A PLATEAU IN THE PLANET POPULATION BELOW TWICE THE SIZE OF EARTH

The Kepler mission has discovered an extraordinary sample of more than 2300 planets with radii ranging from larger than Jupiter to smaller than Earth (Borucki et al. 2011b; Batalha et al. 2012). Cleanly measuring and debiasing this distribution will be one of Kepler's great legacies. Howard et al. (2012, H12 hereafter) took a key step, showing that the planet radius distribution increases substantially with decreasing planet size down to at least 2 R_E. While the distribution of planets of all periods and radii contains a wealth of information, we choose to focus on the smallest planets. Currently, only Kepler is able to make quantitative statements about the occurrence of planets down to 1 R_E.

The occurrence distributions in H12 were based on planet candidates³ detected in the first four months of Kepler photometry (Borucki et al. 2011b). These planet candidates were detected by a sophisticated pipeline developed by the Kepler team Science Operations Center (Twicken et al. 2010; Jenkins et al. 2010a).⁴ Understanding pipeline completeness, the fraction of planets missed by the pipeline as a function of size and period, is a key component to measuring planet occurrence. Pipeline completeness can be assessed by injecting mock dimmings into photometry and measuring the rate at which injected signals are found. The completeness of the official Kepler pipeline has yet to be measured in this manner. This was the key reason why H12 were cautious interpreting planet occurrence under 2 R_E.

In this work, we focus on determining the occurrence of small planets. To maximize our sensitivity to small planets, we restrict our stellar sample to include only the 12,000 stars having the lowest photometric noise in the Kepler survey. We comb through quarters 1–12 (Q1–Q12)—three years of Kepler photometry—with a new algorithm, TERRA, optimized to detect low signal-to-noise (S/N) transit events. We determine TERRA's sensitivity to planets of different periods and radii by injecting synthetic transits into Kepler photometry and measuring the recovery rate as a function of planet period and radius.

We describe our selection of 12,000 low-noise targets in Section 2. We comb their photometry for exoplanet transits with TERRA, introduced in Section 3. We report candidates found with TERRA (Section 4), which we combine with our measurement of pipeline completeness (Section 5) to produce debiased measurements of planet occurrence (Section 6). We offer some comparisons between TERRA planet candidates and those from Batalha et al. (2012) in Section 7 as well as occurrence measured using both catalogs in Section 8. We offer some interpretations of the constant occurrence rate for planets smaller than 2 R_E in Section 9.

We restrict our study to the best 12,000 solar type stars from the perspective of detecting transits by Earth-size planets, hereafter, the "Best12k" sample. For the smallest planets, uncertainty in the occurrence distribution stems largely from pipeline incompleteness due to the low S/N of an Earth-size transit.

Our initial sample begins with the 102,835 stars that were observed during every quarter from Q1–Q9.⁵ From this sample, following H12, we select 73,757 "solar subset" stars that are solar-type G and K having T_eff = 4100–6100 K and log g = 4.0–4.9 (cgs). T_eff and log g values are present in the Kepler Input Catalog (KIC; Brown et al. 2011) which is available online.⁶ Figure 1 shows the KIC-based T_eff and log g values as well as the solar subset. KIC stellar parameters have large uncertainties: σ(log g) ∼ 0.4 dex and σ(T_eff) ∼ 200 K (Brown et al. 2011). As we will discuss in Section 4, we determine stellar parameters for the majority of TERRA planet candidates spectroscopically. For the remaining cases, we use stellar parameters that were determined photometrically, but incorporated a main sequence prior (Batalha et al. 2012). After refining the stellar parameters, we find that 10 of the 129 TERRA planet candidates fall outside of the T_eff = 4100–6100 K and log g = 4.0–4.9 (cgs) solar subset.

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From the 73,757 stars that pass our cuts on log g and T_eff, we choose the 12,000 lowest noise stars. Kepler target stars have a wide range of noise properties, and there are several ways of quantifying photometric noise. The Kepler team computes quantities called CDPP3, CDPP6, and CDPP12, which are measures of the photometric scatter in 3, 6, and 12 hr bins (Jenkins et al. 2010a). Since CDPP varies by quarter, we adopt the maximum 6 hr CDPP over Q1–Q9 as our nominal noise metric. We use the maximum noise level (as opposed to median or mean) because a single quarter of noisy photometry can set a high noise floor for planet detection. One may circumvent this problem by removing noisy regions of photometry, which is a planned upgrade to TERRA. Figure 2 shows the distribution of max(CDPP6) among the 73,757 stars considered for our sample.

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In choosing our sample, we wanted to include stars amenable to the detection of planets as small as 1 R_E. We picked the 12,000 quietest stars based on preliminary completeness estimates. The noisiest star in the Best12k sample has max(CDPP6) of 79.2 ppm. We estimated that the ∼100 ppm transit of an Earth-size planet would be detected at S/N_CDPP ∼ 1.25.⁷ Given that Q1–Q12 contains roughly 1000 days of photometry, we expected to detect a 5 day planet at ${\rm \rm S/N}_{\rm CDPP} \sim 1.25 \times \sqrt{1000/5} \sim 18$ (a strong detection) and to detect a 50 day planet at ${\rm \rm S/N}_{\rm CDPP} \sim 1.25 \times \sqrt{1000/50} \sim 5.6$ (a marginal detection). In our detailed study of completeness, described in Section 5, we find that TERRA recovers most planets down to 1 R_E having P = 5–50 days.

We draw stars from the H12 solar subset for two reasons. First, we may compare our planet occurrence to that of H12 without the complication of varying occurrence with different stellar types. We recognize that subtle differences may exist between the H12 and Best12k stellar samples. One such difference is that the Best12k is noise-limited, while the H12 sample is magnitude-limited. H12 included bright stars with high photometric variability, which are presumably young and/or active stars. Planet formation efficiency could depend on stellar age. Planets may be less common around older stars that formed before the metallicity of the Galaxy was enriched to current levels. This work assesses planet occurrence for a set of stars that are systematically selected to be 3–10 Gyr old by virtue of their reduced magnetic activity.

The second reason for adopting the H12 solar subset is a practical consideration of our completeness study. As shown in Section 5, we parameterize pipeline efficiency as a function of P and R_P. Because M-dwarfs have smaller radii than G-dwarfs, an Earth-size planet dims an M-dwarf more substantially and should be easier for TERRA to detect. Thus, measuring completeness as a function of P, R_P, and R_⋆ (or perhaps P and R_P/R_⋆) is appropriate when analyzing stars of significantly different sizes. Such extensions are beyond the scope of this paper, and we consider stars with R_⋆ ∼ R_☉.

Identifying the smallest transiting planets in Kepler photometry requires a sophisticated automated pipeline. Our pipeline is called "TERRA" and consists of three major components. First, TERRA calibrates photometry in the time domain. Then, TERRA combs the calibrated photometry for periodic, box-shaped signals by evaluating the S/N over a finely-spaced grid in transit period (P), epoch (t₀) and duration (ΔT). Finally, TERRA fits promising signals with a Mandel & Agol (2002) transit model and rejects signals that are not consistent with an exoplanet transit. We review the calibration component in Section 3.1, but refer the reader to Petigura & Marcy (2012) for a detailed description. We present, for the first time, the grid-search and light curve fitting components in Sections 3.2 and 3.3.

3.1. Photometric Calibration

3.2. Grid-based Transit Search

We then search for periodic, box-shaped signals in ensemble-calibrated photometry. Such a search involves evaluating the S/N over a finely sampled grid in period (P), epoch (t₀), and duration (ΔT), i.e.,

$$\begin{equation} {{\rm S/N}} = {{\rm S/N}}(P,t_{0},\Delta T). \end{equation} \tag{ 1 }$$

Our approach is similar to the widely-used BLS algorithm of Kovács et al. (2002) as well as to the TPS component of the Kepler pipeline (Jenkins et al. 2010b). BLS, TPS, and TERRA are all variants of a "matched filter" (North 1963). The way in which such an algorithm searches through P, t₀, and ΔT is up to the programmer. We choose to search first through ΔT (outer loop), then P, and finally t₀ (inner loop).

For computational simplicity, we consider transit durations that are integer numbers of long cadence measurements. Since we search for transits with P = 5–50 days, we try ΔT = [3, 5, 7, 10, 14, 18] long cadence measurements, which span the range of expected transit durations, 1.5–8.8 hr, for G and K dwarf stars.

After choosing ΔT, we compute the mean depth, $\overline{\delta F}(t_{i})$, of a putative transit with duration = ΔT centered at t_i for each cadence. $\overline{\delta F}$ is computed via

$$\begin{equation} \overline{\delta F}(t_{i}) = \sum _{j} F(t_{i-j}) G_{j} \end{equation} \tag{ 2 }$$

where F(t_i) is the median-normalized stellar flux at time t_i and G_j is the jth element of the following kernel

$$\begin{equation} \mathbf {G} = \frac{1}{\Delta T} \left[ \underbrace{\frac{1}{2}, \ldots , \frac{1}{2}}_{\Delta T}, \underbrace{-1, \ldots , -1}_{\Delta T}, \underbrace{\frac{1}{2}, \ldots , \frac{1}{2}}_{\Delta T} \right]. \end{equation} \tag{ 3 }$$

As an example, if ΔT = 3,

$$\begin{equation} \mathbf {G} = \frac{1}{3} \left[ \frac{1}{2}, \frac{1}{2},\frac{1}{2},-1,-1, -1,\frac{1}{2}, \frac{1}{2},\frac{1}{2}\right]. \end{equation} \tag{ 4 }$$

We search over a finely sampled grid of trial periods from 5 to 50 days and epochs ranging from t_start to t_start + P, where t_start is the time of the first photometric observation. For a given (P, t₀, ΔT) there are N_T putative transits with depths $\overline{\delta F}_{i}$, for i = 0, 1, ..., N_T − 1. For each (P, t₀, ΔT) triple, we compute S/N from

$$\begin{equation} {{\rm S/N}} = \frac{\sqrt{N_T}}{\sigma }\ {\rm mean}\ (\overline{\delta F}_i), \end{equation} \tag{ 5 }$$

where σ is a robust estimate (median absolute deviation) of the noise in bins of length ΔT.

For computational efficiency, we employ the "fast folding algorithm" (FFA) of Staelin (1969) as implemented in E. A. Petigura & G. W. Marcy (2013, in preparation). Let P_{cad, 0} be a trial period that is an integer number of long cadence measurements, e.g., P_{cad, 0} = 1000 implies P = 1000 × 29.4 minutes = 20.43 days. Let N_cad = 51413 be the length of the Q1–Q12 time series measured in long cadences. Leveraging the FFA, we compute S/N at the following periods:

$$\begin{equation} P_{{\rm cad},i} = P_{{\rm cad,0}} + \frac{i}{M-1}; \; i=0, 1, \ldots , M-1 \end{equation} \tag{ 6 }$$

where M = N_cad/P_{cad, 0} rounded up to the nearest power of two. In our search from 5 to 50 days, P_{cad, 0} ranges from 245 to 2445, and we evaluate S/N at ∼10⁵ different periods. At each P_cad,i we evaluate S/N for P_{cad, 0} different starting epochs. All told, for each star, we evaluate S/N at ∼10⁹ different combinations of P, t₀, and ΔT.

Due to runtime and memory constraints, we store only one S/N value for each of the trial periods. TERRA stores the maximum S/N at that period for all ΔT and t₀. We refer to this one-dimensional distribution of S/N as the "S/N periodogram," and we show the KIC-3120904 S/N periodogram in Figure 3 as an example. Because we search over many ΔT and t₀ at each trial period, fluctuations often give rise to S/N ∼ 8 events and set the detectability floor in the S/N periodogram. For KIC-3120904, a star not listed in the Batalha et al. (2012) planet catalog, we see an S/N peak of 16.6, which rises clearly above stochastic background.

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If the maximum S/N in the S/N periodogram exceeds 12, we pass that particular (P, t₀, ΔT) on to the "data validation" (DV) step, described in the following section, for additional vetting. We chose 12 as our S/N threshold by trial and error. Note that the median absolute deviation of many samples drawn from a Gaussian distribution is 0.67 times the standard deviation, i.e., σ_MAD = 0.67σ_STD. Therefore, TERRA S/N = 12 corresponds roughly to S/N = 8 in a BLS or TPS search.

Since TERRA only passes the (P, t₀, ΔT) triple with the highest S/N on to DV, TERRA does not detect additional planets with lower S/N due to either smaller size or longer orbital period. As an example of TERRA's insensitivity to small candidates in multi-candidate systems, we show the TERRA S/N periodogram for KIC-5094751 in Figure 4. Batalha et al. (2012) list two candidates belonging to KIC-5094751: KOI-123.01 and KOI-123.02 with P = 6.48 and 21.22 days, respectively. Although the S/N periodogram shows two sets of peaks coming from two distinct candidates, TERRA only identifies the first peak. Automated identification of multi-candidate systems is a planned upgrade for TERRA. Another caveat is that TERRA assumes strict periodicity and struggles to detect low S/N transits with significant transit timing variations, i.e., variations longer than the transit duration.

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3.3. Data Validation

If the S/N periodogram has a maximum S/N peak  >  12, we flag the corresponding (P, t₀, ΔT) for additional vetting. Following the language of the official Kepler pipeline, we refer to these triples as "threshold crossing events" (TCEs), since they have high photometric S/N, but are not necessarily consistent with an exoplanet transit. TERRA vets the TCEs in a step called "DV," again following the nomenclature of the official Kepler pipeline. DV, as implemented in the official Kepler pipeline, is described in Jenkins et al. (2010a). We emphasize that TERRA DV does not depend on the DV component of the Kepler team pipeline.

We show the distribution of maximum S/N for each Best12k star in Figure 5. Among the Best12k stars, 738 have a maximum S/N peak exceeding 12. Adopting S/N = 12 as our threshold balances two competing needs: the desire to recover small planets (low S/N) and the desire to remove as many non-transit events as possible before DV (high S/N). As discussed below, only 129 out of all 738 events with S/N > 12 are consistent with an exoplanet transit, with noise being responsible for the remaining 609. As shown in Figure 5, that number grows rapidly as we lower the S/N threshold. For example, the number of TCEs grows to 3055 with an S/N threshold of 10, dramatically increasing the burden on the DV component.

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A substantial number (347) of TCEs are due to harmonics or subharmonics of TCEs outside of the P = 5–50 day range and are discarded. In order to pass DV, a TCE must also pass a suite of four diagnostic metrics. The metrics are designed to test whether a light curve is consistent with an exoplanet transit. We describe the four metrics in Table 1 along with the criteria the TCE must satisfy in order pass DV. The metrics and cuts were determined by trial and error. We recognize that the TERRA DV metrics and cuts are not optimal and discard a small number of compelling exoplanet candidates, as discussed in Section 7.3. However, since we measure TERRA's completeness by injection and recovery of synthetic transits, the sub-optimal nature of our metrics and cuts is incorporated into our completeness corrections.

Table 1. In Order to Pass the "Data Validation" (DV) Stage, a "Threshold Crossing Event" (TCE) must Pass the Following Suite of Cuts

Name	Description	Value
s2n_out_on_in	Compelling transits have flat out-of-transit light curves. For a TCE with (P, t0, ΔT), we remove the transit region from the light curve and evaluate the S/N of all other ($P^{\prime },t_{0}^{\prime },\Delta T^{\prime }$) triples where P = P' and ΔT = ΔT'. s2n_out_on_in is the ratio of the two highest S/N events.	<0.7
med_on_mean	Since the our definition of S/N (Equation (5)) depends on the arithmetic mean of individual transit depths, outliers occasionally produce high S/N TCEs. For each TCE, we compute a robust S/N, ${\rm med\; {S/N}} = \frac{\sqrt{N_{T}}}{\sigma }\ {\rm median}\ (\overline{\delta F}_{i}).$ med_on_mean is med S/N divided by S/N as defined in Equation (5).	>1.0
autor	We compute the circular autocorrelation of the phase-folded light curve. autor is the ratio of the highest autocorrelation peak (at 0 lag) to the second highest peak and is sensitive to out-of-transit variability.	>1.6
taur	We fit the phase-folded light curve with a Mandel & Agol (2002) model. taur is the ratio of the best fit transit duration to the maximum duration given the KIC stellar parameters and assuming a circular orbit.	<2.0

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Our suite of automated cuts removes all but 145 TCEs. We perform a final round of manual vetting and remove 16 additional TCEs, leaving 129 planet candidates. Most TCEs that we remove manually come from stars with highly non-stationary photometric noise properties. Some stars have small regions of photometry that exceed typical noise levels by a factor of three. We show the S/N periodogram for one such star, KIC-7592977, in Figure 6. Our definition of S/N (Equation (5)) incorporates a single measure of photometric scatter based on the median absolute deviation, which is insensitive to short bursts of high photometric variability. In such stars, fluctuations readily produce S/N ∼ 12 events and raise the detectability floor to S/N ∼ 12, up from S/N ∼ 8 in most stars. We also visually inspect phase-folded light curves for coherent out-of-transit variability, not caught by our automated cuts, and for evidence of a secondary eclipse.

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Out of the 12,000 stars in the Best12k sample, TERRA detected 129 planet candidates achieving S/N > 12 that passed our suite of DV cuts as well as visual inspection. Table 2 in the Appendix lists the 129 planet candidates. We derive planet radii using R_P/R_⋆ (from Mandel–Agol model fits) and R_⋆ from spectroscopy (when available) or broadband photometry.

We obtained spectra for 100 of the 129 stars using HIRES (Vogt et al. 1994) at the Keck I telescope with the standard configuration of the California Planet Survey (Marcy et al. 2008). These spectra have resolution of ∼50,000, at an S/N of 45 pixel⁻¹ at 5500 Å. We determine stellar parameters using a routine called SpecMatch (A. W. Howard et al. 2013, in preparation). In brief, SpecMatch compares a stellar spectrum to a library of ∼800 spectra with T_eff = 3500–7500 K and log g = 2.0–5.0 (determined from LTE spectral modeling). Once the target spectrum and library spectrum are placed on the same wavelength scale, we compute χ², the sum of the squares of the pixel-by-pixel differences in normalized intensity. The weighted mean of the 10 spectra with the lowest χ² values is taken as the final value for the effective temperature, stellar surface gravity, and metallicity. We estimate SpecMatch-derived stellar radii are uncertain to 10% rms, based on tests of stars having known radii from high resolution spectroscopy and asteroseismology.

For 27 stars where spectra are not available, we adopt the photometrically-derived stellar parameters of Batalha et al. (2012). These parameters are taken from the KIC (Brown et al. 2011), but then modified so that they lie on the Yonsei–Yale stellar evolution models of Demarque et al. (2004). The resulting stellar radii have uncertainties of 35% (rms), but can be incorrect by a factor of two or more. As an extreme example, the interpretations of the three planets in the KOI-961 system (Muirhead et al. 2012) changed dramatically when HIRES spectra showed the star to be an M5 dwarf (0.2 R_☉ as opposed to 0.6 R_☉ listed in the KIC). We could not obtain spectra for two stars, KIC-7345248 and KIC-8429668, which were not present in Batalha et al. (2012). We determine stellar parameters for these stars by fitting the KIC photometry to Yonsei–Yale stellar models. We adopt 35% fractional errors on photometrically-derived stellar radii.

Once we determine P and t₀, we fit a Mandel & Agol (2002) model to the phase-folded photometry. Such a model has three free parameters: R_P/R_⋆, the planet to stellar radius ratio; τ, the time for the planet to travel a distance R_⋆ during transit; and b, the impact parameter. In this work, R_P/R_⋆ is the parameter of interest. However, b and R_P/R_⋆ are covariant, i.e., a transit with b approaching unity only traverses the limb of the star, and thus produces a shallower transit depth. In order to account for this covariance, best fit parameters were computed via Markov Chain Monte Carlo (MCMC). We find that the fractional uncertainty on R_P/R_⋆, σ(R_P/R_⋆)/(R_P/R_⋆) can be as high as 10%, but is generally less than 5%. Therefore, the error on R_P due to covariance with b is secondary to the uncertainty on R_⋆.

We show the distribution of TERRA candidates in Figure 7 over the two-dimensional domain of planet radius and orbital period. Our 129 candidates range in size from 6.83 R_E to 0.48 R_E (smaller than Mars). The median TERRA candidate size is 1.58 R_E. In Figure 8, we show the substantial overlap between the TERRA planet sample and those produced by the Kepler team. TERRA recovers 82 candidates listed in Batalha et al. (2012). We discuss the significant overlap between the two works in detail in Section 7. As of 2012 August 8, 10 of our TERRA candidates were listed as false positives in an internal database of Kepler planet candidates maintained by Jason Rowe (Jason Rowe 2012, private communication) and are shown as blue crosses in Figure 8. We do not include these 10 candidates in our subsequent calculation of occurrence. Table 2 lists the KIC identifier, best fit transit parameters, stellar parameters, planet radius, and Kepler team false positive designation of all 129 candidates revealed by the TERRA algorithm. The best fit transit parameters include orbital period, P; time of transit center, t₀; planet to star radius ratio, R_P/R_⋆; time for planet to cross R_⋆ during transit, τ; and impact parameter, b. We list the following stellar properties: effective temperature, T_eff; surface gravity, log g; and stellar radius, R_⋆.

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When measuring the distribution of planets as a function of P and R_P, understanding the number of missed planets is as important as finding planets themselves. H12 accounted for completeness in a rough sense based on S/N considerations. For each star in their sample, they estimated the S/N over a range of P and R_P using CDPP as an estimate of the photometric noise on transit-length timescales. H12 chose to accept only planets with S/N > 10 in a single quarter of photometry for stars brighter than Kp = 15. This metric used CDPP and was a reasonable pass on the data, particularly when the pipeline completeness was unknown. Determining expected S/N from CDPP does not incorporate the real noise characteristics of the photometry, but instead approximates noise on transit timescales as stationary (CDPP assumed to be constant over a quarter) and Gaussian distributed. Moreover, identifying small transiting planets with transit depths comparable to the noise requires a complex, multistage pipeline. Even if the integrated S/N is above some nominal threshold, the possibility of missed planets remains a concern.

We characterize the completeness of our pipeline by performing an extensive suite of injection and recovery experiments. We inject mock transits into raw photometry, run this photometry though the same pipeline used to detect planets, and measure the recovery rate. This simple, albeit brute force, technique captures the idiosyncrasies of the TERRA pipeline that are missed by simple S/N considerations.

We perform 10,000 injection and recovery experiments using the following steps:

• 1.  
  We select a star randomly from the Best12k sample.
• 2.  
  We draw (P, R_P) randomly from log-uniform distributions over 5–50 days and 0.5–16.0 R_E.
• 3.  
  We draw impact parameter and orbital phase randomly from uniform distributions ranging from 0 to 1.
• 4.  
  We generate a Mandel & Agol (2002) model.
• 5.  
  We inject it into the "simple aperture photometry" of the selected star.

We then run the calibration, grid-based search, and DV components of TERRA (Sections 3.1–3.3) on this photometry and calculate the planet recovery rate. We do not, however, perform the visual inspection described in Section 3.3. An injected transit is considered recovered if the following two criteria are met: (1) the highest S/N peak passes all DV cuts and (2) the output period and epoch are consistent with the injected period and epoch to within 0.01 and 0.1 days, respectively.

Figure 9 shows the distribution of recovered simulations as a function of period and radius. Nearly all simulated planets with R_P > 1.4 R_E are recovered, compared to almost none with R_P < 0.7 R_E. Pipeline completeness is determined in small bins in (P, R_P)-space by dividing the number of successfully recovered transits by the total number of injected transits on a bin-by-bin basis. This ratio is TERRA's recovery rate of putative planets within the Best12k sample. Thus, our quoted completeness estimates only pertain to the low photometric noise Best12k sample. Had we selected an even more rarified sample, e.g., the "Best6k," the region of high completeness would extend down toward smaller planets.

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Following H12, we define planet occurrence, f, as the fraction of a defined population of stars having planets within a domain of planet radius and period, including all orbital inclinations. TERRA, however, is only sensitive to one candidate (highest S/N) per system, so we report occurrence as the fraction of stars with one or more planets with P = 5–50 days. Our occurrence measurements apply to the Best12k sample of low-noise, solar-type stars described in Section 2.

In computing planet occurrence in the Best12k sample, we follow the prescription in H12 with minor modifications. Notably, we have accurate measures of detection completeness described in the previous section. In contrast, H12 estimated completeness based on the presumed S/N of the transit signal, suffering both from approximate characterization of photometric noise using CDPP and from poor knowledge of the efficiency of the planet-finding algorithm for all periods and sizes.

For each P–R_P bin, we count the number of planet candidates, n_{pl, cell}. Each planet that transits represents many that do not transit given the orientation of their orbital planes with respect to Kepler's line of sight. Assuming random orbital alignment, each observed planet represents a/R_⋆ total planets when non-transiting geometries are considered. For each cell, we compute the number of augmented planets, n_{pl, aug, cell} = ∑_ia_i/R_{⋆, i}, which accounts for planets with non-transiting geometries. We then use Kepler's third law together with P and M_⋆ to compute a/R_⋆ assuming a circular orbit.⁸

To compute occurrence, we divide the number of stars with planets in a particular cell by the number of stars amenable to the detection of a planet in a given cell, n_{⋆, amen}. This number is just N_⋆ = 12,000 times the completeness, computed in our Monte Carlo study. The debiased fraction of stars with planets per P–R_P bin, f_cell, is given by f_cell = n_{pl, aug, cell}/n_{⋆, amen}. We show f_cell on the P–R_P plane in Figure 10 as a color scale. We also compute d²f_cell/dlog P/dlog R_P, i.e., planet occurrence divided by the logarithmic area of each cell, which is a measure of occurrence which does not depend on bin size. We annotate each P–R_P bin of Figure 10 with the corresponding value of n_{pl, cell}, n_{pl, aug, cell}, f_cell, and d²f_cell/dlog P/dlog R_P.

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Due to the small number of planets in each cell, errors due to counting statistics alone are significant. We compute Poisson errors on n_{pl, cell} for each cell. Errors on n_{pl, aug, cell}, f_cell, and d²f_cell/dlog P/dlog R_P include only the Poisson errors from n_{pl, cell}. There is also shot noise associated with the Monte Carlo completeness correction due to the finite number of simulated planets in each P–R_P cell, but such errors are small compared to errors on n_{pl, cell}. The orbital alignment correction, a/R_⋆, is also uncertain due to imperfect knowledge of stellar radii and orbital separations. We do not include such errors in our occurrence estimates.

Of particular interest is the distribution of planet occurrence with R_P for all periods. We marginalize over P by summing occurrence over all period bins from 5 to 50 days. The distribution of radii shown in Figure 11 shows a rapid rise in occurrence from 8.0 to 2.8 R_E. H12 also observed a rising occurrence of planets down to 2.0 R_E, which they modeled as a power law. Planet occurrence is consistent with a flat distribution from 2.8 to 1.0 R_E, ruling out a continuation of a power law increase in occurrence for planets smaller than 2.0 R_E. We find $15.1^{+1.8}_{-2.7}\%$ of Sun-like stars harbor a 1.0–2.0 R_E planet with P = 5–50 days. Including larger planets, we find that $24.8^{+2.1}_{-3.4}\%$ of stars harbor a planet larger than Earth with P = 5–50 days. Occurrence values assuming a 100% efficient pipeline are shown as gray bars in Figure 11. The red bars show the magnitude of our completeness correction. Even though TERRA detects many planets smaller than 1.0 R_E, we do not report occurrence for planets smaller than Earth since pipeline completeness drops abruptly below 50%.

We show planet occurrence as a function of orbital period in Figure 12. In computing this second marginal distribution, we include radii larger than 1 R_E so that corrections due to incompleteness are small. Again, as in Figure 11, gray bars represent uncorrected occurrence values while red bars show our correction to account for planets that TERRA missed. Planet occurrence rises as orbital period increases from 5.0 to 10.8 days. Above 10.8 days, planet occurrence is nearly constant per logarithmic period bin with a slight indication of a continued rise. This leveling off of the distribution was noted by H12, who considered R_P > 2.0 R_E. We fit the distribution of orbital periods for R_P > 1.0 R_E with two power laws of the form

$$\begin{equation} \frac{d f }{ d \log P} = k_{P} P^{\alpha }, \end{equation} \tag{ 7 }$$

where α and k_P are free parameters. We find best fit values of $k_{P} = 0.185^{+0.043}_{-0.035},\ \alpha =0.16 \pm 0.07$ for P = 5–10.8 days and $k_{P} = 8.4^{+0.9}_{-0.8} \times 10^{-3},\ \alpha =1.35 \pm 0.05$ for P = 10.8–50 days. We note that k_P and α are strongly covariant. Extrapolating the latter fit speculatively to P > 50 days, we find $41.7^{+6.8}_{-5.9}$% of Sun-like stars host a planet 1 R_E or larger with P = 50–500 days.

Here, we compare our candidates to those of Batalha et al. (2012). Candidates were deemed in common if their periods agree to within 0.01 days. We list the union of the TERRA and Batalha et al. (2012) catalogs in Table 3 in the Appendix. Eighty-two candidates appear in both catalogs (Section 7.1), 47 appear in this work only (Section 7.2), and 33 appear in Batalha et al. (2012) only (Section 7.3). We discuss the significant overlap between the two catalogs and explain why some candidates were detected by one pipeline but not the other.

7.1. Candidates in Common

7.2. TERRA Candidates Not in Batalha et al. (2012) Catalog

TERRA revealed 47 planet candidates that did not appear in Batalha et al. (2012). Such candidates are colored blue and red in Figure 8. Many of these new detections likely stem from the fact that we use twice the photometry that was available to Batalha et al. (2012). To get a sense of how additional photometry improves the planet yield of the Kepler pipeline beyond Batalha et al. (2012), we compared the TERRA candidates to the Kepler team KOI list dated 2012 August 8 (Jason Rowe 2012, private communication). The 28 candidates in common between the 2012 August 8 Kepler team sample and this work are colored blue in Figure 8. Of these 28 candidates, 10 are listed as false positives and denoted as crosses in Figure 8.

We announce 37 new planet candidates with respect to Batalha et al. (2012) that were not listed as false positives in the Kepler team sample. These 37 candidates, all with R_P ⪅ 2 R_E, are a subset of those listed in Table 2. As a convenience, we show this subset in Table 4 in the Appendix. We remind the reader that all photometry used in this work is publicly available. We hope that interested readers will fold the photometry on the ephemeris in Table 2 and assess critically whether a planet interpretation is correct. As a quick reference, we have included plots of the transits of the 37 new candidates from Table 4 in Figures 17 and 18 in the Appendix. We do not claim that our additional candidates bring pipeline completeness to unity for planets with R_P ⪅ 2 R_E. As shown in Section 5, our planet sample suffers from significant incompleteness in the same P–R_P space where most of the new candidates emerged.

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7.3. Batalha et al. (2012) Candidates Not in TERRA Catalog

There are 33 planet candidates in the Batalha et al. (2012) catalog from Best12k stars that TERRA missed. Of these, 28 are multi-candidate systems where one component was identified by TERRA. TERRA is currently insensitive to multiple planet systems (as described in Section 3.2). TERRA missed the remaining five (Batalha et al. 2012) candidates for the following reasons:

• 1.  
  2581.01. A bug in the pipeline prevented successful photometric calibration (Section 3.1). This bug affected 19 out of 12,000 stars in the Best12k sample.
• 2.  
  70.01, 111.01, 119.01. Failed one of the automated DV cuts (taur, med_on_mean, and taur, respectively). We examined these three light curves in the fashion described in Section 3.3, and we determined these light curves were consistent with an exoplanet transit. The fact that DV is discarding compelling transit signals decreases TERRA's overall completeness. Computing DV metrics and choosing the optimum cuts is an art. There is room for improvement here.
• 3.  
  KOI-1151.01. Period misidentified in Batalha et al. (2012). In Batalha et al. (2012) KOI-1151.01 is listed with a P = 5.22 days. TERRA found a candidate with P = 10.43 days. Figure 13 shows phase-folded photometry with the TERRA ephemeris. A period of 5.22 days would imply dimmings in regions where the light curve is flat.

We plot the 33 total candidates listed in Batalha et al. (2012), but not found by TERRA in Figure 14. We highlight the five missed candidates that cannot be explained by the fact that they are a lower S/N candidate in a multi-candidate system. TERRA is blind to planets in systems with another planet with higher S/N. Figure 14 shows that most of these missed planets occur at R_P < 1.4 R_E.

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While the TERRA planet occurrence measurement benefits from well-characterized completeness, it does not include the contribution of multis to overall planet occurrence. As discussed in Sections 3.2 and 7.3, TERRA only detects the highest S/N candidate for a given star. Here, we present planet occurrence including multis from Batalha et al. (2012). Thus, the occurrence within a bin, f, in this section should be interpreted as the average number of planets per star with P = 5–50 days. The additional planets from Batalha et al. (2012) raise the occurrence values somewhat over those of the previous section. However, the rise and plateau structure remain the same.

We compute f_cell from the 32 candidates present in Batalha et al. (2012), but not found by TERRA (mislabeled KOI-1151.01 was not included). For clarity, we refer to this separate occurrence calculation as f_{cell, Batalha}. Because the completeness of the Kepler pipeline is unknown, we apply no completeness correction. This assumption of 100% completeness is certainly an overestimate, but we believe that the sensitivity of the Kepler pipeline to multis is nearly complete for R_P > 1.4 R_E. TERRA has > 80% completeness for R_P > 1.4 R_E because planets in that size range with P = 5–50 days around Best12k stars have high S/N. The Kepler pipeline should also be detecting these high S/N candidates. Also, once a KOI is found, the Kepler team reprocesses the light curve for additional transits (Jason Rowe 2012, private communication). Due to this additional scrutiny, we believe that the Kepler completeness for multis is higher than for singles, all else being equal.

We then add f_{cell, Batalha} to f_cell computed in the previous section. We show occurrence computed using TERRA and Batalha et al. (2012) planets as a function of P and R_P in Figure 15 and as a function of only R_P in Figure 16. The 32 additional planets from Batalha et al. (2012) do not change the overall shape of the occurrence distribution: rising from 4.0 to 2.8 R_E and consistent with flat from 2.8 down to 1.0 R_E.

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H12 fit occurrence for R_P > 2 R_E with a power law,

$$\begin{equation} \frac{d f }{ d \log R_{P} } = k_{R} R_{P} ^{\alpha }, \end{equation} \tag{ 8 }$$

finding α = −1.92 ± 0.11 and $k_{R} = 2.9^{+0.5}_{-0.4}$ (Section 3.1 of H12). As a point of comparison, we plot the H12 power law over our combined occurrence distribution in Figure 16. The fit agrees qualitatively for R_P > 2 R_E, but not within errors. We expect the H12 fit to be ∼25% higher than our occurrence measurements since H12 included planets with P  <  50 days (not P = 5–50 days). Additional discrepancies could stem from different characterizations of completeness, reliance on photometric versus spectroscopic measurements of R_⋆, and magnitude-limited, rather than noise-limited, samples.

9.1. TERRA

9.2. Planet Occurrence

9.3. Interpretation

The authors are indebted to Jon Jenkins, Peter Nugent, Howard Isaacson, Rea Kolbl, Eugene Chiang, Jason Rowe, and Stephen Bryson for productive and enlightening conversations that improved this work. We recognize the independent and complementary work of Fressin et al. (2013), who arrive at similar estimates of planet occurrence. We acknowledge salary support for Petigura by the National Science Foundation through the Graduate Research Fellowship Program. This research used resources of the National Energy Research Scientific Computing Center, which is supported by the Office of Science of the U.S. Department of Energy under contract No. DE-AC02-05CH11231. This work made use of NASA's Astrophysics Data System Bibliographic Services as well as the NumPy (Oliphant 2007), SciPy (Jones et al. 2001), h5py (Collette 2008), IPython (Pérez & Granger 2007), and Matplotlib (Hunter 2007) Python modules. Finally, we extend special thanks to those of Hawai'ian ancestry on whose sacred mountain of Mauna Kea we are privileged to be guests. Without their generous hospitality, the Keck observations presented herein would not have been possible.