Transit ephemerides and timing variations from Kepler and K2 to TESS

ABSTRACT

We present a catalogue of transiting exoplanetary systems discovered in Kepler or K2 data that were observed again by Transiting Exoplanet Survey Satellite (TESS). The long baseline spanned by the data enabled us to refine ephemerides and orbital parameters, as well as look for long-term non-linear trends in their transit times indicative of transit timing variations (TTVs). In total, we re-examine 111 Kepler or K2 planets. Of these, 11 were known previously to show TTVs. Additionally, we find evidence for either periodic timing residuals or non-linear trends in the ephemerides of 22 planets for which timing variations were not previously detected. Of particular interest are Kepler-522 b, which shows periodic TTVs of 20 min in amplitude, and K2-348 b, one of two transiting Neptune-sized planets in the system for which TESS data shows long-term second-order trends. For all planets, we report an update to their linear ephemerides with an improvement in the precision of the period by a factor of 2–10.

ephemerides, exoplanets, planets and satellites: detection, planets and satellites: dynamical evolution and stability

1 INTRODUCTION

The Kepler mission centred on a single field in Cygnus over four years from 2009 to 2012 and discovered over half of the approximately 5500 validated exoplanets to date (Borucki et al. 2010; Thompson et al. 2018). When the second of the four reactor wheels broke in 2012, the Kepler spacecraft was repurposed into what became the K2 mission, in which fields along the Ecliptic were observed for stretches of 75 d at a time (Howell et al. 2014). While both Kepler missions were very successful, the ongoing follow-up of these systems depends critically on maintaining up-to-date, accurate ephemerides.

The primary successor to Kepler among space-based exoplanet-hunting missions is the Transiting Exoplanet Survey Satellite (TESS) which was designed to detect small planets orbiting nearby bright stars (Ricker et al. 2014). Unlike the Kepler mission, which continuously looked at a single 110° square of sky for 4 yr, or K2, which monitored multiple equatorial fields for 3 months at a time, TESS examines much larger |$9^{\circ } \times 24^{\circ }$| fields for 27 d ‘sectors’. To date, TESS has re-examined most of the sky anywhere from once to three or more times between 2019 and 2022 including the Kepler field and some of the fields monitored by K2 as can be seen in Fig. 1. This provides the unique opportunity to take a long look at the brighter Kepler and K2 systems over a period spanning nearly a decade.

Figure 1.

Sky coverage of the data set used in this study. The central yellow field raised from the Ecliptic depicts the original Kepler field, while the equatorially spread fields are the K2 fields. K2 observed in or near the Ecliptic plane to minimize the torque due to solar radiation pressure in order to achieve stable pointing with only two remaining reaction wheels. Overlying these are the sectors from TESS Cycles 1–6. From Pyle & Wiessinger (2023).

A notable benefit of taking such a long look is the ability to search for long-term trends in transit times. Such transit timing variations (TTVs) result from dynamical effects, specifically of gravitational interactions among n-body systems that cause the planets to deviate from a Keplerian orbit (Agol et al. 2005; Holman & Murray 2005). These often occur in systems with multiple planets, especially those near orbital mean-motion resonance. One of the first and most famous examples of TTVs was found in Kepler data: the Kepler-9 system, which hosts two Jupiter-sized planets near 2:1 orbital resonance (Holman et al. 2010; Borsato et al. 2014, 2019).

A scoping search for TTVs in the Kepler and K2 fields was initially performed by Mazeh et al. (2013) who found 130 systems that showed evidence of TTVs among approximately 2000 planets and planet candidates. This enabled the creation of a catalogue of transit timing and transit duration variations for the Kepler mission (Holczer et al. 2016). An analysis of planetary masses and eccentricities from these results was performed by Hadden & Lithwick (2016), enabling robust mass calculations for half of the planets.

Through TTV analysis, we can obtain or refine estimates of the planetary mass and orbital eccentricities, and provide insight into their composition that is critical for atmospheric follow-up (Fabrycky et al. 2014; Nespral et al. 2017). TTVs can also be used to provide evidence of further non-transiting planets (Ballard et al. 2011; Nesvorný & Vokrouhlický 2014). A famous example of this is Kepler 46 c, a non-transiting Saturn-mass planet which was inferred from TTV analysis of its Jupiter-sized inner companion, and which is expected to transit in several decades when its orbit has precessed (Saad-Olivera et al. 2017). Non-periodic long-term trends can also indicate other effects such as orbital evolution from disc friction or tidal orbital decay (Birkby et al. 2014; Christ, Montet & Fabrycky 2019).

However, because the length of a TTV dynamic cycle, or super period (SP), can span many years, it is very possible that long-term trends exist that were not spotted in the time frame of the Kepler primary mission, and even more so the shorter K2 observing campaigns. Several of these systems have been followed up more recently by Demory et al. (2020), Ikwut-Ukwa et al. (2020), and Battley et al. (2021), who re-examined planets with known TTVs or with known planets near to orbital resonance. Together, these studies updated the linear ephemerides of 20 systems and significantly reduced the uncertainty in the period for four planets. In addition to enabling TTV studies, TESS observations have proven valuable in refining the linear ephemerides of planets slated for future telescope missions, where propagated uncertainties in archival parameters often span several hours (Thygesen et al. 2023). These studies highlight the utility of re-examining Kepler targets for more in-depth system parametrization and for obtaining updated ephemerides critical for follow-up.

In this paper, we expand on the aforementioned follow-ups by re-examining over a hundred Kepler and K2 systems that were re-observed with TESS with high signal-to-noise (SNR) ratio, and take a more systematic approach to their analysis. We also include TESS data up to and including Cycle 6. We seek to refine linear ephemerides, which help with future observation scheduling and atmospheric studies, as well as to follow-up on systems with known TTVs. We further hope to identify planets for which there is evidence of long-term variation that was not previously observed.

The rest of this paper is structured as follows. Section 2 details the methods of target selection and inclusion in this study, Section 3 discusses the observations from Kepler, K2, and TESS before discussing light-curve fitting and analysis in Section 4. We present our results in Section 6 before discussing the implications of our findings in Section 7.

2 TARGET SELECTION

Most Kepler systems are too faint for detailed follow-up with radial velocity (RV) analysis or ground-based telescopes (Ricker et al. 2014). However TESS has re-observed the Kepler and much of the K2 fields, sometimes multiple times, in its first four years of operation. Christ et al. (2019) estimated that TESS should detect 260 Kepler planets in one sector and 120 planets in two or more sectors, not including K2 planets being re-observed by TESS. While Christ et al. (2019) point out that hot Jupiters will be the easiest planets to re-observe and detect in TESS, it should be possible to detect a handful of smaller Neptune-sized, or even rocky planets.

For this study, targets were initially selected based on whether they appeared in either Kepler or K2, including Kepler Objects of Interest (KOIs), and were re-observed in at least one TESS sector. More specifically, a MAST search was performed on 2022 November 18 for all planets with known periods that were observed within the Kepler or K2 campaigns. Then, the TESS-point Web Tool was used to determine whether the system was or would be observed in Cycles 1–6. For each observed target, the SNR per transit in TESS data was estimated as follows

$$\begin{eqnarray} \mathrm{SNR} = \frac{\delta }{\sigma _{T}} \times \sqrt{\frac{t_{\mathrm{dur}}}{\Delta t}} \end{eqnarray}$$

(1)

where |$\delta$| is the transit depth, |$\sigma _{T}$| is the expected noise level of TESS observations (calculated per target using ticgen; Jaffe & Barclay 2017), |${t_{\mathrm{dur}}}$| is the total transit duration (⁠|${T_{\mathrm{14}}}$|⁠), |$\Delta t$| is the time interval between consecutive data points, and |$t_{\mathrm{dur}}/{\Delta t}$| gives the number of data points per transit.

In the TTV catalogue of KOIs produced by Mazeh et al. (2013) and Holczer et al. (2016), target selection and refinement was grouped into three categories on the basis of SNR. For our study, we restricted ourselves to TESS Objects of Interest (TOIs) that were first observed with either Kepler or K2 for which the estimated SNR per transit in TESS data was |$\ge$| 2.5, following Mazeh et al. (2013), who used this cut-off for Kepler planets. To further examine this choice of value, we explored the detectability of individual TESS transits as a function of SNR, as well as the transit timing uncertainty, with the aim of selecting an SNR cut-off around which TESS transits were either not detectable (as described below) or could offer no improvement on the linear ephemeris of Kepler or K2 data alone. The detection of a transit was determined through cross-correlation as described below in Section 4.3. If a transit was not detected, it was discarded from further analysis, and if more than half of a planet’s TESS transits were not detected, the planet was excluded from further analysis as being too faint to offer an accurate or helpful improvement on the linear ephemeris or evidence of timing trends.

We ultimately decided on an estimated TESS SNR cut-off of 2.5 in keeping with Mazeh et al. (2013), as a significant fraction of TESS transits were undetectable below this limit. Although 241 targets with SNR |$>$| 2.5 fall into the coverage of at least one TESS sector, TESS light curves were only produced for 189 of these. For several long-period planets, no transits fell within the time frame of the sector being observed, leaving 152 planets with at least one full transit covered by TESS. Several of these were excluded during analysis as detailed in Section 4.5 to leave 111 systems fully analysed. These planets can be seen in blue in Fig. 2.

Figure 2.

The top panel shows the original Kepler field and the bottom panel shows the K2 fields. Red star markers indicate systems with TTVs known prior to the study. Green circles indicate multiplanet systems. Blue markers show the systems selected for this study, re-observed with adequate SNR in at least one sector of TESS.

The majority (87) of Kepler and K2 targets in this group were only observed in one TESS sector, though 15 were observed in two sectors and five were observed in three or more sectors. This contrasts with the proportion predicted by Christ et al. (2019) where they found that TESS should detect 260 Kepler planets in one sector and 120 planets in two or more sectors, though their detections relied on counting multiple transits, which enables the detection of lower SNR signals than individual transit fits.

While our SNR threshold tends to favour planets with large radii, our population spans a wide range, with several dozen super-Earths and sub-Neptunes. Six planets have periods greater than 30 d. A period-radius distribution of the final population, along with timing trends uncovered in this analysis is presented in Fig. 3.

Figure 3.

The population of Kepler and K2 planets examined in this work. Systems with periodic timing trends either previously known, or newly identified in this study are marked with stars. Non-periodic second-order trends are marked as either positive curvature or negative curvature following the quadratic term of their best-fitting ephemeris. Active stars and multiplanet systems are also highlighted.

3 OBSERVATIONS

This section briefly details the observations and data reduction performed in this work for each mission.

3.1 Kepler

Kepler’s primary mission, which ran nearly continuously for 4 yr between 2009 and 2013 until the failure of its second reaction wheel, is responsible for the discovery of thousands of planets in the constellations of Cygnus and Lyra, including several Earth-like planets in the habitable zone of their stars (Borucki et al. 2011). It observed most of its targets in 30 min cadence mode, with the exception of 512 pre-selected targets for which it observed with 1 min cadence (Thompson et al. 2016).

Publicly available transit light curves for a pre-selected list of targets were taken from the Mikulski Archive for Space Telescopes (MAST). We utilized the Pre-search Data Conditioned Simple Aperture Photometry (PDC-SAP) light curves extracted by the Kepler science processing pipeline (Jenkins et al. 2010). After applying pixel-level corrections, light curves were extracted using an aperture mask that encloses a subset of pixels that result in the highest SNR flux from a given target. The pipeline then detrends the data, removing instrumental and spacecraft anomalies as well as data artefacts. Further detrending was performed as described in Section 4.

3.2 K2

The K2 mission was an extension of the original Kepler mission that was put into place when two of the original reactor wheels failed for Kepler. By observing along the Ecliptic, and rotating its field of view every three months to continue pointing away from the sun, the effects of solar radiation pressure were minimized, allowing the spacecraft to maintain stability (Howell et al. 2014). In the years 2014–2018, it added significantly to the growing population of confirmed Kepler exoplanets, contributing a couple hundred planets and many more planet candidates.

Each target was observed once for a period of 80 d in one of 20 sectors located along the Ecliptic, some of which overlap as can be seen in the lower half of Fig. 2. K2 observed targets in either long-cadence (30 min) or short-cadence (1 min) mode: both are used for various targets in this study. Several data pipelines are available to extract light curves from K2 images and correct for the significant systematics present in these data due to the spacecraft’s reduced pointing accuracy. For this study, we chose to use the EPIC Variability Extraction and Removal for Exoplanet Science Targets pipeline everest (Luger et al. 2016) which employs pixel-level correlation as described in Deming et al. (2015) to remove systematics and achieve a comparable precision to the original Kepler mission (Luger et al. 2018). While other pipelines such as K2SFF (Vanderburg 2014), or K2SC (Aigrain, Parviainen & Pope 2016) are also effective at correcting for K2 systematics, everest has been shown to be marginally better at reducing K2 systematics across both long- and short-cadence data (Hirano et al. 2018).

3.3 TESS

The entire Kepler field was first re-observed by TESS from 2019 to 2020, primarily in sector 14, with 962 stars pre-selected to receive 2 min cadence data. Parts of it were also observed in sectors 15 and 26. It was then more recently observed in 2021 in sectors 40 and 41, again in sectors 53–55 in 2022, and again in sectors 74–75 in 2024. We used data from TESS years 1–6 in this work, up to and including sector 82 in 2024 September.

The data were reduced and light curves extracted using the Science Processing Operations Center (SPOC) pipeline (Jenkins 2002, 2017; Jenkins et al. 2010, 2016; Smith et al. 2012; Stumpe et al. 2014). We used the 2-min cadence PDC-SAP light curves as the starting point for our analysis, in which most systematic effects have already been corrected including corrections for aperture loss and crowding (Smith et al. 2012; Stumpe et al. 2012, 2014). This analysis is further described in the next section.

4 LIGHT-CURVE ANALYSIS

When analysing the photometric data across two different missions, our aim was to be as self-consistent as possible between Kepler and TESS while acknowledging the comparative limitations of the latter, namely its higher scatter, as can be appreciated in Fig. 4 and more limited observing windows, as well as differences in limb-darkening.

Figure 4.

K2-30 b is a warm Jupiter with an orbital period of 11.6 d. Transit data are phase-folded around the best-fitting period, totalling 17 transits in K2 and 5 transits in TESS, highlighting the expected relative loss of precision of transit time measurements for the latter.

4.1 Overview

An overview of the fitting and analysis process for each planet is as follows. This is then described in greater detail in the following sections. For each target, we started with the higher SNR Kepler or K2 data, using the PDC-SAP light curve for Kepler, and the EVEREST light curve for K2. After a normalization step, cross-correlation was performed on individual transits in order to produce a first rough estimate of the central time of each transit, even in the cases showing large timing variations. These initial transit times were then used to produce a phase-folded transit light curve which was then fit to obtain refined estimates of the global planet and system parameters. These parameters were then held fixed as individual transit times were fit for. A similar procedure was then carried out for TESS transits, using the Kepler/K2 global parameters as priors. Finally, linear ephemerides were fit to the transit times, and tests for timing variability were performed on the |$O-C$| residuals. A summary of this process is shown in Fig. 5 and illustrated in further detail for a high SNR and low SNR planet in Figs A1 and A2, respectively. Each of these steps is described in greater detail below.

Figure 5.

An overview of the process for fitting each planet and its transits. These steps are described in Section 4.1 and shown for a high and low SNR planet, respectively, in Figs A1 and A2.

4.2 Normalization

First, a simple normalization was performed, dividing by the average flux. Then, to account for correlated noise, a third-order polynomial was fit to the out-of-transit light curve. Each polynomial was fit to a window of five transits surrounding the time of conjunction based on published ephemerides. In most cases, a 1.5 transit-duration window around the time of conjunction achieved a masking of the transits. However, for a few cases of planets with large TTVs of magnitudes greater than their transit duration, different individualized windows were adopted.

Once optimization was achieved, we divided the light curve by the best-fitting polynomial. We then checked for any residual correlated noise in the normalized light curve using the out-of-transit data, by examining its autocorrelation function. In most cases no evidence of correlated noise was found. However, in some cases, such as particularly active stars, the third-order polynomial fitting left significant residuals in the out-of-transit light curve. These cases were identified and discarded from the final analysis.

4.3 Cross-correlation

We then cross-correlated the normalized light curve with a template transit model constructed from the published transit parameters (see Table A2). This cross-correlation step, which was performed on the transit in windows of four transit durations (or five transit durations in exceptional cases with large TTVs), was used to obtain an initial estimate of the central time of each transit. This was a necessary pre-requisite for the following step (global planet parameter fitting, see Section 4.4). All times used in the analysis were in |$\mathrm{BJD}_{\mathrm{TDB}}$|⁠, and our reported ephemerides use the same time system.

Transit modelling was implemented in python using the batman package (Kreidberg 2015) with a supersampling factor of 10 for long-cadence data (⁠|$>$|2 min) and no supersampling for short-cadence data. The parameters of the batman transit model were the orbital period P, the time of transit centre |$T_0$|⁠, the planet-to-star radius ratio |$R_p/R_\star$|⁠, the scaled semimajor axis |$a/R_\star$| (where a is the orbital semimajor axis), the orbital inclination i, the orbital eccentricity e, and the longitude of periastron |$\omega$| and the quadratic limb-darkening coefficients |$u_1$| and |$u_2$|⁠. The parameters used to generate the cross-correlation templates were taken from the exoplanet archive, with references given in Table A2.

We used the limb-darkening toolkit (LDTk) of Parviainen & Aigrain (2015) to compute the limb-darkening coefficients |$u_1$| and |$u_2$|⁠, using the stellar parameters from the NASA Exoplanet Archive and the relevant instrument/filter throughput as inputs.

In the absence of TTVs and if the published ephemeris was correct, the centre of the transit should coincide with the centre of the cross-correlation window, and the peak should occur at zero lag. However, in the presence of TTVs or large ephemeris drift (sometimes the case in TESS transits), the peak occurs at non-zero lag. We used the results of this cross-correlation step to account for any significant timing offsets of individual transits before phase-folding the light curve ahead of global planetary parameter fitting in the next step.

An individual transit was considered to be detected if the peak cross-correlation value was |$>$|3 standard deviations above the out-of-transit mean, where the standard deviation was calculated by taking the cross-correlation values between 1 and 2 transit durations on either side of the linear prediction of mid-transit time. For planets with very large TTVs, this window was expanded, while for planets with other nearby transits, these were masked in windows of 1.2 times the duration of the companion’s transit. If the transit was not detected, it was discarded from further analysis. At this point, transits that overlapped with the transits of other companions were also discarded. If more than half of a planet’s TESS transits were not detected, the planet was excluded from any further analysis.

4.4 Planetary parameter fits

Before determining the times of individual transits, we first refined the transit model for each planet by modelling the phase-folded light curves. For each polynomial-normalized transit of a given planet, we subtracted from the time array the time of transit centre obtained at the cross-correlation step (Section 4.3), resulting in a normalized, phase-folded light curve. We then fit for the global planetary parameters which control the transit shape, depth and duration in the Kepler or K2 phase-folded light curve. In total, we fit for P, |$T_0$|⁠, |$R_p/R_\star$|⁠, |$a/R_\star$|⁠, i, |$u_1$|⁠, and |$u_2$|⁠. We chose to fit for cosi since this makes convergence quicker. LDTk was used to generate priors for limb-darkening coefficients, and wide uniform priors were used over other parameters. A window of three transit durations surrounding each transit centre was used. We first optimized the posterior with respect to the parameters using the default minimize function of the scipy.optimize module. When performing optimization, we used the default minimize function of the scipy.optimize module, which employs the Broyden–Fletcher–Goldfarb–Shanno algorithm. We then sampled the full posterior distribution using Markov Chain Monte Carlo (MCMC) implemented with the emcee package (Foreman-Mackey et al. 2013, 2019). We evolved chains of 40 walkers with a thinning factor of 15 with a burn-in of 1000 steps and conservative removal of the first 25 per cent of steps of the production chain. Convergence was achieved when the estimated autocorrelation time |$\tau _{\mathrm{autocor}}$| (checked every 200 steps) was changed by less than 1 per cent and the chain was longer than 200 times the autocorrelation time, resulting in chain lengths of about 30 000–40 000 for planetary parameter fits. The final parameter values were taken to be the median of the posterior distribution, with uncertainties reported as the 68 per cent confidence interval from the 16th to the 84th percentile. We used uninformative, wide uniform priors over all parameters apart from limb darkening, for which we used LDTk to calculate priors over the limb-darkening coefficients u|$_1$| and u|$_2$|⁠. In total, the parameters fit for in this step included P, |$T_0$|⁠,¹  |$R_p/R_\star$|⁠, |$a/R_\star$|⁠, cos|$_i$|⁠, |$u_1$|⁠, and |$u_2$|⁠. Any non-zero lag in maximum cross-correlation signal was applied as an initial timing offset to align transits for a global fit of the planetary parameters. While this process of cross-correlation and global fitting could be repeated recursively, the initial correction resulted in a fit for planetary parameters that was typically consistent with and matched or exceeded the precision of archival parameters.

For TESS, we used the planetary parameters derived from the Kepler or K2 fit as Gaussian priors, and used the TESS response function to generate priors on the limb-darkening coefficients from LDTk and again fit using MCMC. However, we unsurprisingly found that this method rarely resulted in an improvement to the precision of Kepler fitting alone, so planetary parameters derived from the Kepler fit were adopted and fixed and the MCMC was rerun to fit TESS limb-darkening coefficients alone. Although we could have fit the TESS data at the same time as the Kepler or K2 data, this would have vastly increased the computational times due to the larger number of data points and parameters, with no associated improvement on precision. This might change in the future as the number of TESS observations increase.

4.5 Timing fits

We then fit for individual transit times using a window of three transit durations around the initial transit midpoint. Planetary parameters derived in the previous step were kept fixed. A third-order polynomial was used to account for any long-term trends and the only other parameter allowed to vary was |$\delta t$|⁠, the correction to a linear ephemeris. All parameters were fit using MCMC as described above resulting in chains of about 10 000 for individual transit time fits. Residuals were checked to ensure they were Gaussian. Where they were not (sometimes in the case of star-spots or very active stars), the transit was discarded. Of the more than 20 000 transits fit, several hundred were deemed to be outliers and thus discarded by one of the following criteria:

• Divergence from typical scatter. If the |$\left|O-C\right|$| deviation of a given transit from the updated linear ephemeris was more than 3σ away from the median |$\left|O-C\right|$| plus the median error, or if their error was more than 3σ larger than the median error.

• Overlapping transits. For systems containing multiple companions, an overlapping transit was considered a transit for which a window of |$1.2 \times t_{\rm {dur}}$| centred on each transit overlapped. These were removed from further analysis.

• Partial transits. Partial transits of approximately 70 per cent or less were found to be unconstraining to system parameters and TTV trends. If a window of |$1.2 \times t_{\rm {dur}}$| contained fewer data points than |$0.8/\rm {cadence}$|⁠, it was discarded as a likely partial transit.

• No transit detected. For each transit, we examined the cross-correlation of the flux (normalized by out-of-transit fit with a polynomial) with a transit model based on archival parameters. If the peak in correlation was less than three times greater than the standard deviation of out-of-transit correlation lag, the transit was discarded. This was most often the case for TESS transits of smaller planets, lower SNR planets, or transits for which the data were flawed or missing.

• Visual inspection. Every transit was visually inspected, along with its cross-correlation signal, with close attention to those whose timings visually diverged from a clear trend. Residuals to transit fits were plotted below each fit and those too were visually inspected for Gaussian distribution and lack of correlated features. In some cases, a fit was clearly incorrect. This was typically due either to features in the transit, such as star-spots, or erroneous/outlier points in the processed light curves. Rarely, it was a case of a failure of fit of the polynomial, such as in very active stars with large trends, in which case a fit was reattempted placing realistic constraints.

These excluded transits were not used in fitting for planetary parameters, ephemeris updates or in TTV analysis. The final result of this step is a list of |$\mathrm{BJD}_{\mathrm{TDB}}$| transit times for each planet.

5 TRANSIT TIMING ANALYSIS

5.1 Updating the linear ephemeris

All transit times were fit to a linear ephemeris using scipy.optimize. Because of the significantly more numerous points and smaller errors arising from Kepler and K2 data sets compared to those of TESS, Kepler points tended to dominate the trend.

The uncertainty on the predicted mid-transit time (⁠|$T_{\text{pred}}$|⁠) for a given epoch (E) can be estimated using the uncertainties on the reference mid-transit time (⁠|$T_0$|⁠) and the orbital period (P), along with their covariance, estimated using the inverse Hessian matrix reported by the minimize function. The predicted time is given by |$T_{\text{pred}} = T_0 + E \cdot P$|⁠, and the uncertainty is propagated as:

$$\begin{eqnarray} \sigma _{T_{\text{pred}}} = \sqrt{\sigma _{T_0}^2 + E^2 \cdot \sigma _P^2 + 2 \cdot E \cdot \text{Cov}(T_0, P)}, \end{eqnarray}$$

where |$\sigma _{T_0}^2$| and |$\sigma _P^2$| are the variances of |$T_0$| and P, respectively, and |$\text{Cov}(T_0, P)$| is their covariance. This formulation ensures that the uncertainties on |$T_0$| and P are correctly propagated to |$T_{\text{pred}}$|⁠, regardless of the choice of reference epoch. For ease of comparison to literature values, we have chosen our (⁠|$T_0$|⁠) to align with the beginning of the Kepler or K2 missions for corresponding systems. These values are all reported for each planet in Table A2.

5.2 Evaluation of TTVs

In order to identify planets with significant TTVs, we performed a series of statistical tests to characterize the scatter of timing residuals. We then identified three major types of |$O-C$| variation: clear periodic trends, significant scatter, or higher order trends.

• Periodic trends. TTVs in systems of planets near j: j−1 orbital resonance are characterized by quasi-periodic modulation with a period known as the TTV SP |$P_{\mathrm{TTV}}$| relating to the ratio of the orbital periods as

  $$\begin{eqnarray} P_{\mathrm{TTV}} = {\left|\frac{j-1}{P_{\mathrm{inner}}}-\frac{j}{P_{\mathrm{outer}}} \right|}^{-1} \end{eqnarray}$$

  (2)

  from Lithwick, Xie & Wu (2012). We looked for periodic trends in the O − C residuals to the linear fit using the astropy implementation of a Lomb–Scargle (LS) periodogram. We recorded the period corresponding to the peak power of the LS, and estimated the associated false alarm probability (FAP) using the analytical method of Baluev (2008), which offers computational efficiency compared to bootstrap resampling, and provides more pessimistic, conservative estimates. We utilized the default frequency range and oversampling settings provided by the implementation. To enhance sensitivity to long-term trends, we increased the samples-per-peak parameter to 10, ensuring a robust exploration of the frequency domain. For planets with previously known TTVs, the LS power and period are recorded in Table A1. For planets without previously known TTVs, an FAP of 0.05 or lower was taken to indicate significant periodicity.

• Significant scatter. Even if a long-term periodic trend is not obvious, planets might display more scatter in their transit times than could be expected in a purely linear, non-interacting case. While effects from stellar activity and star-spots could result in short-term timing offsets (Nesvorný & Vokrouhlický 2016), another possibility is the chopping effects of planets close to resonance but whose SPs still exceed the span of data collection, so that they appear linear on average. This can be quantified by comparing the scatter of the timing residuals, |$s_{O-C}$| measured as the median absolute deviation (MAD) of the |$O-C$|⁠, and the median uncertainty of these residuals, |$\left|\sigma _{O-C}\right|$|⁠. We used a MAD statistic, using a robust estimate of |$\sigma _{\rm {res}} = 1.483 \times \left|s_{O-C}\right|$|⁠. Following Mazeh et al. (2013), we defined an alarm score, |$\sigma _{\rm {res}}/\sigma _{O-C}$|⁠. A significant scatter is defined as an alarm score greater than 15 (Tamuz, Mazeh & North 2006). Alarm scores were calculated for all data points together as well as for Kepler/K2 and TESS separately, and a planet was listed as displaying significant scatter if either score was greater than 15.

• Long-term TTVs. In some of the systems we analysed, there is no clear periodicity or significant scatter, but the linear ephemeris is a poor fit, indicating the possible presence of orbital evolution or TTVs with periods exceeding the time span of the available data. To quantify the departure from a linear ephemeris we also fit each set of transit times with a second-order polynomial. We compared the linear, quadratic, and cubic fits using the Akaike Information Criterion (AIC): |$\mathrm{AIC} = 2n_{\mathrm{par}}-2\ln (L)$|⁠, where |$n_{\mathrm{par}}$| is the number of model parameters and L is the likelihood function. Given the heavy dominance of Kepler data on model fitting due to its significantly lower timing error, we calculated AICs for each fit type for all data points, as well as for the TESS data points only, using fits to the whole data set. If the difference between the AIC of the quadratic or cubic fit and that of the linear fit was |$>$|1, a planet was flagged as tentatively exhibiting long-term timing trends.

• No evidence of variation. If a planet did not exhibit any of the above behaviours, it was deemed to show no variation. For these planets, we merely report the linear ephemeris to support future observations.

6 RESULTS

Of the 241 total planets with an SNR |$>$| 2.5, and 152 with TESS transits, 111 had detectable transits in both Kepler/K2 and TESS fields that were not considered outliers. Errors on transit times scaled with SNR as can be seen in Fig. 6, with the timing errors of TESS points consistently being significantly larger than those of Kepler as expected. In a large fraction of planets, we were able to use TESS data to refine the ephemerides and improve the precision of the period by a factor of 2–10. A summary of the improvement factors can be seen in Fig. 7. All transit times for each planet and their associated uncertainties are available as Supporting Information.

Figure 6.

The median error on timing residuals of individual transits is plotted versus TESS SNR for each observing program, following the expected trend. As expected, TESS has a significantly higher timing error than Kepler.

Figure 7.

Improvement in precision for the reference time, |$T_0$| and the orbital period P between published ephemerides and the present study. In all panels, the y-axes show the ratio of ‘old’ (previously published) to ‘new’ (this study) uncertainty labelled improvement factor |$\sigma _{\rm {old}}/\sigma _{\rm {new}}$|⁠. The colour scale indicates the orbital periods in the top set of plots and the number of transits initially captured in Kepler or K2 in the bottom set of plots. There is considerable scatter across both parameters.

In almost all cases, using Kepler transit times alone resulted in comparable ephemerides to archival values, which validates our detrending and fitting methods. In a few cases, our analysis of Kepler data alone resulted in greater errors on the linear ephemeris than previously reported values. These were typically systems surrounding highly active stars, as can be seen in Fig. 7. Because we performed no special individualized analysis beyond what is described above for such systems, it is quite possible that our analysis is not sufficient to handle significant variability, star-spots, etc. Examples of the changes to the precision of the period achieved by this work can be seen in Fig. A3.

6.1 Systems with previous TTVs

In our sample, 11 planets had been flagged previously as displaying TTVs. These are presented in Table A1 and shown in Fig. A4. In most cases, these timing trends are also present in our data, such as the clear periodicity of the Kepler-18 system, shown in Fig. 8. This three-planet system, comprised of an inner super-Earth and two outer Neptune-sized planets in near 2:1 orbital resonance, has been well characterized and its TTVs used to measure masses (Cochran et al. 2011; Hadden & Lithwick 2016). However, as can be seen for Kepler-18 c, some TESS transits, while detected, are not very constraining.

Figure 8.

Kepler-18 c and d are previously known to have TTVs. The top panel shows |$O-C$| timing residuals from Kepler and TESS. The bottom-left panel shows the LS periodogram for planet c’s Kepler |$O-C$| and the bottom right panel shows the |$O-C$| for this planet phase-folded around the peak power LS period. The large errors on individual TESS transit times make it unlikely that any further refinement is possible for this system. Plots of residuals and LS periodograms can be seen for all planets with previously known TTVs in Fig. A4.

There are also cases for which periodic trends in Kepler data were not recaptured to statistical significance, as can be seen in Fig. A4 and Table A1. In several cases, such as for Kepler-37 d, this is consistent with other re-analyses that likewise do not find evidence of periodic trends in these systems (Gajdoš, Vaňko & Parimucha 2019). Several high amplitude TTV systems, such as Kepler-9 b and c as well as K2-19 b and c were excluded due to the difficulty of disentangling their overlapping or highly displaced transits. Such a system would require a more individualized approach than is possible through the methods in this study. In some cases with previously known TTVs, the TESS data exhibit a faint second-order trend to the linear average of a system’s periodic behaviour. For example, the single TESS transit captured for Kepler-51 b is significantly offset from the average linear ephemeris of Kepler data alone. In most cases, TESS data are not very useful to constrain to the periodic component of the TTVs. However, even if they do not constrain TTV amplitudes, TESS data do yield a slight improvement on the average linear ephemeris, as can be seen in Fig. 7.

6.2 New periodic TTVs

Several planets that were not previously known to have periodic TTVs show significant evidence of periodic variation in our analysis. An LS FAP of |$<$|0.05 supplied strong evidence of periodicity, which is the case for nine such planets. We then examined each system in detail, noting four cases where we suspect star-spots or stellar variability could be causing the apparent TTVs, and this leaves five new candidate periodic TTV planets. Two of these have very low amplitude (⁠|$<$|4 min) and high scatter, on the same order of magnitude as their amplitude. These are shown in Fig. 9 and described in Table 1.

Figure 9.

New periodic TTVs. Kepler and K2 planets that show periodic trends but not previously identified as having TTVs. In the leftmost column are the O − C residuals to linear ephemerides. In the middle column are the LS periodograms of Kepler/K2 transits that show significant periodicity in the residuals, and in the rightmost column are the timing residuals of Kepler/K2 transits phase-folded around the best-fitting SP. Some of these systems have low amplitude variations with high scatter that are attributable to stellar activity.

Table 1.

New periodic TTVs. Several of these periodic trends are low amplitude, correlate to polynomial fitting, or correspond to a fraction or multiple of their host’s stellar activity cycle. We note these here as well but do not trust these to be TTVs caused by dynamical interactions.

Name	Period	LS period	LS period	FAP	Stellar activity	Comment
	(d)	(epochs)	(d)		(d)
Kepler-493 b	3.00388	129.333	388.501 23	2.02E−04
Kepler-450 b	28.45493	34.483	981.204 36	1.69E−05	20.796
Kepler-522 b	38.58438	20.645	796.580 68	5.86E−07	4.919
Kepler-840 b	2.49578	41.566	103.739 17	6.33E−06		Very low amplitude (⁠|$<$|4 min)
Kepler-628 b	15.45806	20.215	312.485 48	3.00E−03		Very low amplitude (⁠|$<$|4 min)
Kepler-548 b	4.45419	58.919	262.436 28	3.22E−03		Correlated to polynomial fit
Kepler-531 b	29.88495	35.556	1062.576 13	1.57E−06	35.053	Correlated to stellar activity cycles
Kepler-854 b	2.14463	22.838	48.979 26	3.05E−16		Correlated to polynomial fit
Kepler-686 b	1.59475	22.057	35.175 30	1.70E−03		Correlated to polynomial fit

In order to ensure that this observed periodicity was independent from stellar activity, we examined the following: we visually inspected each transit as described above to ensure that transit fits contained no unexpected features and that residuals were Gaussian. We then plotted the out-of-transit average stellar flux over a window of two transit durations against the individual transit |$O-C$| and visually assessed that there was no correlation. We then similarly compared the first and second coefficients of the polynomial fit with the out-of-transit flux against the |$O-C$|⁠, to further check for effects of slope or curvature of the time-series that impact the timing fit and falsely appear as timing periodicity. Where visual inspection showed no correlation, the periodicity was considered robust. Kepler-531 has timing variations on the order of |$\sim$|30 min, however is also noted to be quite active and likely has star-spots, as seen in Fig. A5. A sample fit of a transit that was excluded is shown in the same figure.

In several cases, very high frequency periodicity (cycles lasting fewer than 10 epochs or 30 d) lined up obviously with stellar activity periods or low-order multiples thereof, so we do not report them as planetary TTVs. Kepler-548 b, Kepler-854 b, and Kepler-686 b showed significant periodic signals, however the amplitude of |$O-C$| offsets is less than 4 min, |$O-C$| scatter is high, and |$O-C$| offsets showed a linear correlation to the first-order (linear) coefficient of polynomials fitted to the out-of-transit flux, so it is unlikely that these are planetary timing variations. Of the five remaining systems that showed significant periodicity, two of these (Kepler-450 b and Kepler-522 b) are also around active stars. However, there appeared to be no correlation between their stellar activity cycles and their |$O-C$| ‘SP’, taken to be the frequency peak of the LS periodogram.

6.3 Systems with second-order trends

Of the planets we analysed, 17 show a significant deviation from a linear ephemeris in the form of non-periodic second-order trends as determined through AIC comparison to a linear ephemeris. Of these, eight planets show a decreasing period or ‘negative curvature’ over the observing window and nine show an increasing period or ‘positive curvature’. These planets are shown in Fig. A6 and summarized in Table 2. One of these planets, the hot-Jupiter K2-113 b which has a 5.8 d period, shown in Fig. 10, is found to have a significantly shorter period in TESS data than it had in K2, and best fits a second-order polynomial with negative curvature. This also results in a significant change in the linear ephemeris when derived from both data sets. Several of the planets with second-order trends have known planetary companions, also identified in Table 2.

Figure 10.

K2-113 b is a hot-Jupiter with a 5.8 d period that shows a clear second-order trend in its timings. Blue points are K2 |$O-C$|⁠. Pink crosses are TESS residuals to a second-order fit and orange circles are residuals to a first-order (linear) ephemeris. The lower panel is a river diagram showing the normalized flux of each transit phase-folded around the average linear ephemeris. Other planets that show such timing trends can be seen in Fig. A6.

Table 2.

Planets with second-order trends. These planets show significant evidence of deviation from a linear ephemeris as defined in Section 5. The updated times of conjunction and average periods for all planets are given in Table A2.

Planet	Curvature	Period (d)	Radius (Earth)	Known companions
K2-113 b	Negative	5.817 61	12.14
K2-140 b	Positive	6.569 20	13.563
K2-174 b	Negative	19.562 54	2.6
K2-180 b	Negative	8.865 60	2.466
K2-198 b	Negative	17.042 88	4.189	2
K2-212 b	Positive	9.795 63	2.52
K2-261 b	Negative	11.633 47	9.528
K2-277 b	Negative	6.326 74	2.07	2
K2-295 b	Positive	4.024 89	10.054
K2-348 b	Positive	4.688 61	3.26	1
K2-352 d	Positive	14.871 27	2.23	2
K2-43 b	Positive	3.471 26	4.51	1(+TOI)
K2-77 b	Pegative	8.199 99	2.3	2
K2-79 b	Negative	10.995 28	3.69
Kepler-39 b	Positive	21.087 21	13.9	KOI
Kepler-468 b	Positive	38.478 76	13.34
Kepler-725 b	Negative	39.643 51	10.63

6.4 Large corrections to ephemerides

In several cases, quite significant changes were made to previous ephemerides. These are often longer period planets that had very few transits in Kepler or K2. In some of these cases the uncertainty on the period has actually increased despite the longer baseline. We believe that this is due to an underestimation of the original uncertainty due to the small number of samples.

A further nine planets in our sample show statistically significant timing variation or evidence for non-linear trends according to our criteria, but have fewer than five points in their Kepler or K2 data sets. While our analysis provides a valuable update to their ephemerides, further observation is needed in order to establish the significance of any timing trend. We present the updated ephemerides for these planets in Table A2, but they are not included in our count of planets that show more definitive timing trends.

7 DISCUSSION

Up-to-date linear ephemerides are critical to future observations of planetary systems. Overall, we were able to update the linear ephemerides of 111 Kepler and K2 systems and identify robust timing trends for 22 planets. This approximately matches the incidence of significant timing trends recorded by Mazeh et al. (2013) in the Kepler population. A summary of our improvements to linear ephemerides as well as systems with timing trends can be seen in Fig. 7 and in Table A2. While we obtained mild to moderate improvements in precision for most of the planets in our study, there are a few for which we do not achieve the improvement we might have expected compared to the published archival ephemerides, as predicted by Batalha et al. (2019) and Christ et al. (2019). This is similar to the findings of Battley et al. (2021), which updated the fits for 21 Kepler planets, and Thygesen et al. (2023), which examined a population of K2 planets and updated their ephemerides using TESS observations. For every system where the period uncertainty we report is greater than the archival parameters, we visually inspected each fit to make sure the larger error was not due to a problem with the transit fits, and that the ephemerides reported in Table A2 are robust to the best of our knowledge.

Several systems examined in this study have been recently re-examined by others with TESS data. For example, Kepler-51 b is a planet showing clear periodic TTVs in Kepler data that was recently re-examined by Battley et al. (2021), which found the TESS data points to be relatively unconstraining to the system’s TTVs and planetary masses. We found the TESS transits to be similarly unconstraining to the TTVs, but were able to improve the precision of its average period by an order of magnitude. Ikwut-Ukwa et al. (2020) further identified several K2 systems that would be among the first observed with TESS with a promising potential for follow-up. Of these, we analyse K2-114 b, for which K2 only observed three transits. We have now added 10 additional TESS transits over three sectors which enable us to report much more precise ephemerides, greatly facilitating any future follow-up observations. Another planet, K2-260 b was noted by Thygesen et al. (2023) to have ephemerides that differed greatly from archival ephemerides (well over 3|$\sigma$|⁠). We arrive at ephemerides that are similarly discrepant to these older literature values and in good agreement with their more recent analysis.

One of the most interesting results of our re-examination of the Kepler and K2 fields with TESS is the uncovering of long-term second-order trends. In their study of 22 Kepler planets, Battley et al. (2021) identified one planet, Kepler-2b/HAT-P-7b, where the TESS transit times were offset by several minutes from the Kepler linear ephemerides. The fact that we found similar offsets in a number of other systems over a larger sample strongly indicates that these second-order trends arise from a real evolution of the period, rather than a fitting error. However, these results should still be treated with some caution, particularly where the total number of transits included in the ephemeris fit is small, as the results can be very sensitive to even small outliers.

While the |$O-C$| diagrams for the 17 planets we have identified as displaying second-order trends do not appear periodic in the time frame of this study, one possibility is that the available data span a fraction of a longer TTV super-period due to (sometimes unidentified) additional planet(s). Several of the planets with second-order trends have known companions near to orbital resonance. One such example, the Neptune-sized K2-348 b with an orbital period of 4.6 d, can be seen in Fig. A6. K2-348 also hosts another transiting sub-Neptune, K2-348 c with a 12.6 d orbital period. Another such planet, K2-352 d is a super-Earth with two known super-Earth companions and is notably very close to 4:1 resonance with the recently discovered K2-352 b (de Leon et al. 2021). Finally, K2-43 b is a Neptune-sized planet found to show a positive second-order trend in our analysis. With a 2.2 d period, it lies close to 3:2 resonance with K2-43 c, a sub-Neptune with an 3.4 d period. While we have no evidence that these deviations from linear ephemerides are planetary in nature, follow-up on these systems might be of interest.

It is worth noting that most of the systems for which we detect second-order trends but not periodic TTVs, were first observed by K2 rather than Kepler. Most K2 targets were only observed in one |$\sim$|80 d time frame, during which only a handful of transits were observed. Because K2 data are much more precise than TESS data, the K2 points dominate the ephemeris fit. However, because the K2 points span a short baseline, any errors in the K2 transit times that are significantly larger than the formal uncertainty suggests (due for example, to correlated noise not accounted for by our model), could then bias the ephemeris fit and lead to an erroneous detection of a non-linear trend. In such cases, ongoing TESS observations, as well as future PLATO (PLAnetary Transits and Oscillations of stars) observations, will help to refine the ephemerides (Rauer et al. 2014).

Another possible explanation for the high proportion of K2 systems among those showing second-order trends is that these are dynamically induced TTVs that do not yet cover a full SP. Because K2 observations did not span as long a baseline as Kepler, it is more likely that planets with longer SPs appeared approximately linear in the K2 time frame alone. For either scenario, further observations will ultimately confirm whether the timing trends are real and elucidate their origin or continue to refine the linear ephemeris.

Even where timing trends are not detected, the TESS transit times reported in this study can be used to place limits on the existence, masses and eccentricities of any additional companions. Goldberg et al. (2019) predicted that TESS’s extended mission would allow mass estimates to be updated for 12–25 Kepler planets. It is also worth noting that of the five planets with new periodic trends observed, only one of them (Kepler-450 b) has known companions. If future observations confirm these periodic TTVs, these systems could contain additional planets.

Finally, non-periodic but non-linear timing trends could be caused by orbital evolution due to tidal interaction (Christ et al. 2019). The rate of orbital decay depends on several factors, including the exoplanet’s mass, the star’s mass, and the exoplanet’s orbital separation. The strength of tidal interaction and the time-scale for the resulting orbital decay depends very strongly on the orbital separation. For any tidal evolution to be observable over the |$\sim$|10 yr baseline covered in this study would require considerable fine-tuning, since we expect most of our planets to orbit ‘middle-aged’ field stars, where any tidal interaction strong enough to be observed would have ended long ago, either through the planet’s destruction or engulfment by the star, or because the system reached equilibrium. None the less, interactions with other companions can reduce the orbital separation or drive up the eccentricity or inclination of the orbit, potentially leading to observable tidal evolution at a later stage (Ragozzine & Wolf 2009). For example, we examined Kepler-1658, a hot Jupiter which was recently studied by Vissapragada et al. (2022), which showed evidence for tidal infall using data from TESS and Palomar. We find transit times consistent with this study, with TESS data points appearing to come |$\sim$|10–20 min earlier than predicted by the average linear ephemeris, which can be seen in Fig. A7.

We also plot the spin:orbit commensurability for planets with published rotation periods in Fig. 11. For planets orbiting faster than their star rotates, we would expect tidal interactions to convert orbital to rotational energy, slowing the planet’s orbit and causing it to move closer in to the star. While we do observe a negative curvature to a quadratic ephemeris for three such cases shown in Fig. 11, we also observe a positive curvature to a quadratic ephemeris for three other such planets. Furthermore, the amplitude of the observed timing offsets, which in some cases is several tens of minutes, is likely too high to be due to tidal effects alone (Ragozzine & Wolf 2009). We did not perform any in-depth analysis of tidal interactions in this study, but out results indicate that for high SNR targets, this analysis should be possible.

Figure 11.

Planetary period is plotted on the y-axis versus stellar rotation period on the x-axis, for systems in our sample where the latter is known (from McQuillan, Mazeh & Aigrain 2013). Systems that showed best-fitting second-order polynomial trends to the |$O-C$| in our study are plotted in yellow if they had negative curvature, or green if they had positive curvature. In the case of close-in planetary systems where the star and planet might be tidally interacting, a stellar period |$>$| planetary period might produce a decreasing period (negative curvature) or vice versa. We do not find strong evidence of such correlations here.

8 CONCLUSIONS

We have presented a systematic transit timing analysis of 111 planets detected by Kepler or K2 and re-observed by TESS. We presented new ephemerides, improving on the previously published values by up to an order of magnitude. We also identified five systems with periodic TTVs that were previously unknown and 17 systems displaying long-term trends in their transit times that may indicate ongoing dynamical interaction.

This study clearly demonstrates the power of using TESS data to re-examine Kepler and K2 targets. First of all, our updated ephemerides should prove valuable for any follow-up studies that require observations of the transit at a later date. For several systems, we report significant changes to the periods, or evidence of previously unseen timing trends that must be taken into account when planning future observations. In the coming years, observatories like the JWST and the Extremely Large Telescopes will enable detailed follow-up and characterization of exoplanet atmospheres and composition (Gardner et al. 2006, 2023). Given the busy observing schedules and value of telescope time, achieving precise ephemerides is critical for these studies. These trends also have potentially interesting implications for the dynamics and evolution of those systems.

This study has several limitations that could be revisited in future work. The TESS SNR, used to select the planets to include was a rough estimate. Other studies have detected transits in TESS data for previously known targets with SNR below our cut-off. Deeper looks into the Kepler and K2 catalogue could be possible, especially by stacking or phase-folding TESS transits, such that if TESS observed a large number of transits over an extended period, it should be possible to divide the TESS baseline into segments containing |$>$|1 transits, and fit a mean transit timing offset for each segment. This would be a possible way to search for long-term transit timing trends in low SNR transits (Léger et al. 2009). Furthermore, our strategy of performing a fully automated analysis meant that a number of individual transits as well as some full systems were excluded from the analysis if the automated fits did not converge. This excludes systems with higher stellar activity, star-spots, or many companions, as well as longer period planets for which there are large data gaps leading to too many ‘missed’ transits.

For the systems for which we identified TTVs or long-term timing trends, ongoing TESS observation as well as future observation by (for example) CHEOPS (CHaracterising ExOPlanet Satellite) and PLATO, offer excellent prospects for further confirmation and further characterization of those trends in the future (Rauer et al. 2014; Benz et al. 2021). Although the field of exoplanetary discovery and characterization is relatively young, we are entering an era where thousands of planetary systems will have been observed over the course of several decades, and such studies open doors to more in-depth characterization.

ACKNOWLEDGEMENTS

LK acknowledges funding support from the Clarendon Scholarship. This paper includes data collected by the TESS mission. Funding for the TESS mission is provided by NASA’s Science Mission Directorate. SA acknowledges funding from the European Research Councol (ERC) under the European Unions Horizon 2020 research and innovation programme (grant agreement no. 865624) and from the UK Space Agency and Science and Technology Facilities Council (STFC) under grant ST/R004846/1.

We acknowledge the use of public TESS data from pipelines at the TESS Science Office and at the TESS SPOC. Resources supporting this work were provided by the NASA High-End Computing (HEC) Program through the NASA Advanced Supercomputing (NAS) Division at Ames Research Center for the production of the SPOC data products. This paper includes data collected by the TESS mission that are publicly available from the MAST.

DATA AVAILABILITY

The TESS, Kepler, and K2 data used within this study are hosted and made publicly available by the Mikulski Archive for Space Telescopes (MAST, http://archive.stsci.edu/tess/).

The models and analyses of transit timings were conducted using publicly available open software codes, astropy (Astropy Collaboration 2013), AstroImageJ (Collins et al. 2017), PyTransit (Parviainen 2015), emcee (Foreman-Mackey et al. 2013, 2019), george (Ambikasaran et al. 2014), LDTk (Parviainen & Aigrain 2015), and TTVFast (Deck et al. 2014).

Data files including all transit times and their associated errors are available as Supporting Informaion.

Facilities used: TESS and Kepler

Footnotes

1

We note that the fit for P and |$T_0$| at this step is not strictly necessary and that these are not the final reported values. These were taken as helpful comparisons of our Kepler/K2-only fits to the literature values and uncertainties. Final transit times and resulting ephemerides are drawn from the Timing Fits described in Section 4.5.

APPENDIX A

Figure A1.

The left side of the figure shows the process of fitting Kepler data, and TESS data are shown on the right – in this case for the hot-Jupiter Kepler-8 b, which has an estimated TESS SNR/transit of 15. As described by Section 4.2, (a) and (b) show the raw PDC-SAP flux of Kepler and TESS respectively, while (c) and (d) show the polynomial fit to the out-of-transit PDC-SAP flux performed for each transit. This fit to the out-of-transit data is then used to normalize the flux for each transit. A transit model that uses archival parameters is then cross-correlated to individual transit data. The model is shown in green in (e) and (f) with the archival time of conjunction represented by the vertical red line. The results of the cross-correlation are shown in (g) and (h). These lags are then used to correct any significant transit offsets to facilitate with the global planetary parameter fit shown by the red line in (i) and (j).

Figure A2.

Same as previous figure, but for the lower SNR planet Kepler-74 b, which has an estimated TESS SNR/transit of 4. While TESS transits are sometimes difficult to make out by eye, they are detected in the cross-correlation step and appear when binned.

Figure A3.

Representations of a few of the updated ephemerides for the planets in this work. The red shaded region represents the archival uncertainty on the linear ephemeris (1σ and 3|$\sigma$|⁠) and the blue shaded region shows the ephemerides reported in this study. 1|$\sigma$| is shown by dashed lines in red (archival) and blue (this study). Epoch 0 begins at |$T_0$|⁠, which is typically near to the start of the Kepler mission or respective K2 sector. The first red vertical line represents the launch of JWST in late 2021, and the second vertical line represents the projected start of the PLATO mission in 2026 and finally the blue vertical line represents the start of ARIEL in 2029. Most planets show modest to significant improvement on the timing uncertainty, and several show significant offsets from archival parameters of several hours that would be expected to impact observations.

Figure A4.

Systems with previously identified TTVs. These planets were previously identified to display TTVs through periodic variability or significant scatter. In some cases, error bars on TESS transit times are too large to be considered in any way constraining.

Figure A5.

Example of the diagnostic plots used to verify the origin of cases with TTVs. Kepler-531 b shows significant TTVs with an SP of 35 d as seen in Fig. 9. However, its Kepler PDC-SAP light curve, shown in the top-left panel reveals a clear signature of stellar activity at the same period and McQuillan et al. (2013) report a rotation period of 35.053 d for this star that was replicated by our own periodicity indicator. The top panels show the full (left) and the phase-folded (right) light curve. The three lower panels show a single transit. The panel labelled ‘Fit’ shows the full model plotted in red, while the vertical lines indicate the archival transit-centre (blue) and fit transit-centre (orange). This transit visually appears to have significant short-term variation and outlying data points, possibly from occulted star-spots. This transit and several others from this system were therefore excluded from analysis.

Figure A6.

Systems with non-linear timing trends whose best fit is a second-order polynomial. Residuals to a non-linear trend are indicated with pink crosses. These planets display significant deviations from a linear ephemeris, however, they do not show evidence of periodicity.

Figure A7.

O − C plots for Kepler and K2 planets after fitting for a linear ephemeris using both Kepler/K2 and TESS transits. These are planets for which no non-linear timing trends were detected. In some cases, timing trends look to be visible by eye, but were either not determined to be statistically significant, or there were too few data points to conclude the presence of a trend without further follow-up.

Table A1.

Systems with known TTVs examined in this study. Period (d) is the fit period from this analysis. FAP is false alarm probability as defined in Section 5 and periodicity was deemed significant for an FAP |$<$|0.05 (5 per cent). LS best period is the period corresponding to the peak frequency of an LS periodogram, and is reported in epochs and days. Not all systems with previously identified periodic TTVs were recovered with statistical significance in our analysis.

Name	Period	FAP	LS best	LS best	Mass if known	Source	n TESS	Known
	(d)		period (epochs)	period (d)	(⁠|$M_J$|⁠)		sectors	companions
Kepler-39 b	21.087 21	1.000			20.1	Bonomo et al. (2015)	1
Kepler-18 d	14.858 92	8.14E−17	17.7982	264.461 56	0.047	Hadden & Lithwick (2016)	1	2
Kepler-51 b	45.155 44	3.45E−05	19.3548	873.976 28	0.012	Libby-Roberts et al. (2020)	1	3
Kepler-17 b	1.485 71	0.017	4.0731	6.051 41	2.47	Bonomo et al. (2017)	2
Kepler-37 d	39.792 23	0.838					1	3
Kepler-117 b	18.795 91	0.951			0.094	Bruno et al. (2015)	1	1
KOI-217 b	3.905 08	3.29E−10	8.9689	35.024 46			1
Kepler-412 b	1.720 86	5.92E−14	4.6020	7.919 42	0.927	Bonomo et al. (2017)	3
Kepler-210 c	7.972 53	2.09E−06	95.1351	758.467 26			2	1
Kepler-18 c	7.641 56	1.18E−36	35.0459	267.805 30	0.062	Weiss et al. (2024)	1	2
Kepler-1624 b	3.290 31	0.684					1

Table A2.

Observation log. Updates to linear ephemerides for all the planets and planet candidates examined in this work. Most planets see a modest improvement in ephemerides of a factor of 2–10.

Planet	Archival |$T_0$|	Archival period	Reference	Updated |$T_0$|	Updated period	Covariance	Timing
	BJD−2454833	(d)		BJD−2454833	(d)		trend
K2-100 b	2307.71941 |$\pm$| 0.00027	1.673904 |$\pm$| 4e−07	Barragán et al. (2019)	2307.7199 |$\pm$| 8e−05	1.673903 |$\pm$| 2.26e−07	–1.07e−11	4
K2-105 b	2314.989572 |$\pm$| 0.000444	8.266993 |$\pm$| 7e−06	Castro-González et al. (2022)	2314.98989 |$\pm$| 0.00032	8.267004 |$\pm$| 2.19e−05	–2.99e−09	4
K2-108 b	2312.09609 |$\pm$| 0.00172	4.733999 |$\pm$| 0.000221	Livingston et al. (2018)	2312.09784 |$\pm$| 0.00052	4.733431 |$\pm$| 3.32e−05	–9.73e−09	4
K2-113 b	2565.70433 |$\pm$| 0.0003	5.817685 |$\pm$| 4.1e−05	Livingston et al. (2018)	2565.70452 |$\pm$| 0.00012	5.817608 |$\pm$| 7.63e−06	–4.74e−10	3
K2-114 b	2854.08869 |$\pm$| 0.00016	11.390931 |$\pm$| 3.1e−06	Thygesen et al. (2023)	2854.08889 |$\pm$| 0.00021	11.390937 |$\pm$| 2.88e−06	–4.28e−10	4
K2-115 b	2689.07014 |$\pm$| 0.00017	20.272991 |$\pm$| 5e−06	Thygesen et al. (2023)	2689.07057 |$\pm$| 0.00017	20.272988 |$\pm$| 1.26e−05	–5.42e−10	4
K2-121 b	2310.560581 |$\pm$| 7.8e−05	5.185754 |$\pm$| 6e−07	Castro-González et al. (2022)	2310.56064 |$\pm$| 5e−05	5.185754 |$\pm$| 3.28e−07	–1.19e−11	4
K2-140 b	2755.28509 |$\pm$| 5e−05	6.569188 |$\pm$| 3.1e−05	Korth et al. (2019)	2755.28513 |$\pm$| 5e−05	6.569199 |$\pm$| 3.18e−06	–7.79e−11	3
K2-174 b	2250.77917 |$\pm$| 0.00095	19.562307 |$\pm$| 7.8e−05	Livingston et al. (2019)	2250.78491 |$\pm$| 0.00068	19.562538 |$\pm$| 6.6e−05	–1.6e−08	3
K2-18 b	2431.39144 |$\pm$| 0.00065	32.939623 |$\pm$| 9.5e−05	Sarkis et al. (2018)	2431.39189 |$\pm$| 0.00044	32.939948 |$\pm$| 5.97e−05	–7.4e−09	4
K2-180 b	2656.1566 |$\pm$| 0.00078	8.865663 |$\pm$| 1.1e−05	Thygesen et al. (2023)	2656.15478 |$\pm$| 0.00106	8.865597 |$\pm$| 0.000144	–1.28e−07	3
K2-182 b	2819.79755 |$\pm$| 0.00027	4.73697 |$\pm$| 1.7e−06	Thygesen et al. (2023)	2819.79774 |$\pm$| 0.00025	4.736968 |$\pm$| 1.7e−06	–2.66e−10	4
K2-186 b	2321.8454 |$\pm$| 0.001887	41.474219 |$\pm$| 0.00329	Mayo et al. (2018)	2321.847 |$\pm$| 0.00136	41.473908 |$\pm$| 0.000376	–2.27e−07	4
K2-197 b	2385.18248 |$\pm$| 0.00364	8.357881 |$\pm$| 0.00071	Livingston et al. (2018)	2385.18268 |$\pm$| 0.00073	8.357842 |$\pm$| 8.17e−06	–4.55e−09	4
K2-198 b	2371.5687 |$\pm$| 0.00014	17.042868 |$\pm$| 7.1e−06	Hedges et al. (2019)	2371.56953 |$\pm$| 0.00014	17.042877 |$\pm$| 1.43e−05	–7.5e−10	3
K2-198 d	2380.5759 |$\pm$| 0.001	7.450018 |$\pm$| 5.2e−06	Hedges et al. (2019)	2380.57688 |$\pm$| 0.00032	7.450012 |$\pm$| 1.99e−05	–2.32e−09	4
K2-212 b	2566.64109 |$\pm$| 0.00039	9.795647 |$\pm$| 2.1e−05	Duck et al. (2021)	2566.63986 |$\pm$| 0.00053	9.795626 |$\pm$| 8.29e−05	–2.95e−08	3
K2-232 b	2992.349782 |$\pm$| 9.3e−05	11.168454 |$\pm$| 2.3e−05	Brahm et al. (2018)	2992.3514 |$\pm$| 2e−05	11.16844 |$\pm$| 1.64e−06	–1.29e−11	4
K2-260 b	3061.284876 |$\pm$| 6e−05	2.626698 |$\pm$| 6.6e−07	Thygesen et al. (2023)	3061.2853 |$\pm$| 4e−05	2.626702 |$\pm$| 1.17e−06	–2.54e−11	4
K2-261 b	3318.14392 |$\pm$| 0.00024	11.633468 |$\pm$| 4.4e−06	Thygesen et al. (2023)	3318.14438 |$\pm$| 9e−05	11.633475 |$\pm$| 6.99e−06	–2.28e−10	3
K2-271 b	2316.71271 |$\pm$| 0.00053	8.562421 |$\pm$| 0.000127	Livingston et al. (2018)	2316.71279 |$\pm$| 0.00028	8.562329 |$\pm$| 2.86e−06	–6.19e−10	4
K2-275 b	2309.133959 |$\pm$| 0.000337	3.280963 |$\pm$| 1.6e−06	Castro-González et al. (2022)	2309.13293 |$\pm$| 0.00023	3.280966 |$\pm$| 1.03e−06	–1.61e−10	4
K2-275 c	2309.486716 |$\pm$| 0.000458	8.438839 |$\pm$| 7.5e−06	Castro-González et al. (2022)	2309.48633 |$\pm$| 0.00043	8.438835 |$\pm$| 4.1e−06	–1.39e−09	4
K2-277 b	2470.4771 |$\pm$| 0.001	6.326768 |$\pm$| 1.5e−05	Thygesen et al. (2023)	2470.47697 |$\pm$| 0.0004	6.326743 |$\pm$| 2.36e−05	–3.74e−09	3
K2-284 b	3026.11316 |$\pm$| 0.00043	4.795069 |$\pm$| 8.6e−05	David et al. (2018)	3026.11208 |$\pm$| 0.00033	4.79485 |$\pm$| 9.76e−06	–1.18e−09	4
K2-295 b	2562.41405 |$\pm$| 1e−06	4.024867 |$\pm$| 1.5e−05	Smith et al. (2019)	2562.41378 |$\pm$| 4e−05	4.02489 |$\pm$| 9.06e−07	–1.05e−11	3
K2-30 b	2230.80714 |$\pm$| 0.0001	4.098503 |$\pm$| 1.1e−05	Johnson et al. (2016)	2230.80723 |$\pm$| 6e−05	4.09848 |$\pm$| 1.4e−06	–2.32e−11	4
K2-308 b	2990.2233 |$\pm$| 0.0006	3.38628 |$\pm$| 2e−05	Rampalli et al. (2019)	2990.22239 |$\pm$| 0.00062	3.386263 |$\pm$| 2.65e−05	–8.79e−09	4
K2-321 b	3308.26759 |$\pm$| 0.00064	2.297975 |$\pm$| 1.7e−06	Thygesen et al. (2023)	3308.26454 |$\pm$| 0.00068	2.297933 |$\pm$| 3.01e−05	–1.69e−08	4
K2-334 b	3263.810228 |$\pm$| 0.00055	5.113981 |$\pm$| 6.1e−05	de Leon et al. (2021)	3263.81125 |$\pm$| 0.00023	5.113868 |$\pm$| 1.58e−05	–2.44e−09	4
K2-34 b	2308.351483 |$\pm$| 6.9e−05	2.995635 |$\pm$| 2e−07	Castro-González et al. (2022)	2308.35118 |$\pm$| 3e−05	2.995636 |$\pm$| 1.04e−07	–2.26e−12	4
K2-348 b	3263.716772 |$\pm$| 0.000995	4.688418 |$\pm$| 0.000119	de Leon et al. (2021)	3263.71571 |$\pm$| 0.00036	4.688611 |$\pm$| 2.28e−05	–4.57e−09	3
K2-352 d	3270.622842 |$\pm$| 0.002276	14.871387 |$\pm$| 0.000916	de Leon et al. (2021)	3270.62296 |$\pm$| 0.00096	14.871274 |$\pm$| 0.000106	–4.06e−08	3
K2-43 b	1976.8843 |$\pm$| 0.00071	3.471149 |$\pm$| 0.000104	Hedges et al. (2019)	1976.88313 |$\pm$| 0.00047	3.471261 |$\pm$| 1.16e−05	–1.98e−09	3
K2-45 b	1976.52604 |$\pm$| 0.00018	1.729268 |$\pm$| 6.9e−06	Crossfield et al. (2016)	1976.52629 |$\pm$| 0.00011	1.729259 |$\pm$| 6.6e−07	–1.75e−11	4
K2-77 b	2483.80766 |$\pm$| 0.00099	8.200084 |$\pm$| 8.6e−06	Thygesen et al. (2023)	2483.80577 |$\pm$| 0.00059	8.19999 |$\pm$| 4.53e−05	–1.14e−08	3
K2-79 b	2270.2275 |$\pm$| 0.00084	10.9947 |$\pm$| 0.00047	Bonomo et al. (2023)	2270.22701 |$\pm$| 0.00057	10.995283 |$\pm$| 2.82e−05	–4.48e−09	3
K2-87 b	2239.30232 |$\pm$| 0.00171	9.726618 |$\pm$| 5.5e−05	Livingston et al. (2019)	2239.30156 |$\pm$| 0.0012	9.726638 |$\pm$| 1.91e−05	–1.73e−08	4
Kepler-117 b	145.82204 |$\pm$| 0.00035	18.795923 |$\pm$| 7.5e−06	Bruno et al. (2015)	145.82402 |$\pm$| 0.00058	18.795913 |$\pm$| 1.29e−05	–6.51e−09	1
Kepler-12 b	171.00915 |$\pm$| 1e−05	4.437963 |$\pm$| 6e−07	Esteves, De Mooij & Jayawardhana (2015)	171.00938 |$\pm$| 3e−05	4.437962 |$\pm$| 1.41e−07	–3.15e−12	4
Kepler-15 b	136.328651 |$\pm$| 8.4e−05	4.942782 |$\pm$| 1.3e−06	Endl et al. (2011)	136.32943 |$\pm$| 4e−05	4.942784 |$\pm$| 2.16e−07	–6.55e−12	4
Kepler-1624 b	131.90925 |$\pm$| 0.00183	3.290305 |$\pm$| 4.56e−06	Morton et al. (2016)	131.90861 |$\pm$| 0.00144	3.290306 |$\pm$| 3.6e−06	–5.17e−09	1
Kepler-1656 b	177.2061 |$\pm$| 0.00083	31.562 |$\pm$| 0.011	Brady et al. (2018)	177.20638 |$\pm$| 0.00016	31.578649 |$\pm$| 5.88e−06	–7.78e−10	4
Kepler-17 b	352.678035 |$\pm$| 2.3e−05	1.485711 |$\pm$| 2e−07	Désert et al. (2011)	352.67879 |$\pm$| 1e−05	1.485711 |$\pm$| 2.63e−08	–3.27e−13	1
Kepler-18 c	334.0883 |$\pm$| 0.0023	7.64159 |$\pm$| 3e−05	Cochran et al. (2011)	334.08908 |$\pm$| 0.0002	7.641565 |$\pm$| 1.81e−06	–3.14e−10	1
Kepler-18 d	336.1776 |$\pm$| 0.0013	14.85888 |$\pm$| 4e−05	Cochran et al. (2011)	336.17864 |$\pm$| 0.00022	14.858923 |$\pm$| 3.71e−06	–7.01e−10	1
Kepler-210 c	171.58436 |$\pm$| 0.0002	7.972513 |$\pm$| 3e−06	Rowe et al. (2014)	171.58404 |$\pm$| 0.00021	7.972525 |$\pm$| 1.82e−06	–3.38e−10	1
Kepler-37 d	175.24982 |$\pm$| 0.00013	39.792262 |$\pm$| 6.5e−06	Bonomo et al. (2023)	175.2503 |$\pm$| 0.00024	39.792228 |$\pm$| 1.11e−05	–2.29e−09	1
Kepler-39 b	1109.60708 |$\pm$| 0.00024	21.08721 |$\pm$| 3.7e−05	Bonomo et al. (2015)	1109.60809 |$\pm$| 0.00015	21.087207 |$\pm$| 3.97e−06	–5.19e−10	1
Kepler-412 b	133.02122 |$\pm$| 2e−05	1.72086 |$\pm$| 3e−07	Esteves et al. (2015)	133.02096 |$\pm$| 3e−05	1.720861 |$\pm$| 7.08e−08	–2.12e−12	1
Kepler-422 b	177.25005 |$\pm$| 0.00011	7.891448 |$\pm$| 5e−07	Endl et al. (2014)	177.25005 |$\pm$| 4e−05	7.891448 |$\pm$| 4.34e−07	–1.6e−11	4
Kepler-423 b	133.35481 |$\pm$| 2.1e−05	2.684329 |$\pm$| 7e−08	Gandolfi et al. (2015)	133.35482 |$\pm$| 2e−05	2.684328 |$\pm$| 6.73e−08	–1.18e−12	4
Kepler-424 b	131.7427 |$\pm$| 0.00023	3.311864 |$\pm$| 3.9e−07	Endl et al. (2011)	131.74258 |$\pm$| 0.00011	3.311865 |$\pm$| 4.28e−07	–3.98e−11	4
Kepler-425 b	133.508785 |$\pm$| 3.9e−05	3.797018 |$\pm$| 1.9e−07	Hébrard et al. (2014)	133.5088 |$\pm$| 4e−05	3.797018 |$\pm$| 1.75e−07	–5.74e−12	4
Kepler-426 b	133.631964 |$\pm$| 4.7e−05	3.217519 |$\pm$| 1.9e−07	Hébrard et al. (2014)	133.63208 |$\pm$| 5e−05	3.217518 |$\pm$| 1.84e−07	–7.61e−12	4
Kepler-427 b	137.02207 |$\pm$| 9e−05	10.290994 |$\pm$| 1.1e−06	Hébrard et al. (2014)	137.02205 |$\pm$| 8e−05	10.290994 |$\pm$| 9.96e−07	–7.07e−11	4
Kepler-428 b	170.048008 |$\pm$| 3.2e−05	3.525633 |$\pm$| 1.5e−07	Hébrard et al. (2014)	170.04795 |$\pm$| 3e−05	3.525633 |$\pm$| 1.47e−07	–4.43e−12	4
Kepler-43 b	132.41624 |$\pm$| 2e−05	3.024095 |$\pm$| 6e−07	Esteves et al. (2015)	132.4169 |$\pm$| 5e−05	3.024092 |$\pm$| 1.84e−07	–8.18e−12	4
Kepler-433 b	131.98152 |$\pm$| 0.00015	5.334084 |$\pm$| 1.1e−06	Almenara et al. (2015)	131.98176 |$\pm$| 0.00015	5.334082 |$\pm$| 9.7e−07	–1.27e−10	4
Kepler-435 b	177.64241 |$\pm$| 0.0004	8.600154 |$\pm$| 1.8e−06	Almenara et al. (2015)	177.64231 |$\pm$| 0.00015	8.600155 |$\pm$| 1.55e−06	–2.05e−10	4
Kepler-45 b	170.82176 |$\pm$| 2.102e−05	2.455239 |$\pm$| 4e−06	Johnson et al. (2012)	170.82175 |$\pm$| 4e−05	2.455241 |$\pm$| 1.07e−07	–3.47e−12	4
Kepler-450 b	176.70618 |$\pm$| 17807.0	28.454851 |$\pm$| 2.5e−05	Van Eylen & Albrecht (2015)	176.7058 |$\pm$| 0.00054	28.454927 |$\pm$| 1.55e−05	–7.39e−09	2
Kepler-468 b	151.853006 |$\pm$| 0.000101	38.478757 |$\pm$| 4.85e−06	Morton et al. (2016)	151.85294 |$\pm$| 6e−05	38.478764 |$\pm$| 2.57e−06	–1.2e−10	3
Kepler-470 b	132.8633 |$\pm$| 0.0001	24.669193 |$\pm$| 3e−06	Cañas et al. (2023)	132.86427 |$\pm$| 0.00014	24.669188 |$\pm$| 3.96e−06	–4.91e−10	4
Kepler-471 b	133.174258 |$\pm$| 9.7e−05	5.014235 |$\pm$| 5.9e−07	Morton et al. (2016)	133.17428 |$\pm$| 8e−05	5.014235 |$\pm$| 4.79e−07	–3.33e−11	4
Kepler-485 b	133.669044 |$\pm$| 4.6e−05	3.24326 |$\pm$| 1.8e−07	Morton et al. (2016)	133.669 |$\pm$| 5e−05	3.24326 |$\pm$| 1.81e−07	–7.1e−12	4
Kepler-486 b	148.091528 |$\pm$| 0.000111	30.360447 |$\pm$| 4.24e−06	Morton et al. (2016)	148.09162 |$\pm$| 0.00011	30.360443 |$\pm$| 3.91e−06	–3.42e−10	4
Kepler-487 b	132.385103 |$\pm$| 0.000117	15.358768 |$\pm$| 2.21e−06	Morton et al. (2016)	132.38529 |$\pm$| 0.00012	15.358766 |$\pm$| 2.23e−06	–2.32e−10	4
Kepler-489 b	133.839246 |$\pm$| 0.000116	17.276296 |$\pm$| 2.45e−06	Morton et al. (2016)	133.8393 |$\pm$| 0.00011	17.276293 |$\pm$| 2.17e−06	–2.01e−10	4
Kepler-491 b	137.560674 |$\pm$| 5.3e−05	4.225385 |$\pm$| 2.7e−07	Morton et al. (2016)	137.56077 |$\pm$| 5e−05	4.225384 |$\pm$| 2.6e−07	–1.16e−11	4
Kepler-492 b	142.174098 |$\pm$| 0.000105	11.720123 |$\pm$| 1.54e−06	Morton et al. (2016)	142.17398 |$\pm$| 9e−05	11.720124 |$\pm$| 1.33e−06	–1.02e−10	4
Kepler-493 b	134.711271 |$\pm$| 0.000271	3.003877 |$\pm$| 9.9e−07	Morton et al. (2016)	134.71182 |$\pm$| 0.00011	3.003875 |$\pm$| 3.92e−07	–3.78e−11	2*
Kepler-494 b	132.4689 |$\pm$| 0.0003	8.025118 |$\pm$| 2e−06	Cañas et al. (2023)	132.46909 |$\pm$| 0.00023	8.025118 |$\pm$| 2.3e−06	–4.63e−10	4
Kepler-5 b	122.90144 |$\pm$| 2e−05	3.548466 |$\pm$| 7e−07	Esteves et al. (2015)	122.90145 |$\pm$| 4e−05	3.548466 |$\pm$| 1.7e−07	–5.78e−12	4
Kepler-506 b	172.662603 |$\pm$| 0.000357	6.883406 |$\pm$| 3.15e−06	Morton et al. (2016)	172.66221 |$\pm$| 0.0003	6.883409 |$\pm$| 2.27e−06	–6.36e−10	4
Kepler-51 b	881.5977 |$\pm$| 0.0004	45.154 |$\pm$| 0.0002	Masuda (2014)	881.59123 |$\pm$| 0.00043	45.155441 |$\pm$| 2.21e−05	–8.15e−09	1
Kepler-516 b	174.000979 |$\pm$| 0.00093	24.854624 |$\pm$| 2.82e−05	Morton et al. (2016)	173.99882 |$\pm$| 0.00078	24.854665 |$\pm$| 2.15e−05	–1.47e−08	4
Kepler-522 b	174.833446 |$\pm$| 0.000869	38.584228 |$\pm$| 4.57e−05	Morton et al. (2016)	174.82996 |$\pm$| 0.0005	38.584377 |$\pm$| 2.35e−05	–1.08e−08	2
Kepler-531 b	173.194425 |$\pm$| 0.000896	29.884857 |$\pm$| 3.25e−05	Morton et al. (2016)	173.1915 |$\pm$| 0.0008	29.884954 |$\pm$| 2.67e−05	–1.85e−08	2*
Kepler-546 b	170.326066 |$\pm$| 0.000171	4.14702 |$\pm$| 8.8e−07	Morton et al. (2016)	170.32609 |$\pm$| 0.00017	4.14702 |$\pm$| 8.4e−07	–1.26e−10	4
Kepler-548 b	172.820206 |$\pm$| 6.9e−05	4.454194 |$\pm$| 3.8e−07	Morton et al. (2016)	172.82031 |$\pm$| 6e−05	4.454194 |$\pm$| 3.27e−07	–1.74e−11	2*
Kepler-6 b	121.486528 |$\pm$| 9e−06	3.2347 |$\pm$| 4e−07	Esteves et al. (2015)	121.48646 |$\pm$| 2e−05	3.234699 |$\pm$| 9.43e−08	–1.78e−12	4
Kepler-628 b	173.782 |$\pm$| 0.0002	15.458053 |$\pm$| 3e−06	Cañas et al. (2023)	173.78232 |$\pm$| 0.00014	15.458058 |$\pm$| 2.61e−06	–3.19e−10	2
Kepler-63 b	177.84307 |$\pm$| 5e−05	9.434151 |$\pm$| 1e−06	Sanchis-Ojeda et al. (2013)	177.84324 |$\pm$| 4e−05	9.434153 |$\pm$| 4.8e−07	–1.76e−11	4
Kepler-643 b	177.916757 |$\pm$| 0.000877	16.338896 |$\pm$| 1.83e−05	Morton et al. (2016)	177.92602 |$\pm$| 0.00206	16.33876 |$\pm$| 3.53e−05	–6.97e−08	4
Kepler-670 b	170.967687 |$\pm$| 4.4e−05	2.816505 |$\pm$| 1.5e−07	Morton et al. (2016)	170.96737 |$\pm$| 0.00021	2.816506 |$\pm$| 5.19e−07	–1.06e−10	4
Kepler-686 b	170.648322 |$\pm$| 5.8e−05	1.594745 |$\pm$| 1.1e−07	Morton et al. (2016)	170.64835 |$\pm$| 6e−05	1.594745 |$\pm$| 1.08e−07	–5.38e−12	2
Kepler-695 b	169.949006 |$\pm$| 7.8e−05	3.04033 |$\pm$| 3e−07	Morton et al. (2016)	169.94907 |$\pm$| 8e−05	3.04033 |$\pm$| 2.9e−07	–1.96e−11	4
Kepler-699 b	186.713532 |$\pm$| 0.000147	27.807563 |$\pm$| 5.29	Morton et al. (2016)	186.71366 |$\pm$| 0.00013	27.807562 |$\pm$| 4.44e−06	–4.93e−10	4
Kepler-7 b	134.27687 |$\pm$| 3e−05	4.885489 |$\pm$| 9e−07	Esteves et al. (2015)	134.27683 |$\pm$| 4e−05	4.885489 |$\pm$| 2.04e−07	–6.53e−12	4
Kepler-702 b	176.522503 |$\pm$| 0.000142	10.526294 |$\pm$| 1.9e−06	Morton et al. (2016)	176.52238 |$\pm$| 0.00013	10.526295 |$\pm$| 1.64e−06	–1.86e−10	4
Kepler-714 b	176.623889 |$\pm$| 0.000218	8.098888 |$\pm$| 2.19e−06	Morton et al. (2016)	176.62386 |$\pm$| 0.00022	8.098888 |$\pm$| 2.16e−06	–4.2e−10	4
Kepler-718 b	169.890629 |$\pm$| 5.8e−05	2.05235 |$\pm$| 1.5e−07	Morton et al. (2016)	169.89071 |$\pm$| 5e−05	2.05235 |$\pm$| 1.32e−07	–6.02e−12	4
Kepler-720 b	171.44652 |$\pm$| 0.000125	4.708327 |$\pm$| 7.3e−07	Morton et al. (2016)	171.44636 |$\pm$| 0.00013	4.708327 |$\pm$| 7.52e−07	–8.71e−11	4
Kepler-723 b	169.637529 |$\pm$| 4.6e−05	4.082275 |$\pm$| 2.3e−07	Morton et al. (2016)	169.63752 |$\pm$| 5e−05	4.082275 |$\pm$| 2.19e−07	–8.74e−12	4
Kepler-725 b	206.596256 |$\pm$| 0.000218	39.643178 |$\pm$| 1.11e−05	Morton et al. (2016)	206.58786 |$\pm$| 0.00097	39.643509 |$\pm$| 3.6e−05	–3.43e−08	3
Kepler-731 b	170.678886 |$\pm$| 6.1e−05	3.855604 |$\pm$| 2.9e−07	Morton et al. (2016)	170.67889 |$\pm$| 6e−05	3.855604 |$\pm$| 2.7e−07	–1.37e−11	4
Kepler-74 b	354.56737 |$\pm$| 0.00014	7.340711 |$\pm$| 6e−06	Bonomo et al. (2015)	354.56721 |$\pm$| 9e−05	7.340714 |$\pm$| 7.67e−07	–6.05e−11	4
Kepler-76 b	133.54841 |$\pm$| 1e−05	1.54493 |$\pm$| 4e−07	Esteves et al. (2015)	133.54944 |$\pm$| 4e−05	1.544929 |$\pm$| 8.35e−08	–3.09e−12	4
Kepler-762 b	357.118242 |$\pm$| 0.000176	3.770552 |$\pm$| 9.4e−07	Morton et al. (2016)	357.11811 |$\pm$| 9e−05	3.770553 |$\pm$| 4.87e−07	–3.99e−11	4
Kepler-77 b	262.865727 |$\pm$| 2.9e−05	3.578781 |$\pm$| 2.3e−07	Gandolfi et al. (2013)	262.86647 |$\pm$| 4e−05	3.578781 |$\pm$| 1.6e−07	–5.28e−12	4
Kepler-785 b	178.689372 |$\pm$| 4e−05	1.973761 |$\pm$| 1e−07	Morton et al. (2016)	178.68933 |$\pm$| 4e−05	1.973761 |$\pm$| 1.03e−07	–3.9e−12	4
Kepler-8 b	121.11931 |$\pm$| 2e−05	3.522499 |$\pm$| 7e−07	Esteves et al. (2015)	121.11948 |$\pm$| 4e−05	3.522498 |$\pm$| 1.72e−07	–5.87e−12	4
Kepler-840 b	135.337166 |$\pm$| 0.00016	2.49578 |$\pm$| 4.82e−07	Morton et al. (2016)	135.33706 |$\pm$| 2e−05	2.49578 |$\pm$| 4.71e−08	–6.64e−13	2
Kepler-854 b	133.98434 |$\pm$| 0.00026	2.144633 |$\pm$| 1.1e−07	Kokori et al. (2023)	133.98618 |$\pm$| 2e−05	2.144633 |$\pm$| 5.39e−08	–9.87e−13	2*
Kepler-856 b	360.171476 |$\pm$| 0.000179	8.027681 |$\pm$| 2.01e−06	Morton et al. (2016)	360.17141 |$\pm$| 0.00019	8.027681 |$\pm$| 2.05e−06	–3.26e−10	4
Kepler-93 b	111.29227 |$\pm$| 0.00013	4.72674 |$\pm$| 9.7e−07	Bonomo et al. (2023)	111.29238 |$\pm$| 0.00019	4.726742 |$\pm$| 1.03e−06	–1.67e−10	4
Kepler-96 b	171.01897 |$\pm$| 7.27e−06	16.2385 |$\pm$| 4.8e−06	Marcy et al. (2014)	171.01912 |$\pm$| 0.00024	16.238487 |$\pm$| 4.57e−06	–9.91e−10	4
KOI-217 b	133.414 |$\pm$| 0.0002	3.90512 |$\pm$| 5e−05	Howell et al. (2010)	133.41503 |$\pm$| 4e−05	3.905081 |$\pm$| 1.93e−07	–6.49e−12	1

Notes. Some see noticeably less precise linear ephemerides, which in several cases is due to the finding of significant variation between Kepler and TESS data points. Covariance is reported so that accurate uncertainty can be computed for any observation starting point. A timing trend indicator of 1 indicates TTVs were previously found in the planet. A 2 indicates new periodic TTVs found in this work; 2* indicates planets for which periodicity was found, but it either corresponded to stellar activity timescales, or to polynomial fits, thus more follow-up should be performed to verify if these findings are significant. A flag of 3 indicates a better fit to a second-order polynomial than to a linear ephemeris. A flag of 4 indicates a best fit of a linear ephemeris with no evidence of periodic or higher order behaviour.