VALIDATION OF KEPLER'S MULTIPLE PLANET CANDIDATES. III. LIGHT CURVE ANALYSIS AND ANNOUNCEMENT OF HUNDREDS OF NEW MULTI-PLANET SYSTEMS

Our main objective is to identify FPs and to select candidates found in multi-transiting systems that have a very high probability of being bona fide extrasolar planets. Our strategy was to examine each photometric light curve for signatures of stellar binarity: secondary eclipses, phase curve variations and a comparison of the transit model determination of ρ_⋆ to our classification and modeling of the host star. We also examine the populations of stellar and planetary systems to establish regions of parameter space, namely orbital period and impact parameter, that are most susceptible to contamination from FPs.

Our measured planetary parameters are listed in Table 3 and are based on a transit model fit similar to the description given at the beginning of Section 4, except that we have modeled each PC in a system independently. We start by using the best fit model from the multi-planet model to remove the photometric signature of all transiting candidates except the one we wish to measure. We assumed a circular orbit and fit for ρ_⋆, T0, P, b, R_p/R_⋆, and ρ_c, where ρ_c is the value of ρ_⋆ when a circular orbit is assumed. Thus, each PC provides an independent measurement of ρ_c. If the value of ρ_c is statistically the same for each PC, then the planetary system is consistent with each planet being in a circular orbit around the same host star. We examine the distribution of transit-determined values of ρ_c in Appendix A.

To estimate the posterior distribution on each fitted parameter, we use a MCMC approach similar to the procedure outlined in Ford (2005). To account for the strong correlation between ρ_c, b, and R_p/R_⋆, we use a Gibbs sampler to shuffle the value of parameters for each step of the MCMC procedure and use a control set of parameters to approximate the scale and orientation for the jumping distribution of correlated parameters as outlined in Gregory (2011). An initial control set consists of 2000 chains was generated by a MCMC run where the width of the Gaussian proposal distributions was adjusted to achieve a success rate of ∼25%. Once the success rate for a jump was between 20% and 30%, the width of the Gaussian was fixed for the duration of the calculations. The control set is updated during the MCMC run by adding every second accepted jump proposal parameter set and removing the oldest element. The control set is fixed once an acceptance rate between 20% and 25% is achieved. Any chain that was generated before the proposal sample was fixed was discarded. We found that this method allows the MCMC approach to efficiently sample parameter space even with highly correlated model parameters. We generated four 1,000,000 Markov chains for each PC. The first 20% of each chain was discarded and the remaining sets were combined and used to calculate the median; standard deviation; and 1σ, 2σ, 3σ bounds of the distribution centered on the median of each modeled parameter. Our model fits and uncertainties are reported in Table 3. We use the Markov chains to derive model dependent measurements of the transit depth (T_dep) and transit duration (T_dur). We also convolve the transit model parameters with the stellar parameters (see Section 3) to compute the planetary radius, R_p, and the flux received by the planet relative to the Earth (S). To compute the transit duration, we used Equation (3) from Seager & Mallen-Ornelas (2003) for a circular orbit,

$$\begin{equation} T_{{\rm dur}} = \frac{P}{\pi } \arcsin \left(\frac{R_\star }{a} \left[ \frac{\left(1+\frac{R_{\rm p}}{R_\star }\right)^2- \left(\frac{a}{R_\star }\cos i \right)^2}{1-\cos ^2i} \right]^{1/2} \right), \end{equation} \tag{ 2 }$$

which defines the transit duration as the time from first to last contact. We estimate the ratio of incident flux received by the planet relative to the Earth's incident flux,

$$\begin{equation} S = \left(\frac{R_\star }{R_\odot } \right)^2 \left(\frac{T_{\rm eff}}{T_{{\rm eff}\odot }} \right)^4 \left(\frac{a}{a_{\oplus }} \right)^{-2}, \end{equation} \tag{ 3 }$$

where T_eff is the effective temperature of host star, T_eff☉ is the temperature of the Sun and a is the semi-major axis calculated with Kepler's Second Law using the measured orbital period and estimated stellar mass.

We estimate transit timing variations (TTVs) for each light curve using the best fit models from Sections 4 and 4.1 as a template. Center-of-transit times are measured by selecting data obtained within one transit duration of the predicted center of transit time (thus the time series has a length that is twice the transit duration). If the transit duration is less than two hours, then we select data within two hours of the center-of-transit time. We then refit the transit model but we allow only T0 to vary. The measured center-of-transit time is then compared to the predicted time to produce the observed minus calculated transit time (TTV_n) for each transit, n. If significant variations are detected, we improve the transit model template by compressing and expanding the time interval between measurements by linearly interpolating between timing offsets observed for each transit. The improved template is then used to redetermine the transit times. We report our measured transit-timing variations in Table 4. If fewer than four observations were selected for fitting, we do not report TTV_n. Note that here, t_n is the measured transit time and not the prediction of a linear transit ephemeris (unlike Ford et al. 2011).

Although the vast majority of the TTVs were processed in bulk, some KOIs with large TTVs received individual attention. When the center-of-transit time was shifted substantially away from the predicted transit time, the fitting process failed. An example is KOI-142 (Kepler-88), where the transit times shift by ∼20 hr. For such cases, the previous two transit timing measurements were used to linear extrapolate an estimate of the next transit time to initialize the fitter.

We calculated the reduced chi-square for each transit model. If the value was greater than 2 or less than 0.5, then the fit and photometry were visually inspected. In most cases, it was found that a transit overlapped an instrumental effect, the most common effect being photometric deviations observed after the instrument returned to nominal operations that involved a reorientation of the spacecraft. In these cases, the offending segment of data was excluded and the model fits were repeated. Other cases include models that produced a poor transit fit from convergence to a local minimum, excess scatter from stellar variability, and evidence of a stellar binary in the light curve shape from the presence of a strong occultation or ellipsoidal variations. This level of vetting was performed for both the single and multi-planet population.

From our inspection we discovered that KOI-1134.01 and 1134.02 were both tracing the same EB that had a period of 100 days but had transit depths that were heavily modulated due to third-light contamination that appeared as independent transit candidates in early analysis using only a few months of observations. We have labeled KOI-1134.01 as an EB FP and 1134.02 was labeled as an FA. KOI-1792.02 was found to be an FP with stellar variability mimicking a transit signal. KOI-2048.02 was identified as residuals of the transit fit to KOI-2048.01, thus 2048.02 was labeled as an FA and removed.

The S/N was calculated from depth of the transit using the transit model and noise was estimated by the standard deviation of observations obtained outside of transit and then scaled with a geometric sum to match the transit duration, yielding

$$\begin{equation} {\rm S/N} = \sqrt{N_T} \frac{T_{{\rm dep}}}{\sigma _{OT}}, \end{equation} \tag{ 4 }$$

where T_dep is the transit depth, N_T is the number observations obtained during transit and σ_OT is the standard deviation of out-of-transit observations. The estimation of the S/N assumes that the depth of the transit is uniform, which is a good approximation for small Earth-sized planets with central transits (b = 0). For relatively large planet-to-star radius ratio and/or large impact parameters, our technique will overestimate the S/N, but this has minimal impact on our assessment of PCs. The impact parameter is not well defined for low S/N events, but the transit depth and duration are measurable quantities.

We used S/N estimates for two purposes: to identify FAs and to determine a threshold for planet validation. The KOI catalog has an adopted S/N limit of 7.1 to classify a target as a KOI. FAs are present in the KOI catalog as initially the transit signal was estimated to have a S/N greater than 7.1 (Jenkins et al. 2002), and then as additional observations were gathered the S/N dropped below 7.1. These types of events inform us that validation of transiting planets with a S/N near the KOI threshold has a risk of introducing FAs, which we will now assess.

Using a S/N cut of 7.1 based on the transit depth and visual inspection, 26 KOIs in multis were classified as FAs: KOI-111.04, 439.02, 489.02, 966.02, 989.01, 1070.03, 1134.02, 1198.04, 1312.01, 1316.02, 1408.02, 1576.03, 1639.01, 1639.02, 1792.02, 1940.02, 1961.02, 2048.02, 2160.02, 2188.02, 2224.02, 2261.02, 2339.02, 2473.02, 2533.02, and 2586.02. For 15 systems, only a single PC remained, as indicated by the M1 column in Table 1 when the S/N cut is applied. The objects were considered single-planet systems for statistical counts. The rate of FAs from the single-planet and multi-planet population were both found to be ∼2%. As FAs do not represent real detections (quite the opposite), there is no reason to expect the rates to be predictable or reliable. The KOI creation process has been very inhomogeneous. This has caused the introduction of biases that favor finding and identifying additional PCs once the first candidate in a system has been found. This is especially true because of the notion that the FP rate for multi-planet systems is low. Quantifying this human bias is difficult and is part of our motivation to choose a larger transit S/N cut of 10 for planet validation.

The distribution of the transit S/N is shown in Figure 1. There is a rise in the observed number of PCs from a S/N of 50 to ∼15. The increase is driven by the increase in the number of PCs toward smaller radii and the increase in Kepler targets toward fainter magnitudes. A sharp drop is observed at S/N below 15, which marks the transition where the KOI catalog becomes significantly incomplete. We also inspected all transit candidates with a S/N less than 15 and found convincing transit signals for all candidates with a S/N greater than 10. Based on our observation of the S/N distribution shown in Figure 1 and inspection of the observed transit, we only validate PCs with a S/N greater than 10. We expect a large number of lower S/N candidates in the range of 7.1 to 10 to still be good PCs.

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The primary signature of an EB relative to a transiting planet is the presence of eclipses of different depths due to the difference in surface brightness of the two orbiting stars. The change in depth can be quite dramatic depending on the nature of the two stars. As Kepler photometry has high precision and a spectral bandpass extending to 850 nm, occultations or eclipses can reliably be found for companions with radii similar to Jupiter with temperatures greater than approximately 2000 K. For bright host stars (Kepmag ∼ 10), this limit can be pushed to even cooler temperatures (e.g., TReS-2b; Barclay et al. 2012). As such, a secondary event can be due to a secondary eclipse from a stellar binary, or an occultation when a planet is blocked by the host star. To distinguish between secondary eclipses and occultations, we estimated the expected equilibrium temperature (T_eq) for an orbiting body heated by incident stellar flux and compared it to an estimate of the temperature (T_effp) based on the depth of the occultation. The expectation is that a star, which is self luminous from nuclear fusion, will have a temperature T_effp that is much larger than T_eq. We also test whether the depth of the occultation is consistent with reflected light from a planet by computing the geometric albedo, A_g in the Kepler bandpass. The secondary event is inconsistent with the planet hypothesis if A_g is significantly greater than unity.

Although visual inspection reveals some obvious occultations present in the data, we performed a more thorough search to identify occultations and eclipses. To search for secondary events, the light curve was phased to the orbital period and for each phase point the mean was calculated. Observations that occurred within one transit duration were compared to mean values computed at phases within ±1 transit duration. The difference divided by the standard deviation of observations at all phases was computed and used to identify occultations at any phase outside of transit.

To distinguish between planetary occultations and stellar eclipses, we compared the event depth with the occultation depth expected by a highly radiated exoplanet. We estimate the equilibrium temperature by

$$\begin{equation} T_{\rm eq}= T_{\rm eff}(R_{\star }/2a)^{1/2} [f(1-A_{\rm B})]^{1/4}, \end{equation} \tag{ 5 }$$

where R_⋆ and T_eff are the stellar radius and temperature, a is the semi-major axis, A_B is the planet's Bond albedo, and f is a proxy for atmospheric thermal circulation. To calculate the mean, we assume A_B = 0.1 for highly irradiated planets (Rowe et al. 2006) and f = 1 for efficient heat distribution to the night side. The occultation depth was used to estimate the temperature of the companion (T_effp) using our best estimate of the stellar parameters,

$$\begin{equation} T_{{\rm eff}p}^4 = T_{\rm eff}^4 \frac{R_\star ^2}{R_{\rm p}^2} \frac{F_p}{F_\star }, \end{equation} \tag{ 6 }$$

where F_p/F_⋆ is the ratio of the companion and stellar flux and is equal to the depth of the occultation. We are assuming that the occultation depth observed over the Kepler bandpass is a proxy for the true bolometric flux ratio. We estimate uncertainties in T_eq and T_effp by propagating our determined errors in the stellar parameters from Table 2. We also estimate whether the occultation could be due to reflection rather than thermal emission by estimating the geometric albedo,

$$\begin{equation} A_g=\frac{F_p}{F_{\star }}\frac{a^2}{R^2_p}. \end{equation} \tag{ 7 }$$

In the case that T_effp is greater than T_eq at the 99.7 percentile (3σ) and A_g is greater than 1, we identify the event as a stellar eclipse and the candidate as an EB FP. While unexpected, such a test may classify self-luminous planets (e.g., youth or external forces) as FPs.

A number of FPs were detected through the identification of secondary eclipses (see Section 5.8). The only planet in a multi-planet system with a detected occultation was Kepler-10b with A_g < 1 and T_effp ∼ T_eq. The lack of detected occultations in the multi-planet population is a consequence of the dearth of large, highly irradiated planets in these systems (see Figure 3). For the single-planet population, it is likely a handful of EBs that show occultations are classified as close-in Jupiter-sized PCs heated to ∼2000 K because such planets have an occultation and transit depth similar to an eclipsing low-mass star. The philosophy for the KOI catalog has been to keep a candidate classified as a PC until strong evidence is presented that shows the FP nature of the candidate. A consequence is that a handful of FPs will be misclassified as large (Jupiter-sized) candidates, which has no impact on our validation of multi-planet systems.

The orbital motion of a companion is imprinted in the photometric light curve due to day-night effects (thermal emission and reflectivity), ellipsoidal variations from gravity darkening due to tidal forces, and Doppler boosting from orbital motion. The latter two effects are dependent on the mass of the companion and, when present, can be used to estimate the mass of the companion (e.g., Mazeh et al. 2013).

There are cases when a companion interacts with the stellar surface through magnetic fields and produces star spots that could be misinterpreted as phase linked variations due to a massive companion (e.g., Tau Bootis; Walker et al. 2008). Thus our analysis would label a planet such as Tau Bootis as an FP.

To search for phase-curve variations (only relevant for short period systems), we filtered the data using the same procedure described in Section 4 except we changed the timescale of the polynomial fitter to five days instead of two days as the filter is destructive to astrophysical signals with a similar or longer timescale. This means our initial search is not sensitive to phase-linked variations on these longer timescales. The search was performed by calculating the occultation depth statistic introduced in Section 5.3, which is equivalent to average filtered data with a width equal to the transit duration. The standard deviation of the set of occultation measurements was calculated. This value was compared to the standard deviation of the data. This test determines whether the scatter is Gaussian on transit-duration timescales. When the ratio was found to be greater than two, variability in the phased light curve is detected. In these cases we inspected the light curves and found evidence of coherent EB effects as well as variable stars with fast pulsation or rotation timescales.

As was the case with the occultation search, it is expected that phase-linked variations will be rare because highly irradiated Jupiter-sized objects are rare in the Kepler multi-planet sample. KOI-1731.02 and 1447.02 were found to show phase curves and labeled as FPs. From the single-planet population, phase curves were discovered in KOI-23, 130, 143, 631, 636, 681, and 699. If ellipsoidal variations are source of the signal, then the mass of the companion was estimated, which ranged from ∼0.1 to 0.23 M_☉, indicating that these sources are EB FPs.

The occultation/eclipse search is effective when the orbital period is correctly estimated. In cases when EB eclipses are similar in depth, it is common to have the period off by a factor of two. For most EBs, the depths of the alternating (odd- and even-numbered) transits differ. We search for this odd–even effect by separately modeling the odd- and even-numbered transits where we only allowed R_p/R_⋆ to vary and the other parameters were fixed to their global solution. We used the change in R_p/R_⋆ as a proxy for a change in transit depth. When the change was greater than 3σ, we inspected the transit light curves to insure that the effect was real. For the cases where we noticed spot-crossing-induced variations in the transit depths, the systems were retained as candidates.

From the multi-planet sample, only KOI-966.01 was found to exhibit an odd–even transit effect. Thus, the true orbital period is double the reported KOI value. From the single-planet population, 102 candidates had detected odd–even effects, although this count is incomplete as an FP is not always searched for additional effects or FP signatures in the light curve.

A dominant source of FP planetary transit detections is EBs, or giant planet transits, on background stars that are captured in the aperture of the target star (BEB). These background signals are diluted by the target star and can have the appearance of small-planet transits. In this subsection, we describe the method we use to find KOIs with "clean" centroids, where the measurement is of high quality and there is no indication that the transit is not on the target star. This "clean" centroid standard, described in detail below, is a more stringent centroid standard than that used for PC status (see, for example, Batalha et al. 2013), and gives us confidence that the centroid signal is coincident on the sky with the target star.

We use centroid analysis to identify KOIs that are not clearly on the target star. The centroid method we use is the fit of a point response function (PRF) to the pixel difference image constructed by subtracting an average in-transit image from an average out-of-transit image (Bryson et al. 2013). This centroid method provides an offset from the target star position for each quarter, and the final offset for the KOI is a robust average of the quarterly offsets. We also use data quality metrics that indicate whether the data support the centroid offset measurement. We do not validate a KOI if the centroid offset suggests that the transit is not very close to the target star or if the data quality does not provide confidence that the transit signal is on the target star. This centroiding method does not work for highly saturated targets. Our treatment of saturated targets is described in Section 5.6.1.

We provide here a brief overview of the PRF-difference image centroid method. For details see Bryson et al. (2013). PRF-based centroids are measured on both out-of-transit and difference images quarter by quarter. When the target star is isolated, the centroid of the out-of-transit image gives the position of the target star. Assuming that the transit source is the only source of variability in the aperture, the centroid of the difference image gives the location of the transit signal source. These quarterly centroid measurements are robustly averaged to estimate the target star and centroid signal locations on the sky. These average locations are differenced to provide the average offset of the transit signal from the target. The robust average also provides a 1σ uncertainty per quarter, which is propagated through the robust average and offset calculation to provide an offset uncertainty. An alternative method for estimating the centroid uncertainty is via a bootstrap, using a resampling with replacement of the quarterly centroid measurements. The bootstrap-estimated uncertainty is also propagated through the robust average and offset calculation. We choose the larger of the two offset uncertainties when performing the cuts described below.

Centroid measurements are subject to several systematic errors, caused primarily by errors in the measurement of the Kepler PRF and crowding by background stars. The systematic error due to PRF error is mitigated by computing the offset as the difference between the PRF centroid of the out-of-transit image and the PRF centroid of the transit signal in the difference image. Because the transit source and the target star are near each other on the Kepler focal plane, their PRF errors are very similar so centroid systematic errors due to PRF error approximately cancel. The residual PRF error systematic varies from quarter to quarter and is statistically zero mean, so averaging over quarters further reduces PRF-error-driven centroid systematics. The residual systematics have a statistical standard deviation of less than 01. To account for this systematic error, a constant 0067 is added in quadrature to the final offset uncertainty. This added constant does not, however, eliminate all apparently significant offsets due to systematic error, so we pass any KOI with offsets less than 03, even if that offset is formally statistically significant.

In a few cases, there is a field star in the target's aperture that is brighter than the target. In this case, the centroid of the out-of-transit image is strongly biased by the bright star, and the centroid offsets are invalid. We detect such cases by computing the offset of the out-of-transit image centroid from the catalog position of the target star, and declare the centroid measurement to be invalid if the offset of the out-of-transit image centroid from the target star catalog position is ⩾15.

We classify a KOI as having "clean centroids" if it passes three criteria, described in more detail below: (1) it has a good centroid measurement, (2) that centroid measurement indicates small offsets from the target star, and (3) there is at least a 99% probability that the transit signal is on the target star rather than another known star.

Good centroid measurement. The quality of a centroid measurement is determined by several factors, most notably the transit S/N and systematic error. We do not validate KOIs as planets for which difference images are not available. There are three ways in which a KOI can fail to have a good measurement.

• 1.  
  When the S/N is very low, the measured offset uncertainty can be too large to sufficiently localize the transit signal. When the offset uncertainty is ⩾15, we say that the KOI does not have a good measurement.
• 2.  
  The measured offset of the out-of-transit centroid from the target star's catalog position is ⩾15, indicating that it is likely that the out-of-transit centroid measurement is strongly biased by crowding.
• 3.  
  The quality of the difference image in a quarter is determined by measuring the correlation of the difference image pixels with the fit PRF. If the correlation is less than 0.7, we consider the signal in the difference image too weak to trust the centroid value; otherwise we say that quarter has a good PRF quality. We demand that there be at least three quarters with good PRF quality, or that with two-thirds of the observed quarters have good PRF quality, otherwise we say the KOI does not have a good measurement.

Small offsets. We demand that the measured offsets be close to the target, satisfying both of the following criteria:

• 1.  
  The offset is statistically close, that is the offset is <3σ, or the offset is <03 to allow for small systematic error.
• 2.  
  The offset is smaller than 4''.

Probability ⩾99%. The systematic due to crowding is addressed via forward modeling of the observed pixels based on catalogs and the Kepler PRF (S. T. Bryson & T. D. Morton 2013, in preparation). A synthetic pixel scene is created for each quarter by placing a flux-scaled PRF at the pixel location of every known star close enough to contribute flux to the observed aperture. In this way, a synthetic pixel image modeling the average out-of-transit image is created for each quarter. A synthetic in-transit pixel image is created for each star in the aperture by reducing the flux of that star by the transit depth that best reproduces the overall observed transit depth, accounting for dilution. These images are analyzed for each star via difference-image PRF centroiding just like the observed pixels. The resulting offsets provide a prediction for the transit signal offset from the target star under the hypothesis that the transit occurs on each star in the aperture. The predicted offsets are compared with the observed offsets by inferring the underlying probability distributions. For each star in the aperture, the normalized integral of the product of the observed and modeled distributions provides a relative probability that the transit signal is at the same location as that star, when the modeled depth on that star is less than 100%. An unknown background source is also included as an alternative hypothesis. For details, see S. T. Bryson & T. D. Morton (2013, in preparation). For this paper we assume an underlying Gaussian distribution, which is characterized by the mean and uncertainty of the offset averages. We say a KOI is not clean if its relative probability is less than 0.99.

The KOI is considered clean if the measurement quality, small centroid offset, and probability criteria are all satisfied.

5.6.1. Manual KOI Inspection

Table 5 lists candidates that have newly detected companions inside the photometric aperture by Adams et al. (2012, 2013) and one of us (S.H.). Because these companions were not in the catalog used to compute the probability criterion described in Section 5.6, we give these special attention. Table 5 gives the observed offset of the newly found companion from the target star and the offset of the companion from the measured transit source in units of the centroid uncertainty. When the companion star is more than 3σ from the measured transit source, we consider that companion as ruled out as a source of the transit signal. Because our validation criteria includes the requirement that the centroid source be no more than 3σ from the target star, companions outside of 4σ will not reduce the KOI's probability of being on the target star to less than 99%. We do not consider whether or not the companion is gravitationally bound to the primary star.

When the companion stars are within 3σ of the transit position, the transit signal is not necessarily an FP. However, this indicates that we did not determine which star was the source of the transits. We do not validate such candidates unless we have strong evidence that the nearby star is a bound companion to the Kepler target as described in Section 9 of Paper II.

All of the FP tests described above have been applied to the original sample of 1212 PCs identified as potential multi-planet systems. For each FP, a brief description of the types of FPs detected in multi-planet transiting systems is presented below. The FP disposition was used for a comparison of the single planet, multi-planet, EB, and FP samples. There are three classes of FPs used to describe the nature of the transiting object: (1) Period and epoch (P/T0) collisions where multiple sources show the same orbital period and transit times. Such events can be produced by direct PRF contamination, optical reflections or electronic interference, such as crosstalk (Coughlin et al. 2014); (2) flux FPs, where the photometric light curve shows evidence of an EB; and (3) active-Pixel-Offsets (APOs) FPs, where centroid measurements of the photometric aperture indicate that the source of the transits is due to a source offset from the Kepler target. Categories (2) and (3) are not mutually exclusive. Table 1 gives a breakdown of the PCs for various cuts and number of systems with six candidates, five candidates, four candidates, and so forth.

There were 12 P/T0 collisions detected in the following KOIs: 376.01, 489.01, 989.02, 1119.01, 1196.01, 1231.01, 1231.02, 1747.02, 1803.02, 1806.01, 1944.02, and 2188.01. A secondary eclipse was detected for most of these systems, indicating that the primary sources of P/T0 collisions are EBs. The distribution of P/T0 collisions will not favor transit candidate targets, thus it is expected that the rate of P/T0 collisions is lower for multi-planet sample relative to the single planet sample. KOI-489.02 was flagged a FA with a transit S/N of 6.6, thus the KOI-489 system does not count as a multi-planet system in any of our statistical counts.

KOI-199.02 shows a secondary eclipse with a depth of 50 ppm and an orbital period of 8.8 days. The depth of the secondary eclipse is inconsistent with the planet hypothesis. Derived values from the occultation are A_g ∼ 22, T_eff ∼ 4500 K and T_eq = 1200 K.

KOI-376.01 is a P/T0 collision and shows strong quarterly depth variations due to quarterly variation of contamination. The second candidate, KOI-376.02, has a period of 1.4 days and shows a strong secondary eclipse with an observed depth of 260 ppm. Since the signals observed are likely heavily diluted, the true eclipse depths are likely much deeper.

KOI-379.01 was found to have an additional star within the photometric aperture with a separation of 1''. Centroid analysis points toward the fainter star as the source of the transits, thus this candidate has been flagged as an APO. Centroids analysis of KOI-379.02 is inconclusive to determine which star is the source of the transits and is kept as a PC.

KOI-414.01 shows a secondary eclipse with a depth of 400 ppm. The location of the secondary eclipse indicates that the orbit is non-circular. KOI-414.02 shows a clear centroid offset and was flagged as an APO.

KOI-2671.01 was marked as an FP as a secondary eclipse was detected. The occultation shows the orbit to be eccentric (35 hr offset). KOI-2671.02 has a centroid offset and was labeled as an APO.

KOI-989.01 and KOI-989.02 are the same event with the periods being integer multiples of one another. These two candidates were also flagged as a P/T0 collision. The confusion of these two candidates arose because of strong variations in the quarter transit depths from quarterly dependence of dilution. KOI-989.03 remains as a PC.

KOI-549.01, 549.02, 1196.01, 1231.01, 1378.02, and 2007.01 are flagged as APOs from centroid analysis. KOI-1119.01 is a P/T0 collision and 1119.02 shows strong centroid offsets. KOI-1342.01 shows a small offset of 09 with a significance of 4.1σ. It is therefore considered to be an APO. KOI-2159.02 shows centroid offsets and a secondary eclipse. KOI-1731.02 shows an occultation and phase-linked variations and was labeled as an FP because its transit depth appears to be heavily diluted.

KOI-966.01 shows an odd–even transit effect, but KOI-966.02 was labeled as an FA due the low S/N of the transit event. Thus, the KOI-966 system does not count as a multi for our statistics.

KOI-1447.01 showed a "V" shaped transit event with a depth greater that 15%. While transit-depth is not an indication of the FP nature of the candidate, KOI-1447.02 shows large amplitude phase-linked variations. Thus, KOI-1447.02 is a clear FP, which removes the KOI-1447 system as a multi-planet system. Due to the large transit depth, we classified 1447.01 as an FP.

From the single-planet population of 2482 PCs, 976 were classified as FPs resulting in an FP rate of ∼40%, which, as expected, is in stark contrast to the multi-planet FP rate. In total we found 26 FPs (including P/T0 collisions) in the multi-planet sample of 1167 PCs remaining after the removal of FAs and single planets. A few of the classified FPs had an associated FA, such as KOI-1231.02, and are included in the FP totals for the single planet population. The 26 FPs include cases of two FPs associated with the same target. Candidates that were flagged as having not clean centroids in Table 3 are not FPs and remain as unvalidated PCs. There is no strong evidence to suggest that any one target with not clean centroids is a blend; however, the probability of blends existing within the population is large enough that we cannot validate this sample at the 99th percentile. The 26 FPs are found around 20 systems, with 6 double FP systems, 12 cases of FP associated with a single PC, and 2 cases where 2 PCs and 1 FP where associated with the same target.

The results of the multi-planet disposition are summarized in Table 4 of Paper II, which gives a comparison and breakdown of the expected FP rate. The agreement between the observations and predictions is very good which leads to our conclusion that a vast majority of transiting candidates found in multi-planet systems are genuine planets. After removal of FPs, there are 1129 remaining multi-planet candidates. The next step is to explore this large population of candidates and set additional criteria to reduce the chances that undetected blends still exist. There will be a population of FPs from blends that exist in the Kepler transit sample, but cannot be detected via our methods, for example, a blend from a BEB where the separation from the target source is too small to be detected by centroid motion.

To reduce the number of potential FPs in our validated list of planets in multi-planet systems, we eliminate regions of phase space where we have reduced confidence. For example, we have reduced confidence in the validity of a PC if centroids cannot localize the position of transit to eliminate the chance of a background blend at the 99th percentile. The first requirement is that a candidate has a S/N > 10 as established in Section 5.2. This insures that the multi-planet sample is free of FAs and removes an additional 21 candidates after the removal of FPs and P/T0 collisions.

Using the analysis for Section 3, Section 4, and this section, we are able to use the criteria set out in Paper II to select a population of multi-planet transiting systems that have an FP rate substantially less than 1% (additional details below). Table 1 lists the number of PCs after various cuts and tests are applied. The various cuts are: FA, where either a transit candidate has insufficient S/N (<7.1) or was labeled as a non-transit event such as stellar variability, or was observed to have less than three transits. Col indicates a P/T0 collision. These sources are non-unique by nature so they are classified separately. FP indicates when an FP is identified that is a not a P/T0 collision. An FP can either be an EB masquerading as a PC or a diluted signal where the source of the transits has been localized off the Kepler target. SN: the transit models were used to determine the S/N of the phase folded transit for each candidate. We adopted a threshold of S/N > 10 to consider a transit event. P marks period cuts. We require the orbital period to be greater than 1.6 days due to the increased rate of FP found with shorter orbital periods (see Figure 2 in Section 5.8 and Appendix A of Paper II). b marks when cuts are made based on the measured impact parameter. When a transit is "V"-shaped, there is a larger chance that a FP has been identified. This does not mean that "V"-shaped events are FPs, only that we have less confidence in declaring such objects as planets. The fraction of "V"-shaped signatures that are produced by EBs as opposed to transiting planets is far larger than that for "U" shaped profiles. Our criteria for "V"-shaped transits is that b  +  b_σ + R_p/R_⋆ > 1.00. centr indicates the centroid test as outlined in Section 5.6. A target that does not pass our centroid test is not a statement that the object is an FP/APO, but rather that we had insufficient information to localize the source of the transits on the Kepler target. The column number of multi-pl indicates the total number of multi-planet systems that pass the indicated test. The column number of new multi-pl indicated the total number of multi-planet systems that pass the indicated test and have been previously verified (already assigned a Kepler ID).

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Figure 2 shows the cumulative distribution of orbital periods for candidate multi-planet systems (black), candidate single planet systems (red), FPs KOIs (green), P/T0 collisions (cyan), and EBs from the Kepler Eclipsing Binary catalog (blue).²¹ FA are not considered. The distribution of EBs shows a large population of short period events due to the inclusion of contact binaries. The sample of FPs from the KOI catalog also shows a relatively strong population of FPs at short orbital periods, which is different from the EB population, but there is still a much larger fraction of FPs at shorter orbital periods compared to the PC populations. There are two reasons for this difference. (1) KOIs are selected based on a visual inspection of the photometric transit event, with a requirement that the event has an appearance of a planetary transit. This process heavily reduces the number of contact binaries in the KOI catalog, as a distinct transit that shows a clear ingress and egress are not present. (2) The Kepler transit search algorithm does not conduct searches for events with periods less than 0.5 d, so unless a strong harmonic of the orbital period is detected at a longer period, many short period events will also be missed. Similarly, the distribution of P/T0 collisions, which is dominated by EBs, shows a larger fraction of events with short orbital periods relative to the PC distributions. As articulated in Appendix A of Paper II, the expected abundance of unidentified FPs in multis to planets in multis is far larger at small orbital periods than at large ones. Thus we do not validate PCs with orbital periods less than 1.6 days. We also excluded systems with an orbital period less than 4 days and a S/N < 15, due to concerns of an increased FP rate and the lack of well constrained transit model parameters due to the low S/N of the transit. Eliminating candidates with a P < 1.6 days reduced the sample of 1107 PCs in multis that passed our FP and S/N tests to 1084 PCs as shown by row 6 of Table 1.

Dispositioning of PCs relies on transit models to characterize the orbiting companion. A transit can be "V"-shaped in appearance when the transit duration is comparable to the photometric cadence or we have a grazing transit (b + R_p/R_⋆ ∼ 1). The transit model incorporates the cadence time to convolve the synthetic lightcurve to match observations and allows a quantitative assessment as to whether a grazing transit is observed. A grazing transit also results in increased uncertainty in transit model parameters, which makes assessment of the transit event difficult. From the single-transit population, after the removal of FAs, 16% have grazing or close to grazing transits (b + b_σ + R_p/R_⋆ > 1.00), which drops to 8% after the removal of FPs and P/T0 collisions, which is larger than one would expect based on an isotropic distribution of orbital planes. Almost half of the grazing-transit single PCs have a radius greater than 15 R_⊕, which indicates that a large fraction of this population is likely FPs. However, planetary radius is not a criterion to label a candidate as FP because an upper limit on the radius of a planet is not well established, and measurement uncertainty on "V"-shaped transits precludes making a definitive statement regarding the absolute radius of the transiting object. From the multi-planet population, after the removal of FAs, 3.8% are measured to be grazing or near grazing, which drops to 3.1% after the removal of FPs and T/P0 collisions. Only 1 (5.9%) of the grazing multi-planet candidates has a large inferred planetary radius, KOI-1477.01, which is associated with an EB (KOI-1477.02). It is very likely that KOI-1477.01 is also an EB FP. The lower rate of grazing transits in the multi-planet population leads us to conclude that a large fraction of the grazing transits in the single-planet population are FPs. This means that there is a higher probability of a FP being found when a grazing transit is present, thus we do not validate multi-planet candidates that have b + b_σ + R_p/R_⋆ > 1.00. These KOIs are still good PCs. Application of FP, P/T0, S/N, and cuts based on impact parameter reduces the set of multi-planet candidates for validation to 1054.

Cuts based on S/N, period, and transit shape use the transit models and comparison with the single-planet population to identify regions of model parameter space with reduced confidence in the validity of a PC when the rate of FPs is observed to be larger relative to the multi-planet population. Cuts based on methodology are presented in Section 5.6, where PRF models were used to identify which multi-planet candidates have at least a 99% probability that the transit signal is on the target star rather than another known or unknown star. This statement means that there is a low probability that a blended background transit event is present. After application of our centroid criteria, the number of multi-planet candidates still considered for validation is 851.

As previously mentioned, planet radius is not a criterion for classification of a transiting candidate as an FP. A problem thus arises for large Jupiter-sized PCs, as there is a degeneracy in radius for planets, brown dwarfs, and low-mass stars. Additional dynamical tests were applied to the multi-planet candidates: using Hill's criterion to test for stability of neighboring pairs of PCs, and when $R_{\rm p}+ 2\sigma _{R_{\rm p}} > 9\,R_{\oplus }$, dynamical fits to transit times observed in Q1–Q14 Kepler long cadence data were conducted to determine whether the giant candidates can have masses exceeding 13 M_J, assuming all of the candidates orbit the Kepler target. Details of both tests can be found in Appendix C of Paper II. All candidates that passed the above tests, apart from the special case of KOI-284 (Kepler-132, addressed in Section 9.1 of Paper II), passed the stability tests. Table 3 categorizes candidates as having passed, failed, or being too small to have been tested for mass limits large enough to be stars. The last row of Table 1 gives the number of planetary candidates after all tests have been applied and gives us a total of 851 planets that we validate. From this sample, 60 have been previously validated via other methods, thus we are able to introduce 768 newly validated planets, which roughly doubles the current number of confirmed and validated planets. Planets discovered and confirmed by the Kepler mission currently account for more than half of the known and validated extrasolar planets.

We discuss herein, validated multiple-planet systems that contain a planet that is in the nominal habitable zone of their star. The location of the habitable zone depends on stellar luminosity (and the orbital period range also depends on stellar mass), so we introduce only those planets whose host stars have been characterized spectroscopically in this section.

As we do not have information regarding either the albedo or atmospheric characteristics of these planets (nor of any moons that they might have), we can only make reasonable estimates of the flux of stellar radiation that they intercept, i.e., the amount of insolation that they receive. We therefore quote results in terms of the average solar flux intercepted by Earth, S, which is generally referred to as the solar constant. For the purposes of our tabulation, we list objects that intercept flux less than 1.5 S (comparable to the flux Venus received 1 billion years ago, see Kopparapu et al. 2013 and references therein). For comparison, Venus receives 1.91 S. We don't specify an outer boundary to the HZ because few of the Kepler planets that we have validated have significantly smaller insolation than does Earth, but note that Mars receives an average of 0.43 S. Orbital eccentricity, e, which generally is unknown, affects the flux of stellar radiation that a planet receives, but the change that it induces in the annual average insolation is roughly quadratic in e, and as few of the planets in Kepler's multis have large eccentricities, the magnitude of the change in mean insolation resulting from planetary eccentricity is likely to be small. The spectrum of stellar radiation received by a planet also affects atmospheric and surface temperatures (Kasting et al. 1993), but these variations are small compared to uncertainties in estimated insolation and in-atmospheric properties. Nonetheless, we note the effective temperatures of the stellar hosts to aid in investigations by other researchers. Only six of the planets that we validate intercept less than 150% of the radiation flux encountered by Earth. Two of these orbit KOI-701 (Kepler-62) and have been analyzed in detail by Borucki et al. (2013). The transits for the four new planets that receive less than 1.5 S, KOI-518.03 (Kepler-174 d), 1422.04 (Kepler-296 f) (see Paper II for more detailed discussion), 1430.03 (Kepler-298 d), and 1596.02 (Kepler-309 c), are shown in Figure 4. The planets have nominal radii of 2.19, 1.79, 2.50, and 2.51 R_⊕.

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Our work provides a substantial increase in the number of verified exoplanetary systems and demonstrates the ability of the Kepler mission to probe the statistics of exoplanetary systems with a sample that is relativity clean of FPs. Both transit models and centroid models are used to characterize the photometric data and various tests were used to identify FPs. The rate of FPs was found to be low relative to the single transiting planet population, in quantitative agreement with theory (Papers I and II). This result demonstrates that 851 PCs in multi-planet systems are valid planets at greater than the 99% level. In Appendix A, the multi-planet population was used to investigate the rate of hierarchical blends in multi-planet systems; while no limits on the occurrence rate can be currently set, it was found that the eccentricity distribution of transiting multi-planet systems found by Kepler is significantly different from the planetary distribution found by RV surveys. The list of validated planets presented is reliable, but the sample suffers from both incompleteness and strong biases. Many of the candidates that were not validated in this study are still excellent planetary candidates.

To construct a synthetic population, ρ_⋆ for the primary star was adopted from Table 2 and matching orbital periods for the orbiting planets from Table 3. Thus, a synthetic transiting planet is generated for each planet in our transiting multi-planet sample. A co-eval binary companion is generated for each star and later we decide whether the primary or secondary component is hosting the planet. The companion is not used to estimate the binary fraction, but to determine the change in ρ_c and estimate the number of hierarchical blends. A hierarchical blend occurs when transiting planets are observed around both components of a stellar binary. Masses for a bound companion (M₂) were chosen to be greater than 0.1 M_☉ and less than the primary (M₁) and have a mass ratio (q) distribution,

$$\begin{equation} N(q) \propto q^{n}, \end{equation} \tag{ A2 }$$

where n = −1 would produce a 1/q distribution matching the distribution observed in RV surveys (Trimble 1990). The transit depth from Table 3 together with an estimate of the luminosity of the primary (L₁) and secondary (L₂) stellar components was used to check that the undiluted transit depth around the fainter secondary star would not exceed 50%. This sets a lower limit on the luminosity of the secondary,

$$\begin{equation} L_{2,{\rm min}} = 2 L_1 T_{{\rm dep}}, \end{equation} \tag{ A3 }$$

where T_dep is the transit depth from Table 3. If L₂ < L_{2, min}, then we choose another mass for the stellar companion and repeated until a suitable choice is found for this check. To determine which stellar component the planet would be orbiting in our model, a fitted parameter, binfrac, was used to represent the fraction of planets that are orbiting the fainter, bound companion. For each transiting planet in the system we drew a uniform random number. If that number was less than binfrac, then we adopted ρ_⋆ of the bound companion.

A system has now been constructed that consists of two bound stars, each having a probability of having a transiting planet. To account for eccentricity in the synthetic population, a two-parameter model of the eccentricity distribution based on the beta function, as described in Kipping (2013), was adopted,

$$\begin{equation} P_{\beta } = \frac{1}{\beta (x,y)} e^{x-1} (1-e)^{y-1}, \end{equation} \tag{ A4 }$$

where e was the eccentricity of the planet and x and y were fitted parameters. Other distributions, such as a Rayleigh distribution, could also be used to produce similar results. The distribution P_β was used to draw a value of e for each transiting planet. With the orbital period and ρ_⋆ selected, a/R_⋆ was calculated using Equation (1). The argument of periapsis (ω) was then randomly selected from a uniform distribution and used to determine the star–planet separation, d/R_⋆, during transit. The transit probability was then calculated,

$$\begin{equation} T_{{\rm prob}} = \frac{R_\star }{d}, \end{equation} \tag{ A5 }$$

and a uniform random number from 0 to 1 was drawn. If the random number was greater than T_prob then the choice of ω was rejected and a new value was drawn and the exercise repeated. This process insures that transits occurring near periastron are preferred. The estimate of d/R_⋆ was substituted in Equation (1) to estimate ρ_c,

$$\begin{equation} \left(\frac{d}{R_\star } \right)^3 \simeq \frac{\rho _cG P^2}{3 \pi }. \end{equation} \tag{ A6 }$$

Measurement error was incorporated by choosing a model solution from the MCMC analysis for the corresponding KOI and comparing ρ_c to the median value of ρ_c from all the chains. The difference was added to the synthetic value of ρ_c. To investigate the dependance of the synthetic model on reliability of estimating uncertainties in transit parameters, we used a nuisance parameter, errfrac, to scale the errors on ρ_⋆ as measured by the transit when assuming a circular orbit.

In Figure 5, we plot the binned distribution of dρ_{c, i − j} based on various synthetic populations for comparison to the observations. The cyan line shows a population of planets with only circular orbits and no hierarchical blends; only incorporating measurement error. This model does not match the observations shown in black. The red line was produced by incorporating measurement error and an eccentric distribution of planets from RV planets (Wright et al. 2011) with best-fit parameters from Kipping (2013). This model produced a better fit, but not an ideal fit to the observations. The green line shows a population produced using measurement error, circular orbits and a hierarchical blend rate of 0.5 and N(q)∝q⁻¹. In this scenario, half of the planets are transiting primarily low-mass stellar companions. The blue line shows a population with circular orbits and a hierarchical blend rate of 0.5 and a uniform distribution of q. For both cases with hierarchical blends, we see an overabundance of mismatches in ρ_c between planet pairs, with the strength of the mismatches modulated by the distribution of q.

To measure posterior distributions of the parameters to describe the planet population we use a MCMC routine that uses methods similar to the description found in Section 4.1. In Figure 6 we show distributions for binfrac, n, a, b, and errfrac based on 48,000 chains. Parameters where restricted to binfrac = {0, 0.5}, n = { − 2: 2}, a, b = {0, 10} and errfrac > 0. It is immediately clear that posterior distributions for each fitted parameter are quite broad, but we can draw a few conclusions.

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An extensive search for blended companions based on KECK high-resolution spectra was described in Section 8.1 of Paper II. The sample included 270 multi-planet candidate systems and would be sensitive to blends due to companions that are 2%–3% as bright as the companion and show a RV difference of ∼10 km s⁻¹. From this sample, only one clear blend was found, and in that case (KOI-2311.02) the S/N of the transit was found to be very low, so we do not even consider that system to be a multi-planet candidate. However, based on the one potential blend detection, an estimate of the blend rate is 0.004 ± 0.004 for companion stars that are 2%–3% as bright as the primary. Beyond ∼5 AU, the RV component of the stellar binary will be too small to allow reliable detection of stellar blends, which would account for less than half of companions found in solar-type stars (Raghavan et al. 2010). We double the potential blend rate and take the 3σ upper limit to get a rough estimate of the number of hierarchical blends that could exist. The synthetic population has 1158 planets in 460 systems, so observations suggest that no more than 14 blended systems could exist and either be missed by the spectroscopic survey or have separations large enough that a companion would not be detected. The inclusion of high-resolution imaging observations would allow additional constraints on the number of companions detected at large separations. In Figure 6, all simulations that have less than 14 blended systems (e.g., at least one planet around each stellar component) have been marked in red.

The rate of hierarchical blends, binfrac, was found to be dependent on how well uncertainties are determined for ρ_c. The transit model used a fixed set of limb-darkening parameters, thus we expect our uncertainties to be both underestimated and potentially systematically biased. We do not expect our uncertainties to be underestimated. There is weak (2σ) evidence that there are zero hierarchical blends in our sample based on the measurement uncertainties in Table 3. Limits on the blend rate from spectroscopic analysis also suggests that the uncertainties on ρ_c have been underestimated.

The eccentricity distribution as parameterized by Equation (A4) is relatively independent of binfrac, errfrac, and n. Figure 7 shows the range of allowed eccentricity distributions in black based on the synthetic populations from the MCMC analysis with 1σ uncertainties. The blue line shows the eccentricity distribution based on RV surveys (Wright et al. 2011) based on an analysis by Kipping (2013). A chi-square test of the two distributions gives χ² = 46.2 for 10 samples, which indicates that the two distributions are different with high confidence. The multi-planet eccentricity distribution is more sharply peaked toward zero, thus highly eccentric planets in multi-planet systems are relativity rare compared to the RV sample. It would be interesting to further break down the RV sample to compare the eccentricity distributions of single and multi-planet samples, but it outside the scope of the initial analysis presented here.

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The difference in ρ_c when hierarchical blends are present will be strongly dependent on probability distribution of the underlying mass function. As shown in Figure 5, the distribution of dρ_{c, i − j} will contain an increased number of large mismatches as n decreases. In the case of equal or nearly equal-mass binaries the difference in ρ_c will be indistinguishable from measurement error. The synthetic population model shows that as errfrac is reduced, n pushes toward large values that produce a large number of equal mass binaries. For errfrac = 1, there is no strong measurement of the mass fraction distribution.

The value of ρ_c can also be directly compared to ρ_⋆ (i.e., Tingley et al. 2011) from Table 2. Eccentric planets will be seen when the transit occurs close to pericenter, which decreases the transit duration relative to a circular orbit. This pushes ρ_c toward larger values (a denser star). As stated above, hierarchical blends will also produce a bias toward larger values of ρ_c as Kepler planet host stars are typically close to the main sequence. Measurement error should not introduce any bias. A comparison of ρ_⋆ and ρ_c in a similar manor to the comparison of ρ_c for planet pairs could be carried out, but a strict requirement is that ρ_⋆ is a good estimate of the true mean stellar density, which is not true. In particular, the use of Yale-Yonsei evolution models are known to produce radii too large and hence, densities that are systematically too small for low-mass stars (Plavchan et al. 2014). However, when ρ_⋆ is based on asteroseismology (Huber et al. 2013), such biases are likely better controlled. Figure 8 shows the difference in ρ_⋆ and ρ_c scaled by the uncertainty versus ρ_⋆. The asteroseismology sample is limited to solar-like and evolved stars as the amplitudes of p-mode oscillations scale proportionally to stellar luminosity. The bias toward smaller values of ρ_⋆ for low mass stars with ρ_⋆ based on Yonsei-Yale models can be seen for stars with ρ_⋆ >3 g cm⁻³. From this small sample, there is evidence of a bias of ρ_c being larger that ρ_⋆; however, the sample is too small to draw any inferences on the underlying eccentricity and hierarchical blend population.

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Currently, the sample of multi-planet systems and well characterized stars is too small to draw strong conclusions about the number of hierarchical blends. The strongest constraints currently come from observations that attempt to directly detect a nearby companion that may be gravitationally bound. When additional multi-planet candidates and observations become available, model fits can be repeated to determine the rate of hierarchical blends and, in turn, establish whether orbital planes in binary stars systems are aligned.