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DETECTION OF POTENTIAL TRANSIT SIGNALS IN THE FIRST 12 QUARTERS OF KEPLER MISSION DATA

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Published 2013 April 23 © 2013. The American Astronomical Society. All rights reserved.
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0067-0049/206/1/5

Abstract

We present the results of a search for potential transit signals in the first three years of photometry data acquired by the Kepler mission. The targets of the search include 112,321 targets that were observed over the full interval and an additional 79,992 targets that were observed for a subset of the full interval. From this set of targets we find a total of 11,087 targets that contain at least one signal that meets the Kepler detection criteria: periodicity of the signal, an acceptable signal-to-noise ratio, and three tests that reject false positives. Each target containing at least one detected signal is then searched repeatedly for additional signals, which represent multi-planet systems of transiting planets. When targets with multiple detections are considered, a total of 18,406 potential transiting planet signals are found in the Kepler mission data set. The detected signals are dominated by events with relatively low signal-to-noise ratios and by events with relatively short periods. The distribution of estimated transit depths appears to peak in the range between 20 and 30 parts per million, with a few detections down to fewer than 10 parts per million. The detections exhibit signal-to-noise ratios from 7.1σ, which is the lower cutoff for detections, to over 10,000σ, and periods ranging from 0.5 days, which is the shortest period searched, to 525 days, which is the upper limit of achievable periods given the length of the data set and the requirement that all detections include at least three transits. The detected signals are compared to a set of known transit events in the Kepler field of view, many of which were identified by alternative methods; the comparison shows that the current search recovery rate for targets with known transit events is 98.3%.

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1. INTRODUCTION

We have previously reported (Tenenbaum et al. 2012) on the results of searching the first 218 days of Kepler mission (Borucki et al. 2010) data for potential signals indicative of transiting planets. In the intervening time, there have been two developments in the search for potential exoplanets in the Kepler data set. First, the algorithms used in the Kepler analysis pipeline have undergone dramatic improvements. Second, the data available for searching have expanded from 218 days to 1050.5 days. This massive increase in data volume makes possible searches for exoplanets with much longer orbital periods, as well as searches for extremely small exoplanets with relatively short-period orbits. In this study, we report on the results of searching the current set of Kepler observations with the upgraded analysis pipeline. This study can be considered an update of the previous report (Tenenbaum et al. 2012).

1.1. Kepler Science Data

The operational parameters of the Kepler mission have been extensively reported (Haas et al. 2010). In brief, the Kepler spacecraft is in an Earth-trailing heliocentric orbit with a 372 day period. Its single instrument, the Kepler photometer, points almost constantly at a 115 deg2 region of the sky centered on α = 19h22m40s, δ = +44fdg5. During science operations, photometric data are taken in 29.4 minute integrations, known within Kepler as "long cadences" (as distinguished from "short cadences," which are 1/30 of a "long cadence" and are collected for a small subset of targets). In order to maintain the correct orientation of the solar panels and thermal radiator, the spacecraft rotates about the photometer boresight axis by 90° approximately every 93 days, the interval at a given orientation being referred to as a "quarter." A consequence of this rotation is that each target star is observed each year on four different readout channels on the focal plane. Science acquisition is interrupted for monthly downlinking of pixel data, maneuvering from one quarter's attitude to the next, reaction wheel desaturation (one 29.4 minute sample is lost for this purpose approximately every three days), and a variety of spacecraft anomalies.

The data acquisition period for this analysis begins at 2009 May 12 00:00:00 UTC, ends at 2012 March 28 12:47:26 UTC, and contains 51,412 sample intervals of 29.4 minutes. Of these, 47,588 intervals are dedicated to science data acquisition, the balance of 3824 intervals being consumed by the interruptions listed above. During this period the spacecraft performed 11 axial rotations, resulting in 12 quarters of data.

A total of 192,313 targets were observed by Kepler during the 12 quarters of data acquisition and were subsequently searched for indications of transiting planets. Of those, 112,321 were observed in all 12 quarters; the balance of 79,992 was observed only in a subset of quarters. Figure 1 shows the distribution of targets according to the number of quarters observed. Observation of a target in a subset of quarters can occur for any of three reasons. The most significant cause of limited observation is an onboard electronics failure which occurred on 2010 January 23, one month into Quarter 4; this failure resulted in the subsequent loss of all data from 4 of the 84 CCD readouts on the focal plane (specifically, the four CCD readouts in Module 3). Due to the quarterly rotation of the spacecraft, this failure produced a "blind spot" in the Kepler field of view that moves relative to the target stars, causing a large number of targets to be visible only 75% of the time. Any target that falls onto Module 3 was only observed in 10 out of 12 quarters. The 28,965 stars that were observed for 10 quarters as shown in Figure 1 are mainly due to this effect. A second limitation on the number of quarters for which a target is observed is that the process of target selection and prioritization has evolved over the life of the Kepler mission; targets that are added or removed subsequent to Quarter 1 will not be observed during all quarters. Additionally, a fraction of Kepler's observing capacity is reserved for use by the Kepler Guest Observer and Asteroseismic Science Consortium (KASC) programs; the targets observed in these programs are frequently updated, resulting in a number of targets observed for relatively short intervals. Finally, due to small asymmetries in the construction of the focal plane, a small number of targets cannot be observed in all spacecraft orientations; in some quarters these targets are imaged onto one or another CCD detector, while in some quarters the target images fall between the detectors. In total, 28,826 targets were observed in 8 or fewer quarters; 43,339 targets were observed for 9 or 10 quarters; and 7819 targets were observed for 11 quarters.

Figure 1.

Figure 1. Histogram of number of quarters of observation for all targets. The significant number of targets observed for 10 quarters out of 12 is primarily due to an onboard electronics failure which prevents readout from 4 out of the 84 CCD modules on the focal plane, resulting in a "blind spot" that rotates through the field of view as Kepler rotates about its axis.

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In addition to the aforementioned 192,313 targets that were searched for planets, a total of 2123 known eclipsing binaries were observed but not searched for transiting planet signatures. This was done for operational reasons. The Kepler processing pipeline has limited capacity to identify circumbinary planets because their transit signatures are generally neither periodic nor of constant duration. However, the eclipses of an eclipsing binary system mimic planetary transits with sufficient fidelity to be identified by a transiting planet search (TPS) as potential signals of transiting planets. These known eclipsing binaries were removed to reduce the computational and human burden which would otherwise have been imposed by their false-positive detections.

1.2. Pre-search Processing

The processing of pixel data from the Kepler spacecraft, prior to the search for transiting planet signatures, is summarized elsewhere (Jenkins et al. 2010a). The processing step that has seen the most dramatic change is pre-search data conditioning (PDC). The purpose of PDC is to remove variations in the flux time series that are generated by changes in the spacecraft environment or other systematic effects. The original PDC algorithm determined the systematics by performing a robust least-squares fit of assorted spacecraft engineering variables to each flux time series, and then subtracting the systematics thus determined to yield a conditioned flux time series (Twicken et al. 2010). While such an approach is guaranteed to reduce the bulk rms variation of each target's flux, it can also distort the true stellar variations and can even add variability on timescales of interest for planet searches. Both of these unwanted side effects are driven by the same source: the least-squares fit is removing variability which is coincidentally correlated with some engineering variable, but not causally related.

This unwanted behavior is corrected by applying a Bayesian approach to constrain the fitted amplitudes of systematic error terms which are then removed from the light curves. This process allows the algorithm to deduce "reasonable" values for the correlation of each identified systematic to the light curves, and thus to reject correlations that are wildly out of family. Additionally, the ensemble of target star data across a large number of stars is used to empirically identify systematic trends in the light curves rather than relying upon the available spacecraft engineering data. The algorithm is fully described elsewhere (Smith et al. 2012; M. Stumpe et al. 2013, in preparation).

In addition to the corrections described above, the current PDC algorithm identifies and corrects the signature of a cosmic-ray-related artifact known as a sudden pixel sensitivity dropout (SPSD). An SPSD occurs when a cosmic ray produces a step reduction in the quantum efficiency of a pixel; the reduction is typically of order 1%, and the quantum efficiency partially recovers, typically over a period of hours to days. Because an SPSD bears a superficial resemblance to a transit signature (at least to a computer), efficient removal of SPSDs without inadvertent removal of actual transits is a crucial step in data conditioning for Kepler. Unlike environmental signatures, SPSDs are completely uncorrelated from one target star to another, and thus are removed from the data via a separate algorithm within PDC.

2. TRANSITING PLANET SEARCH

The TPS algorithm is described in some detail in Jenkins (2002) and Jenkins et al. (2010b), as well as Tenenbaum et al. (2012). The improvements in the algorithm since Tenenbaum et al. (2012) are summarized below.

2.1. Edge Detrending of Contiguous Blocks of Flight Data

The algorithm that was previously used to remove trends at the ends of single-quarter data segments was replaced with an algorithm that performs a robust fit of the form:

Equation (1)

where y is the median-corrected flux, x is the sample time normalized to a range from 0 to 1, and P1 through P6 are the parameters of the fit. In words, Equation (1) fits a line plus two exponential edge trends, one at the leading edge of the data region and one at the trailing region, with both the amplitude and the time constant of the exponentials as fit parameters. The form in Equation (1) was found to match the actual edge trends as well as the constrained polynomial fit which had previously been used. The advantages of the reformulated edge-trend removal are: a reduced number of assumptions and/or configuration parameters for the fit, use of the full data segment for the entire fit, robust fitting, and the fact that the new fit does not under any circumstances introduce a polynomial "wave" into the data segment in an attempt to correct the edges (i.e., overfitting). Additionally, whereas in the past the edge detrending was applied only to full quarters of data, in the current implementation it is applied at any time when there was an interruption of data acquisition to change the spacecraft orientation. This was done to mitigate the thermal transients that occur when the spacecraft attitude is changed. Attitude change incidents include all data downlink intervals, plus any transitions into or out of safe mode.

2.2. Detection and Vetoing of Potential Signals

The first step in detection of potential signals is described in Section 2 of Tenenbaum et al. (2012): a wavelet-based, adaptive matched filter is utilized to search for periodic reductions in flux occurring against the non-white, non-stationary background of stellar variability. The significance of such a reduction is known as its multiple event statistic. The multiple event statistic is computed across a two-dimensional grid of signal period and epoch of first transit, and across 14 trial transit pulse durations; the maximum multiple event statistic from this set is captured, along with the combination of period, epoch, and transit pulse duration (henceforth "signal timing") which generated it. A threshold is then applied to the maximum multiple event statistic to reject targets that are unlikely to contain a true transiting planet signature. The threshold value represents a balance between rejecting true positives in the event of an excessively high threshold versus accepting false positives in the event of an excessively low threshold. This balance was extensively explored prior to Kepler launch (Jenkins 2002). Based on these studies, a threshold of 7.1σ was adopted for the multiple event statistic. At this threshold, the probability of detecting an Earth-sized planet that produces 4 transits of a 12th magnitude Sun-like star is approximately 80%; the expected false alarm probability from statistical fluctuations is at the level of 1 false alarm per 600,000 target years of observations, which translates to 1 false alarm detection during the entirety of the nominal Kepler mission. Target stars for which the maximum multiple event statistic falls below the specified detection threshold of 7.1σ are rejected from further analysis. The requirement that the maximum multiple event statistic exceed 7.1σ removes from further consideration 76,668 targets, leaving 115,645 with at least one potential transit signal that lies above this threshold.

The principal weakness of the multiple event statistic calculation is that it cannot discriminate between a true train of transit events (which have uniform depth, duration, and shape to within the precision limits of the instrument) and a chance combination of dissimilar events that coincidentally occur within a flux time series. As an example, consider a flux time series for which the combined differential photometric precision for transit detection is 50 parts per million (PPM) at all times (Christiansen et al. 2012). If the flux time series contains four uniformly spaced transits with uniform depths of 250 ppm, the resulting multiple event statistic for that period and epoch will be 10σ, and will be reported as an above-threshold event by the multiple event statistic calculation. On the other hand, if the four transits are uniformly spaced but do not have uniform depth—for example, if the depths of the four transits are 20 ppm, 30 ppm, 50 ppm, and 900 ppm, respectively—the multiple event statistic for this combination of events will also be 10σ, and will also be reported as an above-threshold event by the multiple event statistic calculation. While the former scenario might be the signature of a transiting planet, the latter clearly is not. Thus, a multiple event statistic that is above the detection threshold is a necessary but not sufficient condition for identifying a potential transiting planet signature. More generally, while the matched filter approach is optimal with respect to rejecting the null hypothesis, it is insufficient for discrimination between competing alternate models. For this reason, once a multiple event statistic above the detection threshold is identified, the event thus detected is subjected to a series of tests that are designed to discriminate between potential transit signatures and heterogeneous combinations of unrelated events. These tests accept the former while vetoing the latter.

2.2.1. Robust Statistic Veto of False Positive Detections

As described above, detections due to transiting planets and false alarm detections can be separated from one another by the requirement that the transits are periodic, of equal duration, and of uniform depth. The multiple event statistic calculation strongly enforces the requirement of periodicity and weakly enforces the requirement of uniform duration, but as described above does not enforce the requirement of uniform depth. The robust statistic veto is complementary to the multiple event statistic test in that it tests each detection for uniformity of transit depth. This is accomplished by constructing a model flux time series with transits in which the transits are represented by square pulses that have the epoch, period, and duration dictated by the signal timing of the multiple event statistic. The model flux time series is fitted to the data, with the transit depth being the only free parameter in the fit. In order to eliminate the effect of stellar variations, both the flux time series and the model transit pulse train are whitened, as described in Jenkins et al. (2010b). A robust fit is utilized in order to reduce the influence of out-of-family samples in the flux values that participate in the fit. The robust statistic, which is the signal-to-noise ratio estimated from the fit, is then used to reject false positives. For a more complete description of the robust statistic, see Appendix A. Specifically, a large value of the robust statistic indicates a detection in which the transits are reasonably uniform in depth and duration, which is characteristic of true transit signatures; a small value indicates that the multiple event statistic has been formed from a combination of heterogeneous transit-like events with unequal depths, which is characteristic of false positives. The robust statistic threshold was selected using the results of an earlier TPS exercise with fewer quarters of data: the robust statistics for targets known to have true-positive transiting planets were compared to those for other stars with multiple event statistics above the threshold of 7.1σ. The value of 6.4σ caused 98% of the former to be accepted, while rejecting 66% of the latter. Increasing the threshold above this value caused an unacceptable number of known true-positive detections to be rejected. Note that this method of tuning the robust statistic threshold implicitly assumes that the latter set of detections is so dominated by false alarms and contains so few true positive detections that it is safe to treat the set as being entirely false alarms; given that over 100,000 target stars produced multiple event statistics in excess of 7.1σ, this seems a safe assumption. In the current TPS run, a threshold of 6.4σ for the robust statistic rejects 79,030 targets, leaving 36,614 targets that require further scrutiny.

2.2.2. χ2 Veto of False Positive Detections

In the second test used for vetoing of false positives, the signal that produced the multiple event statistic is decomposed in two different ways, namely, first into its wavelet scale contributions for each transit and second into its temporal contributions. For a true transit event with the period, epoch, transit duration, and multiple event statistic of the detected signal, and assuming uniform transit depths, it is possible to compute the expected values in each of these decompositions. As shown briefly in Appendix B, and more thoroughly in Seader et al. (2013), the expected component values for each transit are compared to observed values in the construction of two functions, each of which is expected to be distributed according to a χ2 distribution. These functions are then combined with the multiple event statistic of the potential signal, as shown in Appendix B. By requiring that the values of the two resulting discriminators, X(1) and X(2), both exceed 7.0, we veto an additional 25,506 targets, yielding 11,108 targets which contain potential transiting planet signatures. Note that these thresholds were tuned empirically in a manner identical to that used to tune the robust statistic described above. An event that has passed all four tests—multiple event statistic, robust statistic, and χ2 discriminators—is referred to as a threshold crossing event (TCE).

2.3. Iterative Rejection of False Positives and Re-searching of the Flux Time Series

Prior versions of TPS suffered from a significant design weakness: in cases in which the strongest transit-like feature was vetoed, the search of that target would terminate. In this way a strong but low-quality transit-like signal could inadvertently mask a weaker but higher-quality event. This flaw is addressed in the current version of TPS: in the event that an apparent transit is vetoed, TPS goes on to search for additional transit signatures in the same light curve. Because the search of additional periods and epochs can potentially be extremely time consuming, for operational purposes it is necessary to limit the number of iterations of searching which are permitted for a given target and a given trial transit pulse duration. At present the limit is set to 1000 iterations of re-searching. In the analysis reported here, approximately three quarters of all TCEs occurred on the first iteration of the search with the balance TCEs detected on subsequent iterations. The largest number of iterations required to detect a TCE was 404.

2.4. Removal of Non-periodic Transit-like Features

The benefits of the multiple iterations of search, described above, can only be fully exploited in the absence of relatively strong non-astrophysical single events. Such strong events will cause the multiple event statistic to exceed the 7.1σ threshold for large numbers of possible periods: folding a single strong event with a small number of weak events will produce a large multiple event statistic, and there are an extremely large number of period–epoch combinations which will result in such a folding. If this happens, the 1000 iterations of searching can easily be exhausted in the process of eliminating a fraction of the spurious multiple event statistics caused by a single strong event. Such an outcome can be avoided if these strong events are identified and removed prior to folding, but such removals are obviously dangerous: without prior knowledge, a feature in the data that is identified as a non-astrophysical event, and removed, could actually be a strong transit. For this reason, any event removal must be used sparingly. TPS addresses this issue in two ways. First, a minimum number of transits is required for an event to be accepted since the probability of such chance combinations yielding a multiple event statistic over threshold decreases as the number of events folded together increases. At present, the threshold number of transits is 3. Second, the current version of TPS is permitted to remove one, and only one, single event, and only in the case in which the first iteration of planet searching produces a strongest event that exceeds the multiple event statistic threshold of 7.1σ but which is then vetoed by RS, X(1), or X(2). In such a case the strongest single event in the time series is removed, if and only if the strongest single event has an amplitude that is greater than the multiple event statistic threshold multiplied by the square root of the minimum number of transits (7.1 $\sigma \times \sqrt{3}$, or 12.3σ for the current parameter choices). Out of the 11,108 TCEs, 2193 are found on light curves that have had such a feature removed. Additionally, on each target the number of such identifiable features is counted and recorded, regardless of whether any such events are removed. Out of all 192,313 targets, the number that have at least one identifiable strong single event is 46,481. Figure 2 shows the distribution of the number of strong events for targets that have at least one such event. Note that the distribution is strongly peaked toward small numbers of events, implying that it is worth considering the option of using more aggressive removal of features in future TPS runs.

Figure 2.

Figure 2. Distribution of the number of strong features in each flux time series, as defined in the text. The final bin includes overflows: there are a total of 327 targets with 10 features and 4650 with more than 10 features.

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2.5. Limitation on Allowable Transit Duty Cycles

During development of the most recent version of TPS, it was observed that a substantial number of false positives were produced with a short period and a long trial transit pulse duration, which implied that applying a threshold to the ratio of trial transit pulse duration to period (henceforth known as the "transit duty cycle") would allow suppression of a large number of false positive detections. The transit duty cycle for a central transit of the Sun by the Earth is approximately 7.4 × 10−4, which implies that for Earth-analogues the threshold could be set to an extremely low value; however, for a given star, the transit duty cycle is inversely proportional to the semi-major axis of the transiting body's orbit, and thus setting a low threshold for the transit duty cycle will implicitly eliminate sensitivity to short-period planets. For a solar-type star, the minimum TPS search period of 0.5 days would lead to a transit duty cycle of 0.092 for a circular orbit; the TPS duty cycle should therefore be somewhat larger than this value in order to preserve sensitivity to 0.5 day orbits on larger stars and to allow some margin for eccentric orbits. Given these considerations, for the processing run reported on here we limited the transit duty cycle to values below 0.16.

2.6. Detection of Multiple Planet Systems

In Wu et al. (2010), the process for detection of multiple planet systems is described. In brief, for each target star that yields a valid detection as described above, a planet model is fit to the flux time series, using the period and epoch of the TCE as a starting point for the fit; the transit signatures from the fitted planet model are removed from the flux time series; and the residual flux time series is then searched for additional TCEs. The subsequent TCE search is performed using the same TPS algorithm as is used for the initial search. When multiple planet detections are included, the total number of TCEs increases to 18,427.

Following the detection and model fitting described above, an additional set of automated analyses are performed which allow astrophysical false positives, such as background eclipsing binaries, to be ruled out. For the purposes of the discussion below, we will consider only the TCEs for which the additional automated analyses were successfully completed: this set includes 18,406 TCEs falling on 11,087 targets. Of the 21 excluded targets, 19 are non-stellar "super-aperture" targets, for which the automated post-detection analyses cannot be performed, while 2 are conventional Kepler targets for which the automated post-detection analyses failed due to software errors. Each of the excluded targets produced a single TCE.

3. DETECTED SIGNALS OF POTENTIAL TRANSITING PLANETS

Figure 3 shows the epoch and period of the 18,406 detections, with period in days and epoch in Kepler-modified Julian date (KJD), which is the Julian date − 2,454,833.0. While Figure 3 is relatively free of obvious artifacts, there is an evident overabundance of detections at periods of approximately 1 yr. Figure 4 shows the distribution of periods from Figure 3; the overabundance is even clearer here, with 2042 TCEs with periods between 300 and 400 days as compared to 305 TCEs with periods of 200–300 days and 168 TCEs with periods of 400–500 days.

Figure 3.

Figure 3. Epoch and period of the 18,427 TCEs detected in the twelve-quarter TPS run. Periods are in days, epochs are in Kepler-modified Julian date (KJD); see the text for definition.

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Figure 4.

Figure 4. Distribution of TCE periods. The excess of detections at periods close to 1 yr is due to the rotation of a small number of image artifact channels about the focal plane as Kepler rotates about its axis.

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Figure 5 shows the participation of the various detector channels on the Kepler focal plane in TCEs with periods between 300 and 400 days: each sub-image shows one quarter, and the relative intensity of each channel represents the participation, of that channel in that quarter, in the 2042 TCEs. A small number of channels are disproportionately involved in these TCEs, mainly channels that are known to suffer from excess noise due to issues in the read-out electronics (Gilliland et al. 2011; Caldwell et al. 2012). As Kepler rotates each quarter, certain stars will typically be imaged onto one of these misbehaving channels once per year; this will result in detections on those stars with periods of approximately 1 yr. Efforts to manage the excess noise of these channels in Kepler data processing are ongoing.

Figure 5.

Figure 5. Participation of Kepler output channels in TCEs with periods between 300 and 400 days. The sub-plots are all oriented such that modules 2, 3, and 4 are at the top. Each column of sub-plots represents a common roll orientation of the spacecraft (i.e., quarters 1, 5, and 9 all correspond to the same orientation of the spacecraft). The strongest contributions come from Module 17, Output 2, which is known to exhibit temperature-dependent noise artifacts. Other strong contributors shown are Module 9, output 2; Module 13, output 4; and Module 18, output 2. All of these channels are also known to exhibit unusually elevated noise, though not at the level of Module 17, output 2. Note that the pattern and intensity of misbehaving channels repeat annually, giving further evidence that the misbehaviors are driven by the spacecraft thermal environment, which itself repeats annually due to the quarterly change in roll orientations.

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Figure 6 shows the distribution of detections in the plane of orbital period and multiple event statistics. Note that the overabundance of detections at one year is completely dominated by relatively weak signals. Figure 7 shows the distribution of multiple event statistics: on the left is the distribution of 17,547 detections with multiple event statistics less than or equal to 100σ; the right panel shows the same but for the 15,007 detections with multiple event statistics less than or equal to 20σ. Figure 8 shows the distribution of detected periods: on the left are the 5043 detections with periods over 15 days, and on the right are the 13,363 detections with periods less than 15 days. As compared to Figure 6 in Tenenbaum et al. (2012), the right side of Figure 8 is far more strongly peaked toward short periods. Note that, in addition to the excess of detections with periods close to 1 yr, there is a smaller excess of detections with periods of 0.5 yr. This peak is caused by the presence of two high-noise channels that are located symmetrically opposite one another on the focal plane, specifically Module 17, Output 2 and Module 9, Output 2; stars that are imaged onto one of these channels will be imaged onto the other 6 months later.

Figure 6.

Figure 6. Distribution of TCE periods and multiple event statistics.

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Figure 7.

Figure 7. Distribution of multiple event statistics. Left: 17,568 TCEs with multiple event statistic of 100 or lower. Right: 15,018 TCEs with multiple event statistic of 20 or lower.

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Figure 8.

Figure 8. Distribution of periods. Left: 5045 TCEs with periods greater than 15 days, with the data anomaly-driven excess at approximately 1 yr clearly visible. Right: 13,382 TCEs with periods less than 15 days.

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Figure 9 shows the distribution in estimated transit depths. These depths are estimated from the event statistics and the noise properties of each light curve as described in Tenenbaum et al. (2012). The top plot shows the 16,095 signals that have estimated depths of 1000 parts per million (ppm) or less; the bottom plot shows the 8060 cases with estimated depths of 100 ppm or less. These sub-distributions contain 87.4% and 43.8%, respectively, of all the detections in this data set. Compared to the same transit depth ranges in Tenenbaum et al. (2012), we find that in the processing of the first 3 quarters of data the totals were 72.3% and 13.2%, respectively. This increased sensitivity to weaker transits is driven in the main by the vastly increased amount of data collected since the end of Quarter 3.

Figure 9.

Figure 9. Distribution of estimated transit depths. Top: 16,115 signals with estimated depth of 1000 parts per million (ppm) or less; bottom: 8068 signals with 100 ppm or less.

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Figure 10 shows the distribution of transit duty cycles for all detections, where the transit duty cycle is defined to be the ratio of the trial transit pulse duration to the detected period of the transit (effectively, the fraction of the time during which the TCE is in transit). The top plot shows all 18,406 TCEs, while the bottom plot shows the 7721 TCEs with transit duty cycle below 0.04. Figure 11 shows the relationship between period and transit duty cycle for all 18,406 detections. As expected, the relationship is quantized due to the quantization of trial transit pulse durations utilized in the TPS detection algorithm, and as a consequence of this quantization the period and transit duty cycle are inversely proportional for a given trial transit pulse duration. Figure 11 also demonstrates why there is an abundance of events with transit duty cycles of approximately 0.002 shown in Figure 10: this is actually a reflection of the abundance of events with periods near 1 yr, for which the possible transit duty cycles are all in the realm of 1 × 10−4 to 0.002.

Figure 10.

Figure 10. Distribution of transit duty cycles. Top: all TCEs; bottom: 7729 TCEs with transit duty cycle below 0.04.

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Figure 11.

Figure 11. Relationship between period and transit duty cycle for all TCEs. The structure observed is driven by the fact that TPS uses a small number of fixed trial transit pulse durations for its searches, and by the fact that at a given trial transit pulse duration the transit duty cycle is inversely proportional to the TCE period.

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Detailed information on all TCEs that contributed to this analysis can be found at the NASA Exoplanet Archive: http://exoplanetarchive.ipac.caltech.edu/cgi-bin/ExoTables/nph-exotbls?dataset=tce.

3.1. Comparison with Known Kepler Objects of Interest (KOIs)

In order to gauge the performance of TPS as a detector of periodic transit-like phenomena, it is necessary to compare the set of TCEs to a set of known events which can function as a "ground truth." For this purpose, we use the list of Kepler Objects of Interest (KOIs). Out of the current set of KOIs (C. J. Burke et al. 2013, in preparation), we have selected 2630 KOIs that are judged reasonable for comparison to the TCE list; these are KOIs for which the signal-to-noise ratio is high enough to permit detection in TPS, the number of transits which fall within the 12 quarters of Kepler data is 3 or more, and which do not fall on targets that were excluded from TPS processing. The selected set of KOIs includes planet candidates, known astrophysical false positives (mainly eclipsing binaries and background eclipsing binaries), and objects that have not yet been characterized as planetary or non-planetary; for the purpose of the comparison, it is sufficient that each KOI be reasonably expected to produce a TCE.

The comparison of the KOI and TCE lists is complicated by the fact that any target star can have multiple KOIs and/or multiple TCEs, and the multiplicities of the two are obviously not guaranteed to agree. As a first step, we compared the number of TCEs on each KOI target star with the number of KOIs on those stars. The result of this comparison is as follows.

  • 1.  
    A total of 31 KOIs do not have a corresponding TCE.
  • 2.  
    The remaining 2599 KOIs were matched one-for-one by TCEs that occurred on the same target stars.
  • 3.  
    337 KOI target stars produced more TCEs than their known KOIs, resulting in a total of 438 TCEs that fall on KOI targets but are not matched by known KOIs.

3.1.1. Failure to Detect Short-period KOIs Due to Data Artifacts

Subsequent analysis of the KOIs that were not matched by TCEs showed that 21 out of the 31 had relatively short periods, typically under 2 weeks. Figure 12 shows the maximum multiple event statistic as a function of period for a selected target in this group. The period and multiple event statistic of the KOI on this target star is indicated with a marker in the plot. As shown in Figure 12, the multiple event statistic is dramatically and systematically larger for long periods than for short periods, with a gross pattern of the multiple event statistic rising as the square root of the period.

Figure 12.

Figure 12. Maximum multiple event statistic as a function of period for a sample target. In this target, the KOI period of 3.766 days is shown at the marker, with a multiple event statistic of 11.66σ. One or more artifacts in the flux time series are causing the large number of larger multiple event statistic values at longer periods. Because of the 1000 iteration limit on rejecting strong signals and re-searching for better but weaker signals, this KOI is not detected: the 1000 iterations are exhausted before all of the false alarms in the figure can be rejected.

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The root cause of this pattern is a small number of strong transit-like data anomalies which are randomly distributed among the flux time series. During the folding process that results in Figure 12, the anomalies are combined with background noise to produce strong multiple event statistics. For short periods, the number of events folded together is large, thus there are many background noise events combined with a single data anomaly; as a result, the multiple event statistic is relatively small due to the dilution from the many background noise events. For long periods, because the number of noise events is small, the data anomaly is relatively undiluted and the resulting multiple event statistic is relatively large.

A multiple event statistic that is composed of one strong transit-like anomaly and multiple non-transit-like background noise signals will not survive the robust statistic and chi-square vetoes, as it does not match the quantitative signatures of a true transit pulse train which those vetoes require. Unfortunately, as noted above, the ability of TPS to reject large numbers of such false detections in a single light curve has been limited for reasons of computational performance: the 1000 combinations of period and transit epoch which produce the strongest multiple event statistics are searched, after which the search algorithm declares that no transit signatures were found. In the case of a target such as the one selected for Figure 12, the 1000 strongest signals are all at the long-period end of the distribution, and the search iterations are exhausted before the actual signal at 3.766 days is examined.

In the limit where strong transit-like data anomalies are distributed uniformly and randomly throughout the data set, there will inevitably be some targets for which early quarters of data contain no anomalies but later quarters contain one or more. If such a target also contains a short-period, low-intensity transit signature, then the transit signature will be detectable only so long as the data used for the detection were entirely acquired prior to the first anomaly occurrence. This appears to be the case for the 21 instances of short-period, low-intensity KOIs that were not detected by the most recent TPS run. Note that this is one of those unusual situations in which a twelve-quarter data set does not permit detection of a signal which was apparent in a three- or six-quarter data set.

3.1.2. Matching of KOI and TCE Ephemerides

Detection of a TCE on a KOI target is a necessary but not sufficient condition to determine that the TCE is a detection of the KOI. An additional requirement is that the TCE and KOI are referring to the same transit signature. This is typically best determined by matching the ephemerides of the two signatures. For this purpose we use an ephemeris-matching calculation described in Appendix C. The resulting match parameter varies from a value of zero, indicating no match whatsoever, to a value of one, indicating a perfect match within the limits of the Kepler data and data processing algorithm. In the case of a target star that has multiple KOIs and/or multiple TCEs, it is necessary to attempt to correctly match each KOI with the corresponding TCE. A subtlety in this process is that it is at least conceivable that multiple KOIs will be best matched by the same TCE. For example, consider a target that has two KOIs with periods of 0.5 and 1.0 yr, and three TCEs, with periods of 0.5, 0.1, and 0.03144 yr. Depending on the detailed transit timings, it is at least conceivable that the TCE with the 0.5 yr period will be the best match out of the three TCEs for both the 0.5 yr and 1.0 yr period KOIs. In order to ensure that each TCE is paired with one and only one KOI, the following approach is used:

  • 1.  
    compute the ephemeris matches for all nKOI × nTCE possible matches between KOI and TCE,
  • 2.  
    find the best match in that matrix, and pair the corresponding KOI and TCE with one another,
  • 3.  
    eliminate both the KOI and the TCE which have now been paired,
  • 4.  
    repeat the exercise with the remaining (nKOI − 1) × (nTCE − 1) possible matches, and iterate until either the number of TCEs or the number of KOIs on the given target star are exhausted.

Figure 13 shows the value of the ephemeris match between each of the 2599 KOIs and the TCE on that star that provided the closest match. The values in Figure 13 are sorted into descending order. Of the 2599 match values, only 104 are less than 1.0, with 2495 identically equal to 1. Of these 104 cases, 91 are either harmonic mismatches between the TCE and the KOI (especially in cases where the KOI period is under the 0.5 day minimum period used in TPS) or cases in which the KOI timing was determined using only data from early quarters, resulting in errors when extrapolating the timing to the full 12 quarters used in this analysis. The remaining classes of discrepancy between TCE and KOI are as follows.

  • 1.  
    In eight cases, transit timing variations cause confusion for TPS, which is explicitly designed to find periodic transit signatures; this generally results in a tremendous period mismatch between the KOI timing and the TCE, since TPS will usually detect a tiny subset of all transits.
  • 2.  
    In three cases, the KOI and the TCE have inconsistent transit timing signatures, but both signatures appear valid. In each of these cases it is assumed that TPS has identified a heretofore-unknown transit signature on the KOI target, but then failed to detect the known KOI during the multiple-planet search which followed detection of the new TCE. For this reason, these cases are classified as failures of the TPS algorithm to recover the known KOIs.
  • 3.  
    In two cases the KOI timing clearly produces a transit signature and the TCE timing clearly does not.
Figure 13.

Figure 13. Value of the ephemeris-match parameter described in the text across all 2608 TCEs that are matched to known KOIs. Only 113 of the values are not identically equal to 1.

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3.1.3. Conclusion of TCE–KOI Comparison

Out of 2630 KOIs that could be expected to produce TCEs, 44 did not produce TCEs. This includes 31 cases in which there was no TCE and 13 cases in which a TCE was produced but the timing of the TCE did not match the timing of the KOI, even when "near misses" such as harmonic or sub-harmonic detection are taken into account. This yields a KOI recovery rate of 2586 out of 2630, or 98.3%.

3.1.4. Transit Duty Cycle of TCEs Matched to KOIs

Figure 14 shows the distribution of TCE transit duty cycles for the 2495 cases in which the TCE–KOI ephemeris match is identically equal to 1, as well as the distribution for the 2205 cases in which the ephemeris match is identically equal to 1 and the transit duty cycle is below 0.04. When compared to Figure 10, which shows the transit duty cycle for all TCEs, two differences are instantly apparent. First, and least surprisingly, the spike in transit duty cycle values around 0.002 which is visible in Figure 10 is absent from Figure 14. This is because the spike in the former is due to the spurious, anomaly-driven detections at the 1 yr period which are caused by CCD readouts with unusually strong noise properties; these spurious detections are not present in the set of KOIs, thanks to the greater degree of scrutiny on KOIs which allows elimination of such false detections. Second, the KOI transit duty cycle distribution shows a monotonic reduction in the number of KOIs as the transit duty cycle is increased; the TCE distribution shows a reduction from 0.01 to 0.04 transit duty cycle, and an increase from 0.04 to 0.16. Quantitatively, while 58% of all TCE detections in Figure 10 have a transit duty cycle of 0.04 or greater, only 12% of all KOIs in Figure 14 have a transit duty cycle above 0.04. The implication is that the long transit duty cycle TCEs are most likely dominated by false positive detections, and that further reduction in the maximum allowed transit duty cycle from the current value of 0.16 would result in further reduction of the fraction of false positive TCEs, though of course some study would be needed to determine an optimum threshold for the transit duty cycle.

Figure 14.

Figure 14. Distribution of transit duty cycles for TCEs successfully matched with KOIs. Top: 2495 cases in which the ephemeris match is identically equal to 1. Bottom: 2205 cases in which the ephemeris match is identically equal to 1 and the transit duty cycle is less than 0.04.

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4. CONCLUSIONS

The Kepler transiting planet search (TPS) algorithm has been run on 192,313 targets in the Kepler field of view, including 112,321 targets that have been observed near-continuously for the first 12 quarters of the mission. Potential signals of transiting planets were detected on 11,087 of these targets. When subjected to further searches for multiple planets, the total number of detected signals grew to 18,406. Comparison with a known and vetted set of transit-like astrophysical signatures, the Kepler Objects of Interest (KOIs), demonstrates that within the parameter regime of the search algorithm and the KOIs the recovery rate of known events is 98.3%.

Funding for this mission is provided by NASA's Space Mission Directorate. The contributions of Hema Chandrasekaran and Chris Henze have been essential in the studies documented here.

APPENDIX A: CONSTRUCTION OF THE ROBUST STATISTIC VETO

The first step in constructing the robust statistic is to generate the transit model pulse train. This consists of a train of square wave pulses that are positioned at the locations of the transits as determined by the period and epoch associated with the multiple event statistic. Let s be this model pulse train vector. The pulse train s and the data, or flux time series, x are each whitened to eliminate the effect of stellar variations. The whitened model and data vectors, $\tilde{\mathbf {s}}$ and $\tilde{\mathbf {x}}$ (where "~" denotes a whitened vector), are then windowed to remove out-of-transit samples. The resulting whitened, windowed, transit model $\tilde{\mathbf {s}}$ is then robustly fit to the whitened, windowed, data $\tilde{\mathbf {x}}$ to generate a diagonal matrix of fit weights W. The robust statistic (RS) is then calculated as

Equation (A1)

where T denotes the transpose of a vector, and where Equation (A1) is applied only to data samples within the transit windows described above.

In the limit in which the data vector x and the model vector s are well matched in shape and duration, the matrix W will approach the identity matrix and the RS as defined in Equation (A1) will be approximately equal to the multiple event statistic. In reality, the match between data and model is imperfect: the transits in the model vector are represented as square wave pulses rather than true transit shapes, and in general the duration of the trial transit pulse and the true transits will not be identically matched to one another. In studies of known transiting planet systems, this mismatch can lower the RS by about 10% compared to the multiple event statistic.

Now consider a situation in which the multiple event statistic is constructed from folding a single, extremely strong transit-like signature over two or more events that are consistent with statistical fluctuations, which is a typical case of non-uniform-depth events being combined into a multiple event statistic which lies above threshold. Because the fit is performed robustly, the weak transit-like signatures will "out-vote" the strong one, leading to near-unity weights for the weak events and near-zero weights for the strong event. When the weights in this instance are combined with the data and model vectors as shown in Equation (A1), the result will be a low value for RS. It is in this way that the RS permits events with significant transit depth mismatches to be vetoed while preserving events with relatively uniform transit depths.

APPENDIX B: THRESHOLD CROSSING EVENT VETOES USING CHI-SQUARE DISCRIMINATORS

The basic idea behind the construction of the test statistic is to break up the detection statistic into several contributions and compare each observed contribution with what is expected (Allen 2004). Note that with some basic assumptions on the detector noise (namely, that the noise after whitening is zero mean, unit variance, and uncorrelated) the expectation values of test statistics formulated below are independent of whether or not a signal is present in the data, making them ideal discriminators for noise events.

Beginning with the single-event statistic time series z(n):

Equation (B1)

where

Equation (B2)

where M, xi, $\hat{\sigma }_i$, and $\tilde{s}_i$ are defined in Appendix A of Tenenbaum et al. (2012). Qualitatively, the time series $\mathbb {N}(n)$ represents the amplitude of a transit-like signal centered at sample n, $\mathbb {D}(n)$ represents the square of the noise limit for detecting a transit-like signature at sample n; z(n) therefore represents the significance of a transit-like signature detected at sample n. Equation (B2) also defines quantities $\mathbb {N}_i$ and $\mathbb {D}_i$: these are the contributions to $\mathbb {N}$ and $\mathbb {D}$, respectively, from frequency band i. Choosing a particular point in transit duration, period, and epoch space, {D, T, t0}, selects out a set of data samples {A}, one for each transit, that start with the sample corresponding to the epoch t0 and are spaced T samples apart. These samples form a subset of {n}, A⊂{1, 2,..., P}, where P is the number of transits in the data set. The multiple event statistic is then constructed as

Equation (B3)

One version of the χ2 can be constructed by focusing on the wavelet contributions to the single event statistics. If we start now with Equation (B1), we can make the identifications:

Equation (B4)

Equation (B5)

where now the zi(n) are the actual contributions to the SES time series from the ith wavelet component and qi(n) are the corresponding expected contributions. Now the χ2 statistic can be formed:

Equation (B6)

Equation (B7)

Using the previously mentioned noise assumptions, this statistic should be χ2 distributed with M − 1 degrees of freedom; due to leakage between the wavelet components it turns out to be gamma distributed in actual practice. We have a value for this statistic at each n, so we can form a coherent statistic by adding up the points that contribute to the multiple event statistic at times j where jA. This will give us, $\chi _{(1)}^2$,

Equation (B8)

where the Δzij and qij have been introduced for notational convenience. Using the previous assumptions on noise and assuming a perfect match between the signal and template, this statistic is χ2-distributed with P(M − 1) degrees of freedom.

Another version of the χ2 statistics can be constructed by examining the P temporal contributions to the multiple event statistic. To begin, Equation (B3) can be rewritten using the quantities defined for notational convenience:

Equation (B9)

Now, choosing to examine the contributions to the multiple event statistic from each jA,

Equation (B10)

Equation (B11)

where Zj are the actual temporal contributions to the multiple event statistic and the Qj are the expected contributions. Now, $\chi _{(2)}^2$ can be constructed:

Equation (B12)

Equation (B13)

Under the previous noise assumptions, this statistic is χ2-distributed with P − 1 degrees of freedom. Since we have summed over the wavelet contributions prior to computing this statistic it avoids the leakage issue and turns out to be a much more powerful discriminator. Note that dozens of other version of the chi-square veto have been formulated and investigated with real data, and indeed an infinity of such statistics exists. These two versions give us the greatest detection efficiency while simultaneously minimizing the false alarm rate.

The results quoted in what follows are subject to a subtle issue discovered after the Q1–Q12 run was completed. The whitening coefficients in the calculation should be robust against the presence of a signal in the data since they are computed using a moving circular median absolute deviation. However, the χ2 statistics are very sensitive to any signal dependence of the whitening coefficients, however small it may be, due to the way in which they are constructed. The code is now being re-written so that in-transit cadences are first gapped and filled to re-compute the whitening coefficients for use in the χ2 calculation. This should explicitly remove the signal dependence and give us more vetoing power.

Based on analysis of known true-positive and expected false-positive targets, TPS uses the following discriminators in vetoing false-positive detections:

Equation (B14)

In words, the multiple event statistic for a possible detection is divided by the square-root of the reduced chi-square for each of the chi-square statistics computed above, resulting in two discriminators.

APPENDIX C: EPHEMERIS-MATCHING CALCULATION USED IN KOI–TCE COMPARISONS

Consider a TCE that is characterized by its period TTCE, epoch tTCE, and trial transit pulse duration D; on the same target star, consider a KOI that is characterized by its period TKOI and epoch tKOI. The following calculation can be used to determine whether the two ephemerides represent a good match or a poor match in transit timing.

First, of the two periods, define Tshort to be the shorter, and tshort to be the corresponding epoch (i.e., if the KOI has a shorter period, then TshortTKOI and tshorttKOI); define Tlong and tlong to be the period and epoch of the ephemeris with the longer period. The ephemeris matching parameter is the fraction of transits predicted by (Tshort, tshort) which fall within D/2 of one of the transits predicted by (Tlong, tlong).

The reason for using the fraction of short-period transits that are predicted is that there will always be more short-period transits than long-period ones. In the case of an extremely large mismatch in periods between the two ephemerides (for example, a 3 day and a 300 day period), it is possible for all of the longer-period transits to fall close to transits of the shorter period, but the reverse is not true. Thus, in cases of extreme mismatch in period, using the fraction of short-period transits as the metric ensures that the matching parameter has a low value, whereas the fraction of long-period transits that fall near a short-period transit can be large, and thus use of the long-period transits in this way could result in a large value of the matching parameter even though the ephemerides are wildly mismatched.

The duration of the trial transit pulse must be included because the finite pulse width and the finite duration of a real transit result in a family of nearly degenerate (period, epoch) combinations. For example, a data set that contains 3 transits of 13 hr duration at 365 day period would be well matched by a model transit with a 365 day period, but almost equally well by a transit with a 364.9 day period or 365.1 day period. The matching parameter takes this degeneracy into account by requiring that the short-period transits be within one-half of a trial transit duration of the long-period transits. The duration "smearing" is applied to the longer-period ephemeris because, in a case with a huge period mismatch, applying it to the short-period ephemeris could result in duty-cycle problems. For example, consider the match between a 365 day period ephemeris with 13 hr duration and a 1 day period ephemeris. Applying the pulse duration smearing to the short-period ephemeris would result in a duty cycle greater than 0.5; applying the smearing to the long-period ephemeris ensures that such absurd combinations of parameters do not occur.

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