Updated Catalog of Kepler Planet Candidates: Focus on Accuracy and Orbital Periods

In preparing this catalog (Table 1) and throughout our study, we take stellar properties for the hosts of more than 99.7% of the planet candidates from one of three sources. When available, we select parameters from the latest catalog provided by the California Kepler Survey (CKS), ¹² which lists properties of Kepler PC hosts that have both spectral measurements from the Keck I telescope and well-determined Gaia properties, especially distances (Fulton & Petigura 2018). This list includes ∼60% of the hosts of multis, as well as ∼60% of the single-planet hosts brighter than Kepler magnitude Kp = 14.2, but fewer than 6% of the fainter hosts of singles. For stars with Gaia parallaxes/distances that were not included in the CKS sample, we use properties from Berger et al. (2020), which includes ∼95% of the Kepler targets; this list accounts for most of the remaining PC hosts. For stars absent from both catalogs, we use the stellar parameters listed in Kepler DR25 (Thompson et al. 2018).

Parameters for KOI-3206 were obtained from the Gaia online archive. We adopted custom parameters, as described in the following paragraph, for stellar hosts in two binary star systems. No useful data were found for KOIs 2324, 4713, 5226, and 5718, so we have adopted solar parameters with large uncertainties (R_⋆ = 1 ± 1 R_⊙, M_⋆ = 1 ± 1 M_⊙, = 4.5 ± 4.5) for these planet-hosting stars.

Transit depths for both KOI-119 (Kepler-108) and KOI-284 (Kepler-132) suffer substantial dilution due to stellar companions. For the case of KOI-119, we adopt the nominal dilution of 69.9% as reported in Mills & Fabrycky (2017) for their (preferred) mutually inclined solution. Observations indicate that the KOI-284 system consists of two nearly identical stars with a total of four known transiting exoplanets. From orbital stability considerations, the orbital periods of KOI-284.02 (6.41 days) and KOI-284.03 (6.17 days) are inconsistent with these planets orbiting the same star. Thus, KOI-284 represents the special case of a split multi; see Section 3.2 for more details on this system and other Kepler split multis. For computing the planetary parameters presented in this paper, we adopted a dilution of 50% for the KOI-284 transit models, which implies that half of the light in the photometric aperture is due to the companion star.

The calculation of transit models and the preparation of data products is worth reviewing in the context of potential biases and providing motivation for future improvements of the Kepler catalog and its legacy value. For all transit models, we use Presearch Data Conditioning (PDC) lightcurves (Stumpe et al. 2014) as reported in DR25. Observations with a quality flag set for any of bits 1, 2, 3, 4, 6, 7, 9, 13, 15, 16, and 17 were rejected from our analysis for the reasons described in Table 2.3 of the Kepler Archive Manual (Thompson et al. 2016). PDC lightcurves were prepared for transit analysis by detrending using a second-order polynomial filter with a running window of 5 days. The width of the running window was always truncated to avoid gaps larger than 10 long-cadence (30 minute) observations of valid Kepler photometry. Thus, the filter does not cross large gaps in Kepler photometry that arise from monthly data downloads or quarterly spacecraft rotations, and it avoids problems with significant jumps in the reported photometric flux from thermal settling of the spacecraft after attitude adjustments that are not fully captured by PDC (Van Cleve & Caldwell 2016). Observations within one transit duration of the mid-transit time for each observed transit were excluded from the polynomial fit. Thus, the photometric baseline of in-transit data was interpolated based on out-of-transit observations only. Outliers in the detrended data were identified and removed; outliers are defined herein as single long-cadence photometric measurements more than five standard deviations away from the mean after removal of a best-fit transit model.

The DR25 transit models and updated models for this paper use the same software (TRANSITFIT5 transit modeling software; Rowe et al. 2015; Rowe 2016) and techniques for parameter estimation and the calculation of posteriors. Additional details can be found in Rowe et al. (2014, 2015) and Thompson et al. (2018). Briefly, a multiplanet transit model was calculated for each lightcurve and used to isolate the transits for each individual planet in the system by subtracting the model with the depth set to zero for the planet of interest. This lightcurve was then used to fit each KOI separately with a photometric transit model using the analytic quadratic limb-darkening model from Mandel & Agol (2002). Limb-darkening coefficients are based on the tables of Claret & Bloemen (2011) and were fixed to values used in the DR25 KOI catalog. The photometric model parameterization uses the mean stellar density (ρ_⋆), fixed quadratic limb-darkening coefficients, photometric zero-point, mid-transit time (T₀), orbital period (P), impact parameter (b), and scaled planet radius (R_p/R_⋆). Eccentricity was fixed at zero for these models; thus, ρ_⋆ is replaced by ρ_⋆c. The adoption of mean stellar density as a fitted parameter assumes that the mass of the host star is much larger than the combined mass of the transiting planet and any other planet(s) orbiting closer to the star, whether transiting or not. Errors from TTVs for specific systems (see additional discussion below) were corrected by adjusting the observation times based on a linear interpolation of the measured center-of-transit times to create an aligned ephemeris. We calculated the center-of-transit times for each observed transit by fitting two transit durations of Kepler photometric data centered on the predicted or precalculated time of each observed transit seeded with the best-fit transit model and only allowing the mid-transit time to vary. Biases in TTs can be introduced from overlapping transits, as the multiplanet models used for lightcurve preparation do not simultaneously fit the transit parameters and center-of-transit times.

We adopted the Markov Chains calculated for DR25, apart from KOI PCs with large impact parameters (b > 1) and a few other KOIs that required model updates. The DR25 Markov Chain Monte Carlo (MCMC) sampler assumed a noninformative prior for the impact parameter, which works well for nongrazing transits. However, when b > 1, the minimum value of the scaled planetary radius, R_p/R_⋆, required for a planetary transit grows linearly with b. Uniform sampling would result in a bias toward very large impact parameters, which are unphysical for planetary transits of stars. To sample correctly, a prior was introduced to disfavor large impact parameters by deweighting the model likelihood by b² when b > 1 − R_p/R_⋆, i.e., multiplying the prior by b⁻² for such regions of parameter space and not allowing b > 10. We computed new MCMC models for all KOI PCs with b > 1 and used these models to calculate the values presented in Table 1. Some KOIs in our cumulative catalog required updated best-fit models and new MCMC runs to allow for sufficient sampling of low b parameter space. New models are presented for the following KOIs: 1681.02, 1681.03, 1681.04, 2092.03, 2398.01, 2474.01, 2578.01, 2604.01, 2695.01, 2775.01, 2919.01, 2933.01, 3013.01, 3130.01, 3384.01, 3572.01, 3853.01, 4007.01, 4034.02, 4035.01, 4056.01, 4345.01, 4498.01, 4528.01, 4625.01, 4632.01, 4670.01, 4743.01, 4778.01, 4782.02, 4838.02, 4886.01, 4890.01, 5804.01, 5831.01, 6103.02, 6941.01, and 7368.01, as well as for all of the new KOIs listed in Section 2.3.

The development of the Kepler catalog introduced a few quirks that biased some of the reported best-fit parameters. As the Kepler mission progressed, transit modeling techniques were improved and the methodology of reporting parameters changed, such as the choice of reporting either maximum likelihood versus mode or median from Markov Chains without much consistency between iterations. A more insidious consequence from model evolution were biases that resulted from how subsequent-generation models were initialized based on the ancestral adaptation of previous models as a reference starting point. Prior to the Mullally et al. (2015) catalog, each update introduced new models that incorporated new photometry, resulting in longer time-series observations that ideally should produce more accurate measurements of periods and transit depths and overall better fidelity. However, there are two identifiable deficiencies from this approach: a bias toward extreme values of impact parameter due to the nature of Kepler observations and excessive dependence on the average period measured up to the time that the KOI was initially announced when appreciable TTVs occur.

In general, impact parameters are not well measured for Kepler planets. This is due to the low S/N of a majority of Kepler discoveries and the long (30 minutes) cadence, which is comparable to the ingress and egress durations of the typical observed transit. Combined with a potentially simplistic limb-darkening model, it is common to see the posterior distribution of b from the transit model skew toward 0 or 1. From a probabilistic view, the data for most Kepler transits are insufficient to confidently distinguish a model with b = 0 from models with b ∼ 0.5. The evolution of the Kepler models has meant that over time, a large number of models would always be initialized near the boundary of allowable values of b. Using least-squares methods such as Levenberg–Marquardt (e.g., More et al. 1980), which explores the local gradient of the model parameter value, can result in many models remaining near b = 0 when initialized there. The solution to this problem is to recognize that distributions of model parameters are more robust when sampling from estimates of posteriors based on methods such as MCMC. For this paper, we explicitly note how all model parameters are reported in Section 2.5. Additional improvements may result from improved modeling of limb-darkening in future studies.

Figure 2 compares the impact distribution of PCs presented in Table 1 (orange) against that in DR25 (Thompson et al. 2018; green). The DR25 catalog shows a pronounced excess of model fits with b ∼ 0 and b > 1. Our efforts to reinitialize fits and include priors deweighting large impact parameters produce a distribution of impact parameters more consistent with an isotropic inclination distribution (Kipping & Sandford 2017), although there are still excess populations near b = 0 and for b ≳ 1.

Zoom In Zoom Out Reset image size

As recognized by Newton (1687), the orbital period of a planet is, in general, not constant. The orbital periods observed by Kepler are the mean values that were observed during the four years of the primary mission, at least to a good approximation. In the case of systems with large TTVs that were discovered early in the mission, the evolution of the models can result in the period reported in catalogs published by the Kepler mission being only valid for the duration of data used for the initial discovery and characterization. This occurred because the transit models assumed a noninteracting Keplerian orbit. Transit timing variations were handled by explicitly measuring the TT of each individual event, then resampling the time stamps with linear interpolation based on the TT of each event to be aligned. Resampling used the observed-minus-calculated (O − C) values with the calculated TT based on the reported mean period from the best-fit model. As the transit models evolved with each new data release, the TTs and O − C values were measured based on the transit model from the previous catalog. The new TTs were incorporated and the model updated. Since the O − C values were fixed when the transit model parameters were updated, any significant change in the orbital period was captured in the O − C, not the reported transit model period. Our solution to this problem is to calculate the mean orbital period, as observed by Kepler, directly from a straight-line fit to the measured times of each transit. An O − C diagram (O − C versus time) should have no significant slope. If a single planet demonstrated clear TTVs, then TTVs were typically calculated and included for all planets in the system. See Section 2.5 for an explicit description of how each model parameter is reported, Section 2.4 for extended discussion of orbital periods during the Kepler epoch, and Section 4 for an analysis of period variations over longer timescales.

Transit timing variations are commonly observed when two or more planets interact dynamically (Agol et al. 2005; Holman & Murray 2005). Other potential causes of TTVs include stellar binarity and astrophysical effects such as activity and starspots. (The latter two processes do not cause variations in the actual times of transit, but their observational signals can mimic TTVs.) The process by which TTVs have been accounted for in previous catalogs was inhomogeneous and overall ad hoc. The primary criterion for selecting the solution with TTVs included was either the visual identification of TTVs from examination of O − C diagrams or the inclusion of specific KOIs for the detailed study of individual systems. For example, KOI-8298.01 is reported to use TTVs in the model but has a period of less than 0.2 days. The model with TTVs shows a significantly deeper transit, hence the inclusion of TTVs. However, this may indicate either that KOI-8298.01 is not a transiting planet (e.g., a manifestation of stellar variability) or that the apparent times of transit are significantly affected by spot crossings. Thus, the TTV flag is not a definitive indication of whether or not TTVs are present and should not be used as evidence for the validity of a KOI being a true planet. The first digit of the TTV flag merely reports which KOIs have transit models that include TTVs, and the second and third digits pull tabulations from the published TTV catalogs of Holczer et al. (2016) and Kane et al. (2013), respectively. These external catalogs provide excellent assessments of TTVs for most KOIs.

The S/N of the sum of all observed transits was calculated via

where m_i is the calculated flux from the fitted transit model at each observation i with a total of n observations, and σ_i is the standard deviation of the photometric time series data near the time of the i^th transit based on out-of-transit observations using detrended PDC photometry with outliers removed. The model is scaled to have the out-of-transit flux equal to unity. Figure 3 shows the complex dependence of the S/N of the population of exoplanet transit signatures on planetary radius and orbital period. The trends in S/N are further discussed in Section 2.5.

Zoom In Zoom Out Reset image size

Figure 4 compares the mean stellar density based on input stellar parameters to the mean stellar density calculated from our circular orbit transit models. Colors denote the source of the adopted stellar parameters. The left panel shows the complete sample. The right panel shows only PCs with reasonably high values of S/Ns, nongrazing transits, and well-measured ρ_⋆c. The distribution is clearly skewed such that the mean stellar density estimated by fitting the lightcurve and assuming circular orbits is generally larger than that from stellar parameter tables. This trend is expected because detection bias favors planets that transit near the periastron of their orbits. A simple application of Kepler’s second law dictates that for a given impact parameter, transits observed near periastron are shorter than predicted from a circular orbit, and impact parameters should be roughly uniformly distributed for b < 1. The distribution is also biased by impact parameter and dilution (see Section 6.2 of Rowe & Thompson 2015).

Zoom In Zoom Out Reset image size

A visual examination of the right panel in Figure 4 appears to shows a bias in ρ_⋆ between the CKS (Fulton & Petigura 2018) and Berger et al. (2020) samples. It is important to note that our stellar parameters give preference to CKS, then Berger et al. (2020), and use those of DR25 only for targets not appearing in either of the preferred catalogs. The CKS sample was skewed toward the inclusion of multiplanet systems. As shown in Section 3.5, planets in compact multiplanet systems tend to have more circular orbits (Figures 20 and 23). Thus, the observed bias is a selection effect. Figure 17 shows that the measured transit duration distributions do not depend on the choice of stellar parameter catalog in any systematic manner. The vertical blue error bars tend to be large because of the high uncertainties in DR25 stellar parameters. These stars were not characterized in Gaia DR2 or CKS, which suggests that they may have strong stellar blends or other observational challenges. This also explains the larger scatter of the blue points relative to the middle diagonal line.

We pulled planet candidates from a variety of sources, including 9564 KOIs from the cumulative DR25 supplement catalog ¹³ ; ultrashort-period (USP) planets from Sanchis-Ojeda et al. (2014); long-period (P ≳ 1 yr) and transit candidates only observed to transit once, which we refer to as monotransits, from Kawahara & Masuda (2019); the autoregressive planet search from Caceres et al. (2019); PCs from the machine-learning search by Shallue & Vanderburg (2018) that sought additional planet candidates around targets already having two or more PCs; and low-S/N candidates found by revisiting marginal TCEs (Bryson et al. 2021). The inclusion of new catalogs and discoveries yields new KOIs: KOI-1843.03 and 8298-8303 are USP planet candidates (including the three shortest-period PCs in our catalog) from Sanchis-Ojeda et al. (2014); KOI-8304–8335, 1108.04, 4307.02, 408.06, 2525.02, 847.02, 671.05, 3349.02, 693.03, 7194.02, and 1870.02 are long-period planet candidates; KOI-500.06, 351.08, 691.03, 354.03, 191.05, 1165.03, 2248.05, 542.03, 1589.06, 2193.03, 1240.03, 1992.04, 1276.03, 416.05, 1889.03, 4772.04, 2433.08, and 597.04 are from Shallue & Vanderburg (2018) ¹⁴ ; and KOI-4246.03, 4302.02, 8336.01, 8337.01, and 8338.01 are from Bryson et al. (2021).

Finally, KOI-8339–8394, the majority of which we have dispositioned as false positives/false alarms because their S/N < 7.1, are from Caceres et al. (2019). There are 30 potential candidates from Table 5 of Caceres et al. (2019) not included, as we were unable to compute a best-fit model based on the predicted ephemeris. Table 5 of Caceres et al. (2019) also lists 11 identifications around targets with preexisting KOIs, including 8302.01 from Sanchis-Ojeda et al. (2014), that are already in our table, some of which are discussed below because we corrected their periods to the values given in Caceres et al. (2019).

We provide new transit models and parameter posteriors for these new KOIs using the same models and methodology presented in Section 2.2. We corrected the period of KOI-1353.03 (Kepler-289d) to match the reported value in Schmitt et al. (2014) rather than that in the DR25 catalog because the latter was the result of an aliasing problem. One of our new long-period PCs, KOI-3349.02, was previously only observed to transit once (Kawahara & Masuda 2019 and references therein), but we located a second, nearly identical transit in the simple aperture photometry (SAP) lightcurve, showing that it is a duotransit (only two transits observed) planet with P = 805 days.

We revisited the dispositions of KOIs that have Kepler numbers but were classified as FPs in DR25supp. Since there is no peer-reviewed source for the reasoning leading to the dispositions, we do not know whether or not additional observations beyond Kepler photometry and centroids were used. Based on our assessment of TCERT data validation reports downloaded from the NASA Exoplanet Archive, we reverted the following KOIs from FP disposition to PC: KOI-125.01, 129.01 (but see below), 631.01, 3138.02, and 3184.02. DR25supp listed these KOIs as showing evidence of stellar eclipses; however, we found no evidence of a secondary eclipse in the photometry lightcurve, either directly or through measurement of an odd–even effect.

We confirmed that the following KOIs with Kepler numbers are indeed false positives: KOI-3032.01, 126.01 (see Carter et al. 2011), 1416.01, and 1450.01. Their photometric lightcurves show clear evidence of secondary eclipses that are indicative of stellar companions. Despite an analysis of putative TTVs by Hadden & Lithwick (2014), Kepler-37e (KOI-245.04) is a false alarm that was never validated (Barclay et al. 2013).

Table 7 of Carmichael et al. (2019) lists five KOIs, all of which are listed as PCs in DR25supp and three of which have been given Kepler numbers, as having measured masses above the planet/brown dwarf dividing line (see also Carmichael 2023). We have reclassified as false positives all five of these KOIs: KOI-423.01 = Kepler-39 b, KOI-189.01 = Kepler-486 b, KOI-205.01 = Kepler-492 b, KOI-415.01, and KOI-607.01. Note that Kepler-39 b has a mass of ∼20 M_Jupiter, whereas the others have masses of ≳40 M_Jupiter.

Santerne et al. (2016) analyze RV measurements of more than 100 KOIs that were listed as giant planet candidates in one or more of the Kepler project’s catalogs. They present convincing evidence that the following five KOIs, which we would have otherwise classified as planet candidates, are produced by eclipsing binary stars: 129.01, 969.01, 1465.01, 1784.01, and 3787.01. Furthermore, the following three KOIs, which we would have classified as planet candidates had they not failed our upper size limits, are also produced by eclipsing binary stars: 3411.01, 3811.01, and 5745.01. Table 1 dispositions the 13 KOIs listed in this and the previous paragraph with the letter “M” to distinguish them from other false positives.

We undertook a photometric analysis of new candidates presented in Table 5 of Caceres et al. (2019) to assign dispositions based solely on Kepler photometry. Photometry was processed in a manner similar to DR25 (Thompson et al. 2018) by detrending the data with a second-order Savitzky–Golay filter. Data in the transit window as predicted from the reported transit duration, period, and center-of-transit time were excluded from polynomial fits. This means that we did not complete an exhaustive test against false alarms due to in-phase periodicity (e.g., a depth test as described in Coughlin et al. 2016 for uniqueness). We attempted to compute best-fit transit models through χ² minimization. For 33 of the 86 proposed new candidates, the model either failed to converge or returned a fit consistent with a flat line. In these cases, we then ran a box least-squares (BLS; see Kovács et al. 2002) search restricted to ±0.1 days around the reported period, which allowed us to recover three of them. As we could not find evidence of the remaining 30 proposed PCs, we do not include them in our KOI table. The ephemerides reported in Caceres et al. (2019) are only accurate to one Kepler long cadence (∼30 minutes), which, combined with uncertainties in the reported period and barycentric drift, may result in poor recovery with our methods. If the BLS search found a candidate event with P within 0.01 days of the reported event, we adopted the period and center-of-transit times from our localized search. Thus, there is risk that we have not recovered all events as previously reported. With best-fit models, we ran our MCMC algorithm to compute posteriors. All new candidates from this activity that had an S/N determined by our transit models to be less than 7.1 have been flagged as false positives in Table 1. The S/N reported in Table 5 of Caceres et al. (2019) is a detection statistic that appears to not be strongly related to the folded transit S/N that we report (see Equation (1) above). We did not assess the photometric centroids for this sample; however, we noted that there is a substantial mismatch when comparing ρ_⋆c from our transit models to ρ_⋆ from stellar properties catalogs for KOI-8345.01 and KOI-8366.01, and for this reason, we have flagged these two KOIs as false positives.

We investigated several cases in which the Caceres et al. (2019) table listed signals around existing KOIs with periods that are a small integer multiple or fraction of those listed in DR25supp. This investigation led us to revise the orbital periods of two PCs downward by a factor of 2, KOI-6262.01 from P = 0.673 to 0.3365 days and KOI-4777.01 (which was independently identified with the correct period by Cañas et al. 2022) from P = 0.824 to 0.412 days. Additionally, we revised the periods of FPs (both of which were previously identified as having centroid offsets) KOI-4305.01 (from 0.935 to 0.234 days) and KOI-4872.01 (from 1.035 to 0.207 days). Reported detections for KOI-6749, KOI-6984, KOI-2431, KOI-4788, and KOI-2642 are the secondaries; these targets are already FPs.

KOIs have been vetted into new categories using several criteria. Nonetheless, the vast majority of KOIs are dispositioned as one of the two categories: planetary candidates (PCs) and false positives (FPs, which here, as in previous catalogs, include false alarms). False positives are defined as transit-like astrophysical events that are not produced by a planet. Eclipsing binaries are the primary source of false positives. False alarms are spurious detections caused by features in the target star's lightcurve that are not transit-like. False alarms can be caused by stellar variability and/or instrumental systematics. We provide additional vetting criteria based on S/N and planetary radius. A description of all vetting flags is presented in Section 2.5.

Initial FP and PC classification is adopted from the DR25 and DR25supp catalogs. DR25supp is represented by the cumulative catalog from the NASA Exoplanet Archive retrieved on 2022 October 27. If vetting classification is not available in either DR25 or DR25supp, we use DR24 (Coughlin et al. 2016), which includes KOIs from previous searches that were not seen in the TCE search conducted for DR24.

Consistent with previous Kepler catalogs, we generally require a total transit S/N ≥ 7.1 for a KOI to be considered a PC; KOIs below this threshold are considered false alarms. We allowed four exceptions to this rule, KOI-2022.02 = Kepler-349 c, KOI-2034.02 = Kepler-1065 c, KOI-4024.01 = Kepler-1541 b, and KOI-7368.01 = Kepler-1974 b, all of which have been validated as planets and passed our visual inspection—which revealed marginal evidence of a visible transit event in their lightcurves.

The S/N ≥ 7.1 criterion was based on limiting the number of false alarms considered to be PCs to one in the 90 day–2 yr period range per 10⁵ stars searched the Kepler sample in the presence of white noise on 6 hr timescales. However, much of the noise in the Kepler lightcurves is correlated, so multiple false alarms become problematic for S/Ns in the range of 10 ≳ S/N ≥ 7.1, as has been verified through injection tests and reliability studies (Thompson et al. 2018; Hsu et al. 2021). Moreover, low-S/N KOIs do not have sufficient signals to allow for precise centroiding and other vetting procedures used to distinguish astrophysical signals such as eclipsing binaries from transiting planets. Therefore, higher-S/N cuts are needed to obtain the purer (higher-confidence) samples of PCs that are required for some studies, including most of our analyses of the characteristics of the population of Kepler’s planet candidates.

Some KOIs have all transit model parameter uncertainties listed as zero. This indicates that the MCMC computations did not converge. This can happen when the S/N of a transit event is low, or the event is non-transit-like in shape. While we did not consider MCMC convergence when assigning PC or FP (P or F) dispositions, there is a high probability that these KOIs are indeed false alarms. There are 220 KOIs flagged as “P” or “S” without MCMC computed posteriors; 24 are members of candidate multiplanet systems, and a total of 26 are PCs, 14 of which have an S/N > 10.

There is increasing degeneracy when distinguishing between ultracool stars, brown dwarfs, and planets for transiting objects with radii approaching the size of Jupiter that lack mass information. We have flagged KOIs with estimated planetary radii R_p > 21.947 R_⊕ ≈ 2 R_Jupiter, as well as KOIs with period P > 20 days and a 1σ lower limit for the planetary radii R_p that exceed 13.17 R_⊕ ≈ 1.2 R_Jupiter, as likely FPs, but as these boundaries are not precisely defined, KOIs that pass all criteria for being classified as PC apart from size are given the disposition “R.” The following KOIs have Kepler numbers but radii that exceeded our limit on planetary size using nominal stellar parameters: KOI-846.01, 855.01, and 1792.01. There is no strong evidence from Kepler photometry or in the literature that contradicts their status as confirmed planets. Thus, no change in their disposition as PCs is warranted based on our study. The radius of the star KOI-1792 estimated by Fulton & Petigura (2018) is almost three times as large as that estimated by Berger et al. (2020); the former leads to a radius estimate for KOI-1792.01 of R_p = 31.1 R_⊕, so we used the Berger et al. (2020) parameters for this star. The other two cases both appear to be evolved stars (KOI-846 being slightly evolved, whereas KOI-854 is a giant), whose radii are difficult to estimate, so we suspect that the stellar radii have been overestimated, but in these cases, we only have one Gaia-constrained radius estimate (from Berger et al. 2020), so we retained the probably overestimated nominal values of R_⋆ and R_p .

We give new KOIs dispositions based on an assessment of the photometric transit. All new KOIs are assigned a PC status unless they failed to meet our S/N, radius cuts, or visual inspection. Visual inspection includes a determination of whether the observed transit duration is in very rough agreement with the stellar parameters and the phased lightcurve has a transit-like shape. We did not attempt to measure photometric centroids; thus, it is possible that many of the new short-period KOIs from the Caceres et al. (2019) sample (8339–8394) may be background binary blends. As most of the orbital periods from this sample are relatively short, there is a reasonable expectation that the modeled mean stellar density should match stellar tables because the orbits should be nearly circular. For example, KOI-8386.01 is likely a background binary due to the significant order-of-magnitude mismatch in the transit duration and stellar classification, but we nonetheless dispositioned it as a PC, as we have not conducted an analysis of photometric crowding or photometric centroid shift during the transit event. None of the KOIs from the Caceres et al. (2019) sample failed the radius cuts, so all were dispositioned as either “S” (36 cases), “P” (18 cases), or “F” (two cases that were discarded based upon visual inspection of the lightcurve); only five of the PCs have S/N > 12, and only one has S/N > 14.

We visually reexamined the KOIs that were classified as PCs in either DR24 or DR25 but subsequently classified as FPs in DR25supp. Based on our analysis, we reverted the following KOIs back to PCs: 82.06, 198.01, 1693.01, 1796.01, 1902.01, 2306.01, and 2307.01. These KOIs do not exhibit measurable centroid shifts from examination of DR25 or DR25supp vetting reports and show visually identifiable transits that can be modeled with MCMC that meet convergence criteria.

The orbital periods listed in Table 1 are estimated by fitting the Kepler lightcurve assuming a Keplerian, noninteracting orbit and using a limb-darkened transit model. As such, we are estimating something akin to the planet’s mean period over the interval of Kepler observations.

Transit times are the measure of when mid-transit occurs. We define the mid-transit as when the projected distance between the center of the star and the planet is minimized. For a circular orbit, this is equivalent to when the transit model is deepest. We measure TTs and then report TTVs relative to the orbital period and reference time. The latter has typically been the first or mid-observation transit. The transit model is used as a template to measure the center-of-transit time for each transit, and the transit model orbital period from the initial fit is used to calculate the difference between the observed transit and the calculated transit (O − C). If TTVs are present, then the transit model is updated by deTTVing the lightcurve, whereby the time stamps from Kepler observations are adjusted to have all transits aligned. Time-stamp adjustments use a linear interpolation based on the O − C TTs such that the new time stamps have an effective O − C of zero. A new transit model is then fit to the data with updated time stamps.

If TTVs are not properly accounted for, the ingress and egress of sequential transits become misaligned, leading to errors in the reported characteristics of the transit, especially the transit’s duration, depth, and impact parameter. Since the scaled planetary radius (R_p /R_⋆) and transit model mean that stellar density (ρ_⋆c) is correlated with the impact parameter (b) due to geometry and stellar limb-darkening, those parameters also have increased errors. The models based on deTTVed lightcurves better estimate the transit shape, often leading to substantially improved estimates of the abovementioned parameters, especially for b and R_p /R_⋆. However, this process previously led to errors in the reported period that were not accounted for in estimates of period uncertainties.

Transit timing variations are calculated independent of the fitted transit model, and, when possible, initial TTVs were adopted from previous catalogs. Examples where the period reported in previous catalogs is significantly different from the observed mean period (averaged over the time interval during which transits were observed by Kepler) include KOI-142.01 (Kepler-88 b) and KOI-377.01 and .02 (Kepler-9 b and c). Early KOI catalogs used only a few quarters of observations, and the mean period over that time frame differed significantly from estimates using the entire Kepler observational baseline. The model parameters for each previous catalog were used as a seed for the updated model. Thus, the period from a previous catalog was used as the seed for the best fit in the subsequent catalog in which it appears. If additional TTVs (O − C) were measured, the mean period from a previous catalog was held fixed before the new model fit was made. In some cases, the previously reported period did not represent the true mean period in the presence of strong TTVs.

For the catalog presented herein, the mean observed period from Kepler observations is of interest, as it is the best approximation to the long-term mean period that can be straightforwardly estimated for all KOIs apart from those that only transited once (monotransits). The value of P is calculated by applying a correction to the best-fit period, P_bf, from the transit models. For each KOI with four or more measured TTs, a straight-line model is fit to the O − C versus observed (O) values using standard least-squares minimization weighted by measurement uncertainties in O − C values. The slope, m, from the fit gives the correction, with

We report P and propagate the uncertainty in m to compute the orbital period uncertainty thereof for all KOIs in Table 1. The measured O − C values are based on P_bf; thus, any nonzero slope indicates that P_bf does not represent the mean period. This correction specifically addresses the issue of incorrect periods for planets in systems such as Kepler-9 and Kepler-88. For non-TTV planets, there is good agreement between P_bf and P. If a planet has just two or three observed transits, we adopt P from P_bf. To avoid additional inhomogeneities in our catalog, only P is reported.

The incorporation of TTVs in orbital period calculations, as described above, yields more robust estimates of orbital periods of planets with observed TTVs than provided by previous catalogs. For most KOIs, no TTVs are detected, and the revised period is statistically similar to the reported period from DR25. However, there are extreme examples, such as Kepler-9 b and c and Kepler-88 b, where the period changes significantly (∼1 hr) relative to the DR25 catalog. Overall, the distribution of fractional changes in period ratio is non-Gaussian, with substantially larger numbers in the tails of the distribution.

However, we account for TTVs only to the extent that they average out over the time in which Kepler observed the planets to transit. Typically, this cancellation is incomplete for planets with observed TTVs. The periods and uncertainties quoted in Table 1 thus do not account for TTVs that have periodicities that are long compared to the Kepler observations and do not fully account for TTVs with periodicities comparable to the amount of time between Kepler’s first and last observations of transits of a particular planet. Some types of TTVs are of very small amplitude over 4 yr (and thus unlikely to have been detected) but grow to become substantially larger on timescales of decades to centuries, affecting the long-term mean orbital period. These differences in orbital period are important for understanding three-body resonances and producing ephemerides that are accurate far into the future. We discuss these issues in more detail in Section 4.

We searched the lightcurves of the 10 duotransit PCs with P > 730 days to determine whether or not there are Kepler data that can exclude the possibility that additional transit(s) were missed because they occurred during a data gap(s) and the actual period of the planet is either half or one-third of the reported value. Data points with SAP_QUALITY equal to 16 or greater were excluded from the examined lightcurves. The period of KOI-375.01 (=Kepler-1704 b) could, indeed, be half of the reported value of 988.9 days, but Kepler data exclude the possibility of any of the other nine PCs having periods equal to half or one-third of the values listed in Table 1.

Our catalog of Kepler planet candidates is presented in Table 1. We list, from left to right, catalog numbers of the target star and planet (columns (1)–(3)), fundamental transit model parameters (columns (4)–(19)), properties derived from the transit model and Kepler photometry (columns (20)–(38)), parameters that depend on the transit model and stellar parameters (columns (39)–(44)), stellar parameters (columns (45)–(62)), ¹⁵ and vetting dispositions (column (63)). Data in columns (64)–(85) list the same properties as provided in columns (39)–(44) and (46)–(61) with stellar parameters taken from Berger et al. (2020); these columns are left blank if the properties of the target star are not given by Berger et al. (2020).

Table 2 presents our adopted values for astrophysical constants. The mean radii of the Sun (R_⊙) and Earth (R_⊕) are used to calculate the absolute radii of planets (R_p) and incident flux, S, normalized to that of the Earth. The gravitational constant (G) is used to calculate transit durations, the scaled semimajor axis (a/R_⋆) from the mean stellar density (ρ_⋆c), and orbital inclination (i). The astronomical unit (au) and solar effective temperature T_{eff ,⊙} are used in the calculation of the flux incident upon Earth, S_⊕.

For many properties, we report the location of the maximum posterior density and give additional information about the uncertainty. For some properties, these uncertainties are typically well described by a symmetric distribution, so we report the half-width of the 68.27% credible interval, assuming a symmetric distribution. For other properties, the posterior distribution is often significantly asymmetric, so we report separate uncertainties in the positive and negative directions based on the 68.27% credible interval that minimizes the width of the marginal distribution for that parameter. The units for all uncertainties are the same as for the quantities themselves.

Overall, we have updated transit models for more than 1000 KOI systems. Included among these are 121 KOIs that previously had unreasonably large impact parameters. We have updated the dispositions of 11 KOIs from FP to PC and six KOIs from PC to FP. We flagged 361 cases with very weak S/N transit signals as likely false alarms, 100 KOIs with large planet radii that may be due to stellar binaries, and 13 KOIs that previously had a PC flag that now have a mass measurement inconsistent with the exoplanet hypothesis.

We determine the following transit parameters and their uncertainties by fitting the transit models of Mandel & Agol (2002) to Kepler lightcurves assuming circular orbits (and adjusted for TTVs if TTVflag = 1): mean stellar density, ρ_⋆c; orbital period, P; impact parameter, b; and scaled radius, R_p/R_⋆. Limb-darkening coefficients are held fixed. The transit model is used calculate the depth of transit, d_transit, and the S/N. The fitted transit model parameters, together with Equations (2) and (3) from Seager & Mallén-Ornelas (2003), are used to calculate the transit duration measured from first to fourth contact, T_dur, and an alternate measure of transit duration (see items 24–27 in the numbered list below for details). Planetary inclinations are calculated using the formula

• 1.  
  KIC. The Kepler input catalog number of the target star.
• 2.  
  KOI. The Kepler object of interest number, with the integer portion referring to the target star (source of light) and the decimal portion referring to the particular signal, i.e., the putative planet’s transits.
• 3.  
  Kepler ID. The number and letter assigned to verified planets. It is blank if the KOI has not been assigned a Kepler planet number/letter. Kepler IDs do not automatically qualify a KOI to be dispositioned a PC (see Section 2.3) and are supplied for cross-identification purposes only.
• 4.  
  Period, P [days]. Mean orbital period (the period that gives the best fit to the observed TTs). For candidates with only one transit observed, the period is estimated based on transit duration and impact parameter, assuming a circular orbit, and given as a negative value to distinguish it from periods computed for multiple-transit objects. Note that monotransit planet candidates (those observed to transit just once) are not included in any of our analyses, figures, or tabulations that require knowledge of the orbital period but are accounted for in assessing the multiplicity of planetary systems in all cases apart from the analysis presented in Section 3.5.
• 5.  
  σ(P). Uncertainty (half-width of 68.27% confidence interval for the average period during the epoch of Kepler observations) of P. See discussion in Section 2.4. It is based on the uncertainty in the fitted slope of the measured TTs. As discussed in the second and third paragraphs of Section 4, the actual mean orbital periods of Kepler planets over timescales much longer than the 4 yr of Kepler observations can differ from the values given in Table 1 by many times as much as the listed uncertainties. The reported periods and uncertainties for monotransit candidates assume circular orbits and are considered unreliable.
• 6.  
  Epoch, T0 [BJD–2454900]. Time, calculated using the constant period reported in column (4), at which the center of the planetary disk is closest to the center of the stellar disk for the last transit that occurred prior to halfway between the start of the first quarter and the end of the final quarter that Kepler observed the target star in question, whether or not said transit was actually observed by Kepler. This is the mode value from MCMC, calculated using the SciPy kernel density estimator stats.gaussian_kde with default settings (Virtanen et al. 2020). BJD ≡ Julian Date viewed from the barycenter of the solar system.
• 7.  
  σ(T0). Uncertainty in the epoch.
• 8.  
  Planet/star radius ratio, R_p /R_⋆. The mode value from MCMC of the ratio of the planet’s radius to the stellar radius.
• 9.  
  σ₊(R_p /R_⋆). Upward (to higher values) uncertainty of the ratio of the planet’s radius to stellar radius.
• 10.  
  σ₋(R_p /R_⋆). Downward uncertainty of the ratio of the planet’s radius to stellar radius.
• 11.  
  Impact parameter, b. We report the best-fit value. See Section 2.2.
• 12.  
  σ₊(b). Upward uncertainty of b.
• 13.  
  σ₋(b). Downward uncertainty of b.
• 14.  
  Stellar density from lightcurve, ρ_⋆c [g cm⁻³]. The mean stellar density computed from the photometric model. A circular orbit model has been assumed, and a separate fit is done for each planet in a multi. This is the mode value from MCMC.
• 15.  
  σ₊(ρ_⋆c). Upward uncertainty of ρ_⋆c.
• 16.  
  σ₋(ρ_⋆c). Downward uncertainty of ρ_⋆c.
• 17.  
  u₁. First quadratic limb-darkening parameter.
• 18.  
  u₂. Second quadratic limb-darkening parameter.
• 19.  
  TTV flag. A three-digit number preceded by “T” representing the results of three separate searches for TTVs. The first digit is 1 if TTVs have been included in our transit model and 0 otherwise. The second digit refers to results listed in Holczer et al.'s (2016) catalog, with 2 signifying sinusoidal TTVs, 1 polynomial TTVs, and 0 no TTVs found, and a dash means not investigated. The third digit is the overall rating from Kane et al.'s (2013) catalog, with 9 signifying the strongest TTVs, 8 strong TTVs, 7 weak and/or noisy TTVs, and 6 and below no TTVs of interest; we use a dash for KOIs not rated by Kane et al. (2013).
• 20.  
  # transits. Total number of transits observed by Kepler.
• 21.  
  # TTs. Number of transits for which the transit time has been measured. This number is always ≤ the number in the previous column.
• 22.  
  d_transit [ppm]. Depth of transit. Specifically, it is the depth of the transit model evaluated at the mid-transit time assuming a circular orbit. We report the mode of the MCMC distribution.
• 23.  
  σ(d_transit). Uncertainty of d_transit.
• 24.  
  Transit duration, T_dur (hr). Transit duration, measured from first to fourth contact (T_dur ≡ T_1,4), which is the standard measurement for exoplanet transit duration. It is based on transit model parameters (ρ_⋆c, P, R_p /R_⋆, b) using Equation (3) of Seager & Mallén-Ornelas (2003). The mode of the MCMC distribution is reported.
• 25.  
  σ(T_dur). Uncertainty of T_dur.
• 26.  
  Alternate measure of transit duration, T_1.5,3.5 (hr). Mode of (T_1,4 + T_2,3)/2. T_2,3 uses Equation (1) from Seager & Mallén-Ornelas (2003). If b > 1 − (R_p /R_⋆), then T_2,3 is set to 0. Note T_dur > T_1.5,3.5 ≥ 0.5 T_dur.
• 27.  
  σ(T_1.5,3.5). Uncertainty of T_1.5,3.5.
• 28.  
  S/N. The ratio of the signal, S, which is the integral of the transit model over all transits, to the noise, N, estimated as the standard deviation of the photometric lightcurve out of transit. It is calculated assuming a constant period if TTVflag = 0 and incorporating TTVs if the first digit of the TTVflag = 1. S/N ≥ 7.1 is generally necessary (but not sufficient) for a KOI to be dispositioned as a PC in our catalog. Four exceptions to this general requirement are made for KOIs with Kepler numbers whose lightcurves also passed our visual inspection (see Section 2.3).
• 29.  
  MES. Multiple event statistic, as reported in DR25. KOIs that were not listed in DR25 have 0.0 listed in this column.
• 30.  
  Improvement in S/N when transit model allows for TTVs. S/NwTTV–S/NwoTTV. Difference in S/N calculated without TTVs and with TTVs (e.g., Ofir et al. 2018). It is positive for most KOIs, and it is typically large when the preferred fit uses TTVs (since less out-of-transit data obfuscates the signal) and small when it does not.
• 31.  
  . Chi-squared calculated with TTVs based on the best-fit transit model. If the first digit of TTVflag = 0, then photometric uncertainties have been scaled to have = degrees of freedom (DOF). If the first digit of TTVflag = 1, then photometric uncertainties have been scaled to have = DOF.
• 32.  
  . Chi-squared calculated without TTVs. The values are normalized as for .
• 33.  
  Scaled semimajor axis, a/R_⋆. Scaled semimajor axis.
• 34.  
  σ₊(a/R_⋆). Upward uncertainty of the scaled semimajor axis.
• 35.  
  σ₋(a/R_⋆). Downward uncertainty of the scaled semimajor axis.
• 36.  
  Inclination, i [deg]. Inclination of the planet’s orbit relative to the plane of the sky; an edge-on orbit has i = 90°.
• 37.  
  σ₊(i). Upward uncertainty of the inclination. Note that i + σ₊(i) ≤ 90°.
• 38.  
  σ₋(i). Downward uncertainty of the inclination.
• 39.  
  Planet radius, R_p [R_⊕]. Radius of the planet.
• 40.  
  σ₊(R_p ). Upward uncertainty of R_p .
• 41.  
  σ₋(R_p ). Downward uncertainty of R_p .
• 42.  
  (Bolometric) incident flux, S [S_⊕]. Amount of flux intercepted by the planet relative to that intercepted by the Earth.
• 43.  
  σ₊(S). Upward uncertainty of incident flux.
• 44.  
  σ₋(S). Downward uncertainty of incident flux.
• 45.  
  Kepmag. Target star magnitude in the Kepler passband.
• 46.  
  Stellar density, ρ_⋆ [g cm⁻³]. The mean stellar density from the stellar parameter tables (not the estimate derived from the transit model, which is given in column (14)).
• 47.  
  σ₊(ρ_⋆). Upward uncertainty of ρ_⋆.
• 48.  
  σ₋(ρ_⋆). Downward uncertainty of ρ_⋆.
• 49.  
  Stellar temperature, T_eff [K]. The target star’s effective temperature, taken from the stellar properties catalog.
• 50.  
  σ(T_eff). Uncertainty of T_eff.
• 51.  
  Stellar radius, R_⋆ [R_⊙]. Radius of the star as given in the stellar properties catalog.
• 52.  
  σ₊(R_⋆). Upward uncertainty of R_⋆.
• 53.  
  σ₋(R_⋆). Downward uncertainty of R_⋆.
• 54.  
  Stellar mass, M_⋆ [M_⊙]. Mass of the star.
• 55.  
  σ₊(M_⋆). Upward uncertainty of M_⋆.
• 56.  
  σ₋(M_⋆). Downward uncertainty of M_⋆.
• 57.  
  Stellar surface gravity, [cgs]. Surface gravity of the star as given in the stellar properties catalog. For stellar parameters taken from DR25 and Berger et al. (2020), is from isochrone models. For stellar parameters taken from Fulton & Petigura (2018), is based on spectroscopy.
• 58.  
  . Upward uncertainty of .
• 59.  
  . Downward uncertainty of .
• 60.  
  Metallicity [dex]. Metallicity of the target star.
• 61.  
  σ(Metallicity). Uncertainty of the metallicity of the target star.
• 62.  
  Stellar model source. 0 = solar parameters, 1 = DR25, 2 = Berger et al. (2020), 3 = Fulton & Petigura (2018), 4 = special (see Section 2.1). Aside from KOI-1792 (for reasons discussed in Section 2.3) and two binary star systems for which special parameters are used, Fulton & Petigura (2018) values are used where available, and the Berger et al. (2020) values are always preferred over those listed in DR25.
• 63.  
  Dispositions. This four-letter code gives the dispensations (status flags) from this work, DR25supp, DR25, and DR24 in reverse chronological order, with F = false positive (or false alarm), P = planet/candidate, N = not included in specified catalog, S = rejected by us because S/N < 7.1, ¹⁶ M = Jupiter-sized objects for which mass measured via RV clearly exceeds the 13 M_Jupiter limit for classification as a planet that otherwise would have been dispositioned “P” or “R,” and R = radius too large (see Section 2.3) but meets all other criteria. “N” is never applicable to the first column (our catalog provides dispositions for all KOIs listed) and “S,” “M,” and “R” are used exclusively for our dispositions. The seven validated planets (with Kepler numbers listed on NExScI) that failed either the radius cut or the S/N cut but passed visual inspection of the lightcurve were dispositioned “P” (see Section 2.3 for details).
• 64.  
  Values of the same properties reported in columns (39)–(44) and (46)–(61) with stellar parameters taken from Berger et al. (2020). They are all zeros if the host star is not characterized by Berger et al. (2020).

We treat KOIs with dispositions beginning with “S” and “M” as false positives throughout our study; KOIs whose dispositions beginning with “R” are treated as false positives for most purposes but are included together with planet candidates in our study of the distribution of planetary radii (Section 3.3). The total number of planet candidates is 4376, including 35 monotransits; additionally, there are 100 KOIs, including two monotransits, that we vetted as “R.” There are 709 multiple-transiting planet systems, which account for 1791 candidates, including seven monotransits.

Figure 1 displays the sizes and radii of almost all of the planet candidates in our catalog and indicates the multiplicity of the planetary systems in which they are observed. The bulk of the multiplanet candidates have radii R_p ≲ 4 R_⊕ and orbital periods 2 days ≲ P ≲ 100 days. Figure 5 confirms these general observations and demonstrates the paucity of additional transiting planets in systems hosting transiting hot Jupiters (e.g., Steffen et al. 2012) and planets with periods longer than ∼100 days. The period–radius valley separating super-Earths from sub-Neptunes slopes downward from ∼2 R_⊕ at P = 2 days to ∼1.5 R_⊕ at P = 40 days (left panel of Figure 6).

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Figure 3 shows the typical transit S/Ns for planet candidates as a function of R_p and P. The left panel shows the total S/N calculated using Equation (1), and the right panel shows the average S/N per transit. The total S/N has a floor of 7.1 as discussed in Section 2.3 and marks the boundary where the smallest planets can be found for a specified range in orbital period before the population is dominated by false alarms. The rate of false alarms drastically increases for periods close to 1 yr because of the rolling band noise, which is video crosstalk between detectors that produces a slowly drifting band of noise and static starlike artifacts. The rolling band is most prevalent on detectors 22, 26, 44, and 58 and creates a mismatch in the sky estimate leading to a semiperiodic instrumental systematic that can resemble a photometric signature of a transit-like signal (see Table 13 from Van Cleve & Caldwell 2016). The 372 day heliocentric orbit of Kepler and quarterly change of the spacecraft roll results in the rolling band only being present once or twice per year for any given target star. Significant effort was invested to identify and eliminate false alarms with periods centered on 372 days without being overly aggressive against low-S/N Earth-like transits (Thompson et al. 2018). Nonetheless, visual examination of the left panel of Figures 6 and 7 of Thompson et al. (2018) shows an overall increase in planet candidates centered on 372 days. As noted by Burke et al. (2019), extra care and analysis is needed when assessing the statistical properties of planets in this regime, and this analysis would strongly benefit from a directed study of noise properties and follow-up observations from facilities such as the Hubble Space Telescope and Plato. Moreover, low-S/N long-period PCs can also be produced by a few systematic dips in a lightcurve that line up to produce a signal that looks transit-like. Such chance alignments are common for TCEs that appear to transit just three or four times, but their frequency declines with five or more transits (Mullally et al. 2015).

We computed minimum periods of each of the seven monotransit PCs in multis by examining their target’s lightcurve to determine the shortest possible period for which no additional transits would have occurred at times when Kepler obtained good data (i.e., was observing and photometric noise was not excessive). These lower bounds, as well as the (crude) estimates of the orbital periods using the duration and shape of the lightcurve, together with the stellar properties and an assumed circular orbit (the absolute value of the negative P listed for these planets in Table 1) and the periods of the longest-period observed companion planets, are given in Table 3. None of these monotransiting planets in multis have minimum periods significantly shorter than the period estimates from lightcurve analysis, which would be the case if their transverse orbital velocity at the time of transit was larger than that of a planet on a circular orbit with the minimum calculated period. Note that all of the multitransiting PCs in these seven systems, apart from KOI-2525.01 and 4307.01, have been verified as planets and given Kepler numbers.

Figure 7 provides a compact sketch of the architectures of all 709 multiplanet systems discovered by Kepler. Systems are grouped into panels by the number of planets detected and sorted within each panel by the orbital period of the innermost planet. Planetary radii and the presence of detected TTVs are also indicated for planets in systems with four or more PCs. Note that all 81 systems with four or more planets have at least one planet with P < 13.5 days, and only two such systems lack planets with P < 10.4 days. The smallest period ratio of planet candidates plausibly orbiting the same star is 1.1167, between the two small ∼0.75 R_⊕ PCs KOI-3444.04 (P = 14.15 days) and KOI-3444.01 (P = 12.67 days); no other pair of Kepler PCs orbiting the same star has a period ratio smaller than 1.15.

Zoom In Zoom Out Reset image size

Figures 1 and 7 show that there are now two Kepler systems with more than six transiting planet candidates. One of these is the familiar eight-planet KOI-351 (Kepler-90) system (e.g., Lissauer et al. 2014; Shallue & Vanderburg 2018). With the addition of KOI-2433.08 from Shallue & Vanderburg's (2018) list, a second Kepler target now has eight KOIs, which we elaborate upon in the following paragraphs. According to Table 1, the 10 and 15 day planets, which were validated by Rowe et al. (2014) as Kepler-385 b and c, both have S/N ∼ 28. The 56 day planet has S/N = 16.4 and was validated as Kepler-385 d by Armstrong et al. (2021). The S/Ns of the other five KOIs range from 11.1 to 14.3, above the S/N > 10 required as one of many tests for planet validation of PCs in multis by Rowe et al. (2014).

The 0.6 day KOI-2433.05 has disposition FFFF. ¹⁷ Its lightcurve clearly exhibits secondary events indicative of an eclipsing binary. Furthermore, pixel-level data show the periodic dimming to be spatially offset from the target star. Thus, we do not consider this USP KOI further.

The three validated planets plus the 28 day KOI have dispositions PPPP. The disposition of the 6 day KOI is PPNP; its S/N = 12.4. The 86 day KOI has PPFN; its S/N = 11.2. The 3.4 day KOI-2433.08 has PNNN; its S/N = 11.1.

The outer three planet candidates have neighboring pairs with period ratios nominally placing them just wide of first-order mean-motion resonances (MMRs; as does the pair of validated planets orbiting interior to this threesome), increasing the likelihood of them being real planets (Lissauer et al. 2014). Indeed, it is possible that the validated pair of planets is locked within a two-body resonance and that the three outer planets librate within a three-body resonance. Period ratios between the planets all exceed 1.5, which is sufficient for system stability provided the planets have masses typical for their sizes and small eccentricities, which are the norm for Kepler planets in systems of high multiplicity (Figure 20). In sum, we consider KOI-2433 as having seven strong planet candidates. Nonetheless, validation of all of these candidates to well above 99% probability of representing true planets is beyond the scope of this paper.

Figure 9 compares the histograms of singles versus multis as functions of S/N. Note that there are similar numbers of multis and singles over a wide range in S/N, but singles dominate for both the smallest values of S/N and the largest ones. Examining the distributions more quantitatively, Kepler found almost 50% more planet candidates in singles than in multis, but Figure 9 shows that the numbers of singles and multis are nearly equal across the S/N range 25–180. Below S/N = 12 and above S/N = 300, well over twice as many singles as multis have been identified; intermediate ratios are found in transition regions (see Figure 9). The predominance of singles at high S/N is primarily accounted for by the paucity of large planets, especially hot Jupiters, in multis (Latham et al. 2011; Steffen et al. 2012).

Zoom In Zoom Out Reset image size

The excess of singles at low S/N is probably caused by a combination of the following effects, the first two of which are related to Kepler multis being a highly reliable subsample of the Kepler planet candidate population (Lissauer et al. 2011b, 2012, 2014; Latham et al. 2011; Rowe et al. 2014). (1) A larger fraction of singles are false alarms (this was not the situation in 2014 because of the more aggressive search for PCs in multis (Rowe et al. 2014), but probably is the case now because of the more automated procedure used to find and vet KOIs in recent catalogs). (2) Planet candidates with low S/N cannot be tested as rigorously as can PCs with high S/N, so a larger fraction of eclipsing binaries and other FPs are included in the PC list, and low S/N PCs are more common among targets with a single transit-like pattern than around targets with more than one such pattern (Lissauer et al. 2011b; Latham et al. 2011; Lissauer et al. 2012, 2014; Rowe et al. 2014). (3) When multiple planets transit, the detectability of planets other than the highest-S/N candidate by the Kepler pipeline is reduced significantly (Zink et al. 2019), especially if the highest-S/N planet has low S/N, so it is less likely that other transiting planets in the system have been detected. (4) TTVs, which are detected in more than twice as large a fraction of multis compared to singles (Table 1 and Section 3.4), reduce MES, but if they are accounted for in our model fits, they do not reduce S/N. (5) Perhaps real differences between the populations, such as longer-period planets having lower S/N, all other factors being equal, combined with neighboring companion planets typically being within a factor of ∼2–3 in period and planets with longer periods being less likely to transit unless typical relative inclinations of planetary orbits decrease significantly as periods increase. Determining what fraction of the difference is caused by item 3 is important both to a comparison between multis and singles and to understanding the fidelity of the sample of single-planet candidates, but it is beyond the scope of this work.

Searching through photometric time series for transiting planets may yield false positives, a term that conventionally means a real astrophysical signal within the same detector pixels but caused by something other than a planetary transit. For the purpose of computing occurrence rates, a subclass of FPs is signals resulting from transits of real planets that are not hosted by the intended target star, such as the planets observed to transit KOI-119 (Section 2.1). Such blending causes the planetary radius to be significantly underestimated (as the target star is invariably the one providing the most photons absorbed by the pixels), and the stellar host type and other parameters can be incorrect, to the detriment of statistical occurrence efforts. These real planets, which we classify as PCs, should therefore be termed false positives for the purposes of computing planetary occurrence rates. (The number of Kepler planet candidates identified as likely to orbit stars other than the Kepler target star is quite small, and all prime suspects that we know of are discussed within the first half of this subsection.)

In the case of candidate multiplanet systems, it could be that each periodic signal is due to a real planet, but these planets are not orbiting the same star. Recognizing the planets’ reality, we do not use the word “false” here but instead call these systems “split multis.” Although they are real planets, split multis can be a source of contamination for dynamical studies. Given that the orbital periods of planet candidates span a very large range, random planets around different stars will not necessarily appear strange when (mis)interpreted as orbiting the same star. In some fraction of cases, however, we would notice that their periods are too close together to be stable if interpreted as being around the same star. Given the steep dependence of occurrence rates on planet size, split multis are expected to be most common for binary stars of similar luminosities. Therefore, when studying multiple-planet systems, it can be advantageous to restrict one’s attention to a subset of Kepler targets that have been filtered to minimize contamination from binary stars with similar luminosities (e.g., He et al. 2020; Hsu et al. 2021).

Here we discuss a few potential split multis. KOI-284, first listed in the catalog of Borucki et al. (2011b) and introduced here in Section 2.1, includes the planet candidates KOI-284.02 (with orbital period P = 6.415 days) and KOI-284.03 (with P = 6.178 days), both of which appear to be larger than Earth. Lissauer et al. (2011b) noted that if both of these planet candidates represented planets orbiting the same stellar host, then for any reasonable densities, their proximal orbits would lead to dynamical instability on a short timescale. Thus, they do not represent planets in the same planetary system; this system is the prototypical Kepler example of a split multi. Further investigation revealed that the target “star” is actually a binary system with nearly identical components (Lissauer et al. 2012), and both of these PCs were subsequently validated as true planets, one orbiting each star, by Lissauer et al. (2014). These two planetary systems are now known collectively as Kepler-132. There are two additional validated planets, with orbital periods of 18 and 110 days. However, it is not yet known which member of the stellar binary any of the planets orbit, apart from the two 6 day period planets needing to orbit different stars.

Two planet candidates apparent in the lightcurve of KOI-2248, denoted KOI-2248.01 (P = 2.818 days) and KOI-2248.04 (P = 2.646 days), were first listed in the Batalha et al. (2013) catalog, and they were highlighted as a split multi by Fabrycky et al. (2014). Neither of these KOIs has ever been dispositioned as a false positive in any KOI catalog, including our own (Table 1). Two other planet candidates in the system were listed in DR25supp, KOI-2248.02 (P = 9.49 days) and KOI-2248.03 (P = 0.762 days). Shallue & Vanderburg (2018) listed a “new” candidate associated with the same Kepler target and P = 4.745 days that we have included in our catalog and dubbed KOI-2248.05 (Section 2.3). None of the KOIs in the system has yet been verified as a bona fide planet. One member of the pair of planet candidates with similar periods, KOI-2248.04, has S/N = 8.4 in our catalog, which is quite peculiar given that it was identified so early in the Kepler mission, calling its veracity into question. The other member of that nominally unstable pair, KOI-2248.01, has a fairly low DR25 disposition score of 0.895, but we find that it has a respectable S/N = 16.5. The 9.5 day signal, KOI-2248.02, has an unacceptably low S/N = 4.5, so we classify it as a false alarm, probably an alias of KOI-2248.05, which has a period almost exactly half as long.

Thus, KOI-2248 hosts four planet candidates, two of which (both somewhat suspect for other reasons) could not represent planets orbiting the same star, as a system containing both of these putative planets would be dynamically unstable. High-resolution imaging of the target star KOI-2248 has been carried out by Furlan et al. (2017), whose Table 8 (accessed by Vizier, 2018 July 16) lists KOI-2248 with two stellar near neighbors on the sky plane detected by WFC3 (Hubble Space Telescope): one (denoted B) that is 0.169 mag fainter in the F555W filter band (a nearly equal-brightness companion) with a separation of just 0148, so it could well host one of the P ∼ 2.7 day PCs. The other stellar sky-plane neighbor (denoted C) is 3872 away and 5.318 mag fainter, so it is not likely to be the host of either planet. The only survey using WFC3 listed in Table 1 of Furlan et al. (2017) is that by Gilliland et al. (2015) and Cartier et al. (2015), but neither of these latter papers addresses KOI-2248, so we suspect that the results were posted to the Kepler Follow-up Observing Program website but not published in the refereed literature.

Although Table 1 does not list any proximate-period split multis other than those in KOI-284 and KOI-2248, as discussed above, an early draft version of this table listed the planet candidate KOI-521.02 with an orbital period P = 10.82 days, which would place it too close to the Neptune-sized PC KOI-521.01 (P = 10.16 days) for dynamical stability. We therefore reexamined the Kepler lightcurve of this target and found that alternate transits of KOI-521.02 had been missed, so the actual period of this PC is only half of what is listed in previous catalogs, P = 5.41 days. The Q1–Q16 KOI catalog was the first to list KOI-521.02 and dispositioned it as an FP (Mullally et al. 2015). In the subsequent catalogs, DR24, DR25, and DR25supp, it was listed as a PC, albeit one with a weak S/N (initially 8.7 and later 7.3; with the corrected period, it has a more respectable S/N = 9.7). The estimated orbital periods of both KOI-521.01 and 521.02 changed by less than one part in 10⁴ between previous catalogs and were similarly close to these estimates in early drafts of our catalog. The TCE searches for DR24 and DR25 both identified KOI-521.02 with the correct 5.41 day period. However, because this TCE had a period that differed by a factor of 2 from a previously cataloged KOI on the same target, and the orbital phases matched, it was assigned the previous period in both of those PC catalogs.

A systematic search for period aliasing was done in Section 5.4 of the Batalha et al. (2013) catalog paper. However, it is unlikely that the error in the initial period estimate for KOI-521.02, which was not identified as a KOI until a later catalog search, would have been recognized had we not given it special scrutiny because it appeared to be part of a split multi. This example suggests that some nontrivial fraction of low-S/N PCs in our catalog could have listed periods that err by a factor of 2. Several other PCs have had their estimated periods changed by a factor of roughly 2 subsequent to their first appearance in an official Kepler KOI catalog, most notably KOI-730.03, which was initially listed with a period of half its true value (Borucki et al. 2011b) that would have placed it within a 1:1 (co-orbital) resonance with KOI-730.02, but the additional scrutiny that this putative pair of Trojan planets received quickly led to its estimated period being corrected by Lissauer et al. (2011b). Using data from the entire Kepler mission, the DR24 TCE table lists KOI-730.03 with its correct period, with a large MES of 17. So in the calculation below, we do not count it as an alias corrected by dynamical considerations.

Note that PCs with estimated periods (nearly) identical to or a factor of 2 different from the period of another KOI of the same target may suffer from the discarding of data surrounding the transit of the first candidate (Schmitt et al. 2017), a type of aliasing that is not an issue when periods are not commensurate. This effect contributed in the case of KOI-2248.02 but not for KOI-730.03, whose transit phase differed substantially from that of its commensurate sibling. Both KOI-730.03 and KOI-521.02 had their periods adjusted only after receiving additional attention due to apparent co-orbital planets or unstable systems. We do not know of other PCs whose orbital periods were corrected for dynamical reasons.

Motivated by these examples, we next estimate the number of period aliases that likely remain in the sample of multis. The number of PCs with measured periods in multis that have at least one other PC with a measured period is 1781. We took each of the pairs of planets in the same target, 1610 in total, multiplied the lower orbital period by 2, and counted how often that change would make the pair unstable. Following Lissauer et al. (2011b) and Fabrycky et al. (2014), planetary masses were taken as (R_p /R_⊕)^α M_⊕, where α = 3 for R_p < R_⊕ (sub-Earths), α = 2.06 for R_⊕ < R_p < R_Saturn, and finally, M_Saturn for R_p > R_Saturn. Pairs of planets were deemed unstable if their difference in semimajor axis was less than their mutual Hill separation (Gladman 1993). No higher-order accounting of stability (for triples, for instance) was performed for this calculation. Stability requirements for systems of ≥three planets are more restrictive, but cannot be expressed in such simple terms (see, e.g., Petit et al. 2020 and Lissauer & Gavino 2021).

Of these 1610 mock pairs, 281 (17.5%) were unstable. Since only one out of 1610 real pairs of planets was found (via instability) to be an alias, the estimated rate for a planet candidate to have an aliased period is (1/281) ≈ 0.36%. This rate is small enough that we can safely neglect the rate of systems becoming unstable due to multiple PCs being affected by aliases in the same system. Among planets with multiple transits in multitransiting systems, we expect roughly 0.0036 × 1781 ≈ 6 aliased planets, five of which are yet undiscovered, though since this is based on only one such detected system, it should be considered an order-of-magnitude statement. If the systems with just a single transiting planet with a quoted period also have this aliasing problem at the same rate, then 0.0036 × 2550 ≈ 9 are also aliased. Aliasing may occur at a different rate among singles, however, as the S/N distribution differs (Figure 9), and the data set when searching for additional transits needs to be censored in certain areas.

Using a similar approach, we update the estimated rate of split multis from Fabrycky et al. (2014). We choose pairs of (P, M_p /M_⋆) values of all of the planet candidates and determine that 453,281/9,419,770 ≈ 4.8% are unstable. ¹⁸ Having two detected unstable pairs (via the split multi channel) out of 1610 PC pairs, we estimate that 2/0.048 ∼ 42 of our pairs may actually be split multis. Thus, split multis are likely a larger contaminant for the study of planetary system architectures than are period aliases, even though only ∼2.6% of sampled pairs in multis are expected to be planets orbiting different stars.

The above estimates suggest that, among planets in multis, there are ∼7 times as many split multis as period aliases. A factor of 2 comes from twice as many split multis having been identified. A slightly larger factor is because period aliases are more likely to result in instability, since it is fairly common for a pair of planets in the same system to have a period ratio near 2 (Figure 7; see also Fabrycky et al. 2014), but the overall distribution of periods is quite broad (Figure 13), so few pairs of randomly selected planets have a period ratio near unity.

The estimates of the number of period aliases calculated in this section do not account for the aliases found for planets with periods ≲1 day, analogous to those noted in Section 2.3, which are caused by Kepler automated pipeline searches for TCEs being limited to signals with periods >0.5 day. Aliases among USP planets are very unlikely to be found via the techniques discussed in this section because only one planetary system (Kepler-42) is known to have more than a single planet with P < 2.25 days (e.g., Steffen & Farr 2013; Lissauer et al. 2023).

A total of 100 KOIs, none of which are in multis and 73 of which have P > 10 days, are dispositioned as “R” in Table 1, indicating that they were rejected as planet candidates solely due to their estimated size. The placement of our upper bound on the estimated radius of a body that we classify as a planet candidate (Section 2.3) is somewhat arbitrary, so we consider KOIs that are rejected based on size alone together with planet candidates (KOIs vetted as “P”) for all studies presented in this subsection.

Figure 10 contrasts the fractional cumulative distributions of the impact parameters of small and large planets in singles and multis. Small planets are clearly underrepresented for b very close to unity because near-grazing transits of small planets are difficult to detect, and they are quite rare for larger impact parameters because R_p /R_⋆ ≳ b − 1 is a requirement for a transit to be observable. Therefore, when comparing radius distributions, we only consider planets with estimated impact parameters b + σ₊(b) < 0.95. (Because we report planet sizes using the modes of the distributions, we do not need to adopt the stricter b < 0.8 cutoff used by Petigura 2020.) We also exclude from our analysis in this subsection those candidates with S/N < 12 because the population of low-S/N candidates has more FPs and typically higher fractional uncertainties on estimates of R_p ; this cut removes the four PCs around targets for which solar parameters were assumed, and for which planetary radii are especially poorly estimated. Planets that are excluded from the counts by these cuts are nevertheless included when determining the multiplicity of the system in which companion planets that meet these criteria reside.

Zoom In Zoom Out Reset image size

Figures 11 and 12 compare the size distributions of ensembles of planets in systems of different multiplicity. In all cases, we lump together systems with three or more PCs to have adequate numbers of PCs for statistically robust results.

Zoom In Zoom Out Reset image size

Giant planets are more common among Kepler singles than among Kepler multis (Latham et al. 2011). Nonetheless, when the cumulative size distributions for singles and planets in multis are normalized to unity at R_p = 5 R_⊕ (Figure 11), the curves for planets up to this size are very similar. Over the size range R_p < 5 R_⊕, the differences between the distributions are of marginal statistical significance, and for R_p < 2.5 R_⊕, the distributions of singles and planets in multis are consistent with being drawn from the same population.

There are, however, various biases in the discovery of planet candidates in multis as opposed to singles that could allow the actual distributions of transiting planets with R_p < 2.5 R_⊕ to have some size dependence. The biases that make the size distribution of small planets appear more prevalent in singles than in multis include: detecting a PC in a Kepler lightcurve reduced the amount of data used by the Kepler pipeline to search for additional PCs and thereby lowered the efficiency of detecting any other transiting planets associated with the same target star, reducing the probability of finding more PCs, especially small ones (Zink et al. 2019); geometric factors imply that a larger fraction of planets of a given orbital period transit large stars than small stars, and small planets are difficult to detect around large stars. By contrast, less photometrically noisy, brighter, and/or smaller stars make all (but especially small) planets easier to detect, yielding a bias toward detecting multiple small PCs around the best target stars. Searching for additional planet candidates has at times been more aggressive for targets with at least one candidate already identified, and some searches for transiting planet signatures, such as that of Shallue & Vanderburg (2018), have focused exclusively on lightcurves of targets already known to possess planet candidates.

Plotting a similar comparison between the size distributions of planets in systems with two planet candidates versus those in higher-multiplicity systems (right panel in Figure 11) shows a slight excess of super-Earth-sized planets relative to (sub-)Neptunes in systems with three or more PCs, with the distribution of two-planet systems lying between those for single planets and those for high-multiplicity systems. While the differences between singles and multis over the entire range in radii are highly significant, both the Kolmogorov–Smirnov (K-S) test and the Anderson–Darling (AD) test show marginally significant differences between 2's and 3+'s. These marginal differences persist when restricting to the range R_p < 5 R_⊕, but such differences are no longer evident if restricting to R_p < 2.5 R_⊕, even though the majority of PCs in multis are smaller than 2.5 R_⊕.

The divergence between the curves in the left panel of Figure 11 comes in gradually as R_p increases. There is no sign of divergence below 4 R_⊕, but it is plainly there above 5 R_⊕. Nonetheless, it is clear that multis are deficient relative to singles for planets that are larger than Neptune. Note that because Kepler has detected far more small planets than large ones, errors in estimates of R_p could well mean that a small fraction of ∼3 R_⊕ PCs with overestimated sizes contribute a substantial fraction of the population of apparently Neptune-sized objects, and the actual transition between the multis-rich population of “small” planets that are ≲1% H/He by mass and the multis-poor population of gas-rich planets may occur closer to 4 R_⊕. Indeed, although this transition appears to occur at a somewhat larger radius than that of the radius cliff, which is the sharp reduction in the overall occurrence rate of Kepler PCs observed near 3 R_⊕ (Kite et al. 2019; Hsu et al. 2021) that manifests as a reduction in slope in all of the curves in Figure 11, the break in the ratio of the number of multis to that of singles could be coincident with the radius cliff.

Figure 12 compares cumulative size distributions of large planet candidates (and KOIs that we dispositioned as R because they failed our upper radius cutoff) in multis versus singles and in two-PC systems versus higher-multiplicity systems. The size distributions of singles and multis are essentially indistinguishable over the range 4.5–10 R_⊕, but the number of PCs in multis in this range is only a little more than half the number in singles, whereas similar numbers are present for smaller planets. Very few PCs in multis have R_p > 12 R_⊕ ≈ 1.1 R_Jupiter, but plenty of singles have radii larger than 12 R_⊕ (e.g., Santerne et al. 2016). The divergence of the curve for singles with P > 10 days from that for all singles shows that almost half of the members of the plotted population with R_p > 12 R_⊕ orbit within the period range of inflated hot Jupiters. Most of the excess at longer periods (as well as some at short periods) is probably caused by false positives, which could be nearby (on the sky plane) eclipsing binaries or transits of the target star by ultracool dwarfs that are too faint to show an occultation (sometimes referred to as a secondary eclipse) deeper than can be explained by heating of the dayside by radiation from the primary star or which travel on sufficiently eccentric and inclined orbits that no such occultation occurs. This was our motivation for classifying KOIs that otherwise would have been considered PCs that have P > 20 days and R_p > 1.2 R_Jupiter ≈ 13 R_⊕, i.e., significantly, albeit not substantially, larger than this boundary, with the disposition “R.” Further investigation of KOIs vetted “R” is a topic worthy of investigation by observers interested in small stars within eclipsing binary systems but is beyond the scope of this work.

Zoom In Zoom Out Reset image size

Overall, there appears to be an abundant population of planets with sizes less than 3 R_⊕ (in agreement with Kite et al. 2019; Hsu et al. 2021), 44% of which are in multis. Kepler found a much less abundant population of giant planets; 29% within the size range 5 R_⊕ < R_p < 10 R_⊕ reside in multis, whereas the population of larger objects is dominated by singles that include hot Jupiters and FPs. Boundaries are smoothed over by a combination of radius errors and true fuzziness. Note that the size range from 5 to 10 R_⊕ includes a huge diversity of planets from superpuffs (currently only known in systems with multiple transiting planets because no cool, very low-mass/low-density Kepler transiting planets have RV mass measurements) to exo-Jupiters that are more enriched in heavy elements than the prototype present in our solar system. However, all planets in this size range are clearly gas-rich, with H/He abundances by mass of the same order as or larger than that of astrophysical metals.

We now compare the distributions of the orbital periods of Kepler single-planet systems, planets in two-planet systems, and planets in systems of higher multiplicity. As in Section 3.3, we consider only planet candidates with S/N > 12 but nonetheless count objects that are classified as PCs yet are rejected from these samples due to an S/N below this threshold in determining the multiplicity of a planetary system. However, here we do not include monotransit PCs and KOIs with dispositions of R in our sample, and apart from the portion of our analysis wherein we compare the period distributions of different ranges of PC sizes, we do not impose any restriction on estimated impact parameters.

Figure 13 displays the orbital period distributions of single Kepler PCs, planets in two-planet systems, and PCs in systems with three or more transiting planets. The vast majority of PCs within multis, ∼90%, have orbital periods between 2 and 100 days, whereas 78% of singles are in that same period range (10% of singles have periods less than 2 days, and ∼12% have periods longer than 100 days). The difference is even larger when restricting to the shortest periods (P < 1.6 days), at which only ∼20% of the Kepler PCs have known transiting siblings. A K-S test shows highly significant differences between the distributions of singles and multis (p-value ≈ 10⁻⁶), but one comparing two-planet systems with higher multiplicity fails to show significance, with p-value ≈ 0.27. These results reinforce the findings of Lissauer et al. (2014) and Rowe et al. (2014). Geometric factors reduce the probability of longer-period planetary siblings of a transiting planet also transiting when viewed from the solar system. The tendency of hot Jupiters to lack nearby companions (Figure 5) contributes to the smallness of the fraction of short-period planets residing in multis; larger period ratios for neighboring PCs with short periods (perhaps due to tidal decay of the orbits of very short period planets) and tides driving very short period PCs to the host star’s equatorial plane also likely contribute. See Appendix B of Lissauer et al. (2014) for a more comprehensive discussion of possible additional causes of the paucity of very short period planets in multis.

Zoom In Zoom Out Reset image size

The normalized distribution of orbital periods of planets with exactly one companion PC is very similar to that of planets in higher-multiplicity systems for periods up to ∼20 days (Figure 13). The fraction of PCs with two or more companions having periods in the range 20 days < P < 80 days is larger than that of PCs possessing one transiting sibling, with a compensating deficit of PCs with multiple companions for P > 80 days, but despite this visual divergence, the overall differences between the two period distributions are not statistically significant (neither the K-S test nor the AD test allows us to reject the hypothesis that the curves were drawn from the same distribution at the 95% confidence level). Nonetheless, since the two-PC systems are intermediate between single PCs and high-multiplicity systems, there may well be real differences that are obscured by the small number statistics. He et al. (2019) compute conditional occurrence rates of an additional putative planet as a function of both the period and radius of the detected and putative planets.

Figure 14 shows the period distributions of various subsets of Kepler PCs that have been grouped by planet size and system multiplicity. Of the eight curves in Figure 14, seven conform to the following trends. For a given size range, singles have a broader period distribution (more planets at both very short and very long periods) than do multis, with the cumulative fractions crossing “near” 10 days. For all size ranges considered in multis and for nongiant singles, larger planets tend to have longer periods. The exceptional curve is for giant PCs in singles, a larger fraction of which are detected at short periods than is the case for mid-sized PCs in singles. The tendency to observe smaller transiting planets at shorter periods is consistent with observational biases. Mid-sized PC with P ≲ 2 days are quite uncommon (the hot Neptune desert), consistent with H/He envelopes being stripped from close-in planets with M_p ≲ 20 M_⊕.

Zoom In Zoom Out Reset image size

Figure 15 shows the distributions of orbital periods of planet candidates with and without TTVs, with the PCs also being categorized as singles or members of a multi. Here, we consider PCs to have TTVs if our lightcurve fitting prefers solutions with TTVs, they are listed as having TTVs in the Holczer et al. (2016) catalog, and/or they are classified as having strong or moderate TTVs by Kane et al. (2013; 8 or 9 overall rating). ¹⁹ Note that only 7% of single PCs with S/N > 12 have TTVs that meet our criteria, whereas 15%–20% of such planets in multis have TTVs. ²⁰ This result is consistent with Xie et al. (2014), who found TTV rates that grow with transit multiplicity. For both singles and multis with P < 200 days, we find that TTVs are more likely to be observed in planets having longer periods, with the period differences being larger in singles than in multis. For self-similar (scale-invariant) planetary system architectures, both TTV amplitudes and near-resonant superperiods increase linearly with orbital period, so the dependence of the fraction of PCs with TTVs on orbital period may be explained in whole or in part by observational selection effects (small TTV amplitudes for the shortest-period planets and small observational baselines compared to TTV superperiods for long-period planets).

Zoom In Zoom Out Reset image size

We next examine the “cumulative” fraction of planet candidates with inner companions as a function of planetary orbital period. The green curves in Figure 16 show , the fraction of transiting PCs with periods less than the value specified on the horizontal axis that have at least one transiting sibling on an interior orbit. This fraction can be calculated from the formula

where N^P<Period gives the total number of PCs with P < Period and represents the number of such PCs having one or more transiting companions with a period shorter than its own period. The resulting distribution (in green) increases at any orbital period where a PC has an inner sibling, and (once its value exceeds zero) decreases where a PC lacking inner companions is added to the count, since the denominator accounts for all PCs, including singles and PCs within multis with no inner neighbors.

Zoom In Zoom Out Reset image size

Similarly, we examine the fractions of PCs with outer companions. The orange curves in Figure 16 show , the fraction of transiting planets with at least one transiting sibling with P > Period, which is given by

The terms on the right-hand side of Equation (5) are defined analogously to those in Equation (4), but note that in this case, planets are added to the distribution starting from the longest period and moving to the shortest. To zeroth order, the solid orange and green curves appear to be mirror images of one another, albeit a bit stretched out toward longer periods. The symmetry and “reflection” near 10 days and somewhat broader shape at long periods all mimic trends visible in the right panel of Figure 13.

The solid curves in Figure 16 can be compared to subsamples (dashed curves) that show the fraction of large planets (R_p > 5 R_⊕) that have transiting siblings, where siblings of any size are included. It is well known that giant transiting planets, especially hot Jupiters, rarely have companions that also transit (Latham et al. 2011; Steffen et al. 2012). The dashed curves in Figure 16 quantify the differences, showing (among other things) that large PCs are only one-third as likely to have outer companions and two-thirds as likely to have inner companions as are Kepler PCs overall (note that no effort has been made to attempt to correct these numbers for possible detection biases). Although large planets overall are substantially less likely to have outer companions, large and small planets with P ≳ 30 days are equally likely to have one or more outer companions.

The improved estimates of stellar, planetary, and transit parameters in our new catalog (Table 1) enable us to characterize the eccentricity distribution of various subsets of the Kepler planet candidates using the distribution of period-normalized transit durations. As accurate period estimates are required for this analysis, monotransits (planet candidates with only a single detected transit) were excluded from the analysis in this section. Our analysis presented below shows that there is no evidence for changes in the eccentricity distribution of Kepler PCs as a function of the host star effective temperatures over the range from 4000 to 6200 K. Similarly, we see no dependence on period for the eccentricity distribution for PCs with orbital periods greater than 6 days. In contrast, we show that there are significant differences in the eccentricity distribution as function of planet size and the number of PCs detected by Kepler in a given system. We find marginal evidence suggestive of a trend for planets with a known companion within a factor of 2.04 in orbital period to have a smaller eccentricity than more widely spaced planets. Below, we describe the specific comparisons and statistical tests performed to support these findings.

In order to compare the eccentricity distributions of various subsets of Kepler planet candidates, we begin by measuring the transit duration for each PC. We calculate the posterior mode of the average of the full transit duration, T_1,4, and the duration of the “flat bottom” portion of the transit, T_2,3, normalized by the analogous predicted transit duration for the measured orbital period and host star properties, assuming a circular orbit and central transit (b = 0). Using the notation introduced in Section 2, the normalized transit duration, τ, is given by

For planet candidates with b ≥ 1 − R_p /R_⋆, T_2,3 = 0, and we exclude these cases from our analysis due to difficulty in precisely constraining physical parameters for grazing and near-grazing transits. The particular definition of the normalized transit duration given in Equation (6) was chosen for robustness in measurement, as well as its independence (to first order) of the value of R_p /R_⋆. By defining τ relative to the duration for b = 0 (rather than the best estimate of b), we avoid a dependence on b, which is advantageous because estimates of the value of impact parameters can have significant measurement uncertainties (Section 2.2 and Table 1).

Planets with τ > 1 (when accounting for measurement uncertainties) must transit while the planet–star separation exceeds the semimajor axis, and there can be an interesting constraint on the pericenter direction ω (e.g., Dawson & Johnson 2012). In most cases, τ is comparable to or less than unity, and there is only a minimal constraint on the marginal distribution of ω, since the transit duration could be shortened due to either the planet being near pericenter or the impact parameter b ≠ 0 (or both).

Star–planet orbital planes are isotropically distributed and thus randomly oriented relative to Kepler’s line of sight; hence, the intrinsic distributions of the impact parameters (within the range b ≲ 1) and the pericenter angles (of all planets, not just transiting ones) are nearly uniform. Therefore, the distribution of τ provides a useful probe of the eccentricity distribution of a population of transiting planets (Ford et al. 2008). In practice, the distribution of observed impact parameters is affected by detection biases, since shorter transits that occur for larger b provide less time in transit to accumulate signal. Fortunately, this is a relatively modest effect for most PCs, since the changes in transit duration, and hence transit signal, are typically small, and most detected PCs have an S/N much greater than required for detection.

Multiple studies have begun to characterize the eccentricity distribution based on the observed period-normalized transit duration (τ) distribution. Moorhead et al. (2011), Kane et al. (2012), Plavchan et al. (2014), and Xie et al. (2016) focused on using the τ distribution to characterize the eccentricity distribution (as opposed to modeling the several observed properties at once, as in, e.g., Mulders et al. 2018; Zhu et al. 2018; He et al. 2019; Sandford et al. 2019; He et al. 2020; MacDonald et al. 2020; Yang et al. 2020). The strength of conclusions from these early studies was limited due to the uncertainty in the host star density. While random measurement errors can be easily incorporated into the analysis (Ford et al. 2008), the potential for systematic error is more concerning. For example, Moorhead et al. (2011) showed a trend for the τ distribution to broaden with increasing host star temperature, which could be attributed to either the eccentricity distribution changing with host star temperature or the errors in host star densities increasing for stars that have had time to evolve far from the zero-age main sequence. Such concerns helped to motivate follow-up campaigns to characterize host star properties using high-resolution spectroscopy (Fulton & Petigura 2018) and more detailed stellar modeling (Berger et al. 2020), both of which incorporate improved distance measurements from Gaia.

To assess the effects of potential systematic errors in the host star density on the eccentricity distribution, we focus our analysis on planets whose host star properties are available from either Fulton & Petigura (2018) or Berger et al. (2020), both of which represent dramatic improvements of stellar properties from those derived using photometric data (e.g., the KIC). (Asteroseimic densities are expected to be even more accurate but are available for only a substantially smaller subset of planet candidates; Van Eylen & Albrecht 2015; Van Eylen et al. 2019.) While the stellar properties estimated using high-resolution spectroscopy (Fulton & Petigura 2018) are likely more accurate than those with lower-resolution spectroscopy, they are not available for a substantial fraction of the stars in the full sample (e.g., most faint stars with a single known transiting planets and stars whose planets were identified after the CKS survey). However, the Fulton & Petigura (2018) properties are available for a substantial majority of the hosts of systems with multiple transiting planets.

Figure 17 compares the cumulative distribution of normalized transit durations (τ values) based on using the host star properties from Fulton & Petigura (2018; CKS) and Berger et al. (2020). The orange and green curves are for an overlap sample for which both sets of stellar parameters are available. The thick blue curve shows the results using the Berger et al. (2020) properties for the full sample. (The result for all transiting planet candidates for which CKS parameters are available is not plotted because it is visually indistinguishable from the overlap sample using stellar properties from CKS.) The distributions of τ values for planets using host star properties from Fulton & Petigura (2018) show small, but statistically significant, differences from those using the stellar properties of Berger et al. (2020). Most of this difference arises due to the fact that the CKS sample is significantly less complete than the Berger et al. (2020) sample, particularly for targets with a single transiting planet candidate, and the distributions using the overlap sample are statistically indistinguishable.

Zoom In Zoom Out Reset image size

In order to minimize the risk of systematic biases, we perform nearly all of our subsequent analyses of the τ distribution based on stellar properties from Berger et al. (2020), even when parameters from Fulton & Petigura (2018) are available. The single exception is for comparing the distributions between multiple transiting planet systems of differing multiplicity, where we make comparisons using each set of stellar parameters (since the CKS sample has much better completeness for hosts of multiplanet systems than for single-planet hosts). We also applied cuts so that our subsequent analysis only includes PCs with a measured orbital period, transit S/N ≥ 12 (so other transit parameters are well measured), impact parameter b < 1 − R_p /R_⋆, host star temperature between 4000 and 6600 K, planet radii estimated to be smaller than 12.5 R_⊕, and reported uncertainty in the stellar density of less than 25%.

Having defined a sample of 2762 planets for which systematic biases should be minimal, we performed several checks. We verified that there were no significant differences in the τ distribution if we increased the thresholds for the minimum transit S/N to be included in our primary analysis. Similarly, we confirmed that using a fixed maximum impact parameter of 0.95 or greater would not affect our conclusions.

Motivated by Moorhead et al. (2011), we divided the sample into four bins based on host star temperature (Figure 18) and performed a four-sample AD test of the null hypothesis that each subsample was drawn from the same distribution. (The AD test is usually more powerful than the more common K-S test. The K-S test is most sensitive to a shift of the distribution but significantly less sensitive to differences in the shape of the wings.) The AD p-value was <3 × 10⁻⁵. The highest-temperature bin is clearly the most discrepant from the other three. If we exclude this bin, the p-value from a three-sample AD test is 0.016, much less extreme, but still low enough to reject, at the level comparable to 2.4σ, the null hypothesis that the τ distribution is the same for the three remaining subsets of host stars with 4000 K ≤ T_eff ≤ 6200 K.

Zoom In Zoom Out Reset image size

Moorhead et al. (2011) also saw significant differences between planets orbiting stars with temperatures above and below the Kraft break (≈6200 K). (Main-sequence stars with effective temperatures greater than the Kraft break have negligible convective envelopes, resulting in dramatically reduced tidal dissipation in the star compared to cooler main-sequence stars.) However, uncertainty in stellar parameters led them to focus on stars with T_eff ≤ 5100 K, rather than risk confusing changes in the τ distribution due to the eccentricity distribution with those caused by measurement uncertainties and selection effects. Thanks to improved stellar parameters made possible by Gaia distance measurements, we no longer find differences in the τ distribution for stars cooler than the Kraft break and see a more significant difference between the τ distribution of planets with stars on either side of the Kraft break.

These differences are unlikely to be due to uncertainties in stellar properties for stars with 6200 K ≤ T_eff ≤ 6600 K, since imposing strict cuts on uncertainty in stellar densities has a minimal effect on this subset. The observed differences could be partially due to the reduced efficiency of tidal dissipation in hotter stars with no significant convective envelope. However, we caution that the differences do not go away, even if we focus only on planets with periods greater than 10 days, for which tidal effects during the main sequence would be small.

Therefore, subsequent analyses of the τ distribution are focused on planets with host stars with 4000 K ≤ T_eff ≤ 6200 K, so as to minimize the risk of systematic bias (e.g., due to small planets being more readily detectable around bright stars). These restrictions leave us with 2485 planets. For the analyses below, we only consider those PCs that passed all of these cuts in determining system multiplicity. However, we do consider PCs that were rejected by these cuts in determining the smallest period ratio to a neighboring planet in Section 3.5.5.

3.5.1. Fitting the Eccentricity Distribution

We attempted to fit multiple simple analytic models to the observed τ distribution. For each analytic model, we generate simulated populations of planets assuming a uniform distribution of and , where is based on the value reported in Table 1 for each planet. We sample the planet–star radius ratio, orbital periods, and stellar densities from those of the observed sample and add measurement noise based on the reported uncertainties (parameterized as mixture of two half-Gaussians). We weight each simulated planet by its geometric transit probability, which accounts for the dependence on the e and ω drawn for each planet. (However, this does not account for how differences in transit duration affect the detection probability conditional on the planet transiting, as done in He et al. 2019, 2020). We find that a Rayleigh distribution of eccentricities that is truncated to be less than 1, which is best fit with a Rayleigh parameter of 0.053, is not sufficient to reproduce the observed τ distribution (see Figure 19, red dashed curve). Using a small Rayleigh parameter underpredicts the number of extreme τ values, while using a large Rayleigh parameter results in too broad a distribution. This led us to consider a continuous mixture of Rayleigh distributions, where the weights for each Rayleigh parameter are proportional to a Rayleigh distribution (with Rayleigh parameter 0.043), i.e., a Rayleigh of Rayleighs, as in Section 6.1.2 of Lissauer et al. 2011b. While this results in a slight improvement in the fit for long-duration transits, it does not significantly improve the fit for τ in the range of 0.8–1.0 (see the left panel of Figure 19, green dotted–dashed curve).

Zoom In Zoom Out Reset image size

More detailed modeling of the joint probability density function (PDF) of multiplicity, inclination, and eccentricity distributions suggests that the joint PDF is not simply the product of the distribution for each quantity individually (Millholland et al. 2017; He et al. 2019, 2020; Yang et al. 2020). Therefore, we do not attempt to perform a detailed characterization of the uncertainties in such a model.

While the transit duration distribution can provide a powerful constraint on parameters given an assumed functional form for the eccentricity distribution, it has much less statistical power for comparing different functional forms for the eccentricity distribution. This is illustrated by the substantial differences between the two histograms in the right panel of Figure 19, despite the very similar predictions of the two models for the transit durations (dashed and dotted–dashed curves in left panel).

3.5.2. Transit Duration versus Multiplicity

Long-term orbital stability and planet formation models suggest that the eccentricity and mutual inclination distributions of planets depend on the multiplicity of their host planetary system (Pu & Wu 2015; Bartram et al. 2021; Gratia & Lissauer 2021). Indeed, previous studies of the observed transit duration ratio distribution find evidence that the mutual inclination distribution decreases with the multiplicity of the inner planetary system (He et al. 2020; Yang et al. 2020). Therefore, in Figure 20, we compare the τ distributions for (i) systems with a single known transiting planet candidate (“singles”), (ii) systems with two known transiting PCs (“doubles”), (iii) systems with three known transiting PCs (“triples”), and (iv) systems with four or more known transiting PCs (“high multiplicity”).

Zoom In Zoom Out Reset image size

Both the K-S and AD tests strongly reject the null hypotheses that the τ distributions of planets smaller than 5 R_⊕ are the same when comparing singles and doubles (p_KS ≈ 3.3 × 10⁻⁴, p_AD ≈ 1.5 × 10⁻⁵). Comparing all multiples to singles, both the K-S and AD tests very strongly reject the null hypothesis that the τ distributions are the same (p < 10⁻¹⁴), confirming the result that Xie et al. (2016) obtained for a smaller sample of Kepler PCs whose host stars had been characterized using spectra obtained by LAMOST. When comparing the τ distributions of PCs in doubles to those of PCs in higher-multiplicity systems, the p-values for the K-S and AD tests are ≈1.3 × 10⁻⁵ and ≈1.4 × 10⁻⁶, respectively. (If using the CKS stellar properties, then the p-value for the K-S test decreases to ≈2.4 × 10⁻⁶, but the p-value for the AD test does not change significantly.) The significance is strengthened by the clear pattern of the τ distribution becoming more highly concentrated near unity as one moves from singles to doubles to higher-multiplicity systems. When comparing the τ distributions of PCs in triples to those of PCs in systems with ≥four detected PCs (using CKS stellar parameters), the p-values for the K-S and AD tests are ≈0.01 and ≈0.009, respectively.

Note that the observed transit duration distributions for multiple-planet systems are subject to complex observational biases due to a combination of transit S/N and geometric transit probabilities that can be correlated within a planetary system. Properly accounting for these biases requires a full forward model that accounts for the joint distribution of planet sizes, orbital periods, eccentricities, and inclinations. For example, He et al. (2019, 2020) find that the mutual inclination distribution is narrower for systems with more detected transiting planets. However, a narrow distribution of mutual inclinations does not imply a narrow distribution of impact parameters. Even small mutual inclinations can cause significant Δb between planets (by construction, since Kepler detects planets with b ≲ 1). Further, the Δb can have either a positive or negative sign. Indeed, we have verified that the distribution of maximum-likelihood estimates of bs is not more narrowly peaked for widely spaced multiple-PC systems than for closely spaced multiplanet systems (nor that of systems with only a single detected PC). Therefore, we conclude that the difference in normalized transit duration distribution shown in Figure 20 is primarily due to differences in the eccentricity distributions between systems with one, two, or more detected transiting PCs. The strength of these differences is consistent with the change in eccentricity distribution as a function of the number of detected transiting planets resulting primarily from the constraints imposed by long-term orbital stability (He et al. 2020).

3.5.3. Transit Duration versus Orbital Period

We next compare the τ distribution for planets as a function of orbital period. Dividing the sample at 11.8 days (near the median orbital period of our sample), the p-values for the K-S and AD tests are ≈0.092 and 0.045, respectively. Figure 21 partitions the sample into five subsets with boundaries at 2, 6, 12, and 24 days, for which the five-sample AD test p-value is 0.15. The biggest difference across subsets is that the two shortest-period subsets have the sharpest τ distributions (implying lower eccentricities). If we combine PCs with periods of 0–2 and 2–6 days, then the p-value for a four-sample AD test is 0.08. If we perform a two-sample AD test for PCs with periods of 0–6 days and PCs with period in 6–1200 days, then the p-value for an AD test is 0.012. It is tempting to attribute the differences in transit duration as primarily due to the increased efficiency of orbital circularization for small orbital periods. However, we caution that the size distribution of the PCs with P < 6 days is weighted toward significantly smaller values than the size distribution of PCs with larger periods, since Kepler has greater sensitivity for detecting planets at shorter orbital periods.

Zoom In Zoom Out Reset image size

3.5.4. Transit Duration versus Planet Size

Splitting the cumulative τ distributions for planet candidates by planet radius at 2.16 R_⊕ (near the median planet size of our sample), the p-value for the K-S test is 0.15, and the p-value for the AD test is 0.07. Thus, neither of these tests finds statistically significant differences between the two samples.

For the left panel of Figure 22, we divide the population into subsets based on theoretically motivated size bins with boundaries of 0.5, 1.0, 1.6, 1.8, 2.7, 5.0, 9.0, and 12.5 R_⊕. Visual inspection shows a similarity between the distributions of the two smallest size bins; the three middle size bins also appear to have similar size distributions to one another, as do the distributions for the two largest size bins. Furthermore, statistical tests do not find any significant differences among the distributions within any of these three subsets of planet size bins.

Zoom In Zoom Out Reset image size

Thus, we combine PCs into three size bins, 0.5–1.6, 1.6–5, and 5–12.5 R_⊕, in the right panel of Figure 22. The hypothesis that all three of these bins have the same underlying distributions can be strongly rejected, with p-value ∼ 10⁻¹¹ (using a three-sample AD test), and comparing any pair of these three results in a p-value of <10⁻³ (K-S tests) or <10⁻⁴ (AD tests), showing highly significant evidence for differences in the eccentricity distributions between these three broad size bins. Further supporting this interpretation, PCs with R_p ≤ 1.6 R_⊕ have a particularly concentrated τ distribution (i.e., small eccentricities), while that for PCs with R_p > 5 R_⊕ has a relatively large tail (i.e., larger eccentricities), with PCs 1.6 R_⊕ < R_p ≤ 5 R_⊕ having an intermediate distribution of τ values.

These differences are consistent with the observation that planet sizes (and likely masses) are correlated within a planetary system (Ciardi et al. 2013; Millholland et al. 2017; Weiss et al. 2018; He et al. 2020), together with the theoretical idea that more massive planets excite larger eccentricities in neighboring planets via planet–planet scattering and/or secular perturbations (Laskar 2000; Ford & Rasio 2008; Johansen et al. 2012; Pu & Wu 2015). However, they run counter to (although do not necessarily conflict with) the expectation that within a given planetary system, equipartition of the angular momentum deficit would typically lead to more massive planets having smaller eccentricities (Lissauer 1995). To contrast the eccentricities of large and small PCs within the same system, we looked at all systems with two or more PCs with 6 days < P < 1200 days (the lower limit being chosen to minimize the effects of tidal damping; see Figure 21) that meet our criteria for analysis in this section and found that in 136 out of 273 cases (50%), the largest such planet has a larger value of τ than the smallest planet. This implies that there is not a strong preference for PCs within a given inner planetary system to have different eccentricity distributions (after one removes planets potentially affected by tidal circularization). These conclusions are not affected by restricting the sample to PCs with orbital periods greater than 8 days (rather than 6 days).

3.5.5. Transit Duration versus Spacing between Orbits

We divided planets in multiple-planet systems (excluding the split multis KOI-284 and KOI-2248; see Section 3.2) into two or four nearly equal-size subsets based upon their period ratios with their nearest detected neighbor in . The period ratio boundaries are the first three quartiles of the period ratio distribution, 1.65, 2.06, and 2.94. Very closely spaced planet pairs typically need to have small eccentricities to avoid close encounters (with potential for exceptions related to resonances). As expected, the period-normalized transit duration distribution is more sharply peaked for PCs with a nearby neighbor. However, there was not a statistically significant difference in the τ distribution between these subsets of PCs based on performing a two- or four-sample AD test using our nominal sample of PCs. However, if, as shown in Figure 23, we exclude planets with orbital periods less than 8 days (intentionally larger than 6 days to be confident that we exclude all PCs that are likely to be significantly affected by tidal circularization), then the two-sample AD test rejects the null hypothesis that the observed normalized transit duration distributions for the different subsets of planets (based on period ratio to nearest detected neighbor) are drawn from the same underlying distribution with a p-value of 0.037.

Zoom In Zoom Out Reset image size

We also compared the duration distributions of planet candidates in multiple-planet systems near first-order MMRs with those of other PCs in multiple-planet systems and did not find statistically significant differences. However, we note that the number of PCs near resonances is small (214), so nontrivial differences in the distributions may become evident when larger samples become available for study (e.g., combining Kepler, K2, TESS, and PLATO data).

Precession causes the time interval between successive periapse passages to differ from the time interval required to travel 360° in azimuth (which is the reason that anomalistic periods differ from orbital periods). This non-Keplerian behavior leads to eccentric planets spending a little less time at some radial distances between successive transits, resulting in variations in time intervals between successive transits that average out only over timescales much longer than that of Kepler observations (Agol et al. 2005).

The eccentricity of a planet near resonance can be viewed as a superposition of its free and forced eccentricity vectors in the (, ) plane. The precession of the forced eccentricity is relatively rapid and accounts for some of the TTVs that are found among Kepler planets; its effects are partially accounted for in period estimates. The precession of the free eccentricity is much slower. It is another source of TTVs, which we refer to as secular TTVs. However, if Δϖ T_obs/P ≪ 1, i.e., precession is ≪2π during Kepler’s prime mission, then the secular TTVs will not be recognized and thus cannot be removed from/accounted for in calculations of the planet’s period. Secular precession of the planets’ free eccentricities is unrelated to resonances and generally has a period much longer than the baseline of the Kepler observations, so it is not accounted for in our estimates of mean periods and the uncertainties thereof (nor was it in previous Kepler planet candidate catalogs). Consider a planet with eccentricity e ≪ 1 whose longitude of periapse precesses by Δϖ ≪ 1 rad orbit^–1 and that transits near periapse. During the interval of time between successive periapse passages, the planet moves through 2π + Δϖ rad. As the planet completes one radial oscillation during this interval, its average angular velocity is equal to the long-term average value. However, from Kepler’s second law, we know a line from the star to the planet sweeps out angle at a rate 1 + e times as fast near periapse. Thus, the (measured) time interval between successive transits, P, is related to the mean orbital period and the difference between the longitude of periapse and that of the transit midpoint by

As the cosine term in Equation (7) integrates to zero, this effect averages out over one precession period of the eccentricity. Nonetheless, the long-term mean orbital period can differ significantly from the mean period between successive transits during the Kepler era, even for planets having TTVs that are too small to be observable during the epoch of Kepler observations.

As planets in Kepler multis typically have eccentricities of one to several percent (Jontof-Hutter et al. 2016; Xie et al. 2016), , but can nonetheless be much larger than the formal uncertainties in measured orbital periods. The difference between these periods is so small that it can be ignored for our comparisons of the period distributions of various subsets of the Kepler sample (Section 3.4) and for most studies of the period ratios of two planets. However, the (ill-quantified) errors in our estimates of are important for studies of the distribution of three-body resonances (J. J. Lissauer et al. 2024, in preparation).

TTVs have been used for detailed dynamical analyses of a small fraction of the Kepler multiplanet systems. Most of the publications presenting these studies list osculating orbital periods at an epoch near the midpoint of the Kepler data. For several landmark systems, we integrated numerous (in most cases, 101) samples of system parameters from the MCMC chains deduced via photodynamical or TTV analyses of the Kepler data to compute estimated TTs during and after the Kepler epoch. Table 4 compares orbital periods (in most cases, at an epoch near the midpoint of Kepler observations) from the dynamical solutions to long-term average periods of these planets that we computed by integrating these dynamically solved systems, our standard estimates of the orbital periods obtained via a best constant-period fit to observed TTs (Table 1), and (where available) to the average orbital periods estimated in the sinusoidal fits of Holczer et al. (2016). The values of “P at epoch” in Table 4 for the planets of Kepler-29 and Kepler-60 represent osculating orbital periods at the epochs listed in the subsections below, following our dynamical fits to the TTs reported by Rowe & Thompson (2015).

Table 4. Orbital Periods, in Days, of Selected Well-studied Planets Estimated Using Various Methods

Comparison of Estimates of Orbital Periods of Select Planets
KOI	Kepler-	P (Table 1)	P (Holczer et al. 2016)	P at Epoch
157.06	11 b	10.304031 ± 0.000026	⋯	10.30260 ± 0.00027	10.30391 ± 0.00004
157.01	11 c	13.024917 ± 0.000018	13.0249115 ± 0.0000014	13.02555 ± 0.00018	13.02507 ± 0.00004
157.02	11 d	22.687141 ± 0.000037	⋯	22.68546 ± 0.00037	22.68708 ± 0.00003
157.03	11 e	31.995517 ± 0.000067	31.9954254 ± 0.0000009	31.99834 ± 0.00052	31.99555 ± 0.00004
157.04	11 f	46.68587 ± 0.00027	46.6857474 ± 0.0000036	46.6933 ± 0.0018	46.6855 ± 0.0005
157.05	11 g	118.37857 ± 0.00025	⋯	118.38089 ± 0.00057	118.3782 ± 0.0005
277.02	36 b	13.84899 ± 0.00034	13.848692 ± 0.000006	13.849194 ± 0.00004	13.848063 ± 0.0002
277.01	36 c	16.231949 ± 0.00026	16.232080 ± 0.000001	16.231774 ± 0.00002	16.232628 ± 0.0002
500.05	80 f	0.9867860 ± 0.0000013	⋯	0.9867873 ± 0.0000044	0.9867862 ± 0.0000012
500.03	80 d	3.0721523 ± 0.0000058	⋯	3.07253 ± 0.00029	3.0721293 ± 0.0000086
500.04	80 e	4.6453934 ± 0.0000084	⋯	4.64474 ± 0.00022	4.645410 ± 0.000014
500.01	80 b	7.0535287 ± 0.0000094	7.0535152 ± 0.0000013	7.05357 ± 0.00036	7.053570 ± 0.000025
500.02	80 c	9.521646 ± 0.000014	9.5216221 ± 0.0000013	9.52330 ± 0.00030	9.521525 ± 0.000047
500.06	80 g	14.64538 ± 0.00011	⋯	14.6503 ± 0.0018	14.6457 ± 0.0013
730.04	223 b	7.384456 ± 0.000072	⋯	7.38449 ± 0.00022	7.3845 ± 0.0002
730.02	223 c	9.848211 ± 0.000082	⋯	9.84564 ± 0.00052	9.84934 ± 0.00014
730.01	223 d	14.78701 ± 0.00018	14.7869296 ± 0.0000095	14.78869 ± 0.00029	14.7841 ± 0.0002
730.03	223 e	19.72434 ± 0.00033	19.725722 ± 0.000018	19.72567 ± 0.00055	19.7256 ± 0.0007
738.01	29 b	10.339236 ± 0.000056	⋯	10.33842 ± 0.00029	10.336927 ± 0.000025
738.02	29 c	13.28712 ± 0.00011	⋯	13.28841 ± 0.00053	13.290961 ± 0.000037
1474.01	419 b	69.7262 ± 0.0012	69.7281819 ± 0.0000004	69.7960 ± 0.0042	69.7869 ± 0.0454
2086.01	60 b	7.132950 ± 0.000041	⋯	7.13335 ± 0.00013	7.1325157 ± 0.000025
2086.02	60 c	8.918977 ± 0.000041	8.91867 ± 0.00020	8.91866 ± 0.00018	8.919029 ± 0.000004
2086.03	60 d	11.89825 ± 0.00010	⋯	11.89810 ± 0.00020	11.899566 ± 0.000070

Note. From left to right, the columns give the values presented in Table 1 of this work, Table 6 of Holczer et al. (2016), period at epoch (typically near the mid-time of Kepler observations) from dynamical fits to TTs, and long-term (averaged over the same intervals used for ordering the samples to select representative systems shown in Figures 25–31; 10⁴ yr for Kepler-419 b, 10⁶ days ≈ 2738 yr for Kepler-11's planets, 100 yr for Kepler-80, 1000 yr for planets in other systems) average periods computed by integrating the systems using samples of the initial conditions at epochs obtained from dynamical studies.

Download table as:  ASCIITypeset image

Period variations for one or more of the planets in each of these dynamically active multiplanet systems are shown in Figures 25–31. Each panel within these figures shows three samples, which we selected by ordering the solutions by the long-term average period of the first KOI found in the system (KOI number ending in .01), then selecting the median member of the list (usually the sample with the 51st longest value, in black), one with that planet’s period roughly one standard deviation shorter than the mean (17th sample, in red), and one with this planet’s mean period one standard deviation longer than the mean (85th sample, in blue).

Zoom In Zoom Out Reset image size

All of the systems that we analyze in this subsection have strong interactions among all or most of the known planets, leading to substantial TTVs. Three of these systems, Kepler-11, Kepler-36, and Kepler-419, do not have any planets known to be librating within orbital resonances. One system, Kepler-29, has two planets that are locked in a 9:7 (second-order) MMR. The other three systems considered below, Kepler-60, Kepler-80, and Kepler-223, each have three or more planets locked into a resonant chain that includes (zeroth-order) three-body resonances and probably first-order two-body MMRs (in some cases, it is not known whether or not the two-body resonance variables are librating).

To estimate the long-term periods of all known transiting planets in the abovementioned seven Kepler planetary systems, we ran simulations with an embedded Runge–Kutta Prince–Dormand integrator (Prince & Dormand 1981; gsl_odeiv2_step_rk8pd within the GNU Scientific Library, Gough 2009). From the TTs (more precisely, the times near inferior conjunction when the distance between the center of the planet and that of the star projected onto the plane of the sky reaches a minimum, as we typically do not have sufficient information on the impact parameter to know whether or not a transit actually occurs) simulated over the specified interval (typically 1000 yr) beginning with the start of Kepler observations, we determine the best-fitting linear ephemeris. The transit periods thus derived are given in the final column of Table 4. Due to numerical dissipation, our nonsymplectic code causes the orbital periods to decrease slightly; in a run for 1 million days with only the 10 day planet Kepler-11 b, P_b decreased by −1.16 × 10⁻⁶ day. For simulations with all six planets of Kepler-11, P_b averages a loss of −5.8 × 10⁻⁷ days, which is a systematic bias but is dwarfed by the statistical uncertainty of ∼4 × 10⁻⁵ days for that system. (We would advocate using a symplectic algorithm for integrations longer than those reported here.)

We present the results of our modeling in the rightmost columns of Table 4 and Figures 25–31. Note the differences in the mean period and period at the epoch of the various solutions that fit the observed data, as well as the differences of the periods averaged over 4 yr when taken starting at differing times for three representative solutions for each of the planets.

4.2.1. Kepler-11 = KOI-157

4.2.2. Kepler-36 = KOI-277

Kepler-36 has two planets, each more than four times as massive as Earth, on orbits moderately close to the 7:6 MMR. The planets are not in a low-order MMR, but their proximal orbits lead to strong dynamical interactions. The outer planet, which is a bit less than twice as massive as the inner one, nonetheless has 15 times the volume. Some planetary parameters allowed by short-term fits to the Kepler data in the system discovery paper by Carter et al. (2012) were subsequently eliminated by imposing the requirement for long-term stability (Deck et al. 2012).

We performed a photodynamical analysis of the data using the PhotoDynamical Multiplanet Model (PhoDyMM; D. Ragozzine et al. 2024, in preparation), similar to other analyses in the past (e.g., MacDonald et al. 2021). Each of the 101 samples from the posterior distribution was then integrated using REBOUND’s WHFast integrator (Rein & Tamayo 2015) with a time step of 2% of the inner planet’s orbit for 10⁶ inner planet orbits. Inspection of the final orbital states for all 101 systems showed that they were still on stable orbits very similar to those of the current system. While our integrations are substantially shorter than those of Carter et al. (2012), they are still much longer than the typical Lyapunov time of 10 yr (Deck et al. 2012). Therefore, it seems likely that the change in stability properties results from having a more accurate set of initial conditions that were derived from the full Kepler lightcurve, which included an additional year of short-cadence data.

In analogy with Figure 25, Figure 26 shows orbital period evolution for three of the 101 simulations, selected as described in the caption. Note that the orbital periods of both planets averaged over 4 yr intervals vary by almost 10 times as much as the uncertainties in their orbital periods measured during the Kepler epoch.

Zoom In Zoom Out Reset image size

4.2.3. Kepler-80 = KOI-500

Kepler-80 harbors six known transiting planets. The innermost is a USP planet that is (at least on the timescales relevant here) dynamically detached from its siblings. The middle four planets, with periods of 3.1, 4.6, 7.1, and 9.5 days, were studied extensively by MacDonald et al. (2016). Two years later, a sixth transiting planet, with P = 14.6 days, was discovered by Shallue & Vanderburg (2018).

All six planets were included in a full photodynamical analysis by MacDonald et al. (2021). This photodynamical analysis was performed using PhoDyMM (D. Ragozzine et al. 2024, in preparation), with all six masses allowed to float, but all the longitudes of ascending node were fixed at 0°. The middle four planets form a resonance chain, with each neighboring pair having a period ratio ∼1%–3% larger than either 3/2 or 4/3 and each neighboring threesome librating within a three-body resonance. The orbital period of the outer planet suggests that it, too, is a member of the resonance chain. However, transits of the outer planet are much shallower, and only 14% of the fits to the Kepler data by MacDonald et al. (2021) find it to be librating within a resonance.

As for Kepler-36, 101 samples were taken from the posterior distribution from PhoDyMM. Each of these mock systems was integrated for 100 yr (nearly 4 × 10⁴ orbits of the innermost planet). The 9th–14th numerical rows of Table 4 list the mean and dispersion of the periods of each planet in this sample at epoch T_epoch= 800.0 days and averaged over 100 yr. Figure 27 shows period variations of each of the planets derived from our integrations that used three samples from the posteriors calculated by D. Ragozzine et al. (2024, in preparation). The 4 yr averaged periods of the five planets in the resonance chain have fractional variations of ∼10⁻⁵–10⁻⁴.

Zoom In Zoom Out Reset image size

4.2.4. Kepler-223 = KOI-730

Kepler-223 is a system of four planets in MMR, where the three-body angles are consistent with libration, and the TTVs have a few-hour amplitude with a timescale of a few years (Mills et al. 2016). The planets have higher eccentricities than typical for closely spaced planetary systems, and the period ratios of the planets are extremely close to the ratios of small integers, with departures of order 0.1% or less (Lissauer et al. 2011b; Mills et al. 2016).

To understand how the observed TTVs manifest on a timescale beyond the Kepler time series, we draw samples from the posterior of the photodynamic fits to the data. Mills et al. (2016) produced a sample of solutions that ensure that the two Laplace angles, one relating the inner three planets and the other relating the outer three planets, librate with small amplitude over a 100 yr span and called it the C3 posterior. Values of their osculating orbital periods for Kepler-223's planets are given at T_epoch= 800.0 days.

Mills et al. (2016) ran N-body simulations of 300 systems from the C3 posterior for 10⁷ yr, with outputs every 10⁴ yr, and found all of them to be stable. We inspected the osculating semimajor axes over that time. In many cases, the oscillations in semimajor axes on 100 yr timescales are small but then grow by a factor of several or even an order of magnitude and simultaneously lose the libration of the three-body angles, indicating that these solutions were not in secular steady state.

We found that only 12 systems out of the 300 in the C3 posterior seemed to keep the same semimajor axis behavior from the first 100 yr for all 10⁷ yr. If the other trajectories were taken as models of the observed system, then the system must have been observed at a special time, which violates the Copernican Principle and thus is highly unlikely. Therefore, we consider that the 12 continually librating solutions likely represent better models of the system, and we restrict our attention to them for our analysis of long-term orbital period variations of this system.

The long-term (1000 yr) mean periods of each of Kepler-223's four planets and the standard deviations thereof are listed in Table 4. Figure 28 shows the period variations of three of these possible realizations, with black representing the system with the median mean period of KOI-730.01 (we broke the degeneracy caused by the even number of systems in the sample by selecting the one with a period closer to the mean of the 12 samples) and blue and red showing the sample systems for which KOI-730.01 has the second-shortest and second-longest mean values, respectively.

Zoom In Zoom Out Reset image size

On shorter timescales, due to orbital fluctuations, the period ratios cross back and forth across the nominal period ratios of 4:3 and 3:2—this is true when computing the ratios with either osculating periods (Mills et al. 2016, their Extended Data Figure 5) or modeled transit periods. The 4 yr average periods vary by a few parts in 10⁻⁴, and the period ratio for the inner pair of planets drops slightly below 4/3 for some portions of the simulated interval in each of the 12 samples. In contrast, the ratio of periods of the middle pair of planets stays slightly above these small integer ratios.

4.2.5. Kepler-29 = KOI-738

We took 101 samples from the TTV posteriors of Kepler-29 from Jontof-Hutter et al. (2021) and simulated TTs over 10⁶ days. The system contains two sub-Neptunes on proximate, dynamically interacting orbits with orbital periods close to the 9:7 MMR (Migaszewski et al. 2017).

In Figure 29, we see that Kepler observed Kepler-29 when the transit-to-transit periods of Kepler-29 b and c were near their extrema. Four-year mean periods vary by a few parts in 10⁻⁴, with the periods of the two planets being highly anticorrelated. The ratio of orbital periods over the Kepler baseline is 1.2851, slightly (about 1 part in 2000) less than that of the 9:7 commensurability (1.285714). The average period ratios over 1000 yr of the samples shown in Figure 29 are 1.285812 (17th), 1.285754 (51st), and 1.285729 (85th), respectively, all just above commensurability. Indeed, much of the variation averages out on decadal timescales, but 4 yr average period ratios oscillate about 9/7, and the long-term average period ratio in this system is several standard deviations away from the value implied by the orbital periods for the planets listed in our catalog.

Zoom In Zoom Out Reset image size

Holczer et al. (2016) found strong TTVs for both planets in KOI-738, but each planet’s TTs were fit by a polynomial rather than a sine wave, so mean orbital periods were not estimated. This fit was likely selected because the orbital periods of the planets oscillate with a periodicity of about 10 yr, and their TTVs during the 4 yr of the Kepler mission resemble parabolas; see the panels on the left in Figure 29.

4.2.6. Kepler-419 = KOI-1474

Two planets are known to orbit Kepler-419: a transiting inner planet (Table 1) on a quite eccentric orbit (e ≈ 0.8) and a nontransiting super-Jupiter that has a period almost 10 times as long but nonetheless induces large TTVs because of its high mass and the large eccentricity of the transiting planet’s orbit. The system is well characterized by both RV and TTV analysis (see Table 5 for the properties of the nontransiting planet). The transiting planet, Kepler-419 b, displays a quintessential “photoeccentric effect” (Dawson et al. 2012), wherein a lower bound on the eccentricity of a planet can be estimated from the shape and duration of the transit lightcurve. The period ratio of these two planets is ∼9.657.

Table 4 lists the value of the osculating orbital period for Kepler-419 b at T_BJD = 2454958 from Almenara et al. (2018). We took the 4044 posterior samples of the system parameters from the combined RV/photodynamical fits of Almenara et al. (2018) and ran simulations for 10,000 yr. For each of these samples, we computed a long-term orbital period by taking a linear fit to simulated TTs. We tabulated the average and standard deviation of these 4044 long-term periods. The transit-to-transit interval of the inner planet and its 4 yr running average for three of the samples are plotted in Figure 30. Note that the plots in this figure cover longer timescales than those for other planets considered in this subsection because the longer orbital periods of the planets lead to larger characteristic time intervals over which major variations are observed.

Zoom In Zoom Out Reset image size

At the present epoch, Kepler-419 b is close to periapse when its transits. The dips in orbital period near 2700 and 7300 yr occur when the planet passes through apoapse near the time when it transits. This illustrates the variations predicted from Equation (7) with ϖ ≪ 1.

We examined the intervals between times when the center of the nontransiting planet Kepler-419 c was closest to the sky-projected location of its host star Kepler-419 and closer to the solar system than is Kepler-419. These intervals varied with the same periodicity and opposite phase as the corresponding variations of the averaged transit-to-transit period of Kepler-419 b. However, unlike its transiting companion (right panel of Figure 30), this planet’s variations are nearly sinusoidal. These behaviors are consistent with Almenara et al.'s (2018) findings that the periapse locations of the two planets oscillate about antialignment with a small amplitude and that the orbital eccentricity of Kepler-419 c is small (e < 0.2), whereas Kepler-419 b has e ≈ 0.8.

4.2.7. Kepler-60 = KOI-2086

The Kepler-60 system contains three confirmed planets with periods between 7 and 12 days, with both neighboring pairs orbiting near first-order MMRs. The lightcurve also reveals an unverified planet candidate with an orbital period of 336 days that we do not consider in our analysis of the inner threesome.

We took 101 samples from the TTV posteriors of Kepler-60 from Jontof-Hutter et al. (2021). From simulations of these 101 samples, we estimated uncertainties on the period ratios of Kepler-60 c/b (5:4) and Kepler d/c (4:3) given the 17th, 51st, and 85th period ratio after averaging period ratios over 1000 yr simulations and sorting. We found that the inner pair has a period ratio of 1.250473 ± 0.000015, and the outer pair has a period ratio of 1.334176 ± 0.000027. Averaged over 4 yr, none of the neighboring planet pairs dropped below the ratio of small integers signifying their resonance in any of the three samples plotted in Figure 31.

Zoom In Zoom Out Reset image size

We have made special efforts, both in data analysis (Section 2.4) and theoretically (Section 4), to improve the accuracy of planetary orbital periods to aid in ephemeris predictions and dynamical studies. The values of P listed in Table 1 are generally of equal or higher accuracy than those in previous tabulations, with estimates for many of the planets exhibiting TTVs that are significantly improved. Fractional uncertainties quoted for the orbital periods of the majority of planet candidates listed in recent Kepler catalogs, including those presented herein, are <10⁻⁵, with values of ∼10⁻⁶ (corresponding to 2 minutes per 4 yr) being typical. These small uncertainties suggest (in some cases misleadingly) that ephemeris predictions for most Kepler planets are robust for decades to come. However, the tabulated periods and quoted uncertainties are for the mean times between midpoints of successive transits during the time interval in which transits were observed and do not reflect possible long-timescale TTVs (Section 4).

TTVs produce errors in estimates of some planets’ orbital periods that need to be accounted for in certain dynamical investigations and ephemeris predictions. Periodic sinusoidal TTVs with timescales that are short compared to the interval of Kepler observations largely average out and do not produce significant errors in estimates of orbital periods. TTVs with timescales comparable to the 4 yr interval of Kepler observations have been fit for dozens of Kepler planet candidates to estimate long-term average orbital periods by Holczer et al. (2016), and more detailed dynamical models have been used to estimate long-term average periods of a small number of well-studied planets, including the seven systems presented herein (Section 4.2, Figures 25–31). These figures show that in many cases, the 4 yr average transit-to-transit orbital period deviates from the long-term average orbital period by a factor many times as large as the formal uncertainty of the 4 yr average. Most Kepler planets that show large TTVs are near mean-motion orbital resonances with other planets. The largest effect for planets moderately close to two-body resonances is due to rotation of the forced eccentricity vector by resonant perturbations; the timescale of this precession for most Kepler planets is short compared to the 4 yr of Kepler observations, so the variations tend to average out.

Libration of planets locked in resonances typically occurs on timescales longer than the Kepler baseline, but most Kepler planets do not appear to be resonantly locked. The more general but smaller (during the era of Kepler observations) effect is caused by secular precession of the planets’ free eccentricities, which usually takes much longer than the 4 yr baseline of the Kepler observations to complete a revolution, so it is not accounted for in estimates of mean periods or uncertainties. This precession causes a discrepancy between Kepler-era mean orbital period and long-term mean orbital period to exist even for planets having TTVs that are too small to be observable during the epoch of Kepler observations, and the magnitude of this discrepancy increases with the eccentricity of the planet’s orbit. Thus, some systems/planets that do not show TTVs, for which an observer might assume that the linear ephemeris is reliable, could have deviations on longer timescales, although for most planets without clear TTVs, the Kepler predictions are likely to be very good. The main concern is systems where the variations on 4 yr timescales are too small to be detected but the amplitude of the decadal timescale TTVs is substantial. Because of the very complicated and poorly quantified selection biases of the Kepler sample, as well as small number statistics applying to some classes of dynamical configurations, we do not attempt to quantify the number of systems that exhibit the types of behavior seen within these systems. Nonetheless, it should thus be kept in mind that the actual uncertainties in future TTs have broad (albeit low) tails that are not captured by tabulated uncertainties and are growing in length and height with time until additional transits are observed.

We identify multiplanet candidates that have periods that are too close to each other to remain stable, which we use to estimate the percentage of apparent multiplanet systems wherein the planets are distributed between two blended stars as ∼2.6%. Similarly, we use an error of a factor of 2 in the estimated period of a planet candidate that we identified by stability considerations to estimate the number of orbital periods that are aliases of the true periods to be ∼0.36% (Section 3.2). Other evidence suggests that a somewhat larger fraction of planets with estimated orbital periods of ≲1 day actually complete two or more complete orbits within the period listed in catalogs (Section 2.3).

We find that the vast majority of PCs with low S/N are candidate single planets rather than being in multiplanet systems (multis). In contrast, for those with moderate S/N, there are similar numbers of PCs in multis and singles (Figure 9). Since the fraction of actual planetary detections that are in multis probably is similar for PCs with low S/N and those with moderate S/N, we suspect that a substantial fraction of these low-S/N single-planet candidates are false positives.

An early Kepler result was that the fraction of large planets among systems with multiple transiting planets is smaller than among lone transiting planets (Latham et al. 2011). A few years later it was pointed out that multis are more concentrated to the period range 1.6–100 days than are singles (Lissauer et al. 2014; Rowe et al. 2014).

The size distributions of singles and multis are quite similar over the range ∼0.5–3 R_⊕ (Figure 11), although there is a hint of a larger fraction of planets in multis below the radius valley at ∼1.7 R_⊕. The size distributions of singles and multis are also indistinguishable over the range ∼5–10 R_⊕ (Figure 12), although the fraction of planets with sizes 5–10 R_⊕ that are found in multis is only about two-thirds that among small planets; planets in the 5–10 R_⊕ size interval range from low-mass superpuffs with tens of percent H/He by mass to cool giant planets hundreds or even thousands of times as massive as the Earth; mature brown dwarfs and the smallest main-sequence stars also fall within this size range. The transition between this multis/singles abundance ratio is gradual in the range of 3–5 R_⊕, which may imply a fuzzy boundary or simply be the result of errors in estimated planetary sizes (primarily small planets having sizes overestimated, since there are far more small planets than large ones). Placing the boundary between “small” and “large” planets near the size of Neptune is consistent with the results of the contemporaneous study of Ghezzi et al. (2021), who investigated correlations between stellar metallicity and maximum radius of observed transiting planets. Above 10 R_⊕, the number of multis drops off steeply relative to singles, with few planets of radius R_p > 12 R_⊕ found in multitransiting systems. The concentration of the largest planets in singles is partly due to inflated hot Jupiters rarely having close companions, but the singles also appear to have inflated Jupiters at longer periods—perhaps on eccentric orbits that bring them close. Alternatively, it may be that most PCs significantly larger than Jupiter with P > 10 days are FPs caused by eclipsing binary stars. Both the size distribution and the period distribution of planets in two-planet systems are intermediate between the distributions of single planets and those in systems with more than two transiting planets.

We analyze the distributions of normalized transit durations (Equation (6)) to confirm the previous result that single transiting planets are more likely to have high eccentricity than are planets in multiply transiting systems. We extend this result by demonstrating that planets in systems with two transiting planets are typically more eccentric than those in systems with three transiting planets, and the orbits of PCs in systems with four or more transiting planets tend to be even less elongated (Figure 20).

Planets with orbital periods P > 6 days are typically more eccentric than short-period transiting planets. In contrast, we find no other clear trends in the eccentricity distribution with orbital period (Figure 21).

Transiting planets in the rocky size range (R_p < 1.6 R_⊕) have a lower average e than do sub-Neptunes and Neptunes, which in turn are typically less eccentric than planets with R_p > 5 R_⊕ (Figure 22). However, no such trend exists within the population of planets in systems with both large and small transiting planets.

It has been more than a decade since Kepler ceased its collection of data from its prime field of view. Nevertheless, the list of Kepler planet candidates remains the largest and most homogeneous collection of small and mid-sized exoplanets known. Our new catalog contributes to the understanding of Kepler planet candidates, especially with our focus on the information-rich systems with multiple transiting planets. Improvements in estimates of orbital periods and a better understanding of processes that alter apparent orbital periods on a variety of timescales advance our understanding of planetary dynamics and improve ephemerides for prediction of future transits. More accurate impact parameters, identification of correlations with multiplicity, and identifying trends with eccentricities also provide new avenues for research into the formation and evolution of planetary systems.