KEPLER-10 c: A 2.2 EARTH RADIUS TRANSITING PLANET IN A MULTIPLE SYSTEM

The Kepler mission has recently made public a catalog of all transiting planet candidates identified during the first four months of observation by the spacecraft (Borucki et al. 2011b). Included in this list of 1235 objects are nearly 300 in the category of super-Earths (defined here as having radii in the range 1.25 R_⊕ < R_p < 2 R_⊕), and several dozen of Earth-size (R_p < 1.25 R_⊕). The wealth of new information promises to revolutionize our knowledge of extrasolar planets. Although strictly speaking these are still only candidates since confirmation by spectroscopic or other means is not yet in hand, expectations are high that the rate of false positives in this list is relatively small (see Borucki et al. 2011b; Morton & Johnson 2011). Consequently, results from this sample concerning the general properties of exoplanets have already begun to emerge, including studies of the architecture and dynamics of multiple transiting systems (Lissauer et al. 2011b), an investigation of the statistical distribution of eccentricities (Moorhead et al. 2011), and first estimates of the rate of occurrence of planets larger than 2 R_⊕ with orbital periods up to 50 days (A. W. Howard et al. 2011, in preparation), among others.

For good reasons the confirmation or "validation" of small transiting planets¹⁸ (Earth-size or super-Earth-size) has attracted considerable attention, but has proven to be non-trivial in many cases because of the difficulty of detecting the tiny radial-velocity (RV) signatures that these objects cause on their parent stars, as exemplified by the cases of CoRoT-7 b (Léger et al. 2009), Kepler-9 d (Torres et al. 2011), and Kepler-11 g (Lissauer et al. 2011a). In fact, such spectroscopic signals are often too small to detect with current instrumentation, and the planetary nature of the candidate must be established statistically, as in the latter two cases.

The smallest planet discovered to date, Kepler-10 b, was announced recently by Batalha et al. (2011), and is the Kepler mission's first rocky planet. It has a measured radius of 1.416^+0.033_−0.036 R_⊕ and a mass of 4.6^+1.2_−1.3 M_⊕, leading to a mean density of 8.8^+2.1_−2.9 g cm⁻³ which implies a significant iron mass fraction (Batalha et al. 2011). Its parent star, Kepler-10 (KIC 11904151, 2MASS 119024305+5014286), is relatively bright among the Kepler targets (Kepler magnitude Kp = 10.96) and displays two periodic signals with periods of 0.84 days and 45.3 days, and flux decrements (ignoring limb darkening) of 152 ± 4 ppm and 376 ± 9 ppm, respectively (Batalha et al. 2011). The extensive observations that followed the detection of these signals are documented in detail by those authors, and include the difficult measurement of the reflex RV motion of the star with a semi-amplitude of only 3.3^+0.8_−1.0 m s⁻¹ and a period that is consistent with the shorter signal. As is customary also in ground-based searches for transiting planets, the shapes of the spectral lines were examined carefully to rule out changes of similar amplitude correlating with orbital phase that might indicate a false positive, such as a background eclipsing binary (EB) blended with the target, or an EB physically associated with it. However, the precision of the measurements (bisector spans) compared to the small RV amplitude did not allow such changes to be ruled out unambiguously. False positive scenarios were explored with the aid of BLENDER, a technique that models the transit light curves to test a wide range of blend configurations (Torres et al. 2011), and it was found that the overwhelming majority of them can be rejected. This and other evidence presented by Batalha et al. (2011) allowed the planetary nature of Kepler-10 b to be established with very high confidence.

This was not the case, however, for the 45 day period signal referred to as KOI-072.02 (Kepler Object of Interest 72.02), which is the subject of this paper. No significant RV signal was detected at this period, and only an upper limit on its amplitude could be placed. Using BLENDER, Batalha et al. (2011) were able to rule out a large fraction of the blend scenarios involving circular orbits (including hierarchical triples), but eccentric orbits were not explored because of the increased complexity of the problem and the much larger space of parameters for false positives. While circular orbits are a reasonable assumption for Kepler-10 b because of the strong effects of tidal forces at close range, this is not true for KOI-072.02 on account of its much longer orbital period (see, e.g., Mazeh 2008); eccentric orbits cannot be ruled out.

This provides the motivation for the present work, in which we set out to examine all viable astrophysical false positive scenarios for KOI-072.02 with the goal of validating it as a bona fide planet. In addition to improvements in the BLENDER modeling, we bring to bear new near-infrared observations obtained with the Spitzer Space Telescope in which the transits are clearly detected as well as the complete arsenal of follow-up observations gathered by the Kepler team, including high-resolution adaptive optics (AO) imaging and speckle interferometry, high-resolution spectroscopy, and an analysis based on the Kepler observations themselves of the difference images in and out of transit for positional displacements (centroid motion). All of these observations combined with the strong constraints provided by BLENDER significantly limit the kinds of blends that remain possible, and as we describe below they allow us to claim with very high confidence that KOI-072.02 is indeed a planet. Its estimated radius is approximately 60% of that of Neptune. With this, Kepler-10 becomes the mission's third confirmed multi-planet system (after Kepler-9 and Kepler-11; Holman et al. 2010; Lissauer et al. 2011a) containing a transiting super-Earth-size planet and at least one larger planet that also transits.

We begin with a brief recapitulation of the BLENDER technique, including recent improvements. We then present the Warm Spitzer observations at 4.5 μm that help rule out many blends, and we summarize additional constraints available from other observations. This is followed by the application of BLENDER to KOI-072.02 in order to identify all blend scenarios that can mimic the Kepler transit light curve. Next we combine this information with the other constraints and carry out a statistical assessment of the false alarm rate (FAR) for the planet hypothesis, leading to the validation of the candidate as Kepler-10 c. We conclude with a discussion of the possible constitution of the new planet in the light of current models, and the significance of this type of validation.

The detailed morphology of a transit light curve (length of ingress/egress, total duration) contains important information that can be used to reject many false positive scenarios, producing brightness variations that do not quite have the right shape even though they may well match the observed transit depth (see, e.g., Snellen et al. 2009). BLENDER (Torres et al. 2004, 2011) takes advantage of this to explore a very large range of scenarios, including background or foreground eclipsing binaries blended with the target as well as eclipsing binaries physically associated with the target in a hierarchical triple configuration. Following the notation introduced by Torres et al. (2011), the objects composing the binary are referred to as the "secondary" and "tertiary," and the candidate is the "primary." The tertiary can be either a star (including a white dwarf) or a planet, and the secondary can be a main-sequence star or a (background) giant.

With the help of model isochrones to set the stellar properties, BLENDER simulates blend light curves resulting from the flux of the eclipsing pair diluted by the brighter target (and any additional stars that may fall within the photometric aperture). Each simulated light curve is compared with the Kepler observations in a χ² sense to identify which of them result in acceptable fits (to be defined later). The parameters varied during the simulations are the mass of the secondary star (M₂), the mass of the tertiary (M₃, or its radius R₃ if a planet), the impact parameter (b), the relative linear distance (d) between the eclipsing pair and the target, and the relative duration (D/D_circ) of the transit compared to the duration for a circular orbit (see below). For convenience, the relative linear distance is parameterized in terms of the difference in distance modulus, Δδ, where Δδ = 5log (d_EB/d_KOI). In the case of hierarchical triple configurations, the isochrone for the binary is assumed to be the same as for the primary (metallicity of [Fe/H] = −0.15 and a nominal age of 11.9 Gyr; see Batalha et al. 2011), whereas for background blends we have adopted for the binary a representative 3 Gyr isochrone of solar metallicity although these parameters have a minimal impact on the results. For full details of the technique we refer the reader to the references above. Three recent changes and improvements that are especially relevant to the application to KOI-072.02 are described next.

• 1.  
  The relatively long orbital period of KOI-072.02 (45.3 days) precludes us from assuming that the eccentricity (e) is zero, as we were able to suppose in previous applications of BLENDER to Kepler-9 d and Kepler-10 b, which have periods of 1.59 and 0.84 days, respectively. The reason that matters is that the duration of the transit is set, among other factors, by the size of the secondary star. Eccentricity can alter the speed of the tertiary around the secondary, making it slower or faster than in the circular case depending on the orientation of the orbit (longitude of periastron, ω). Given a fixed (measured) duration, blends with smaller or larger secondary stars than in the circular case may still provide satisfactory fits to the light curve, effectively increasing the pool of potential false positives.BLENDER now takes this into account, although rather than using e and ω, which are the natural variables employed in the binary light curve generating routine at the core of BLENDER (see Torres et al. 2011) as parameters, a more convenient variable that captures the effects of both is the duration relative to a circular orbit. Following Winn (2010), this may be expressed as $D/D_{\rm circ} \approx \sqrt{1-e^2}/(1+e\sin \omega)$. Operationally, then, we vary D/D_circ over wide ranges as we explore different blend scenarios, and for each value, we infer the corresponding values of e and ω. In practice, in order to solve for {e, ω} from D/D_circ it is only necessary to consider the limiting cases with ω = 90° and 270°, corresponding to transits occurring at periastron and apastron, respectively, since these are the orientations resulting in the minimum and maximum durations for a given eccentricity. Other combinations of e and ω will lead to intermediate relative durations that are already sampled in our D/D_circ grid. It is worth noting that use of only these two values of ω leads to predicted secondary eclipses in the simulated light curves that are always located at phase 0.5, whereas secondary eclipses in the real data might be present at any phase. For our purposes, this is of no consequence, as KOI-072.02 has already had its light curve screened for secondary eclipses at any phase that might betray a false positive as part of the vetting process. No such features are present down to the 100 ppm level. Thus, any simulated light curves from BLENDER that display a significant secondary eclipse will yield poor fits no matter where the secondary eclipse happens to be, and will lead to the rejection of that particular blend scenario.
• 2.  
  For each false positive configuration BLENDER can predict the overall photometric color of the blend, for comparison with the measured color index of the candidate as reported in the Kepler Input Catalog (KIC; Brown et al. 2011). A color index such as Kp − K_s, where Kp is the Kepler magnitude and K_s is derived from the Two Micron All Sky Survey catalog, provides a reasonable compromise between wavelength leverage and the precision of the index. The latter varies typically between 0.015 and 0.030 mag depending on the passband and the brightness of the star (see Brown et al. 2011). We consider a particular blend to be rejected when its predicted color deviates from the KIC value by more than three times the error of the latter. As it turns out, color is a particularly effective way of rejecting blends that include secondary stars of a different spectral type than the primary, such as those that become possible when allowing for eccentric orbits.
• 3.  
  Recent refinements in the resolution of the BLENDER simulations to better explore parameter space, in addition to the inclusion of eccentricity (or D/D_circ) as an extra variable, have increased the complexity of the problem as well as the computing time (by nearly two orders of magnitude) compared to the relatively simple case of circular orbits. The number of different parameter combinations examined with BLENDER (and corresponding light curve fits) can approach 7 × 10⁸ in some cases. Consequently, the simulations are now performed on the Pleiades cluster at the NASA Advanced Supercomputing Division, located at the Ames Research Center (California), typically on 1024 processors running in parallel. For convenience hierarchical triple configurations (four parameters) and background/foreground blends (five parameters) are studied separately, each for the two separate cases of stellar and planetary tertiaries (for a total of four grids). One additional fit is carried out using a true transiting planet model to provide a reference for the quality of the false positive fits in the other grids.

The discriminating value of the shape information contained in the light curves, mentioned at the beginning of this section, is highlighted by our BLENDER results for Kepler-10 b, as described by Batalha et al. (2011). In that study it was found that all background EB configurations with stellar tertiaries yield very poor fits to the Kepler light curve and are easily rejected. The underlying reason is that all such blend models predict obvious brightness changes out of eclipse (ellipsoidal variations) with an amplitude that is not seen in the data, and which are a consequence of the very short orbital period.¹⁹ Hierarchical triple scenarios were also excluded based on joint constraints from BLENDER and other follow-up observations. The only configurations providing suitable alternatives to the true planet scenario involved stars in the foreground or background of the target that are orbited by a larger transiting planet. The considerable reduction in the blend frequency (BF) from the exclusion of all background eclipsing binaries led to a false alarm probability low enough to validate Kepler-10 b with a very high level of confidence, independently of any spectroscopic evidence. This remarkable result speaks to the power of BLENDER when combined with all other observational constraints. It also assumes considerable significance for Kepler-10 b, given that it was not possible to provide separate proof of the planetary nature of this signal in the Batalha et al. (2011) study from an examination of the bisector spans. The scatter of the bisector span measurements (10.5 m s⁻¹) was three times larger than the RV semi-amplitude (3.3 m s⁻¹), rendering them inconclusive.

The situation regarding the BLENDER analysis of the KOI-072.02 signal in the Batalha et al. (2011) study was very different: the orbital period is much longer, and ellipsoidal variations are predicted to be negligible, so that background eclipsing binaries with stellar tertiaries remain viable blends. This, and the added complication from eccentric orbits, hindered the efforts of those authors to validate this candidate. With the benefit of the enhancements in BLENDER described above, we are now in a better position to approach this problem anew.

As follow-up observations provide important constraints that are complementary to those supplied by BLENDER, and play an important role in determining the FAR for the planetary nature of KOI-072.02 (Section 6), we describe those first below, beginning with our new near-infrared Spitzer observations.

3.1. Warm Spitzer Observations of KOI-072.02

KOI-072.02 was observed during two transits with the IRAC instrument on the Spitzer Space Telescope (Werner et al. 2004; Fazio et al. 2004) at 4.5 μm (program ID 60028). The observations were obtained on UT 2010 August 30 and November 15, with each visit lasting approximately 15 hr 10 minutes. The data were gathered in full-frame mode (256 × 256 pixels) with an exposure time of 6.0 s per image, which resulted in approximately a 7.1 s cadence and yielded 7700 images per visit.

The method we used to produce photometric time series from the images is described by Désert et al. (2009). It consists of finding the centroid position of the stellar point-spread function (PSF) and performing aperture photometry using a circular aperture on individual exposures. The images used are the Basic Calibrated Data delivered by the Spitzer archive. These files are corrected for dark current, flat fielding, and detector nonlinearity, and are converted to flux units. We converted the pixel intensities to electrons using the information on the detector gain and exposure time provided in the FITS headers. This facilitates the evaluation of the photometric errors. We extracted the UTC-based Julian date for each image from the FITS header (keyword DATE_OBS) and corrected to mid-exposure. We converted to TDB-based barycentric Julian dates using the UTC2BJD²⁰ procedure developed by Eastman et al. (2010). This program uses the JPL Horizons ephemeris to estimate the position of the Spitzer spacecraft during the observations. We then corrected for transient pixels in each individual image using a 20 point sliding median filter of the pixel intensity versus time. To do so, we compared each pixel's intensity to the median of the 10 preceding and 10 following exposures at the same pixel position, and we replaced outliers greater than 4σ with their median value. The fraction of all pixels we corrected is 0.02% for the first visit and 0.06% for the second.

The centroid position of the stellar PSF was determined using the DAOPHOT-related procedures GCNTRD, from the IDL Astronomy Library.²¹ We applied the APER routine to perform aperture photometry with a circular aperture of variable radius, using a range of radii between 1.5 and 8 pixels in steps of 0.5. The propagated uncertainties were derived as a function of the aperture radius, and we adopted the aperture providing the smallest errors. We found that the transit depths and errors varied only weakly with aperture radius for all light curves analyzed in this project. The optimal aperture was found to have a radius of 4.0 pixels.

We estimated the background by examining a histogram of counts from the full array. We fit a Gaussian curve to the central region of this distribution (ignoring bins with high counts, which correspond to pixels containing stars), and we adopted the center of this Gaussian as the value of the residual background intensity. As seen already in previous Warm Spitzer observations (Deming et al. 2011; Beerer et al. 2011), we found that the background varies by 20% between three distinct levels from image to image, and displays a ramp-like behavior as function of time. The contribution of the background to the total flux from the stars is low for both observations, from 0.1% to 0.55% depending on the image. Therefore, photometric errors are not dominated by fluctuations in the background. We used a sliding median filter to select and trim outliers in flux and position greater than 5σ, representing 1.6% and 1.3% of the data for the first and second visits, respectively. We also discarded the first half-hour's worth of observations, which is affected by significant telescope jitter before stabilization. The final number of photometric measurements used is 7277 and 7362.

The raw time series are presented in the top panel of Figure 1. We find that the point-to-point scatter in the photometry gives a typical signal-to-noise ratio (S/N) of 280 per image, which corresponds to 90% of the theoretical S/N. Therefore, the noise is dominated by Poisson statistics.

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3.2. Analysis of the Warm Spitzer Light Curves and Results

3.3. Additional Observational Constraints on Possible False Positives

The Kepler photometry used here is the same as employed in the work of Batalha et al. (2011) and was collected between 2009 May 2 and 2010 January 9. These dates correspond to Kepler Quarter 0 (first nine days of commissioning data) through the first month of Quarter 4. For this study we used only the long-cadence observations (10,870 measurements) obtained by the spacecraft at regular intervals of about 29.4 minutes. All blend models generated with BLENDER were integrated over this time interval for comparison with the measurements. The original data have been de-trended for this work by removing a first-order polynomial, and then applying median filtering with a two-day wide sliding window. Observations that occur during transits were masked and did not contribute to the median calculation. Because this sliding window is considerably shorter than the 45.3 day orbital period, any ellipsoidal variations present in the original data should be largely preserved, although in any case they are expected to be very small for binaries with periods as long as this. We adopted also the ephemeris of mid-transit for KOI-072.02 as reported by Batalha et al. (2011), which is T_c[BJD] = 2, 454, 971.6761 + N × 45.29485 days, where N is the number of cycles from the reference epoch.

Because it is relatively bright (Kp = 10.96), Kepler-10 was also observed by the mission with a shorter cadence of approximately 1 minute for a period of several months to allow an asteroseismic characterization of the star. A total of 19 oscillation frequencies were detected, and enabled a very precise determination of the mean stellar density. When combined with stellar evolution models and a spectroscopic determination of the effective temperature and chemical composition, the resulting parameters for the star are very well determined. Kepler-10 is relatively old (>7.4 Gyr) but is otherwise quite similar to the Sun, with a temperature of T_eff = 5627 ± 44 K, a mass and radius of M_⋆ = 0.895 ± 0.060 M_☉ and R_⋆ = 1.056 ± 0.021 R_☉, and a composition [Fe/H] = −0.15 ± 0.06 slightly below solar (Batalha et al. 2011).

As indicated earlier we considered four general scenarios for false positives: chance alignments (a pair of background/foreground eclipsing objects) and hierarchical triple systems, each with tertiaries that can be either stars or planets. The free parameters were varied over the following ranges: secondary mass M₂ between 0.10 and 1.40 M_☉, in steps of 0.02 M_☉; tertiary mass M₃ between 0.10 and M₂, also in steps of 0.02 M_☉; tertiary radius R₃ between 0.06 and 2.00 R_Jup in steps of 0.02 R_Jup; impact parameter b between 0.00 and 1.00 in steps of 0.05; relative duration D/D_circ between 0.2 and 4.6 in steps of 0.2, corresponding to eccentricities up to 0.92 and values of ω of 90° and 270° (see Section 2); and relative distance Δδ (distance modulus difference) between −5.0 and +9.0 in steps of 0.5 mag, except for hierarchical triple configurations, for which Δδ = 0.

The goodness of the fit of each of the large number of synthetic light curves generated by BLENDER is quantified here by computing the χ² statistic and comparing it with that of the best planet model fit. The difference can be assigned a significance level (or FAR) that depends on the number of free parameters of the problem. For example, for a blend scenario corresponding to a hierarchical triple system (4 degrees of freedom), a trial model giving a worse fit than the planet solution by Δχ² = 4.72 is statistically different at the 1σ level, assuming Gaussian errors (see, e.g., Press et al. 1992). A fit that is worse by Δχ² = 16.3 is different at the 3σ level. Hierarchical triple blends giving poorer fits than this are considered here to be ruled out by the Kepler photometry. For background/foreground scenarios (5 degrees of freedom) the 3σ blend rejection level is Δχ² = 18.2.

4.1. BLENDER Results

In this section, we describe the simulations carried out for the four general blend configurations mentioned above. Although the secondaries for the background scenarios can in principle also be evolved stars (giants), as opposed to main-sequence stars, we consistently found that the transit light curves generated by such systems give a very poor match to the observations because they do not have the right shape (the ingress/egress phases are too long). Therefore, we restricted our exploration of parameter space to main-sequence stars only.

An additional possibility for a false positive may stem from an error in the determination of the orbital period. If the true period were twice the nominal value, alternating transit events would correspond to primary and secondary eclipses, implicating a blended EB. The primary and secondary eclipses would often (but not always) be of different depth. As part of the vetting process for each candidate, the Kepler team examines the even-numbered and odd-numbered events to look for differences in depth that may indicate a false positive of this kind. As described by Batalha et al. (2011), no significant differences were found for KOI-072.02 beyond the 2σ level, where σ represents the uncertainty in the transit depth (9 ppm). Nevertheless, as the possibility still exists that the components of the EB are identical, experiments were run with BLENDER to examine the transit shape produced by such scenarios, and it was found that the ingress and egress phases are always much too long compared to the observations, as expected for two equal-size stars eclipsing each other. Thus, these scenarios are easily ruled out as well.

4.1.1. Background Eclipsing Binaries (Star+Star)

The simulations with BLENDER indicate that few background blend scenarios with stellar tertiaries are able to mimic the transit features in the light curve at an acceptable level, and they all correspond to somewhat eccentric orbits. In Figure 2, we show the goodness of fit of these scenarios, with the small closed 3σ contour representing the region of parameter space within which the fits are satisfactory, according to the criteria given above. Only blends with secondary masses M₂ larger than about 1.3 M_☉ are allowed, and the EB can only be within a small range of distances behind the target (4.0 ≲ Δδ ≲ 4.7) for the dilution effect to be just right, such that the corresponding apparent brightness difference ΔKp is between 2.5 and 3.5 mag (see figure). The best among these blend models (located near the center of the contour) provides a fit that is about 2.1σ worse than a planet model (but still acceptable), and is shown in the top panel of Figure 3 compared against the planet model. The tertiary stars in these blends are constrained to be very small, between 0.10 and 0.16 M_☉.

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That most blends involving background eclipsing binaries can be ruled out may appear somewhat surprising and is worth investigating. Indeed, for a given measured transit depth d_tran, a blend can only reproduce the light curve if it contributes at least a fraction d_tran of the total flux collected in the Kepler aperture. Thus, one would expect that binaries as faint as ΔKp = −2.5log (d_tran) ≈ 8.6 mag relative to the target should be able to match that amount of dimming if they were totally eclipsed (see, e.g., Morton & Johnson 2011), and furthermore, that the measured duration could also be reproduced by a large range of secondary sizes with an appropriate combination of orbital eccentricity and ω. Yet we find that no blends fainter than ΔKp = 3.5 give tolerable fits to the light curve (see Figure 2). A visual understanding of the underlying reason for this may be seen in the bottom panel of Figure 3, in which we show a blend model that one would naively expect should be able to match the observations, according to the crude recipe described above. This particular blend scenario is marked with a cross in Figure 2, and corresponds to Δδ = 5 and M₂ = 1.0 M_☉, resulting in a magnitude difference of ΔKp = 5.6 for the EB relative to the target. While this model does yield a good match to the measured depth, and even the total duration, it does not perform nearly as well in the ingress/egress phases, which are too long when compared against the observations. The quality of this fit relative to the best planet fit, which can also be seen in the figure, corresponds to a 10.1σ difference, and therefore BLENDER rejects it. Thus, the reason that most blends of this class can be ruled out is ultimately the high precision of the Kepler light curves, which provides a very strong constraint on the shape of the transit light curve, and in particular on the size ratio between the secondary and tertiary, which sets the duration of the ingress and egress phases.

4.1.2. Background/Foreground Star+Planet Pairs

There is a very broad range of blends consisting of a background or foreground star transited by a planet (as opposed to a star) that are found by BLENDER to give satisfactory fits to the data, as shown in Figure 4. These viable blends occupy the area below the 3σ contour represented with a thick white line. Secondary stars of all spectral types (masses) are permitted, in principle, although in practice other constraints described below eliminate a substantial fraction of them. All of these blends involve secondary+tertiary pairs that are within 4 mag of the target in the Kepler passband (diagonal dashed line in the figure). The tertiary sizes in these blends range from 0.42 R_Jup to 1.84 R_Jup.

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Our Warm Spitzer observations set a lower limit of about 0.77 M_☉ for the secondary masses of these blends, as described earlier; scenarios involving redder stars would result in transits at 4.5 μm significantly deeper than we observe (i.e., deeper than the measured depth + 3σ). This exclusion region is indicated by the shaded area. Additionally, blends that are much brighter than ΔKp = 2 would most likely have been detected spectroscopically (see Batalha et al. 2011), so we consider those to be ruled out as well. We indicate this with the green hatched region in the lower right-hand side of the figure. Finally, the colors of the background/foreground configurations simulated with BLENDER provide a further constraint which is represented by the blue hatched area on the lower left of the figure. This swath of parameter space is excluded because the blends are significantly redder than the color index measured for Kepler-10 (Kp − K_s = 1.465 ± 0.029), by more than three times the uncertainty in the observed index. As a result of these complementary constraints, the only section of parameter space remaining for viable blends involving star+planet pairs is the area under the 3σ contour and limited from below and on the left by the hatched areas (color and brightness conditions) and shaded area (Spitzer constraint), respectively. All of these blends have the eclipsing pair behind the target (foreground scenarios are all ruled out).

We note that in this star+planet blend scenario, white dwarfs can also act as tertiaries as long as they are cooler than the secondaries so that they do not lead to deep occultation events that would have been seen in the light curve of KOI-072.02. The above range of tertiary radii (0.42 R_Jup to 1.84 R_Jup) excludes essentially all cool carbon–oxygen and oxygen–neon white dwarfs more massive than about 0.4 M_☉, as these are smaller than the lower limit set by BLENDER, which corresponds to 4.7 R_⊕ (see, e.g., Panei et al. 2000). Low-mass helium-core or oxygen-core white dwarfs that are the product of common-envelope evolution in binary stars can be considerably larger in size, although they appear to be very rare. The Kepler mission itself has uncovered only three examples to date (Rowe et al. 2010; Carter et al. 2011); however, all of them are very hot (T_eff > 10,000 K) and produce deep and unmistakable flat-bottomed occultation signals. Model calculations such as those of Panei et al. (2007) show that as these helium-core white dwarfs cool, their radii quickly become Earth-size or smaller. Therefore, we do not consider white dwarfs to be a significant source of blends for KOI-072.02.

4.1.3. Hierarchical Triple Scenarios (Star+Star and Star+Planet Blends)

Eclipsing binaries composed of two stars physically associated with the target are clearly ruled out by BLENDER, as they produce very poor fits to the Kepler light curves. For cases in which the tertiaries are planets, viable scenarios identified by BLENDER span a range of secondary masses and tertiary radii within the 3σ contour shown in Figure 5. Most of these configurations turn out to involve eccentric orbits, with transit durations longer than those corresponding to circular orbits along with secondary stars that are smaller than the primary (see Figure 6). Once again other observational constraints are very complementary, and in this case they are sufficient to exclude all of these blends. For example, the shaded area of parameter space to the left of 0.77 M_☉ is eliminated by the Spitzer observations, as described earlier. The constraint on the Kp − K_s color (hatched area on the left) is partly redundant with the NIR observations, but extends to slightly larger secondary masses. And finally, the spectroscopic constraint removes the remaining scenarios corresponding to higher-mass (brighter) secondaries.

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We conclude that of all the hierarchical triple blend scenarios that are capable of precisely reproducing the detailed shape of the Kepler transit light curve, none would have escaped detection by one or more of our follow-up efforts, including NIR Spitzer observations, high-resolution spectroscopy, or absolute photometry (colors).²² This highlights the importance of these types of constraints for validating Kepler candidates, given that blends involving physically associated stars would generally be spatially unresolved by our high-resolution imaging with AO or speckle interferometry, and they would typically also be below the sensitivity limits of our centroid motion analysis, so that they would not be detected by those means. Therefore, the only blends we need to be concerned about for KOI-072.02 are those consisting of stars in the background of the target that are orbited by other stars or by transiting planets.

In order to estimate the frequency of the blend scenarios (i.e., background configurations) that remain possible after applying BLENDER and all other observational constraints, we follow a procedure similar to that described by Torres et al. (2011) for Kepler-9 d. We appeal to the Besançon Galactic structure models of Robin et al. (2003) to predict the number density of background stars of each spectral type (mass) and brightness around Kepler-10, in half-magnitude bins, and we make use of estimates of the frequencies of transiting planets and of eclipsing binaries from recent studies by the Kepler team to infer the number density of blends. Using constraints from our high-resolution imaging (specifically, the sensitivity curves presented by Batalha et al. 2011, their Figure 9) we calculate the area around the target within which blends would go undetected, and with this the expected number of blends.²³

The recent release by Borucki et al. (2011b) of a list of 1235 candidate transiting planets (KOIs) from Kepler provides a means to estimate planet frequencies needed for our calculations, with significant advantages over the calculations of Torres et al. (2011) for Kepler-9 d, which were based on the earlier list of candidates published by Borucki et al. (2011a). The sample is now not only much larger, but also the knowledge of the rate of false positives for Kepler is much improved, and that rate is believed to be relatively small (20%–40% depending on the level of vetting of the candidate, according to Borucki et al. 2011a; less than 10% according to the recent study by Morton & Johnson 2011). Thus, our results will not be significantly affected by the assumption that all of the candidates are planets (see also below). An additional assumption we make is that this census is largely complete. Among these candidates we count a total of 267 having radii in the range allowed by BLENDER for the tertiaries of viable blends (i.e., between 0.42 and 1.84 R_Jup). With the total number of Kepler targets being 156,453 (Borucki et al. 2011b), the relevant frequency of transiting planets for our blend calculation is f_planet = 267/156, 453 = 0.0017. Slawson et al. (2011) have recently published a catalog of the 2165 eclipsing binaries found in the Kepler field, from the first four months of observation. Only the 1225 detached systems among these are considered here, since binaries in the category of semi-detached, overcontact, or ellipsoidal variables would not produce light curves with a shape consistent with a transit. The frequency of eclipsing binaries for our purposes is then f_EB = 1225/156, 453 = 0.0078.

Table 1 presents the results of our calculation of the frequency of blends, separately for background blends with stellar tertiaries (eclipsing binaries) and with planetary tertiaries. Columns 1 and 2 give the Kp magnitude range of each bin and the magnitude difference ΔKp relative to the target, calculated at the upper edge of each bin. Column 3 reports the mean number density of stars per square degree obtained from the Besançon models, for stars in the mass range allowed by BLENDER as shown in Figure 2. In Column 4, we list the maximum angular separation ρ_max at which stars in the corresponding magnitude bin would go undetected in our imaging observations, taken from the information in the work of Batalha et al. (2011). The product of the area implied by this radius and the stellar densities in the previous column give the number of stars in the appropriate mass range, listed in Column 6 in units of 10⁻⁶. Multiplying these figures by the frequency of eclipsing binaries f_EB then gives the number of background star+star blends in Column 7. A similar calculation for the background star+planet blends, making use of f_planet, is presented in Columns 7–10. We sum up the contributions from each magnitude bin at the bottom of Columns 6 and 10. The total number of blends we expect a priori (BF) is given in the last line of the table by adding these two values together, and is BF = 1.62 × 10⁻⁸. The calculations show that background blends consisting of star+planet pairs contribute to this frequency about three times more than background eclipsing binaries.

While we have assumed up to now that any companions to KOI-072.02 within ΔKp = 2 mag of the target would have been seen spectroscopically, we note that relaxing this condition to a much more conservative ΔKp = 1 has no effect at all on the contribution from eclipsing binaries, and a negligible effect on the contribution of star+planet scenarios.

To obtain a Bayesian estimate of the probability that KOI-072.02 is indeed a planet as opposed to a false positive (or equivalently, the "FAR") we follow the general methodology of Torres et al. (2011) and compare the a priori likelihoods of blends and of planets: FAR = BF/PF. If the a priori BF is sufficiently small compared the planet frequency (PF), we consider the planet validated. Our a priori blend frequencies above correspond to false positive scenarios giving fits to the light curve that are within 3σ of the best planet fit. We use a similar criterion to estimate the a priori PF by counting the KOIs in the Borucki et al. (2011b) sample that have radii within 3σ of the best fit from a planet model (R_p = 2.227^+0.052_−0.057 R_⊕; see Table 2 below). We find that 157 among the 1235 KOIs are in this radius range (2.06–2.38 R_⊕), giving PF = 157/156, 453 = 0.0010. This results in a FAR for KOI-072.02 of FAR = 1.6 × 10⁻⁵, which is so small that it allows us to validate the candidate with a very high level of confidence. The planet is designated Kepler-10 c.

This result rests heavily on the a priori frequency of planets from the Kepler mission, derived from the assumption that all 1235 candidates reported by Borucki et al. (2011b) are indeed planets rather than false positives. If we were to be as pessimistic as to assume that as many as 90% of the small-size candidates are actually false positives (a similar rate of false positives as is typically found in ground-based surveys for transiting planets), and at the same time that all of the larger-size candidates that come into the BF calculation are real planets (thereby maximizing BF and minimizing PF), the FAR would be 10 times larger than before, or 1.6 × 10⁻⁴. This is still a very small number, and our conclusion regarding validation is unchanged. We note that a rate of false positives as high as 90% yields a PF that is strongly inconsistent not only with the expectations of Borucki et al. (2011b) and Morton & Johnson (2011), but also with the independent results of ground-based Doppler surveys as reported by Howard et al. (2010).

In the above calculations we have implicitly assumed similar period distributions for planets of all sizes and for eclipsing binaries. However, it is conceivable that the results could change if the period distribution of planets such as Kepler-10 c were significantly different from the one for larger planets that go into the BF calculations, or from the one for EBs (which have a smaller contribution to BF; see Table 1). Therefore, as a further test, we considered the impact of restricting the periods to be within an arbitrary factor of two of the Kepler-10 c period of 45.3 days, both in our BF calculations and for the a priori estimate of the PF. We find that the planet frequencies are reduced by a factor of 4.5, and the EB frequency by a factor of 10.4, and as a result the FAR for KOI-072.02 is FAR = 1.4 × 10⁻⁵, which is about the same as before. Thus, our conclusions are robust against assumptions about the period distributions.

Finally, our FAR is conservative in the sense that we have not accounted for the flatness (coplanarity) of the Kepler-10 system. Only a small fraction of single transiting planets with periods as long as 45 days orbiting background stars (i.e., those acting as blends) are likely to transit, a priori, whereas a planet of this period such as Kepler-10 c is much more likely to transit if it is coplanar with Kepler-10 b. Taking this into account would boost the PF and decrease the FAR by as much as an order of magnitude (see, e.g., Beatty & Seager 2010). Coplanarity in multiple systems is in fact supported by the large number of multiple transiting system candidates found by Kepler (Borucki et al. 2011b; Latham et al. 2011), and their mutual inclinations seem to be small (1°–5°; Lissauer et al. 2011b). Therefore, we consider our estimate of the FAR for Kepler-10 c to be conservative.

The stellar, orbital, and planetary parameters inferred for the system as determined by Batalha et al. (2011) are summarized in Table 2, to which we add the transit duration. The small formal uncertainty in the planetary radius (∼2.4%) derives from the relatively high precision of the stellar radius, which is based on asteroseismic constraints on the mean density of the star. With its radius of about 2.2 R_⊕, Kepler-10 c is among the smallest exoplanets discovered to date. The mass is undetermined as the Doppler signature has not been detected. Nevertheless, Batalha et al. (2011) placed a constraint on it based on the distribution of masses resulting from the Markov Chain Monte Carlo (MCMC) fitting procedure they applied to the existing RV measurements of Kepler-10. Their conservative 3σ upper limit for the mass is 20 M_⊕. The corresponding maximum mean density is 10 g cm⁻³.

Given a precise radius measurement and mass upper limit of 20 M_⊕, some minimal constraints can be placed on the composition of Kepler-10 c. Using the models of Fortney et al. (2007), we find that an Earth-like rock-iron composition is only possible at ∼20 M_⊕. Lower masses would require a depletion in iron compared to rock, or more likely an enrichment in low-density volatiles such as water and/or H₂/He gas. A 50/50 rock/water composition yields 2.23 R_⊕ at 7 M_⊕. Still lower masses are possible with an H₂/He gas envelope. Using models presented in Lissauer et al. (2011a), a planet with a rock/iron core and a 5% H₂/He atmosphere (by mass) matches the measured radius of Kepler-10 c at only 3 M_⊕. A massive 20 M_⊕ core should have attained an H₂/He envelope, and it would appear to be stable at Kepler-10 c's relatively modest irradiation level, which would lead to a planetary radius dramatically larger than 2.23 R_⊕. This would tend to favor a scenario where Kepler-10 c is more akin to GJ 1214b (Charbonneau et al. 2009; Bean et al. 2010; Nettelmann et al. 2010; Désert et al. 2011c) and Kepler-11 b and Kepler-11 f, which are all below 7 M_⊕ and enriched in volatiles.

The well-measured inclinations of both Kepler-10 b and Kepler-10 c allow us to put a weak constraint on the true mutual inclination (ϕ_bc) between the orbital planes of the two planets. Although the relative orientation in the plane of the sky (i.e., the mutual nodal angle) is unknown, the different impact parameters and resulting apparent inclinations place a lower limit on ϕ_bc. As discussed by Ragozzine & Holman (2010), the geometric limits to the mutual inclination are given by |i_b − i_c| ⩽ ϕ_bc ⩽ i_b + i_c, where i_b = 844^+1.1_−1.6 (Batalha et al. 2011) and i_c = 8965^+0.09_−0.12 (Table 2) are the usual inclinations with respect to the line of sight. Assuming a random orientation of the lines of nodes (which does not account for the a priori knowledge that both planets are transiting), the mutual inclination is constrained to be in the interval 525 ⩽ ϕ_bc ⩽ 17405, with the most likely values being at the extremes of this distribution. Making the reasonable supposition of non-retrograde orbits, a mutual inclination close to the lower limit of about 5° is most likely for these planets. A more detailed probabilistic argument requires making assumptions about the number of planets in the Kepler-10 system.

This mutual inclination is on the high end of the distribution inferred for other Kepler multiple candidate systems (1°–5°) by Lissauer et al. (2011b). If this mutual inclination is typical for planets in this system, then it is relatively likely (depending on the orbital period) that other planets, if present, are not transiting. When considering the set of Kepler candidates in multiple systems that have periods less than 125 days, the ratio of periods between Kepler-10 c and Kepler-10 b (which is 54.1) is by far the highest of all period ratios of neighboring pairs of Kepler candidates (the next highest being 23.4), and is even higher than the period ratios between non-neighboring planets. Clearly, there is room for multiple additional planets between Kepler-10 b and Kepler-10 c. The preponderance of tightly packed Kepler multiple candidate systems suggests that additional planets may exist, and these may be revealed in the future with more detailed transit timing variation measurements.

Kepler-10 c is the first Kepler target observed with Warm Spitzer with the aim of testing the wavelength dependence of the transit depth. This is currently the only facility available that has the capability of detecting such shallow transits at wavelengths that are sufficiently separated from the Kepler passband to be helpful. In this case the observations were successful, and the transit at 4.5 μm is shown to have virtually the same depth as in the optical. This places a very strong constraint on the color of potential blends, which are restricted to have secondaries of similar spectral type as the primary star.

The detailed analysis of the Kepler photometry with BLENDER combined with constraints from other observations eliminates the vast majority of possible blend scenarios. This includes most background eclipsing binaries (leaving only a small range of possible spectral types and relative fluxes for the secondaries), most of the scenarios involving chance alignments with a star transited by a larger planet, and all possible hierarchical triple configurations. The latter are among the most difficult to detect observationally since they are typically spatially unresolved. The key factors that have allowed this, and made possible the validation of the planet, are the high precision of the Kepler photometry, the relatively short ingress and egress phases (which places strong constraints on the size ratio between the secondary and tertiary), and the near equatorial orientation, resulting in a relatively flat transit that leaves less freedom for the parameters of the eclipsing binaries. We expect BLENDER to be similarly effective for other Kepler candidates that show similar features in their light curves.

Kepler-10 c along with Kepler-9 d and Kepler-11 g are examples of transiting planets that have not received the usual confirmation by dynamical means that previous discoveries have enjoyed (including essentially all ground-based discoveries), in which either the Doppler signature is detected unambiguously (and verified by the lack of bisector span variations), or transit timing variations in a multiple system are directly measured (as in Kepler-9 b and c as well as the five inner planets of the Kepler-11 system). Instead, the planets in those three cases have been validated statistically, with a Bayesian approach to estimate the probability that the transit signals are due to a planet rather than a false positive. This probability has been computed by first estimating the a priori likelihood of a false positive, and then comparing it with the a priori chance of having observed a true planet. In the three cases mentioned above the ratio of the false positive to planet likelihoods is small enough that the planetary nature of the signal is established with a very high degree of confidence. For Kepler-10 c the FAR is 1.6 × 10⁻⁵.

The recent work of Morton & Johnson (2011) has provided a means of assessing a rough FAR for Kepler candidates as a function of the depth of the transit signal and the brightness of the object. As noted also by those authors, while these estimates are extremely valuable for statistical studies, the validation of candidates on an individual basis with a sufficiently high degree of confidence will usually require a much more detailed analysis of false positives, such as we have performed here. Masses for these objects (other than upper limits) may of course be difficult or impractical to determine in many cases, but it is worth keeping in mind that some of the most exciting candidates to be discovered by Kepler will be in this category, namely, Earth-size planets in the habitable zones of their parent stars. Except for stars of late spectral type, the RV signals will generally be very challenging to detect with the sensitivity of current instrumentation. Thus, statistical validation of planets is likely to play an important role for Kepler in the years to come.

Funding for this Discovery mission is provided by NASA's Science Mission Directorate. This research has made use of the facilities at the NASA Advanced Supercomputing Division (NASA Ames Research Center), and is based also on observations made with the Spitzer Space Telescope which is operated by the Jet Propulsion Laboratory, California Institute of Technology under a contract with NASA. Support for this work was provided by NASA through an award issued by JPL/Caltech. We thank Mukremin Kilic and Rosanne Di Stefano for helpful discussions about white dwarfs, and the anonymous referee for constructive comments.

Facilities: Kepler - The Kepler Mission, Spitzer - Spitzer Space Telescope satellite, Keck:I (HIRES) - KECK I Telescope, Hale - Palomar Observatory's 5.1m Hale Telescope, WIYN:0.9m - WIYN 0.9m Observatory at Kitt Peak National Observatory