The California-Kepler Survey. III. A Gap in the Radius Distribution of Small Planets

TL;DR

This study uses precise measurements from the California-Kepler Survey to identify a significant gap in the size distribution of small, close-in exoplanets, revealing two distinct populations separated by a size gap.

Contribution

It provides the first detailed detection of a size gap at 1.5-2.0 R⊕ in small planets, supporting a two-regime model of planet composition and formation.

Findings

A factor of ≥2 deficit at 1.5-2.0 R⊕ in planet occurrence.

Two size regimes: <1.5 R⊕ and 2.0-3.0 R⊕, with few planets in between.

Similar intrinsic frequencies for both regimes after detection efficiency correction.

Abstract

The size of a planet is an observable property directly connected to the physics of its formation and evolution. We used precise radius measurements from the California-Kepler Survey (CKS) to study the size distribution of 2025 $\textit{Kepler}$ planets in fine detail. We detect a factor of $\geq$2 deficit in the occurrence rate distribution at 1.5-2.0 R$_{\oplus}$. This gap splits the population of close-in ($P$ < 100 d) small planets into two size regimes: R$_P$ < 1.5 R$_{\oplus}$ and R$_P$ = 2.0-3.0 R$_{\oplus}$, with few planets in between. Planets in these two regimes have nearly the same intrinsic frequency based on occurrence measurements that account for planet detection efficiencies. The paucity of planets between 1.5 and 2.0 R$_{\oplus}$ supports the emerging picture that close-in planets smaller than Neptune are composed of rocky cores measuring 1.5 R$_{\oplus}$ or smaller with varying amounts of low-density gas that determine their total sizes.

Tables

Table 1. Table 1 Depth of the Gap

Filter	$V_A$
Full CKS sample	0.746
False positives removed	0.742
Kp	0.686
	0.572
d	0.498
Giant stars removed	0.507
$T_{\mathrm{eff}}$ = 4700–6500 K	0.483

Table 2. Table 2 Planet Detection Statistics

Planet	$P$	$R_P$	SNR	Detection probability	Transit probability	Weight
candidate	d	$R_{\oplus }$	$m_i$	$p_{\det }$	$p_{\det }$	$1/w_i$
K00002.01	2.20	13.41	750.22	1.00	0.14	6.94
K00003.01	4.89	5.11	877.10	1.00	0.05	20.14
K00007.01	3.21	4.13	146.38	1.00	0.11	8.88
K00010.01	3.52	13.39	914.62	1.00	0.09	11.06
K00017.01	3.23	15.04	1212.38	1.00	0.11	9.40
K00018.01	3.55	13.94	820.96	1.00	0.10	9.58
K00020.01	4.44	21.41	1469.42	1.00	0.10	10.15
K00022.01	7.89	14.20	1085.97	1.00	0.06	17.98
K00041.01	12.82	2.37	37.15	0.98	0.05	22.37
K00041.02	6.89	1.35	15.04	0.91	0.07	15.98

Table 3. Table 3 Planet Occurrence

Radius bin	Number of planets per star
$R_{\oplus }$	$f_{\mathrm{bin}}$ for d
1.16–1.29	$0.078\pm 0.017$
1.29–1.43	$0.08\pm 0.013$
1.43–1.59	$0.053\pm 0.011$
1.59–1.77	$0.0334\pm 0.0092$
1.77–1.97	$0.05\pm 0.01$
1.97–2.19	$0.086\pm 0.016$
2.19–2.43	$0.098\pm 0.016$
2.43–2.70	$0.077\pm 0.016$
2.70–3.00	$0.053\pm 0.012$
3.00–3.33	$0.0316\pm 0.0089$
3.33–3.70	$0.0242\pm 0.0066$
3.70–4.12	$0.0094\pm 0.0057$
4.12–4.57	$0.0056\pm 0.0034$
4.57–5.08	$0.0037\pm 0.0031$
5.08–5.65	$0.0066\pm 0.0048$
5.65–6.27	$0.005\pm 0.003$
6.27–6.97	$0.0\pm inf$
6.97–7.75	$0.0019\pm 0.0029$
7.75–8.61	$0.0044\pm 0.0034$
8.61–9.56	$0.00022\pm 0.00032$
9.56–10.63	$0.001\pm 0.0015$
10.63–11.81	$0.00035\pm 0.00053$
11.81–13.12	$0.00104\pm 0.00094$
13.12–14.58	$0.0038\pm 0.0021$
14.58–16.20	$0.00084\pm 0.00066$
16.20–18.00	$0.0003\pm 0.0004$

Table 4. Table 4 Spline Fit

Node Location	Best-fit Value	1 $\sigma$ Credible Interval
$R_{\oplus }$	($f_{\mathrm{bin}}$)	($f_{\mathrm{bin}}$)
1.3	0.078	fixed
1.5	0.051	$0.05\pm 0.02$
1.9	0.030	$0.03\pm 0.02$
2.4	0.116	$0.11\pm 0.01$
3.0	0.043	$0.044\pm 0.005$
4.5	0.0050	$0.005\pm 0.002$
11.0	0.00050	$0.0005\pm 0.0003$

Table 5. Table 5 Occurrence Rate Comparison

Radius Interval	Period Interval	This Work11Uncertainties do not include the scaling factors derived in Appendix C	H122,6,72,6,7footnotemark:	P133,63,6footnotemark:	F1344Fressin et al. (2013)	M1555Mulders et al. (2015)
$R_{\oplus }$	(days)	($f_{\mathrm{bin}}$ %)	($f_{\mathrm{bin}}$ %)	($f_{\mathrm{bin}}$ %)	($f_{\mathrm{bin}}$ %)	($f_{\mathrm{bin}}$ %)
1.4–2.8		$43.1\pm 2.2$	$\cdots$	$32.8\pm 1.4$	$35.0\pm 2.8$88Periods shorter than 85 days	$26.7\pm 1.7$88Periods shorter than 85 days
2–2.8		$19.4\pm 1.4$	$9.0\pm 1.5$	$18.6\pm 1.6$	$17.5\pm 1.6$	$12.8\pm 0.5$
2–4		$25.4\pm 1.6$	$13.0\pm 0.8$	$16.6\pm 1.8$	$18.3\pm 1.3$	$18.6\pm 0.6$
2–4		$36.6\pm 2.2$	$\cdots$	$24.1\pm 2.3$	$24.0\pm 2.2$88Periods shorter than 85 days	$22.9\pm 0.8$88Periods shorter than 85 days

Table 6. Table C.1 Bin Uncertainty Scaling Factors

Radius bin	Scaling Factor
$R_{\oplus }$
0.50–0.56	2.82
0.56–0.62	2.50
0.62–0.69	2.30
0.69–0.76	2.54
0.76–0.85	2.35
0.85–0.94	2.09
0.94–1.05	1.92
1.05–1.16	1.95
1.16–1.29	1.89
1.29–1.43	1.46
1.43–1.59	1.65
1.59–1.77	1.81
1.77–1.97	1.38
1.97–2.19	1.50
2.19–2.43	1.39
2.43–2.70	1.58
2.70–3.00	1.48
3.00–3.33	1.58
3.33–3.70	1.25
3.70–4.12	1.48
4.12–4.57	1.47
4.57–5.08	1.46
5.08–5.65	1.63
5.65–6.27	1.45
6.27–6.97	1.50
6.97–7.75	1.52
7.75–8.61	1.34
8.61–9.56	1.44
9.56–10.63	1.46
10.63–11.81	1.52
11.81–13.12	1.57
13.12–14.58	1.36
14.58–16.20	1.35
16.20–18.00	1.45
18.00–20.00	1.44

Equations

$$\frac{R_{\star }}{R_{\odot }}>10^{0.00025(T_{\mathrm{eff}}/\mathrm{K}-5500)+0.20},$$

$$m_i={\left(\frac{R_P}{R_{\star ,i}}\right)}^2\sqrt{\frac{T_{\mathrm{obs},\text{i}}}{P}}\left(\frac{1}{{\mathrm{CDPP}}_{\mathrm{dur},\text{i}}}\right),$$

$$C(m_i;k,\theta ,l)=\Gamma (k){\int }_0^{\frac{m_i-l}{\theta }}{t}^{k-1}{e}^{-t}dt,$$

$$w_i=\frac{1}{(p_{\det }\cdot p_{\mathrm{tr}})},$$

$$p_{\det }=\frac{1}{N_{\star }}\sum _i^{N_{\star }}C(m_i).$$

$$f_{\mathrm{bin}}=\frac{1}{N_{\star }}\sum _{i=1}^{n_{\mathrm{pl},\mathrm{bin}}}{w}_i.$$

$$\varphi (x)=\frac{1}{N_{\star }}\sum _{i=1}^{n_{\mathrm{pl}}}{w}_i\cdot K(x-x_i,{\sigma }_{x,i}).$$

Full text

1. Introduction

NASA’s Kepler space telescope enabled the discovery of over 4000 transiting planet candidates111NASA Exoplanet Archive, 2/27/2017,222 The false positive probability for the majority of the Kepler candidates is 5–10% (Morton & Johnson 2011). opened the door to detailed studies of exoplanet demographics. One of the first surprises to arise from studies of the newly revealed sample of planets was the multitude of planets with radii smaller than Neptune but larger than Earth ($R_{P}$=1.0–3.9 $R_{\earth}$, Batalha et al. 2013). Our solar system has no example of these intermediate planets, yet they are by far the most common in the Kepler sample (Howard et al. 2012; Fressin et al. 2013; Petigura et al. 2013b; Youdin 2011; Christiansen et al. 2015; Dressing & Charbonneau 2015; Morton & Swift 2014).

A key early question of the Kepler mission was whether these sub-Neptune-size planets are predominantly rocky or possess low-density envelopes that contribute significantly to the planet’s overall size. The radial velocity (RV) follow-up effort of the Kepler project focused on 22 stars hosting one or more sub-Neptunes (Marcy et al. 2014). In addition, detailed modeling of transit timing variations (TTVs) provided mass constraints for a large number of systems in specific architectures (e.g., Wu & Lithwick 2013; Hadden & Lithwick 2014, 2016). The resulting mass measurements revealed that most planets larger than 1.6 $R_{\earth}$ have low densities that were inconsistent with purely rocky compositions, and instead required gaseous envelopes (Weiss & Marcy 2014; Rogers 2015).

The distinction between rocky and gaseous planets reflects the typical core sizes of planets as well as the physical mechanisms by which planets acquire (and lose) gaseous envelopes. The densities of planets with radii smaller than $\sim$1.6 $R_{\earth}$ are generally consistent with a purely rocky composition (Weiss & Marcy 2014; Rogers 2015) and their radius distribution likely reflects their initial core sizes. However, a small amount of H/He gas added to a roughly Earth-size rocky core can substantially increase planet size, without significantly increasing planet mass. For this reason, it has been suggested that the radii of sub-Neptune-size planets, along with knowledge of the irradiation history, would be sufficient to estimate bulk composition without additional information (Lopez & Fortney 2013; Wolfgang & Lopez 2015).

The large number of planets smaller than Neptune discovered by the Kepler mission was unexpected given prevailing theories of planet formation, which were developed to explain the distribution of giant planets (Ida & Lin 2004; Mordasini et al. 2009). These theories predicted that planets should either fail to accrete enough material to become super-Earths, or they would grow quickly, accreting all of the gas in their feeding zones growing to massive, gas-rich giant planets. Modern formation models are now able to reproduce the observed population of super-Earths (Hansen & Murray 2012; Mordasini et al. 2012; Alibert et al. 2013; Chiang & Laughlin 2013; Lee et al. 2014; Chatterjee & Tan 2014; Coleman & Nelson 2014; Raymond & Cossou 2014; Lee & Chiang 2016). Many of these new models can be corroborated by measuring the bulk properties of individual planets and the typical properties of the population.

As formation models continue to be refined, the role of atmospheric erosion on these short-period planets is becoming more apparent. Several authors have predicted the existence of a “photoevaporation valley” in the distribution of planet radii (e.g., Owen & Wu 2013; Lopez & Fortney 2014; Jin et al. 2014; Chen & Rogers 2016; Lopez & Rice 2016).

Photoevaporation models predict that there should be a dearth of intermediate sub-Neptune size planets orbiting in highly irradiated environments. The mass of H/He in the envelope must be finely tuned to produce a planet in this intermediate size range. Planets with too little gas in their envelopes are stripped to bare, rocky cores by the radiation from their host stars. In general, the radii of bare, rocky cores versus planets with a few percent by mass H/He envelopes depend on many uncertain variables such as the initial core mass distribution and the insolation flux received by the planet. A rift in the distribution of small planet radii is a common result of the planet formation models that include photoevaporation.

Owen & Wu (2013) provided tentative observational evidence for such a feature in the radius distribution of Kepler planets. They observed a bimodal structure in the planet radius distribution, particularly when the planet sample was split into subsamples with low and high integrated X-ray exposure histories. However, the relatively large planet radius uncertainties in Owen & Wu (2013) diluted the gap and reduced its statistical significance. Their study also considered the number distribution of planets, and was not corrected for completeness as we do below. Such corrections mitigate sample bias and allow for the recovery of the underlying planet distribution from the observed one.

Here, we examine a sample of planets orbiting stars with precisely measured radii from the California-Kepler Survey (CKS; see Petigura et al. (2017) and Johnson et al. (2017)). We use the precise stellar radii to update the planet radii, bringing the distribution of planet radii into sharper focus and revealing a gap between 1.5 and 2.0 $R_{\earth}$.

This paper is structured as follows. In §2 we discuss our stellar and planetary samples. We describe our methods for correcting for pipeline search sensitivity and transit probabilities in §3. In §4 we examine the one-dimensional marginalized radius distribution and also two-dimensional distributions of planet radius as a function of orbital period, stellar radius, and insolation flux. We discuss potential explanations for the observed planet radius gap in §5 and finish with some concluding remarks in §6.

2. Sample of Planets

2.1. California Kepler Survey

For this work we adopt the stellar sample and the measured stellar parameters from the CKS program (Petigura et al. 2017, hereafter Paper I). The measured values of $T_{\mathrm{eff}}$, $\log g$, and [Fe/H] are based on a detailed spectroscopic characterization of \IfEqCasenstars-cksnstars-cks 1305 ncand-cks2025 med-srad-frac-err11% med-smass-frac-err 4% cks-rp-frac-err-median12% gamma-k$k=17.56$ gamma-l$l=1.00$ (fixed) gamma-theta$\theta=0.49$ logistic-b$b=9.54$ logistic-c$c=10.09$ logistic-d$d=0.90$ num-sim-planets45000 num-planets-filtered900 num-stars-inj3840 error-bin-fudge1.5–2.9 nstars-occ36,075 ncand-sample900 occ-ratio$0.8\pm 0.2$ kstest0.003 adtest0.012 tr-prob-sm6% tr-prob-lg3% det-prob-sm86% det-prob-lg96% [] Kepler Object of Interest (KOI) host stars using observations from Keck/HIRES (Vogt et al. 1994). In Johnson et al. (2017, hereafter Paper II), we associated those stellar parameters from Paper I to Dartmouth isochrones (Dotter et al. 2008) to derive improved stellar radii and masses, allowing us to recalculate planetary radii using the light curve parameters from Mullally et al. (2015), hereafter “Q16”. Median uncertainties in stellar radius improve from 25% (Huber et al. 2014) to \IfEqCasemed-srad-frac-errnstars-cks 1305 ncand-cks2025 med-srad-frac-err11% med-smass-frac-err 4% cks-rp-frac-err-median12% gamma-k$k=17.56$ gamma-l$l=1.00$ (fixed) gamma-theta$\theta=0.49$ logistic-b$b=9.54$ logistic-c$c=10.09$ logistic-d$d=0.90$ num-sim-planets45000 num-planets-filtered900 num-stars-inj3840 error-bin-fudge1.5–2.9 nstars-occ36,075 ncand-sample900 occ-ratio$0.8\pm 0.2$ kstest0.003 adtest0.012 tr-prob-sm6% tr-prob-lg3% det-prob-sm86% det-prob-lg96% [] after our CKS spectroscopic analysis. Stellar mass uncertainties improve from 14% to \IfEqCasemed-smass-frac-errnstars-cks 1305 ncand-cks2025 med-srad-frac-err11% med-smass-frac-err 4% cks-rp-frac-err-median12% gamma-k$k=17.56$ gamma-l$l=1.00$ (fixed) gamma-theta$\theta=0.49$ logistic-b$b=9.54$ logistic-c$c=10.09$ logistic-d$d=0.90$ num-sim-planets45000 num-planets-filtered900 num-stars-inj3840 error-bin-fudge1.5–2.9 nstars-occ36,075 ncand-sample900 occ-ratio$0.8\pm 0.2$ kstest0.003 adtest0.012 tr-prob-sm6% tr-prob-lg3% det-prob-sm86% det-prob-lg96% [] in the Paper II catalog. This leads to median uncertainties in planet radii of \IfEqCasecks-rp-frac-err-mediannstars-cks 1305 ncand-cks2025 med-srad-frac-err11% med-smass-frac-err 4% cks-rp-frac-err-median12% gamma-k$k=17.56$ gamma-l$l=1.00$ (fixed) gamma-theta$\theta=0.49$ logistic-b$b=9.54$ logistic-c$c=10.09$ logistic-d$d=0.90$ num-sim-planets45000 num-planets-filtered900 num-stars-inj3840 error-bin-fudge1.5–2.9 nstars-occ36,075 ncand-sample900 occ-ratio$0.8\pm 0.2$ kstest0.003 adtest0.012 tr-prob-sm6% tr-prob-lg3% det-prob-sm86% det-prob-lg96% [] which enable the detection of finer structures in the planet radius distribution.

2.2. Sample Selection

The CKS stellar sample was constructed to address a variety of science topics (Paper I). The core sample is a magnitude-limited set of KOIs (Kp $<$ 14.2). Additional fainter stars were added to include habitable zone planets, ultra-short-period planets, and multi-planet systems. Here, we enumerate a list of cuts in parameter space designed to create a sample of planets with well-measured radii and with well-quantified detection completeness. The primary goal is to determine anew the occurrence of planets as a function of planet radius, with greater reliability than was previously possible.

We start by removing planet candidates deemed false positives in Paper I. The Paper I false positive designations were determined using the false positive probabilities calculated by Morton & Johnson (2011); Morton (2012); Morton et al. (2016), the Kepler team’s designation available on the NASA Exoplanet Archive, and a search for secondary lines in the HIRES spectra (Kolbl et al. 2015) as well as any other information available in the literature for individual KOIs. Next, we restrict our sample to only the magnitude-limited portion of the larger CKS sample (Kp $<14.2$).

The planet-to-star radius ratio ($R_{P}/R_{\star}$) becomes uncertain at high impact parameters ($b$) due to degeneracies with limb-darkening. We excluded KOIs with $b>0.7$ to minimize the impact of grazing geometries. We experimented other thresholds in $b$ and found that our results are relatively insensititve to $b<$ 0.6, 0.7, or 0.8, with the trade-off of smaller sample size with decreasing threshold in $b$.

We removed planets with orbital periods longer than 100 days in order to avoid domains of low completeness (especially for planets smaller than about 4 $R_{\earth}$) and low transit probability.

We also excised planets orbiting evolved stars since they have somewhat lower detectability and less certain radii. This was implemented using an ad hoc temperature-dependent stellar radius filter,

[TABLE]

which is plotted in Figure 1. We also restricted our sample to planets orbiting stars within the temperature range where we can extract precise stellar parameters from our high resolution optical spectra (6500–4700 K). Finally, we accounted for uncertainties in the completeness corrections caused by systematic and random measurement errors in the simulations, described in Appendix C.

The multiple filters purify the CKS sample of stars and planets and are summarized in Figure 2. We assessed the impact of filters on the depth of the planet radius valley using an ad hoc metric $V_{A}$. This quantity is defined as the ratio of the number of planets with radii of 1.64–1.97 $R_{\earth}$ (the bottom of the valley) to the average number of planets with radii of 1.2–1.44 $R_{\earth}$ or 2.16–2.62 $R_{\earth}$(the peaks of the distrubtion immediately outside of the valley). The radius limits for the calculation of $V_{A}$ were chosen so that $V_{A}=1$ for a log-uniform distribution of planets with radii between 1.2 $R_{\earth}$ and 2.62 $R_{\earth}$. Smaller values of $V_{A}$ denote a deeper valley. The values of $V_{A}$ after applying each successive filter are tabulated in Table 1.

Furlan et al. (2017) compiled a catalog of KOI host stars that were observed using a collection of high-resolution imaging facilities (Lillo-Box et al. 2012, 2014; Horch et al. 2012, 2014; Everett et al. 2015; Gilliland et al. 2015; Cartier et al. 2015; Wang et al. 2015a, b; Adams et al. 2012, 2013; Dressing et al. 2014; Law et al. 2014; Baranec et al. 2016; Howell et al. 2011). Many of the 1902 KOIs in the Furlan et al. (2017) catalog also appear in our sample. We investigated removing KOI hosts with known companions or large dilution corrections but found no significant changes to the shape of the distribution. Since only a subset of our KOIs were observed by Furlan et al. (2017) and it is difficult to determine the binarity of the parent stellar population for occurrence calculations, we chose not to filter our planet catalog using the results of high-resolution imaging. However, many of these stars may have already been identified as false positives in the Paper I catalog and therefore removed from our final sample of planets.

We investigated the impact of our apparent magnitude cut by examinging the size distribution for three ranges of Kp (Figure 3). For these tests we applied all of the filters described in this section except the Kp $<14.2$ magnitude cut. We found that the planet radius distribution for Kp $<13.5$ is statistically indistinguishable from the radius distribution for planets orbiting stars with $13.5<$ Kp $\leq 14.2$. An Anderson-Darling test (Anderson & Darling 1952; Scholz & Stephens 1987) predicts that the two distributions were drawn from the same parent population with a p-value of 0.6. However, the radius distribution of planets orbiting host stars with $\textit{Kp}\geq 14.2$ is visually and statistically different (p-value < 0.0004). This is somewhat expected given the non-systematic target selection for both the initial Kepler target stars and the stars observed in the CKS survey. Stars with Kp $>14.2$ were only observed in the CKS program because they were hosts to multi-planet systems, habitable-zone candidates, ultra-short period planets, or other special cases. Targets fainter than $\textit{Kp}>14.0$ were observed by Kepler only if their stellar and noise properties indicated that there was a high probability of the detection of small planets (Batalha et al. 2010). These non-uniform Kepler target selection effects motivate our choice to exclude faint stars. The final distributions of planet radii do not depend on the Kp $<14.2$ or Kp $<14.0$ (p-value $>$ 0.95) choice. But there are 153 planet candidates with $14.0<$Kp $<14.2$ so we choose to include those additional candidates to maximize the statistical power of the final sample.

The two distinct peaks separated by a valley (Figure 2) are apparent in the initial number distribution of planet radii and the final distribution after the filters are applied. The depth of the valley increases as we apply these filters, suggesting that the purity of the planet sample improves with filter application. Note that the filters act on the stellar characteristics and are agnostic to planet radius.

Figure 4 shows histograms of the stellar radii and planet-to-star radius ratios ($R_{P}/R_{\star}$) for the filtered sample stars. These two distributions are both unimodel. This demonstrates that the bimodality of the planet radius distribution is not an artifact of the stellar sample or the light curve fitting used to measure $R_{P}/R_{\star}$.

3. Completeness Corrections

To recover the underlying planet radius distribution from the observed distribution we made completeness corrections to compensate for decreasing detectability of planets with small radii and/or long orbital periods.

An additional complication associated with the completeness corrections in this work is that the stellar properties of the planet-hosting stars come from a different source and have higher precision than the stellar properties for the full set of Kepler target stars. We explore the additional uncertainties introduced by this fact by running a suite of simulated transit surveys described in Appendix C. We inflate the uncertainties on the histogram bin heights by the scaling factors listed in Table C.1 to account for these effects.

3.1. Pipeline Efficiency

We followed the procedure described in Christiansen et al. (2016) using the results from their injection-recovery experiments (Christiansen et al. 2015). They injected about ten-thousand transit signals into the raw pixel data and processed the results with version 9.1 of the official Kepler pipeline (Jenkins et al. 2010). These completeness tests were used to identify combinations of transit light curve parameters that could be recovered by the Kepler pipeline for a given sample of target stars. They injected signals onto both target stars and neighboring pixels to quantify the pipeline’s ability to identify astrophysical false positives. We assumed that our sample is free of the vast majority of false positives so we only considered injections of transits onto the target stars. We only considered injections on stars that would have been included in the CKS sample and would not be removed by the filters described in §2.2. Namely, we considered injected impact parameters less than 0.7, injected periods shorter than 100 days, Kp $\leq 14.2$, 4700 K $<T_{\mathrm{eff}}<$ 6500 K, and stellar radii compatible with Equation 1 based on the values in the Stellar17 catalog333https://archive.stsci.edu/kepler/stellar17/search.php prepared by the Kepler stellar parameters working group (Mathur et al. 2016). This leaves a total of \IfEqCasenum-stars-injnstars-cks 1305 ncand-cks2025 med-srad-frac-err11% med-smass-frac-err 4% cks-rp-frac-err-median12% gamma-k$k=17.56$ gamma-l$l=1.00$ (fixed) gamma-theta$\theta=0.49$ logistic-b$b=9.54$ logistic-c$c=10.09$ logistic-d$d=0.90$ num-sim-planets45000 num-planets-filtered900 num-stars-inj3840 error-bin-fudge1.5–2.9 nstars-occ36,075 ncand-sample900 occ-ratio$0.8\pm 0.2$ kstest0.003 adtest0.012 tr-prob-sm6% tr-prob-lg3% det-prob-sm86% det-prob-lg96% [] synthetic transit signals injected onto the target pixels of \IfEqCasenum-stars-injnstars-cks 1305 ncand-cks2025 med-srad-frac-err11% med-smass-frac-err 4% cks-rp-frac-err-median12% gamma-k$k=17.56$ gamma-l$l=1.00$ (fixed) gamma-theta$\theta=0.49$ logistic-b$b=9.54$ logistic-c$c=10.09$ logistic-d$d=0.90$ num-sim-planets45000 num-planets-filtered900 num-stars-inj3840 error-bin-fudge1.5–2.9 nstars-occ36,075 ncand-sample900 occ-ratio$0.8\pm 0.2$ kstest0.003 adtest0.012 tr-prob-sm6% tr-prob-lg3% det-prob-sm86% det-prob-lg96% [] stars observed by Kepler. We also apply these same filters to the stars in the Stellar17 catalog. The number of stars remaining after the filters are applied is the number of stars observed by Kepler that could have led to detections of planets that would be present in our filtered planet catalog ($N_{\star}=$ \IfEqCasenstars-occnstars-cks 1305 ncand-cks2025 med-srad-frac-err11% med-smass-frac-err 4% cks-rp-frac-err-median12% gamma-k$k=17.56$ gamma-l$l=1.00$ (fixed) gamma-theta$\theta=0.49$ logistic-b$b=9.54$ logistic-c$c=10.09$ logistic-d$d=0.90$ num-sim-planets45000 num-planets-filtered900 num-stars-inj3840 error-bin-fudge1.5–2.9 nstars-occ36,075 ncand-sample900 occ-ratio$0.8\pm 0.2$ kstest0.003 adtest0.012 tr-prob-sm6% tr-prob-lg3% det-prob-sm86% det-prob-lg96% []). We calculated the fraction of injected signals recovered as a function of injected signal-to-noise as

[TABLE]

where $R_{P}$ and $P$ are the radius and period of the particular injected planet. $R_{\star,i}$ is the stellar radius for the $i^{\rm th}$ star in the Stellar17 catalog, $T_{\rm obs,\emph{i}}$ is the amount of time that the particular star was observed, and $\rm{CDPP}_{\rm dur,\emph{i}}$ is the Combined Differential Photometric Precision (CDDP, Koch et al. 2010) value for each star extrapolated to the transit duration for each injection. We fit a 2nd order polynomial in $1/\sqrt{d}$ to the $d=3$, 6, and 12-hour CDPP values for each star to perform the extrapolation (Sinukoff et al. 2013).

We fit a $\Gamma$ cumulative distribution function (CDF) to the recovery fraction vs. injected ($m_{i}$) of the form

[TABLE]

to derive the average pipeline efficiency. $C(m_{i})$ is the probability that a signal with a given value of $m_{i}$ would actually be detected by the Kepler transit search pipeline. In practice we used the scipy.stats.gammacdf(t, k, l, $\theta$) function in SciPy version 0.18.1. Using the lmfit Python package (Newville et al. 2014) to minimize the residuals we found best-fit values of \IfEqCasegamma-knstars-cks 1305 ncand-cks2025 med-srad-frac-err11% med-smass-frac-err 4% cks-rp-frac-err-median12% gamma-k$k=17.56$ gamma-l$l=1.00$ (fixed) gamma-theta$\theta=0.49$ logistic-b$b=9.54$ logistic-c$c=10.09$ logistic-d$d=0.90$ num-sim-planets45000 num-planets-filtered900 num-stars-inj3840 error-bin-fudge1.5–2.9 nstars-occ36,075 ncand-sample900 occ-ratio$0.8\pm 0.2$ kstest0.003 adtest0.012 tr-prob-sm6% tr-prob-lg3% det-prob-sm86% det-prob-lg96% [], \IfEqCasegamma-lnstars-cks 1305 ncand-cks2025 med-srad-frac-err11% med-smass-frac-err 4% cks-rp-frac-err-median12% gamma-k$k=17.56$ gamma-l$l=1.00$ (fixed) gamma-theta$\theta=0.49$ logistic-b$b=9.54$ logistic-c$c=10.09$ logistic-d$d=0.90$ num-sim-planets45000 num-planets-filtered900 num-stars-inj3840 error-bin-fudge1.5–2.9 nstars-occ36,075 ncand-sample900 occ-ratio$0.8\pm 0.2$ kstest0.003 adtest0.012 tr-prob-sm6% tr-prob-lg3% det-prob-sm86% det-prob-lg96% [], and \IfEqCasegamma-thetanstars-cks 1305 ncand-cks2025 med-srad-frac-err11% med-smass-frac-err 4% cks-rp-frac-err-median12% gamma-k$k=17.56$ gamma-l$l=1.00$ (fixed) gamma-theta$\theta=0.49$ logistic-b$b=9.54$ logistic-c$c=10.09$ logistic-d$d=0.90$ num-sim-planets45000 num-planets-filtered900 num-stars-inj3840 error-bin-fudge1.5–2.9 nstars-occ36,075 ncand-sample900 occ-ratio$0.8\pm 0.2$ kstest0.003 adtest0.012 tr-prob-sm6% tr-prob-lg3% det-prob-sm86% det-prob-lg96% []. Figure 5 shows the fraction of injections recovered as a function of $m_{i}$ and our model for pipeline efficiency.

Our pipeline efficiency curve is $\sim$15-25% lower than the efficiency as a function of the Kepler multi-event statistic (MES) derived in (Christiansen et al. 2015) for their FGK subsample. The difference can be explained by the fact that the MES is estimated in the Kepler pipeline during a multidimensional grid search. In most cases, the search grid is not fine enough to find the exact period and transit time for a given planet candidate. Since the grid search doesn’t find the best-fit transit model it generally underestimates the SNR ($m_{i}$) by a factor of $\sim$25% (Petigura et al., in preparation).

3.2. Survey Sensitivity