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

Jack J. Lissauer Space Science & Astrobiology Division
MS 245-3
NASA Ames Research Center
Moffett Field, CA 94035, USA  Jason F. Rowe Department of Physics and
Astronomy
Bishops University
2600 Rue College
Sherbrooke, QC J1M 1Z7, Canada  Daniel Jontof-Hutter Department of Physics
University of the Pacific
601 Pacific Avenue
Stockton, CA 95211, USA  Daniel C. Fabrycky Department of Astronomy and
Astrophysics
University of Chicago
5640 South Ellis Avenue
Chicago, IL 60637, USA  Eric B. Ford Department of Astronomy and Astrophysics
525 Davey Laboratory
The Pennsylvania State University
University Park, PA 16802, USA  Center for Exoplanets & Habitable Worlds
525 Davey Laboratory
The Pennsylvania State University
University Park, PA 16802, USA  Department for Astrostatistics
525 Davey Laboratory
The Pennsylvania State University
University Park, PA 16802, USA  Institute for Computational & Data Sciences
The Pennsylvania State University
University Park, PA 16802, USA  Darin Ragozzine Department of Physics and
Astronomy
Brigham Young University
Provo, UT 84602, USA  Jason H. Steffen Department of Physics and Astronomy
University of Nevada Las Vegas
4505 South Maryland Parkway
Las Vegas, NV 89154, USA  Kadri M. Nizam Department of Physics
University of the Pacific
601 Pacific Avenue
Stockton, CA 95211, USA  Department of Astronomy and Astrophysics
525 Davey Laboratory
The Pennsylvania State University
University Park, PA 16802, USA

(Received 2022 May 10; Revised 2023 September 30; Accepted 2023 October 2)

###### Abstract

We present a new catalog of Kepler planet candidates that prioritizes accuracy
of planetary dispositions and properties over uniformity. This catalog
contains 4376 transiting planet candidates, including 1791 residing within 709
multi-planet systems, and provides the best parameters available for a large
sample of Kepler planet candidates. We also provide a second set of stellar
and planetary properties for transiting candidates that are uniformly-derived
for use in occurrence rates studies. Estimates of orbital periods have been
improved, but as in previous catalogs, our tabulated values for period
uncertainties do not fully account for transit timing variations (TTVs). We
show that many planets are likely to have TTVs with long periodicities caused
by various processes, including orbital precession, and that such TTVs imply
that ephemerides of Kepler planets are not as accurate on multi-decadal
timescales as predicted by the small formal errors (typically 1 part in 106
and rarely $>10^{-5}$) in the planets’ measured mean orbital periods during
the Kepler epoch. Analysis of normalized transit durations implies that
eccentricities of planets are anti-correlated with the number of companion
transiting planets. Our primary catalog lists all known Kepler planet
candidates that orbit and transit only one star; for completeness, we also
provide an abbreviated listing of the properties of the two dozen non-
transiting planets that have been identified around stars that host transiting
planets discovered by Kepler.

Unified Astronomy Thesaurus concepts: Exoplanets (498); Exoplanet catalogs
(488); Transit photometry (1709); Exoplanet dynamics (490); Planetary theory
(1258)

††journal: The Planetary Science Journal††facilities: Kepler space telescope

##  1 Introduction

NASA’s Kepler spacecraft monitored a single star field for four years during
its prime mission, with a duty cycle of almost 90%. The principal objective of
the Kepler Mission was to take a statistical census of planets having orbital
periods of up to $\sim$ 1 year. The Kepler Project released eight catalogs of
planet candidates found during the mission (Borucki et al., 2011a, b; Batalha
et al., 2013; Burke et al., 2014; Rowe et al., 2015; Mullally et al., 2015;
Coughlin et al., 2016; Thompson et al., 2018). Each new catalog used more
sophisticated methods, and aside from the last one, each used more Kepler data
and listed more planet candidates than its predecessor. The Project’s latter
catalogs employed successively more automated procedures.

The primary goal of the Kepler Project’s final catalog of planetary candidates
(Thompson et al., 2018), often referred to as DR25 (which is an abbreviation
of Data Release 25), was to produce a listing of planet candidates (PCs) found
and vetted in a well-defined and reproducible manner for the exoplanet
community to use as input for studies of planet occurrence rates (e.g., Hsu et
al. 2019). As such, the data were processed in a highly automated manner, with
uniformity and reproducibility prioritized over using all available
information to identify and classify each individual potential planet
signature. Previously found planet candidates that were not identified by the
final search for transit-like patterns do not appear in the Thompson et al.
(2018) catalog. Furthermore, their vetting of individual candidates did not
include the hands-on treatment (e.g., examination of lightcurves and centroid
shifts by eye) that was a feature of the first six Kepler planet catalogs.

The neglect of hands-on vetting from previous catalogs ended up excluding
dozens of Kepler planets, (some of which are well-known and were verified with
high confidence) from Thompson et al. (2018)’s list of Kepler Objects of
Interest (KOIs; the integer number refers to the target star, whereas the
digits after the decimal point refer to the putative planet), and led to
classifying others as false positives (FPs) rather than as viable planet
candidates. One particularly relevant example is that the Thompson et al.
(2018) catalog is more biased against planets exhibiting significant transit
timing variations (TTVs) than are previous catalogs. This limitation leads to
a bias against planet pairs with near-resonant orbits, as foreseen by García-
Melendo & López-Morales (2011). An unpublished analysis of strategies for
detecting planets with large TTVs by A. Moorhead & one of us (EBF) found that
most such planets would naturally be detected by applying standard transit
search algorithms to $\sim$one-year-long subsets of the data, since large
near-resonant TTVs typically accumulate over multi-year timescales. Therefore,
the use of candidates from multiple searches conducted with different amounts
of data reduces the bias against planets with substantial TTVs. This
improvement in sensitivity to planets with large TTVs (i.e., TTVs comparable
to or exceeding the transit duration) is most significant for planets that
would, in the absence of TTVs, likely be detected using only 13 or 16 months
of data (the durations searched for the second and third Kepler PC catalog
releases).

Our new catalog is more analogous to the Kepler Project’s final cumulative
catalog, DR25supp, which was not presented in a refereed
publication111Documentation for DR25supp is available at:
https://exoplanetarchive.ipac.caltech.edu/docs/PurposeOfKOITable.html#q1-q17_sup_dr25.,
than to the DR25 catalog. Both the DR25supp catalog and our Table 1 differ
from the DR25 catalog in two key respects: the use of manual vetting and
including KOIs from multiple sources, including previous Kepler project
catalogs. For DR25supp, the cumulative DR24 catalog was combined with the DR25
catalog and the Kepler False Positive Working Group re-dispositioned all KOIs
whose dispositions were disputed in the last six project catalogs (listed as a
planet candidate in at least one catalog but listed as a false positive in at
least one other); apart from dispositions, the most recent properties were
listed. We began with the DR25supp catalog, added other KOIs from various
sources, and manually vetted select KOIs, as described in Section 2.3.

The Kepler mission defined a threshold crossing event (TCE) as a periodic
signature with multiple event statistic (MES) at least 7.1, where the MES is
effectively the signal-to-noise ratio of the putative planetary transits in
the folded lightcurve, as measured by the Kepler pipeline. The threshold of
MES $\geq 7.1$ was chosen to keep the expected number of KOIs resulting from
white noise small (see §2.3 for details). The Kepler pipeline looked for
planet candidates in the lightcurve of an individual target star one by one,
beginning with the signal possessing the largest MES and then performing the
search again on a lightcurve from which data at the times of the transits of
this KOI were removed. The process was repeated until no additional signal
with MES $\geq 7.1$ was found. Because data were removed from the lightcurve
prior to searching for additional candidates, the search for multis (systems
with multiple transiting candidates) is less complete than that for singles
(systems with only a single transiting candidate). This bias against multis is
analyzed quantitatively by Zink et al. (2019).

Various groups have published lists of additional Kepler planet candidates;
see §2.3 for details. Together with those appearing in the official Kepler
planet candidate tabulations, the total number of Kepler planet candidates
listed in one or more catalogs is $\sim$ 5000, although several hundred of
these have subsequently been re-classified as FPs. More than 2700 of these
planet candidates have been verified (either confirmed using radial velocity
data or via TTVs, or statistically validated as having a high likelihood of
being true planets) and assigned official Kepler planet designations such as
Kepler-11 g (Lissauer et al., 2011a).

We assembled our catalog using data from the final Kepler project planet
candidate catalog (Thompson et al., 2018), previous planet candidate catalogs
produced by the Kepler project, and planet candidate lists from other groups
(§2.3). We incorporated the improved stellar properties derived using distance
measurements by ESA’s Gaia spacecraft (Gaia Collaboration et al., 2018; Berger
et al., 2020) and, where available, spectroscopic measurements taken with the
Keck I telescope (Petigura et al., 2017; Fulton & Petigura, 2018). Figure 1
shows the distribution of orbital period vs. radius for our Kepler planetary
candidates based on the work reported herein.

Our goals in compiling a new catalog of Kepler planet candidates are to
provide a comprehensive listing of KOIs with significantly more accurate
vetting and to give improved estimates of planet properties. As described in
more detail in Section 2, our new listing relies on more homogeneous and
robust techniques to compute planetary parameters, removing previous biases
such as the dependence on orbital period estimates of planets exhibiting TTVs
with the amount of data analyzed when they were first announced. We also list
more planet properties and use more robust techniques to compute the values
and uncertainties of estimated planetary characteristics.

The primary advantages of using our new catalog are: We present the most
complete listing of Kepler planet candidates to date, based on the Kepler
project’s catalogs, community efforts, and our own analysis. We have provided
initial dispositions for new KOIs in our sample. We have also revisited
dispositions for those KOIs that were dispositioned as FP in DR25supp despite
being listed as a PC in DR24 or DR25, or having a Kepler number according to
NASA Exoplanet Science Institute (NExScI). We provide additional disposition
cuts based on S/N, mass measured from radial velocity variations, and derived
planetary radius, $R_{p}$.

The parameter values listed in our catalog (Table 1) are more accurate than
those in previous catalogs, with significant effort to systematically and
uniformly improve transit models and calculations of posteriors for model
parameters, including corrections for bias in impact parameter and the mean-
stellar density computed from the photometric model, $\rho_{\star{\rm c}}$
(Gilbert et al., 2022). Orbital periods have been revisited and in some cases
recomputed to address the complications of transit timing variations by
providing the best fit constant period to transit times observed by Kepler. We
investigated multiplanet systems with suspiciously close-period planet
candidates and corrected period-aliasing. Significant improvements in stellar
parameters from ground based followup and the parallax survey from Gaia have
been incorporated (Fulton & Petigura, 2018; Berger et al., 2020). This has
allowed for planetary parameters to be derived in a more uniform manner than
in other cumulative catalogs. We include a concise description of all
planetary characteristics listed (see §2.5) to allow for the community to
maximize the combined knowledge of exoplanets in the Kepler field-of-view.

For ease of reading, we often refer to planet candidates simply as “planets”.
Planets that are the sole transiting candidate of their host star are referred
to as “singles”, whereas the term “multis” is used for both systems with more
than one transiting planets and individual planets in such systems.

We present our catalog of Kepler planet candidates in Section 2. In Section 3,
we characterize the sample of planet candidates, compare the ensemble of
planet candidates in multis to those in singles, compare planets in two-planet
systems with those in higher-multiplicity multis, and quantify the reliability
of the sample of multis as representing true planetary systems. We investigate
the distribution orbital eccentricity for various subsets of Kepler planet
candidates (§3.5), improving on previous studies thanks to the enhanced
accuracy of the stellar densities and impact parameters in our catalog. We
find significant changes in the eccentricity distribution as a function of the
inferred size of the planet candidates (§3.5.4) and the number of Kepler
planet candidates detected around a given host star (§3.5.2). We also find
significant differences between planet candidates with orbital periods less
than 6 days and those with longer orbital periods (§3.5.3). We consider
various factors that can lead to orbital period variations of Kepler planets
(Section 4). We show that planets on eccentric orbits have variations in the
times between successive transits on time scales much longer than the 4-year
duration of the Kepler mission (§4.1). Section 4.2 analyses long-term
variations in mean orbital periods of planets within several Kepler systems
showing significant TTVs that have been solved for dynamically. We conclude
the main text by summarizing our principal results in Section 5.

We select which objects to include in our catalog of planet candidates and
list their properties to maximize accuracy on an object-by-object basis.
Therefore our selection criteria and various planetary properties are not
homogeneous, and _our planet candidate list is not appropriate for use as
input for planetary occurrence rate calculations_. Nonetheless, some aspects
of our derivation of planetary properties provide estimates that are more
accurate and at least as uniform as those found in previous studies.
Therefore, we also present a second set of planetary properties using a
uniformly-derived set of stellar parameters in Table 1 and outline a process
for utilizing some of the information tabulated therein for studies of
occurrence rates in Appendix A. Our primary planet candidate catalog (Table 1)
is restricted to transiting planets orbiting just one star. An abbreviated
catalog of non-transiting planets found (using TTVs and/or radial velocity
measurements) around stars with transiting Kepler planets is provided in
Appendix B.

##  2 Planet Catalog

In this section, we introduce our catalog of Kepler planet candidates and
describe the calculation of properties of the planets, as well as the sources
used for characterizing their host stars. Figure 1 displays the radius vs.
period distribution of the planets in our Kepler catalog and highlights the
abundance of multiplanet systems discovered. Multiplanet systems provide a
special opportunity to study the potentially rich dynamical history of
exoplanet formation and evolution. Our construction and review of the Kepler
exoplanet catalog focuses on orbital periods and the prevalence of multi-
planet discoveries.

We discuss stellar parameters in §2.1, transit models in §2.2, candidate
selection and catalog unification in §2.3 and the calculation and
interpretation of orbital periods in §2.4. Our planet catalog is presented in
§2.5. Weaknesses of this catalog and its previous incarnations are discussed,
including the impact of transit timing variations, relationships between the
historical Kepler catalogs, and the non-uniformity of candidate selection and
biases of some derived properties. Section 2.6 focuses in on the KOI-2433
system, which now has seven planet candidates.

###  2.1 Input Stellar Properties

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 California Kepler Survey (CKS)222Stellar parameters for KOI-1792
were chosen from Berger et al. (2020) despite the availability of CKS
parameters for that target for reasons specified in §2.3., which lists
properties of Kepler planet hosts that have both spectral measurements from
the Keck I telescope and well-determined Gaia properties, especially distances
(Fulton & Petigura, 2018). This list includes $\sim 60$% of the hosts of
multis as well as $\sim 60$% 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 $\sim 95\%$ of the
Kepler targets; this list accounts for most of the remaining planet 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, 5718, so we have adopted solar parameters with large uncertainties
($R_{\star}$ = $1\pm 1$ R⊙, $M_{\star}=1\pm 1$ M⊙, $\log{g}$= $4.5\pm 4.5$)
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 the
KOI-284 system consists of 2 nearly identical stars with a total of 4 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 §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.

###  2.2 Transit Models

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 bit 1,2,3,4,6,7,9,13,15,16,17 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 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 1 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); Thompson et al.
(2018). Briefly, a multi-planet 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
($\rho_{\star}$), fixed quadratic limb-darkening coefficients, photometric
zero point, the mid-transit time ($T_{0}$), orbital period ($P$), impact
parameter ($b$) and scaled planet radius ($R_{\rm p}/R_{\star}$). Eccentricity
was fixed at zero for these models; thus, $\rho_{\star}$ is replaced by
$\rho_{\star{\rm c}}$. The adoption of mean stellar density as a fitted
parameter assumes 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 measured center of transit times (TTs) 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 pre-calculated 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
multi-planet models used for lightcurve preparation do not simultaneously fit
transit parameters and center-of-transit times.

Figure 1: Orbital period vs. radius of Kepler’s planetary candidates. Those
planets that are the only candidate for their given star are represented by
black dots, those in two-planet systems as dark blue circles, members of
three-planet systems as green triangles, those in four-planet systems as light
blue squares, in systems of five PCs as yellow five-pointed stars, with six
PCs as orange six-pointed stars, the seven PCs associated with KOI-2433 as
pink seven-pointed stars, and the eight planets orbiting KOI-351 (Kepler-90)
as red eight-pointed stars. The legend lists the number of stars hosting each
multiplicity. The planetary candidates are listed in Table 1 (First letter of
disposition = “P”). Non-transiting planets (listed in Table 5) and
circumbinary planets are not included. Planet candidates only observed to
transit once (mono-transits) are not plotted because their orbital periods are
highly uncertain (see item #4 in the list presented in Section 2.5), but they
are accounted for in the multiplicity designation of their companion planets
and the total numbers of systems of each multiplicity given in the lower right
of the figure. KOI-846.01, with a radius $R_{p}=30.043$ R⊕ (see §2.3), falls
outside the plotting window. All planets to the right of the dotted gray
vertical line at $P=730$ days (as well as some of the planets with shorter
periods) transited only twice during the Kepler mission, and therefore were
not detected by the standard Kepler pipeline, which required a minimum of
three transits for a detection. It is immediately apparent that there is a
paucity of giant planets in multi-planet systems, especially giants with short
orbital periods $P<15$ days. The upward slope in the lower envelope of the
plotted points is caused by the low S/N of small transiting planets with long
orbital periods, for which few transits occurred during the time intervals
that Kepler observed. Adapted from a previous figure generously provided by
Rebekah Dawson.

We adopted 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 non-informative
prior for the impact parameter, which works well for non-grazing transits.
However, when $b>1$, the minimum value of scaled planetary radius, $R_{\rm
p}/R_{\star}$, required for a planetary transit grows linearly with $b$.
Uniform sampling would result in a bias towards 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 de-weighting the
model likelihood by $b^{2}$ when $b>1-$$R_{\rm p}/R_{\star}$, 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,
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
modelling techniques were improved and the methodology of reporting parameters
changed, such as the choice of reporting either maximum likelihood vs. 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, transit-depths
and overall better fidelity. However, there are two identifiable deficiencies
from this approach: a bias towards 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-minute) 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 towards 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 $b\sim 0.5$. 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-square 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 the 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 §2.5. Additional
improvements may result from improved modelling 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 $b\sim 0$ and $b>1$. Our efforts to
reinitialize fits and include priors de-weighting large impact parameters
produces a distribution of impact parameters more consistent with an isotropic
inclination distribution (Kipping & Sandford, 2016), although there are still
excess populations near $b=0$ and for $b\gtrsim 1$.

Figure 2: Comparison of the impact parameter ($b$) distributions for planet
candidates from this work (orange) to the values listed for planet candidates
in DR25 (green). Only planetary candidates with $b<1.2$ are shown; PCs with
$b>1.2$ represent $0.5\%$ of our sample and $1.4\%$ of the DR25 PCs.
Restricting the samples to those KOIs that are classified PCs in both catalogs
does not substantially alter either of the distributions.

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 4-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 non-interacting Keplerian orbit.
TTVs were handled by explicitly measuring the transit time of each individual
event, then re-sampling 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 transit time based on the reported mean
period from the best fit model. As the transit models evolved with each new
data release, the transit times 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 vs. 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 §2.5 for an explicit description
of how each model parameter is reported, §2.4 for extended discussion of
orbital periods during the Kepler epoch, and §4 for an analysis of period
variations over longer time scales.

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 star spots. (The latter two processes do not cause
variations in acutal 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 criteria for
selecting the solution with TTVs included was either the visual identification
of TTVs from examination of O-C diagrams or for the inclusion with 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 less than 0.2 days. The
model with TTVs shows a significantly deeper transit, hence inclusion of TTVs.
However, this may indicate that either 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. (2019),
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

${\rm S/N}=\sqrt{\sum^{n}_{i=1}\left(\frac{m_{i}-1}{\sigma}\right)^{2}},$ (1)

where $m_{i}$ is the calculated flux from the fitted transit model at each
observation $i$ with a total of $n$ observations, and $\sigma$ is the standard
deviation 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 §2.5.

Figure 3: Median signal to noise ratio for Kepler planet candidates as a
function of location in the period-radius plane. Each square represents a
factor of $\approx 1.1$ in radius and $\approx 1.2$ in period. The left panel
shows the total S/N, and the right panel shows the average (mean) S/N per
transit. The average S/N was calculated by dividing the total S/N by the
square root of the number of observed transits for each planet candidate. Note
the differences between the color scales of the two panels. The number of
planets represented in each colored square ranges from 1 to 32.

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 good S/N, non-
grazing transits and well measured $\rho_{\star{\rm c}}$. 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 parameters tables. This trend is expected because detection bias
favors planets that transit near the periastron of their orbits. A simple
application of Kepler’s 2nd 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
§6.2 of Rowe & Thompson 2015).

Figure 4: Comparison of the mean stellar density from our circular orbit
transit models ($\rho_{\star{\rm c}}$) to the mean stellar density from our
adopted stellar parameters ($\rho_{\star}$). The red diagonal line is for
$\rho_{\star}$= $\rho_{\star{\rm c}}$, and the two parallel darker red lines
show a bias of 10% in $\rho_{\star}$. The colored markers note the source of
stellar parameter: blue = DR25 (Thompson et al., 2018), green = Berger et al.
(2020) and orange = CKS (Fulton & Petigura, 2018). The left panel shows all
planetary candidates from our sample with 0.1 $<$ $\rho_{\star}$$<$ 10 g/cm3.
The right panel is restricted to planetary candidates that have S/N $>$ 10,
$b$ $<$ 0.9, and uncertainty (average of $\sigma_{+}(\rho_{\star{\rm c}})$ and
$\sigma_{-}(\rho_{\star{\rm c}})$) of less than 20%. Uncertainties are half-
widths of the 68.27% credible interval ($\pm 1\sigma$).

A visual examination of the right panel in Fig. 4 appears to shows a bias in
$\rho_{\star}$ 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 towards the inclusion of multiplanet systems. As shown in §3.5, planets
in compact multiplanet systems tend to have more circular orbits (Figs. 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
parameters 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.

###  2.3 Planet Candidate Selection

We pulled planet candidates from a variety of sources, including 9564 KOIs
from the cumulative DR25 supplement catalog333Retrieved from the NASA
Exoplanet Archive on 2022/10/27., ultra-short period (USP) planets from
Sanchis-Ojeda et al. (2014), long period ($P\gtrsim 1$ year) and transit
candidates only observed to transit once, which we refer to as mono-transits,
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
2 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, 8298-8303 are ultra-short period planet candidates (including the
3 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, 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, 597.04 are from
Shallue & Vanderburg (2018)444We classified 2061.03 from the machine learning
search as an FP because we recognized that it resulted from a poor masking of
2061.01, due to large TTVs. We did this by using Quasiperiodic Automated
Transit Search (QATS, Carter & Agol 2013), determining an approximate
ephemeris $T_{n}[BJD]=2454949.27+n\times 14.097+0.12\times\cos(2\pi/P_{\rm
ttv}(t-2455154))$, where $P_{\rm ttv}=1160$ days., 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
their predicted ephemeris. Table 5 of Caceres et al. (2019) also lists 11
identifications around targets with pre-existing 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 §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
SAP (Simple Aperature Photometry) lightcurve, showing it is a duo-transit
(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 lightcures 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
$\sim 20$ MJupiter, whereas the others have masses $\gtrsim 40$ MJupiter.

Santerne et al. (2016) analyze radial velocity 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 been
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 thirteen 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., depth-test as described in Coughlin et al. 2016
for uniqueness). We attempted to compute best fit transit models through
$\chi^{2}$ 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 BLS (box least squares; see Kovács et al. 2002)
search restricted to $\pm 0.1$ days around the reported period, which allowed
us to recover 3 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
($\sim$ 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 a S/N as 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 $\rho_{\star{\rm c}}$ from our transit
models to $\rho_{\star}$ 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 two, KOI-6262.01 from
$P=0.673$ days to 0.3365 days and KOI-4777.01 (which was independently
identified with the correct period by Cañas et al. 2022) from $P=0.824$ days
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 day). Reported detections for
KOI-6749, KOI-6984, KOI-2431, KOI-4788, 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 (PC) and False Positives (FP, which here, as in previous
catalogs, includes 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 2022/10/27. 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, generally we require the total
transit S/N $\geq 7.1$ 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 both
validated as planets and passed our visual inspection—which revealed marginal
evidence of a visible transit event in their lightcurves.

The S/N $\geq 7.1$ criterion was based on limiting the number of false alarms
considered to be PCs to 1 in the 90 day – 2 year period range per $10^{5}$
stars searched the Kepler sample in the presence of white noise on 6-hour
timescales. However, much of the noise in Kepler lightcurves is correlated, so
multiple false alarms become problematic for S/N in the range of $10\gtrsim$
S/N $\geq 7.1$, as has been verified through injection tests and reliability
studies (Thompson et al., 2018; Hsu et al., 2019). 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 (EBs) 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 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 a S/N $>$ 10.

There is increasing degeneracy when distinguishing between ultra-cool stars,
brown dwarfs and planets for transiting objects with radii approaching the
size of Jupiter that lack mass information. We have flagged KOIs with an
estimated planetary radii $R_{p}>21.947$ R${}_{\oplus}\approx 2$ RJupiter, as
well as KOIs with period $P>20$ days and a $1\sigma$ lower limit for the
planetary radii $R_{p}$ that exceeds 13.17 R${}_{\oplus}\approx 1.2$ RJupiter,
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 a confirmed
planet. Thus, no change in their disposition as PC 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 $R_{\star}$ 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
determination whether 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 reasonable expectation
that the modelled 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
either “S” (36 cases), “P” (18 cases) or “F” (2 cases that were discarded
based upon visual inspection of the lightcurve); only 5 of the PCs have S/N
$>12$, and only 1 has S/N $>14$.

We visually re-examined 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, 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.

###  2.4 Orbital Periods During the Kepler Epoch

The orbital periods listed in Table 1 are estimated by fitting the Kepler
lightcurve assuming a Keplerian, non-interacting 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 project distance between the center of the star and planet
is minimized. For a circular orbit, this is equivalent to when the transit
model is deepest. We measure transit times (TT) and then report transit timing
variations 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 transit times 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 the impact
parameter. Since the scaled planetary radius ($R_{p}/R_{\star}$) and transit
model mean stellar density ($\rho_{\star{\rm c}}$) are 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 above mentioned parameters, especially for $b$ and
$R_{p}/R_{\star}$. However, this process previously led to errors in the
reported period that were not accounted for in estimates of period
uncertainties.

TTVs 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 are 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 (mono-transits). 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 transit times, a straight line
model is fit to the Observed-Calculated (O-C) vs. 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

$P=P_{bf}(1+m).$ (2)

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 non-zero 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 2 or 3 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 ($\sim 1$ hour) 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 four years (and
thus unlikely to have been detected), but grow to become substantially larger
on time scales 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 duo-transit 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 9 PCs having
periods equal to half or one-third of the values listed in Table 1.

###  2.5 Unified Planet Candidate Catalog

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 (4 – 19), properties derived from the
transit model and Kepler photometry (columns 20 – 38), parameters that depend
on the transit model and stellar parameters (39 – 44), stellar parameters (45
– 62)555Columns 14 – 16 list stellar parameters derived from the transit
model. Data in column 45 are taken from DR25. Data in columns 46 – 61 are
taken from the source specified in column 62., and vetting dispositions (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 properties of the target star are not given by
Berger et al. (2020).

Table 2 presents our adopted values for astrophysical constants. The mean
radius of the Sun (R⊙) and Earth (R⊕) are used to calculate the absolute
radius of planets ($R_{\rm p}$) and incident flux, $S$, normalized to that of
the Earth. The gravitational constant (G) is used to calculate transit
durations, the scaled semimajor axis ($a/R_{\star}$) from the mean stellar
density ($\rho_{\star{\rm c}}$) and orbital inclination ($i$). The
astronomical unit (AU) and solar effective temperature $T_{\rm eff}$,⊙ 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 6 KOIs from PC to FP. We flagged 361 cases with very weak S/N transit
signals as likely false-alarms, flagged 100 KOIs with large planet radii that
may be due to stellar binaries and flagged 13 KOIs that previously had a PC
flag that now have a mass measurement inconsistent with the exoplanet
hypothesis.

KOI | KIC | Period [d] | T0 [MJD] | $R_{\rm p}/R_{\star}$ | $b$ |  $\rho_{\star{\rm c}}$[g/cm3] |  u1 |  u2
---|---|---|---|---|---|---|---|---
Disp | Kepler-Name |  $R_{\rm p}$[R⊕] |  $d_{\rm transit}$ [ppm] |  $T_{\rm dur}$ [h] |  $T_{1.5}$ [h] |  #TTobs | #TT | TTVflag
| $a/R_{\star}$ |  $i$ [deg] |  $S$ [S⊕] | S/N | MES |  $\Delta$S/Nttv |  S/NwTTV |  S/NwoTTV
| kepmag |  $\rho_{\star}$[g/cm3] |  $T_{\rm eff}$[K] |  $R_{\star}$[R⊙] |  $M_{\star}$[M⊙] | $\log{g}$ | [Fe/H] | sparflag
1.01 | 11446443 |  2.47061338 $\pm$ 0.00000002 |  787.064865 $\pm$ 0.000009 |  0.123865 $\pm$ ${}^{0.000066}_{0.000062}$ |  0.8179 $\pm$ ${}^{0.0006}_{0.0006}$ |  1.83652 $\pm$ ${}^{0.00623}_{0.00849}$ | 0.3860 | 0.2724
PPPP | Kepler-1 b |  14.102 $\pm$ ${}^{0.392}_{0.30100}$ |  14230.5 $\pm$ 4.6 |  1.743025 $\pm$ 0.001158 |  1.226357 $\pm$ 0.001031 | 431 | 431 | 000
|  8.4001 $\pm$ ${}^{0.0095}_{0.01300}$ |  84.4130 $\pm$ ${}^{0.0096}_{0.01400}$ |  873.86 $\pm$ ${}^{55.88}_{39.91}$ | 4360.4 | 6468.0 | -0.19 | 7070.7 | 7323.0
| 11.338 |  1.21942 $\pm$ ${}^{0.06243}_{0.06146}$ |  5815 $\pm$ 66 |  1.04 $\pm$ ${}^{0.02}_{0.02}$ |  0.99 $\pm$ ${}^{0.03}_{0.03}$ |  4.390 $\pm$ ${}^{0.078}_{0.07800}$ |  0.013 $\pm$ 0.041 | 2
2.01 | 10666592 |  2.20473545 $\pm$ 0.00000005 |  788.535435 $\pm$ 0.000017 |  0.075182 $\pm$ ${}^{0.000010}_{0.000008}$ |  0.2242 $\pm$ ${}^{0.1585}_{0.2163}$ |  0.40810 $\pm$ ${}^{0.00024}_{0.00030}$ | 0.3043 | 0.3129
PPPP | Kepler-2 b |  16.180 $\pm$ ${}^{0.389}_{0.357}$ |  6668.3 $\pm$ 1.4 |  3.874104 $\pm$ 0.000522 |  3.599230 $\pm$ 0.000472 | 602 | 602 | 006
|  4.7156 $\pm$ ${}^{0.0009}_{0.00120}$ |  89.9596 $\pm$ ${}^{0.0323}_{0.23440}$ |  4145.89 $\pm$ ${}^{182.66}_{182.66}$ | 5952.0 | 3862.2 | 13.12 | 20695.8 | 20905.0
| 10.463 |  0.27881 $\pm$ ${}^{0.00877}_{0.00691}$ |  6447 $\pm$ 64 |  1.97 $\pm$ ${}^{0.04}_{0.04}$ |  1.51 $\pm$ ${}^{0.03}_{0.02}$ |  4.150 $\pm$ ${}^{0.110}_{0.11}$ |  0.192 $\pm$ 0.042 | 2
3.01 | 10748390 |  4.88780321 $\pm$ 0.00000024 |  786.095757 $\pm$ 0.000081 |  0.057856 $\pm$ ${}^{0.000451}_{0.000123}$ |  0.0539 $\pm$ ${}^{0.1855}_{0.0410}$ |  3.69279 $\pm$ ${}^{0.03943}_{0.28590}$ | 0.6403 | 0.1021
PPPP | Kepler-3 b |  4.914 $\pm$ ${}^{0.080}_{0.08800}$ |  4315.6 $\pm$ 6.5 |  2.363853 $\pm$ 0.005403 |  2.232651 $\pm$ 0.002984 | 184 | 184 | 007
|  16.6988 $\pm$ ${}^{0.0811}_{0.40530}$ |  89.7905 $\pm$ ${}^{0.1773}_{0.62850}$ |  85.27 $\pm$ ${}^{4.84}_{4.44}$ | 862.1 | 2034.6 | 0.02 | 4445.0 | 4449.0
| 9.174 |  2.30908 $\pm$ ${}^{0.14896}_{0.09863}$ |  4541 $\pm$ 54 |  0.77 $\pm$ ${}^{0.01}_{0.01}$ |  0.77 $\pm$ ${}^{0.03}_{0.02}$ |  4.542 $\pm$ ${}^{0.023}_{0.01400}$ |  0.410 $\pm$ 0.084 | 1
4.01 | 3861595 |  3.84937138 $\pm$ 0.00000116 |  787.262929 $\pm$ 0.000525 |  0.039604 $\pm$ ${}^{0.000723}_{0.001286}$ |  0.9147 $\pm$ ${}^{0.0161}_{0.0346}$ |  0.22168 $\pm$ ${}^{0.12742}_{0.06371}$ | 0.3150 | 0.3029
PPPP | Kepler-1658 b |  13.073 $\pm$ ${}^{0.563}_{0.56300}$ |  1297.6 $\pm$ 14.4 |  2.638645 $\pm$ 0.091166 |  2.107915 $\pm$ 0.040755 | 279 | 279 | 001
|  5.5613 $\pm$ ${}^{0.8861}_{0.36920}$ |  80.7504 $\pm$ ${}^{1.4116}_{0.94100}$ |  5158.05 $\pm$ ${}^{587.69}_{452.07}$ | 132.7 | 235.6 | -0.27 | 6624.3 | 6706.0
| 11.432 |  0.08682 $\pm$ ${}^{0.00798}_{0.00818}$ |  6776 $\pm$ 139 |  3.05 $\pm$ ${}^{0.09}_{0.08}$ |  1.78 $\pm$ ${}^{0.06}_{0.14}$ |  3.716 $\pm$ ${}^{0.026}_{0.03700}$ |  -0.076 $\pm$ 0.082 | 1
5.01 | 8554498 |  4.78032745 $\pm$ 0.00000087 |  787.803497 $\pm$ 0.000159 |  0.036536 $\pm$ ${}^{0.000218}_{0.000187}$ |  0.9517 $\pm$ ${}^{0.0017}_{0.0020}$ |  0.34568 $\pm$ ${}^{0.01766}_{0.01494}$ | 0.3746 | 0.28
PPPP | —– |  8.699 $\pm$ ${}^{0.496}_{0.41900}$ |  959.3 $\pm$ 4.2 |  2.019873 $\pm$ 0.015371 |  1.382730 $\pm$ 0.018526 | 282 | 282 | 002
|  7.4764 $\pm$ ${}^{0.1228}_{0.11340}$ |  82.6917 $\pm$ ${}^{0.1306}_{0.13060}$ |  1386.71 $\pm$ ${}^{142.44}_{129.49}$ | 380.8 | 360.2 | -0.50 | 5139.5 | 5446.0
| 11.665 |  0.18222 $\pm$ ${}^{0.02956}_{0.02649}$ |  5936 $\pm$ 109 |  2.19 $\pm$ ${}^{0.11}_{0.10}$ |  1.37 $\pm$ ${}^{0.13}_{0.10}$ |  3.894 $\pm$ ${}^{0.049}_{0.05400}$ |  0.235 $\pm$ 0.158 | 1
5.02 | 8554498 |  7.05201465 $\pm$ 0.00010277 |  785.646252 $\pm$ 0.030691 |  0.003162 $\pm$ ${}^{0.000379}_{0.000506}$ |  0.7272 $\pm$ ${}^{0.0001}_{0.6674}$ |  0.34677 $\pm$ ${}^{1.71561}_{0.34312}$ | 0.3746 | 0.28
FFNN | —– |  0.739 $\pm$ ${}^{0.130}_{0.11100}$ |  10.8 $\pm$ 2.6 |  4.343656 $\pm$ 1.115429 |  4.272998 $\pm$ 1.116031 | 191 | 187 | 000
|  11.5844 $\pm$ ${}^{4.3004}_{4.73040}$ |  89.4505 $\pm$ ${}^{0.5491}_{3.56930}$ |  826.72 $\pm$ ${}^{81.43}_{81.43}$ | 6.2 | 0.0 | -9.27 | 6127.5 | 6329.0
| 11.665 |  0.18222 $\pm$ ${}^{0.02956}_{0.02649}$ |  5936 $\pm$ 109 |  2.19 $\pm$ ${}^{0.11}_{0.10}$ |  1.37 $\pm$ ${}^{0.13}_{0.10}$ |  3.894 $\pm$ ${}^{0.049}_{0.05400}$ |  0.235 $\pm$ 0.158 | 1
6.01 | 3248033 |  1.33410696 $\pm$ 0.00000319 |  788.451898 $\pm$ 0.001062 |  0.010121 $\pm$ ${}^{0.000348}_{0.000174}$ |  0.1530 $\pm$ ${}^{0.3525}_{0.1443}$ |  1.23481 $\pm$ ${}^{0.11145}_{0.41795}$ | 0.3306 | 0.3007
FFFF | —– |  1.443 $\pm$ ${}^{0.066}_{0.05100}$ |  120.0 $\pm$ 3.7 |  2.106019 $\pm$ 0.035052 |  2.081369 $\pm$ 0.033207 | 804 | 803 | 100
|  4.8806 $\pm$ ${}^{0.1619}_{0.62050}$ |  89.0266 $\pm$ ${}^{0.9732}_{5.83920}$ |  3811.78 $\pm$ ${}^{191.84}_{177.08}$ | 43.0 | 21.5 | -8.89 | 16014.0 | 16685.3
| 12.161 |  0.79984 $\pm$ ${}^{0.04527}_{0.04859}$ |  6344 $\pm$ 66 |  1.30 $\pm$ ${}^{0.03}_{0.03}$ |  1.20 $\pm$ ${}^{0.04}_{0.02}$ |  4.323 $\pm$ ${}^{0.060}_{0.06}$ |  0.041 $\pm$ 0.042 | 2
7.01 | 11853905 |  3.21366892 $\pm$ 0.00000100 |  786.114673 $\pm$ 0.000297 |  0.024505 $\pm$ ${}^{0.000138}_{0.000078}$ |  0.0210 $\pm$ ${}^{0.2160}_{0.0161}$ |  0.46369 $\pm$ ${}^{0.00795}_{0.03497}$ | 0.4025 | 0.2645
PPPP | Kepler-4 b |  4.050 $\pm$ ${}^{0.122}_{0.11300}$ |  727.6 $\pm$ 2.7 |  3.986002 $\pm$ 0.010266 |  3.887977 $\pm$ 0.008271 | 338 | 338 | 005
|  6.3260 $\pm$ ${}^{0.0402}_{0.16090}$ |  89.4047 $\pm$ ${}^{0.5456}_{1.63690}$ |  1191.33 $\pm$ ${}^{61.85}_{73.09}$ | 346.8 | 294.5 | -0.44 | 10568.0 | 10881.0
| 12.211 |  0.47490 $\pm$ ${}^{0.02319}_{0.03213}$ |  5833 $\pm$ 64 |  1.51 $\pm$ ${}^{0.04}_{0.04}$ |  1.15 $\pm$ ${}^{0.05}_{0.05}$ |  4.117 $\pm$ ${}^{0.106}_{0.10600}$ |  0.171 $\pm$ 0.042 | 2
8.01 | 5903312 |  1.16015301 $\pm$ 0.00000161 |  787.921021 $\pm$ 0.000720 |  0.012011 $\pm$ ${}^{0.000350}_{0.000275}$ |  0.7483 $\pm$ ${}^{0.0001}_{0.5935}$ |  4.27560 $\pm$ ${}^{0.32135}_{1.46902}$ | 0.3792 | 0.2764
FFFF | —– |  1.230 $\pm$ ${}^{0.052}_{0.04400}$ |  169.6 $\pm$ 4.5 |  1.324100 $\pm$ 0.020189 |  1.305910 $\pm$ 0.018947 | 1153 | 1152 | 100
|  6.7340 $\pm$ ${}^{0.1781}_{0.89060}$ |  89.3221 $\pm$ ${}^{0.6774}_{4.33550}$ |  2024.98 $\pm$ ${}^{93.44}_{110.43}$ | 48.4 | 36.6 | -8.99 | 15328.0 | 16106.1
| 12.450 |  1.73088 $\pm$ ${}^{0.04162}_{0.07127}$ |  5883 $\pm$ 66 |  0.94 $\pm$ ${}^{0.02}_{0.02}$ |  1.00 $\pm$ ${}^{0.02}_{0.03}$ |  4.460 $\pm$ ${}^{0.044}_{0.04400}$ |  -0.063 $\pm$ 0.042 | 2
Table 1: Abbreviated Catalog of planet candidates. The positions of the KOI Disposition and Kepler-Name columns have been moved relative to the electronic table to facilitate viewing the examples presented in the typeset manuscript. The tabulated information is described in the numbered list in Section 2.5, where the numbers refer to the column numbers in the electronic table. (Alternative values of stellar and planetary parameters listed in Columns 63-84 are not presented in this abbreviated sample.) Name | Value | units
---|---|---
R⊙ | 6.957$\times$108 | m
R⊕ | 6.371$\times$106 | m
G | 6.674$\times$10-11 | m3 kg-1 s-2
au | 1.4956$\times$1011 | m
$T_{\rm eff}$,⊙ | 5778 | K
Table 2: Adopted astrophysical constants for derived planetary parameters. The
mean radius of the Sun (R⊙), Earth (R⊕), the gravitational constant (G),
astronomical unit (au), and solar effective temperature ($T_{\rm eff}$,⊙) are
used to calculate absolute planetary radii, orbital inclinations and incident
flux.

We determine the following transit parameters and their uncertainties by
fitting transit models of Mandel & Agol (2002) to Kepler lightcurves assuming
circular orbits (and adjusted for TTVs if TTVflag = 1): mean stellar density,
$\rho_{\star{\rm c}}$; orbital period, $P$; impact parameter, $b$; scaled
radius, $R_{\rm p}/R_{\star}$. Limb-darkening coefficients are held fixed. The
transit model is used calculate the depth of transit, $d_{\rm transit}$, and
the signal to noise ratio, S/N. The fitted transit model parameters, together
with equations (3) and (2) from Seager & Mallén-Ornelas (2003), are used to
calculate transit duration measured from $1^{\rm st}$ to $4^{\rm th}$ contact,
$T_{\rm dur}$, 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:

$i=\arccos{\left(b\frac{R_{\star}}{a}\right)}.$ (3)

  1. 1.

KIC: The Kepler input catalog number of the target star.

  2. 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. 3.

Kepler ID: The number and letter assigned to verified planets. Blank if the
KOI has not been assigned a Kepler planet number/letter. Kepler IDs are do not
automatically qualify a KOI to be dispositioned a PC (see Section 2.3) and are
supplied for cross-identification purposes only.

  4. 4.

Period, $P$ [days]: Mean orbital period (the period that gives the best fit to
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 mono-transit 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 multiplicity of planetary systems in all cases.

  5. 5.

$\sigma(P)$: 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. Based on the uncertainty in 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 time scales much longer than the four
years 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 mono-transit candidates assume circular orbits and are
considered unreliable.

  6. 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 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 $\equiv$ Julian Date viewed from the
barycenter of the Solar System.

  7. 7.

$\sigma({\rm T0})$: Uncertainty in the epoch.

  8. 8.

Planet/star radius ratio, $R_{p}/R_{\star}$: This is the mode value from MCMC
of the ratio of the planet’s radius to the stellar radius.

  9. 9.

$\sigma_{+}(R_{p}/R_{\star})$: Upwards (to higher values) uncertainty of the
ratio of planet’s radius to stellar radius.

  10. 10.

$\sigma_{-}(R_{p}/R_{\star})$: Downwards uncertainty of the ratio of planet’s
radius to stellar radius.

  11. 11.

Impact parameter, $b$: We report the best fit value. See Section 2.2.

  12. 12.

$\sigma_{+}(b)$: Upwards uncertainty of $b$.

  13. 13.

$\sigma_{-}(b)$: Downwards uncertainty of $b$.

  14. 14.

Stellar density from lightcurve, $\rho_{\star{\rm c}}$ [g/cm3]: 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. 15.

$\sigma_{+}(\rho_{\star{\rm c}})$: Upwards uncertainty of $\rho_{\star{\rm
c}}$.

  16. 16.

$\sigma_{-}(\rho_{\star{\rm c}})$: Downwards uncertainty of $\rho_{\star{\rm
c}}$.

  17. 17.

$u_{1}$: First quadratic limb-darkening parameter.

  18. 18.

$u_{2}$: Second quadratic limb-darkening parameter.

  19. 19.

TTV flag: A three digit number preceded by ‘T’ represents the results of three
separate searches for TTVs. The first digit is 1 if TTVs have been included in
our transit model, 0 otherwise. The second digit refers to results listed in
Holczer et al. (2016)’s catalog, with 2 signifying sinusoidal TTVs, 1
polynomial TTVs, 0 means no TTVs found, and - means not investigated. The
third digit is the overall rating from Kane et al. (2019) 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 ‘-’ for KOIs not rated by Kane et al.
(2019).

  20. 20.

# transits: Total number of transits observed by Kepler.

  21. 21.

# TTs: Number of transits for which the transit time has been measured. This
number is always $\leq$ the number in the previous column.

  22. 22.

$d_{\rm transit}$ [ppm]: Depth of transit. Specifically, 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. 23.

$\sigma(d_{\rm transit})$: Uncertainty of $d_{\rm transit}$.

  24. 24.

Transit duration, $T_{\rm dur}$ [hr]: Transit duration, measured from first
contact to fourth contact ($T_{\rm dur}\equiv$ T1,4), which is the standard
measurement for exoplanet transit duration. Based on transit model parameters
($\rho_{\star{\rm c}}$, $P$, $R_{p}/R_{\star}$, $b$), using equation (3) of
Seager & Mallén-Ornelas (2003). The mode of the MCMC distribution is reported.

  25. 25.

$\sigma(T_{\rm dur})$: Uncertainty of $T_{\rm dur}$.

  26. 26.

Alternate measure of transit duration, $T_{1.5,3.5}$ [hr]: Mode of (T1,4 \+
T${}_{2,3})/2$. T2,3 uses equation (2) from Seager & Mallén-Ornelas (2003). If
$b>1-(R_{p}/R_{\star})$ then T2,3 is set to zero. Note $T_{\rm
dur}>T_{1.5,3.5}\geq 0.5~{}T_{\rm dur}$.

  27. 27.

$\sigma(T_{1.5,3.5})$: Uncertainty of $T_{1.5,3.5}$.

  28. 28.

Signal to noise ratio, 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. Calculated
assuming a constant period if TTVflag=0, and incorporating TTVs if the first
digit of the TTVflag = 1. S/N $\geq$ 7.1 is a 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. 29.

MES: Multiple event statistic, as reported in DR25. KOIs that were not listed
in DR25 have ‘0.0’ listed in this column.

  30. 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). Positive for most KOIs; typically large when the preferred fit uses
TTVs (since less out-of-transit data obfuscates the signal) and small when it
does not.

  31. 31.

$\chi^{2}_{\rm wttv}$: 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 $\chi^{2}_{\rm wottv}$ = DOF (degrees
of freedom). If the first digit of TTVflag = 1, then photometric uncertainties
have been scaled to have $\chi^{2}_{\rm wttv}$ = DOF.

  32. 32.

$\chi^{2}_{\rm wottv}$: Chi-squared calculated without TTVs. The values are
normalized as for $\chi^{2}_{\rm wttv}$.

  33. 33.

Scaled semimajor axis, $a/R_{\star}$: Scaled semimajor axis.

  34. 34.

$\sigma_{+}(a/R_{\star})$: Upward uncertainty of the scaled semimajor axis.

  35. 35.

$\sigma_{-}(a/R_{\star})$: Downward uncertainty of the scaled semimajor axis.

  36. 36.

Inclination, $i[^{\circ}]$: Inclination of the planet’s orbit relative to the
plane of the sky; an edge-on orbit has $i=90^{\circ}$.

  37. 37.

$\sigma_{+}(i)$: Upwards uncertainty of the inclination. Note that
$i+\sigma_{+}(i)\leq 90^{\circ}$.

  38. 38.

$\sigma_{-}(i)$: Downwards uncertainty of the inclination.

  39. 39.

Planet radius, $R_{p}$ [R⊕]: Radius of the planet.

  40. 40.

$\sigma_{+}(R_{p})$: Upwards uncertainty of $R_{p}$.

  41. 41.

$\sigma_{-}(R_{p})$: Downwards uncertainty of $R_{p}$.

  42. 42.

(Bolometric) Incident flux, $S$ [S${}_{\oplus}]$: Amount of flux intercepted
by the planet relative to that intercepted by the Earth.

  43. 43.

$\sigma_{+}(S)$: Upward uncertainty of incident flux.

  44. 44.

$\sigma_{-}(S)$: Downward uncertainty of incident flux.

  45. 45.

Kepmag: Target star magnitude in the Kepler passband.

  46. 46.

Stellar density, $\rho_{\star}$ [g/cm3]: The mean stellar density from the
stellar parameter tables. (Not the estimate derived from the transit model,
which is given in Column 14).

  47. 47.

$\sigma_{+}(\rho_{\star})$: Upwards uncertainty of $\rho_{\star}$.

  48. 48.

$\sigma_{-}(\rho_{\star})$: Downwards uncertainty of $\rho_{\star}$.

  49. 49.

Stellar temperature, $T_{\rm eff}$ [K]: The target star’s effective
temperature, taken from the stellar properties catalog.

  50. 50.

$\sigma(T_{\rm eff})$: Uncertainty of $T_{\rm eff}$.

  51. 51.

Stellar radius, $R_{\star}$ [R⊙]: Radius of the star as given in the stellar
properties catalog.

  52. 52.

$\sigma_{+}(R_{\star})$: Upwards uncertainty of $R_{\star}$.

  53. 53.

$\sigma_{-}(R_{\star})$: Downwards uncertainty of $R_{\star}$.

  54. 54.

Stellar mass, $M_{\star}$ [M⊙]: Mass of the star.

  55. 55.

$\sigma_{+}(M_{\star})$: Upwards uncertainty of $M_{\star}$.

  56. 56.

$\sigma_{-}(M_{\star})$: Downwards uncertainty of $M_{\star}$.

  57. 57.

Stellar surface gravity, $\log{g}$ [cgs]: Surface gravity of the star as given
in the stellar properties catalog. For stellar parameters taken from DR25 and
Berger et al. (2020), $\log{g}$ is from isochrone models. For stellar
parameters taken from Fulton & Petigura (2018), $\log{g}$ is based on
spectroscopy.

  58. 58.

$\sigma_{+}(\log{g})$: Upwards uncertainty of $\log{g}$.

  59. 59.

$\sigma_{-}(\log{g})$: Downwards uncertainty of $\log{g}$.

  60. 60.

Metallicity, [dex]: Metallicity of the target star.

  61. 61.

$\sigma$(Metallicity): Uncertainty of the metallicity of the target star.

  62. 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. 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^{(}$666Note, the
rounding done to produce the S/N numbers displayed in Column 31 of this table
is done _after_ assignment of ‘S’ in the flags. ), M = Jupiter-sized objects
for which mass measured via RV clearly exceeds the 13 MJupiter limit for
classification as a planet but 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 – 85. Values of the same properties reported in columns 39 – 44 and 46 – 61
with stellar parameters taken from Berger et al. (2020). All zeros if host
star 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 (§3.3). The
total number of planet candidates is 4376, including 35 mono-transits;
additionally, there are 100 KOIs, including 2 mono-transits, that we vetted as
‘R’. There are 709 multiple transiting planet systems, which account for 1791
candidates, including 7 mono-transits.

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 multi-planet candidates have radii
$R_{p}\lesssim 4$ R⊕ and orbital periods $2\lesssim P\lesssim 100$ 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 $\sim 100$
days. The period-radius valley separating super-Earths from sub-Neptunes
slopes downward from $\sim 2$ R⊕ at $P=2$ days to $\sim 1.5$ R⊕ at $P=40$ days
(left panel of Figure 6).

Figure 5: The observed multiplicity of transiting planets from the Kepler
sample as a function of period and radius, with individual squares
representing a factor of $10^{3/8}\approx 2.371$ in $P$ and $12.5^{1/6}\approx
1.523$ in $R_{p}$. Black dots represent individual Kepler planets and black
stars represent the terrestrial planets in our Solar System. The colored
squares give the multiplicity fraction for each area that contains at least 4
planets. A planet is considered to be part of a multiplanet system if more
than one transiting planet candidate is seen in the photometric lightcurve. If
a square were to contain four planets, three of which were from multi-planet
systems, then the multi-planet fraction would be equal to 0.75. The
multiplicity fraction shows a paucity in multiplanets observed for hot
jupiters, ultra-short periods ($P<1$ day) and long period planets ($P>100$
days), with an observed increase in multiplicity as the planet radius
decreases in the well-populated population with periods from 1 – 40 days.

Figure 3 shows the distribution of transit S/N for planet candidates. 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 §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-year because of the rolling band noise, which is video
cross-talk between detectors that produces a slowly drifting band of noise and
static star-like artifacts. Rolling-band is most prevalent on detectors 22,
26, 44 and 58 and creates a mismatch in the sky estimate leading to a semi-
periodic 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 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 our Figure 6 and Figure 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 HST 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 3 or 4
times, but their frequency declines with 5 or more transits (Mullally et al.,
2018).

Figure 6: The Kepler planet population showing radius ($R_{p}$) vs. period
($P$) on the left and radius vs. incident flux (S⊕) on the right. The points
have been colored by the radius of the host star ($R_{\star}$). Solar System
planets are marked with black stars, and the conservative habitable zone for
an Earth-like planet around a Sun-like star (Kopparapu et al., 2014) is shown
by the hashed green lines.

We computed minimum periods of each of the 7 mono-transit 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
longest-period observed companion planet, are given in Table 3. None of these
mono-transiting 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 multi-transiting PCs in these seven systems, apart from KOI-2525.01
and 4307.01, have been verified as planets and given Kepler numbers.

KOI |  $P$ [minimum] |  $P$ [estimate] | system |  $P$ [neighbor]
---|---|---|---|---
| (days) | (days) | multiplicity | (days)
$435.02$ | $528.5$ | $934$ | 6 | $62.30$
$671.05$ | $691.5$ | $4865$ | 5 | $16.26$
$693.03$ | $588.0$ | $719$ | 3 | $28.78$
$1108.04$ | $507.0$ | $1289$ | 4 | $18.93$
$1870.02$ | $490.0$ | $550$ | 2 | $7.96$
$2525.02$ | $418.5$ | $562$ | 2 | $57.29$
$4307.02$ | $483.0$ | $993$ | 2 | $160.85$
Table 3:  The minimum orbital period of each of the mono-transit planet
candidates in multi-planet systems based on Kepler lightcurve coverage (the
smallest value, stepping by 0.5 days, in which the data could not rule out a
second transit) is listed together with the orbital period estimated from the
transit duration and shape and the stellar properties, system multiplicity and
the period of the transiting planet orbiting immediately interior to it. The
uncertainties of the estimated periods are poorly-quantified and likely to be
substantial.

Figure 7 provides a compact sketch of the architectures of all 709 multi-
planet systems discovered by Kepler. Systems are grouped into panels by 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 4 or more PCs. Note that all 81 systems
with 4 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 $\sim 0.75$ 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.

Figure 7:  These plots show the orbital periods of each of the planets in
every Kepler multi-planet system. The three panels show systems of differing
multiplicity. Within each panel, all symbols along a given vertical line
represent planets belonging to the same star, and the systems are ordered
horizontally according to the orbital period of the innermost planet. The top
panel presents systems with two (red) or three (blue) detected transiting
planets, the middle panel systems with four transiting planets, and the bottom
panel systems with at least five transiting planets. The top panel does not
plot KOI-1843.03 due to its short orbital period, $0.177$ day, which is
$<40\%$ as long as that of any other Kepler PC in a multi; however, its two
companion planets are represented by blue points touching the vertical axis.
In the middle and bottom panels, the symbol size is proportional to planetary
radius, although in the middle panel planets larger than 5 R⊕ are shown as if
$R_{p}=5$ R⊕, and the systems are labeled by KOI number at the bottom, with
red numbers being used for systems with 6 – 8 transiting planets. Colors of
planets in the lower two panels denote the TTV flag given by Kane et al.
(2019) – light yellow is 8 or 9 (moderate or strong TTVs), dark green is 7
(probable TTVs), and black is no TTVs (note, however, that recently-identified
PCs were not examined by Kane et al. 2019). The periods of mono-transit
objects are constrained to be greater than the plotted values by not showing a
second transit in the recorded data; see Table 3.

###  2.6 KOI-2433: A Candidate Seven-Planet System

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 8-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 (2018)’s 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 & c, both have S/N $\sim$
28\. The 56 day planet has an S/N = 16.4 and was validated as Kepler-385 d by
Armstrong et al. (2021). The S/N of the other five KOIs range from 11.1 –
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 FFFF777See the item #63 in the
numbered list in §2.5 for an explanation of the dispositions given in the last
column of Table 1.. 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 ultra-short period 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 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 are 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 (Fig. 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.

##  3 Characteristics of the Planet Population: Multis vs. Singles

Kepler found far more multiple planet candidate systems (multis) than would be
the case if candidates were randomly distributed among target stars (Lissauer
et al., 2011b; Latham et al., 2011). Lissauer et al. (2012) presented a
statistical analysis that combined the large numbers of multis observed by
Kepler that were listed in Borucki et al. 2011b (as modified by Lissauer et
al. 2011b) together with the assumption that false positives are nearly
randomly distributed among Kepler targets to demonstrate that the fidelity of
Kepler multiple planet candidates is far higher than that for singles.
Lissauer et al. (2014) expanded upon the statistical analysis of Lissauer et
al. (2012) and developed techniques that Rowe et al. (2014) used to validate
more than 700 of Kepler’s multiple planet candidates as true planets.

The distribution of stellar host magnitudes for singles and multis are shown
in Figure 8. We compare the S/N distributions of Kepler singles and multis in
§3.1 and estimate the fraction of apparent Kepler multis in Table 1 that do
not represent planets orbiting the same star in §3.2.

Figure 8: Histogram of Kepler planet hosting star magnitudes, with hosts of a
single PC in blue and stellar hosts of multis in red. All PCs that were
observed by Kepler are represented in this plot.

Many groups, dating back to the aforementioned Lissauer et al. (2011b) and
Latham et al. (2011), have compared characteristics of the distributions of
single planet candidates in the Kepler sample with those of individual planets
in multis. We compare size distributions (in §3.3) and orbital period
distributions (in §3.4) for ensembles of planets in singles to those within
multis and between planets in two-planet systems to those in systems of higher
multiplicity. The distributions of normalized transit durations and
eccentricities of subsets of planets segregated by multiplicity, orbital
period, size and orbital spacing are presented and analyzed in §3.5.

Up through Subsection 3.4, all PCs are accounted for in the determination of
the multiplicity of a planetary system. For our analysis of planetary size
distributions (§3.3), we treat KOIs dispositioned as “R” as if they were PCs,
but do not count PCs with high impact parameter or S/N $<12$. In §3.4, we omit
PCs with S/N $<12$ as well as those with only one observed transit. In §3.5,
we employ somewhat different restrictions described in the introduction to
that subsection.

###  3.1 Signal to Noise Distributions & Reliability of the Sample

Figure 9 compares the histograms of singles vs. 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).

Figure 9: Number of singles (blue) and multis (red) as functions of S/N. The
dotted red curve is restricted to the lowest S/N planet in each multi-planet
system. All PCs that were observed by Kepler are represented. See Fig. 3 for a
plot of how typical S/N varies with planetary radius and orbital period.

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 (Latham et
al., 2011; Lissauer et al., 2011b, 2012, 2014; Rowe et al., 2014): (1) A
larger fraction of singles being false alarms (this wasn’t 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 EBs and other FPs are included in the PC list, since they are more
common among targets with a single transit-like pattern than around targets
with more than one such pattern (Latham et al., 2011; Lissauer et al., 2011b,
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
§3.4), reduce MES, but if they are accounted for in our model fits, they don’t
reduce S/N; and, perhaps, (5) 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
$\sim 2-3$ in period and planets less likely to transit at long periods unless
typical relative inclinations of planetary orbits decrease significantly as
periods increase. Determining what fraction of the difference is caused by (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.

###  3.2 Split Multis & Orbital Period Aliases

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 false positives 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 (§2.1). Such blending causes the planetary radius to be
significantly under-estimated (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 multi-planet 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 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., Hsu et al., 2019; He et al.,
2020).

Here we discuss a few potential split multis. KOI-284, first listed in the
catalog of Borucki et al. (2011b) and introduced here in §2.1, includes the
planet candidates KOI-284.02 (with orbital period is $P=6.415$ 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
time scale. 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 (§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 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, 16 July 2018)
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 magnitude fainter
in the F555W filter band (a nearly-equal brightness companion) with a
separation of just 0.148 arcsec, so it could well host one of the $P\sim 2.7$
day PCs. The other stellar sky-plane neighbor (denoted C) is 3.872 arcsec away
and 5.318 magnitudes 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 KFOP (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-size PC
KOI-521.01 ($P=10.16$ days) for dynamical stability. We therefore re-examined
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 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 1 part in $10^{4}$ 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 two 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 §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 two. Several other PCs have had
their estimated periods changed by a factor of roughly two 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). By the end of the mission, the DR24 TCE table,
KOI-730.03 was found 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 two
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 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 that 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 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${}_{\oplus})^{\alpha}$M⊕, where $\alpha=3$ for $R_{p}<$ R⊕ (sub-
Earths), $\alpha=2.06$ for R${}_{\oplus}<R_{p}<$ RSaturn, and finally, MSaturn
for $R_{p}>$ RSaturn. Pairs of planets were deemed unstable if their
difference in semimajor axis was less than $2\sqrt{3}$ 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 $\geq 3$ 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 1 out of
1610 real pairs of planets were found (via instability) to be an alias, then
the estimated rate for a planet candidate to have an aliased period is (1/281)
$\approx 0.36$%. 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 multi-transiting
systems, we expect roughly $0.0036\times 1781\approx 6$ aliased planets, $5$
of which are yet-undiscovered, though since this is based on only $1$ 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 $0.0036\times 2550\approx 9$
are also aliased. Aliasing may occur at a different rate among singles,
however, as the signal to noise distribution differs (Fig. 9), and the dataset
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 ($P$, $M_{p}/M_{\star}$) values of
all of the planet candidates, and determine that 453,281/9,419,770 $\approx
4.8$% are unstable888Fabrycky et al. (2014) restricted their choice to pairs
of planet candidates in multis. Applying that prescription to our data set
results in $88,758/1,590,436\approx 5.6$% being unstable, a slightly larger
percentage since the period distribution of PCs overall is broader than that
of PCs in multis (Fig. 13). This yields an estimate of $\sim 36$ split multis.
We prefer using all PCs because split multis can include planets that are
single and/or those that are in multis.. Having two detected unstable pairs
(via the split multi channel) out of 1610 PC pairs, we estimate that
$2/0.048\sim 42$ 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 $\sim 2.6$% of sampled
pairs in multis are expected to be planets orbiting different stars.

The above estimates suggest that, among planets in multis, there are $\sim 7$
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’s
fairly common for a pair of planets in the same system to have a period ratio
near two (Fig. 7; see also Fabrycky et al. 2014), but the overall distribution
of periods is quite broad (Fig. 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 $\lesssim 1$ 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 ultra-short period 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).

###  3.3 Size Distribution

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 (§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 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 $R_{p}/R_{\star}\gtrsim b-1$ is a requirement
for a transit to be observable. Therefore, when comparing radius
distributions, we only consider planets with estimated impact parameters
$b+\sigma_{+}(b)<$ 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 4 PCs around targets for which solar parameters
were assumed, 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.

Figure 10: The cumulative impact parameter distributions of various subsets of
planet candidates, as well as KOIs rejected solely because they are too large
(dispositioned as R in Table 1), that were observed by Kepler. Red curves
represent planet candidates in multis, blue curves single planet candidates;
small planets are shown by solid lines and large planets by dashed lines.
There are 6 small ($R_{p}<5$ R⊕) and 54 large singles that have $b>1$. In
multis, there are 2 small and 1 large planets that have $b>1$. PCs with S/N
$<12$ are not counted for the CDFs, but they are considered when determining
the multiplicity of the systems in which the counted planets reside.

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 planets to have adequate numbers of planet candidates for
statistically robust results.

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 the
size range $R_{p}<5$ R⊕, 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 planets 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. Those tilting the
size distribution of small planets towards singles include: detecting a
transiting planet in a Kepler lightcurve reduced the amount of data used by
the Kepler pipeline to search for additional planets and thereby lowered the
efficiency of detecting any other transiting planets associated with the same
target star, reducing the probability of finding more planets, 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 towards detecting multiple
small planets 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.

Figure 11: Normalized cumulative distribution function (CDF) of planetary
radii of planets for specified system multiplicity. The panel on the left
compares singles with planets in multis, whereas the panel on the right
compares planets in two-planet systems with those in systems of higher
multiplicity. We normalized the CDFs to unity at $R_{p}=5$ R⊕. Candidates with
S/N $<12$, or $b+\sigma_{+}(b)\geq$ 0.95 are excluded from this radius
distribution, although planets are considered to be in multis even if all of
their companions fail to meet one or both of these cuts. These plots include
KOIs rejected because they are too large (dispositioned as R in Table 1). Only
the portion of the distributions with $R_{p}<30$ R⊕ are shown, although larger
bodies that satisfy our criteria are included in the computation of the
numbers given within brackets. No candidates in multis satisfying our upper
limit on impact parameter have $R_{p}>30$ R⊕, whereas 1 singles with status P
and 12 with status R are larger than this value and not shown. (The green
triangle with $P\approx 150$ days in Fig. 1 represents KOI-1426.03, which has
V-shaped transits and $b>1$.)

Plotting a similar comparison between the size distributions of planets in
systems with two planet candidates vs. those in higher-multiplicity systems
(the panel on the right in Figure 11) shows a slight excess of super-Earth
size 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 test and the Anderson-Darling 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 planets in multis are smaller than 2.5 R⊕.

The divergence between the curves in the left panel of Fig. 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 $\sim
3$ R⊕ planets with overestimated sizes contribute a substantial fraction of
the population of apparently Neptune-size objects, and the actual transition
between the multis-rich population of “small” planets that are $\lesssim 1$%
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 planets observed near 3 R⊕
(Kite et al., 2019; Hsu et al., 2019) that manifests as a reduction in slope
in all of the curves in Fig. 11, the break in the ratio of number of multis to
that of singles could be coincident with the radius cliff.

Figure 12 compares cumulative size distributions of large planets (and KOIs
that we dispositioned as R because they failed our upper radius cutoff) in
multis vs. singles and in two-planet systems vs. higher multiplicity systems.
The size distributions of singles and multis are essentially indistinguishable
over the range 4.5 – 10 R⊕, but the number of planets 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 planets in multis have
$R_{p}>12~{}$R${}_{\oplus}\approx$ 1.1 RJupiter, 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${}_{\rm Jupiter}\approx 13$ 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.

Figure 12: Cumulative distribution function of planetary radii comparing
“large” planets in multiplanet systems with similarly-sized planets in
singles, as well as to planets in singles with the additional constraint $P>$
10 days; all curves are normalized to unity at 10 R⊕. The four panels, upper
left, upper right, lower left and lower right, consider planets larger than 3,
4, 4.5 and 5 R⊕, respectively. The numbers in square brackets represent
planets (plus KOIs rejected solely because they are too large and therefore
dispositioned as “R” in Table 1) larger than the minimum value represented in
the particular panel (no upper size cutoff). This plot uses the same criteria
for inclusion as used in Fig. 11.

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. 2019), 44% of which
are in multis. Kepler found a much less abundant population of giant planets;
29% within the size range 5 R${}_{\oplus}<R_{p}<10$ R⊕ reside in multis,
whereas the population of larger objects is dominated by singles that includes
HJs and FPs. Boundaries are smoothed over by a combination of radius errors
and true fuzziness. Note that the size range from 5 – 10 R⊕ includes a huge
diversity of planets from super-puffs (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.

###  3.4 Period Distribution

We now compare the distributions of the orbital periods of Kepler single
planet systems, that of planets in two-planet systems, and planets in systems
of higher multiplicity. As in §3.3, we consider only planets with S/N $>12$,
but nonetheless count objects that are classified as planet candidates yet are
rejected from these samples due to S/N below this threshold in determining the
multiplicity of a planetary system. However, here we do not include mono-
transit 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 planetary sizes, we do not impose any restriction on
estimated impact parameters.

Figure 13 displays the orbital period distributions of single Kepler PCs, that
of planets in two-planet systems, and PCs in systems with three or more
transiting planets. The vast majority of planets within multis, $\sim 90\%$,
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
$\sim 12\%$ have periods longer than 100 days). The difference is even larger
restricting to the shortest periods ($P<1.6$ days), at which only $\sim 20\%$
of the Kepler PCs have known transiting siblings. A KS test shows highly-
significant differences between the distributions of singles and multis
($p$-value $\approx 10^{-6}$) and one comparing two-planet systems with higher
multiplicity shows significance with $p$-value $\approx 0.27$. These results