A HIGH FALSE POSITIVE RATE FOR KEPLER PLANETARY CANDIDATES OF GIANT STARS USING ASTERODENSITY PROFILING^*

From the several thousand Kepler Objects of Interest (KOIs) known at the time of writing, we focus on those KOIs with asteroseismically determined stellar densities. While asteroseismology is not a requisite for conducting AP, it is usually the most accurate and precise measurement available and is often referred to as the "gold standard" (Bastien et al. 2013). Huber et al. (2013) recently provided a homogeneous catalog of asteroseismically determined stellar densities for 77 planet-candidate host stars. This sample, including 107 planetary candidates, forms the input catalog for our work. We subsequently refer to the independent measure of the stellar density used in our AP analysis as ρ_{⋆, astero}.

In this work, we limit our sample to only those host stars with a single transiting planet candidate. The rationale for this choice is two-fold. First, one of the objectives of our work is to provide insights into the FPR between the dwarf and giant planet-hosting stars and since multi-planet systems are known to have a very low FPR (Lissauer et al. 2012, 2014), the single KOIs represent the more unknown subset. The second reason is that we propose that the single KOIs have a higher a priori probability of exhibiting large AP discrepancies than the multiple planet systems. This choice can be understood by considering the six known AP effects discussed recently in Kipping (2014a).

• 1.  
  PE effect. Orbit of the transiting body is non-circular; causes ρ_{⋆, obs}>ρ_{⋆, astero} if 0 < ω < π and ρ_{⋆, obs}<ρ_{⋆, astero} if π < ω < 2π.
• 2.  
  PB effect. A background, foreground, or associated star dilutes the transit depth; causes ρ_{⋆, obs}<ρ_{⋆, astero}.
• 3.  
  Photo-timing (PT) effect. Unaccounted TTVs affect the composite transit light curve; causes ρ_{⋆, obs}<ρ_{⋆, astero}.
• 4.  
  Photo-duration (PD) effect. Unaccounted transit duration variations (TDVs) affect the composite transit light curve; causes ρ_{⋆, obs}<ρ_{⋆, astero}.
• 5.  
  Photo-spot (PS) effect. Unocculted starspots behave like an anti-blend, enhancing the transit depth; causes ρ_{⋆, obs}>ρ_{⋆, astero}.
• 6.  
  Photo-mass (PM) effect. The mass of the transiting body is significant and one cannot assume M_transiter ≪ M_⋆; causes ρ_{⋆, obs}>ρ_{⋆, astero}.

Although we direct the reader to Kipping (2014a) for exact formulae and details of each AP effect, we point out that the PT, PD, PS, and PM effects are all generally much weaker (typically ≲ 10⁻¹–10⁰ effect) than the PB and PE effects (typically ≲ 10¹–10³). Since large AP discrepancies are the most interesting to study, one should expect such variations to be caused by either the PB or PE effects. Multi-planet systems certainly have a low a priori probability of exhibiting the PB effect, since the FPR is known to be very low (Lissauer et al. 2012, 2014). They are also unlikely to have planets on large eccentricities in order to be dynamically stable, which means we expect low PE effects. For these reasons, we argue that single KOIs have a higher a priori probability of exhibiting large and thus interesting AP discrepancies.

An added bonus of studying the single KOIs exclusively is that they are less probable, a priori, to exhibit TTVs and thus the PT effect. Physically speaking, this is because they are less likely to have nearby planets near mean motion resonance inducing large perturbations (Agol et al. 2005; Holman et al. 2005). Mazeh et al. (2013) recently reported that multi-transiting KOIs exhibit significant TTVs in 120 out of 894 cases (≃13%), whereas single-transiting KOIs show significant TTVs in just 23 out of 1066 cases (≃2%). Since periodic TDVs are usually associated with periodic TTVs (Kipping 2009; Nesvorný et al. 2013), then the a priori probability of the PD effect is also significantly less. On this basis, we argue that any large observed AP variations found in this sample are likely due to either (1) the PE effect, (2) the PB effect, or (3) the target with asteroseismology modes detected is not the same as the target with the transiting body.

In the Huber et al. (2013), there are 43 KOIs in single transiting systems. Of these, we exclude two objects for different reasons. KOI-42.01 shows strong TTVs (Van Eylen et al. 2013) and so falls in that small 2% category of dynamically active single KOIs. In addition, we also exclude KOI-981.01 as our attempts to fit the light curve were unable to retrieve a converged, unimodal ephemeris due to excessive correlated noise. This leaves us with 41 KOIs for our survey, which are listed in the first column of Table 1. Note that we include KOIs regardless as to whether they have been dispositioned as a false-positive or not, since the theory of AP is general for any eclipsing body, not just planets (Kipping 2014a).

Table 1. Fitted Transit Light Curve Parameters for 41 Single KOIs with Asteroseismology

KOI	RP/R⋆	log10(ρ⋆ (kg m−3))	b	P (days)	q1	q2
1.01	$0.1258_{-0.0015}^{+0.0020}$	$3.1853_{-0.0096}^{+0.0134}$	$0.8442_{-0.0027}^{+0.0025}$	$2.4706123_{-0.0000024}^{+0.0000023}$	$0.344_{-0.034}^{+0.037}$	$0.31_{-0.22}^{+0.29}$
2.01	$0.077738_{-0.000012}^{+0.000012}$	$2.44165_{-0.00058}^{+0.00058}$	$0.50033_{-0.00078}^{+0.00079}$	$2.204735409_{-0.000000015}^{+0.000000015}$	$0.2712_{-0.0014}^{+0.0014}$	$0.3366_{-0.0028}^{+0.0028}$
7.01	$0.02505_{-0.00024}^{+0.00023}$	$2.593_{-0.051}^{+0.055}$	$0.351_{-0.142}^{+0.087}$	$3.2136701_{-0.00000065}^{+0.00000065}$	$0.385_{-0.032}^{+0.034}$	$0.333_{-0.043}^{+0.046}$
64.01	$0.03933_{-0.00057}^{+0.00096}$	$2.463_{-0.021}^{+0.022}$	$0.9380_{-0.0033}^{+0.0033}$	$1.95108246_{-0.00000026}^{+0.00000026}$	$0.474_{-0.075}^{+0.068}$	$0.15_{-0.11}^{+0.21}$
69.01	$0.01508_{-0.00013}^{+0.00014}$	$3.160_{-0.051}^{+0.051}$	$0.346_{-0.129}^{+0.086}$	$4.72673879_{-0.00000053}^{+0.00000053}$	$0.358_{-0.023}^{+0.025}$	$0.386_{-0.039}^{+0.041}$
75.01	$0.03979_{-0.00012}^{+0.00012}$	$1.8538_{-0.0099}^{+0.0105}$	$0.6808_{-0.0068}^{+0.0063}$	$105.881608_{-0.000034}^{+0.000034}$	$0.311_{-0.010}^{+0.011}$	$0.559_{-0.040}^{+0.041}$
87.01	$0.0215_{-0.0012}^{+0.0017}$	$3.47_{-0.46}^{+0.35}$	$0.67_{-0.48}^{+0.19}$	$289.86442_{-0.00087}^{+0.00088}$	$0.41_{-0.16}^{+0.32}$	$0.16_{-0.12}^{+0.35}$
97.01	$0.082950_{-0.000083}^{+0.000084}$	$2.3688_{-0.0033}^{+0.0032}$	$0.5574_{-0.0036}^{+0.0037}$	$4.88548901_{-0.00000018}^{+0.00000018}$	$0.3062_{-0.0085}^{+0.0088}$	$0.354_{-0.015}^{+0.016}$
98.01	$0.045650_{-0.000077}^{+0.000076}$	$2.2252_{-0.0080}^{+0.0082}$	$0.5893_{-0.0075}^{+0.0072}$	$6.79012304_{-0.00000060}^{+0.00000061}$	$0.2760_{-0.0093}^{+0.0096}$	$0.350_{-0.025}^{+0.026}$
107.01	$0.01993_{-0.00022}^{+0.00052}$	$2.679_{-0.145}^{+0.064}$	$0.33_{-0.20}^{+0.21}$	$7.2569658_{-0.0000043}^{+0.0000043}$	$0.392_{-0.075}^{+0.090}$	$0.35_{-0.10}^{+0.11}$
113.01	$0.75_{-0.28}^{+0.17}$	$3.703_{-0.067}^{+0.048}$	$1.56_{-0.30}^{+0.18}$	$386.5980986_{-0.0000017}^{+0.0000007}$	$0.609_{-0.083}^{+0.089}$	$0.15_{-0.11}^{+0.22}$
118.01	$0.01549_{-0.00028}^{+0.00056}$	$2.964_{-0.237}^{+0.077}$	$0.34_{-0.26}^{+0.28}$	$24.993233_{-0.000042}^{+0.000040}$	$0.29_{-0.12}^{+0.15}$	$0.44_{-0.25}^{+0.31}$
122.01	$0.02061_{-0.00010}^{+0.00014}$	$3.246_{-0.046}^{+0.012}$	$0.13_{-0.12}^{+0.16}$	$11.5230707_{-0.0000040}^{+0.0000041}$	$0.369_{-0.060}^{+0.067}$	$0.314_{-0.079}^{+0.087}$
257.01	$0.02371_{-0.00050}^{+0.00062}$	$2.985_{-0.031}^{+0.033}$	$0.8621_{-0.0081}^{+0.0073}$	$6.8834063_{-0.0000012}^{+0.0000012}$	$0.298_{-0.023}^{+0.022}$	$0.47_{-0.28}^{+0.28}$
263.01	$0.01466_{-0.00096}^{+0.00081}$	$3.03_{-0.29}^{+0.36}$	$0.70_{-0.042}^{+0.12}$	$20.719416_{-0.000029}^{+0.000028}$	$0.38_{-0.13}^{+0.30}$	$0.29_{-0.22}^{+0.41}$
268.01	$0.02063_{-0.00015}^{+0.00028}$	$2.691_{-0.118}^{+0.041}$	$0.26_{-0.20}^{+0.21}$	$110.37849_{-0.00013}^{+0.00013}$	$0.339_{-0.061}^{+0.072}$	$0.146_{-0.080}^{+0.082}$
269.01	$0.01056_{-0.00033}^{+0.00036}$	$2.05_{-0.22}^{+0.23}$	$0.825_{-0.089}^{+0.053}$	$18.011628_{-0.000029}^{+0.000030}$	$0.375_{-0.059}^{+0.067}$	$0.045_{-0.038}^{+0.070}$
273.01	$0.0214_{-0.0011}^{+0.0017}$	$2.959_{-0.080}^{+0.083}$	$0.9325_{-0.0100}^{+0.0086}$	$10.5737625_{-0.0000071}^{+0.0000072}$	$0.608_{-0.092}^{+0.083}$	$0.49_{-0.35}^{+0.34}$
276.01	$0.01978_{-0.00074}^{+0.00083}$	$3.30_{-0.22}^{+0.23}$	$0.59_{-0.36}^{+0.14}$	$41.746004_{-0.000032}^{+0.000032}$	$0.396_{-0.087}^{+0.143}$	$0.26_{-0.15}^{+0.22}$
280.01	$0.01952_{-0.00031}^{+0.00045}$	$3.036_{-0.060}^{+0.074}$	$0.866_{-0.018}^{+0.013}$	$11.8728958_{-0.0000044}^{+0.0000044}$	$0.363_{-0.043}^{+0.044}$	$0.18_{-0.14}^{+0.24}$
281.01	$0.01695_{-0.00069}^{+0.00058}$	$2.21_{-0.19}^{+0.25}$	$0.62_{-0.32}^{+0.11}$	$19.556609_{-0.000023}^{+0.000023}$	$0.392_{-0.070}^{+0.091}$	$0.32_{-0.15}^{+0.17}$
288.01	$0.01377_{-0.00031}^{+0.00035}$	$2.40_{-0.14}^{+0.14}$	$0.48_{-0.27}^{+0.14}$	$10.2753113_{-0.0000055}^{+0.0000055}$	$0.334_{-0.047}^{+0.055}$	$0.40_{-0.10}^{+0.12}$
319.01	$0.0485_{-0.0019}^{+0.0019}$	$2.344_{-0.027}^{+0.025}$	$0.9050_{-0.0042}^{+0.0041}$	$46.151110_{-0.000026}^{+0.000026}$	$0.430_{-0.036}^{+0.039}$	$0.58_{-0.39}^{+0.30}$
975.01	$0.007573_{-0.000019}^{+0.000019}$	$2.7520_{-0.0061}^{+0.0045}$	$0.178_{-0.018}^{+0.023}$	$2.78581687_{-0.00000063}^{+0.00000057}$	$0.280_{-0.029}^{+0.031}$	$0.425_{-0.053}^{+0.057}$
1282.01	$0.04729_{-0.00072}^{+0.00105}$	$2.636_{-0.110}^{+0.037}$	$0.23_{-0.18}^{+0.22}$	$30.863933_{-0.000080}^{+0.000078}$	$0.36_{-0.14}^{+0.20}$	$0.48_{-0.20}^{+0.28}$
1537.01	$0.00720_{-0.00032}^{+0.00071}$	$2.45957_{-0.383943}^{+0.120555}$	$0.43_{-0.29}^{+0.32}$	$10.191592_{-0.000049}^{+0.000045}$	$0.47_{-0.20}^{+0.25}$	$0.66_{-0.31}^{+0.25}$
1618.01	$0.00514_{-0.00013}^{+0.00014}$	$2.734_{-0.147}^{+0.046}$	$0.23_{-0.20}^{+0.28}$	$2.3643709_{-0.0000062}^{+0.0000064}$	$0.25_{-0.16}^{+0.25}$	$0.25_{-0.20}^{+0.43}$
1621.01	$0.01199_{-0.00031}^{+0.00048}$	$2.998_{-0.326}^{+0.080}$	$0.36_{-0.27}^{+0.34}$	$20.310507_{-0.000054}^{+0.000053}$	$0.27_{-0.14}^{+0.29}$	$0.18_{-0.16}^{+0.36}$
1890.01	$0.00954_{-0.00014}^{+0.00042}$	$2.679_{-0.264}^{+0.056}$	$0.30_{-0.22}^{+0.33}$	$4.3364290_{-0.0000038}^{+0.0000034}$	$0.38_{-0.11}^{+0.15}$	$0.37_{-0.20}^{+0.23}$
1924.01	$0.004264_{-0.000056}^{+0.000065}$	$1.815_{-0.017}^{+0.014}$	$0.341_{-0.032}^{+0.035}$	$2.1191569_{-0.0000021}^{+0.0000023}$	$0.986_{-0.013}^{+0.010}$	$0.99955_{-0.00160}^{+0.00045}$
1962.01	$0.03700_{-0.00092}^{+0.00137}$	$4.064_{-0.221}^{+0.066}$	$0.31_{-0.23}^{+0.29}$	$32.858685_{-0.000055}^{+0.000051}$	$0.36_{-0.17}^{+0.31}$	$0.39_{-0.26}^{+0.34}$
371.01	$0.40_{-0.29}^{+0.45}$	$2.85_{-0.12}^{+0.14}$	$1.36_{-0.30}^{+0.45}$	$498.3915_{-0.0012}^{+0.0012}$	$0.33_{-0.25}^{+0.30}$	$0.66_{-0.37}^{+0.25}$
674.01	$0.03772_{-0.00020}^{+0.00029}$	$2.274_{-0.038}^{+0.013}$	$0.15_{-0.11}^{+0.13}$	$16.338893_{-0.000013}^{+0.000013}$	$0.356_{-0.055}^{+0.062}$	$0.511_{-0.079}^{+0.092}$
1222.01	$0.00513_{-0.00035}^{+0.00049}$	$2.74_{-0.41}^{+0.18}$	$0.41_{-0.28}^{+0.35}$	$4.285768_{-0.000061}^{+0.000046}$	$0.54_{-0.34}^{+0.32}$	$0.53_{-0.35}^{+0.33}$
1230.01	$0.07859_{-0.00020}^{+0.00019}$	$1.9996_{-0.0082}^{+0.0086}$	$0.317_{-0.021}^{+0.019}$	$165.739391_{-0.000084}^{+0.000080}$	$0.390_{-0.017}^{+0.019}$	$0.441_{-0.021}^{+0.021}$
1299.01	$0.02855_{-0.00028}^{+0.00057}$	$2.118_{-0.084}^{+0.044}$	$0.28_{-0.17}^{+0.16}$	$52.500934_{-0.000049}^{+0.000048}$	$0.287_{-0.033}^{+0.036}$	$0.744_{-0.083}^{+0.095}$
1314.01	$0.01180_{-0.00031}^{+0.00068}$	$1.965_{-0.291}^{+0.093}$	$0.38_{-0.29}^{+0.30}$	$8.575116_{-0.000031}^{+0.000030}$	$0.54_{-0.20}^{+0.26}$	$0.32_{-0.19}^{+0.28}$
1894.01	$0.01739_{-0.00039}^{+0.00087}$	$1.79_{-0.23}^{+0.10}$	$0.41_{-0.31}^{+0.25}$	$5.2879067_{-0.0000085}^{+0.0000085}$	$0.361_{-0.092}^{+0.115}$	$0.51_{-0.15}^{+0.21}$
2133.01	$0.019429_{-0.000066}^{+0.000109}$	$1.6382_{-0.0348}^{+0.0066}$	$0.093_{-0.093}^{+0.160}$	$6.2467332_{-0.0000046}^{+0.0000046}$	$0.977_{-0.044}^{+0.022}$	$0.044_{-0.025}^{+0.019}$
2481.01	$0.01460_{-0.00050}^{+0.00092}$	$1.70_{-0.31}^{+0.11}$	$0.39_{-0.27}^{+0.30}$	$33.84513_{-0.00052}^{+0.00086}$	$0.24_{-0.11}^{+0.25}$	$0.85_{-0.24}^{+0.11}$
2640.01	$0.01619_{-0.00043}^{+0.00064}$	$2.33_{-0.43}^{+0.16}$	$0.49_{-0.37}^{+0.29}$	$33.17354_{-0.00014}^{+0.00014}$	$0.104_{-0.072}^{+0.118}$	$0.52_{-0.33}^{+0.33}$

Note. Rows above the horizontal line have log g > 3.7 and those below have log g ⩽ 3.7.

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We here describe the procedure used to detrend and fit the Kepler transit light curves. We first downloaded all available light curves spanning quarters 1–16 for each object from the Mikulski Archive for Space Telescopes database.³ Where available we use the SC data over the long cadence (LC) and we always use the Simple Aperture Photometry time series. We exclude all data greater than three transit durations on either side of the times of transit minimum.

For each KOI, our goal is to derive posterior distributions for the fitted parameters in a Bayesian framework. This is achieved by coupling a Bayesian regression routine to a transit light curve model. To model the transits, we make the same idealized assumptions as Seager & Mallen-Ornélas (2003): spherical planet, spherical star, circular orbits, no blended light, opaque planet, etc. The transit light curves are generated using the Mandel & Agol (2002) algorithm described by seven free parameters with the following priors.

• 1.  
  log₁₀(ρ_{⋆, obs} (kg m⁻³)). log-base-ten of the observed stellar density. Uniform prior from 0 < log₁₀(ρ_{⋆, obs} (kg m⁻³)) < 6.
• 2.  
  (R_P/R_⋆) = p. Ratio of the planetary candidate's radius to the star's radius. Uniform prior 0 < (R_P/R_⋆) < 1.
• 3.  
  b. Impact parameter of the transit. Uniform prior 0 < b < 2.
• 4.  
  P. Orbital period of the planet. Uniform prior $(\bar{P}-1\,(\mathrm{{\rm days}}))<P<(\bar{P}+1\,({{\rm days}}))$, where $\bar{P}$ is the period reported by Borucki et al. (2011).
• 5.  
  τ. Time of transit minimum. Uniform prior $(\bar{\tau }-1\,(\mathrm{d}))<\tau <(\bar{\tau }+1(\,\mathrm{d}))$, where $\bar{\tau }$ is the period reported by Borucki et al. (2011).
• 6.  
  q₁. First modified quadratic limb darkening coefficient defined in Kipping (2013a). Uniform prior 0 < q₁ < 1.
• 7.  
  q₂. Second modified quadratic limb darkening coefficient defined in Kipping (2013a). Uniform prior 0 < q₂ < 1.

We highlight that the stellar density is fitted uniformly in log-space, which can also be thought of as a Jeffreys prior in ρ_{⋆, obs} (Jeffreys 1946). We choose a Jeffreys prior for this term since it can span several orders of magnitude and it is generally considered the most uninformative prior choice possible. We also highlight that the limb darkening coefficients use the uninformative priors proposed in Kipping (2013a), which both improves the sampling efficiency and ensures complete coverage of the physically permissible prior volume. LC data are resampled to account for smearing using N_resam = 30 and the technique described in Kipping (2010).

In general, our Bayesian regression routine is a Markov Chain Monte Carlo (MCMC) algorithm (see Gregory 2005 for the use of MCMC in uncertainty estimates) using the Metropolis–Hastings rule (Metropolis et al. 1953; Hastings 1970). We use a Gaussian likelihood function in all fits and the seven parameters are allowed to vary with jump sizes tuned to be between 10% and 100% of the parameter uncertainties with the goal of ≃ 10%–40% of trials accepted. For each trial, we first compute the trial model and then determine the best fitting linear slope for each transit epoch which matches the observations and the trial model. This is achieved using a least squares linear minimization routine at every trial and naturally for cases where there are many transits and SC data, the computational time to achieve this is significant (see Kundurthy et al. 2013 for a previous example of this technique). Nevertheless, this approach essentially detrends the data simultaneously to fitting the actual transit model and thus the parameter uncertainties are more realistic.

In certain cases, visual inspection of the light curves revealed that linear detrending was insufficient to adequately correct the photometry, since substantial curvature existed in the out-of-transit baseline data. These cases were usually, but not exclusively, long-period KOIs, since such transits have longer transit durations and thus the baseline can cover several days. In these cases, we opted to use a more sophisticated detrending algorithm devised by the Hunt for Exomoons with Kepler project (Kipping et al. 2012a) known as CoFiAM (see Kipping et al. 2013 for details). This algorithm is well suited for longer-period planets and works on a Fourier-basis to guarantee that the transit profile remains undisturbed by the detrending procedure. The detrended light curves are then fitted using a multimodal nested algorithm, MultiNest (Feroz et al. 2008, 2009), rather than MCMC, since MultiNest is both more expedient in low-dimensional space and requires no tuning of jump sizes.

It is important to note that the regression algorithms are both well-established Bayesian Monte Carlo routines with identical input priors and thus the inferred posterior distributions are statistically equivalent. In cases where linear detrending is sufficient, one should expect consistent results between CoFiAM plus MultiNest versus linear detrending plus MCMC. This was verified in the example of KOI-273.01, where we infer a nearly identical a posteriori distribution for ρ_{⋆, obs}, as shown in Figure 1.

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We provide all of the fitted transit parameters for the 41 KOIs in our sample in Table 1, except for τ as this is the least relevant term for our study (available upon request). Table 2 provides a direct comparison of ρ_{⋆, obs} and ρ_{⋆, astero}, which is essentially the act of AP (this is also visualized in Figure 2). As discussed in Section 2.1, any significant AP discrepancies in this sample are likely due to either the PE effect, the PB effect, or that the transiting body in fact orbits an alternative star to the asteroseismically measured target. For each of these three possible explanations, we can quantify a relevant descriptive parameter. For the PE effect, the minimum eccentricity, e_min, naturally falls out of the AP expressions and we use Equation (39) of Kipping (2014a), the results of which are shown in Column 4 of Table 2. Similarly, for the PB effect we use Equation (17) of Kipping (2014a), with results shown in Column 5 of Table 2, provided b < (1 − p) which is a underlying assumption for the derivation of the PB effect.

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Table 2. Asterodensity Profiling Parameters for the 41 KOIs in our Sample

KOI		ρ⋆, obs (kg m−3)	ρ⋆, astero (kg m−3)		emin	B	ρ⋆, alt, max (kg m−3)		Classification
1.01		$1532_{-34}^{+48}$	1530 ± 30		$0.0074_{-0.0051}^{+0.0081}$	$0.991_{-0.087}^{+0.086}$	$2636_{-55}^{+72}$		N
2.01		$276.47_{-0.37}^{+0.37}$	271.2 ± 3.2		$0.0064_{-0.0038}^{+0.0040}$	$0.969_{-0.019}^{+0.019}$	$1539.1_{-3.8}^{+3.7}$		N
7.01		$392_{-43}^{+53}$	427 ± 13		$0.034_{-0.023}^{+0.036}$	$1.13_{-0.19}^{+0.21}$	$11600_{-2200}^{+3000}$		N
64.01		$290_{-14}^{+15}$	123.8 ± 3.3		$0.277_{-0.017}^{+0.018}$	$0.192_{-0.016}^{+0.017}$	$753_{-55}^{+58}$		PE/FP
69.01		$1440_{-160}^{+180}$	1640 ± 10		$0.042_{-0.030}^{+0.039}$	$1.19_{-0.18}^{+0.21}$	$89000_{-17000}^{+22000}$		N
75.01		$71.4_{-1.6}^{+1.7}$	115.2 ± 4.0		$0.158_{-0.014}^{+0.014}$	$2.13_{-0.15}^{+0.16}$	$678_{-28}^{+31}$		FP/PE
87.01		$2900_{-1900}^{+3600}$	1458 ± 30		$0.26_{-0.19}^{+0.20}$	$0.37_{-0.24}^{+1.42}$	$66000_{-56000}^{+230000}$		N
97.01		$233.8_{-1.8}^{+1.8}$	245 ± 15		$0.018_{-0.013}^{+0.018}$	$1.08_{-0.11}^{+0.12}$	$1119_{-15}^{+15}$		N
98.01		$168_{-3.1}^{+3.2}$	224 ± 14		$0.096_{-0.022}^{+0.021}$	$1.56_{-0.16}^{+0.17}$	$1614_{-53}^{+56}$		PB/PE/FP
107.01		$478_{-136}^{+75}$	427 ± 32		$0.065_{-0.043}^{+0.048}$	$0.86_{-0.17}^{+0.52}$	$19900_{-9100}^{+6200}$		N
113.01*		$5050_{-720}^{+580}$	382 ± 24		$0.696_{-0.028}^{+0.021}$	-	-		-
118.01		$920_{-390}^{+180}$	581 ± 30		$0.170_{-0.109}^{+0.048}$	$0.53_{-0.12}^{+0.60}$	$55000_{-35000}^{+21000}$		PE/FP
122.01		$1760_{-177}^{+48}$	540 ± 19		$0.372_{-0.029}^{+0.015}$	$0.201_{-0.014}^{+0.029}$	$77700_{-13500}^{+3700}$		PE/FP
257.01		$966_{-67}^{+76}$	990 ± 34		$0.019_{-0.013}^{+0.021}$	$1.04_{-0.13}^{+0.14}$	$8500_{-1000}^{+1300}$		N
263.01		$1070_{-520}^{+1400}$	378 ± 11		$0.33_{-0.21}^{+0.22}$	$0.24_{-0.16}^{+0.35}$	$37000_{-26000}^{+147000}$		PE/FP
268.01		$491_{-117}^{+49}$	662 ± 21		$0.100_{-0.033}^{+0.088}$	$1.53_{-0.20}^{+0.73}$	$20500_{-7900}^{+3900}$		PB/PE/FP
269.01		$113_{-44}^{+79}$	605 ± 18		$0.51_{-0.14}^{+0.11}$	$12.1_{-7.2}^{+20.9}$	$3700_{-2200}^{+6300}$		FP/PE
273.01		$910_{-150}^{+190}$	1193 ± 25		$0.091_{-0.056}^{+0.061}$	$1.65_{-0.50}^{+0.81}$	$4600_{-1200}^{+1700}$		N
276.01		$2010_{-790}^{+1440}$	898 ± 32		$0.26_{-0.16}^{+0.16}$	$0.32_{-0.16}^{+0.32}$	$61000_{-37000}^{+104000}$		PE/FP
280.01		$1090_{-140}^{+200}$	1281 ± 27		$0.059_{-0.039}^{+0.045}$	$1.29_{-0.30}^{+0.33}$	$12200_{-2700}^{+4400}$		PB/PE/FP
281.01		$160_{-57}^{+124}$	459 ± 15		$0.34_{-0.18}^{+0.12}$	$4.6_{-2.6}^{+5.1}$	$5600_{-3100}^{+10600}$		FP/PE
288.01		$249_{-69}^{+92}$	221.2 ± 2.7		$0.081_{-0.057}^{+0.068}$	$0.85_{-0.29}^{+0.49}$	$15100_{-6900}^{+11900}$		N
319.01		$221_{-13}^{+13}$	206 ± 15		$0.027_{-0.019}^{+0.028}$	$0.87_{-0.15}^{+0.18}$	$637_{-45}^{+50}$		N
975.01		$564.9_{-7.9}^{+5.8}$	288.6 ± 8.7		$0.220_{-0.010}^{+0.010}$	$0.405_{-0.017}^{+0.018}$	$105800_{-2600}^{+1900}$		PE/FP
1282.01		$432_{-97}^{+39}$	392 ± 21		$0.048_{-0.031}^{+0.034}$	$0.88_{-0.13}^{+0.40}$	$5750_{-2110}^{+860}$		N
1537.01		$288_{-169}^{+92}$	314 ± 11		$0.078_{-0.053}^{+0.235}$	$1.12_{-0.35}^{+2.67}$	$49000_{-41000}^{+33000}$		N
1618.01		$542_{-155}^{+60}$	524 ± 13		$0.040_{-0.027}^{+0.061}$	$0.96_{-0.13}^{+0.55}$	$179000_{-86000}^{+29000}$		N
1621.01		$1000_{-530}^{+200}$	244 ± 14		$0.436_{-0.182}^{+0.051}$	$0.150_{-0.034}^{+0.251}$	$86000_{-65000}^{+34000}$		PE/FP
1890.01		$478_{-218}^{+66}$	450 ± 17		$0.060_{-0.035}^{0.121}$	$0.93_{-0.16}^{+1.21}$	$60000_{-41000}^{+16000}$		N
1924.01		$65.3_{-2.5}^{+2.2}$	131.9 ± 4.3		$0.230_{-0.015}^{+0.016}$	$2.58_{-0.16}^{+0.18}$	$25900_{-2200}^{+2000}$		FP/PE
1962.01		$11600_{-4600}^{+1900}$	477 ± 45		$0.785_{-0.073}^{+0.023}$	$0.0126_{-0.0028}^{+0.0106}$	$207000_{-124000}^{+59000}$		PE/FP
371.01*		$710_{-170}^{+270}$	66.1 ± 1.6		$0.658_{-0.054}^{+0.057}$	-	-		-
674.01		$187.8_{-15.7}^{+5.6}$	84.0 ± 3.7		$0.259_{-0.027}^{+0.019}$	$0.325_{-0.026}^{+0.041}$	$3510_{-520}^{+190}$		PE/FP
1222.01		$540_{-340}^{+270}$	38.5 ± 1.9		$0.707_{-0.211}^{+0.063}$	$0.029_{-0.012}^{+0.070}$	-		FP
1230.01		$99.9_{-1.9}^{+2.0}$	7.091 ± 0.067		$0.7073_{-0.0035}^{+0.0036}$	$0.02216_{-0.00057}^{+0.00057}$	$646_{-22}^{+23}$		PE/FP
1299.01		$131_{-23}^{+14}$	26.50 ± 0.49		$0.488_{-0.051}^{+0.025}$	$0.109_{-0.013}^{+0.032}$	$3410_{-1000}^{+700}$		PE/FP
1314.01		$92_{-45}^{+22}$	42.33 ± 0.82		$0.259_{-0.154}^{+0.062}$	$0.345_{-0.086}^{+0.510}$	$7900_{-5700}^{+3900}$		N
1894.01		$62_{-26}^{+16}$	36.5 ± 1.3		$0.180_{-0.123}^{+0.069}$	$0.49_{-0.13}^{+0.54}$	$2900_{-1900}^{+1700}$		PE/FP
2133.01		$43.47_{-3.35}^{+0.67}$	6.81 ± 0.32		$0.547_{-0.019}^{+0.013}$	$0.0813_{-0.0061}^{+0.0092}$	$2113_{-297}^{+49}$		FP
2481.01		$50_{-26}^{+14}$	1.999 ± 0.096		$0.791_{-0.104}^{+0.030}$	$0.0128_{-0.0037}^{0.018}$	$3200_{-2400}^{+1600}$		FP
2640.01		$212_{-133}^{+97}$	4.29 ± 0.10		$0.862_{-0.113}^{+0.029}$	$0.0051_{-0.0020}^{+0.0127}$	$10200_{-8400}^{+9800}$		FP

Notes. ρ_{⋆, astero} values come from Huber et al. (2013). Columns 4–6 are computed using Equations (39) and (17) of Kipping (2014a) and Equation (2) of this work, respectively. Classification denotes AP which can explain the observations, where "N" denotes no AP effect required. KOIs with a "*" imply that b > (1 − p) in more than half the posteriors samples, meaning the AP equations become invalid. Rows above the horizontal line have log g > 3.7 and those below have log g ⩽ 3.7.

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If the transiting body orbits a different star in the same aperture, then we know (1) the transit is diluted and so the PB effect is occurring (2) we do not know the "true" stellar density. However, in the case of the PB effect, Kipping (2014a) showed that the observed density can only be altered up to a minimum limit defined as (Equation (13) of Kipping 2014a):

$$\begin{eqnarray} &&\left (\frac{\rho _{\star,\mathrm{obs}}}{\rho _{\star,\mathrm{true}}} \right) \ge \left (\frac{ 2 p_{\mathrm{obs}} \left(1+\sqrt{1-b_{\mathrm{obs}}^2}\right)}{(1+p_{\mathrm{obs}})^2 - b_{\mathrm{obs}}^2} \right)^{3/2}, \end{eqnarray} \tag{ 1 }$$

where p_obs and b_obs are the observed ratio-of-radii and impact parameter, respectively. In the case where ρ_{⋆, true} is unknown then, we can re-write this as

$$\begin{eqnarray} &&\rho _{\star,\mathrm{alt,max}} = \rho _{\star,\mathrm{obs}} \left (\frac{ 2 p_{\mathrm{obs}} \left(1+\sqrt{1-b_{\mathrm{obs}}^2}\right)}{(1+p_{\mathrm{obs}})^2 - b_{\mathrm{obs}}^2} \right)^{-3/2}. \end{eqnarray} \tag{ 2 }$$

Equation (2) reveals the maximum allowed density of the alternative star. Since it is only an upper limit, only cases where ρ_{⋆, alt, max} is very low allow us to exclude this scenario. We calculate this term for every KOI in Column 6 of Table 2. In the last column, we provide a plausible list of which of these three effects, or no effect at all ("N"), can explain the observation. Note that objects for which B < 1 (where B is the blend factor defined in Kipping 2014a) correspond to an anti-blend, which could be the consequence of the PS effect (Kipping 2014a). Nevertheless, the PS effect is expected to cause AP deviations ≲ 10% and thus generally does not have a significant impact. Additionally, although the PE effect can explain a diverse range of AP observations, some cases can be rejected as being due to eccentricity if the periastron passage goes inside the star. We define and compute this using:

$$\begin{eqnarray} (r_{\mathrm{peri}}/R_{\star }) &\le (a/R_{\star }) (1-e_{\mathrm{min}}), \nonumber \\ \qquad &\le \left (\frac{\rho _{\star,\mathrm{astero}} G P^2}{3\pi }\right)^{1/3} (1-e_{\mathrm{min}}). \end{eqnarray} \tag{ 3 }$$

As evident in Table 2, our AP survey of 41 single KOIs reveals numerous strong discrepancies between ρ_{⋆, obs} and ρ_{⋆, astero}. By plotting the observed discrepancies as a function of log g (values taken from Huber et al. 2013), one immediately identifies an apparent split in the distribution at the boundary of log g = 3.7 (see Figure 3). We choose log g = 3.7 as our split since this is a reasonable proxy for the boundary between dwarfs and giants/sub-giants, plus all of the observed densities below this limit have an overestimated value. Performing an Anderson–Darling (A–D) test (Anderson & Darling 1952) between these two populations reveals a 0.009% chance they are drawn from the same underlying distribution.

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The evidence for a significant divide is supported by further analysis too. We decided to investigate how well our measured (ρ_{⋆, obs}/ρ_{⋆, astero}) distribution matches that which would be expected if the PE effect alone was responsible for the observations. The motivation for this is that the PE effect is capable of explaining the greatest range of measurements (Kipping 2014a). We therefore generated a synthetic population of (ρ_{⋆, obs}/ρ_{⋆, astero}) measurements using:

$$\begin{eqnarray} \left (\frac{\rho _{\star,\mathrm{obs}}}{\rho _{\star,\mathrm{true}}} \right) &= \frac{(1+e\sin \omega)^3}{(1-e^2)^{3/2}}. \end{eqnarray} \tag{ 4 }$$

Using the code ECCSAMPLES (Kipping 2014b), we draw random e and ω samples from the joint probability distribution P(e, ω|object  known  to  transit) to generate our synthetic PE population. ECCSAMPLES assumes a Beta distribution for the underlying probability distribution of e, P(e) with shape parameters a and b. In our simulation, we adopt a = 0.867 and b = 3.03, which has been shown to provide an excellent match to the observed eccentricity distribution from RV surveys (Kipping 2013b). The final synthetic (ρ_{⋆, obs}/ρ_{⋆, astero}) distribution is shown in Figure 4 as a gray histogram. Overlaying the observed distributions for the dwarfs and giants immediately demonstrates how different the two samples are. The dwarf sample is fully compatible with the PE effect, with an A–D test giving a p-value of 77.1%. In contrast, the giants are grossly incompatible with the PE effect, with an A–D p-value of 0.003%, or 4.2σ.

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In considering this puzzling observation, we devised four possible hypotheses that could reconcile this split.

• 1.  
  The larger pulsations of the giants induce significant time-correlated noise in the folded light curve, which subsequently skews the ρ_{⋆, obs} determination.
• 2.  
  The asteroseismically determined densities systematically underestimate the stellar density for giant stars.
• 3.  
  Companions to giant stars are highly eccentric and have a dramatically different eccentricity distribution than dwarf stars.
• 4.  
  A large fraction of the KOIs associated with giant stars in fact orbit a different star within the aperture—cases we define as false-positives.

Hypothesis 1 can be tested by first quantifying the degree of time-correlated noise in the data. The timescale of this spurious noise must have dominant power at ν_max, the frequency of maximum asteroseismology power. For many of the targets, Huber et al. (2013) directly provide ν_max and where unavailable we use Equation (10) of Kjeldsen & Bedding (1995). We then tried cross-correlating the (ρ_{⋆, obs}/ρ_{⋆, astero}) measurements to the transit duration normalized by this timescale. Performing an A–D test about the median of this new variable, as we did with log g before, finds no significant split with a p-value of 3%. We also repeated this exercise using transit depth normalized by the amplitude of the maximum pulsation (computed using Equation (8) of Kjeldsen & Bedding 1995) and this yields a p-value of 14%. If time-correlated noise was genuinely responsible, one should expect these p-values should be lower than that found when using log g as the variable. Finally, we note that the median period of the giant sample is 25 days and thus the number of transits stacked together is typically large. This folding effect reduces the effect of time-correlated noise as $N_{\mathrm{transits}}^{-1/2}$ (Pont et al. 2006), further detracting from hypothesis 1. We therefore conclude hypothesis 1 is an improbable explanation for the observed distribution.

Hypothesis 2 seems improbable on the basis that giant stars yield the largest pulsation amplitudes and timescales. Further, the density of the star is typically the most precisely determined parameter from asteroseismology, directly related to the frequency splitting, Δν via (Ulrich 1986):

$$\begin{eqnarray} \Delta \nu &= \frac{ (M_{\star }/M_{\odot })^{1/2} }{ (R_{\star }/R_{\odot })^{3/2} } \Delta \nu _{\odot }. \end{eqnarray} \tag{ 5 }$$

The possibility that companions to giant stars are highly eccentric, hypothesis 3, has no direct physical motivation. The median minimum eccentricity required to explain this observation is 0.60. This appears inconsistent with the planets detected log g ⩽ 3.7 stars from RVs, for which the median eccentricity is much lower at 0.129 (see www.exoplanets.org; Wright et al. 2011). Secondly, even though the PE effect is expected to yield a small overestimate bias (Kipping 2014a), the fact that none of the objects have an underestimation effect is improbable. Finally, 4 of the 10 giants (KOI-1222.01, KOI-2133.01, KOI-2481.01, and KOI-2640.01) cannot possibly be explained by the PE effect, since this requires a periastron passage inside the star, which we deem unphysical. We therefore find the super-eccentric planets hypothesis strongly disfavored.

By deduction, this leaves us with hypothesis 4 as the only viable explanation. As discussed in Section 4, this appears consistent with independent arguments regarding the FPR for this sample. Given the small number statistics involved with a sample of just 10 giant star KOIs, the precision to which the associated FPR can be measured is naturally low. However, from Figure 4, we estimate that 3 of the 10 giant star KOIs have (ρ_{⋆, obs}/ρ_{⋆, astero}) compatible with the synthetic PE effect population. The other seven have sufficiently high (ρ_{⋆, obs}/ρ_{⋆, astero}) measurements that they appear incompatible with the synthetic photo-eccentric population. On this basis, we estimate FPR ≃ (70 ± 30)%. We further note that the FPR can easily be seen to be at least FPR≳ (40 ± 20)%, on the basis that 4 of the 10 KOIs are classified unambiguously as false-positives in Table 2, since they would have to be so eccentric they would pass inside or contact the star.

Recently, Lillo-Box et al. (2013) claimed to confirm the planetary nature of KOI-2133.01, or Kepler-91b, which is an object in our sample. If this KOI was genuinely a planet, it would seem to be a counter-example to our conclusion of a high FPR for giant host stars. Further, in Table 2 we identify KOI-2133.01 as an unambiguous FP using AP. For these reasons, it is important that we investigate this apparent discrepancy.

The key reason why we identified this object as a false-positive is because ρ_{⋆, obs} is so much larger than ρ_{⋆, astero} that the orbit would have to be highly eccentric, such that $(r_{\mathrm{peri}}/R_{\star })=1.10_{-0.05}^{+0.06}$ i.e., the planet is essentially in contact with the star. Specifically, we have ρ_{⋆, astero} = 6.81 ± 0.032 kg m⁻³ but $\rho _{\star,\mathrm{obs}}=43.47_{-3.35}^{+0.67}$ kg m⁻³, which may be equivalently expressed in terms of the semi-major axis using Kepler's Third Law as $(a/R_{\star })_{\mathrm{obs}} = 4.476_{-0.118}^{+0.023}$. Critically, Lillo-Box et al. (2013) find a much lower observed stellar density, which is more compatible with ρ_{⋆, astero} and thus does not require a highly eccentric planet. They report $\rho _{\star,\mathrm{obs}}=7.1_{-1.9}^{+0.7}$ kg m⁻³, which is equivalent to $(a/R_{\star })_{\mathrm{obs}} = 2.32_{-0.22}^{+0.07}$.

With two dramatically different light curve determinations of ρ_{⋆, obs}, or (a/R_⋆)_obs, it remains unclear which solution is correct. Fortunately, two additional independent studies have also computed light curve solutions for this object, namely Burke et al. (2013) and Esteves et al. (2013). In the case of Burke et al. (2013), Table 1 reports (a/R_⋆) = 4.346 (no associated uncertainty reported), which is within 1.1σ of our solution but >25σ discrepant to that of Lillo-Box et al. (2013). In the case of Esteves et al. (2013), the authors report $(a/R_{\star }) = 4.51_{-0.26}^{+0.12}$, which is in excellent agreement with our solution (<1σ) and inconsistent with that of Lillo-Box et al. (2013) (>8σ). We note that Lillo-Box et al. (2013) cite Tenenbaum et al. (2013) as finding (a/R_⋆) = (2.64  ±  0.23), however this value is not actually listed anywhere in Tenenbaum et al. (2013) and the authors have stated this is not a result from their paper (P. Tenenbaum 2014, private communication).

Additionally, Esteves et al. (2013) also identified KOI-2133.01 as a false-positive using a completely different technique than us. They reported strong phase variations indicative of reflected light and ellipsoidal variations. However, if (a/R_⋆) ≃ 4.5, the amplitude of the variations is so great that KOI-2133.01 must be self-luminous and thus a false-positive. Lillo-Box et al. (2013) remark on this but since their (a/R_⋆) value is much lower, the phase variations can be explained by reflected light without KOI-2133.01 being self-luminous.

It should therefore be clear that the planetary nature of KOI-2133.01 hangs primarily on whether (a/R_⋆) ≃ 4.5, in which case it is a false-positive, or (a/R_⋆) ≃ 2.3, in which case it can be a planet. With three independent measurements by ourselves, Burke et al. (2013), and Esteves et al. (2013) in agreement versus one study finding the planet scenario (Lillo-Box et al. 2013), the current consensus would favor the false-positive scenario. However, we would encourage multiple independent groups to study this light curve in order to resolve this important question.

In this work, using the novel technique of AP in isolation, we demonstrate that the FPR of Kepler planetary candidates associated with stars of log g ⩽ 3.7 (i.e., the giants and sub-giants) is FPR ≃ 70% ± 30% (see Section 3.2). Due to the small-number statistics of our giant-star sample, we prefer the interpretation of a merely "high" false-positive rate, rather than an explicit numerical value. In contrast, we find no compelling evidence for a non-zero FPR of the log g > 3.7 sample (i.e., the dwarfs). This latter result is consistent with the low FPR for Kepler dwarfs reported by Morton & Johnson (2011) and Fressin et al. (2013) of ≲ 10%. However, we are aware of no previous studies characterizing the FPR for Kepler's giant star population.

Although there are no explicit studies regarding Kepler's giant star FPR, numerous other works indicate our result is not a surprise. For example, the population of exoplanets discovered using the RV technique provides some useful insights. In Figure 5, it is apparent that there is a paucity of planets detected with periods below 100 days for host stars with log g < 3.7 (1/87), where the data come from www.exoplanets.org (Wright et al. 2011). Eight of the 10 giant planetary candidates studied in our sample have periods below 100 days though, and therefore seem to occupy a parameter space where RVs predict a low occurrence rate. To investigate this further, we estimated the approximate RV amplitudes of the 10 KOIs. Approximate masses were estimated from the radii using the empirical relation of Weiss & Marcy (2014) for planets below 4 Earth radii and a simple power-law interpolation through the www.exoplanets.org catalog (Wright et al. 2011) for larger worlds, capping masses off at 1.3 M_J. The estimated RV amplitudes are shown as triangles in Figure 5. This reveals that indeed five of the KOIs occupy a region where genuine planets are very rare. This lends credence to the hypothesis that the FPR for our sample is high.

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Another useful insight comes from the number of multiple transiting planet systems detected between the dwarfs and giants, since the FPR of multiple planet systems is known to be very low (Lissauer et al. 2012, 2014) at FPR≲1%. Using the catalog available at http://exoplanetarchive.ipac.caltech.edu, we count that 1.8% of host stars with log g ⩽ 3.7 and with KOIs not dispositioned as a false-positive reside in multiple transiting KOI systems. In contrast, doing the same for the log g > 3.7 sample yields 16.9% of the objects. In other words, a Kepler star identified to have transiting planetary candidates is nearly 10 times more likely to have multiple candidates if it is a dwarf rather than a giant. This test is not definitive since the number of multi-planet systems orbiting giant stars may genuinely be much lower, but equally it can be explained by an order of magnitude higher FPR for the giants.

We argue that our sample of single KOIs associated with giant stars is largely unbiased, since the only selection criterion is the presence of detectable oscillation modes. This criterion implies that the target (1) exhibits large oscillation modes and (2) is bright enough for these modes to be detected. Among the population of giant stars, point (1) introduces no significant bias, since the amplitude of maximum oscillation power is enhanced for all low log g targets via $\sim T_{\mathrm{eff}}^2/\log g$ (Kjeldsen & Bedding 1995). The second point implies we are biased toward brighter targets and so our population may be closer than giant stars without asteroseismic detections. If anything, targets further away may have a higher FPR due to an increased probability of chance alignment. We also note that oscillations may be damped for short-period binaries (Gaulme et al. 2014), however such systems are easily identified via large ellipsoidal variations and beaming effects and unlikely to be a significant source of bias in our sample either.

Our AP survey concludes that many of the giants are false-positives, which specifically is defined as meaning that the transiting body is actually eclipsing a different star. This implies that we should expect many of these KOIs to have potentially detectable companions using adaptive optics (AO) imaging. To investigate this, we use the database of AO images acquired for 715 KOIs by Law et al. (2013). Of these 715, just 19 targets have log g ⩽ 3.7 which include the 10 giant star KOIs in our survey (log g estimates taken from Huber et al. 2014). Law et al. (2013) report no detections of companions for any of these 19 KOIs, with typical limits of Δm ≃ 6 from ≈015 to 25. However, AO imaging is less constraining for the giants since they are intrinsically brighter and thus in a magnitude-limited survey like Kepler will have to be at greater distance from the observer. Using the stellar parameters of Huber et al. (2014), we estimate that the dwarf KOIs (log g > 3.7) have a median distance of $d=820_{-400}^{+420}$ pc, whereas the giants (log g ⩽ 3.7) have $d=1500_{-700}^{+2200}$ pc. We therefore do not consider the lack of AO detections to be incompatible with our result.

This work demonstrates the unique power of the relatively new technique of AP. While AP is usually associated with the goal of constraining the orbital eccentricities of exoplanets (Kipping et al. 2012b; Dawson & Johnson 2012), we here verify that the method is also a powerful tool in vetting planetary candidates using photometry alone (Tingley et al. 2011; Kipping 2014a). In this work, we considered just 41 KOIs, but future studies with hundreds or thousands of objects would be able to realistically measure the eccentricity distribution using AP alone. Ensemble studies require targets with homogeneously and accurately derived stellar densities and we encourage work in this area to provide AP for a larger sample of targets for future applications.

Future space-based transit survey missions, such as TESS⁴ (Ricker et al. 2010) and PLATO (Rauer et al. 2013), will also surely benefit from using AP in both planet validation and characterization and our work highlights the value of accurate stellar parameters for such surveys.