CHARACTERIZING THE COOL KOIs. III. KOI 961: A SMALL STAR WITH LARGE PROPER MOTION AND THREE SMALL PLANETS

Digital photometric measurements of KOI 961 include griz photometry from the KIC (Batalha et al. 2010) and Two Micron All Sky Survey (2MASS) JHK photometry (Cutri et al. 2003). Digital photometric measurements of Barnard's Star include UVBRI photometry by Koen et al. (2010) and 2MASS JHK photometry. The 2MASS J- and H-band magnitudes of Barnard's Star are near the saturation limit (Skrutskie et al. 2006) and produce colors that are inconsistent with the spectral type from Reid et al. (1995). With this in mind, we used the J_CIT- and H_CIT-band measurements of Barnard's Star by Leggett (1992), which we converted to 2MASS J and H magnitudes using the transformations of Carpenter (2001).

We obtained Johnson B- and V-band and Cousins R_C-band measurements of KOI 961 with the Hereford Arizona Observatory (HAO) 11 inch telescope on UT 2011 November 17 using standard star fields from Landolt (1992) for calibration. Three Landolt (1992) star fields were used containing a total of 45 stars with accurate B- and V-band photometry, and 40 stars with accurate R_C-band photometry. We acquired 108 B- and V-band measurements of reference stars, and 94 R_C-band measurements to calibrate the measurements of KOI 961. Gary (2007) describes the calibration procedures for HAO photometric measurements. The measurements and uncertainties are listed in Table 1, along with available 2MASS J, H, and K_S measurements.

Figures 1 and 2 show J − K_S versus V − K_S and J − H versus H − K_S for a sample of nearby M dwarfs, with Barnard's Star and KOI 961 indicated. The V − K_S color is sensitive to both effective temperature and metallicity (Leggett 1992; Johnson & Apps 2009). In Johnson et al. (2011, hereafter Paper 2) we identified the metallicity sensitivity of J − K_S. The J − H versus H − K_S color–color regime has also been shown to distinguish metallicities and ages of M dwarfs, and we include the population designations of Leggett (1992) in Figure 2: young disk (YD), young–old disk (Y/O), old disk (OD), old disk-halo (O/H), and halo (H).

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Barnard's Star and KOI 961 show remarkable agreement in both color–color plots. This indicates that KOI 961 and Barnard's Star have similar effective temperatures and metallicities, and are metal-poor compared to the Solar neighborhood, consistent with the K-band spectroscopic measurements in Paper 1 and by RA11. KOI 961 has a slightly bluer V − K_S color (0.33 ± 0.04), which can be due to either higher effective temperature or lower metallicity. Medium-resolution optical spectroscopy of the stars indicates the latter, and we discuss this in the following section.

We observed KOI 961 on UT 2011 July 22 with the Double Spectrograph on the Palomar Observatory 200 inch Hale Telescope: a two-channel (red/blue), low-to-medium resolution spectrograph (Oke & Gunn 1982). We used settings that provided spectral coverage from 5300 Å to 10000 Å in the red channel, with 10 Å wide resolution elements. At the time, the efficiency of the red channel limited the usable spectral coverage to 5300–8000 Å. The blue channel exposure contains significantly less signal, so we do not include those data. We observed the white dwarf standard BD +284211 at the same airmass to calibrate the throughput and telluric absorption in the raw spectrum. We used a theoretical spectrum of BD +284211 from Stone (1977) for the calibration. We used available dome lamps for flat fielding and wavelength calibration, and reduced the data using the NOAO doslit package written in IRAF. The final spectrum has a signal-to-noise ratio of roughly 350 at 6500 Å.

Dawson & De Robertis (2004) combined spectra of Barnard's Star taken over many wavelength regimes in order to accurately measure the star's bolometric luminosity. We show a region of their combined spectrum and our spectrum of KOI 961 in Figure 3. The spectra are strikingly similar, aided by the fact that they have similar resolution. To quantitatively compare them, we calculated the spectroscopic indices of the Palomar-Michigan State University Nearby-Star Spectroscopic Survey (PMSU; e.g., Reid et al. 1995; Hawley et al. 1997; Gizis et al. 2002). We used the band definitions and index formula introduced by Reid et al. (1995), who also report the indices for Barnard's Star.

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Table 2 lists the PSMU spectroscopic indices for Barnard's Star from Reid et al. (1995), our calculations using the Dawson & De Robertis (2004) spectrum of Barnard's Star, and our calculations using our spectrum of KOI 961. The difference between our calculated indices for Barnard's Star and the Reid et al. (1995) values is likely due to the spectra having different resolution. The standard deviation in the difference between our calculated indices of Barnard's Star and the PMSU indices is 0.05, which we used to estimate the uncertainty in the indices.

KOI 961 has very similar CaH indices to Barnard's Star, but systematically higher TiO indices, indicating weaker TiO absorption. TiO absorption has long been the standard for spectral typing of M dwarfs, and is sensitive to stellar effective temperature (e.g., Reid et al. 1995). However, Lépine et al. (2007) showed that TiO is sensitive to both temperature and metallicity, and showed that CaH is a better diagnostic for effective temperature and spectral type. The relationship between TiO and CaH for nearby stars, Barnard's Star, and KOI 961 is illustrated in Figure 4, showing CaH2 + CaH3 versus TiO5 (see Figures 1, 3, 6, and 8 of Lépine et al. 2007). KOI 961 and Barnard's Star have similar CaH2 + CaH3 values within their uncertainties, indicating similar effective temperature. The position of KOI 961 to the right of Barnard's Star implies lower metallicity, consistent with the slightly bluer V − K color.

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The Lépine et al. (2007) TiO and CaH relations are diagnostics for estimating the relative differences in effective temperature and metallicity, and useful for distinguishing dwarfs from subdwarfs. They were not calibrated to empirical metallicity or effective temperate measurements. For calibrated measurements, we used the K-band spectroscopic indices of RA11.

Figure 5 shows the near-infrared spectra of KOI 961 and Barnard's Star from the observational programs of Paper 1 and RA11. Both spectra were obtained with TripleSpec on the Palomar Observatory 200 inch Hale Telescope (Herter et al. 2008). We reduced the KOI 961 spectrum with the SpexTool program, modified for Palomar TripleSpec (Cushing et al. 2004; M. Cushing 2011, private communication), and used spectra of nearby A0V stars for telluric removal and flux calibration with the xtellcor package (Vacca et al. 2003). The RA11 spectrum was reduced using a custom package written by P.S.M., based on the reduction package for the TripleSpec Externally Dispersed Interferometer (TEDI) instrument, described in Muirhead et al. (2011a). We supplemented the Barnard's Star spectrum with an additional observation taken on UT 2011 October 19. The supplemental observation was taken using an observing sequence allowing for more reliable Y-band data reduction.

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Figure 5 shows spectra for Y, J, H, and K bands. The throughput calibration in Y and J bands is incomplete, so we continuum normalized by fitting a quadratic function to the spectrum within each of the bands and then dividing by the fit. Regions between the bands are heavily obscured by telluric lines. The strong correspondence between the stars' spectral shapes and line depths further reinforces the conclusion that they have similar effective temperatures and metallicities.

We obtained high-resolution optical spectra of KOI 961 with the HIgh Resolution Echelle Spectrometer (HIRES) on the Keck I Telescope (Vogt et al. 1994) on UT 2011 October 9. We exposed for 1060 s using settings that achieved a resolving power of 55,000 from 3630 to 8900 Å, and a signal-to-noise ratio of 25 at 6400 Å.

Barnard's Star is currently monitored for orbiting exoplanets as part of the California Planet Search, a precision radial-velocity survey using an iodine absorption cell for calibration (e.g., Johnson et al. 2010b). We combined all of the spectra taken of Barnard's Star over the course of the survey using regions that are not contaminated by molecular iodine absorption. The result is a spectrum with an average signal-to-noise ratio of roughly 2500.

Figure 6 shows the HIRES spectra of KOI 961 and Barnard's Star in molecular bands corresponding to select PMSU indices, including Hα. The correspondence is remarkable; however, unlike the moderate-resolution spectrum in Figure 3, the HIRES spectra are continuum normalized within each order using a quadratic fit. This will hide differences in the bulk depths of molecular bands which are more apparent at moderate resolution. Nevertheless, the striking similarity attests to the presence of similar molecular opacities, which are a strong function of abundance and temperature in M dwarfs. Like the moderate-resolution spectra, neither star shows Hα emission. West et al. (2004) and West et al. (2008) showed that the fraction of mid-to-late M dwarfs with Hα emission decreases with height above the galactic plane, and the lack of Hα in either star indicates that they are part of an old, high disk-scale-height population, consistent with their low metallicities. West et al. (2008) estimate the activity lifetime of M4 dwarfs to be 4.5 ±^0.5_1.0 Gyr, providing a lower limit on the ages of KOI 961 and Barnard's Star.

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2.4.1. Projected Rotational Velocity

Browning et al. (2010) report an upper limit to the projected rotational velocity, or Vsin (i), of Barnard's Star of <2.5 km s⁻¹ by measuring the broadening of HIRES spectra obtained by the California Planet Search. This value represents the lowest Vsin (i) measurable given the resolving power of HIRES. KOI 961 shows no appreciable rotational broadening compared to Barnard's Star, indicating similar slow rotation. To quantify the rotational broadening, we followed the approach of Browning et al. (2010).

For each of the spectral regions in Figure 6, we cross-correlated the spectrum of Barnard's Star with a rotationally broadened version of itself using a convolution kernel of specified Vsin (i). We did this for several Vsin (i) values from 0.1 to 10 km s⁻¹. We fitted Gaussian functions to the central peaks of the resulting cross-correlation functions (XCFs) and recorded the full width at half-maximum (FWHM) of each fit. Figure 7 shows the resulting XCF FWHM versus Vsin (i) of the rotational kernels for the six spectral regions (solid blue lines). This provides a calibration to convert XCF FWHM into Vsin (i) for each spectral region.

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We then cross-correlated the spectrum of Barnard's Star with KOI 961 and interpolate that value onto the FWHM–Vsin (i) curves for each spectral region (dashed red lines in Figure 7). The mean Vsin (i) for the six regions is 2.9 km s⁻¹, and the standard deviation is 0.4 km s⁻¹, showing that both Barnard's Star and KOI 961 are slowly rotating. We caution that this Vsin (i) value should be treated as an upper limit rather than an absolute measurement, since this is near the resolving power of HIRES, and we estimate that upper limit to be 1σ higher than the mean: 3.3 km s⁻¹.

Assuming that the inclination of the rotational axis of the star is 90°, and the 0.17 R_☉ stellar radius calculated in the following section, we calculated the rotation period of KOI 961 to be greater than 2.6 days. Such slow rotation is consistent with KOI 961 being an older mid-M dwarf (>1–2 Gyr; Figure 12 from Irwin et al. 2011a); however, the low Vsin (i) measurement does not constrain the age as well as the lack of Hα (>4.5 Gyr), or the kinematic properties (see Section 3).

2.4.2. Absolute Radial Velocity

Unlike most known M dwarfs, Barnard's Star is bright enough to have a reliable parallax measurement from the Hipparcos satellite (van Leeuwen 2007), providing absolute photometric magnitudes for the bands described in Section 2.1. Delfosse et al. (2000) derived mass–luminosity relations for M dwarfs using nearby M dwarf eclipsing binary systems; of their relations, the $M_{K_{\rm CIT}}$–mass relation has the lowest residuals with the least dependence on stellar metallicity (see Figure 3 from their paper). We calculated $M_{K_{\rm CIT}}$ by combining the K_CIT measurement reported in Leggett (1992) with the Hipparcos parallax. Using the empirical $M_{K_{\rm CIT}}$–mass relation of Delfosse et al. (2000), we calculated a stellar mass of 0.158 M_☉ for Barnard's Star.

Delfosse et al. (2000) do not report the systematic uncertainty in the $M_{K_{\rm CIT}}$–mass relation. To estimate the uncertainty in the relation, we calculated the root mean square of the residuals between $M_{K_{\rm CIT}}$ of the calibration stars and those of the relation, and found it to be 0.18 mag. This is much larger than the uncertainty in $M_{K_{\rm CIT}}$ of Barnard's Star and will dominate the uncertainty in its measured mass.

We estimate the uncertainty in the mass of Barnard's Star using a Monte Carlo approach. We recalculated the mass of Barnard's Star for 500 instances of $M_{K_{\rm CIT}}$, each with noise added, drawn from a Gaussian distribution of standard deviation of 0.18 mag. The resulting standard deviation of the mass distribution is 0.013 M_☉.

The radius of Barnard's Star has been measured previously using optical long-baseline interferometry. Lane et al. (2001) reported a radius of 0.201 ± 0.008 R_☉ for Barnard's Star using observations with the Palomar Testbed Interferometer. The uncertainty in the stellar radius comes from a combination of measurement errors, uncertainty in the angular size of the calibrator stars, and uncertainties in the adopted limb-darkening parameters. Ségransan et al. (2003) recalculated the radius using a different set of limb-darkening coefficients, and reported a radius of 0.196 ± 0.008 R_☉ for Barnard's Star, consistent with Lane et al. (2001) to within the quoted uncertainty. This shows that the interferometrically measured stellar diameter is robust against assumptions about the limb darkening and strengthens our confidence that this radius measurement represents an accurate measurement of the true radius of Barnard's Star.

Finally, Dawson & De Robertis (2004) combined and calibrated available spectra of Barnard's Star across optical and infrared wavelengths to accurately measure its luminosity ((3.46 ± 0.17) × 10⁻³ L_☉), which, when combined with the radius measurement, provides an empirical measurement of the star's effective temperature (3134 ± 102 K). This agrees remarkably well with the effective temperature measured using K band in RA11 of 3266 ± 29 K. The quoted uncertainty for the effective temperature measured in RA11 only accounts for measurement errors in the K-band spectrum, and not systematic effects from errors in the calibration.

To estimate the difference in mass, radius, and luminosity between Barnard's Star and KOI 961, we interpolated the K-band measurements of stellar effective temperature and metallicity onto the 5 Gyr isochrones of Baraffe et al. (1998). We use the Baraffe et al. (1998) models because, unlike other sets of evolutionary isochrones, they include grid points with stellar effective temperatures below 3200 K for sub-Solar metallicity. The Baraffe et al. (1998) isochrones are available in two metallicities, [M/H] = 0.0 and −0.5, which bracket the measured metallicities of Barnard's Star and KOI 961. The [M/H] = −0.5 isochrones are only available for a mixing-length parameter (α) of 1 and for a Helium abundance ([Y]) of 0.25, so we choose the [M/H] = 0.0 isochrones with those same parameters. To test the effect of age on the parameter estimation, we repeated the calculations using 10 Gyr isochrones. The resulting stellar parameters were only marginally different compared to the uncertainties, a consequence of the slow evolution of M dwarfs.

Figure 8 shows stellar radius and stellar mass versus effective temperature for the 5 Gyr isochrones of Baraffe et al. (1998) and the interpolated location of Barnard's Star and KOI 961 based on the K-band measurements. We applied the difference in mass, radius, and luminosity between the model interpolants to the empirical measurements of Barnard's Star to estimate the values for KOI 961.

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To estimate the uncertainties in KOI 961's parameters, we interpolated onto the isochrones 500 Monte Carlo simulations of the K-band T_eff and [M/H] measurements, each with random noise added. The noise is drawn from a normal distribution with standard deviation of 100 K for T_eff and 0.12 dex for [M/H]. We then calculated the standard deviation of the distribution in the resultant radii, masses, and luminosities. We chose to inflate the measurement uncertainties in T_eff to 100 K to account for potential errors in the calibration. For [M/H], we used an uncertainty of 0.12 based on the calibration uncertainty calculated in Rojas-Ayala et al. (2010). This provides uncertainties in the interpolated masses, radii, and luminosities of KOI 961 and Barnard's Star, which, when added in quadrature, provide the uncertainty in the difference between the stars' parameters. We then added in quadrature the uncertainties in the differences to the empirical measurements of Barnard's Star, which provides the uncertainty of KOI 961's parameters. Table 4 lists the empirical measurements of Barnard's Star and our estimated values for the stellar parameters of KOI 961 and their uncertainties.

Table 4. Barnard's Star and KOI 961 Stellar Parameters

Parameter	Barnard's Star	KOI 961
	(empirically measured)	(this work)
R⋆	0.199 ± 0.006 R☉a	0.17 ± 0.04 R☉
M⋆	0.158 ± 0.013 M☉b	0.13 ± 0.05 M☉
L⋆	(3.46 ± 0.17) × 10−3 L☉c	2.40 × 10−3.0 ± 0.3 L☉d
Teff	3134 ± 102 Kc	3068 ± 174 K
MK, CIT	8.21 ± 0.03e,f	8.55 ± 0.35
$M_{K_{\rm S}}$	8.22 ± 0.03e,g	8.53 ± 0.35
d	1.824 ± 0.005 pcg	38.7 ± 6.3 pc
U	−134 ± 1 km s−1	50 ±10 km s−1
V	18 ± 1 km s−1	−67 ± 1 km s−1
W	27 ± 1 km s−1	−59 ± 8 km s−1
Vsin (i)	<2.5 km s−1h	<3.3 km s−1

Notes. ^aWeighed mean of Lane et al. (2001, 0.201 ± 0.008 R_☉) and Ségransan et al. (2003, 0.196 ± 0.008 R_☉). ^bCalculated using the M_{K, CIT}-mass relations of Delfosse et al. (2000). ^cDawson & De Robertis (2004). ^dSince luminosity is a strong power of the effective temperature and radius, the distribution of interpolated Monte Carlo luminosities is asymmetric. Therefore, we quote the error in the inferred luminosity of KOI 961 in the exponent, equivalent to the standard deviation of the log of the resultant luminosity distribution. ^eParallax from van Leeuwen (2007). ^fLeggett (1992). ^g2MASS, Cutri et al. (2003). ^hBrowning et al. (2010).

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We calculated the space motion of KOI 961 by combining the proper motion measurements of Lépine & Shara (2005) with our estimated distance and measured radial velocity. Figure 9 shows a Toomre diagram of nearby G-, K-, and M-type primary stars with measured trigonometric parallaxes greater than 100 mas. Using the population definitions of Fuhrmann (2004), it is clear that both Barnard's Star and KOI 961 have kinematic properties consistent with the "thick disk," containing stars with ages of 7–13 Gyr (Feltzing & Bensby 2008). We acknowledge that the presence of a thick disk in the Milky Way is controversial; however, the difference in stellar kinematics as a function of disk scale height of the galactic disk is firmly established (e.g., Bovy et al. 2011). The kinematic properties, metallicities, Hα emission, and Vsin(i) measurements of KOI 961 and Barnard's Star are all consistent with an old, high disk-scale-height stellar population.

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As a consistency check of our methods, we performed an identical perturbation analysis on GJ 1214, a low-mass star with a transiting super-Earth: GJ 1214 b (Charbonneau et al. 2009). Carter et al. (2011b) measured multiple transit light curves of GJ 1214 b and calculated the mean density of the star using the fitted transit parameters (e.g., Seager & Mallén-Ornelas 2003). GJ 1214 has a parallax measurement by van Altena et al. (1995) and 2MASS JHK photometry. Combining the mass–M_K relation of Delfosse et al. (2000) with the transit-derived stellar density, Carter et al. (2011b) reported a stellar mass of 0.157 ± 0.012 M_☉ and a stellar radius of 0.210 ± 0.007 R_☉ for GJ 1214 (Method A from their paper). The stellar radius was larger than estimated using photometric relationships from the Baraffe et al. (1998) evolutionary isochrones, which they also report as R_⋆ = 0.179 ± 0.006 R_☉ (Method B from their paper).

RA11 report a stellar effective temperature of 3245 ± 31 K and metallicity ([M/H]) of 0.15 ± 0.12 for GJ 1214 using the K-band spectroscopic indices. With an effective temperature similar to Barnard's Star and KOI 961, as well as available K-band measurements, GJ 1214 provides a valuable opportunity to test our perturbation analysis.¹⁴

We proceed assuming Solar metallicity for GJ 1214 since we cannot interpolate the super-Solar K-band value of [M/H] = +0.15 onto the isochrones of Baraffe et al. (1998). Using the same perturbation procedure used for KOI 961, and the K-band effective temperature measurement, we calculate a stellar mass of 0.17 ± 0.05 M_☉ and a stellar radius of 0.21 ± 0.04 R_☉ for GJ 1214. The values are remarkably consistent with the results of Method A by Carter et al. (2011b) to within our estimate uncertainties. Incorporating the K-band metallicity measurement of GJ 1214 would result in a larger radius had we used super-Solar-metallicity isochrones, but not by greater than ΔR_⋆ = 0.1 R_☉ (see Figure 8): still well within the estimated uncertainty. We conclude that the perturbation analysis is an effective method for determining accurate stellar parameters of M dwarfs in the absence of parallax measurements.

Photometric signals that appear to be transiting planets may be caused by astrophysical scenarios other than a planet orbiting the target star. In this work, we consider a false positive to be any scenario different from a planet transiting a star in the KOI 961 system. This may be either an eclipsing binary or a transiting planet around a chance-aligned background star or an eclipsing binary that is physically bound to KOI 961 in a hierarchical triple system. Each of these scenarios involves the presence of another star within the Kepler aperture of KOI 961.

We constrain the properties of such a star using archival and recently acquired imaging data, which we show in Figure 12. The large proper motion of KOI 961 allows us to use digitized images from the Palomar Observatory Sky Survey (POSS; Minkowski & Abell 1963; Reid et al. 1991) to rule out the presence of any contaminating object up to 4.8 mag fainter than KOI 961 in the blue-sensitive plates, anywhere within the Kepler apertures used in the Borucki et al. (2011) data release. We also use both publicly available J-band UKIRT data (Lawrence et al. 2007) and a HIRES guider camera image we obtained using Keck I (equivalent of combined R- and V-band response), which allows us to constrain the presence of blends within a few arcseconds of the current position of KOI 961.

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In addition, the HIRES spectrum of KOI 961 constrains the presence of any potentially blending companion through the absence of noticeably contaminating lines. The symmetry of the cross-correlation profile of the spectrum with that of Barnard's Star reveals that KOI 961 has no companion of comparable brightness (ΔV ∼ 1; or mass ratio q ∼ 0.75) with velocity separation >2 km s⁻¹ (e.g., Howard et al. 2010).¹⁵

Using the methods of Morton & Johnson (2011), which use simulations of Galactic structure and stellar evolution to estimate the sky density of background stars, we determine that there is a probability of about 0.13 for there to be a chance-aligned star undetected by the POSS images or the Keck guider image anywhere within the Kepler apertures used for the Borucki et al. (2011) data release. To determine the extent to which the above observations constrain hierarchical triple scenarios, we simulate a population of stellar binary companions with mass ratio, period, and eccentricity distributions typical of local multiplicity surveys (Raghavan et al. 2010; Tokovinin et al. 2010), and assign magnitudes to each companion based on the models of Baraffe et al. (1998). We then place each simulated companion at a random point on its orbit (using a uniform distribution of mean anomaly) and use the resulting projected separation (assuming a distance of 40 pc), relative velocity, and magnitude difference to determine whether it would be observable given our observations. We find that 59% of companions to KOI 961 would have eluded detection. Assuming an overall 40% binary fraction for M dwarfs (e.g., Fischer & Marcy 1992; Covey et al. 2008), this yields a probability of 0.24 that KOI 961 has an undetected companion.

Calculating the FPP for a particular transit signal requires comparing the probabilities of astrophysical false-positive scenarios with an assumed probability of the planet hypothesis. To this end, we calculate the probability that each of the potentially blending scenarios discussed above might be an eclipsing system mimicking a planet-like transit signal around KOI 961.

Direct application of the exact results of Morton & Johnson (2011) is not appropriate for these signals, because the transit shape does not immediately rule out grazing eclipses. As we show in Section 10, the 29.4 minute observing cadence of the public Kepler data strongly affects the apparent shape of the light curve, including its depth (see Figure 10). Instead, we extend the methods of Morton & Johnson (2011) tailored to the specifics of the KOI 961 system.

The FPP for any of the three signals may then be written as follows:

$$\begin{equation} {\rm FPP} = 1 - \frac{\pi _{\rm pl} \mathcal L_{\rm pl}(\delta,T)}{\displaystyle \sum _i \pi _i \mathcal L_i(\delta,T)}, \end{equation} \tag{ 1 }$$

where the index i represents each of the false-positive scenarios and the planet hypothesis, $\mathcal L_i(\delta,T)$ is the probability density function (PDF) for the given scenario evaluated at the specific depth and duration of the observed signal (the "likelihood"), and π_i is the a priori probability of each scenario existing.

The a priori probability for each scenario has two components: first, that the particular blending star configuration exists; second, that it eclipses and can mimic a planet around KOI 961. For the background eclipsing binary (BGEB) scenario, we assume that a chance-aligned star has a 50% chance of being a binary, and a 10% chance of that binary being on a short-period orbit (Raghavan et al. 2010), using <50 days as a definition of "short period" to allow convenient comparison to planet occurrence studies. Using 0.18 as a conservative average eclipse probability (corresponding to the grazing eclipse probability of two 0.5 R_☉ stars orbiting each other with a 1.2 day period), this gives a probability of 0.13 × 0.50 × 0.1 × 0.18 = 1.1 × 10⁻³ that there is a BGEB within the Kepler apertures of KOI 961. In addition, the binary must not show a secondary eclipse in the diluted light curve. We determine by inspection that a signal with a depth of 5 × 10⁻⁴ would have been detectable by Kepler, and so we use that as a conservative limit for the maximum depth of any putative diluted secondary eclipse. We find that this disqualifies 39% of BGEBs, resulting in π_BGEB = 6.9 × 10⁻⁴.

To estimate the prior probability of a hierarchical triple eclipsing binary (HEB) system, we assume that a potential companion to KOI 961 has a 40% chance of having a companion of its own, a 20% chance of that companion being on a short orbit (we conservatively use a higher short-orbit probability for a hierarchical system than for an isolated binary motivated by the observation that many close binaries have distant companions; Tokovinin et al. 2006), and an eclipse probability of 0.10 (corresponding to two 0.15 R_☉ stars in a 1.2 day orbit). Furthermore, we simulate a population of HEBs according to a flat mass ratio distribution and find that only 2.3% of HEB scenarios have diluted secondary eclipses shallower than 5 × 10⁻⁴. This results in π_HEB = 0.24 × 0.40 × 0.20 × 0.10 × 0.023 = 4.4 × 10⁻⁵.

For the background transiting planet hypothesis (BGpl), we assume an overall 40% occurrence rate for planets with periods <50 days (Howard et al. 2010, 2011; Bonfils et al. 2011), and an average transit probability of 0.13 (corresponding to a planet orbiting a 0.5 R_☉ star in a 1.2 day period), giving π_BGpl = 6.8 × 10⁻³. These same assumptions (except for a transit probability of 0.07 corresponding to the radius of KOI 961) yield an a priori probability for the bona fide KOI 961 transiting planet hypothesis of π_pl = 0.40 × 0.07 = 0.028. We note that we have assumed isotropically oriented orbits for all the eclipse and transit probabilities we have used. This may not necessarily be the case for multiple planet systems (orbits may be coplanar) or for hierarchical systems (orbits of a hierarchical binary may be aligned with the plane of the planets), but assuming isotropic orientation for all orbital planes is a conservative approach.

To calculate the likelihood factors for each of the scenarios, we simulate the expected population of light curves following Mandel & Agol (2002), allowing for all impact parameters (including grazing eclipses), including limb darkening, and accounting for the effects of the 29.4 minute observing cadence. In contrast to the BLENDER procedure (Torres 2011; Fressin et al. 2011a, 2011b), we do not fit false-positive light curves to the data; instead we choose the scenarios to simulate according to their a priori probability of existence, which we determine using the methodology of Morton & Johnson (2011). This allows us to directly extract the joint PDF of duration and depth for each scenario, taking care to include only those light curves with undetectable diluted secondary eclipses. We also create the same distribution for the planet scenarios, assuming a population of planets with a radius distribution dN/dR_p ∼ R⁻²_p from 0.4 to 20 R_⊕, and accounting for the mass and radius errors of KOI 961. The result is exactly analogous to Figure 5 of Morton & Johnson (2011) extended into the duration dimension. "Duration" and "depth" for our purposes here are determined by inspection, not by any fitting procedure; "duration" is the total time the transit model flux spends below unity, and "depth" is simply the minimum value of the model light curve during primary transit. These values correspond to what one would naively call duration and depth from inspecting a real light curve by eye; they have no direct physical significance except for providing a convenient, model-independent way to parameterize a light curve. We then evaluate the PDFs at the duration and depth of each of the three candidate signals (integrating over errors in both dimensions) to obtain each of the likelihood factors (i.e., the "probability of the data given the model"). The simulations we use to create the PDFs are all based on the period of KOI 961.01, so in order to evaluate the likelihoods for 961.02 and 961.03 at the proper locations we scale their true durations by a factor of $(P/1.21{\rm \,d{\rm ays}})^{1/3}$. Figure 13 illustrates each of these PDFs and shows the likelihood value for each candidate for each scenario.

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Table 6 lists the normalized probabilities ($\pi _i \mathcal L_i/\sum \pi _i \mathcal L_i$) for each of the false-positive scenarios for each of the candidates as well as the overall FPP. The total FPPs for 961.01, 961.02, and 961.03 are 6.7 × 10⁻⁴ (∼1 in 1500), 1.4 × 10⁻³ (∼1 in 700), and 3.4 × 10⁻³ (∼1 in 290). We note that these calculations have used conservative assumptions at every stage. In particular, we have ignored the fact that the ensemble evidence of the Kepler planet sample points to coplanarity being common among close-in, tightly packed planetary systems (Lissauer et al. 2011); if we properly accounted for this effect, the planet prior for each of the candidates would increase (corresponding to an increased transit probability), leading to correspondingly lower FPPs. We therefore declare all three candidates to be statistically validated planets in the KOI 961 system, according to the standards set in the literature by the Kepler team.

We note that even though we have demonstrated all three transit signals to be of planetary origin in the KOI 961 system, there is still some probability of their radii being underestimated due to the presence of diluting flux within the aperture. Specifically, there are three sub-scenarios that are encompassed by the "planet in the KOI 961 system" scenario.

• a.  
  KOI 961.0x orbits KOI 961, which is a single star.
• b.  
  KOI 961.0x orbits the primary component of KOI 961, which is an unresolved binary star system.
• c.  
  KOI 961.0x orbits the secondary component of KOI 961, which is an unresolved binary star system.

We calculated in the previous subsection that there was a 24% chance of KOI 961 having an undetected binary companion, giving scenario (a) a probability of 0.76. We then assume that if KOI 961 is a binary, there is equal probability for a planet to be orbiting either component, giving both (b) and (c) probabilities of 0.12. Using the same binary companion simulations discussed above, we calculate the effect this has on the radii of the planets, according to the following:

$$\begin{equation} R_{\rm p,true} = R_{\rm p, inferred} (1 + 10^{-0.4 \Delta M_{\rm Kep}}) ^ {\frac{1}{2}}, \end{equation} \tag{ 2 }$$

where R_{p, true} is the true planet radius, R_{p, inferred} is the radius inferred assuming the star is single, and ΔM_Kep is the magnitude difference between the two components (which would be negative if the planet is orbiting the fainter star). As each binary companion in our simulation has an assigned Kepler magnitude, we can turn this putative population into a probability distribution function for the true planet radius, which we plot in Figure 14 for KOI 961.01. 73% of the probability mass remains below 1 R_⊕, compared to 84% of probability mass for the (a) scenario probability distribution alone. The planets 961.02 and 961.03 have analogous probability distributions, so we conclude that the possibility of a binary companion does not alter our conclusions of the sub-Earth-sized nature of the KOI 961 planets. The resulting uncertainties in the planet radii due to mischaracterization are similar to the quoted uncertainties in Table 5, which were calculated assuming scenario (a) and dominated by errors in the stellar parameters. Therefore, we proceed assuming the uncertainties in Table 5.

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