A STUDY OF THE SHORTEST-PERIOD PLANETS FOUND WITH KEPLER

To carry out an independent search for the shortest-period planets, we used the Kepler long-cadence time-series photometric data (30 minute samples) obtained between quarters 0 and 16. A list was prepared of all ≈200,000 target stars for which photometry is available for at least one quarter, and the version 5.0 FITS files, which were available for all quarters, were downloaded from the STScI MAST Web site. We used the data that had been processed with the PDC-MAP algorithm (Stumpe et al. 2012; Smith et al. 2012), which is designed to remove many instrumental artifacts from the time series while preserving any astrophysical variability. We also made use of the time series of the measured X and Y coordinates of the stellar images (the "centroids") in order to test for false positives (see Section 3.2). Finally, we used the information in the file headers to obtain estimates of the combined differential photometric precision (CDPP). The CDPP is a statistic determined by the Kepler team's data analysis pipeline that is intended to represent the effective photometric noise level for detecting a transit with a duration of six hours (Christiansen et al. 2012). We used this quantity in our analysis of survey completeness (see Section 5.1).

For estimates of basic stellar properties including not only radii, but also masses and effective temperatures, we relied on the catalog of Huber et al. (2014). This catalog is based on a compilation of photospheric properties derived from many different sources. Although it is not a homogeneous catalog, it likely provides the most accurate stellar parameters that are currently available. Stars for which only broadband photometry is available have radii that could be uncertain by up to ∼40%, while stars for which spectroscopic or even astroseismic constraints are available have radius uncertainties as small as 10%.

In our study of short-period planets, we elected to use a Fourier transform ("FT") search. Since this is different from the standard algorithm for transit searching—the Box Least Squares ("BLS") algorithm (Kovács et al. 2002)—it seems appropriate to provide some justification for our choice. The BLS is designed to have the greatest efficiency for transits with a duration that is short in comparison to the orbital period. The Fourier spectrum of an idealized transit light curve has a peak at the orbital period and a series of strong harmonics. By using a matched filter, the BLS algorithm effectively sums all of the higher harmonics into a single detection statistic, which seems like a superior approach.

However, the standard BLS has a few drawbacks. One is that the BLS spectrum includes peaks at multiples of the orbital period and at multiples of the orbital frequency, thereby complicating attempts to ascertain the correct period. In addition, we have found that the standard BLS algorithm produces spurious signals at periods that are integer multiples of the Kepler sampling period of ≈0.02 day. These spurious peaks constitute a highly significant noise background in searches for planets with periods ≲0.5 day. These spurious peaks can be partially suppressed by pre-whitening the data, i.e., attempting to remove the non-transit astrophysical variability prior to computing the BLS spectrum, but the introduction of such a step complicates the search.

A key advantage of the FT method is that FTs can be computed so quickly that it was practical to repeat the search of the entire database several times while the code was being developed (see for example the simulations by Kondratiev et al. 2009). Although the FT of a transit signal has power that is divided among several harmonics, the number of significant harmonics below the Nyquist limit declines as the orbital period is decreased, and the FT is therefore quite sensitive to the shortest-period planets. The ratio of transit duration to period, or duty cycle, varies as P^−2/3 and is as large as 20% for USP planets, in which case the efficacy of the FT search is nearly equivalent to that of the BLS. Furthermore, it is straightforward to detect a peak in the FT and its equally spaced harmonics, either by means of an automated algorithm or by eye. The absence of any subharmonics is a useful and important property of true planet transits as opposed to background blended binaries (which often produce subharmonics due to the difference in depth between the primary and secondary eclipses).

Regardless of the justification, it is often worthwhile to carry out searches with different techniques. Thus far, the independent searches of the Kepler database have utilized the BLS technique (Ofir & Dreizler 2013; Huang et al. 2013; Petigura et al. 2013; Jackson et al. 2013), and for this reason alone it seemed worthwhile to take a different approach. In the end, though, the proof of the effectiveness of an FT search lies in what is found, and in this work we demonstrate empirically that the FT is a powerful tool for finding short-period planets.

Figure 1 is a flow chart illustrating the numbers of the ∼200,000 Kepler stars that survived each stage of our search program. The PDC-MAP long-cadence time series data from each quarter was divided by the median flux of that quarter. We removed outliers with flux levels 50% above the mean (likely due to cosmic rays), and all of the quarterly time series were stitched together into a single file. We further cleaned the data using a moving-mean filter with a length of three days. Using a filter to clean the data does not affect our ability to detect high frequency signals as long as the length of the window captures a few cycles of the target signal, and, at the same time, increases the sensitivity to signals at intermediate frequencies, since it removes long-term trends that have the potential to increase the FT power at these frequencies. To prepare the data for the application of the fast Fourier transform (FFT), gaps in the time series were filled by repeating the flux of the data point immediately prior to the gap. The mean flux was then subtracted, giving a zero-mean, evenly spaced time series. The FFT was then evaluated in the conventional manner all the way up to the Nyquist limit, i.e., the number of frequencies (n_f) in the transform is equal to half the number of data points.

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We searched for the presence of at least one peak that is more than four times the local mean level in the Fourier amplitude spectrum (square root of the power spectrum), with a frequency higher than 1 cycle day⁻¹. The local mean level was estimated using the closest n_f/250 frequencies, a number close to 120 for stars with 16 quarters of data. (This number is large enough to ensure that the interval used to estimate the local mean level is much wider than the FFT peaks, and that any possible contribution to the mean local level due to the peak is much smaller than the true local mean level.) In order for the target to be considered further, we also required that the FT exhibit at least one additional harmonic that stands out at least three times the local mean. For the selected objects, we increased the accuracy of the peak frequencies by calculating the FFT of a new flux series that was built from the original series by adding as many zeroes as needed to multiply the length of the array by a factor of 20 (a process known as "frequency oversampling").

Approximately 15,000 objects were selected by virtue of having at least one significant high-frequency peak and a harmonic. These objects underwent both visual and automated inspection.

First, all the Fourier spectra were examined by eye. Those that showed peaks at several harmonics with slowly monotonically decreasing amplitudes, all of which were higher than 1 cycle day⁻¹, were selected for further study (see Figure 2 for some examples of the FT spectra). This process resulted in 240 objects worthy of attention, and was effective in quickly identifying the targets with the highest signal-to-noise ratios (S/Ns) and periods shorter than 12–16 hr, such as Kepler-78b (Sanchis-Ojeda et al. 2013a).

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Second, our automatic vetting codes were used to try to identify and reject unwanted sources such as long-period planets, eclipsing binaries, and pulsators. The initial step in this process was to find all significant FT peaks with an amplitude 4 or more times the local noise level. Next, all objects with more than 10 significant frequencies between 1 and 10 cycles day⁻¹, a clear sign that the system is a pulsator or long-period planet and not a short-period transiting planet, were rejected. In other cases where frequencies lower than 1 cycle day⁻¹ were detected and successfully identified as the true periodicity of the signal, the sources were also rejected. Application of these filters reduced the number of candidates from 15,000 to ∼9000.

The surviving candidates then underwent a time-domain analysis. To remove the slow flux variations caused by starspots and stellar rotation, a moving-mean filter was applied to the flux series, with a width in time equal to the candidate orbital period. The data were then folded with that period, and the light curve was fitted with three models: a simple transit model, a sinusoid with the candidate period, and a sinusoid with twice the candidate period. For a candidate to survive this step, the transit signal needed to be detected with S/N ⩾ 7, and the transit model needed to provide a fit better than the fits of either of the sinusoidal models. This automated analysis reduced the number of objects from 9000 to ∼3500.

The folded light curves were then inspected by eye. We found that in many cases the automated pipeline was still passing through some longer-period planets and pulsators. However, with only 3500 objects, it was straightforward to reduce the list further through visual inspection of the folded light curves. We selected only those light curves with plausible transit-like features, including transit depths shallower than 5 to 10%, to reduce the list of surviving candidates along this pathway to 375.

We then combined the following lists: the 240 candidates from visual inspection of the FTs; the 375 candidates from the automated pipeline followed by visual inspection of the folded light curves; the 109 candidates with P < 1 day in the KOI list (as of 2014 January; Akeson et al. 2013); and the 28 candidates with P < 1 day that emerged from the independent searches of Ofir & Dreizler (2013), Huang et al. (2013), and Jackson et al. (2013). There was substantial overlap among these various lists (see Figure 3). The result was a list of 471 individual candidates. The next step was to subject these candidates to selected standard Kepler tests for false positives (Batalha et al. 2010).

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The first in this series of tests was the image centroid test which was used to reject those systems wherein the apparent motion of the stellar image during transits was large enough to rule out the Kepler target as the main source of the photometric variations. To carry out this test, the time series comprising the row (X) image centroid estimates and that comprising the column (Y) estimates were each processed with the moving-mean filter that was used for the flux time series. Each filtered time series was folded with the candidate period. The degree of correlation between the flux deviations and the centroid deviations was then calculated by computing the values of dY/df and dX/df, where dX and dY represent, respectively, the in versus out of transit change in row or column pixel position, and df represents the in versus out of transit change in relative flux. These gradients were multiplied by the Kepler plate scale of 4 arcsec pixel⁻¹ in order to estimate the location of the flux-variable source with respect to the center of light of the sources within the photometric aperture (Jenkins et al. 2010; Bryson et al. 2013).

A significant centroid shift implies the presence of more than one star in the Kepler aperture, but does not necessarily imply that the object should be classified as a false positive; even if the intended Kepler target is indeed the source of the photometric variations, the steady light of other nearby stars would cause the centroid to move during transits. Ideally, one would use the measured centroid shift and the known positions and mean fluxes of all the stars within the Kepler aperture to pinpoint the variable star. In the present case, though, the best available images (from the UKIRT Kepler field survey⁵) only allow us to differentiate fainter companion stars outside of a limiting separation that is target-dependent. For example, a star 2 mag fainter than the target star may be difficult to distinguish from the target star at separations less than an angle as small as 1'' or as large as 3'' (Wang et al. 2014).

Our procedure was to rely on the fact that very large centroid shifts almost always indicate a false positive, since the Kepler target is, by construction, intended to be the dominant source of light in the photometric aperture. The distribution of positions relative to the center of light among the objects under study may be reasonably modeled by the superposition of a uniform distribution out to distances beyond 40'' and a Gaussian distribution centered at the center of light and having a width in each orthogonal coordinate characterized by a standard deviation close to 05. Based on this, we discarded those sources where the best-fit distance from the flux-variable source to the center of light exceeded 15 and a 3σ lower bound exceeded 1''. This test assures that the "radius of confusion" (the maximum distance from the target star where a relevant background binary could be hiding) for almost all of the candidates is smaller than 2''. This is illustrated in the upper left panel of Figure 4. It was shown by Morton & Johnson (2011) that by shrinking the radius of confusion to this level, the probability of false positives due to background binaries is reduced to of the order of 5%. This centroid test eliminated half of the remaining candidates, confirming that short-period eclipsing binaries in the background are an important source of false positives.

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The second in this series of tests was a check for any statistically significant (>3σ) differences between the depths of the odd- and even-numbered transits. Such a difference would reveal the candidate to be an eclipsing binary with twice the nominal period (see lower left panel of Figure 4). Approximately 40% of the objects were removed on this basis.

Third, there were five cases in which primary eclipses with a depth in the range 0.5%–2% were accompanied by secondary eclipses 5 to 10 times shallower, similar to what is expected for a high-albedo hot Jupiter in such a short period orbit. However, the orbits of all five objects were found to be synchronized with the rotation of their host stars, and a close inspection revealed that in all cases the secondary's brightness temperature (inferred from the secondary eclipses) was higher than what would be expected from a planet (assuming zero albedo and inefficient transfer of heat to the nightside), both of which are signs that these objects are probably low-mass stars rather than planets (see Demory & Seager 2011).

Finally, we eliminated every candidate with a transit S/N less than 12, after judging that such detections are too weak to put meaningful constraints on false positive scenarios, and also out of concern that the pipeline might not be complete at low S/Ns (as confirmed in Section 5.1). This eliminated 22 more candidates. Within the remaining systems were two special cases, KIC 12557548 and KOI-2700 (Rappaport et al. 2012, 2014), which have asymmetric transit profiles that have been interpreted as signs that these planets are emitting dusty comet-like tails. Since the relationship between transit depth and planet radius is questionable for these objects, they were removed from our sample at this stage.

Using a trial period obtained from the FT, we folded each light curve, binned it in phase, and fitted a simple trapezoidal model representing the convolution of the transit profile and the 30 minute sampling function. The free parameters were the duration, depth, the ingress (or egress) time, and the time of the midpoint.

With the best-fitting model in hand, the orbital period was recalculated according to the following procedure. For most of the candidates, individual transit events could not be detected with a sufficiently high S/N to obtain individual transit times. Instead, we selected a sequence of time intervals each spanning many transits, and the data from each interval were used to construct a folded light curve with a decent S/N. To decide how long the time intervals should be, we considered S/N_q = S/N$/\sqrt{q}$, where S/N is the signal-to-noise ratio of the transit in the full-mission folded light curve and q is the total number of quarters of data available. For the cases with S/N_q < 10, the intervals were chosen to span two quarters. For 10 < S/N_q < 20, the intervals spanned one quarter. For 20 < S/N_q < 30, the intervals spanned one-third of a quarter (approximately one month), and finally for S/N_q > 30 the intervals spanned one-sixth of a quarter (approximately half a month). The trapezoidal model was fitted to each light curve to determine a mean epoch. For these fits, all parameters except the mean epoch were held fixed at the values determined for the full-mission folded light curve. Then, the collection of mean transit epochs was used to recalculate the orbital period. The formal uncertainty of each mean epoch was calculated using Δχ² = 1, where χ² was normalized to be equal to the number of degrees of freedom. We then fitted a linear function to the mean epochs. In this manner the final transit ephemeris was determined. We examined the residuals, and did not find any cases of significant transit-timing variations. This finding is consistent with the prior work by Steffen & Farr (2013) who noted that such short-period planets tend to avoid near mean-motion resonances with other planets.

We then returned to the original time series and repeated the process of filtering out variability on timescales longer than the orbital period, including starspot-induced variations. In this instance, the refined orbital period P and a slightly different procedure were used. For each data point f₀ taken at time t, a linear function of time was fitted to the out-of-transit data points at times t_j within P/2 of t₀. Then, f₀(t) was replaced by f₀(t) − f_fit(t) + 1, where f_fit was the best-fitting linear function. Illustrative filtered and folded light curves are shown in Figure 5.

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Further analysis was restricted to data from quarters 2–16, since quarters 0 and 1 were of shorter duration than the other quarters, and the data seem to have suffered more from instrumental artifacts. For each system, the data were folded with period P and then binned to reduce the data volume and to enhance the statistics. The bin duration was 2 minutes unless this resulted in fewer than 240 bins, in which case the bin duration was set to P/240 (as short as 1 minute for orbital periods of 4 hr). The trapezoidal model was fitted to this binned light curve. Due to the long cadence 30 minute time averaging, the effective ingress duration of the transit of a typical short-period small planet (with a transit duration of one to two hours) will then be slightly longer than 30 minutes. However, in the case of a background binary, the ingress time can be much longer, up to half the duration of the entire event (see the upper right panel of Figure 4). Once the best-fitting parameters were found, the allowed region in parameter space was defined with a Markov Chain Monte Carlo (MCMC) routine, which used the Gibbs sampler and the Metropolis–Hastings algorithm, and a likelihood proportional to exp(−χ²/2) (see, e.g., Appendix A of Tegmark et al. 2004; or Holman et al. 2006). We found that five of the USP candidates had ingress times significantly longer than 45 minutes (with >3σ confidence). These were removed from the candidate list.

With the list now reduced to 109 candidates, final transit parameters were determined. Our final model included a transit, an occultation, and an orbital phase modulation. For the transit model we used an inverse boxcar (zero ingress time, and zero limb darkening) for computational efficiency. More realistic transit models are not justified given the relatively low S/N and the effects of convolution with the 30 minute sampling function. The parameters were the midtransit time t₀, the transit depth δ_tran ≡ (R_p/R_⋆)², and the transit duration. The orbital period was held fixed, and a circular orbit was assumed.

The occultation (i.e., secondary eclipse) model f_occ(t) was also a boxcar dip, centered at an orbital phase of 0.5, and with a total duration set equal to that of the transit model. The only free parameter was the occultation depth δ_occ. The orbital phase modulations were modeled as sinusoids with phases and periods appropriate for ellipsoidal light variations (ELVs), illumination effects (representing both reflected and reprocessed stellar radiation) and Doppler boosting (DB). Expressed in terms of orbital phase ϕ = (t − t_c)/P, these components are

$$\begin{equation} {-} A_{\rm ELV}\cos (4\pi \phi) - A_{\rm ill}\cos (2\pi \phi) + A_{\rm DB} \sin (2\pi \phi), \end{equation} \tag{ 1 }$$

respectively, where t_c is the time of the transit center. The final parameter in the model is an overall flux multiplier, since only the relative flux values are significant and there is always an uncertainty associated with the normalization of the data. In the plots to follow, the normalizations of the model and the data were chosen to set the flux to unity during the occultations when only the star is visible. For comparison with the data, the model was evaluated every 30 s, and the resulting values were averaged in 29.4 minute bins to match the time averaging of the Kepler data.

Finally, we determined the allowed ranges for the model parameters using an MCMC algorithm. The errors in the flux data points in a given folded and binned light curve were set to be equal and defined by the condition χ² = N_dof. No ELVs were found with amplitudes at or exceeding the >3σ confidence level; hence, all of the candidates passed this test. Secondary eclipses were detected in seven cases with amplitudes consistent with those of USP planets. In three cases, an illumination component was detected while the corresponding secondary eclipses were not detected: δ_occ/A_ill < 2 with 3σ confidence. For one of the three the DB component was also detected (the only system that did not past that test). Those three candidates were eliminated from the list because only a background binary with twice the nominal period could plausibly be responsible for the out-of-eclipse variability without producing a secondary eclipse (see the lower right panel of Figure 4). We can also compare our results with some literature values. We obtained a secondary eclipse of 7.5 ± 1.4 ppm for Kepler-10b, a bit smaller than the value found by Fogtmann-Schulz et al. (2014) of 9.9 ± 1.0 ppm, using only the SC data, but slightly larger than the original value reported by Batalha et al. (2011) of 5.8 ± 2.5.

Based on the measured transit depth, and the stellar radius from Huber et al. (2014), we calculated the implied planet radius. Based on our results for Kepler-78b (Sanchis-Ojeda et al. 2013a), we added a 20% systematic uncertainty to the marginalized distribution of transit depths to account for the lack of limb darkening in our transit model. In almost all cases the uncertainty in the planet radius was still dominated by the uncertainty in the stellar radius.

One of the limitations of our transit analysis is that we could not readily obtain a value for the scaled semi-major axis (a/R_⋆), which can ordinarily be obtained when fitting high-S/N transit light curves, and is needed for computing planet occurrence rates due to its role as the inverse geometric transit probability. We can obtain a good estimate using the approximate relation:

$$\begin{equation} \frac{a}{R_\star } \approx \frac{P}{\pi T }\sqrt{1-b^2}, \end{equation} \tag{ 2 }$$

where P is the known orbital period, T is the total duration of the transit, and b is the impact parameter. For each of the values of T obtained in the MCMC analysis, we evaluate the expression on the right-hand side using a value of b drawn from a uniform distribution over the interval 0.0 to 0.9. The latter limit was not set to 1.0 because the transits in very high impact parameter cases are very short in duration and therefore very difficult to detect. Since the uncertainty in a/R_⋆, estimated as the standard deviation of the final distribution, is large and dominated by the wide range of allowed b values, the systematic error induced by using Equation (2) can be neglected.

Within our final list of 106 USP planet candidates, 8 candidates emerged uniquely from our search, including Kepler-78b (see Figure 3 for a diagram describing the origin of all 106 candidates). Another 10 objects were flagged by the Kepler pipeline but were marked as false positives in the updated KOI list (Akeson et al. 2013), because, in at least some of the cases, the pipeline gave the wrong orbital period (the pipeline is not intended to work with periods below 12 hr). The transit light curves of these 18 candidates are shown in Figure 5, and the transit parameters are given in Table 1. An examination of the KOI list shows that 27 KOI planet candidates with orbital periods shorter than 1 day were excluded in our analysis, either because of a low S/N (8 KOIs), or because those candidates did not pass our tests intended to exclude false positives (19 KOIs). Similarly, 7 planet candidates proposed by Ofir & Dreizler (2013), Huang et al. (2013), or Jackson et al. (2013) did not appear in the KOI list, and also did not pass our tests. This information is summarized in Table 2.

Table 1. Characteristics of the 106 USP Planet Candidates Discovered in the Kepler Data

KIC #	KOI #	mKep	Teff (K)	Logg	R⋆ (R☉)	Porb (days)	t0 (BJD-2454900)	Depth (ppm)	Duration (hr)	a/R⋆	Rp (R⊕)
2711597	4746.01	14.5	$4861^{+ 150}_{-129}$	$4.57^{+ 0.05}_{-0.05}$	$0.72^{+ 0.08}_{-0.07}$	0.49020979	64.289540	$92^{+ 6}_{ -6}$	$0.910^{+ 0.055}_{-0.055}$	$3.63^{+ 0.48}_{-0.96}$	$0.76^{+ 0.12}_{-0.11}$
2718885	0.00	14.7	$5597^{+ 160}_{-142}$	$4.55^{+ 0.03}_{-0.28}$	$0.84^{+ 0.35}_{-0.07}$	0.19733350	64.716193	$96^{+ 14}_{ -14}$	$0.372^{+ 0.058}_{-0.029}$	$3.43^{+ 0.66}_{-0.94}$	$0.94^{+ 0.34}_{-0.17}$
3112129	4144.01	14.4	$6096^{+ 144}_{-201}$	$4.46^{+ 0.05}_{-0.31}$	$1.02^{+ 0.48}_{-0.09}$	0.48765719	64.677406	$107^{+ 6}_{ -6}$	$0.899^{+ 0.045}_{-0.048}$	$3.67^{+ 0.45}_{-0.97}$	$1.21^{+ 0.50}_{-0.21}$
4665571	2393.00	14.9	$4894^{+ 149}_{-128}$	$4.60^{+ 0.04}_{-0.06}$	$0.70^{+ 0.09}_{-0.06}$	0.76669043	64.430840	$259^{+ 13}_{ -12}$	$0.983^{+ 0.046}_{-0.046}$	$5.28^{+ 0.64}_{-1.40}$	$1.24^{+ 0.19}_{-0.17}$
4929299	4018.01	14.8	$5761^{+ 191}_{-164}$	$4.33^{+ 0.17}_{-0.23}$	$1.05^{+ 0.38}_{-0.19}$	0.43436157	285.738945	$192^{+ 8}_{ -8}$	$1.076^{+ 0.045}_{-0.044}$	$2.74^{+ 0.32}_{-0.73}$	$1.60^{+ 0.56}_{-0.35}$
5480884	4841.01	16.0	$4607^{+ 123}_{-132}$	$4.73^{+ 0.06}_{-0.03}$	$0.52^{+ 0.03}_{-0.04}$	0.23624272	102.883560	$261^{+ 15}_{ -16}$	$0.844^{+ 0.049}_{-0.046}$	$1.89^{+ 0.24}_{-0.50}$	$0.90^{+ 0.11}_{-0.11}$
5955905	0.00	15.0	$6762^{+ 183}_{-283}$	$4.26^{+ 0.11}_{-0.32}$	$1.38^{+ 0.79}_{-0.25}$	0.51705775	1207.057671	$978^{+ 24}_{ -24}$	$0.753^{+ 0.019}_{-0.018}$	$4.67^{+ 0.51}_{-1.25}$	$4.84^{+ 2.54}_{-1.13}$
6359893	0.00	15.9	$6197^{+ 183}_{-226}$	$4.45^{+ 0.05}_{-0.30}$	$1.03^{+ 0.47}_{-0.10}$	0.18043173	285.389235	$258^{+ 23}_{ -23}$	$0.385^{+ 0.043}_{-0.032}$	$3.09^{+ 0.53}_{-0.81}$	$1.90^{+ 0.76}_{-0.34}$
6525946	2093.03	15.4	$6487^{+ 187}_{-257}$	$4.40^{+ 0.06}_{-0.31}$	$1.12^{+ 0.56}_{-0.12}$	0.49607106	285.534133	$124^{+ 7}_{ -7}$	$1.424^{+ 0.059}_{-0.059}$	$2.36^{+ 0.27}_{-0.63}$	$1.42^{+ 0.63}_{-0.26}$
8435766	0.00	11.6	$5085^{+ 106}_{ -93}$	$4.60^{+ 0.02}_{-0.06}$	$0.74^{+ 0.06}_{-0.03}$	0.35500753	53.604848	$229^{+ 1}_{ -1}$	$0.719^{+ 0.004}_{-0.004}$	$3.37^{+ 0.37}_{-0.90}$	$1.23^{+ 0.14}_{-0.14}$
9642018	4430.01	15.5	$5275^{+ 169}_{-142}$	$4.59^{+ 0.03}_{-0.16}$	$0.79^{+ 0.19}_{-0.06}$	0.25255594	64.516303	$223^{+ 23}_{ -20}$	$0.438^{+ 0.074}_{-0.051}$	$3.74^{+ 0.81}_{-0.96}$	$1.32^{+ 0.30}_{-0.20}$
9825174	2880.01	15.9	$5618^{+ 164}_{-144}$	$4.49^{+ 0.06}_{-0.27}$	$0.87^{+ 0.36}_{-0.09}$	0.74092963	102.695775	$210^{+ 15}_{ -14}$	$0.894^{+ 0.069}_{-0.065}$	$5.56^{+ 0.80}_{-1.46}$	$1.44^{+ 0.53}_{-0.25}$
10006641	4469.01	14.3	$4981^{+ 156}_{-127}$	$4.60^{+ 0.03}_{-0.08}$	$0.73^{+ 0.11}_{-0.06}$	0.44708779	64.506995	$76^{+ 5}_{ -5}$	$0.773^{+ 0.051}_{-0.050}$	$3.89^{+ 0.53}_{-1.02}$	$0.70^{+ 0.12}_{-0.10}$
10527135	2622.01	15.3	$5430^{+ 174}_{-145}$	$4.54^{+ 0.03}_{-0.25}$	$0.84^{+ 0.32}_{-0.07}$	0.32868757	64.454984	$165^{+ 9}_{ -9}$	$0.767^{+ 0.041}_{-0.043}$	$2.90^{+ 0.37}_{-0.76}$	$1.23^{+ 0.41}_{-0.20}$
10585738	3032.01	15.7	$5343^{+ 176}_{-156}$	$4.54^{+ 0.03}_{-0.23}$	$0.89^{+ 0.28}_{-0.06}$	0.63642490	64.688541	$196^{+ 18}_{ -17}$	$0.947^{+ 0.079}_{-0.082}$	$4.51^{+ 0.70}_{-1.17}$	$1.41^{+ 0.39}_{-0.22}$
11187332	0.00	15.2	$5790^{+ 160}_{-172}$	$4.53^{+ 0.03}_{-0.28}$	$0.90^{+ 0.35}_{-0.07}$	0.30598391	64.826889	$126^{+ 13}_{ -14}$	$0.369^{+ 0.045}_{-0.027}$	$5.42^{+ 0.94}_{-1.43}$	$1.15^{+ 0.39}_{-0.19}$
11453930	0.00	13.1	$6823^{+ 165}_{-242}$	$4.21^{+ 0.12}_{-0.34}$	$1.49^{+ 0.92}_{-0.29}$	0.22924638	64.512355	$37^{+ 3}_{ -4}$	$0.367^{+ 0.046}_{-0.025}$	$4.08^{+ 0.71}_{-1.07}$	$1.03^{+ 0.58}_{-0.26}$
11550689	0.00	14.6	$4168^{+ 124}_{-135}$	$4.65^{+ 0.06}_{-0.03}$	$0.60^{+ 0.05}_{-0.07}$	0.30160087	64.744663	$412^{+ 6}_{ -6}$	$0.806^{+ 0.014}_{-0.014}$	$2.55^{+ 0.27}_{-0.68}$	$1.31^{+ 0.18}_{-0.19}$
1717722	3145.02	15.7	$4812^{+ 145}_{-133}$	$4.61^{+ 0.03}_{-0.06}$	$0.71^{+ 0.07}_{-0.06}$	0.97730809	102.820321	$266^{+ 21}_{ -22}$	$1.249^{+ 0.109}_{-0.095}$	$5.22^{+ 0.81}_{-1.36}$	$1.27^{+ 0.19}_{-0.18}$
3444588	1202.01	15.9	$4894^{+ 607}_{-904}$	$4.60^{+ 0.10}_{-0.10}$	$0.72^{+ 0.11}_{-0.11}$	0.92831093	64.945619	$397^{+ 21}_{ -21}$	$1.104^{+ 0.055}_{-0.054}$	$5.70^{+ 0.69}_{-1.51}$	$1.55^{+ 0.30}_{-0.28}$
4055304	2119.01	14.1	$5203^{+ 171}_{-134}$	$4.51^{+ 0.05}_{-0.19}$	$0.86^{+ 0.24}_{-0.07}$	0.57103885	64.653190	$243^{+ 5}_{ -5}$	$0.978^{+ 0.019}_{-0.020}$	$3.98^{+ 0.43}_{-1.07}$	$1.52^{+ 0.39}_{-0.23}$
4144576	2202.01	14.1	$5285^{+ 169}_{-138}$	$4.59^{+ 0.03}_{-0.16}$	$0.78^{+ 0.20}_{-0.06}$	0.81316598	64.826536	$162^{+ 6}_{ -5}$	$1.237^{+ 0.045}_{-0.044}$	$4.47^{+ 0.50}_{-1.19}$	$1.12^{+ 0.26}_{-0.16}$
4852528	500.05	14.8	$4040^{+ 64}_{-175}$	$4.70^{+ 0.07}_{-0.07}$	$0.57^{+ 0.07}_{-0.07}$	0.98678600	64.196497	$295^{+ 11}_{ -11}$	$1.238^{+ 0.041}_{-0.041}$	$5.42^{+ 0.60}_{-1.44}$	$1.06^{+ 0.18}_{-0.17}$
5040077	3065.01	14.6	$5837^{+ 192}_{-169}$	$4.50^{+ 0.07}_{-0.28}$	$0.84^{+ 0.37}_{-0.09}$	0.89638348	65.296606	$133^{+ 15}_{ -14}$	$0.965^{+ 0.098}_{-0.094}$	$6.19^{+ 1.07}_{-1.60}$	$1.11^{+ 0.43}_{-0.21}$
5095635	2607.01	14.5	$5883^{+ 160}_{-165}$	$4.53^{+ 0.03}_{-0.27}$	$0.89^{+ 0.33}_{-0.07}$	0.75445863	64.210363	$237^{+ 16}_{ -12}$	$0.561^{+ 0.037}_{-0.047}$	$9.11^{+ 1.28}_{-2.38}$	$1.57^{+ 0.50}_{-0.25}$
5175986	2708.01	15.9	$4790^{+ 159}_{-132}$	$4.60^{+ 0.03}_{-0.06}$	$0.73^{+ 0.07}_{-0.05}$	0.86838702	103.275858	$511^{+ 19}_{ -18}$	$0.923^{+ 0.033}_{-0.034}$	$6.40^{+ 0.72}_{-1.70}$	$1.80^{+ 0.25}_{-0.24}$
5340878	4199.01	14.3	$5166^{+ 148}_{-136}$	$4.65^{+ 0.03}_{-0.10}$	$0.68^{+ 0.11}_{-0.05}$	0.53991700	64.548229	$96^{+ 6}_{ -6}$	$1.040^{+ 0.057}_{-0.059}$	$3.51^{+ 0.45}_{-0.93}$	$0.74^{+ 0.12}_{-0.10}$
5513012	2668.01	14.2	$5596^{+ 155}_{-144}$	$4.59^{+ 0.03}_{-0.25}$	$0.78^{+ 0.31}_{-0.06}$	0.67933587	65.178693	$246^{+ 6}_{ -7}$	$1.066^{+ 0.030}_{-0.028}$	$4.33^{+ 0.48}_{-1.16}$	$1.39^{+ 0.48}_{-0.22}$
5642620	2882.02	15.4	$4467^{+ 165}_{-160}$	$4.67^{+ 0.04}_{-0.06}$	$0.61^{+ 0.07}_{-0.05}$	0.49339463	285.612655	$270^{+ 16}_{ -15}$	$0.870^{+ 0.052}_{-0.051}$	$3.82^{+ 0.50}_{-1.01}$	$1.10^{+ 0.17}_{-0.15}$
5942808	2250.02	15.6	$4922^{+ 195}_{-151}$	$4.50^{+ 0.07}_{-0.65}$	$0.82^{+ 0.69}_{-0.07}$	0.62628076	285.358414	$418^{+ 15}_{ -16}$	$1.011^{+ 0.038}_{-0.036}$	$4.20^{+ 0.48}_{-1.12}$	$1.96^{+ 1.41}_{-0.36}$
5972334	191.03	15.0	$5700^{+ 167}_{-154}$	$4.51^{+ 0.05}_{-0.28}$	$0.87^{+ 0.37}_{-0.08}$	0.70862545	64.944413	$180^{+ 8}_{ -9}$	$1.366^{+ 0.056}_{-0.057}$	$3.52^{+ 0.41}_{-0.94}$	$1.33^{+ 0.49}_{-0.23}$
6129524	2886.01	15.9	$5260^{+ 213}_{-163}$	$4.59^{+ 0.03}_{-0.17}$	$0.78^{+ 0.22}_{-0.06}$	0.88183821	285.152838	$268^{+ 17}_{ -16}$	$1.121^{+ 0.071}_{-0.076}$	$5.31^{+ 0.72}_{-1.40}$	$1.44^{+ 0.37}_{-0.22}$
6183511	2542.01	15.5	$3339^{+ 50}_{ -51}$	$4.96^{+ 0.06}_{-0.12}$	$0.29^{+ 0.08}_{-0.05}$	0.72733112	64.641934	$368^{+ 23}_{ -22}$	$0.773^{+ 0.044}_{-0.047}$	$6.36^{+ 0.82}_{-1.67}$	$0.61^{+ 0.17}_{-0.13}$
6265792	2753.01	13.6	$6002^{+ 153}_{-178}$	$4.36^{+ 0.10}_{-0.27}$	$1.09^{+ 0.45}_{-0.14}$	0.93512027	65.197130	$64^{+ 4}_{ -4}$	$1.383^{+ 0.074}_{-0.073}$	$4.57^{+ 0.57}_{-1.21}$	$0.98^{+ 0.37}_{-0.19}$
6294819	2852.01	15.9	$5966^{+ 178}_{-211}$	$4.49^{+ 0.04}_{-0.30}$	$0.96^{+ 0.41}_{-0.09}$	0.67565200	285.617217	$193^{+ 13}_{ -12}$	$1.702^{+ 0.085}_{-0.096}$	$2.69^{+ 0.33}_{-0.71}$	$1.52^{+ 0.56}_{-0.26}$
6310636	1688.01	14.5	$5738^{+ 183}_{-147}$	$4.49^{+ 0.09}_{-0.29}$	$0.82^{+ 0.38}_{-0.09}$	0.92103467	65.028092	$94^{+ 5}_{ -5}$	$2.328^{+ 0.076}_{-0.072}$	$2.69^{+ 0.30}_{-0.72}$	$0.90^{+ 0.37}_{-0.17}$
6362874	1128.01	13.5	$5487^{+ 100}_{-111}$	$4.56^{+ 0.01}_{-0.12}$	$0.83^{+ 0.12}_{-0.03}$	0.97486662	54.379852	$193^{+ 2}_{ -2}$	$1.448^{+ 0.018}_{-0.018}$	$4.59^{+ 0.49}_{-1.23}$	$1.30^{+ 0.19}_{-0.16}$
6607286	1239.01	15.0	$6108^{+ 159}_{-234}$	$4.47^{+ 0.04}_{-0.29}$	$1.03^{+ 0.43}_{-0.10}$	0.78327657	285.814763	$264^{+ 7}_{ -7}$	$1.489^{+ 0.030}_{-0.030}$	$3.58^{+ 0.38}_{-0.96}$	$1.90^{+ 0.71}_{-0.32}$
6607644	4159.01	14.5	$5406^{+ 170}_{-144}$	$4.58^{+ 0.02}_{-0.24}$	$0.82^{+ 0.28}_{-0.06}$	0.97190650	65.184987	$64^{+ 5}_{ -5}$	$1.489^{+ 0.090}_{-0.088}$	$4.40^{+ 0.58}_{-1.16}$	$0.75^{+ 0.22}_{-0.12}$
6666233	2306.01	14.8	$3878^{+ 76}_{ -75}$	$4.73^{+ 0.06}_{-0.09}$	$0.52^{+ 0.07}_{-0.05}$	0.51240853	64.726189	$321^{+ 9}_{ -9}$	$0.997^{+ 0.025}_{-0.027}$	$3.50^{+ 0.38}_{-0.94}$	$1.02^{+ 0.17}_{-0.15}$
6697756	2798.01	14.1	$4374^{+ 126}_{-137}$	$4.64^{+ 0.06}_{-0.03}$	$0.61^{+ 0.05}_{-0.06}$	0.91614360	65.021825	$93^{+ 5}_{ -5}$	$1.025^{+ 0.057}_{-0.054}$	$6.03^{+ 0.76}_{-1.59}$	$0.64^{+ 0.09}_{-0.09}$
6755944	4072.01	13.4	$6103^{+ 150}_{-177}$	$4.41^{+ 0.08}_{-0.27}$	$1.02^{+ 0.39}_{-0.12}$	0.69297791	53.714011	$60^{+ 3}_{ -3}$	$1.014^{+ 0.054}_{-0.054}$	$4.63^{+ 0.57}_{-1.22}$	$0.89^{+ 0.31}_{-0.16}$
6867588	2571.01	14.4	$5203^{+ 177}_{-135}$	$4.59^{+ 0.03}_{-0.14}$	$0.78^{+ 0.17}_{-0.06}$	0.82628363	64.444070	$131^{+ 6}_{ -6}$	$0.866^{+ 0.049}_{-0.043}$	$6.44^{+ 0.81}_{-1.71}$	$1.00^{+ 0.21}_{-0.14}$
6934291	1367.01	15.1	$5076^{+ 161}_{-131}$	$4.61^{+ 0.03}_{-0.10}$	$0.73^{+ 0.12}_{-0.05}$	0.56785704	64.223769	$338^{+ 8}_{ -8}$	$0.906^{+ 0.023}_{-0.023}$	$4.27^{+ 0.46}_{-1.14}$	$1.48^{+ 0.26}_{-0.20}$
6964929	2756.01	14.8	$5957^{+ 154}_{-187}$	$4.51^{+ 0.04}_{-0.27}$	$0.94^{+ 0.34}_{-0.08}$	0.66502914	64.655142	$99^{+ 5}_{ -5}$	$1.541^{+ 0.055}_{-0.052}$	$2.93^{+ 0.33}_{-0.78}$	$1.06^{+ 0.34}_{-0.17}$
6974658	2925.01	13.5	$5552^{+ 160}_{-135}$	$4.35^{+ 0.15}_{-0.24}$	$1.03^{+ 0.38}_{-0.17}$	0.71653130	64.560833	$90^{+ 17}_{ -13}$	$0.475^{+ 0.121}_{-0.096}$	$9.60^{+ 3.29}_{-2.49}$	$1.10^{+ 0.41}_{-0.25}$
7102227	1360.03	15.6	$5153^{+ 170}_{-143}$	$4.60^{+ 0.03}_{-0.12}$	$0.76^{+ 0.14}_{-0.06}$	0.76401932	64.458108	$131^{+ 14}_{ -13}$	$0.991^{+ 0.101}_{-0.116}$	$5.16^{+ 0.93}_{-1.32}$	$0.97^{+ 0.18}_{-0.14}$
7605093	2817.01	15.8	$5238^{+ 202}_{-169}$	$4.60^{+ 0.03}_{-0.13}$	$0.75^{+ 0.16}_{-0.06}$	0.63400313	102.935144	$338^{+ 21}_{ -21}$	$0.821^{+ 0.047}_{-0.050}$	$5.23^{+ 0.66}_{-1.38}$	$1.54^{+ 0.32}_{-0.23}$
7749002	4325.01	15.7	$5936^{+ 171}_{-204}$	$4.50^{+ 0.04}_{-0.31}$	$0.96^{+ 0.42}_{-0.08}$	0.60992303	102.621977	$143^{+ 11}_{ -11}$	$1.118^{+ 0.084}_{-0.075}$	$3.65^{+ 0.52}_{-0.95}$	$1.32^{+ 0.49}_{-0.22}$
7826620	4268.01	15.2	$4294^{+ 130}_{-146}$	$4.74^{+ 0.07}_{-0.04}$	$0.50^{+ 0.04}_{-0.05}$	0.84990436	64.675047	$124^{+ 11}_{ -10}$	$0.984^{+ 0.082}_{-0.087}$	$5.80^{+ 0.89}_{-1.51}$	$0.61^{+ 0.09}_{-0.09}$
7907808	4109.01	14.5	$4968^{+ 149}_{-126}$	$3.92^{+ 0.63}_{-0.40}$	$1.75^{+ 1.32}_{-0.97}$	0.65594057	64.444338	$64^{+ 4}_{ -4}$	$1.175^{+ 0.075}_{-0.073}$	$3.76^{+ 0.50}_{-0.99}$	$1.53^{+ 1.16}_{-0.86}$
8235924	2347.01	14.9	$3972^{+ 65}_{ -60}$	$4.69^{+ 0.06}_{-0.06}$	$0.56^{+ 0.05}_{-0.05}$	0.58800041	64.471042	$300^{+ 8}_{ -8}$	$0.994^{+ 0.026}_{-0.026}$	$4.03^{+ 0.44}_{-1.08}$	$1.06^{+ 0.15}_{-0.14}$
8278371	1150.01	13.3	$5735^{+ 107}_{-115}$	$4.34^{+ 0.11}_{-0.12}$	$1.10^{+ 0.19}_{-0.13}$	0.67737578	54.074797	$71^{+ 2}_{ -2}$	$1.701^{+ 0.039}_{-0.042}$	$2.71^{+ 0.29}_{-0.73}$	$1.02^{+ 0.20}_{-0.17}$
8558011	577.02	14.4	$5244^{+ 215}_{-169}$	$4.45^{+ 0.10}_{-0.47}$	$0.89^{+ 0.68}_{-0.11}$	0.63816268	103.039136	$111^{+ 8}_{ -8}$	$1.078^{+ 0.069}_{-0.063}$	$3.98^{+ 0.53}_{-1.05}$	$1.08^{+ 0.72}_{-0.22}$
8561063	961.02	15.9	$3200^{+ 174}_{-174}$	$5.09^{+ 0.20}_{-0.20}$	$0.17^{+ 0.04}_{ 0.03}$	0.45328732	64.603393	$1885^{+ 28}_{ -38}$	$0.385^{+ 0.010}_{-0.007}$	$8.01^{+ 0.87}_{-2.14}$	$0.92^{+ 0.15}_{-0.12}$
8804845	2039.02	14.3	$5614^{+ 147}_{-142}$	$4.56^{+ 0.03}_{-0.22}$	$0.83^{+ 0.27}_{-0.06}$	0.76213044	64.617466	$63^{+ 6}_{ -6}$	$1.015^{+ 0.097}_{-0.097}$	$5.01^{+ 0.84}_{-1.29}$	$0.75^{+ 0.21}_{-0.12}$
8895758	3106.01	15.4	$5853^{+ 177}_{-210}$	$4.46^{+ 0.05}_{-0.31}$	$1.01^{+ 0.47}_{-0.10}$	0.96896529	285.390115	$114^{+ 11}_{ -12}$	$1.364^{+ 0.119}_{-0.101}$	$4.73^{+ 0.72}_{-1.23}$	$1.23^{+ 0.50}_{-0.22}$
8947520	2517.01	14.5	$5854^{+ 179}_{-145}$	$4.44^{+ 0.13}_{-0.30}$	$0.87^{+ 0.41}_{-0.12}$	0.96852469	64.512671	$118^{+ 7}_{ -7}$	$1.155^{+ 0.060}_{-0.063}$	$5.68^{+ 0.71}_{-1.50}$	$1.06^{+ 0.46}_{-0.22}$
9092504	2716.01	15.8	$5693^{+ 178}_{-156}$	$4.57^{+ 0.03}_{-0.28}$	$0.80^{+ 0.34}_{-0.06}$	0.96286621	102.676246	$310^{+ 13}_{ -13}$	$1.438^{+ 0.056}_{-0.055}$	$4.55^{+ 0.52}_{-1.21}$	$1.61^{+ 0.58}_{-0.26}$
9149789	2874.01	14.9	$5461^{+ 171}_{-137}$	$4.59^{+ 0.03}_{-0.18}$	$0.77^{+ 0.23}_{-0.06}$	0.35251639	64.663443	$163^{+ 7}_{ -8}$	$0.864^{+ 0.042}_{-0.042}$	$2.76^{+ 0.33}_{-0.73}$	$1.11^{+ 0.30}_{-0.17}$
9221517	2281.01	13.8	$5178^{+ 172}_{-136}$	$4.56^{+ 0.03}_{-0.15}$	$0.82^{+ 0.19}_{-0.06}$	0.76985432	64.749343	$106^{+ 4}_{ -4}$	$0.905^{+ 0.038}_{-0.037}$	$5.77^{+ 0.67}_{-1.53}$	$0.95^{+ 0.20}_{-0.14}$
9388479	936.02	15.1	$3581^{+ 65}_{ -50}$	$4.81^{+ 0.08}_{-0.08}$	$0.44^{+ 0.06}_{-0.06}$	0.89304119	64.861219	$746^{+ 11}_{ -11}$	$0.987^{+ 0.014}_{-0.014}$	$6.17^{+ 0.66}_{-1.65}$	$1.30^{+ 0.23}_{-0.22}$
9456281	4207.01	15.4	$4239^{+ 123}_{-138}$	$4.70^{+ 0.07}_{-0.04}$	$0.54^{+ 0.04}_{-0.06}$	0.70194345	65.003065	$216^{+ 32}_{ -25}$	$0.526^{+ 0.099}_{-0.086}$	$8.68^{+ 2.31}_{-2.23}$	$0.85^{+ 0.13}_{-0.13}$
9472074	2735.01	15.6	$5154^{+ 172}_{-140}$	$4.59^{+ 0.02}_{-0.14}$	$0.80^{+ 0.14}_{-0.05}$	0.55884253	64.576962	$284^{+ 17}_{ -16}$	$0.708^{+ 0.045}_{-0.047}$	$5.33^{+ 0.71}_{-1.40}$	$1.51^{+ 0.27}_{-0.20}$
9473078	2079.01	13.0	$5582^{+ 161}_{-153}$	$4.31^{+ 0.15}_{-0.23}$	$1.16^{+ 0.42}_{-0.20}$	0.69384332	54.078837	$46^{+ 2}_{ -2}$	$1.355^{+ 0.052}_{-0.051}$	$3.47^{+ 0.40}_{-0.92}$	$0.87^{+ 0.31}_{-0.19}$
9580167	2548.01	15.0	$4865^{+ 183}_{-160}$	$4.61^{+ 0.03}_{-0.07}$	$0.70^{+ 0.09}_{-0.06}$	0.82715274	285.928326	$270^{+ 34}_{ -33}$	$1.114^{+ 0.156}_{-0.123}$	$4.87^{+ 0.98}_{-1.27}$	$1.27^{+ 0.21}_{-0.19}$
9787239	952.05	15.8	$3727^{+ 104}_{ -64}$	$4.76^{+ 0.08}_{-0.08}$	$0.50^{+ 0.06}_{-0.06}$	0.74295778	64.794040	$221^{+ 21}_{ -21}$	$1.004^{+ 0.100}_{-0.101}$	$4.94^{+ 0.84}_{-1.27}$	$0.80^{+ 0.14}_{-0.13}$
10024051	2409.01	14.9	$5256^{+ 168}_{-139}$	$4.61^{+ 0.03}_{-0.12}$	$0.72^{+ 0.15}_{-0.05}$	0.57736948	64.270976	$395^{+ 8}_{ -8}$	$0.997^{+ 0.019}_{-0.019}$	$3.94^{+ 0.42}_{-1.06}$	$1.60^{+ 0.33}_{-0.23}$
10028535	2493.01	15.3	$5166^{+ 185}_{-144}$	$4.58^{+ 0.02}_{-0.13}$	$0.79^{+ 0.15}_{-0.06}$	0.66308666	64.978343	$321^{+ 12}_{ -12}$	$0.991^{+ 0.033}_{-0.034}$	$4.55^{+ 0.51}_{-1.21}$	$1.58^{+ 0.30}_{-0.22}$
10281221	3913.01	15.0	$6263^{+ 169}_{-211}$	$4.46^{+ 0.05}_{-0.28}$	$1.01^{+ 0.40}_{-0.10}$	0.58289571	64.620720	$581^{+ 11}_{ -11}$	$0.712^{+ 0.017}_{-0.015}$	$5.57^{+ 0.60}_{-1.49}$	$2.75^{+ 0.97}_{-0.46}$
10319385	1169.01	13.2	$5676^{+ 100}_{-117}$	$4.53^{+ 0.02}_{-0.14}$	$0.92^{+ 0.15}_{-0.03}$	0.68920948	53.191352	$187^{+ 2}_{ -2}$	$1.456^{+ 0.020}_{-0.019}$	$3.22^{+ 0.34}_{-0.86}$	$1.41^{+ 0.22}_{-0.18}$
10468885	2589.01	15.6	$5177^{+ 178}_{-137}$	$4.58^{+ 0.02}_{-0.15}$	$0.80^{+ 0.17}_{-0.05}$	0.66407444	65.075194	$221^{+ 19}_{ -19}$	$0.958^{+ 0.078}_{-0.085}$	$4.66^{+ 0.72}_{-1.21}$	$1.34^{+ 0.27}_{-0.20}$
10604521	2797.01	15.8	$6173^{+ 140}_{-276}$	$4.43^{+ 0.05}_{-0.32}$	$1.11^{+ 0.59}_{-0.12}$	0.86812153	285.335843	$245^{+ 13}_{ -13}$	$1.398^{+ 0.053}_{-0.051}$	$4.22^{+ 0.48}_{-1.13}$	$1.98^{+ 0.92}_{-0.36}$
10647452	4366.01	15.6	$5377^{+ 223}_{-170}$	$4.59^{+ 0.04}_{-0.17}$	$0.74^{+ 0.23}_{-0.06}$	0.76295078	285.410004	$197^{+ 18}_{ -17}$	$0.865^{+ 0.092}_{-0.074}$	$5.83^{+ 1.00}_{-1.50}$	$1.18^{+ 0.33}_{-0.19}$
10975146	1300.01	14.3	$4441^{+ 142}_{-137}$	$4.71^{+ 0.05}_{-0.03}$	$0.53^{+ 0.04}_{-0.04}$	0.63133223	64.418552	$423^{+ 5}_{ -5}$	$0.993^{+ 0.013}_{-0.012}$	$4.33^{+ 0.47}_{-1.16}$	$1.18^{+ 0.15}_{-0.15}$
11030475	2248.03	15.5	$5290^{+ 173}_{-142}$	$4.60^{+ 0.03}_{-0.15}$	$0.75^{+ 0.19}_{-0.06}$	0.76196246	64.259895	$182^{+ 11}_{ -10}$	$1.123^{+ 0.059}_{-0.060}$	$4.59^{+ 0.57}_{-1.22}$	$1.13^{+ 0.26}_{-0.17}$
11197853	2813.01	13.6	$5143^{+ 164}_{-141}$	$4.64^{+ 0.05}_{-1.10}$	$0.62^{+ 1.88}_{-0.04}$	0.69846257	53.675039	$120^{+ 10}_{ -9}$	$0.732^{+ 0.066}_{-0.066}$	$6.38^{+ 1.02}_{-1.65}$	$0.84^{+ 2.12}_{-0.18}$
11246161	2796.01	14.8	$6108^{+ 143}_{-196}$	$4.47^{+ 0.05}_{-0.27}$	$1.00^{+ 0.38}_{-0.09}$	0.53740984	64.495864	$93^{+ 6}_{ -6}$	$1.037^{+ 0.065}_{-0.062}$	$3.49^{+ 0.46}_{-0.92}$	$1.09^{+ 0.36}_{-0.18}$
11401182	1428.01	14.6	$4911^{+ 157}_{-128}$	$4.53^{+ 0.06}_{-0.65}$	$0.78^{+ 0.11}_{-0.07}$	0.92785963	64.122301	$492^{+ 5}_{ -5}$	$1.224^{+ 0.013}_{-0.014}$	$5.17^{+ 0.55}_{-1.39}$	$1.90^{+ 0.31}_{-0.27}$
11547505	1655.01	13.8	$5902^{+ 155}_{-168}$	$4.44^{+ 0.07}_{-0.27}$	$0.99^{+ 0.39}_{-0.11}$	0.93846561	63.909869	$206^{+ 5}_{ -5}$	$1.176^{+ 0.024}_{-0.025}$	$5.43^{+ 0.58}_{-1.46}$	$1.60^{+ 0.57}_{-0.28}$
11600889	1442.01	12.5	$5626^{+ 92}_{-124}$	$4.40^{+ 0.06}_{-0.15}$	$1.07^{+ 0.21}_{-0.09}$	0.66931017	53.745653	$122^{+ 2}_{ -2}$	$1.461^{+ 0.020}_{-0.022}$	$3.12^{+ 0.33}_{-0.84}$	$1.31^{+ 0.25}_{-0.19}$
11752632	2492.01	13.8	$6060^{+ 145}_{-170}$	$4.51^{+ 0.04}_{-0.28}$	$0.93^{+ 0.36}_{-0.08}$	0.98493890	64.350365	$75^{+ 4}_{ -4}$	$1.828^{+ 0.062}_{-0.067}$	$3.67^{+ 0.41}_{-0.98}$	$0.91^{+ 0.31}_{-0.15}$
11870545	1510.01	15.9	$4924^{+ 156}_{-129}$	$4.60^{+ 0.03}_{-0.07}$	$0.74^{+ 0.08}_{-0.06}$	0.83996160	102.954242	$521^{+ 19}_{ -20}$	$1.118^{+ 0.042}_{-0.039}$	$5.10^{+ 0.58}_{-1.36}$	$1.84^{+ 0.27}_{-0.25}$
11904151	72.01	11.0	$5627^{+ 44}_{ -44}$	$4.34^{+ 0.03}_{-0.03}$	$1.06^{+ 0.02}_{-0.02}$	0.83749060	53.687909	$177^{+ 1}_{ -1}$	$1.634^{+ 0.010}_{-0.009}$	$3.49^{+ 0.38}_{-0.94}$	$1.54^{+ 0.15}_{-0.16}$
12170648	2875.01	14.8	$5180^{+ 168}_{-138}$	$4.55^{+ 0.05}_{-0.13}$	$0.78^{+ 0.16}_{-0.07}$	0.29969767	64.717608	$344^{+ 11}_{ -11}$	$0.643^{+ 0.028}_{-0.028}$	$3.16^{+ 0.37}_{-0.84}$	$1.61^{+ 0.33}_{-0.24}$
12265786	4595.01	15.5	$4985^{+ 152}_{-135}$	$4.68^{+ 0.03}_{-0.09}$	$0.63^{+ 0.09}_{-0.04}$	0.59701811	64.713396	$268^{+ 41}_{ -33}$	$0.464^{+ 0.076}_{-0.074}$	$8.41^{+ 2.06}_{-2.17}$	$1.15^{+ 0.19}_{-0.17}$
12405333	3009.01	15.2	$5231^{+ 183}_{-140}$	$4.48^{+ 0.08}_{-0.27}$	$0.86^{+ 0.30}_{-0.09}$	0.76486301	64.368844	$154^{+ 13}_{ -13}$	$1.017^{+ 0.084}_{-0.085}$	$5.05^{+ 0.78}_{-1.31}$	$1.20^{+ 0.38}_{-0.21}$
3834322	2763.01	15.4	$4787^{+ 144}_{-128}$	$4.61^{+ 0.03}_{-0.06}$	$0.71^{+ 0.07}_{-0.05}$	0.49843515	64.569826	$237^{+ 13}_{ -12}$	$0.828^{+ 0.045}_{-0.043}$	$4.06^{+ 0.51}_{-1.07}$	$1.20^{+ 0.17}_{-0.16}$
5008501	1666.01	14.5	$6444^{+ 174}_{-227}$	$4.38^{+ 0.06}_{-0.30}$	$1.20^{+ 0.63}_{-0.15}$	0.96802576	64.756733	$203^{+ 9}_{ -9}$	$0.707^{+ 0.034}_{-0.035}$	$9.29^{+ 1.12}_{-2.47}$	$1.94^{+ 0.91}_{-0.38}$
5080636	1843.00	14.4	$3584^{+ 69}_{ -54}$	$4.80^{+ 0.06}_{-0.09}$	$0.45^{+ 0.08}_{-0.05}$	0.17689162	64.552254	$155^{+ 8}_{ -8}$	$0.581^{+ 0.027}_{-0.030}$	$2.07^{+ 0.25}_{-0.55}$	$0.61^{+ 0.12}_{-0.10}$
5773121	4002.01	15.0	$5396^{+ 155}_{-133}$	$4.49^{+ 0.05}_{-0.23}$	$0.93^{+ 0.30}_{-0.08}$	0.52417582	64.601474	$240^{+ 11}_{ -12}$	$0.709^{+ 0.043}_{-0.038}$	$4.98^{+ 0.65}_{-1.31}$	$1.63^{+ 0.47}_{-0.26}$
5980208	2742.01	15.0	$4258^{+ 120}_{-146}$	$4.63^{+ 0.06}_{-0.02}$	$0.62^{+ 0.04}_{-0.07}$	0.78916057	64.505441	$225^{+ 12}_{ -11}$	$0.834^{+ 0.044}_{-0.046}$	$6.41^{+ 0.80}_{-1.69}$	$1.00^{+ 0.14}_{-0.15}$
6750902	3980.01	14.7	$5888^{+ 159}_{-160}$	$4.39^{+ 0.11}_{-0.25}$	$1.01^{+ 0.36}_{-0.14}$	0.46933249	64.742598	$316^{+ 11}_{ -10}$	$0.743^{+ 0.029}_{-0.031}$	$4.30^{+ 0.49}_{-1.15}$	$2.00^{+ 0.67}_{-0.38}$
7051984	2879.01	12.8	$5653^{+ 165}_{-145}$	$3.85^{+ 0.44}_{-0.22}$	$2.18^{+ 0.87}_{-1.03}$	0.33906981	53.680375	$47^{+ 1}_{ -1}$	$0.987^{+ 0.033}_{-0.032}$	$2.33^{+ 0.26}_{-0.62}$	$1.61^{+ 0.70}_{-0.76}$
7269881	2916.01	14.4	$5088^{+ 160}_{-130}$	$4.58^{+ 0.04}_{-0.09}$	$0.74^{+ 0.12}_{-0.06}$	0.30693783	64.635907	$134^{+ 9}_{ -9}$	$0.464^{+ 0.043}_{-0.036}$	$4.39^{+ 0.70}_{-1.14}$	$0.95^{+ 0.17}_{-0.14}$
7582691	4419.01	15.2	$3939^{+ 61}_{ -51}$	$4.74^{+ 0.06}_{-0.06}$	$0.53^{+ 0.05}_{-0.05}$	0.25981944	285.437765	$281^{+ 30}_{ -26}$	$0.423^{+ 0.053}_{-0.052}$	$4.05^{+ 0.81}_{-1.04}$	$0.96^{+ 0.14}_{-0.14}$
8189801	2480.01	15.7	$3990^{+ 84}_{ -66}$	$4.69^{+ 0.06}_{-0.11}$	$0.55^{+ 0.09}_{-0.05}$	0.66682749	64.706275	$539^{+ 26}_{ -25}$	$0.712^{+ 0.038}_{-0.037}$	$6.33^{+ 0.80}_{-1.67}$	$1.42^{+ 0.25}_{-0.21}$
8416523	4441.01	14.9	$5017^{+ 160}_{-132}$	$4.59^{+ 0.03}_{-0.09}$	$0.75^{+ 0.11}_{-0.06}$	0.34184200	64.318576	$279^{+ 6}_{ -6}$	$0.856^{+ 0.019}_{-0.019}$	$2.72^{+ 0.29}_{-0.73}$	$1.39^{+ 0.22}_{-0.19}$
9353742	3867.01	14.9	$5825^{+ 157}_{-180}$	$4.52^{+ 0.03}_{-0.30}$	$0.92^{+ 0.38}_{-0.07}$	0.93875053	64.682273	$272^{+ 7}_{ -7}$	$1.240^{+ 0.028}_{-0.027}$	$5.15^{+ 0.56}_{-1.38}$	$1.73^{+ 0.62}_{-0.28}$
9467404	2717.01	12.4	$6437^{+ 151}_{-205}$	$4.25^{+ 0.14}_{-0.31}$	$1.33^{+ 0.77}_{-0.26}$	0.92990607	53.162180	$105^{+ 1}_{ -1}$	$1.601^{+ 0.019}_{-0.020}$	$3.96^{+ 0.42}_{-1.06}$	$1.53^{+ 0.82}_{-0.37}$
9475552	2694.01	15.0	$4818^{+ 142}_{-108}$	$4.48^{+ 0.09}_{-0.89}$	$0.86^{+ 1.90}_{-0.08}$	0.84338039	64.122461	$337^{+ 7}_{ -7}$	$1.253^{+ 0.026}_{-0.026}$	$4.58^{+ 0.49}_{-1.22}$	$1.89^{+ 3.60}_{-0.40}$
9873254	717.00	13.4	$5665^{+ 105}_{-127}$	$4.39^{+ 0.07}_{-0.17}$	$1.09^{+ 0.25}_{-0.09}$	0.90037004	53.519695	$34^{+ 3}_{ -3}$	$1.255^{+ 0.100}_{-0.099}$	$4.81^{+ 0.71}_{-1.26}$	$0.72^{+ 0.16}_{-0.11}$
9885417	3246.01	12.4	$4854^{+ 130}_{-107}$	$3.94^{+ 0.62}_{-0.43}$	$1.75^{+ 1.40}_{-0.96}$	0.68996781	53.595490	$113^{+ 2}_{ -2}$	$1.042^{+ 0.024}_{-0.023}$	$4.51^{+ 0.49}_{-1.21}$	$2.03^{+ 1.62}_{-1.14}$
9967771	1875.02	14.5	$5953^{+ 158}_{-175}$	$4.49^{+ 0.05}_{-0.28}$	$0.93^{+ 0.37}_{-0.09}$	0.53835457	64.648809	$185^{+ 5}_{ -5}$	$1.344^{+ 0.029}_{-0.029}$	$2.73^{+ 0.29}_{-0.73}$	$1.43^{+ 0.51}_{-0.24}$
10186945	4070.01	14.5	$5080^{+ 178}_{-137}$	$4.53^{+ 0.04}_{-0.13}$	$0.83^{+ 0.16}_{-0.06}$	0.39683023	64.395900	$170^{+ 7}_{ -7}$	$0.753^{+ 0.033}_{-0.033}$	$3.58^{+ 0.42}_{-0.95}$	$1.20^{+ 0.23}_{-0.17}$
12115188	2396.01	14.7	$5529^{+ 158}_{-142}$	$4.47^{+ 0.06}_{-0.26}$	$0.94^{+ 0.35}_{-0.09}$	0.49543428	64.477019	$281^{+ 6}_{ -6}$	$1.111^{+ 0.023}_{-0.026}$	$3.04^{+ 0.33}_{-0.81}$	$1.77^{+ 0.59}_{-0.29}$

Notes. Stellar parameters obtained from the compilation by Huber et al. (2014). Transit parameters based on an analysis of the folded transit light curve. The first 18 objects are new planet candidates. These objects either emerged from our search for the first time, or appear on the KOI list as false positives but that we consider to be viable candidates. After the horizontal line are the 69 KOIs that passed our false positive tests (Akeson et al. 2013). Finally, after the next horizontal line are 19 other candidates that were discovered by other authors (Ofir & Dreizler 2013; Huang et al. 2013; Jackson et al. 2013). Planets discovered transiting stars with other KOI planets with no KOI number assigned are labeled with the KOI of the star followed by a dot and two zeros.

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Figure 6 shows the radii and orbital periods of the 106 USP candidates. In this figure, we have identified the planets orbiting G and K dwarf stars (4100 K <T_eff < 6100 K, 4.0 < log g < 4.9), which represent the subset of stars where most of the planets are detected. One striking feature of this group is the relative scarcity of planets larger than 2 R_⊕. This finding cannot be easily attributed to a selection effect, because larger planets are easier to detect than small planets. This is in strong contrast to the planet population with periods in the range 2–100 days, for which planets with size 2–4 R_⊕ ("sub-Neptunes") are just as common as planets with radius 1–2 R_⊕ (Howard et al. 2012; Fressin et al. 2013).

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The explanation that comes immediately to mind is that gaseous planets are missing from the USP candidate list because such planets would have lost most of their gaseous envelopes to photoevaporation (Watson et al. 1981; Lammer et al. 2003; Baraffe et al. 2004; Murray-Clay et al. 2009; Valencia et al. 2010; Sanz-Forcada et al. 2011; Lopez et al. 2012; Lopez & Fortney 2013a; Owen & Wu 2013; Kurokawa & Kaltenegger 2013). Indeed, the levels of irradiation suffered by these planets (see Figure 6) are high enough to remove the entire envelope from a gaseous planet for a wide range of mass loss efficiencies and core masses (Lopez & Fortney 2013a). If that is the case, then the USPs may be a combination of rocky planets and formerly gaseous planets that lost their atmospheres. In fact, Lopez & Fortney (2013a) predicted that planets smaller than Neptunes with incident fluxes larger than a hundred times the Earth's incident flux would end their lives mostly as rocky planets smaller than 2 R_⊕.

In the observed radius distribution, 95% of the candidates have a radius smaller than 1.9 R_⊕. If we assume that the USP candidate list contains only rocky planets (because gas would have been evaporated), and that it represents a fair sampling of the full range of possible sizes of rocky planets, then this finding can be interpreted as a direct measurement of the maximum possible size of a rocky planet. The value of 1.9 R_⊕ is in agreement with a limiting size of 1.75 R_⊕ quoted by Lopez & Fortney (2013b) based on the complexity of forming planets larger than this limit with no significant H/He envelopes. Weiss & Marcy (2014) also note that planets smaller than 1.5 R_⊕ seem to have, on average, densities similar to that of Earth, which could be a sign that they are mostly rocky. It would be interesting to firm up our empirical determination of the limiting size of rocky planets after the estimates of the stellar radii have been improved using spectroscopy or other means.

There are also two USP candidates with implied planet radii larger than 2 R_⊕. They both belong to the group of six USP candidates that orbit stars hotter than 6250 K. The probability of this occurring by chance is only 0.3%, and it is intriguing that these hotter stars are distinguished from the cooler stars by the lack of a convective envelope (Pinsonneault et al. 2001). Perhaps the weaker tidal friction associated with the lack of a convective envelope has allowed these planets to survive tidal decay, and, furthermore, their core masses could be large enough that some of their gaseous envelopes have been retained (see Owen & Wu 2013 and references therein).

For the shortest-period planets, meaningful lower limits on the mean densities can be established from the mere requirement that they orbit outside of their Roche limiting distances. We have calculated these limiting densities for all the planet candidates with orbital periods shorter than 10 hr orbiting G and K dwarfs, using the simplified expression (Rappaport et al. 2013)

$$\begin{equation} \rho _{\rm p}\, [{\rm g\, cm}^{-3}] \ge \left(\frac{11.3\, {\rm hr}}{P_{\rm orb}}\right)^2, \end{equation} \tag{ 3 }$$

which is based on the assumption that the planet's central density is no more than twice the mean density. From this lower limit on the density, and the best-fit estimated planet radius, we were able to calculate a lower limit on the planet mass. The results are shown in Figure 6. Including the USP sample, it is now possible to place constraints on the mean density of about 10 additional terrestrial-sized planets. For a few of them, namely, those with orbital periods shorter than 5–6 hr, the minimum mean density is large enough that the planets are likely to be rocky. The caveats here are that we have not established individual false positive rates for most of these planets; and without spectroscopically determined properties for the host star, the uncertainties in the planet properties are large.

Our search procedure was designed mainly to detect a sizable sample of new USP candidates, and in this respect it was successful, having yielded 18 new candidates. However, the procedure was complicated enough that it is not straightforward to perform quantitative tests for completeness. At least there are good signs that the completeness is high, based on a comparison between the yield of our pipeline, and the overlap with the various other searches that have been conducted with the Kepler data (see Figure 3).

We have attempted to measure the completeness of the automated portion of our search by injecting simulated transits into real Kepler light curves and then determining whether they are detected by our pipeline analysis. A similar inject-and-recover test for longer-period planets was recently performed by Petigura et al. (2013). A more realistic simulation in which the signatures of background binaries and other false positive sources, as well as planet transits, are injected into pixel level data, was beyond the scope of our study.

To start, we selected a set of 105,300 Kepler target stars consisting of G dwarfs and K dwarfs with m_Kep < 16. A star was then chosen at random from this set, an orbital period was chosen from a distribution with equal probabilities in equal logarithmic intervals over the range 4 hr to 24 hr, a planet radius was chosen from the same type of distribution, but over the size range 0.84 R_⊕ to 4 R_⊕, and a transit impact parameter was chosen at random from the interval 0 to 1. Given the period and the stellar properties, the orbital radius could be estimated from the relation R_⋆/a = (3π/(GP²ρ_⋆)). The duration of each transit was then calculated using Equation (2), and the planet radius and stellar radius were used to compute the transit depth (R_p/R_⋆)². Simulated transit signals were then constructed using the trapezoidal model described in Section 4.2, which takes into account the 30 minute cadence of the observations, and injected into the mean-normalized PDC-MAP light curve for the selected star.

This procedure, starting with the random selection of a star, was repeated, on average, twice for each of the 105,000 stars in the list, and all of the resulting flux time series were processed with the part of the automated pipeline corresponding to the first four boxes in Figure 1 as used in our actual planet search. In brief, a three day moving-mean filter was applied to the light curve, and the FT was calculated. The planet was considered further if the correct frequency or an integral multiple of the correct frequency was detected in the Fourier spectrum. Given a detection, a filtered folded light curve was constructed, using the detected period, and the planet was rejected if the S/N of the transit dip was lower than 7 or if a sinusoid yielded a better fit than a transit profile. The final step was to remove those objects for which the transit signals have an S/N lower than 12 at the real orbital period (which may differ from the S/N at the detected orbital period).

We did not attempt to simulate the visual-inspection portions of the pipeline, as this would be prohibitively time-consuming (this issue is discussed further at the beginning of Section 6). The false positive tests also were not performed, since the transit shapes should be similar to the injected transit shapes.

The results of these numerical experiments are shown in Figure 7. The left portion of the figure shows the dependence of the completeness on period and planet radius. Evidently, the pipeline was able to detect nearly all planets larger than 2 R_⊕. The efficiency is also seen to generally rise at shorter periods. This may be simply explained, at least in part, by the fact that more transits occur for shorter-period planets over the duration of the observations, which usually results in a higher total S/N than that of longer-period planets of the same size and with the same type of host star.

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The right side of Figure 7 gives information relating to the efficiency of the pipeline as a function of the "fiducial S/N" of the transit signals. We calculated the fiducial S/N of the folded transit light curve based on the average over the available quarters of the CDPP noise levels that were reported in the headers of the FITS files (see Section 2.1; Howard et al. 2012) according to

$$\begin{equation} {\rm S/N}_{\rm CDPP} \equiv \frac{\delta _{\rm tran}}{\sigma _{\rm CDPP, 6\, hr}}\sqrt{\frac{T\,[{\rm hr}] (84.4\,q\, {\rm [days]})}{(6\,{\rm hr}) P_{\rm orb}{\rm \, [days]}}}, \end{equation} \tag{ 4 }$$

where q is the number of quarters of data, the factor 84.4 represents the average number of effective days of observations per quarter, and the factor of 6 hr is introduced because the CDPP is reported for a 6 hr timescale. This definition of fiducial S/N is convenient because the level of noise in the light curve depends on the filter used, but the CDPP noise is easily accessible through the Kepler FITS files. As seen in the upper right panel of Figure 7, our original S/N > 12 cut (see Section 3.2) does not perfectly translate into S/N_CDPP > 12. There is a small fraction (a few hundred) of simulated planets that were detected with S/N_CDPP < 12, and there were also some planets with S/N_CDPP > 12 that were missed by the pipeline. The efficiency increases from nearly zero at S/N_CDPP = 10 (due to our S/N > 12 cut) to for example 90% at S/N_CDPP = 25 (also in Figure 7).

This "ramp" in detection efficiency is reminiscent of the work of Fressin et al. (2013), who proposed that the detection efficiency of the Kepler transit search pipeline rises as a nearly linear function of S/N_CDPP from essentially zero at S/N_CDPP = 7 to approximately 100% at S/N_CDPP = 16. As one might have expected, our FT-based pipeline seems to be somewhat less efficient at detecting low-S/N planets than the Kepler pipeline. The likely reason is that the significance of the FT-based detection is spread over numerous harmonics, and since we used only two harmonics to detect signals, some of the lowest-S/N planets are not detectable. Evidence for this can be seen in the lower right panel of Figure 7, where the detection efficiency has been calculated for orbital periods shorter and longer than 12 hr. The pipeline detects many more low-S/N planets with short orbital periods, because their signals are spread over fewer harmonics. This may be the reason why most of our newly detected planet candidates have short orbital periods, while the candidates missed by our search and discovered by the Kepler pipeline tend to have longer orbital periods.

We now investigate the issue of false positives. One approach would be to pursue the validation or confirmation of all 106 objects, most definitively by direct mass measurement (as was done for Kepler-78; Howard et al. 2013; Pepe et al. 2013) or, somewhat less definitively, by the lack of ELVs and by statistical arguments (as was done for KOI-1843.03; Rappaport et al. 2013). We have found that a simpler argument is available for the USP candidates, based on the empirically high probability that USP planets are found in compact, coplanar, multi-planet systems.

It has already been shown that the false positive rate for a transit candidate that is found in association with other transit candidates of different periods around the same Kepler target is much lower than the false positive rate of a transit candidate that is found in isolation (Lissauer et al. 2012, 2014). The reason is that background binary stars (an important source of false positives) are expected to be spread randomly over the Kepler field of view, whereas the planet candidates are found to be clustered around a relatively smaller sample of stars. This same line of argument can be extended to the USP candidates. To be specific, in what follows, we will restrict our attention to candidates detected with our pipeline and with G- and K-type hosts. For convenience we will also define a "short-period planet" (SP planet) as a planet with a period in the range 1–50 days, as opposed to USP planets with P < 1 day.

There are 10 stars hosting USP candidates in our sample of G and K stars that also host at least one transiting SP candidate. The orbital periods and planet sizes for these 10 systems are displayed in Figure 8. These planet candidates are likely to be genuine planets, based on the statistical arguments by Lissauer et al. (2012). In order for this argument to work, false positives with orbital periods shorter than one day should be found less frequently orbiting multiplanet systems (Lissauer et al. 2014). In particular, if we focus only on the active KOI list (Akeson et al. 2013), we evaluate the fraction of false positives detected for G- and K-type hosts with P < 1 day as being 6 times lower for stars that have at least one active SP planet candidate than for those that do not. Since all of our 10 USP candidates accompanied by transiting SP candidates have also passed conservative false positive tests, we feel justified in assuming that they have a high probability of being genuine planets, which we will take to be ≃ 100%. Next, we show that the existence of these 10 systems implies the existence of a larger number of transiting USP planets which have non-transiting SP planets. This is because the geometric probability of transits decreases with orbital distance, and is lower for the SP planet than for the USP planet.

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The transit probability is given by R_*/a∝P^−2/3 for stars of the same size. We can derive an effective transit probability for a range of orbital periods by integrating P^−2/3 against the period distribution of planets, f(P). For the latter we utilize Equation (8) of Howard et al. (2012):

$$\begin{eqnarray} &&f(P) \propto P^{\beta -1} \left(1-e^{-(P/P_0)^\gamma }\right), \end{eqnarray} \tag{ 5 }$$

where, for their case which includes all planet sizes, P₀ ≃ 4.8 days, β ≃ 0.52 and γ ≃ 2.4 (Howard et al. 2012). We then compute the effective transit probability, $\mathcal {P}$

$$\begin{eqnarray} &&\mathcal {P}(P_1 \rightarrow P_2) \propto \left(\int _{P_1}^{P_2} f(P)\, P^{-2/3} \,{ dP} \left /\right. \int _{P_1}^{P_2} f(P) { dP}\right). \nonumber\\ \end{eqnarray} \tag{ 6 }$$

When we carry this integral out numerically for the ranges P = 1/6 → 1 day and P = 1 → 50 days, we find

$$\begin{eqnarray} &&\frac{\mathcal {P}(1/6 \rightarrow 1)}{\mathcal {P}(1 \rightarrow 50)} \simeq 6.9. \end{eqnarray} \tag{ 7 }$$

This indicates that the effective transit probability is ∼ 7 times higher in the USP range than in the SP range. If the distribution of SP planets is unaffected by the presence of a USP planet, this then implies that if we find $10\pm \sqrt{10}$ USP planets with transiting SP planets, there should be a total of 69 ± 22 USP planets with SP planet companions.⁶ We have found 69 USP candidates orbiting G and K dwarfs, which is fully consistent with the idea that essentially all USP planets are in systems with other SP planets, whether they are seen transiting or not. This also suggests that the number of false positives among the 69 USP planet candidates is likely to be quite low.

We do acknowledge that part of the above argument is somewhat circular since we have extrapolated the expression for f(P) from Howard et al. (2012), which was only derived for planets with P > 1 day and all sizes, into the USP range and for small planets. We do, however, find numerically that the result is rather insensitive to the choice of γ at least over the range 2 < γ < 5, γ being the parameter which largely dictates the slope in the short-period range. Moreover, it turns out that the Howard et al. (2012) extraplolated slope for f(P) is essentially the same for their smallest planets as compared with all planets. Thus, at a minimum, this very plausible period distribution function provides an interesting self-consistency check.

In summary, it is very clear that the 10 USP candidates with transiting SP companions imply the existence of a much larger number of USP candidates with non-transiting SP planets—and we indeed find them.

Interestingly, Schlaufman et al. (2010) predicted that a large family of terrestrial planets would be found with Kepler with periods shorter than one day, and that they would be accompanied by other planets in their systems. This was based on the fact that the dynamical interactions between planets can bring a close-in planet even closer to its star, where tidal dissipation can further shrink the orbit. However, it is unclear if planets with orbital periods as long as 50 days could have had any dynamical influence on the USP, particularly since these planets are also quite small (see Figure 8). What is clear, though, is that the USPs are very different from hot Jupiters, which have been found to have a very low probability of having additional companions with periods <50 days (Steffen et al. 2012).

We have argued that the list of 106 USP candidates has a low false positive rate, and that we have a good understanding of the completeness of our pipeline for a wide range of orbital periods and planet radii. We can now use the estimate of the completeness to estimate the occurrence rate (Petigura et al. 2013). Out of the 106 USP planet candidates, 97 were detected by our automatic pipeline (see Figure 1; the 97 candidates belong to the box with ∼3500 candidates), although 7 of them were dropped in the subsequent visual inspection of the light curves, giving a final list of 90 USP candidates. This low fraction of dropped planets (7 out of 97) gives us confidence that our visual inspection procedure is fairly robust. In order to compare our results with the simulation, we should minimize the effects caused by this visual inspection step on the final sample of planets, since this step was not taken into account in the injection/recovery simulations. We decided to use the 97 USP planet candidates in our group of 106 that were identified by our automatic pipeline prior to the visual inspection step, and acknowledge that a few additional planetary candidates could have been misclassified in this step.

We again concentrate on the 105,300 G and K dwarfs with m_Kep < 16, the same stars used in our simulation of multiplanet systems. There are a total of 69 USP planet candidates detected by our pipeline orbiting the G and K dwarfs with orbital periods ranging from 4 to 24 hr and radii ranging from 0.84 to 4 R_⊕ (see Figure 7). For each planet, we selected the 400 injected planets closest in orbital period and planet radius, where the distance to the ith detected planet with coordinates [R_i, P_i] is calculated with the following expression:

$$\begin{equation} d^2([R, P], [R_i, P_i]) = \left(\frac{(P-P_i)}{P_i}\right)^2 + \left(\frac{(R-R_i)}{R_i}\right)^2. \end{equation} \tag{ 8 }$$

This formula ensures that the positions of the 400 simulated planets are selected from a circle in log-log space. We calculated the local completeness rate C_i as the fraction of planets detected among those 400 simulated planets. The contribution of each planet to the final occurrence rate is f_i ≡ (a/R_⋆)/(C_iN_⋆), where N_⋆ is the total number of stars searched for planets and the factor a/R_⋆ is the inverse transit probability. Given a range of orbital periods and planet radii, we sum all the contributions from all n_pl planets detected in that range to estimate the final occurrence rate f. Following the recipe given by Howard et al. (2012), we then define the effective number of stars searched as $n_{\rm \star, {\rm eff}} \equiv n_{\rm pl}/f$ and compute the uncertainties in f from the 15.9 and 84.1 percent levels in the cumulative binomial distribution that results from drawing n_pl planets from $n_{\rm *, {\rm eff}}$ stars.

Figure 9 shows the resulting occurrence rates for four logarithmic intervals in the orbital period (marginalized over radius) and for nine logarithmic intervals in the planet radius (marginalized over period). The occurrence rate increases with period, and is consistent with a power law, as had already been seen for longer orbital periods (Howard et al. 2012; Fressin et al. 2013; Petigura et al. 2013). As a function of radius, the occurrence rate changes by less than a factor of ∼2 over the interval 0.84–1.68 R_⊕, and drops sharply for larger radii. At ultra-short orbital periods, Earth-size planets are more common around these stars than sub-Neptunes or any other type of larger planets. Integrating over all periods and sizes, we find that there are 5.5 ± 0.5 planets per thousand stars (G and K dwarfs), with orbital periods in the range 4–24 hr and radii larger than 0.84 R_⊕.

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As mentioned in Section 4.3, the lack of planets larger than 2 R_⊕ may be interpreted as a consequence of the extremely strong illumination in the tight USP orbits (see, e.g., Owen & Wu 2013 and references therein). We can compare our results to occurrence rates calculated at longer orbital periods, to try to detect trends that might support or refute this interpretation. We use the occurrence rates measured by Fressin et al. (2013) for planets as small as Earth and for periods of 0.8–85 days. To allow a direct comparison, we divided our 50 planet candidates with orbital periods shorter than 0.8 days into three groups according to size: Earths, super-Earths and sub-Neptunes (see Figure 10), and calculated the occurrence rate for each group. The results are shown in Figure 10.

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A lack of sub-Neptunes is evident in Figure 10 at the shortest orbital periods; in fact, sub-Neptunes are significantly less common than super-Earths for orbital periods shorter than 5 days and possibly even up to 10 days. This absence of sub-Neptunes had been predicted by different simulations of mass loss of close-in planets (Baraffe et al. 2004; Valencia et al. 2010; Lopez & Fortney 2013a; Owen & Wu 2013). There also seems to be a modest excess of USP Earth-size planets relative to the occurrence rate of super-Earths, which could very well be the cores of the transmuted sub-Neptunes. A more detailed statistical analysis is required to assess the significance of this excess, and it would be convenient to update the occurrence rate at longer orbital periods with new data.

It is still possible that any mechanism that caused these planets to have such tight orbits is less efficient for sub-Neptunes than for smaller planets. Obtaining the masses of some of these planets in order to measure their densities and compositions would provide some helpful context. Obtaining precise stellar radii (via spectroscopy) and hence planet radii for a uniform sample of close-in planets, could reveal additional signatures of photoevaporation.

We have also investigated the dependence of the occurrence rate on the type of star, to the extent that this is possible with the Kepler data. If ultra-short period planets had an occurrence rate independent of stellar temperature, we would expect to find most planets orbiting Sun-like stars (with T_eff ≈ 5800 K), since these stars are the most common type of star in the Kepler target catalog, and they dominate the number of USP-searchable stars. However, the host stars of the detected USP candidates tend to be cooler, with temperatures closer to T_eff ≈ 5000 K (such as Kepler-78b). This might be a clue that planets with orbital periods shorter than one day are more common around cooler stars. In order to test this, we first calculate the occurrence rate for the hotter stars in our sample (G dwarfs, 5100–6100 K). Using all such stars with m_kep < 16, we have 48 USP candidates around 89,000 stars. Thus the occurrence rate for such planets is 5.1 ± 0.7 planets for every thousand stars. The same calculation for the cooler stars (K dwarfs, 4100–5100 K) yields 8.3 ± 1.8 planets per thousand stars, based on 21 USP candidates orbiting 16,200 stars. The difference between the occurrence rates for these two spectral classes is only marginally significant (at the 1.7σ level).

It would be interesting if USP planets are more common around K dwarfs than G dwarfs. A similar temperature trend in the occurrence rate of planets with 1–50 day periods was found by Howard et al. (2012), although this finding has been challenged by Fressin et al. (2013) based on the statistics of planets with even longer orbital periods. This collection of results suggests that the dependence of occurrence rate on the type of star is most pronounced for the shortest period planets. Indeed, since planet formation depends on the temperature of the materials in the protoplanetary disk, and since the protoplanetary disks around cooler stars are cooler at a given orbital distance, one could imagine scenarios in which cooler stars have a higher abundance of short-period planets. In this case, it will be worth extending our study to slightly longer orbital periods to find the transition point at which K and G dwarf stars have the same number of planets, and also to extend this study to include stars of spectral types F and M.

Even though our sample of USP planets transiting F and M dwarfs is somewhat limited, we can use it to put very useful bounds on the occurrence rate of such planets. For that we repeated the completeness calculation using F and M hosts, and repeated all the steps of the computation. For F dwarfs (6100–7100 K), we find 9 planet candidates within a sample of 48,000 stars, giving an occurrence rate of 1.5 ± 0.5 planets per thousand stars. For M dwarfs (3100–4100 K), we find 6 planet candidates out of a sample of 3537 stars, yielding a rate of 11 ± 4 planets per thousand stars. This represents further evidence that cooler stars are more likely to host USP planets. If we now combine the USP planet occurrence rates for all four spectral classes of host star (M, K, G, and F) the decreasing trend in occurrence rate with increasing T_eff becomes quite significant. These results are summarized in Table 3.