THE FALSE POSITIVE RATE OF KEPLER AND THE OCCURRENCE OF PLANETS

The first part of our analysis is the calculation of the false positive rate for Kepler targets. We consider here all Kepler targets that have been observed in at least one out of the first six quarters of operation of the spacecraft (Q1–Q6),⁴ to be consistent with the selection imposed on the Batalha et al. (2013) list, which comes to a total of 156,453 stars. For each of these stars, identified by a unique Kepler Input Catalog (KIC) number (Brown et al. 2011), we have available their estimated mass, surface gravity log g, and brightness in the Kepler bandpass (Kp magnitude) as listed in the KIC.⁵ Additionally, we have their CDPP values on different timescales and for different quarters, provided in the Mikulski Archive for Space Telescopes (MAST) at STScI.⁶ The CDPP allows us to estimate the detectability of each type of false positive scenario.

We perform Monte Carlo simulations specific to each Kepler target to compute the number of blends of different kinds that we expect. The calculations are based on realistic assumptions about the properties of objects acting as blends, informed estimates about the frequencies of such objects either in the background or physically associated with the target, the detectability of the signals produced by these blends for the particular star in question based on its CDPP, and also on the ability to reject blends based on constraints provided by the centroid motion analysis or the presence of significant secondary eclipses. Blended stars beyond a certain angular distance from the target can be detected in the standard vetting procedures carried out by the Kepler team because they cause changes in the flux-weighted centroid position, depending on the properties of the signal. This constraint, and how we estimate it for each star, is described more fully in the Appendix. The inclusion of detectability and the constraint from centroid motion analysis are included in order to emulate the vetting process of Kepler to a reasonable degree.

The second part of our analysis seeks to determine the rate of occurrence of planets of different sizes. For this we compare the frequencies and distributions of simulated false positive from the previous step with those of the real sample of KOIs given by Batalha et al. (2013). We interpret the differences between these two distributions as due to true detectable planets. In order to obtain the actual rate of occurrence of planets, it is necessary to correct for the fact that the KOI list includes only planets with orbital orientations such that they transit, and for the fact that some fraction of transiting planets are not detectable by Kepler. We determine these corrections by means of Monte Carlo simulations of planets orbiting the Kepler targets, assuming initial distributions of planets as a function of planet size and orbital period (Howard et al. 2012). We then compare our simulated population of detectable transiting planets with that of KOIs minus our simulated false positive population, and we adjust our initial assumptions for the planetary distributions. We proceed iteratively to obtain the occurrence of planets that provides the best match to the KOI list. This analysis permits us to also study the dependence of the occurrence rates on properties such as the stellar mass.

For the purpose of interpreting both the false positive rates and the frequencies of planets, we have chosen to separate planets into five different size (radius R_p) ranges.

• 1.  
  Giant planets: 6 R_⊕ < R_p ⩽ 22 R_⊕.
• 2.  
  Large Neptunes: 4 R_⊕ < R_p ⩽ 6 R_⊕.
• 3.  
  Small Neptunes: 2 R_⊕ < R_p ⩽ 4 R_⊕.
• 4.  
  Super-Earths: 1.25 R_⊕ < R_p ⩽ 2 R_⊕.
• 5.  
  Earths: 0.8 R_⊕ < R_p ⩽ 1.25 R_⊕.

An astrophysical false positive that is not ruled out by centroid motion analysis and that would produce a signal detectable by Kepler with a transit depth corresponding to a planet in any of the above categories is counted as a viable blend, able to mimic a true planet in that size range. These five classes were selected as a compromise between the number of KOIs in each group and the nomenclature proposed by the Kepler team (Borucki et al. 2011; Batalha et al. 2013). In particular, we have subdivided the 2–6 R_⊕ category used by the Kepler team into "small Neptunes" and "large Neptunes." Howard et al. (2012) chose the same separation in their recent statistical study of the frequencies of planets larger than 2 R_⊕, in which they claimed that the smallest planets (small Neptunes) are more common among later-type stars, but did not find the same trend for larger planets. Our detailed modeling of the planet detectability for each KOI, along with our new estimates of the false positive rates among small planets, enables us to extend the study to two smaller classes of planets, and to investigate whether the claims by Howard et al. (2012) hold for objects as small as the Earth.

We begin by simulating the most obvious type of false positive involving a background (or foreground) eclipsing binary in the photometric aperture of the target star, whose eclipses are attenuated by the light of the target and reduced to a shallow transit-like event. We model the stellar background of each of the 156,453 KIC targets using the Besançon model of the Galaxy (Robin et al. 2003).⁷ We simulate 1 deg² areas at the center of each of the 21 modules comprising the focal plane of Kepler (each module containing two CCDs), and record all stars (of any luminosity class) down to 27th magnitude in the R band, which is sufficiently close to Kepler's Kp band for our purposes. This limit of R = 27 corresponds to the faintest eclipsing binary that can mimic a transit by an Earth-size planet, after dilution by the target. We assign to each Kepler target a background star drawn randomly from the simulated area for the module it is located on, and we assign also a coefficient N_bs to represent the average number of background stars down to 27th magnitude in the 1 deg² region around the target.

Only a fraction of these background stars will be eclipsing binaries, and we take this fraction from results of the Kepler mission itself, reported in the work of Slawson et al. (2011). The catalog compiled by these authors provides not only the overall rate of occurrence of eclipsing binaries (1.4%, defined as the number of eclipsing binaries found by Kepler divided by the number of Kepler targets), but also their eclipse depth and period distributions. In practice, we adopt a somewhat reduced frequency of eclipsing binaries F_eb = 0.79% that excludes contact systems, as these generally cannot mimic a true planetary transit because the flux varies almost continuously throughout the orbital cycle. We consider only detached, semi-detached, and unclassified eclipsing binaries from the Slawson et al. (2011) catalog. Furthermore, we apply two additional corrections to this overall frequency that depend on the properties of the background star. The first factor, C_bf, adjusts the eclipsing binary frequency according to spectral type, as earlier-type stars have been found to have a significantly higher binary frequency than later-type stars. We interpolate this correction from Figure 12 of Raghavan et al. (2010). The second factor, C_geom, accounts for the increased chance that a larger star in the background would be eclipsed by a companion of a given period, from geometrical considerations. This correction factor is unity for a star of 1 R_☉, and increases linearly with radius. We note that the scaling law we have chosen does not significantly impact the overall occurrence of eclipsing binaries established by Slawson et al. (2011), as the median radius of a Kepler target (0.988 R_☉) is very close to 1 R_☉. The size of each of our simulated background stars is provided by the Besançon model.

Next, we assign a companion star to each of the 156,453 background stars drawn from the Besançon model, with properties (eclipse depth and orbital period) taken randomly from the Slawson et al. (2011) catalog. We then check whether the transit-like signal produced by this companion has the proper depth to mimic a planet with a radius in the size category under consideration (Section 2.2), after dilution by the light of the Kepler target (which depends on the known brightness of the target and of the background star). If it does not, we reject it as a possible blend. Next we compute the S/N of the signal as described in the Appendix, and we determine whether this false positive would be detectable by the Kepler pipeline as follows. For each KIC target observed by Kepler between Q1 and Q6 we compute the S/N as explained in the Appendix, and with Equation (1) we derive D_r. We then draw a random number between 0 and 1 and compare it with D_r; if larger, we consider the false positive signal to be detectable. We also estimate the fraction of the N_bs stars that would be missed in the vetting carried out by the Kepler team because they do not induce a measurable centroid motion. This contribution, F_cent, is simply the fraction of background stars interior to the exclusion limit set by the centroid analysis. As we describe in more detail in the Appendix, the size of this exclusion region is itself mainly a function of the S/N.

Based on all of the above, we compute the number N_beb of background (or foreground) eclipsing binaries in the Kepler field that could mimic the signal of a transiting planet in a given size range as

$$\begin{equation*} N_{\rm beb} = \sum _{i=1}^{N_{\rm targ}} N_{\rm bs}\, F_{\rm eb}\, C_{\rm bf}\, C_{\rm geom}\, T_{\rm depth}\, T_{\rm det}\, T_{\rm sec}\,F_{\rm cent}\,, \end{equation*}$$

where F terms represent fractions, C terms represent corrections to the eclipsing binary frequency, and T terms are either 0 or 1.

• 1.  
  N_targ is the number of Kepler targets observed for at least one quarter during the Q1–Q6 observing interval considered here.
• 2.  
  N_bs is the average number of background stars down to magnitude R = 27 in a 1 deg² area around the target star.
• 3.  
  F_eb is the frequency of eclipsing binaries, defined as the fraction of eclipsing binaries (excluding contact systems) found by Kepler divided by the number of Kepler targets, and is 0.79% for this work.
• 4.  
  C_bf is a correction to the binary occurrence rate F_eb that accounts for the dependence on stellar mass (or spectral type), following Raghavan et al. (2010).
• 5.  
  C_geom is a correction to the geometric probability of eclipse that takes into account the dependence on the size of the background star.
• 6.  
  T_depth is 1 if the signal produced by the background eclipsing binary has a depth corresponding to a planet in the size range under consideration, after accounting for dilution by the Kepler target, or 0 otherwise.
• 7.  
  T_det is 1 if the signal produced by the background eclipsing binary is detectable by the Kepler pipeline, or 0 otherwise.
• 8.  
  T_sec is 1 if the background eclipsing binary does not show a secondary eclipse detectable by the Kepler pipeline, or 0 otherwise. We note that we have also set T_sec to 1 if the simulated secondary eclipses have a depth within 3σ of that of the primary eclipse, as this could potentially be accepted by the pipeline as a transiting object, with a period that is half of the binary period.
• 9.  
  F_cent is the fraction of the N_bs background stars that would show no significant centroid shifts, i.e., that are interior to the angular exclusion region that is estimated for each star, as explained in the Appendix.

A second type of astrophysical scenario that can reproduce the shape of a small transiting planet consists of a larger planet transiting a star in the background (or foreground) of the Kepler target. We proceed in a similar way as for background eclipsing binaries, except that instead of assigning a stellar companion to each background star, we assign a random planetary companion drawn from the actual list of KOIs by Batalha et al. (2013). However, there are three factors that prevent us from using this list as an unbiased sample of transiting planets: (1) the list is incomplete, as the shallowest and longest period signals are only detectable among the Kepler targets in the most favorable cases; (2) the list contains an undetermined number of false positives; and (3) the occurrence of planets of different sizes may be correlated with the spectral type of the host star (see Howard et al. 2012).

While it is fairly straightforward to quantify the incompleteness by modeling the detectability of signals for the individual KOIs (see the Appendix for a description of the detection model we use), the biases (2) and (3) are more difficult to estimate. We adopt a bootstrap approach in which we first determine the false positive rate and the frequencies for the larger planets among the KOIs, and we then proceed to study successively smaller planets. The only false positive sources for the larger planet candidates involve even larger (that is, stellar) objects, for which the frequency and distributions of the periods and eclipse depths are well known (i.e., from the work of Slawson et al. 2011). Comparing the population of large-planet KOIs (of the "giant planet" class previously defined) to the one of simulated false positives that can mimic their signals enables us to obtain the false positive correction factor (2). This, in turn, allows us to investigate whether the frequency of large planets is correlated with spectral type (see Section 6), and therefore to address bias (3) above. As larger planets transiting background stars are potential false positive sources for smaller planets, once we have understood the giant planet population we may proceed to estimate the false positive rates and occurrence of each of the smaller planet classes in order of decreasing size, applying corrections for the biases in the same way as described above.

We express the number N_btp of background stars in the Kepler field with larger transiting planets able to mimic the signal of a true transiting planet in a given size class as

$$\begin{equation*} N_{\rm btp} = \sum _{i=1}^{N_{\rm targ}} N_{\rm bs}\, F_{\rm tp}\, C_{\rm tpf}\, C_{\rm geom}\, T_{\rm depth}\, T_{\rm det}\, F_{\rm cent}\,, \end{equation*}$$

where N_targ, N_bs, C_geom, T_depth, T_det, and F_cent have similar meanings as before, and

• 1.  
  F_tp is the frequency of transiting planets in the size class under consideration, computed as the fraction of suitable KOIs found by Kepler divided by the number of non-giant Kepler targets (defined as those with KIC values of log g > 3.6, following Brown et al. 2011)⁸;
• 2.  
  C_tpf is a correction factor to the transiting planet frequency F_tp that accounts for the three biases mentioned above: incompleteness, the false positive rate among the KOIs of Batalha et al. (2013), and the potential dependence of F_tp on the spectral type of the background star.

An eclipsing binary gravitationally bound to the target star (forming a hierarchical triple system) may also mimic a transiting planet signal since its eclipses can be greatly attenuated by the light of the target. The treatment of this case is somewhat different from the one of background binaries, though, as the occurrence rate of intruding stars eclipsed by others is independent of the Galactic stellar population. To simulate these scenarios, we require knowledge of the frequency of triple systems, which we adopt from the results of a volume-limited survey of the multiplicity of solar-type stars by Raghavan et al. (2010). They found that 8% of their sample stars are triples, and another 3% have higher multiplicity. We therefore adopt a total frequency of 8% + 3% = 11%, since additional components beyond three merely produce extra dilution, which is small in any case compared to that of the Kepler target itself (assumed to be the brighter star). Of all possible triple configurations, we consider only those in which the tertiary star eclipses the close secondary, with the primary star being farther removed (hierarchical structure). The two other configurations involving a secondary star eclipsing the primary or the tertiary eclipsing the primary would typically not result in the target being promoted to a KOI, as the eclipses would be either very deep or "V"-shaped. Thus, we adopt one-third of 11% as the relevant frequency of triple- and higher-multiplicity systems.

We proceed to simulate this type of false positive by assigning to each Kepler target a companion star ("secondary") with a random mass ratio q (relative to the target), eccentricity e, and orbital period P drawn from the distributions presented by Raghavan et al. (2010), which are uniform in q and e, and lognormal in P. We further infer the radius and brightness of the secondary in the Kp band from a representative 3 Gyr solar-metallicity isochrone from the Padova series of Girardi et al. (2000). We then assign to each of these secondaries an eclipsing companion, with properties (period and eclipse depth) taken from the catalog of Kepler eclipsing binaries by Slawson et al. (2011), as we did for background eclipsing binaries in Section 3.1. We additionally assign a mass ratio and eccentricity to the tertiary from the above distributions.

To determine whether each of these simulated triple configurations constitutes a viable blend, we perform the following four tests: (1) we check whether the triple system would be dynamically stable, to first order, using the condition for stability given by Holman & Wiegert (1999); (2) we check whether the resulting depth of the eclipse, after accounting for the light of the target, corresponds to the depth of a transiting planet in the size range under consideration; (3) we determine whether the transit-like signal would be detectable, with the same signal-to-noise criterion used earlier; and (4) we check whether the eclipsing binary would be angularly separated enough from the target to induce a centroid shift. For this last test, we assign a random orbital phase to the secondary in its orbit around the target, along with a random inclination angle from an isotropic distribution, a random longitude of periastron from a uniform distribution, and a random eccentricity drawn from the distribution reported by Raghavan et al. (2010). The semimajor axis is a function of the known masses and the orbital period. The final ingredient is the distance to the system, which we compute using the apparent magnitude of the Kepler target as listed in the KIC and the absolute magnitudes of the three stars from the Padova isochrone, ignoring extinction. If the resulting angular separation is smaller than the radius of the exclusion region from centroid motion analysis, we count this as a viable blend.

The number N_ceb of companion eclipsing binary configurations in the Kepler field that could mimic the signal of a transiting planet in each size category is given by

$$\begin{equation*} N_{\rm ceb} =\sum _{i=1}^{N_{\rm targ}} F_{{\rm eb/triple}}\, C_{\rm geom}\, T_{\rm stab}\, T_{\rm depth}\, T_{\rm det}\, T_{\rm sec}\, T_{\rm cent}\,, \end{equation*}$$

in which N_targ, C_geom, T_depth, T_sec, and T_det represent the same quantities as in previous sections, while

• 1.  
  F_eb/triple is the frequency of eclipsing binaries in triple systems (or systems of higher multiplicity) in which the secondary star is eclipsed by the tertiary. As indicated earlier, we adopt a frequency of eclipsing binaries in the solar neighborhood of F_eb = 0.79%, from the Kepler catalog by Slawson et al. (2011). The frequency of stars in binaries and higher-multiplicity systems has been given by Raghavan et al. (2010) as F_bin = 44% (where we use the notation "bin" to mean "non-single"). Therefore, the chance of a random pair of stars to be eclipsing is F_eb/F_bin = 0.0079/0.44 = 0.018. As mentioned earlier, we will assume here that one-third of the triple star configurations (with a frequency F_triple of 11%, according to Raghavan et al. 2010) have the secondary and tertiary as the close pair of the hierarchical system. F_eb/triple is then equal to F_eb/F_bin × 1/3 × F_triple = 0.018 × 1/3 × 0.11 = 0.00066. The two other configurations in which the secondary or tertiary is eclipsing the primary would merely correspond to target binaries slightly diluted by the light of a companion, and we assume here that those configurations would not have passed the Kepler vetting procedure.
• 2.  
  T_stab is 1 if the triple system is dynamically stable, and 0 otherwise.
• 3.  
  T_cent is 1 if the triple system does not induce a significant centroid motion, and 0 otherwise.

Planets transiting a physical stellar companion to a Kepler target can also produce a signal that will mimic that of a smaller planet around the target itself. One possible point of view regarding these scenarios is to not consider them a false positive, as a planet is still present in the system, only not of the size anticipated from the depth of the transit signal. However, since one of the goals of the present work is to quantify the rate of occurrence of planets in specific size categories, a configuration of this kind would lead to the incorrect classification of the object as belonging to a smaller planet class, biasing the rates of occurrence. For this reason, we consider these scenarios to be a legitimate false positive.

To quantify them we proceed in a similar way as for companion eclipsing binaries in the preceding section, assigning a bound stellar companion to each target in accordance with the frequency and known distributions of binary properties from Raghavan et al. (2010), and assigning a transiting planet to this bound companion using the KOI list by Batalha et al. (2013). Corrections to the rates of occurrence F_tp are as described in Section 3.2. The total number of false positives of this kind among the Kepler targets is then

$$\begin{equation*} N_{\rm ctp} = \sum _{i=1}^{N_{\rm targ}} F_{\rm bin}\, F_{\rm tp}\, C_{\rm geom}\, C_{\rm tpf}\, T_{\rm depth}\, T_{\rm det}\, T_{\rm cent}\, T_{\rm stab}\,, \end{equation*}$$

where F_bin is the frequency of non-single stars in the solar neighborhood (44%; Raghavan et al. 2010) and the remaining symbols have the same meaning as before.

Adding up the contributions from this section and the preceding three, the total number of false positives in the Kepler field due to physically bound or background/foreground stars eclipsed by a smaller star or by a planet is N_beb + N_btp + N_ceb + N_ctp.

Because of their small size, brown dwarfs transiting a star can produce signals that are very similar to those of giant planets. Therefore, they constitute a potential source of false positives, not only when directly orbiting the target but also when eclipsing a star blended with the target (physically associated or not). However, because of their larger mass and non-negligible luminosity, other evidence would normally betray the presence of a brown dwarf, such as ellipsoidal variations in the light curve (for short periods), secondary eclipses, or even measurable velocity variations induced on the target star. Previous Doppler searches have shown that the population of brown dwarf companions to solar-type stars is significantly smaller than that of true Jupiter-mass planets. Based on this, we expect the incidence of brown dwarfs as false positives to be negligible, and we do not consider them here.

A white dwarf transiting a star can easily mimic the signal of a true Earth-size planet, since their sizes are comparable. Their masses, on the other hand, are of course very different. Some theoretical predictions suggest that binary stars consisting of a white dwarf and a main-sequence star may be as frequent as main-sequence binary stars (see, e.g., Farmer & Agol 2003). However, evidence so far from the Kepler mission seems to indicate that these predictions may be off by orders of magnitude. For example, none of the Earth-size candidates that have been monitored spectroscopically to determine their radial velocities (RVs) have shown any indication that the companion is as massive as a white dwarf. Also, while transiting white dwarfs with suitable periods would be expected to produce many easily detectable gravitational lensing events if their frequency were as high as predicted by Farmer & Agol (2003), no such magnification events compatible with lensing have been detected so far in the Kepler photometry (J. Jenkins 2012, private communication). For these reasons, we conclude that the relative number of eclipsing white dwarfs among the 196 Earth-size KOIs is negligible in comparison to other contributing sources of false positives.

To illustrate the process of false positive estimation for planets of Earth size (0.8 R_⊕ < R_p ⩽ 1.25 R_⊕), we present here the calculation for a single case of a false positive involving a larger planet transiting a star physically bound to the Kepler target (Section 3.4), which happens to produce a signal corresponding to an Earth-size planet.

In this case selected at random, the target star (KIC 3453569) has a mass of 0.73 M_☉, a brightness of Kp = 15.34 in Kepler band, and a 3 hr CDPP of 133.1 ppm. As indicated above, the star has a 44% a priori chance of being a multiple system based on the work of Raghavan et al. (2010). We assign to the companion a mass and an orbital period drawn randomly from the corresponding distributions reported by these authors, with values of 0.54 M_☉ and 67.6 yr, corresponding to a semimajor axis of a = 18.0 AU.

This companion has an a priori chance F_tp of having a transiting planet equal to the number of KOIs (orbiting non-giant stars) divided by the number of non-giant stars (F_tp = 2, 302/132, 756 = 0.017) observed by Kepler in at least one quarter during the Q1–Q6 interval. We assign to this companion star a transiting planet drawn randomly from the Batalha et al. (2013) list of KOIs, with a radius of 1.81 R_⊕ (a super-Earth) and an orbital period of 13.7 days. The diluted depth of the signal of this transiting planet results in a 140 ppm dimming of the combined flux of the two stars during the transit, that could be incorrectly interpreted as an Earth-size planet (0.91 R_⊕) transiting the target star. We set T_depth = 1, as this example pertains to false positives mimicking transits of Earth-size planets. Because of the smaller radius of the stellar companion (R_comp = 0.707 R_☉, determined with the help of our representative 3 Gyr, solar-metallicity isochrone), that star has a correspondingly smaller chance that the planet we have assigned to orbit it will actually transit at the given period, compared to the median ∼1 R_☉ Kepler target. The corresponding correction factor is then C_geom = 0.707.

The correction C_tpf to the rate of occurrence of transiting planets acting as blends is the most complicated to estimate, as it relies in part on similar calculations carried out sequentially from the larger categories of planets, starting with the giant planets, as described earlier. For this example, we make use of the results of those calculations presented in Section 5. The C_tpf factor corrects the occurrence rate of planets in the 1.25–2 R_⊕ super-Earth category (since this is the relevant class for the particular blended planet we have simulated) for three distinct biases in the KOI list of Batalha et al. (2013), due to incompleteness, false positives, and the possible correlation of frequency with spectral type.

For our simulated planet with R_p = 1.81 R_⊕ and P = 13.7 days, the incompleteness contribution to C_tpf was estimated by computing the S/N of the signals generated by a planet of this size transiting each individual Kepler target, using the CDPP of each target. We find that the transit signal could only be detected around 54.1% of the targets, from which the incompleteness boost is simply 1/0.541 = 1.848. The contribution to C_tpf from the false positive bias was taken from Table 1 of Section 4 below, which summarizes the results from the bootstrap analysis mentioned earlier. That analysis indicates that the false positive rate for super-Earths is 8.8%, i.e., 91.2% of the KOIs in the 1.25–2 R_⊕ are transited by true planets. Finally, the third contributor to C_tpf addresses the possibility that small Neptunes from the KOI list acting as blends may be more common, for example, around stars of later spectral types than earlier spectral types. Our analysis in Section 6.4 in fact suggests that super-Earths have a uniform occurrence as a function of spectral type, and we account for this absence of correlation by setting a value of 1.0 for the spectral-type dependence correction factor. Combining the three factors just described, we arrive at a value of C_tpf = 1.848 × 0.912 × 1.0 = 1.685.

To ascertain whether this particular false positive could have been detected by Kepler, we compute its S/N as explained in the Appendix, in terms of the size of the small Neptune relative to the physically bound companion, the dilution factor from the target, the CDPP of the target, the number of transits observed (from the duration interval for the particular Kepler target divided by the period of the KOI), and the transit duration as reported by Batalha et al. (2013). The result, ${\rm {\rm S/N}} = 9.44$, is above the threshold (determined using the detection recovery rate studied in Section 5.1), so the signal would be detectable and we set T_det = 1.

With the above S/N, we may also estimate the angular size of the exclusion region outside of which the multi-quarter centroid motion analysis would rule out a blend. The value we obtain for the present example with the prescription given in the Appendix is 11. To establish whether the companion star is inside or outside of this region, we compute its angular separation from the target as follows. First, we estimate the distance to the system from the apparent magnitude of the target (Kp = 15.34) and the absolute magnitudes of the two stars, read off from our representative isochrone according to their masses of 0.73 and 0.54 M_☉. Ignoring extinction, we obtain a distance of 730 pc. Next we place the companion at a random position in its orbit around the target, using the known semimajor axis of 18.0 AU and the random values of other relevant orbital elements (eccentricity, inclination angle, and longitude of periastron) as reported above. With this, the angular separation is 002. This is about a factor of 50 smaller than the limit from centroid motion analysis, so we conclude this false positive would not be detected in this way (i.e., it remains a viable blend). We therefore set T_cent = 1.

Finally, the formalism by Holman & Wiegert (1999) indicates that, to first order, a hierarchical triple system like the one in this example would be dynamically unstable if the companion star were within 1.06 AU of the target. As their actual mean separation is much larger (a = 18.0 AU), we consider the system to be stable and we set T_stab = 1.

Given the results above, this case is therefore a viable false positive that could be interpreted as a transiting Earth-size planet. The chance that this would happen for the particular Kepler target we have selected is given by the product F_bin × F_tp × C_geom × C_tpf × T_depth × T_det × T_cent × T_stab = 44% × 0.017 × 0.707 × (1.848 × 0.912 × 1.0) × 1 × 1 × 1 × 1 = 0.0089. We performed similar simulations for each of the other Kepler targets observed between Q1 and Q6, and found that larger planets transiting unseen companions to the targets are the dominant type of blend in this size class, and should account for a total of 16.3 false positives that might be confused with Earth-size planets. Doing the same for all other types of blends that might mimic Earth-size planets leads to an estimate of 24.1 false positives among the Kepler targets.

Burke et al. (2012) have reported that a significant fraction of the Kepler targets have not actually been searched for transit signals down to the official S/N threshold of 7.1. This is due to the fact that spurious detections with S/N over 7.1 can mask real planet signals with lower S/N in the same light curve. Pont et al. (2006) have shown that time-correlated noise features in photometric time series can produce spurious detections well over this threshold, and that this has significant implications for the yield of transit surveys. We note also that the vetting procedure that led to the published KOI list of Batalha et al. (2013) involves human intervention at various stages, and is likely to have missed some low-S/N candidates for a variety of reasons. Therefore, the assumption that most signals with ${\rm {\rm S/N}} > 7.1$ have been detected and are present in the KOI list is probably optimistic. Nevertheless, this hypothesis is useful in that it sets lower limits for the planet occurrence rates, and an upper limit for the false positive rates. For example, the overall false positive rate (all planet classes) we obtain when following the a priori detection rate is 14.9%, compared to our lower rate of 9.4% when using a more accurate model given below.

The clearest evidence that the Kepler team has missed a significant fraction of the low-S/N candidates is seen in Figure 1. This figure displays the distribution of the S/Ns of the actual KOIs, and compares it with the S/Ns for our simulated population of false positives and true planets. To provide for smoother distributions we have convolved the individual S/Ns with a Gaussian with a width corresponding to 20% of each S/N (a kernel density estimation technique). The S/Ns for the KOIs presented by Batalha et al. (2013) were originally computed based on observations from Q1 to Q8. For consistency with our simulations, which only use Q1–Q6, we have therefore degraded the published S/Ns accordingly. Also shown in the figure is the S/N distribution for the KOIs computed with the prescription described in the Appendix based on the CDPP (Christiansen et al. 2012). Several conclusions can be drawn. One is that there is very good agreement between the S/Ns computed by us from the CDPP (red line) and those presented by Batalha et al. (2013; adjusted to Q1–Q6; black line). Importantly, this validates the CDPP-based procedure used in this paper to determine the detectability of a signal. It also suggests that the KOI distribution contains useful information on the actual signal recovery rate of Kepler. Second, we note that a small number of KOIs (∼70) have S/Ns (either computed from the CDPP or adjusted to Q1–Q6) that are actually below the nominal threshold of 7.1. Most of these lower S/Ns are values we degraded from Q1–Q8 to Q1–Q6, and others are for KOIs that were not originally found by the Kepler pipeline, but were instead identified later by further examination of systems already containing one or more candidates. Third and most importantly, the peak of the S/N distribution from our simulations (green line in the figure), which by construction match the size and period distributions of the KOIs and use the a priori detection model, is shifted to smaller values than the one for KOIs. This suggests that the a priori detection model described in Section 3 in which 50% of the signals with ${\rm {\rm S/N}} > 7.1$ (and 99.9% of the signals ${\rm {\rm S/N}} > 10.1$) have been detected as KOIs is not quite accurate, and indicates the detection model requires modification.

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It is not possible to adjust the S/N-dependent recovery rate simultaneously with the occurrence rates of planets in our simulations, as the two are highly degenerate. However, we find that we are able to reach convergence if we assume the following.

• 1.  
  The recovery rate is represented by a monotonically rising function, rather than a fixed threshold, increasing from zero at some low S/N to 100% for some higher S/N.
• 2.  
  The model for the recovery rate function should allow for a good match of the distribution of S/Ns separately for each of our five planet classes.

A number of simple models were tried, and we used the Bayesian Information Criterion (BIC) to compare them and make a choice: BIC = χ² + kln n. In this expression, χ² was computed in the usual way by comparing our simulated S/Ns for the population of false positives and true planets with the one for the KOIs; k is the number of free parameters of the model, and n is the number of bins in the histogram of the S/N distributions. We considered the five planet categories at the same time and computed the BIC by summing up the corresponding χ² values, with n therefore being the sum of the number of bins for all classes. The model that provides the best BIC involves a simple linear ramp for the recovery rate between S/Ns of 6 and 16; in other words, no transit signal with an S/N below 6 is recovered, and every transit signal is recovered over 16. Figure 2 (to be compared with Figure 1) shows the much better agreement between the S/N distribution of our simulated population of false positives and planets, and the S/Ns for the KOIs. We adopt this detection model for the remainder of the paper.

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Planet occurrence rates and false positive rates are interdependent. As described earlier, the bootstrap approach we have adopted to determine those properties for the different planet classes begins with the giant planets, as only larger objects (stars) with well understood properties can mimic their signals. The process then continues with smaller planets in a sequential fashion.

The frequencies of giant planets per period bin were adjusted until their distribution added to that of false positives reproduces the period spectrum of actual KOIs. This is illustrated in Figure 3, where the simulated and actual period distributions (green and dotted black lines) match on average, though not in detail because of the statistical nature of the simulations. The frequency of false positives (orange line) is seen to increase somewhat toward shorter periods, peaking at P ∼ 3–4 days. Similar adjustments to the frequencies by period bin have been performed successively for large Neptunes, small Neptunes, super-Earths, and Earth-size planets.

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We find that the overall frequency of giant planets (planets per star) in orbits with periods up to 418 days is close to 5.2%. Figure 4 (top panel) displays the distribution of simulated false positives for this class of planets as a function of the mass of the host star (orange curve). After adding to these the simulated planets, we obtain the green curve that represents stars that either have a true transiting giant planet or that constitute a false positive mimicking a planet in this category. The comparison with the actual distribution of KOIs (black dotted curve) shows significant differences, with a Kolmogorov–Smirnov (K-S) probability of 0.7%. This suggests a possible correlation between the occurrence of giant planets and spectral type (mass), whereas our simulations have assumed none. In particular, the simulations produce an excess of giant planets around late-type stars, implying that in reality there may be a deficit for M dwarfs. The opposite seems to be true for G and K stars. Doppler surveys have also shown a dependence of the rates of occurrence of giant planets with spectral type. For example, Johnson et al. (2010) reported a roughly linear increase as a function of stellar mass, with an estimate of about 3% for M dwarfs. We find a similar figure of 3.6% ± 1.7%. As is the case for the RV surveys, our frequency increases for G and K stars to 6.1%  ±  0.9%, but then reverses for F (and warmer) stars to 4.3% ± 1.0%, while the Doppler results suggest a higher frequency approaching 10%. We point out, however, that there are rather important differences between the two samples: (1) the estimates from RV surveys extend out to orbital separations of 2.5 AU, while our study is only reasonably complete to periods of ∼400 days (1.06 AU for a solar-type star); (2) the samples are based on two very different characteristics: the planet mass for RV surveys, and planet radius for Kepler; and (3) there may well be a significant correlation between the period distribution of planets and their host star spectral type. Indeed, the results from the study of planets orbiting A-type stars by Bowler et al. (2010) provides an explanation for the apparent discrepancy between RV and Kepler results for the occurrence of giant planets orbiting hot stars: Figure 1 of the above paper shows that no Doppler planets have been discovered with semimajor axes under 0.6 AU for stellar masses over 1.5 M_☉, creating a "planet desert" in that region. Thus, the Doppler surveys find the more common longer-period planets around the hotter stars, and Kepler has found the rarer planets close-in.

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Averaged over all spectral types, the frequency of giant planets up to orbital periods of 418 days is approximately 5.2% (Table 3).

Our simulations result in an overall frequency of large Neptunes with periods up to 418 days of approximately 3.2%. The distribution in terms of host star mass is shown in Figure 5, and indicates a good match to the distribution of the large Neptune-size KOIs from Batalha et al. (2013; K-S probability of 23.3%). We conclude that there is no significant dependence of the occurrence rate of planets in this class with the spectral type of the host star.

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The overall rate of occurrence (planets per star) of small Neptunes rises significantly compared to that of larger planets, reaching 31% out to periods of 245 days. Due to small number statistics, we are unable to provide reliable estimates for periods as long as those considered for the larger planets (up to 418 days). The logarithmic distribution of sizes we have assumed within each planet category allows for a satisfactory fit to the actual KOI distributions in each class (with separate K-S probabilities above 5%), with the exception of the small Neptunes. As noted earlier, the increase in planet occurrence toward smaller radii for these objects is very steep (Figure 7). We find that dividing the small Neptunes into two subclasses (two radius bins of the same logarithmic size: 2–2.8 R_⊕ and 2.8–4 R_⊕) we are able to obtain a much closer match to the KOI population (K-S probability of 6%) with similar logarithmic distributions within each sub-bin as assumed before.

In our analysis, we have deliberately chosen the size range for small Neptunes to be the same as that adopted by Howard et al. (2012), to facilitate the comparison with an interesting result they reported in a study based on the first three quarters of Kepler data. They found that small Neptunes are more common around late-type stars than early-type stars, and that the chance, f(T_eff), that a star has a planet in the 2–4 R_⊕ range depends roughly linearly on its temperature. Specifically, they proposed f(T_eff) = f₀ + k_T(T_eff − 5100 K)/1000 K, valid over the temperature range from 3600 K to 7100 K, with coefficients f₀ = 0.165 ± 0.011 and k_T = −0.081 ± 0.011.

In addition to the use in the present work of the considerably expanded KOI list from Batalha et al. (2013) (which is roughly twice the size of the KOI list from Borucki et al. 2011 used by Howard et al. 2012), there are a number of other significant differences between our analysis and theirs including the fact that we account for false positives, and we use a different model for the detection efficiency of Kepler. It is of considerable interest, therefore, to see if their result still holds, as it could provide important insights into the process of planet formation and/or migration.

We first repeated our analysis as before, with no assumed dependence of the planet frequencies on the spectral type of the host star, but we adjusted other assumptions to match those of Howard et al. (2012). Instead of our modified detection model (linear ramp; Section 5.1), we adopted a fixed S/N threshold of 10, as they did. Also, rather than assuming logarithmic distributions within each of our sub-bins (which Howard et al. 2012 also considered), we assigned to each planet a radius equal to the value corresponding to the center of the bin from Howard et al. (2012), as they did. Adopting the same linear relation f(T_eff) proposed by Howard et al. (2012), a least-squares fit to the frequencies from our simulations as a function of host star effective temperature yielded the coefficients f₀ = 0.170 and k_T = −0.082, in very good agreement with their results.

However, returning to the assumptions of our work in this paper (linear ramp detection model, logarithmic/quadratic size distribution, and false positive corrections), we find that the correlation between planet occurrence and spectral type (or equivalently T_eff, or mass) for the small Neptunes all but disappears (Figure 6): a K-S test indicates that the KOI distribution and our simulated population of false positives and true planets (with the assumption of no mass dependence) are not significantly different (false alarm probability = 11.4%). Thus, we do not confirm the Howard et al. (2012) finding.

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A number of results from our simulations help to explain why we see no dependence of the planet frequency on spectral type, whereas Howard et al. (2012) did. One is that the median S/N for small Neptunes (detectable or not) in our simulated sample is 12.5, a value for which we have shown that the KOI list is likely incomplete (see Section 5.1). This means that a significant number of small Neptunes in the Kepler field have not been recovered as KOIs, especially those transiting larger stars. Second, we find that the distribution of sizes inside the small Neptune class rises sharply toward smaller radii. This is shown in Figure 7. Since these more numerous smaller planets are easier to detect around late-type stars, this artificially boosts the occurrence of planets around such stars. Third, the false positive rate is slightly higher for later-type stars, again resulting in a higher planet occurrence around those stars, if not corrected for.

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According to our simulations the overall average number of super-Earths per star out to periods of 145 days is close to 30%. The distribution of host star masses for the super-Earths is shown in Figure 8. While there is a hint that planets of this size may be less common around M dwarfs than around hotter stars, a K-S test indicates that the simulated and real distributions are not significantly different (false alarm probability of 4.9%).

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As indicated in Table 3, the overall rate of occurrence (average number of planets per star) we find for Earth-size planets is 18.4%, for orbital periods up to 85 days. Similarly to the case for larger planets, our simulated population of false positives and Earth-size planets is a good match to the KOIs in this class, without the need to invoke any dependence on the mass of the host star (see Figure 9).

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Among the Earth-size planets that we have randomly assigned to KIC target stars in our simulations, we find that approximately 23% have S/Ns above 7.1, but only about 10% would be actually detected according to our ramp model for the Kepler recovery rate. These are perhaps the most interesting objects from a scientific point of view. Our results also indicate that 12.3% of the Earth-size KOIs are false positives (Table 1). This fraction is small enough to allow statistical analyses based on the KOI sample, but is too large to claim that any individual Earth-size KOI is a bona fide planet without further examination. Ruling out the possibility of a false positive is of critical importance for the goal of confidently detecting the first Earth-size planets in the habitable zone of their parent star.

On the basis of our simulations we may predict the kinds of false positives that can most easily mimic an Earth-size transit, so that observational follow-up efforts may be better focused toward the validation of the planetary nature of such a signal. Figure 10 shows a histogram of the different kinds of false positives that result in photometric signals similar to Earth-size transiting planets, as a function of their magnitude difference compared to the Kepler target.

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There are two dominant sources of false positives for this class of signals. One is background eclipsing binaries, most of which are expected to be between 8 and 10 mag fainter than the Kepler target in the Kp passband, and some will be even fainter. The most effective way of ruling out background eclipsing binaries is by placing tight limits on the presence of such contaminants as a function of angular separation from the target. In previous planet validations with BLENDER (e.g., Fressin et al. 2011, 2012b; Cochran et al. 2011; Borucki et al. 2012) the constraints from ground-based high spatial resolution adaptive optics imaging have played a crucial role in excluding many background stars beyond a fraction of an arcsecond from the target. However, these observations typically only reach magnitude differences up to 8–9 mag (e.g., Batalha et al. 2011), and such dim sources can only be detected at considerably larger angular separations of several arcsecond. Any closer companions of this brightness would be missed. Since background eclipsing binaries mimicking an Earth-size transit can be fainter still, other more powerful space-based resources may be needed in some cases such as choronography or imaging with the Hubble Space Telescope.

Another major contributor to false positives, according to Figure 10, is larger planets transiting a physically bound companion star. In this case, the angular separations from the target are significantly smaller than for background binaries, and imaging is of relatively little help. Nevertheless, considerable power to rule out such blends can be gained from high-S/N spectroscopic observations in the optical or near-infrared, which can provide useful limits on the presence of very close companions in the form of maximum companion brightness as a function of RV difference compared to the target.

In this work, we have adopted the use of the CDPP, which is an empirical measure of the effective noise seen by transits as a function of their duration (Jenkins et al. 2010b; Christiansen et al. 2012). The CDPP is obtained as a time series for each star for each of 14 trial transit durations ranging from 1.5 hr to 15 hr as a by-product of the search for transiting planets by the Kepler Transiting Planet Search (TPS) pipeline. TPS characterizes the power spectral density of each observed flux time series and calculates the expected S/N of the reference transit pulse at each time step. The CDPP time series are obtained by dividing the reference transit depth by the S/N time series, thereby allowing the S/N to be calculated easily for any depth transit of the given duration. We use the rms CDPP calculated across each quarter of Kepler observations, and take the median value for each star across Q1 through Q6, interpolating across the 14 CDPP transit durations to estimate the CDPP for each simulated transit or false positive eclipse duration.

The measured CDPP is empirical and accounts for the three known sources of noise: Poisson errors from the number of photons received, which depends on star brightness, the stellar variability noise due to stellar surface physics including spots, turbulence (e.g., granulation), acoustic p-modes and magnetic effects, and the residual instrumental effects.

The signal-to-noise ratio of a transit is defined as

$$\begin{equation} {\rm {\rm S/N}} = \frac{\delta }{{\rm CDPP}_{\rm eff}} \sqrt{\frac{t_{\rm obs} f}{P}}, \end{equation} \tag{ A1 }$$

where δ is the photometric depth of the signal and is computed as $\delta =R_p^2/R_{\star }^2$ for a transiting planet of radius R_p transiting a star of radius R_⋆, or as

$$\begin{equation} \delta = \frac{R_{\rm ecl}^2}{R_{\rm blend}^2} \frac{F_{\rm blend}}{F_{\rm blend} +F_{\rm KIC}} \end{equation} \tag{ A2 }$$

for a blend involving an object eclipsing a blending star in the photometric aperture of the KIC target. In the above expression, F_blend/(F_blend + F_KIC) is the contribution in the Kepler bandpass of the flux of the blending star in the photometric aperture normalized to the sum of the blending star and KIC target fluxes. The symbol t_obs is the duration of the Kepler observations from Q1 to Q6, f is target-specific fraction of the total time the target was observed, and P is the orbital period of the transiting object. The transit duration depends on the mass, size, and period of the eclipsing object, all of which are known from our simulations. Assumptions on the eccentricity have some impact on the S/N through the duration. However, the duration enters only as the square root, so any errors are reduced by a factor of two.

The most useful observational constraint available to rule out false positives that does not require additional observations is obtained by measuring the photocenter displacement during the transit. If the transit signal is due to a diluted eclipse of another star in the same photometric aperture as the target, there will generally be a shift in the position of the photocenter that occurs during the transit, as the neighboring star contributes less of the total flux at those times.

Of the ∼2300 KOIs in the cumulative catalog by Batalha et al. (2013), the 1023 new ones that were added to the prior list from Borucki et al. (2011) were subjected to a multi-quarter centroid motion analysis by the Kepler team. This analysis provides a maximum angular separation that corresponds to a 3σ limit beyond which a false positive would have been identified.

In their study, Morton & Johnson (2011) assumed that this exclusion radius scales linearly with the flux from the star and the transit depth, with a lower limit they set at 2'' and an upper limit at 64. The sample of 1023 new KOIs with multi-quarter centroid analysis enables us to re-examine the model of MJ11, and test the strength of the proposed correlations between the centroid exclusion radius and those two parameters (stellar magnitude and transit depth). We also studied the dependencies of the centroid exclusion radius with several other characteristics related to the star and the transit detection.

• 1.  
  Spectral type. By virtue of their different ages, levels of activity, and rotation periods, stars of different spectral types may show different noise patterns that can impact the centroid exclusion radius.
• 2.  
  Galactic latitude. Kepler targets located closer to the Galactic plane are likely to have more contamination from background stars in their aperture, which could directly impact the centroid exclusion radius.
• 3.  
  Noise level. We investigated whether the centroid exclusion radius is correlated with the CDPP of each KOI.
• 4.  
  Transit signal-to-noise ratio. As described in the first part of this Appendix, this parameter is correlated with both the CDPP and the transit depth, along with the number of transits and the transit duration.

For this paper we require a prescription for predicting the approximate multi-quarter centroid exclusion radius for each Kepler target to be used in our simulations. It is sufficient for our purposes to be able to predict a reasonable range for this quantity. Figure 12 shows that the centroid exclusion radius has a large scatter, regardless of which parameter we display it against. Median values in appropriate bins do not appear to show any correlation with the spectral type (or equivalently stellar mass M_⋆), Galactic latitude, or the CDPP. The centroid exclusion radius shows only a weak correlation with the Kepler magnitude, but different from the linear correlation with the flux proposed by MJ11: there is little variation except for the faintest bin (Kp > 15.8 mag), which is likely due to the higher background noise level, and for the brightest stars (Kp < 12 mag), due to the fact that Kepler stars saturate below a magnitude of about 11.5 (Batalha et al. 2013). The clearest correlation in the centroid exclusion radius is with the transit depth (R_p/R_⋆)² (with a Pearson correlation coefficient of −0.1), and with the transit S/N (with a Pearson correlation coefficient of −0.08), which is of course highly correlated with the transit depth (Equation (A1)).

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In order to make use of this correlation with the transit S/N for our simulations, and at the same time to account for the large scatter present in that correlation, we proceeded as follows. We first computed the S/N of each false positive scenario we simulated using Equation (A1), and we then selected a random exclusion radius from the sub-sample of the 1023 KOIs having an S/N within 10% of the one of this false positive. This is the exclusion radius we used for our emulation of the Kepler vetting procedure.