

1457

641.

194.

24-0

.15

0.27

3.0±

0.1

664.

0220

6b7.

7820

664.

0320

6d23

.442

8

409

6521

045

41.0

110

0b12

.816

Mar

1458

251.

494.

13+

0.02

0.29

2.8

0.1

41.0

210

0c6.

887

41.0

310

0d35

.333

435

6946

199

1359

.01

37.1

01R

1459

850.

854.

53-0

.51

0.4

5.9

0.1

1359

.02

104.

8202

438

6948

054

869.

0124

5b7.

4902

R14

5100

0.80

4.56

-0.0

30.

414.

60.

1

869.

0224

5d36

.277

1

869.

0324

5c17

.460

8

440

6949

061

1960

.01

343b

8.96

86R

1458

071.

434.

18-0

.14

0.29

1.8

0.5

1960

.02

343c

23.2

218

441

6949

607

870.

0128

b5.

9123

R14

4633

0.67

4.65

+0.

340.

564.

50.

1

870.

0228

c8.

9858

Ste

12

444

7032

421

1747

.01

20.5

585

R14

5658

0.89

4.54

+0.

070.

394.

60.

1

1747

.02

0.56

73

445

7033

233

2339

.01

2.03

23R

1446

660.

684.

64+

0.38

0.61

3.5

0.1

2339

.02

65.1

900

448

7115

785

672.

0120

9b41

.749

9R

1455

130.

944.

47+

0.01

0.37

3.4

0.1

– 31 –aA

ssig

ned

num

ber

inth

eK

eple

rco

nfirm

edpla

net

sequen

ce,

e.g.

,K

eple

r-20

6c.

aM

ar14

=M

arcy

etal

.(2

014)

,Ste

12=

Ste

ffen

etal

.(2

012)

,R

14=

Row

eet

al.

(201

4).

– 32 –

10

5

0

-5

Red

uced

Pro

per M

otio

n (H

K(0

) = K

0 + 5

log(µ

) )

1.20.80.40.0-0.4(J-K)0

1

23

4

5

67

8

910

11

1213

14151617

18

19

20

21

22

2324

25

26

27

28

29

30

32

3334

3536

3738

39

40

4142

10

5

0

-5

MK

1.20.80.40.0-0.4(J-K)0

1

23

4

5

6 7

8

910

11

1213 1415

1617

18

1920

21

22

2324

25

26

2728

29

30

32

33

34 35

363738

39

40

4142

Fig. 1.— Left: Hertzsprung-Russell (HR) diagram absolute magnitude MK(0) vs (J-K)0,

both corrected for interstellar extinction. Plotted lines show predicted loci for 10Gyr age

solar metallicity stars (- - -) and 3Gyr age metal-poor ([Fe/H]=-2.5) stars (-··-) from Dart-

mouth Stellar Evolution models (Dotter et al. 2008). Right: RPM diagram, same targets

plotted. Horizontal line separates main sequence and sub-giants, giants and super-giants.

Star numbers are from Table 1. Color coding denotes main sequence (red), white dwarfs

(blue), super-giants (black).

– 33 –

10

5

0

-5

MK(0

)

10 5 0 -5 -10Reduced Proper Motion (HK(0) = K(0) + 5log(µ) )

-2

-1

0

1

2

Resid

ual (

mag

nitu

de)

1

234

5

67

8

910

11

121314151617

1819

2021

22

2324

25

26

2728

29

3031

32

33

34353637

38

39

40

4142

123

45

6

7 8910

11

12

13

1415161718

19

20

21

22

2324

25

2627

28

29

30

31

32

33

34

35

36

37

3839

404142

a = 1.51 ± 0.12b = 0.90 ± 0.02

Fig. 2.— A linear mapping between MK(0) and HK(0) using HST parallaxes and proper

motions for targets scattered over the entire sky. RMS residual is 0.7 mag. The linear fit

(MK(0) = a + b HK(0)) coefficient errors are 1-σ. Stellar classifications range from white

dwarfs to Cepheids, as listed in Table 1.

– 34 –

Kepler ID = 7031732

Fig. 3.— Left: KID 7031732 in a crowded field. Image from Digital Sky Survey via Aladin.

Middle: Postage stamp for KID 7031732. Right: Optimum aperture for KID 7031732.

– 35 –

-20

-10

0

10

20

mill

ipix

el

158401582015800157801576015740JD -2440000

1

2

3

45

6

7

8

9

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23

2425

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34

35

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4344

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4950

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656667686970

7172

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9596

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456

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959697

1

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11121314

151617

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23

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34

353637383940

4142

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47

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6061

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6364

6566676869707172

737475

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959697

12345678

9

1011121314151617181920212223242526

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85868788899091929394959697 123456789101112131415161718192021222324

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2728293031323334353637383940414243444546474849505152535455565758596061626364

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X

Fig. 4.— x and y residuals as a function of time for the Q10-only four parameter modeling

from Section 4.1. The residual clumps from left to right are ’plates’ 1–9, the epochs of the

averaged normal points. Stars are labeled with a running number from 1 to 97. The residuals

exhibit significant time-dependency. Regarding two of the stars with more extreme behavior,

neither star 9 (=KID 6363534) nor star 27 (=KID 6606001) is a high-proper motion object.

– 36 –

STScI POSS2UKSTU_Red 19:26:34.92 +41:44:27

30" 2.698’ x 2.404’

N

EPowered by Aladin

2MASS.J.980601N_JI0640256

30" 2.675’ x 2.388’

N

EPowered by Aladin

STScI POSS2UKSTU_Blue 19:27:08.16 +42:02:44

30" 2.698’ x 2.404’

N

EPowered by Aladin

Star 9! Star 9!

Star 27! Star 27!

Fig. 5.— Top: Star 27 (POSS-J on left, 2MASS on right) obviously with a close compan-

ion that confused the first moment centering. Bottom: Star 9, similarly illustrated. No

companion to Star 9 is detected. The positional shift is assumed instrumental.

– 37 –

-20

-10

0

10

20

mas

1518015160151401512015100JD -2440000

1

23

4

6791011

12

13

14

16

1718

19

20

21

22

23

24

25

27

28

29

30

31

32

33

34

3536

37

38

39

40

414243

45

46

47

48

49

50

51

52

53

54

55

565759

60

61

62

63

6466

67

6869

7172737475767879

8081

82

83

8485

86

88

89

9091

9293

94

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100

101

102103

104

105

106107

1

23

4

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12

13

14

16

1718

192021

22

23

24

25

27

28

29

30

31

32

33

34

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38

39

40

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45

46

47

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52

53

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61

62

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7172737475767879

80

81

82

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63

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8283848586

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929394959699100101102103

104105106107 1

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4

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12

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101102103104

105

106107

1

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6 7910

11

1213

1416

1718

19

20

21

22

23

24

25

27

28

29

30

31

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101

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1

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67

910

11

1213

1416

1718

19

20

21

22

2324

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28

29

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7172737475 76787980

81

82

83

8485

86 88

89

9091

9293

94

959699

100

101

102103104

105

106107

-20

-10

0

10

20

mas

1518015160151401512015100

123

4

5

6

79101112

13

1416

171819

20

2122

2324

25

27

2829

30

31

32

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565759

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72

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73747576

7879

80

8182

8384858688

8990919293

94

95

96

99100101102

103

104

105106107

1234

5

6

79101112

13

1416171819

20

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25

272829

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838485868889909192

93

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100101102

103

104

105

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5

6

7

9

1011 1213

14

16

17

18

19

20

21

22

2324

25

27

2829

3031

32

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44

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47

48

49

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606162

63

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686971

72

73

7475 76

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81

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83848586

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909192

93

94

95

96

99

100101

102

103

104

105

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123

4

5

6

7

9

1011 1213

1416

17

18

19

20

21

22

2324

25

27

2829

3031

32

33343536

3738

39

4041

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44

45

46

47

48

49

50

51

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53 54

55

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59

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63

64

66

67

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7980 81

82

83848586

88

89

909192

93

94

95

96

99

100101

102

103

104

105

106

107

X Residuals

S1Q3 FOVC Residuals

-20

-10

0

10

20

mas

1526015240152201520015180JD -2440000

123

4

5

6

7

8

9

101112

13

15

16

17

18

19

2021

22

23

24

25

2627

28

29

3031

32

33

343536

37

39

40

4143

44

45

4647

48

49

50

5152

53

54

55

5657

58

596061

62

636465

6667

6869

70

72

74

757678

79

80

82

83

8485

86

87

88

89

90

92

93

9495

96

99

101

102

103

104

105

106

107

123

4

5

6

7

8

9

101112

1315

16

17

18

19

202122

23

24

25

2627

28

29

30

31

32

33343536

37

39

404143

44

45

4647

48

49

50

5152

53

54

55

56

57

58

596061

62

636465

6667

6869

70

72

73

74

757678

79

8082

83

8485

86

87

88

89

90

91

92

93

9495

96

97

99

101

102

103

104

105

106

107123

4

5

67

8

9

101112

1315

16

17

18

19202122

23

24

252627

28

29

3031

32

33343536

37

39

404143

44

45

4647

48

49

50

5152

53

54

55

5657

58

596061626364

65

6667

6869

70

72

73

74757678

798082

83

8485

86

87

88

89

90

91

92

93

9495

96

97

98

99

101

102

103

104

105

106

107123

4

5

67

8

9

101112

1315

1617

18

19202122

23

24

252627

28

293031

32

33343536

37

39

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44

45

4647

48

49

50

515253

54

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57

58

59606162636465

6667

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70

72

73

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79808283

8485

86

87

88

89

90

91

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7072737475767879808283848586

878889909192939495 9697

9899101102103104105106107 1

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101102103104105106107 1

23

456

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111213

15161718

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232425

2627

28

29

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4445464748

49

50515253

545556

5758596061

626364

6566

67

6869

70

727374757678

79

8082838485

86

87

88

89

909192

939495 969798

99

101102

103104105

106107

-20

-10

0

10

20

mas

1526015240152201520015180

1

23

4

567

8

910111213

15161718192021222324252627

28

29303132333435

3637

3940414344

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69

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82

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87

88

89

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96

979899101

102103

104105

106107

1

23

4

567

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910111213

15161718192021222324252627

28293031323334353637394041434445

46474849

5051

525354555657

58

596061626364

65

66676869

70

72737475

76787980

82

83848586

87

88

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106107

1

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67

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7879808283848586

87

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979899101102103104105106107

1

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104105106107 1 23 45

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123

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83848586

87

88

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102103104

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4

56

7

8910111213

15

16

17

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25

2627

28293031

32

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75

767879

80

82

83848586

87

88

89

90

91

9293949596

9798

99

101

102103104

105

106107

X Residuals

S2Q4 FOVC Residuals

-20

-10

0

10

20

mas

1536015340153201530015280JD -2440000

1234567891011121314151617181920212223242526272829303132333435363738394041434445464748495051525354555657585960616263646566676869707273747576787980818283848586

87888990919293949596979899100101102103104105106107

12345678910

11121314151617181920212223242526272829303132333435363738394041434445464748495051525354555657585960616263646566676869707273747576787980818283848586

87888990919293949596979899100101102

103104105106107

12345678910

1112131415161718192021222324252627282930313233343536373839404143444546474849505152

535455565758596061626364656667686970727374757678798081828384858687888990919293949596979899100101

102103104105106107

1234567891011121314151617181920212223242526272829303132333435363738394041

43444546474849505152

53545556575859606162636465666768697072737475767879

8081

8283848586878889

909192939495969798

99100101102103104105106107 123

4

5678

910

11

1213141516171819202122232425262728293031323334353637383940414344454647484950515253

545556

57

585960616263646566

676869

7072737475767879

80818283848586

87888990919293949596979899100101102103104105106107

1

2

3

4

56

7

8

9

10

1112

13

14

15

16

17

18

19

2021

22

2324

252627

28

29

3031

32

33343536

37

38

3940

41

4344454647

48

49

50

51

52

53

54

5556

57

5859

60616263646566

67

6869

70

72

7374

7576

78

79

80

81

82

83

84

8586

87

88

89

90

91929394

95

96

97

98

99

100101

102

103104

105

106107

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

252627

28

29

30

31

32

33

3435

36

37

38

3940

41

4344

4546

47

48

49

50

51

52

53

54

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57

58

59

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6263646566

67

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70

72

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75

76

78

79

80

81

82

83

84

85

86

87

88

89

90

9192

9394

95

96

97

98

99

100101

102

103104

105

106

107

1

2

35

6

7

9

10

11

12

13

14

15

16

18

20

21

22

23

24

25

2627

29

31

32

33

3435

36

37

38

3940

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45

46

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52

53

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62

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75

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80

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83

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85

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87

88

89

90

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95

96

97

98

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100101

102

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106

107

2

35

6

9

10

11

12

13

14

15

18

20

21

22

23

24

25

2627

29

31

32

33

3435

36

37

3940

41

4344

45

46

47

49

50

52

53

55

56

58

59

60

61

62

63

64

65

66

686972

7374

75

76

79

80

81

84

85

86

87

88

90

91

92

95

96

97

98

99

100

101

102

103104

106

107-20

-10

0

10

20

mas

1536015340153201530015280

123456789101112131415

161718

192021222324252627

28293031

32333435363738

39

404143

44

45464748495051525354555657585960616263646566676869

707273747576787980818283848586

87888990919293949596979899100

101

102103104105106107 123

456789

101112131415161718192021222324252627

282930313233343536373839404143444546474849505152535455565758596061626364656667686970727374757678798081828384858687888990919293949596979899100

101

102103104105106107 123

456789101112131415161718192021222324252627

282930313233343536373839404143444546474849505152535455565758596061626364656667686970727374757678798081828384858687888990919293949596979899100

101102103104105106107 12345678910111213141516

1718192021222324252627282930313233343536373839404143444546474849505152535455565758596061626364656667686970727374757678798081828384858687888990919293949596979899100

101102103104105106107 12345678910111213141516

1718192021222324252627282930313233343536373839404143444546474849505152535455565758596061626364656667686970727374757678798081828384858687888990919293949596979899

100101102103104105106107 12

3

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90

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100

101

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1

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12

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52

53

54

55

5657

58

59

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626364

65

666768

697072

73

7475

76

7879

8081

82

83

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89

90

91

92

93

94

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99

100

101

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104

105

10610712

3

4

567

8

9

10

1112

13

14

15

16

17

18

19

2021

22

23

24

25

2627

28

29

30

31

32

3334

35

3637

383940

41434445

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48

4950

51

52

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55

56

57

58

59

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626364

65

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73

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75

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82

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88

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91

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93

94

95

96

9798

99

100

101

102103

105

106107

X Residuals

S3Q5 FOVC Residuals

-20

-10

0

10

20

mas

1546015440154201540015380JD -2440000

1

2

3

45

6

78910

11

12

131415

16

1718

19

20

2122

232425

2627

28

29

30

31

32

33

34

35

363738

39

4041

42

43

44

45464748

49

5051

52

53

54

55

56

575859

6061

626364

6566

67

6869

707172737475

767778

79

80

81

82838485

86

87

88

89

90

9192

939495

96

9798

99100

101

102103

104105

106107

1

2

3

4

5

6

7

891011121314

1516

171819

20

2122

232425

26272829

30

31

3233

34

35

363738

39

4041

42

43

4445

46474849505152

53

54

55565758

596061

626364

65

66

67

6869

707172737475

767778

79

80

81

82

838485

86

87

88

89909192

93

949596979899

100

101

102103

104

1051061071

2

34

5

6

78910

11121314

1516

171819

20212223242526272829

30

31

32333435363738

39

4041

42

434445

46474849505152

53

5455565758596061

626364

65

66676869707172737475

76777879

80

8182

83848586

87

88

89909192

93

9495969798

99

100

101102103

104

105106107 1

2345

67

891011121314151617181920212223242526272829

30

313233

3435363738

39404142

4344454647484950515253

5455565758596061626364

65

66676869707172737475 7677787980

818283848586

8788

8990919293949596979899

100101

102103104105106107 123 4567891011121314151617181920212223242526272829

3031323334353637383940414243444546474849505152535455565758596061626364

6566676869707172737475 767778798081

828384858687 888990919293

949596979899100101102103104

105106107 123 456789 10111213141516171819202122232425262728

29303132 3334353637383940414243444546474849505152535455565758596061626364656667686970717273

7475 767778798081828384858687 888990919293

949596979899100101102103

104

105106107 1

23 456

789 1011

1213141516171819202122232425262728293031

32 33343536373839404142

4344454647484950515253

545556

57585960616263 6465

6667686970717273

747576777879

80

81828384858687

888990919293

9495 96979899

100101

102103

104

105106107

12

3

456

789 1011

1213141516

1718192021222324252627

2829

3031

3233

3435363738

39

40414243

4445

464748495051

52

53

5455

565758596061

6263 6465

66

67

6869

707172737475

767778

79

80

81828384

8586

87

88

8990

919293 9495

96979899

100

101102103

104

1051061071

23

45

6789 10

11

1213 141516

1718192021222324252627

2829

3031

32

3334

35363738

39

404142

43

4445

464748495051

52

53

5455

56

575859

60616263 64

65

66

67

6869

70717273

7475

76777879

80

81828384

8586

87

88

8990

919293 9495

9697

9899100

101

102103

104

105106107

-20

-10

0

10

20

mas

1546015440154201540015380

12

34

5

6

78910111213141516

1718192021

22232425262728293031

32

33343536

37

38

39

4041424344

454647

48

49

50515253

54

555657

585960616263

64

6566

67

68

69707172

7374757677

7879

80

818283848586

87

888990919293

94

95

96979899100

101

102

103

104

105106107 1234

5678910

11121314

15161718192021

2223242526272829303132

3334353637

38

39

404142434445464748

49

50515253

54

555657

58596061626364

65666768

697071727374757677

7879

80

81828384858687

888990919293

94

9596979899100101

102

103104

10510610712345678910

11121314

151617181920212223242526272829303132

333435363738

39404142434445464748

4950515253

54555657

585960616263

646566

676869707172737475 7677787980

81828384858687

888990919293

949596979899100101

102103104

10510610712345678910

1112131415161718192021

2223242526272829303132333435363738

394041424344454647484950515253

5455565758596061626364

6566676869707172737475 7677787980

81828384858687888990919293949596979899100101102

103104105106107 123 4567891011121314151617181920212223242526272829303132333435363738

39404142434445464748495051525354555657585960616263646566676869707172737475 767778798081828384858687 888990919293949596979899100101102103104105106107 123 456789 1011121314151617181920212223242526272829303132 33343536373839404142434445464748495051525354555657

5859606162636465

66676869707172737475 767778798081828384858687 888990919293949596979899100101102103104105106107 123 456 7

89 1011

121314151617181920212223242526272829303132 33343536373839404142434445464748

4950515253

54555657

585960616263 6465

66676869707172737475 76

77787980

8182838485868788899091929394

95 96979899100101102

103104105106107 12

3 45

67

8910

11 121314

151617181920212223242526272829303132

333435363738

39404142

4344

45464748

4950515253

54

555657

5859

6061626364

65

6667

68

69707172737475

7677

7879

80818283848586

8788899091929394

9596

979899100101

102

103104

105106107 1 2

3 456

789

1011

1213

141516171819202122

23242526272829

303132

333435363738

39

404142

4344

45464748

49

50515253

54

555657

58

5960616263

64

65

66

6768

69707172737475

7677

7879

80

81828384858687

8889

90919293

94

9596

979899100101

102

103104

105106107

X Residuals

S0Q6 FOVC Residuals

-20

-10

0

10

20

mas

1582015800157801576015740JD -2440000

1 1 1 1 1 1 1 1 12 2 2 2 2 2 2 2 233 3 3 3 3 3 3 3

4

4

4 4 4 4 4

4 4

5 5 55

5 5 5 5 56 6 6 6 6 6 6 6 6

77 7 7 7 7 7 7 7

8

88 8 8 8 8

8 8

9

99 9 9 9 9

9 910 10 10 10 10 10 10 10 10

11 11 11 11 11 11 11 11 11

12 12 12 12 12 12 12 12 12

13 13 13 13 13 13 13 13 13

1414 14 14 14 14 14 14 14

15

1515 15 15

15 1515 15

16 16 16 16 16 16 16 16 1617 17 17 17 17 17 17 17 1718 18 18 18 18 18 18 18 18

1919

19 19 19 19 19 19 1920

20 20 20 2020

20 20 2021

2121 21 21 21 21 21 21

22 22 22 22 22 22 22 22 2223

23 23 23 23 23 23 23 2324 24 24 24 24 24 24 24 2425

2525 25 25

25 25 25 25

2626

26 26 26 26 26 26 26

27

2727 27 27 27 27 27 27

28

2828 28 28 28 28

28 28

29

2929 29 29 29 29

29 293030 30 30 30 30 30 30 30

31 31 3131 31 31 31 31 31

3232

32 32 32 32 3232 32

33 33 33 33 33 33 33 33 33

3434 34 34 34 34 34 34 3435 35 35 35 35 35 35 35 35

3636

36 36 36 36 36 36 36

37

3737 37 37 37 37 37 37

3838

3838

38 38 38 38 3839

3939 39 39 39 39 39 39

40 40 4040

40 40 40 40 40

41 4141

4141 41 41 41 4142 42 42 42 42 42 42 42 42

43 4343

4343 43 43 43 43

44

4444 44 44 44 44

44 4445 45 45 45 45 45 45 45 4546

4646 46 46

4646 46 4647

4747 47 47

4747 47 47

4848

48 48 48 48 4848 4849 49 49 49 49 49 49 49 49

5050

50 50 50 50 5050 50

51

51 51 51 51 51 5151 5152 52 52 52 52 52 52 52 52

5353

53 53 53 53 5353 53

54

5454

54 54 54 5454 5455 55 55 55 55 55 55 55 55

5656 56 56 56 56 56 56 565757 57 57 57 57 57 57 57

5858 58 58 58 58 58 58 58

59

5959 59 59

5959 59 59

6060 60

60 6060

60 60 6061

61 6161 61

6161 61 6162

62 62 62 62 62 62 62 62

63

6363 63 63 63 63 63 63

64

6464 64 64 64 64

64 64

65

6565

65 6565

65 65 6566 66 66 66 66 66 66 66 6667 67 67 67 67 67 67 67 6768

6868 68 68 68 68 68 68

69

6969 69 69 69 69 69 69

7070 70 70 70 70 70

70 70

71 71 71 71 71 71 71 71 717272 72 72 72 72 72 72 72

7373 73 73 73 73 73

73 73

74 74 74 74 74 74 7474 74

75 75 75 75 75 75 75 75 757676 76 76 76 76 76 76 76

7777 77 77 77 77 77 77 7778 78 78 78 78 78 78 78 78

79

7979 79 79 79 79

79 798080 80

80 8080

80 80 80

8181

8181

81 81 8181 8182

82 8282 82 82 82

82 82

83

8383

83 8383

8383 83

84 84 84 84 84 84 84 84 8485

85 85 85 85 85 85 85 85

86

8686

86 86 86 8686 8687 87 87 87 87 87 87 87 87

88

88

8888

88 88 8888 8889

8989 89 89 89 89

89 899090

90 90 90 90 90 90 9091 91 91 91 91 91 91 91 91

9292 92 92 92 92 92

92 9293

93 9393 93 93 93 93 93

9494 94 94 94 94 94 94 949595 95 95 95 95 95 95 95

96

9696 96 96 96 96

96 9697

97 97 97 97 97 9797 97

9898 98 98 98 98 98

98 98

99

9999 99 99 99 99 99 99

100100

100100

100 100 100 100 100101101 101 101 101 101 101 101 101102 102 102 102 102 102 102 102 102

103 103103 103 103 103 103 103 103

104 104 104104 104

104104 104 104

105105 105

105 105 105 105105 105

106

106106 106 106 106 106 106 106107 107 107 107 107 107 107 107 107

-20

-10

0

10

20

mas

1582015800157801576015740

11 1 1 1 1 1 1 12 2 2 2 2 2 2 2 2

3

33 3 3 3 3

3 3

4

44 4 4 4 4

4 45 5 5 5 5 5 5 5 5

66

6 6 6 6 6 6 67 7 7 7 7 7 7 7 78

8 8 8 88 8 8 8

9 9 9 9 9 9 9 9 910 10 10 10 10 10 10 10 1011

11 11 11 11 11 11 11 11

12 12 12 12 12 12 12 12 12

1313 13 13 13 13 13 13 13

14 14 14 14 14 14 14 14 1415

15 15 15 15 15 15 15 15

16 16 16 16 16 16 16 16 161717 17 17 17 17 17 17 17

1818 18 18 18 18 18 18 1819 19 19 19 19 19 19 19 19

2020 20 20 20 20 20 20 20

2121

21 21 21 21 2121 21

22 22 22 22 22 22 22 22 22

2323

23 23 23 23 23 23 2324 24 24 24 24 24 24 24 2425 25 25 25 25 25 25 25 2526 26 26 26 26 26 26 26 2627 27 27 27 27 27 27 27 2728 28 28 28 28 28 28 28 2829 29 29 29 29 29 29 29 2930

30 30 30 30 30 30 30 3031 31 31 31 31 31 31 31 31

32

3232 32 32 32 32

32 32

33 33 33 33 33 33 33 33 3334 34 34 34 34 34 34 34 3435 35 35 35 35 35 35 35 3536

36 36 36 36 36 36 36 3637

37 37 37 37 37 37 37 37

3838 38 38 38 38 38 38 38

39

3939 39

39 39 3939 39

4040 40 40 40 40 40 40 40

41 41 41 41 41 41 41 41 4142

42 42 42 42 42 42 42 42

4343

43 43 43 43 4343 43

44

4444 44 44 44 44

44 44

45 45 45 45 45 45 45 45 4546 46 46 46 46 46 46 46 46

4747 47 47 47 47 47 47 474848 48 48 48 48 48 48 48

4949

49 4949

4949

49 49

50 50 50 50 50 50 50 50 5051 51 51 51 51 51 51 51 5152 52 52 52 52 52 52 52 5253

53 53 53 5353 53 53 53

5454 54 54 54 54 54 54 5455 55 55 55 55 55 55 55 555656 56 56 56 56 56 56 56

5757 57 57 57 57 57 57 5758 58 58 58 58 58 58 58 58

59 59 59 59 59 59 59 59 5960 60 60 60 60 60 60 60 60

61 61 61 61 61 61 61 61 6162 62 62 62 62 62 62 62 6263 63 63 63 63 63 63 63 6364

64 64 64 64 64 64 64 6465 65 65 65 65 65 65 65 6566 66 66 66 66 66 66 66 6667 67 67 67 67 67 67 67 67

6868

68 68 68 68 6868 68

69 69 69 69 69 69 69 69 6970

70 70 70 70 70 70 70 7071 71 71 71 71 71 71 71 7172

72 72 72 72 72 72 72 7273 73 73 73 73 73 73 73 73

74 74 74 74 74 74 74 74 74

75 75 75 75 75 75 75 75 75

76 7676 76 76

76 76 76 76

77 77 77 77 77 77 77 77 7778

7878 78 78 78 78 78 78

7979

79 79 79 79 79 79 7980 80 80 80 80 80 80 80 8081 81 81 81 81 81 81 81 8182 82 82 82 82 82 82 82 8283 83 83 83 83 83 83 83 8384 84 84 84 84 84 84 84 8485

85 85 85 85 85 85 85 8586 86 86 86 86 86 86 86 86

8787

87 87 87 87 8787 87

88 88 88 88 88 88 88 88 8889 89 89 89 89 89 89 89 8990 90 90 90 90 90 90 90 9091 91 91 91 91 91 91 91 9192

92 92 92 92 92 92 92 929393 93 93 93 93 93 93 93

9494 94 94 94 94 94 94 9495 95 95 95 95 95 95 95 9596 96 96 96 96 96 96 96 96

97 97 97 97 97 97 97 97 979898 98 98 98 98 98 98 98

99 99 99 99 99 99 99 99 99

100 100 100 100 100 100 100 100 100

101101

101 101 101 101 101101 101102 102 102 102 102 102 102 102 102

103 103 103 103 103 103 103103 103

104104

104 104104 104 104

104 104

105 105 105 105 105 105 105 105 105

106106 106 106 106 106 106 106 106

107 107 107 107 107 107 107 107 107

X Residuals

S0Q10 FOVC Residuals

-20

-10

0

10

20 m

as

157401572015700156801566015640JD -2440000

1

25678

910111213141516

171819

202122

23242550

51

52

535455565758596061

6263646566

67

6869

70

7273747578798081

82

8384858687888990

91929394

95979899

100101

102

103104

105

106107 125678910111213141516

17181920212223242550515253545556575859606162636465666768697072737475787980818283848586

878889909192939495979899100101102103104105106107 125678910111213141516171819202122232425505152

53545556575859606162636465666768697072737475787980818283848586

878889909192939495979899100101102103104105106107 125

6789

10111213141516

171819202122232425505152

53545556575859606162636465666768697072737475787980818283848586

878889

909192939495979899100101102103104105106107 125

6789

10111213141516171819

20212223242550515253545556575859606162636465666768697072737475787980818283848586

878889909192939495979899100101102103104105106107

12567

8

9101112

1314

1516

171819202122232425505152535455565758

596061626364656667686970727374757879

80

818283848586878889

909192939495979899100101102103104105106107

1

25

6

7

8

9

10

1112

13

14

15

16

17

18

19

2021

22

23242550

51

52

53

54

5556

57

58

59

60616263646566

67

6869

70

72

7374

75

78

79

80

81

82

83

84

85

86

87

88

89

90

91929394

9597

98

99

100101

102

103104

105

106107

1

25

6

7

9

10

1112

13

14

15

16

17

18

19

20

21

22

23

24

25

50

51

52

53

54

5556

57

58

59

6061

6263

64

6566

67

6869

70

72

7374

75

78

79

80

81

82

83

84

85

86

87

88

89

90

9192

9394

95

97

98

99

100101

102

103104

105

106

10725

6

7

9

10

1112

13

14

15

16

18

20

21

22

23

24

25

50

52

53

54

55

56

57

58

59

6061

62

63

64

6566

686972

7374

7579

80

81

83

84

85

86

87

88

89

90

91

92

93

95

97

98

99

100101

102

103104

106

107

25

6

7

9

10

1112

13

14

15

18

20

21

22

23

24

25

50

52

53

55

56

58

59

6061

62

63

64

6566

686972

7374

7579

80

81

83

84

85

86

87

88

89

90

91

92

95

97

98

99

100

101

102

103104

106

-20

-10

0

10

20

mas

157401572015700156801566015640

1

25

678

910111213141516

1718

19202122232425505152

53

54555657585960616263646566

67

6869

70727374757879

80

818283848586

87

8889909192939495979899

100

101

102103

104

105106107 1

256789

1011121314151617181920212223242550515253545556575859606162636465666768697072737475787980818283848586878889909192939495979899100

101

102103104105106107 12

56789

1011121314151617181920212223242550515253545556575859606162636465666768697072737475787980818283848586878889909192939495979899100

101

102103104105106107 125678910111213141516

17181920212223242550515253545556575859

606162636465666768697072737475787980818283848586878889909192939495979899100101102103104105106107 1

2567891011121314151617181920212223242550515253545556575859

60616263646566676869707273747578798081

82838485868788899091

92939495979899100101102103104

105106107 1256789101112131415

1617181920212223242550515253545556575859606162636465666768697072737475787980818283848586878889909192939495979899100101102103104105106107

12567

89

10111213

1415

161718

192021

22

2324

25

505152

5354

55

565758596061

626364

656667686970727374757879808182

83

8485868788

89

90

91

929394959798

99

100101102103

104

10510610712567

8

9

10

1112

13

14

15

16

1718

19

2021

22

2324

25

5051

52

5354

55

5657

58

59

6061

6263

64

65

6667

6869

7072

73

74757879

80

8182

83

848586

87

88

89

90

91

92

939495

9798

99

100

101

102103

104

105

106107

1

25

67

8

9

10

1112

13

14

15

16

1718

19

2021

22

23

24

25

50

51

52

53

54

55

56

57

58

59

6061

6263

64

65

66

67

68

69

7072

73

74757879

80

8182

83

848586

87

88

89

90

91

92

93

9495

9798

99

100

101

102103

104

105

106107

1

2567

8

9

10

11

12

13

14

15

16

1718

19

2021

22

23

24

25

50

51

52

53

54

55

56

57

58

59

6061

6263

64

65

66

67

68

69

70

72

73

74

75

7879

80

8182

83

848586

87

88

89

90

91

92

93

9495

9798

99

100

101

102103

105

106107

X Residuals

S3Q9 FOVC Residuals

-20

-10

0

10

20

mas

1554015520155001548015460JD -2440000

1

23

4

67

8

91011

12

13

14

15

16

1718

192021

22

23

24

25

2627

28

29

30

31

32

33

3435

36

37

38

39

40

41

43

45

46

47

48

49

50

51

52

53

54

55

5657

58

59

60

61

62

63

64

6566

67

6869

70

7273747576

78

79

8081

82

83

84

8586

87

88

89

9091

9293

94

95

96

97

98

99

100

101

102103

104

105

106107

1

23

4

67

8

910

11

12

13

14

15

16

1718

192021

22

23

24

25

2627

28

29

30

31

32

33

3435

36

37

38

39

40

4143

45

46

47

48

49

50

51

52

53

54

55

5657

58

59

60

61

62

63

64

6566

67

68

69

70

7273747576

78

79

8081

82

83

848586

87

88

89

9091

9293

94

9596

97

98

99

100

101

102103

104

105

106107

1

23

4

67

8

91011

12

1314

15

16

1718

19202122

23

24

25

2627

28

29

30

31

32

33

343536

37

38

39

40

4143

44

45

46

47

48

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53

54

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58

59

60

61

62

63

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1554015520155001548015460

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13

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25

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58

59

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65

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72

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79

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5051525354555657

58

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6566

67686970

72

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79

80

8182

838485868788

8990919293

94

95

96

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102

103

104

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4

5

6

789101112

13

141516

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20

21222324

25

26272829

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33

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5051525354555657

58

59

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6566

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70

72

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82838485868788

8990919293

94

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2526272829

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33343536

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72

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79808182838485868788

89909192939495

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17181920212223242526272829303132

3334353637383940414344454647484950

51525354555657585960616263646566

676869707273

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78798081828384858687

888990919293949596979899100101102

103104105106107

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5

678

9

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19

20

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25

2627

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303132

33343536373839

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47

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50

5152

5354

55

56575859606162

6364

6566676869

7072

73

7475 7678

798081

82

8384858687 8889

909192

93

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100101102

103

104

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123

4

5

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9

10111213

14

1516

1718

19

20

21

22

2324

25

2627

2829

3031

3233343536

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46

47

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50

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63

646566676869

7072

73

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82

8384858687 8889

909192

93

9495

96

979899

100101102

103

104

105

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123

4

5

678

9

10111213

141516

1718

19

20

21

22

2324

25

2627

2829

3031

32

333435363738

394041434445

46

47

48

49

50

51

5253 54

55

565758

59606162

63

64656667

6869

7072

73

7475 7678

7980 81

82

8384858687 88

89

909192

93

94

9596

979899

100101102

103

104

105

106

107

X Residuals

S1Q7 FOVC Residuals

-20

-10

0

10

20

mas

1564015620156001558015560JD -2440000

123

4

5

67

8

9

101112

13

14

15

1617

18

19

20

21

22

2324

25

2627

28

29

3031

32

33

343536

37

38

39

40

4143

44

45

4647

48

49

50

5152

53

54

55

5657

58

596061

62

636465

66

67

6869

70

72

74

757678

7980

8182

83

8485

86

87

88

89

90

92

93

9495

96

97

99

100

101

102

103104

105

106

107

1234

5

67

8

9

101112

13

14

15

1617

18

1920

21

22

2324

25

2627

28

29

3031

32

33

343536

37

38

39

404143

44

45

4647

48

49

50

5152

53

54

55

5657

58

596061

62636465

6667

6869

70

72

73

74757678

79808182

83

8485

86

87

88

89

90

91

92

93

9495

96

97

98

99

100

101

102

103104

105

106

107

1234

5

6 7

8

9

101112

1314

15

1617

18

192021

22

2324

252627

28

29

3031

32

33343536

3738

39

404143

44

454647

48

49

50

5152

53

54555657

58

59606162636465

6667

6869

7072

73

747576787980818283

8485

86

87

88

89

90

91

92

939495

96

9798

99100

101102

103104105

106

107

123456 78910111213141516171819202122

232425262728293031

323334353637383940414344454647484950515253

54555657

5859606162636465666768697072

7374757678798081828384858687

888990919293949596

9798

99100101102103104105

106107

123456 78910111213141516171819

2021

22232425262728

2930313233343536373839404143

444546474849

505152535455565758596061626364656667

686970

7273747576787980818283848586

87888990919293949596979899100101102103104105106107 123

456

7891011 12131415161718

19

2021222324252627

28

29

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49

50515253

5455565758596061626364

6566676869

70

727374757678

7980

8182838485

86

8788

89

90919293949596

9798

99100101102103

104105106107

-20

-10

0

10

20

mas

1564015620156001558015560

1

23

45

67

8

91011

121314151617181920212223242526

27

28

29303132333435

3637383940

41

43444546474849

5051

5253

54555657

58

596061626364

65

66676869

70

72737475

76

78798081

82

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87

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104105

106107

1

23

45

67

8

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12131415161718192021222324252627

2829303132333435363738394041434445464748495051

5253

54555657

58

596061626364

65

66676869

70

7273747576

78798081

8283848586

87

88

899091

92939495

96

979899100101102103

104105106107

1

23456 7

8

910111213141516171819202122232425

262728

293031323334353637383940

4143444546474849505152

5354555657

58

596061626364

65

66676869

70

72737475767879808182838485868788

89909192939495

96979899100101102103104105106107 1234

56 7891011121314151617181920212223242526272829303132333435363738394041434445

464748495051525354

55565758596061626364656667

6869707273747576787980818283848586

8788899091929394959697989910010110210310410510610712345

67

891011121314

151617181920

212223242526272829303132333435363738

394041434445

4647484950515253

5455565758596061626364656667

68

69707273747576

78798081

8283848586

87

88

8990919293949596979899

100

101102103104105106107

123

45

6

7

891011

121314

15

16171819

2021222324252627

2829303132

333435363738

394041434445

46474849

50515253545556575859606162

6364

656667

68

6970

72737475

7678798081

8283848586

87

88

8990

9192939495

96979899

100

101

102103104105

106107

X Residuals

S2Q8 FOVC Residuals

-20

-10

0

10

20

mas

162001618016160161401612016100JD -2440000

1 1 1 1 1 1 1 1 12 2

2 2 2 2 2 2 23 3 3 3 33 3 3 3

4

4

4 4 44

44 4

5 55 5

5 5 5 5 56 6 6 6 66 6 6 6

77

7 7 7 7 7 7 7

88

8 8 88 8 8 8

9

9

9 9 99 9 9 910 10 10 10 10 10 10 10 10

1111 11

11 11 11 11 11 1112 12 12 12 12 12 12 12 12

13 1313 13 13 13 13 13 13

1414

14 14 14 1414 14 14

15

15

15 15 1515 15 15 15

16 16 16 16 16 16 16 16 1617 1717 17 17 17 17 17 17

1818 18 18 18 18 18 18 18

1919

1919 19 19 19 19 19

2020

20 20 2020 20 20 20

2121

21 21 21 21 21 21 2122 22

22 22 22 22 22 22 2223

23 23 23 23 23 23 23 232424 24 24 24 24 24 24 24

2525

25 25 2525 25 25 25

2626

26 26 26 26 26 26 26

27

27

2727 27 27

27 27 2728

28 28 28 28 28 28 28 28

29

29

2929 29

2929 29 2930 30 30 30 30

30 30 30 30

31 3131

31 31 31 31 31 3132

32 32 32 32 32 32 32 3233 33 33 33 33 33 33 33 33

3434

34 34 34 3434 34 34

35 35 35 35 35 35 35 35 35

3636

36 36 36 3636 36 36

37

37

37 37 37 3737 37 37

3838

38 38 38 38 38 38 38

39

3939 39 39 39 39 39 3940 4040 40 40 40 40 40 40

41 4141 41 41 41 41 41 41

43 4343

43 43 43 43 43 43

44

4444 44 44 44 44 44 44

45 45 45 45 45 45 45 45 454646

46 46 4646 46 46 4647

4747 47

4747 47 47 47

48

48 48 48 48 48 48 48 4849 4949 49 49 49 49 49 49

50

50 50 50 50 50 50 50 50

51

51 51 51 51 51 51 51 5152 5252 52 52 52 52 52 52

53

5353

53 53 5353 53 53

54

54

5454 54 54

54 54 545555 55 55 55 55 55 55 55

5656 56 56 56 56 56 56 56

5757

57 57 57 5757 57 57

5858

58 58 5858 58 58 58

59

59

59 5959

59 59 59 596060

60 6060

60 60 60 6061

6161 61

6161 61 61 61

6262

62 62 62 62 62 62 62

6363

63 63 6363 63 63 63

64

64

64 64 6464

64 64 64

65

6565 65

65

65 65 65 6566 66 66 66 66 66 66 66 6667

67 67 67 67 67 67 67 67

6868

6868 68 68

68 68 68

6969

6969 69 69

69 69 6970

70 70 70 70 70 70 70 7072 7272 72 72 72

72 72 7273

73 73 73 73 73 73 73 73

7474 74 74 74 74 74 74 74

7575 75 75 75 75 75 75 75

76 7676 76 76 76

76 76 7678 78 78 78 7878 78 78 78

7979

79 79 7979 79 79 79

80 80 80 8080

80 80 80 80

81

81

8181 81 81

81 81 8182

82 82 82 82 82 82 82 8283

83

83 8383

83 83 83 8384 84 84 84 84 84 84 84 84

8585

85 85 85 8585 85 85

86

86

8686 86 86

86 86 8687

87 87 87 87 87 87 87 87

88

88

8888 88 88

88 88 88

89

8989 89 89 89 89 89 89

9090

90 90 9090 90 90 90

9191 91 91 91 91 91 91 91

92

92 92 92 92 92 92 92 9293

93 93 93 93 93 93 93 9394 94

94 94 94 94 94 94 9495 95 95 95 95 95 95 95 95

96

9696 96 96 96

96 96 9697

97 97 97 97 97 97 97 97

98

98 98 98 98 98 98 98 98

99

99

99 99 9999

99 99 99

100100

100100 100 100

100 100 100101101 101 101 101 101 101 101 101

102 102102 102 102 102 102 102 102

103103

103103 103 103 103 103 103

104 104 104 104104

104 104 104 104105

105 105 105 105 105 105 105 105

106106

106 106 106 106106 106 106

107 107 107 107 107 107 107 107 107

-20

-10

0

10

20

mas

162001618016160161401612016100

11 1 1 1 1 1 1 1

2 2 2 2 2 2 2 2 2

3

33 3 3 3 3 3 3

44

4 4 4 4 4 4 45 5 5 5 5 5 5 5 5

66

6 6 6 6 6 6 67 7 7 7 7 7 7 7 78

8 8 88

8 8 8 89 9 9 9 9 9 9 9 910 10 10 10 10 10 10 10 1011

11 11 1111

11 11 11 1112

12 12 12 12 12 12 12 12

1313

13 13 13 13 13 13 1314 14 14 14 14 14 14 14 1415 15 15 15 15 15 15 15 15

16 16 16 16 16 16 16 16 1617

17 17 17 17 17 17 17 17

1818 18 18 18 18 18 18 1819 19 19 19 19 19 19 19 19

2020

20 20 20 20 20 20 20

2121

21 21 21 21 21 21 21

22 22 22 22 22 22 22 22 22

2323

23 23 23 23 23 23 23

24 24 24 24 24 24 24 24 2425 25 25 25 25 25 25 25 2526 26 26 26 26 26 26 26 2627 27 27 27 27 27 27 27 2728 28 28 28 28 28 28 28 2829 29 29 29 29 29 29 29 2930

30 30 30 30 30 30 30 303131 31 31 31 31 31 31 31

32

3232 32 32 32

32 32 32

33 33 33 33 33 33 33 33 3334 34 34 34 34 34 34 34 3435 35 35 35 35 35 35 35 35

3636 36 36 36 36 36 36 36

3737

37 37 37 37 37 37 37

3838

38 38 38 38 38 38 38

39

39

39 39 39 3939

3939

4040 40 40 40 40 40 40 40

41 41 41 41 41 41 41 41 41

4343

43 43 43 43 43 43 43

44

4444 44 44 44

44 44 44

45 45 45 45 45 45 45 45 4546 46 46 46 46 46 46 46 46

4747 47 47 47 47 47 47 474848 48 48 48 48 48 48 48

49

4949

4949 49

4949

49

50 50 50 50 50 50 50 50 505151 51 51 51 51 51 51 5152 52 52 52 52 52 52 52 525353 53 53

5353 53 53 53

5454

54 54 54 54 54 54 5455 55 55 55 55 55 55 55 555656 56 56 56

56 56 56 5657

57 57 57 57 57 57 57 5758 58 58 5858

58 58 58 5859 59 59 59 59 59 59 59 5960 60 60 60 60 60 60 60 60

61 61 61 61 61 61 61 61 6162 62 62 62 62 62 62 62 6263 63 63 63 63 63 63 63 6364

64 64 64 64 64 64 64 6465 65 65 6565

65 65 65 6566 66 66 66 66 66 66 66 66

6767

67 67 67 67 67 67 67

6868

68 68 68 6868

6868

69 69 69 69 69 69 69 69 6970

70 70 70 70 70 70 70 7072

72 72 72 72 72 72 72 72

7373

73 73 73 73 73 73 73

74 7474 74 74 74 74 74 74

7575

75 75 75 75 75 75 75

7676

76 76 76 76 76 76 76

7878

78 78 78 78 78 78 78

7979

79 79 79 79 79 79 7980 80 80 80 80 80 80 80 8081 81 81 81 81 81 81 81 8182

82 82 82 82 82 82 82 8283 83 83 83 83 83 83 83 8384 84 84 84 84 84 84 84 8485

85 85 85 85 85 85 85 8586 86 86 86 86 86 86 86 86

8787

87 87 87 8787 87 87

88 88 88 88 88 88 88 88 8889 89 89 89 89 89 89 89 89

90 90 90 90 90 90 90 90 9091 91 91 91 91 91 91 91 9192

92 92 92 92 92 92 92 929393 93 93 93 93 93 93 93

94 94 94 94 94 94 94 94 9495 95 95 95 95 95 95 95 9596 96 96 96 96 96 96 96 9697 97 97 97 97 97 97 97 9798

98 98 98 98 98 98 98 98

99 99 99 99 99 99 99 99 99

100100

100 100 100 100 100 100 100

101

101101 101 101 101

101 101 101102 102 102 102 102 102 102 102 102

103 103103 103

103 103 103 103 103

104

104

104104

104 104104

104104

105 105 105 105 105 105 105 105 105106

106 106 106 106 106 106 106 106107 107 107 107 107 107 107 107 107

X Residuals

S0Q14 FOVC Residuals

-20

-10

0

10

20

mas

16000159801596015940JD -2440000

123

4

5

67

8

9

101112

13

14

15

16

17

18

1920

21

22

24

25

26

27

28

29

3031

33343536

37

38

39

4041

43

44

45

4647

48

49

50

5152

53

54

55

56

57

58

596061

62

636465

66

67

6869

70

72

74757678

79808182

83

8485

86

87

88

89

90

92

93

949596

99

100

101

102

103

104

105

106

107

123

4

5

67

8

9

101112

13

14

15

16

17

18

1920

21

22

23

24

25

2627

28

29

3031

32

333435

36

37

38

39

404143

44

454647

48

49

50

5152

53

54

55

56

57

58

596061

62

636465

6667

6869

70

72

73

74757678

79808182

83

8485

86

8788

89

90

91

92

93

949596

9798

99

100

101

102

103

104

105

106

107

1234

5

67

8

9

101112

1314

15

16

1718

19202122

23

24

252627

28

293031

32

333435

36

3738

39

404143

44

454647

48

49

50

5152

53

54

55

56

57

58

596061

62

636465

6667

6869

70

72

73

74757678

7980818283

8485

868788

89

90

91

92

93949596

9798

99100

101

102

103104

105

106

107

1234567

8

910111213141516171819202122

232425262728293031

32

3334

35

36373839

404143

44

454647484950515253

54555657

58

5960616263646566

67686970

72

73

747576

7879808182838485868788

8990

91

92

939495

96

9798

99100

101102103104105106107

12345678910111213141516171819

2021

222324252627282930313233

34

353637383940414344

45

464748

495051525354555657

585960616263646566

676869707273

7475

76787980818283848586

87888990

9192939495

96

979899100101102103

104105106107

1 234

56 78910111213 14

1516171819

2021

22232425262728

29

3031323334

3536

37383940414344

45464748

49

50515253 54555657585960616263 646566

67

6869

707273

7475

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8081

8283848586

87

8889909192939495 9697

98

99100101102103

104105106107

123

4

56

7

8

9

1011

1213 1415161718

19

20

2122

232425

2627

28

29

30

31

32333435 363738

394041

4344

45464748

49

5051 5253

545556

5758

596061

6263 64

6566

67

68

69

70

72

73

74

75

7678

79

80

81

82

83

8485

86

87

88

89

9091

9293 9495 9697

98

99100

101

102103

104105106107

-20

-10

0

10

20

mas

16000159801596015940

1

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X Residuals

S2Q12 FOVC Residuals

-20

-10

0

10

20

mas

1592015900158801586015840JD -2440000

1

23

4

67

8

91011

12

13

14

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1718

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22

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2627

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12

1314

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93

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104

105

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107

X Residuals

S1Q11 FOVC Residuals

-20

-10

0

10

20

mas

1610016080160601604016020JD -2440000

12345678910111213141516

171819

202122232425262728

29

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48

4950

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82

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91929394

9596979899100101102103104

105106107

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87888990919293949596979899100101102103104105106107 12

345678910111213141516

17181920212223242526272829303132333435363738394041434445464748495051525354555657585960616263646566676869707273747576787980818283848586

87888990919293949596979899100101102103104105106107

12345678910

11121314151617181920212223242526272829303132333435363738394041434445464748495051525354555657585960616263646566676869707273747576787980818283848586

87888990919293949596

979899100101102103104105106107 12

345678910

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8283848586

87888990919293949596

979899100101102103104105106107

1

2

3

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7

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67

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80

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84

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90

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1

2

3

4

5

6

7

8

9

10

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17

18

19

20

21

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2324

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28

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3940

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57

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6061

6263

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67

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70

72

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75

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3

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2

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0

10

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106

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X Residuals

S3Q13 FOVC Residuals

Channel 42 Season 1

Channel 43 Season 2

Channel 44 Season 3

Channel 41 Season 0

Fig. 6.— As in Figure 4, x and y residuals as a function of time for four parameter modeling of

a sample of KIC-identified giants in Channels 41–44, but for 11 Quarters. Top row, Quarters

3–6; middle row, Quarters 7–10; bottom row, Quarters 11–14. Top half of each box contains

x residuals; y below. The y-axis range within each coordinate half-box is ±20 millipixels

with an x-axis range between 80 and 90 days. The residuals exhibit time-dependency within

each Quarter that correlates with focal plane temperature changes. Season 0 appears to have

a larger fraction of stable astrometry.

– 38 –

-20

-10

0

10

20

mill

ipix

el

1536015340153201530015280JD-2440000

4

4

4

44

444

45

555

5 5555 10

10101010

1010 10 103434

34 343434 3434

3462

62

62

62

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62

67

67

67

67

67

6767

77

77

77

77

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7777 77

77

9696

969696

96 9696 96103103

103103

103103103 103

103106

106106

106

106

106

106106

106

X Residuals

Fig. 7.— Selected x residuals as a function of time for a four parameter modeling (Equations

3–4) of 127 stars in Channel 26 from Season 3, Quarter 5. The star numbers are identified

with Kepler IDs in Table 2. Note that stars [4, ..., 62] are more astrometrically stable than

stars [67, ..., 106].

– 39 –

4

3

2

1

0

ΔΚ

s Mag

nitu

de C

ontra

st

1.61.41.21.00.80.60.40.2r [arcsec]

Star 4 Star 67 <Good> <Bad>

Fig. 8.— Normalized K-band contrast curves for stars 4 and 67, along with average contrast

curves for stars 4 through 62 (Good) and 67 through 106 (Bad). Note that while star 4 is

more astrometrically stable than star 67, they have virtually identical contrast curves.

– 40 –

-20

-10

0

10

20

mill

ipix

el

158401582015800157801576015740JD -2440000

1

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1112131415161718192021222324252627282930

3132333435

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4142434445464748495051525354

55565758596061626364

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85868788899091929394959697

X

Fig. 9.— x and y residuals as a function of time for the Q10-only four parameter modeling

from Section 4.1. The residual clumps from left to right are ’plates’ 3–7 first seen in Figure 4.

Stars are labeled with a running number from 1 to 97. The residuals exhibit far less time-

dependency. Stars 9 and 27 continue to exhibit unmodeled behavior.

– 41 –

200

150

100

50

0

Num

ber

-4 -2 0 2 4Residual (miillipixel)

Y Residualsσ = 0.8 millipixelN = 485

200

150

100

50

0

Num

ber

X Residualsσ = 0.4 millipixelN = 485

Fig. 10.— Histograms of x and y residuals for the Q10-only four parameter modeling of only

plates 3–7 (Figure 9) from Section 4.1. The residuals are well-characterized with Gaussians

with 1-σ dispersions as indicated.

– 42 –

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ipix

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X

Fig. 11.— x and y residuals in millipixels as a function of time from the full Schmidt 14

parameter modeling from Section 4.3. The residual clumps on the left hand side (Channel

21) from left to right are ’plates’ 3–7. Stars are labeled with a running number from 1 to 97.

Note the scale change along the y-axis, a range five times larger than that in Figure 9. The

residuals exhibit extreme time-dependency. Star 27 continues to show unmodeled behavior

in both channels. Virtually all stars in Channel 37 (right) exhibit unmodeled behavior.

– 43 –

-600

-400

-200

0

200

400

600

CCD

row

[pix

el]

-600 -400 -200 0 200 400 600CCD column [pixel]

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757677 78

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8384

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89

90

9192

9394

959697

100 mas

Fig. 12.— Average vector residuals in milliseconds of arc (scale at lower left) as a function

of position within Channel 37 from the full Schmidt 14 parameter modeling of Channel 21

and Channel 37 described in Section 4.3. All positions have been re-origined to the CCD

center. Note the extreme variation in vector length and position angle over small spatial

scales, for example the grouping consisting of stars 22 through 34 (row∼50, column∼-200).

Comparing Channel 21 with Channel 37 demonstrates serious astrometric sytematics on very

small spatial scales.

– 44 –

600

400

200

0

-200

-400

Y (p

ixel

)

6004002000-200-400 X (pixel)

5 mas

Fig. 13.— Average vector residuals in milliseconds of arc as a function of position within

Channel 21 from the full Schmidt 14 parameter modeling of three Season 0 observation sets

described in Section 5.3. Other than a strong tendency for larger residuals in the y direction,

the pattern is satisfactorily random.

– 45 –

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30

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sure

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imag

nitu

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446447448449

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451

452

0! 100! 200! 300! 400!Star Number!

Fig. 14.— Measured photometric dispersion (mf standard deviation) over 2.1 y with a nine-

day cadence for each star modeled in Section 5.3. Giants (1–100) exhibit the highest overall

variability. Other groups are the hot star sample (101–199), mid-range Teff (201–299), K-

M star sample (300–350), and KOI sample (400–452). The trends to smaller photometric

variation with number within each sample group (as defined in Section 5.1) may be a function

of position within Channel 21. Lowest variations are nearer (x,y)=(0,1000), highest nearer

(x,y)=(1000,0).

– 46 –

16

14

12

10

8

6

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0

x re

sidua

l [m

illip

ixel

]

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mas

Fig. 15.— Average x residual from the Section 5.5 modeling plotted against average mf ,

scale in millipixel on the left, mas on the right. The solid curve is a quadratic fit to average

reference star residuals resulting from the modeling in Section 5.3. Also plotted are upper

and lower bounds within which 99% of the Section 5.3 reference stars are expected to fall.

These stars are plotted with smallest ID numbers. The KOI are plotted with larger symbols.

The nine confirmed planetary system host stars are plotted with large bold symbols. All

KOI ID numbers are from Table 4. Clearly KOI 426 and 452 are astrometrically peculiar,

and the nine planetary system host stars behave as expected.

– 47 –

6

4

2

0

-2

-4

-6

Red

uced

Pro

per M

otio

n (H

K(0

) = K

(0) +

5lo

g(µ

) )

1.20.80.40.0-0.4(J-K)0 (2Mass)

Kepler Ch21 Field6

4

2

0

-2

-4

-6

Redu

ced

Prop

er M

otio

n (H

K(0

) = K

(0) +

5lo

g(µ

) )

1.20.80.40.0-0.4(J-K)0 (2Mass)

401

402

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451452

Kepler Ch21 Fieldwith KOI

Fig. 16.— Left: RPM from the results of modeling Kepler, UCAC4, and PPMXL data.

The average HK(0) and (J-K)0 ±1σ errors are indicated in the lower left. That error is

reduced by a factor of three compared to an RPM derived by averaging proper motions

from UCAC4 and PPMXL. The heavy tilted line is the location of the main sequence in the

Figure 1 RPM derived from all sky HST proper motions. Note the vertical offset in HK(0)

discussed in the text. Right: RPM with a ∆HK(0) = -1.0 correction, containing the KOI

also shifted. Plotted numbers are from Table 4. The horizontal dotted line represents a

rough demarkation between giant and dwarf stars.

– 48 –

8

6

4

2

0

-2

-4

-6

MK

(0)

1.21.00.80.60.40.20.0-0.2(J-K)0

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451452

[Fe/H] =-2.5, Age = 3 Gyr [Fe/H] = 0.0, Age = 10 Gyr

Fig. 17.— An HR diagram for the Section 5.5 reference stars and the Table 4 KOIs. MK(0)

are calculated using the Figure 2 linear mapping between MK(0) and HK(0), after applying

the Section 5.8 systematic correction to HK(0). KOI labeling is the same as in Figure 16,

right. Lines show predicted loci for 10Gyr age solar metallicity stars (- - -) and 3Gyr age

metal-poor stars (-··-) from Dartmouth Stellar Evolution models (Dotter et al. 2008). Note

the large number of KOI sub-giants.

– 49 –

5.0

4.5

4.0

3.5

3.0

2.5

2.0

log

g

4.0 3.9 3.8 3.7 3.6log Teff

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452

[Fe/H] =-2.5, Age = 3 Gyr [Fe/H] = 0.0, Age = 10 Gyr

Fig. 18.— A theoretical HR diagram for the Section 5.5 astrometric reference stars and the

Table 4 KOI. Data are from the MAST. KOI labeling is the same as in Figure 17. The

main sequence and giant branch are apparent. Lines show predicted loci for 10Gyr age solar

metallicity stars (- - -) and 3Gyr age metal-poor stars (-··-) from Dartmouth Stellar Evolution

models (Dotter et al. 2008). Compared with Figure 17, existing log g values apparently do

not readily identify sub-giants.