PLANET OCCURRENCE WITHIN 0.25 AU OF SOLAR-TYPE STARS FROM KEPLER^*

To measure planet occurrence, we compare the number of detected planets having some set of properties (radii, orbital periods, etc.) to the set of stars from which planets with those properties could have been reliably detected. Errors in either the number of planets detected or the number of stars surveyed corrupt the planet occurrence measurement. We adopt the philosophy that it is preferable to suffer higher Poisson errors from considering fewer planets and stars than the difficult-to-quantify systematic errors caused by studying a larger number of planets and stars with more poorly determined detection completeness.

We begin our winnowing of target stars with the Kepler Input Catalog (KIC; Brown et al. 2011; Kepler Mission Team 2009). In this paper, we include only planet candidates found in three data segments ("Quarters") labeled Q0, Q1, and Q2, for which all photometry is published (Borucki et al. 2011b). Q0 was data commissioning (2009 May 2–11), Q1 includes data from 2009 May 13 to June 15, and Q2 includes data from 2009 June 15 to September 17. The segments had durations of 9.7, 33.5, and 93 days, respectively. Kepler achieved a duty cycle of greater than 90%, which almost completely eliminated window function effects (von Braun et al. 2009). A total of 156,097 long-cadence targets (30 minute integrations) were observed in Q1, and 166,247 targets were observed in Q2, with the targets in Q2 being nearly a superset of those in Q1. In this paper, we consider only the "exoplanet target stars," of which there were 153,196 observed during Q2, and which are used for the statistics presented here (Batalha et al. 2010). (The remaining Kepler targets in Q2 were evolved stars, not suitable for sensitive planet detection.) The few percent changes in the planet search target stars are not significant here as Q2 data dominate the planet detectability. The KIC contains stellar T_eff and radii (R_⋆) that are based on four visible-light magnitudes (g, r, i, z) and a fifth, D51, calibrated with model atmospheres and JHK IR magnitudes (Brown et al. 2011).

The photometric calibrations yield T_eff reliable to ±135 K (rms) and surface gravity log g reliable to ±0.25 dex (rms), based on a comparison of KIC values to results of high-resolution spectra obtained with the Keck I telescope and LTE analysis (Brown et al. 2011). Stellar radii are estimated from T_eff and log g and carry an uncertainty of 0.13 dex, i.e., 35% rms (Brown et al. 2011). There is a concern that values of log g for subgiants are systematically overestimated, leading to stellar radii that are smaller than their true radii perhaps by as much as a factor of two. One should be concerned that a magnitude-limited survey such as Kepler may favor slightly evolved stars, implying systematic underestimates of stellar radii, an effect worth considering at the interpretation stage of this work. The quoted planet radii may be too small by as much as a factor of two for evolved stars. We adopt these KIC values for stellar T_eff and R_⋆ from the KIC and their associated uncertainties, following Borucki et al. (2011b). The stellar metallicities are poorly known. The KIC is available on the Multi-Mission Archive at the Space Telescope Science Institute (MAST) Web site.³⁰

In this paper, we primarily consider stars having properties in the core of the Kepler mission, namely, bright solar-type main-sequence stars. Specifically, we consider only Kepler target stars within this domain of the H-R diagram: T_eff = 4100–6100 K, log g = 4.0–4.9, and Kepler magnitude Kp < 15 mag (Table 1). These parameters select for the brightest half of the GK-type target stars (the other half being fainter, Kp > 15 mag), as shown in Figure 1. The goal is to limit our study to main-sequence GK stars well characterized in the KIC (Brown et al. 2011) and to provide a stellar sample that is a close match to that of Howard et al. (2010), offering an opportunity for a comparison of the radii and masses from the two surveys. The brightness limit of Kp < 15 promotes high photometric signal-to-noise ratios (SNRs), needed to detect the smaller planets. These three criteria in T_eff, log g, and Kp seem, at first glance, to be quite modest, representing the core target stars in the Kepler mission. Yet these three stellar criteria yield a subsample of only 58,041 target stars, roughly one-third of the total Kepler sample. Except for our study of planet occurrence as a function of T_eff (Section 4) and our discussion of hot Jupiters, we consider only this subset of Kepler stars and the associated planet candidates detected among them. The goal, described in more detail below, is to establish a subset of Kepler target stars for which the detection efficiency of planets (of specified radius and orbital period) is close to 100%.

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We further restrict the Kepler stellar sample by including only those stars with high enough photometric quality to permit detection of planets of a specified radius and orbital period. To begin, we consider differential domains in the two-dimensional space of planet radius and orbital period. For each differential domain, only a subset of the Kepler target stars has sufficient photometric quality to permit detection of such a planet. In effect, the survey for such specific planets is carried out only among those stars having photometric quality so high that the transit signals stand out easily (literally by eye). For photometric quality, we adopt the metric of the SNR of the transit signal integrated over a 90 day photometric time series. We define SNR to be the transit depth divided by the uncertainty in that depth due to photometric noise (to be defined quantitatively below).

We set a threshold, SNR > 10, which is higher than that (SNR > 7.0) adopted by Borucki et al. (2011b), lending our study an even higher standard of detection. Thus, we restrict our sample of stars so strongly that planets of a specified radius and orbital period are rarely, if ever, missed by the "Transiting Planet Search" (TPS; Jenkins et al. 2010b) pipeline. Moreover, we base our SNR criterion on just a single 90 day quarter of Kepler photometry. This conservatively demands that the photometric pipeline detect transits only during a single pointing of the telescope. (The CCD pixels that a particular star falls on change quarterly as Kepler is rolled by 90° to maintain solar illumination.) As noted in Borucki et al. (2011b), the photometric pipeline does not yet have the capability to stitch together multiple quarters of photometry and search for transits. In contrast, the SNR quoted in Borucki et al. (2011b) was based on the totality of photometry, Q0–Q5 (approximately one year in duration). Thus, we are setting a threshold that is considerably more stringent than in Borucki et al. (2011b), i.e., including target stars of the quietest photometric behavior.

Finally, we restrict our study to orbital periods under 50 days. All criteria by which Kepler stars are retained in our study are given in Table 1. As demonstrated below, these restrictions on SNR > 10 and on orbital period (P < 50 days) yield a final subsample of Kepler targets for which very few planet candidates will be missed by the current Kepler photometric pipeline as the transit signals both overwhelm the noise and repeat multiple times (for P < 50 days).

We explored the adoption of two measures of photometric SNR for each Kepler star, one taken directly from Borucki et al. (2011b) and the other using the so-called Combined Differential Photometric Precision (σ_CDPP), which is the empirical rms noise in bins of a specified time interval, coming from the Kepler pipeline. Actually, Borucki et al. (2011b) derived their SNR values from σ_CDPP, integrated over all transits in Q0–Q5. We employed the measured σ_CDPP for time intervals of 3 hr and compared the resulting SNR from Borucki et al. (2011b) for transits to those we computed from the basic σ_CDPP. These values agreed well (understandably, accounting for the use of a total SNR from all five quarters in Borucki et al. 2011b). Thus, we adopted the basic 3 hr σ_CDPP for each target star as the origin of our noise measure.

Each Kepler target star has its own measured rms noise level, σ_CDPP. Typical 3 hr σ_CDPP values are 30–300 parts per million (ppm), as shown in Figure 1 of Jenkins et al. (2010c), albeit for 6 hr time bins. Clearly, the photometrically noisiest target stars are less amenable to the detection of small planets, which we treat below. The noise has three sources. One is simply Poisson errors from the finite number of photons received, dependent on the star's brightness, causing fainter stars to have higher σ_CDPP. This photon-limited photometric noise is represented by the lower envelope of the noise as a function of magnitude in Figure 1 of Jenkins et al. (2010c). A second noise source stems from stellar surface physics including spots, convective overshoot and turbulence (granulation), acoustic p-modes, and magnetic effects arising from plage regions and reconnection events. A third noise source stems from excess image motion in Q0, Q1, and Q2 stemming from the use of variable guide stars that have now been dropped. In Q2, the presence of bulk drift corrected by four re-pointings of the bore sight, plus a safe mode followed by an unusually large thermal recovery, also contributed. The measured σ_CDPP accounts for all such sources, as well as any unmentioned since it is an empirical measure.

Using σ_CDPP for each target star, we define SNR integrated over all transits as

$$\begin{equation} \mathrm{{\rm S/N}} = \frac{\delta }{\sigma _\mathrm{CDPP}}\sqrt{\frac{n_{\mathrm{tr}} \cdot t_\mathrm{dur}}{3\,\mathrm{hr}}}. \end{equation} \tag{ 1 }$$

Here, δ = R²_p/R_⋆² is the photometric depth of a central transit of a planet of radius R_p transiting a star of radius R_⋆, n_tr is the number of transits observed in a 90 day quarter, t_dur is the transit duration, and the factor of 3 hr accounts for the duration over which σ_CDPP was measured. We include only those stars yielding SNR > 10, for a given specified transit depth and orbital period. The threshold imposes such a stringent selection of target stars that few planets are missed by the Kepler TPS pipeline. Our planet occurrence analysis below assumes that (nearly) all planets with R_p > 2R_⊕ that meet the above SNR criteria have been detected by the Kepler pipeline and are included in Borucki et al. (2011b). The Kepler team is currently engaged in a considerable study of the completeness of the Kepler pipeline by injecting simulated transit signals into pipeline at the CCD pixel level and measuring the recovery rate of those signals as a function of SNR and other parameters. In advance of the results of this major numerical experiment, we demonstrate detection completeness of SNR > 10 signals in two ways.

First, Figure 2 shows the SNR of detected transits as a function of Kepler Object of Interest (KOI) number. The Kepler photometry and TPS pipeline detect planet candidates over the course of months as data arrive. There is a learning curve involved with this process, as both software matures and human intervention is tuned (Rowe et al. 2010). As a result, the obvious (high SNR) planet candidates are issued low KOI numbers as they are detected early in the mission. The shallower transits, relative to noise, are identified later, as they require more data, and are issued larger KOI numbers. Thus, KOI number is a rough proxy for the time required to accumulate enough photometry to identify the planet candidate. Among the KOIs 1050–1600, much less vetting was done, and indeed we rejected five planet candidates (KOIs 1187.01, 1227.01, 1387.01, 1391.01, and 1465.01) reported in Borucki et al. (2011b) based on both V-shaped light curves and at least one other property indicating a likely eclipsing binary.

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Figure 2 shows that the early KOIs, 1–1050, had a wide range of SNR values spanning 7–1000, as the first transit signals had a variety of depths. KOIs 400–1000 correspond to pipeline detections of transit planet candidates around target stars as faint as 15th mag and fainter. The more recent transit identifications of KOIs 1050–1600 exhibit far fewer transits with SNR > 20 (90 days), and about half of these new KOIs have SNR < 10, below our threshold for inclusion. Apparently most newly identified KOIs have SNR < 20, and few planets remain to be found with P < 50 days and SNR > 20. Figure 2 suggests that the great majority of planet candidates with P < 50 days and SNR > 10 have already been identified by the Kepler pipeline. This apparent asymptotic success in the detection of SNR > 10 transits is enabled by our orbital period limit of 50 days, which is considerably less than the duration of a quarter (90 days). The current Kepler pipeline for identifying transits within a single 90 day quarter is more robust than the multi-quarter transit search. For such short periods, at least two transits typically occur within one quarter. Moreover, when such planet candidates appear during another quarter, the short-period planets are quickly confirmed. We suspect that for periods greater than 90 days, many more planet candidates are yet to be identified by Kepler. Thus, this study restricts itself to P < 50 days in part because of the demonstrated completeness of detection for such short periods.

We examined the light curves themselves for a second demonstration of nearly complete detection efficiency of planet candidates with P < 50 days, R_p > 2 R_⊕, and SNR > 10. Figure 3 shows four representative light curves of planet candidates whose properties are listed in Table 2. All four have small radii of 2–3 R_⊕ and "long" periods of 30–50 days, the most difficult domain for planet detection in this study (the lower right corner of Figure 4, discussed below). The SNR values are near the threshold value of ∼10; in fact, one planet candidate (KOI 592.01) has an SNR of 9.7 and is therefore conservatively excluded from this study. The four light curves show how clearly such transits stand out, indicating the high detection completeness of planets down to 2 R_⊕ and P < 50 days for the SNR > 10 threshold we adopted.

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We adopt the Kepler planet candidates and their orbital periods and planet radii from Table 2 of Borucki et al. (2011b), with two exceptions. First, we exclude the five KOIs noted above that are likely to be false positives. Second, we exclude KOIs that orbit "unclassified" KIC stars (identified with "T_eff Flag" = 1 in Table 1 of Borucki et al. 2011b). We measure planet occurrence only around stars with well-defined stellar parameters from the KIC.

To summarize the Borucki et al. (2011b) results, photometry at roughly 100 ppm levels in 29.4 minute integrations allows detection of repeated, brief drops in stellar brightness caused by planet transits across the star. The technical specifics of the instrument, photometry, and transit detection are described in Borucki et al. (2011a), Koch et al. (2010a), Jenkins et al. (2010c), Jenkins et al. (2010b), and Caldwell et al. (2010). We begin the identification of planet candidates based on those revealed in public Kepler photometric data (Q0–Q2). This data release contains 997 stars with a total of 1235 planetary candidates that show transit-like signatures, all with some follow-up work that could not rule out the planet hypothesis (Gautier et al. 2010). Borucki et al. (2011b) include three planets discovered in the Kepler field before launch: TrES-2b (O'Donovan et al. 2006), HAT-P-7b (Pál et al. 2008), and HAT-P-11b (Bakos et al. 2010). We are including only those planet candidates that meet two SNR standards: they must have SNR > 10 in one quarter alone and they must have SNR > 7 in all quarters. The former standard should guarantee the latter, but this double standard reinforces the quality of the planet candidates.

As this data release contains 136 days of photometric data, with only a few small windows of downtime, most planet candidates with periods under 50 days have exhibited two or more transits. The multiple transits for P < 50 days offer relatively secure candidates, periods, and radii, provided by the repeated transit light curves. For P < 40 days, Kepler has detected typically three or more transits in the publicly available data. Moreover, in Borucki et al. (2011b), the periods, radii, and ephemerides are based on the full set of Kepler data obtained in Q0–Q5, constituting over one year of photometric data. Thus, planet candidates with periods under 50 days are securely detected with multiple transits. They have improved SNR in the light curves from the full set of data available to the Kepler team, offering excellent verification, radii, and periods for short-period planets.

We expect that some of the planet candidates reported in Borucki et al. (2011b) are actually false positives. These would be mostly background eclipsing binaries diluted by the foreground target star. They may also be background stars orbited by a transiting planet of larger radius, but diluted by the light of the foreground star mimicking a smaller planet. False positives can also occur from gravitationally bound companion stars that are eclipsing binaries or have larger transiting planets. We expect that false-positive probabilities will be estimated for many planet candidates in Borucki et al. (2011b) using "BLENDER" (Torres et al. 2011).

In the meantime, the false-positive rate has been estimated carefully by Morton & Johnson (2011b). They find the false-positive probability for candidates that pass the standard vetting gates to be less than 10% and normally closer to 5%. In particular, the Kepler vetting process included a difference analysis between CCD images taken in and out of transit, allowing direct detection of the pixel that contains the eclipsing binary, if any. This vetting process found that ∼12% of the original planet candidates were indeed eclipsing binaries in neighboring pixels, and these were deemed false positives and removed from Table 2 of Borucki et al. (2011b). This process leaves only the 1 pixel itself, with a half-width of 2 arcsec within which any eclipsing binary must reside. As 12% of the planet candidates had an eclipsing binary within the ∼10 pixels total of the photometric aperture, the rate of eclipsing binaries hidden behind the remaining 1 pixel is likely to be ∼1.2%, a small probability of false positives. The bound, hierarchical eclipsing binaries were estimated by Morton & Johnson (2011b), finding that another few percent may be such false positives, yielding a total false-positive probability of ∼5%–10%. Morton & Johnson (2011b) note that the false-positive probability depends on transit depth δ, galactic latitude b, and Kp. Using their "detailed framework" and computing the false-positive probability for each of the 438 planet candidates among our "solar subset" (Table 1), we estimate that ∼22 planet candidates are actually false positives.³¹ The resulting false-positive rate of 5% is on the low end of the 5%–10% estimate above because we restricted our stellar sample to bright main-sequence stars and planet sample to R_p > 2 R_⊕. We do not expect this low false-positive rate to substantially impact the statistical results below.

Nearly all of the KOIs reported in Borucki et al. (2011b) are formally "planet candidates," absent planet validation (Torres et al. 2011), or mass determination (Borucki et al. 2010; Koch et al. 2010b; Dunham et al. 2010; Latham et al. 2010; Jenkins et al. 2010a; Holman et al. 2010; Batalha et al. 2011; Lissauer et al. 2011a). For simplicity, we will refer to all KOIs as "planets," bearing in mind that a small percentage will turn out to be false positives.

Planet occurrence varies by three orders of magnitude in the radius–period plane (Figure 4). To isolate the dependence on these parameters, we first considered planet occurrence as a function of planet radius, marginalizing over all planets with P < 50 days. We computed occurrence using Equation (2) for cells with the ranges of radii in Figure 4 but for all periods less than 50 days. This is equivalent to summing the occurrence values in Figure 4 along rows of cells to obtain the occurrence for all planets in a radius interval with P < 50 days. The resulting distribution of planet radii (Figure 5) increases substantially with decreasing R_p.

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We modeled this distribution of planet occurrence with planet radius as a power law of the form

$$\begin{equation} \frac{df(R)}{d\log R} = k_R R^{\alpha }. \end{equation} \tag{ 4 }$$

Here, df(R)/dlog R is the mean number of planets having P < 50 days per star in a log₁₀ radius interval centered on R (in R_⊕), k_R is a normalization constant, and α is the power-law exponent. To estimate these parameters, we used measurements from the 2–22.7 R_⊕ bins because of incompleteness at smaller radii and a lack of planets at larger radii. We fit Equation (4) using a maximum likelihood method (Johnson et al. 2010). Each radius interval contains an estimate of the planet fraction, F_i = df(R_i)/dlog R, based on a number of planet detections made from among an effective number of target stars, such that the probability of F_i is given by the binomial distribution

$$\begin{equation} p(F_i|n_{\rm pl}, n_{\rm nd}) = F_i^{n_{\rm pl}} (1-F_i)^{n_{\rm nd}}, \end{equation} \tag{ 5 }$$

where n_pl is the number of planets detected in a specified radius interval (marginalized over period), n_nd ≡ n_pl/f_cell − n_pl is the effective number of non-detections per radius interval, and f_cell is the estimate of planet occurrence over the marginalized radius interval obtained from Equation (2). The planet fraction varies as a function of the mean planet radius R_{p, i} in each bin, and the best-fitting parameters can be obtained by maximizing the probability of all bins using the model in Equation (4):

$$\begin{equation} \mathcal {L} = \prod _{i=1}^{n_{\rm bin}} p(F(R_{{\rm p},i})). \end{equation} \tag{ 6 }$$

In practice the likelihood becomes vanishingly small away from the best-fitting parameters, so we evaluate the logarithm of the likelihood

$$\begin{eqnarray} \ln {\mathcal {L}} &=& \sum _{i=1}^{n_{\rm bin}} \ln {p(F(R_{{\rm p},i}))} \\ \nonumber &=& \sum _{i=1}^{n_{\rm bin}} n_{{\rm pl}, i} (\ln {k_R} + \alpha \ln {R_{{\rm p},i}}) + n_{{\rm nd}, i} \ln {\big(1-k_R R_{{\rm p},i}^\alpha \big)}. \end{eqnarray} \tag{ 7 }$$

We calculate $\ln {\mathcal {L}}$ over a uniform grid in k_R and α. The resulting posterior probability distribution is strongly covariant in α and k_R. Marginalizing over each parameter, we find α = −1.92  ±  0.11 and k_R = 2.9^+0.5_{− 0.4}, where the best-fit values are the median of the marginalized one-dimensional parameter distributions and the error bars are the 15.9 and 84.1 percentile levels.

Howard et al. (2010) found a power-law planet mass function, $df/d\log M = k^{\prime }M^{\alpha ^{\prime }}$, with k' = 0.39^+0.27_{− 0.16} and α' = −0.48^+0.12_{− 0.14} for periods P < 50 days and masses M_psin i = 3–1000 M_⊕. We explore planet densities and the mapping of R_p to M_psin i in Section 5.

We computed planet occurrence as a function of orbital period using Equation (2). We considered this period dependence for ranges of planet radii (R_p = 2–4, 4–8, and 8–32 R_⊕). This is equivalent to summing the occurrence values in Figure 4 along two adjacent columns of cells to obtain the occurrence for all planets in specified radius ranges. Figure 6 shows that planet occurrence increases substantially with increasing orbital period, particularly for the smallest planets with R_p = 2–4 R_⊕.

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For P < 2 days, planets of all radii in our study (>2 R_⊕) are extremely rare with an occurrence of <0.001 planets per star. Extending to slightly longer orbital periods, hot Jupiters (P < 10 days, R_p = 8–32 R_⊕) are also rare in the Kepler survey. We measure an occurrence of only 0.004  ±  0.001 planets per star, as listed in Table 4. That occurrence value is based on Kp < 15 and the other restrictions that define the "solar subset" (Table 1). Expanding our stellar sample out to Kp < 16, but keeping the other selection criteria constant, we find a hot Jupiter occurrence of 0.005 ± 0.001 planets per star. This fraction is more robust as it is less sensitive to Poisson errors and our concern about detection incompleteness for Kp > 15 vanishes for hot Jupiters that typically produce SNR > 1000 signals. Marcy et al. (2005a) found an occurrence of 0.012 ± 0.001 for hot Jupiters (a < 0.1 AU, P ≲ 12 days) around FGK dwarfs in the solar neighborhood (within 50 pc). Thus, the occurrence of hot Jupiters in the Kepler field is only 40% that in the solar neighborhood. One might worry that our definition of R_p > 8 R_⊕ excludes some hot Jupiters detected by RV surveys. For Kp < 16 and the same T_eff and log g criteria, we find an occurrence of 0.0076 ± 0.0013 for R_p > 5.6, which is still 40% lower than the RV measurement.

However, we do see modest evidence among the Kepler giant planets of the pileup of hot Jupiters at orbital periods near 3 days (Figures 4 and 6) as is dramatically obvious from Doppler surveys of stars in the solar neighborhood (Marcy et al. 2008; Wright et al. 2009). These massive, close-in planets are detected with high completeness by both Doppler and Kepler techniques (including the geometrical factor for Kepler), so the different occurrence values are real. We are unable to explain this difference, although a paucity of metal-rich stars in the Kepler sample is one possible explanation. Unfortunately, the metallicities of Kepler stars from KIC photometry are inadequate to test this hypothesis (Brown et al. 2011). A future spectroscopic study of Kepler stars with LTE analysis similar to Valenti & Fischer (2005) offers a possible test. In addition to the metallicity difference, the stellar populations may have different T_eff distributions, despite having similar T_eff ranges. Johnson et al. (2010) found that giant planet occurrence correlates with both stellar metallicity and stellar mass (for which T_eff is a proxy). A full study of the occurrence of hot Jupiters is beyond the scope of this paper, but we note that other photometric surveys for transiting hot Jupiters orbiting stars outside of the stellar neighborhood have measured reduced planet occurrence (Gilliland et al. 2000; Weldrake et al. 2008; Gould et al. 2006).

The occurrence of smaller planets with radii R_p = 2–4 R_⊕ rises substantially with increasing P out to ∼10 days and then rises slowly or plateaus when viewed in a log–log plot (orange histogram, top panel of Figure 6). Out to 50 days we estimate an occurrence of 0.130 ± 0.008 planets per star. Small planets in this radius range account for approximately three-quarters of the planets in our study, corrected for incompleteness.

The occurrence distributions in the top panel of Figure 6 have shapes that are more complicated than simple power laws. Occurrence falls off rapidly at short periods. We fit each of these distributions to a power law with an exponential cutoff,

$$\begin{equation} \frac{df(P)}{d\log P} = k_P P^{\beta } (1-e^{-(P/P_0)^\gamma }). \end{equation} \tag{ 8 }$$

This function behaves like a power law with exponent β and normalization k_P for P ≫ P₀. For periods P (in days) near and below the cutoff period P₀, f(P) falls off exponentially. The sharpness of this transition is governed by γ. Thus, the parameters of Equation (8) measure the slope of the power-law planet occurrence distribution for "longer" orbital periods, as well as the transition period and sharpness of that transition.

As shown in Figure 7, we fit Equation (8) to the four ranges of radii shown in Figure 6 (top panel) and list the best-fit parameters in Table 5. We note that β > 0 for all planet radii considered, i.e., planet occurrence increases with log P. For the largest planets (R_p = 8–32 R_⊕), β = 0.37 ± 0.35 is consistent with the power-law occurrence distribution derived by Cumming et al. (2008) for gas-giant planets with periods of 2–2000 days, df∝M^{−0.31  ±  0.2}P^{0.26  ±  0.1} dlog M dlog P.

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P₀ and γ can be interpreted as tracers of the migration and stopping mechanisms that deposited planets at the closest orbital distances. With decreasing planet radius, P₀ increases and γ decreases, shifting the cutoff period outward and making the transition less sharp. Thus, gas-giant planets (R_p = 8–32 R_⊕) on average migrate closer to their host stars (P₀ is small), and the stopping mechanism is abrupt (γ is large). On the other hand, the smallest planets in our study have a distribution of orbital distances (and periods) with a characteristic stopping distance farther out and a less abrupt falloff close-in.

The normalization constant k_P is highly correlated with the other parameters of Equation (8). A more robust normalization is provided by the requirement that the integrated occurrence to P = 50 days is given in Table 4.

In the previous section, we considered only GK stars with properties consistent with those listed in Table 1. In particular, only stars with T_eff = 4100–6100 K were used to compute planet occurrence. Here, we expand this range to 3600–7100 K and measure occurrence as a function of T_eff. This expanded set includes stars as cool as M0 and as hot as F2. For T_eff outside of this range there are too few stars to compute occurrence with reasonable errors. We use the same cuts on brightness (Kp < 15) and gravity (log g = 4.0–4.9) as before. We also used the photometric noise σ_CDPP values (as before) to compute the fraction of target stars around which each detected planet could have been detected with SNR ⩾ 10. This ensures that planet detectability down to sizes of 2 R_⊕ will be close to 100%, for all of these included target stars independent of their T_eff.

We computed planet occurrence using the same techniques as in the previous section, namely, Equation (2). We subdivided the stars and their associated planets into 500 K bins of T_eff. We further subdivided the sample by planet radius, considering different ranges of R_p (2–4, 4–8, 8–32, and 2–32 R_⊕) separately. In summary, we computed planet occurrence as a function of T_eff for several ranges of R_p, and in all cases we considered all planets with P < 50 days.

Figure 8 shows these occurrence measurements as a function of T_eff. Most strikingly, occurrence is inversely correlated with T_eff for small planets with R_p = 2–4 R_⊕. Fitting the occurrence of these small planets in the T_eff bins shown in Figure 8, we find that a model linear in T_eff,

$$\begin{equation} f(T_{\rm eff}) = f_0 + k_T \left(\frac{T_{\rm eff} - 5100\ \mathrm{K}}{1000\ \mathrm{K}}\right), \end{equation} \tag{ 9 }$$

fits the data well. Using linear least-squares, the best-fit coefficients are f₀ = 0.165 ± 0.011 and k_T = −0.081 ± 0.011 and the relation is valid over T_eff = 3600–7100 K. We adopted a linear model because it is simple and provides a satisfactory fit with a reduced χ² of 1.03. However, we caution that the occurrence measurements in the three coolest bins have relatively large errors and are consistent with a flat occurrence rate, independent of T_eff.

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The occurrence of planets with radii larger than 4 R_⊕ does not appear to correlate with T_eff (Figure 8), although detecting such a dependence would be challenging given the lower occurrence of these planets and the associated small number statistics in our restricted sample.

The correlation between the occurrence of 2–4 R_⊕ planets and T_eff is striking. In this subsection, we consider three possible sources of error and/or bias that could have spuriously produced this result. First, we rule out random errors in the occurrence measurements or in the stellar parameters in the KIC. Next, we consider a systematic bias in R_⋆ but conclude that any such bias will be too small to cause the correlation. Finally, we consider a systematic metallicity bias as a function of T_eff. While we consider this unlikely, we cannot rule it out as the cause of the observed correlation.

4.2.1. Random Errors

4.2.2. Systematic R_⋆ Bias?

4.2.3. Systematic Metallicity Bias?

We begin by considering the known planets with R_p < 8 R_⊕ and M_p > 0.1 M_⊕. This range of parameters selects planets smaller than Saturn and as large as or larger than Mars. Figure 9 shows all such planets with good mass and radius measurements from our solar system and other systems. Theoretical calculations of Kepler-10b (Batalha et al. 2011) based on its mass and radius (4.5 M_⊕ and 1.4 R_⊕) suggest a rock/iron composition with little or no water. Corot-7b has a radius of 1.7 R_⊕ (Léger et al. 2009). Queloz et al. (2009) measured a mass of 4.8 M_⊕ for this planet, implying a density of 5.6 g cm⁻³ and a rocky composition. However, the mass and density have remained controversial. Independent mass determinations based on the same spot-contaminated Doppler data yield masses that vary by a factor of two to three (Pont et al. 2011; Hatzes et al. 2010; Ferraz-Mello et al. 2011). We adopt the mass estimate of 1–4 M_⊕ from Pont et al. (2011), which implies a wide range of possible compositions and also marginally favors a water/ice-dominated planet. GJ 1214b is a less dense super-Earth orbiting an M dwarf. The planet has been modeled as a solid core surrounded by H/He/H₂O and may be intermediate in composition between ice giants like Uranus and Neptune and a 50% water planet (Nettelmann et al. 2011). The discovery of the six coplanar planets orbiting Kepler-11 added five planets with measured masses (from transit-timing variations) to Figure 9 (Lissauer et al. 2011a). The remaining exoplanets in Figure 9 all have masses greater than Neptune's (17 M_⊕) and densities less than 2 g cm⁻³: Kepler-4b (Borucki et al. 2010), Gl 436b (Maness et al. 2007; Gillon et al. 2007; Torres et al. 2008), HAT-P-11b (Bakos et al. 2010), HAT-P-26b (Hartman et al. 2011), Corot-8b (Bordé et al. 2010), and HD 149026b (Sato et al. 2005; Torres et al. 2008).

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Figure 9 shows that among known planets their radii increase with planet mass faster than do any of the theoretical curves representing solid compositions of iron, rock, or ice. This rapid increase in radius with mass suggests that planets of higher mass contain larger fractional amounts of H/He gas. The slope increases markedly for masses above 4.5 M_⊕, indicating that above that planet mass the contribution of gas is common, even for these close-in planets. Apparently planets above 4.5 M_⊕ are rarely solid. We suspect that for planets orbiting beyond 0.1 AU where collisional stripping of the outer envelope is less energetic and common, the occurrence of gaseous components will be greater.

Fortney et al. (2007b) modeled solid exoplanets composed of pure water ("ice"), rock (Mg₂SiO₄), iron, and binary admixtures. Their models include no gas component and are shown as gray lines in Figure 9. Adding gas to any of the models increases R_p and decreases ρ (Adams et al. 2008). Thus, planets below and to the right of the ice contour (Figure 9, lower panel) have low densities due to a gas component. Planets above the ice contour contain increasing fractions of rock and iron, depending on the specific system. Compositional details matter greatly for specific systems, but for our simple purpose we make the crude approximation that planets with R_p ≲ 3 R_⊕ that have ρ ≳ 4 g cm⁻³ are composed substantially of refractory materials (usually rock in the form of silicates and iron/nickel). These planets may have some water and gas, but those components do not dominate the planet's composition as they do for Uranus, Neptune, and larger planets.

The Eta-Earth Survey measured planet occurrence as a function of M_psin i in a volume-limited sample of 166 G and K dwarfs using Doppler measurements from Keck-HIRES. The stars have a nearly unbiased metallicity distribution and are chromospherically quiet to enable high Doppler precision. In all, 35 planets were detected around 24 of the 166 stars, including super-Earths and Neptune-mass planets (Howard et al. 2009, 2011a, 2011b). Correcting for inhomogeneous sensitivity at the lowest planet masses, Howard et al. (2010) measured increasing planet occurrence with decreasing mass over five planet mass domains, M_psin i = 3–10, 10–30, 30–100, 100–300, 300–1000 M_⊕, spanning super-Earths to Jupiter-mass planets. This study was restricted to planets with P < 50 days.

We mapped the planet radius distribution from Kepler (Figure 4, including planets down to 1 R_⊕) onto mass (M_psin i) using toy density functions, ρ(R_p). These single-valued functions map all planets of a particular radius, R_p, onto a planet mass M_p = 4πρ(R_p)R³_p/3. Of course, real planets exhibit far more diversity in radii for a given mass owing to different admixtures of primarily iron/nickel, rock, water, and gas. Nevertheless, the models allow us to check if average masses associated with Kepler radii are consistent with Doppler measurements.

As part of this numerical experiment we converted M_p to M_psin i for each simulated planet using random orbital orientations (inclinations i drawn randomly from a probability distribution function proportional to sin i.) Our simulated M_psin i distributions account for the transit probabilities of planets detected by Kepler and the detection incompleteness for planets with small radii. That is, the simulated M_psin i distributions reflect the true distribution of planet radii (Section 3.1).

Figure 10 shows simulated M_psin i distributions assuming several toy density functions. These distributions are binned in the same M_psin i intervals as in the Howard et al. (2010) study. In the left column ρ(R_p) = ρ₀, where ρ₀ is a constant. From bottom to top, we considered four densities, ρ₀ = 0.4, 1.35, 1.63, and 5.5 g cm⁻³ (the bulk densities of HAT-P-26b, Jupiter, Neptune, and Earth). We are most interested in the densities of small planets, so we make comparisons in the two lowest mass bins for which Eta-Earth Survey measurements are available, M_psin i = 3–10 and 10–30 M_⊕. In these bins, the predicted occurrence from Kepler is too small by 1.5σ–2σ compared with the Eta-Earth Survey measurements for the three lowest constant density models, ρ₀ = 0.4, 1.35, and 1.63 g cm⁻³. Kepler predicts fewer small planets than the Eta-Earth Survey measured. The simulated M_psin i distribution matches the observed M_psin i distribution well for an assumed density, ρ = 5.5 g cm⁻³. While this model is clearly unphysical when extended over the entire radius range, consistency in the two low-mass bins suggests that the small planets have higher densities.

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We explored slightly more complicated density functions in the right column of Figure 10. These functions are piecewise constant density models, with density rising to 4.0, 5.5, and 8.8 g cm⁻³ for small radii, as depicted in the sub-panels of Figure 10. (Kepler-10b has a density of 8.8 g cm⁻³; Batalha et al. 2011.) We find the greatest consistency between the synthetic and measured mass distributions for two density models. One (model h) is shown in the upper right panel of Figure 10, which has ρ = 8.8, 5.5, 1.64, 1.33 g cm⁻³ for R_p = 1–1.4, 1.4–3.0, 3.0–6.0, and >6.0 R_⊕, respectively. This model has a high density (8.8 g cm⁻³) for the smallest planets but successively smaller densities for larger planets, approximately consistent with the densities of known planets in Figure 9. The other successful model (g) has a density of 4 g cm⁻³ for the smallest planets, with declining densities for larger planets, qualitatively similar to the previous model (h). This model (g) also yields a predicted distribution of M_psin i that agrees well with the observed distribution of M_psin i. Thus, it too is viable. Both successful models, g and h, are characterized by a high density for the smallest planets of 1–3 R_⊕. We tried a variety of piecewise constant density functions and found that all models that achieved consistency (<1σ difference in the 3–10 and 10–30 M_⊕ bins) have ρ ≳ 4 g cm⁻³ for R_p ≲ 3 R_⊕.

The mapping of radius to mass offers circumstantial evidence that a substantial population of small planets detected by Kepler have high densities. Rocky composition for the smallest planets supports the core-accretion model of planet formation (Pollack et al. 1996; Lissauer et al. 2009; Movshovitz et al. 2010). But we caution again that the stellar populations of the Kepler and Doppler surveys may be quite different. Planet multiplicity also makes this an especially challenging comparison. We computed the simulated Kepler mass distributions (black histograms in Figure 10) based on occurrence measured as the average number of planets per star, while the Doppler results from the Eta-Earth Survey (red points in Figure 10) computed occurrence as the fraction of stars hosting at least one planet in the specified M_psin i interval. This difference is based on the intrinsic limitations of each approach. To infer the fraction of stars with at least one planet from a transit survey requires an assumption about the mutual inclinations (Lissauer et al. 2011b). For Doppler surveys, it is significantly easier to determine if a particular star has at least one planet down to some specified mass limit, but it is much more difficult to be sure that all planets orbiting a star have been detected down to that same mass limit (Howard et al. 2010). Finally, we note that no planets at the extreme of our proposed high-density regime (R_p ∼ 3 R_⊕ and ρ ∼ 4 g cm⁻³) have been detected (Figure 10). To date all detected planets with R_p > 2 R_⊕ have ρ < 2 g cm⁻³. We conclude that while this technique offers qualitative support for rising density with decreasing planet size, in practice extracting firm quantitative conclusions is difficult because of the intrinsic differences between Doppler and transit searches.

We have attempted to measure pristine properties of planets that can be compared with, and can inform, theories of the formation, dynamical evolution, and interior structures of planets. We have built upon the unprecedented compendium of over 1200 planet candidates found by the historic Kepler mission (Borucki et al. 2011b). One goal here was to measure planet occurrence—the number of planets per star having particular orbital periods and planet radii—by minimizing the deleterious effects of detection efficiencies that are a function of planet properties, notably radius and orbital period.

Our treatment of the vast numbers of target stars and transiting planet candidates involved careful accounting of two important effects. First, only planets whose orbital planes are nearly aligned to Kepler's line of sight will transit their host star, leaving many planets undetected. We applied the standard geometrical correction for the small probability, R_⋆/a in Equation (2), that the orbital plane is sufficiently aligned to cause a transit. In counting planets, we assumed that for each detected planet candidate there are actually a/R_⋆ planets, on average, at all inclinations. Second, only planets whose transits produce photometric signals exceeding some SNR threshold will be reliably detected. For each possible planet radius and orbital period, we carefully identified the subset of the Kepler target stars a priori around which such planets could be detected with high probability. We adopted a threshold SNR of 10 for the transit signal in a single 90 day quarter of data, thereby limiting both the target stars and the planet detections with this SNR threshold. To be included, a target star must have a radius and photometric noise that allowed a planet detection with SNR > 10, i.e., a transit depth 10 times greater than the uncertainty in the mean depth from noise. Such restricted target stars offer a high probability that planets will be detected.

We further selected Kepler target stars having a specific range of T_eff, log g, and brightness to ensure a well-defined sample of stars. We consider only bright target stars (Kp < 15). We ignore all other Kepler target stars and their associated planets. Remarkably, this a priori selection of Kepler target stars immediately yields a sample of only ∼58,000 stars (and fewer when accounting for requisite photometric noise), not the full 156,000 stars. For most of the paper, we restricted the sample to main-sequence G and K stars (log g = 4.0–4.9, T_eff = 4100–6100 K) to permit comparison with similar Sun-like stars in the Eta-Earth Survey. This selection of Kepler target stars for a given planet radius and orbital period crucially leaves only a subset of stars in the "sub-survey" for those planet properties. Importantly, for planets with small radii (near 2 R_⊕) and long periods (near 50 days), only some 36,000–49,000 stars are amenable to detection of such difficult-to-detect planets, as shown in the annotations in the lower left corners of the cells in Figure 4. By counting planets and dividing by the number of appropriate stars that could have permitted their secure detection, we computed the planets per star for a specific planet radius and orbital period (within a specified delta in each quantity).

It is worth describing the differences between this paper and Borucki et al. (2011b) resulting from differing goals and methods. The primary propose of Borucki et al. (2011b) was to summarize the results of the Kepler observations and to act as a guide to the tables of data. The number of new planets announced in Borucki et al. (2011b) is more than twice the number known before Kepler (even when allowing for a false-positive rate of ∼5%; Morton & Johnson 2011b). Borucki et al. (2011b) considered the number distributions of all planets detected by Kepler, independent of the properties of their host stars (T_eff, log g, Kp, σ_CDPP). They also computed the "intrinsic frequencies" of planetary candidates, a close cousin of our planet occurrence measurements, and plotted these frequencies as a function of T_eff.

The results in this paper are derived directly from the planets announced in Borucki et al. (2011b) and from stellar parameters in the KIC (Brown et al. 2011). We measure the occurrence distributions of planets orbiting bright, main-sequence G and K stars, which represent only a third of the stars observed by Kepler and considered in Borucki et al. (2011b). Our desire for high detection completeness compelled us to consider only robustly detected planets satisfying R_p > 2 R_⊕, P < 50 days, SNR > 10 in 90 days of photometry, and stars with Kp < 15. This selection of stars and planets facilitated comparison with the Eta-Earth Survey (Howard et al. 2010), which focused on the Doppler detection of planets orbiting G and K dwarfs with P < 50 days. In this paper, we measured the detailed patterns of planet occurrence as a function of R_p, P, and T_eff only for that subset of stars and interpreted these distributions in the context of planet formation, evolution, and composition.

Borucki et al. (2011b) chose to compute intrinsic frequencies in small domains of semimajor axis and planet radius, while we work in a space of orbital periods and planet radii. There are trade-offs with these choices. We chose to work in period space because Kepler directly measures orbital periods and translating to semimajor axes requires either assumed stellar masses or radii. On the other hand, by working in small domains of semimajor axis, Borucki et al. (2011b) compensate for this by considering the range of orbital periods and transit durations that contribute to each domain for the range of masses and radii among the target stars. In this paper, we applied a binary detection criterion of SNR > 10 for 90 days of photometry (approximately one quarter). Borucki et al. (2011b) adopted a detection criterion of SNR > 7 for the 136 days of Q0–Q2, with corrections for the probability of low-SNR detections (e.g., 7σ detections are only recognized 50% of the time).

Figure 4 shows graphically some of the key features of close-in planet occurrence. The number of planets per star varies by three orders of magnitude in the radius–period plane (Figure 4) that spans periods less than 50 days and planet radii less than 32 R_⊕. Planet occurrence increases toward smaller radii (see Figure 5) down to our completeness limit of 2 R_⊕, with a power-law dependence given by df(R)/dlog₁₀R = k_RR^α, where df(R)/dlog₁₀R is the number of planets per star, for planets with P < 50 days in a log₁₀ radius interval centered on R (in R_⊕), k_R = 2.9^+0.5_{− 0.4}, and α = −1.92 ± 0.11. This is a remarkable result, showing that from planets larger than Jupiter to those only twice the radius of Earth planet occurrence rises rapidly by nearly two orders of magnitude. This rise with smaller size is consistent with, and supports the measured rise of, the planet occurrence with decreasing planet mass found by Howard et al. (2010). The increased occurrence of small planets seen in both studies supports the core-accretion theory for planet formation (Pollack et al. 1996).

Planet occurrence also increases with orbital period (Figure 6) in equal intervals of log P as $f(P) = k_P P^{\beta } (1-e^{-(P/P_0)^\gamma })$, with coefficients that all depend on planet radius, and both β and γ being positive. This functional form traces the steep rise in planet occurrence near a cutoff period, P₀. Below P₀ planets are rare, but for longer periods the planet occurrence distribution rises modestly with a power-law dependence. We find that P₀ and γ (which governs how steep the occurrence falloff is below P₀) depend on planet radius. The smaller planets, R_p = 2–4 R_⊕, have P₀ ∼ 7 days, while larger planets have P₀ ∼ 2 days. Further, γ is larger for planets with R_p > 4 R_⊕, making the falloff in planet occurrence more abrupt below P₀. The trends suggest that the mechanisms that caused the planets to migrate and stop at close orbital distances depend on planet size. Alternatively, if a substantial number of small close-in planets formed by in situ accretion, then our measurements trace the contours of this process (Raymond et al. 2008).

This period dependence of planet occurrence seems to contradict the results from Doppler surveys of exoplanets, for which we find a pileup of planets at periods of 3 days and a nearly flat distribution of planets for longer periods, out to periods of 1 yr (Wright et al. 2009). The key difference is that Kepler is sensitive to much smaller planets (in radius and mass) than were Doppler surveys, especially beyond 0.1 AU. To be sure, Kepler suffers a geometrical decline in detectability as R_⋆/a, but we have corrected for this trivially. Such a correction is more difficult for Doppler surveys that have less uniform detectability from star to star.

Another difference in the period distributions between Kepler and Doppler surveys is in the pileup of hot Jupiters at orbital periods near 3 days (Figures 4 and 6). The Kepler-detected planets show a modest pileup, while for single planets in Doppler surveys the pileup is a factor of three above the background occurrence at other periods (Marcy et al. 2008; Wright et al. 2009). This different planet occurrence for hot Jupiters appears to be real and may be due to fewer metal-rich stars in the Kepler sample that are located 50–200 pc above the Galactic plane, or different stellar mass distributions in the magnitude-limited and volume-limited surveys. The Kepler field has a greater admixture of thick-disk stars (that are metal-poor with [Fe/H] ≈ −0.5) to thin-disk stars than do the Doppler target stars. Other photometric surveys have noted that hot Jupiter occurrence appears to vary with stellar population. Gilliland et al. (2000) found no planets in a Hubble Space Telescope survey of the globular cluster 47 Tucanae and estimated a hot Jupiter occurrence that is an order of magnitude lower than in the solar neighborhood. Similarly, Weldrake et al. (2008) found no planets in the ω Centauri globular cluster and found the occurrence of hot Jupiters (P = 1–5 days) to be less than 0.0017 planets per star. Gould et al. (2006) found an occurrence of 0.003^+0.004_{− 0.002} hot Jupiters per star for P = 3–5 days, based on the magnitude-limited OGLE-III survey in the bulge of the Galaxy, which is compatible with our results from Kepler, 0.005 ± 0.001 planets per star for R_p= 8–32 R_⊕, P < 50 days, and Kp < 16.

We further find that planets larger than 16 R_⊕ (1.5 R_Jup) are extremely rare. Such inflated planets are also rare among transiting planets detected from the ground (see, e.g., the mass–radius diagram for gas-giant planets in Bakos et al. 2011). For several Gyr-old planets, theoretical mass–radius curves show a maximum near R_p ≈ 13 R_⊕ ≈ 1.2 R_Jup (Fortney et al. 2007b). Larger planets are typically young or close-in and inflated by one of several proposed mechanisms (e.g., Batygin & Stevenson 2010; Laughlin et al. 2011; Burrows et al. 2007).

We also note some interesting morphology in the two-dimensional occurrence domain of planet radius and orbital period (Figure 4). There is a ridge of higher planet occurrence for super-Earths and Neptunes, similar to that identified in Howard et al. (2010). The ridge appears to be diagonal when plotting either M_p or R_p versus P extending from a period and radius of 3 days and 2 R_⊕ (lower left) to a period and radius of 50 days and 4 R_⊕. This ridge can be seen by direct inspection of Figure 4, both by the density of the dots and by the colors. The upper envelope of red boxes (indicating high planet occurrence) extends along a diagonal from lower left to upper right. This ridge conveys some key information about the formation and perhaps dynamical evolution or migration of the 2–4 R_⊕ planets.

The paucity of close-in Neptune-mass planets (M_psin i = 10–100 M_⊕, P < 20 days) seen in Howard et al. (2010) is not as clearly visible in the Kepler data. In particular, the "top" of this desert (M_psin i = 100 M_⊕, or the radius equivalent) is not as clear. A further study of Kepler stars to fainter magnitudes of Kp = 16 may shed light on this desert. The overall planet occurrence for GK stars and periods less than 50 days, listed in Table 3, shows that planets of 2–4 R_⊕ are 0.130 ± 0.008 planets per star. This agrees well with the planet occurrence of 3–30 M_⊕ planets found by Howard et al. (2010) of 15⁺⁵_{− 4}%. The planet occurrence for all planet radii from 2 to 32 R_⊕ is only 16.5%, again in agreement with Howard et al. (2010) and Cumming et al. (2008). We find little support for the suggestion of planet occurrences of super-Earths and Neptunes (M_psin i = 3–30 M_⊕) of 30% ± 10% (Mayor et al. 2009) for P < 50 days.

We also measured planet occurrence as a function of T_eff of the host star, a proxy for stellar mass. For the smallest planets, 2–4 R_⊕, the results show a nearly linear rise in planet occurrence with smaller stellar mass. One may wonder if this rise might be caused by some systematic error due to poor values of T_eff or R_⋆ in the KIC. Such a systematic error seems nearly impossible, as the KIC values of T_eff are accurate to 135 K (rms) and in any case the T_eff values certainly vary monotonically with the true value of T_eff even if one imagines some large systematic error in the KIC values of T_eff. Thus, the increase in planet occurrence with smaller T_eff  and hence smaller stellar mass appears to be real. Again, we emphasize that the SNR = 10 criterion for a Kepler target star to be included in our survey implies that the detection efficiency is close to unity for all stars, from 7100 K to 3600 K, for R > 2 R_⊕. Examination of Figure 8 shows that even if one ignores the coolest and hottest stars, the increase of planet occurrence persists robustly. Thus, it appears that the number of planets per star increases by a factor of seven from stars of 1.5 M_☉ to stars of 0.4 M_☉ (T_eff = 7100–3600 K), with all of that T_eff dependence coming from the smallest planets, 2–4 R_⊕. This high occurrence of close-in small planets around low-mass stars represents significant information about the formation mechanisms of planets of 2–4 R_⊕.

We considered the possibility that this correlation is due to a systematic metallicity bias that depends on T_eff. That is, cool stars are relatively nearby, close to the galactic plane, and have higher metallicities, while hot stars are on average more distant, at greater heights above the galactic plane, and have lower metallicities. In this scenario, low metallicity is the driving force behind lower planet occurrence at higher T_eff. Using the Besancon galactic model, we estimate that metallicities may vary by ∼0.1 dex as a function of T_eff, but the dependence need not be monotonic because the median age varies with T_eff. It would be remarkable if such a modest difference in metallicity could cause a factor of seven difference in close-in planet occurrence. Unfortunately, due to the poor [Fe/H] measurements in the KIC, we are unable to measure the occurrence of planets as a function of [Fe/H]. Note, however, that either result has profound implications for planet formation: the occurrence of 2–4 R_⊕ planets depends strongly on stellar properties, T_eff or [Fe/H].

Sub-Neptune-size and jovian planets appear to have opposite trends in occurrence as a function of M_⋆. We showed that the occurrence of 2–4 R_⊕ planets decreases by a factor of seven with M_⋆ over ∼0.4–1.5 M_☉ (T_eff = 3600–7100 K). Johnson et al. (2010) measured the occurrence of giant planets as a function of M_⋆ and [Fe/H] and found a positive correlation with both quantities. That is, the occurrence of giant planets increases with increasing M_⋆ over the range ∼0.3–1.9 M_☉. Their study considered only giant planets that produce K > 20 m s⁻¹ Doppler signals and orbit within 2.5 AU. Subgiants with M_⋆ = 1.4–1.9 M_☉ have the highest rate of giant planet occurrence in their study. However, most of these planets orbit at ∼1–2 AU, with almost no planets inside of P = 50 day orbits (Bowler et al. 2010). Close-in planets of all sizes larger than 2 R_⊕ appear to be rare around the most massive stars accessible to transit and Doppler surveys.

Population synthesis models of planet formation by core accretion simulate the growth and migration of planet embryos embedded in a protoplanetary disk of gas and dust. Among their key predictions is the distribution of planet mass or radius as a function of orbital distance. Early versions of these models (Ida & Lin 2004; Alibert et al. 2005; Mordasini et al. 2009a) were tuned to match the distribution of giant planets detected by RV (Cumming et al. 2008; Udry & Santos 2007) by decreasing the rate of Type I migration compared to theoretical predictions (Ward 1997; Tanaka et al. 2002). The simulations predicted that planet occurrence rises with decreasing planet mass. But most of the low-mass planets resided in orbits near or beyond the ice line at ∼2–3 AU. These models also robustly predicted a "planet desert," a region of parameter space nearly devoid of planets. Planets with M_psin i ≈ 1–20 M_⊕ and a ≲ 1 AU were predicted to be extremely rare because producing such planets requires the gas disk to dissipate while one of two faster processes was happening, Type II migration or runaway gas accretion. Meanwhile, the models predicted that planets with masses above the desert, M > 20 M_⊕, but residing inside of ∼1 AU would exhibit a nearly constant distribution with mass.

Howard et al. (2010) demonstrated that the observed distribution of close-in planets (P < 50 days) exhibited quite different properties from those predicted by population synthesis. The predicted planet desert is actually populated by the highest planet occurrence of any region of the mass-period parameter space yet probed (the "ridge" noted above). The planet mass function rises steeply with decreasing planet mass, in contradiction to the expected nearly constant distribution with mass outside of the desert. From Kepler, we also see many planets populating the predicted desert (Figure 4) and a planet radius distribution that rises steeply with decreasing planet size (tracking the mass distribution). The latest versions of the population synthesis models (Ida & Lin 2010; Alibert et al. 2011) offer improvements, including non-isothermal treatment of the disk (Paardekooper et al. 2010) and multiple, interacting planet embryos per simulation. But they still predict a planet desert (albeit partially filled in). The contours of planet occurrence in Figure 4 offer rich detail to which future refinements of these models can be tuned. Alternatively, the distribution of observed planets may be strongly shaped by processes that take place after the gas clears, namely, planet–planet scattering (e.g., Ford et al. 2005; Ford & Rasio 2008; Chatterjee et al. 2008; Raymond et al. 2009), secular and resonant migration (e.g., Lithwick & Wu 2011; Wu & Lithwick 2011), and planetesimal migration and growth (e.g., Kirsh et al. 2009; Capobianco et al. 2011; Walsh & Morbidelli 2011). If these processes strongly shape the final planet distributions, then the planet distributions from population synthesis models (which truncate when the gas clears) will form the input to additional simulations that model post-disk effects and hope to match the currently observed planet distributions.

Current planet formation theory must also adapt to account for remarkable orbital properties of exoplanets. Not included here is an analysis of the orbital eccentricities that span the range e = 0–0.93 (e.g., Marcy et al. 2005b; Udry & Santos 2007; Moorhead et al. 2011), and the close-in "hot Jupiters" show a wide distribution of inclinations relative to the equatorial plane of the host star (Winn et al. 2010, 2011; Triaud et al. 2010; Morton & Johnson 2011a). Thus, standard planet formation theory probably requires additional planet–planet gravitational interactions to explain these non-circular and non-coplanar orbits (Chatterjee et al. 2011; Wu & Lithwick 2011).

We strongly advocate for an improved catalog of stellar parameters for the ∼1000 Kepler planet host stars and a comparably sized control sample. Our occurrence measurements and their interpretations would be strengthened by an improved knowledge of R_⋆, log g, [Fe/H], and T_eff. The R_⋆ values from the KIC are only known to 35% (rms), which leads to a proportionally large uncertainty in R_p. We saw that hot Jupiters have a significantly lower occurrence in the Kepler sample than in RV surveys. We were unable to test whether this is due to differing metallicities of the host stars because [Fe/H] is poorly measured in the KIC. Similarly, we are unable to completely rule out a metallicity gradient with height above the galactic plane as the underlying cause of the observed sevenfold decrease in the occurrence of 2–4 R_⊕ planets with increasing T_eff.

Finally, we note that Figure 3 shows representative planets having R_p ∼ 2.5 R_⊕ and P < 50 days, all of which reach SNR ∼ 20 in four quarters of Kepler photometry (and SNR ∼ 10 in one quarter). If we consider the SNR for planets of radius 1 R_⊕, the transit depth is six times shallower, implying total SNR values near SNR = 20/6 = 3.3. Thus, planets of 1 R_⊕, even in short periods under 50 days, would not reach the threshold SNR for meriting a secure detection with current data in hand. For planets of 1 R_⊕ to reach SNR ∼ 6.6, Kepler must acquire four times more data, i.e., five years total, still constituting a marginal detection. Clearly an extended mission of an additional ∼3 yr is needed to bring planets of 1 R_⊕ to SNR > 7.